We quantified the temperature depression that can be achieved by evaporative cooling. There exist many empirical equations to directly calculate the wet-bulb temperature (θ_wb [°C] or T_wb [K]) from the dry-bulb temperature (θ_db [°C] or T_db [K]) and the relative humidity of the moist air (φ_a [%]) (Stull, 2011). However, we introduced an error in predicting the wet-bulb temperature with most empirical equations. This error can amount to 0.3°C–0.6°C (see Supplementary Material). Therefore, these 'direct' equations do not provide adequate sensitivity for estimating the design parameters of an evaporative cooler. In the present study, we calculated T_wb from T_db, and φ_a using an iterative approach. This approach was based on the psychrometric constant γ [K⁻¹], which is often used to construct psychrometric tables (Jensen et al., 1990; Simões-Moreira, 1999; Allen et al., 2005). This approach converted the temperature depression (T_wb-T_db) into a vapor pressure deficit (p_v,sat(T_wb) - p_v,a(T_db)) through well-established empirical coefficients for estimating vapor pressure from temperature and quantifying the latent energy (Sadeghi et al., 2013). p v T d b − p v , s a t T w b T d b , φ a + γ P a t m T d b − T w b = 0 (1)

Here, p_v is the ambient vapor pressure at the dry-bulb temperature [Pa], p_v,sat is the saturated vapor pressure [Pa] at the wet-bulb temperature, and P_atm is the total atmospheric pressure [Pa], which is composed out of the partial pressure of vapor (p_v) and of dry air. The angle brackets indicate here the main dependencies to other parameters. Here, γ is the psychrometric constant (γ = c_p,d/(ε. L_v ^ref) ≈ 0.65 × 10⁻³ K⁻¹), derived from the latent heat of vaporization (L_v ^ref=2.5 MJ kg_l ⁻¹), specific heat capacity of dry air at constant pressure (c_p,d = 1006 J kg_d ⁻¹ K⁻¹), and the ratio of molecular weights of water vapor and dry air (ε = 0.622 kg_v kg_d ⁻¹). This value is not entirely constant but remains fairly constant until wet-bulb temperatures of 40 °C (Simões-Moreira, 1999).

The saturated vapor pressure was derived as a function of temperature using Eq. 2, commonly referred to as Teten’s equation (Allen et al., 1998). p v , s a t = 610.8 × exp 17.27 θ θ + 237.3 (2)

Here, θ represents the dry or wet-bulb temperature [°C], corresponding to ambient or saturated conditions. The ambient vapor pressure (p_v [Pa]) was calculated by computing p_v,sat at T_db and multiplying this term by the relative humidity of the air (φ_a).

We solved Eq. 1 iteratively via the saturated vapor pressure p_v,sat at wet-bulb temperature conditions to determine the wet-bulb temperature at different dry-bulb temperature and relative humidity values. The iterative process, underlying assumptions, initial guess, and margin of error are detailed in the Supplementary Material.

The maximal temperature depression achieved by evaporative cooling (∆T_ev [K]) was calculated by subtracting wet-bulb temperature from dry-bulb temperature, using Eq. 3. Δ T e v = T d b − T w b T d b , φ a (3)

We quantified the maximal gain in the postharvest life of fruit (∆PL_fr [days]) corresponding to this maximal temperature depression that can be achieved by evaporative cooling. We evaluated this gain at different combinations of T_db and φ_a. The maximal gain in postharvest life corresponded to the additional days gained by storing the fruit at wet-bulb temperature instead of ambient dry-bulb temperature conditions. These are the additional remaining days in a product’s postharvest life when it is still marketable. We thus compared storing fruit inside an idealized evaporative cooler versus outside in the natural shade. This gain was thus the difference in postharvest life caused by the maximal temperature depression obtained from evaporative cooling. The postharvest life (PL_fr [days]) was calculated at each temperature using first-order kinetic models for respiration-driven quality decay for every fruit. These models are detailed in the Supplementary Material. We quantified the postharvest life for selected fruits: apple, banana, mango, and tomato. The model calculated the evolution of an overall fruit quality index, which is affected by the temperature. This temperature corresponded to T_db for the postharvest life calculation in the absence of evaporative cooling and T_wb for a case where fruits were subject to evaporative cooling and achieved the maximal temperature reduction that was possible. The maximal gain in postharvest life (∆PL_fr [days]) was computed using Eq. 4. Δ P L f r = P L f r T w b T d b , φ a − P L f r T d b (4)

Note that ∆PL_fr is the maximal gain in postharvest life achieved under ideal conditions. However, ∆PL_fr might be lower as the wet-bulb temperature is not always reached in evaporative cooling (Defraeye et al., 2022). Additionally, the gain in postharvest life was only calculated based on the reduction in temperature due to evaporative cooling. Our current calculation does not account for the increase in postharvest life due to the reduced moisture loss due to increased relative humidity in the evaporative cooler. This reduced moisture loss in the evaporative cooler also helps to preserve the food longer, by reducing wilting.

We calculated and mapped the gain in postharvest life that evaporative cooling gives at every location in India and Nigeria for each month at a resolution of approximately 30 km. In this way, we identified the most promising regions to deploy the technology. We also developed a similar map for the entire world for the four seasons: summer, autumn, winter, and spring. The details are given in the Supplementary Material. We combined the following data for calculating the wet-bulb temperature and the gain in postharvest life: local monthly climate data, psychometrics, information on postharvest life for each crop, and kinetic rate law modeling. As a benchmark, we calculated the postharvest life of fruit after harvest, when the fruit was stored under ambient conditions in the shade (T_db, φ_a). Then we calculated the postharvest life after harvest when the fruit was stored in an idealized evaporative cooler at wet-bulb temperature conditions.

We calculated how much fruit that an amount of water can cool by evaporation. We know how much energy was extracted from the air and the horticultural products when 1 L of water (m_l = 1 kg) was evaporated. This energy equalled the latent heat needed for evaporating 1 L of water (E_lat = m_l. L_v ^ref). The energy was extracted from the air surrounding the evaporative cooler and the stored food. Here we assumed idealized evaporative cooling, so just an energy conversion of sensible heat into latent heat in the air. The enthalpy of the air remained constant, so it was an isenthalpic process. For an ideal isenthalpic evaporative cooling process, the wet-bulb temperature was reached.

We calculated theoretically how many kilograms of fruit 1 L of water can cool down from its initial temperature (T_ini [K]) to the wet-bulb temperature (T_wb [K]). T_wb was the lowest temperature that could be reached by evaporative cooling. The following equation quantifies the energy needed to achieve this temperature reduction in the fruit by evaporative cooling (E_ec,fr [J]): E e c , f r = m f r c p , f r T i n i − T w b (5)

With m_fr the mass of fruit [kg], c_p,fr the specific heat capacity of the fruit [J kg⁻¹ K⁻¹], which is for apple fruit 3800 J kg⁻¹K⁻¹. We assumed that all the energy needed to evaporate 1 L of water was extracted from the fruit, not the air. In that case, we had an ideally efficient evaporative cooling process. We wrote the following equation when we assumed that the fruit is initially equilibrated at the dry-bulb temperature of the environment: E e c , f r = m f r c p , f r T d b − T w b = E l a t = m l L v r e f = 1 k g × 2,500,000 J k g (6) m f r = 2,500,000 c p , f r T d b − T w b (7)

This equation calculates the amount of fruit that can be cooled by evaporative cooling. This amount is a function of the ambient dry-bulb temperature and relative humidity. The fruit’s specific heat is the only fruit-dependent parameter in this equation. The heat of respiration was not factored in here. Note that highly respiring fruit will slightly change the heat balance during evaporative cooling and reduce the amount of fruit 1 L of water can cool down.

We quantified the cooling power or cooling capacity of an evaporative cooler. As an example, we took a charcoal cooler blanket we developed (Defraeye et al., 2022). We assumed one square meter of a charcoal blanket (A_ec = 1 m²). The charcoal absorbed a certain amount of liquid water that can be evaporated (m_l,evap). This amount of water was estimated as: m l , e v a p = A e c D e c w P M , e c = A e c D e c 1 − ϕ 0 , b u l k w P M = A e c D e c 1 − ϕ 0 , b u l k X P M w s (8)

Here, D_ec is the thickness of the walls of the evaporative cooler so equivalent to the thickness of a cooling pad in several evaporative coolers [m], w_PM,ec is the bulk moisture content of the porous material (charcoal) in the evaporative cooler [kg m⁻³], ϕ_0,bulk is the bulk porosity (or macroporosity) of the cooler, so the air space between the charcoal pieces, w_PM is the moisture content of the charcoal pieces (not saturated) [kg m⁻³], X_PM is the dry-base moisture content of the charcoal [kg kg⁻¹], and w_s is the dry-base density of the charcoal pieces [kg m⁻³].

The cooling capacity of the cooler, which holds this amount of water (m_l,evap), was calculated based on the latent heat (2.5 MJ kg⁻¹ of water) and the time interval over which the amount of water is fully evaporated (Δt), so the evaporative cooling heat flow (Q_evap [W]). The average cooling power over this period (P_ec [W]) could be estimated as: P e c = Q e v a p = L v r e f G v , e v a p = L v r e f m l , e v a p A e c , D e c , ϕ 0 , b u l k , X P M Δ t (9)

Note, however, that the evaporation rate (G_v,evap [kg s⁻¹]) is not linear over this period. Thereby, the cooling capacity will also vary over time. With this equation, we only quantified the average cooling capacity. In addition, we cannot guarantee that this cooling capacity can be reached as this implies the amount of water can be evaporated in that amount of time. The time frames assume the airflow conditions are such that this amount can be evaporated.

Using the bulk moisture content and the thickness of the cooler walls, we got the cooling power as a function of the time interval and the cooler wall thickness: P e c = L v r e f m v , e v a p D e c , A e c , ϕ 0 , m a c r o , X w Δ t = L v r e f w P M , e c A e c D e c Δ t (10)

We quantified the average cooling power of this wall of an evaporative cooler based on our measurements of the charcoal properties and the charcoal cooler we developed previously (Defraeye et al., 2022). We assumed a charcoal cooler with a bulk porosity of 61% (ϕ_0,bulk), charcoal with a dry-base density of 442 kg m⁻³ (w_s), and a dry-base moisture content after wetting of about 20%. The moisture content of the porous material equals 88 kg m⁻³ (w_PM). These values gave a bulk moisture content (w_PM,ec) of 34 kg m⁻³.

We should design the thickness of the cooling wall of an evaporative cooler to reach optimal cooling efficiency under the local airflow conditions at that specific location. The accessible surface area of the charcoal pieces for evaporation is relevant here when we assume air can flow through the porous structure. This implies a sufficient open porous stacking of the charcoal pieces and a permeable textile membrane. If the size or surface area for mass exchange is too low, air will not be saturated. Then the air temperature will not reach the wet-bulb temperature. In the Supplementary Material, we calculated the optimal thickness of the walls of an evaporative cooler (D_ec [m]): D e c = U ⋅ ρ a ⋅ ε C M T C U ⋅ P a t m ⋅ A s f ϕ 0 , b u l k , D c c (11)

Here, A_sf is the surface area of the evaporative material for mass exchange per volume unit of the cooler [m² m⁻³], U is the airspeed [m s⁻¹], ρ_a is the air density [kg m⁻³], ε is the ratio of molecular weight of water and dry air [kg_v kg_d ⁻¹], CMTC is the convective mass transfer coefficient at the air-material interface [s m⁻¹], D_cc is the spherical piece size (e.g., charcoal) [m] and P_atm is the atmospheric pressure [Pa]. This simplified equation is only valid under steady-state conditions with no external heat exchange of the system with the environment, so an adiabatic system. The equation also assumed that the air becomes fully saturated after moving through the evaporative cooler. This case implies idealized evaporative cooling.

We already know how much evaporative cooling can maximally reduce air temperature. We can get the maximal temperature reduction (∆T_ev) when the wet-bulb temperature is reached. The wet-bulb temperature is the theoretical lower limit to which we can cool down the air and the food products. This maximal temperature drop depends on the dry-bulb temperature and the relative humidity of the environment. In principle, psychometric charts contain this information (Figure 1). However, such a psychometrics-lookup exercise is required then for each environmental condition to determine the evaporative cooling temperature depression. Consequently, it can be time-consuming to look up the cooling potential for each month of the harvest season for different potential sites in a country.

Therefore, we present the following design charts for engineers developing evaporative coolers (Figure 2; Figure 3). They present the following parameters as a function of the ambient dry-bulb temperature and relative humidity:

- The wet-bulb temperature (T_wb), so the lowest temperature that can be reached by pure evaporative cooling (Figures 2A, C, E). This value is essential to determine for a certain food to which extent its ideal storage temperature can be reached.

From these results in Figure 2, we can conclude that for a certain humidity, the warmer the region gets, the higher the maximal temperature depression becomes (Figure 2D). This increase is more pronounced in drier areas. The increase in temperature depression with increasing dry-bulb temperature at a constant humidity is almost linear (Figure 2D). The drier the region becomes for a certain dry-bulb temperature, the higher the maximal temperature depression becomes (Figure 2B). The increased temperature depression with decreasing relative humidity is also almost linear (Figure 2B). From Figure 2E, we see that in warm climates (T_db > 25°C), it will be unlikely to reach the ideal storage conditions for several fruits unless the air is very dry. Still, fruit can be stored well below the ambient conditions found under the natural shade. However, the fruit temperature will still be above the recommended conditions. Such storage under lower temperatures will anyway lead to longer shelf life. So even if the ideal temperature is not reached, evaporative cooling provides an advantage. The presented design charts (Figures 2E, F) enable faster identification of the wet-bulb temperature and temperature depression than the traditional psychometric charts since they only depict the wet-bulb temperature. These design charts emulate in fact the different points that otherwise needed to be looked up.

From the results in Figure 3, we see that the temperature reduction induced by evaporative cooling can lead to several days of additional postharvest life. Here the gain in postharvest life implies the additional days gained by storing the fruit in the evaporative cooler at the wet-bulb temperature conditions, as opposed to ambient storage at dry-bulb temperature conditions. The gain in additional postharvest life using passive cooling is remarkable. Even for temperate climates (T_db = 20°C and φ_a = 50%), the postharvest life gain is roughly 2–15 days for the fruits considered. However, this wet-bulb temperature will not be reached in an evaporative cooler due to additional heat sources (e.g., radiation) and non-optimal evaporative cooling. The temperature will also not remain constant throughout the day. Therefore these charts represent the best-case scenario. Nevertheless, adding several days of storage can help farmers sell more of their produce on the markets, leading to less food loss and a higher income for the farmers.

We tackle a crucial bottleneck in evaporative cooling here: how to scope for regions where evaporative cooling can be successfully applied (Verploegen, 2021b; Verploegen, 2021c). The evaporative cooling potential and its impact on extending the postharvest life of fresh food at a certain location in the world need to be assessed before deploying evaporative cooling. If evaporative cooling is deployed in a region that does not add postharvest life, due to insufficient cooling, farmers will cease to use it and lose trust in the technology. Tools exist to qualitatively assess this cooling potential (Verploegen, 2021b). Our design charts can also identify the temperature depression that can be reached for a certain location worldwide. Using our charts, we can select a specific season or month based on the local environmental conditions (Figure 2; Figure 3). We took the next step forward and calculated and mapped the gain in postharvest life that evaporative cooling gives at every location in India or Nigeria for each month of the year at a resolution of approximately 30 km (section 3.2.1). Furthermore, we developed a similar map for the entire world for four seasons (section 3.2.2). Note that these calculations are based on theoretical estimates, and in the future, a detailed comparison with experimental data is advised.

We aim to quantify how much fruit we can cool with 1 L of water by evaporative cooling. This resulting amount of fruit is depicted in Figure 14 as a function of the dry-bulb temperature and relative humidity for apple fruit. In addition, results for mango, tomato, and banana fruit are also shown on the right-hand side, but the differences were very small.

From these results, we see that 1 L of water can cool down a considerable amount of fruit to the wet-bulb temperature. Note, however, that the wet-bulb temperature might not always be the optimal storage temperature for the fruit. Let us consider a temperature reduction of about 10 °C from the dry-bulb temperature to the wet-bulb temperature. This temperature reduction is achievable by evaporative cooling (Figure 2). Typical conditions in which the wet-bulb temperature drops by 10 °C are, for example, a dry-bulb temperature of about 30 °C and relative humidity of about 40%. When we apply Eq.7, 1 L (or kg) of water can cool down 66 kg of apple fruit by 10 °C. For the same temperature conditions (T_db = 30 °C and φ_a = 40%), we can cool down 71 kg of banana fruit, 68 kg of mango fruit, and 62 kg of tomato fruit by 10 °C ideally. As such, the type of fruit that can be cooled down by 1 L of water does not change so much. Note that these calculations do not consider additional heating by the heat of respiration.

This amount of fruit is the limit to which we can cool as we consider a perfect heat extraction from the fruit with no cooling of the air and no additional heat losses. Even if the cooling potential of this 1 L of water is lower due to these additional losses, the evaporative cooling capacity of water is huge. One liter of water can easily cool down over tenfold of its weight in fruit by 10 °C. Once the fruit is cool, the evaporation of water just needs to maintain the right temperature. The evaporation rate then will strongly depend on various conditions, including the temperature and humidity of the environment.

We aim to quantify the cooling power of an evaporative cooler. We quantify it for a unit cell of a wall of the cooler, so one square meter of wall. We quantify the average cooling power for this unit cell over a specific time. The unit cell is based on our measurements of the charcoal properties and the charcoal cooler we developed previously (Defraeye et al., 2022) but can also be done for other cooling pads. Other cooling pads will then have a different porosity, thickness and moisture content of the pieces of material, which will affect the cooling power. We quantified this average cooling power in Figure 15 as a function of the period over which this water is evaporated (Δt) and the thickness of the walls of a cooler (D_ec).

From these results, we see that larger time intervals that would be used for cooling rapidly decrease the cooling power of the evaporative cooler (Figure 15A). A cooler with a thicker wall has a higher cooling capacity for the same time interval. For example, for a time of operation of 5 h, a cooler with a wall thickness of 20 cm has a two times the higher cooling capacity than a cooler with 10 cm walls.

The moisture content of unsaturated charcoal pieces (w_PM) varies strongly depending on the wetting time. We investigated the influence of charcoal moisture content on the cooling power. We varied this parameter between 50 kg m⁻³ and 400 kg m⁻³. Our findings show that the moisture content of charcoal has a large influence on the cooling power, which is logical as more water is available for evaporation (Figure 15C).

A larger macroporosity decreases the cooling capacity of the evaporative cooler (Figure 15B). This is because as the air space between the charcoal pieces (so higher bulk porosity) increases, the bulk moisture content (w_PM,ec) decreases. In turn, there is less water available for evaporation, thus a lower cooling capacity. Increasing the thickness of the cooler wall and reducing the bulk porosity can lead to a higher cooling capacity of the evaporative cooler. When engineers decide which changes in design parameters should be implemented, important factors to consider are the practicability of implementation and the water use efficiency. Thicker cooler walls would require more water to be supplied to the cooler.

In summary, we can reach a cooling power of several hundred Watts with one square meter of cooler, which is substantial. We only quantified the average cooling capacity over a specific time interval. We can not, however, guarantee that this cooling capacity can be reached as this implies the amount of water can be evaporated in that amount of time. We cannot control the time interval over which the cooling occurs, as this depends on the environmental conditions and the airspeed, among others. Our time frames assume the conditions are such that this amount can be evaporated. Note also that this cooling power is not directly transmitted to the product to be cooled. Finally, note that the heat of respiration produced by the fruit can introduce additional heat to be removed. For example, the heat of respiration for bananas is about 55 W tonne⁻¹ at 13°C and 119 W tonne⁻¹ at 20 °C (Kader, 1996). Compared to the cooling power of a cooling blanket of several square meters, these amounts are rather small.

We plot a design chart for calculating the optimal thickness of the walls of the evaporative cooler, given a certain bulk porosity and airspeed (Figure 16). An optimal wall thickness implies that, under the local airflow conditions at that specific location, the maximal cooling efficiency is reached. The airspeed through the cooler is defined by the local airflow conditions. We assume one square meter of a wall of the evaporative cooler (A_ec = 1 m²) and a spherical charcoal piece size (D_cc) of 80 mm. We consider airspeeds ranging between 0.01 and 5 m s⁻¹.

From these results, we see that in regions where airspeeds are low and are typically varying during the day around these low values, optimal performance will be difficult to achieve. The reason is that with a slight change of airspeed, the optimal wall thickness largely varies. As an example, for an evaporative cooler with a bulk porosity of 40% operating in a region with an airspeed of 0.1 m s⁻¹, the optimal blanket thickness would be 76 mm. The optimal blanket thickness would be 350 mm for an evaporative cooler with a bulk porosity of 40% operating in a region with an airspeed of 0.5 m s⁻¹. Once built, we can not change the wall thickness or bulk porosity. That way, the performance will just vary strongly with airspeed. However, we find that the optimal wall thickness becomes much more independent of the bulk porosity at higher airspeeds. For example, a 50 mm thick cooler requires about the same bulk porosity for airspeeds above 1 m s⁻¹.

Note that the optimal wall thickness is independent of T_db and φ_a. This implies that the environmental conditions will not determine the required optimal wall thickness. As such, a cooler with an optimized wall thickness for specific airflow conditions will perform similarly in different climates. We can, thereby, design a cooler based on local wind conditions and deploy it in regions with similar airflow conditions around the world.

Such design charts are useful to evaluate for a certain region (country, urban, rural) or location (free field, building roof) with a certain airspeed how thick the walls of an evaporative (charcoal) cooler should be. This optimal wall thickness implies optimal efficiency, so the lowest possible temperature and a high cooling capacity. Note that our quantification of the optimal wall thickness was based on analytical equations. Future steps should be to verify this experimentally or with physics-based simulations. The reason is that the complex transient heat and mass balances cannot be captured fully with simplified analytical equations.