A relationship between power and evaporation rate

An evaporation-driven engine placed just above the water surface is powered by absorbing water at a high chemical potential, μ s , and releasing it at a lower chemical potential, μ e , to the atmosphere, μ a (μ s > μ e > μ a ; Fig. 1e). For a reversible and isothermal engine, the power output depends on E and the work done per mole of evaporating water w = μ s –μ e . However, one cannot simply multiply existing E data by w, as the energy conversion process alters the evaporation rate. Therefore, predicting the power available from natural evaporation requires a relationship between w and E.

E is affected by w in two ways. First, the chemical potential drop w across the engine results in a reduction in water vapor pressure across the engine, which reduces the mass transport. In the case of an ideal gas20, w is − RT s ln(α), where R is the molar gas constant, T s is the temperature of the surface, and α is the ratio of the vapor pressures above and below the engine. Note that the air immediately above the water surface is saturated with water vapor, therefore the ratio α is also the relative humidity at the top of the engine (in dimensionless units 0.00–1.00)21. We can rewrite α as follows:

$$\alpha \left( w \right) = {e^{\frac{{ - w}}{{R{T_s}}}}}$$ (1)

Because the evaporation rate depends on the vapor pressure deficit between the engine surface and the atmosphere, an increase in w causes a reduction in evaporation rate. Second, the total energy required to evaporate water and extract energy from an evaporation-driven engine is the sum of the latent heat L and the work energy w. We define the ratio of this total energy to the unperturbed case as β:

$$\beta \left( w \right) = \frac{{L + w}}{L}$$ (2)

Here, L is the molar latent heat of vaporization of water in J/mol. Thus, β represents the energy penalty for evaporating water through an evaporation-driven engine versus the case with no engine. Consequently, w affects the energy balance between net radiation and heat loss due to convection and evaporation, because some portion of the energy from net radiation is now removed from the system as work.

Using parameters α and β, it is possible to derive a model that predicts the evaporation rate and power generated from it. Note that w can be dynamically adjusted during operation by varying the resistance of the load so that the water responsive material in the engine must exert a larger force on the load. Thus, it is possible to control α and β. At steady-state, the net radiation leaves the engine surface via convection, evaporation (i.e., latent heat), and power generation. The convective heat flux is proportional to the temperature difference between the engine surface and the atmosphere, whereas the latent heat flux is proportional to the difference in vapor pressures between the engine surface and the atmosphere. The magnitudes of these two energy fluxes also depend on the transport characteristics of the air, which is primarily determined by turbulence and wind speed. Using these relationships, we derived an equation that relates the latent heat flux, F, to α and β (Methods):

$$F = \frac{{\alpha \Delta }}{{\alpha \beta \Delta + \gamma }}\left( {I + \frac{\gamma }{{\alpha \Delta }}f\left( u \right)\left( {\alpha - {\rm RH}} \right){p_a}} \right)$$ (3)

Here, f(u) is the convective mass transport coefficient of water vapor as a function of wind speed u, I is the net radiation, Δ is the slope of the saturation vapor pressure versus. temperature curve, γ is the psychrometric constant, RH is the relative humidity of the air, and p a is the saturated vapor pressure of water at the air temperature. Equation (3) shows that evaporation occurs even when the net radiation is zero, as long as the relative humidity of the air is less than α. Under this condition, the remaining term in the parenthesis can be viewed as the drying power of the sub-saturated atmosphere.

Once F is calculated, the evaporation rate E can be obtained from the relationship F = ELρM v , where ρ and M v , are the respective liquid density and molecular weight of water. Finally, the areal power density W is given by W = Fw/L.

Power generation and evaporative losses vary with weather conditions

Figures 2a, b illustrates predictions for W and E as a function of α(w, T s ) for a range of RH values at conditions representative of typical mild weather conditions. As α is lowered from unity (w = 0), the surface temperature rises while E gradually falls (Fig. 2b, c). This gradual increase in surface temperature results in a proportional increase in convective heat losses C. Evaporation ultimately stops at a certain α value, at which point heat is released mostly as convective heat C. Importantly, W peaks at an optimal α value (i.e., an optimal w that maximizes the power density for given weather conditions). Interestingly, E at optimal power density is approximately half the open water E (α = 1) under the same weather conditions (Supplementary Fig. 1).

Fig. 2 Steady-state power generation and effects on evaporative losses. a Energy fluxes, b evaporation rates, and c surface temperatures are calculated as a function of α(w,T s ) for weather conditions of 200 W m−2 I, 16 °C T a , 101.3 kPa P, and 2.7 m s−1 (6 mph) u at 5 values of RH (mild conditions). d Maximum energy flux and e water saved from evaporation as a function of RH at cool (pale, 12 °C, 150 W m−2), mild (neutral, 16 °C, 200 W m−2), and warm (dark, 20 °C, 250 W m−2) weather conditions and three wind speeds: 1.8 (4 mph, solid), 2.7 (6 mph, dashed), and 3.6 m s−1 (8 mph, dotted) Full size image

To better understand which weather variables most influence the optimal power density, we plot the optimal power densities and corresponding evaporation rate reductions as a function of relative humidity for a range of weather conditions (Fig. 2d, e). Interestingly, we find that the optimal power density varies weakly with wind speed, and increases strongly with decreasing atmospheric relative humidity. We also find that the potential water savings increases with increasing wind speed and decreasing relative humidity. The results suggest power densities of up to 15 W m−2 and parallel evaporation rate reductions up to 7.5 mm H 2 O per day at some of the warmest and driest conditions. Note that these conditions vary over time and geography. For example, the distribution of daily relative humidity values at Daggett-Barstow, California shows that the days where the relative humidity falls below 40% occurs about 65% of the time (Supplementary Fig. 2). Therefore, one has to take into account the variability of weather conditions to determine the average power available.

Using regional meteorological data22, our model can now provide insight into the distribution of power densities available. By calculating maximum daily W and averaging it across an entire year, we generate a 5′ resolution map of power density and parallel water savings across the contiguous USA (Fig. 3 and Supplementary Fig. 3). These maps suggest average annual power densities and corresponding water savings up to 10.49 W m−2 and 5.9 mm H 2 O per day, respectively. These maximums are located at Needles Airport in California, only 11 km from Goose Lake and 47 km from Lake Havasu. This result is particularly striking since the locations of peak power potential and water savings occur simultaneously in the US Southwest, a region that frequently suffers from water scarcity. As a point of reference, the current mean total area power densities for current US wind and photovoltaic installations are 2.90 and 8.06 W m−2, respectively23, 24.

Fig. 3 Maps of power generation from natural evaporation and water savings. a Maximum power density available and b total decrease in evaporation rate due to power harvesting potentially available from open water surfaces across the contiguous United States of America. Maps calculated using the data22 across 934 weather stations to calculate W Max and corresponding ΔE at each location from eq. (3) with natural neighbor interpolation and linear extrapolation to generate a 5′ resolution map Full size image

The data in Fig. 3 allow us to predict the total power and water savings potentially available from lakes and reservoirs in the US via a database of open water bodies25. By identifying the location and surface area of each open water body, we predict the potential annual mean power output and corresponding water savings available at each water body if it was covered entirely with an ideal evaporation driven engine. Our analysis reveals that 325 GW (2.85 million MWh per year) is potentially available by covering lakes and reservoirs larger than 0.1 km2 across the contiguous US (excluding the Great Lakes). Additionally, an additional 96.4 billion cubic meters of water could be recovered each year due to lower evaporation rates. Our results shown in Table 1 indicate that potential power available exceeds demand in 15 of 47 US states studied26, and saves more freshwater than consumed in 7 of those 15 US states27. The summary results of all US states studied can be found in Supplementary Table 1.

Table 1 US States where the potential power available due to evaporation from open water surface area exceeds the net energy generation rate Full size table

Potential effects of feedback between the engine and the atmosphere

Our estimates of steady state evaporation rates and power do not currently consider potential changes in atmospheric conditions due to the reduction in evaporation rates. This can be viewed as a feedback interaction between the engine and the atmosphere. Such feedback mechanisms can be critical to distributed renewable energy systems. For example, atmospheric feedback imposes limits to the maximum power generation of wind turbines28, 29. Therefore, it is important to consider potential feedback effects in our model.

One potential feedback pathway is caused by the changes on the atmosphere due to covering lakes and reservoirs with evaporation-driven engines. The evaporation-driven engine reduces the evaporation rate while increasing the rate of convective heat loss (due to higher surface temperatures). This shift of energy from evaporation to convection mimics the conditions seen when moist soils become dry, where higher convective heat fluxes warm the air due to reduced water availability for evaporation. Previous studies30,31,32,33,34,35,36 show that as previously moist soil become drier, the atmosphere becomes more arid, consistently shifting toward higher air temperatures and lower relative humidities37, 38. These changes contribute toward a reduction in cloud cover39, 40 (i.e., an increase in net radiation). Individually, these changes would increase the potential for evaporation that could result in power densities greater than those for fixed weather conditions, as seen in eq. (3).

Another feedback pathway is to expand the total available area for evaporation driven engines. This could be due to artificially creating new reservoirs. This would have the opposite effect; with more open water surfaces made available, more evaporation would occur, leading to reduced air temperature and increased humidity. Such feedback has been shown in studies involving large-scale changes in land-use (e.g., urbanization, irrigation)41, 42. This would result in power densities lower than those for fixed weather conditions.

However, the magnitude of these feedback pathways is likely to be small for the daily mean temperature and would primarily modify temperature extremes43. Globally, any changes that could occur in the atmosphere is small since ocean evaporation dominates total global evaporation and the resulting temperature and humidity responses44, 45. Locally, feedback effects will also be small if the dimensions covered by an engine are below 500 km (ref. 46). This is due to the important role of horizontal heat and moisture transport in the atmosphere that couples neighboring regions. Therefore, we are neglecting potential feedback effects, as they would not drastically affect our estimates.

Control of power output under varying weather conditions

While the model described by eq. (3) allows estimating power density and its dependence on meteorological variables, the ability to predict variability of power from evaporation at short timescales is limited due to the approximation that the net heat storage in the body of water is negligible. Evaluating this variability is crucial to understand the potential of evaporation as a renewable energy source since many renewable energy technologies suffer from intermittent availability.

To explore the variability of power from evaporation, we incorporate heat storage in the body of water below an evaporation driven engine into the energy balance among net radiation, evaporation, convection, and power generation. To approximate the heat storage, we assume a simple mixed-layer water body with density ρ, specific heat capacity c w , and mixed-layer depth d (i.e., the epilimnion; typically, at least 5 m deep for lakes larger than 1 km2)47, 48. The energy balance is then given by (Methods, Supplementary Fig. 4):

$$\rho d{c_{\rm{w}}}\frac{{\partial {T_{\rm{s}}}}}{{\partial t}} = I - \beta F - C$$ (4)

Here, the rate of heat storage is balanced by incoming net radiation (I) and outgoing convective heat losses (C) and the sum of latent heat flux (F) and power output (W). Note that βF = F + W. Thus, eq. (4) allows us to predict the water temperature T s , the latent heat flux F, and the power density W as a function of the chemical potential drop w and changing weather conditions over time. Importantly, w can be independently controlled. This feature might allow us to control power generation, potentially mitigating the effect of changing weather conditions.

To demonstrate this, we develop a control system that adjusts w to match a power demand target over time (Supplementary Fig. 5). We set the system’s power demand to that of three major U.S. energy markets in 2010 (South-East Central California49, North Central Texas50, and New York City51) along with their respective varying typical weather conditions22. Because the power output of an evaporation driven engine scales with area, we are interested in relative variations in power demand over time rather than absolute values. Thus, we normalize each power demand curve to a target annual mean power density. Figure 4 illustrates the results of a simulation year in California with a target annual mean power demand of 2 W m−2. The results show that power generation matches demand 95% of the time, exhibiting some shortages on winter days where net radiation is low and relative humidity is higher. Supplementary Fig. 6 illustrates results for Texas (93% match) and New York (67% match).

Fig. 4 Matching variable demand by controlling power output via heat storage. Results for the final year of a simulation run for South-East Central California from Daggett-Barstow, California. From inside-out: Hourly (1) I (yellow, W m−2), (2) RH (blue, %), (3) T a (red, °C), (4) u (cyan, m s−1), (5) W PD (gray, W m−2) and predicted W O (green dots, W m−2). Clockwise from the top-right are 3-day samples of hourly W PD (gray, W m−2) and predicted W O (green dots, W m−2) for January, May, August, and November. Despite the variability of power demand and weather, power generation matches demand 95% of the time. Meteorological data22 and power demand data49 are from publically available databases. Annual data are evenly divided by hourly data Full size image

As the annual mean power demand increases, the frequency of power shortages increases despite an increase in the mean power generation. Figures 5a, b illustrates this aspect by comparing the 2 W m−2 case to a 10 W m−2 case in California. As this comparison shows, the 10 W m−2 case suffers from more power shortages during cooler months and is only able to match demand 48% of the time. However, some power generation still occurs during these cooler months resulting in the system’s annual generation-to-demand ratio to climb above 80%.

Fig. 5 The relationship between reliability and average power output. The demand (gray line) and generation (dots) for (a) 2 W m−2 and (b) 10 W m−2 annual average demand targets for the final simulation year in California. In a, generation matches demand 95% of the time with 99% annual generation to demand ratio. In b, generation matches demand 48% of the time with 71% annual generation to demand ratio. c Predicted average power generation as a function of target power demand for California (circles), Texas (triangles), and New York (squares) test locations. The overlaid contour map is the resulting generation to demand ratios at each power demand condition for that specific average power generation. These simulations predict that the maximal generation is 2.4, 5.1, and 8.4 W m−2 for the respective New York, Texas, and California test locations Full size image

To better understand the relationship between generation and demand, we repeat these calculations for a range of mean power demands. Figure 5c plots mean generation versus mean demand at each test location along with a generation-to-demand ratio heat map (see also Supplementary Fig. 7 for water savings versus mean demand). As demand increases, the system eventually saturates and provides no more additional generation. These simulations predict a maximum generation of 2.4, 5.1 and 8.4 W m−2 for the respective New York, Texas, and California test locations. Compared to the map in Fig. 3a, the control system delivers at least 85% of the power generation predicted by eq. (3) (2.8, 5.3, and 8.4 W m−2 for the respective New York, Texas, and California locations). Importantly, as the imposed generation target is reduced, the reliability of the system to match power demand increases.