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Mathematics 2021, 9, 33 16 of 31 The energy balance equation is [72,73]: h ∂ T wh α ∂T w ∂ y y=0 Eams [74] predicted the water evaporation rate as shown in Equation (20). uw ∂z = αw The solution of Equation (18) requires the following boundary condition [73]: (18) (19) =−α ∂Tw s ∂ y 2 y2 =0 + λWj ρ s c p , a Wj = ε ∗ (Ps − Pv)M/2πRTn Knudson constant of evaporation [75] ε∗ is written in terms of the coefficient of ∂ 2 T wh ∂y2 2 evaporation: whereas the coefficient of evaporation is calculated using [74]: ε∗= 2ε (21) 2−ε 2πRTs Ts ε=h∗ M ρvλ2 (22) Considering the evaporation affecting the heat transfer coefficient h∗, the concept of wet bulb coefficient of heat transfer as proposed by MacLaine-Cross and Banks [76] can be used to correct the heat transfer coefficient. It can be written as: eλ h∗=h 1+C (23) pa where e is derived from the wet-bulb coefficient, given by [76]: e = ωmax − ωmin (24) Tmax − Tmin 3.4. Dehumidifier/Condenser Section The efficacy of the condenser, modelled as a counter-flow concentric heat exchanger, is computed using NTU (Number of Transfer Units) method [77–79] which is used to calculate the rate of heat transfer in heat exchangers when there is an insufficient data to evaluate the Log-Mean-Temperature-Difference [77–79]. Therefore, in NTU method, the effectiveness of the heat exchanger is calculated which can be converted into the actual heat transfer of the heat exchanger. It also involves the calculation of minimum heat capacity (Cmin) and maximum heat capacity (Cmax) which can be either heat capacity of the cold fluid or the hot fluid depending on whichever can be minimum or maximum. The evaluation of U is carried out using the thermal resistance diagram by computing the convective heat transfer coefficient of the humidifier and feed water. Once the actual heat transfer is calculated, it can be used to evaluate the outlet temperatures of both steam and finally, the mass balance can give the quantity of the condensed fresh water. This methodological framework is presented in a stepwise format as in Figure 4. (20)PDF Image | Humidification Dehumidification Solar Desalination System
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