1 Purpose, Scope, and Success Criteria of the Application Whitepaper (Including Reality Lock Criteria)

1.1 Goal

The primary goal of this application whitepaper is to deploy the Volume Particle (VP) theory as a “designable engineering system” to: (1) obtain renewable energy electrical output from ESS (Black Copper based Solar Thermal Collection/Storage), (2) use that electricity to perform Water Purification (Desalination/Purification) and Pt Catalyst based Hydrogen/Reaction Applications, and (3) demonstrate that the entire process is reproducible under energy/mass conservation laws and thermodynamic constraints. This whitepaper aims not to introduce an “idea” but to simultaneously provide the following four deliverables:

1. Design: Module structure (dimensions/materials), power bus (DC), control/measurement, and safety design.

2. Estimated Results: Specifying input conditions (solar irradiance, area, feed water salinity, etc.) and calculating outputs (Wh/day, L/day, g-H\(_2\)/day, etc.) as ranges.

3. Physical Basis: Specifying thermodynamics/electrochemistry/transport equations and constraints (upper/lower bounds) to exclude impossible claims (e.g., zero separation energy, violation of heat engine efficiency limits) and mathematically grounding the possible design space.

4. Reproducibility Package: Defining measurement items, data formats, calculation procedures, uncertainty evaluation, and standards for open-source design files/measurement logs/analysis codes, organized in a form that allows for DOI assignment upon completion of the whitepaper.

1.2 Scope and System Boundary

Since this is an application phase whitepaper, it does not restate the “entire VP theory” but uses only the minimum definitions necessary for design/verification. The system boundary is presented as a figure (block diagram), but mathematically it has the following energy/material boundaries.

1.2.0.1 (1) Include.

1. ESS Module: Solar radiation input \(\to\) Collection/Thermal Storage state \(\to\) Electrical output (DC Bus).

2. Water Purification (Desalination/Purification) Module: Separating feed water into product and brine/concentrate via electric fields (including electromagnetic waves) or electrochemical processes (e.g., ED/CDI/RO assistance).

3. Hydrogen/Catalyst Module: Generating H\(_2\) via water splitting (if necessary) using ESS electricity, and supplying it to a Pt catalyst reactor (hydrogen activation/hydrogenation/reduction, etc.) to evaluate reaction performance.

4. Integrated Control/Measurement: Power measurement (Voltage/Current/Accumulated Wh), Thermal measurement (Temperature field/Stored heat), Water quality measurement (Conductivity/Salinity/TDS), Gas measurement (Flow rate/Pressure), Safety (Corrosion/Gas/Electric Shock/Overheating).

1.2.0.2 (2) Exclude.

1. Detailed licensing design and certification tests at the national/large plant level (however, requirements lists and safety design principles are included).

2. Full quantum many-body simulation (this whitepaper focuses on engineering reproducibility/measurement).

3. Claims that directly conflict with thermodynamics, such as “seawater desalination with zero energy” (if such results appear, we define/measure/re-examine hidden inputs and require external reproduction tests first).

1.2.0.3 (3) Definition of “Free” in this Whitepaper.

In this whitepaper, “free” is restricted to the meaning of driving with renewable energy (solar) without grid electricity costs. That is, material/fabrication/maintenance costs exist, and since water purification (desalination) and water splitting physically require energy input, it does not mean “zero energy”.

1.3 Core Physical Constraints (Required Upper/Lower Bounds) and Mathematical Definitions

This whitepaper grounds reality (the entity) in “measurable conserved quantities”. Therefore, all designs, estimates, and success judgments must pass the following mathematical definitions and constraints.

1.3.1 Energy Conservation (1st Law) and Overall ESS Efficiency

Solar radiation input energy is defined as: \[E_{\mathrm{in}} \;=\; \int_{t_0}^{t_1} G(t)\,A_{\mathrm{col}}\,dt \label{eq:Ein_solar}\] Here, \(G(t)\) is the solar irradiance perpendicular to the collection surface (W/m\(^2\)), and \(A_{\mathrm{col}}\) is the effective collection area (m\(^2\)).

Electrical output energy is defined as: \[E_{\mathrm{out}} \;=\; \int_{t_0}^{t_1} V(t)\,I(t)\,dt \label{eq:Eout_elec}\] Here, \(V(t)\) is the DC bus voltage (V), and \(I(t)\) is the current (A).

The overall efficiency is: \[\eta_{\mathrm{total}} \;=\; \frac{E_{\mathrm{out}}}{E_{\mathrm{in}}} \label{eq:eta_total}\] and must satisfy \(\eta_{\mathrm{total}}\le 1\) by measurement definition. If \(\eta_{\mathrm{total}}>1\) appears in actual measurement, it implies one of the following exists: (i) missing \(E_{\mathrm{in}}\) definition (area/angle/irradiance), (ii) error in \(E_{\mathrm{out}}\) calculation (power meter integration/waveform), (iii) hidden paths of external energy input (electricity/fuel/heat), or (iv) error in setting the time window (\(t_0,t_1\)). Thus, it is treated not as success but as a re-verification condition.

1.3.2 Heat Engine Upper Limit (Carnot Limit) and Separation of Conversion Efficiency

If the ESS performs “Heat \(\to\) Work (Electricity)” conversion via a heat engine (or its equivalent system), the conversion efficiency \(\eta_{\mathrm{conv}}\) has a Carnot efficiency upper limit. \[\eta_{\mathrm{conv}} \;\le\; \eta_{\mathrm{Carnot}} \;=\; 1-\frac{T_c}{T_h}, \qquad (T_h > T_c) \label{eq:carnot}\] Here, \(T_h\) is the absolute temperature (K) of the high-temperature heat source (rotor/hot side), and \(T_c\) is the absolute temperature (K) of the low-temperature heat sink (ambient/cold side). The application whitepaper must separate and report the following: \[\eta_{\mathrm{total}}=\eta_{\mathrm{opt}}\cdot \eta_{\mathrm{th}}\cdot \eta_{\mathrm{conv}}\cdot \eta_{\mathrm{elec}}, \label{eq:eta_factorization}\] Here, \(\eta_{\mathrm{opt}}\) denotes optical absorption, \(\eta_{\mathrm{th}}\) thermal storage/insulation, and \(\eta_{\mathrm{elec}}\) power electronics/wiring losses. If this decomposition is not fixed, “efficiency” figures are incomparable and reproducibility cannot be ensured.

1.3.3 Performance Definition in Water Purification (Desalination): Mass Balance and Specific Energy Consumption (SEC)

Water purification (desalination) is defined by at least the following three indicators. Let feed flow rate be \(Q_f\) (m\(^3\)/s), product flow rate be \(Q_p\), concentrate flow rate be \(Q_c\), and salt concentrations (e.g., TDS) be \(C_f, C_p, C_c\) (kg/m\(^3\)) respectively. The mass balance is: \[\begin{aligned} Q_f &= Q_p + Q_c, \label{eq:mass_balance_flow}\\ Q_f C_f &\approx Q_p C_p + Q_c C_c \quad (\text{when salt loss/precipitation is negligible}). \label{eq:mass_balance_salt}\end{aligned}\] Rejection ratio (salt removal performance) is: \[R \;=\; 1-\frac{C_p}{C_f}, \label{eq:removal_ratio}\] Recovery ratio (product water ratio) is: \[Y \;=\; \frac{Q_p}{Q_f}. \label{eq:recovery_ratio}\] The key indicator for electrical energy consumption is Specific Energy Consumption (SEC), defined as: \[\mathrm{SEC} \;=\; \frac{E_{\mathrm{desal}}}{V_p} \;=\; \frac{\int V(t)I(t)\,dt}{\int Q_p(t)\,dt}, \qquad \left[\mathrm{kWh/m^3}\right] \label{eq:sec}\] From the perspective of grounding reality, desalination technologies must be compared in the 3D performance space of (i) \(R\), (ii) \(Y\), and (iii) SEC. Presenting only one is not judged as success.

1.3.4 Definition of Water Splitting/Hydrogen Production: Faraday’s Law and Actual Measurement

Given electrolysis current \(I(t)\), the ideal number of moles of hydrogen produced is given by Faraday’s law: \[n_{H_2,\mathrm{ideal}} \;=\; \frac{1}{2F}\int_{t_0}^{t_1} I(t)\,dt, \label{eq:faraday_ideal}\] Here, \(F\) is the Faraday constant. The actual produced moles are defined by introducing Faraday efficiency \(\eta_F \in [0,1]\): \[n_{H_2} \;=\; \eta_F\, n_{H_2,\mathrm{ideal}} \;=\; \frac{\eta_F}{2F}\int I(t)\,dt \label{eq:faraday_real}\] Mass production is converted as \(m_{H_2}=n_{H_2}M_{H_2}\) (where \(M_{H_2}\) is the molar mass). The application whitepaper must cross-verify \(n_{H_2}\) via (i) power integration (Wh) and (ii) gas flow measurement (or water displacement) to measure \(\eta_F\). Presenting only power or only gas volume does not fix reproducibility/reality.

1.4 Success Criteria: PASS/HOLD/FAIL Rules

Success judgment in this whitepaper is fixed primarily by measurement results based on conserved quantities and passing constraint conditions, not by “VP variables (Amplitude/Alignment)”. Secondary judgment is made based on how well VP variables predict/explain the measured quantities.

1.4.1 Primary Success Judgment (Engineering Reality): Passing Energy/Mass Balance

1.4.1.1 (ESS) Criteria.

1. Energy Balance Closure: Satisfies \(\eta_{\mathrm{total}} \le 1\), and does not exceed it even including measurement uncertainty (irradiance sensor, area, power meter integration).

2. Reproducibility: Report the mean and standard deviation of \(E_{\mathrm{out}}\) for at least \(N\ge 3\) repetitions under identical conditions (irradiance range, ambient temperature range, same module), and record variation causes (clouds/wind speed/load) as metadata.

3. Safety: Specify the operation window free from risks of overheating, electric shock, and fire (temperature upper limit, insulation/grounding, cutoff).

1.4.1.2 (Water Purification/Desalination) Criteria.

1. Mass Balance Consistency: [eq:mass_balance_flow]–[eq:mass_balance_salt] hold within measurement error.

2. Water Quality Performance: \(R\) and \(C_p\) satisfy the target (separate criteria based on target feed water type).

3. Energy Performance: Calculate SEC and present the target SEC range. (Since complete seawater desalination and brackish water purification have different target SECs, the feed range is fixed in the text.)

1.4.1.3 (Electrolysis/Catalyst) Criteria.

1. Hydrogen Production Measurement: [eq:faraday_real] matches actual gas flow measurement, reporting \(\eta_F\).

2. Energy-Hydrogen Indicator: Report \(E_{\mathrm{elec}}/m_{H_2}\) (kWh/kg-H\(_2\)).

3. Catalyst Performance: Quantify the change in reaction rate (selecting at least one standard reaction) before/after the Pt reactor.

1.4.2 Secondary Success Judgment (Theoretical Reality): Predictive/Explanatory Power of VP Variables

“Reality Lock” of VP theory is defined as established when the following two conditions are met simultaneously.

1. Predictability of Observations: VP variables (Amplitude/Alignment) provide a quantitative prediction formula for at least one key KPI among ESS output ON/OFF, purification separation performance (\(R,Y,\mathrm{SEC}\)), or electrolysis performance (\(\eta_F\), kWh/kg), and report the error range in independent experimental data (data not used for training).

2. Fixing Variable Mapping: VP variables must correspond (or be calibrated) to measurable physical quantities. Example: Fix the functional relationship (even if empirical) between amplitude threshold \(\pm 250\) fm and electric field strength \(|E|\), temperature \(T\), charge density/potential, or current density.

If these two conditions are not met, VP variables remain as “internal coordinates (latent variables for explanation)”, and the application whitepaper maintains an engineering performance focus (PASS/HOLD), but judges the theoretical reality as HOLD.

1.5 Reality Lock Strategy: Separation of “Intrinsic Amplitude” and “Operational Amplitude” and Calibration Layer

To prevent contradictions arising when the term “amplitude” is mixed across different levels, this whitepaper separates and fixes amplitude into at least two types.

1.5.1 Intrinsic Amplitude: Whitepaper LOCK Formula

The intrinsic amplitude of element/species \(X\) is defined as: \[r_{\mathrm{eff}}^{(0)}(X)=\frac{r_{\mathrm{vac}}}{\sqrt{P_{\mathrm{idx}}(X)}}, \qquad P_{\mathrm{idx}}(X)=\frac{\dfrac{Z_X}{r_{\mathrm{cov}}(X)^2}}{\dfrac{Z_H}{r_{\mathrm{cov}}(H)^2}} \label{eq:reff0_lock}\] This is a canonical value representing “reference compressibility of the species”, and it is not assumed that this value itself is identical to the operational amplitude observed in experiments.

1.5.2 Operational Amplitude: Calibratable State Variable

Let \(A_{\mathrm{op}}\) be the amplitude observed/assumed during operation. Its ratio to the intrinsic amplitude is defined as: \[\gamma(X;\,\Theta)=\frac{A_{\mathrm{op}}(X;\,\Theta)}{r_{\mathrm{eff}}^{(0)}(X)} \label{eq:gamma_def}\] Here, \(\Theta\) is the set of environmental/driving conditions, for example: \[\Theta=\{T,\;|E|,\;f,\;|\nabla E|,\;\Phi,\;\dot{m},\;\text{surface/solution state},\;\text{structure (tube/slit)}\}\] (\(f\) is frequency, \(\Phi\) is potential, \(\dot{m}\) is flow rate/mass flow rate). The application whitepaper treats \(\gamma\) as a key parameter of the calibration layer, and estimates \(\gamma(\Theta)\) from data to fix the reality of “amplitude”.

1.5.3 Materialization of Alignment: Dimensionless Order Parameter and Measurement Proxy

Alignment is defined as a dimensionless order parameter \(S\in[0,1]\). \[S = \frac{\langle P_2(\cos\theta)\rangle - \langle P_2(\cos\theta)\rangle_{\mathrm{iso}}}{1-\langle P_2(\cos\theta)\rangle_{\mathrm{iso}}} \quad\text{with}\quad P_2(x)=\frac{1}{2}(3x^2-1), \label{eq:order_parameter}\] Here, \(\theta\) is the angle between the “alignment axis” and the local direction (e.g., polarization/structural axis). Since \(\langle P_2\rangle_{\mathrm{iso}}=0\) in the isotropic state, it can be simplified to \(S=\langle P_2(\cos\theta)\rangle\). Since it is difficult to directly measure \(\theta\) in experiments, this whitepaper fixes the proxy measurement of \(S\) as one of the following:

1. Optical anisotropy (polarization/birefringence) based indicator,

2. Electrical anisotropy (permittivity tensor/impedance anisotropy) based indicator,

3. Flow/Structure based indicator (channel directionality, turbulence intensity, concentration polarization stability).

If the selected indicator (including instrument/formula) is not specified, alignment is unreproducible and excluded from success judgment.

1.6 Replication Plan: DOI-Ready Package Standard

This whitepaper assumes DOI assignment upon completion, and the reproducibility package has the following items as mandatory components. (Even if design drawings/CAD files are not immediately distributed, minimal text standards and measurement logs must be publicly available.)

1.6.1 Fixing Experimental Protocol

Each module records the protocol using the same template.

1. Target KPI and Success Criteria (PASS/HOLD/FAIL).

2. Input condition ranges (irradiance, ambient temp, feed salinity, flow rate, electric field strength/frequency, etc.).

3. Instrument list (Model name/Accuracy/Calibration date/Sampling period).

4. Data format (CSV/JSON), Time standard (UTC or Local), Unit standard (SI).

5. Number of repetitions \(N\) and statistical reporting (Mean, Standard Deviation, Confidence Interval).

1.6.2 Dataset and Analysis Code Standards

1. Raw Data: Sensor raw data (Irradiance, Temp, Voltage, Current, Conductivity, Flow, Gas Flow).

2. Metadata: Device version, Wiring/Load, Weather summary, Experimenter/Date, Calibration info.

3. Analysis Code:

   • Integration logic for [eq:Ein_solar]–[eq:Eout_elec],

   • SEC calculation logic for [eq:sec],

   • Hydrogen calculation logic for [eq:faraday_real],

   • Uncertainty propagation (Sensor error, Integration error) calculation.

4. Result Reproduction Command: Specify execution order and dependencies so that tables/graphs are reproduced with a single command in the same environment.

1.6.3 Versioning, Checksum, and DOI Preparation

1. Assign version tags (e.g., v1.0.0) to whitepaper text, drawings, data, and code.

2. Record the checksum (e.g., SHA256) of the release bundle before DOI assignment to enable identity verification.

3. DOI metadata includes: Title, Author (ORCID optional), Version, License, Abstract, Keywords, Data/Code links, Verification Status (PASS/HOLD/FAIL).

1.7 Conclusion of This Section: “Reality” Fixed in Major Topic 1

The reality of this application whitepaper is fixed as follows.

1. The primary criterion for reality is measurement based on conserved quantities: \(E_{\mathrm{in}},E_{\mathrm{out}}\), \(R,Y,\mathrm{SEC}\), \(m_{H_2},\eta_F\).

2. VP Amplitude/Alignment are treated as calibratable latent variables, separating \(r_{\mathrm{eff}}^{(0)}\) and \(A_{\mathrm{op}}\) and connecting them with \(\gamma(\Theta)\).

3. Success judgment operates under PASS/HOLD/FAIL rules, where PASS satisfies “Engineering Performance + Reproducibility” first, and then theoretical reality is judged by “VP Predictability”.

2 Summary of VP Amplitude (LOCK) Formulas and Redefinition of “Operational Amplitude \(A_{\mathrm{op}}\)” (Bulk/Surface/Solution Separation)

2.1 Purpose of this Section and the Need for “Redefinition”

The purpose of this section is to fix the following two aspects in a mathematically complete form.

1. Summarize the Amplitude (Effective Amplitude) LOCK Formula presented in the Volume I whitepaper, including units, definitions, and back-calculation formulas.

2. Redefine the “amplitude” repeatedly used in applications (ESS/Water/Catalyst) and simulations (e.g., \(\pm 250\sim 280\) fm, \(\pm 300\) fm) as the Operational Amplitude \(A_{\mathrm{op}}\).

3. Explicitly distinguish the physical layers of Bulk/Surface/Solution even when using the same symbol (fm), thereby fixing the reality so that “whitepaper calculations” and “application simulations” do not directly conflict.

2.1.0.1 Core Point (Fixing the Reality).

The LOCK formula in Volume I calculates the Intrinsic (Canonical) Amplitude, whereas the amplitude used in applications/simulations is the Operational (Environmental) Amplitude that actually appears during operation under external conditions (temperature, electric field/EM waves, surface, solution, flow, structure). Therefore, in this section, we separate amplitude into at least two types: \[\begin{aligned} r_{\mathrm{eff}}^{(0)}(X) &: \text{Intrinsic (Canonical) Effective Amplitude (Calculated via LOCK Formula)}, \\ A_{\mathrm{op}}(X;\Theta) &: \text{Operational (Environmental/Driving) Amplitude (State Variable for Design)}.\end{aligned}\] Here, \(\Theta\) is the set of operating conditions (temperature, electric field strength/frequency, surface state, solution composition, flow rate, etc.).

2.2 Units, Symbols, and Constants (Fixing Reproducibility)

Throughout this whitepaper, SI units are the standard, but VP amplitude is expressed in fm. \[\begin{aligned} 1~\mathrm{fm} &= 10^{-15}~\mathrm{m}, \\ 1~\mathrm{pm} &= 10^{-12}~\mathrm{m}.\end{aligned}\]

2.2.0.1 Canonical scale separation (v0.2).

To ensure inheritance from the theoretical framework (v0.1.2), we distinguish the oscillation scale from the spatial extent: \[L_{\mathrm{quant}} \approx 2\pi^2 \cdot r_{\mathrm{vac}} \approx 4854\,\mathrm{fm}. \label{eq:scale_lock_Lquant_rvac}\] While \(L_{\mathrm{quant}}\) defines the lattice geometry, the application layer (Catalysis/ESS) is governed by the Reference Amplitude \(r_{\mathrm{vac}}=245.9\,\mathrm{fm}\). The previous notation in early drafts incorrectly assigned the quantum-diameter value to \(r_{\mathrm{vac}}\). In v0.2, \(r_{\mathrm{vac}}\) is strictly reserved for the amplitude scale.

• Reference Amplitude (\(r_{\mathrm{vac}}\)): 245.9 fm (Governs energy-driven interactions)

• Quantum Size (\(L_{\mathrm{quant}}\)): 4854 fm (Governs spatial lattice geometry)

The symbols used in this section are as follows:

• \(X\): Species. Includes atoms, molecules, ions, and surface states (effective species).

• \(Z_X\): “VP Effective Charge Number” assigned to species \(X\) (default is atomic number or sum).

• \(r_{\mathrm{vac}}\): Reference Amplitude (used in Whitepaper LOCK). Treated as a normalization constant. In this application whitepaper, the default value is set to \(r_{\mathrm{vac}}=245.9\,\mathrm{fm}\).

• \(r_{\mathrm{cov}}(X)\): Standard Covalent Radius (pm). Standard value for bulk atoms. We use \(r_{\mathrm{cov}}(H)=31\) pm as the reference.

• \(r_{\mathrm{conf}}^{(\alpha)}(X;\Theta)\): Effective Confinement Radius (pm) in environment \(\alpha\). \(\alpha\in\{\mathrm{bulk},\mathrm{surf},\mathrm{sol}\}\) (Bulk/Surface/Solution).

• \(P_{\mathrm{raw}}\), \(P_{\mathrm{idx}}\): Pressure Index (defined below).

• \(r_{\mathrm{eff}}^{(\alpha)}(X;\Theta)\): Effective Amplitude (fm) in environment \(\alpha\).

• \(A_{\mathrm{op}}^{(\alpha)}(X;\Theta)\): Operational Amplitude (fm) in environment \(\alpha\).

2.3 LOCK Formula: Intrinsic (Canonical) Effective Amplitude \(r_{\mathrm{eff}}^{(0)}\)

The LOCK formula in the Volume I whitepaper calculates the effective amplitude via the “Pressure Index”. This application whitepaper fixes this structure to allow Definition-Calculation-Back-calculation.

2.3.1 Raw Pressure and Pressure Index

The standard (canonical) definition is: \[P_{\mathrm{raw}}(X) \;=\; \frac{Z_X}{r_{\mathrm{cov}}(X)^2}, \qquad \left[ P_{\mathrm{raw}} \right]=\mathrm{pm^{-2}}. \label{eq:Praw_cov}\] The normalized pressure index relative to hydrogen is: \[P_{\mathrm{idx}}(X) \;=\; \frac{P_{\mathrm{raw}}(X)}{P_{\mathrm{raw}}(H)} \;=\; \frac{\dfrac{Z_X}{r_{\mathrm{cov}}(X)^2}}{\dfrac{Z_H}{r_{\mathrm{cov}}(H)^2}} \;=\; Z_X\left(\frac{r_{\mathrm{cov}}(H)}{r_{\mathrm{cov}}(X)}\right)^2 \quad\text{(Default $Z_H=1$)}. \label{eq:Pidx_cov}\] Thus, \(P_{\mathrm{idx}}(X)\) is dimensionless.

2.3.2 LOCK Amplitude Equation

The LOCK amplitude equation is: \[r_{\mathrm{eff}}^{(0)}(X) \;=\; \frac{r_{\mathrm{vac}}}{\sqrt{P_{\mathrm{idx}}(X)}} \;=\; r_{\mathrm{vac}}\, \sqrt{\frac{P_{\mathrm{raw}}(H)}{P_{\mathrm{raw}}(X)}}. \label{eq:reff0_lock_alt}\] Here, \(r_{\mathrm{vac}}\) is treated as a normalization (anchor) constant, not as an upper limit like a “maximum vacuum value”. That is, under certain environmental/driving conditions, \(P_{\mathrm{idx}}<1\) is possible, and in that case, \(r_{\mathrm{eff}}>r_{\mathrm{vac}}\) is allowed by [eq:reff0_lock]. (In the application phase, this corresponds to “weakened confinement/deconfinement”.)

2.3.3 Inverse Calculation of LOCK Formula (For Verification)

Equation [eq:reff0_lock] can be inverted as follows: \[P_{\mathrm{idx}}(X) \;=\; \left(\frac{r_{\mathrm{vac}}}{r_{\mathrm{eff}}^{(0)}(X)}\right)^2. \label{eq:Pidx_from_reff}\] This inversion is used for numerical verification of tables within the whitepaper. For example, if \(r_{\mathrm{eff}}^{(0)}(H)=r_{\mathrm{vac}}\) at \(X=H\), then \(P_{\mathrm{idx}}(H)=1\) holds via [eq:Pidx_from_reff]. This is the Self-Consistency Check 1 of this section.

2.4 Definition of Species \(X\): \(Z_X\) Rules for Atoms/Molecules/Ions (Fixing Reproducibility)

In the application phase, not only “atoms” but also “molecules, ions, and surface phases (effective species)” appear. This whitepaper fixes the basic rules for \(Z_X\) as follows.

2.4.1 Atom

For an atom \(X\), the atomic number is used. \[Z_X = Z_{\mathrm{atomic}}(X).\]

2.4.2 Molecule

For a molecule \(X\), the sum of the atomic numbers of its constituent atoms is the default. \[Z_X = \sum_{i\in X} Z_i. \label{eq:Z_molecule}\] This definition is used when treating the “entire molecule as a single effective rotor/effective species”. (When calculating for each atom inside a molecule separately, use the atomic unit \(Z_i\) instead of [eq:Z_molecule].)

2.4.3 Ion

Since the atomic number \(Z\) itself does not change for an ion, the basic rule remains: \[Z_X = Z_{\mathrm{atomic}}(X).\] The charge state of the ion (e.g., Na\(^+\), Cl\(^-\)) is reflected not in \(Z\) but through the Effective Confinement Radius in Solution \(r_{\mathrm{conf}}^{(\mathrm{sol})}\) (hydration, double layer, dielectric environment). In other words, the core redefinition of this section is that “ion charge is handled in \(r_{\mathrm{conf}}\), not \(Z\)”. (If necessary, an advanced extension of \(Z_X \to Z_X^{\ast}\) (effective charge model) can be added, but for reproducibility in this Application Whitepaper v1.0, the basic rule is fixed.)

2.5 Bulk/Surface/Solution Separation: Extending Covalent Radius \(r_{\mathrm{cov}}\) to “Effective Confinement Radius \(r_{\mathrm{conf}}\)”

The primary reason why amplitude values observed (or set) in application simulations differ from the whitepaper’s \(r_{\mathrm{eff}}^{(0)}\) is mostly because the “Environment” (Surface/Solution/Interface) and “Drive” (Electric Field/Temperature/Flow) change the confinement length scale. To capture this mathematically, this section extends \(r_{\mathrm{cov}}\) to the environment-dependent \(r_{\mathrm{conf}}^{(\alpha)}\).

2.5.1 Definition of Environment Index \(\alpha\)

The environment index is: \[\alpha \in \{\mathrm{bulk},\mathrm{surf},\mathrm{sol}\}\] Each represents:

• \(\mathrm{bulk}\): Bulk (Inside crystal/Inside homogeneous solid) standard.

• \(\mathrm{surf}\): Surface (Metal surface, Oxide surface, Nano-coating surface, Catalyst surface).

• \(\mathrm{sol}\): Solution (State affected by hydration/double layer/ionic strength).

2.5.2 Effective Confinement Radius Model

The effective confinement radius is defined in the following general form: \[r_{\mathrm{conf}}^{(\alpha)}(X;\Theta) \;=\; r_{\mathrm{cov}}(X)\cdot \kappa^{(\alpha)}(X;\Theta), \qquad \kappa^{(\alpha)}(X;\Theta)>0, \label{eq:rconf_kappa}\] Here, \(\kappa^{(\alpha)}\) is the confinement scale factor depending on environment/drive.

2.5.2.1 (Bulk) \(\alpha=\mathrm{bulk}\).

In the bulk standard: \[\kappa^{(\mathrm{bulk})}(X;\Theta)\equiv 1 \quad\Rightarrow\quad r_{\mathrm{conf}}^{(\mathrm{bulk})}(X)=r_{\mathrm{cov}}(X). \label{eq:kappa_bulk}\]

2.5.2.2 (Surface) \(\alpha=\mathrm{surf}\).

At the surface, since coordination number reduction, electron density tail (spill-out), and surface potential/work function changes can act to weaken confinement, we assume: \[\kappa^{(\mathrm{surf})}(X;\Theta)\ge 1 \label{eq:kappa_surf_ge1}\] as the basic assumption. (However, if confinement is strengthened at oxide/insulator interfaces, \(\kappa^{(\mathrm{surf})}<1\) is allowed, but it must be fixed with experimental evidence.)

2.5.2.3 (Solution) \(\alpha=\mathrm{sol}\).

In solution, hydration radius, ionic strength (screening), and permittivity changes determine the effective length. We bundle this into a single coefficient: \[\kappa^{(\mathrm{sol})}(X;\Theta) \;=\; \kappa^{(\mathrm{sol})}\!\bigl(X;\,T,\,I,\,\varepsilon_r,\,\mathrm{pH},\,\ldots\bigr) \label{eq:kappa_sol_args}\] To ensure reproducibility, at least \((T,I,\varepsilon_r)\) must be recorded as metadata. (Here \(I\) is ionic strength, \(\varepsilon_r\) is relative permittivity.)

2.5.3 Environmental Effective Pressure Index and Environmental Effective Amplitude

The raw pressure in the environment is: \[P_{\mathrm{raw}}^{(\alpha)}(X;\Theta) \;=\; \frac{Z_X}{\bigl(r_{\mathrm{conf}}^{(\alpha)}(X;\Theta)\bigr)^2} \qquad [\mathrm{pm^{-2}}], \label{eq:Praw_env}\] The environmental pressure index is normalized by the bulk value of reference hydrogen (H): \[P_{\mathrm{idx}}^{(\alpha)}(X;\Theta) \;=\; \frac{P_{\mathrm{raw}}^{(\alpha)}(X;\Theta)}{P_{\mathrm{raw}}(H)} \;=\; \frac{\dfrac{Z_X}{(r_{\mathrm{conf}}^{(\alpha)}(X;\Theta))^2}}{\dfrac{1}{r_{\mathrm{cov}}(H)^2}} \;=\; Z_X\left(\frac{r_{\mathrm{cov}}(H)}{r_{\mathrm{conf}}^{(\alpha)}(X;\Theta)}\right)^2 \label{eq:Pidx_env}\] The environmental effective amplitude is: \[r_{\mathrm{eff}}^{(\alpha)}(X;\Theta) \;=\; \frac{r_{\mathrm{vac}}}{\sqrt{P_{\mathrm{idx}}^{(\alpha)}(X;\Theta)}}. \label{eq:reff_env}\]

2.5.3.1 Verification (Self-Consistency Check 2).

In [eq:Pidx_env]–[eq:reff_env], if \(X=H\) and \(\alpha=\mathrm{bulk}\), then \(r_{\mathrm{conf}}^{(\mathrm{bulk})}(H)=r_{\mathrm{cov}}(H)\) and \(Z_H=1\). Thus, \(P_{\mathrm{idx}}^{(\mathrm{bulk})}(H)=1\) and \(r_{\mathrm{eff}}^{(\mathrm{bulk})}(H)=r_{\mathrm{vac}}\) hold.

2.6 Redefinition of Operational Amplitude \(A_{\mathrm{op}}\): “Effective Amplitude + Driving Excitation”

Since what is directly observed/controlled in the application (device) phase is usually a “state excited by driving”, this whitepaper defines the Operational Amplitude as follows.

2.6.1 Definition of Operational Amplitude

The Operational Amplitude is defined as the Environmental Effective Amplitude multiplied by a driving excitation factor \(\chi\). \[A_{\mathrm{op}}^{(\alpha)}(X;\Theta) \;=\; \chi^{(\alpha)}(X;\Theta)\, r_{\mathrm{eff}}^{(\alpha)}(X;\Theta), \qquad \chi^{(\alpha)}(X;\Theta)>0. \label{eq:aop_def}\] Here, \(\chi\) is a dimensionless coefficient summarizing the excitation caused by “external field/temperature/flow/structure”.

2.6.2 Physics-based Parameterization of Excitation Factor \(\chi\) (Standard for Reproducibility)

For reproducibility in this Application Whitepaper v1.0, \(\chi\) is parameterized as the following dimensionless combination: \[\chi^{(\alpha)}(X;\Theta) \;=\; \sqrt{1+\Xi^{(\alpha)}(X;\Theta)}, \qquad \Xi^{(\alpha)}(X;\Theta)\ge 0, \label{eq:chi_Xi}\] \[\begin{aligned} \Xi^{(\alpha)}(X;\Theta) &= c_T^{(\alpha)}(X)\left(\frac{T-T_0}{T_0}\right)\\ &\quad + c_E^{(\alpha)}(X)\left(\frac{|E|}{E_0}\right)^2\\ &\quad + c_f^{(\alpha)}(X)\left(\frac{f}{f_0}\right)^2\\ &\quad + c_J^{(\alpha)}(X)\left(\frac{J}{J_0}\right)^2\\ &\quad + c_{\dot{m}}^{(\alpha)}(X)\left(\dot{m}/\dot{m}_0\right)^2\\ &\quad + \cdots \end{aligned} \label{eq:Xi_param}\] Here,

• \(T\) is absolute temperature (K), \(T_0\) is reference temperature (K),

• \(|E|\) is electric field magnitude (V/m), \(E_0\) is reference electric field,

• \(f\) is EM/driving frequency (Hz), \(f_0\) is reference frequency,

• \(J\) is current density (A/m\(^2\)), \(J_0\) is reference current density,

• \(\dot{m}\) is mass flow rate (kg/s), \(\dot{m}_0\) is reference mass flow rate,

• \(c_{\bullet}^{(\alpha)}(X)\) are dimensionless coefficients (targets for calibration),

• “\(\cdots\)” represents terms added only when necessary (e.g., pressure, potential, shear rate).

Important: Equation [eq:Xi_param] fixes the physics-based “form”. That is, the electric field effect is proportional to energy density (\(|E|^2\)), and frequency/flow terms are squared to represent resonance/mixing effects by default. The coefficients \(c_{\bullet}^{(\alpha)}(X)\) are estimated from experimental data, and the estimation procedure is fixed in §2.8.

2.6.3 Location of Stable Window (Operational Amplitude Window)

The “Stable Window” repeatedly appearing in application simulations (e.g., \(\pm 250\sim 280\) fm, threshold \(\pm 250\) fm, collapse \(\pm 300\) fm) is treated as a condition for the Operational Amplitude \(A_{\mathrm{op}}\) defined in this section, not for the Intrinsic Amplitude \(r_{\mathrm{eff}}^{(0)}\). That is, \[\text{Stable Operation Condition}:\quad A_{\mathrm{op}}^{(\alpha)}(X;\Theta)\in [A_-,A_+] \label{eq:stable_window}\] The numerical values of \(A_-, A_+\) are fixed by measurement/calibration for each system. (e.g., ESS output ON/OFF threshold, sharp change point in purification separation performance, catalyst reaction onset point.)

2.7 Back-calculation Formulas (Key to Reality Lock): Estimating \(P_{\mathrm{idx}}^{(\alpha)}\) and \(r_{\mathrm{conf}}^{(\alpha)}\) from Observed \(A_{\mathrm{op}}\)

The most important tool for “Fixing the Reality” in this section is Back-calculation. That is, given the Operational Amplitude \(A_{\mathrm{op}}\), we inversely calculate the “Environmental Effective Pressure Index” and “Effective Confinement Radius” to quantify at which level (Bulk/Surface/Solution) and how the confinement change occurred.

2.7.1 \(A_{\mathrm{op}} \to r_{\mathrm{eff}}^{(\alpha)}\)

From definition [eq:aop_def]: \[r_{\mathrm{eff}}^{(\alpha)}(X;\Theta) \;=\; \frac{A_{\mathrm{op}}^{(\alpha)}(X;\Theta)}{\chi^{(\alpha)}(X;\Theta)}. \label{eq:reff_from_aop}\]

2.7.2 \(r_{\mathrm{eff}}^{(\alpha)} \to P_{\mathrm{idx}}^{(\alpha)}\)

Inverting [eq:reff_env]: \[P_{\mathrm{idx}}^{(\alpha)}(X;\Theta) \;=\; \left(\frac{r_{\mathrm{vac}}}{r_{\mathrm{eff}}^{(\alpha)}(X;\Theta)}\right)^2 \;=\; \left(\frac{r_{\mathrm{vac}}\ \chi^{(\alpha)}(X;\Theta)}{A_{\mathrm{op}}^{(\alpha)}(X;\Theta)}\right)^2. \label{eq:pidx_from_aop}\] This is the formula to directly calculate the pressure index implied by the “Operational Amplitude obtained from simulation/experiment”.

2.7.3 \(P_{\mathrm{idx}}^{(\alpha)} \to r_{\mathrm{conf}}^{(\alpha)}\)

Solving [eq:Pidx_env] for \(r_{\mathrm{conf}}^{(\alpha)}\): \[r_{\mathrm{conf}}^{(\alpha)}(X;\Theta) \;=\; r_{\mathrm{cov}}(H)\sqrt{\frac{Z_X}{P_{\mathrm{idx}}^{(\alpha)}(X;\Theta)}}. \label{eq:rconf_from_pidx}\] Alternatively, substituting [eq:pidx_from_aop]: \[r_{\mathrm{conf}}^{(\alpha)}(X;\Theta) \;=\; r_{\mathrm{cov}}(H)\sqrt{Z_X}\, \frac{A_{\mathrm{op}}^{(\alpha)}(X;\Theta)}{r_{\mathrm{vac}}\ \chi^{(\alpha)}(X;\Theta)}. \label{eq:rconf_from_aop}\] Important: Equation [eq:rconf_from_aop] allows us to interpret “Operational Amplitude” by reducing it to the Environmental Confinement Length Scale, rather than just explaining it. That is, by calculating \[\kappa^{(\alpha)}(X;\Theta) \;=\; \frac{r_{\mathrm{conf}}^{(\alpha)}(X;\Theta)}{r_{\mathrm{cov}}(X)} \label{eq:kappa_from_rconf}\] relative to the bulk standard (covalent radius), the degree to which confinement is weakened (or strengthened) on the surface/solution is fixed as a numerical value.

2.7.3.1 Verification (Self-Consistency Check 3).

At \(X=H\), \(Z_X=1\), \(r_{\mathrm{cov}}(H)=31\) pm. In the reference state, if \(\chi=1\) and \(A_{\mathrm{op}}=r_{\mathrm{vac}}\), then [eq:rconf_from_aop] yields \(r_{\mathrm{conf}}=31\) pm, restoring the reference.

2.8 Reproducibility Provision: Calculation Procedure (Algorithm) and Data Schema (For DOI Issuance)

All formulas in this section must be reproducible in the form of “Data \(\to\) Calculation \(\to\) Result”. Therefore, considering DOI issuance, the minimum reproducibility package is fixed as follows.

2.8.1 Mandatory Input Data (Minimum Schema)

Each experiment/simulation record must include at least the following fields:

1. Identifier: Experiment ID, Module ID, Date/Time (Timestamp), Version Tag.

2. Canonical constants: amplitude_ref (fm) \(=245.9\) and quantum_extent (fm) \(=4854\). These fields lock the scale separation (\(r_{\mathrm{vac}}\) vs. \(L_{\mathrm{quant}}\)) for DOI-level reproducibility.

3. Species Info: \(X\) Name, Type (atom/molecule/ion/surface-state), \(Z_X\), \(r_{\mathrm{cov}}(X)\) (pm).

4. Environment: \(\alpha\in\{\mathrm{bulk},\mathrm{surf},\mathrm{sol}\}\). If solution, \(T,I,\varepsilon_r\). If surface, surface phase metadata (Oxide/Nano-coating/Crystal plane, etc.).

5. Drive: Time functions of \(T(t)\), \(|E|(t)\), \(f(t)\), \(J(t)\), \(\dot{m}(t)\), etc., as available.

6. Observed/Estimated Amplitude: \(A_{\mathrm{op}}^{(\alpha)}(t)\) (or measurement proxy variables and conversion formulas corresponding to \(A_{\mathrm{op}}\)).

Note: If \(A_{\mathrm{op}}\) cannot be directly measured in the actual device, the “Proxy variable” and “Conversion Rule” (e.g., conversion formula from Voltage/Current/Impedance/Optical Anisotropy) must be recorded together. Without this record, DOI reproducibility requirements cannot be met.

2.8.2 Calculation Algorithm (Fixed Sequence)

For each record, calculate in the following order:

1. Fix Reference Constants: \(r_{\mathrm{vac}}=245.9\,\mathrm{fm}\), \(r_{\mathrm{cov}}(H)=31\) pm.

2. Calculate \(\chi^{(\alpha)}(X;\Theta)\): Use [eq:chi_Xi]–[eq:Xi_param]. In initial versions: (i) Set \(\chi=1\) for reference experiments. (ii) Estimate \(c_{\bullet}^{(\alpha)}(X)\) from subsequent calibration data.

3. Calculate \(r_{\mathrm{eff}}^{(\alpha)}\): [eq:reff_from_aop].

4. Calculate \(P_{\mathrm{idx}}^{(\alpha)}\): [eq:pidx_from_aop].

5. Calculate \(r_{\mathrm{conf}}^{(\alpha)}\): [eq:rconf_from_aop].

6. Calculate \(\kappa^{(\alpha)}\): [eq:kappa_from_rconf].

7. Check Verification Conditions:

   • \(P_{\mathrm{idx}}^{(\alpha)}>0\), \(r_{\mathrm{conf}}^{(\alpha)}>0\).

   • Check if \((H,\mathrm{bulk})\) satisfies [eq:kappa_bulk] in reference experiments.

   • Check if the variance of \(\kappa\) and \(A_{\mathrm{op}}\) in repeated measurements under identical conditions is within the allowable range.

2.8.3 Calibration Rules: Minimum Requirements for \(\chi\) Coefficient Estimation

Since \(\chi\) is the key coefficient representing “Driving Excitation” in this section, at least the following calibration rules are established.

1. Define at least one reference state for each environment \(\alpha\) (e.g., one for surface, one for solution).

2. Fix \(T_0, E_0, f_0, J_0, \dot{m}_0\) such that \(\Xi=0\) in the reference state.

3. For the same species \(X\), sweep at least 3 driving condition points (e.g., \(|E|\) or \(T\)) and fit the coefficients \(c_{\bullet}^{(\alpha)}(X)\) of [eq:chi_Xi] from the observed \(A_{\mathrm{op}}\).

4. Report the prediction error in conditions not used for fitting (Hold-out).

2.8.4 Release Configuration for DOI (Mandatory List)

For DOI issuance after whitepaper completion, the release bundle related to this section includes at least:

1. Text body (LaTeX), Formula/Symbol definition files.

2. Input Data (raw) and Metadata (meta): CSV/JSON, etc.

3. Calculation Code: Script implementing [eq:reff_from_aop]–[eq:kappa_from_rconf] as is (Version fixed).

4. Reproduction Execution Procedure: Configured so that tables/graphs are reproduced with a single command or notebook.

5. Checksum (e.g., SHA256) and Version Tag (v1.0.0, etc.).

2.9 Conclusion of This Section: Fixing the Reality of “Amplitude” (Summary)

The following have been fixed in this section:

1. The LOCK formula [eq:reff0_lock] provides the Intrinsic (Canonical) Amplitude \(r_{\mathrm{eff}}^{(0)}\).

2. The amplitude used in applications/simulations is defined as the Operational Amplitude \(A_{\mathrm{op}}\), which is the product of the environmental effective amplitude \(r_{\mathrm{eff}}^{(\alpha)}\) and the driving excitation \(\chi\) [eq:aop_def].

3. The difference between Bulk/Surface/Solution is absorbed not by \(r_{\mathrm{cov}}\) but by the Effective Confinement Radius \(r_{\mathrm{conf}}^{(\alpha)}\). The back-calculation formula [eq:rconf_from_aop] allows direct estimation of the environmental confinement length from the operational amplitude of experiments/simulations.

4. For DOI-level reproducibility, the minimum data schema and calculation algorithm have been fixed in this section.

3 Reinterpretation of Simulations: Meaning and Limits of Amplitude/Alignment Variables Used in ESS, Water Purification, and Pt Catalyst

3.1 Goal of this Section (Why Reinterpretation is Needed)

The goal of this section is to reinterpret, based on physics, the ESS, Water Purification, and Pt Catalyst simulations already performed (or presented) in the application phase, in order to: (1) Identify what the “Amplitude” and “Alignment” used in the simulations actually represent (The Entity), (2) Clarify which physical laws/constraints were reflected and which were not (The Limits), (3) Present in a mathematically complete form what is required to fix them reproducibly using experimental measurements (Voltage/Current/Temperature/Salinity/Flow Rate/Gas Flow Rate, etc.).

This section fixes the following principles:

1. The “Amplitude” in simulations is not identified with the Reference Amplitude (LOCK) of Volume I, but is interpreted as the Operational Amplitude \(A_{\mathrm{op}}\) (defined in §3.2).

2. “Alignment” is interpreted as a dimensionless order parameter \(S\in[0,1]\) and must be connected to a measurable proxy variable.

3. Success criteria for ESS/Water/Catalyst must first pass Energy/Mass Balance and Thermodynamic Upper/Lower Bounds.

4. For reproducibility (DOI after whitepaper completion), inputs, outputs, algorithms, and uncertainties are fixed in this section.

3.2 Redefinition of Common State Variables: Operational Amplitude \(A_{\mathrm{op}}\) and Alignment \(S\)

This section fixes the two axes commonly used in the three module simulations (ESS, Water, Catalyst) as follows.

3.2.1 Operational Amplitude \(A_{\mathrm{op}}\)

Operational Amplitude \(A_{\mathrm{op}}(t)\) is defined as a state variable representing “how much the electron (or VP degree of freedom) is excited from the reference state under driving/environmental conditions”. Operational Amplitude is decomposed into the Reference (Environmental) Effective Amplitude \(r_{\mathrm{eff}}\) and the Excitation Factor \(\chi\). \[A_{\mathrm{op}}^{(\alpha)}(X;\Theta,t) \;=\; \chi^{(\alpha)}(X;\Theta,t)\, r_{\mathrm{eff}}^{(\alpha)}(X;\Theta,t), \qquad \chi>0, \label{eq:aop_common}\] Here,

• \(\alpha\in\{\mathrm{bulk},\mathrm{surf},\mathrm{sol}\}\) is the bulk/surface/solution environment,

• \(X\) is the species or effective species (e.g., surface state),

• \(\Theta\) is the set of operating conditions (Temperature \(T\), Electric Field \(|E|\), Frequency \(f\), Current Density \(J\), Flow Rate \(\dot{m}\), Solution Composition, etc.),

• \(r_{\mathrm{eff}}^{(\alpha)}\) is the environmental effective amplitude,

• \(\chi\) is the dimensionless summary coefficient of driving excitation (Electric Field/Temperature/Flow/Structure).

Important: The “\(\pm 250\sim 280\) fm stability window” and “\(\pm 300\) fm collapse” mentioned in application/simulation documents are interpreted as conditions for the Operational Amplitude \(A_{\mathrm{op}}\), not the Reference Amplitude.

3.2.2 Alignment \(S\)

Alignment is defined as a dimensionless order parameter \(S\in[0,1]\). For reproducibility, this whitepaper adopts the following standard definition based on the 2nd Legendre polynomial: \[S(t)=\left\langle P_2(\cos\theta)\right\rangle \;=\; \left\langle \frac{1}{2}\left(3\cos^2\theta-1\right)\right\rangle, \label{eq:S_def}\] Here, \(\theta\) is the angle between the system’s “alignment axis” (e.g., electric field direction, flow direction, structural axis) and the local degree of freedom (e.g., polarization/structural direction). Since it is difficult to measure \(\theta\) directly in experiments, a proxy variable for \(S\) and a conversion formula must be fixed. (e.g., Optical Anisotropy, Impedance Anisotropy, or Flow Stability Index.)

3.3 Mathematically Fixing the Common “Gate (Threshold) Structure” of Simulations

The application simulations (ESS, Water, Catalyst) commonly possess a Gate Structure: “Stability/Output/Separation/Reaction occurs when the amplitude falls within a certain window and the alignment is above a threshold.” This is mathematically fixed as follows.

3.3.1 Stable Gate

The stable gate function \(\mathcal{G}_{\mathrm{st}}\) is defined as: \[\mathcal{G}_{\mathrm{st}}(t) \;=\; H\!\left(A_{\mathrm{op}}(t)-A_{\min}\right)\, H\!\left(A_{\max}-A_{\mathrm{op}}(t)\right)\, H\!\left(S(t)-S_{\min}\right), \label{eq:gate_stable}\] Here, \(H(\cdot)\) is the Heaviside step function: \[H(x)= \begin{cases} 1, & x\ge 0,\\ 0, & x<0. \end{cases}\] \(A_{\min}, A_{\max}\) are the stable amplitude window bounds, and \(S_{\min}\) is the stable alignment lower bound. Typical values often used in simulations are \[A_{\min}\approx 250~\mathrm{fm},\quad A_{\max}\approx 280~\mathrm{fm},\quad S_{\min}\approx 0.70, \label{eq:typical_stable_numbers}\] but in the application whitepaper, these must be re-fixed by actual measurement.

3.3.2 Breakdown/Activation Gate

The gate \(\mathcal{G}_{\mathrm{act}}\), representing breakdown or reaction activation (e.g., bond dissociation), is defined as: \[\mathcal{G}_{\mathrm{act}}(t) \;=\; H\!\left(A_{\mathrm{op}}(t)-A_{\mathrm{br}}\right)\, H\!\left(S_{\mathrm{br}}-S(t)\right), \label{eq:gate_activation}\] Here, \(A_{\mathrm{br}}\) is the breakdown amplitude threshold, and \(S_{\mathrm{br}}\) is the breakdown alignment threshold. Typical values often appearing in simulations are \[A_{\mathrm{br}}\approx 300~\mathrm{fm}, \qquad S_{\mathrm{br}}\approx 0.60\sim 0.70, \label{eq:typical_break_numbers}\] and these must also be fixed by measurement.

3.3.3 Mapping Observables to Gates

The gate structure can model the observable \(Y(t)\) (Output/Reaction Rate/Separation Performance) as follows: \[Y(t)=Y_{\mathrm{base}}(t)\;+\;\Delta Y_{\mathrm{st}}(t)\,\mathcal{G}_{\mathrm{st}}(t)\;+\;\Delta Y_{\mathrm{act}}(t)\,\mathcal{G}_{\mathrm{act}}(t), \label{eq:observable_from_gates}\] Here, \(Y_{\mathrm{base}}\) is the base level when gates are closed (OFF/Low Reaction/Low Separation), \(\Delta Y_{\mathrm{st}}\) is the increment when the stable gate is open, and \(\Delta Y_{\mathrm{act}}\) is the additional change caused by activation/breakdown events. This expression is advantageous for reproducibility as it maintains “Amplitude/Alignment” as latent variables while mathematically fixing their connection to observables.

3.4 ESS Simulation Reinterpretation: Meaning of Amplitude/Alignment, Observables, and Limits

3.4.1 Meaning of Amplitude/Alignment Variables Used in ESS

In ESS simulations, amplitude \(A_{\mathrm{op}}\) was used as the condition for the initiation (ON) and maintenance (stable output) of the “electric field/charge separation state”. Alignment \(S\) was used as a threshold variable determining the “stability of the electric field/rotor state” or the “persistence of output”. Thus, the gate in ESS is interpreted as: \[\text{ESS Output ON Condition}:\quad \mathcal{G}_{\mathrm{st}}(t)=1 \quad\Rightarrow\quad P_{\mathrm{out}}(t)>0, \label{eq:ess_on_condition}\] Here, \(P_{\mathrm{out}}(t)=V(t)I(t)\) is the DC power.

3.4.2 Mathematical Completeness of ESS Observables (Electrical Output) and Energy Balance

The key observables for ESS simulations are: \[\begin{aligned} P_{\mathrm{out}}(t) &= V(t)\,I(t), \label{eq:ess_power}\\ E_{\mathrm{out}} &= \int_{t_0}^{t_1} V(t)I(t)\,dt, \label{eq:ess_energy_out}\\ E_{\mathrm{in}} &= \int_{t_0}^{t_1} G(t)A_{\mathrm{col}}\,dt, \label{eq:ess_energy_in}\\ \eta_{\mathrm{total}} &= \frac{E_{\mathrm{out}}}{E_{\mathrm{in}}}\le 1. \label{eq:ess_eta_total}\end{aligned}\] From the perspective of Fixing the Reality, all ESS efficiency claims are recognized only under definition [eq:ess_eta_total]. If a simulation presents only \(P_{\mathrm{out}}\) without calculating \(E_{\mathrm{in}}\) over the same time window, the result is judged as usable for “preliminary output curves” but HOLD for “efficiency”.

3.4.3 Connecting ESS Amplitude \(A_{\mathrm{op}}\) to Measurable Physical Quantities (Calibration Layer)

If \(A_{\mathrm{op}}\) cannot be directly measured in the actual device, ESS defines \(A_{\mathrm{op}}\) through the following “Calibration Layer”: \[A_{\mathrm{op}}(t)=f_{\mathrm{ESS}}\!\left(T_h(t),T_c(t),V(t),I(t),R_{\mathrm{load}},\ldots\right), \label{eq:aop_ess_proxy}\] Here, \(T_h\) is the hot-side temperature, \(T_c\) is the cold-side (ambient/cooling part) temperature, and \(R_{\mathrm{load}}\) is the load resistance. For reproducibility, this application whitepaper fixes \(f_{\mathrm{ESS}}\) to the following minimum form.

3.4.3.1 (Minimum Calibration Form: Monotonic Mapping).

Under the assumption that the output initiation threshold is governed by \(T_h\): \[A_{\mathrm{op}}(t)=A_{\min} + (A_{\max}-A_{\min})\,\sigma\!\left(\frac{T_h(t)-T_{\mathrm{th}}}{\Delta T}\right), \label{eq:aop_vs_Th}\] Here, \(\sigma(x)=\frac{1}{1+e^{-x}}\) is the logistic function, \(T_{\mathrm{th}}\) is the threshold temperature (use value from simulation as initial if available), and \(\Delta T\) is the transition width (calibration target). Equation [eq:aop_vs_Th] fixes the simulation structure where “\(A_{\mathrm{op}}\) increases sharply at a critical temperature” to a measurable temperature curve.

3.4.3.2 (Limits).

While [eq:aop_vs_Th] is structurally valid, this mapping alone does not guarantee that “Heat \(\to\) Electricity conversion efficiency” does not violate the Carnot limit. Therefore, ESS must report [eq:ess_eta_total] and (if it is a heat engine) the Carnot limit \[\eta_{\mathrm{conv}}\le 1-\frac{T_c}{T_h} \label{eq:ess_carnot}\] together. (If not reported, the simulation is valid only as a gate model but HOLD as a physical efficiency model.)

3.5 Water Purification (Desalination) Simulation Reinterpretation: Meaning of Amplitude/Alignment, Transport Equations, and Limits

3.5.1 Meaning of Amplitude in Water Purification Simulations

Water purification simulations assume that “Water (H\(_2\)O)” and “Ions (Na\(^+\), Cl\(^-\))” have different amplitude bands, and this difference leads to separation. This application whitepaper reinterprets this to fix the reality as follows.

3.5.1.1 Reinterpretation (Fixing the Reality).

Amplitude \(A_{\mathrm{op}}^{(\mathrm{sol})}(X)\) in water purification is interpreted as a “latent variable representing the effective mobility or effective response of species \(X\) in solution”. That is, a large \(A_{\mathrm{op}}\) does not mean “it sinks because it is heavy”, but rather that the response (movement, polarization, adsorption) under electric field/driving changes.

3.5.2 Minimum Physics-Based Model for Water: Nernst–Planck Transport + Convection (Mandatory)

Water purification is physically a “transport problem”, and ion transport is minimally described by the Nernst–Planck equation. The molar flux (or flux proportional to mass flux) of ion \(i\) is defined as: \[\mathbf{J}_i \;=\; -D_i\nabla c_i \;-\; z_i \mu_i c_i \nabla\phi \;+\; c_i \mathbf{u}, \label{eq:nernst_planck}\] Here,

• \(c_i\) is concentration,

• \(D_i\) is diffusion coefficient,

• \(z_i\) is charge number,

• \(\mu_i\) is mobility,

• \(\phi\) is electric potential,

• \(\mathbf{u}\) is fluid velocity field (convection).

Concentration evolves via the continuity equation: \[\frac{\partial c_i}{\partial t} + \nabla\cdot \mathbf{J}_i = R_i, \label{eq:continuity_np}\] Here, \(R_i\) is the generation/consumption term at reaction/adsorption/electrode sites (can be \(R_i\neq 0\) near electrodes/walls even without membranes).

3.5.2.1 Intervention of Amplitude/Alignment (Calibration Layer).

Amplitude/Alignment in water simulations are fixed to intervene in the coefficients of [eq:nernst_planck]. That is, \[\begin{aligned} \mu_i &= \mu_i\!\left(A_{\mathrm{op}}^{(\mathrm{sol})}(i),\,S^{(\mathrm{sol})}\right), \label{eq:mobility_vs_aop}\\ D_i &= D_i\!\left(A_{\mathrm{op}}^{(\mathrm{sol})}(i),\,S^{(\mathrm{sol})}\right), \label{eq:diffusion_vs_aop}\end{aligned}\] and the relationship between \(\mu\) and \(D\) is constrained by the Einstein relation: \[D_i = \mu_i\,k_B T \quad (\text{Use consistent coefficient forms if adopting charge-inclusive definitions}). \label{eq:einstein_relation}\] Thus, the assumption that “amplitudes are different” is converted to “mobility/diffusion are different”, which is verified by measurable conductivity/current/concentration distributions.

3.5.3 Separation Condition (Time/Length Scales): Residence Time and Separation Time

Let \(L\) be the representative length of the purification device (tube/channel), \(u\) be the mean flow velocity, and \(\tau_{\mathrm{res}}=L/u\) be the residence time. Let the electromigration drift velocity of ion \(i\) be \(v_{i,\mathrm{drift}}=\mu_i |E|\). The representative drift distance is: \[\ell_{i,\mathrm{drift}} \approx v_{i,\mathrm{drift}}\,\tau_{\mathrm{res}} = \mu_i |E|\frac{L}{u}. \label{eq:drift_length}\] The mixing length due to diffusion is: \[\ell_{i,\mathrm{diff}} \approx \sqrt{2D_i \tau_{\mathrm{res}}} = \sqrt{2D_i \frac{L}{u}}. \label{eq:diff_length}\] For separation to occur, with respect to the characteristic length \(\ell_{\mathrm{sep}}\) in the separation direction of the structure (slit/branch), conditions like: \[\ell_{i,\mathrm{drift}} \gtrsim \ell_{\mathrm{sep}} \quad\text{and}\quad \ell_{i,\mathrm{diff}} \ll \ell_{\mathrm{sep}} \label{eq:separation_inequalities}\] are required. In other words, a “long tube” should be interpreted not just for gravity, but to increase \(\tau_{\mathrm{res}}\) to increase [eq:drift_length] and induce separation relative to [eq:diff_length].

3.5.4 Limits of Water Purification Simulations (Mandatory Check)

For water purification simulations to be recognized as physically “complete technology”, the following must be met:

1. Energy Balance: Desalination Energy Consumption (SEC) must be calculated via [eq:sec]. Even if claiming “zero external power”, structural head/flow loss/field generation energy are Hidden Inputs and must be measured.

2. Mass Balance: Equations [eq:mass_balance_flow]–[eq:mass_balance_salt] must hold within measurement error.

3. Conversion to Drive (EM Wave): If using electric fields/EM waves instead of magnets, specify how \(|E|, f, \nabla |E|\) form the electromigration term in [eq:nernst_planck] (Electrodes/Potentials/Boundary Conditions).

4. Realization of Amplitude/Alignment: \(A_{\mathrm{op}}\) and \(S\) must be calibrated via conversion rules to measurable quantities like conductivity, current, impedance, or concentration distribution. Otherwise, they remain latent variables (HOLD).

3.6 Pt Catalyst Simulation Reinterpretation: Meaning of Amplitude/Alignment, Reaction Rate Model, and Limits

3.6.1 Meaning of Amplitude/Alignment Used in Pt Catalyst

In Pt catalyst simulations, amplitude was used as a state variable representing “bond stability of reactants (H\(_2\))” and “rearrangement on the surface”. This application whitepaper reinterprets this as Effective Barrier Change on the Reaction Coordinate to fix the reality.

3.6.1.1 Reinterpretation (Fixing the Reality).

The reaction rate constant \(k\) of a catalytic reaction is generally governed by the activation free energy \(\Delta G^\ddagger\): \[k(T) = \nu \exp\!\left(-\frac{\Delta G^\ddagger}{k_B T}\right), \label{eq:arrhenius_like}\] Here, \(\nu\) is the attempt frequency (pre-exponential factor). VP simulation amplitude/alignment variables can be included as latent variables that modulate \(\Delta G^\ddagger\): \[\Delta G^\ddagger = \Delta G^\ddagger_0 - \Lambda_A\,g_A\!\left(A_{\mathrm{op}}^{(\mathrm{surf})}\right) - \Lambda_S\,g_S\!\left(S^{(\mathrm{surf})}\right), \label{eq:barrier_modulated}\] Here, \(\Lambda_A,\Lambda_S\ge 0\) are coupling coefficients, and \(g_A,g_S\) are dimensionless monotonic functions. Thus, the simulation statement “bond breaks when amplitude exceeds threshold” becomes physically interpretable as “reaction rate surges due to barrier reduction”.

3.6.2 Coupling Threshold-Based Gates to Reaction Rate (Mathematical Fix)

To couple the threshold concept frequently used in simulations (e.g., \(A_{\mathrm{br}}\approx 300\) fm) to the reaction rate, \(g_A\) is fixed as a step or smooth transition function. For example: \[g_A(A)=\sigma\!\left(\frac{A-A_{\mathrm{br}}}{\Delta A}\right), \qquad \sigma(x)=\frac{1}{1+e^{-x}}, \label{eq:gA_logistic}\] \[g_S(S)=\sigma\!\left(\frac{S-S_{\min}}{\Delta S}\right). \label{eq:gS_logistic}\] Then, from [eq:barrier_modulated]–[eq:arrhenius_like], the structure where \(k\) surges in the region where \(A_{\mathrm{op}}\) exceeds the threshold is mathematically reproduced.

3.6.3 Connection with Electrochemistry (HER/Electrolysis) (ESS Electrical Application)

In the application phase, Pt catalysts can be connected not just to simple surface reactions but also to Water Electrolysis/Hydrogen Evolution Reaction (HER). The minimum model for electrochemical reaction rate is the Butler–Volmer equation: \[j = j_0 \left[ \exp\!\left(\frac{\alpha_a F\eta}{RT}\right) - \exp\!\left(-\frac{\alpha_c F\eta}{RT}\right) \right], \label{eq:butler_volmer}\] Here, \(j\) is current density, \(j_0\) is exchange current density, \(\eta\) is overpotential, \(\alpha_a,\alpha_c\) are transfer coefficients, \(F\) is Faraday constant, and \(R\) is gas constant. VP simulation amplitude/alignment can enter as \(j_0\) or an effective overpotential term as follows: \[j_0 = j_{0,\mathrm{base}}\; h_A\!\left(A_{\mathrm{op}}^{(\mathrm{surf})}\right)\; h_S\!\left(S^{(\mathrm{surf})}\right), \label{eq:j0_modulated}\] Here, \(h_A, h_S\) are positive functions and must be calibrated with experimental data (Current-Voltage curves). Fixing this connection reproduces the entire “ESS Electricity \(\to\) Electrolysis \(\to\) Pt Catalyst” chain within a measurable electrochemical framework.

3.6.4 Limits of Pt Catalyst Simulations (Mandatory Check)

1. Idealization of Heat/Friction/Fluid: If idealizations like zero friction or elasticity of 1 are included, the results are valid as “Qualitative Mechanisms” but cannot be used immediately for quantitative performance (Reaction Rate, Yield).

2. Realization of Amplitude: If \(A_{\mathrm{op}}\) is not calibrated to measurable quantities like adsorption energy, barrier, or \(j_0\), the “Amplitude Threshold” is irreproducible in experiments.

3. Separation of Hydrogen Production vs. Activation: The process of making H\(_2\) (Electrolysis) and splitting or activating H\(_2\) (Pt Catalyst) are physically different stages and must be reported separately in energy/mass balance.

3.7 Common Limits of the Three Simulations and Solutions in “Application Whitepaper Design”

3.7.1 Limit 1: Amplitude/Alignment are Latent Variables, so without Calibration to Observables, they are not “Real entities”

\(A_{\mathrm{op}}\) and \(S\) are difficult to measure directly. Therefore, Proxy Variables and Calibration Functions (Mappings) must be fixed. This application whitepaper sets the following minimum calibration rules:

1. ESS: Fix monotonic mapping \(A_{\mathrm{op}}\leftrightarrow (T_h, V, I)\) (e.g., [eq:aop_vs_Th]) and calibrate \(\Delta T\) with repeated identical condition experiments.

2. Water: Connect \(A_{\mathrm{op}}\leftrightarrow (\mu_i, D_i)\) ([eq:mobility_vs_aop]–[eq:diffusion_vs_aop]) and calibrate with conductivity/current/concentration distributions.

3. Pt: Connect \(A_{\mathrm{op}},S \leftrightarrow (\Delta G^\ddagger, j_0)\) ([eq:barrier_modulated], [eq:j0_modulated]) and calibrate with reaction rates/electrochemical curves.

3.7.2 Limit 2: Without Fixing Energy/Mass Balance (Conservation Laws), “Performance Figures” are Incomparable

Performance must be fixed via: [eq:ess_eta_total] Overall Efficiency for ESS, [eq:sec] SEC and [eq:mass_balance_salt] Mass Balance for Water, and [eq:faraday_real] Faraday-based Mass Balance (Charge \(\to\) Hydrogen Amount) for Electrolysis. If these definitions are missing, simulation results can be used for “Relative Comparison” but are judged HOLD for engineering performance claims.

3.7.3 Limit 3: If Values Violate Thermodynamic Upper/Lower Bounds, it is not Success but a trigger for Re-verification of “Definition/Measurement/Hidden Inputs”

If results violate the Heat Engine Limit ([eq:ess_carnot]), Minimum Desalination Energy Lower Limit, or show \(E_{\mathrm{out}}>E_{\mathrm{in}}\), it is more likely due to (1) Missing Input Definition, (2) Integration Error, (3) Hidden Inputs (Electricity/Heat/Pressure), or (4) Time Window Error, rather than “New Physics”. The application whitepaper classifies such cases not as FAIL but as Re-verification (HOLD), requiring cross-measurement (independent sensors, external lab reproduction) first.

3.8 Reproducibility Provision (For DOI Issuance): Experiment/Simulation Log Schema for this Section

Although this section is “Simulation Reinterpretation”, the following log schema is fixed for DOI-level reproducibility.

3.8.1 Common Log (All Modules)

1. Time Axis: Sampling period \(\Delta t\), Timestamp (UTC or Local specified).

2. Input: \(T(t)\), \(|E|(t)\), \(f(t)\), \(J(t)\), \(\dot{m}(t)\), Feed Concentration \(C_f(t)\), etc.

3. Output: \(V(t),I(t)\) for ESS; \(C_p(t),C_c(t),Q_p(t),Q_c(t)\) for Water; \(j(t),\eta(t), Q_{H_2}(t)\) for Catalyst/Electrolysis.

4. Proxy Variables: If \(A_{\mathrm{op}}(t)\) and \(S(t)\) cannot be measured directly, record the proxy variables and conversion formulas (version fixed) together.

5. Integrated Results: Final KPIs like \(E_{\mathrm{in}}, E_{\mathrm{out}}, \mathrm{SEC}, m_{H_2}, \eta_F\).

3.8.2 Reproduction Execution Procedure (Fixed)

1. Perform integration of [eq:ess_energy_out], [eq:ess_energy_in] using raw logs as input.

2. For Water, calculate SEC via [eq:sec] and calculate [eq:removal_ratio]–[eq:recovery_ratio].

3. For Electrolysis, cross-verify Faraday-based [eq:faraday_real] with actual gas measurement to calculate \(\eta_F\).

4. Calibration Layer: Apply the function corresponding to the module ([eq:aop_vs_Th], [eq:mobility_vs_aop], or [eq:j0_modulated]) to estimate \(A_{\mathrm{op}}, S\) and record gate open/close events based on [eq:gate_stable]–[eq:gate_activation].

3.9 Conclusion of This Section (Summary): Final Fixing of Meaning and Limits

1. The “Amplitude/Alignment” in ESS, Water, and Pt Catalyst simulations commonly implement a Gate Structure (ON/OFF, Separation/No-Separation, Reaction/No-Reaction).

2. Contradictions are removed by reinterpreting Amplitude as Operational Amplitude \(A_{\mathrm{op}}\), not Reference (LOCK) Amplitude.

3. Alignment is fixed as a dimensionless order parameter \(S\), but it is not a real entity without proxy variables and conversion formulas.

4. To claim quantitative performance based on physics, calibration with Energy Balance for ESS, Nernst–Planck Transport and SEC for Water, and Barrier/Electrochemistry (Butler–Volmer) for Catalyst is mandatory. Missing this results in a HOLD judgment.

5. For DOI-level reproducibility, this section has fixed the common log schema and reproduction execution procedure.

4 ESS (Black Copper Based) Module: Thermal Balance, Carnot Limit, Realistic Power Conversion Options (TEG/Stirling/ORC), and Design Parameters

4.1 Purpose of this Section and Definition of ESS “Reality”

4.1.0.1 Scale separation note for energy mapping.

When a VP-based energy-density interpretation is used in this ESS module, the geometric volume scale is fixed by the quantum diameter \(L_{\mathrm{quant}}=4854\,\mathrm{fm}\) (lattice cell extent), while charge separation and conversion efficiency are governed by the amplitude scale \(r_{\mathrm{vac}}=245.9\,\mathrm{fm}\) (amplitude-driven interactions). This separation prevents misusing the quantum extent as an interaction-amplitude proxy. The purpose of this section is to organize the ESS (Black Copper based solar thermal collection/storage) module so that it can be designed based on Thermal Balance (Energy Conservation), to explicitly state Thermodynamic Upper Limits including the Carnot Limit, and to present realistic power conversion options (TEG/Stirling/ORC) in a mathematically complete form.

In this application whitepaper, the “Reality (Ground Truth)” of the ESS is fixed by the following measurement definitions:

1. Solar Input Energy: \[E_{\mathrm{in}} \;=\; \int_{t_0}^{t_1} G(t)\,A_{\mathrm{col}}\,dt \qquad [\mathrm{J}] \ \text{or}\ [\mathrm{Wh}], \label{eq:ess_Ein}\] Here, \(G(t)\) is the effective solar irradiance on the collector surface (W/m\(^2\)), and \(A_{\mathrm{col}}\) is the collector area (m\(^2\)).

2. Electrical Output Energy: \[E_{\mathrm{out}} \;=\; \int_{t_0}^{t_1} V(t)\,I(t)\,dt \qquad [\mathrm{J}] \ \text{or}\ [\mathrm{Wh}], \label{eq:ess_Eout}\] Here, \(V(t)\) is the DC output voltage (V), and \(I(t)\) is the output current (A).

3. Overall Efficiency (Definition): \[\eta_{\mathrm{total}} \;=\; \frac{E_{\mathrm{out}}}{E_{\mathrm{in}}}, \qquad 0\le \eta_{\mathrm{total}} \le 1. \label{eq:ess_eta_total_def}\] Verification Rule: If \(\eta_{\mathrm{total}}>1\) appears in actual measurement, it is not a success but signifies potential errors in input/output definition, measurement/integration, or the existence of hidden inputs. Thus, it is treated as Re-verification (HOLD).

4.2 ESS System Boundary and Components

The ESS module is defined as consisting of the following four sub-blocks:

1. Collection Unit (Including Black Copper Selective Absorber): Solar radiation input \(\to\) Absorbed heat generation.

2. Thermal Storage Unit (Heat Storage Medium): Stores absorbed heat, supplies heat during night/fluctuations.

3. Power Conversion Unit (Realistic Options): Converts stored heat into electrical work (TEG/Stirling/ORC).

4. Power Electronics/Output Unit: DC bus stabilization, load matching, measurement/protection.

The representative temperatures used in this section are fixed as follows:

• \(T_a(t)\): Ambient (Environmental) Temperature (K).

• \(T_s(t)\): Representative Temperature of Thermal Storage Unit (Storage Medium) (K).

• \(T_h(t)\): Hot-side Temperature of Power Conversion Unit (K).

• \(T_c(t)\): Cold-side Temperature of Power Conversion Unit (K).

Generally, in design, it can be approximated as \(T_h(t)\approx T_s(t)\), but since \(T_h<T_s\) may occur due to contact thermal resistance or heat exchanger performance, this application whitepaper treats \(T_s\) and \(T_h\) as distinct measurement channels in principle.

4.3 Optical-Thermal Model of Black Copper Based Collection Unit (Selective Absorber)

4.3.1 Definition of Optical Absorption (Input Heat Flux)

The effective heat flux (absorbed heat rate) absorbed by the collection unit is defined as: \[\dot{Q}_{\mathrm{abs}}(t) \;=\; \eta_{\mathrm{opt}}(t)\,G(t)\,A_{\mathrm{col}}, \qquad [\mathrm{W}], \label{eq:Qabs}\] Here, the optical efficiency \(\eta_{\mathrm{opt}}(t)\) is fixed as the product of: \[\eta_{\mathrm{opt}}(t) \;=\; \tau_{\mathrm{cov}}(t)\,\alpha_{\mathrm{sol}}(t)\,K_\theta(t), \label{eq:eta_opt}\]

• \(\tau_{\mathrm{cov}}\): Transmittance (dimensionless) when a cover (glass/polymer/vacuum tube, etc.) is present.

• \(\alpha_{\mathrm{sol}}\): Solar spectrum average absorptance of Black Copper (including selective coating) (dimensionless).

• \(K_\theta\): Incidence Angle Modifier, \(0<K_\theta\le 1\).

Reproducibility Requirement: \(\alpha_{\mathrm{sol}}\) and (if possible) thermal emittance \(\varepsilon_{\mathrm{th}}\) must be estimated via spectral measurement of samples or standard measurement methods and recorded as metadata. (If measurement is impossible, this whitepaper sets \(\alpha_{\mathrm{sol}}\) as a design parameter and performs sensitivity analysis.)

4.3.2 Steady-State Check Formula: Absorbed Heat Must Exceed Loss at Target Temperature

To accumulate heat or supply heat to the power conversion unit at the target temperature \(T_s=T_s^\ast\), \[\dot{Q}_{\mathrm{abs}}(t) \;>\; \dot{Q}_{\mathrm{loss}}(t) + \dot{Q}_{\mathrm{conv}}(t) \quad\text{(During heating or power generation)}, \label{eq:steady_condition}\] is required. Here, \(\dot{Q}_{\mathrm{conv}}\) is the heat flow rate transferred to the power conversion unit (Heat Engine/TEG input). In the heating-only section (Generation OFF), we set \(\dot{Q}_{\mathrm{conv}}=0\).

4.4 Dynamic Thermal Balance Equation of Thermal Storage Unit

4.4.1 Thermal Capacity and Stored Energy

The effective thermal capacity of the thermal storage unit is defined as: \[C_{\mathrm{th}} \;=\; m_{\mathrm{st}}\,c_{p,\mathrm{st}} \qquad [\mathrm{J/K}], \label{eq:Cth}\] Here, \(m_{\mathrm{st}}\) is the mass of the storage medium (kg), and \(c_{p,\mathrm{st}}\) is the specific heat at constant pressure (J/kg/K). The stored thermal energy at temperature \(T_s(t)\) (relative to reference \(T_{\mathrm{ref}}\)) is: \[E_{\mathrm{st}}(t) \;=\; \int_{T_{\mathrm{ref}}}^{T_s(t)} C_{\mathrm{th}}(T)\,dT. \label{eq:Est_general}\] For reproducibility in this Whitepaper v1.0, we approximate \(C_{\mathrm{th}}\) as a constant: \[E_{\mathrm{st}}(t) \;\approx\; C_{\mathrm{th}}\,(T_s(t)-T_{\mathrm{ref}}). \label{eq:Est_linear}\] (Temperature dependence of specific heat is extended in the Appendix.)

4.4.2 Dynamic Thermal Balance of Thermal Storage Unit (Differential Equation)

Energy conservation of the thermal storage unit is fixed by the following differential equation: \[C_{\mathrm{th}}\frac{dT_s}{dt} \;=\; \dot{Q}_{\mathrm{abs}}(t) \;-\; \dot{Q}_{\mathrm{loss}}(t) \;-\; \dot{Q}_{\mathrm{conv}}(t), \label{eq:ess_heat_ode}\] Each term is defined as:

• \(\dot{Q}_{\mathrm{abs}}(t)\): Absorbed heat from [eq:Qabs].

• \(\dot{Q}_{\mathrm{loss}}(t)\): Heat loss to the outside (including radiation/convection/conduction).

• \(\dot{Q}_{\mathrm{conv}}(t)\): Heat transferred to the power conversion unit (Generation input heat).

4.4.3 Heat Loss Model (Radiation + Convection/Conduction)

Heat loss is fixed to include at least the following two components: \[\dot{Q}_{\mathrm{loss}}(t) \;=\; \dot{Q}_{\mathrm{rad}}(t) \;+\; \dot{Q}_{\mathrm{UA}}(t). \label{eq:Qloss_split}\]

4.4.3.1 (1) Radiation Loss.

Let the thermal emittance of the Black Copper selective absorber be \(\varepsilon_{\mathrm{th}}\) and the effective radiation area be \(A_{\mathrm{rad}}\). Then: \[\dot{Q}_{\mathrm{rad}}(t) \;=\; \varepsilon_{\mathrm{th}}\,\sigma\,A_{\mathrm{rad}} \left(T_{\mathrm{surf}}(t)^4 - T_a(t)^4\right), \label{eq:Qrad}\] Here, \(\sigma=5.670374419\times 10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}\) is the Stefan-Boltzmann constant, and \(T_{\mathrm{surf}}\) is the surface temperature facing the outside (K). (Since \(T_{\mathrm{surf}}<T_s\) may occur due to insulation/vacuum/cover in design, it is recommended to measure \(T_s\) and \(T_{\mathrm{surf}}\) separately in experiments.)

4.4.3.2 (2) Convection + Conduction (UA Model).

Let the overall heat transfer coefficient through insulation/cover/frame be \(U_{\mathrm{loss}}\) and the effective heat loss area be \(A_{\mathrm{loss}}\). Then: \[\dot{Q}_{\mathrm{UA}}(t) \;=\; U_{\mathrm{loss}}\,A_{\mathrm{loss}}\,(T_{\mathrm{surf}}(t)-T_a(t)). \label{eq:QUA}\] \(U_{\mathrm{loss}}\) can be fixed as a series thermal resistance of “conduction + convection”. Given insulation thermal conductivity \(k_{\mathrm{ins}}\) (W/m/K), thickness \(L_{\mathrm{ins}}\) (m), and external convection coefficient \(h_{\mathrm{out}}\) (W/m\(^2\)/K): \[\frac{1}{U_{\mathrm{loss}}} \;=\; \frac{L_{\mathrm{ins}}}{k_{\mathrm{ins}}} \;+\; \frac{1}{h_{\mathrm{out}}}. \label{eq:Uloss_series}\] (If a cover/vacuum layer is added, add thermal resistance terms in the same way.)

4.4.4 Inverse Estimation of \(C_{\mathrm{th}}\) and \(U_{\mathrm{loss}}\) via Nighttime (No Solar) Cooling Curve (Core Verification/Reproducibility)

In the no-solar section (\(G(t)=0\)), if we select a time when power generation is also stopped (\(\dot{Q}_{\mathrm{conv}}(t)=0\)), [eq:ess_heat_ode] becomes: \[C_{\mathrm{th}}\frac{dT_s}{dt} \;=\; -\dot{Q}_{\mathrm{rad}}(t)-\dot{Q}_{\mathrm{UA}}(t), \label{eq:cooling_ode}\] That is: \[C_{\mathrm{th}}\frac{dT_s}{dt} \;=\; -\varepsilon_{\mathrm{th}}\,\sigma\,A_{\mathrm{rad}}\left(T_{\mathrm{surf}}^4 - T_a^4\right) -U_{\mathrm{loss}}A_{\mathrm{loss}}(T_{\mathrm{surf}}-T_a). \label{eq:cooling_ode_full}\] In experiments, record \(T_s(t)\), \(T_{\mathrm{surf}}(t)\), \(T_a(t)\), and calculate the derivative \(dT_s/dt\) using central difference. For time \(t_k\): \[\left.\frac{dT_s}{dt}\right|_{t=t_k} \;\approx\; \frac{T_s(t_{k+1})-T_s(t_{k-1})}{2\Delta t}, \label{eq:central_diff}\] where \(\Delta t=t_{k}-t_{k-1}\). By fitting \((C_{\mathrm{th}}, U_{\mathrm{loss}}, \varepsilon_{\mathrm{th}})\) to [eq:cooling_ode_full] using least squares, the thermal storage/insulation performance is fixed based on measurement. This inverse estimation must be included as a verification tool to exclude hidden inputs when “efficiency appears excessive”.

4.5 Carnot Limit and Upper Bound of ESS Output (Verification Formula)

4.5.1 Heat Engine Upper Limit (Carnot Efficiency)

If the power conversion unit is a heat engine (or a system equivalent to Heat \(\to\) Work conversion), the conversion efficiency has a Carnot efficiency upper limit. \[\eta_{\mathrm{Carnot}}(t) \;=\; 1-\frac{T_c(t)}{T_h(t)}, \qquad (T_h(t) > T_c(t)). \label{eq:carnot_eff}\]

4.5.2 Absolute Upper Bound Verification: Output Power Cannot Exceed Absorbed Heat \(\times\) Carnot Limit

When the heat transferred to the power conversion unit is \(\dot{Q}_{\mathrm{conv}}(t)\), the electrical output power is: \[P_{\mathrm{out}}(t) \;\le\; \eta_{\mathrm{Carnot}}(t)\,\dot{Q}_{\mathrm{conv}}(t) \;\le\; \eta_{\mathrm{Carnot}}(t)\,\dot{Q}_{\mathrm{abs}}(t). \label{eq:power_upper_bound}\] Verification Rule: If a section violating [eq:power_upper_bound] is found in measurement or simulation, that section is not classified as success but as a subject for Re-verification of Measurement/Definition/Hidden Input.

4.5.3 Example Verification (Procedure, not Value): Maximum Possible Power at Target Temperature

As an example, if \(T_h=145^\circ\mathrm{C}=418\,\mathrm{K}\) and \(T_c=27^\circ\mathrm{C}=300\,\mathrm{K}\): \[\eta_{\mathrm{Carnot}} \;=\; 1-\frac{300}{418} \;\approx\; 0.282. \label{eq:carnot_example}\] Also, setting \(G=800\,\mathrm{W/m^2}\), \(A_{\mathrm{col}}=0.5\,\mathrm{m^2}\), \(\eta_{\mathrm{opt}}=\alpha_{\mathrm{sol}}=0.982\), and \(K_\theta=\tau_{\mathrm{cov}}=1\): \[\dot{Q}_{\mathrm{abs}} \;=\; 0.982 \times 800 \times 0.5 \;\approx\; 393\,\mathrm{W}. \label{eq:Qabs_example}\] At this time, the absolute upper bound is: \[P_{\mathrm{out}} \;\le\; 0.282 \times 393 \;\approx\; 111\,\mathrm{W}. \label{eq:Pmax_example}\] Interpretation: This value is an ideal upper bound assuming “zero loss” and “Carnot achievement”. Therefore, if 100 W class output is observed, it implies one of the following: (i) losses are extremely small, (ii) actual absorbed heat is larger (area/irradiance/concentration), (iii) hidden inputs exist, or (iv) measurement definition is incomplete. Thus, the application whitepaper must verify this with actual measurements via [eq:ess_Ein]–[eq:ess_eta_total_def].

4.6 General Formula for Power Conversion Unit (Common to All Options)

Let \(\dot{Q}_{\mathrm{conv}}(t)\) be the heat flow transferred to the power conversion unit, \(\eta_{\mathrm{conv}}(t)\) be the Heat \(\to\) Electricity conversion efficiency, and \(\eta_{\mathrm{elec}}(t)\) be the electrical efficiency of power electronics/generator/wiring. The electrical output power is: \[P_{\mathrm{out}}(t) \;=\; \eta_{\mathrm{elec}}(t)\,\eta_{\mathrm{conv}}(t)\,\dot{Q}_{\mathrm{conv}}(t), \label{eq:Pout_general}\] and from [eq:power_upper_bound]: \[\eta_{\mathrm{conv}}(t)\;\le\;\eta_{\mathrm{Carnot}}(t) \quad\text{(For heat engine equivalent conversion)}. \label{eq:conv_le_carnot}\]

Also, the heat input to the power conversion unit is limited by the thermal storage temperature and the heat exchanger. Let the heat exchanger equivalent thermal resistance be \(R_{\mathrm{th,hx}}\) (K/W): \[\dot{Q}_{\mathrm{conv}}(t) \;=\; \frac{T_s(t)-T_h(t)}{R_{\mathrm{th,hx}}} \;=\; U_{\mathrm{hx}}A_{\mathrm{hx}}\,(T_s(t)-T_h(t)), \label{eq:Qconv_hx}\] Here, \(U_{\mathrm{hx}}A_{\mathrm{hx}}=1/R_{\mathrm{th,hx}}\) is the overall heat transfer coefficient product (UA) of the heat exchanger. In actual design, \(R_{\mathrm{th,hx}}\) is minimized (UA maximized) so that \(T_h\) follows \(T_s\).

4.7 Realistic Power Conversion Option 1: Thermoelectric Generator (TEG)

4.7.1 TEG Basic Equations (Voltage, Heat Flow, Output)

Let the effective Seebeck coefficient of the thermoelectric module be \(S_{\mathrm{TE}}\) (V/K), electrical resistance be \(R_{\mathrm{TE}}\) (\(\Omega\)), and thermal conductance be \(K_{\mathrm{TE}}\) (W/K). Let the hot-side/cold-side temperatures be \(T_h, T_c\) respectively (\(T_h>T_c\)). When current \(I\) flows, the module terminal voltage is: \[V \;=\; S_{\mathrm{TE}}(T_h-T_c) - I R_{\mathrm{TE}}. \label{eq:teg_voltage}\] The heat flow absorbed at the hot side is: \[\dot{Q}_h \;=\; S_{\mathrm{TE}}\,I\,T_h \;-\; \frac{1}{2}I^2 R_{\mathrm{TE}} \;+\; K_{\mathrm{TE}}(T_h-T_c), \label{eq:teg_Qh}\] The heat flow rejected at the cold side is: \[\dot{Q}_c \;=\; S_{\mathrm{TE}}\,I\,T_c \;+\; \frac{1}{2}I^2 R_{\mathrm{TE}} \;+\; K_{\mathrm{TE}}(T_h-T_c). \label{eq:teg_Qc}\] The electrical output power is: \[P_{\mathrm{TE}} \;=\; VI \;=\; S_{\mathrm{TE}}I(T_h-T_c) - I^2 R_{\mathrm{TE}} \;=\; \dot{Q}_h - \dot{Q}_c. \label{eq:teg_power}\] The thermoelectric efficiency is: \[\eta_{\mathrm{TE}} \;=\; \frac{P_{\mathrm{TE}}}{\dot{Q}_h}. \label{eq:teg_eff}\] Verification: Since [eq:teg_Qh]–[eq:teg_power] include energy conservation, fitting \((S_{\mathrm{TE}}, R_{\mathrm{TE}}, K_{\mathrm{TE}})\) from experimental data allows reproduction of module behavior.

4.7.2 Maximum Power Condition (Load Matching) and Maximum Output

Let load resistance be \(R_L\). The circuit equation is \(V=IR_L\). Combined with [eq:teg_voltage]: \[I \;=\; \frac{S_{\mathrm{TE}}(T_h-T_c)}{R_{\mathrm{TE}}+R_L}. \label{eq:teg_current_load}\] The output is: \[P_{\mathrm{TE}} \;=\; I^2 R_L \;=\; \frac{S_{\mathrm{TE}}^2 (T_h-T_c)^2 R_L}{(R_{\mathrm{TE}}+R_L)^2}. \label{eq:teg_power_load}\] \(P_{\mathrm{TE}}\) is maximized at \(R_L=R_{\mathrm{TE}}\) (Matching): \[P_{\mathrm{TE,max}} \;=\; \frac{S_{\mathrm{TE}}^2 (T_h-T_c)^2}{4R_{\mathrm{TE}}}. \label{eq:teg_power_max}\] Design Parameters: In TEG design, \(S_{\mathrm{TE}}\) is determined by materials/modules, \(R_{\mathrm{TE}}\) by number of elements/area/wiring, while cooling design determines \(T_c\) and thermal storage/exchange design determines \(T_h\).

4.7.3 Material Performance Figure of Merit (\(ZT\)) and Maximum Efficiency (Mathematical Fix)

The figure of merit for thermoelectric materials is defined as: \[ZT \;=\; \frac{S_{\mathrm{TE}}^2 \, \bar{T}}{R_{\mathrm{TE}}K_{\mathrm{TE}}}, \qquad \bar{T}=\frac{T_h+T_c}{2}, \label{eq:ZT_def}\] The theoretical maximum efficiency (ideal condition) is fixed as: \[\eta_{\mathrm{TE,max}} \;=\; \frac{T_h-T_c}{T_h}\; \frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+\dfrac{T_c}{T_h}}. \label{eq:etaTEmax}\] Verification: Equation [eq:etaTEmax] always satisfies \(\eta_{\mathrm{TE,max}}<\eta_{\mathrm{Carnot}}\). Therefore, while TEG-based ESS has the advantage of “simple structure”, efficiency is limited if \(ZT\) is low. Thus, if the target output is large, expansion of area (absorbed heat) or number of modules (parallel) is required.

4.8 Realistic Power Conversion Option 2: Stirling Engine + Generator

4.8.1 Work and Efficiency in Ideal Stirling Cycle (Perfect Regenerator) (Mathematical Completeness)

The ideal Stirling cycle consists of two isothermal processes and two isochoric regeneration processes. Let \(n\) be moles of working gas, \(R_g\) be gas constant (denoted \(R_g\) to avoid confusion), \(T_h\) be high isothermal temperature, \(T_c\) be low isothermal temperature, and the volume ratio (compression ratio) in isothermal processes be: \[r_V \;=\; \frac{V_{\max}}{V_{\min}} \;>\; 1 \label{eq:stirling_rV}\] The heat absorbed and work done in isothermal expansion are: \[Q_{\mathrm{in}} \;=\; W_{\mathrm{exp}} \;=\; nR_g T_h \ln r_V, \label{eq:stirling_Qin}\] The heat rejected and work done in isothermal compression are: \[Q_{\mathrm{out}} \;=\; W_{\mathrm{comp}} \;=\; nR_g T_c \ln r_V. \label{eq:stirling_Qout}\] (Regeneration processes are ideally assumed to have no heat exchange with the outside and zero external work.) Thus, net work per cycle is: \[W_{\mathrm{net}} \;=\; Q_{\mathrm{in}}-Q_{\mathrm{out}} \;=\; nR_g (T_h-T_c)\ln r_V, \label{eq:stirling_Wnet}\] Efficiency is: \[\eta_{\mathrm{St,ideal}} \;=\; \frac{W_{\mathrm{net}}}{Q_{\mathrm{in}}} \;=\; 1-\frac{T_c}{T_h} \;=\; \eta_{\mathrm{Carnot}}. \label{eq:stirling_eta_ideal}\] That is, the ideal Stirling reaches Carnot efficiency. Verification: In actual devices, due to regenerator losses and mechanical losses, efficiency must be lower than [eq:stirling_eta_ideal].

4.8.2 Efficiency Decomposition of Real Stirling (For Reproducibility)

The effective electrical efficiency of a real Stirling is fixed as the product: \[\eta_{\mathrm{St,real}} \;=\; \eta_{\mathrm{Carnot}}\, \eta_{\mathrm{reg}}\, \eta_{\mathrm{mech}}\, \eta_{\mathrm{gen}}, \label{eq:stirling_eta_real}\] Here,

• \(\eta_{\mathrm{reg}}\): Regenerator effectiveness (0–1).

• \(\eta_{\mathrm{mech}}\): Efficiency including mechanical losses (friction/leakage) (0–1).

• \(\eta_{\mathrm{gen}}\): Generator/Rectification/Power Electronics efficiency (0–1).

Thus, output power is: \[P_{\mathrm{St}}(t) \;=\; \eta_{\mathrm{St,real}}(t)\,\dot{Q}_{\mathrm{conv}}(t). \label{eq:stirling_power}\]

4.8.3 Cycle Frequency and Output (Design Parameters)

If the Stirling engine performs \(f_{\mathrm{cyc}}\) (Hz) cycles per second, the power (mechanical output) is: \[P_{\mathrm{mech}} \;=\; W_{\mathrm{net}}\;f_{\mathrm{cyc}} \;=\; nR_g (T_h-T_c)\ln r_V\; f_{\mathrm{cyc}}. \label{eq:stirling_power_mech}\] In actual design, \(n\) can be converted from mean pressure \(p_m\) and mean volume \(\bar{V}\): \[n \;=\; \frac{p_m \bar{V}}{R_g \bar{T}}, \qquad \bar{T}=\frac{T_h+T_c}{2}, \label{eq:stirling_n_from_pV}\] Thus, [eq:stirling_power_mech] has \((p_m, \bar{V}, r_V, f_{\mathrm{cyc}}, T_h, T_c)\) as design parameters.

4.9 Realistic Power Conversion Option 3: Organic Rankine Cycle (ORC)

4.9.1 Rankine Cycle State Point Definition (1–4)

The basic 4 states of ORC are fixed as follows:

• State 1: Condenser Outlet (Pump Inlet), Pressure \(p_1\), Enthalpy \(h_1\).

• State 2: Pump Outlet (Evaporator Inlet), Pressure \(p_2\), Enthalpy \(h_2\).

• State 3: Evaporator Outlet (Turbine Inlet), Pressure \(p_3\), Enthalpy \(h_3\).

• State 4: Turbine Outlet (Condenser Inlet), Pressure \(p_4\), Enthalpy \(h_4\).

Let mass flow rate be \(\dot{m}_{\mathrm{wf}}\) (kg/s) (working fluid).

4.9.2 Heat/Work Balance and Efficiency (Mathematical Completeness)

Pump power: \[\dot{W}_p \;=\; \dot{m}_{\mathrm{wf}}(h_2-h_1), \label{eq:orc_pump_work}\] Turbine power: \[\dot{W}_t \;=\; \dot{m}_{\mathrm{wf}}(h_3-h_4), \label{eq:orc_turb_work}\] Evaporator heat input: \[\dot{Q}_{\mathrm{in,orc}} \;=\; \dot{m}_{\mathrm{wf}}(h_3-h_2), \label{eq:orc_Qin}\] Condenser heat rejection: \[\dot{Q}_{\mathrm{out,orc}} \;=\; \dot{m}_{\mathrm{wf}}(h_4-h_1). \label{eq:orc_Qout}\] Net power: \[\dot{W}_{\mathrm{net}} \;=\; \dot{W}_t-\dot{W}_p \;=\; \dot{m}_{\mathrm{wf}}\left[(h_3-h_4)-(h_2-h_1)\right]. \label{eq:orc_Wnet}\] Thermal efficiency: \[\eta_{\mathrm{ORC}} \;=\; \frac{\dot{W}_{\mathrm{net}}}{\dot{Q}_{\mathrm{in,orc}}} \;=\; \frac{(h_3-h_4)-(h_2-h_1)}{h_3-h_2}. \label{eq:orc_eta}\]

4.9.3 Inclusion of Non-Idealities (Isentropic Efficiency) (For Reproducibility)

Let turbine isentropic efficiency be \(\eta_t\) and pump isentropic efficiency be \(\eta_p\). Let \(h_{4s}, h_{2s}\) be enthalpies at isentropic states. \[\begin{aligned} \eta_t &= \frac{h_3-h_4}{h_3-h_{4s}}, \label{eq:eta_turb}\\ \eta_p &= \frac{h_{2s}-h_1}{h_2-h_1}. \label{eq:eta_pump}\end{aligned}\] Thus, actual \(h_4, h_2\) are fixed as: \[\begin{aligned} h_4 &= h_3 - \eta_t (h_3-h_{4s}), \label{eq:h4_real}\\ h_2 &= h_1 + \frac{h_{2s}-h_1}{\eta_p}. \label{eq:h2_real}\end{aligned}\] With this, ORC prediction is complete given “Working Fluid Property Tables (or Equation of State)”. Reproducibility Requirement: If ORC is adopted in this whitepaper, the source of working fluid property data (Table/Library Version) and state point calculation code must be included in the DOI package.

4.10 Power Electronics/Output Design (Load Matching, DC Bus Stabilization)

ESS output voltage/current characteristics vary greatly depending on the power conversion option. Therefore, this application whitepaper fixes the following power electronics principles.

4.10.1 DC Bus Target and Output Normalization

Define the target DC bus voltage as \(V_{\mathrm{bus}}\). Normalize output to \(V_{\mathrm{bus}}\) using DC-DC or Rectification+DC-DC. Use [eq:ess_Eout] for electrical energy integrated on the bus (Measurement Definition).

4.10.2 Maximum Power Point Tracking (MPPT) or Load Matching

For TEG, since \(R_L=R_{\mathrm{TE}}\) is optimal in [eq:teg_power_load], the DC-DC converter controls the input impedance to satisfy: \[R_{\mathrm{in,eq}} \;\approx\; R_{\mathrm{TE}} \label{eq:mppt_match}\] For Stirling/ORC, generator output may be AC, so bus stabilization is performed in DC-DC after rectification, and generator voltage/frequency fluctuations are recorded as metadata.

4.10.3 Energy Buffer (Short-term Stabilization) Definition

To prepare for output fluctuations (Clouds/Wind/Temperature transitions), a capacitor \(C_{\mathrm{bus}}\) or small battery/supercapacitor can be placed on the bus. Considering only the capacitor, stored energy is: \[E_{\mathrm{cap}} \;=\; \frac{1}{2}C_{\mathrm{bus}}V_{\mathrm{bus}}^2. \label{eq:cap_energy}\] Verification: Since the buffer is a “Time Rearrangement” device, not “Energy Production”, the difference in buffer initial/final stored energy must be included in \(\eta_{\mathrm{total}}\) calculation.

4.11 ESS Design Parameters (Minimum Design Variable Set) and Initial Settings

4.11.1 Minimum Design Variable Set

To defining the ESS module reproducibly in this application whitepaper, at least the following parameter set \(\Pi_{\mathrm{ESS}}\) is fixed: \[\Pi_{\mathrm{ESS}}= \left\{ A_{\mathrm{col}}, \alpha_{\mathrm{sol}}, \tau_{\mathrm{cov}}, K_\theta, A_{\mathrm{rad}}, \varepsilon_{\mathrm{th}}, A_{\mathrm{loss}}, k_{\mathrm{ins}}, L_{\mathrm{ins}}, h_{\mathrm{out}}, m_{\mathrm{st}}, c_{p,\mathrm{st}}, U_{\mathrm{hx}}A_{\mathrm{hx}}, T_{\mathrm{th}}, V_{\mathrm{bus}} \right\}. \label{eq:pi_ess}\] The meaning of each parameter follows the definitions earlier in this section. \(T_{\mathrm{th}}\) is the “Output Initiation Threshold Temperature”, recording the transition point observed in experiments.

4.11.2 Initial Settings (Document-Based Prototype Values as “Initial Guess” Only)

Since the application design of this whitepaper must be fixed based on measurements, values presented in documents/simulations are used only as initial guesses. For example, values like: \[A_{\mathrm{col}} = 0.5~\mathrm{m^2},\qquad \alpha_{\mathrm{sol}} \ge 0.982,\qquad T_{\mathrm{th}} \approx 145^\circ\mathrm{C} \label{eq:initial_from_doc}\] are recorded as “Design Starting Points” and re-fixed through verification (Irradiance/Temperature/Power Integration).

4.12 Verification (Mandatory): Procedure to Confirm ESS Output Claims Align with Physical Limits and Energy Balance

This subsection mathematically fixes the verification procedure to fix “Claims” as “Reality”.

4.12.1 Procedure 1: Integrate \(E_{\mathrm{in}}\) and \(E_{\mathrm{out}}\) over the Same Time Window

1. Set the same time window \([t_0,t_1]\) (e.g., 1 day, or 6 hours of output section).

2. Calculate \(E_{\mathrm{in}}\) by integrating [eq:ess_Ein] with irradiance \(G(t)\) and area \(A_{\mathrm{col}}\).

3. Calculate \(E_{\mathrm{out}}\) by integrating [eq:ess_Eout] with voltage/current.

4. Calculate \(\eta_{\mathrm{total}}\) via [eq:ess_eta_total_def].

4.12.2 Procedure 2: Independent Verification of “Actual Heat Extraction” from Thermal Storage Temperature Drop

If the thermal storage temperature changed by \(\Delta T_s\) during the generation time window, the thermal storage energy change is (via [eq:Est_linear]): \[\Delta E_{\mathrm{st}} \;=\; C_{\mathrm{th}}\Delta T_s, \label{eq:deltaEst}\] Integrating [eq:ess_heat_ode]: \[\Delta E_{\mathrm{st}} \;=\; \int_{t_0}^{t_1}\dot{Q}_{\mathrm{abs}}(t)\,dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{loss}}(t)\,dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{conv}}(t)\,dt. \label{eq:heat_balance_integral}\] Thus, the heat energy transferred to the power conversion unit (Extracted Heat) is: \[E_{\mathrm{conv}} \;=\; \int_{t_0}^{t_1}\dot{Q}_{\mathrm{conv}}(t)\,dt \;=\; \int_{t_0}^{t_1}\dot{Q}_{\mathrm{abs}}(t)\,dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{loss}}(t)\,dt -\Delta E_{\mathrm{st}}. \label{eq:Econv_from_balance}\] And electrical output is, via [eq:Pout_general]: \[E_{\mathrm{out}} \;=\; \int_{t_0}^{t_1}P_{\mathrm{out}}(t)\,dt \;=\; \int_{t_0}^{t_1}\eta_{\mathrm{elec}}(t)\eta_{\mathrm{conv}}(t)\dot{Q}_{\mathrm{conv}}(t)\,dt. \label{eq:Eout_from_conv}\] Core Verification: Independently estimate \(E_{\mathrm{conv}}\) via [eq:Econv_from_balance] and confirm that \(E_{\mathrm{out}} \le E_{\mathrm{conv}}\) and \(E_{\mathrm{out}} \le E_{\mathrm{in}}\) are simultaneously satisfied.

4.13 Reproducibility Provision (For DOI Issuance): ESS Data/Code/Protocol

4.13.1 Mandatory Measurement Channels and Accuracy (Minimum Requirements)

For ESS verification (reproducibility), record at least the following channels:

1. \(G(t)\): Irradiance (W/m\(^2\)), Sampling period \(\Delta t_G\).

2. \(T_a(t)\): Ambient Temp (K), \(\Delta t_T\).

3. \(T_s(t)\): Thermal Storage Temp (K), \(\Delta t_T\).

4. \(T_h(t), T_c(t)\): Power Conversion Hot/Cold Temp (K), \(\Delta t_T\).

5. \(V(t), I(t)\): Output Voltage/Current, \(\Delta t_V=\Delta t_I\).

6. (Optional) \(T_{\mathrm{surf}}(t)\): Surface Temp (K) or Cover Outer Temp (K).

7. (Optional) Wind Speed/Humidity (Causes of convection loss variation).

Accuracy Requirement: Record accuracy, calibration date, and installation location (including thermal contact method) of each sensor as metadata. Without this, DOI reproducibility requirements cannot be met.

4.13.2 Data Format (Fixed Schema)

Each log record is fixed as CSV (or JSON with same structure) containing at least: \[\{\texttt{t},\,\texttt{G},\,\texttt{Ta},\,\texttt{Ts},\,\texttt{Th},\,\texttt{Tc},\,\texttt{V},\,\texttt{I}\}. \label{eq:ess_schema_set}\] Units are fixed as: \[t[\mathrm{s}],\; G[\mathrm{W/m^2}],\; T[\mathrm{K}],\; V[\mathrm{V}],\; I[\mathrm{A}]\]

4.13.3 Analysis Code (Mandatory Outputs and Calculation Sequence)

For DOI issuance, analysis code implementing the formulas in this section must generate at least:

1. \(E_{\mathrm{in}}\) Calculation: Integration of [eq:ess_Ein].

2. \(E_{\mathrm{out}}\) Calculation: Integration of [eq:ess_Eout].

3. \(\eta_{\mathrm{total}}\) Calculation: [eq:ess_eta_total_def].

4. Cooling Fitting: Inverse estimation of \((C_{\mathrm{th}}, U_{\mathrm{loss}}, \varepsilon_{\mathrm{th}})\) using [eq:cooling_ode_full].

5. Carnot Verification: Flag for violation of [eq:carnot_eff] and [eq:power_upper_bound].

Also, the release must include a version tag (v1.0.0, etc.) and checksum (SHA256) to enable identity verification.

4.14 Conclusion of This Section (Summary): Fixing ESS as a Designable “Thermal-Electrical System”

1. ESS can claim “Output” only on the thermal balance of [eq:ess_heat_ode], and overall efficiency is defined by [eq:ess_eta_total_def].

2. The power conversion unit must pass the Carnot Limit [eq:carnot_eff] and Upper Bound [eq:power_upper_bound].

3. Mathematical models for realistic power conversion options TEG (Thermoelectric), Stirling, and ORC are presented in complete forms, and each option must be calibrated with measurement data.

4. For DOI reproducibility, mandatory measurement channels, data schema, and calculation code outputs are fixed in this section.

5 Water Purification Module (EM Wave/Electric Field Based): Seawater vs. Brackish Water Technology Branching (RO/ED/CDI/Thermal-MD) and Physical Basis

5.1 Purpose of this Section and Definition of “Purification” Reality (Fixed Measurable KPIs)

The purpose of this section is to make the design of a module that “purifies (cleans/desalinates)” water using electricity (or heat) obtained from the ESS physically feasible within a physically possible range. Specifically, the condition “based on electromagnetic waves/electric fields” is engineeringly translated to cover both: (1) Electric Field Based Desalination (Driven by ED/CDI/RO) and (2) Thermal Based Desalination (Thermal-Membrane Distillation, MD). We mathematically fix that the technology choice differs for Seawater versus Brackish Water (Low-salinity Groundwater/Weak Brine).

In this application whitepaper, the “Reality (Ground Truth)” of purification must be fixed by the following KPIs:

1. Salt Rejection (or Removal Performance): \[R \;=\; 1-\frac{C_p}{C_f}, \label{eq:desal_R}\] Here, \(C_f\) is the feed salt concentration, and \(C_p\) is the permeate/product salt concentration.

2. Recovery Ratio (Product Water Ratio): \[Y \;=\; \frac{Q_p}{Q_f}, \label{eq:desal_Y}\] Here, \(Q_f\) is the feed flow rate, and \(Q_p\) is the product flow rate.

3. Specific Energy Consumption (SEC): \[\mathrm{SEC} \;=\; \frac{E_{\mathrm{desal}}}{V_p} \;=\; \frac{\int_{t_0}^{t_1} P_{\mathrm{desal}}(t)\,dt}{\int_{t_0}^{t_1} Q_p(t)\,dt}, \qquad \left[\mathrm{J/m^3}\right]\ \text{or}\ \left[\mathrm{kWh/m^3}\right]. \label{eq:desal_SEC}\] Here, \(P_{\mathrm{desal}}\) is the net electric power (or thermal power) supplied to the purification module. Verification Rule: “Zero grid input” is possible, but \(\mathrm{SEC}=0\) is physically impossible. Therefore, SEC must always be calculated and verified with energy balance (including hidden inputs: pumps/head/flow loss/field generation).

Also, Mass Balance (Flow and Salt Balance) must be reported as mandatory. \[\begin{aligned} Q_f &= Q_p + Q_c, \label{eq:desal_flow_balance}\\ Q_f C_f &\approx Q_p C_p + Q_c C_c, \label{eq:desal_salt_balance}\end{aligned}\] Here, \(Q_c, C_c\) are the flow rate and salt concentration of the concentrate/brine. If [eq:desal_flow_balance]–[eq:desal_salt_balance] do not hold, it implies measurement/definition/sampling errors or precipitation/adsorption (hidden paths), so it is treated as Re-verification (HOLD) rather than “Success”.

5.2 Seawater vs. Brackish Water: Physical Branching Rules for Technology Selection (Fixed Decision Algorithm)

Purification technology selection is determined not by “preference” but by Feed Salinity, Target Water Quality, and Available Energy Form (Electricity/Heat). In this application whitepaper, we let the Total Dissolved Solids (TDS) of the feed be \(C_f\) (kg/m\(^3\)) and fix the technology selection with the following branching rules.

5.2.1 Feed Classification (Fixed Salinity Bins)

\[\text{Brackish Water}:\ C_f \le C_{\mathrm{LB}}, \qquad \text{Seawater}:\ C_f \ge C_{\mathrm{SW}}, \qquad \text{Intermediate}:\ C_{\mathrm{LB}} < C_f < C_{\mathrm{SW}}. \label{eq:salinity_bins}\] Here, \(C_{\mathrm{LB}}\) and \(C_{\mathrm{SW}}\) are boundary values chosen by the designer. In this Whitepaper v1.0, for reproducibility, default values (initial values) are set as: \[C_{\mathrm{LB}} = 3~\mathrm{kg/m^3}\ (3~\mathrm{g/L}), \qquad C_{\mathrm{SW}} = 35~\mathrm{kg/m^3}\ (35~\mathrm{g/L}). \label{eq:salinity_default}\] (These values are not “norms” but initial values to specify branch points. They can be changed according to project goals, but changes must be fixed in the text.)

5.2.2 Definition of Technology Selector Function (Branching Function)

Let purification technology be \(\mathcal{T}\in\{\mathrm{RO},\mathrm{ED},\mathrm{CDI},\mathrm{MD}\}\). Define the selector function \(\Phi\) as: \[\mathcal{T} \;=\; \Phi(C_f,\ R^\ast,\ Y^\ast,\ \mathcal{E}), \label{eq:tech_selector}\] Here, \(R^\ast\) is target rejection, \(Y^\ast\) is target recovery, and \(\mathcal{E}\in\{\text{Elec-Dominant},\ \text{Heat-Dominant},\ \text{Mixed}\}\) is the available energy form.

The basic branching rules of this whitepaper are fixed as follows:

1. If \(C_f \ge C_{\mathrm{SW}}\) (Seawater) and Elec-Dominant: RO is the default choice (Seawater Reverse Osmosis, SWRO).

2. If \(C_f \le C_{\mathrm{LB}}\) (Brackish) and Elec-Dominant: ED or CDI is prioritized (Electric Field Based).

3. If \(C_f\) is very high (Concentrate/Brine) or Heat-Dominant: Branch to Thermal Membrane Distillation (MD) or Thermal Evaporation/Crystallization.

4. Intermediate zone is finally selected based on target water quality (\(R^\ast\)), recovery (\(Y^\ast\)), contaminants (Organics/Turbidity/Hardness), and ESS output characteristics (Voltage/Current/Heat). Selection basis must be verified by the Physical Models (SEC, Upper/Lower Bounds) of this section.

5.3 Thermodynamic Lower Bound of Desalination: Fixing Minimum Separation Work (Reversible Minimum Energy)

Since purification (especially desalination) is a process of separating a mixture, a thermodynamic reversible minimum work exists. Presenting this lower bound and verifying that all design SECs are above it is the core of this section.

5.3.1 General Definition: Minimum Separation Work \(W_{\min}\)

When separating a mixture (feed) into product and concentrate at temperature \(T\) and pressure \(P\), the reversible minimum work is defined as the change in Gibbs free energy. \[W_{\min} \;=\; \Delta G \;=\; G_{\mathrm{out}} - G_{\mathrm{in}}. \label{eq:Wmin_DG}\] In terms of mole numbers: \[\Delta G \;=\; \sum_{k} n_k^{(\mathrm{out})}\mu_k^{(\mathrm{out})} \;-\; \sum_{k} n_k^{(\mathrm{in})}\mu_k^{(\mathrm{in})}, \label{eq:DG_mu}\] Here, \(k\) is component (water, ions, etc.), \(n_k\) is moles, and \(\mu_k\) is chemical potential. Thus, the SEC of an actual process has the following lower bound: \[\mathrm{SEC} \;\ge\; \mathrm{SEC}_{\min} \;\equiv\; \frac{W_{\min}}{V_p}. \label{eq:SEC_min_bound}\]

5.3.2 \(W_{\min}\) in Ideal Solution Approximation (Mathematically Complete Form)

In an ideal solution, the chemical potential of each component is: \[\mu_k = \mu_k^\circ + RT\ln x_k, \label{eq:mu_ideal}\] Here, \(x_k\) is the mole fraction. Thus, the free energy of mixing (mixing term) is: \[G_{\mathrm{mix}} = RT\sum_k n_k \ln x_k. \label{eq:Gmix}\] The minimum work of separation is given by subtracting the “mixing term of input stream” from the “sum of mixing terms of output streams”. \[W_{\min} = RT\left[ \sum_{s\in\{p,c\}}\ \sum_k n_{k,s}\ln x_{k,s} \;-\; \sum_k n_{k,f}\ln x_{k,f} \right], \label{eq:Wmin_ideal}\] Here, \(f\) is feed, \(p\) is product, and \(c\) is concentrate. (This equation is complete for ideal solutions. Since actual seawater has non-ideality (activity coefficients, osmotic coefficients), the application whitepaper treats [eq:Wmin_ideal] as a Lower Bound of the Lower Bound, and actual SEC must always be greater than this in verification.)

5.3.3 Practical Lower Bound Verification Formula: Osmotic Pressure Based Integral (Common for RO/Membrane)

Since “pressure or chemical potential difference” is the driving force in membrane processes (RO/MD), the lower bound can also be expressed as an integral form using osmotic pressure \(\pi(C)\). Assuming perfect salt rejection (\(C_p\approx 0\)) and recovery \(Y\), the concentrate concentration is ideally: \[C_c \approx \frac{C_f}{1-Y}. \label{eq:Cc_ideal}\] Using osmotic pressure \(\pi(C)\) (exact form determined by activity/osmotic coefficients), the reversible minimum work (per unit product volume) is fixed by the following integral: \[\mathrm{SEC}_{\min} \;=\; \int_{0}^{Y} \frac{\pi\!\left(\dfrac{C_f}{1-y}\right) - \pi(C_p)}{1-y}\,dy, \qquad (C_p\approx 0 \Rightarrow \pi(C_p)\approx 0). \label{eq:SECmin_integral}\] Verification Usage: [eq:SECmin_integral] is used as a criterion to quickly judge whether the design SEC violates the thermodynamic lower bound.

5.4 Common Physics of EM Wave/Electric Field Based Purification: Ion Transport Equations and Length/Time Scales

5.4.1 Nernst–Planck Ion Transport (Mandatory Minimum Model)

Electric field based purification (ED/CDI and field-assisted processes) governs ion transport. Flux of ion \(i\) is fixed by the following Nernst–Planck equation: \[\mathbf{J}_i \;=\; -D_i\nabla c_i \;-\; \frac{z_i D_i F}{RT}\,c_i\nabla \phi \;+\; c_i \mathbf{u}, \label{eq:NP}\] Here, \(c_i\) is concentration, \(D_i\) is diffusion coefficient, \(z_i\) is charge number, \(\phi\) is electric potential, \(\mathbf{u}\) is fluid velocity field, and \(F\) is Faraday constant. The continuity equation is: \[\frac{\partial c_i}{\partial t} + \nabla\cdot \mathbf{J}_i = R_i, \label{eq:NP_continuity}\] Here, \(R_i\) is the source term for electrode reaction/adsorption/precipitation.

5.4.2 Length/Time Scales: Fixing Meaning of “Long Tube” via Residence Time instead of Gravity

Electric field based separation is determined not by “gravity” but by the competition between electromigration and diffusion/mixing. Let representative length be \(L\), mean flow velocity be \(u\), and residence time be \(\tau_{\mathrm{res}}=L/u\). The representative migration distance is: \[\ell_{i,\mathrm{mig}} \;\approx\; v_{i,\mathrm{mig}}\,\tau_{\mathrm{res}} \;=\; \left(\frac{z_i D_i F}{RT}|E|\right)\frac{L}{u}, \label{eq:migration_length}\] The mixing length due to diffusion is: \[\ell_{i,\mathrm{diff}} \;\approx\; \sqrt{2D_i \tau_{\mathrm{res}}} \;=\; \sqrt{2D_i\frac{L}{u}}. \label{eq:diffusion_length2}\] For separation (or concentration difference maintenance) to hold, relative to the characteristic length \(\ell_{\mathrm{sep}}\) in the separation axis, we need: \[\ell_{i,\mathrm{mig}} \gtrsim \ell_{\mathrm{sep}} \quad\text{and}\quad \ell_{i,\mathrm{diff}} \ll \ell_{\mathrm{sep}} \label{eq:sep_condition_np}\] Therefore, “long tube” design is fixed as a design to increase \(\tau_{\mathrm{res}}\) to increase [eq:migration_length], not for gravity separation.

5.4.3 Minimum Definition of EM Wave (AC) Drive: Electric Field RMS and Frequency

Since the user stated that “EM waves are more stable than magnets”, this whitepaper covers electric field drive as DC or AC. In the case of AC electric field, define: \[E_{\mathrm{rms}}=\sqrt{\frac{1}{T_{\mathrm{per}}}\int_{0}^{T_{\mathrm{per}}} E(t)^2\,dt}, \qquad T_{\mathrm{per}}=\frac{1}{f}, \label{eq:Erms}\] In transport equations (especially migration term), approximations using time-resolved \(E(t)\) at low frequencies and equivalent driving force \(E_{\mathrm{rms}}\) at high frequencies are allowed. However, at high frequencies, dielectric loss (heating) and electrode polarization are added, so the application whitepaper must measure conversion of power to heat (heating) and include it in the energy balance when using AC drive.

5.5 Technology Option A: Reverse Osmosis (RO) — Physical Model for Default Seawater (High Salinity) Choice

5.5.1 Osmotic Pressure and Minimum Required Pressure

Water flux in RO is proportional to the pressure difference minus the osmotic pressure difference. Letting osmotic pressure be \(\pi(C)\): \[\Delta P_{\min} \;\ge\; \Delta \pi \;=\; \pi(C_m)-\pi(C_p), \label{eq:ro_dPmin}\] Here, \(C_m\) is membrane surface concentration (including concentration polarization), and \(C_p\) is permeate concentration. Targeting perfect salt rejection, \(C_p\approx 0\), so \(\pi(C_p)\approx 0\).

5.5.2 Membrane Flux Model (Water/Salt) and Permeate Quality

RO water flux (water flow per area) is fixed by the linear driving force model: \[J_w \;=\; A_w\left(\Delta P-\Delta \pi\right), \qquad \left[J_w\right]=\mathrm{m^3/(m^2\,s)}, \label{eq:ro_Jw}\] Here, \(A_w\) is water permeability coefficient (permeance). Salt flux is: \[J_s \;=\; B_s\left(C_m-C_p\right), \qquad \left[J_s\right]=\mathrm{kg/(m^2\,s)}, \label{eq:ro_Js}\] Here, \(B_s\) is salt permeability coefficient. Permeate concentration is fixed from the ratio of salt/water flux: \[C_p \;\approx\; \frac{J_s}{J_w} \label{eq:ro_Cp}\] (Exact models include diffusion/convection within the membrane, but v1.0 adopts [eq:ro_Cp] as the minimum model.)

5.5.3 Recovery and Concentrate Concentration (Based on Salt Balance)

With recovery \(Y=Q_p/Q_f\) and perfect salt rejection (\(C_p\approx 0\)) approximation, concentrate concentration is given by [eq:Cc_ideal]. Thus, in seawater, as \(Y\) increases, \(C_c\) and \(\pi(C_c)\) increase, increasing required pressure and SEC. This is the physical reason why seawater RO typically selects intermediate recovery (e.g., 0.4–0.6). (This whitepaper does not force a specific value but sets \(Y\) as a design variable and suggests an optimal range via SEC sensitivity analysis.)

5.5.4 RO Power Consumption (Pump) and SEC: Complete Formula Including Energy Recovery

Let high pressure pump pressure rise be \(\Delta P\), pump efficiency be \(\eta_p\), and Energy Recovery Device (ERD) effective recovery efficiency be \(\eta_{\mathrm{ERD}}\). With feed flow \(Q_f\) and concentrate flow \(Q_c=Q_f(1-Y)\), high pressure pump power is: \[P_{\mathrm{pump}} \;=\; \frac{\Delta P\,Q_f}{\eta_p}. \label{eq:ro_Ppump}\] The power (equivalent) recovered by ERD from hydraulic energy of concentrate (ideally \(\Delta P\,Q_c\)) is defined as: \[P_{\mathrm{rec}} \;=\; \eta_{\mathrm{ERD}}\,\Delta P\,Q_c \;=\; \eta_{\mathrm{ERD}}\,\Delta P\,Q_f(1-Y) \label{eq:ro_Prec}\] Thus, net external power is: \[P_{\mathrm{net}} \;=\; P_{\mathrm{pump}}-P_{\mathrm{rec}}+P_{\mathrm{aux}}, \label{eq:ro_Pnet}\] Here, \(P_{\mathrm{aux}}\) is auxiliary power for pre-treatment/low-pressure pump/control (measured or estimated by designer). Dividing by product flow \(Q_p=YQ_f\), RO SEC is fixed by the complete formula: \[\mathrm{SEC}_{\mathrm{RO}} \;=\; \frac{P_{\mathrm{net}}}{Q_p} \;=\; \frac{\Delta P}{Y}\left(\frac{1}{\eta_p}-\eta_{\mathrm{ERD}}(1-Y)\right) \;+\; \frac{P_{\mathrm{aux}}}{YQ_f}. \label{eq:ro_SEC}\] Verification: [eq:ro_SEC] reduces to simple pump formula if \(\eta_{\mathrm{ERD}}=0\), and reflects larger recoverable energy at lower recovery if \(\eta_{\mathrm{ERD}}>0\). Also, [eq:ro_SEC] cannot be smaller than the lower bound of [eq:SEC_min_bound]. (If smaller, it implies errors in \(\Delta P\) definition, flow measurement, ERD estimation, or existence of hidden inputs.)

5.6 Technology Option B: Electrodialysis (ED) — Physical Model for Prioritized Brackish (Electric Field Based) Choice

5.6.1 Core of ED: Ions Transported by Current (Faraday’s Law)

Salt removal through membranes in ED is determined by the amount of charge carried by current. When current \(I(t)\) flows, ideally charge \(dQ=I\,dt\) is connected to migrated ion moles. Based on monovalent ions (charge number \(|z|=1\)), ideal removal moles are: \[dn_{\mathrm{salt,ideal}} \;=\; \frac{I(t)}{F}\,dt. \label{eq:ed_dn}\] Actually, introducing current efficiency (or charge efficiency) \(\eta_I \in [0,1]\): \[dn_{\mathrm{salt}} \;=\; \eta_I\,\frac{I(t)}{F}\,dt. \label{eq:ed_dn_real}\] Thus, removed salt moles in time window \([t_0,t_1]\) are: \[n_{\mathrm{salt}} \;=\; \eta_I \int_{t_0}^{t_1}\frac{I(t)}{F}\,dt. \label{eq:ed_nsalt}\] This is the core equation fixing the “Physical Reality” of ED via current integration.

5.6.2 Stack Voltage (Ohmic Loss + Membrane + Electrode): Minimum Complete Model

Decompose total stack voltage of ED as: \[V_{\mathrm{stack}}(t) \;=\; V_{\mathrm{ohm}}(t) + V_{\mathrm{mem}}(t) + V_{\mathrm{elec}}(t), \label{eq:ed_Vstack}\] In v1.0 minimum model, Ohmic loss with equivalent resistance \(R_{\mathrm{stack}}\) is: \[V_{\mathrm{ohm}}(t) = I(t)\,R_{\mathrm{stack}}. \label{eq:ed_Vohm}\] Since membrane potential/concentration potential varies with concentration: \[V_{\mathrm{mem}}(t)=V_{\mathrm{mem}}\!\left(\{c_i(t)\}\right) \label{eq:ed_Vmem}\] In initial design, conservatively set as constant or linear approximation. Electrode overpotential term \(V_{\mathrm{elec}}\) is determined by electrode reaction and current density, calibrated by measured values (Current-Voltage curve) in v1.0.

5.6.3 ED SEC (Complete Formula) and Interpretation Based on Salt Removal

Since ED instantaneous power is \(P(t)=V_{\mathrm{stack}}(t)I(t)\), electric energy input is: \[E_{\mathrm{ED}}=\int_{t_0}^{t_1}V_{\mathrm{stack}}(t)I(t)\,dt. \label{eq:Eed}\] Dividing by product volume \(V_p=\int Q_p dt\): \[\mathrm{SEC}_{\mathrm{ED}}=\frac{E_{\mathrm{ED}}}{V_p}. \label{eq:SECed}\] ED can also be interpreted as “Energy per Salt Removal”. Using removed salt moles [eq:ed_nsalt]: \[\frac{E_{\mathrm{ED}}}{n_{\mathrm{salt}}} \;=\; \frac{\int V_{\mathrm{stack}}I\,dt}{\eta_I\int (I/F)\,dt} \;=\; \frac{F}{\eta_I}\cdot \frac{\int V_{\mathrm{stack}}I\,dt}{\int I\,dt}. \label{eq:E_per_mol_salt}\] That is, energy efficiency improves as average stack voltage (current weighted average) decreases and \(\eta_I\) increases. This is the physical reason why ED becomes advantageous in brackish water, whereas in seawater, resistance/concentration polarization increases \(V_{\mathrm{stack}}\) and losses.

5.7 Technology Option C: Capacitive Deionization (CDI) — Physical Model for Brackish/Purification (Electric Field Based)

5.7.1 Core of CDI: Charge Storage and Ion Adsorption (Charge Efficiency)

Charge \(Q\) stored in CDI electrodes is connected to ion adsorption amount. Introducing charge efficiency (or charge-to-salt removal efficiency) \(\Lambda \in [0,1]\), removal moles of monovalent ions (e.g., Na\(^+\), Cl\(^-\) pair in NaCl) are: \[n_{\mathrm{salt}} \;=\; \Lambda \frac{Q}{F}, \qquad Q=\int_{t_0}^{t_1} I(t)\,dt. \label{eq:cdi_nsalt}\] That is, CDI salt removal is fixed by Current Integration. (This equation has the same structure as ED’s [eq:ed_nsalt]; the difference is whether the device is “Membrane Based” or “Electrode Adsorption Based”.)

5.7.2 Capacitor Energy and Recovery (Charge/Discharge) Complete Formula

Energy required to charge an ideal capacitor to voltage \(V\) is: \[E_{\mathrm{cap}}=\frac{1}{2}C V^2. \label{eq:cdi_Ecap}\] Since actual CDI voltage varies with time and resistive losses exist, electric energy input is defined as: \[E_{\mathrm{CDI,in}}=\int_{t_0}^{t_1} V(t)I(t)\,dt \label{eq:cdi_Ein}\] Letting energy recovery efficiency during discharge be \(\eta_{\mathrm{rec}}\in[0,1]\), net consumed energy is: \[E_{\mathrm{CDI,net}}=E_{\mathrm{CDI,in}}-\eta_{\mathrm{rec}}E_{\mathrm{CDI,out}}, \label{eq:cdi_Enet}\] Here, \(E_{\mathrm{CDI,out}}\) is the recoverable electric energy integration during discharge (defined by measurement). Thus, CDI SEC is: \[\mathrm{SEC}_{\mathrm{CDI}}=\frac{E_{\mathrm{CDI,net}}}{V_p}. \label{eq:SECcdi}\] Verification: Assuming \(\eta_{\mathrm{rec}}=1\) underestimates SEC, so the application whitepaper must measure discharge energy recovery to fix \(\eta_{\mathrm{rec}}\) empirically.

5.7.3 Effective Scope of CDI (Physical Basis)

Since CDI has limited “Charge Storage Capacity (Capacitance) on Electrodes”, treating larger salt amounts (i.e., \(C_f Q_f\)) requires frequent regeneration (discharge) or expansion of electrode area/volume. Therefore, this whitepaper sets CDI as a primary candidate under the condition: \[C_f \le C_{\mathrm{LB}} \quad\text{(Brackish Water)}, \label{eq:cdi_prefer}\] And at seawater levels (\(C_f\approx C_{\mathrm{SW}}\)), CDI is not claimed as a standalone technology, but allowed only for limited roles like (1) Pre-treatment/Purification or (2) RO assistance (Brine treatment). (Violating this increases regeneration losses and required power scale drastically, making it physically disadvantageous.)

5.8 Technology Option D: Thermal-Membrane Distillation (MD) — Physical Model for “Heat-Dominant” or High Salinity/Brine Treatment

5.8.1 MD Driving Force: Vapor Pressure Difference (Temperature + Salinity)

Membrane Distillation (MD) is a process where not liquid water but Water Vapor passes through pores of a hydrophobic membrane. The driving force is the water vapor partial pressure difference \(\Delta p_v\) on both sides. \[J_v \;=\; C_m \left(p_{v,h}-p_{v,c}\right), \label{eq:md_flux}\] Here, \(J_v\) is vapor flux (kg/m\(^2\)/s), \(C_m\) is membrane mass transfer coefficient, and \(p_{v,h}, p_{v,c}\) are water vapor partial pressures at hot/cold sides respectively.

Letting saturation vapor pressure of pure water be \(p_{\mathrm{sat}}(T)\), in saline solution, due to activity (or water activity) \(a_w\in(0,1]\): \[p_v(T,C)\;=\;a_w(T,C)\,p_{\mathrm{sat}}(T). \label{eq:md_pv}\] Thus, higher salinity reduces \(a_w\), lowering vapor pressure, which acts to decrease MD flux. (However, since osmotic pressure barrier is not dominant as in RO, MD can be selected for high salinity/brine treatment.)

5.8.2 Thermal Balance and Thermal SEC (STEC) Complete Formula

Let MD product mass flow be \(\dot{m}_p\) (kg/s). Required heat (minimum) includes latent heat \(h_{fg}\) and sensible heat. Letting heat input rate be \(\dot{Q}_{\mathrm{MD}}\): \[\dot{Q}_{\mathrm{MD}} \;\ge\; \dot{m}_p\,h_{fg}(T_h) \;+\; \dot{m}_p\,c_{p,w}\,(T_h-T_{\mathrm{ref}}), \label{eq:md_heat_min}\] Here, \(c_{p,w}\) is specific heat of water, \(T_{\mathrm{ref}}\) is reference temperature. Actually, more heat is needed due to membrane/module heat loss and temperature polarization. Letting effective thermal efficiency be \(\eta_{\mathrm{th,MD}}\in(0,1]\): \[\dot{Q}_{\mathrm{MD}} \;=\; \frac{\dot{m}_p\,h_{fg,\mathrm{eff}}}{\eta_{\mathrm{th,MD}}}, \label{eq:md_heat_eff}\] is fixed (\(h_{fg,\mathrm{eff}}\) is effective latent heat including sensible heat). Thus, Specific Thermal Energy Consumption (STEC) is defined as: \[\mathrm{STEC}_{\mathrm{MD}} \;=\; \frac{\int \dot{Q}_{\mathrm{MD}}(t)\,dt}{\int Q_p(t)\,dt} \qquad \left[\mathrm{J/m^3}\right] \label{eq:md_stec}\] Also, since MD has electrical auxiliary power for pumps/fans, electric SEC is reported separately via [eq:desal_SEC], and total energy is presented as (Electric + Heat).

5.8.3 Coupling with ESS (Physical Basis for Heat-Dominant Scenario)

If ESS can provide thermal storage (including nighttime), MD offers a path to utilize solar heat directly for “Water Production” instead of “Electricity Production”. That is, the path: \[\text{Solar} \;\to\; \text{Thermal Storage} \;\to\; \text{MD} \;\to\; \text{Freshwater}\] does not directly suffer from Carnot limit (as it does not convert heat to work). However, thermal balance of [eq:md_stec] still applies, and “Free” is used only to mean no omission in Energy Balance, not zero source cost.

5.9 Linking VP Amplitude/Alignment Variables to Purification Process (Fixing Internal Consistency of Application Whitepaper)

This application whitepaper maintains VP Amplitude/Alignment variables as “Explainable Latent Variables” but connects them to physical cores (transport coefficients) of purification processes to make them verifiable.

5.9.1 Core Connection: Amplitude/Alignment \(\to\) Mobility/Diffusion/Conductivity

Key observable properties in electric field based purification are ion mobility/diffusion/conductivity. This whitepaper parameterizes coefficient \(D_i\) in [eq:NP] with VP variables. \[D_i \;=\; D_{i,0}\; \psi_A\!\left(A_{\mathrm{op}}^{(\mathrm{sol})}(i)\right)\; \psi_S\!\left(S^{(\mathrm{sol})}\right), \label{eq:D_vp}\] Here, \(D_{i,0}\) is reference diffusion coefficient, \(\psi_A,\psi_S\) are dimensionless positive functions, calibrated with experimental data. Then, the coefficient of migration term in [eq:NP], \(\dfrac{z_i D_i F}{RT}\), changes automatically, making the statement “Large Amplitude/High Alignment” verifiable via Current/Conductivity/Concentration Distribution.

5.9.2 Measurement Proxy for Alignment \(S\) (Fixing Reproducibility in Purification)

Since it is difficult to measure \(S\) directly in purification devices, this whitepaper mandates selecting and fixing one of the following:

1. Impedance Anisotropy Index: Define \(S\) as ratio of impedance or permittivity in channel direction vs. normal direction.

2. Flow Stability Index: Map turbulence intensity (\(\mathrm{Re}\), fluctuation velocity RMS) or concentration polarization stability index to \(S\).

3. Optical Anisotropy Index: Define proxy value of alignment using polarization/scattering (if possible).

If a proxy variable and conversion formula are not selected, \(S\) does not meet DOI reproducibility requirements and is excluded from purification performance (KPI) interpretation (HOLD).

5.10 Integrated Design Procedure (Design Parameters per Technology and Calculation Sequence: Reproducible Algorithm)

5.10.1 Fixing Inputs (Design Requirements)

Purification module design inputs are fixed as the set: \[\Pi_{\mathrm{desal}}= \left\{ C_f,\ C_p^\ast,\ R^\ast,\ Y^\ast,\ Q_f^\ast,\ T,\ \mathcal{E},\ P_{\mathrm{ESS}}(t),\ E_{\mathrm{ESS}}(t) \right\}, \label{eq:Pi_desal}\] Here, \(C_p^\ast\) is target product concentration, \(Q_f^\ast\) is target treatment flow rate, and \(P_{\mathrm{ESS}}(t), E_{\mathrm{ESS}}(t)\) are power/energy profiles available from ESS.

5.10.2 Fixing Outputs (Design Results)

Design outputs are fixed as the set: \[\Omega_{\mathrm{desal}}= \left\{ \mathcal{T},\ A_{\mathrm{mem}},\ \Delta P,\ V_{\mathrm{stack}},\ I,\ V_{\mathrm{app}},\ \mathrm{SEC},\ R,\ Y,\ C_c,\ Q_p \right\}, \label{eq:Omega_desal}\] Here, \(A_{\mathrm{mem}}\) is effective membrane (or electrode) area, \(\Delta P\) is RO pressure, \(V_{\mathrm{stack}}\) is ED stack voltage, \(I\) is ED current, and \(V_{\mathrm{app}}\) is CDI applied voltage.

5.10.3 Design Algorithm (Fixed Sequence)

1. Feed Classification: Determine \(C_f\) bin via [eq:salinity_bins]–[eq:salinity_default].

2. Candidate Selection: Generate candidate set \(\{\mathcal{T}\}\) via [eq:tech_selector] and branching rules.

3. Initial Design Calculation with Minimum Physical Models for each technology:

   • RO: Calculate \((A_{\mathrm{mem}}, \Delta P, \mathrm{SEC})\) via [eq:ro_Jw]–[eq:ro_SEC].

   • ED: Calculate \((V_{\mathrm{stack}}, I, \mathrm{SEC})\) via [eq:ed_nsalt]–[eq:SECed].

   • CDI: Calculate \((C, V_{\mathrm{app}}, \mathrm{SEC})\) via [eq:cdi_nsalt]–[eq:SECcdi].

   • MD: Calculate \((A_{\mathrm{mem}}, \mathrm{STEC}, \mathrm{SEC}_{\mathrm{elec}})\) via [eq:md_flux]–[eq:md_stec].

4. Thermodynamic Lower Bound Check: Confirm \(\mathrm{SEC}\ge \mathrm{SEC}_{\min}\) via [eq:SEC_min_bound] or [eq:SECmin_integral] for all candidates.

5. ESS Matching: Evaluate if the process can operate continuously with ESS power/energy profiles. (e.g., RO requires continuous pump power, ED/CDI are relatively flexible to fluctuations, MD allows thermal basis.)

6. Final Selection: Finalize \(\mathcal{T}\) based on achievement of target KPIs (\(R^\ast, Y^\ast, C_p^\ast\)) and minimization of SEC.

7. Demonstration Design: Fix measurement items/logs/uncertainty/repetition count (next section).

5.11 Reproducibility Provision (For DOI Issuance): Purification Module Protocol/Data/Code Standards

5.11.1 Mandatory Measurement Channels (Common for Purification Modules)

For DOI reproducibility of purification modules, record at least:

1. Flow Rate: \(Q_f(t), Q_p(t), Q_c(t)\) (m\(^3\)/s).

2. Salinity/Conductivity: \(C_f(t), C_p(t), C_c(t)\) or Conductivity (including conversion formula).

3. Power: \(V(t), I(t)\) or Thermal Power (based on temperature/flow).

4. Temperature: Feed/Product/Concentrate and internal device representative temperature.

5. Pressure (RO/MD): \(\Delta P(t)\) or pressure at each point.

6. (Optional) pH, Turbidity, Hardness (Tracking scaling/fouling causes).

5.11.2 Data Schema (Fixed)

Logs are fixed as CSV (or equivalent JSON) containing at least: \[\{\texttt{t},\,\texttt{Qf},\,\texttt{Qp},\,\texttt{Qc},\,\texttt{Cf},\,\texttt{Cp},\,\texttt{Cc},\,\texttt{T},\,\texttt{V},\,\texttt{I},\,\texttt{dP}\}. \label{eq:desal_schema_set}\] Units fixed as: \[t[\mathrm{s}],\; Q[\mathrm{m^3/s}],\; C[\mathrm{kg/m^3}],\; T[\mathrm{K}],\; V[\mathrm{V}],\; I[\mathrm{A}],\; \Delta P[\mathrm{Pa}]\] If conductivity is used, include conversion formula \(C=\Gamma(\kappa,T)\) (calibration curve) in DOI package.

5.11.3 Analysis Code Outputs (Mandatory)

DOI package analysis code must automatically generate at least:

1. \(R(t)\), \(Y(t)\): [eq:desal_R], [eq:desal_Y].

2. SEC: [eq:desal_SEC] (Electric), also [eq:md_stec] (Thermal) for MD.

3. Mass Balance Check: Residual time curve of [eq:desal_flow_balance]–[eq:desal_salt_balance].

4. Thermodynamic Lower Bound Check: Ratio relative to [eq:SEC_min_bound] or [eq:SECmin_integral].

5. Design Parameter Calculation per Technology (RO: [eq:ro_SEC], ED: [eq:SECed], CDI: [eq:SECcdi]).

Release must include version tag and checksum (SHA256) for identity verification.

5.12 Conclusion of This Section (Summary): Fixing “Reality Branching” of Electric Field Based Purification with Physics

1. Purification reality is fixed by \((R, Y, \mathrm{SEC})\) and Mass Balance; “Zero Energy” is not allowed.

2. Seawater vs. Brackish is classified by salinity bin [eq:salinity_bins], fixing branching rules: RO for Seawater, ED/CDI for Brackish, and MD for Heat-Dominant or High Salinity/Brine.

3. All technologies cannot violate thermodynamic lower bounds [eq:SEC_min_bound] or [eq:SECmin_integral]. Violation is not Success but Re-verification (HOLD).

4. Electric field based processes are physically grounded by Nernst–Planck transport [eq:NP] and length/time scale conditions [eq:sep_condition_np].

5. For DOI reproducibility, mandatory measurement channels, data schema, and analysis code outputs are fixed in this section.

6 Catalyst Module: Pt-Based Hydrogen Dissociation/Electrolysis/Reactor Design (Mainstream Catalysis Science Mapping + VP Variable Mapping)

6.1 Purpose of this Section and System Boundary (Entity Fixing)

The purpose of this section is to organize the “Catalyst Module”—which utilizes electricity (or heat) provided by the ESS to (1) produce hydrogen (H\(_2\)) via electrolysis, (2) dissociate/activate generated hydrogen on Pt surfaces, and (3) perform target reactions in a reactor—so that it can be designed based on Mainstream Catalysis Science (Surface Science/Reaction Kinetics/Electrochemistry). Simultaneously, it aims to map VP variables (Operational Amplitude \(A_{\mathrm{op}}\), Alignment \(S\)) to Measurable Catalyst Variables (Adsorption, Activation Barrier, Exchange Current Density, TOF, etc.) to fix them in a reproducible manner.

In this application whitepaper, the system boundary of the catalyst module is fixed to the following three blocks:

1. Electrolysis Block (Electrolyzer): Water \(\to\) H\(_2\) + O\(_2\) (Electrical Energy \(\to\) Chemical Energy).

2. Pt Activation Block (Pt Activator): H\(_2\)(g) \(\rightleftharpoons\) 2H\(^*\) (Dissociative Adsorption/Recombination).

3. Reactor Block (Reactor): Substrate (Reactant) + H\(^*\) (or H\(_2\)) \(\to\) Product (Hydrogenation/Reduction, etc.).

Important (Entity Fixing Principle): The Pt catalyst is not a device that “creates” H\(_2\), but a device that “dissociates/activates H\(_2\) to accelerate reactions”. H\(_2\) generation must be included in the mass/energy balance strictly via electrolysis (or external supply).

6.2 Success Criteria KPI (Fixing with Reproducible Observables)

Success judgment of the catalyst module is primarily fixed not by VP variables (Amplitude/Alignment) themselves, but by the following Measurable KPIs.

6.2.1 Electrolysis (Hydrogen Production) KPI

1. Hydrogen Production Molar/Mass Flow Rate: \[\dot{n}_{H_2} \ [\mathrm{mol/s}], \qquad \dot{m}_{H_2}=M_{H_2}\dot{n}_{H_2}\ [\mathrm{kg/s}], \label{eq:kpi_nH2_mH2}\]

2. Faraday Efficiency (Charge Efficiency) \(\eta_F\in[0,1]\): \[\eta_F \;=\; \frac{2F\,\dot{n}_{H_2}}{I}, \label{eq:kpi_faraday_eff}\]

3. Specific Electrical Energy Consumption (Hydrogen Basis): \[\mathrm{SEC}_{H_2} \;=\; \frac{P_{\mathrm{elec}}}{\dot{m}_{H_2}} \qquad \left[\mathrm{W}\middle/\mathrm{kg\,s^{-1}}\right] \;=\; \left[\mathrm{J/kg}\right] \;\;\text{or}\;\; \left[\mathrm{kWh/kg}\right], \label{eq:kpi_sech2}\]

4. Electricity \(\to\) Chemical Efficiency (e.g., LHV Basis): \[\eta_{\mathrm{LHV}} \;=\; \frac{\dot{m}_{H_2}\,\mathrm{LHV}_{H_2}}{P_{\mathrm{elec}}} \;\in\;[0,1], \label{eq:kpi_etaLHV}\] Here, \(\mathrm{LHV}_{H_2}\) is the Lower Heating Value of hydrogen (J/kg).

6.2.2 Catalytic Reactor KPI

The target reaction is represented by a single reaction where a generic reactant \(A\) converts to product \(B\). \[A + \nu_{H_2}H_2 \;\to\; B, \label{eq:generic_hydrogenation}\] Here, \(\nu_{H_2}\) is the stoichiometric coefficient (moles of hydrogen required / moles of reactant). KPIs are fixed as follows:

1. Conversion: \[X_A \;=\; 1-\frac{F_{A,\mathrm{out}}}{F_{A,\mathrm{in}}}, \label{eq:kpi_conversion}\]

2. Selectivity (Selectivity for \(B\) in case of multiple products): \[S_B \;=\; \frac{\nu_A\left(F_{B,\mathrm{out}}-F_{B,\mathrm{in}}\right)} {F_{A,\mathrm{in}}-F_{A,\mathrm{out}}}, \label{eq:kpi_selectivity}\]

3. Space Time Yield (Based on catalyst mass \(W_{\mathrm{cat}}\)): \[\mathrm{STY}_B \;=\; \frac{\dot{n}_B}{W_{\mathrm{cat}}} \qquad \left[\mathrm{mol/(s\cdot kg_{cat})}\right], \label{eq:kpi_sty}\]

4. Turnover Frequency (TOF): \[\mathrm{TOF} \;=\; \frac{\dot{n}_B}{N_{\mathrm{site,eff}}} \qquad \left[\mathrm{s^{-1}}\right], \label{eq:kpi_tof}\] Here, \(N_{\mathrm{site,eff}}\) is the effective number of active sites in moles (mol-site) or count (fixed upon definition selection).

6.3 Electrolysis (Electrolyzer) Design: Faraday’s Law, Voltage Decomposition, Energy Balance (Mathematical Completeness)

6.3.1 Hydrogen Production Rate: Faraday’s Law (Including Steady State)

When current \(I(t)\) flows in electrolysis, the number of hydrogen moles produced is: \[n_{H_2}(t_0,t_1) \;=\; \frac{\eta_F}{2F}\int_{t_0}^{t_1} I(t)\,dt. \label{eq:faraday_integral}\] In steady state (constant current) where \(I(t)=I\): \[\dot{n}_{H_2} \;=\; \frac{\eta_F}{2F}\,I, \qquad I \;=\; \frac{2F}{\eta_F}\,\dot{n}_{H_2}. \label{eq:faraday_steady}\] Thus, given a target hydrogen production rate \(\dot{n}_{H_2}^\ast\), the required current is uniquely fixed by [eq:faraday_steady].

6.3.2 Electrolysis Cell Voltage Decomposition (Reversible + Overpotential + Ohmic Loss)

The terminal voltage of the electrolysis cell is fixed by decomposing it as follows: \[V_{\mathrm{cell}} \;=\; V_{\mathrm{rev}} + \eta_{\mathrm{an}} + \eta_{\mathrm{cat}} + I R_{\mathrm{ohm}}, \label{eq:Vcell_decomp}\] Here,

• \(V_{\mathrm{rev}}\) is the reversible (thermodynamic) voltage,

• \(\eta_{\mathrm{an}}\) is the anode (OER) overpotential,

• \(\eta_{\mathrm{cat}}\) is the cathode (HER) overpotential (Pt applicable),

• \(R_{\mathrm{ohm}}\) is the equivalent Ohmic resistance of membrane/electrolyte/current collector/wiring.

Reversible voltage is defined by the Nernst equation. Fixing the reaction as: \[H_2O(l) \;\to\; H_2(g) + \frac{1}{2}O_2(g) \label{eq:water_splitting_half}\] Approximating water activity as \(a_{H_2O}\approx 1\), and letting partial pressures of product gases be \(p_{H_2}, p_{O_2}\): \[V_{\mathrm{rev}}(T,p_{H_2},p_{O_2}) \;=\; V^\circ(T) +\frac{RT}{2F}\ln\!\left(\frac{p_{H_2}\,p_{O_2}^{1/2}}{a_{H_2O}}\right) \;\approx\; V^\circ(T)+\frac{RT}{2F}\ln\!\left(p_{H_2}\,p_{O_2}^{1/2}\right). \label{eq:Vrev_nernst}\] Here, \(V^\circ(T)\) is the standard reversible voltage (function of temperature). In Application Whitepaper v1.0, \(V^\circ(T)\) is taken from external standards (table/function), and the used function/version is fixed in the DOI package.

6.3.3 Current Density and Electrode Area (Design Variables)

Let the effective area of the electrode (or MEA) be \(A_{\mathrm{elec}}\) and current density be \(j\): \[I=jA_{\mathrm{elec}}. \label{eq:I_jA}\] Combining with [eq:faraday_steady] for target production rate \(\dot{n}_{H_2}^\ast\), the required area is: \[A_{\mathrm{elec}} \;=\; \frac{2F}{\eta_F}\cdot \frac{\dot{n}_{H_2}^\ast}{j}. \label{eq:Aelec_required}\] That is, selecting \(j\) determines \(A_{\mathrm{elec}}\). (In application design, choosing \(j\) too high increases overpotential/thermal load, while choosing it too low increases area/cost, so optimization is needed. Optimization is performed by empirically calibrating each term of [eq:Vcell_decomp].)

6.3.4 Electrical Energy Consumption and Heat Generation (Energy Balance of Electrolysis)

Electrolysis power is: \[P_{\mathrm{elec}} = V_{\mathrm{cell}} I, \label{eq:Pelec}\] Thus, power consumption per unit mass of hydrogen (instantaneous value) is: \[\mathrm{SEC}_{H_2} \;=\; \frac{V_{\mathrm{cell}}I}{\dot{m}_{H_2}} \;=\; \frac{V_{\mathrm{cell}}I}{M_{H_2}\dot{n}_{H_2}} \;=\; \frac{V_{\mathrm{cell}}I}{M_{H_2}\left(\eta_F I/(2F)\right)} \;=\; \frac{2F}{\eta_F M_{H_2}}\,V_{\mathrm{cell}}. \label{eq:SEC_H2_from_V}\] Core Conclusion: In steady state, \(\mathrm{SEC}_{H_2}\) is essentially determined by \(V_{\mathrm{cell}}\), and current \(I\) is a scale variable to match the target production.

Heat generated in the electrolysis cell (Definition) is fixed as the remainder after excluding the electrical input converted to chemical energy (hydrogen). Selecting hydrogen formation enthalpy or LHV/HHV basis, for example, LHV-based heat generation rate is defined as: \[\dot{Q}_{\mathrm{gen}} \;=\; P_{\mathrm{elec}} - \dot{m}_{H_2}\mathrm{LHV}_{H_2}. \label{eq:electrolysis_heat_gen}\] Since this is the entity determining the cooling design (heat exchanger, coolant flow) of the electrolysis stack, it must be recorded for DOI reproduction.

6.4 Hydrogen Dissociation/Recombination on Pt Surface: Surface Science Based Minimum Model (Mathematical Completeness)

6.4.1 Surface Reaction Equation (Active Site Notation) and Coverage Definition

Denote Pt surface active site as “\(*\)”. Hydrogen dissociative adsorption (reversible) is fixed as: \[H_2(g) + 2* \;\rightleftharpoons\; 2H*. \label{eq:H2_dissociation}\] Surface coverage is defined as: \[\theta_H \;=\; \frac{N_{H*}}{N_{\mathrm{site}}}, \qquad \theta_* \;=\; \frac{N_*}{N_{\mathrm{site}}}, \qquad \theta_H+\theta_* = 1 \label{eq:coverage_def}\] Here, \(N_{\mathrm{site}}\) is the total number of active sites (or mol-site), \(N_{H*}\) is the number of sites occupied by hydrogen, and \(N_*\) is the number of empty sites. (If other adsorbed species exist, extend to \(\theta_H+\theta_*+\theta_{\mathrm{others}}=1\), but in v1.0 of this section, the 2-component system of [eq:coverage_def] is the basis to fix the hydrogen activation mechanism.)

6.4.2 Dissociation/Recombination Rate Equations (Langmuir–Hinshelwood Minimum Form)

Dissociation (forward) rate and recombination (reverse) rate are fixed as: \[\begin{aligned} r_{\mathrm{d}} &= k_{\mathrm{d}}(T)\,p_{H_2}\,\theta_*^2, \label{eq:rd}\\ r_{\mathrm{r}} &= k_{\mathrm{r}}(T)\,\theta_H^2, \label{eq:rr}\end{aligned}\] Here,

• \(r_{\mathrm{d}}, r_{\mathrm{r}}\) are reaction rates per unit area (or per unit active site),

• \(p_{H_2}\) is gaseous hydrogen partial pressure,

• \(k_{\mathrm{d}}(T), k_{\mathrm{r}}(T)\) are temperature-dependent rate constants.

Net dissociation (surface H* generation) rate is defined as: \[r_{H*} \;=\; 2\left(r_{\mathrm{d}}-r_{\mathrm{r}}\right) \label{eq:rHstar_net}\] (Factor 2 because 2 H* are generated per dissociation event).

6.4.3 Coverage at Quasi-Equilibrium (Complete Analytical Solution)

When dissociation/recombination reaches quasi-equilibrium, \(r_{\mathrm{d}}=r_{\mathrm{r}}\), so from [eq:rd]–[eq:rr]: \[k_{\mathrm{d}}(T)\,p_{H_2}\,\theta_*^2 \;=\; k_{\mathrm{r}}(T)\,\theta_H^2 \;\Rightarrow\; \frac{\theta_H}{\theta_*} \;=\; \sqrt{\frac{k_{\mathrm{d}}(T)}{k_{\mathrm{r}}(T)}\,p_{H_2}}. \label{eq:theta_ratio}\] Defining equilibrium constant (adsorption equilibrium constant) as: \[K_H(T) \;=\; \frac{k_{\mathrm{d}}(T)}{k_{\mathrm{r}}(T)} \label{eq:KH_def}\] Then: \[\frac{\theta_H}{\theta_*} \;=\; \sqrt{K_H(T)\,p_{H_2}}. \label{eq:theta_ratio2}\] Also using \(\theta_* = 1-\theta_H\) from [eq:coverage_def], we obtain the complete analytical solution: \[\frac{\theta_H}{1-\theta_H}=\sqrt{K_H p_{H_2}} \;\Rightarrow\; \theta_H \;=\; \frac{\sqrt{K_H p_{H_2}}}{1+\sqrt{K_H p_{H_2}}} \label{eq:thetaH_solution}\] This equation quantitatively fixes that (i) \(\theta_H\) increases as \(p_{H_2}\) increases, and (ii) \(\theta_H\) increases as \(K_H\) increases (adsorption becomes favorable).

6.4.4 Temperature Dependence of Rate Constants (TST Based)

Assuming \(k_{\mathrm{d}}(T)\) and \(k_{\mathrm{r}}(T)\) are governed by activation free energy, the Transition State Theory (TST) form is fixed as: \[k(T) \;=\; \nu(T)\,\exp\!\left(-\frac{\Delta G^\ddagger(T)}{k_B T}\right), \label{eq:tst_general}\] Here, \(\nu(T)\) is attempt frequency (pre-exponential), \(\Delta G^\ddagger\) is activation free energy, and \(k_B\) is Boltzmann constant. This definition serves as the basis for fixing how VP variables enter into \(\Delta G^\ddagger\) in the subsequent VP Variable Mapping (§6.6).

6.5 Reactor Design: Mass Balance, Rate Equation, Catalyst Amount/Volume Estimation (Mathematical Completeness)

6.5.1 General Rate Equation: Langmuir–Hinshelwood Form Including Hydrogen Activation

The minimum mechanism assuming substrate \(A\) adsorbs on Pt surface to become \(A^*\), reacts with surface hydrogen \(H^*\), and product \(B\) desorbs, is fixed as: \[\begin{aligned} A(g) + * &\rightleftharpoons A*, \label{eq:A_adsorption}\\ H_2(g) + 2* &\rightleftharpoons 2H*, \label{eq:H2_adsorption_again}\\ A* + \nu_H H* &\to B* + (\nu_H+1)*, \label{eq:surface_reaction}\\ B* &\rightleftharpoons B(g)+*. \label{eq:B_desorption}\end{aligned}\] Assuming surface reaction [eq:surface_reaction] is the Rate Determining Step (RDS), the reaction rate per unit active site (proportional to TOF) can be fixed as the product form: \[r_A \;=\; k_{\mathrm{surf}}(T)\,\theta_A\,\theta_H^{\nu_H}, \label{eq:rate_LH}\] Here, \(\theta_A\) is \(A^*\) coverage, \(\theta_H\) is hydrogen coverage from [eq:thetaH_solution], and \(\nu_H\) is the effective order of H* participating in surface reaction. \(k_{\mathrm{surf}}(T)\) is fixed from [eq:tst_general] as: \[k_{\mathrm{surf}}(T) \;=\; \nu_{\mathrm{surf}}(T)\,\exp\!\left(-\frac{\Delta G^\ddagger_{\mathrm{surf}}(T)}{k_B T}\right) \label{eq:ksurf_tst}\]

6.5.2 Effective Number of Active Sites and Effective Reaction Rate (Including Alignment/Activity Fraction)

In real catalysts, not all geometric surfaces may be in the same active state. Therefore, the effective number of active sites is fixed as: \[N_{\mathrm{site,eff}} \;=\; f_{\mathrm{act}}(S^{(\mathrm{surf})})\,N_{\mathrm{site}}, \qquad f_{\mathrm{act}}(S^{(\mathrm{surf})}) = \left(S^{(\mathrm{surf})}\right)^m, \quad m\ge 1, \label{eq:effective_sites}\] Here, \(S^{(\mathrm{surf})}\in[0,1]\) is the surface alignment (or order parameter representing active state fraction), and \(m\) is an exponent calibrated experimentally. Thus, the molar reaction rate of the entire catalyst (or reactor) is: \[\dot{n}_A \;=\; r_A \, N_{\mathrm{site,eff}} \;=\; r_A \, f_{\mathrm{act}}(S^{(\mathrm{surf})})\,N_{\mathrm{site}}. \label{eq:overall_rate_sites}\] This is the minimum connection rule where VP alignment \(S\) enters as the “effective number of active sites”.

6.5.3 Plug Flow Reactor (PFR) Design Equation (Catalyst Mass Basis)

Let catalyst mass be \(W\) (kg-cat) and molar flow rate be \(F_A\) (mol/s). Mass balance in PFR is: \[\frac{dF_A}{dW} = -r_A, \label{eq:pfr_mass_balance}\] Here, \(r_A\) must be defined as “reaction rate per catalyst mass”. To write the reaction rate of [eq:overall_rate_sites] per catalyst mass, define total active sites per catalyst mass as \(\Gamma_{\mathrm{site}}\) (mol-site/kg-cat): \[N_{\mathrm{site}} = \Gamma_{\mathrm{site}} W, \label{eq:Nsite_gamma}\] Thus, reaction rate per catalyst mass can be fixed as: \[r_A \;=\; \left[\mathrm{TOF}(T,p)\right]\, f_{\mathrm{act}}(S^{(\mathrm{surf})})\, \Gamma_{\mathrm{site}}\, \theta_A\,\theta_H^{\nu_H}, \label{eq:rA_per_mass}\] (TOF may be defined including \(k_{\mathrm{surf}}\) and coverages of [eq:rate_LH], but the definition must be fixed in the DOI package). Catalyst mass satisfying target conversion \(X_A^\ast\) is determined by integrating [eq:pfr_mass_balance] with: \[X_A^\ast = 1-\frac{F_{A,\mathrm{out}}}{F_{A,\mathrm{in}}}, \label{eq:XA_target}\] For example, if \(r_A\) can be approximated as constant (or average value \(\bar{r}_A\)): \[W \;\approx\; \frac{F_{A,\mathrm{in}}-F_{A,\mathrm{out}}}{\bar{r}_A} \;=\; \frac{F_{A,\mathrm{in}}X_A^\ast}{\bar{r}_A}. \label{eq:W_simple}\] In the application whitepaper, \(W\) is calculated by numerical integration when concentration dependence of \(r_A\) is included, and the code/parameters are fixed in the DOI package.

6.5.4 Mass Transfer Limits (External/Internal Diffusion) and Effectiveness Factor (Mathematical Completeness)

Catalytic reactions can be limited by mass transfer as well as kinetics. To exclude or quantify this, the following indices are fixed.

6.5.4.1 (1) External (Film) Transfer.

Let bulk concentration of reactant \(A\) in gas phase be \(C_{A,b}\), surface concentration be \(C_{A,s}\), external mass transfer coefficient be \(k_g\) (m/s), and external surface area of catalyst (per volume) be \(a_s\) (m\(^2\)/m\(^3\)). External transfer flux is: \[r_{A,\mathrm{mt}} \;=\; k_g a_s (C_{A,b}-C_{A,s}). \label{eq:external_mt}\] If reaction is fast, \(C_{A,s}\) decreases, and external transfer becomes the bottleneck. Therefore, in design, \(C_{A,s}\) is determined such that reaction rate \(r_{A,\mathrm{rxn}}\) matches [eq:external_mt], and \(k_g\) is calibrated with flow/Reynolds number based correlations (Correlation selection fixed in DOI).

6.5.4.2 (2) Internal (Pore) Diffusion: Thiele Modulus and Effectiveness Factor.

Approximating catalyst particle as a sphere with radius \(R_p\), effective diffusion coefficient \(D_{\mathrm{eff}}\) (m\(^2\)/s), and intrinsic rate constant \(k_{\mathrm{int}}\) (1/s) for first-order reaction approximation, Thiele modulus is: \[\phi = R_p\sqrt{\frac{k_{\mathrm{int}}}{D_{\mathrm{eff}}}}. \label{eq:thiele}\] Effectiveness factor of spherical particle is fixed as: \[\eta_{\mathrm{eff}} \;=\; \frac{3}{\phi^2}\left(\phi\coth\phi - 1\right), \label{eq:effectiveness}\] Thus, observed reaction rate is: \[r_{A,\mathrm{obs}} = \eta_{\mathrm{eff}}\,r_{A,\mathrm{int}} \label{eq:obs_rate_eta}\] If \(\eta_{\mathrm{eff}}\approx 1\), internal diffusion limit is judged weak. (If diffusion limit is strong, structural/pore design to reduce \(R_p\) or increase \(D_{\mathrm{eff}}\) is needed.)

6.6 Mainstream Catalysis Science Correspondence Table and VP Variable Mapping (Core: Fixing with Empirically Calibratable Functions)

6.6.0.1 Physics mapping: amplitude-governed barrier modulation.

In the application layer, the catalytic barrier modulation is treated as a function of the operational amplitude \(A_{\mathrm{op}}\) normalized by the intrinsic Reference Amplitude \(r_{\mathrm{vac}}=245.9\,\mathrm{fm}\): \[\widehat{A}_{\mathrm{op}}^{(\mathrm{surf})}(t)\;\equiv\;\frac{A_{\mathrm{op}}^{(\mathrm{surf})}(t)}{r_{\mathrm{vac}}}. \label{eq:Aop_hat_surf}\] The spatial lattice geometry scale \(L_{\mathrm{quant}}=4854\,\mathrm{fm}\) does not directly control \(\Delta G^\ddagger\); it is reserved for geometric volume/packing interpretations.

6.6.1 Definition of VP Variables for Catalyst Version (Surface/Electrode)

VP variables used in this section are limited to surface (catalyst) and electrode (electrolysis) environments and fixed as: \[A_{\mathrm{op}}^{(\mathrm{surf})}(t),\quad S^{(\mathrm{surf})}(t) \label{eq:vp_surface_vars}\] Definition Principles:

1. \(A_{\mathrm{op}}^{(\mathrm{surf})}\) is a latent variable for “Excited state affecting surface reaction barrier or electrochemical exchange current density”.

2. \(S^{(\mathrm{surf})}\) is a dimensionless variable representing “Effective fraction of active sites contributing to reaction” or “Order of surface state”.

This whitepaper does not assume direct measurement of \(A_{\mathrm{op}}^{(\mathrm{surf})}, S^{(\mathrm{surf})}\), but ensures reproducibility by calibrating the Mapping Functions to mainstream variables below using experimental data.

6.6.2 Standard Form of Mapping Functions (Including Gate/Smooth Transition, Completely Defined)

To reproduce threshold-based (gate) structures, the following normalization functions are fixed.

6.6.2.1 (1) Logistic Transition Function.

\[\sigma(x)=\frac{1}{1+\exp(-x)}. \label{eq:sigmoid_def}\]

6.6.2.2 (2) Amplitude Based Activity Function.

\[g_A\!\left(A_{\mathrm{op}}^{(\mathrm{surf})}\right) \;=\; \sigma\!\left(\frac{A_{\mathrm{op}}^{(\mathrm{surf})}-A_{\mathrm{th}}}{\Delta A}\right), \label{eq:gA_def}\] Here, \(A_{\mathrm{th}}\) is activation transition (threshold) amplitude, and \(\Delta A\) is transition width.

6.6.2.3 (3) Alignment Based Activity Function.

\[g_S\!\left(S^{(\mathrm{surf})}\right) \;=\; \sigma\!\left(\frac{S^{(\mathrm{surf})}-S_{\mathrm{th}}}{\Delta S}\right), \label{eq:gS_def}\] Here, \(S_{\mathrm{th}}\) is alignment transition (threshold) value, and \(\Delta S\) is transition width.

6.6.2.4 (4) Active Site Fraction Function.

\[f_{\mathrm{act}}\!\left(S^{(\mathrm{surf})}\right)=\left(S^{(\mathrm{surf})}\right)^m,\qquad m\ge 1, \label{eq:fact_def}\] This is identical to [eq:effective_sites], and \(m\) is estimated from data.

6.6.3 VP Mapping to Mainstream Catalyst Variables (Activation Barrier, Adsorption, TOF)

Key mainstream variables for catalytic reactions are (i) Activation Free Energy \(\Delta G^\ddagger\), (ii) Adsorption Free Energy \(\Delta G_{\mathrm{ads}}\), (iii) Active Site Fraction, and (iv) TOF. This whitepaper fixes their mapping to VP variables as follows.

6.6.3.1 (1) Activation Free Energy Mapping.

\[\Delta G^\ddagger_{\mathrm{surf}}(T) \;=\; \Delta G^\ddagger_{0}(T) -\Lambda_A\,g_A\!\left(A_{\mathrm{op}}^{(\mathrm{surf})}\right) -\Lambda_S\,g_S\!\left(S^{(\mathrm{surf})}\right), \qquad \Lambda_A\ge 0,\ \Lambda_S\ge 0. \label{eq:DGddagger_mapping}\] Thus, rate constant of [eq:ksurf_tst] becomes: \[k_{\mathrm{surf}}(T) \;=\; \nu_{\mathrm{surf}}(T)\, \exp\!\left( -\frac{\Delta G^\ddagger_{0}(T)}{k_B T} \right)\, \exp\!\left( \frac{\Lambda_A\,g_A(A_{\mathrm{op}}^{(\mathrm{surf})})+\Lambda_S\,g_S(S^{(\mathrm{surf})})}{k_B T} \right). \label{eq:ksurf_mapping}\] This equation reproduces the structure where “reaction increases sharply at threshold amplitude/alignment” within mainstream kinetics in a complete form.

6.6.3.2 (2) Adsorption Free Energy Mapping (Optional).

Equilibrium constant of hydrogen or substrate adsorption is expressed by adsorption free energy. For example, setting equilibrium constant of hydrogen dissociative adsorption [eq:KH_def] as: \[K_H(T) = \exp\!\left(-\frac{\Delta G_{\mathrm{ads},H}(T)}{RT}\right) \label{eq:KH_from_DGads}\] Apply VP mapping to adsorption free energy: \[\Delta G_{\mathrm{ads},H}(T) \;=\; \Delta G_{\mathrm{ads},H,0}(T) -\Gamma_A\,g_A\!\left(A_{\mathrm{op}}^{(\mathrm{surf})}\right) -\Gamma_S\,g_S\!\left(S^{(\mathrm{surf})}\right), \qquad \Gamma_A\ge 0,\ \Gamma_S\ge 0. \label{eq:DGads_mapping}\] Then \(\theta_H\), from [eq:thetaH_solution] and [eq:KH_from_DGads], becomes a complete model varying with \((A_{\mathrm{op}}^{(\mathrm{surf})}, S^{(\mathrm{surf})})\).

6.6.3.3 (3) Effective TOF and Macroscopic Reaction Rate.

Combining [eq:overall_rate_sites], [eq:ksurf_mapping], and [eq:fact_def], the macroscopic reaction rate is fixed as: \[\dot{n}_B \;=\; \Gamma_{\mathrm{site}}W_{\mathrm{cat}}\, \left(S^{(\mathrm{surf})}\right)^m\, \nu_{\mathrm{surf}}(T)\, \exp\!\left(-\frac{\Delta G^\ddagger_{0}(T)}{k_B T}\right)\, \exp\!\left( \frac{\Lambda_A g_A(A_{\mathrm{op}}^{(\mathrm{surf})})+\Lambda_S g_S(S^{(\mathrm{surf})})}{k_B T} \right)\, \theta_A\,\theta_H^{\nu_H}. \label{eq:macro_rate_complete}\] This equation is a complete structure including (i) Catalyst Amount \(W_{\mathrm{cat}}\), (ii) Active Site Density \(\Gamma_{\mathrm{site}}\), (iii) Effective Site Fraction by Alignment, (iv) Barrier Reduction by Amplitude/Alignment, and (v) Surface Coverage, and can be calibrated with experimental data.

6.6.4 Mainstream Variables in Electrochemistry (HER) and VP Mapping (Exchange Current Density \(j_0\))

When Pt is used as electrolysis cathode (HER) catalyst, mainstream electrochemistry is fixed by Butler–Volmer equation. \[j = j_0 \left[ \exp\!\left(\frac{\alpha_a F\eta}{RT}\right) - \exp\!\left(-\frac{\alpha_c F\eta}{RT}\right) \right], \label{eq:BV_again}\] Here, \(j\) is current density, \(\eta\) is overpotential, \(\alpha_a, \alpha_c\) are transfer coefficients. This whitepaper fixes the minimum mapping where VP variables enter into exchange current density \(j_0\) as: \[j_0 \;=\; j_{0,0}(T)\, \left(S^{(\mathrm{surf})}\right)^m\, \exp\!\left(\beta_A\,g_A(A_{\mathrm{op}}^{(\mathrm{surf})})+\beta_S\,g_S(S^{(\mathrm{surf})})\right), \label{eq:j0_mapping}\] Here, \(j_{0,0}(T)\) is reference exchange current density (function of temperature), \(\beta_A, \beta_S\) are dimensionless coupling coefficients (calibration targets). With this, \((\beta_A, \beta_S, m, A_{\mathrm{th}}, \Delta A, S_{\mathrm{th}}, \Delta S)\) can be estimated from electrochemical data (\(j\)–\(\eta\) curve), fixing the reality of VP variables based on measurement.

6.6.5 Mainstream Catalysis Science Correspondence Table (Complete Definition and Mapping Rules)

Table 1 summarizes the correspondence between “Mainstream Variables” and “VP Variables” fixed in this section. (The table is a summary, but each formula is completely defined in the text.)

6.7 Catalyst Module Design Parameters (Minimum Design Variable Set) and System Integration

6.7.1 Minimum Design Variable Set (Fixing Reproducibility)

The design variable set for the catalyst module is fixed as: \[\Pi_{\mathrm{cat}}= \left\{\begin{aligned} &\dot{n}_{H_2}^\ast,\ \eta_F,\ j,\ A_{\mathrm{elec}},\ V_{\mathrm{cell}},\ R_{\mathrm{ohm}},\ \eta_{\mathrm{an}},\ \eta_{\mathrm{cat}},\ T,\ p_{H_2},\ p_{O_2},\ W_{\mathrm{cat}},\\ &\Gamma_{\mathrm{site}},\ R_p,\ D_{\mathrm{eff}},\ k_g,\ a_s,\ A_{\mathrm{th}},\Delta A,\ S_{\mathrm{th}},\Delta S,\ m,\ \Lambda_A,\Lambda_S,\Gamma_A,\Gamma_S,\beta_A,\beta_S \end{aligned}\right\}. \label{eq:Pi_cat}\] Each term enters directly into the equations defined in this section, and values are fixed by calibrating with experimental/simulation data.

6.7.2 Matching with ESS Electricity (Power Budget and Operation Mode)

Letting the power profile provided by ESS be \(P_{\mathrm{ESS}}(t)\), the power required by the electrolysis block is \(P_{\mathrm{elec}}(t)=V_{\mathrm{cell}}(t)I(t)\). Thus, the real-time power constraint is fixed as: \[P_{\mathrm{elec}}(t) \;\le\; P_{\mathrm{ESS}}(t) \label{eq:power_constraint}\] In sections where power is insufficient, duty operation (intermittent operation) or buffer (battery/capacitor) can be used for compensation. However, if a buffer is used, the change in stored energy of the buffer must be included in the total energy balance. If the buffer is a capacitor, stored energy is fixed same as [eq:cap_energy]: \[E_{\mathrm{cap}} = \frac{1}{2}C_{\mathrm{bus}}V_{\mathrm{bus}}^2 \label{eq:cap_energy_cat}\] and the difference between initial/final states is recorded.

6.8 Reproducibility Provision (For DOI Issuance): Catalyst Module Protocol/Data/Code Standards

6.8.1 Mandatory Measurement Channels (Common for Electrolysis + Catalyst + Reactor)

For DOI-level reproducibility, the catalyst module records at least the following channels:

1. Electrolysis: \(V_{\mathrm{cell}}(t)\), \(I(t)\), \(T_{\mathrm{stack}}(t)\), Coolant flow/temperature (if present), Pressure (gas side).

2. Hydrogen: \(p_{H_2}(t)\), Flow Rate \(\dot{V}_{H_2}(t)\) or Mass Flow Rate \(\dot{m}_{H_2}(t)\), (if possible) Composition (Moisture/Impurities).

3. Reactor: Inlet/Outlet Flow Rate (Gas/Liquid), \(T_{\mathrm{rxn}}(t)\), \(p_{\mathrm{rxn}}(t)\), Composition (e.g., \(F_{A}, F_{B}\) calculated from GC/LC).

4. Catalyst Properties (Pre/Post Experiment): Catalyst mass \(W_{\mathrm{cat}}\), Particle size \(R_p\), Surface area (BET, etc.) or basis for active site density estimation.

Also, for safety, H\(_2\) leakage/ventilation status (sensor or test record) is recorded as metadata.

6.8.2 Data Schema (Fixed)

Logs are fixed as CSV (or equivalent JSON) containing at least: \[\{\texttt{t},\texttt{Vcell},\texttt{I},\texttt{Tstack},\texttt{pH2},\texttt{pO2},\texttt{mH2},\texttt{Trxn},\texttt{prxn},\texttt{FAin},\texttt{FAout},\texttt{FBout}\}. \label{eq:cat_schema_set}\] Units are fixed as: \[t[\mathrm{s}],\; V[\mathrm{V}],\; I[\mathrm{A}],\; T[\mathrm{K}],\; p[\mathrm{Pa}],\; \dot{m}_{H_2}[\mathrm{kg/s}],\; F[\mathrm{mol/s}].\] When recording gas flow as volumetric flow, the standard state conversion formula and applied reference state are fixed in the DOI package.

6.8.3 Analysis Code (Mandatory Outputs)

The analysis code must automatically generate at least the following outputs:

1. Electrolysis: \[\dot{n}_{H_2}(t)=\frac{\eta_F I(t)}{2F},\quad \eta_F(t)=\frac{2F\dot{n}_{H_2}(t)}{I(t)},\quad \mathrm{SEC}_{H_2}(t)=\frac{2F}{\eta_F M_{H_2}}V_{\mathrm{cell}}(t) \label{eq:code_electrolysis_outputs}\]

2. Reactor: \[X_A(t)=1-\frac{F_{A,\mathrm{out}}(t)}{F_{A,\mathrm{in}}(t)},\qquad \mathrm{STY}_B(t)=\frac{\dot{n}_B(t)}{W_{\mathrm{cat}}},\qquad \mathrm{TOF}(t)=\frac{\dot{n}_B(t)}{N_{\mathrm{site,eff}}} \label{eq:code_reactor_outputs}\]

3. Mass Transfer/Diffusion Check: \[\phi = R_p\sqrt{\frac{k_{\mathrm{int}}}{D_{\mathrm{eff}}}},\qquad \eta_{\mathrm{eff}}=\frac{3}{\phi^2}\left(\phi\coth\phi-1\right) \label{eq:code_diffusion_outputs}\]

4. VP Mapping Parameter Fitting: Using \[\begin{aligned} \Delta G^\ddagger_{\mathrm{surf}} &= \Delta G^\ddagger_{0}-\Lambda_A g_A(A_{\mathrm{op}})-\Lambda_S g_S(S),\\ j_0 &= j_{0,0}(T)(S)^m\exp(\beta_A g_A(A_{\mathrm{op}})+\beta_S g_S(S)) \end{aligned} \label{eq:code_vp_fit_outputs}\] Estimate \((\Lambda_A,\allowbreak\Lambda_S,\allowbreak\beta_A,\allowbreak\beta_S,\allowbreak m,\allowbreak A_{\mathrm{th}},\allowbreak \Delta A,\allowbreak S_{\mathrm{th}},\allowbreak \Delta S)\) by least squares fitting to experimental data (Rate-Temp, \(j\)–\(\eta\) curves), and report prediction error on hold-out data.

6.9 Conclusion of This Section (Summary): Fixing Pt Catalyst Module in a “Measure-Calibrate-Reproduce” Form

1. The catalyst module system boundary is separated into Electrolysis (H\(_2\) Generation), Pt Activation (Dissociative Adsorption), and Reactor (Hydrogenation/Reduction), with mass/energy balance of each step fixed as measurement KPIs.

2. Electrolysis is fixed with a complete model using Faraday’s Law and Voltage Decomposition, and [eq:SEC_H2_from_V] explicitly states that specific energy consumption \(\mathrm{SEC}_{H_2}\) is determined by \(V_{\mathrm{cell}}\).

3. Hydrogen dissociation/recombination on Pt surface is completed with the Langmuir–Hinshelwood minimum model, providing the analytical solution for coverage [eq:thetaH_solution].

4. Reactor design is completed with PFR balance and active-site based reaction rate, fixed to verify mass transfer/internal diffusion limits using Thiele modulus and effectiveness factor.

5. VP variables (\(A_{\mathrm{op}}^{(\mathrm{surf})}, S^{(\mathrm{surf})}\)) are fixed with standard mapping functions [eq:gA_def]–[eq:j0_mapping] to Activation Barrier/Adsorption/Exchange Current Density/TOF, securing a reproducibility frame calibratable with experimental data.

6. For DOI issuance, mandatory measurement channels, data schema, and analysis code outputs are fixed in this section.

7 ESS-Water-Catalyst Integrated System Architecture (Energy Flow, Control, Safety, Night Operation)

7.1 Purpose of this Section and Definition of “Reality” for Integrated System

The purpose of this section is to connect the ESS (Black Copper based collection/storage and generation), Water Purification (Desalination/Cleaning), and Pt Catalyst (Electrolysis/Hydrogen Activation/Reactor) into a Single Integrated System, thereby: (1) Enabling operation based on Conservation Equations of Energy (Heat/Electricity) and Mass (Water/Salt/Gas) Flows, (2) Mathematically fixing the Control Structure (High-Low Level Control, Scheduling, State Machine), (3) Completing Safety (Electric Shock/Overheat/Overpressure/Hydrogen Risk/Water Safety) with interlocks and constraints, and (4) Making the Night Operation (Thermal Storage/Electric Buffer/Direct Thermal MD) scenario designable within physically feasible ranges.

The “Reality (Ground Truth)” of the integrated system is fixed first by the following measurable KPIs, rather than VP variables (Amplitude/Alignment):

1. ESS Energy Balance: \(E_{\mathrm{in}}, E_{\mathrm{out}}\) and \(\eta_{\mathrm{total}}=E_{\mathrm{out}}/E_{\mathrm{in}}\).

2. Purification Performance: Rejection \(R\), Recovery \(Y\), Specific Energy Consumption \(\mathrm{SEC}\).

3. Electrolysis/Hydrogen: \(\dot{n}_{H_2}\), \(\eta_F\), \(\mathrm{SEC}_{H_2}\).

4. (Optional) Catalytic Reactor: Conversion \(X_A\), Selectivity \(S_B\), STY/TOF.

Verification Principle: No matter what “High Efficiency” or “Free” claims are made in the integrated system, the Energy/Mass Conservation Equations of this section must be integrated over the same time window to show no violation. If there is a violation, it is not Success but Re-verification of Definition/Measurement/Hidden Inputs (HOLD).

7.2 Integrated System Block Diagram (Module Decomposition) and Signal/Energy Interfaces

7.2.1 Block Decomposition (Mandatory 5 Blocks)

The integrated system is fixed to the following 5 blocks:

1. ESS-Thermal Block (Collection/Storage): Solar \(\to\) Stored Heat (State \(T_s\) or \(E_{\mathrm{th}}\)).

2. ESS-Electrical Block (Generation/Power Electronics): Stored Heat \(\to\) DC Bus Power \(P_{\mathrm{gen}}\).

3. Electric Bus/Buffer Block (DC Link + Storage): \(V_{\mathrm{bus}}\), \(E_{\mathrm{buf}}\) (Battery/Supercap).

4. Water Purification Block (RO/ED/CDI/MD): Uses electricity or heat to produce \(Q_p, C_p\).

5. Catalyst Block (Electrolysis + Pt + Reactor): Generates \(H_2\) with electricity, activates with Pt, performs reaction.

7.2.2 Interface Variables (Energy/Mass/Control Signals)

Interface variables between each block are fixed as follows:

7.2.2.1 (1) Thermal Interface.

• ESS Thermal State: \(T_s(t)\) (K) or \(E_{\mathrm{th}}(t)\) (J).

• Generation Heat Input: \(\dot{Q}_{\mathrm{to\,gen}}(t)\) (W).

• MD Heat Input (Direct Thermal Operation): \(\dot{Q}_{\mathrm{to\,MD}}(t)\) (W).

7.2.2.2 (2) Electrical Interface (DC Bus).

• Bus Voltage/Current: \(V_{\mathrm{bus}}(t)\), \(I_{\mathrm{bus}}(t)\).

• Generation Power: \(P_{\mathrm{gen}}(t)\).

• Purification Consumption Power: \(P_{\mathrm{des}}(t)\).

• Electrolysis Consumption Power: \(P_{\mathrm{el}}(t)\).

• Load/Auxiliary Power: \(P_{\mathrm{aux}}(t)\).

• Buffer Charge/Discharge Power: \(P_{\mathrm{ch}}(t)\), \(P_{\mathrm{dis}}(t)\).

7.2.2.3 (3) Mass Interface.

• Purification: \((Q_f, C_f)\to (Q_p, C_p) + (Q_c, C_c)\).

• Electrolysis/Catalyst: Water Input \(\to H_2\) and (for electrolysis) \(O_2\) generation, Reactor In/Out Molar Flow \(F_k\).

7.2.2.4 (4) Control Signals.

• Generation Control: Target Power \(P_{\mathrm{gen,sp}}\) or Heat Flow \(\dot{Q}_{\mathrm{to\,gen,sp}}\).

• Purification Control: RO \(\Delta P_{\mathrm{sp}}\) or \(Q_{p,\mathrm{sp}}\), ED \(I_{\mathrm{ED,sp}}\), CDI \(V_{\mathrm{CDI,sp}}\), MD \(\dot{Q}_{\mathrm{to\,MD,sp}}\) or \(T_{h,\mathrm{sp}}\).

• Electrolysis Control: \(I_{\mathrm{el,sp}}\) or \(\dot{n}_{H_2,\mathrm{sp}}\).

7.3 Integrated Energy Flow: Thermal Balance + Electrical Balance (Mathematical Completeness)

7.3.1 Definition of Thermal Storage State

Let effective thermal capacity of storage be \(C_{\mathrm{th}}\) (J/K). Stored thermal energy relative to reference temperature \(T_{\mathrm{ref}}\) is defined as: \[E_{\mathrm{th}}(t) \;=\; C_{\mathrm{th}}\bigl(T_s(t)-T_{\mathrm{ref}}\bigr) \qquad [\mathrm{J}] \label{eq:Eth_def}\] Thus: \[\frac{dE_{\mathrm{th}}}{dt}=C_{\mathrm{th}}\frac{dT_s}{dt}. \label{eq:dEth_dt}\]

7.3.2 Integrated Thermal Balance (Generation + Direct Thermal MD Included)

Integrated thermal balance is fixed by the following differential equation: \[\frac{dE_{\mathrm{th}}}{dt} \;=\; \dot{Q}_{\mathrm{abs}}(t) -\dot{Q}_{\mathrm{loss}}(t) -\dot{Q}_{\mathrm{to\,gen}}(t) -\dot{Q}_{\mathrm{to\,MD}}(t) -\dot{Q}_{\mathrm{dump}}(t). \label{eq:thermal_balance_int}\] Here,

• \(\dot{Q}_{\mathrm{abs}}(t)=\eta_{\mathrm{opt}}(t)\,G(t)\,A_{\mathrm{col}}\) (Absorbed Heat).

• \(\dot{Q}_{\mathrm{loss}}(t)\) (Radiation/Convection/Conduction Loss).

• \(\dot{Q}_{\mathrm{to\,gen}}(t)\) (Heat transferred to Generation Unit).

• \(\dot{Q}_{\mathrm{to\,MD}}(t)\) (Heat directly supplied to MD).

• \(\dot{Q}_{\mathrm{dump}}(t)\) (Dump/Heat Rejection for Overheat Protection).

All terms are defined as positive. Verification: [eq:thermal_balance_int] must be integrated over time window \([t_0,t_1]\) to calculate thermal balance closure (residual). \[\Delta E_{\mathrm{th}} = \int_{t_0}^{t_1}\dot{Q}_{\mathrm{abs}}dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{loss}}dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{to\,gen}}dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{to\,MD}}dt -\int_{t_0}^{t_1}\dot{Q}_{\mathrm{dump}}dt. \label{eq:thermal_closure}\] If the residual is large (exceeding sensor error range), it implies incorrect heat loss model/temperature measurement/heat exchanger estimation, so re-verify.

7.3.3 Generation Output (Heat \(\to\) Electricity) and Carnot Upper Limit (Integrated Check)

Generation electrical output power is fixed as: \[P_{\mathrm{gen}}(t) \;=\; \eta_{\mathrm{elec}}(t)\,\eta_{\mathrm{conv}}(t)\,\dot{Q}_{\mathrm{to\,gen}}(t), \label{eq:Pgen}\] Here, \(\eta_{\mathrm{elec}}\) is power electronics/generator efficiency, \(\eta_{\mathrm{conv}}\) is Heat \(\to\) Work conversion efficiency. For heat engine equivalent conversion: \[\eta_{\mathrm{conv}}(t)\;\le\;\eta_{\mathrm{Carnot}}(t) \;=\; 1-\frac{T_c(t)}{T_h(t)}. \label{eq:carnot_int}\] Thus, the absolute upper bound check formula is: \[P_{\mathrm{gen}}(t)\;\le\;\eta_{\mathrm{elec}}(t)\,\eta_{\mathrm{Carnot}}(t)\,\dot{Q}_{\mathrm{to\,gen}}(t). \label{eq:Pgen_bound}\] Verification Rule: If violation of [eq:Pgen_bound] is observed, it is not Success but Re-verification of Measurement/Definition/Hidden Inputs (HOLD).

7.3.4 Power Balance Including Electrical Bus/Buffer (Instantaneous)

Instantaneous power balance of DC bus is fixed as follows (External grid input is set to 0): \[P_{\mathrm{gen}}(t) + P_{\mathrm{dis}}(t) \;=\; P_{\mathrm{des}}(t) + P_{\mathrm{el}}(t) + P_{\mathrm{aux}}(t) + P_{\mathrm{ch}}(t) + P_{\mathrm{curt}}(t), \label{eq:power_balance_dc}\] Here,

• \(P_{\mathrm{des}}\): Purification (RO/ED/CDI, etc.) consumption power (pump/electrode).

• \(P_{\mathrm{el}}\): Electrolysis stack power.

• \(P_{\mathrm{aux}}\): Auxiliary power for control/measurement/valve/fan.

• \(P_{\mathrm{ch}}, P_{\mathrm{dis}}\): Buffer charge/discharge power (both defined positive).

• \(P_{\mathrm{curt}}\): Output curtailment/dump/cut-tail (unused generation power).

Definition Consistency: By including \(P_{\mathrm{curt}}\) in [eq:power_balance_dc], power balance always closes even when generation is possible but load is absent or cut off due to safety limits.

7.3.5 Buffer (Battery/Supercap) Energy State and SOC Equation

Let buffer stored energy be \(E_{\mathrm{buf}}(t)\), charge efficiency \(\eta_{\mathrm{ch}}\in(0,1]\), discharge efficiency \(\eta_{\mathrm{dis}}\in(0,1]\). \[\frac{dE_{\mathrm{buf}}}{dt} \;=\; \eta_{\mathrm{ch}}\,P_{\mathrm{ch}}(t) -\frac{1}{\eta_{\mathrm{dis}}}\,P_{\mathrm{dis}}(t) -P_{\mathrm{buf,loss}}(t), \label{eq:buffer_energy_ode}\] Here, \(P_{\mathrm{buf,loss}}\) is self-discharge/internal resistance loss (fixed by measurement if needed). SOC (State of Charge) is defined as: \[\mathrm{SOC}(t)=\frac{E_{\mathrm{buf}}(t)}{E_{\mathrm{buf,max}}}\in[0,1] \label{eq:SOC_def}\] Buffer safe operation constraints are fixed as: \[E_{\mathrm{buf,min}}\le E_{\mathrm{buf}}(t)\le E_{\mathrm{buf,max}}, \qquad 0\le P_{\mathrm{ch}}(t)\le P_{\mathrm{ch,max}}, \qquad 0\le P_{\mathrm{dis}}(t)\le P_{\mathrm{dis,max}} \label{eq:buffer_constraints}\]

7.4 Integrated Mass Flow: Purification (Water/Salt) + Hydrogen (Electrolysis/Gas) Balance

7.4.1 Integrated Definition of Purification (Water/Salt) Balance

The purification module has the following state variables as functions of time: \[Q_f(t),\,Q_p(t),\,Q_c(t),\qquad C_f(t),\,C_p(t),\,C_c(t).\] Flow balance: \[Q_f(t)=Q_p(t)+Q_c(t) \label{eq:water_flow_balance_time}\] Salt balance (if precipitation/adsorption is small or calibrated separately): \[Q_f(t)C_f(t)\approx Q_p(t)C_p(t)+Q_c(t)C_c(t). \label{eq:salt_balance_time}\] Daily (or evaluation window \([t_0,t_1]\)) product volume: \[V_p=\int_{t_0}^{t_1} Q_p(t)\,dt, \label{eq:Vp_int}\] Rejection and Recovery: \[R(t)=1-\frac{C_p(t)}{C_f(t)},\qquad Y(t)=\frac{Q_p(t)}{Q_f(t)}. \label{eq:R_Y_time}\] From integrated system perspective, SEC for purification electricity consumption is: \[\mathrm{SEC}_{\mathrm{des}} = \frac{\int_{t_0}^{t_1} P_{\mathrm{des}}(t)\,dt}{\int_{t_0}^{t_1} Q_p(t)\,dt}. \label{eq:SEC_des_int}\] For direct thermal MD, Specific Thermal Energy Consumption (STEC) is fixed separately as: \[\mathrm{STEC}_{\mathrm{MD}} = \frac{\int_{t_0}^{t_1} \dot{Q}_{\mathrm{to\,MD}}(t)\,dt}{\int_{t_0}^{t_1} Q_p(t)\,dt} \label{eq:STEC_MD_int}\] and electrical auxiliary power is included in [eq:SEC_des_int].

7.4.2 Electrolysis (Hydrogen) Balance and Integrated Energy Metrics

From electrolysis stack current \(I_{\mathrm{el}}(t)\), hydrogen production molar flow rate by Faraday’s law is: \[\dot{n}_{H_2}(t)=\frac{\eta_F(t)}{2F}\,I_{\mathrm{el}}(t), \label{eq:nH2dot}\] Hydrogen moles produced in time window: \[n_{H_2}=\int_{t_0}^{t_1}\dot{n}_{H_2}(t)\,dt. \label{eq:nH2_int}\] Electrolysis power: \[P_{\mathrm{el}}(t)=V_{\mathrm{cell}}(t)\,I_{\mathrm{el}}(t), \label{eq:Pel}\] Specific Electrical Energy Consumption per hydrogen mass: \[\mathrm{SEC}_{H_2} = \frac{\int_{t_0}^{t_1} P_{\mathrm{el}}(t)\,dt}{\int_{t_0}^{t_1}\dot{m}_{H_2}(t)\,dt}. \label{eq:SEC_H2_int}\] Verification: [eq:SEC_H2_int] must fix \(\eta_F\) by cross-verification of electrical integration and gas flow/mass measurement (or water displacement).

7.5 Integrated Performance Metrics (Relative to Water/Hydrogen/Solar Input) and Upper Bound Check

7.5.1 Water/Hydrogen Productivity Relative to Solar Input

Letting solar input energy be: \[E_{\mathrm{in}}=\int_{t_0}^{t_1}G(t)A_{\mathrm{col}}\,dt \label{eq:Ein_system}\] Useful productivity metrics of the integrated system are fixed as: \[\begin{aligned} \Pi_{\mathrm{water}} &=\frac{V_p}{E_{\mathrm{in}}} \qquad\left[\mathrm{m^3/J}\right]\ \text{or}\ \left[\mathrm{L/kWh_{solar}}\right], \label{eq:Pi_water}\\ \Pi_{H_2} &=\frac{m_{H_2}}{E_{\mathrm{in}}} \qquad\left[\mathrm{kg/J}\right]\ \text{or}\ \left[\mathrm{g/kWh_{solar}}\right]. \label{eq:Pi_H2}\end{aligned}\] These metrics are not for claiming “Free”, but are Reproducible Integrated KPIs to compare how water/hydrogen production varies with system design (RO vs ED vs MD, Electrolysis Duty, etc.) given solar input.

7.5.2 Integrated Electrical Efficiency (Ratio of ESS Generation Delivered to Load)

Define electrical energy generated by ESS as: \[E_{\mathrm{out}}=\int_{t_0}^{t_1}P_{\mathrm{gen}}(t)\,dt \label{eq:Eout_system}\] And energy actually used for useful load (Purification + Electrolysis + Mandatory Aux) as: \[E_{\mathrm{use}}=\int_{t_0}^{t_1}\bigl(P_{\mathrm{des}}(t)+P_{\mathrm{el}}(t)+P_{\mathrm{aux,req}}(t)\bigr)\,dt \label{eq:Euse}\] (\(P_{\mathrm{aux,req}}\) is minimum auxiliary power required for safety/control). Then electrical utilization rate is fixed as: \[\eta_{\mathrm{util}}=\frac{E_{\mathrm{use}}}{E_{\mathrm{out}}} \label{eq:eta_util}\] Interpretation: Low \(\eta_{\mathrm{util}}\) means much power is discarded due to load schedule/buffer/safety limits even if generation is sufficient. This is an “Integrated Control/Scheduling Issue” rather than an “Efficiency Issue”, so it is improved by control design in this section.

7.6 Control Architecture: Hierarchical Control (High-Level EMS + Low-Level Loops) and State Machine

7.6.1 Control Layers (3 Levels) Definition

Integrated control is fixed to 3 levels for reproducibility and safety.

1. Level 0 (Safety Interlock): Immediate cutoff (Hardware/Firmware) upon sensor limit violation.

2. Level 1 (Low-Level Loop Control): Tracking Voltage/Current/Pressure/Flow/Temperature of each module (PI or with limits).

3. Level 2 (High-Level Energy Management/Scheduler, EMS): Day/Night Mode, Load Allocation, Target Production Tracking.

Principle: Level 0 always overrides Level 1 and 2. If Level 0 trips, all setpoints are forced to safe state.

7.6.2 State Machine (FSM) Definition: Operation Modes and Transition Conditions

Integrated operation mode is fixed as a Finite State Machine (FSM). State set is defined as: \[\mathcal{S}= \{\mathrm{SAFE\_OFF},\ \mathrm{STARTUP},\ \mathrm{DAY\_CHARGE},\ \mathrm{DAY\_RUN},\ \mathrm{NIGHT\_RUN},\ \mathrm{SHUTDOWN},\ \mathrm{FAULT}\} \label{eq:state_set}\]

7.6.2.1 State Meanings.

• \(\mathrm{SAFE\_OFF}\): All high energy loads OFF, measurement/recording only.

• \(\mathrm{STARTUP}\): Sensor/Valve/Pump initial check, Bus stabilization, Leak check.

• \(\mathrm{DAY\_CHARGE}\): Daytime collection priority, Generation/Load limited until thermal storage target reached.

• \(\mathrm{DAY\_RUN}\): Daytime integrated operation (Purification/Electrolysis parallel or priority operation).

• \(\mathrm{NIGHT\_RUN}\): Night operation (Storage based Generation or Direct Thermal MD + Limited Electrolysis/Purification).

• \(\mathrm{SHUTDOWN}\): Safe stop (including cooling/depressurization/purging), Log saving.

• \(\mathrm{FAULT}\): Interlock trip, Forced cutoff and safety actions.

7.6.2.2 Transition Conditions (Mathematically Fixed).

For solar based Day/Night judgment, define: \[u_{\mathrm{sun}}(t)=H\!\left(G(t)-G_{\min}\right) \label{eq:usun}\] (\(G_{\min}\) is threshold for effective collection). For thermal state based night operation availability, define: \[u_{\mathrm{th}}(t)=H\!\left(E_{\mathrm{th}}(t)-E_{\mathrm{th,min}}\right) \label{eq:uth}\] For buffer based electrical operation availability, define: \[u_{\mathrm{buf}}(t)=H\!\left(E_{\mathrm{buf}}(t)-E_{\mathrm{buf,min}}\right) \label{eq:ubuf}\]

Transition rules are fixed as:

1. \(\mathrm{SAFE\_OFF}\to\mathrm{STARTUP}\): \[\text{Run Command} \land \text{Interlock Normal} \land V_{\mathrm{bus}}\in[V_{\min},V_{\max}] \label{eq:transition1}\]

2. \(\mathrm{STARTUP}\to\mathrm{DAY\_CHARGE}\): \[u_{\mathrm{sun}}(t)=1 \land \text{Sensor/Valve Check OK} \label{eq:transition2}\]

3. \(\mathrm{STARTUP}\to\mathrm{NIGHT\_RUN}\): \[u_{\mathrm{sun}}(t)=0 \land \bigl(u_{\mathrm{th}}(t)=1 \ \lor\ u_{\mathrm{buf}}(t)=1\bigr) \label{eq:transition3}\]

4. \(\mathrm{DAY\_CHARGE}\to\mathrm{DAY\_RUN}\): \[E_{\mathrm{th}}(t)\ge E_{\mathrm{th,target}} \ \lor\ \text{Load Priority Needed (e.g., Water Shortage Alarm)} \label{eq:transition4}\]

5. \(\mathrm{DAY\_RUN}\to\mathrm{NIGHT\_RUN}\): \[u_{\mathrm{sun}}(t)=0 \land \bigl(u_{\mathrm{th}}(t)=1 \ \lor\ u_{\mathrm{buf}}(t)=1\bigr) \label{eq:transition5}\]

6. \(\mathrm{DAY\_RUN}\to\mathrm{SHUTDOWN}\) or \(\mathrm{NIGHT\_RUN}\to\mathrm{SHUTDOWN}\): \[u_{\mathrm{sun}}(t)=0 \land u_{\mathrm{th}}(t)=0 \land u_{\mathrm{buf}}(t)=0 \label{eq:transition6}\]

7. Any State \(\to \mathrm{FAULT}\): \[\text{Interlock Trip} \ (\text{Overheat/Overpressure/Leak/Shock/Overcurrent, etc.}) \label{eq:transition_fault}\]

7.7 High-Level Energy Management (EMS): Load Allocation, Target Production, Night Schedule

7.7.1 Mathematical Formulation of Load Allocation Problem (Including Constraints)

Define control variables determined by EMS at time \(t\) (Load Power and Heat Allocation) as: \[\mathbf{u}(t)=\bigl[P_{\mathrm{des}}(t),\,P_{\mathrm{el}}(t),\,P_{\mathrm{ch}}(t),\,P_{\mathrm{dis}}(t),\,\dot{Q}_{\mathrm{to\,gen}}(t),\,\dot{Q}_{\mathrm{to\,MD}}(t)\bigr] \label{eq:ems_u}\] Constraints must be satisfied:

7.7.1.1 (Power Constraint).

\[P_{\mathrm{gen}}(t)+P_{\mathrm{dis}}(t) \ge P_{\mathrm{des}}(t)+P_{\mathrm{el}}(t)+P_{\mathrm{aux}}(t)+P_{\mathrm{ch}}(t), \label{eq:ems_power_constraint}\] Here, \(P_{\mathrm{gen}}\) is determined by [eq:Pgen].

7.7.1.2 (Heat Constraint).

\[\dot{Q}_{\mathrm{to\,gen}}(t)+\dot{Q}_{\mathrm{to\,MD}}(t)+\dot{Q}_{\mathrm{dump}}(t) \le \dot{Q}_{\mathrm{abs}}(t) - \dot{Q}_{\mathrm{loss}}(t) - \frac{dE_{\mathrm{th}}}{dt}, \label{eq:ems_heat_constraint}\] Or equivalently satisfy [eq:thermal_balance_int].

7.7.1.3 (Safety Constraint).

Bus voltage, pressure, temperature, gas concentration, etc. must satisfy safe ranges (details fixed in §7.9).

7.7.1.4 (Production Target).

Given daily target product water \(V_p^\ast\) and target hydrogen mass \(m_{H_2}^\ast\), define deficits as: \[D_w(t)=\max\!\left(0,\ V_p^\ast-\int_{t_{\mathrm{day}}}^{t}Q_p(\tau)\,d\tau\right), \qquad D_h(t)=\max\!\left(0,\ m_{H_2}^\ast-\int_{t_{\mathrm{day}}}^{t}\dot{m}_{H_2}(\tau)\,d\tau\right) \label{eq:deficits}\] (\(t_{\mathrm{day}}\) is start time of the day). EMS allocates load to reduce \(D_w, D_h\) towards 0.

7.7.2 Reproducible Priority-Based Allocation Rules (Closed-Loop Control)

Application Whitepaper v1.0 does not enforce complex optimization (MPC) but fixes reproducible closed-loop (rule-based) allocation as default. Define available power as: \[P_{\mathrm{av}}(t)=P_{\mathrm{gen}}(t)+P_{\mathrm{dis,max}}(t)-P_{\mathrm{aux,req}}(t) \label{eq:Pav}\] Let water/hydrogen priority weights be \(w_w, w_h\ge 0\). Define deficit-based weights as: \[\tilde{w}_w(t)=w_w\frac{D_w(t)}{D_w(t)+\epsilon},\qquad \tilde{w}_h(t)=w_h\frac{D_h(t)}{D_h(t)+\epsilon}, \label{eq:adaptive_weights}\] (\(\epsilon>0\) is small constant to prevent division by zero). Then target power allocated to purification and electrolysis is fixed as: \[P_{\mathrm{des,sp}}(t)=\frac{\tilde{w}_w(t)}{\tilde{w}_w(t)+\tilde{w}_h(t)}\,P_{\mathrm{av}}(t), \qquad P_{\mathrm{el,sp}}(t)=\frac{\tilde{w}_h(t)}{\tilde{w}_w(t)+\tilde{w}_h(t)}\,P_{\mathrm{av}}(t), \label{eq:power_split}\] Apply saturation to reflect allowable range of each module. \[P_{\mathrm{des,cmd}}(t)=\mathrm{sat}\!\left(P_{\mathrm{des,sp}}(t),\,0,\,P_{\mathrm{des,max}}\right), \qquad P_{\mathrm{el,cmd}}(t)=\mathrm{sat}\!\left(P_{\mathrm{el,sp}}(t),\,0,\,P_{\mathrm{el,max}}\right), \label{eq:saturation}\] Here, \[\mathrm{sat}(x,x_{\min},x_{\max})=\min(\max(x,x_{\min}),x_{\max}).\] Remaining power is fixed for buffer charging: \[P_{\mathrm{ch,cmd}}(t)=\mathrm{sat}\!\left(P_{\mathrm{gen}}(t)-P_{\mathrm{des,cmd}}(t)-P_{\mathrm{el,cmd}}(t)-P_{\mathrm{aux}}(t),\,0,\,P_{\mathrm{ch,max}}\right) \label{eq:charge_rule}\] If still remaining, treat as \(P_{\mathrm{curt}}\).

7.7.3 Night Schedule (Thermal Storage Usage and Runtime) Calculation

Maximum runtime for continued generation using thermal storage at night (\(u_{\mathrm{sun}}=0\)), assuming average generation heat input \(\bar{Q}_{\mathrm{to\,gen}}\), is fixed as: \[t_{\mathrm{night,max}} \;\approx\; \frac{E_{\mathrm{th}}(t_{\mathrm{sunset}})-E_{\mathrm{th,min}}}{\bar{Q}_{\mathrm{to\,gen}}} \label{eq:night_runtime}\] (\(t_{\mathrm{sunset}}\) is sunset time). For MD direct thermal operation, the upper limit of water producible from available heat at night is, with effective latent heat \(h_{fg,\mathrm{eff}}\) and thermal utilization efficiency \(\eta_{\mathrm{th,MD}}\): \[m_{p,\mathrm{max}} \;\approx\; \eta_{\mathrm{th,MD}}\frac{E_{\mathrm{th}}(t_{\mathrm{sunset}})-E_{\mathrm{th,min}}}{h_{fg,\mathrm{eff}}}, \qquad V_{p,\mathrm{max}}=\frac{m_{p,\mathrm{max}}}{\rho_w}. \label{eq:md_night_water}\] Thus, at night, EMS selects between: \[\text{(Process needing Electricity)}\ \Rightarrow\ \dot{Q}_{\mathrm{to\,gen}}>0, \qquad \text{(Direct Thermal Process)}\ \Rightarrow\ \dot{Q}_{\mathrm{to\,MD}}>0 \label{eq:night_choice}\] depending on which path reduces the deficits (\(D_w, D_h\)) more. Default policy of Application Whitepaper v1.0 is fixed as:

1. If Water Target Unmet and MD possible: Priority to \(\dot{Q}_{\mathrm{to\,MD}}\) at night (Maximize Water Production).

2. If Hydrogen Target Unmet and Electrolysis needed: Limited electrolysis with \(\dot{Q}_{\mathrm{to\,gen}}\) at night.

3. If both targets sufficient: Safety Cooling/Standby at night (Minimize heat loss, prioritize system life).

7.8 Low-Level Loop Control (Level 1): Complete Definition of Module Setpoint Tracking Control

7.8.1 Common PI Controller (Discrete Time) Definition

With sampling period \(\Delta t\), error \(e(k)=y_{\mathrm{sp}}(k)-y(k)\), fix control input \(u(k)\) as PI: \[\begin{aligned} u(k) &= \mathrm{sat}\!\left(u_0 + K_p e(k) + K_i \sum_{j=0}^{k} e(j)\Delta t,\ u_{\min},u_{\max}\right), \label{eq:PI}\\ \sum_{j=0}^{k} e(j)\Delta t &\leftarrow \mathrm{antiwindup}(\cdot)\ \text{(Limit integral term at saturation)}. \label{eq:antiwindup}\end{aligned}\] This PI definition is commonly applied to all low-level loops (pump/current/voltage/valve) to fix control implementation at DOI reproducibility level.

7.8.2 Purification Module Control

7.8.2.1 (RO) Pressure/Flow Control.

Key MV of RO is pump speed (or valve) controlling pressure \(\Delta P\). \[e_{\Delta P}(k)=\Delta P_{\mathrm{sp}}(k)-\Delta P(k), \qquad u_{\mathrm{pump}}(k)=\mathrm{PI}(e_{\Delta P}(k)), \label{eq:ro_pressure_control}\] Here, \(u_{\mathrm{pump}}\) is Pump PWM/Frequency/Valve Opening, etc. If product quality deviates from target (\(C_p^\ast\)) (e.g., increased salt leakage), high-level EMS adjusts \(\Delta P_{\mathrm{sp}}\) or recovery \(Y_{\mathrm{sp}}\) conservatively.

7.8.2.2 (ED) Current Control (With Voltage Limit).

Since current determines salt removal in ED, current control is basic. \[e_I(k)=I_{\mathrm{ED,sp}}(k)-I_{\mathrm{ED}}(k), \qquad V_{\mathrm{ED,cmd}}(k)=\mathrm{PI}(e_I(k)). \label{eq:ed_current_control}\] However, set voltage limit for membrane/electrode protection. \[V_{\mathrm{ED,cmd}}(k)\le V_{\mathrm{ED,max}}, \label{eq:ed_voltage_limit}\] If limit reached, EMS lowers \(I_{\mathrm{ED,sp}}\) (High-Low level interoperation so current target does not violate safety).

7.8.2.3 (CDI) Voltage Control + Charge/Discharge Cycle Control.

Since CDI typically performs charge/discharge by applying voltage: \[e_V(k)=V_{\mathrm{CDI,sp}}(k)-V_{\mathrm{CDI}}(k), \qquad u_{\mathrm{CDI}}(k)=\mathrm{PI}(e_V(k)). \label{eq:cdi_voltage_control}\] Charge/Discharge mode is switched periodically by high-level FSM/EMS. Switching rule is fixed based on salt removal amount or time (e.g., \(Q\) integration). \[Q_{\mathrm{CDI}}(t)=\int I_{\mathrm{CDI}}(t)\,dt, \qquad Q_{\mathrm{CDI}} \ge Q_{\mathrm{th}} \Rightarrow \text{Switch to Discharge Mode}. \label{eq:cdi_switch_rule}\]

7.8.2.4 (MD) Temperature/Heat Flow Control (Direct Thermal).

MV of MD is heat exchanger flow or heat valve, controlling hot side temperature \(T_{h,\mathrm{MD}}\) or \(\dot{Q}_{\mathrm{to\,MD}}\). \[e_T(k)=T_{h,\mathrm{sp}}(k)-T_{h,\mathrm{MD}}(k), \qquad u_{\mathrm{hx}}(k)=\mathrm{PI}(e_T(k)). \label{eq:md_temp_control}\]

7.8.3 Electrolysis/Catalyst Module Control

7.8.3.1 Electrolysis Current Control (Hydrogen Rate Target).

Required current for target hydrogen rate \(\dot{n}_{H_2,\mathrm{sp}}\): \[I_{\mathrm{el,sp}}(t)=\frac{2F}{\eta_F(t)}\,\dot{n}_{H_2,\mathrm{sp}}(t). \label{eq:Iel_sp}\] Low-level loop performs current control. \[e_I(k)=I_{\mathrm{el,sp}}(k)-I_{\mathrm{el}}(k), \qquad V_{\mathrm{el,cmd}}(k)=\mathrm{PI}(e_I(k)), \label{eq:electrolyzer_current_loop}\] However, voltage/temperature limits are set for stack protection. \[V_{\mathrm{cell}}(k)\le V_{\mathrm{cell,max}}, \qquad T_{\mathrm{stack}}(k)\le T_{\mathrm{stack,max}}. \label{eq:stack_limits}\] Upon limit violation, EMS forces \(I_{\mathrm{el,sp}}\to 0\) immediately (at Level 0 or Level 2).

7.8.3.2 Catalytic Reactor Control (Temperature/Flow).

Catalytic reactor typically controls temperature \(T_{\mathrm{rxn}}\) and flow rate. \[e_T(k)=T_{\mathrm{rxn,sp}}(k)-T_{\mathrm{rxn}}(k), \qquad u_{\mathrm{heater/flow}}(k)=\mathrm{PI}(e_T(k)). \label{eq:reactor_temp_control}\] Since hydrogen supply must be limited against excess/shortage for safety, upper/lower limits of hydrogen flow are fixed as: \[\dot{n}_{H_2,\min}\le \dot{n}_{H_2,\mathrm{feed}} \le \dot{n}_{H_2,\max} \label{eq:H2_feed_limits}\] and interlocked with leakage/ventilation status.

7.9 Safety Architecture (Level 0 Interlock): Hazards, Constraints, Forced Actions

7.9.1 Hazard Classification

Major hazards of the integrated system are fixed to 6 types:

1. High Temperature/Burn/Fire: Thermal storage/Generation/MD high temperature.

2. High Pressure: RO high pressure line, Gas lines (H\(_2\), O\(_2\)), Tanks.

3. Electric Shock/Overcurrent: DC Bus, Electrolysis Stack, ED Stack.

4. Hydrogen Risk: Leakage, Explosion Limit, Flashback, Lack of Ventilation, Oxygen Mixing.

5. Water Safety: Product Contamination (Salt Leak, Metal Elution), Sterilization Failure.

6. Corrosion/Scaling: Seawater/Brine, Electrode/Membrane Damage, Water Leak.

7.9.2 Fixed Safety Constraints (Safe Set)

Safe state set \(\mathcal{X}_{\mathrm{safe}}\) is fixed by the following inequalities: \[\begin{aligned} T_s(t) &\le T_{s,\max}, \label{eq:safe_Ts}\\ T_h(t) &\le T_{h,\max},\quad T_c(t)\le T_{c,\max}, \label{eq:safe_ThTc}\\ V_{\mathrm{bus}}(t) &\in [V_{\min},V_{\max}], \label{eq:safe_Vbus}\\ I_{\mathrm{bus}}(t) &\le I_{\max}, \label{eq:safe_Ibus}\\ \Delta P_{\mathrm{RO}}(t) &\le \Delta P_{\max}, \label{eq:safe_RO_pressure}\\ p_{H_2}(t) &\le p_{H_2,\max},\quad p_{O_2}(t)\le p_{O_2,\max}, \label{eq:safe_gas_pressure}\\ c_{H_2,\mathrm{leak}}(t) &\le c_{H_2,\mathrm{alarm}}, \label{eq:safe_H2_leak}\\ C_p(t) &\le C_{p,\max}\quad (\text{Product Water Quality Limit}) \label{eq:safe_water_quality}\end{aligned}\] Here, \(c_{H_2,\mathrm{leak}}\) is hydrogen concentration (or alarm indicator) measured by leak sensor. Each threshold (\(T_{s,\max}\), etc.) is determined by design/component rating/regulation, and Application Whitepaper must fix Definition and Measurement Location as metadata.

7.9.3 Interlock Trip and Forced Actions (Standardization of Safety Actions)

Interlock trip is fixed by the following logic: \[\mathrm{TRIP}(t)= \bigvee_{i}\ H\!\left(x_i(t)-x_{i,\max}\right) \ \ \vee\ \ \bigvee_{j}\ H\!\left(x_{j,\min}-x_j(t)\right), \label{eq:trip_logic}\] Here, \(x_i\) is state variable of constraint. If \(\mathrm{TRIP}(t)=1\), state transitions immediately to \(\mathrm{FAULT}\), and forced actions are fixed in the following order (for Reproducibility/Safety):

1. Electrical Cutoff: \(P_{\mathrm{des}}\to 0\), \(P_{\mathrm{el}}\to 0\) (Load Relay/Contactor OFF).

2. Generation Limit: \(P_{\mathrm{gen,sp}}\to 0\) or \(\dot{Q}_{\mathrm{to\,gen}}\to 0\) (Heat Input Cutoff).

3. Overheat Protection: Activate \(\dot{Q}_{\mathrm{dump}}\) (Heat Rejection/Shutter/Bypass).

4. Gas Safety: Close H\(_2\)/O\(_2\) line isolation valves, ON Purge/Ventilation Fan if needed.

5. High Pressure Safety: Stop RO Pump and Open Depressurization Valve (Defined Safety Discharge Path).

6. Log: Fix and save Trip Cause Channel and data of preceding 60–300 s as “Event Log”.

Important: Safety actions are implementation specifications, not “descriptions”. Execution delay (e.g., \(<100\) ms), priority, and return conditions (Manual Reset, etc.) are further fixed in the Design Section of the Application Whitepaper.

7.10 Night Operation Architecture (Thermal Storage/Buffer/Direct Thermal) and Design Basis per Mode

7.10.1 3 Basic Modes of Night Operation

Night operation is fixed to 3 modes depending on available energy form:

1. N1: Storage \(\to\) Generation \(\to\) Load (Electric Drive): \[\dot{Q}_{\mathrm{to\,gen}}>0,\quad P_{\mathrm{gen}}>0,\quad P_{\mathrm{des}}>0\ \text{or}\ P_{\mathrm{el}}>0.\]

2. N2: Storage \(\to\) MD Direct Thermal (Thermal Drive): \[\dot{Q}_{\mathrm{to\,MD}}>0,\quad P_{\mathrm{gen}} \approx 0\ (\text{or Aux Only}).\]

3. N3: Buffer Discharge Drive (Electric Buffer): \[P_{\mathrm{dis}}>0,\quad P_{\mathrm{gen}}=0,\quad P_{\mathrm{des}}>0\ \text{or}\ P_{\mathrm{el}}>0.\]

7.10.2 Night Mode Selection Rules (Entity Based)

Night mode selection depends on deficit (\(D_w, D_h\)) and availability (\(E_{\mathrm{th}}, E_{\mathrm{buf}}\)). Basic rules of Application Whitepaper v1.0 are fixed as: \[\begin{aligned} &\text{(Water Priority)}\quad D_w(t)>0 \land \text{MD Possible} \Rightarrow \mathrm{N2}, \label{eq:night_rule1}\\ &\text{(Hydrogen Priority)}\quad D_h(t)>0 \land E_{\mathrm{th}}(t)>E_{\mathrm{th,min}} \Rightarrow \mathrm{N1}, \label{eq:night_rule2}\\ &\text{(Buffer Usage)}\quad D_h(t)>0 \land E_{\mathrm{buf}}(t)>E_{\mathrm{buf,min}} \Rightarrow \mathrm{N3}, \label{eq:night_rule3}\\ &\text{(End)}\quad D_w(t)=0 \land D_h(t)=0 \Rightarrow \mathrm{SHUTDOWN}\ \text{or Low Power Standby}. \label{eq:night_rule4}\end{aligned}\] Verification: Results of night operation (Product Water/Hydrogen Amount) cannot physically exceed upper limits of [eq:night_runtime] or [eq:md_night_water]. If excess appears, it is judged as Hidden Input or Measurement/Definition Error and re-verified.

7.11 Connection of VP Variables (Operational Amplitude/Alignment) to Integrated Control (Optional, but Fixed Reproducibly)

This section primarily fixes the entity of integrated architecture with conserved quantities, but for user’s goal (entity comparison based on amplitude formula), we selectively fix how VP variables (\(A_{\mathrm{op}}, S\)) can enter into integrated control in a “Measurable/Calibratable Form”.

7.11.1 VP Gate Based Operation Permission Function

For each module \(m\in\{\mathrm{ESS},\mathrm{DES},\mathrm{CAT}\}\), define stable gate of operational amplitude and alignment as: \[\mathcal{G}_m(t) = H\!\left(A_{\mathrm{op},m}(t)-A_{m,\min}\right) H\!\left(A_{m,\max}-A_{\mathrm{op},m}(t)\right) H\!\left(S_m(t)-S_{m,\min}\right) \label{eq:vp_gate_module}\] Then EMS can limit module command power as: \[P_{m,\mathrm{cmd}}(t)\leftarrow P_{m,\mathrm{cmd}}(t)\,\mathcal{G}_m(t). \label{eq:vp_gate_power_limit}\] Reproducibility Requirement: \(A_{\mathrm{op},m}\) and \(S_m\) are not direct measurements, but conversion formulas (Calibration Layer) from proxy variables (Voltage/Current/Temp/Impedance/Conductivity, etc.) of each module must be fixed in the DOI package. Without conversion formulas, [eq:vp_gate_power_limit] is not applied (HOLD).

7.12 Reproducibility Provision (For DOI Issuance): Integrated Log/Protocol/Verification Procedure

7.12.1 Integrated Log Schema (Single Time Series + Event Log)

For DOI-level reproducibility, integrated system logs are fixed to at least two types.

7.12.1.1 (1) Single Time Series Log (Mandatory).

Fixed as CSV (or equivalent JSON) containing the following fields with sampling period \(\Delta t\): \[\begin{aligned} \{\ & \texttt{t},\texttt{G},\texttt{Ta},\texttt{Ts},\texttt{Th},\texttt{Tc},\texttt{Qabs},\texttt{Qloss},\texttt{Qgen},\texttt{Qmd},\texttt{Eth},\\ &\texttt{Vbus},\texttt{Ibus},\texttt{Pgen},\texttt{Pdes},\texttt{Pel},\texttt{Paux},\texttt{Pch},\texttt{Pdis},\texttt{Ebuf},\texttt{SOC},\\ &\texttt{Qf},\texttt{Qp},\texttt{Qc},\texttt{Cf},\texttt{Cp},\texttt{Cc},\texttt{dP},\\ &\texttt{Vcell},\texttt{Iel},\texttt{Tstack},\texttt{pH2},\texttt{mH2},\\ &\texttt{state},\texttt{Pdes\_sp},\texttt{Pel\_sp},\texttt{Qgen\_sp},\texttt{Qmd\_sp} \ \}. \end{aligned} \label{eq:integrated_schema}\] Units fixed to SI, derivative terms (Qabs, etc.) recorded with formula version.

7.12.1.2 (2) Event Log (Mandatory).

Record timestamp and cause channel of Interlock Trip/State Transition/Mode Change. \[\{\texttt{t\_event},\ \texttt{event\_type},\ \texttt{trigger\_channels},\ \texttt{pre\_window\_hash},\ \texttt{post\_window\_hash}\}. \label{eq:event_log}\] Hash is optional item for verification of data integrity (anti-tampering) before/after event.

7.12.2 Integrated Verification Procedure (Closure Tests of Energy/Mass Balance)

Integrated verification is fixed to always perform these 3 steps:

1. Thermal Balance Closure: Calculate residual of [eq:thermal_closure], check if within allowable range including sensor error propagation.

2. Power Balance Closure: Check [eq:power_balance_dc] on time axis, verify identity with integrated energy.

3. Mass Balance Closure: Purification [eq:water_flow_balance_time]–[eq:salt_balance_time], Hydrogen [eq:nH2dot] and Gas Flow Measurement Cross-Verification to fix \(\eta_F\).

If these 3 steps are not passed, integrated system performance figures do not meet DOI reproducibility requirements, so result is classified as “Verification Failed (HOLD)” not “Success”.

7.12.3 Reproduction Simulation (Digital Twin) Minimum Model and Input Files

Digital Twin (Reproduction Simulation) of integrated system is fixed to at least the following equation set:

1. Thermal Dynamics: [eq:thermal_balance_int]

2. Generation Output: [eq:Pgen] and Carnot Upper Limit [eq:Pgen_bound]

3. Power Balance and Buffer Dynamics: [eq:power_balance_dc], [eq:buffer_energy_ode]

4. Purification KPI Calculation: [eq:R_Y_time], [eq:SEC_des_int], (MD) [eq:STEC_MD_int]

5. Hydrogen KPI Calculation: [eq:nH2dot], [eq:SEC_H2_int]

6. EMS Rules: [eq:power_split]–[eq:charge_rule], Night Rules [eq:night_rule1]–[eq:night_rule4]

Input files include at least: \[\{\texttt{weather\_G(t)},\texttt{Ta(t)},\texttt{Vp\_target},\texttt{mH2\_target},\texttt{params\_ESS},\texttt{params\_desal},\texttt{params\_el}\}. \label{eq:twin_inputs}\] Including these inputs and code version (hash) in DOI package allows third parties to reproduce identical results.

7.13 Conclusion of This Section (Summary): Fixing Integrated System Completely from Conserved Quantities-Control-Safety to Night Operation

1. Fixed ESS-Water-Catalyst connection into a single conserved quantity system centered on Integrated Thermal Balance [eq:thermal_balance_int] and Power Balance [eq:power_balance_dc].

2. Fixed Control as Hierarchical Structure of Safety Interlock (Level 0), Low-Level Loop (Level 1), High-Level EMS/FSM (Level 2), mathematically defining State Machine and Transition Conditions [eq:state_set]–[eq:transition_fault].

3. Fixed Night Operation to 3 Modes: Storage-based Generation (N1), Direct Thermal MD (N2), Buffer Discharge (N3), defining selection rules based on Water/Hydrogen deficit and availability.

4. Fixed Safety Implementation Specifications with Constraint Set [eq:safe_Ts]–[eq:safe_water_quality], Trip Logic [eq:trip_logic], and Forced Action Procedures.

5. For DOI issuance, completed Integrated Log Schema, Event Log, Balance Closure Verification, and Digital Twin Minimum Model in this section.

8 Back-calculation Performance Prediction (Rough Estimate): kWh/day \(\rightarrow\) L/day(Water), g/day(H\(_2\)), Catalyst Throughput

8.1 Purpose of this Section and Meaning of “Rough Estimate” (Verifiable Back-calculation)

The purpose of this section is to provide mathematically complete conversion formulas to back-calculate the electrical energy (or thermal energy) that ESS can provide per day into Water Production, Hydrogen Production, and Catalyst Throughput. Here, “Rough Estimate” means a linear scalable first-order prediction within the range where Specific Energy Consumption (SEC) and Efficiency (e.g., \(\eta_F\)) of each process are set as design parameters.

Verification Principle: The back-calculation results in this section are strictly based on conservation equations. Once empirical data (measurement data) is secured, the parameters in this section (SEC, \(V_{\mathrm{cell}}\), \(\eta_F\), auxiliary power, etc.) must be re-fixed with measured values to ensure DOI reproducibility.

8.2 Input Variables, Constants, Units (Fixing Reproducibility)

8.2.1 Daily Energy Input (Electricity/Heat from ESS)

Energy available from ESS per day (24 h) is defined as: \[\begin{aligned} E_{\mathrm{elec,day}} &:\ \text{Daily Electrical Energy} \qquad [\mathrm{kWh/day}], \label{eq:inv_Eelec_day}\\ E_{\mathrm{th,day}} &:\ \text{Daily Thermal Energy (Directly usable from thermal storage)} \qquad [\mathrm{kWh_{th}/day}]. \label{eq:inv_Eth_day}\end{aligned}\] (Since the title of this section is \(kWh/day\), [eq:inv_Eelec_day] is fundamental, but [eq:inv_Eth_day] is also defined for direct thermal paths like MD.)

8.2.2 Daily Energy Split (Decomposition into Water/Electrolysis/Auxiliary)

Since electrical energy is distributed by load, dimensionless split ratios are defined as: \[f_{\mathrm{des}} + f_{\mathrm{el}} + f_{\mathrm{aux}} + f_{\mathrm{buf}} + f_{\mathrm{curt}} = 1, \qquad f_i \in [0,1]. \label{eq:inv_fsum}\] Each term means: \[\begin{aligned} f_{\mathrm{des}} &: \text{Ratio of electricity sent to purification (RO/ED/CDI, etc.)},\\ f_{\mathrm{el}} &: \text{Ratio of electricity sent to electrolysis (hydrogen production)},\\ f_{\mathrm{aux}} &: \text{Ratio of mandatory auxiliary power (Control/Pump/Valve/Ventilation, etc.)},\\ f_{\mathrm{buf}} &: \text{Ratio including buffer charge/discharge loss and operation},\\ f_{\mathrm{curt}} &: \text{Ratio discarded unused (Curtailment)}. \end{aligned}\] Thus, the daily electrical energy actually input to each load is: \[\begin{aligned} E_{\mathrm{des,day}} &= f_{\mathrm{des}}\,E_{\mathrm{elec,day}}, \label{eq:inv_Edes_day}\\ E_{\mathrm{el,day}} &= f_{\mathrm{el}}\,E_{\mathrm{elec,day}}, \label{eq:inv_Eel_day}\\ E_{\mathrm{aux,day}} &= f_{\mathrm{aux}}\,E_{\mathrm{elec,day}}. \label{eq:inv_Eaux_day}\end{aligned}\] For MD direct thermal coupling, thermal split ratio is defined similarly: \[E_{\mathrm{MD,day}} = f_{\mathrm{MD}}\,E_{\mathrm{th,day}}, \qquad f_{\mathrm{MD}}\in[0,1]. \label{eq:inv_EMD_day}\]

8.2.3 Fixed Constants (Numerical Values Fixed for DOI Reproduction)

Physical constants and unit conversions used in this section are fixed as: \[\begin{aligned} 1~\mathrm{kWh} &= 3.6\times 10^{6}~\mathrm{J}, \label{eq:inv_kWh_to_J}\\ 1~\mathrm{m^3} &= 1000~\mathrm{L}, \label{eq:inv_m3_to_L}\\ F &= 96485.33212~\mathrm{C/mol}, \label{eq:inv_F}\\ M_{H_2} &= 2.01588~\mathrm{g/mol}, \label{eq:inv_MH2}\\ M_{\mathrm{Pt}} &= 195.084~\mathrm{g/mol}, \label{eq:inv_MPt}\\ \rho_w &\approx 1000~\mathrm{kg/m^3}\quad(\text{Water density, used for L conversion}). \label{eq:inv_rho_w}\end{aligned}\]

8.3 Purification Back-calculation: \(kWh/day \rightarrow L/day\) (Common Complete Formula for RO/ED/CDI/MD)

8.3.1 Common Definition of Purification: Back-calculating Product Volume from SEC

Defining Specific Energy Consumption of purification process (Electricity basis) as: \[\mathrm{SEC}_{\mathrm{des}} \;=\; \frac{E_{\mathrm{des,day}}}{V_{p,\mathrm{day}}} \qquad \left[\mathrm{kWh/m^3}\right] \label{eq:inv_SEC_def}\] Conversely, product water volume is: \[V_{p,\mathrm{day}} \;=\; \frac{E_{\mathrm{des,day}}}{\mathrm{SEC}_{\mathrm{des}}} \qquad \left[\mathrm{m^3/day}\right] \label{eq:inv_Vp_m3}\] Converting to L/day using [eq:inv_m3_to_L]: \[L_{p,\mathrm{day}} \;=\; 1000\,V_{p,\mathrm{day}} \;=\; 1000\,\frac{E_{\mathrm{des,day}}}{\mathrm{SEC}_{\mathrm{des}}} \qquad \left[\mathrm{L/day}\right]. \label{eq:inv_Lp_day}\] Therefore, the core of back-calculation is selecting (or measuring) \(\mathrm{SEC}_{\mathrm{des}}\) appropriate for the technology (RO/ED/CDI) and feed salinity.

8.3.2 Defining SEC per Technology as Separate Variables (Reflecting Seawater vs Brackish Branching)

This whitepaper fixes electrical SEC per technology as separate variables: \[\begin{aligned} \mathrm{SEC}_{\mathrm{RO}} &:\ \text{Reverse Osmosis (RO) Electrical SEC} \qquad [\mathrm{kWh/m^3}], \label{eq:inv_SEC_RO}\\ \mathrm{SEC}_{\mathrm{ED}} &:\ \text{Electrodialysis (ED) Electrical SEC} \qquad [\mathrm{kWh/m^3}], \label{eq:inv_SEC_ED}\\ \mathrm{SEC}_{\mathrm{CDI}} &:\ \text{CDI Electrical SEC} \qquad [\mathrm{kWh/m^3}]. \label{eq:inv_SEC_CDI}\end{aligned}\] Thus, for technology selection \(\mathcal{T}\in\{\mathrm{RO},\mathrm{ED},\mathrm{CDI}\}\): \[L_{p,\mathrm{day}}(\mathcal{T}) \;=\; 1000\,\frac{E_{\mathrm{des,day}}}{\mathrm{SEC}_{\mathcal{T}}}. \label{eq:inv_Lp_bytech}\]

8.3.2.1 (Design Range Example for Rough Estimate; Replace with Measurement in Verification).

For initial application design “Rough Estimate”, SEC is selected from ranges below for sensitivity analysis (Values are initial values, not norms). \[\begin{aligned} \mathrm{SEC}_{\mathrm{RO}} &\in [3,\,6]\quad \text{(Example range for Seawater RO)}, \label{eq:inv_SEC_RO_range}\\ \mathrm{SEC}_{\mathrm{ED}} &\in [0.5,\,3]\quad \text{(Example range for Brackish ED)}, \label{eq:inv_SEC_ED_range}\\ \mathrm{SEC}_{\mathrm{CDI}} &\in [0.2,\,2]\quad \text{(Example range for Brackish CDI)}. \label{eq:inv_SEC_CDI_range}\end{aligned}\] (Since SEC varies with feed salinity, recovery, pre-treatment, membrane/electrode state, operating pressure/current, it must be re-fixed with measured SEC in DOI version.)

8.3.3 Direct Thermal MD Back-calculation (Thermal kWh\(_{th}\)/day \(\rightarrow\) L/day)

Since MD uses heat directly, defining Specific Thermal Energy Consumption (STEC) as: \[\mathrm{STEC}_{\mathrm{MD}}=\frac{E_{\mathrm{MD,day}}}{V_{p,\mathrm{day}}} \qquad \left[\mathrm{kWh_{th}/m^3}\right] \label{eq:inv_STEC_def}\] Product water is back-calculated as: \[L_{p,\mathrm{day}}(\mathrm{MD}) \;=\; 1000\,\frac{E_{\mathrm{MD,day}}}{\mathrm{STEC}_{\mathrm{MD}}} \qquad [\mathrm{L/day}] \label{eq:inv_Lp_MD}\] Since MD may require electrical auxiliary power (pumps/fans), [eq:inv_Lp_MD] and [eq:inv_Lp_bytech] are reported simultaneously from total energy perspective.

8.3.4 “Production per 1 kWh/day” Conversion Factor for Purification (Scaling Rule)

Setting \(E_{\mathrm{des,day}}=1~\mathrm{kWh/day}\) in [eq:inv_Lp_day]: \[\frac{L_{p,\mathrm{day}}}{E_{\mathrm{des,day}}} \;=\; \frac{1000}{\mathrm{SEC}_{\mathrm{des}}} \qquad \left[\frac{\mathrm{L/day}}{\mathrm{kWh/day}}\right]. \label{eq:inv_water_per_kWh}\] That is, (Purification production is linear to electrical energy), and given actual \(E_{\mathrm{des,day}}\), it can be immediately back-calculated by multiplying with [eq:inv_water_per_kWh].

8.4 Hydrogen Back-calculation: \(kWh/day \rightarrow g/day\) (Faraday’s Law + Voltage Based Complete Formula)

8.4.1 Connection of Faraday’s Law and Electrical Energy (Step-by-step Derivation)

Hydrogen moles produced in electrolysis (including Faraday efficiency \(\eta_F\)) are: \[n_{H_2} \;=\; \frac{\eta_F}{2F}\int_{t_0}^{t_1} I(t)\,dt \label{eq:inv_faraday}\] Approximating cell voltage as nearly constant \(V_{\mathrm{cell}}(t)\approx V_{\mathrm{cell}}\) in operation range, electrical energy (Joules) is: \[E_{\mathrm{el,J}} \;=\; \int_{t_0}^{t_1} V_{\mathrm{cell}}\,I(t)\,dt \;=\; V_{\mathrm{cell}}\int_{t_0}^{t_1} I(t)\,dt \label{eq:inv_EJ}\] Thus, \[\int_{t_0}^{t_1} I(t)\,dt \;=\; \frac{E_{\mathrm{el,J}}}{V_{\mathrm{cell}}}. \label{eq:inv_charge_from_energy}\] Substituting into [eq:inv_faraday]: \[n_{H_2} \;=\; \frac{\eta_F}{2F}\,\frac{E_{\mathrm{el,J}}}{V_{\mathrm{cell}}}. \label{eq:inv_nH2_from_EJ}\] Converting to mass (grams): \[m_{H_2}[\mathrm{g}] \;=\; M_{H_2}\,n_{H_2} \;=\; M_{H_2}\frac{\eta_F}{2F}\,\frac{E_{\mathrm{el,J}}}{V_{\mathrm{cell}}}. \label{eq:inv_mH2_from_EJ}\] When electrical energy is given in kWh, using [eq:inv_kWh_to_J]: \[E_{\mathrm{el,J}} \;=\; \left(3.6\times 10^6\right)\,E_{\mathrm{el,day}} \qquad (E_{\mathrm{el,day}}[\mathrm{kWh/day}]). \label{eq:inv_EJ_from_kWh}\] Thus, daily hydrogen production (g/day) is given by the complete formula: \[m_{H_2,\mathrm{day}}[\mathrm{g/day}] \;=\; \left(\frac{3.6\times 10^6\,M_{H_2}}{2F}\right)\, \frac{\eta_F}{V_{\mathrm{cell}}}\, E_{\mathrm{el,day}}[\mathrm{kWh/day}]. \label{eq:inv_mH2_day_general}\] The constant in parenthesis is fixed numerically by [eq:inv_F]–[eq:inv_MH2]: \[\frac{3.6\times 10^6\,M_{H_2}}{2F} \;=\; \frac{3.6\times 10^6\times 2.01588}{2\times 96485.33212} \;\approx\; 37.6076. \label{eq:inv_const_37p6}\] So finally: \[m_{H_2,\mathrm{day}}[\mathrm{g/day}] \;\approx\; 37.6076\, \frac{\eta_F}{V_{\mathrm{cell}}[\mathrm{V}]}\, E_{\mathrm{el,day}}[\mathrm{kWh/day}]. \label{eq:inv_mH2_day_final}\] This equation is the core of back-calculation. Given \((\eta_F, V_{\mathrm{cell}})\), the conversion \(kWh/day\rightarrow g/day\) is complete.

8.4.2 Hydrogen Production Conversion Factor per \(1~kWh/day\)

Setting \(E_{\mathrm{el,day}}=1\) in [eq:inv_mH2_day_final]: \[\frac{m_{H_2,\mathrm{day}}}{E_{\mathrm{el,day}}} \;\approx\; 37.6076\,\frac{\eta_F}{V_{\mathrm{cell}}} \qquad \left[\frac{\mathrm{g/day}}{\mathrm{kWh/day}}\right]. \label{eq:inv_h2_per_kWh}\] For example, if \(\eta_F=0.95\) and \(V_{\mathrm{cell}}=2.0\) V: \[\frac{m_{H_2,\mathrm{day}}}{E_{\mathrm{el,day}}} \;\approx\; 37.6076\times \frac{0.95}{2.0} \;\approx\; 17.86\ \left[\frac{\mathrm{g}}{\mathrm{kWh}}\right]. \label{eq:inv_h2_example_perkWh}\]

8.4.3 Electrolysis Energy Intensity from Reverse Perspective: \(kWh/kg\) Complete Formula

Inverting [eq:inv_mH2_day_final], the electrical energy required to obtain 1 kg of hydrogen (Ideal back-calculation, actual includes \(V_{\mathrm{cell}}\) and \(\eta_F\)) is: \[\mathrm{SEC}_{H_2}[\mathrm{kWh/kg}] \;\equiv\; \frac{E_{\mathrm{el,day}}}{m_{H_2,\mathrm{day}}/1000} \;\approx\; \frac{1000}{37.6076}\,\frac{V_{\mathrm{cell}}}{\eta_F} \;\approx\; 26.5904\,\frac{V_{\mathrm{cell}}}{\eta_F}. \label{eq:inv_SEC_H2}\] For example, if \(V_{\mathrm{cell}}=2.0\) V, \(\eta_F=0.95\): \[\mathrm{SEC}_{H_2}\approx 26.5904\times \frac{2.0}{0.95}\approx 55.98~\mathrm{kWh/kg}.\] (This is useful as a “Rough Estimate”. In DOI version, it is calculated by integrating actual operation \(V_{\mathrm{cell}}(t)\) via [eq:inv_EJ].)

8.4.4 Water Consumption for Electrolysis and Oxygen Byproduct (For Integrated System Verification)

Electrolysis stoichiometry is: \[2H_2O \;\rightarrow\; 2H_2 + O_2 \label{eq:inv_water_split_stoich}\] Thus, \[n_{H_2O} = n_{H_2}, \qquad n_{O_2} = \frac{1}{2}n_{H_2}. \label{eq:inv_stoich_moles}\] Water consumption mass with \(M_{H_2O}=18.01528\) g/mol is: \[m_{H_2O} = 18.01528\,n_{H_2} \label{eq:inv_mH2O}\] Using [eq:inv_MH2]: \[\frac{m_{H_2O}}{m_{H_2}} = \frac{18.01528}{2.01588} \approx 8.93. \label{eq:inv_water_to_h2_ratio}\] That is, approximately: \[m_{H_2O}[\mathrm{kg/day}] \approx 8.93\, m_{H_2}[\mathrm{kg/day}] \approx 9\, m_{H_2}[\mathrm{kg/day}]. \label{eq:inv_water_need_rule}\] Therefore, in the integrated system, the purification module must provide at least: \[L_{H_2O,\mathrm{need}} \approx 9\, m_{H_2}[\mathrm{kg/day}]\times 1000\ [\mathrm{L/m^3}]\times \frac{1}{\rho_w} \approx 9\, m_{H_2}[\mathrm{kg/day}]\ \mathrm{L/day} \label{eq:inv_water_need_L}\] of freshwater (or water quality suitable for electrolysis) for electrolysis use. (Generally this value is very small compared to purification production, but must be included as DOI verification item.)

8.5 Catalyst Throughput Back-calculation: Minimum of Hydrogen Supply Limit vs Catalyst Reaction Limit (Complete Formula)

8.5.1 Throughput Definition (Based on Generic Reaction \(A \rightarrow B\))

Catalyst throughput is defined as daily production of a specific product \(B\). Let the generic reaction consuming hydrogen be: \[A + \nu_{H_2}H_2 \rightarrow B \label{eq:inv_reaction_generic}\] where \(\nu_{H_2}\) is moles of hydrogen required to produce 1 mole of product.

8.5.2 (1) Hydrogen Supply Limited (H\(_2\)-limited) Throughput Back-calculation

Let utilization rate of electrolytically produced hydrogen actually input/consumed in catalytic reaction be \(u_{H_2}\in[0,1]\). Then daily hydrogen moles available for catalytic reaction is: \[n_{H_2,\mathrm{use}} \;=\; u_{H_2}\,n_{H_2,\mathrm{day}} \;=\; u_{H_2}\,\frac{m_{H_2,\mathrm{day}}}{M_{H_2}}. \label{eq:inv_nH2_use}\] Thus, considering only hydrogen supply limit (assuming reactor/catalyst is fast enough), the upper limit of product molar throughput is: \[n_{B,\mathrm{H2\mbox{-}lim}} \;=\; \frac{n_{H_2,\mathrm{use}}}{\nu_{H_2}} \;=\; \frac{u_{H_2}}{\nu_{H_2}}\frac{m_{H_2,\mathrm{day}}}{M_{H_2}} \qquad [\mathrm{mol/day}]. \label{eq:inv_nB_H2lim}\] With product molar mass \(M_B\) (g/mol), mass throughput is: \[m_{B,\mathrm{H2\mbox{-}lim}}[\mathrm{g/day}] \;=\; M_B\,n_{B,\mathrm{H2\mbox{-}lim}} \;=\; M_B\frac{u_{H_2}}{\nu_{H_2}}\frac{m_{H_2,\mathrm{day}}}{M_{H_2}}. \label{eq:inv_mB_H2lim}\] Since \(m_{H_2,\mathrm{day}}\) is already back-calculated from \(E_{\mathrm{el,day}}\) by [eq:inv_mH2_day_final], \(m_{B,\mathrm{H2\mbox{-}lim}}\) is also linear to \(kWh/day\).

8.5.3 (2) Catalyst Reaction Limited (Kinetics-limited) Throughput Back-calculation: TOF + Effective Active Sites

Let catalyst (total mass including support) be \(W_{\mathrm{cat}}\) (g), and Pt mass fraction in catalyst be \(w_{\mathrm{Pt}}\in[0,1]\). Then Pt mass is \(m_{\mathrm{Pt}}=w_{\mathrm{Pt}}W_{\mathrm{cat}}\) (g), and total Pt moles is \(m_{\mathrm{Pt}}/M_{\mathrm{Pt}}\). Let Pt dispersion (fraction of Pt atoms exposed on surface) be \(D\in[0,1]\). Surface active site moles (= Upper limit of effective site moles) is: \[N_{\mathrm{site}}[\mathrm{mol\mbox{-}site}] \;=\; D\,\frac{m_{\mathrm{Pt}}}{M_{\mathrm{Pt}}} \;=\; D\,\frac{w_{\mathrm{Pt}}W_{\mathrm{cat}}}{M_{\mathrm{Pt}}}. \label{eq:inv_Nsite}\] Introducing effective active site fraction \(f_{\mathrm{act}}\in[0,1]\) due to surface state/alignment and effective coefficient \(\phi_{\mathrm{cov}}\in[0,1]\) integrating coverage/mass transfer/reaction conditions (e.g., \(\phi_{\mathrm{cov}}=\theta_A\theta_H^{\nu_H}\) combined with mass transfer effectiveness factor), effective active sites are: \[N_{\mathrm{site,eff}} = f_{\mathrm{act}}\,\phi_{\mathrm{cov}}\,N_{\mathrm{site}}. \label{eq:inv_Nsite_eff}\] Defining TOF (product moles per site per second) as \(\mathrm{TOF}\) (s\(^{-1}\)), daily product molar throughput limit (kinetics limit) of the catalyst is: \[n_{B,\mathrm{kin\mbox{-}lim}} \;=\; \left(\mathrm{TOF}\right)\,N_{\mathrm{site,eff}}\,(86400) \;=\; 86400\,\mathrm{TOF}\,f_{\mathrm{act}}\phi_{\mathrm{cov}}\,D\,\frac{w_{\mathrm{Pt}}W_{\mathrm{cat}}}{M_{\mathrm{Pt}}}. \label{eq:inv_nB_kinlim}\] Mass throughput is: \[m_{B,\mathrm{kin\mbox{-}lim}}[\mathrm{g/day}] \;=\; M_B\,n_{B,\mathrm{kin\mbox{-}lim}}. \label{eq:inv_mB_kinlim}\] Interpretation: [eq:inv_nB_kinlim] is determined by catalyst amount (\(W_{\mathrm{cat}}\)), Pt content (\(w_{\mathrm{Pt}}\)), dispersion (\(D\)), TOF, and effective coefficients (\(f_{\mathrm{act}},\phi_{\mathrm{cov}}\)), and is not directly connected to electrical energy (\(kWh/day\)). Therefore, in the integrated system, the smaller of Hydrogen Supply and Catalyst Reaction Capacity determines the actual throughput.

8.5.4 (3) Final Throughput (Actual): Minimum of Two Limits

Actual daily throughput is fixed by the minimum rule: \[\begin{aligned} n_{B,\mathrm{day}} &= \min\!\left(n_{B,\mathrm{H2\mbox{-}lim}},\ n_{B,\mathrm{kin\mbox{-}lim}}\right), \label{eq:inv_nB_final}\\ m_{B,\mathrm{day}} &= M_B\,n_{B,\mathrm{day}}. \label{eq:inv_mB_final}\end{aligned}\] Also, hydrogen consumption is verified as: \[n_{H_2,\mathrm{cons}} = \nu_{H_2}\,n_{B,\mathrm{day}} \label{eq:inv_H2_consumption}\] and must satisfy \(n_{H_2,\mathrm{cons}}\le n_{H_2,\mathrm{use}}\).

8.6 Integrated Back-calculation Examples (Rough Estimate): “1 kWh/day Basis” and “10 kWh/day Scenario”

8.6.1 1 kWh/day Basis Conversion (Scaling Standard)

8.6.1.1 (Water Purification).

By [eq:inv_water_per_kWh]: \[\boxed{ \ \frac{L_{p,\mathrm{day}}}{E_{\mathrm{des,day}}} = \frac{1000}{\mathrm{SEC}_{\mathrm{des}}}\ }\] e.g., if \(\mathrm{SEC}_{\mathrm{des}}=4\), then \(250~\mathrm{L/day}\) per \(1~\mathrm{kWh/day}\).

8.6.1.2 (Hydrogen).

By [eq:inv_h2_per_kWh]: \[\boxed{ \ \frac{m_{H_2,\mathrm{day}}}{E_{\mathrm{el,day}}} \approx 37.6076\,\frac{\eta_F}{V_{\mathrm{cell}}}\ }\] e.g., if \(\eta_F=0.95\), \(V_{\mathrm{cell}}=2.0\), then \(17.86~\mathrm{g/day}\) per \(1~\mathrm{kWh/day}\).

8.6.1.3 (Catalyst Throughput; when H\(_2\) supply limited).

Combining [eq:inv_mB_H2lim] and hydrogen conversion above: \[\frac{m_{B,\mathrm{H2\mbox{-}lim}}}{E_{\mathrm{el,day}}} \;\approx\; M_B\frac{u_{H_2}}{\nu_{H_2}} \frac{1}{M_{H_2}} \left(37.6076\,\frac{\eta_F}{V_{\mathrm{cell}}}\right) \qquad \left[\frac{\mathrm{g/day}}{\mathrm{kWh/day}}\right]. \label{eq:inv_product_per_kWh}\] That is, the “g/kWh” conversion factor for product is completely determined by \(M_B\), \(\nu_{H_2}\), \(u_{H_2}\), \(\eta_F\), \(V_{\mathrm{cell}}\).

8.6.2 10 kWh/day Example Scenario (For Design Sensing; Parameters Replaceable)

Assume the following (all Rough Estimates, replaced by measured values in DOI version): \[\begin{aligned} &E_{\mathrm{elec,day}} = 10~\mathrm{kWh/day}, \label{eq:inv_ex_E10}\\ &f_{\mathrm{des}}=0.6,\quad f_{\mathrm{el}}=0.4,\quad f_{\mathrm{aux}}=0\ (\text{Simple example; actually }f_{\mathrm{aux}}>0), \label{eq:inv_ex_split}\\ &\mathrm{SEC}_{\mathrm{des}}=4~\mathrm{kWh/m^3}\quad(\text{e.g., Seawater RO estimate}), \label{eq:inv_ex_SEC}\\ &\eta_F=0.95,\quad V_{\mathrm{cell}}=2.0~\mathrm{V}, \label{eq:inv_ex_el}\\ &\nu_{H_2}=1,\quad u_{H_2}=0.9,\quad M_B=100~\mathrm{g/mol}. \label{eq:inv_ex_reaction}\end{aligned}\] Then \[\begin{aligned} E_{\mathrm{des,day}} &= 0.6\times 10 = 6~\mathrm{kWh/day}, \label{eq:inv_ex_Edes}\\ E_{\mathrm{el,day}} &= 0.4\times 10 = 4~\mathrm{kWh/day}. \label{eq:inv_ex_Eel}\end{aligned}\]

8.6.2.1 (Water Production).

From [eq:inv_Lp_day]: \[L_{p,\mathrm{day}} = 1000\frac{6}{4} = 1500~\mathrm{L/day}. \label{eq:inv_ex_Lwater}\]

8.6.2.2 (Hydrogen Production).

From [eq:inv_mH2_day_final]: \[m_{H_2,\mathrm{day}} \approx 37.6076\frac{0.95}{2.0}\times 4 \approx 71.4~\mathrm{g/day}. \label{eq:inv_ex_gH2}\] Moles \(n_{H_2}=m_{H_2}/M_{H_2}\): \[n_{H_2,\mathrm{day}} \approx \frac{71.4}{2.01588} \approx 35.4~\mathrm{mol/day}. \label{eq:inv_ex_molH2}\]

8.6.2.3 (Water Consumption for Electrolysis).

From [eq:inv_water_need_L]: \[L_{H_2O,\mathrm{need}} \approx 9\times m_{H_2}[\mathrm{kg/day}] = 9\times 0.0714 \approx 0.64~\mathrm{L/day}, \label{eq:inv_ex_water_for_el}\] Only a tiny fraction of 1500 L/day water production is needed for electrolysis (Water quality condition verified separately).

8.6.2.4 (Catalyst Throughput: Hydrogen Supply Limited Example).

From [eq:inv_mB_H2lim]: \[m_{B,\mathrm{H2\mbox{-}lim}} = 100\cdot \frac{0.9}{1}\cdot \frac{71.4}{2.01588} \approx 3190~\mathrm{g/day} \approx 3.19~\mathrm{kg/day}. \label{eq:inv_ex_product}\] For this to be actual throughput, catalyst reaction limit ([eq:inv_nB_kinlim]) must be sufficiently larger; otherwise, catalyst becomes the bottleneck by [eq:inv_nB_final].

8.7 Reproducibility Provision (For DOI): Input Parameters, Calculation Sequence, Output Items (Complete Spec)

8.7.1 Mandatory Input Parameter Set (Minimum)

Minimum input set for reproducing back-calculation in this section is fixed as: \[\Pi_{\mathrm{inv}}= \left\{ E_{\mathrm{elec,day}},\ f_{\mathrm{des}},f_{\mathrm{el}},f_{\mathrm{aux}}, \mathrm{SEC}_{\mathcal{T}}, \eta_F,\ V_{\mathrm{cell}}, u_{H_2},\nu_{H_2},M_B, W_{\mathrm{cat}},w_{\mathrm{Pt}},D,\mathrm{TOF},f_{\mathrm{act}},\phi_{\mathrm{cov}} \right\}. \label{eq:inv_input_set}\] Each parameter is defined in the text, and DOI package includes units and measurement/estimation basis as metadata.

8.7.2 Calculation Sequence (Algorithm; Deterministic/Reproducible)

Executing the following sequence deterministically reproduces the same result.

1. Electrical Energy Decomposition: [eq:inv_Edes_day], [eq:inv_Eel_day].

2. Water Production: \[L_{p,\mathrm{day}} = 1000\,\frac{E_{\mathrm{des,day}}}{\mathrm{SEC}_{\mathcal{T}}}.\]

3. Hydrogen Production: Calculate \(m_{H_2,\mathrm{day}}\) via [eq:inv_mH2_day_final], \(n_{H_2,\mathrm{use}}\) via [eq:inv_nH2_use].

4. Hydrogen Supply Limited Throughput: [eq:inv_nB_H2lim], [eq:inv_mB_H2lim].

5. Catalyst Reaction Limited Throughput: [eq:inv_nB_kinlim], [eq:inv_mB_kinlim].

6. Actual Throughput: [eq:inv_nB_final], [eq:inv_mB_final].

7. Integrated Verification: Confirm \(n_{H_2,\mathrm{cons}}=\nu_{H_2}n_{B,\mathrm{day}} \le u_{H_2}n_{H_2,\mathrm{day}}\).

8.7.3 Output Items (Mandatory Reporting)

For DOI reporting, output items are fixed as: \[\begin{aligned} &L_{p,\mathrm{day}}[\mathrm{L/day}],\quad m_{H_2,\mathrm{day}}[\mathrm{g/day}],\quad n_{H_2,\mathrm{day}}[\mathrm{mol/day}], \label{eq:inv_outputs1}\\ &m_{B,\mathrm{H2\mbox{-}lim}}[\mathrm{g/day}],\quad m_{B,\mathrm{kin\mbox{-}lim}}[\mathrm{g/day}],\quad m_{B,\mathrm{day}}[\mathrm{g/day}], \label{eq:inv_outputs2}\\ &L_{H_2O,\mathrm{need}}[\mathrm{L/day}] \ \text{(Water for electrolysis)},\quad \mathrm{SEC}_{H_2}[\mathrm{kWh/kg}] \ \text{(Electrolysis Energy Intensity)}. \label{eq:inv_outputs3}\end{aligned}\]

8.7.4 Uncertainty (Quantification of Rough Estimate): First-Order Error Propagation (Complete Formula)

To provide confidence intervals for “Rough Estimates”, first-order error propagation is fixed. For example, for water production [eq:inv_Lp_day], since \(L = 1000 E/\mathrm{SEC}\), relative uncertainty (assuming independent errors) is: \[\left(\frac{\sigma_L}{L}\right)^2 \approx \left(\frac{\sigma_E}{E}\right)^2 + \left(\frac{\sigma_{\mathrm{SEC}}}{\mathrm{SEC}}\right)^2. \label{eq:inv_unc_water}\] For hydrogen production [eq:inv_mH2_day_final], since \(m \propto (\eta_F/V_{\mathrm{cell}})E\): \[\left(\frac{\sigma_m}{m}\right)^2 \approx \left(\frac{\sigma_E}{E}\right)^2 + \left(\frac{\sigma_{\eta_F}}{\eta_F}\right)^2 + \left(\frac{\sigma_{V}}{V_{\mathrm{cell}}}\right)^2. \label{eq:inv_unc_h2}\] Including these formulas in the DOI package generates reproducible error bars for back-calculation results.

8.8 Conclusion of This Section (Summary): Back-calculation Immediately Yields Linear Scale Output Once “SEC/Voltage/Efficiency” is Fixed

1. Water production is completed by [eq:inv_Lp_day], with core parameter \(\mathrm{SEC}_{\mathrm{des}}\).

2. Hydrogen production is completed by [eq:inv_mH2_day_final], with core parameters \((\eta_F, V_{\mathrm{cell}})\).

3. Catalyst throughput is fixed as the minimum of Hydrogen Supply Limit [eq:inv_mB_H2lim] and Catalyst Reaction Limit [eq:inv_mB_kinlim] via [eq:inv_mB_final].

4. For DOI reproducibility, Input Set [eq:inv_input_set], Calculation Sequence, Output Items, and Uncertainty Propagation Formulas are fixed.

9 Experiment/Measurement Based Verification Plan: Establishing “Amplitude \(\leftrightarrow\) Observable” Calibration Layer and Step-by-Step Validation Roadmap

9.1 Purpose of this Section: Falsifiable (Verifiable) Empirical Design for “Entity Fixing”

The purpose of this section is to mathematically fix an experiment/measurement based roadmap to verify whether the VP theory’s Operational Amplitude \(A_{\mathrm{op}}\) and Alignment Variable \(S\) are connected to actual observables in a reproducible manner across ESS, Purification, Electrolysis/Catalyst, and Integrated Operation.

In this application whitepaper, “Entity Fixing” is defined as established when the following three conditions are met simultaneously:

1. Conservation Closure: Heat/Power/Mass balances close within sensor error ranges.

2. Predictability: Under identical inputs (feed salinity, flow, current/voltage, temperature, operation mode), the VP-based model explains observations via predictive (hold-out) performance.

3. Identifiability: “Amplitude” is distinguished from other factors (flow, temperature, simple power increase) and identified independently, with its estimation reported along with uncertainty (confidence interval).

The Core Verification Question is fixed as follows: \[\begin{aligned} \textbf{Q:}\quad &\text{Does the model introducing } \Bigl(A_{\mathrm{op}},S\Bigr) \\ &\text{(1) without violating conservation laws, (2) compared to mainstream models,}\\ &\text{(3) significantly reduce prediction error in independent validation data?} \end{aligned} \label{eq:core_validation_question}\] If the answer to this question is not “Yes”, the VP entity in the application phase is not fixed (i.e., cannot be claimed in a DOI package), and the cause is decomposed according to the diagnostic procedure in this section (§9.8).

9.2 Statistical Formulation of Verification: Null/Alternative Hypotheses and Success Criteria

Let the integrated system observation vector be \(\mathbf{y}(t)\in\mathbb{R}^{n_y}\), control input be \(\mathbf{u}(t)\in\mathbb{R}^{n_u}\), and exogenous state (environment/feed/load conditions) be \(\mathbf{x}(t)\in\mathbb{R}^{n_x}\). (e.g., \(\mathbf{y}\) includes \(V_{\mathrm{bus}}, P_{\mathrm{gen}}, Q_p, C_p, \dot{m}_{H_2}\), etc.)

9.2.1 Mainstream Model (Baseline) vs. VP-Augmented Model (Target)

Let the mainstream (baseline) model be \(\mathcal{M}_0\) and the model including VP variables be \(\mathcal{M}_1\). \[\begin{aligned} \mathcal{M}_0:\quad &\mathbf{y}(t)=\mathbf{f}_0\!\left(\mathbf{u}(t),\mathbf{x}(t);\boldsymbol{\theta}_0\right)+\boldsymbol{\varepsilon}(t), \label{eq:M0}\\ \mathcal{M}_1:\quad &\mathbf{y}(t)=\mathbf{f}_0\!\left(\mathbf{u}(t),\mathbf{x}(t);\boldsymbol{\theta}_0\right) +\mathbf{f}_{\mathrm{VP}}\!\left(\mathbf{A}_{\mathrm{op}}(t),\mathbf{S}(t),\mathbf{u}(t),\mathbf{x}(t);\boldsymbol{\theta}_1\right) +\boldsymbol{\varepsilon}(t), \label{eq:M1}\end{aligned}\] Here, \[\boldsymbol{\varepsilon}(t)\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma})\] is fixed as the default noise model. (If non-normality/autocorrelation is confirmed, the noise model is updated in the DOI package, but the update rule must be specified.)

9.2.2 Null/Alternative Hypotheses

\[\begin{aligned} H_0:&\quad \mathbf{f}_{\mathrm{VP}} \equiv \mathbf{0}\ \ \text{(VP terms add no predictive power)}, \label{eq:H0}\\ H_1:&\quad \mathbf{f}_{\mathrm{VP}} \not\equiv \mathbf{0}\ \ \text{(VP terms provide independent predictive power)}. \label{eq:H1}\end{aligned}\]

9.2.3 Success Criteria (Quantitative Rules)

Verification requires both predictability and conservation compliance, not just “Right/Wrong”. Thus, success judgment must satisfy both of the following 2 conditions:

1. Prediction Error Reduction (Hold-out): In validation set \(\mathcal{D}_{\mathrm{val}}\): \[\Delta_{\mathrm{RMSE}} \;\equiv\; \frac{\mathrm{RMSE}(\mathcal{M}_0)-\mathrm{RMSE}(\mathcal{M}_1)}{\mathrm{RMSE}(\mathcal{M}_0)} \;\ge\; \Delta_{\min}, \label{eq:delta_rmse}\] Here, \[\mathrm{RMSE}(\mathcal{M}) = \sqrt{\frac{1}{N}\sum_{k=1}^{N}\left\|\mathbf{y}_k-\hat{\mathbf{y}}_k^{(\mathcal{M})}\right\|_2^2} \label{eq:rmse_def}\] and \(\Delta_{\min}\) is the minimum improvement rate defined by the project (e.g., 0.1), fixed in the DOI package.

2. Conservation Closure (Integration over Same Time Window): Residuals of heat/power/mass balances must be within allowable ranges. Taking power balance residual as an example (integrated formula [eq:power_balance_dc] in §7): \[r_P(t) = P_{\mathrm{gen}}(t)+P_{\mathrm{dis}}(t) -\Bigl(P_{\mathrm{des}}(t)+P_{\mathrm{el}}(t)+P_{\mathrm{aux}}(t)+P_{\mathrm{ch}}(t)+P_{\mathrm{curt}}(t)\Bigr), \label{eq:power_residual}\] and the normalized residual over the time window is: \[\bar{r}_P = \frac{\int_{t_0}^{t_1} |r_P(t)|\,dt}{\int_{t_0}^{t_1} \left(P_{\mathrm{gen}}(t)+P_{\mathrm{dis}}(t)\right)dt} \;\le\; \delta_{P}, \label{eq:power_residual_norm}\] Here, \(\delta_P\) is the allowable residual upper bound (e.g., 1–5%) fixed in DOI. Heat/Salt/Hydrogen balances are fixed with the same residual criteria.

Note: Even if [eq:delta_rmse] is satisfied, violation of [eq:power_residual_norm] means failure. (Sensor/definition/hidden inputs must be resolved first; this is the minimum condition for “Entity Fixing”.)

9.3 Mathematical Definition of “Amplitude \(\leftrightarrow\) Observable” Calibration Layer (2 Stages: Forward/Inverse)

9.3.1 Operational Form of Amplitude and Alignment Variables (Vectorization)

Operational amplitude is fixed as a vector decomposed by domain (Bulk/Solution/Surface). \[\mathbf{A}_{\mathrm{op}}(t) = \begin{bmatrix} A_{\mathrm{op}}^{(\mathrm{bulk})}(t)\\ A_{\mathrm{op}}^{(\mathrm{sol})}(t)\\ A_{\mathrm{op}}^{(\mathrm{surf})}(t) \end{bmatrix}, \qquad \mathbf{S}(t) = \begin{bmatrix} S^{(\mathrm{bulk})}(t)\\ S^{(\mathrm{sol})}(t)\\ S^{(\mathrm{surf})}(t) \end{bmatrix}. \label{eq:Aop_S_vector}\] Theoretical formulas for each component (LOCK formula and \(A_{\mathrm{op}}\) redefinition) are assumed fixed in §2. In this section, we do not trust those calculations as is, but build a structure calibratable by measurement.

9.3.2 Forward Calibration Layer: \(A_{\mathrm{op}}\rightarrow\) Observation Prediction

The forward calibration layer quantifies “how amplitude enters into mainstream properties/performance coefficients”. This is fixed as the parameterized forward model: \[\hat{\mathbf{y}}(t) = \mathbf{h}\!\left(\mathbf{u}(t),\mathbf{x}(t);\boldsymbol{\theta}_0\right) +\mathbf{g}\!\left(\mathbf{A}_{\mathrm{op}}(t),\mathbf{S}(t),\mathbf{u}(t),\mathbf{x}(t);\boldsymbol{\theta}_{\mathrm{cal}}\right), \label{eq:forward_model}\] Here,

• \(\mathbf{h}\): Prediction term from mainstream physics (Membrane Flux, Butler–Volmer, Thermal Balance, etc.),

• \(\mathbf{g}\): VP-augmented term (Term where operational amplitude/alignment modifies properties/barriers/efficiencies),

• \(\boldsymbol{\theta}_{\mathrm{cal}}\): Mapping parameters to be calibrated (e.g., \(\Lambda_A, \beta_A, m\) in §6).

Adopting this structure makes VP validity falsifiable by whether \(\mathbf{g}\) adds predictive power.

9.3.3 Inverse Calibration Layer: Observation \(\rightarrow \hat{A}_{\mathrm{op}}\) Estimation

Since amplitude is a latent variable, not direct measurement, we define estimated amplitude \(\hat{\mathbf{A}}_{\mathrm{op}}(t)\) inversely from observables. Inverse model is fixed as the optimization: \[\hat{\mathbf{A}}_{\mathrm{op}}(t) = \arg\min_{\mathbf{A}\in\mathcal{A}} \left\| \mathbf{W}^{1/2} \Bigl(\mathbf{y}(t)-\mathbf{h}(\mathbf{u}(t),\mathbf{x}(t))-\mathbf{g}(\mathbf{A},\mathbf{S}(t),\mathbf{u}(t),\mathbf{x}(t))\Bigr) \right\|_2^2 +\lambda\,\mathcal{R}(\mathbf{A}), \label{eq:inverse_estimation}\] Here,

• \(\mathbf{W}\): Weights based on observation uncertainty (Diagonal or Inverse Covariance),

• \(\mathcal{A}\): Allowable set of amplitude (e.g., Physical range \([A_{\min}, A_{\max}]\)),

• \(\mathcal{R}(\mathbf{A})\): Regularization (e.g., Temporal smoothness, suppressing excessive fluctuation),

• \(\lambda\ge 0\): Regularization strength (Calibration target).

Including [eq:inverse_estimation] in the DOI package enables reproduction of identical \(\hat{\mathbf{A}}_{\mathrm{op}}\) from identical data.

9.3.4 State-Space (Dynamic) Calibration for Time-Continuous Measurement (Optional but Recommended)

If amplitude varies with time (Night/Day, Load Fluctuation), the following state-space model is the standard. \[\begin{aligned} \mathbf{A}_{k+1} &= \mathbf{f}_A(\mathbf{A}_k,\mathbf{u}_k,\mathbf{x}_k)+\mathbf{w}_k, \label{eq:ssm_A}\\ \mathbf{y}_k &= \mathbf{h}(\mathbf{u}_k,\mathbf{x}_k)+\mathbf{g}(\mathbf{A}_k,\mathbf{S}_k,\mathbf{u}_k,\mathbf{x}_k)+\mathbf{v}_k, \label{eq:ssm_y}\end{aligned}\] Here, \(k\) is discrete time index, \(\mathbf{w}_k, \mathbf{v}_k\) are noises. In this case, Extended Kalman Filter (EKF) or Particle Filter (PF) can estimate \(\mathbf{A}_k\), and filter selection/implementation version are fixed in DOI.

9.4 Identifiability Conditions (Mandatory): Design to Separate “Amplitude Effect” from Other Factors

9.4.1 Jacobian Rank Condition (Local Identifiability)

When calibration parameters \(\boldsymbol{\theta}_{\mathrm{cal}}\) and amplitude \(\mathbf{A}\) are simultaneously unknown, local identifiability requires the Jacobian of the observation equation to have sufficient rank. Writing the observation model as \(\mathbf{y}=\mathbf{F}(\mathbf{A},\boldsymbol{\theta}_{\mathrm{cal}})\): \[\mathbf{J} = \begin{bmatrix} \dfrac{\partial \mathbf{F}}{\partial \mathbf{A}} & \dfrac{\partial \mathbf{F}}{\partial \boldsymbol{\theta}_{\mathrm{cal}}} \end{bmatrix} \label{eq:jacobian_def}\] must satisfy \[\mathrm{rank}(\mathbf{J})\ \ge\ \dim(\mathbf{A})+\dim(\boldsymbol{\theta}_{\mathrm{cal}}). \label{eq:rank_condition}\] To satisfy this, experiments require multi-condition data where inputs affecting amplitude (Voltage/Current/Frequency/Temperature/Salinity/Flow, etc.) are varied independently, not fixed to a single condition.

9.4.2 Minimum Requirements for Experimental Design (Factor Decomposition)

At least the following factors must be separated:

1. Net Power Effect vs Amplitude Effect: Check performance change by varying waveform/frequency/current density distribution at same \(E_{\mathrm{day}}\).

2. Temperature Effect vs Amplitude Effect: Separate by varying only temperature at same amplitude condition (or vice versa).

3. Flow/Residence Time Effect vs Amplitude Effect: Separate by varying flow rate at same electric field/current condition (Common for Purification/Reactor).

4. Salinity/Composition Effect vs Amplitude Effect: Separate influence of solution conductivity/ionic strength change on amplitude estimation.

Implementing this via Design of Experiments (DoE), letting factor vector be \(\mathbf{p}=[p_1,\dots,p_d]\) with \(L_i\) levels for each factor \(p_i\), the number of full factorial experiments is: \[N_{\mathrm{full}}=\prod_{i=1}^{d} L_i \label{eq:full_factorial}\] If realistic constraints exist, use fractional factorial (specify resolution) or Latin Hypercube Sampling (LHS), but sampling rules and seed must be fixed in the DOI package.

9.5 “Observabilization” of Amplitude: Definition of Domain-Specific Measurable Proxies

This subsection fixes the rules to build the calibration layer with measurable proxy variables, under the constraint that \(\mathbf{A}_{\mathrm{op}}, \mathbf{S}\) cannot be directly measured. Proxies must be (1) measureable by sensors, (2) clear in unit/calibration, and (3) stably reproducible in repeated experiments.

9.5.1 Common: Proxy Vector \(\mathbf{z}(t)\) and Input to Amplitude Estimator

Define proxy variable vector as \(\mathbf{z}(t)\in\mathbb{R}^{n_z}\), and fix amplitude estimator as: \[\hat{\mathbf{A}}_{\mathrm{op}}(t)=\mathbf{G}\!\left(\mathbf{z}(t);\boldsymbol{\phi}\right) \label{eq:A_from_proxy}\] Here, \(\mathbf{G}\) is calibrated conversion formula (linear/nonlinear possible), \(\boldsymbol{\phi}\) are estimation parameters (calibration target). Principle: \(\mathbf{z}(t)\) must be included in the integrated log schema of §7.12 or saved in a separate proxy log with same timestamps.

9.5.2 ESS (Bulk) Proxy: Observables of Heat/Radiation/Electrical Conversion

Minimum proxies for ESS (Bulk) are fixed as:

1. Stored Thermal State: \(T_s(t)\) or \(E_{\mathrm{th}}(t)\) (Already [eq:Eth_def]).

2. Heat Flow: \(\dot{Q}_{\mathrm{to\,gen}}(t)\), \(\dot{Q}_{\mathrm{abs}}(t)\).

3. Generation Output: \(P_{\mathrm{gen}}(t)\) and Efficiency Estimate \(\hat{\eta}_{\mathrm{conv}}(t)=P_{\mathrm{gen}}/\dot{Q}_{\mathrm{to\,gen}}\).

4. (Optional) Surface Radiation Proxy: Surface Temperature Field (Multi-point), or Radiometer/IR Camera based Radiation Flux (if absolute calibration possible).

Example of proxy-based definition of bulk amplitude (Pre-calibration prototype) is set as: \[z^{(\mathrm{bulk})}_1(t)=\frac{P_{\mathrm{gen}}(t)}{\dot{Q}_{\mathrm{to\,gen}}(t)},\qquad z^{(\mathrm{bulk})}_2(t)=T_s(t), \label{eq:proxy_bulk_examples}\] And \[\hat{A}_{\mathrm{op}}^{(\mathrm{bulk})}(t) =\phi_{b,0}+\phi_{b,1}\,z^{(\mathrm{bulk})}_1(t)+\phi_{b,2}\,z^{(\mathrm{bulk})}_2(t) \label{eq:A_bulk_linear_proxy}\] is set as initial form (Nonlinear extension possible with empirical data). Note: [eq:A_bulk_linear_proxy] is not a definition but an initial model. In final DOI version, it must be calibrated to satisfy (i) Energy Balance Closure, and (ii) Predictability Improvement.

9.5.3 Purification (Solution) Proxy: Conductivity/Impedance/IV/Concentration Response

In Purification (Solution), “Ion Transport” is core, so fix the following proxies:

1. Conductivity or Salinity: \(\kappa(t)\) or \(C(t)\) (including conversion curve).

2. Electrical Proxies: \(V(t), I(t)\), Current Efficiency Estimate (ED/CDI) or Membrane Pressure (RO).

3. (Recommended) Impedance Spectroscopy (EIS) Proxies: Real/Imaginary parts of Impedance \(Z(\omega)\) per frequency.

4. Dynamic Proxy: Response Time Constant of \(C_p(t)\) or \(Q_p(t)\) to Step Input (Voltage/Current/Flow).

For example, if equivalent resistance \(R_{\mathrm{sol}}\) can be estimated from EIS: \[z^{(\mathrm{sol})}_1(t)=\frac{1}{R_{\mathrm{sol}}(t)},\qquad z^{(\mathrm{sol})}_2(t)=\kappa(t),\qquad z^{(\mathrm{sol})}_3(t)=\frac{C_f(t)-C_p(t)}{C_f(t)} \label{eq:proxy_sol_examples}\] Then solution amplitude is estimated as: \[\hat{A}_{\mathrm{op}}^{(\mathrm{sol})}(t)=\mathbf{G}_{\mathrm{sol}}\!\left(\mathbf{z}^{(\mathrm{sol})}(t);\boldsymbol{\phi}_{\mathrm{sol}}\right) \label{eq:A_sol_proxy}\] \(\mathbf{G}_{\mathrm{sol}}\) can be linear, logistic, or neural network, but for reproducibility and interpretability in v1.0, the following form is standard: \[\hat{A}_{\mathrm{op}}^{(\mathrm{sol})}(t) = \phi_{s,0} +\sum_{i}\phi_{s,i}\,z^{(\mathrm{sol})}_i(t) +\sum_{i<j}\phi_{s,ij}\,z^{(\mathrm{sol})}_i(t)z^{(\mathrm{sol})}_j(t), \label{eq:G_sol_quadratic}\] i.e., fixed up to 2nd order polynomial (including interactions), extended only in DOI version updates if necessary.

9.5.4 Catalyst/Electrolysis (Surface) Proxy: \(j_0\), Tafel, EIS, TOF, Activation Energy

Surface (Catalyst/Electrolysis) amplitude and alignment are observationalized by mainstream variables like Activation Barrier or Exchange Current Density. Minimum proxies are fixed as:

1. Electrolysis: \(V_{\mathrm{cell}}(t)\), \(I_{\mathrm{el}}(t)\), \(T_{\mathrm{stack}}(t)\), \(\eta_F(t)\).

2. Electrolysis (Recommended): Polarization curve \(j\)–\(\eta\) and Low Overpotential Region Slope.

3. Surface Electrochemistry (Recommended): Estimate \(R_{\mathrm{ct}}\) (Charge Transfer Resistance) from EIS.

4. Catalytic Reaction: Product Molar Flow \(\dot{n}_B\), Conversion \(X_A\), Selectivity \(S_B\).

5. Catalyst Temp Dependence: Estimate Effective Activation Energy \(E_a\) via Arrhenius Plot.

For example, if \(R_{\mathrm{ct}}\) is obtained in HER, approximately: \[j_0 \propto \frac{1}{R_{\mathrm{ct}}} \label{eq:j0_from_Rct}\] Use this relation (exact constant is calibration target) as proxy. Thus construct: \[z^{(\mathrm{surf})}_1(t)=\frac{1}{R_{\mathrm{ct}}(t)},\qquad z^{(\mathrm{surf})}_2(t)=\eta_F(t),\qquad z^{(\mathrm{surf})}_3(t)=\mathrm{TOF}(t) \label{eq:proxy_surf_examples}\] And estimate as: \[\hat{A}_{\mathrm{op}}^{(\mathrm{surf})}(t)=\mathbf{G}_{\mathrm{surf}}\!\left(\mathbf{z}^{(\mathrm{surf})}(t);\boldsymbol{\phi}_{\mathrm{surf}}\right) \label{eq:A_surf_proxy}\]

9.6 Calibration Parameter Estimation: Weighted Least Squares/MLE/Bayesian (Complete Form)

9.6.1 Weighted Least Squares (Default) Objective Function

The basic method to estimate calibration parameters \(\boldsymbol{\theta}_{\mathrm{cal}}\) of [eq:forward_model] from dataset \(\mathcal{D}=\{(\mathbf{y}_k,\mathbf{u}_k,\mathbf{x}_k)\}_{k=1}^{N}\) is Weighted Least Squares. \[\hat{\boldsymbol{\theta}}_{\mathrm{cal}} = \arg\min_{\boldsymbol{\theta}_{\mathrm{cal}}\in\Theta} \sum_{k=1}^{N} \left(\mathbf{y}_k-\hat{\mathbf{y}}_k(\boldsymbol{\theta}_{\mathrm{cal}})\right)^{\mathsf{T}} \mathbf{W}_k \left(\mathbf{y}_k-\hat{\mathbf{y}}_k(\boldsymbol{\theta}_{\mathrm{cal}})\right), \label{eq:wls_obj}\] Here, \(\Theta\) includes physical constraints (e.g., Efficiency \(\in[0,1]\), Fraction \(\in[0,1]\), Positive Resistance, etc.).

9.6.2 Equivalence with Maximum Likelihood (Normal Noise)

If noise is normal and \(\mathbf{W}_k=\mathbf{\Sigma}^{-1}\), [eq:wls_obj] is equivalent to Maximum Likelihood Estimation (MLE): \[\hat{\boldsymbol{\theta}}_{\mathrm{cal}} = \arg\max_{\boldsymbol{\theta}_{\mathrm{cal}}} \prod_{k=1}^{N} \mathcal{N}\!\left(\mathbf{y}_k\ \middle|\ \hat{\mathbf{y}}_k(\boldsymbol{\theta}_{\mathrm{cal}}),\mathbf{\Sigma}\right). \label{eq:mle}\]

9.6.3 Bayesian Calibration (Standard for Providing Uncertainty)

Since “Uncertainty” is mandatory in DOI reproducibility, Bayesian calibration with priors on calibration parameters is recommended as standard. \[p(\boldsymbol{\theta}_{\mathrm{cal}}|\mathcal{D}) \propto p(\mathcal{D}|\boldsymbol{\theta}_{\mathrm{cal}})\,p(\boldsymbol{\theta}_{\mathrm{cal}}), \label{eq:bayes_posterior}\] For example, parameters needing positive constraints (Resistance, Dispersion, etc.) are fixed with Log-Normal priors, and fractions (Efficiency, Dispersion, Utilization) with Beta priors for reproducibility. Form and hyperparameters of priors are fixed in DOI package.

9.6.4 Calibration Performance Metrics (Goodness of Fit + Generalization) Fixed

After calibration, separate Train/Validation sets and report:

1. Training Error: \(\mathrm{RMSE}_{\mathrm{train}}\)

2. Validation Error: \(\mathrm{RMSE}_{\mathrm{val}}\)

3. Generalization Gap: \[\Delta_{\mathrm{gen}}=\mathrm{RMSE}_{\mathrm{val}}-\mathrm{RMSE}_{\mathrm{train}} \label{eq:gen_gap}\]

4. Conservation Residuals (Heat/Power/Mass): e.g. [eq:power_residual_norm]

If \(\Delta_{\mathrm{gen}}\) is large, overfitting is likely, so simplify model (e.g., maintain [eq:G_sol_quadratic]) or increase data.

9.7 Step-by-Step Experiment/Validation Roadmap (Single Module \(\rightarrow\) Partial Integrated \(\rightarrow\) Fully Integrated)

This subsection fixes experiments into a **Step-by-Step Gate** structure to decompose “what is reality” at each stage. Each stage has Input, Mandatory Measurement, Verification Formula, Pass Criteria.

9.7.1 Stage 0: Measurement Infrastructure Setup and Metrology Verification

9.7.1.1 Goal.

Verify first if sensor/measurement system is reliable enough to close conservation equations. This is a precondition for VP verification.

9.7.1.2 Mandatory Measurements (Minimum).

• Power: Voltage/Current simultaneous sampling (Same timestamp), Integrated Energy Verification.

• Heat: Multi-point Temperature, Heat Flow Estimation (Flow\(\times c_p \times \Delta T\)), Heat Loss Estimation.

• Flow/Salinity: \(Q_f,Q_p,Q_c\) and \(C_f,C_p,C_c\) (Purification Module), Gas Flow/Composition (Hydrogen/Oxygen).

9.7.1.3 Pass Criteria (Example).

\[\bar{r}_P\le \delta_P,\quad \bar{r}_Q\le \delta_Q,\quad \bar{r}_{\mathrm{salt}}\le \delta_{\mathrm{salt}}, \label{eq:stage0_criteria}\] Here, \(\bar{r}_Q, \bar{r}_{\mathrm{salt}}\) are normalized residuals of flow/salt balance, following closure test rules in §7.12.

9.7.2 Stage 1: Single Module Verification (Driven by External Reference Energy) + Amplitude Proxy Calibration

9.7.2.1 Goal.

Verify ESS/Purification/Electrolysis/Catalyst individually, and obtain initial parameters for \(\mathbf{z}\leftrightarrow \hat{\mathbf{A}}_{\mathrm{op}}\) calibration layer in each domain.

9.7.2.2 (1) Purification Standalone (External Power).

Apply \(P_{\mathrm{des}}\) from external source to fix \(\mathrm{SEC}_{\mathrm{des}}\), \(R\), \(Y\) empirically. \[\mathrm{SEC}_{\mathrm{des}}=\frac{\int P_{\mathrm{des}}dt}{\int Q_p dt},\quad R=1-\frac{C_p}{C_f},\quad Y=\frac{Q_p}{Q_f}. \label{eq:stage1_desal_kpi}\] Simultaneously record \(\mathbf{z}^{(\mathrm{sol})}\) (EIS/Conductivity/Response Time) to calibrate \(\boldsymbol{\phi}_{\mathrm{sol}}\) of [eq:G_sol_quadratic].

9.7.2.3 (2) Electrolysis Standalone (External Power).

Fix electrolysis state via polarization curve and \(\eta_F\) measurement. \[\eta_F=\frac{2F\dot{n}_{H_2}}{I_{\mathrm{el}}}, \qquad \mathrm{SEC}_{H_2}=\frac{\int V_{\mathrm{cell}}I_{\mathrm{el}}dt}{\int \dot{m}_{H_2}dt}. \label{eq:stage1_el_kpi}\] Simultaneously record surface proxies like \(R_{\mathrm{ct}}\), Tafel slope, \(j_0\) to calibrate [eq:A_surf_proxy] and mapping parameters (e.g., \(\beta_A, \beta_S\)) of §6.

9.7.2.4 (3) Catalytic Reactor Standalone (External H\(_2\)).

Supply bottled or standard hydrogen to fix TOF/Activation Energy of catalytic reaction itself. Fit Arrhenius form in temperature sweep: \[\mathrm{TOF}(T)=\mathrm{TOF}_0\exp\!\left(-\frac{E_a}{RT}\right) \label{eq:arrhenius_tof}\] to estimate \(E_a\), and verify absence of mass transfer limits (Thiele modulus, Effectiveness factor in §6).

9.7.2.5 Pass Criteria.

Standalone verification passes not by “Good Values” but by satisfying:

1. Conservation (Power/Mass) closure satisfied at Stage 0 level.

2. KPI variation in repeated conditions within allowable range (e.g., CV \(\le\) specified value).

3. Proxy-based \(\hat{A}_{\mathrm{op}}\) shows monotonic or consistent response to input factor changes (No abnormal jumps/drifts).

9.7.3 Stage 2: 2-Module Partial Integration (ESS\(\to\)Purification, ESS\(\to\)Electrolysis, Purification\(\to\)Electrolysis)

9.7.3.1 Goal.

Verify energy interfaces (DC Bus, Heat Exchange) actually work, and SEC/Efficiency calibrated in standalone are maintained in integration.

9.7.3.2 (A) ESS\(\to\)Purification.

Drive purification with ESS power, verify back-calculation: \[E_{\mathrm{elec,day}} \Rightarrow L_{p,\mathrm{day}} \label{eq:stage2_E_to_water}\] ( §8 ) matches measurement. Core pass criteria are Power Balance Closure and SEC Maintenance. \[\left|\frac{\mathrm{SEC}_{\mathrm{des,int}}-\mathrm{SEC}_{\mathrm{des,standalone}}}{\mathrm{SEC}_{\mathrm{des,standalone}}}\right| \le \delta_{\mathrm{SEC,des}}. \label{eq:stage2_SEC_match}\]

9.7.3.3 (B) ESS\(\to\)Electrolysis.

Drive electrolysis with ESS power, verify back-calculation: \[E_{\mathrm{el,day}} \Rightarrow m_{H_2,\mathrm{day}} \label{eq:stage2_E_to_h2}\] ( [eq:inv_mH2_day_final] in §8 ) matches measurement. Core is measurement fixing of \(\eta_F\) and stability under bus fluctuation.

9.7.3.4 (C) Purification\(\to\)Electrolysis (Water Quality Coupling).

When water for electrolysis comes from purification module, separate effect of water quality (Conductivity/Impurities) on electrolysis performance (\(V_{\mathrm{cell}}, \eta_F\)). Measure: \[V_{\mathrm{cell}}=V_{\mathrm{cell}}(C_p,\text{Impurities},T,\ldots) \label{eq:stage2_water_quality_effect}\] by changing only input water quality under same electrolysis condition, and include this term as covariate in calibration layer (Do not confuse with Amplitude Effect).

9.7.4 Stage 3: Electrolysis\(\to\)Catalyst (Hydrogen Coupling) + Throughput Minimum Rule Verification

9.7.4.1 Goal.

Experimentally judge whether hydrogen from electrolysis dominates catalyst throughput (Hydrogen Limited) or catalyst reaction dominates (Kinetics Limited). This step fixes the Minimum Value Rule of §8 to reality.

9.7.4.2 Verification Rule.

Daily throughput \(m_{B,\mathrm{day}}\) must satisfy: \[m_{B,\mathrm{day}}=\min\!\left(m_{B,\mathrm{H2\mbox{-}lim}},\ m_{B,\mathrm{kin\mbox{-}lim}}\right) \label{eq:stage3_min_rule}\] Calculate each term via [eq:inv_mB_H2lim], [eq:inv_mB_kinlim] in §8. In experiment, intentionally modulate \(m_{H_2,\mathrm{day}}\) (Current setpoint change) and check scaling of \(m_{B,\mathrm{day}}\).

1. If Hydrogen Limited: \(m_{B,\mathrm{day}}\propto m_{H_2,\mathrm{day}}\) (Linear).

2. If Kinetics Limited: \(m_{B,\mathrm{day}}\) saturates despite increase in \(m_{H_2,\mathrm{day}}\).

This judgment is fixed by regression, not “feeling”. \[m_{B,\mathrm{day}} = a\,m_{H_2,\mathrm{day}} + b, \label{eq:stage3_regression}\] If slope \(a\) is statistically significant (confidence interval excludes 0), and compared with saturation model (e.g., Michaelis–Menten form): \[m_{B,\mathrm{day}} = \frac{m_{\max} m_{H_2,\mathrm{day}}}{K+m_{H_2,\mathrm{day}}} \label{eq:stage3_saturation}\] Fix which limit is dominant via Model Selection (AIC/BIC or Bayesian Comparison).

9.7.5 Stage 4: ESS-Purification-Electrolysis-Catalyst Full Integration (Daytime) + Calibration Layer Verification

9.7.5.1 Goal.

In daytime integrated operation, simultaneously perform: (1) Conservation Closure, (2) Agreement with Back-calculation Prediction (§8), (3) Predictability Comparison of \(\mathcal{M}_0\) vs \(\mathcal{M}_1\), to fix “Reality”.

9.7.5.2 Core Verification Procedure.

1. Collect integrated logs with schema identical to [eq:integrated_schema].

2. With calibration parameters fixed from standalone/partial integration (No Re-training), evaluate prediction performance on validation set (New Day/New Load Profile).

3. If [eq:delta_rmse] and [eq:power_residual_norm] are simultaneously satisfied, judge as “Daytime Integrated Reality Fixed”.

9.7.6 Stage 5: Night Operation Verification (Storage/Buffer/Direct Thermal) + Time-Varying Amplitude Verification

9.7.6.1 Goal.

Verify validity of calibration layer when amplitude varies with time in Night Operation Modes (N1/N2/N3; §7). Especially, stable operation of State-Space Estimation [eq:ssm_A]–[eq:ssm_y] must be verified.

9.7.6.2 Mandatory Verification.

1. Night Upper Limit Check: \[\text{Must not exceed } t_{\mathrm{night,max}}\ \text{(or }V_{p,\mathrm{max}}\text{) upper limit.}\] (Refer to [eq:night_runtime], [eq:md_night_water] in §7)

2. Residual Continuity Before/After Night Transition: \[\lim_{t\to t_{\mathrm{switch}}^-}\bar{r}_P(t)\ \approx\ \lim_{t\to t_{\mathrm{switch}}^+}\bar{r}_P(t) \label{eq:night_residual_continuity}\]

3. No Drift/Divergence of Amplitude Estimator (e.g., \(\hat{A}_{\mathrm{op}}\) does not repeatedly exit allowable range \(\mathcal{A}\)).

9.8 Decision Tree (Cause Decomposition): Procedure to Distinguish “Full Physics” vs “Whitepaper” vs “Measurement”

To decompose the user’s core question (What is Reality) experimentally, fix the diagnostic procedure upon failure (or mismatch) as a Decision Tree.

9.8.1 Branch 1: Conservation Closure

\[\text{If any of }\bar{r}_P,\bar{r}_Q,\bar{r}_{\mathrm{salt}},\bar{r}_{H_2}\ \text{exceeds criteria}\Rightarrow \text{(A) Resolve Measurement/Definition Issues First}. \label{eq:tree_first}\] That is, if conservation laws do not close, no model comparison can discuss “Reality”.

9.8.2 Branch 2: Standalone Module Reproducibility

\[\text{If KPI variation is large in Standalone Verification}\Rightarrow \text{(B) Device/Process Stability (Fouling, Scaling, Leak) Issue}. \label{eq:tree_second}\] In this case, basic process control of device must be fixed before VP/Mainstream Model.

9.8.3 Branch 3: Is Mainstream Model Sufficient? (Additional Explanatory Power Evaluation)

If conservation closes and standalone reproducibility is secured, compare \(\mathcal{M}_0\) and \(\mathcal{M}_1\) via [eq:delta_rmse]. \[\Delta_{\mathrm{RMSE}} < \Delta_{\min}\Rightarrow \text{(C) Additional explanatory power of VP term is insufficient (or Proxy/Calibration Layer inappropriate)}. \label{eq:tree_third}\] Tasks: (i) Redesign Proxy \(\mathbf{z}\), (ii) Strengthen Identifiability (DoE Expansion), (iii) Simplify/Regularize Model. Do not immediately conclude “Theory is Wrong”.

9.8.4 Branch 4: VP Term Valid but Fails in Specific Module

Even if \(\mathcal{M}_1\) shows improvement in integration, it may fail in specific modules. Fix this by Module-wise Residual Decomposition. Split observation vector by module \(\mathbf{y}=[\mathbf{y}^{(\mathrm{ESS})},\mathbf{y}^{(\mathrm{DES})},\mathbf{y}^{(\mathrm{CAT})}]\) and calculate module RMSE. \[\mathrm{RMSE}^{(m)}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left\|\mathbf{y}^{(m)}_k-\hat{\mathbf{y}}^{(m)}_k\right\|_2^2}, \qquad m\in\{\mathrm{ESS,DES,CAT}\}. \label{eq:module_rmse}\] If no improvement only in specific \(m\), separate possibilities: Inappropriate Proxy/Calibration Layer for that module, or VP variables actually have no effect in that module. Fix conclusion.

9.9 Reproducibility Provision (For DOI): Data/Code/Metadata Standards and Experimental Protocol

9.9.1 Data Package Structure (Deterministic Structure)

DOI Package is fixed to at least:

1. README.md: Experiment Purpose, Date, Device Config, Safety Checklist, Sensor List.

2. metadata.json: Sensor Model/Serial/Calibration Date, Unit, Position, Sampling Period, Sync Method.

3. raw/: Raw Logs (Before Conversion).

4. processed/: Logs after Unit Unification & Missing Handling (Whitepaper Schema).

5. calibration/: Calibration Parameters, Priors, Fitting Results (Posterior Samples or Estimate+Covariance).

6. code/: Analysis Code (Version Tag, Hash Included).

7. results/: Figures/Tables/Summary Metrics & Auto-generated Report.

All files record Checksum (SHA256) for integrity verification.

9.9.2 Common Rules of Experimental Protocol (Randomization/Repetition/Blind)

Fix following rules as DOI Protocol:

1. Repetition: Minimum \(n_{\mathrm{rep}}\) repetitions for same condition (Value set by project, fixed in DOI).

2. Randomization: Randomize order of conditions in multi-condition experiments, record seed.

3. Blind (If possible): Hide condition labels during data preprocessing/fitting (or automate) to minimize subjective selection.

4. Missing/Outlier Rules: Fix criteria for missing handling/outlier removal beforehand (Prohibit post-hoc adjustment).

9.9.3 Calibration Layer Versioning (Control of Amplitude Definition/Proxy/Mapping Changes)

Since changing Amplitude (LOCK Formula) or Proxy Definition changes result interpretation, fix versioning rule: \[\text{Calibration Layer Version} = (\text{Amplitude Def Version},\ \text{Proxy Def Version},\ \text{Mapping Function Version}) \label{eq:cal_version_tuple}\] And for each version change, must record:

1. Reason for Change (Physics/Measurement/Statistics),

2. Impact Range (Which Module KPI affected?),

3. Retraining Necessity (Parameter portability).

9.10 Conclusion of This Section (Summary): Reality of Amplitude is Fixed Only by “Calibration Layer + Conservation + Predictability”

1. VP verification is formulated as [eq:core_validation_question], requiring both Predictability Improvement ([eq:delta_rmse]) and Conservation Closure ([eq:power_residual_norm]).

2. “Amplitude \(\leftrightarrow\) Observable” Calibration Layer is completed by Forward Model ([eq:forward_model]) and Inverse Estimation ([eq:inverse_estimation]), using State-Space ([eq:ssm_A]–[eq:ssm_y]) for time-varying cases.

3. Identifiability is verified by Jacobian Rank Condition ([eq:rank_condition]), designing DoE to satisfy it.

4. Validation proceeds via Stage 0–5 Roadmap, decomposing/fixing “Reality” in order of Standalone \(\to\) Partial Integration \(\to\) Full Integration \(\to\) Night Operation.

5. For DOI reproducibility, Data/Code/Metadata Structure, Randomization/Repetition Rules, and Calibration Layer Versioning Rules are fixed.

10 Cost, Procurement, Open Source (Free Distribution) Strategy, and Risk Management

10.1 Purpose and Scope of This Section (Fixing the Distinction Between “Free Distribution” and “Cost=0”)

The purpose of this section is to disclose the ESS–Purification–Electrolysis/Catalyst integrated system in a freely distributable (Open Source) form, while simultaneously fixing the following four aspects as reproducible specifications:

1. Cost: Fixing CapEx/OpEx/TCO and LCOW/LCOH/LCOB formulas and input parameters.

2. Procurement: Fixing BOM/Specs/Substitute Rules/Receiving QC for each module.

3. Open Source: Fixing Deliverables Scope, Repository Structure, License Stack, and DOI Packaging Rules.

4. Risk: Quantifying Safety/Technology/Procurement/Regulatory/Data (Reproducibility) risks and fixing mitigation/verification procedures.

10.1.0.1 Definition (Mandatory).

“Free Distribution” means the distribution cost (license fee) for design/document/code/data schema is 0. The physical cost of hardware fabrication/operation is not 0. Therefore, in this whitepaper, the following definitions are fixed: \[C_{\mathrm{license}}=0, \qquad C_{\mathrm{cap}}\ge 0,\quad C_{\mathrm{op}}\ge 0. \label{eq:def_free_vs_physical_cost}\]

10.2 Cost Model (Mathematical Completeness): CapEx + OpEx + Replacement + Levelized Cost

10.2.1 CapEx (Initial Capital Expenditure) Model: BOM Sum + Mfg/Install/Test + Contingency

For component index \(i=1,\dots,N\), with quantity \(q_i\) and unit cost \(c_i\), fix BOM cost as: \[C_{\mathrm{BOM}}=\sum_{i=1}^{N} q_i c_i. \label{eq:cap_bom}\] CapEx is fixed by the complete formula: \[C_{\mathrm{cap}} = C_{\mathrm{BOM}} +C_{\mathrm{mfg}} +C_{\mathrm{install}} +C_{\mathrm{test}} +C_{\mathrm{ship}} +C_{\mathrm{cont}}. \label{eq:cap_total}\] Contingency is fixed as a risk-based ratio (§10.5.4). \[C_{\mathrm{cont}} = \alpha_{\mathrm{cont}} \left( C_{\mathrm{BOM}}+C_{\mathrm{mfg}}+C_{\mathrm{install}}+C_{\mathrm{test}}+C_{\mathrm{ship}} \right), \qquad \alpha_{\mathrm{cont}}\in[0,1). \label{eq:cap_cont}\]

10.2.2 OpEx (Annual Operating Expenditure) Model: Fixed + Variable + Replacement (Cycle Model)

For lifetime \(N_{\mathrm{life}}\) years, OpEx in year \(y\) is fixed as: \[C_{\mathrm{op}}(y) = C_{\mathrm{fixed}}(y) +C_{\mathrm{var}}(y) +C_{\mathrm{rep}}(y) +C_{\mathrm{waste}}(y). \label{eq:op_total}\] For replacement item \(j\) with cycle \(T_j\) years and cost \(C_j\), cycle-based replacement cost is: \[C_{\mathrm{rep}}(y) = \sum_{j} \mathbf{1}\!\left( y\in\{T_j,2T_j,3T_j,\dots\}\right)\,C_j, \label{eq:op_replacement}\] where \(\mathbf{1}(\cdot)\) is the indicator function.

10.2.3 Net Present Value (NPV) and Annualization (CRF) Fixation

With discount rate \(r>0\), the sum of present values of costs is fixed as: \[\mathrm{NPV}_{\mathrm{cost}} = C_{\mathrm{cap}} +\sum_{y=1}^{N_{\mathrm{life}}}\frac{C_{\mathrm{op}}(y)}{(1+r)^y} -\frac{S_{\mathrm{salv}}}{(1+r)^{N_{\mathrm{life}}}}. \label{eq:npv_cost}\] Capital Recovery Factor (CRF) is fixed as: \[\mathrm{CRF}(r,N_{\mathrm{life}}) = \frac{r(1+r)^{N_{\mathrm{life}}}}{(1+r)^{N_{\mathrm{life}}}-1} \label{eq:crf}\] and Annualized Cost is fixed as: \[C_{\mathrm{ann}} = \mathrm{CRF}(r,N_{\mathrm{life}})\,C_{\mathrm{cap}} +\bar{C}_{\mathrm{op}} \label{eq:annualized_cost}\] Here, \(\bar{C}_{\mathrm{op}}\) is the average annual OpEx (or representative year OpEx), and the selection rule (averaging window, availability reflection) must be fixed in the DOI package.

10.2.4 Levelized Cost (LCOW/LCOH/LCOB) Complete Definition

Let availability (annual operation ratio) be \(a\in[0,1]\). Given daily production from §8 as \(L_{p,\mathrm{day}}\)[L/day], \(m_{H_2,\mathrm{day}}\)[g/day], \(m_{B,\mathrm{day}}\)[g/day]. Annual production is fixed as: \[\begin{aligned} V_{p,\mathrm{yr}}[\mathrm{m^3/yr}] &= a\cdot 365\cdot \frac{L_{p,\mathrm{day}}}{1000}, \label{eq:cost_Vp_year}\\ m_{H_2,\mathrm{yr}}[\mathrm{kg/yr}] &= a\cdot 365\cdot \frac{m_{H_2,\mathrm{day}}}{1000}, \label{eq:cost_mH2_year}\\ m_{B,\mathrm{yr}}[\mathrm{kg/yr}] &= a\cdot 365\cdot \frac{m_{B,\mathrm{day}}}{1000}. \label{eq:cost_mB_year}\end{aligned}\] Then Levelized Costs are fixed as: \[\begin{aligned} \mathrm{LCOW} &=\frac{C_{\mathrm{ann}}}{V_{p,\mathrm{yr}}} \qquad\left[\mathrm{currency/m^3}\right], \label{eq:LCOW}\\ \mathrm{LCOH} &=\frac{C_{\mathrm{ann}}}{m_{H_2,\mathrm{yr}}} \qquad\left[\mathrm{currency/kg_{H_2}}\right], \label{eq:LCOH}\\ \mathrm{LCOB} &=\frac{C_{\mathrm{ann}}}{m_{B,\mathrm{yr}}} \qquad\left[\mathrm{currency/kg_{B}}\right]. \label{eq:LCOB}\end{aligned}\]

10.2.4.1 Cost Allocation for Multiple Outputs (Mandatory Rule).

When producing Water/Hydrogen/Product \(B\) simultaneously, cost allocation rules must be fixed in DOI. As v1.0 default rule, Energy Allocation is used. From annual integrated energy: \[E_{\mathrm{des,yr}}=\int P_{\mathrm{des}}(t)\,dt,\qquad E_{\mathrm{el,yr}}=\int P_{\mathrm{el}}(t)\,dt \label{eq:cost_energy_integrals}\] Weights are defined as: \[w_{\mathrm{des}} = \frac{E_{\mathrm{des,yr}}}{E_{\mathrm{des,yr}}+E_{\mathrm{el,yr}}+\epsilon}, \qquad w_{\mathrm{el}} = \frac{E_{\mathrm{el,yr}}}{E_{\mathrm{des,yr}}+E_{\mathrm{el,yr}}+\epsilon}, \label{eq:cost_weights}\] (\(\epsilon>0\) prevents division by zero). Separated Annualized Costs are fixed as: \[C_{\mathrm{ann}}^{(\mathrm{water})}=w_{\mathrm{des}}C_{\mathrm{ann}},\qquad C_{\mathrm{ann}}^{(H_2)}=w_{\mathrm{el}}C_{\mathrm{ann}} \label{eq:cost_alloc}\] Then Separated Levelized Costs are reported as: \[\mathrm{LCOW}^{\star}=\frac{C_{\mathrm{ann}}^{(\mathrm{water})}}{V_{p,\mathrm{yr}}},\qquad \mathrm{LCOH}^{\star}=\frac{C_{\mathrm{ann}}^{(H_2)}}{m_{H_2,\mathrm{yr}}} \label{eq:cost_star}\] (For catalyst product \(B\), separating reactor module cost to calculate LCOB is the principle; separation rules are fixed in DOI after design confirmation.)

10.3 Procurement Strategy: Modular BOM Standardization, Specifications, Receiving QC, Substitute Rules

10.3.1 Module Decomposition Based BOM Standard

BOM is standardized by module decomposition. \[\mathrm{BOM} = \mathrm{BOM}_{\mathrm{ESS}} \cup \mathrm{BOM}_{\mathrm{BUS}} \cup \mathrm{BOM}_{\mathrm{DES}} \cup \mathrm{BOM}_{\mathrm{EL}} \cup \mathrm{BOM}_{\mathrm{CAT}} \cup \mathrm{BOM}_{\mathrm{SAFE}} \cup \mathrm{BOM}_{\mathrm{INST}}. \label{eq:bom_modules}\] Each BOM item mandatorily includes the following fields: \[\{\texttt{PartID},\texttt{Spec},\texttt{Qty},\texttt{UnitCost}, \texttt{Alt1},\texttt{Alt2},\texttt{LeadTime}, \texttt{IncomingQC},\texttt{HazardClass}\}. \label{eq:bom_fields}\]

10.3.2 Purchasing Specification (Spec) Minimum Fields (Removing Brand Dependency)

All core parts are defined by Specifications, not Brand/Model names. Spec fields are fixed as: \[\begin{aligned} \mathrm{Spec}(i)=\{&\texttt{rated\_voltage/current/pressure/temp},\ \texttt{materials},\ \texttt{interfaces},\\ &\texttt{accuracy(if\ sensor)},\ \texttt{proof\_test},\ \texttt{acceptance\_criteria},\ \texttt{traceability}\}. \end{aligned} \label{eq:spec_fields}\]

10.3.3 Receiving QC Standard (Safety/Measurement Priority)

Receiving QC is part of “Entity Fixing (Reproducibility/Safety)”, not just “Purchasing”. Minimum Receiving QC items are fixed as:

1. Safety Rating Check (Pressure rating/Heat resistance/Insulation/Breaker).

2. Leak/Proof Pressure (Pressure parts), Insulation/Dielectric (Electric parts), Function Test (Pump/Valve).

3. Sensor Calibration Check (Certificate or Reference Comparison) and Serial Recording.

4. Substitute (Alt1/Alt2) Compatibility (Interface/Rating) Check.

\[\mathrm{IncomingQC}_{\min} = \{\text{Safety rating},\ \text{Leak/Proof},\ \text{Calibration},\ \text{Alt compatibility}\}. \label{eq:incoming_qc}\]

10.4 Open Source (Free Distribution) Strategy: Deliverables Scope, Repository Structure, License Stack, DOI Packaging

10.4.1 Free Distribution Deliverables Scope (Mandatory Set)

The set of free distribution deliverables is fixed as: \[\mathcal{A}_{\mathrm{oss}} = \left\{\begin{aligned} &\text{Whitepaper(TeX)},\ \text{Design(CAD/PCB)},\ \text{BOM Template},\\ &\text{Control/Analysis Code},\ \text{Data Schema},\ \text{Test/Inspection Protocol},\\ &\text{Safety Docs(Checklist/Interlock)} \end{aligned}\right\}. \label{eq:oss_assets}\]

10.4.2 Repository Structure (Fixed Vertical List)

Repository structure is fixed as vertical (tree) (Checksum included per release):

repo/
  whitepaper/
  design/
    cad/
    pcb/
    bom/
  firmware/
  control/
  analysis/
  datasets/
    schema/
    example/
  safety/
  tests/
  releases/
  licenses/
  metadata/

Checksum of all files is recorded per release. \[\texttt{SHA256SUMS.txt}=\{\mathrm{sha256}(\text{each file})\}. \label{eq:sha256sums}\]

10.4.3 License Stack (Document/Code/Hardware/Data Separation)

Licenses are fixed separately by deliverable type. \[\mathcal{L} = \{ \mathcal{L}_{\mathrm{doc}}, \mathcal{L}_{\mathrm{code}}, \mathcal{L}_{\mathrm{hw}}, \mathcal{L}_{\mathrm{data}} \}. \label{eq:license_set}\] Selection of each license (e.g., CC family/OSI license/Open Hardware License) is decided by project policy, but at DOI issuance, (i) License Text, (ii) Scope, (iii) Exceptions (3rd Party Material) must be included in the package.

10.4.4 DOI Packaging Rules (Deterministic Reproduction Execution)

DOI package must include:

1. Commit Hash and Document Version (Whitepaper/Code/Schema).

2. Parameter Standard File (Appendix §12).

3. Reproduction Execution Script (Single command to generate tables/figures).

4. Random Seed Fixing (For all processes involving randomization).

Random seed is fixed by rule: \[\forall\ \text{Random Process}:\ \mathrm{seed}=s\ \text{used identically}. \label{eq:seed_rule}\]

10.5 Risk Management: Quantitative Scoring, FMEA (RPN), Mitigation/Verification, Contingency Estimation

10.5.1 Risk Scoring (Probability\(\times\)Impact) and Grade Thresholds

For risk item \(k\), define probability \(P_k\in[0,1]\) and impact (cost equivalent) \(I_k\ge 0\), fixing risk score as: \[R_k=P_k I_k \label{eq:risk_score}\] Grade thresholds are fixed as: \[\text{Grade}(R_k)= \begin{cases} \mathrm{Low}, & 0\le R_k < R_1,\\ \mathrm{Med}, & R_1\le R_k < R_2,\\ \mathrm{High},& R_2\le R_k, \end{cases} \label{eq:risk_grade}\] \(R_1, R_2\) are fixed numerically in DOI.

10.5.2 FMEA: RPN (Severity\(\times\)Occurrence\(\times\)Detection) Fixation

For FMEA, set \(S_k, O_k, D_k\in\{1,\dots,10\}\) and fix RPN as: \[\mathrm{RPN}_k=S_k O_k D_k \label{eq:rpn}\] Operating rule is fixed as: \[\mathrm{RPN}_k \ge \mathrm{RPN}_{\mathrm{crit}} \Rightarrow \text{Design Change/Interlock Strengthen/Stop Operation}. \label{eq:rpn_stop}\]

10.5.3 Risk Register Standard Fields

Risk register mandatorily includes: \[\{\texttt{RiskID},\texttt{Category},\texttt{Cause},\texttt{Effect}, \texttt{P},\texttt{I},\texttt{R}, \texttt{S},\texttt{O},\texttt{D},\texttt{RPN}, \texttt{Mitigation},\texttt{VerificationTest},\texttt{Owner},\texttt{Status}\}. \label{eq:risk_fields}\]

10.5.4 Risk-Based Contingency Estimation (No Arbitrary Contingency)

Let sum of expected risk losses be: \[L_{\mathrm{exp}}=\sum_{k} R_k \label{eq:expected_loss}\] Fix CapEx contingency ratio as: \[\alpha_{\mathrm{cont}} = \min\!\left( \alpha_{\max}, \frac{L_{\mathrm{exp}}}{C_{\mathrm{BOM}}+C_{\mathrm{mfg}}+C_{\mathrm{install}}} \right), \label{eq:alpha_cont}\] Here, \(\alpha_{\max}\) is upper limit (fixed numerically in project, included in DOI). This fixes the quantitative link where contingency increases as risk increases.

10.6 Conclusion of This Section

This section has fixed Cost ([eq:cap_total]–[eq:LCOB]), Procurement ([eq:bom_fields]–[eq:incoming_qc]), Open Source ([eq:oss_assets]–[eq:seed_rule]), and Risk ([eq:risk_score]–[eq:alpha_cont]) into reproducible mathematical/procedural specifications. Thus, in the DOI package, cost and risk can be reproduced with the same calculation procedure by changing only unit costs/replacement cycles/availability/risk inputs.

11 Constants, Units, and Calculation Log

11.1 Unit System Principles (SI Fixed) and Conversion Formulas

This whitepaper fixes the SI unit system as the default. For operational convenience, kWh and L may be used, but all calculations are performed after converting to SI. \[\begin{aligned} 1~\mathrm{kWh} &= 3.6\times 10^{6}~\mathrm{J}, \label{eq:app_kwh_to_j}\\ 1~\mathrm{m^3} &= 1000~\mathrm{L}. \label{eq:app_m3_to_L}\end{aligned}\]

11.2 Mandatory Physical Constants (Representative; Include Source/Version in DOI Metadata)

\[\begin{aligned} F &= 96485.33212~\mathrm{C/mol}, \label{eq:app_F}\\ R &= 8.314462618~\mathrm{J/(mol\cdot K)}, \label{eq:app_R}\\ M_{H_2} &= 2.01588~\mathrm{g/mol}, \label{eq:app_MH2}\\ M_{H_2O} &= 18.01528~\mathrm{g/mol}, \label{eq:app_MH2O}\\ \rho_w &\approx 1000~\mathrm{kg/m^3}. \label{eq:app_rho_w}\end{aligned}\]

11.3 Summary of Key Conversion Formulas (Purification/Electrolysis)

Purification Specific Energy Consumption (SEC) is fixed as: \[\mathrm{SEC}_{\mathrm{des}}[\mathrm{kWh/m^3}] = \frac{E_{\mathrm{des}}[\mathrm{kWh}]}{V_p[\mathrm{m^3}]} \label{eq:app_sec_des}\] For electrolysis, the conversion \(kWh/day \to g/day\) (based on Faraday’s Law) uses the complete formula: \[m_{H_2,\mathrm{day}}[\mathrm{g/day}] \approx 37.6076\, \frac{\eta_F}{V_{\mathrm{cell}}[\mathrm{V}]}\, E_{\mathrm{el,day}}[\mathrm{kWh/day}]. \label{eq:app_h2_from_kwh}\] Water consumption for electrolysis (stoichiometry) is fixed as: \[m_{H_2O} \approx \frac{M_{H_2O}}{M_{H_2}}\,m_{H_2} \approx 8.93\,m_{H_2} \label{eq:app_water_ratio}\]

11.4 Calculation Log Standard (Auditability + DOI Reproducibility)

Every calculation result \(z\) must be recorded with the following 4 elements: \[\mathrm{CalcLog}(z) = \{ \texttt{formula\_id}, \texttt{inputs}, \texttt{units}, \texttt{code\_hash} \}. \label{eq:app_calc_log_fields}\] formula_id is fixed to the equation number or unique ID in this whitepaper. inputs includes determinants such as data range, filters, and seeds. code_hash is fixed to the commit hash or SHA256 of the DOI package.

12 Dataset and Parameter Standards

12.1 Dataset Schema (Mandatory): Time Series + Event Log

The integrated log consists of (1) a Single Time Series Table and (2) an Event Log Table. The schema version is fixed as: \[\texttt{schema\_version} = \texttt{MAJOR.MINOR.PATCH}. \label{eq:app_schema_version}\] Incrementing MAJOR represents a change that breaks reproducibility (e.g., changing field meanings or units) and is principally prohibited in the DOI version.

12.2 Naming Convention (Mandatory): domain_quantity_qualifier

All variable names are fixed by the following rule: \[\texttt{name} = \texttt{domain\_quantity\_qualifier}. \label{eq:app_naming_rule}\] Examples: \[\texttt{bus\_V},\ \texttt{bus\_I},\ \texttt{ess\_Ts},\ \texttt{des\_Qp},\ \texttt{des\_Cp},\ \texttt{el\_Vcell},\ \texttt{el\_I},\ \texttt{cat\_nB}.\] Units are not mixed within the data but are fixed in separate metadata (units.json) (No mixed units allowed).

12.3 Parameter Standard (Mandatory Fields): Value/Unit/Basis/Uncertainty

Parameter \(\theta\) is fixed as a table with the following fields: \[\mathrm{Param}(\theta) = \{ \texttt{name}, \texttt{value}, \texttt{unit}, \texttt{method}, \texttt{source}, \texttt{uncertainty} \}. \label{eq:app_param_fields}\] method is fixed to one of {measured, fit, literature, assumed}. source records the filename, experiment ID, or commit hash. uncertainty includes standard deviation or confidence interval in principle.

13 Safety Checklist

13.1 Checklist Operation Rules (Mandatory)

1. All items are checked Yes/No, and evidence (measurement value/photo/log) is recorded.

2. If any item is checked No, operation start is prohibited (Re-check after correction).

3. Interlock tests include actual trips and are recorded in the event log.

13.2 Design Safety

1. (Electrical) Do DC bus rating, fuse/breaker/contactor ratings satisfy derating under maximum operating conditions?

2. (Grounding) Is grounding/equipotential bonding/leakage protection included in the design?

3. (Thermal) Is insulation/guard/warning label included at maximum temperature points?

4. (Pressure) Is overpressure protection (relief/rupture disk) and a safety discharge path included?

5. (Hydrogen) Is leak sensor/ventilation/flashback arrestor/ignition source control included?

6. (Water Quality) Is monitoring and cutoff/discharge path for product contamination (salt leak/metal elution) included?

13.3 Assembly/Piping/Wiring Safety

1. (Wiring) Do cable spec/crimping/insulation/fixing match specifications?

2. (Piping) Do wetted material/seal/torque match specifications?

3. (Leak) Was a water/gas leak test performed and criteria met (with record)?

4. (Sensor) Is installation position/direction/calibration status confirmed (Serial/Calibration Date recorded)?

13.4 Commissioning Safety

1. (Interlock) Were overheat/overpressure/leak/overcurrent trips actually tested (with log)?

2. (Balance) Do power/heat/mass balances close within allowable residuals?

3. (Purification) Are product conductivity/salinity below criteria (Auto discharge/cutoff confirmed upon excess)?

4. (Electrolysis) Is gas separation/discharge safe and \(\eta_F\) calculated from measurement?

13.5 Normal Operation Check (Daily/Weekly)

1. (Hydrogen) Leak alarm 0, Ventilation normal, Ignition source controlled?

2. (Temp/Pressure) All states within safe operating window (High/Low alarms normal)?

3. (Purification) No worsening trend in conductivity/salinity/pressure drop (Early warning of fouling)?

4. (Electrolysis) No rising trend in \(V_{\mathrm{cell}}\) or stack temperature anomaly?

13.6 Maintenance Check

1. (Replacement) Are membrane/filter/electrode/sensor replacement cycles managed (Replacement record included)?

2. (Calibration) Is instrument calibration periodically updated (Certificate included)?

3. (Cleaning) Is CIP procedure documented and performance recovered after cleaning (Log included)?

13.7 Emergency Response Check

1. (Stop) Is manual E-stop position/operation confirmed?

2. (Hydrogen) Are cutoff/ventilation/ignition isolation procedures ready for leakage?

3. (Fire/Overheat) Are heat dump/cutoff procedures ready for overheating?

4. (Contamination) Are immediate discharge/isolation/re-test procedures ready for suspected product contamination?