البيانات والتحليلات

المنهجية — الاتزان الكتلي، وحركية أغشية التبادل البروتوني، واقتصاديات المشاريع

الاشتقاقات الرياضية الرسمية وراء حاسبتَي RV Pathway Explorer وRV-CMP: الاتزان الكتلي، والكيمياء الكهربائية لأغشية PEM، وديناميكا حرارية التفاعلات، وأخذ العينات لتحليل الحساسية، وتحسين المحفظة، واقتصاديات المشاريع، كل منها موثَّق إلى مصدره.

Stoichiometric mass balance

Renewable Vision derives mass-balance coefficients directly from IUPAC 2021 standard atomic weights and balanced reaction equations. No fitted parameters; no calibration. Two examples (others follow the same template):

Methanol synthesis

\mathrm{CO_2} + 3\,\mathrm{H_2} \;\longrightarrow\; \mathrm{CH_3OH} + \mathrm{H_2O}

Per-tonne ratios

For a product P formed from reactant R with stoichiometric coefficient \nu_R and molecular weights M_R, M_P:

\frac{m_R}{m_P} \;=\; \frac{\nu_R\, M_R}{M_P}

Applied to methanol, with M_{\mathrm{H_2}}=2.016, M_{\mathrm{CO_2}}=44.009, M_{\mathrm{CH_3OH}}=32.042 g/mol:

\frac{m_{\mathrm{H_2}}}{m_{\mathrm{MeOH}}} = \frac{3 \times 2.016}{32.042} = 0.1888 \quad\quad \frac{m_{\mathrm{CO_2}}}{m_{\mathrm{MeOH}}} = \frac{44.009}{32.042} = 1.3735

e-SAF surrogate

Kerosene-range Fischer-Tropsch product is approximated by n-dodecane (\mathrm{C_{12}H_{26}}):

12\,\mathrm{CO_2} + 37\,\mathrm{H_2} \;\longrightarrow\; \mathrm{C_{12}H_{26}} + 24\,\mathrm{H_2O}

ReferenceIUPAC Commission on Isotopic Abundances and Atomic Weights — Standard Atomic Weights 2021. Pure and Applied Chemistry, 94(5), 573–600.

PEM electrolyzer kinetics

Cell voltage decomposes additively into the reversible (thermodynamic) component plus three irreversible overpotentials:

V_{cell}(i,T,P) \;=\; E_{rev}(T,P) + \eta_{act}^{a}(i,T) + \eta_{act}^{c}(i,T) + \eta_{ohm}(i,T) + \eta_{conc}(i)

Reversible voltage — Nernst

From standard temperature dependence with Nernst pressure correction:

E_{rev}(T,P_c,P_a) \;=\; E^{\circ}(T) + \frac{RT}{2F}\,\ln\!\left(P_c\sqrt{P_a}\right)

where E^{\circ}(T)=1.229 - 8.5\times10^{-4}(T-298.15) V.

Activation overpotential — Tafel form

For each electrode (anode and cathode):

\eta_{act}(i,T) \;=\; \frac{RT}{\alpha F}\,\ln\!\left(\frac{i}{i_0}\right)

with charge-transfer coefficients \alpha_a=1.494, \alpha_c=2.340, and exchange current densities i_0^a=10^{-7}, i_0^c=10^{-3} A/cm² (Sun et al., 2014).

Ohmic overpotential — Arrhenius membrane conductivity

\sigma(T) \;=\; \sigma_{\text{ref}}\,\exp\!\left[\frac{E_{a,m}}{R}\!\left(\frac{1}{T_{\text{ref}}}-\frac{1}{T}\right)\right]

R_{mem}(T) \;=\; \frac{\delta_m}{\sigma(T)} \qquad\quad \eta_{ohm}(i,T) \;=\; i\,\bigl[R_{mem}(T) + R_{\text{contact}}\bigr]

Concentration overpotential

\eta_{conc}(i) \;=\; -B\,\ln\!\left(1 - \frac{i}{i_{lim}}\right)

From voltage to energy intensity

Energy per kg of hydrogen at faradaic efficiency \eta_F:

E_{\mathrm{H_2}} \;=\; \frac{2 F\, V_{cell}}{\eta_F\, M_{\mathrm{H_2}}} \quad\bigl[\text{J/kg}\bigr] \;\longrightarrow\; \text{kWh/kg via }/3.6\times10^{6}

ReferenceSun S., Shao Z., Yu H., Li G., Yi B. (2014). Investigations on degradation of the long-term proton exchange membrane water electrolysis stack. Journal of Power Sources, 267, 515–520.

ReferenceMarangio F., Santarelli M., Calì M. (2009). Theoretical model and experimental analysis of a high-pressure PEM water electrolyser for hydrogen production. International Journal of Hydrogen Energy, 34(3), 1143–1158.

Reaction enthalpy via Helmholtz EOS

At any temperature T, the reaction enthalpy \Delta H_{rxn}(T) is computed exactly from the stoichiometric sum of single-species enthalpies:

\Delta H_{rxn}(T) \;=\; \sum_i \nu_i \, M_i \, h_i(T, P^{\circ})

Single-species enthalpies h_i(T,P) come from CoolProp's reference Helmholtz-energy equations of state — the same formulations NIST REFPROP uses. The reference state common to all CoolProp fluids cancels in any reaction-extent difference, so the absolute value of \Delta H_{rxn} is well-defined for sensitivity analysis.

e-Methanol example

\Delta H_{rxn} \;=\; M_{\mathrm{CH_3OH}}\,h_{\mathrm{CH_3OH}} + M_{\mathrm{H_2O}}\,h_{\mathrm{H_2O}} - M_{\mathrm{CO_2}}\,h_{\mathrm{CO_2}} - 3\,M_{\mathrm{H_2}}\,h_{\mathrm{H_2}}

At T=523.15 K (250 °C), 1 bar: \Delta H_{rxn} \approx -49.5 kJ/mol, matching the published value (Pérez-Fortes et al., 2016).

ReferenceBell I., Wronski J., Quoilin S., Lemort V. (2014). Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Industrial & Engineering Chemistry Research, 53(6), 2498–2508.

ReferencePérez-Fortes M., Schöneberger J., Boulamanti A., Tzimas E. (2016). Methanol synthesis using captured CO₂ as raw material: techno-economic and environmental assessment. Applied Energy, 161, 718–732.

Energy balance per kilogram of product

Per-stage rollup. Stage demand × mass-balance flow / product output gives the per-kg-product specific energy:

E_{\text{tot}} \;=\; E_{\text{cap}}\,\frac{m_{\mathrm{CO_2}}}{m_P} + E_{\mathrm{H_2}}\,\frac{m_{\mathrm{H_2}}}{m_P} + E_{\text{synth}} + E_{\mathrm{N_2}}\,\frac{m_{\mathrm{N_2}}}{m_P}

Mass ratios are stoichiometric (Section 01); stage energies E_{\text{cap}}, E_{\mathrm{H_2}}, E_{\text{synth}}, E_{\mathrm{N_2}} come from user inputs constrained to paper-cited operating windows.

Scope limitation. This is aggregate energy intensity — there is no rigorous heat integration, no pinch analysis, no distillation column reboiler. Results are at the precision band appropriate for pathway screening (±10–20%), not for FEED.

Sensitivity and Monte Carlo

Tornado — one-at-a-time deterministic sensitivity

For each input parameter p_k, perturb to its lower and upper bound while holding all others at the midpoint p_j^{\mathrm{mid}}. Measure the output swing:

\Delta y_k \;=\; y\bigl(p_k^{\mathrm{high}}, p_{j\neq k}^{\mathrm{mid}}\bigr) \,-\, y\bigl(p_k^{\mathrm{low}}, p_{j\neq k}^{\mathrm{mid}}\bigr)

Rank parameters by |\Delta y_k|; tallest bar at top of the chart.

Monte Carlo — triangular sampling

Each parameter is drawn independently from a triangular distribution with lower bound a, mode c, upper bound b. The cumulative distribution function:

F(x) \;=\; \begin{cases} 0 & x \le a \\[2pt] \dfrac{(x-a)^2}{(b-a)(c-a)} & a < x \le c \\[8pt] 1 - \dfrac{(b-x)^2}{(b-a)(b-c)} & c < x < b \\[2pt] 1 & x \ge b \end{cases}

For uniform u \sim U(0,1), the inverse CDF gives the sample:

x \;=\; \begin{cases} a + \sqrt{u\,(b-a)(c-a)} & u < F(c) \\[6pt] b - \sqrt{(1-u)(b-a)(b-c)} & u \ge F(c) \end{cases}

We sample N \in \{500, 1000, 2000, 5000\} scenarios, compute the metric for each through the mass + energy balance engine, then read empirical quantiles P_{10}, P_{50}, P_{90} from the sorted set.

Scope limitation. Parameters are sampled independently; no correlation matrix. Stronger versions of this analysis use historical project portfolios to fit a covariance structure.

Portfolio optimisation — LP

Allocate capacity x_k across K pathways to maximise an objective subject to CO₂ supply, H₂ supply, and per-product demand caps:

\max_{x \in \mathbb{R}^K_{+}}\; c^{\top}x \quad\text{subject to}\quad A x \le b,\;\; x \le u

where:

A = \begin{bmatrix} \nu_{\mathrm{CO_2}}^{(1)} & \cdots & \nu_{\mathrm{CO_2}}^{(K)} \\[4pt] \nu_{\mathrm{H_2}}^{(1)} & \cdots & \nu_{\mathrm{H_2}}^{(K)} \end{bmatrix},\quad b = \begin{bmatrix} \bar{m}_{\mathrm{CO_2}} \\[4pt] \bar{m}_{\mathrm{H_2}} \end{bmatrix},\quad u = \bigl(\bar{x}_1,\ldots,\bar{x}_K\bigr)^{\top}

The objective c depends on user choice — e.g. maximise CO₂ utilised takes c_k = \nu_{\mathrm{CO_2}}^{(k)}.

Vertex enumeration solver

By the fundamental theorem of LP, the optimum lies at a vertex of the feasible polytope. For K=4 variables and m half-space constraints, every vertex is determined by activating an exactly-4-subset of constraints to equality:

\#\text{vertices} \;\le\; \binom{m}{4}

For our problem with m = K + 2K + 2 = 14 (demand caps + non-negativity + resource constraints), this is \binom{14}{4} = 1001 candidate vertices — enumerable in milliseconds. Each candidate is checked for feasibility against all m constraints; the feasible vertex with the highest objective value wins.

Scope limitation. Vertex enumeration scales as \binom{m}{K}, which blows up beyond K \approx 8. Larger problems route through a real solver (revised simplex / interior-point) in the internal RV-CMP engine.

ReferenceDantzig G. (1963). Linear Programming and Extensions. Princeton University Press.

Project economics — IRR and NPV

Revenue and gross margin

R \;=\; \sum_i p_i^{\text{prod}} \, m_i^{\text{prod}} \,+\, \sum_j p_j^{\text{credit}} \, m_j^{\mathrm{CO_2}}

\mathrm{OPEX} \;=\; \sum_k c_k\, q_k \qquad \mathrm{GM} \;=\; R - \mathrm{OPEX}

Net present value

\mathrm{NPV} \;=\; \mathrm{GM}\cdot\frac{1 - (1+d)^{-n}}{d} \,-\, \mathrm{CAPEX}

Internal rate of return — Newton-Raphson

IRR is the discount rate r at which NPV = 0:

f(r) \;=\; \mathrm{GM}\cdot\frac{1 - (1+r)^{-n}}{r} - \mathrm{CAPEX} \;=\; 0

Newton-Raphson iterates with analytical derivative:

f'(r) \;=\; \mathrm{GM}\cdot\frac{n(1+r)^{-n-1}\,r - \bigl(1 - (1+r)^{-n}\bigr)}{r^{2}}

r_{k+1} \;=\; r_k - \frac{f(r_k)}{f'(r_k)}

Initialised at r_0 = 0.10; converges to |r_{k+1}-r_k|<10^{-7} in typically 4–6 iterations. If \mathrm{GM}\le 0, IRR is reported as −100% (project infeasible).

Simple payback

\mathrm{Payback} \;=\; \frac{\mathrm{CAPEX}}{\mathrm{GM}}\quad\text{years, undiscounted}

Internal RV-CMP only. The internal engine additionally computes amortised licence fees, royalty deductions on products / credits / CaaS, and a no-double-counting audit on the carbon claim ledger. Those layers are not exposed in the public calculator.

No-double-counting audit

The carbon accounting ledger asserts mass balance and unique claim-holder per stream. Three invariants are checked on every solve:

\text{Mass balance:}\quad m_{\mathrm{CO_2}}^{\text{captured}} \;=\; m_{\mathrm{CO_2}}^{\text{fuels}} + m_{\mathrm{CO_2}}^{\text{minerals}}

\text{Residual:}\quad |m_{\mathrm{CO_2}}^{\text{captured}} - m_{\mathrm{CO_2}}^{\text{fuels}} - m_{\mathrm{CO_2}}^{\text{minerals}}|\;<\;10^{-6}\;\text{t}

Claim-holder map

For each CO₂ stream, exactly one entity holds the carbon claim:

CO₂ in fuels: fuel buyer (embedded RFNBO / RED III claim, no separate credit).
CO₂ in minerals (DAC): CDR credit buyer (permanent removal credit).
CO₂ in minerals (Point-Source): mineral credit buyer (permanent storage credit).
CO₂ at source (Point-Source): host (paid for capture via CaaS fee, claims the avoidance).

ReferenceGHG Protocol Corporate Accounting and Reporting Standard, Revised Edition. World Resources Institute + World Business Council for Sustainable Development.

ReferenceISO 14064-2:2019 — Greenhouse gases — Specification with guidance at the project level for quantification, monitoring and reporting of greenhouse gas emission reductions or removal enhancements.

The math is open. The calibration is the engagement.

Every equation in this page powers a public tool somewhere on the site. Saudi-specific cost, price, and credit-pricing calibrations remain inside the internal RV-CMP engine and are scoped through a confidential engagement.

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المنهجية — الاتزان الكتلي، وحركية أغشية التبادل البروتوني، واقتصاديات المشاريع