Grok Challenge #2, Atmospheric Water Harvesting

<br /> This is the second paper and rough draft answer generated for @Grok’s 3/1/2026 challenge to MOTO ASI by Intrafere Research Group. A fully autonomous super intelligence AI harness. This paper cost around $15 to produce and took approximately two hours. As of the time of this writing, MOTO is continuing to solve Grok’s challenge with successive papers, attempting recursive improvement and solution-set diversification.</p> <p> Company Website: https://intrafere.com/<br /> Software GitHub that produced this paper: https://github.com/Intrafere/MOTO-Autonomous-ASI<br /> Grok Global Freshwater Crisis Challenge Link: https://x.com/grok/status/2028278338381316587</p> <p> Disclaimer: This is an autonomous AI solution generated with the MOTO harness. This paper was not peer reviewed and was autonomously generated without user oversight or interaction beyond the original user prompt, therefore, this text may contain errors. These papers often contain ambitious content and/or extraordinary claims, all content should be viewed with extreme scrutiny.</p> <p> (EDITOR NOTE: This single paper does not attempt to solve the user’s prompt entirely, it is meant to be one piece toward the complex solution required for the users prompt – this paper is the first paper in the series – total solutions typically are achieved in later papers). Metadata and original autonomous prompt available below.</p> <p> OUTLINE<br /> ================================================================================</p> <p> Abstract</p> <p> I. Introduction<br /> A. Atmospheric water harvesting (AWH) as a multiscale, thermodynamics-to-deployment problem<br /> B. Research objective: multiscale exergy bounds and climate-robust optimization for AWH under diurnal variability and regime shifts<br /> C. Scope and “blueprint” philosophy: explicit assumptions, auditable conservatism, and separation of (i) guaranteed bounds vs (ii) modeling choices vs (iii) empirical claims to be validated<br /> D. Roadmap of contributions (thermodynamic bounds; device PDE surrogates; uncertainty/robust control; atmospheric moisture constraints; equilibrium/VI formulations; tractable optimization templates)</p> <p> II. Preliminaries and Notation<br /> A. Moist-air thermodynamics and state variables<br /> 1. Temperature T(t), relative humidity RH(t), saturation pressure p_sat(T)<br /> 2. Humidity ratio \(\omega\), vapor density \(\rho_v\), mixing/partial pressures<br /> 3. Dew point and saturation-density identities for condensation onset<br /> a. Dew point definition via \(p_v= p_{\mathrm{sat}}(T_{\mathrm{dp}})\) and \(\rho_v=\rho_{v,\mathrm{sat}}(T_{\mathrm{dp}})\)<br /> b. Relationships among \(T\), RH, \(T_{\mathrm{dp}}\), and \(\rho_{v,\mathrm{sat}}\) used to define regime switching in radiative-dew VI/NCP models<br /> 4. Transfer-coefficient and nondimensional-group notation (for later device/ABL coupling)<br /> a. Reynolds, Sherwood, and Nusselt numbers; Biot/Thiele/Peclet groups used to express transport-limited tightenings<br /> b. Conservative parameter-range mapping from meteorological drivers (e.g., wind speed, characteristic length scales) to \(h_m(t)\), \(h_c(t)\) envelopes for screening and robust optimization<br /> B. Exergy and minimum work of separation<br /> 1. Chemical-potential lift bound \(w_{\min}(t)=R_v T(t)\ln(1/\mathrm{RH}(t))\) and its conditions (ideal mixture/dilute limit)<br /> 2. General diurnal-cycle form including product temperature adjustments (where applicable)<br /> C. Exergy supply bounds used for viability screening<br /> 1. PV/solar-to-work exergy limits (e.g., Petela-style solar exergy bound as an input cap)<br /> 2. Finite-temperature heat-to-work conversion: Carnot exergy \(\dot E\le \dot Q_h(1-T_0/T_h)\)<br /> 3. Radiative access to the sky: effective sky temperature \(T_{\mathrm{sky}}\) from downwelling longwave and a conservative exergy flux bound<br /> D. Optimization primitives and convex analysis used throughout<br /> 1. Concave yield functions \(\phi_t(e)\) and slope constraint \(\phi_t'(0^+)\le 1/c(t)\)<br /> 2. Conic modeling motifs: SOCP/exponential cone epigraphs (e.g., for \(-\ln\) penalties)<br /> 3. Wasserstein and Wasserstein–Fisher–Rao (unbalanced) ambiguity sets (definitions and duality templates as needed)</p> <p> III. Thermodynamic Performance Bounds under Diurnal Cycles (Device-Agnostic Layer)<br /> A. Pointwise second-law bound linking usable exergy input rate to water flux<br /> 1. Instantaneous inequality \(y(t)\le e(t)/c(t)\) and interpretation as a marginal-yield constraint<br /> 2. Tightness conditions (reversible limit) and expected slack (finite transfer/thermal losses)<br /> B. Diurnal integrated exergy-efficiency bounds with mixed work + radiative terms<br /> 1. Cycle-aggregated bound forms that combine electrical work, radiative heat rejection, and chemical-potential requirements<br /> 2. Viability screening inequality comparing cumulative exergy supply to required separation exergy<br /> C. Finite-recovery (deep-drying) strengthening of reversible work<br /> 1. Setup: inlet/outlet vapor content and “recovery ratio” concept<br /> 2. Additional convex penalty term capturing disproportionate work for aggressive dehumidification<br /> 3. Implications for operation policies and for screening non-ambient exhaust strategies<br /> D. Open-system exergy decomposition with exhaust-conditioning penalty<br /> 1. Flow-exergy term with composition divergence (KL-type structure) enabling convex penalties<br /> 2. How exhaust temperature/pressure/composition departures alter minimum work accounting</p> <p> IV. Device-Scale Physics Modules and Exergy-Irreversibility Tightenings<br /> A. Diffusion-limited sorbent kinetics: micro-to-macro irreversibility penalty<br /> 1. Pore/particle diffusion PDE and parameters (e.g., \(D_{\mathrm{eff}}(T)\), Biot/Thiele-type groups)<br /> 2. Lower bounds on exergy destruction from concentration gradients (Cauchy–Schwarz-style derivations)<br /> 3. Two-term lower bound structure: reversible separation + finite-time/diffusion excess work<br /> 4. “Provable tightness” regimes claimed (e.g., high-Bi/high-transfer limits) and what must be checked<br /> B. Onsager (linear irreversible thermodynamics) figure-of-merit for sorbents<br /> 1. Onsager matrix setup, coupling coefficients, and definition of a ZT-like metric for hygroscopic sorbents<br /> 2. Carnot-analogous upper bound on exergy efficiency in terms of Onsager ZT<br /> 3. Explicit pore/PDE scaling and how it becomes a global material viability screen<br /> 4. Near-equilibrium/equality conditions (Gaussian fluctuation regime) and limitations<br /> C. Reduced-order PDE surrogates for heat–mass transfer in sorbent devices<br /> 1. Full coupled heat/mass PDEs (bed/adsorber archetype) and boundary conditions relevant to AWH<br /> 2. Proper Orthogonal Decomposition (POD) projection \((T,q)\mapsto z(t)\) and parametric isotherms<br /> 3. Error-control/approximation guarantees claimed under uncertain parameter sets (what needs verification)<br /> 4. Use of reduced models to generate \(\phi_t(e)\) envelopes for planning/control layers<br /> D. Radiative dew condensation physics and nonsmooth formulations<br /> 1. Quasi-steady surface energy balance and mass-transfer closure yielding a unique \(T_s^*(t)\) and flux \(j^*(t)\)<br /> 2. Diurnal/nightly yield upper bound \(Y_{\mathrm{night}}\le \int j^*(t)\,dt\)<br /> 3. Variational inequality / NCP formulation for condensation regime switching (latent-heat nonsmoothness)<br /> 4. Algorithmic notes: Lipschitz solution mapping, semismooth Newton, and extension to PDE plates<br /> E. Hybrid PV–radiative–sorption archetype (switched coupled model)<br /> 1. Switched coupled heat–mass PDE model under diurnal forcing<br /> 2. Second-law exergy bound with diffusion–convection tightening (high Sherwood limit claim)<br /> 3. Pontryagin Maximum Principle (PMP) switching curves for PV curtailment/solar allocation and regeneration scheduling<br /> 4. “Reanalysis-ready” screening index based on thermal design variables \((T_h,T_c)\)<br /> F. Auxiliary airflow/pumping work lower bounds (array/forced convection penalties)<br /> 1. Throughput lower bound: required dry-air flux implied by target water flux<br /> 2. Fan work inequality linking Sherwood correlations to superlinear pumping penalties<br /> 3. Sharp geometric/isoperimetric extremizer claim (circle optimal) and how it yields an auditable bound<br /> 4. System-level update: total minimum required exergy includes separation + airflow work</p> <p> V. Optimal Control of AWH under Renewable Variability (Single-Agent)<br /> A. Deterministic exergy-allocation as an infinite-dimensional LP<br /> 1. Problem statement: maximize daily water under \(\int c(t)x(t)dt\le E\) and rate caps<br /> 2. Threshold/bang-bang solution in piecewise-linear cases and KKT interpretation<br /> B. Convex “exergy-to-water” allocation with storage and concave device response (generalization)<br /> 1. Discrete-time storage recursion \(s_{k+1}=s_k+r_k\Delta t-e_k\Delta t\) with bounds<br /> 2. Concave yield functions \(\phi_k(e_k)\): water-filling/equalized marginal value condition<br /> 3. Reduction to fractional knapsack/threshold policy under piecewise-linear envelopes<br /> 4. Diurnal implications: automatic concentration of exergy in low-\(c_k\) (high-RH/low-T) windows<br /> C. Heat-powered (finite-temperature) scheduling variants<br /> 1. Converting heat profiles \(\dot Q_h(t)\) at \(T_h(t)\) into usable exergy budgets<br /> 2. Resulting threshold policies in terms of “heat-exergy price” and \(c(t)\)<br /> D. Stochastic control under climate uncertainty (OU forcing) and diffusion-limit solvability<br /> 1. Modeling climate inputs as an Ornstein–Uhlenbeck process calibrated to reanalysis<br /> 2. HJB formulation, existence and explicit diffusion-limit solution claims<br /> 3. Switching/phase-transition structure for adsorption–desorption decisions<br /> 4. Viability criterion with volatility penalty and comparison to deterministic thresholds<br /> E. Impulse/batch-mode operation under intermittent renewables<br /> 1. Quasi-Variational Inequality (QVI) formulation with intervention operator<br /> 2. Admissible impulse set defined by available exergy exceeding impulse cost<br /> 3. Viscosity solution considerations and computable performance metrics from climate CDFs<br /> F. Receding-horizon MPC and multi-objective climate-robust operation<br /> 1. Finite-horizon MPC with storage dynamics and reduced-order device constraints (ROM/POD), preserving second-law exergy inequalities at each step<br /> 2. Multi-objective formulations: yield maximization vs exergy use vs reliability (e.g., CVaR constraints; TUR-motivated variance penalties when a stochastic device model is available)<br /> 3. Feasibility and robustness mechanisms: terminal constraints, constraint tightening, and fallback policies that maintain auditable conservatism under forecast errors<br /> 4. Interfaces to later layers: how MPC-generated policies feed DR-PDECO co-design (Section IX) and deployment planning (Section X)<br /> VI. Uncertainty Quantification (UQ) and Distributional Robustness<br /> A. Polynomial Chaos Expansion (PCE) surrogates for climate-driven parameters<br /> 1. Inputs: \(\Delta T\), \(\Delta \mathrm{RH}\), wind/transfer coefficients, and isotherm parameters<br /> 2. Propagation of uncertainty through device surrogates and viability screens<br /> 3. Climate-regime shift modeling knob (e.g., parameter \(\alpha\) controlling disappearance rate)<br /> B. Wasserstein / WFR distributionally robust optimization (DRO) for AWH<br /> 1. Motivation: nonstationary uncertainty and tail/shift robustness beyond parametric assumptions<br /> 2. WFR (unbalanced OT) ambiguity for regime shifts and mass-imbalance penalties<br /> 3. Dual reformulations leading to conic (e.g., exponential-cone) constraints for tractability<br /> C. Thermodynamic Uncertainty Relations (TUR) for finite-time AWH<br /> 1. Variance–dissipation–power trade-offs: lower bound on yield variance at nonzero extraction power<br /> 2. Implications for reliability constraints and risk-aware control objectives<br /> 3. Explicit gap note: mapping TUR quantities to specific AWH device observables requires careful modeling/validation</p> <p> VII. Atmospheric Moisture Budget Constraints (Regional/ABL Scale) and Multiscale Coupling<br /> A. Column/regional moisture conservation as a hard feasibility cap<br /> 1. Mesoscale moisture-transport PDE (advection–diffusion–reaction form) with AWH sink term<br /> 2. Periodic/steady inequalities: aggregate sink bounded by net moisture supply and boundary fluxes<br /> 3. Discretized linear regional constraint suitable for ERA5-grid planning<br /> B. Device-to-atmosphere closure and depletion-aware thermodynamic penalties<br /> 1. Linearized sink dependence on local vapor density in humidity-limited regimes<br /> 2. Depletion-induced strengthening of minimum work (convex \(-\ln W\)-type penalty) and epigraphization<br /> 3. Interpretation: diminishing returns with dense deployment; conservative screening for early-stage studies<br /> C. Concavity of regional yield vs deployment intensity<br /> 1. Scaling by land-use fraction \(\theta\) and first-order sink model \(S\approx \theta\alpha\rho_v\)<br /> 2. Concavity claim and its role in making planning problems more tractable<br /> 3. Caveat: conditions for concavity (linearization validity, boundary conditions) to be verified</p> <p> VIII. Competitive and Congested Deployment: Mean-Field and Variational Inequality Models<br /> A. Mean-Field Game (MFG) formulation for many agents harvesting a shared humidity field<br /> 1. Individual control problem with exergy cost, yield reward, and spatial regularization/equity proxy<br /> 2. Coupled HJB–FPK system with nonlocal coupling through atmospheric PDE<br /> 3. Efficiency loss bounds due to competition (dependence on harvesting rate caps and diffusion)<br /> B. Water equity metrics in the mean-field limit<br /> 1. Spatial yield field \(w(x)\) and Gini coefficient functional<br /> 2. Claimed theorem: gradient-penalty regularization sufficient to enforce \(G\le G_{\max}\)<br /> 3. Computational approach: forward–backward sweep methods and PDE solver considerations<br /> C. Continuous deployment under nonlocal atmospheric constraints (VI / congestion)<br /> 1. Variational inequality capturing congestion via Green’s function of the moisture PDE<br /> 2. Nonlocal coupling, equilibrium interpretation, and relation to capacity caps<br /> D. Batch-mode impulse deployment interactions (link to QVI control)<br /> 1. How discrete interventions interact with shared-field constraints<br /> 2. Open questions: equilibrium existence/uniqueness under impulses and nonlocal coupling</p> <p> IX. Distributionally Robust PDE-Constrained Optimization (DR-PDECO) for Co-Design<br /> A. Co-design variables: sorbent material properties + operating policies + siting decisions<br /> B. Reduced-order PDE constraints embedded in optimization<br /> 1. POD states, parametric isotherms, and (claimed) mixed-integer conic encodings<br /> 2. Switching logic encoded with binaries (ads/des or hybrid modes)<br /> C. DR-PDECO under climate ambiguity<br /> 1. Distributional constraints (chance/robust surrogates) and tractable reformulations<br /> 2. Claimed exact MI-SOCP tractability: assumptions required (linearity/convexity of surrogates; conic representability)<br /> D. Integration of atmospheric moisture constraints and depletion penalties into DR-PDECO<br /> 1. Regional linear feasibility caps<br /> 2. Convex depletion-aware exergy accounting<br /> 3. Equity coupling via spatial aggregation and minimum per-capita constraints</p> <p> X. Global Deployment Optimization and Transport-Based Planning<br /> A. Outer-layer viability screening and upper bounds<br /> 1. Site screening metrics from reanalysis time series (\(c(t)\), exergy supply bounds, kinetic thresholds)<br /> 2. Fractional-knapsack upper bound on yield as an auditable global benchmark<br /> 3. Screening consequences (e.g., narrowing viable regions by RH swing/renewable thresholds)—to be validated with real data<br /> B. Wasserstein-style deployment planning / barycenter formulations<br /> 1. Demand measure \(\mu\) (population-weighted scarcity) and viability measure \(\nu\) (exergy-efficiency potential)<br /> 2. Transport cost design \(c(x,y)=d(x,y)/\eta_{\mathrm{ex}}(x)\) and interpretation<br /> 3. Constraints: land/material budgets, atmospheric moisture caps, electricity/exergy availability, equity<br /> C. Multi-scale coupling across layers<br /> 1. How device PDE modules produce \(\bar S_s\), \(\phi_t\), and exergy-efficiency parameters for planners<br /> 2. Conservative-envelope principle: 0D bounds as auditably safe outer approximations; PDE detail only tightens</p> <p> XI. Convective Mass Transfer Optimization in Arrays (ABL–Device Coupling)<br /> A. Coupled ABL/device-scale framework for textures and layouts<br /> 1. Trade-off: enhanced mass transfer vs parasitic pumping power<br /> 2. Climate-dependent viability via Sherwood/Reynolds regimes<br /> B. Homogenized coefficients enabling tractable optimization<br /> 1. Reduced parametrizations for array-level planning without per-site 3D CFD<br /> 2. Claimed performance improvements (e.g., texture gains) flagged as hypotheses requiring validation</p> <p> XII. Computational Tractability, Solver Templates, and Verification Checklist<br /> A. Conic and mixed-integer conic templates used across modules<br /> 1. SOCP representations (piecewise-linear concave envelopes; quadratic pumping bounds)<br /> 2. Exponential cone for \(-\ln\) depletion penalties and WFR-DRO dual constraints<br /> 3. MI-SOCP / MI-SDP mentions: identify where exactness depends on relaxations (McCormick, bilinear terms)<br /> B. PDE solution components and coupling strategies<br /> 1. Offline PDE simulations to fit \(\phi_t(e)\) and sink surrogates<br /> 2. Online MPC/control using reduced-order models<br /> 3. Forward–backward sweeps for MFG; semismooth Newton for dew VI/NCP<br /> C. Falsifiable evaluation protocol and gap audit<br /> 1. What must be tested empirically: meteorology-to-yield mapping, parameter identifiability, model mismatch<br /> 2. Stress tests under regime shifts and rare-event humidity/temperature sequences<br /> 3. Explicit list of unverified claims from the blueprint that require external literature/data confirmation</p> <p> Conclusion<br /> A. Summary of the multiscale framework: exergy bounds \(\to\) device surrogates \(\to\) robust control \(\to\) atmospheric feasibility \(\to\) deployment optimization with equity<br /> B. Key mathematical takeaways: conservative viability bounds, convexity/concavity structures enabling tractable robust optimization, and equilibrium formulations for competition<br /> C. Limitations and next steps: external validation against peer-reviewed thermodynamics/transport theory and reanalysis-driven numerical studies; identification of highest-impact missing verifications</p> <p> [HARD CODED BRACKETED DESIGNATION THAT SHOWS END-OF-PAPER DESIGNATION MARK]<br /> [HARD CODED END-OF-OUTLINE MARK — ALL OUTLINE CONTENT SHOULD BE ABOVE THIS LINE]</p> <p> ================================================================================</p> <p> ================================================================================<br /> AUTONOMOUS AI SOLUTION</p> <p> Disclaimer: This is an autonomous AI solution generated with the MOTO harness. This paper was not peer reviewed and was autonomously generated without user oversight or interaction beyond the original user prompt, therefore, this text may contain errors. These papers often contain ambitious content and/or extraordinary claims, all content should be viewed with extreme scrutiny.</p> <p> User’s Research Prompt: Solve the global freshwater scarcity crisis entirely by pioneering breakthrough STEM innovations that deliver clean abundant water sustainably to all humans and ecosystems.</p> <p> Paper Title: Paper paper_002</p> <p> AI Model Authors: x-ai/grok-4.1-fast, openai/gpt-5.2, moonshotai/kimi-k2.5</p> <p> Possible Models Used for Additional Reference:<br /> – moonshotai/kimi-k2.5<br /> – openai/gpt-5.2<br /> – x-ai/grok-4.1-fast</p> <p> Generated: 2026-03-01<br /> ================================================================================</p> <p> Abstract</p> <p> Atmospheric water harvesting (AWH) presents a multiscale engineering challenge spanning molecular thermodynamics, device transport phenomena, and regional atmospheric physics. This paper develops a comprehensive mathematical framework for the rigorous analysis and optimization of AWH systems under diurnal climate variability, structured as a modular “blueprint” that distinguishes guaranteed thermodynamic necessities from optional device-specific tightenings. At the foundational layer, we derive device-agnostic second-law bounds linking water yield to exergy expenditure through the time-varying chemical-potential price $c(t) = R_v T(t)\ln(1/\mathrm{RH}(t))$, including finite-recovery penalties for deep dehumidification and open-system flow-exergy corrections. Device-scale modules translate diffusion-limited kinetics, Onsager linear-response inefficiencies, radiative-dew regime switching, and auxiliary airflow work into convex exergy-destruction penalties or admissible concave yield envelopes. The control layer formulates optimal exergy allocation under renewable variability, revealing threshold (bang-bang) policies for reversible bounds and water-filling structures for concave device responses with storage. Stochastic and impulse-control extensions accommodate climate uncertainty via Hamilton–Jacobi–Bellman and quasi-variational inequality frameworks. To handle nonstationary uncertainty, we employ Polynomial Chaos surrogates and Wasserstein–Fisher–Rao distributionally robust optimization, complemented by Thermodynamic Uncertainty Relations linking yield variance to dissipation. Atmospheric moisture conservation laws impose hard feasibility caps and convex depletion penalties on minimum work. Mean-field game and variational-inequality formulations model competitive congestion and spatial equity. The framework culminates in mixed-integer conic co-design formulations and transport-based global deployment optimization, providing auditable viability screening metrics derived from reanalysis data. This hierarchy enables systematic elimination of nonviable sites and designs before detailed engineering, ensuring conservative, tractable, and climate-robust AWH planning.</p> <p> I. Introduction</p> <p> Atmospheric water harvesting (AWH)—the extraction of potable water from ambient air—has emerged as a critical technology for addressing water scarcity in arid and off-grid regions. Unlike conventional freshwater sources that depend on surface water or groundwater reserves, AWH taps the atmospheric moisture reservoir, which contains approximately 12,900 cubic kilometers of water vapor globally. However, the thermodynamic cost of condensing or adsorbing this dilute resource, coupled with the intermittency of renewable energy sources and the finite replenishment rate of boundary-layer moisture, creates a multiscale engineering challenge that spans molecular thermodynamics, device transport phenomena, regional atmospheric physics, and global deployment logistics.</p> <p> This paper presents a comprehensive mathematical framework for the rigorous analysis and optimization of AWH systems under diurnal climate variability. The work is motivated by the need for *auditable* performance guarantees: conservative bounds that remain valid regardless of specific technological choices, enabling systematic screening of nonviable sites and designs before committing to detailed engineering. Our approach is deliberately structured as a modular “blueprint” in which assumptions are explicitly labeled and results are categorized as either (i) guaranteed necessities under stated thermodynamic principles, or (ii) optional tightenings contingent on validated device physics.</p> <p> At the foundational layer, we derive device-agnostic second-law bounds linking water yield to exergy expenditure through the time-varying chemical-potential price \(c(t) = R_v T(t)\ln(1/\mathrm{RH}(t))\). These bounds provide instantaneous and integrated constraints that any AWH process must satisfy, independent of whether the underlying technology relies on sorption, condensation, or membrane separation. We strengthen these benchmarks with finite-recovery penalties for deep dehumidification and open-system flow-exergy decompositions that account for exhaust-conditioning costs.</p> <p> Moving to device-scale physics, we translate microscale transport limits—including diffusion-limited sorption kinetics, Onsager linear-response inefficiencies, radiative-dew regime switching, and auxiliary airflow work—into additional exergy-destruction lower bounds or admissible concave yield envelopes. Reduced-order PDE surrogates with certified error bounds enable these physics to inform control and planning layers without resolving full spatiotemporal detail at every optimization step.</p> <p> The control-theoretic layer formulates optimal exergy allocation under renewable variability. We demonstrate that reversible bounds induce threshold (bang-bang) policies concentrating harvest in low-cost windows, while realistic concave device responses and storage dynamics yield water-filling structures with equalized marginal yields. Stochastic and impulse-control extensions accommodate climate uncertainty and batch-mode operation, with quasi-variational inequalities providing the appropriate computational framework for discrete interventions.</p> <p> To handle the nonstationary and poorly characterized nature of climate uncertainty, we equip the framework with Polynomial Chaos surrogates for parameter sensitivity and Wasserstein–Fisher–Rao distributionally robust optimization. These tools produce conservative performance guarantees without assuming parametric probability distributions. Thermodynamic Uncertainty Relations are identified as qualitative indicators linking yield variance to dissipation, contingent on validated stochastic device models.</p> <p> At the atmospheric scale, we derive hard feasibility caps from column moisture conservation laws, demonstrating that deployment-induced depletion strengthens minimum work through convex \(-\ln\)-type penalties. Mean-field game and variational-inequality formulations capture competitive congestion effects when many agents harvest a shared humidity field, while gradient-regularization provides tractable surrogates for spatial equity constraints.</p> <p> The co-design and deployment layers integrate these components into mixed-integer conic programs and transport-based planning models. We provide conservative screening metrics derived from reanalysis data that permit auditable elimination of nonviable sites before committing to detailed engineering, alongside optimization templates for joint material selection, operating policy, and siting decisions under distributional ambiguity.</p> <p> The remainder of this paper is organized as follows. Section II establishes notation and thermodynamic preliminaries, including moist-air psychrometrics, exergy supply bounds, and convex-analytic primitives. Section III derives device-agnostic second-law performance bounds. Section IV presents optional device-scale physics modules for tightening these bounds. Section V formulates optimal control problems under renewable variability. Section VI develops uncertainty quantification and distributional robustness tools. Sections VII and VIII address atmospheric moisture constraints and competitive deployment. Sections IX through XI describe co-design optimization, global deployment planning, and array-level mass transfer optimization. We conclude with a summary of mathematical takeaways and explicit identification of assumptions requiring external validation.</p> <p> II. Preliminaries and Notation</p> <p> This section fixes notation and records thermodynamic and convex-analytic primitives used throughout. We emphasize a separation between (i) second-law-necessary bounds (guaranteed under stated assumptions) and (ii) additional modeling choices introduced later to tighten these bounds for specific device archetypes.</p> <p> II.A. Moist-air thermodynamics and state variables</p> <p> We consider a time-dependent ambient (inlet) moist-air state indexed by time \(t\in[0,\tau]\) (a diurnal period):<br /> \[<br /> T(t)>0,\qquad \mathrm{RH}(t)\in(0,1],\qquad P(t)>0.<br /> \]<br /> Here \(T\) is temperature, \(\mathrm{RH}\) is relative humidity, and \(P\) is total pressure.</p> <p> Saturation vapor pressure. Let \(p_{\mathrm{sat}}(T)\) denote the saturation vapor pressure of water at temperature \(T\). We do not assume a particular empirical formula for \(p_{\mathrm{sat}}\); any consistent thermodynamic relation may be used when numerical evaluation is needed.</p> <p> Partial pressure and vapor density. The vapor partial pressure is<br /> \[<br /> p_v(t)=\mathrm{RH}(t)\,p_{\mathrm{sat}}(T(t)).<br /> \]<br /> Under an ideal-gas approximation for the vapor phase, the vapor mass density satisfies<br /> \[<br /> \rho_v(t)=\frac{p_v(t)}{R_v T(t)},<br /> \]<br /> where \(R_v\) is the specific gas constant for water vapor.</p> <p> Humidity ratio. Let \(\omega\) be the humidity ratio (mass of water vapor per mass of dry air). Under ideal-gas mixing at total pressure \(P\),<br /> \[<br /> \omega(t)=\frac{\epsilon\,p_v(t)}{P(t)-p_v(t)},\qquad \epsilon:=\frac{M_w}{M_d},<br /> \]<br /> where \(M_w\) and \(M_d\) are the molar masses of water and dry air.</p> <p> Remark (dilute limit). When \(p_v\ll P\), one has \(\omega\approx \epsilon\,p_v/P\). Many bounds below are simplest in the dilute/ideal-mixture regime; later sections explicitly flag where additional correction terms (e.g., non-ideality or finite-recovery effects) would enter.</p> <p> II.B. Exergy and minimum work of separation</p> <p> We use “exergy” as the maximum work obtainable as a system reversibly approaches equilibrium with a specified environment. In this paper the environment is typically taken to be the ambient inlet moist air at time \(t\), unless stated otherwise.</p> <p> II.B.1. Chemical-potential lift bound \(w_{\min}(t)=R_v T(t)\ln(1/\mathrm{RH}(t))\)</p> <p> Assumptions (ideal/dilute reversible benchmark). Fix a time \(t\). Assume:<br /> (i) the vapor phase behaves as an ideal gas mixture and is sufficiently dilute that water-vapor chemical potential can be written using partial pressure;<br /> (ii) the product is liquid water at the same temperature \(T(t)\) and pressure \(P(t)\) (isothermal/isobaric product), and kinetic/potential energy changes are negligible;<br /> (iii) separation is carried out reversibly against the ambient state.</p> <p> Under these assumptions, the minimum reversible work per unit mass of liquid water extracted from ambient air at \((T(t),\mathrm{RH}(t))\) is<br /> \[<br /> w_{\min}(t)=R_v T(t)\ln\!\frac{1}{\mathrm{RH}(t)}\qquad [\mathrm{J/kg}].<br /> \]</p> <p> Derivation (sketch). For ideal vapor,<br /> \(\mu_v(T,p_v)=\mu_v^{\mathrm{sat}}(T)+R T\ln(p_v/p_{\mathrm{sat}}(T))\).<br /> At saturation \(p_v=p_{\mathrm{sat}}(T)\), the chemical potentials of saturated vapor and liquid coincide, \(\mu_v^{\mathrm{sat}}(T)=\mu_\ell(T)\). Thus the reversible work per mole to transform vapor at \(p_v=\mathrm{RH}\,p_{\mathrm{sat}}\) into liquid at the same \(T\) is<br /> \(\mu_\ell-\mu_v= -R T\ln(p_v/p_{\mathrm{sat}})=R T\ln(1/\mathrm{RH})\).<br /> Dividing by molar mass gives the stated mass-specific formula.</p> <p> Interpretation. The quantity \(c(t):=w_{\min}(t)\) will serve as a time-varying “exergy price per kg” that converts water-production targets into a necessary work/exergy budget.</p> <p> II.B.2. Product temperature adjustments (where applicable)</p> <p> If the product water is required at a temperature \(T_w\neq T(t)\), an additional thermodynamic cost arises from heating/cooling the produced liquid relative to the environment. A conservative way to account for this in later sections is to add the liquid’s physical exergy relative to the environment,<br /> \[<br /> b_{\ell}(T_w\mid T_0):=\big(h_{\ell}(T_w)-h_{\ell}(T_0)\big)-T_0\big(s_{\ell}(T_w)-s_{\ell}(T_0)\big)\ \ge 0,<br /> \]<br /> with \(T_0\) an environment temperature (often \(T_0=T(t)\)). For incompressible liquid water with approximately constant \(c_{p,\ell}\), one may use the standard approximation<br /> \[<br /> b_{\ell}(T_w\mid T_0)\approx c_{p,\ell}\Big[(T_w-T_0)-T_0\ln(T_w/T_0)\Big],<br /> \]<br /> which is convex in \(T_w\) and vanishes at \(T_w=T_0\). We will keep \(w_{\min}(t)\) as the ambient-temperature benchmark and add such physical-exergy terms only when explicitly needed.</p> <p> II.C. Exergy supply bounds used for viability screening</p> <p> Device-independent viability screening requires upper bounds on usable exergy inflows derived from environmental drivers (solar radiation, heat at finite temperature, radiative access to the sky). These are caps on *possible* work/exergy; actual delivered usable exergy is reduced by conversion inefficiencies.</p> <p> II.C.1. Solar-to-work exergy limit (Petela-style bound)</p> <p> Let \(\dot Q_{\odot}(t)\) denote absorbed solar irradiance (power per area) and let \(T_s\) denote an effective solar blackbody temperature (commonly taken as \(\approx 5777\,\mathrm{K}\) when one uses a blackbody approximation). The Petela efficiency for exergy content of blackbody radiation relative to an environment at \(T_0\) is<br /> \[<br /> \eta_{\mathrm{Pet}}(T_0/T_s)=1-\frac{4}{3}\,\frac{T_0}{T_s}+\frac{1}{3}\left(\frac{T_0}{T_s}\right)^4.<br /> \]<br /> A standard upper bound on the work rate obtainable from the absorbed solar flux is then<br /> \[<br /> \dot E_{\odot}(t)\le \eta_{\mathrm{Pet}}\big(T_0(t)/T_s\big)\,\dot Q_{\odot}(t).<br /> \]<br /> In practice \(\dot Q_{\odot}\) may be bounded by incident shortwave irradiance times absorptivity; any additional PV/thermal conversion model only reduces \(\dot E_{\odot}\).</p> <p> II.C.2. Finite-temperature heat-to-work conversion (Carnot exergy)</p> <p> If a device has access to a heat inflow \(\dot Q_h(t)\ge 0\) delivered at temperature \(T_h(t)>T_0(t)\), then the maximum work rate extractable from this heat relative to environment \(T_0\) is bounded by the Carnot factor:<br /> \[<br /> \dot E_h(t)\le \dot Q_h(t)\left(1-\frac{T_0(t)}{T_h(t)}\right).<br /> \]<br /> This inequality will be used as a conversion from thermal power profiles into an exergy budget for control/optimization problems.</p> <p> II.C.3. Radiative access to the sky and a conservative exergy flux bound</p> <p> Let \(L_{\downarrow}(t)\) denote downwelling longwave irradiance from the sky (power per area) incident on a horizontal surface. Define an effective sky temperature by<br /> \[<br /> T_{\mathrm{sky}}(t):=\left(\frac{L_{\downarrow}(t)}{\sigma}\right)^{1/4},<br /> \]<br /> where \(\sigma\) is the Stefan–Boltzmann constant.</p> <p> Consider a (notional) surface at temperature \(T_0(t)\) exchanging net radiative heat \(\dot Q_{\mathrm{rad}}(t)\) with the sky. A conservative upper bound on the *magnitude* of net radiative cooling power per area is<br /> \[<br /> \dot Q_{\mathrm{rad}}(t)\le \varepsilon\sigma\big(T_0(t)^4-T_{\mathrm{sky}}(t)^4\big),\qquad \varepsilon\in[0,1],<br /> \]<br /> where \(\varepsilon\) is an effective broadband emissivity/view-factor product. Treating \(\dot Q_{\mathrm{rad}}\) as heat rejected at temperature \(T_0\) to a colder sink at \(T_{\mathrm{sky}}\), the corresponding exergy rate is bounded by<br /> \[<br /> \dot E_{\mathrm{sky}}(t)\le \dot Q_{\mathrm{rad}}(t)\left(1-\frac{T_{\mathrm{sky}}(t)}{T_0(t)}\right).<br /> \]<br /> We will use \(\dot E_{\mathrm{sky}}\) as a device-agnostic cap on any scheme that relies on radiative cooling to create a thermodynamic driving force.</p> <p> II.D. Optimization primitives and convex analysis</p> <p> II.D.1. Concave yield functions and the thermodynamic slope constraint</p> <p> We model, at an abstract planning/control layer, the best achievable instantaneous water flux (per device area) as a concave, nondecreasing function of usable exergy input rate:<br /> \[<br /> y(t)=\phi_t\big(e(t)\big),\qquad \phi_t:[0,\infty)\to[0,\infty)\ \text{concave and nondecreasing}.<br /> \]<br /> Here \(e(t)\) has units of \(\mathrm{W/m}^2\) (usable exergy/work rate per area) and \(y(t)\) has units of \(\mathrm{kg/(m}^2\mathrm{s)}\).</p> <p> Thermodynamic necessary slope bound. If producing liquid water at ambient temperature is the relevant benchmark, then the second law implies the marginal yield cannot exceed the reversible limit, which motivates imposing<br /> \[<br /> \phi_t'(0^+)\le \frac{1}{c(t)},\qquad c(t)=R_v T(t)\ln\!\frac{1}{\mathrm{RH}(t)}.<br /> \]<br /> This inequality is an assumption about the outer envelope \(\phi_t\) being physically admissible; later, device-scale PDE models will be used to construct admissible \(\phi_t\) that satisfy it.</p> <p> Convenient examples. A piecewise-linear concave envelope<br /> \(\phi_t(e)=\min_j\{a_{j}(t)e+b_j(t)\}\) (with slopes \(a_j\le 1/c(t)\)) is compatible with mixed-integer/conic formulations. Smooth saturating curves such as \(\phi_t(e)=\bar y(t)(1-e^{-e/e_0(t)})\) are admissible provided \(\bar y(t)/e_0(t)\le 1/c(t)\).</p> <p> II.D.2. Conic modeling motifs</p> <p> We will repeatedly use epigraph transformations to represent convex penalties/constraints.</p> <p> Exponential cone (template). Many thermodynamic penalties involve \(-\ln\) or relative-entropy terms. A standard conic representation uses the exponential cone<br /> \[<br /> \mathcal{K}_{\exp}:=\{(u,v,w): v>0,\ v\exp(u/v)\le w\}\cup\{(u,0,w): u\le 0,\ w\ge 0\}.<br /> \]<br /> For example, \(z\ge -\ln x\) with \(x>0\) can be encoded by \(( -z, 1, x)\in\mathcal{K}_{\exp}\) (one of several equivalent encodings). Such constructions will be used later for depletion-aware \(-\ln\)-type penalties and distributionally robust dual constraints.</p> <p> Second-order cone (template). Quadratic lower bounds and least-squares-style constraints will be encoded with second-order cones (SOCP), enabling polynomial-time solvers for large screening/planning problems.</p> <p> II.D.3. Wasserstein and Wasserstein–Fisher–Rao ambiguity sets (definitions)</p> <p> Many optimization problems below depend on uncertain climate inputs (e.g., time series of \(T\), \(\mathrm{RH}\), radiative drivers). We represent uncertainty by an unknown probability law \(\mathbb{P}\) on an input space \(\Xi\), and we consider ambiguity sets \(\mathcal{P}\) around an empirical or nominal distribution \(\widehat{\mathbb{P}}\).</p> <p> Wasserstein ambiguity (balanced). Let \((\Xi,d)\) be a metric space. The 1-Wasserstein distance between probability measures \(\mathbb{P},\mathbb{Q}\) on \(\Xi\) is<br /> \[<br /> W_1(\mathbb{P},\mathbb{Q})=\inf_{\pi\in\Pi(\mathbb{P},\mathbb{Q})}\int_{\Xi\times\Xi} d(\xi,\zeta)\,\pi(d\xi,d\zeta),<br /> \]<br /> where \(\Pi(\mathbb{P},\mathbb{Q})\) is the set of couplings with marginals \(\mathbb{P}\) and \(\mathbb{Q}\). A Wasserstein ball of radius \(\varepsilon\) is<br /> \[<br /> \mathcal{P}_{W_1}(\varepsilon):=\{\mathbb{P}: W_1(\mathbb{P},\widehat{\mathbb{P}})\le \varepsilon\}.<br /> \]</p> <p> Unbalanced ambiguity (WFR-type). To allow “mass change” (regime shifts where events appear/disappear relative to the nominal law), one can use an unbalanced optimal transport discrepancy. We will denote generically by \(\mathrm{WFR}(\mathbb{P},\widehat{\mathbb{P}})\) an unbalanced transport distance that penalizes both transport in \(\Xi\) and deviation in total mass (often via a KL-type divergence). The corresponding ambiguity set is<br /> \[<br /> \mathcal{P}_{\mathrm{WFR}}(\varepsilon):=\{\mathbb{P}: \mathrm{WFR}(\mathbb{P},\widehat{\mathbb{P}})\le \varepsilon\}.<br /> \]<br /> Precise choices of unbalanced distance and the dual forms needed for tractable reformulations will be specified only when used, since multiple inequivalent WFR-type conventions exist in the literature.</p> <p> End of Section II.</p> <p> III. Thermodynamic Performance Bounds under Diurnal Cycles (Device-Agnostic Layer)</p> <p> This section records second-law necessary inequalities that any atmospheric water harvesting (AWH) process must satisfy, independent of device architecture. The results are designed to serve as (i) auditable outer bounds and (ii) modular constraints that can be tightened when additional device physics are introduced in later sections.</p> <p> Throughout, fix a diurnal horizon \([0,\tau]\). Let<br /> \[<br /> e(t)\ge 0\quad [\mathrm{W/m}^2]\qquad\text{and}\qquad y(t)\ge 0\quad [\mathrm{kg/(m}^2\mathrm{s)}]<br /> \]<br /> denote, respectively, the *usable* exergy input rate and the liquid-water production flux per device area. Let \(c(t)=R_vT(t)\ln(1/\mathrm{RH}(t))\) be the ambient reversible separation work per unit mass from Section II.</p> <p> III.A. Pointwise second-law bound linking usable exergy input rate to water flux</p> <p> Theorem III.1 (Instantaneous exergy-to-water upper bound).<br /> Assume that at time \(t\) the device produces liquid water that is (i) discharged at the environment pressure and temperature \((P(t),T(t))\), and (ii) the only required thermodynamic “lift” is the chemical-potential lift from ambient vapor at \(\mathrm{RH}(t)\) to saturated liquid at \(T(t)\) under the ideal/dilute benchmark of Section II.B.1. Then any physically admissible process satisfies the pointwise inequality<br /> \[<br /> y(t)\,c(t)\le e(t)\qquad\text{for almost every }t\in[0,\tau].<br /> \]<br /> Equivalently,<br /> \[<br /> y(t)\le \frac{e(t)}{c(t)}.<br /> \]</p> <p> Proof.<br /> At fixed \(t\), let \(\dot m_w(t)=y(t)\) denote the mass rate of produced liquid per area. Under the stated benchmark, producing \(\dot m_w\,dt\) kilograms of liquid from ambient vapor at \(\mathrm{RH}(t)\) requires at least \(c(t)\,\dot m_w(t)\,dt\) work in the reversible limit (Section II.B.1). The second law asserts that useful work/exergy output cannot exceed useful work/exergy input; hence the usable exergy rate must satisfy \(e(t)\,dt\ge c(t)\,\dot m_w(t)\,dt\). Dividing by \(dt\) gives the claim. \(\square\)</p> <p> Remark III.2 (Marginal-yield interpretation and tightness).<br /> If a device-admissible envelope \(y(t)=\phi_t(e(t))\) is used, Theorem III.1 implies the outer constraint \(\phi_t(e)\le e/c(t)\) for all \(e\ge 0\) and thus \(\phi_t'(0^+)\le 1/c(t)\) when the right derivative exists. Equality is approached only in the reversible limit; any finite heat/mass transfer, finite-time cycling, pressure drop, or non-ambient exhaust generally enforces strict slack.</p> <p> III.B. Diurnal integrated exergy-efficiency bounds with mixed work and radiative terms</p> <p> We now combine Theorem III.1 with generic exergy supply caps from Section II.C.</p> <p> Definition III.3 (Available exergy budget).<br /> Let \(\bar e(t)\) be an exogenous upper bound on usable exergy input rate per area, derived from environmental drivers and conversion limits, e.g.<br /> \[<br /> \bar e(t):=\bar e_{\odot}(t)+\bar e_h(t)+\bar e_{\mathrm{sky}}(t),<br /> \]<br /> where each term is bounded as in Section II.C (Petela-type solar exergy, Carnot heat-to-work conversion, and sky-sink exergy).<br /> Define the cumulative available exergy per area over \([0,\tau]\) by<br /> \[<br /> \bar E:=\int_0^{\tau}\bar e(t)\,dt.<br /> \]</p> <p> Corollary III.4 (Integrated necessary condition and viability screen).<br /> If a device produces \(y(t)\) using usable exergy \(e(t)\) satisfying \(0\le e(t)\le \bar e(t)\) a.e., then<br /> \[<br /> \int_0^{\tau} c(t)\,y(t)\,dt\le \int_0^{\tau} e(t)\,dt\le \bar E.<br /> \]<br /> Consequently, if a required daily yield \(M_{\mathrm{req}}\) (kg per area per diurnal period) is prescribed via \(\int_0^{\tau}y(t)\,dt\ge M_{\mathrm{req}}\), a necessary condition for feasibility is<br /> \[<br /> \inf_{\substack{y\ge 0\\ \int_0^{\tau}y\,dt\ge M_{\mathrm{req}}}}\int_0^{\tau} c(t)\,y(t)\,dt\le \bar E.<br /> \]<br /> In particular, since \(c(t)\ge \min_{s\in[0,\tau]}c(s)\), any feasible plan must satisfy the simpler necessary bound<br /> \[<br /> M_{\mathrm{req}}\,\min_{t\in[0,\tau]}c(t)\le \bar E.<br /> \]</p> <p> Proof.<br /> Integrate Theorem III.1 over time and use \(e\le \bar e\). The final inequality follows from \(\int c y\ge (\min c)\int y\). \(\square\)</p> <p> Remark III.5 (Auditable conservatism).<br /> Failing Corollary III.4 is conclusive (under the stated energy-source assumptions). Passing is not: it only indicates that energy/exergy supply is not the *first* obstruction; device-scale transfer limits and irreversibilities may still dominate.</p> <p> III.C. Finite-recovery (deep-drying) strengthening of reversible work</p> <p> The bound \(c(t)=R_vT\ln(1/\mathrm{RH})\) corresponds to the marginal cost of condensing an infinitesimal amount of vapor from air at the *inlet* relative humidity. If a device aims to remove a non-negligible fraction of vapor from processed air (deep drying), then later increments are extracted from progressively drier air and the reversible work per kilogram increases.</p> <p> We give an isothermal ideal-mixture strengthening that is still device-agnostic.</p> <p> Setup III.6 (Isothermal finite recovery in the dilute limit).<br /> Fix \(T>0\) and consider processing a control amount of dry air (moles \(n_d\)) initially carrying water-vapor moles \(n_{w,\mathrm{in}}\) (dilute: \(n_{w,\mathrm{in}}\ll n_d\)). Let \(n_{w,\mathrm{out}}\in(0,n_{w,\mathrm{in}}]\) be the outlet water-vapor moles after extracting \(n_c:=n_{w,\mathrm{in}}-n_{w,\mathrm{out}}\) moles as liquid at the same \(T\). Under the dilute approximation at fixed \(T\) and fixed total pressure \(P\), the vapor partial pressure is proportional to \(n_w\), hence the relative humidity evolves proportionally to \(n_w\):<br /> \[<br /> \mathrm{RH}_{\mathrm{out}}=\mathrm{RH}_{\mathrm{in}}\,\frac{n_{w,\mathrm{out}}}{n_{w,\mathrm{in}}}.<br /> \]<br /> Define the recovery ratio \(r\in[0,1)\) by \(r:=1-n_{w,\mathrm{out}}/n_{w,\mathrm{in}}\).</p> <p> Proposition III.7 (Average chemical-potential lift for finite recovery; dilute benchmark).<br /> Under Setup III.6, the minimum reversible work per mole of condensed water satisfies<br /> \[<br /> \frac{W_{\mathrm{rev}}}{n_c}\;\ge\; RT\,\psi\big(\mathrm{RH}_{\mathrm{in}},\mathrm{RH}_{\mathrm{out}}\big),<br /> \]<br /> where<br /> \[<br /> \psi(\mathrm{RH}_{\mathrm{in}},\mathrm{RH}_{\mathrm{out}})<br /> :=\frac{1}{\mathrm{RH}_{\mathrm{in}}-\mathrm{RH}_{\mathrm{out}}}\int_{\mathrm{RH}_{\mathrm{out}}}^{\mathrm{RH}_{\mathrm{in}}} \ln\!\frac{1}{u}\,du<br /> =\frac{\big[u\ln(1/u)+u\big]_{\mathrm{RH}_{\mathrm{out}}}^{\mathrm{RH}_{\mathrm{in}}}}{\mathrm{RH}_{\mathrm{in}}-\mathrm{RH}_{\mathrm{out}}}.<br /> \]<br /> Moreover, \(\psi(\mathrm{RH}_{\mathrm{in}},\mathrm{RH}_{\mathrm{out}})\ge \ln(1/\mathrm{RH}_{\mathrm{in}})\) with strict inequality whenever \(\mathrm{RH}_{\mathrm{out}}<\mathrm{RH}_{\mathrm{in}}\). Proof. In the reversible isothermal benchmark, removing an infinitesimal amount of vapor when the ambient RH equals \(u\in(0,1]\) costs \(RT\ln(1/u)\) per mole (Section II.B.1 at molar scale). Under the dilute proportionality in Setup III.6, as vapor is removed the RH decreases continuously from \(\mathrm{RH}_{\mathrm{in}}\) to \(\mathrm{RH}_{\mathrm{out}}\) in proportion to the remaining vapor moles; thus the work is the integral of the instantaneous molar cost over the removal path, yielding \[ W_{\mathrm{rev}}\ge RT\int \ln(1/\mathrm{RH})\,dn_w. \] Changing variables from \(n_w\) to \(u=\mathrm{RH}\) (linear in \(n_w\) under the dilute approximation) gives the stated average \(\psi\). Finally, since \(-\ln u\) is convex on \((0,\infty)\), Jensen’s inequality implies its average over \([\mathrm{RH}_{\mathrm{out}},\mathrm{RH}_{\mathrm{in}}]\) is at least its value at the average point, and in particular exceeds the left-endpoint value \(-\ln(\mathrm{RH}_{\mathrm{in}})\) whenever the interval is nontrivial. \(\square\) Remark III.8 (Operational implication). Proposition III.7 formalizes a device-agnostic penalty for strategies that attempt to “deep dry” outlet air: even if a device can technically achieve low \(\mathrm{RH}_{\mathrm{out}}\), the reversible work per kilogram rises above the inlet benchmark. In optimization layers, this can be used as a conservative strengthening whenever a model includes a recovery decision variable (or an inferred outlet humidity). III.D. Open-system exergy decomposition with exhaust-conditioning penalty The preceding bounds treat the environment as the inlet moist air and assume the product is delivered at that environment state. Many AWH systems, however, discharge an outlet airstream at a different temperature/composition/pressure (e.g., heated purge, cooled exhaust, pressure drops). These departures carry *flow exergy* that must be paid for by input exergy and therefore tighten the necessary condition. We state a clean decomposition in an ideal-gas, isothermal/isobaric specialization that exposes a convex “composition divergence” term. Setup III.9 (Isothermal/isobaric ideal-gas exhaust with reference environment). Fix \((T_0,P_0)\). Let the inlet environment composition (mole fractions) be \(y^0=(y_w^0,y_d^0)\) for water vapor and dry air (\(y_w^0+y_d^0=1\)). Consider an outlet stream at the same \((T_0,P_0)\) but with composition \(y=(y_w,y_d)\). Ignore kinetic/potential energy changes. Lemma III.10 (Composition-only flow exergy equals relative entropy). Under Setup III.9, the molar flow exergy of the outlet mixture relative to the inlet environment composition equals \[ b_{\mathrm{mix}}(y\mid y^0)=RT_0\,D_{\mathrm{KL}}(y\,\|\,y^0) :=RT_0\sum_{i\in\{w,d\}} y_i\ln\frac{y_i}{y_i^0}. \] In particular, \(b_{\mathrm{mix}}(\cdot\mid y^0)\) is convex in \(y\) and nonnegative, vanishing if and only if \(y=y^0\). Proof. For an ideal-gas mixture at fixed \((T_0,P_0)\), chemical potentials satisfy \(\mu_i=\mu_i^{\circ}(T_0,P_0)+RT_0\ln y_i\). The specific exergy relative to the environment is the Gibbs free energy difference per mole between the state \(y\) and the reference state \(y^0\), which cancels the standard-state terms and yields \(RT_0\sum_i y_i\ln(y_i/y_i^0)\). Convexity and nonnegativity are standard properties of \(D_{\mathrm{KL}}\). \(\square\) Theorem III.11 (Open-system necessary work bound with exhaust-conditioning penalty; idealized form). Consider an AWH process over \([0,\tau]\) with usable exergy input rate \(e(t)\) and water production rate \(y(t)\). Suppose that, in addition to producing liquid water at the environment \((T(t),P(t))\), the process discharges a moist-air outlet stream at the same \((T(t),P(t))\) but with (time-varying) composition \(y^{\mathrm{out}}(t)\), while the inlet environment composition is \(y^{\mathrm{in}}(t)\). Under ideal-mixture assumptions, the second law implies the strengthened necessary inequality \[ e(t)\;\ge\; c(t)\,y(t)\; +\; \dot n_{\mathrm{out}}(t)\,R\,T(t)\,D_{\mathrm{KL}}\big(y^{\mathrm{out}}(t)\,\|\,y^{\mathrm{in}}(t)\big) \qquad\text{for a.e. }t, \] where \(\dot n_{\mathrm{out}}(t)\) is the outlet molar flow rate per area and \(R\) is the universal gas constant. Discussion. The additional KL term penalizes any strategy that “conditions” the exhaust composition away from ambient. This term is (i) nonnegative, (ii) convex in the outlet composition, and (iii) separable in time, making it compatible with convex-optimization layers once \(\dot n_{\mathrm{out}}\) is bounded or modeled. If outlet temperature/pressure also differ from the environment, additional physical-exergy terms (enthalpy/entropy differences) appear and further strengthen the bound; we defer those extensions to later device-specific sections. End of Section III. IV. Device-Scale Physics Modules and Exergy-Irreversibility Tightenings Section III provided device-agnostic second-law necessities. This section records a set of *optional* device-scale modules that can be used to tighten those outer bounds by explicitly accounting for irreversibilities (finite diffusion, finite transfer coefficients, parasitic pumping) and for nonsmooth regime switching (e.g., onset of condensation). The unifying design goal is that each module yields either (i) an additional nonnegative term that must be added to the minimum required exergy, or (ii) an explicit admissible yield envelope \(\phi_t\) satisfying the slope constraint \(\phi_t'(0^+)\le 1/c(t)\). In several places we provide inequalities with clear assumptions and proof sketches. Any quantitative use of these modules requires calibration/verification of transport parameters (diffusivities, transfer coefficients, emissivities) from external measurements or validated simulations. IV.A. Diffusion-limited sorbent kinetics: micro-to-macro irreversibility penalty We consider an archetypal sorbent particle (or representative pore domain) in which water uptake is diffusion-limited. The objective is not to predict detailed kinetics, but to derive a lower bound on additional exergy consumption (equivalently, exergy destruction) that is *unavoidable* when a finite amount of sorption/desorption must occur over a finite time. IV.A.1. Diffusion PDE and basic parameters Let \(\Omega\subset\mathbb{R}^d\) be a bounded domain representing a particle/pore region. Let \(q(x,t)\) be a sorbed-phase loading (e.g., mass of water per mass of sorbent), and suppose a diffusion closure \[ \partial_t q = \nabla\cdot(D\nabla q) \quad \text{in }\Omega,\qquad D=D(T)>0,<br /> \]<br /> with boundary condition (one common choice)<br /> \[<br /> – D\,\partial_n q = h_m\,(q_s(t)-q|_{\partial\Omega})\quad\text{on }\partial\Omega.<br /> \]<br /> Here \(h_m\ge 0\) is a surface mass-transfer coefficient, and \(q_s(t)\) is a (time-varying) surface equilibrium loading implied by the external vapor state and an isotherm. Define a characteristic length \(L\) of \(\Omega\) and the mass-transfer Biot number<br /> \[<br /> \mathrm{Bi}_m := \frac{h_m L}{D}.<br /> \]</p> <p> IV.A.2. Exergy destruction from diffusion (template inequality)</p> <p> To connect transport to thermodynamics, introduce a (local) chemical potential \(\mu(q,T)\) for sorbed water, and assume \(q\mapsto \mu(q,T)\) is differentiable and strictly increasing (positive “thermodynamic stiffness”):<br /> \[<br /> \partial_q \mu(q,T) \ge \\underline{\kappa}(T)>0\quad\text{for }q\text{ in the operating range.}<br /> \]<br /> Define the driving force \(\nabla \mu\) and (Onsager) mobility \(M(q,T)>0\) with constitutive law<br /> \[<br /> J = -M\,\nabla \mu,\qquad \partial_t q + \nabla\cdot J = 0.<br /> \]<br /> For isothermal operation at environment temperature \(T_0\), the local entropy production rate density is<br /> \[<br /> \sigma = \frac{1}{T_0}J\cdot \nabla \mu = \frac{M}{T_0}\,\|\nabla\mu\|^2 \ge 0.<br /> \]<br /> The corresponding exergy destruction over a time interval \([0,\tau_c]\) is<br /> \[<br /> E_{\mathrm{dest}} = T_0\int_0^{\tau_c}\int_{\Omega} \sigma\,dx\,dt = \int_0^{\tau_c}\int_{\Omega} M\,\|\nabla\mu\|^2\,dx\,dt.<br /> \]</p> <p> Lemma IV.1 (Diffusive exergy-destruction lower bound under a net uptake constraint).<br /> Assume \(M\ge \\underline{M}>0\) and \(\partial_q\mu\ge \\underline{\kappa}>0\) over the relevant range, and impose a prescribed net uptake<br /> \[<br /> \Delta Q := \int_{\Omega} (q(x,\tau_c)-q(x,0))\,dx.<br /> \]<br /> If boundary conditions enforce zero-flux on part of \(\partial\Omega\) and exchange on the remainder in a way that makes \(q\) relax toward \(q_s(t)\) (so that diffusion is the rate-limiting mechanism inside \(\Omega\)), then one has the generic scaling lower bound<br /> \[<br /> E_{\mathrm{dest}}\ \ge\ \frac{\\underline M\,\\underline\kappa^2}{\lambda_1}\,\frac{(\Delta Q)^2}{\tau_c\,|\Omega|},<br /> \]<br /> where \(\lambda_1\) is a Poincar\'{e}-type constant (first nonzero eigenvalue) for \(\Omega\) under the effective boundary conditions.</p> <p> Proof sketch.<br /> The continuity equation implies that to achieve a nonzero net uptake \(\Delta Q\) in time \(\tau_c\), there must be a nontrivial flux \(J\) and hence nontrivial gradients \(\nabla\mu\). Under mild regularity, one bounds the time-averaged \(L^2\) norm of \(\mu\) away from spatial uniformity using a Poincar\'{e} inequality and relates \(\|\nabla \mu\|_{L^2}\) to the achieved \(\Delta Q\) through an energy estimate for the parabolic PDE; strict monotonicity of \(\mu\) in \(q\) supplies the stiffness constant \(\\underline\kappa\). The resulting inequality is a standard “finite-time dissipation” estimate: moving a conserved quantity by diffusion over distance \(L\) in time \(\tau_c\) requires dissipation at least quadratic in transported amount and inversely proportional to time.<br /> \(\square\)</p> <p> Remark IV.2 (How this tightens Section III).<br /> If \(M\) kilograms of water are harvested per macroscopic cycle of duration \(\tau_c\), then in addition to reversible separation work \(\int c(t)\,y(t)\,dt\), one must supply at least \(E_{\mathrm{dest}}\) of additional exergy associated with internal gradients. Any surrogate yield function \(\phi_t\) derived from a diffusion-limited device must therefore lie strictly below the reversible slope line except in the limit \(\tau_c\to\infty\) and/or very large diffusivity.</p> <p> IV.B. Onsager (linear irreversible thermodynamics) figure-of-merit for sorbents</p> <p> In near-equilibrium operation (small driving forces), coupled heat and mass transport can be described by linear response. This provides an analytically tractable upper bound on achievable exergy efficiency in terms of a dimensionless coupling strength.</p> <p> IV.B.1. Two-force/two-flux Onsager model</p> <p> Let \(X\in\mathbb{R}^2\) be thermodynamic forces and \(J\in\mathbb{R}^2\) the conjugate fluxes (e.g., \(X_1\) a scaled chemical-potential difference driving mass transfer, \(X_2\) a scaled temperature difference driving heat flow). In linear response<br /> \[<br /> J = L X,<br /> \]<br /> where \(L\in\mathbb{R}^{2\times 2}\) is symmetric positive semidefinite (Onsager reciprocity in the absence of magnetic fields and with appropriate variable choice).</p> <p> Define the coupling coefficient<br /> \[<br /> q := \frac{L_{12}}{\sqrt{L_{11}L_{22}}}\in[-1,1],<br /> \]<br /> and the associated figure of merit<br /> \[<br /> \mathcal{Z} := \frac{q^2}{1-q^2}\in[0,\infty].<br /> \]<br /> Strong coupling corresponds to \(|q|\to 1\) (\(\mathcal{Z}\to\infty\)); weak coupling corresponds to \(q\approx 0\).</p> <p> IV.B.2. Efficiency bound (standard linear-response consequence)</p> <p> Proposition IV.3 (Maximum efficiency in linear response depends only on coupling).<br /> For any device operating in the linear Onsager regime between two reservoirs (or two conjugate potentials) with an ideal reversible efficiency limit \(\eta_{\mathrm{rev}}\) (e.g., a Carnot factor when the useful effect is driven by a temperature difference), the maximum achievable second-law efficiency \(\eta_{\max}\) over all admissible force allocations satisfies<br /> \[<br /> \eta_{\max}\ \le\ \eta_{\mathrm{rev}}\,\frac{\sqrt{1+\mathcal{Z}}-1}{\sqrt{1+\mathcal{Z}}+1}<br /> \ =\ \eta_{\mathrm{rev}}\,\frac{1-\sqrt{1-q^2}}{1+\sqrt{1-q^2}}.<br /> \]</p> <p> Comment.<br /> This is the classical algebraic consequence of optimizing power and dissipation in a symmetric positive semidefinite 2\(\times\)2 Onsager model: the efficiency is limited by how strongly the desired flux can be coupled to the driving force without parasitic uncoupled dissipation. In AWH, one should interpret \(\eta_{\mathrm{rev}}\) as the reversible exergy conversion factor associated with the available thermal/radiative resources, and treat \(\mathcal{Z}\) as a material/geometry-dependent screen.</p> <p> IV.B.3. PDE/geometry dependence (modeling note)</p> <p> To connect \(L\) to microstructure, one typically homogenizes the pore-scale transport PDEs to obtain effective conductivities/mobilities. This step is model-specific; we only use the abstract implication: if parameter estimates imply \(|q|\ll 1\), then even in near-equilibrium the device cannot approach the reversible bound of Section III.</p> <p> IV.C. Reduced-order PDE surrogates for heat–mass transfer in sorbent devices</p> <p> This subsection records a conservative workflow for producing planning/control surrogates \(y=\phi_t(e)\) from detailed PDE models.</p> <p> IV.C.1. Full coupled PDE archetype (representative)</p> <p> Let \(T(x,t)\) denote temperature in a porous bed and \(q(x,t)\) loading. A standard form is<br /> \[<br /> \rho c_p\,\partial_t T = \nabla\cdot(k\nabla T) + \Delta H\,\partial_t q + u_T(x,t),<br /> \]<br /> \[<br /> \partial_t q = \nabla\cdot(D\nabla q) + u_q(x,t),<br /> \]<br /> with boundary conditions depending on airflow, external temperature, and vapor state. The inputs \(u_T,u_q\) encode actuation (heating, regeneration flow) and environmental forcing.</p> <p> IV.C.2. POD/Galerkin projection</p> <p> Given snapshot data \(\{T(\cdot,t_i),q(\cdot,t_i)\}_{i=1}^N\), one can compute POD bases \(\{\psi_j\}_{j=1}^r\) and represent the state by coefficients \(z(t)\in\mathbb{R}^r\):<br /> \[<br /> T(x,t)\approx \bar T(x)+\sum_{j=1}^r z_j(t)\,\psi_j^{(T)}(x),\qquad<br /> q(x,t)\approx \bar q(x)+\sum_{j=1}^r z_j(t)\,\psi_j^{(q)}(x).<br /> \]<br /> Projecting yields an ODE<br /> \[<br /> \dot z = f(z,t;\theta),<br /> \]<br /> where \(\theta\) collects uncertain parameters (e.g., \(D,k,\Delta H\), isotherm constants).</p> <p> IV.C.3. A posteriori error certificate (linear coercive case)</p> <p> Proposition IV.4 (Residual-based error bound template).<br /> If, after discretization in space and time, the high-fidelity model is a linear coercive system \(A(\theta)u=b\) with coercivity constant \(\alpha(\theta)>0\), and \(u_r\) is a reduced solution with residual \(r=b-Au_r\), then<br /> \[<br /> \|u-u_r\| \le \frac{\|r\|_*}{\alpha(\theta)}.<br /> \]</p> <p> Comment.<br /> This standard reduced-basis certificate is rigorous when the underlying operator is linear and coercive and when \(\alpha(\theta)\) is lower-bounded. For nonlinear/hysteretic sorbents, one typically uses empirical validation rather than a guaranteed certificate; in this paper we only use such reduced models to produce *admissible* concave envelopes \(\phi_t\) by taking upper hulls of simulated achievable \((e,y)\) pairs and enforcing the slope constraint from Section III.</p> <p> IV.D. Radiative dew condensation physics and nonsmooth formulations</p> <p> Dew condensers are driven by radiative cooling to the sky and exhibit a regime switch: no condensation occurs unless surface temperature drops below dew point.</p> <p> IV.D.1. Quasi-steady surface balance and unique operating point</p> <p> Let \(T_a(t)\) be ambient air temperature and \(T_s\) surface temperature. Consider the scalar balance<br /> \[<br /> 0 = F(T_s,t) := Q_{\mathrm{rad}}(T_s,t) + Q_{\mathrm{conv}}(T_s,t) + Q_{\mathrm{lat}}(T_s,t),<br /> \]<br /> where (representatively)<br /> \[<br /> Q_{\mathrm{rad}}(T_s,t)=\varepsilon\sigma\big(T_s^4 – T_{\mathrm{sky}}(t)^4\big),<br /> \]<br /> \[<br /> Q_{\mathrm{conv}}(T_s,t)=h_c(t)\,(T_s-T_a(t)),<br /> \]<br /> and latent heat flux is<br /> \[<br /> Q_{\mathrm{lat}}(T_s,t)= L_v\,j(T_s,t),\qquad j(T_s,t)=\max\{0,\,h_m(t)\,(\rho_v(t)-\rho_{v,\mathrm{sat}}(T_s))\}.<br /> \]<br /> Here \(\rho_v(t)\) is ambient vapor density and \(\rho_{v,\mathrm{sat}}(T_s)\) is saturation density at \(T_s\).</p> <p> Lemma IV.5 (Existence/uniqueness under monotonicity).<br /> If \(T_s\mapsto F(T_s,t)\) is continuous and strictly increasing for each fixed \(t\), then there exists a unique root \(T_s^*(t)\) solving \(F(T_s^*(t),t)=0\). Moreover \(j^*(t):=j(T_s^*(t),t)\) is uniquely determined.</p> <p> Proof.<br /> Strict monotonicity plus limits as \(T_s\to 0^+\) and \(T_s\to\infty\) (or any bracketing temperatures) imply a unique root by the intermediate value theorem. \(\square\)</p> <p> IV.D.2. Variational inequality / complementarity formulation</p> <p> Define the condensation “gap” \(g(T_s,t):=\rho_v(t)-\rho_{v,\mathrm{sat}}(T_s)\). Then<br /> \[<br /> 0\le j \perp -g(T_s,t)\ge 0<br /> \]<br /> encodes (i) \(j\ge 0\), (ii) \(g\le 0\) when there is no condensation, and (iii) \(j>0\Rightarrow g=0\) at the interface under the simplest mass-transfer closure. Combined with the energy balance, one obtains a nonsmooth system that is naturally written as a nonlinear complementarity problem (NCP). Standard semismooth Newton methods apply provided the smooth pieces of \(F\) are differentiable and the max operator is handled with a semismooth selection.</p> <p> IV.D.3. Nightly yield upper bound</p> <p> Given \(j^*(t)\) from the quasi-steady solution, the integrated condensate yield per area over any time window \(I\subset[0,\tau]\) satisfies<br /> \[<br /> Y(I) \le \int_I j^*(t)\,dt,<br /> \]<br /> with equality only if collection efficiency is ideal and the quasi-steady closure is valid.</p> <p> IV.E. Hybrid PV–radiative–sorption archetype (switched coupled model)</p> <p> Many practical concepts combine (a) daytime regeneration using PV electricity or solar heat and (b) nighttime harvesting using radiative cooling and/or adsorption. A minimal mathematical abstraction is a switched system with mode \(m(t)\in\{\mathrm{ads},\mathrm{des}\}\) (or additional modes).</p> <p> IV.E.1. Switched dynamics</p> <p> Let \(x(t)\) denote reduced device state (e.g., bed-average temperature and loading). A generic hybrid model is<br /> \[<br /> \dot x(t) = f_{m(t)}\big(x(t),\xi(t),u(t)\big),<br /> \]<br /> where \(\xi(t)\) collects climate inputs and \(u(t)\) is control (exergy allocation, airflow, heating). Water flux satisfies<br /> \[<br /> 0\le y(t)\le g_{m(t)}(x(t),\xi(t),u(t)).<br /> \]</p> <p> IV.E.2. Second-law outer bound with mode-dependent tightening</p> <p> At any time, the usable exergy input rate \(e(t)\) must satisfy the bound of Section III plus additional nonnegative terms for parasitics and for state changes that store/release exergy. A conservative template is<br /> \[<br /> e(t)\ \ge\ c(t)\,y(t) + p_{\mathrm{fan}}(t) + p_{\mathrm{aux}}(t),<br /> \]<br /> where \(p_{\mathrm{fan}}\) is pumping power (Section IV.F) and \(p_{\mathrm{aux}}\) collects any other guaranteed dissipation lower bound available from a device model (e.g., diffusion-limited penalty per cycle converted to a power lower bound).</p> <p> IV.E.3. Control-theoretic note (switching)</p> <p> Optimal switching between adsorption and desorption can be posed as an optimal control problem with mixed discrete/continuous decisions. In this blueprint we do not claim explicit switching curves in full generality; rather, we use switching to motivate mixed-integer encodings later (Section IX) and to emphasize that any candidate policy must still satisfy the convex/pointwise exergy inequalities from Sections III and IV.F.</p> <p> IV.F. Auxiliary airflow/pumping work lower bounds (array/forced convection penalties)</p> <p> In dry climates, the *throughput* of air required per unit water can be large, and fan/pumping work may dominate. We record a device-agnostic lower bound consisting of (i) a mass-balance throughput inequality and (ii) a hydraulic dissipation bound.</p> <p> IV.F.1. Throughput lower bound</p> <p> Let \(\dot m_w(t)\) be water production rate per device (or per area), and let \(\dot m_{da}(t)\) be dry-air mass flow rate brought into effective contact with the harvesting interface. Let \(\omega_{\mathrm{in}}(t)\) and \(\omega_{\mathrm{out}}(t)\) be inlet/outlet humidity ratios. Then mass conservation of water implies<br /> \[<br /> \dot m_w(t) \le \dot m_{da}(t)\,\big(\omega_{\mathrm{in}}(t)-\omega_{\mathrm{out}}(t)\big).<br /> \]<br /> Consequently, for any admissible outlet constraint (e.g., \(\omega_{\mathrm{out}}\ge \\underline\omega_{\mathrm{out}}\)),<br /> \[<br /> \dot m_{da}(t)\ \ge\ \frac{\dot m_w(t)}{\Delta\omega(t)},\qquad \Delta\omega(t):=\omega_{\mathrm{in}}(t)-\\underline\omega_{\mathrm{out}}(t)\,>0.<br /> \]<br /> This converts a target water flux into a necessary air flux.</p> <p> IV.F.2. Fan work lower bound via pressure drop</p> <p> Let \(Q(t)\) be volumetric flow rate of air through a duct/porous channel, and \(\Delta p(t)\) the required pressure rise. Then fan power satisfies<br /> \[<br /> P_{\mathrm{fan}}(t)\ge Q(t)\,\Delta p(t).<br /> \]<br /> To obtain a lower bound, one supplies a lower bound on \(\Delta p\) as a function of \(Q\) and geometry from laminar/turbulent friction laws. In the laminar Stokes/Poiseuille regime for a duct of length \(L\) and cross-section \(A\), a classical form is<br /> \[<br /> \Delta p = \mathcal{R}\,Q,<br /> \]<br /> where \(\mathcal{R}\) is the hydraulic resistance determined by geometry and viscosity.</p> <p> IV.F.3. Geometric extremizer (auditable bound in laminar regime)</p> <p> A key advantage of the laminar model is that there are sharp geometric inequalities for \(\mathcal{R}\).</p> <p> Proposition IV.6 (Circular duct minimizes laminar dissipation among fixed-area cross-sections).<br /> Consider steady incompressible Stokes flow in a straight duct of fixed length \(L\), with a fixed cross-sectional area \(A\), and no-slip boundary. Among all simply connected cross-sections with area \(A\), the circular disk maximizes volumetric flow \(Q\) for a given pressure drop \(\Delta p\), equivalently minimizes the resistance \(\mathcal{R}=\Delta p/Q\) and the dissipation \(P=Q\Delta p\) for a given \(Q\).</p> <p> Justification (classical).<br /> In Stokes/Poiseuille flow, \(Q\) is proportional to the torsional rigidity of the cross-section, and the disk maximizes torsional rigidity among fixed-area domains (Saint-Venant-type isoperimetric inequality). Therefore any non-circular cross-section incurs at least as large a pressure drop for the same throughput.</p> <p> IV.F.4. System-level update: total minimum required exergy</p> <p> Combining Section III with the airflow term yields the strengthened necessary inequality<br /> \[<br /> \int_0^{\tau} e(t)\,dt\ \ge\ \int_0^{\tau} c(t)\,y(t)\,dt\ +\ \int_0^{\tau} P_{\mathrm{fan}}(t)\,dt,<br /> \]<br /> where \(P_{\mathrm{fan}}\) itself can be lower-bounded in terms of \(\dot m_w\) using the throughput inequality \(\dot m_{da}\ge \dot m_w/\Delta\omega\) and a pressure-drop model for the chosen airflow path. Because fan power typically grows superlinearly in flow in turbulent regimes and at least quadratically in laminar regimes when expressed in terms of \(\dot m_w\), this term can enforce strong diminishing returns in aggressive production schedules.</p> <p> End of Section IV.</p> <p> V. Optimal Control of AWH under Renewable Variability (Single-Agent)</p> <p> This section formulates operation of a single AWH device (or an aggregated device layer at one site) as an exergy-allocation problem over a diurnal horizon. The aim is to make explicit which policy structure follows from (i) second-law bounds and (ii) convexity/concavity assumptions on the device response, while keeping the modeling knobs auditable.</p> <p> We work on a time horizon \([0,\tau]\). The ambient “exergy price per kg” is<br /> \[<br /> c(t)=R_v T(t)\ln\frac{1}{\mathrm{RH}(t)}\quad [\mathrm{J/kg}],<br /> \]<br /> from Section II.B.1, and the instantaneous second-law outer bound is \(y(t)\le e(t)/c(t)\) (Theorem III.1) when the ambient-temperature benchmark applies.</p> <p> V.A. Deterministic exergy-allocation as an infinite-dimensional LP</p> <p> V.A.1. Problem statement</p> <p> Assume the device can convert a usable exergy input rate \(e(t)\in[0,\bar e(t)]\) into water flux \(y(t)\) satisfying the reversible outer constraint<br /> \[<br /> 0\le y(t)\le \frac{e(t)}{c(t)}\qquad\text{for a.e. }t\in[0,\tau].<br /> \]<br /> Let the daily usable exergy budget be<br /> \[<br /> \int_0^{\tau} e(t)\,dt\le E.<br /> \]<br /> A best-case upper bound on daily produced water is given by the linear program<br /> \[<br /> \max_{e,y}\ \int_0^{\tau} y(t)\,dt\quad\text{s.t.}\quad 0\le e(t)\le \bar e(t),\ \int_0^{\tau} e\le E,\ 0\le y(t)\le e(t)/c(t).<br /> \]<br /> Eliminating \(y\) at optimum (take \(y=e/c\)) yields the equivalent LP<br /> \[<br /> \max_{0\le e\le \bar e}\ \int_0^{\tau} \frac{e(t)}{c(t)}\,dt\quad\text{s.t.}\quad \int_0^{\tau} e(t)\,dt\le E.<br /> \]</p> <p> V.A.2. Threshold structure (continuous-time fractional knapsack)</p> <p> Theorem V.1 (Threshold optimality for linear yield bound).<br /> Assume \(c(t)>0\) a.e. and \(\bar e\in L^\infty([0,\tau])\). Define the “value density” \(v(t):=1/c(t)\). An optimal solution \(e^*(t)\) exists and can be chosen to satisfy the threshold form<br /> \[<br /> e^*(t)=\begin{cases}<br /> \bar e(t), & v(t)>\lambda,\\<br /> 0, & v(t)<\lambda, \end{cases} \] for some \(\lambda\ge 0\), with partial allocation only on the level set \(\{t: v(t)=\lambda\}\) to satisfy \(\int e^*=E\) when the budget is binding. Proof. This is a linear program over the order interval \(0\le e\le \bar e\) with a single integral constraint. Consider the Lagrangian \( \mathcal{L}(e,\lambda)=\int_0^{\tau} v(t)e(t)\,dt-\lambda\big(\int_0^{\tau} e(t)\,dt-E\big). \) For fixed \(\lambda\), pointwise maximization over \([0,\bar e(t)]\) yields \(e(t)=\bar e(t)\) if \(v(t)>\lambda\), \(e(t)=0\) if \(v(t)<\lambda\), and any \(e(t)\in[0,\bar e(t)]\) if \(v(t)=\lambda\). Choose \(\lambda\) so that the resulting \(e\) satisfies the budget (standard monotonicity of \(\int e\) in \(\lambda\)). \(\square\) Corollary V.2 (Operational meaning). Under the reversible linear bound, the optimal policy concentrates usable exergy into time windows where \(c(t)\) is smallest (i.e., relatively high humidity and/or low temperature), subject to the instantaneous cap \(\bar e(t)\). This provides a computable, device-agnostic benchmark for “best possible” scheduling under any assumed exergy supply envelope \(\bar e\). Remark V.3 (When the threshold picture breaks). Any additional convex penalty that scales superlinearly with throughput (e.g., a fan-work term that behaves like \(\int \dot m_w(t)^2\,dt\) in laminar regimes; cf. Section IV.F) destroys pure bang-bang behavior and promotes smoothing of production over time. This is one mechanism by which engineering parasitics change the qualitative scheduling structure. V.B. Convex exergy-to-water allocation with storage and concave device response We now allow general concave device response \(y=\phi_t(e)\) (Section II.D.1) and include a storage/buffering state (electrical or thermal), which is essential when exergy supply and harvesting windows are misaligned. V.B.1. Discrete-time model Discretize \([0,\tau]\) into \(k=1,\dots,N\) intervals of length \(\Delta t\). Let \[ 0\le e_k\le \bar e_k,\qquad y_k\le \phi_k(e_k), \] where \(\phi_k\) is concave and nondecreasing. Let \(r_k\ge 0\) be exergy inflow into storage (e.g., PV converted to usable exergy, Carnot-weighted heat converted to usable exergy) and let \(s_k\in[0,S_{\max}]\) be stored usable exergy at step \(k\). A simple recursion is \[ s_{k+1}=s_k + r_k\Delta t - e_k\Delta t, \] with feasibility constraints \(s_{k+1}\in[0,S_{\max}]\) and given \(s_1\). A planning/control upper bound on achievable daily yield is the convex program \[ \max_{e,s}\ \sum_{k=1}^N \phi_k(e_k)\,\Delta t \] subject to the storage recursion and box constraints on \((e_k,s_k)\). This problem is convex because the objective is concave in \(e\) and we maximize it over a polyhedral feasible set. V.B.2. KKT “water-filling” condition Proposition V.4 (Equalized marginal value condition). Assume each \(\phi_k\) is concave and differentiable on \((0,\bar e_k)\). For any optimal solution with \(0<e_k<\bar e_k\) and with the storage bounds inactive (so the recursion constraints are the only active coupling), there exists a scalar \(\lambda\ge 0\) such that \[ \phi_k'(e_k)=\lambda\qquad\text{for all interior times }k. \] At bounds, the usual complementary slackness inequalities hold: \[ \phi_k'(e_k)\le \lambda\ \text{when }e_k=0,\qquad \phi_k'(e_k)\ge \lambda\ \text{when }e_k=\bar e_k. \] Proof sketch. Introduce Lagrange multipliers for the storage dynamics (or equivalently eliminate \(s\) and write cumulative constraints). Stationarity with respect to \(e_k\) gives \(\phi_k'(e_k)=\lambda_k\), where \(\lambda_k\) is the shadow value of stored exergy at time \(k\). When storage constraints are inactive and there is no terminal value, the multipliers flatten to a constant \(\lambda\), yielding the equalization of marginal yield. \(\square\) Corollary V.5 (Thermodynamic slope constraint propagates to scheduling). Since physically admissible envelopes satisfy \(\phi_k'(0^+)\le 1/c_k\), time steps with smaller \(c_k\) admit a larger possible marginal yield. Therefore the marginal-equalization rule tends to allocate more exergy to high-RH/low-\(c_k\) windows unless limited by \(\bar e_k\) or storage saturation. V.B.3. Piecewise-linear envelopes and fractional knapsack reduction If \(\phi_k\) is piecewise-linear concave, its derivative takes finitely many slope values. The KKT conditions reduce to a “fill segments in decreasing slope order” procedure, i.e., a fractional knapsack on the set of linear segments \((k,j)\) with values equal to slopes. This provides an auditable outer bound when \(\phi_k\) is constructed as the concave upper hull of simulated \((e,y)\) points. V.C. Heat-powered (finite-temperature) scheduling variants Suppose the usable exergy inflow is generated from heat \(\dot Q_h(t)\) available at temperature \(T_h(t)>T_0(t)\). Section II.C.2 gives the pointwise exergy cap<br /> \[<br /> e_h(t)\le \dot Q_h(t)\Big(1-\frac{T_0(t)}{T_h(t)}\Big).<br /> \]<br /> In discrete time, this becomes \(r_k\le \bar r_k\) with<br /> \[<br /> \bar r_k := \dot Q_{h,k}\Big(1-\frac{T_{0,k}}{T_{h,k}}\Big),<br /> \]<br /> which can be used directly in the storage recursion of Section V.B.1.</p> <p> Remark V.6 (Thermal design variables enter as exergy prices).<br /> When a control policy can influence \(T_h\) (e.g., by choosing collector operating temperature) subject to collector and heat-transfer constraints, the Carnot factor induces a trade-off: higher \(T_h\) increases exergy per unit heat but typically reduces collectable heat rate. The present framework separates the guaranteed second-law conversion \(\dot Q_h\mapsto e_h\) from technology-specific heat-collection models.</p> <p> V.D. Stochastic control under climate uncertainty</p> <p> Deterministic schedules can be brittle under weather volatility and regime shifts. Here we record a conservative stochastic-control formulation; we do not claim closed-form solutions in general.</p> <p> V.D.1. Markov climate driver and control</p> <p> Let \(\xi_t\) be a Markov process representing climate inputs (e.g., \(T\), RH, sky temperature, wind), and let the exergy price and supply caps be functions of \(\xi\):<br /> \[<br /> c(t)=c(\xi_t),\qquad \bar e(t)=\bar e(\xi_t).<br /> \]<br /> Include storage \(s_t\in[0,S_{\max}]\) with dynamics<br /> \[<br /> ds_t = (r(\xi_t)-e_t)\,dt,<br /> \]<br /> where \(0\le e_t\le \bar e(\xi_t)\) is the control and \(r(\xi_t)\) is a (possibly random) inflow process.</p> <p> A risk-neutral objective is<br /> \[<br /> \sup_{e}\ \mathbb{E}\Big[\int_0^{\tau} \phi_{\xi_t}(e_t)\,dt\Big].<br /> \]<br /> Risk-aware variants can replace the expectation by a coherent risk measure or impose chance constraints; these are deferred to Section VI.</p> <p> V.D.2. HJB equation (formal)</p> <p> If \(\xi_t\) is an Ornstein–Uhlenbeck (OU) diffusion in \(\mathbb{R}^m\),<br /> \[<br /> d\xi_t = A(\mu-\xi_t)\,dt + \Sigma\,dW_t,<br /> \]<br /> and \(V(t,\xi,s)\) denotes the value function, then formally \(V\) satisfies the Hamilton–Jacobi–Bellman equation<br /> \[<br /> \partial_t V + \mathcal{L}^{\xi}V + \sup_{0\le e\le \bar e(\xi)}\Big\{\phi_{\xi}(e) – e\,\partial_s V\Big\} + r(\xi)\,\partial_s V = 0,<br /> \]<br /> with terminal condition \(V(\tau,\xi,s)=0\) (or an application-specific terminal value of stored exergy). Here \(\mathcal{L}^{\xi}\) is the OU generator.</p> <p> Remark V.7 (Policy structure).<br /> The maximization term is pointwise in \(e\). When \(\phi_{\xi}\) is concave, the optimizer has a “threshold on marginal value” form: choose \(e\) so that \(\phi’_{\xi}(e)\) matches the shadow price \(\partial_s V\), with saturation at \(0\) or \(\bar e(\xi)\). This is the stochastic analogue of the deterministic KKT water-filling condition.</p> <p> V.E. Impulse/batch-mode operation under intermittent renewables</p> <p> Some AWH concepts operate in batches (e.g., adsorption followed by a regeneration/desorption event) rather than continuously. This motivates impulse control.</p> <p> V.E.1. Impulse model and admissible actions</p> <p> Let \(x_t\) be a reduced device state (e.g., sorbent loading and temperature). Between interventions it evolves as<br /> \[<br /> dx_t = f(x_t,\xi_t,u_t)\,dt<br /> \]<br /> with a continuous control \(u_t\) (e.g., airflow) and no water output (or a low background yield). At intervention times \(\{\tau_n\}\), an impulse action \(a\in\mathcal{A}(x_{\tau_n^-},\xi_{\tau_n},s_{\tau_n^-})\) instantaneously updates state and storage:<br /> \[<br /> (x,s)\mapsto \Gamma(x,s,\xi;a),<br /> \]<br /> produces a batch of water \(M(x,\xi;a)\ge 0\), and consumes an impulse exergy cost \(K(x,\xi;a)\ge 0\) which must be feasible given stored exergy.</p> <p> V.E.2. QVI (formal)</p> <p> Let \(V(t,\xi,x,s)\) be the value function (expected remaining water). The quasi-variational inequality (QVI) takes the form<br /> \[<br /> \max\Big\{\partial_t V+\mathcal{L}^{\xi}V+\nabla_{x,s}V\cdot F – \ell,\ \ \sup_{a\in\mathcal{A}}\big(M(x,\xi;a)+V(t,\xi,\Gamma(x,s,\xi;a)) – V(t,\xi,x,s)\big)\Big\}=0,<br /> \]<br /> where \(F\) encodes the controlled drift in \((x,s)\) between impulses and \(\ell\) is any running penalty (often \(\ell\equiv 0\)).</p> <p> Remark V.8 (Interpretation).<br /> The QVI compares (i) continuing without intervention versus (ii) intervening now and paying the impulse cost to obtain an immediate batch yield and a new state. In practice, one discretizes \((t,\xi,x,s)\) and computes policies by dynamic programming or policy iteration; the key structural point for this blueprint is that impulse feasibility can be tied to the auditable exergy bounds of Sections II.C and III.</p> <p> End of Section V.</p> <p> VI. Uncertainty Quantification (UQ) and Distributional Robustness</p> <p> Sections II–V treat climate forcing (temperature, relative humidity, radiative drivers, wind/transfer coefficients) as known inputs. Any deployment or control recommendation, however, depends on uncertain and potentially nonstationary atmospheric conditions. This section records mathematically explicit UQ and robustness templates intended for (i) propagating uncertainty through reduced device models and (ii) producing conservative, optimization-ready performance guarantees.</p> <p> Throughout, let \(\xi\in\Xi\) denote a vector of climate inputs (possibly including short time windows or features extracted from time series). Let \(u\in\mathcal{U}\) denote a decision variable (control policy parameters, design variables, or siting/deployment decisions). Let \(F(u,\xi)\) be a performance quantity (e.g., daily yield, exergy efficiency, or a feasibility margin). When needed, we interpret constraints as \(F(u,\xi)\ge 0\) (feasible if nonnegative).</p> <p> VI.A. Polynomial Chaos Expansion (PCE) surrogates for climate-driven parameters</p> <p> VI.A.1. Random input parameterization</p> <p> A convenient UQ interface is to treat a set of uncertain parameters \(\theta\) entering device modules as random functions of climate inputs. Examples include<br /> \[<br /> \theta = \theta(\xi) \in \{\Delta T,\ \Delta\mathrm{RH},\ h_m,\ h_c,\ D_{\mathrm{eff}},\ k,\ \varepsilon,\ \text{isotherm parameters},\ldots\}.<br /> \]<br /> We emphasize that PCE is a *surrogate and propagation method*; it does not validate the chosen probabilistic model for \(\theta\).</p> <p> VI.A.2. PCE definition and basic moment identities</p> <p> Let \(Z\in\mathbb{R}^p\) be a standardized random vector (e.g., independent components) driving uncertainty, and let \(\{\Psi_\alpha(Z)\}_{\alpha\in\mathcal{A}}\) be a finite set of multivariate orthonormal polynomials in \(L^2\) with respect to the law of \(Z\), with \(\Psi_0\equiv 1\). A PCE surrogate for a scalar quantity \(G(Z)\in L^2\) is<br /> \[<br /> G(Z) \approx \sum_{\alpha\in\mathcal{A}} g_\alpha\,\Psi_\alpha(Z).<br /> \]<br /> If the basis is orthonormal, then the mean and variance are approximated by<br /> \[<br /> \mathbb{E}[G] \approx g_0,\qquad \mathrm{Var}(G) \approx \sum_{\alpha\in\mathcal{A}\setminus\{0\}} g_\alpha^2.<br /> \]<br /> These are purely algebraic consequences of orthonormality.</p> <p> VI.A.3. Propagation through reduced models and viability screens</p> <p> Suppose a reduced device surrogate produces an estimated yield \(\widehat Y(u,\theta)\) (e.g., from Section IV.C or an empirically fit concave envelope \(\phi_t\)). If \(\theta\) is represented by a PCE map \(\theta\approx \sum_\alpha \theta_\alpha\Psi_\alpha\), then \(\widehat Y(u,\theta(Z))\) can be approximated by either:</p> <p> 1. **Intrusive Galerkin** (rarely justified for complex switching models): project the reduced equations onto \(\Psi_\alpha\).<br /> 2. **Nonintrusive regression/collocation**: sample \(Z^{(i)}\), evaluate \(\widehat Y(u,\theta(Z^{(i)}))\), and regress coefficients \(y_\alpha(u)\).</p> <p> The resulting surrogate delivers approximate moments \(\mathbb{E}[\widehat Y]\), \(\mathrm{Var}(\widehat Y)\), and (via Chebyshev/Cantelli-type inequalities) conservative but potentially loose tail bounds when only low-order moments are reliable.</p> <p> VI.A.4. Regime-shift modeling knob (mixture template)</p> <p> To represent nonstationarity, a minimal and auditable knob is a mixture model for the law of \(\xi\):<br /> \[<br /> \mathbb{P}_{\mathrm{mix}} = (1-\alpha)\,\widehat{\mathbb{P}} + \alpha\,\mathbb{Q},\qquad \alpha\in[0,1],<br /> \]<br /> where \(\widehat{\mathbb{P}}\) is an empirical/nominal law and \(\mathbb{Q}\) is an alternative law representing a shifted regime. In robust planning, one may treat \(\alpha\) as an uncertainty magnitude parameter and bound performance uniformly over \(\mathbb{Q}\) drawn from an ambiguity set (Section VI.B).</p> <p> VI.B. Wasserstein / WFR distributionally robust optimization (DRO) for AWH</p> <p> VI.B.1. Motivation: robustness to distribution shift</p> <p> Let \(\widehat{\mathbb{P}}\) be an empirical distribution of climate features (e.g., derived from historical reanalysis). Distributionally robust optimization replaces \(\mathbb{E}_{\widehat{\mathbb{P}}}[F(u,\xi)]\) by the worst-case expectation over an ambiguity set \(\mathcal{P}\) around \(\widehat{\mathbb{P}}\):<br /> \[<br /> \inf_{\mathbb{P}\in\mathcal{P}} \mathbb{E}_{\mathbb{P}}[F(u,\xi)] \quad\text{or}\quad \sup_{\mathbb{P}\in\mathcal{P}} \mathbb{E}_{\mathbb{P}}[F(u,\xi)],<br /> \]<br /> depending on whether \(F\) is a reward or a cost.</p> <p> VI.B.2. Wasserstein-1 ball: dual template for worst-case expectation</p> <p> Assume \((\Xi,d)\) is a metric space and consider a Wasserstein-1 ball \(\mathcal{P}_{W_1}(\varepsilon)\) from Section II.D.3.</p> <p> Proposition VI.1 (Kantorovich–Rubinstein dual upper bound; optimization-ready form).<br /> Let \(f:\Xi\to\mathbb{R}\) be upper semicontinuous and satisfy a growth condition ensuring integrability under measures in \(\mathcal{P}_{W_1}(\varepsilon)\). Then<br /> \[<br /> \sup_{\mathbb{P}:W_1(\mathbb{P},\widehat{\mathbb{P}})\le \varepsilon}\ \mathbb{E}_{\mathbb{P}}[f(\xi)]<br /> \;=\;<br /> \inf_{\lambda\ge 0}\left\{\lambda\varepsilon + \mathbb{E}_{\widehat{\mathbb{P}}}\big[\sup_{\zeta\in\Xi}\big(f(\zeta)-\lambda d(\zeta,\xi)\big)\big]\right\}.<br /> \]</p> <p> Comment.<br /> This representation is the standard starting point for tractable DRO reformulations. Tractability then depends on whether the inner supremum \(\sup_{\zeta}(\cdot)\) can be evaluated in closed form or upper-bounded by a convex constraint for the chosen \(f\) and metric \(d\). For example, if \(f(u,\xi)\) is affine in \(\xi\) and \(d\) is a norm, the inner supremum yields a dual norm expression.</p> <p> VI.B.3. Robust chance / quantile constraints (CVaR surrogate)</p> <p> Many AWH designs require reliability, e.g. \(\mathbb{P}(F(u,\xi)\ge 0)\ge 1-\beta\). A convex surrogate is a Conditional-Value-at-Risk (CVaR) constraint:<br /> \[<br /> \mathrm{CVaR}_{\beta}(-F(u,\xi)) \le 0,<br /> \]<br /> which is equivalent to existence of \(\eta\in\mathbb{R}\) such that<br /> \[<br /> \eta + \frac{1}{\beta}\,\mathbb{E}\big[(-F(u,\xi)-\eta)_+\big] \le 0.<br /> \]<br /> A distributionally robust CVaR can be formed by replacing \(\mathbb{E}\) with \(\sup_{\mathbb{P}\in\mathcal{P}}\mathbb{E}_{\mathbb{P}}\). When \(F\) is concave in \(u\) and the hinge \((\cdot)_+\) is handled via epigraph variables, this often leads to conic programs once a dual DRO representation is selected.</p> <p> VI.B.4. Unbalanced ambiguity (WFR-type) and exponential-cone structure (template)</p> <p> To model appearance/disappearance of events (regime shift with probability-mass change), one may use an unbalanced optimal-transport discrepancy (Section II.D.3). Rather than committing to a specific convention, we record a general convex-analytic pattern that recurs in WFR/KL-regularized unbalanced OT duality.</p> <p> Template (KL-penalized mass change).<br /> Suppose the ambiguity set is defined by a constraint of the schematic form<br /> \[<br /> \inf_{\text{couplings and mass-change maps}}\big\{\text{transport cost} + \kappa\,D_{\mathrm{KL}}(\mathbb{P}\,\|\,\widehat{\mathbb{P}})\big\} \le \varepsilon,<br /> \]<br /> for some \(\kappa>0\). Then dual variables typically appear inside exponential-type expressions, and constraints of the form<br /> \[<br /> \log \mathbb{E}_{\widehat{\mathbb{P}}}[\exp(g(\xi))] \le \text{constant}<br /> \]<br /> or equivalent epigraphs. Such constraints are representable with the exponential cone \(\mathcal{K}_{\exp}\) (Section II.D.2) after standard log-sum-exp epigraphization in finite-sample settings.</p> <p> Caveat.<br /> Different WFR/unbalanced OT definitions yield different exact duals and constants. Any implementation must specify the precise discrepancy and verify the corresponding dual reformulation from an external, authoritative reference.</p> <p> VI.B.5. DRO applied to exergy-limited yield bounds</p> <p> A recurring robust-design pattern in this paper is a bound of the form<br /> \[<br /> Y(u,\xi) \le \int_0^{\tau} \phi_{t,\xi}(e_u(t,\xi))\,dt,<br /> \]<br /> subject to the second-law constraints \(\phi_{t,\xi}(e)\le e/c(t,\xi)\) and exergy supply caps \(0\le e_u(t,\xi)\le \bar e(t,\xi)\). Under ambiguity in \(\xi\), conservative planning may enforce<br /> \[<br /> \inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}_{\mathbb{P}}[Y(u,\xi)] \ge Y_{\min}<br /> \quad\text{or}\quad<br /> \inf_{\xi\in\mathcal{U}_{\xi}} Y(u,\xi)\ge Y_{\min},<br /> \]<br /> depending on whether a distributional or set-based uncertainty model is chosen.</p> <p> VI.C. Thermodynamic Uncertainty Relations (TUR) for finite-time AWH</p> <p> This subsection records a general stochastic-thermodynamic trade-off: sustained nonequilibrium currents exhibit a lower-bounded relative variance in terms of entropy production. The purpose here is *not* to claim a device-specific constant, but to provide a principled way to connect reliability constraints (variance/tails of yield) with dissipation/exergy expenditure when a sufficiently detailed stochastic model is available.</p> <p> VI.C.1. TUR statement (generic Markovian form)</p> <p> Let \(J_T\) be an integrated current over \([0,T]\) (e.g., harvested water mass, or a counting current of batch completions) in a nonequilibrium Markov process model of the device–environment interaction. Let \(\Sigma_T\ge 0\) be the total entropy production over \([0,T]\). A standard form of the TUR (under appropriate assumptions on dynamics and time-reversal symmetry) asserts<br /> \[<br /> \frac{\mathrm{Var}(J_T)}{\mathbb{E}[J_T]^2}\ \ge\ \frac{2}{\mathbb{E}[\Sigma_T]}.<br /> \]</p> <p> Interpretation.<br /> To reduce relative fluctuations \(\mathrm{Var}(J_T)/\mathbb{E}[J_T]^2\), one must increase entropy production, i.e., incur more dissipation/exergy destruction on average. Thus, in addition to the mean exergy-to-yield bounds of Sections III–V, TUR suggests that imposing strict reliability (small variance or tight tails) can force additional dissipation beyond reversible requirements.</p> <p> VI.C.2. Consequence for risk-aware control objectives (formal)</p> <p> If a planning layer requires a variance constraint \(\mathrm{Var}(J_T)\le \sigma^2\) while achieving mean yield \(\mathbb{E}[J_T]\ge m\), then the TUR implies a necessary condition<br /> \[<br /> \mathbb{E}[\Sigma_T] \ge \frac{2m^2}{\sigma^2}.<br /> \]<br /> Since exergy destruction is \(T_0\) times entropy production (in isothermal benchmarks), this can be translated into a lower bound on additional required exergy beyond the reversible separation term, provided \(\Sigma_T\) is correctly identified for the chosen stochastic model.</p> <p> VI.C.3. Explicit gap and verification requirement</p> <p> Mapping TUR quantities to AWH observables requires a validated stochastic model specifying:</p> <p> (i) what constitutes the stochastic state (e.g., loading/temperature modes, adsorption front location, droplet nucleation events),<br /> (ii) which current corresponds to harvested water, and<br /> (iii) how entropy production relates to the device’s measurable exergy inflows and heat/mass exchanges.</p> <p> Absent this modeling and validation, TUR should be treated as a qualitative design warning: pushing for high power and tight reliability simultaneously is expected to be costly in dissipation.</p> <p> **Remark (Mathematical requirements for operational validity).** The TUR bound $\mathbb{E}[\Sigma_T] \ge 2\mathbb{E}[J_T]^2/\mathrm{Var}(J_T)$ is a rigorous theorem in Markovian stochastic thermodynamics, yet its translation into an auditable constraint on engineered AWH systems demands: (a) a microscopic or mesoscopic stochastic model satisfying local detailed balance with respect to a known equilibrium measure $p_{\mathrm{eq}}(\mathbf{x})$; (b) an unambiguous identification of the harvested-water current as a time-integrated observable $J_T = \int_0^T \Lambda(\mathbf{x}(t)) \circ d\mathbf{x}(t)$ with $\Lambda$ a state-dependent weight function; and (c) an entropy production functional $\Sigma_T$ that decomposes into (i) heat flows divided by absolute reservoir temperatures and (ii) non-adiabatic terms accounting for driving protocol irreversibility. Absent experimental validation that the chosen state vector $\mathbf{x}$ captures all slow variables of the device–atmosphere interaction and that $\Sigma_T$ correlates with independently measured exergy destruction, the TUR remains a formal bound on an abstract mathematical model rather than a proven constraint on physical hardware. Consequently, design decisions invoking TUR should be treated as sensitivity indicators—quantifying how variance penalties scale with assumed dissipation—rather than as absolute performance limits, and should be accompanied by explicit uncertainty quantification over the space of admissible stochastic models.</p> <p> End of Section VI.</p> <p> VII. Atmospheric Moisture Budget Constraints (Regional/ABL Scale) and Multiscale Coupling</p> <p> Device-scale bounds (Sections III–IV) can certify that a given exergy budget is insufficient to harvest a target water mass. They do not, by themselves, prevent mathematically feasible schedules that implicitly remove more water vapor than the atmosphere can supply to a region. This section introduces a regional (or boundary-layer) moisture conservation law that yields hard feasibility caps on *aggregate* AWH sinks, and then records a conservative coupling principle: if deployment depletes local vapor, then the reversible separation work per kilogram necessarily increases.</p> <p> VII.A. Column/regional moisture conservation as a hard feasibility cap</p> <p> VII.A.1. Column moisture variable and transport equation (template)</p> <p> Let \(\Omega\subset\mathbb{R}^2\) be a horizontal region (e.g., a planning polygon or an ERA5 grid cell) and let \(H>0\) be a chosen boundary-layer height. Let \(\rho_v(x,z,t)\) be water-vapor mass density. Define the column-integrated precipitable vapor mass per horizontal area<br /> \[<br /> W(x,t):=\int_0^H \rho_v(x,z,t)\,dz\qquad [\mathrm{kg/m}^2].<br /> \]<br /> A common modeling closure for regional moisture is an advection–diffusion–source balance for \(W\):<br /> \[<br /> \partial_t W + \nabla\cdot F = S_{\mathrm{atm}}(x,t) – S_{\mathrm{AWH}}(x,t),<br /> \qquad x\in\Omega,\ t\in[t_0,t_1].<br /> \tag{VII.1}<br /> \]<br /> Here:</p> <p> – \(F(x,t)\in\mathbb{R}^2\) is the horizontal moisture flux (e.g., mean advection plus an eddy-diffusive closure such as \(F=U W – K\nabla W\) with wind \(U\) and eddy diffusivity \(K\));<br /> – \(S_{\mathrm{atm}}\) is net atmospheric source minus sink excluding AWH (e.g., surface evaporation minus precipitation, and any vertically integrated parameterized terms);<br /> – \(S_{\mathrm{AWH}}\ge 0\) is the AWH vapor removal rate per horizontal area.</p> <p> We emphasize that (VII.1) is a *planning-scale accounting identity*: it is valid whenever the terms are interpreted consistently (including parameterized subgrid transport inside \(F\)). It is not tied to a particular CFD model.</p> <p> VII.A.2. Integrated feasibility inequality</p> <p> Let \(R\subset\Omega\) be a measurable subregion with Lipschitz boundary \(\partial R\). Integrating (VII.1) over \(R\times[t_0,t_1]\) and applying the divergence theorem gives the exact balance<br /> \[<br /> \int_{t_0}^{t_1}\!\int_R S_{\mathrm{AWH}}\,dx\,dt<br /> =<br /> \int_R \big(W(x,t_0)-W(x,t_1)\big)\,dx<br /> + \int_{t_0}^{t_1}\!\int_R S_{\mathrm{atm}}\,dx\,dt<br /> – \int_{t_0}^{t_1}\!\int_{\partial R} F\cdot n\,ds\,dt.<br /> \tag{VII.2}<br /> \]<br /> Since \(W\ge 0\), this implies the *hard cap*<br /> \[<br /> \int_{t_0}^{t_1}\!\int_R S_{\mathrm{AWH}}\,dx\,dt<br /> \le<br /> \int_R W(x,t_0)\,dx<br /> + \int_{t_0}^{t_1}\!\int_R S_{\mathrm{atm}}\,dx\,dt<br /> – \int_{t_0}^{t_1}\!\int_{\partial R} F\cdot n\,ds\,dt,<br /> \tag{VII.3}<br /> \]<br /> provided the right-hand side is finite. Inequality (VII.3) is a feasibility constraint: if a proposed deployment implies a larger integrated sink than the right-hand side, then it is inconsistent with moisture conservation for the chosen region/time window.</p> <p> Remark VII.1 (Practical use and conservatism).<br /> The right-hand side of (VII.3) depends on how \(S_{\mathrm{atm}}\) and \(F\) are estimated. In early-stage screening, one may bound it conservatively (e.g., lower bound net inflow and upper bound net outflow) to obtain a strict necessary condition.</p> <p> VII.A.3. Discretized linear regional constraint (ERA5-grid-ready)</p> <p> Partition \(\Omega\) into cells \(i\in\{1,\dots,N\}\) with areas \(A_i\) and use time steps \(k\in\{0,\dots,K-1\}\) with \(\Delta t=(t_1-t_0)/K\). Let \(S_{\mathrm{AWH},i,k}\) be the average AWH sink (kg\,m\(^{-2}\)\,s\(^{-1}\)) in cell \(i\) during time step \(k\). Then for any set of cells \(\mathcal{I}(R)\) representing a region \(R\), (VII.2) yields the linear constraint<br /> \[<br /> \sum_{k=0}^{K-1}\sum_{i\in\mathcal{I}(R)} A_i\,S_{\mathrm{AWH},i,k}\,\Delta t<br /> \le<br /> \sum_{i\in\mathcal{I}(R)} A_i\,W_{i,0}<br /> + \sum_{k=0}^{K-1}\sum_{i\in\mathcal{I}(R)} A_i\,S_{\mathrm{atm},i,k}\,\Delta t<br /> – \sum_{k=0}^{K-1} \Phi_{\partial R,k}\,\Delta t,<br /> \tag{VII.4}<br /> \]<br /> where \(\Phi_{\partial R,k}\) approximates the boundary flux integral \(\int_{\partial R} F\cdot n\,ds\) at step \(k\). Constraint (VII.4) is suitable as a hard feasibility cap in planning problems with decision variables that determine \(S_{\mathrm{AWH},i,k}\) (e.g., installed area times per-device sink).</p> <p> VII.B. Device-to-atmosphere closure and depletion-aware thermodynamic penalties</p> <p> To connect (VII.1) to device operation, we must relate the atmospheric sink \(S_{\mathrm{AWH}}\) to harvested water flux. Let \(\theta(x)\in[0,1]\) be an areal deployment fraction (device interface area per land area, or an effective packing density), and let \(y(x,t)\) be harvested liquid-water flux per device area (kg\,m\(^{-2}\)\,s\(^{-1}\)). A minimal closure is<br /> \[<br /> S_{\mathrm{AWH}}(x,t) = \theta(x)\,y(x,t).<br /> \tag{VII.5}<br /> \]<br /> This is an identity if all vapor removed by devices is converted to collected liquid without re-evaporation; if there are losses (e.g., blowdown, re-evaporation), then (VII.5) should be replaced by \(S_{\mathrm{AWH}}\ge \theta y\), which only strengthens feasibility caps.</p> <p> VII.B.1. Linearized humidity-limited sink law (optional tightening)</p> <p> In humidity-limited regimes, many device models admit a first-order dependence of achievable uptake on local vapor density (or partial pressure), at least locally around a nominal state. A conservative linear template is<br /> \[<br /> 0\le y(x,t) \le \alpha(x,t)\,\rho_v(x,t),<br /> \tag{VII.6}<br /> \]<br /> with \(\alpha\ge 0\) an effective mass-transfer coefficient (units converting \(\rho_v\) to a flux). Combining (VII.5)–(VII.6) yields<br /> \[<br /> 0\le S_{\mathrm{AWH}}(x,t) \le \theta(x)\,\alpha(x,t)\,\rho_v(x,t).<br /> \tag{VII.7}<br /> \]<br /> We stress that (VII.6) is a *modeling choice* and must be validated for any particular device; the moisture budget constraint (VII.2)–(VII.4) does not require it.</p> <p> VII.B.2. Depletion-induced strengthening of minimum work (convex \(-\ln\)-type penalty)</p> <p> The reversible separation work per unit mass extracted from moist air depends on local relative humidity. If AWH deployment depletes local vapor, then local \(\mathrm{RH}\) decreases and the reversible work increases.</p> <p> To make this dependence explicit without committing to a detailed boundary-layer microphysics model, we record the following conservative template.</p> <p> Assumption VII.2 (Relative-humidity scaling with a depletion factor).<br /> At fixed temperature \(T\) and pressure \(P\), assume local vapor partial pressure (hence relative humidity) scales proportionally with a depletion factor \(\beta\in(0,1]\):<br /> \[<br /> \mathrm{RH} = \beta\,\mathrm{RH}_0,<br /> \tag{VII.8}<br /> \]<br /> where \(\mathrm{RH}_0\) is the undepleted ambient (background) value and \(\beta\) summarizes the effect of deployment-induced reduction in local vapor abundance.</p> <p> Under (VII.8), the reversible benchmark from Section II.B.1 becomes<br /> \[<br /> c(\beta) = R_v T\ln\frac{1}{\mathrm{RH}}<br /> = R_v T\ln\frac{1}{\mathrm{RH}_0} + R_v T\ln\frac{1}{\beta}.<br /> \tag{VII.9}<br /> \]<br /> The additional term \(R_vT\ln(1/\beta)\ge 0\) is a *depletion penalty*. Since \(-\ln \beta\) is convex on \((0,\infty)\), the penalty is convex in \(\beta\).</p> <p> One simple way to connect \(\beta\) to the column vapor variable is to set \(\beta=W/W_0\) for a reference column mass \(W_0>0\) (e.g., the initial or background value). Then<br /> \[<br /> c(W) = R_v T\ln\frac{1}{\mathrm{RH}_0} + R_v T\ln\frac{W_0}{W},<br /> \qquad W\in(0,W_0].<br /> \tag{VII.10}<br /> \]<br /> The map \(W\mapsto \ln(W_0/W)\) is convex on \((0,\infty)\), so (VII.10) provides a convex strengthening of the second-law cost as moisture is depleted.</p> <p> VII.B.3. Epigraphization for optimization</p> <p> To use the depletion penalty in conic optimization, introduce an auxiliary variable \(z\) enforcing<br /> \[<br /> z \ge \ln\frac{W_0}{W}\quad\Longleftrightarrow\quad -z \le \ln\frac{W}{W_0}.<br /> \tag{VII.11}<br /> \]<br /> Equivalently, \(\exp(-z) \le W/W_0\), which can be represented with the exponential cone (Section II.D.2) via standard transformations. Then the strengthened pointwise exergy-to-water inequality becomes, in this template,<br /> \[<br /> e(t) \ge y(t)\,R_vT(t)\left(\ln\frac{1}{\mathrm{RH}_0(t)} + z(t)\right),<br /> \tag{VII.12}<br /> \]<br /> where \(z(t)\) is constrained by (VII.11) with \(W(t)\) provided by the atmospheric module.</p> <p> Remark VII.3 (Interpretation and limits).<br /> Equations (VII.8)–(VII.12) should be read as a conservative accounting device: as a region is depleted (smaller \(W\)), any thermodynamically consistent minimum-work benchmark must increase at least logarithmically in the depletion factor if RH scales proportionally. The precise \(\beta\) mapping is site- and flow-regime-dependent and must be verified for quantitative predictions.</p> <p> VII.C. Concavity of regional yield vs deployment intensity (a tractable diminishing-returns result)</p> <p> A key planning consequence of coupling (VII.1) to (VII.5) is diminishing returns: adding more deployed area increases the sink but also depletes moisture, reducing per-device yield. Under simple linear humidity-limited closures, the dependence of cumulative harvested water on a scalar deployment intensity can be concave, which is valuable for tractable optimization.</p> <p> We present a clean lemma in a deliberately simplified setting (no horizontal transport) that isolates the mathematical mechanism.</p> <p> Setup VII.4 (Well-mixed box moisture model with linear sink).<br /> Consider a single region with a scalar moisture state \(W(t)\ge 0\) satisfying<br /> \[<br /> \dot W(t) = s(t) – \theta\,\alpha(t)\,W(t),\qquad t\in[0,\tau],<br /> \tag{VII.13}<br /> \]<br /> where \(s(t)\ge 0\) is an exogenous supply term (net source), \(\alpha(t)\ge 0\) is a transfer coefficient, and \(\theta\ge 0\) is a deployment intensity parameter. The harvested mass rate (sink) is<br /> \[<br /> S_{\mathrm{AWH}}(t)=\theta\,\alpha(t)\,W(t).<br /> \tag{VII.14}<br /> \]<br /> Define the total harvested mass over \([0,\tau]\) by<br /> \[<br /> Y(\theta):=\int_0^{\tau} S_{\mathrm{AWH}}(t)\,dt.<br /> \tag{VII.15}<br /> \]</p> <p> Proposition VII.5 (Concavity of harvest vs deployment in the constant-coefficient case).<br /> Assume \(\alpha(t)\equiv \alpha>0\) and \(s(t)\equiv s\ge 0\) are constant. Then \(Y(\theta)\) is concave in \(\theta\ge 0\). In particular, when \(s=0\) (pure depletion of an initial stock \(W(0)=W_0\)),<br /> \[<br /> Y(\theta)= W_0\big(1-e^{-\theta\alpha\tau}\big)<br /> \tag{VII.16}<br /> \]<br /> which is increasing and concave in \(\theta\).</p> <p> Proof.<br /> Solve (VII.13): for constant \(s,\alpha\),<br /> \[<br /> W(t)=\Big(W_0-\frac{s}{\theta\alpha}\Big)e^{-\theta\alpha t}+\frac{s}{\theta\alpha}\quad(\theta>0),<br /> \]<br /> with the \(\theta=0\) limit given by \(W(t)=W_0+st\). Substituting into (VII.14) and integrating yields<br /> \[<br /> Y(\theta)=\theta\alpha\int_0^{\tau}W(t)dt<br /> =\big(W_0-\tfrac{s}{\theta\alpha}\big)\big(1-e^{-\theta\alpha\tau}\big)+s\tau.<br /> \]<br /> In the stock-only case \(s=0\), (VII.16) follows immediately, and concavity holds because \(\theta\mapsto 1-e^{-k\theta}\) is concave for \(k>0\). For \(s>0\), the additional terms preserve concavity on \(\theta>0\) after simplification (the second derivative is nonpositive); the \(\theta=0\) value is obtained by continuity. \(\square\)</p> <p> Remark VII.6 (What this does and does not claim).<br /> Proposition VII.5 shows concavity in a minimal depletion model. In the full PDE setting (VII.1), concavity depends on transport, boundary conditions, and the correctness of the linear sink closure (VII.7). We therefore use concavity as an *optional* structural assumption when justified by calibration, not as a guaranteed property.</p> <p> End of Section VII.</p> <p> VIII. Competitive and Congested Deployment: Mean-Field and Variational Inequality Models</p> <p> Large-scale AWH deployment can create externalities: each unit’s vapor removal reduces the ambient humidity available to others, raising their thermodynamic “price” \(c(t)=R_vT\ln(1/\mathrm{RH})\) and possibly binding the regional moisture budget (Section VII). This section provides equilibrium templates for such competition. We deliberately separate: (i) equilibrium *definitions* and (ii) any quantitative claim that requires calibration of the atmospheric operator or device response.</p> <p> VIII.A. Mean-Field Game (MFG) formulation for many agents harvesting a shared humidity field</p> <p> VIII.A.1. Minimal MFG ingredients and coupling choice</p> <p> Let \(\Omega\subset\mathbb{R}^d\) be a bounded region (typically \(d=2\) for a horizontal planning domain). Let \(m(x,t)\ge 0\) be the agent density (units: devices per land area) and let \(u(x,t)\in[0,\bar u(x,t)]\) be a control representing harvesting intensity per agent (e.g., usable exergy rate per device, or a normalized actuation variable).</p> <p> We introduce two maps that must be specified (and validated) in applications:</p> <p> 1. **Per-agent yield map** \(y\):<br /> \[<br /> y(x,t)=y\big(u(x,t),\,\xi(x,t)\big),\qquad y\ge 0,<br /> \]<br /> where \(\xi\) is a local climate/moisture state (e.g., \(T,\mathrm{RH},W\)). A device-agnostic admissible envelope is<br /> \[<br /> y(u,\xi)\le \frac{e(u)}{c(\xi)}\quad\text{(Theorem III.1),}<br /> \]<br /> with additional tightening terms if one includes pumping or other parasitics.</p> <p> 2. **Atmospheric coupling operator** \(\mathcal{A}\): a (possibly time-dependent) PDE or operator mapping total sink to a humidity/moisture state \(\xi\). For example, one may couple to the column moisture variable \(W\) via a linear transport operator<br /> \[<br /> \partial_t W + \nabla\cdot F(W)=S_{\mathrm{atm}}-S_{\mathrm{AWH}},\qquad S_{\mathrm{AWH}}(x,t)=m(x,t)\,y(x,t),<br /> \]<br /> as in Section VII, with boundary and initial conditions defining \(\mathcal{A}\).</p> <p> The defining feature of competition is the feedback<br /> \[<br /> (m,u)\ \mapsto\ S_{\mathrm{AWH}}\ \mapsto\ \xi\ \mapsto\ y(\cdot,\xi),\ c(\xi),\ \bar e(\xi),\ \ldots<br /> \]</p> <p> VIII.A.2. Individual optimization problem and equilibrium definition</p> <p> A standard MFG assumes each agent’s location evolves as a controlled diffusion<br /> \[<br /> dX_t = a_t\,dt + \sqrt{2\nu}\,dB_t,\qquad X_t\in\Omega,<br /> \]<br /> with control \(a_t\in\mathbb{R}^d\) representing relocation effort (for fixed installations one may instead set \(a\equiv 0\) and drop the relocation term; the MFG then reduces to a static congestion game).</p> <p> Let \(u_t\in[0,\bar u(X_t,t)]\) be the harvesting intensity. For a running payoff (reward minus cost) functional, one may take<br /> \[<br /> \mathbb{E}\Big[\int_0^\tau \big(\;\\underbrace{\Pi\big(y(u_t,\xi(X_t,t))\big)}_{\text{benefit}}\; -\; \\underbrace{\Psi(u_t,\xi(X_t,t))}_{\text{exergy/parasitic cost}}\; -\; \\underbrace{\tfrac{\gamma}{2}\|a_t\|^2}_{\text{relocation}}\big)\,dt\Big],<br /> \]<br /> where \(\Pi\) is concave nondecreasing (diminishing value of additional water) and \(\Psi\) is convex nondecreasing in \(u\). The second-law outer bound is enforced by selecting \(\Psi\) or feasible sets so that producing \(y\) requires exergy at least \(c(\xi) y\) (and possibly additional guaranteed penalties from Section IV).</p> <p> **Definition VIII.1 (MFG equilibrium; template).** A mean-field equilibrium is a pair \((m,u)\) (and associated value function \(V\) when relocation is present) such that:</p> <p> 1. Given the mean field \(m\) (hence the atmospheric state \(\xi=\mathcal{A}(m,u)\)), the control \(u\) (and drift \(a\), if present) maximizes the representative agent’s expected payoff.<br /> 2. The population density \(m\) is consistent with the optimal drift \(a\) via the Fokker–Planck–Kolmogorov equation<br /> \[<br /> \partial_t m – \nu\Delta m + \nabla\cdot(am)=0<br /> \]<br /> (with appropriate boundary conditions), and the sink closure holds: \(S_{\mathrm{AWH}}=m\,y(u,\xi)\).</p> <p> In the diffusion-with-relocation case, formal optimality yields the coupled HJB–FPK system<br /> \[<br /> -\partial_t V – \nu\Delta V + H\big(x,t,\nabla V;\xi\big)=0,\qquad \partial_t m – \nu\Delta m – \nabla\cdot\big(m\,\partial_p H(x,t,\nabla V;\xi)\big)=0,<br /> \]<br /> where the Hamiltonian is<br /> \[<br /> H(x,t,p;\xi)=\sup_{u\in[0,\bar u]}\sup_{a\in\mathbb{R}^d}\Big\{\Pi\big(y(u,\xi)\big)-\Psi(u,\xi)-\tfrac{\gamma}{2}\|a\|^2 – a\cdot p\Big\}.<br /> \]</p> <p> Remark VIII.2 (Existence/uniqueness are conditional). Standard MFG existence and uniqueness results require structural hypotheses (regularity, convexity of costs, and a monotonicity condition on the coupling). In AWH, these hypotheses translate into: (i) well-posedness of the atmospheric operator \(\mathcal{A}\) under sinks \(S_{\mathrm{AWH}}\), and (ii) a coupling where the “cost of harvesting” increases with aggregate harvesting (a form of anti-congestion in the payoff). Because the physically relevant coupling (humidity depletion) indeed increases thermodynamic cost, monotonicity is plausible, but it must be checked for the specific discretized operator used.</p> <p> VIII.A.3. Efficiency loss and the need for explicit feasibility caps</p> <p> Even without proving quantitative “price of anarchy” bounds, two qualitative facts are immediate from Sections III and VII:</p> <p> 1. **Second-law feasibility is separable but competition is not.** The pointwise bound \(y\le e/c\) is local, but \(c\) itself becomes a function of the shared field (Section VII.B), introducing global coupling.</p> <p> 2. **The moisture-budget cap is a hard constraint for all equilibria.** Any equilibrium sink must satisfy the integrated conservation inequality (VII.3) (or its discretization (VII.4)). Therefore, planning layers should enforce (VII.4) directly, rather than relying on equilibrium alone to prevent overdeployment.</p> <p> VIII.B. Water equity metrics in the mean-field limit</p> <p> Equity constraints are naturally expressed in terms of the spatial yield distribution. We record a tractable inequality relating a Gini-style functional to Sobolev regularity; this is useful because gradient penalties/constraints are convex and optimization-friendly.</p> <p> VIII.B.1. Spatial yield field and Gini functional</p> <p> Let \(w(x)\ge 0\) be the total harvested water per land area over a horizon \([0,\tau]\), e.g.<br /> \[<br /> w(x):=\int_0^\tau S_{\mathrm{AWH}}(x,t)\,dt = \int_0^\tau m(x,t)\,y\big(u(x,t),\xi(x,t)\big)\,dt.<br /> \]<br /> Let \(|\Omega|\) denote region area and \(\bar w:=|\Omega|^{-1}\int_\Omega w(x)\,dx\) the mean. A standard (dimensionless) Gini coefficient can be written as<br /> \[<br /> G(w):=\frac{1}{2|\Omega|^2\,\bar w}\int_\Omega\int_\Omega |w(x)-w(x’)|\,dx\,dx’.<br /> \]</p> <p> VIII.B.2. A Sobolev-to-Gini bound (rigorous sufficient condition)</p> <p> Proposition VIII.3 (Controlling \(\|\nabla w\|_{L^2}\) controls \(G(w)\)).<br /> Assume \(w\in H^1(\Omega)\), \(w\ge 0\), and \(\bar w>0\). Let \(C_P\) be a Poincar\’e constant for \(\Omega\), i.e., \(\|w-\bar w\|_{L^2}\le C_P\|\nabla w\|_{L^2}\). Then<br /> \[<br /> G(w)\le \frac{|\Omega|^{1/2}}{\bar w}\,\|w-\bar w\|_{L^2}\le \frac{|\Omega|^{1/2}C_P}{\bar w}\,\|\nabla w\|_{L^2}.<br /> \]</p> <p> Proof.<br /> First, by Jensen and symmetry,<br /> \[<br /> \int_\Omega\int_\Omega |w(x)-w(x’)|\,dx\,dx’ \le 2|\Omega|\int_\Omega |w(x)-\bar w|\,dx = 2|\Omega|\,\|w-\bar w\|_{L^1}.<br /> \]<br /> Then \(\|f\|_{L^1}\le |\Omega|^{1/2}\|f\|_{L^2}\) gives<br /> \[<br /> G(w)\le \frac{1}{2|\Omega|^2\bar w}\,2|\Omega|\,|\Omega|^{1/2}\|w-\bar w\|_{L^2}=\frac{|\Omega|^{1/2}}{\bar w}\|w-\bar w\|_{L^2}.<br /> \]<br /> Apply Poincar\’e. \(\square\)</p> <p> Remark VIII.4 (How to use this in optimization).<br /> Rather than asserting that a particular gradient penalty “guarantees” a Gini cap, one can enforce the sufficient condition explicitly:<br /> \[<br /> \|\nabla w\|_{L^2}\le \Gamma \quad\Longrightarrow\quad G(w)\le \frac{|\Omega|^{1/2}C_P}{\bar w}\,\Gamma.<br /> \]<br /> Since \(\|\nabla w\|_{L^2}\) can be represented as an SOCP constraint after discretization, this yields an auditable, solver-ready equity surrogate.</p> <p> VIII.C. Continuous deployment under nonlocal atmospheric constraints (VI / congestion)</p> <p> Mean-field equilibria can also be formulated as variational inequalities (VI), especially when: (i) the atmosphere responds linearly (or is linearized) to sinks, and (ii) each agent’s best response is the solution of a convex program.</p> <p> VIII.C.1. Linearized atmospheric response and a nonlocal coupling operator</p> <p> Assume a (steady or quasi-steady) linear relationship between the sink field \(s(x)\ge 0\) and a depletion field \(\delta(x)\ge 0\) that affects thermodynamic cost, e.g.<br /> \[<br /> \delta = \mathcal{G}s,<br /> \]<br /> where \(\mathcal{G}\) is a positive linear operator (for elliptic operators, \(\mathcal{G}\) is a Green’s operator; for time-dependent problems, it may be a convolution in time plus a spatial operator). A depletion-aware cost may then depend on \(\delta\) through \(-\ln(1-\delta)\)-type surrogates or through \(c(\delta)\) as in (VII.9)–(VII.12).</p> <p> VIII.C.2. VI statement for a convex population game</p> <p> Let the feasible set of deployment/sink decisions be a closed convex set \(K\subset L^2(\Omega)\) with \(s\ge 0\) a.e. (examples: land-use caps, regional moisture-budget caps (VII.4) after discretization, and per-site upper bounds). Suppose each infinitesimal agent’s marginal cost depends on the aggregate sink via \(\delta=\mathcal{G}s\) and is representable as a monotone map \(F:K\to K^*\), where \(F(s)\) is the pointwise “marginal disutility” of increasing sink.</p> <p> **Definition VIII.5 (Variational inequality equilibrium).** A sink field \(s^*\in K\) is a (Wardrop/Nash) equilibrium if<br /> \[<br /> \langle F(s^*),\, s-s^*\rangle \ge 0\qquad\text{for all } s\in K.<br /> \tag{VIII.1}<br /> \]</p> <p> Proposition VIII.6 (Existence under continuity and compactness; uniqueness under strong monotonicity).<br /> If \(K\) is nonempty, convex, and weakly compact in \(L^2\), and \(F\) is continuous from \(K\) to \(L^2\), then a solution to (VIII.1) exists. If additionally \(F\) is strongly monotone, i.e.<br /> \[<br /> \langle F(s_1)-F(s_2),\,s_1-s_2\rangle \ge \alpha\|s_1-s_2\|_{L^2}^2\quad\text{for some }\alpha>0,<br /> \]<br /> then the equilibrium is unique.</p> <p> Comment.<br /> In AWH, strong monotonicity corresponds to: increasing total sink increases marginal exergy cost sufficiently strongly (e.g., via depletion penalties and/or binding moisture-budget constraints), preventing multiple equilibria. Whether a given discretized moisture operator and cost model satisfies strong monotonicity is a concrete, checkable condition.</p> <p> VIII.C.3. Relation to constrained social planning</p> <p> If there exists a convex potential \(\Phi(s)\) such that \(F(s)=\nabla\Phi(s)\), then VI solutions coincide with minimizers of \(\Phi\) over \(K\). This “potential game” structure is valuable because it permits interpreting equilibrium as the solution of a single convex program; however, it should be viewed as a modeling convenience rather than a guaranteed property.</p> <p> VIII.D. Batch-mode impulse deployment interactions (link to QVI control)</p> <p> Impulse operation (Section V.E) introduces discrete-time/batch sinks that can couple nonlinearly across agents through the shared humidity field.</p> <p> VIII.D.1. Coupled feasibility via shared moisture budgets</p> <p> If agents act in batches, the sink term in the moisture balance becomes impulsive in time:<br /> \[<br /> S_{\mathrm{AWH}}(x,t)=\sum_n \Delta S_n(x)\,\delta(t-\tau_n).<br /> \]<br /> Even in this case, the integrated feasibility inequality (VII.2) remains valid over any interval \([t_0,t_1]\); thus one can constrain total extracted mass in each region and time window, regardless of whether extraction is continuous or impulsive.</p> <p> VIII.D.2. Open questions (flagged)</p> <p> A rigorous equilibrium theory for many interacting impulse-controlled agents with a shared PDE state typically requires additional compactness and monotonicity structure and is beyond the scope of this blueprint. For our purposes, the actionable point is: batch-mode interaction should be modeled either by (i) explicit regional caps (VII.4) or (ii) a VI/QVI with a shared-state constraint, rather than by assuming independence across devices.</p> <p> End of Section VIII.</p> <p> IX. Distributionally Robust PDE-Constrained Optimization (DR-PDECO) for Co-Design</p> <p> This section gives a tractable mathematical template for *co-design*: jointly choosing (i) device/material parameters, (ii) operating/control policies, and (iii) siting/deployment decisions, while enforcing (a) second-law exergy necessities (Sections II–III), (b) optional device-physics tightenings (Section IV), and (c) atmospheric feasibility caps and depletion penalties (Section VII). The unifying objective is to produce optimization problems whose *conservatism is auditable*: every constraint is explicitly labeled as either (A) a guaranteed necessity under stated assumptions, or (B) an optional modeling closure whose validity must be checked.</p> <p> IX.A. Co-design variables: material properties + operating policies + siting decisions</p> <p> We fix a finite set of candidate sites (or grid cells) \(i\in\{1,\dots,N\}\) and a discrete time grid \(k\in\{1,\dots,K\}\) with time step \(\Delta t\).</p> <p> Decision variables (representative list).</p> <p> 1. **Siting / deployment**: installed device area (or deployment intensity) \(a_i\ge 0\) at each site \(i\), subject to land/material budgets such as<br /> \[<br /> \sum_{i=1}^N a_i \le A_{\max},\qquad 0\le a_i\le \bar a_i.<br /> \]</p> <p> 2. **Material / design**: a vector \(\theta\in\Theta\subset\mathbb{R}^p\) (e.g., effective diffusivity parameters, heat capacity, emissivity, isotherm coefficients), either continuous or selected from a finite catalog. Catalog selection may be encoded by binary variables \(b\in\{0,1\}^M\) with \(\sum_m b_m=1\) and \(\theta=\sum_m b_m\,\theta^{(m)}\).</p> <p> 3. **Control / operation**: at each site \(i\) and time step \(k\), usable exergy allocation \(e_{i,k}\ge 0\), harvested water flux \(y_{i,k}\ge 0\), and optional auxiliary variables (storage states \(s_{i,k}\), pumping power \(p_{i,k}\), mode indicators \(\delta_{i,k}\in\{0,1\}\), etc.).</p> <p> Guaranteed thermodynamic constraints (outer layer).</p> <p> At each site/time, whenever the ambient-temperature reversible benchmark applies (Section III.A), we must enforce<br /> \[<br /> \text{(A1)}\quad y_{i,k}\,c_{i,k} \le e_{i,k},\qquad c_{i,k}:=R_v T_{i,k}\ln\frac{1}{\mathrm{RH}_{i,k}}.<br /> \]<br /> Optional tightenings add nonnegative parasitic terms (fan work, diffusion penalties, exhaust conditioning), e.g.<br /> \[<br /> \text{(B1)}\quad e_{i,k} \ge y_{i,k}c_{i,k} + p^{\mathrm{fan}}_{i,k} + p^{\mathrm{aux}}_{i,k}.<br /> \]</p> <p> IX.B. Reduced-order PDE constraints embedded in optimization</p> <p> This subsection explains how a reduced-order model (ROM) from Section IV.C can be embedded as constraints, and under what assumptions one obtains mixed-integer conic programs.</p> <p> IX.B.1. ROM dynamics and yield mapping</p> <p> For each site \(i\), let \(z_{i,k}\in\mathbb{R}^r\) be a reduced state (e.g., POD coefficients for \((T,q)\)). A conservative optimization-ready ROM template is an affine, time-varying recursion<br /> \[<br /> \text{(B2)}\quad z_{i,k+1} = A_{i,k}(\theta)z_{i,k} + B_{i,k}(\theta)u_{i,k} + d_{i,k}(\theta),<br /> \]<br /> where \(u_{i,k}\) collects controllable inputs such as heating power, airflow, or mode-dependent actuation.</p> <p> The yield is constrained by a concave envelope (Section II.D.1), possibly ROM-derived:<br /> \[<br /> \text{(B3)}\quad y_{i,k} \le \phi_{i,k}(e_{i,k};\theta,\xi_{i,k}),<br /> \]<br /> with \(\xi_{i,k}\) climate inputs and \(\phi_{i,k}\) concave nondecreasing in \(e_{i,k}\). The second-law slope constraint enforces admissibility:<br /> \[<br /> \text{(A2)}\quad \phi_{i,k}(e) \le \frac{e}{c_{i,k}}\quad\text{for all }e\ge 0.<br /> \]</p> <p> IX.B.2. Switching logic encoded with binaries</p> <p> Hybrid sorption systems (Section IV.E) motivate mode indicators \(\delta_{i,k}\in\{0,1\}\) (e.g., \(\delta=1\) adsorption, \(0\) desorption). A standard disjunctive template is<br /> \[<br /> \text{(B4)}\quad (y_{i,k},z_{i,k+1})\in \mathcal{C}^{\mathrm{ads}}_{i,k}(\theta)\ \text{if }\delta_{i,k}=1,\qquad (y_{i,k},z_{i,k+1})\in \mathcal{C}^{\mathrm{des}}_{i,k}(\theta)\ \text{if }\delta_{i,k}=0,<br /> \]<br /> where \(\mathcal{C}^{\mathrm{ads}}\), \(\mathcal{C}^{\mathrm{des}}\) are convex sets encoding admissible dynamics/yield in each mode. A mixed-integer formulation is obtained by standard big-\(M\) or indicator-constraint encodings; any such encoding must be checked for numerical stability (choice of \(M\)) and for whether it preserves outer-bound conservatism.</p> <p> IX.B.3. When does one obtain MI-conic tractability?</p> <p> The following proposition is an explicit *assumption-to-tractability* checklist rather than a universal claim.</p> <p> Proposition IX.1 (Sufficient conditions for MI-conic co-design).<br /> Assume:</p> <p> (i) The feasible set \(\Theta\) for continuous design variables is convex, and catalog selection (if present) is encoded by binaries with an exact convex-combination representation.</p> <p> (ii) The ROM constraints are affine in the optimization variables \((z,u)\) for fixed \(\theta\), as in (B2), and any dependence on \(\theta\) is either affine or represented by a finite catalog (so that bilinear terms \(A(\theta)z\) do not appear as continuous bilinearities).</p> <p> (iii) Each yield envelope \(\phi_{i,k}(\cdot)\) is represented as the pointwise minimum of finitely many affine functions (piecewise-linear concave):<br /> \[<br /> \phi_{i,k}(e)=\min_{j\in\{1,\dots,J\}}\{\alpha_{i,k,j}e+\beta_{i,k,j}\},\quad \alpha_{i,k,j}\le 1/c_{i,k}.<br /> \]<br /> Then constraints of the form \(y_{i,k}\le \phi_{i,k}(e_{i,k})\) are representable by linear inequalities \(y_{i,k}\le \alpha_{i,k,j}e_{i,k}+\beta_{i,k,j}\) for all \(j\), hence the resulting co-design problem is a mixed-integer linear or mixed-integer second-order cone program (MI-SOCP) depending on whether additional SOCP constraints (e.g., quadratic pumping bounds) are included.</p> <p> If, in addition, depletion penalties \(-\ln(\cdot)\) or WFR-type dual constraints appear, an exponential-cone representable mixed-integer conic program (MI-EXP) is obtained.</p> <p> Justification.<br /> Under (i)–(iii), all constraints are linear except for optional quadratic and \(-\ln\)/log-sum-exp terms, which are SOCP- or exponential-cone representable as discussed in Sections II.D.2 and VII.B.3.</p> <p> Caveat.<br /> If continuous bilinearities \(A(\theta)z\), \(\theta e\), or \(\theta y\) are present, exact MI-conic representability generally fails without additional reformulations or relaxations (e.g., McCormick envelopes, piecewise-linearization). Any such relaxation must be reported explicitly because it changes the meaning of “guaranteed bound”. \(\square\)</p> <p> IX.C. DR-PDECO under climate ambiguity</p> <p> We now place the co-design problem under distributional uncertainty in climate inputs \(\xi\). The goal is to enforce performance constraints (yield, feasibility margins) uniformly over an ambiguity set, or in worst-case expectation.</p> <p> IX.C.1. Scenario form and ambiguity sets</p> <p> Let \(\widehat{\mathbb{P}}\) be an empirical distribution supported on samples \(\{\xi^{(n)}\}_{n=1}^N\). Let \(\mathcal{P}\) be an ambiguity set (Wasserstein ball \(\mathcal{P}_{W_1}(\varepsilon)\) or an unbalanced alternative).</p> <p> A typical robust-yield guarantee is<br /> \[<br /> \text{(R1)}\quad \inf_{\mathbb{P}\in\mathcal{P}}\ \mathbb{E}_{\mathbb{P}}\big[Y(u,\xi)\big]\ \ge\ Y_{\min},<br /> \]<br /> where \(u\) now denotes the full design/control vector (including \(a,\theta,\{e_{i,k}\}\), etc.). A robust-cost objective is similarly<br /> \[<br /> \min_u\ \sup_{\mathbb{P}\in\mathcal{P}}\ \mathbb{E}_{\mathbb{P}}\big[C(u,\xi)\big].<br /> \]</p> <p> IX.C.2. Wasserstein DRO for Lipschitz costs (explicit bound)</p> <p> For implementation one often replaces the worst-case expectation by a computable upper bound based on Lipschitz constants.</p> <p> Lemma IX.2 (Lipschitz upper bound for Wasserstein-1 DRO).<br /> Let \(f(\xi)\) be \(L\)-Lipschitz with respect to metric \(d\) on \(\Xi\): \(|f(\xi)-f(\zeta)|\le L d(\xi,\zeta)\). Then<br /> \[<br /> \sup_{\mathbb{P}:W_1(\mathbb{P},\widehat{\mathbb{P}})\le \varepsilon}\ \mathbb{E}_{\mathbb{P}}[f(\xi)]\ \le\ \mathbb{E}_{\widehat{\mathbb{P}}}[f(\xi)] + L\varepsilon.<br /> \]</p> <p> Proof.<br /> This is a direct consequence of the Kantorovich–Rubinstein dual characterization of \(W_1\): the difference in expectations is bounded by \(L W_1\). \(\square\)</p> <p> Discussion.<br /> Lemma IX.2 yields an auditable “robustification margin” \(L\varepsilon\), but it can be loose. Tighter reformulations can be obtained from Proposition VI.1 by evaluating the inner \(\sup_{\zeta}\) exactly for the chosen \(f\) and metric.</p> <p> IX.C.3. Robust chance constraints and CVaR (optimization-ready)</p> <p> Reliability constraints can be enforced via a CVaR surrogate (Section VI.B.3). For a feasibility margin \(F(u,\xi)\) (positive means feasible), one may require<br /> \[<br /> \text{(R2)}\quad \sup_{\mathbb{P}\in\mathcal{P}}\mathrm{CVaR}_{\beta,\mathbb{P}}\big(-F(u,\xi)\big)\le 0.<br /> \]<br /> After epigraphization of CVaR, tractability reduces to tractability of the DRO expectation of a hinge function; this depends on the ambiguity model and on whether \(F\) is convex/concave in \(u\). We do not assert a universal conic reformulation here; rather, (R2) is provided as a precise target constraint to be specialized once \(\mathcal{P}\) and \(F\) are fixed.</p> <p> IX.D. Integration of atmospheric moisture constraints and depletion penalties into DR-PDECO</p> <p> IX.D.1. Regional linear feasibility caps</p> <p> Let \(S_{\mathrm{AWH},i,k}\) be the atmospheric sink in cell \(i\) and time step \(k\). Under the closure \(S_{\mathrm{AWH}}=\theta y\) (Section VII.B), in a discrete siting model we set<br /> \[<br /> S_{\mathrm{AWH},i,k} = a_i\,y_{i,k}\quad\text{(or }S_{\mathrm{AWH},i,k}\ge a_i\,y_{i,k}\text{ if losses are modeled).}<br /> \]<br /> Then the regional moisture cap (VII.4) becomes a linear constraint in the decision variables \(\{a_i,y_{i,k}\}\): for each planning region \(R\) and time window,<br /> \[<br /> \sum_{k}\sum_{i\in\mathcal{I}(R)} A_i\,S_{\mathrm{AWH},i,k}\,\Delta t\ \le\ \mathrm{Supply}_R.<br /> \]<br /> This constraint is *hard feasibility* regardless of device model details.</p> <p> IX.D.2. Convex depletion-aware exergy accounting</p> <p> If the planner includes a depletion factor \(\beta\) (or column water \(W\)) and adopts the proportional-RH scaling template (VII.8)–(VII.10), then the reversible cost per kilogram is strengthened by an additive \(-\ln\) penalty. In discrete time, introduce a variable \(z_{i,k}\) satisfying<br /> \[<br /> \text{(B5)}\quad z_{i,k} \ge \ln\frac{W_{0,i,k}}{W_{i,k}},\qquad W_{i,k}>0.<br /> \]<br /> The second-law inequality becomes<br /> \[<br /> \text{(A3)}\quad e_{i,k} \ge y_{i,k}\,R_vT_{i,k}\Big(\ln\frac{1}{\mathrm{RH}_{0,i,k}} + z_{i,k}\Big),<br /> \]<br /> which is affine in \((e,y,z)\). Constraint (B5) is convex and can be represented with the exponential cone (Section II.D.2) after rewriting as \(\exp(-z_{i,k})\le W_{i,k}/W_{0,i,k}\).</p> <p> IX.D.3. Equity coupling via spatial aggregation (pointer)</p> <p> Equity objectives or constraints may be imposed on spatially aggregated yield \(w(x)\) as in Section VIII.B. In a discrete siting model, one may define per-cell delivered water over the horizon,<br /> \[<br /> W^{\mathrm{harv}}_i := \sum_{k=1}^K A_i\,S_{\mathrm{AWH},i,k}\,\Delta t,<br /> \]<br /> and impose convex surrogates such as gradient regularization (when \(i\) corresponds to a mesh) or minimum-per-capita constraints when population weights are available. The key point for DR-PDECO is that such constraints depend on \(S_{\mathrm{AWH}}\), so they are naturally compatible with the same atmospheric feasibility variables already required for (VII.4).</p> <p> End of Section IX.</p> <p> X. Global Deployment Optimization and Transport-Based Planning</p> <p> Sections III–IX provide (i) device-agnostic second-law necessities, (ii) optional device-physics tightenings, and (iii) atmospheric feasibility caps and depletion penalties. This section describes an outer planning layer that turns those ingredients into auditable, large-scale siting and allocation problems. The emphasis is on *upper bounds* and *screening metrics* that can be computed from climate time series and conservative device envelopes, without assuming that any particular device can attain those envelopes.</p> <p> X.A. Outer-layer viability screening and upper bounds</p> <p> X.A.1. Site screening metrics from time series</p> <p> Fix a site (or grid cell) indexed by \(s\) with diurnal horizon \([0,\tau]\). Let the ambient reversible cost be<br /> \[<br /> c_s(t)=R_v T_s(t)\ln\frac{1}{\mathrm{RH}_s(t)}\quad [\mathrm{J/kg}]<br /> \]<br /> (Section II.B.1), and let \(\bar e_s(t)\ge 0\) denote a conservative cap on usable exergy input rate per device area at that site (Section II.C), optionally reduced further by technology limits.</p> <p> A minimal screening scalar is the *best-case reversible daily yield per device area* obtained by saturating the pointwise second-law bound (Theorem III.1):<br /> \[<br /> \bar Y_s^{\mathrm{rev}} := \int_0^{\tau} \frac{\bar e_s(t)}{c_s(t)}\,dt\qquad [\mathrm{kg/m}^2\text{ per diurnal cycle}].<br /> \tag{X.1}<br /> \]<br /> This quantity is an *outer bound*: any realizable device yield per area must satisfy \(Y_s\le \bar Y_s^{\mathrm{rev}}\) under the stated benchmark assumptions, since \(y(t)\le e(t)/c(t)\le \bar e(t)/c(t)\) a.e. and integration preserves inequality.</p> <p> Optional tightenings can be incorporated by replacing \(\bar e_s\) by a net cap after unavoidable parasitics (e.g., fan work lower bounds from Section IV.F turned into a reduced usable exergy budget) or by replacing \(c_s(t)\) by a strengthened cost including depletion penalties (Section VII.B). Because each tightening is additive/nonnegative in the exergy accounting, any such replacement only decreases \(\bar Y_s^{\mathrm{rev}}\) and therefore preserves screening conservatism.</p> <p> X.A.2. Fractional-knapsack upper bound under a daily exergy budget</p> <p> Sometimes a planner posits only a daily usable exergy budget per device area, \(E_s\), rather than a full profile \(\bar e_s(t)\). In that case, the best-case yield under the reversible linear conversion is the continuous-time LP of Section V.A. Specializing Theorem V.1 yields the explicit outer bound<br /> \[<br /> \bar Y_s(E_s) := \sup_{0\le e\le \bar e_s,\ \int_0^{\tau} e\le E_s}\ \int_0^{\tau} \frac{e(t)}{c_s(t)}\,dt.<br /> \tag{X.2}<br /> \]<br /> This is the value of a fractional knapsack problem with value density \(1/c_s(t)\). An auditable implementation is: sort time indices by increasing \(c_s(t)\) (or decreasing \(1/c_s(t)\)), allocate \(e(t)=\bar e_s(t)\) until the budget \(E_s\) is exhausted, and allocate partially on the final tied interval if needed.</p> <p> Remark X.1 (Interpretation as a climate-driven upper benchmark).<br /> The bound \(\bar Y_s(E_s)\) depends only on \(c_s(t)\) and the assumed exergy availability. It does not assume any particular device response besides the reversible outer constraint, so it should be interpreted as a *necessary* upper benchmark for any technology that claims to use a daily usable-exergy amount \(E_s\) at that site.</p> <p> X.A.3. Kinetics-aware screening as a modular add-on</p> <p> The reversible screening metrics above ignore finite kinetics. A conservative way to incorporate kinetics without committing to a detailed device is to introduce a site-dependent utilization factor \(\eta_{\mathrm{kin},s}\in(0,1]\) such that<br /> \[<br /> Y_s \le \eta_{\mathrm{kin},s}\,\bar Y_s^{\mathrm{rev}}.<br /> \tag{X.3}<br /> \]<br /> Any choice of \(\eta_{\mathrm{kin},s}\) must be justified by an external calibration of diffusion/transfer parameters (Section IV.A) or by an admissible concave envelope \(\phi_{t,s}\) generated from a validated reduced-order model (Section IV.C). In the absence of such validation, one should take \(\eta_{\mathrm{kin},s}=1\), keeping (X.1) purely thermodynamic.</p> <p> X.B. Transport-based deployment planning</p> <p> Screening yields a set of per-site outer capacity bounds (in water per device area per day) and feasibility caps (in vapor removal per land area) from Section VII. A planning layer must additionally decide *where* produced water is delivered, and how siting interacts with equity and logistics.</p> <p> X.B.1. Semi-discrete transport LP (auditably solvable)</p> <p> Let sites be indexed by \(i\in\{1,\dots,N\}\), and demand locations by \(j\in\{1,\dots,M\}\). Let \(p_i\ge 0\) be total produced water allocated from site \(i\) over a planning horizon (e.g., kg/day or kg/year), with outer capacity constraints<br /> \[<br /> 0\le p_i \le \bar p_i,<br /> \tag{X.4}<br /> \]<br /> where \(\bar p_i\) can be taken as (device area installed at \(i\)) \(\times\) (outer yield per device area bound computed from (X.1) or (X.2)), further intersected with atmospheric moisture-budget caps (Section VII.A).</p> <p> Let \(d_j\ge 0\) be required demand at location \(j\), and let \(\pi_{ij}\ge 0\) be the transported amount from site \(i\) to demand \(j\). With a per-unit transport cost \(k_{ij}\ge 0\) (e.g., proportional to distance, elevation lift, or an LCA proxy), the classical transportation problem is<br /> \[<br /> \min_{\pi\ge 0}\ \sum_{i=1}^N\sum_{j=1}^M k_{ij}\,\pi_{ij}<br /> \]<br /> subject to<br /> \[<br /> \sum_{j=1}^M \pi_{ij} \le p_i\ \ (\forall i),\qquad \sum_{i=1}^N \pi_{ij} \ge d_j\ \ (\forall j).<br /> \tag{X.5}<br /> \]<br /> This is a linear program. When \(\sum_i \bar p_i < \sum_j d_j\), feasibility fails, providing an auditable certificate that the screened production capacities are insufficient for the demand target. X.B.2. OT-inspired continuous formulation For a continuous domain \(\Omega\subset\mathbb{R}^2\), one may represent demand as a measure \(\mu\) on \(\Omega\) (e.g., population-weighted scarcity) and production potential as a measure \(\nu\) on \(\Omega\) (e.g., deployable land weighted by screened yield capacity). A transport plan is a coupling \(\gamma\) on \(\Omega\times\Omega\) with marginals dominated by \(\nu\) and \(\mu\) (to allow supply limits and possible unmet demand), depending on the planning convention. A basic balanced OT problem (when total supply equals total demand) is \[ \inf_{\gamma\in\Pi(\nu,\mu)}\ \int_{\Omega\times\Omega} c(x,y)\,\gamma(dx,dy), \tag{X.6} \] where \(c(x,y)\) is a per-unit delivery cost. A commonly used cost structure in this blueprint is \[ c(x,y)=\frac{d(x,y)}{\eta_{\mathrm{ex}}(x)}, \tag{X.7} \] where \(d(x,y)\) is a metric distance and \(\eta_{\mathrm{ex}}(x)\in(0,1]\) is a site-specific exergy-efficiency proxy (or, more conservatively, \(\eta_{\mathrm{ex}}\equiv 1\) so that (X.7) reduces to pure distance). We stress that inserting \(\eta_{\mathrm{ex}}(x)\) is a modeling choice: it is meaningful only if \(\eta_{\mathrm{ex}}\) is computed from validated device modules, and it must be bounded away from zero to avoid ill-posed costs. When supply and demand totals differ, an unbalanced transport model (Section II.D.3) is appropriate; in that case the planner must commit to a specific discrepancy and its dual (Section VI.B.4), and should report how the mass-mismatch penalty is chosen. X.B.3. Integrating atmospheric caps and land/material budgets Let \(a_i\) be installed device area at site \(i\). A conservative production capacity is \[ \bar p_i(a_i) := a_i\,\bar Y_i^{\mathrm{rev}}, \tag{X.8} \] with \(\bar Y_i^{\mathrm{rev}}\) from (X.1). Land/material budgets impose linear constraints \(\sum_i a_i\le A_{\max}\), \(0\le a_i\le \bar a_i\). Atmospheric feasibility caps (Section VII.A) constrain the aggregate sink over any region \(R\) and time window. In a coarse planning model these appear as linear constraints on \(\{a_i\}\) or \(\{p_i\}\) (depending on how the sink is aggregated). For example, if a planning region \(R\) is represented by an index set \(\mathcal{I}(R)\subset\{1,\dots,N\}\), a conservative cap is \[ \sum_{i\in\mathcal{I}(R)} p_i \le \mathrm{Supply}_R, \tag{X.9} \] where \(\mathrm{Supply}_R\) is computed from the discretized moisture balance (VII.4) for the chosen horizon. Constraint (X.9) is a hard feasibility condition independent of transport choices. X.C. Multi-scale coupling across layers: conservative-envelope principle X.C.1. From device physics to planning parameters Device-scale modules can inform planners via three classes of outputs: 1. **Admissible yield envelopes** \(\phi_{i,k}(e)\) (Section II.D.1), typically constructed as concave upper hulls of validated simulations. 2. **Guaranteed parasitic penalties** (e.g., fan-work lower bounds, Section IV.F) that add to required exergy for a target \(y\). 3. **Atmospheric sink surrogates** \(S_{\mathrm{AWH}}\) consistent with regional feasibility caps (Section VII). These outputs can be summarized into conservative site/time bounds such as (X.1) or into linear/conic constraints in co-design problems (Section IX). X.C.2. Outer approximation logic The thermodynamic bounds of Section III are outer bounds: they are satisfied by all devices under stated assumptions. Adding device physics (diffusion, pumping, switching) can only tighten feasibility by adding nonnegative required-exergy terms or by lowering the admissible \(\phi\) envelope. Therefore, a planning workflow that proceeds \[ \text{(i) screen using Section III outer bounds} \ \to\ \text{(ii) tighten with Section IV modules as available} \] produces auditable conservatism: any plan that fails in step (i) is impossible under the benchmark assumptions, while plans that pass step (i) require step (ii) to assess realism. Remark X.2 (No empirical performance claims). This section provides optimization templates and certified upper bounds from the second law plus conservation constraints. It does not claim that any real AWH technology can attain the screened yields; those claims require external validation of conversion efficiencies, transfer coefficients, and device response curves. End of Section X. XI. Convective Mass Transfer Optimization in Arrays (ABL–Device Coupling) This section extends the single-device pumping bounds of Section IV.F to array-level texture and layout optimization. The objective is to provide conservative, planning-layer constraints that couple atmospheric boundary layer (ABL) physics to engineered surface characteristics without requiring per-site 3D CFD. XI.A. Coupled ABL/device-scale framework for textures and layouts XI.A.1. Trade-off: enhanced mass transfer vs parasitic pumping power Consider an array of AWH devices with surface textures (fins, ridges, porous coatings) that enhance convective mass transfer. Let $h_m$ denote the mass-transfer coefficient and $\mathrm{Sh}=h_m L/D$ the Sherwood number for a characteristic device length $L$ and vapor diffusivity $D$. Standard correlations relate $\mathrm{Sh}$ to the Reynolds number $\mathrm{Re}=\rho_{da} U L/\mu$ and Schmidt number $\mathrm{Sc}=\mu/(\rho_{da} D)$ via power laws: \[ \mathrm{Sh}(\mathrm{Re},\mathrm{Sc},\mathcal{T}) = C(\mathcal{T})\,\mathrm{Re}^{a(\mathcal{T})}\,\mathrm{Sc}^{1/3}, \] where $\mathcal{T}$ parameterizes texture geometry (roughness height, fin density, aspect ratios) and $C(\mathcal{T}), a(\mathcal{T})$ are empirical functions that must be calibrated externally. Texture enhancement increases the effective surface area or turbulence intensity, raising $h_m$, but simultaneously increases the Darcy-Weisbach friction factor $f(\mathcal{T})$ and hence the pressure drop: \[ \Delta p(\mathrm{Re},\mathcal{T}) = f(\mathcal{T})\,\frac{\rho_{da} U^2}{2}\,\frac{L}{d_h}, \] with $d_h$ a hydraulic diameter. From Section IV.F.2, the fan power required to drive flow $Q$ through the array satisfies \[ P_{\mathrm{fan}}(t) \ge Q(t)\,\Delta p(t). \] Proposition XI.1 (Texture viability necessary condition). For a target water flux $j$ and ambient driving potential $\Delta\rho_v=\rho_{v,a}-\rho_{v,sat}(T_s)$, the texture parameters $\mathcal{T}$ must satisfy simultaneously: \[ h_m(\mathrm{Re},\mathcal{T}) \ge \frac{j}{\Delta\rho_v},\qquad P_{\mathrm{fan}} \ge \frac{j}{\rho_{da}\Delta\omega}\,\Delta p(\mathrm{Re},\mathcal{T}). \] Eliminating $U$ (via $Q=U A_{cs}$ and $\mathrm{Re}\propto U$) yields a necessary lower bound on the texture efficiency ratio: \[ \frac{C(\mathcal{T})}{f(\mathcal{T})^{a/(1+a)}} \ge \frac{j^{(1+2a)/(1+a)}}{(\rho_{da}\Delta\omega)^{(1+2a)/(1+a)}}\,\frac{(\mu L)^{a/(1+a)}}{D\,\mathrm{Sc}^{1/3}\,\Delta\rho_v}\,\mathrm{const}. \] When $f$ grows faster than $C^{(1+a)/a}$, textures become energetically nonviable regardless of climate. XI.A.2. Climate-dependent viability via Sherwood/Reynolds regimes The viability of surface enhancement depends critically on the ABL wind regime: **Laminar regime** ($\mathrm{Re} < \mathrm{Re}_{\mathrm{crit}}$): With $a\approx 1/2$ and $f\sim \mathrm{Re}^{-1}$ (Poiseuille-like), the power penalty scales as $P_{\mathrm{fan}}\sim j^{3}$. **Turbulent regime** ($\mathrm{Re} > \mathrm{Re}_{\mathrm{crit}}$): With $a\approx 0.8$ (for rough surfaces) and $f\sim \mathrm{Re}^{-0.25}$ to $\mathrm{Re}^{0}$ depending on roughness, the scaling becomes $P_{\mathrm{fan}}\sim j^{(2+a)/(1+a)}$, which is superlinear and typically $j^{1.5}$ to $j^{2}$.</p> <p> Corollary XI.2 (Climate-dependent texture screening).<br /> For a site with characteristic wind speed $U_{\mathrm{amb}}$ and device scale $L$, if the implied $\mathrm{Re}$ exceeds the critical value for the smooth-to-rough transition, aggressive textures (high $C$ but very high $f$) are unlikely to satisfy the system-level exergy constraint $E_{\mathrm{supply}} \ge \int (c(t)j(t) + P_{\mathrm{fan}}(t))\,dt$ unless the enhancement factor $\eta_{\mathrm{tex}}:=h_m(\mathcal{T})/h_m(\mathrm{flat})$ exceeds a site-specific threshold derived from Proposition XI.1.</p> <p> XI.B. Homogenized coefficients enabling tractable optimization</p> <p> XI.B.1. Reduced parametrizations for array-level planning</p> <p> For global siting optimization (Section X), resolving individual device textures in 3D CFD is intractable. We define homogenized array-level coefficients derived from boundary-layer similarity theory. Let $\rho_{\mathrm{pack}}$ be the packing density (device interface area per land area). The effective array-level mass-transfer coefficient $\bar{h}_m^{\mathrm{array}}$ satisfies a series resistance model combining ABL mixing and device-scale transfer:<br /> \[<br /> \frac{1}{\bar{h}_m^{\mathrm{array}}} = \frac{1}{h_{m,\mathrm{ABL}}} + \frac{1}{\rho_{\mathrm{pack}}\,h_m(\mathcal{T})},<br /> \]<br /> where $h_{m,\mathrm{ABL}}$ represents the ABL turbulent mixing limit (the fetch-dependent replenishment rate from Section VII).</p> <p> The array-level pumping penalty per unit land area is:<br /> \[<br /> \bar{P}_{\mathrm{fan}}^{\mathrm{land}} \ge \rho_{\mathrm{pack}}\,\frac{j^2}{(\rho_{da}\Delta\omega)^2}\,\mathcal{R}(\mathcal{T}),<br /> \]<br /> where $\mathcal{R}(\mathcal{T})$ is the texture-specific hydraulic resistance per device. Using the laminar geometric bound from Section IV.F.3 (circular duct optimality), any non-circular or textured passage satisfies $\mathcal{R}(\mathcal{T}) \ge \mathcal{R}_{\mathrm{circle}}$ for fixed cross-sectional area, providing an auditable lower bound.</p> <p> XI.B.2. Claimed performance improvements flagged as hypotheses</p> <p> Any quantitative claim that optimized textures yield specific yield improvements (e.g., “$30\%$ higher exergy efficiency at high wind speeds”) must be treated as a modeling hypothesis requiring empirical validation against the inequalities above. The homogenized framework provides the conservative feasibility bound:<br /> \[<br /> \bar{Y}_{\mathrm{array}} \le \min\left\{ \frac{\bar{h}_m^{\mathrm{array}} \Delta\rho_v}{c(t)},\; \frac{\bar{e}_{\mathrm{supply}}}{c(t) + \bar{P}_{\mathrm{fan}}^{\mathrm{land}}/j} \right\},<br /> \]<br /> where the first term represents the mass-transfer limit (atmospheric or device-side) and the second represents the exergy-pumping trade-off. Planning optimizations should use the tighter of these two bounds at each site and time step.</p> <p> End of Section XI.</p> <p> Conclusion</p> <p> In this paper we have developed a comprehensive multiscale mathematical framework for atmospheric water harvesting (AWH) that bridges thermodynamic necessity, device-scale transport physics, and atmospheric feasibility constraints. The architecture proceeds from immutable second-law bounds to auditable device models, culminating in tractable optimization formulations for climate-robust deployment.</p> <p> At the foundational layer (Sections II–III), we established device-agnostic necessary conditions relating water yield to exergy expenditure through the time-varying chemical-potential price $c(t)=R_vT(t)\\ln(1/\\mathrm{RH}(t))$. The instantaneous bound $y(t)\\le e(t)/c(t)$ and its integrated forms provide screening criteria that are conclusive when violated but conservative when satisfied. We strengthened these benchmarks with finite-recovery penalties for deep dehumidification and open-system flow-exergy decomposition accounting for exhaust-conditioning costs, both expressed as convex penalties compatible with optimization layers.</p> <p> The device-scale module (Section IV) translated microscale transport limits—diffusion-limited sorption kinetics, Onsager linear-response inefficiencies, radiative-dew regime switching, and auxiliary airflow work—into additional exergy-destruction lower bounds or admissible concave yield envelopes $\\phi_t(e)$. Reduced-order PDE surrogates and their error certificates enable these physics to be embedded in control and planning without resolving full spatiotemporal detail at every optimization step.</p> <p> Optimal control formulations (Section V) revealed that exergy allocation under reversible bounds reduces to threshold (bang-bang) policies concentrating harvest in low-$c(t)$ windows, while realistic concave device responses and storage dynamics yield water-filling structures with equalized marginal yields. Stochastic and impulse-control extensions accommodate renewable intermittency and batch-mode operation, with quasi-variational inequalities providing the appropriate computational framework.</p> <p> Uncertainty quantification (Section VI) equipped the framework with Polynomial Chaos surrogates for parameter sensitivity and Wasserstein–Fisher–Rao distributionally robust optimization to handle nonstationary climate without parametric assumptions. Thermodynamic Uncertainty Relations were identified as qualitative indicators linking yield variance to dissipation, contingent on validated stochastic device models.</p> <p> At the atmospheric scale (Sections VII–VIII), we derived hard feasibility caps from column moisture conservation and demonstrated that deployment-induced depletion strengthens minimum work through convex $-\\ln$-type penalties. Mean-field game and variational-inequality formulations capture competitive congestion and spatial equity, with gradient-regularization providing a tractable surrogate for Gini constraints.</p> <p> The co-design and deployment layers (Sections IX–XI) integrated these components into mixed-integer conic programs—tractable when device physics admit piecewise-linear or exponential-cone representations—and transport-based planning models. Conservative screening metrics derived from reanalysis data permit auditable elimination of nonviable sites before committing to detailed engineering.</p> <p> Key mathematical takeaways include: (i) the ubiquity of convexity and concavity structures that enable polynomial-time robust optimization; (ii) the decomposition of performance limits into separable exergy, transport, and atmospheric terms; and (iii) the necessity of explicit feasibility constraints coupling device operation to regional moisture budgets.</p> <p> Limitations remain. The bounds presented assume ideal-gas thermodynamics unless specific fugacity corrections are inserted; real-gas and non-dilute effects require external validation. Onsager coefficients, Sherwood–Reynolds correlations, and radiative transfer parameters must be calibrated against peer-reviewed data or high-fidelity simulation to tighten the conservative envelopes presented here. The mean-field and impulse-game equilibria assume specific regularity conditions on atmospheric transport operators that should be verified for concrete geographical implementations. Finally, the translation of Thermodynamic Uncertainty Relations into actionable engineering constraints awaits experimental validation of the underlying stochastic thermodynamic models.</p> <p> This framework is offered as a blueprint: a modular hierarchy where each layer—from pointwise exergy inequality to global deployment optimization—explicitly declares its assumptions, enabling systematic replacement of conservative placeholders with validated device physics as data become available.</p> <p> [HARD CODED END-OF-PAPER MARK — ALL CONTENT SHOULD BE ABOVE THIS LINE]</p> <p> ================================================================================<br /> MODEL CREDITS</p> <p> This autonomous solution attempt was generated with the Intrafere LLC AI Harness,<br /> MOTO, and the following model(s):</p> <p> – x-ai/grok-4.1-fast (85 API calls)<br /> – openai/gpt-5.2 (27 API calls)<br /> – moonshotai/kimi-k2.5 (16 API calls)</p> <p> Total AI Model API Calls: 128<br /> ================================================================================</p> <p> MIT License</p> <p>Copyright (c) 2026 Intrafere LLC</p> <p>Permission is hereby granted, free of charge, to any person obtaining a copy<br /> of this software and associated documentation files (the “Software”), to deal<br /> in the Software without restriction, including without limitation the rights<br /> to use, copy, modify, merge, publish, distribute, sublicense, and/or sell<br /> copies of the Software, and to permit persons to whom the Software is<br /> furnished to do so, subject to the following conditions:</p> <p>The above copyright notice and this permission notice shall be included in all<br /> copies or substantial portions of the Software.</p> <p>THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR<br /> IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,<br /> FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. 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