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# On the Power Available from Evaporation for a Range of Weather Conditions

Abstract

a. Increasing demand for environmentally-friendly and low-cost energy production. b. Evaporation is one of the most significant forms of mass and energy transport in Earth’s climate. c. Hygrovoltaic generators produce electrical energy by harvesting evaporation d. Hygrovoltaic devices may allow accessing this resource of renewable energy. e. Desire for a model to predict the energy potential of hydrodynamics / evaporation f. Unified energy generation and storage

# Introduction

Sherlock Holmes understood: “It is a capital mistake,” he said, “to theorise before one has data.” Data are the lifeblood of science, the foundation of innovation. Behind every great discovery is a pile of data; but, crucially, it should not be too far behind. [Rephrase] a. Energy Technologies, alternatives and the introduction of hygrodynamic materials b. Heat Engine on Hygrodynamic heat cycle (examples include the Drinking Bird) c. Interest to model the potential of hydrovoltaic devices for grid scale applications d. Figure 1 (Devices: Drinking Bird, Our composites, Other Materials, hygrometers) Now how about an equation

# The Penman Energy Balance and Steady State Theory

Explain transient energy balance R=E+K+S+G (Elimination of S and G)  R=E+K Introduce Penman Combination Formula Bowen’s Ratio B=E+K Empirical E, f(u), e(T) Define es, ea, ed Define f(u) Δ approximation Δ=(es-ea)/(Ts-Ta )≈∂e/∂T @ a Penman’s final result

$$\label{eqn:penman-balance} E=Δ/(Δ+γ) (R+γ/Δ E_a ).$$

Reformulate Penman Formula for power production Define β(w) and develop β(w)=L'/L=(L+w)/L Define ρ(w) and develop ρ(w)=e^(-w/(Rg Ts )) Modified Penman formula E=ρΔ/(βρΔ+γ) (R+γ/βρΔ E_a ) Figure 2A-C. Surface temperatures (A), evaporation rates (B), and power per unit area (C) are calculated as a function of surface relative humidity ρ(w) for weather conditions of 200 W/m^2 global horizontal irradiance, 18 °C air temperature, 760 Torr air pressure, and 6 mph wind speed at 5 values of RH (mild conditions). Surface relative humidity ρ(w) depends on the amount of load w (work done per unit amount of water evaporated). Figure 2D. Maximum power available from evaporation at different weather conditions. Power per unit area is plotted for cool (blue, 12 °C, 150 W/m^2), mild (green, 16 °C, 200 W/m^2), and warm (orange, 20 °C, 250 W/m^2) weather conditions. Calculations were carried out for three wind speeds, 3.5 mph (solid), 5.5 mph (dashed), and 7.5 mph (dotted). Wind speeds selected as 25, 50, and 75 percentile values for wind speeds along a Rayleigh Distribution (σ=2). Expand about figure 2 graphs. Figure 3A. Maximum power available from evaporation across the continental US. Maximum power density was calculated by using mean R, TA, P, u, and RH to solving for ∂ρ/∂W=0 from TMY3 data across 218 Class I weather stations. Data provided by NSRDB. Figure 3B. Total decrease in evaporation rate due to power harvesting. This decrease was estimated by determining the evaporation rates for both ρ=1 and ∂ρ/∂W=0 using mean R, TA, P, u, and RH from TMY3 data across 218 Class I weather stations. Data provided by NSRDB. Talk about figure 3 graphs High energy regions. Scaling estimates for lakes in the SW (Roosevelt Lake, AZ could produce ~50 GW) Water conservation point. (Roosevelt Lake, AZ could save ~261 ML of water per day) Thermodynamic Efficiency Estimate for η

# The Total Energy Balance and Non-Steady State Theory

Reintroduce energy balance R=E+K+S+G Return storage term estimation (minimal mass losses) Clear explanation of assumptions and calculation method (large body of water for S. Deep but not too deep for conduction. Insulated for zero G. Mention of ODE solver) Sup. Figure 1 (Graph of Transient Relaxation) Table of τ vs. (u,TA,R,RH,d,&ρ) Sup. Figure 2 (Comparison of Penman (E,TS,W) to Related SS Transient (E,T_S,W)) Explain deviation away from Penman (linear estimation of dP/dT for Penman) Thermodynamic Efficiency Model for Transient η (Incorporated Capacity Factor)