Improving Thermal Distribution in Water-Cooled PV Modules and Its Effect on RO Permeate Recovery

: Among the most notable emerging hybrid technologies for water treatment are those that combine reverse osmosis (RO) membrane systems with alternative energy sources such as solar photovoltaic (PV). Solar PV modules can enable systems disconnected from the electricity grid, and in some locations can also be used for water heating as photovoltaic-thermal (PVT) units, a process in which water removes heat from the PV module, increasing its electrical generation efﬁciency. When combined with RO, the higher temperature feed water can increase RO permeate ﬂux, improving recovery but decreasing the rejection of dissolved salts. Although the decrease in efﬁciency of PV modules at higher temperatures is a well-known issue, this is usually under conditions of uniform temperature. However, the temperature distribution in water-cooled PV modules is usually not uniform and, given the anisotropy of the distribution and electrical connection of the PV cells in the module, this factor has not been the focus of much study. In this context, a PVT unit that focuses on increasing the output water temperature with a high global heat transfer coefﬁcient will not necessarily be the most electrically efﬁcient system. This study experimentally assesses several proposed heat-exchange conﬁgurations for PVT systems where the PV modules are cooled by forced convective water ﬂow. A simulation model of PVT performance is then validated and used to predict the productivity of the PVT-RO coupling, both in terms of electrical generation and permeate ﬂux of the hybrid system under different conditions. The results suggest that water-cooled PV modules have several potential applications for off-grid and remote water treatment, as well as water transportation systems.


Introduction
In response to growing concerns about the impact of climate change on vital resources, experts have proposed the concept of a water-energy-food security nexus, which is one of the most accepted ways to understand and approach sustainable development [1]. This concept, first introduced at the 2008 World Economic Forum, states that there is an inherent interconnection between these three sectors (water, energy and food), and therefore the actions taken in one of them can have an effect in one or both other sectors [2]. One of the most promising technologies that fall under this framework is renewable energy-powered reverse osmosis (RO) for water desalination. RO can provide fresh water year-round by using alternative water sources, such as seawater or brackish water, helping mitigate the water scarcity problem [3]. However, energy consumption is the main cost driver for RO, representing 44% of operating costs. Large energy requirements coupled with high energy Although the decrease in the efficiency of PV modules at higher temperatures is a well-known issue, this has traditionally been investigated under conditions of uniform temperature. However, temperature distribution in water-cooled PV modules is usually not uniform [16] and, given the anisotropy of the distribution and electrical connection of the PV cells in the module, this factor has not been the focus of much study and is not well understood. In this context, a PVT unit that focuses on increasing the output water temperature with a high global heat transfer coefficient will not necessarily be the most electrically efficient system. Given that important efforts are currently underway to power RO desalination plants with renewable energy, and that increasing the solar energy efficiency of PVT modules is desirable, this study experimentally assesses several proposed heat-exchange configurations for PVT systems where the PV modules are actively cooled by forced convective water flow. Emphasis is made on understanding the flow characteristics that lead to better temperature distribution and larger efficiency gains compared to modules without active cooling. A simulation model of PVT performance is then validated and used to predict the productivity of the PVT-RO coupling, both in terms of electrical generation and permeate flux of the hybrid system under different conditions. The RO simulation assumes a 1D non-isothermal model which calculates the variation in water properties (i.e., density and viscosity based on concentration, pressure, and temperature) along with the membrane permeance at different operating temperatures to predict permeate production. Hence, this paper presents the first study of a RO desalination system operated with PVT cooling which considers the effect of varying temperature at the RO feed. The insights gained are applicable for environmental conditions with high solar irradiation, high ambient temperature and low humidity, such as those in northwest Mexico.

Materials and Methods
A PVT module was designed and constructed, incorporating 16 fluid inlets/outlets at distinct locations on the back of the PV module (functioning as a heat exchanger), which are used for cooling via forced convection with water as the cooling fluid. Forced convection (as opposed to natural convection) is selected as the cooling strategy because RO systems already require a pumping system for their operation, so implementing this strategy does not require additional pumping equipment. Nonetheless, forced convection does involve an increase in pressure losses within the heat exchanger on the back of the PV module, which in turn requires more pumping energy. There is therefore a trade-off when implementing forced convection as the active cooling method, i.e., between the additional energy generated from the PV efficiency increase and the additional energy consumption due to pressure losses. Experimental data is collected by measuring the electrical and thermal performance of the PVT module under two different flow rates (1 and 2 L min −1 ) and 11 different inlet/outlet heat exchange configurations as described in Appendix A. The collected data is used to validate a numerical model, which is then used to predict the daily water production of a PVT-RO plant under different combinations of flow rate and heat exchange configuration, for distinct weather conditions. Detailed explanations of the experimental and numerical methods employed are described in the following sections.

Characterisation of the PV Module
Commercial PV modules (AXITEC AC-270-P polycrystalline cells) were used for the experimental tests. Table 1 shows the technical specifications for these modules, according to the manufacturer, at standard test conditions (1000 W m −2 solar irradiation and 25 • C). The configuration of the PV cells in the module is distributed in 3 groups of 20 cells connected in series. Each group is formed by two columns of 10 cells in series, and the groups are connected in series with each other by means of diodes. This forms a rectangular arrangement of 6 × 10 cells in the panel, as shown in Figure 1. The arrangement of the PV cells is a characteristic that impacts how cooling may affect any efficiency increases, this due to the dependence of electricity generation between the groups. The group with the cell or cells with highest resistance (due to a higher temperature) may activate the bypass diode and will limit the amount of energy produced by the array. Hence, if there is a large temperature gradient, it should theoretically affect the least number of groups if it is aligned with the groups (i.e., in the horizontal direction).  The configuration of the PV cells in the module is distributed in 3 groups of 20 cells connected in series. Each group is formed by two columns of 10 cells in series, and the groups are connected in series with each other by means of diodes. This forms a rectangular arrangement of 6 × 10 cells in the panel, as shown in Figure 1. The arrangement of the PV cells is a characteristic that impacts how cooling may affect any efficiency increases, this due to the dependence of electricity generation between the groups. The group with the cell or cells with highest resistance (due to a higher temperature) may activate the bypass diode and will limit the amount of energy produced by the array. Hence, if there is a large temperature gradient, it should theoretically affect the least number of groups if it is aligned with the groups (i.e., in the horizontal direction).  The solar modules are electrically characterised by running a power output test under different electric charging conditions, through the capacitor charging method for an I-V curve tracer [21], as depicted in Figure 2.
This test consists of connecting the PV module to a large capacitor C with a Switch (SW 1 ) and to two sensor probes (A, V), both connected to an oscilloscope. The capacitor is Water 2021, 13, 229 5 of 31 then discharged through a large resistor R by activating SW 2 and disconnecting SW 1 . This cycle is repeated several times for statistical analysis. The data collected from this test are then used to trace the characteristic I-V curve for the PV module. This characterisation was conducted at a solar irradiation value of 1000 W m −2 and an average module temperature of 59.2 • C.
The solar modules are electrically characterised by running a power output test under different electric charging conditions, through the capacitor charging method for an I-V curve tracer [21], as depicted in Figure 2.
This test consists of connecting the PV module to a large capacitor C with a Switch (SW1) and to two sensor probes (A, V), both connected to an oscilloscope. The capacitor is then discharged through a large resistor R by activating SW2 and disconnecting SW1. This cycle is repeated several times for statistical analysis. The data collected from this test are then used to trace the characteristic I-V curve for the PV module. This characterisation was conducted at a solar irradiation value of 1000 W m −2 and an average module temperature of 59.2 °C. The relationship between the electrical efficiency of the module and its operating temperature is determined experimentally at a uniform temperature and constant solar irradiation. This is achieved by cooling the PV module using ice, distributed over the surface of the panel. The temperature is monitored until it reaches 10 °C; then, the ice and excess moisture are removed, and finally the open circuit voltage and short circuit current of the PV panel are measured. The measurements are continued while the temperature of the panel increases due to the warm ambient conditions, with data collected in triplicate each time an increment of 5 °C is detected.
The electrical efficiency of the PV module is calculated using the Fill Factor (FF), which is the ratio of maximum obtainable power to the product of open circuit voltage and short circuit current. The calculation of the efficiency (η) considers the energy produced with respect to the received solar energy. For this work, the efficiency is determined by: where Gs is the solar irradiation, Voc is the open-circuit voltage, Isc is the short circuit current, and APV is the active area of the PV module.

PVT Module Design
A heat exchanger is adapted into the PV module, consisting of a rectangular enclosure of expanded PVC mounted on the back of the module. This enclosure has the same width and length dimensions as the module (see Table 1), and a proposed thickness of 27 mm. Preliminary calculations indicated that a fluid layer thickness of 50 mm or less would yield better heat exchange between the cooling fluid and the PV module, so the thickness of the aluminium frame was used for simplicity. The relationship between the electrical efficiency of the module and its operating temperature is determined experimentally at a uniform temperature and constant solar irradiation. This is achieved by cooling the PV module using ice, distributed over the surface of the panel. The temperature is monitored until it reaches 10 • C; then, the ice and excess moisture are removed, and finally the open circuit voltage and short circuit current of the PV panel are measured. The measurements are continued while the temperature of the panel increases due to the warm ambient conditions, with data collected in triplicate each time an increment of 5 • C is detected.
The electrical efficiency of the PV module is calculated using the Fill Factor (FF), which is the ratio of maximum obtainable power to the product of open circuit voltage and short circuit current. The calculation of the efficiency (η) considers the energy produced with respect to the received solar energy. For this work, the efficiency is determined by: where G s is the solar irradiation, V oc is the open-circuit voltage, I sc is the short circuit current, and A PV is the active area of the PV module.

PVT Module Design
A heat exchanger is adapted into the PV module, consisting of a rectangular enclosure of expanded PVC mounted on the back of the module. This enclosure has the same width and length dimensions as the module (see Table 1), and a proposed thickness of 27 mm. Preliminary calculations indicated that a fluid layer thickness of 50 mm or less would yield better heat exchange between the cooling fluid and the PV module, so the thickness of the aluminium frame was used for simplicity.
The design of the heat exchanger includes 16 valves that can work both as inlets and outlets, to allow experimenting with different configurations of forced convective flow of cooling water. This proposed PVT module design with multiple inlets and outlets and an unobstructed enclosure allows experimental versatility for testing different configurations within the same piece of equipment. This particular design was chosen as opposed to the implementation of fixed defined channels within the exchanger, given that such a design would require a different heat exchanger to be constructed for each configuration. Moreover, the proposed design allows the study of the effect of recirculation zones, as well as the effect of volumetric flow rate of cooling fluid on the flow pattern.
A total of 11 configurations are analysed in this paper in terms of heat removal, electrical generation and pressure drop. The location of the valves is indicated in Figure 3. The 11 analysed configurations are presented in Table 2, and one of them (B4) is graphically depicted in Figure 4 as an example. The graphical representation of the remaining configurations can be found in Appendix A. All the configuration schematics are accompanied by the expected flow lines. As the work presented in this paper does not include fluid dynamics simulations, the flow lines presented in Figure 4 are only for illustrative purposes, and are estimated based on the corresponding experimental thermographic image for each configuration. The configurations are classified into three subgroups, named by the manner in which the fluid enters the heat exchanger: from the bottom (B), from a single valve (S), and laterally (L).
The design of the heat exchanger includes 16 valves that can work both as inlets and outlets, to allow experimenting with different configurations of forced convective flow of cooling water. This proposed PVT module design with multiple inlets and outlets and an unobstructed enclosure allows experimental versatility for testing different configurations within the same piece of equipment. This particular design was chosen as opposed to the implementation of fixed defined channels within the exchanger, given that such a design would require a different heat exchanger to be constructed for each configuration. Moreover, the proposed design allows the study of the effect of recirculation zones, as well as the effect of volumetric flow rate of cooling fluid on the flow pattern.
A total of 11 configurations are analysed in this paper in terms of heat removal, electrical generation and pressure drop. The location of the valves is indicated in Figure  3. The 11 analysed configurations are presented in Table 2, and one of them (B4) is graphically depicted in Figure 4 as an example. The graphical representation of the remaining configurations can be found in Appendix A. All the configuration schematics are accompanied by the expected flow lines. As the work presented in this paper does not include fluid dynamics simulations, the flow lines presented in Figure 4 are only for illustrative purposes, and are estimated based on the corresponding experimental thermographic image for each configuration. The configurations are classified into three subgroups, named by the manner in which the fluid enters the heat exchanger: from the bottom (B), from a single valve (S), and laterally (L).

PVT Module Performance Analysis
To analyse and characterise the thermal and electric performance, each proposed heat exchange configuration is tested in five PVT modules simultaneously while recording the weather conditions in terms of solar irradiation (Gs), wind speed (va), relative humidity (HR), and ambient temperature (Ta). A regular (unmodified) PV module without a cooling system is also tested under the same conditions for comparison purposes. Each configuration test is done under two different cooling water flows (1 and 2 L min −1 ). The cooling water is obtained from a nearby well, with an average salinity (wb) of 500 ppm. The cooling water temperature is kept constant and all the tests are repeated three times. Figure 5a shows a photograph of the experiment, whereas Figure 5b shows a schematic of the experimental setup: a low-pressure (LP) pump makes water flow from a storage tank to the PVT modules, cooling them as they generate electricity, before it is finally discharged to another storage tank. The variables measured are the electrical power generated (Pe), the temperatures of the PV module (TPV), the water inlet and outlet temperature at the panel (Twi and Two, respectively), the conductivity of the cool water, and the pressure losses across the heat exchanger. The photovoltaic panel operates at maximum power conditions by connecting a micro-inverter to the electrical output of the panel. The micro-inverter injects the generated energy to the power grid, which draws a current and acts as an electrical load.

PVT Module Performance Analysis
To analyse and characterise the thermal and electric performance, each proposed heat exchange configuration is tested in five PVT modules simultaneously while recording the weather conditions in terms of solar irradiation (G s ), wind speed (v a ), relative humidity (H R ), and ambient temperature (T a ). A regular (unmodified) PV module without a cooling system is also tested under the same conditions for comparison purposes. Each configuration test is done under two different cooling water flows (1 and 2 L min −1 ). The cooling water is obtained from a nearby well, with an average salinity (w b ) of 500 ppm. The cooling water temperature is kept constant and all the tests are repeated three times. Figure 5a shows a photograph of the experiment, whereas Figure 5b shows a schematic of the experimental setup: a low-pressure (LP) pump makes water flow from a storage tank to the PVT modules, cooling them as they generate electricity, before it is finally discharged to another storage tank. The variables measured are the electrical power generated (P e ), the temperatures of the PV module (T PV ), the water inlet and outlet temperature at the panel (T wi and T wo , respectively), the conductivity of the cool water, and the pressure losses across the heat exchanger. The photovoltaic panel operates at maximum power conditions by connecting a micro-inverter to the electrical output of the panel. The micro-inverter injects the generated energy to the power grid, which draws a current and acts as an electrical load. The environmental conditions (Gs, va, HR and Ta) are monitored by a weather station. The water temperature and salinity at the PVT module inlet and outlet are measured using thermocouples and an electrical conductivity meter, respectively, and the cooling/feed water flow rate (Qw) is manipulated using a bypass valve. The pressure losses are estimated from the total flow rate passing through the LP pump (Qw plus the bypass) using the characteristic curve of the pump as supplied by the manufacturer, shown in Appendix B.
The PV module temperature distribution for each configuration is measured using a thermographic camera (model Flir i5) with a thermal sensitivity of <0.1 °C at 25 °C. The The environmental conditions (G s , v a , H R and T a ) are monitored by a weather station. The water temperature and salinity at the PVT module inlet and outlet are measured using thermocouples and an electrical conductivity meter, respectively, and the cooling/feed water flow rate (Q w ) is manipulated using a bypass valve. The pressure losses are estimated from the total flow rate passing through the LP pump (Q w plus the bypass) using the characteristic curve of the pump as supplied by the manufacturer, shown in Appendix B.
The PV module temperature distribution for each configuration is measured using a thermographic camera (model Flir i5) with a thermal sensitivity of <0.1 • C at 25 • C. The collected temperature data from each thermal image are then normalised, and the median, minimum, maximum, and standard deviation of the temperatures on the module surface are determined from these data. The heat removed from the module by the cooling water (q w ) is calculated using the following expression: where ρ w and C pw are the density and heat capacity of the cooling water, respectively. Cooling water properties are estimated using correlations reported by Sharqawy et al. [22] for density, heat capacity [23], and viscosity [24]. The heat removed by the heat exchanger (q x ) is modelled by: where A px is the PV module area in contact with the cooling water (the heat exchange area), ∆T lm is the logarithmic mean temperature difference between the module and the cooling water, and U is the global heat transfer coefficient for the heat exchanger. It is considered that the heat removed from the module is the only heat absorbed by the cooling water, hence q w = q x , such that U can be obtained using Equations (2) and (3) for each heat exchange configuration: As a non-uniform temperature distribution will lead to a decrease in electrical efficiency [16], we define the efficiency drop (∆η) as the difference between the efficiency expected at the mean temperature of the PV module (η u ) and the measured electrical efficiency under forced convective cooling (η c ): The uniform efficiency at the mean temperature (η u ) is estimated from the experimental data obtained following the methodology described in Section 2.1. Finally, the relationship between efficiency drop is correlated to G s , T PV , U, Q w and the standard deviation of the PV module temperature distribution (σ T ). The average values of ∆η, Q w , U, σ T and maximum PVT module temperature (T PV,max ) are quantified for each test. These values are compared with a theoretical distribution of probabilities for continuous quantitative variables, using the t-student parametric distribution [25].
A linear regression between the parameters (Q w , U, σ T and T PV,max ) and the efficiency drop (∆η) is also performed to determine the influence of these parameters on the output performance of the system. This analysis assists in determining which parameter has a stronger effect on the electrical efficiency. If the Pearson correlation coefficient between ∆η and either Q w , U, σ T or T PV,max is less than −0.7, this would suggest that increasing the value of that parameter would improve the efficiency of the PVT module (by decreasing the efficiency drop), according to factorial simplicity index theory of Kaiser [26].

PVT-RO Simulation
A mathematical model of the combined PVT-RO system depicted in Figure 6 is implemented in MATLAB (Mathworks). The PVT-RO model (see Figure 7) consists of a solar PVT module, a high-pressure (HP) pump, and a RO membrane module. Some outputs of the PVT model (Q b and P e ) are inputs to the HP pump model that determines the RO feed pressure (p). The RO model then takes inputs from both the PVT and HP pump models to determine the permeate variables (Q p and w p ). All models are considered to operate under quasi-steady state, under the assumption that any dynamic changes to the operating conditions (changes in weather, location of the sun, etc.) occur at a slower time scale than necessary to reach thermal equilibrium in both the PVT and the RO modules.

PVT-RO Simulation
A mathematical model of the combined PVT-RO system depicted in Figure 6 is implemented in MATLAB (Mathworks). The PVT-RO model (see Figure 7) consists of a solar PVT module, a high-pressure (HP) pump, and a RO membrane module. Some outputs of the PVT model (Qb and Pe) are inputs to the HP pump model that determines the RO feed pressure (p). The RO model then takes inputs from both the PVT and HP pump models to determine the permeate variables (Qp and wp). All models are considered to operate under quasi-steady state, under the assumption that any dynamic changes to the operating conditions (changes in weather, location of the sun, etc.) occur at a slower time scale than necessary to reach thermal equilibrium in both the PVT and the RO modules.

Solar PVT Module
The PVT model consists of a system of non-linear equations, solved using a Newton-Raphson iterative method. The energy balance around the PVT module is given by: where EPV is the rate of net energy input into the PVT module, qs is the energy input from solar irradiation, qa is the thermal energy removed by convective heat transfer with the wind, qr is the cooling via radiative heat transfer with the sky, and Peo is the electrical power output of the PVT module. The terms on the right-hand side of Equation (6) are estimated by Equation (3) and by: where Gbeam and Gdif are the beam and diffuse solar irradiation, αs and APV are the solar

Solar PVT Module
The PVT model consists of a system of non-linear equations, solved using a Newton-Raphson iterative method. The energy balance around the PVT module is given by: where E PV is the rate of net energy input into the PVT module, q s is the energy input from solar irradiation, q a is the thermal energy removed by convective heat transfer with the wind, q r is the cooling via radiative heat transfer with the sky, and P eo is the electrical power output of the PVT module. The terms on the right-hand side of Equation (6) are estimated by Equation (3) and by: where G beam and G dif are the beam and diffuse solar irradiation, α s and A PV are the solar absorptivity and the active area of the PV module, f θ is the angular dependence of solar absorptance [27], θ is the incidence angle between the PV module and the sun, h a is the wind heat transfer coefficient, A pa is the module area in contact with the wind, σ is the Stefan-Boltzmann constant, T sky is the sky temperature, and ε is the emissivity of the PV module. Further details and auxiliary expressions used in computing the values used for Equation (7) through Equation (10) are provided in Appendix C.
For the cooling water, the energy balance considers only the heat exchange with the panel that leads to an increase in water temperature, and the sensible heat absorbed by the water as given by Equation (2) The energy balance for the cooling water is then: where E w is the rate of net energy input into the cooling water. At steady state, both rates of energy input (E PV and E w ) should be equal to zero, leading to a system of two non-linear equations: These non-linear equations represent the energy balances around the PV module-heat exchanger system (i.e., the PVT module). The weather conditions and the inlet cooling water flow properties are inputs to this system, and the outputs consist of the outlet water flow properties and electrical power (P e ), as depicted in Figure 7.
However, not all of the electrical power generated by the PVT module (P eo ) is available for the HP pump. This because a non-negligible amount of power is required to force the flow of cooling water across the PVT heat exchanger. In order to account for this, the low-pressure pumping power (P LP ) is estimated as: where ∆p PVT is the pressure drop across the heat exchanger in the PVT module and η LP is the energy efficiency of the LP pump (assumed as 75% for a fit-for-purpose pump). The power available for the HP pumps is then taken to be:

HP Pump
The HP pump mathematical model is a series of multilinear regressions fitted to the technical datasheet of a HP DC pump (SunPumps SIJ 3.1-1500P-225 BL). The inputs to this model are the PV power output (P e ) and the cooling water outlet flow rate (Q wo ). The model calculates the maximum water pressure (p) that can be provided by the HP pump to the RO module under those conditions. More details are provided in Appendix C.

RO Membrane Module
The RO modelling in this work largely follows the approach of Toh et al. [28] and Bartholomew & Mauter [29], but with the additional complexity of allowing for the fluid properties (density, viscosity, heat capacity) and membrane properties (water and salt permeance) to vary with temperature as well as with salinity. At its core, the RO model is a combination of two simpler models: a non-linear algebraic equation system for determining the local permeate flux, and a system of non-linear ordinary differential equations (ODEs) describing the evolution of the unidimensional profiles of the variables (flow velocity, concentration, temperature, and pressure) along the membrane module, as depicted in Figure 8. The former is solved via the simple fixed-point iteration method [30], while the latter is solved by a Runge-Kutta type algorithm.
The RO modelling in this work largely follows the approach of Toh et al. [28] and Bartholomew & Mauter [29], but with the additional complexity of allowing for the fluid properties (density, viscosity, heat capacity) and membrane properties (water and salt permeance) to vary with temperature as well as with salinity. At its core, the RO model is a combination of two simpler models: a non-linear algebraic equation system for determining the local permeate flux, and a system of non-linear ordinary differential equations (ODEs) describing the evolution of the unidimensional profiles of the variables (flow velocity, concentration, temperature, and pressure) along the membrane module, as depicted in Figure 8. The former is solved via the simple fixed-point iteration method [30], while the latter is solved by a Runge-Kutta type algorithm. The local volumetric permeate flux (J v ) at each point along the RO membrane module (x-direction) is estimated using the Kedem-Katchalsky-Merten equation [31]: where ∆p tm is the transmembrane pressure difference between the feed and permeate channels, π m and π p are the osmotic pressure on the feed and permeate side of the RO membrane, µ is the fluid viscosity and R m is the membrane resistance. The effect of changes in water temperature is explicitly considered to affect the fluid viscosity, as well as the osmotic pressure following the van't Hoff equation for NaCl: where ϕ is the osmotic coefficient, R g is the universal gas constant and M s is the molar mass of NaCl. The temperature dependence of osmotic pressure is explicit in Equation (16), but it is also implicit as ϕ and ρ are also considered to be temperature dependent [22]. The effect of temperature on the membrane resistance and salt permeance were determined experimentally for a commercial DuPont BW30 membrane. A non-linear system arises from the coupling of permeate flux and membrane surface salinity (w m ) due to the effect of concentration polarisation [28]: where Г is the concentration polarisation modulus and k mt is the mass transfer coefficient on the membrane surface. Finally, the model describing the profiles along the membrane module is based on mass and energy balances under steady-state non-isothermal conditions, and neglecting longitudinal dispersion. This results in the following system of ODEs: where x is the direction along the membrane module length, the subscript b represents the feed channel bulk conditions, h ch is the membrane channel height (taken to be equal to the spacer thickness), is the feed channel void fraction (the volume other than the spacer mesh), d h is the hydraulic diameter of the spacer-filled feed channel, f is the Fanning friction factor, R obs is the local observed rejection, and ∆H s,r = 66.2 kJ kg −1 is the specific enthalpy change of solution for NaCl at reference conditions of T r = 25 • C and p r = 1 atm. The transmembrane temperature-enthalpy change (∆H T,tm ), and transmembrane pressure-enthalpy change (∆H p,tm ), are respectively defined as follows: Equations (18)- (21) are integrated along the membrane module length (see Figure 8) to obtain the total permeate flow (Q p ) and permeate salinity (w p ). Further details regarding the estimation of permeate flow and auxiliary equations are presented in Appendix C.
All simulations assume a quasi-steady state using data from a local weather station in Ciudad Obregon, Mexico (27 • 29 35.2 N 109 • 58 10.7 W), for summer (24 July 2018), autumn (20 October 2019) and winter (3 January 2020) conditions. The weather data for these dates and location is presented in Appendix D. Three simulation scenarios are considered: Scenario 1 "Without cooling" emulates the electrical generation of a regular PV-RO plant for comparison; Scenario 2 "Continuous cooling" emulates the performance of a PVT-RO plant incorporating the different heat exchange configurations considered in this paper; and Scenario 3 "Max production" selects the best performance out of scenarios 1 and 2 depending on the time of the day and environmental conditions, which results in the maximum possible permeate production for the day under consideration. All scenarios assume the same dimensions and characteristics of the experimental PVT module, and the simulated RO membrane is based on the BW30-8040 DuPont FilmTec module for brackish water [32]. The parameter values considered are summarised in Table 3.
To simulate the behaviour of the PVT-RO system under the different cooling configurations in Scenario 2, the simulation parameters are adjusted to match each heat exchange configuration experimental data as follows: (1) the experimental value of U is used as an input to the PVT model, and (2) the dependency of ∆η on the operating conditions (G s , T PV and Q w ) and cooling configuration is incorporated. Moreover, to guarantee that 1 and 2 L min −1 are flowing through each of the modules while keeping the same feed flow rate into the RO module (Q b ), half of the feed flow rate is assumed to bypass the PV panels at Q w = 1 L min −1 . Table 3. Simulated PVT-RO unit dimensions and characteristics.

PV Module Characterisation
Experimental tests showed that the behaviour of the PV modules varies from the data reported by the manufacturer. This is due to manufacturing defects, so it is recommended to generate a specific I-V curve for the PV panel used under real conditions. A PV module efficiency loss of 0.7% for each 10 • C increase in uniform temperature was observed. Figure 9 shows the I-V curve obtained in the field at T PV = 59.2 • C and G s = 1000 W m −2 . The real efficiency obtained is 16.02%, about 0.58% lower than that reported by the manufacturer (see Table 1). Regarding the effect of temperature, Figure 10 shows that the electrical efficiency significantly decreases due to increases in temperature. A significant (95% confidence) negative correlation is observed for the relationship between these variables (r = 0.92, p < 0.001). The degree of goodness of fit was high (R 2 = 0.85) for the linear regression model: where TPV is the uniform PV module temperature in °C. Regarding the effect of temperature, Figure 10 shows that the electrical efficiency significantly decreases due to increases in temperature. A significant (95% confidence) negative correlation is observed for the relationship between these variables (r = 0.92, p < 0.001). The degree of goodness of fit was high (R 2 = 0.85) for the linear regression model: where T PV is the uniform PV module temperature in • C.
Regarding the effect of temperature, Figure 10 shows that the electrical efficiency significantly decreases due to increases in temperature. A significant (95% confidence) negative correlation is observed for the relationship between these variables (r = 0.92, p < 0.001). The degree of goodness of fit was high (R 2 = 0.85) for the linear regression model: where TPV is the uniform PV module temperature in °C. Figure 10. Effect of temperature on efficiency for the PV module.

Experimental Results of the PV Module Cooling Configurations
In order to remove the biases and effects introduced by the different environmental conditions at the time of testing each of the different configurations, the temperature distribution for each configuration was first normalised as follows:

Experimental Results of the PV Module Cooling Configurations
In order to remove the biases and effects introduced by the different environmental conditions at the time of testing each of the different configurations, the temperature distribution for each configuration was first normalised as follows: where the subscript n represents the normalised temperature, NC refers to the module that is not cooled, and C refers to the water-cooled module. The overbars represent the arithmetic mean of the temperature for the corresponding module. Subtracting the temperature of the cooled module from that of the non-cooled module removes some of the bias associated with higher or lower solar irradiation during measurement. By this definition, the normalised temperature should have an arithmetic mean of 1. With respect to the temperature distribution in the PVT, Figure 11 shows the ranges of normalised temperatures for each heat exchange configuration, indicating the maximum and minimum temperatures, as well as the sizes of the central interquartiles and the median. This information can help determine which configurations lead to a more uniform temperature distribution. Group B has more uniform and compact ranges than groups S and L. Conversely, group S presents the largest ranges of temperatures, particularly configurations S1 and S3.
The L group shows the configuration with the smallest normalised temperature range (L2), although there is more variability in the ranges within the L group, this compared to the B group. The configurations with the smallest normalised temperature range are the ones that lead to a more uniform temperature, being this the condition with ∆η close to zero.
The results of the experimental measurements of efficiency drop for the PVT modules are shown in Figure 12. In the graphs in this figure, ∆η is plotted against maximum temperature (T PV,max ), overall heat transfer coefficient (U) and the standard deviation of the temperature distribution (σ T ). In addition, the two values of water flow rate (Q w ) for each configuration are joined by a line, with a circle indicating the larger flow rate of Q w = 2 L min −1 . of normalised temperatures for each heat exchange configuration, indicating the maximum and minimum temperatures, as well as the sizes of the central interquartiles and the median. This information can help determine which configurations lead to a more uniform temperature distribution. Group B has more uniform and compact ranges than groups S and L. Conversely, group S presents the largest ranges of temperatures, particularly configurations S1 and S3. The L group shows the configuration with the smallest normalised temperature range (L2), although there is more variability in the ranges within the L group, this compared to the B group. The configurations with the smallest normalised temperature range are the ones that lead to a more uniform temperature, being this the condition with Δη close to zero.
The results of the experimental measurements of efficiency drop for the PVT modules are shown in Figure 12. In the graphs in this figure, Δη is plotted against maximum temperature (TPV,max), overall heat transfer coefficient (U) and the standard deviation of the temperature distribution (σT). In addition, the two values of water flow rate (Qw) for each configuration are joined by a line, with a circle indicating the larger flow rate of Qw = 2 L min −1 .
Configuration types B, S and L are indicated by similar colour groups in Figure 12. This is done to visualise the similarities between the members of each group. In the case of group B, it can be seen that the configurations are grouped on the lower part of each graph, where Δη is lower and therefore closer to the efficiency of the photovoltaic panel at uniform temperature. This suggests that configurations belonging to group B result in  best performance in terms of temperature distribution, leading to lower efficiency drop. It can also be seen in Figure 12b that, for all configurations, increasing Qw reduced the efficiency drop and increased U. However, Figure 12a,c indicate that a larger temperature range (i.e., a larger TPV,max) or a larger temperature variability (i.e., a larger σT) do not necessarily lead to larger efficiency drops in the configurations tested. This is particularly evident for configurations B1, L2 and L3, for which TPV,max and σT increase as Qw is increased, but Δη is reduced. Nevertheless, if the analysis is focused on the overall data, a general tendency can be seen that Δη is larger for configurations with larger σT and larger TPV,max. On the flipside, this general tendency is not observed for the relationship between Δη and U. These observations are corroborated by the overall Pearson correlation coefficients presented in Table 4, and suggest that, although in general a better temperature distribution and lower temperature range will lead to less efficiency losses, for any particular configuration heat removal drives efficiency drop.  Configuration types B, S and L are indicated by similar colour groups in Figure 12. This is done to visualise the similarities between the members of each group. In the case of group B, it can be seen that the configurations are grouped on the lower part of each graph, where ∆η is lower and therefore closer to the efficiency of the photovoltaic panel at uniform temperature. This suggests that configurations belonging to group B result in best performance in terms of temperature distribution, leading to lower efficiency drop. It can also be seen in Figure 12b that, for all configurations, increasing Q w reduced the efficiency drop and increased U.
However, Figure 12a,c indicate that a larger temperature range (i.e., a larger T PV,max ) or a larger temperature variability (i.e., a larger σ T ) do not necessarily lead to larger efficiency drops in the configurations tested. This is particularly evident for configurations B1, L2 and L3, for which T PV,max and σ T increase as Q w is increased, but ∆η is reduced. Nevertheless, if the analysis is focused on the overall data, a general tendency can be seen that ∆η is larger for configurations with larger σ T and larger T PV,max . On the flipside, this general tendency is not observed for the relationship between ∆η and U. These observations are corroborated by the overall Pearson correlation coefficients presented in Table 4, and suggest that, although in general a better temperature distribution and lower temperature range will lead to less efficiency losses, for any particular configuration heat removal drives efficiency drop. Table 4. Pearson correlation coefficients between efficiency drop (∆η) and each of the studied input parameters, for each configuration and for the overall data set. Highly significant correlation coefficients are highlighted in bold. To quantify the trends observed in Figure 12, the statistical relationships between the variables in that figure are presented in Table 4, which shows the Pearson correlation coefficients between the efficiency drop for each configuration and the operating parameters studied, as well as the overall correlations. It can be seen that Q w has a negative correlation with ∆η in all configurations and overall. This means that increasing feed flow rate generally led to an increase in electrical efficiency, confirming the trends observed in Figure 12b. Configurations B4, S1, S2, and L2 show a correlation coefficient magnitude ≥ 0.7, which classifies as highly significant behaviour according to the factorial simplicity index.

Pearson Correlation Coefficient between ∆η and:
On the other hand, the heat transfer coefficient (U) did not show a highly significant behaviour in affecting the efficiency drop overall. Although for most configurations an increase in U was slightly correlated with a decrease in efficiency drop (a trend also observed in Figure 12b), the statistical results suggest that increasing heat transfer efficiency will not necessarily lead to higher energy generation. The results of configurations with a positive correlation may be related with the input cooling fluid being distributed vertically with less uniform heat extraction inside the PVT module. Since the cooling fluid residence time is low, this is probably due to the cooling fluid being directed to vertical outlets placed partially aligned with the inputs. The negative signs may be attributed to the fact that the temperature of the cooling fluid is increased more along flow lines with a longer residence time inside the PVT module. This is due to the fact that the direction of the water changes, as inputs are not aligned with outputs, leading to recirculation zones.
As regards the effect of σ T on the efficiency drop, the L3 configuration presents a correlation coefficient ≤−0.7 (r = −0.92). This result is probably due to the multiple inlets and outlets being aligned horizontally and the PV cells in the module being wired vertically and in parallel with each other, thus the temperature variation along each series of PV cells is minimal. The negative correlation can be explained by the fact that a higher temperature variability is mostly associated with a larger water temperature increase due to more cooling of the PV module, hence reducing the efficiency drop. On the other extreme, the B3 configuration has a correlation coefficient ≥0.7. This could indicate that the fluid tends flow largely in the vertical direction, resulting in a large temperature difference along the PV cells wired in series, therefore reducing the electrical efficiency in the photovoltaic panel.
The correlation coefficients for T PV,max show that configuration B1 has a highly significant coefficient ≤−0.7 (r = −0.82). This can probably be attributed to the symmetric vertical flow of cooling water. Configurations B3 and L3 also show highly significant correlation coefficients ≥0.7. This is probably because the multiple outlets are geometrically spaced from each other.

PVT-RO Modelling Results
The experimental data for heat transfer (U) and efficiency drop is incorporated into the PVT-RO model described in Section 2.4. This in order to estimate the energy generated by the PVT modules under different cooling configurations and without forced cooling, as well as the energy available to be used in the RO desalination process (E e ). The model then uses these results to predict the volume of water produced by the PVT-RO hybrid system (Vol p ) and the salinity of the permeate (w p ). Simulation results for two representative dates are presented in Tables 5 and 6.
The model results for 20 October (Table 5) show that B1 has the best performance of all configurations tested under continuous cooling at Q w = 1 L min −1 , both in terms of energy and permeate production. For the tests under continuous cooling at Q w = 2 L min −1 , the energy available is reduced in comparison with the lower flow rate, and therefore the production of permeate water is also reduced. The model results for 24 July (Table 6) also show that B1 results in the best performance for both Q w = 1 L min −1 and Q w = 2 L min −1 . An important factor to be considered in addition to the amount of water produced is the salinity of the permeate flow, as this has implications in terms of the potential applications for PVT-RO. Observed rejection for all the configurations tested range around 92% to 95% in the summer, when the lowest values are observed. Importantly, the configuration with the largest water production (B1) presents the lowest rejection (92%) and largest permeate salinity (39 ppm) from a feed water salinity of 500 ppm. This is expected, as Figure 12b shows that configuration B1 yields the largest heat transfer coefficient, and hence the highest RO feed water temperature. A higher temperature feed is known to reduce salt rejection as well as increasing water permeance for RO and other osmotic separation membranes [19,34].
Another effect that can be observed in Figure 12 is a lower permeate salinity at the larger cooling water flow rate of Q w = 2 L min −1 . This effect can be related to the lower available pumping energy at this higher cooling rate. Although in general, for all configurations, operating at the larger flow rate results in a lower efficiency drop (see Figure 13), the larger pressure drop across the PVT heat exchanger results in less energy available (E e ) for the RO HP pump. This in turn leads to a lower operating pressure for the RO module, which results in a lower permeate flux, but also in less concentration polarisation as predicted by Equation (17). This latter effect also leads to a lower membrane surface salinity, and thus less salt passage through the RO membrane and ultimately lower permeate salinity. Figure 13 shows the variation in water production under the three scenarios simulated, that is, under continuous cooling, without cooling, and under maximum production conditions. This latter scenario uses cooling only when it is predicted to result in greater water production than without cooling. These data are presented for the day in autumn for configuration B1 at Q w = 1 L min −1 (Figure 13a) and at Q w = 2 L min −1 (Figure 13b). For the lower flow rate, the scenario without cooling yields more permeate water during the early morning and late afternoon hours, but the cooling scenario results in greater permeate production for most of the day. Therefore, a hybrid operation consisting of turning the cooling system on and off is proposed to maximise the generation of electricity and hence maximising the water production. For this case, maximum production would be achieved if the cooling system is engaged only when the increase in energy generation due to the increase in efficiency overcomes the trade-off with energy losses due to pressure drop in the heat exchanger, which occurs roughly between 9:00 h and 15:30 h. On the other hand, for the larger water flow rate, the cooled scenario never overcomes the increased energy losses due to pressure drop, so for that case it is not convenient to engage the cooling system on that particular autumn day. The data presented in Tables 5 and 6 and Figure 13 are for simulation results using weather measurements from the autumn and summer seasons. Nonetheless, simulations were also carried out using data from the winter season (3 January 2020). However, the winter simulations indicated less permeate production under forced cooling regardless of the water flow rate. This because the power required to overcome pressure losses in the PVT heat exchanger is larger than the gains in energy production, resulting in less available power for the HP pump. Hence, the winter data are not presented in this paper.

Conclusions
The results presented in this paper confirm that it is possible to achieve larger PV energy production as well as more RO permeate by cooling the PV modules using the RO feed water, achieving the expected synergies. However, pumping energy is required to force the flow of cooling water across the heat exchange surface with the PV module, which presents a trade-off that limits the conditions under which it is advisable to operate this cooling. This because the efficiency gains may not be sufficient to cover the required pumping energy, resulting in less energy available to operate the RO unit despite larger energy generation by the PV module.
The results of the statistical analysis of the operating parameters for the PVT-RO system yield some insights into the characteristics of the proposed cooling configurations that result in greater electrical efficiency gains. The main objective of the forced convective cooling through heat exchange should be to remove as much thermal energy as possible without incurring in significant efficiency drop due to a non-uniform temperature distribution. Although higher heat transfer coefficients may maximise the efficiency for a particular configuration, this should not be the only consideration.
On the other hand, the way in which the fluid enters the PVT module was seen to be one of the main drivers of efficiency drop, with configurations from group B leading to better performance. This group was characterised by symmetric vertical flow feed and multiple inputs close to each other. This can be related to PV cells being grouped in series by the manufacturer in vertical direction. Moreover, the cooling configurations that presented the best electrical performance were those that forced the fluid to circulate in a continuous direction from inlet to outlet, either horizontal or vertical, preventing recirculation. Future complementary investigations are recommended, in which the implementation of flow channels inside the heat exchanger is contemplated, as well as the optimisation of cooling water flow rate.
In addition, the PVT water outlet temperature was shown to influence the RO system, so it is imperative to consider the PVT-RO system as a whole when selecting the The data presented in Tables 5 and 6 and Figure 13 are for simulation results using weather measurements from the autumn and summer seasons. Nonetheless, simulations were also carried out using data from the winter season (3 January 2020). However, the winter simulations indicated less permeate production under forced cooling regardless of the water flow rate. This because the power required to overcome pressure losses in the PVT heat exchanger is larger than the gains in energy production, resulting in less available power for the HP pump. Hence, the winter data are not presented in this paper.

Conclusions
The results presented in this paper confirm that it is possible to achieve larger PV energy production as well as more RO permeate by cooling the PV modules using the RO feed water, achieving the expected synergies. However, pumping energy is required to force the flow of cooling water across the heat exchange surface with the PV module, which presents a trade-off that limits the conditions under which it is advisable to operate this cooling. This because the efficiency gains may not be sufficient to cover the required pumping energy, resulting in less energy available to operate the RO unit despite larger energy generation by the PV module.
The results of the statistical analysis of the operating parameters for the PVT-RO system yield some insights into the characteristics of the proposed cooling configurations that result in greater electrical efficiency gains. The main objective of the forced convective cooling through heat exchange should be to remove as much thermal energy as possible without incurring in significant efficiency drop due to a non-uniform temperature distribution. Although higher heat transfer coefficients may maximise the efficiency for a particular configuration, this should not be the only consideration.
On the other hand, the way in which the fluid enters the PVT module was seen to be one of the main drivers of efficiency drop, with configurations from group B leading to better performance. This group was characterised by symmetric vertical flow feed and multiple inputs close to each other. This can be related to PV cells being grouped in series by the manufacturer in vertical direction. Moreover, the cooling configurations that presented the best electrical performance were those that forced the fluid to circulate in a continuous direction from inlet to outlet, either horizontal or vertical, preventing recirculation. Future complementary investigations are recommended, in which the implementation of flow channels inside the heat exchanger is contemplated, as well as the optimisation of cooling water flow rate.
In addition, the PVT water outlet temperature was shown to influence the RO system, so it is imperative to consider the PVT-RO system as a whole when selecting the cooling configuration to be used. Thus, it can be concluded that out of the configurations tested, the characteristics of the B1 configuration are the best suited for the production of permeate water under the conditions proposed by this study. This configuration resulted in the highest percentage increase in permeate water compared to not using a cooling system, leading to a predicted 16.7% increase in production during summer and a 27% increase in production during autumn, growing to a 27.5% increase in autumn when only engaging the cooling system when the conditions for an increase in production is observed.
Although increasing the cooling water flow rate generally leads to a decrease in efficiency drop in the PV modules, the energy losses due to pressure drop in the heat exchanger also increase substantially, such that the gains in efficiency do not compensate for the energy losses. For this reason, it is recommended to run the system at lower cooling water flow rates. It is important to point out that the present investigation did not determine the optimal water flow rate for the proposed system. Hence, this is an area of opportunity for future studies. Nevertheless, the results are conclusive for lower cooling water flow rates in the ranges of the proposed experiment.
The decrease in rejection and higher permeate salinity when using feed water to cool the PV modules may be a limitation for applications with high salt content, such as seawater RO for which feed salinity is usually around 35,000 ppm. For that application, a relatively low rejection of 92% may lead to an unacceptably high permeate salinity. Therefore, the relevance of PVT-RO should be analysed according to the specific case and need. In addition, for seawater applications the feed osmotic pressure is higher, which would result in a reduction in the number of hours for which the system can be operated due to low energy generation at lower irradiation conditions. Conversely, if longer operating hours are a requirement, a larger capital investment in solar modules or batteries would be required to allow the operation at times of lower solar irradiation. Other potential applications for PV module cooling include pumping bore water in remote locations, for which the slight increase in water temperature may not be a significant issue.
In general, forced convective cooling is more likely to be beneficial under conditions of high solar irradiation (>900 W m −2 ) and high ambient temperature (>35 • C), such as those experienced in the summer in dry arid regions similar to northwest Mexico. In those regions, it is very likely that cooling will be beneficial for most of the summer daylight hours. However, in the spring and autumn, the times of the day for which cooling is beneficial are reduced, and cooling is basically of negligible use in the winter. As the weather conditions vary significantly with geographic location, a more detailed technoeconomic case study that considers expected weather patterns is recommended in order to determine whether this strategy would be economically beneficial for a particular location.

Patents
The design of the heat exchanger system used to cool the photovoltaic panel using RO feed water, for the purpose of water desalination, has been filed as a patent application to the Mexican Institute of Intellectual Property (IMPI). This application for intellectual property protection, in the industrial design modality, was received by the local IMPI office on December 19, 2019 and is currently pending assessment.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
This section presents thermographic images for each of the heat exchange configurations tested in this work, as well as schematic depiction of the expected flow lines. These are presented in Figure A1 for group B, in Figure A2 for group S, and in Figure A3 for group L. The images show areas with temperatures ranging from 25 to 69 • C. However, a scale with a temperature range of 30 to 45 • C is used to better visualise this range with a wider colour scale. The reader is referred to Table 2 for the description of each configuration. Figure A1. Diagrams of expected flow lines for group B configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. Water 2021, 13, x FOR PEER REVIEW 26 of 33 Figure A2. Diagrams of expected flow lines for group S configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. Figure A3. Diagrams of expected flow lines for group L configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. Figure A2. Diagrams of expected flow lines for group S configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration.
Water 2021, 13, x FOR PEER REVIEW 26 of 33 Figure A2. Diagrams of expected flow lines for group S configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. Figure A3. Diagrams of expected flow lines for group L configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. Figure A3. Diagrams of expected flow lines for group L configurations, alongside their respective thermographic images. The reader is referred to Table 2 for the description of each configuration. 1 a 1,1 a 2 where p is the fluid outlet pressure in psi, V pump is the voltage required by the pump in V, Q w is the volumetric flow rate L min −1 , and P e and is the electrical power available in W. The values of the coefficients for this multilinear fit are given in Table A1. The logarithmic mean temperature difference between the PV module and the cooling water (∆T lm ) used in Equations (3) and (4) is given by: For the experiments described in Section 2.3, solar irradiation (G s ) data are obtained from pyranometer measurements. However, the pyranometer measures the solar radiation on a horizontal plane. Therefore, assuming that the angle of incidence between the solar beam irradiation (G beam ) and the horizontal plane is β, the following holds: Diffuse irradiation (G dif ) in Equations (7) and (A3) is estimated as proposed by Boland et al. [35]: where k t is the hourly clearness index, defined in terms of the extraterrestrial radiation on a horizontal surface (H 0 ): The value of H 0 depends on the month of the year in question [27,36]. The angular dependence of solar absorptance (f θ ) in Equation (7) is given by [27]: f θ = 1 − 1.59 × 10 −3 θ + 2.73 × 10 −4 θ 2 − 2.3 × 10 −5 θ 3 + 9.02 × 10 −7 θ 4 −1.8 × 10 −8 θ 5 + 1.77 × 10 −10 θ 6 + 6.99 × 10 −13 θ 7 (A6) where θ is the incidence angle between the solar PV module and the solar beam irradiation, in degrees. The wind heat transfer coefficient (h a ) in Equation (8) is given by [27]: where h a is given in W m −2 K −1 , and v a in m s −1 . The sky temperature (T sky ) in Equation (9) is given by [27]: T sky = T a 0.71 + 0.0056T dp + 7.3 × 10 −5 T 2 dp + 0.013 cos 15π(t − 12) 180 1/4 (A8) where T sky and T a are in K, T dp is the dew point temperature in • C, and t is the time of the day in hours since midnight. The dew point temperature (T dp ) in Equation (A8) can be obtained using Antoine equation parameters for the vapour pressure of water, such that: where T dp and T a are in • C, R H is given as a percentage, and the coefficient values for water vapour are B = 1730.63 and C = 233.426 [37]. The mass transfer coefficient in Equation (17) depends on the geometry and flow conditions inside the membrane module, and can be obtained from empirical equations that correlate the Reynolds (Re) and Schmidt (Sc) numbers to the Sherwood number (Sh). Schock and Miquel [33] give the following correlation for typical RO spacer-filled (FilmTec) spiral wound membrane module: Sh = 0.065Re 0.875 Sc 0. 25 (A10) where the dimensionless numbers are defined as: where D is the solute diffusivity. The bulk fluid velocity in the feed channel of the RO module is defined as: where A m is the membrane area and L is the RO module length. The membrane resistance (R m ) in Equation (15) is slightly temperature dependent and can be determined from DI water permeation data by varying the feed pressure, a process that is widely reported in the literature [38][39][40]. This process is repeated for several temperature values, and R m is then fitted to a linear dependency on temperature based on experimental data: where T b is given in K, and R m is given in m −1 . This fit was obtained experimentally for the particular membrane modelled, which in the case of this paper is a BW30 (DuPont) TFC RO membrane. Salt passage through the membrane is modelled using the membrane intrinsic rejection, defined in terms of the salinity mass fraction on either side of the membrane, that is: On the other hand, the observed rejection (R obs ) in Equations (19) and (20) is defined in terms of the bulk and permeate salinity mass fractions, that is: These two rejection definitions are related through the concentration polarisation modulus: Similarly to membrane resistance, intrinsic rejection is also slightly temperature dependent. This relationship is determined experimentally for several temperature and salinity values, and R int is then fitted to a linear dependency on temperature based on the experimental data: where T b is given in • C.

Appendix D
This section presents the weather data sets used for the prediction of permeate water production for the proposed PVT-RO system, as well as the resulting simulated average PVT outlet water temperatures. The input data were sourced from a local weather station in Ciudad Obregon, Mexico (27 • 29 35.2 N 109 • 58 10.7 W). These data are presented in Table A2 for summer conditions (24 July 2018), in Table A3 for autumn conditions (20 October 2019), and in Table A4 for winter conditions (3 January 2020).  Table A5 shows the effect of the PVT module configuration on the temperature of the cooling water exiting the module. The inlet water temperatures are 28.7 • C and 20 • C on 24 July 2018 and 20 October 2019, respectively. The water temperatures for 3 January 2020 are not included as implementing cooling on that day did not result in increased permeate production for the PVT-RO system.