Solar water heater

Perhaps second in importance to the humidifier is the solar collector, that captures the thermal energy required to drive the HDH process. For this analysis, it was assumed that a small-scale HDH system would use a typical flat plate solar collector, as they have been used extensively in other HDH and desalination systems (Elminshawy et al. 2015). Therefore, a simple standard double-glazed flat plate solar water collector was modelled as the heat source for the HDH system. Hence, the Hottel-Whillier equations (Equations (1)–(12)) (Duffie and Beckman 1980) were used in order to mathematically model the collector.

As such, the outlet water temperature from the collector can be calculated from Equation (1). $T_{w,\mathrm{out},\mathrm{col}}=T_{w,\mathrm{in},\mathrm{col}}+{\displaystyle \frac{{\dot{Q}}_{u,\mathrm{col}}}{{\dot{m}}_{w,\mathrm{in},\mathrm{col}}{c}_{\mathrm{pw},\mathrm{col}}}}$(1) 

Where the rate of useful energy gained by water, associated with Equation (1), can be determined by solving Equations (2) and (3), simultaneously. ${\dot{Q}}_{u,\mathrm{col}}=A_{\mathrm{col}}[G_{\mathrm{abs}}-U_{\mathrm{col}}(T_{p,\mathrm{col}}-T_{\mathrm{amb}})]$(2) $T_{p,\mathrm{col}}=T_{w,\mathrm{in},\mathrm{col}}+{\displaystyle \frac{{\dot{Q}}_{u,\mathrm{col}}/A_{\mathrm{col}}}{U_{\mathrm{col}}{F}_{R,\mathrm{col}}}(1-F_{R,\mathrm{col}})}$(3) 

In Equation (3), the heat removal factor (F_R,col), can subsequently be determined from Equation (4). $F_{R,\mathrm{col}}={\displaystyle \frac{{\dot{m}}_{w,\mathrm{in},\mathrm{col}}{c}_{\mathrm{pw},\mathrm{col}}}{A_{\mathrm{col}}{U}_{\mathrm{col}}}\left[1-e^{{\displaystyle \frac{-A_{\mathrm{col}}{U}_{\mathrm{col}}{F}_{\mathrm{col}}^{\mathrm{\prime }}}{{\dot{m}}_{w,\mathrm{in},\mathrm{col}}{c}_{\mathrm{pw},\mathrm{col}}}}}\right]}$(4) where F′_col is the collector efficiency factor defined by Equation (5). $F_{\mathrm{col}}^{\mathrm{\prime }}={\displaystyle \frac{1/U_{\mathrm{col}}}{w\left[{\displaystyle \frac{1}{U_{\mathrm{col}}(2S_{\tau }F+D_{o,\tau ,\mathrm{col}})}+{\displaystyle \frac{1}{C_B}+{\displaystyle \frac{1}{h_{i,\tau ,\mathrm{col}}\pi D_{i,\tau ,\mathrm{col}}}}}}\right]}}$(5) 

And where F is the fin efficiency given by Equation (6). $F={\displaystyle \frac{1}{m{L}_{\tau }}\left[{\displaystyle \frac{\exp (m{L}_{\tau })-\exp (-m{L}_{\tau })}{\exp (m{L}_{\tau })+\exp (-m{L}_{\tau })}}\right]}$(6) taking the constant m to be defined by Equation (7). $m=\sqrt{U_{\mathrm{col}}/k_p{l}_p}$(7) 

Now the overall collector heat loss coefficient (U_col) is the summation of the bottom, edge and top heat loss coefficients, where the bottom and edge heat loss coefficient can be calculated from Equations (8) and (9), respectively. $U_{\mathrm{bottom}}={\displaystyle \frac{k_{\mathrm{ins}}}{l_{\mathrm{ins}}}}$(8) $U_{\mathrm{edge}}=U_{\mathrm{edge}}^{\mathrm{\prime }}\left({\displaystyle \frac{A_{\mathrm{pr}}}{A_{\mathrm{col}}}}\right)$(9) where U′ is the edge overall loss coefficient and was assumed to be 0.5.

And the top loss coefficient can be determined from Equation (10) (Agarwal and Larson 1981). $\begin{array}{rl}{U}_{\mathrm{top}}& ={\left[{\displaystyle \frac{N_{\mathrm{gl},\mathrm{col}}}{{\displaystyle \frac{Z}{T_{p,\mathrm{col}}}{\left({\displaystyle \frac{T_{p,\mathrm{col}}-T_{\mathrm{amb}}}{N_{\mathrm{gl},\mathrm{col}}+B}}\right)}^{0.33}}}+{\displaystyle \frac{1}{h_{\mathrm{wd}}}}}\right]}^{-1}\\ & +{\displaystyle \frac{\sigma (T_{p,\mathrm{col}+}{T}_{\mathrm{amb}})(T_{p,\mathrm{col}}^2+T_{\mathrm{amb}}^2)}{{[{\epsilon }_{p,\mathrm{col}}+0.5N_{\mathrm{gl},\mathrm{col}}(1-{\epsilon }_{p,\mathrm{col}})]}^{-1}+{\displaystyle \frac{2N_{\mathrm{gl},\mathrm{col}}+B-1}{{\epsilon }_{\mathrm{gl},\mathrm{col}}}-N_{\mathrm{gl},\mathrm{col}}}}}\end{array}$(10) where N_gl,col is the number of glass covers, and B and Z are empirical variables, which are given in Equations (11) and (12). $Z=250[1-0.0044(s-90)]$(11) $B=[1-0.04h_{\mathrm{wd}}+0.0005h_{\mathrm{wd}}^2][1+0.091N_{\mathrm{gl},\mathrm{col}}]$(12) 

Condenser and economiser

As noted previously, the saline water passes through both a condenser and economiser before entering the solar water heater. The condenser was modelled as a gas–liquid single pass shell and tube heat exchanger, as these have been previously used as condensers in HDH desalination systems (Al-Sahali and Ettouney 2008). For a single pass shell and tube heat exchanger, the effectiveness-NTU method can be used to determine the conditions of the outlet fluids, as shown in Equations (13) and (14) (Holman 2010). $T_{a,\mathrm{out},\mathrm{cond}}=T_{a,\mathrm{in},\mathrm{cond}}+{\displaystyle \frac{{\dot{Q}}_{\mathrm{NTU}}}{{\dot{m}}_{a,\mathrm{in},\mathrm{cond}}{c}_{\mathrm{pa},\mathrm{in},\mathrm{cond}}}}$(13) $T_{w,\mathrm{out},\mathrm{cond}}=T_{w,\mathrm{in},\mathrm{cond}}-{\displaystyle \frac{{\dot{Q}}_{\mathrm{NTU}}}{{\dot{m}}_{w,\mathrm{in},\mathrm{cond}}{c}_{\mathrm{pw},\mathrm{in},\mathrm{cond}}}}$(14) where the actual heat transfer rate, Q̇_NTU, is the product of effectiveness and maximum possible energy gain, as presented in Equation (15). ${\dot{Q}}_{\mathrm{NTU}}=E{\dot{Q}}_{\max }$(15) 

And where Equation (16) can compute maximum possible energy gained and the effectiveness for a heat exchanger that involves phase change is given in Equation (17). ${\dot{Q}}_{\max }=(\dot{m}{c}_p{)}_{\min }(T_{a,\mathrm{in},\mathrm{cond}}-T_{w,\mathrm{in},\mathrm{cond}})$(16) taking (ṁc_p)_min as the smaller of the heat capacity rate of air (ṁ_ac_pa) and heat capacity ratio of water (ṁ_wc_pw). $E_{\mathrm{cond}}=1-\exp (-\mathrm{NTU})$(17) where Equation (18) gives the NTU. $\mathrm{NTU}={\displaystyle \frac{U_{\mathrm{cond}}{A}_{\mathrm{cond}}}{{(\dot{m}{c}_p)}_{\min }}}$(18) and, the overall heat transfer coefficient of the condenser can be determined by Equation (19) (Holman 2010). $U_{\mathrm{cond}}={\displaystyle \frac{1}{{\displaystyle \frac{1}{h_a}+{\displaystyle \frac{A_{i,\tau }\ln \left[\frac{r_{o,\tau }}{r_{i,\tau }}\right]}{2\pi k_{\tau }{L}_{\tau }}+{\displaystyle \frac{A_{i,\tau }}{A_{o,\tau }}{\displaystyle \frac{1}{h_w}}}}}}}$(19) 

For economiser, a liquid–liquid single pass shell and tube heat exchanger is modelled. Similar to the model for the condenser, the effectiveness-NTU method was used to determine the outlet conditions of the fluids from the economiser, as shown in Equation (20), noting that there is no phase change. $E_{\mathrm{econ}}=2{\left\{1+c+\sqrt{1+c^2}{\displaystyle \frac{1+\exp \left[-\mathrm{NTU}\sqrt{1+c^2}\right]}{1-\exp \left[-\mathrm{NTU}\sqrt{1+c^2}\right]}}\right\}}^{-1}$(20) 

Where, c is the capacity ratio, as given in Equation (21). $c={\displaystyle \frac{{(\dot{m}{c}_p)}_{\min }}{{(\dot{m}{c}_p)}_{\max }}}$(21) 

And (ṁc_p)_max is the larger of the heat capacity rates of air and water.

Humidifier

As shown in , the study utilised a novel humidifier, consisting of a series of horizontal trays with a cascading arrangement. As has been mentioned, two main types of interaction occur between the air and the water, counter-current and crossflow. The counter-current interaction flow occurs where water flows along the base and air flows above the water surface. The crossflow interaction occurs when the air passes through the falling water sheet.

In modelling the behaviour of the humidifier, the mass and energy conservation, as well as the correlations of heat and mass transfer, determine the outlet conditions of both air and water streams. For this analysis, steady state conditions were assumed, hence the conservation of energy is given by Equation (22). ${\dot{m}}_{a,\mathrm{hmd}}({\hslash }_{a,\mathrm{in},\mathrm{hmd}-}{\hslash }_{a,\mathrm{out},\mathrm{hmd}})={\dot{m}}_{w,\mathrm{out},\mathrm{hmd}}{\hslash }_{w,\mathrm{out},\mathrm{hmd}}-{\dot{m}}_{w,\mathrm{in},\mathrm{hmd}}{\hslash }_{w,\mathrm{in},\mathrm{hmd}}$(22) where the mass flow rate of discharge water from the humidifier can be calculated from Equation (23). ${\dot{m}}_{w,\mathrm{out},\mathrm{hmd}}={\dot{m}}_{w,\mathrm{in},\mathrm{hmd}}-{\dot{m}}_{\mathrm{ev}}$(23) 

And the rate of evaporation can be computed from Equation (24) (Cengel et al. 2002). ${\dot{m}}_{\mathrm{ev}}=j{\displaystyle \frac{A_{\mathrm{hmd}}}{R_v}\left[{\displaystyle \frac{P_{\mathrm{sat},f}}{T_f}-{\displaystyle \frac{P_{\mathrm{sat},\mathrm{hmd}}}{T_{a,\mathrm{hmd}}}}}\right]}$(24) 

The mass transfer coefficient (j), associated in Equation (24) can be determined as a function of the Sherwood number as given in Equation (25) (Holman 2010). $j={\displaystyle \frac{\mathrm{Sh}\times {\lambda }_{a-w}}{L_{\mathrm{ch}}}}$(25) where the mass diffusion rate for air–water systems (λ_a-w) can be determined from Equation (26) (Marrero and Mason 1972). ${\lambda }_{a-w}=1.87\times {10}^{-10}{\displaystyle \frac{T^{2.072}}{P}}$(26) 

The total rate of heat transfer between air and water is the summation of convective, evaporative and radiative heat transfer. Due to the system operating at moderate temperatures, the radiative heat transfer is neglected. The convective and evaporative heat transfer rates can be determined by Equations (27) and (28), respectively. ${\dot{Q}}_{\mathrm{conv}}=h_{\mathrm{conv}}{A}_{\mathrm{hmd}}(T_{w,\mathrm{in}}-T_{a,\mathrm{in}})$(27) ${\dot{Q}}_{\mathrm{ev}}={\dot{m}}_{\mathrm{ev}}{\hslash }_{\mathrm{fg}}$(28) 

The coefficient of convective heat transfer (h_conv) can be determined as a function of the Nusselt number as given in Equation (29). $h_{\mathrm{conv}}={\displaystyle \frac{\mathrm{Nu}\times k_a}{L_{\mathrm{ch}}}}$(29) 

A stand-alone cascade humidifier

The performance of a humidifier can be evaluated based on the conditions of the exhaust air and the total amount of evaporation that occurs within it. Therefore, a sensitivity analysis was performed by varying the fluid conditions and operating parameters to determine the temperature of the exhaust air as well as the evaporation rate. shows the variables examined, their range of variation, and their nominal values.

As the heat and mass transfer rates are directly related to the mechanism of interaction between the two fluids, changing the air and water mass flow rates transforms the flow regimes. Hence, the effects of both the air and water flow rates needed to be taken into consideration. In particular, increasing the airflow rate at a constant water flow rate can result in the flow regime potentially changing from a ‘stable’ sheet to a ‘flapping’ or broken ‘sheet’ type flow (Enayatollahi et al. 2017b). As shown in , increasing the airflow rate under the conditions of a ‘stable sheet’, where the heat and mass transfer intensities are comparatively weak, initially reduces the temperatures of both exiting air and water streams but increases the total rate of evaporation.

However, further increases to the airflow rate transform the water flow regime to a flapping mode which enhances the heat and mass transfer processes. This results in a higher rate of evaporation and consequently larger temperature drop of the water stream. This is tied to the fact that increasing the airflow rate increases the Reynolds number of air as well as the moisture-carrying capacity of the air stream and thus improves the transfer phenomena. These enhance the total rate of evaporation, and consequently increase the temperature change of the water stream in the humidifier. Conversely, increasing the airflow rate reduces the temperature change of the air stream in the humidifier due to larger mass of air supplied. Once the exhaust air temperature reaches its maximum possible temperature at around 0.015 m³/s (for a given evaporation area, water and air inlet temperature and water flow rate), increasing the airflow rate reduces the exhaust air temperature due to larger quantities of air supplied.

At lower flow rates of water, increasing the airflow rate, results in transformation of the flow from a stable to broken regime, in which the heat and mass transfer processes are weaker. Therefore, as shown in the evaporation rate as well as the outlet temperature of air reduces whereas the discharge water temperature increases with a change of flow regime at an airflow rate of approximately 0.014 m³/s. However, the broken flow regime occurs at higher flow rates of air where the Reynolds number of the air is larger. Therefore, while the flow regime is broken, increasing the airflow rate enhances the intensities of heat and mass transfer. This initially increases the temperature of exhaust air, however as illustrated in the temperature of exhaust air drops for flow rates of above approximately 0.03 m³/s. This can be explained by the fact that the extracted heat from the water stream is transferred to a larger mass of air.

Looking at lower airflow rates, where the crossflow interaction is in a stable regime, as shown in , the temperature of the exhaust air, discharge water and evaporation rate increase with increasing the water flow rate. However, the increase is less significant to the discharge water temperature. Increasing the water flow rate while the inlet water temperature is maintained constant and the flow regime is stable (low airflow rates) increases the mass flow ratio, which directly enhances the transfer processes. As a consequence, the temperature of the exhaust air and the evaporation rate increase with increasing the water flow rate, whereas the temperature of the discharge water tends to decrease. Increasing the water flow rate however, reduces the temperature drop across the water stream as a larger mass of water loses heat to the air stream.

Now, considering the effects of water flow rate, it is apparent that increasing the water flow rate, improves the transfer processes by interacting with the air stream on one hand, and on the other hand, by increasing the input heat to the system. Hence, as shown in the temperature of the discharge water decreases whereas the exhaust air temperature increases with increasing the water flow rate. Further increasing the water flow rate transforms the flow to a flapping mode, which enhances the transfer process significantly. Therefore, as illustrated in , at a water flow rate of around 2.65 × 10⁻⁵ m³/s, a marked increase can be seen in the outlet air temperature and the evaporation rate.

At higher water flow rates, a larger mass of water loses heat to the airflow, hence, the discharge water temperature increases when the water flow rate is increased beyond approximately 3.1 × 10⁻⁵ m³/s. However, the temperature of inlet water constrains the temperature of the exhaust air, and as a result, the moisture-carrying capacity and the evaporation rate. Hence, the exhaust air temperature and the evaporation rate increase only slightly by increasing the water flow rate above 3.1 × 10⁻⁵ m³/s.

Finally, as shown in enlarging the area of the humidifier increases the evaporation rate and the temperature of the exiting moist air, while decreasing the temperature of the discharge water. The area of evaporation can be increased by adding additional trays in the humidifier, which also increases the number of crossflow contacts in the system. In this respect, the temperature of the inlet water constrains the temperature and moisture-carrying capacity of the air and as a result, the evaporation rate. However, as shown in , increasing the area of evaporation beyond 0.45m² does not provide a significant increase in the evaporation rate and hence, the outlet temperatures of air and water change only slightly.

On the whole, the model of the humidifier shows an effective transfer phenomenon in the bioinspired cascading configuration, which justifies its application in the HDH desalination systems.

Solar HDH desalination system with cascading humidifier

In this section, the performance of the novel cascading humidifier in the solar desalination system, shown in , is examined. The design parameters for the cascading humidifier and the other parts of the system are given in .

In order to be able to design and develop an HDH desalination system for a particular location, the sensitivity of the climatic and operational parameters on the rate of production needs to be evaluated. shows the range and the nominal values of the examined parameters.

As shown in , increasing the radiation received by the solar water heater increases the evaporation in the humidifier. This is because it puts extra energy into the water stream, which consequently increases the production of the system.

As illustrated in , increasing the airflow rate initially increases the system production, but above 0.8 × 10⁻² m³/s this begins to decrease. However, further increasing the airflow rate above 1.2 × 10⁻² m³/s increases the yield of the system as a result of a change in the flow regime of the water sheet. Increasing the airflow rate, increases the moisture carrying capacity of the air stream. Additionally, the formation of the ‘flapping’ regime, due to increased airflow, develops a turbulent flow, and thus significantly enhances the intensities of heat and mass transfer and consequently the potable water production.

Now, looking at the effects of increasing the water flow rate, it can be concluded that; increasing the water flow rate enhances the cooling process in the dehumidifier, but weakens the preheating and heating processes in the economiser and the solar water collector. Taking both effects into consideration, increasing the water flow rate, increases the production of the system very gradually. Further, increases in the flow rate of water transform the flow regime from broken to flapping sheets, which enhances the transfer phenomena in the humidifier. This, as discussed previously, increases the evaporation rate but at the same time reduces the temperature of the discharge water from the humidifier. Therefore, as shown in , the effect of increasing the water flow rate on the rate of production was found to be very small.

Finally, the variation of potable production versus the number of cascade trays in the humidifier is shown in . Adding more trays in the humidifier, increases the area of evaporation and the number of crossflow interactions, which increase the total yield of the system. However, the temperature of the exhaust air from the humidifier and consequently the rate of evaporation is limited by the inlet water temperature to the humidifier. Therefore, once the evaporation rate in the humidifier is maximized, increasing the number of trays (or evaporation area) does not increase production unless additional energy is added to the system.

As mentioned earlier, two interaction mechanisms exist in the cascading humidifier, a crossflow, that is due to air crossing through the falling water sheets, and a counterflow flow configuration that is simply the flow of air and water in a horizontal rectangular duct in opposite directions. To be able to compare the performance of the proposed cascading humidifier, a simple humidifier, consist of a long square duct with water flowing at the bottom of the channel and air flowing over water in a counterflow configuration was mathematically modelled. shows the comparison of the performance of the proposed cascading humidifier, and the simple square duct humidifier. Under the same operational condition, the application of the cascade humidifier can increase the production by nearly 40% while reducing the required area by 75%.