3. The numerical modelling of a basin-type forced circulation solar still with enhanced water recovery

In this study, the heat and mass transfer relationships in the forced circulation solar still with enhanced water recovery will be developed. Then, this numerical modeling will be validated by comparing its results with those from the experimental model.

The forced circulation solar still has been chosen in this study for several reasons. Compared with other types of solar powered distillation systems such as the solar multistage flash distillation, solar vapor compression, solar powered reverse osmosis, solar powered electrodialysis and solar membrane distillation systems, solar stills represent simple, yet mature technology.

The low efficiencies of a conventional solar still may be overcome by changing the principle of operation as follows:


#### 3.1. The development of the heat and mass transfer relationships in a forced circulation solar still

Figure 2 shows a schematic diagram of the forced circulation solar still with enhanced water recovery. The air flow having a temperature of Tfin and moisture content win enters the still and is heated up. It absorbs the vapor from the basin water and exits the still at a temperature of

Factors Affecting the Yield of Solar Distillation Systems and Measures to Improve Productivities http://dx.doi.org/10.5772/intechopen.75593 157

Figure 2. Schematic diagram of a forced circulation solar still with enhanced water recovery.

Mg dTg

Mw dTw

with enhanced water recovery

156 Desalination and Water Treatment

nology.

operation as follows:

of water

solar still

Mb dTb

by comparing its results with those from the experimental model.

mass transfer in the still, leading to higher outputs

the air-vapor mixture entering the still

water available, the more effective this condensing process will be

dt <sup>¼</sup> <sup>τ</sup>bQT <sup>þ</sup> qcw <sup>þ</sup> qew <sup>þ</sup> qrw- qra <sup>þ</sup> qca

dt <sup>¼</sup> <sup>τ</sup>wQT � qcw <sup>þ</sup> qew <sup>þ</sup> qrw <sup>þ</sup> qw�<sup>b</sup>

3. The numerical modelling of a basin-type forced circulation solar still

In this study, the heat and mass transfer relationships in the forced circulation solar still with enhanced water recovery will be developed. Then, this numerical modeling will be validated

The forced circulation solar still has been chosen in this study for several reasons. Compared with other types of solar powered distillation systems such as the solar multistage flash distillation, solar vapor compression, solar powered reverse osmosis, solar powered electrodialysis and solar membrane distillation systems, solar stills represent simple, yet mature tech-

The low efficiencies of a conventional solar still may be overcome by changing the principle of

• Using air as an intermediate medium and substituting forced convection for natural convection to increase the heat coefficients in the still, resulting in increased evaporation

• Replacing saturated air in the standard still by "drier" air to increase the potential for

• Circulating the air-vapor mixture from the standard still to external water-cooled condensers to gain efficiency from a lower condensing temperature. The cooler the cooling

• Recovering some of the heat extracted in the condensing process and using it to preheat

• Substituting the condensing area of the flat sheet covers in the standard still by the external condenser with much larger heat exchange areas to increase condensation efficiencies

3.1. The development of the heat and mass transfer relationships in a forced circulation

Figure 2 shows a schematic diagram of the forced circulation solar still with enhanced water recovery. The air flow having a temperature of Tfin and moisture content win enters the still and is heated up. It absorbs the vapor from the basin water and exits the still at a temperature of

: (1c)

: (2c)

dt <sup>¼</sup> <sup>τ</sup>bQT <sup>þ</sup> qw�<sup>b</sup> � qb: (3c)

Tfout and moisture content wout. This air flow goes through the dehumidifying coil, which acts a condenser. The hot air-vapor mixture from the still is passed over the coil and attached fins, while the cooling water runs inside the coil.

The hot air-vapor mixture losses heat to the cooling water and subsequently cools down. When the temperature of the mixture falls below its dew point temperature, the condensation process starts. The air exits the condenser at a temperature of Tc-out and a moisture content of wc-out. Some of the heat extracted from the air flow will be recovered in the preheater, since the air flow goes through it before going back to the still.

The heat and mass transfer relationships in this still can be seen from Figure 3. From this figure, the energy and mass balances for the glass, for the flow in the still, for the basin water and for the basin are

$$
\eta\_{c\xi\xi} + \eta\_{rw} + \alpha\_{\xi} Q\_T = \left(\eta\_{ra} + \eta\_{ca}\right) + M\_{\xi} \frac{dT\_{\xi}}{dt} \tag{11}
$$

$$q\_{ew} + q\_{cwf} = q\_{cfg} + m\_f(h\_{out} - h\_{in}) + M\_f \frac{dT\_f}{dt} \tag{12}$$

$$m\_{ew} = \frac{q\_{ew}}{\hbar\_{\circ}} = m\_f(w\_{out} - w\_{in}) + m\_{ew-\circ} \tag{13}$$

$$a\_w Q\_T' = q\_{cwf} + q\_{ew} + q\_{rw} + q\_{w-b} + M\_w \frac{dT\_w}{dt} \tag{14}$$

$$
\alpha\_b \mathbf{Q}\_T'' + q\_{w-b} = q\_b + M\_b \frac{dT\_b}{dt} \tag{15}
$$

Figure 3. The heat and mass transfer process in a forced circulation solar still.

qcwf is the convective heat transfer rate between the basin water and the flow (in W/m<sup>2</sup> ). In principle, the blower used to transport the air should have the lowest possible energy consumption. The heat transfer process in the still may be natural convection or combined natural and forced convection. In this model, the heat coefficient in the still is calculated by using the forced and natural convection relations separately, and the larger one is chosen. The Grashof and the Reynolds number are first calculated [13]:

$$Gr = \frac{g' \Delta T L^3}{v^2} \tag{16}$$

V = air flow velocity, in m/s.

water and the flow can be derived from

where Pr ¼ is the Prandtl number,

water and the flow, respectively.

qrw is the radiative heat transfer rates (in W/m<sup>2</sup>

qcfg is the convective heat transfer rate (W/m<sup>2</sup>

be calculated by (6).

by

Dh = the hydraulic diameter of the still, defined as Dh <sup>=</sup> <sup>4</sup>∗ð Þ flow area

qcwf ¼ 0:884 Tð Þþ <sup>w</sup> � Tf

Nu <sup>¼</sup> hcwf <sup>L</sup>

to achieve a similar equation to Dunkle's expression [2] with Tg replaced by Tf:

If the forced convection dominates, the relation between Nu and Re is given by [4]

qcwf <sup>¼</sup> <sup>3</sup>:<sup>908</sup> <sup>V</sup>

qew is the evaporative heat transfer and the radiative heat transfer rates (in W/m2

qcfg <sup>¼</sup> <sup>2</sup>:<sup>785</sup> <sup>V</sup><sup>0</sup>:<sup>8</sup>

the ambient surroundings, computed from by using Eqs. (7) and (8), respectively.

where V is the air flow velocity (m/sec) and Ls is the still length (m). qca and qra are the convective and radiative heat transfer rates (in W/m2

Nu <sup>¼</sup> hcwf Dh

wetted perimeter:

Factors Affecting the Yield of Solar Distillation Systems and Measures to Improve Productivities

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:<sup>075</sup> ð Þ Gr:Pr <sup>1</sup>=<sup>3</sup> (18)

http://dx.doi.org/10.5772/intechopen.75593

159

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:<sup>664</sup> � Re1=<sup>2</sup> � Pr<sup>1</sup>=<sup>3</sup> (20a)

� � (20b)

) between the basin water and the cover and can

� � (21)

) between the cover and

) between the flow and the cover given by

ð Þ Tw-Tf (19)

) between the

Then, if the natural convection dominates, the convective heat transfer rate between the basin

ð Þ Tw � Tf ð Þ Tw þ 273:15 268 x 103 � pw

� � " #<sup>1</sup>=<sup>3</sup>

where pw and pf are partial pressures (in Pa) of water vapor at the temperatures of the basin

Considering Tw = 50�C and Tf = 40�C and introducing the corresponding air properties into (20a), the convective heat transfer rate between the basin water and the flow can be computed

> Dh � �<sup>1</sup>=<sup>2</sup>

basin water and the air flow and can be approximated by (5) with Tg and pg replaced by Tf and pr.

L<sup>0</sup>:<sup>2</sup> s

!

TW � Tf

Tf � Tg

$$\text{Re} = \frac{VD\_h}{v} \tag{17}$$

where:

L = average spacing between the water surface and the cover, in m.


V = air flow velocity, in m/s.

Dh = the hydraulic diameter of the still, defined as Dh <sup>=</sup> <sup>4</sup>∗ð Þ flow area wetted perimeter:

Then, if the natural convection dominates, the convective heat transfer rate between the basin water and the flow can be derived from

$$Nu = \frac{h\_{cwf}L}{k} = 0.075 \text{ (Gr.Pr)}^{1/3} \tag{18}$$

where Pr ¼ is the Prandtl number,

qcwf is the convective heat transfer rate between the basin water and the flow (in W/m<sup>2</sup>

Gr <sup>¼</sup> <sup>g</sup><sup>0</sup>

Re <sup>¼</sup> VDh

ΔTL<sup>3</sup>

; for air <sup>0</sup>

= 1/T.

<sup>v</sup><sup>2</sup> (16)

<sup>v</sup> (17)

and the Reynolds number are first calculated [13]:

Figure 3. The heat and mass transfer process in a forced circulation solar still.

g = gravitational constant, 9.81 m/s<sup>2</sup>

v = kinematic viscosity, in m/s2

β' = volumetric coefficient of expansion, in K�<sup>1</sup>

L = average spacing between the water surface and the cover, in m.

ΔT = temperature difference between the water and the cover, in K.

.

.

where:

158 Desalination and Water Treatment

principle, the blower used to transport the air should have the lowest possible energy consumption. The heat transfer process in the still may be natural convection or combined natural and forced convection. In this model, the heat coefficient in the still is calculated by using the forced and natural convection relations separately, and the larger one is chosen. The Grashof

). In

to achieve a similar equation to Dunkle's expression [2] with Tg replaced by Tf:

$$\text{qcwf} = 0.884 \left[ (\text{T}\_{\text{w}} - \text{T}\_{\text{f}}) + \frac{(\text{T}\_{\text{w}} - \text{T}\_{\text{f}})(\text{T}\_{\text{w}} + 273.15)}{(268 \times 10^{3} - \text{p}\_{\text{w}})} \right]^{1/3} (\text{Tw-Tf}) \tag{19}$$

where pw and pf are partial pressures (in Pa) of water vapor at the temperatures of the basin water and the flow, respectively.

If the forced convection dominates, the relation between Nu and Re is given by [4]

$$Nu = \frac{h\_{\rm cutoff} D\_h}{k} = 0.664 \times \text{Re}^{1/2} \times \text{Pr}^{1/3} \tag{20a}$$

Considering Tw = 50�C and Tf = 40�C and introducing the corresponding air properties into (20a), the convective heat transfer rate between the basin water and the flow can be computed by

$$\eta\_{cwf} = 3.908 \left( \frac{V}{D\_h} \right)^{1/2} \left( T\_W - T\_f \right) \tag{20b}$$

qew is the evaporative heat transfer and the radiative heat transfer rates (in W/m2 ) between the basin water and the air flow and can be approximated by (5) with Tg and pg replaced by Tf and pr.

qrw is the radiative heat transfer rates (in W/m<sup>2</sup> ) between the basin water and the cover and can be calculated by (6).

qcfg is the convective heat transfer rate (W/m<sup>2</sup> ) between the flow and the cover given by

$$\eta\_{c\sharp\mathfrak{g}} = 2.785 \left( \frac{V^{0.8}}{L\_s^{0.2}} \right) \left( T\_f - T\_\mathfrak{g} \right) \tag{21}$$

where V is the air flow velocity (m/sec) and Ls is the still length (m).

qca and qra are the convective and radiative heat transfer rates (in W/m2 ) between the cover and the ambient surroundings, computed from by using Eqs. (7) and (8), respectively.

qw-b and qb are the heat transfer rates (in W/m<sup>2</sup> ) between the water and the basin and between the basin and the ambient surroundings and can be calculated from Eqs. (9) and (10), respectively.

Nu <sup>¼</sup> hcon�gLc

Using the properties of the air at Tf = 40�C, one can achieve

3.2. The performance of the condenser and preheater

Lc = the length of the cover, in m; Lc = Ls

k = thermal conductivity, in W/m K g = gravitational constant, 9.81 m/s<sup>2</sup>

β = the slope of the cover, in degree

r = the air density, in kg/m3

μ = absolute viscosity, in Pa.s

Tw, Tf, wout and Tb, can be solved.

where:

in �K

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:<sup>943</sup> <sup>g</sup>2sin hfgL<sup>3</sup>

ΔT = the difference between the dew point temperature of the flow and the cover temperature,

qcon�<sup>g</sup> <sup>¼</sup> <sup>70</sup>:<sup>93</sup> sin<sup>β</sup>

Therefore, using the five equations from Eqs. (11) and (12), the five unknown parameters, Tg,

The theory of the performance of dehumidifying and of heating coils has been developed and is presented in [14, 15]. However, an explicit procedure for calculating the performance of dehumidifying coils was not available in these references. Therefore, the modeling of the performance of the condenser and the preheater in this simulation program was derived from the handbook and the standard. The calculation procedures for the psychrometric properties of humid air were given in [14]. A detailed description of the procedures for modeling the

The procedure for modeling the performance of the preheating coil involves (i) calculating the overall coefficient of heat transfer for the coil, (ii) calculating the effectiveness of the coil and

The procedure for modeling the performance of the dehumidifying coil involves using an iterative process to find a consistent set of temperature and humidity values, subject to the

performance of the preheater and dehumidifying coils in solar still is described in [15].

then (iii) computing the temperatures of the air and cooling water leaving the coil.

constraints imposed by the performance characteristics of the dehumidifying coil.

ΔTLc � �<sup>0</sup>:<sup>25</sup> c

http://dx.doi.org/10.5772/intechopen.75593

(24)

161

(25)

μ k ΔT !<sup>1</sup>=<sup>4</sup>

Factors Affecting the Yield of Solar Distillation Systems and Measures to Improve Productivities

QT is the total horizontal solar radiation incident on the still, in W/m<sup>2</sup> .

Q0 <sup>T</sup> is the total solar radiation incident on the water surface, after transmittance through the cover, in W/m<sup>2</sup> .

Q<sup>00</sup> <sup>T</sup> is the total solar radiation incident on the basin, after transmittance through the basin water, in W/m<sup>2</sup> .

mf is the mass rate of the air flow, in kg/s.

mew is the mass rate pf the evaporation from the basin water to the air flow, in kg/s.

<sup>g</sup>, <sup>w</sup> and <sup>b</sup> are the solar absorptance values of the cover, of the water and of the basin, respectively.

Mg, Mw, Mf and Mb are the heat capacities of the unit area of the cover, of the water, of the air in the still and of the basin, in J/m<sup>2</sup> �C.

Tg, Tw, Tf and Tb are, respectively, the temperatures of the cover, water, air in the still and the basin, in �C.

Hfg is the latent heat of vaporization of water at the temperature Tf, in J/kg.

win and wout are the moisture contents of the air-vapor mixture at the inlet and outlet of the still, in kg/kg.

hin and hout are the enthalpies of the still inlet and outlet air, in J/kg. Assuming that the air in the still is reasonably well mixed, the enthalpy of the still outlet hout can be calculated as a function of the temperature Tf as follows:

$$\text{float} = (\text{Tf} + \text{wout} \times (2501 + 1.805 \text{Tf})) \times 103 \quad \text{(J/kg)}\tag{22}$$

The amount of the distillate water collected inside the still will depend on the temperatures of the air and the cover. Water will condense on the cover surface only when the dew point temperature of the air flow, Tfd, is higher than the cover temperature, Tg. In this case, the amount of the distillate water collected from the cover, mew-g (in kg/s.m2 ), can be calculated from

$$
\dot{m}\_{ew-g} = \frac{q\_{cm-g}}{h\_{\%}} \quad \text{(kg/s.m $^2$ )}\tag{23}
$$

hfg is the latent heat of vaporization of water at the temperature, Tf, in J/kg.

qcon-g = hcon-g(Tf–Tg) is the condensate heat transfer rate between the flow and the cover (in W/m<sup>2</sup> ). Using the Nusselt number in condensing:

Factors Affecting the Yield of Solar Distillation Systems and Measures to Improve Productivities http://dx.doi.org/10.5772/intechopen.75593 161

$$Nu = \frac{h\_{\text{con}-g}L\_c}{k} = 0.943 \left(\frac{g^2 \sin h\_{\text{fg}}L\_c^3}{\mu \, k \, \Delta T}\right)^{1/4} \tag{24}$$

where:

qw-b and qb are the heat transfer rates (in W/m<sup>2</sup>

QT is the total horizontal solar radiation incident on the still, in W/m<sup>2</sup>

tively.

Q0

Q<sup>00</sup>

cover, in W/m<sup>2</sup>

160 Desalination and Water Treatment

water, in W/m<sup>2</sup>

basin, in �C.

still, in kg/kg.

W/m<sup>2</sup>

.

.

mf is the mass rate of the air flow, in kg/s.

the still and of the basin, in J/m<sup>2</sup> �C.

function of the temperature Tf as follows:

distillate water collected from the cover, mew-g (in kg/s.m2

). Using the Nusselt number in condensing:

) between the water and the basin and between

.

the basin and the ambient surroundings and can be calculated from Eqs. (9) and (10), respec-

<sup>T</sup> is the total solar radiation incident on the water surface, after transmittance through the

<sup>T</sup> is the total solar radiation incident on the basin, after transmittance through the basin

<sup>g</sup>, <sup>w</sup> and <sup>b</sup> are the solar absorptance values of the cover, of the water and of the basin, respectively.

Mg, Mw, Mf and Mb are the heat capacities of the unit area of the cover, of the water, of the air in

Tg, Tw, Tf and Tb are, respectively, the temperatures of the cover, water, air in the still and the

win and wout are the moisture contents of the air-vapor mixture at the inlet and outlet of the

hin and hout are the enthalpies of the still inlet and outlet air, in J/kg. Assuming that the air in the still is reasonably well mixed, the enthalpy of the still outlet hout can be calculated as a

The amount of the distillate water collected inside the still will depend on the temperatures of the air and the cover. Water will condense on the cover surface only when the dew point temperature of the air flow, Tfd, is higher than the cover temperature, Tg. In this case, the amount of the

<sup>m</sup>\_ ew�<sup>g</sup> <sup>¼</sup> qcon�<sup>g</sup>

hfg is the latent heat of vaporization of water at the temperature, Tf, in J/kg.

hfg

qcon-g = hcon-g(Tf–Tg) is the condensate heat transfer rate between the flow and the cover (in

hout ¼ ðTf þ wout � ð Þ 2501 þ 1:805Tf Þ � 103 Jð Þ =kg (22)

), can be calculated from

kg=s:m<sup>2</sup> (23)

mew is the mass rate pf the evaporation from the basin water to the air flow, in kg/s.

Hfg is the latent heat of vaporization of water at the temperature Tf, in J/kg.

Lc = the length of the cover, in m; Lc = Ls

k = thermal conductivity, in W/m K


ΔT = the difference between the dew point temperature of the flow and the cover temperature, in �K

μ = absolute viscosity, in Pa.s

Using the properties of the air at Tf = 40�C, one can achieve

$$q\_{con-g} = 70.93 \left(\frac{\sin \beta}{\Delta T L\_c}\right)^{0.25} \tag{25}$$

Therefore, using the five equations from Eqs. (11) and (12), the five unknown parameters, Tg, Tw, Tf, wout and Tb, can be solved.

#### 3.2. The performance of the condenser and preheater

The theory of the performance of dehumidifying and of heating coils has been developed and is presented in [14, 15]. However, an explicit procedure for calculating the performance of dehumidifying coils was not available in these references. Therefore, the modeling of the performance of the condenser and the preheater in this simulation program was derived from the handbook and the standard. The calculation procedures for the psychrometric properties of humid air were given in [14]. A detailed description of the procedures for modeling the performance of the preheater and dehumidifying coils in solar still is described in [15].

The procedure for modeling the performance of the preheating coil involves (i) calculating the overall coefficient of heat transfer for the coil, (ii) calculating the effectiveness of the coil and then (iii) computing the temperatures of the air and cooling water leaving the coil.

The procedure for modeling the performance of the dehumidifying coil involves using an iterative process to find a consistent set of temperature and humidity values, subject to the constraints imposed by the performance characteristics of the dehumidifying coil.
