3.1 Developing of the relationships of heat and mass transfer in an active solar distillation system

Figure 2 presents a schematic diagram of an active solar distillation system with enhanced water recovery condenser. The airflow entering the solar still with a temperature of Tfin and moisture content win is heated up. After absorbing the vapor from the basin water, the airflow exits the solar still at a temperature of Tfout and moisture content wout. Then the hot air-vapor mixture passes through the dehumidifying coil, acting as a condenser. The dehumidifying coil with cooling water running inside will cool the hot air-vapor mixture down and condense the vapor from the mixture to produce the distillate. The air after passing the condenser has Tc-out and wc-out. The airflow continues passing through the preheater before going back to the still; hence, part of its heat will be extracted to recover in the preheater.

The heat and mass transfer relationships in this still can be seen in Figure 3. The heat and mass transfer is mainly similar to that of the conventional solar still, except the energy reflecting from the basin water through convection qcw and evaporation qew will go into the flowing air first (defined as qcwf and qew) instead of going directly to glass as in the conventional case. Then, the flowing air (the flow) will release part of its energy to the glass through convection qcfg. The gained energy of the flow, mainly from the basin water, will increase both latent heat (hout – hin) and sensible heat Mf dTf dt of the flow.

From Figure 3, the energy and mass balances for the glass, for the flow in the still, for the basin water, and for the basin are

$$q\_{\rm g\rm g} + q\_{rw} + a\_{\rm g}Q\_{T} = \left(q\_{ra} + q\_{at}\right) + M\_{\rm g}\frac{dT\_{\rm g}}{dt} \tag{18}$$

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

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

$$q\_{ew} + q\_{cuf} = q\_{cfg} + m\_f(h\_{out} - h) + M\_f \frac{dT\_f}{dt} \tag{19}$$

$$m\_{\epsilon w} = \frac{q\_{\epsilon w}}{h\_{\text{f\!g}}} = m\_f(w\_{out} - w) + m\_{\epsilon w - \text{g}} \tag{20}$$

Nu <sup>¼</sup> hcwf <sup>L</sup>

In order to have the same format with Dunkle's expression [11], Tf is replaced

� � " #1=<sup>3</sup>

where pf and pw are, respectively, the partial water vapor pressures at the

Considering Tw = 50°C and Tf = 40°C and introducing the corresponding air properties into Eq. (27), the convective heat transfer rate between the basin water

> Dh � �<sup>1</sup>=<sup>2</sup>

If the forced convection dominates, the relation between Nu and Re is

ð Þ Tw � Tf ð Þ Tw þ 273:15 <sup>268</sup> � <sup>10</sup><sup>3</sup> � pw

<sup>α</sup> being the Prandtl number.

The Mathematical Model of Basin-Type Solar Distillation Systems

temperatures of the flow and the basin water, in Pa.

Nu <sup>¼</sup> hcwfDh

qcwf <sup>¼</sup> <sup>3</sup>:<sup>91</sup> <sup>V</sup>

is calculated using formula (5) with pf and Tf are replaced for pg and Tg.

qcfg <sup>¼</sup> <sup>2</sup>:<sup>8</sup> <sup>V</sup><sup>4</sup>=<sup>5</sup>

with V the airflow velocity (m/s) and Ls are the still length (m).

QT: the global solar irradiation dropping on the still (W/m2

L<sup>1</sup>=<sup>5</sup> s

qca and qra are, respectively, the heat transfer rates by convection and radiation

from the basin to the ambient around the still and computed from formulae (9) and

Q"T: global solar irradiation dropping on the basin, after transmitting through

mew: the mass rate evaporating from the basin water to the airflow, in kg/s. αb, αw,and α<sup>g</sup> : solar absorption ratios of the basin, of the water and of the glass

Q'T: the global solar irradiation dropping on the water, after transmitting

) from the glass and the ambient around the still, calculated from formulae

!

qcwf ¼ 0:884 Tð Þþ <sup>w</sup> � Tf

DOI: http://dx.doi.org/10.5772/intechopen.83228

and the flow can be computed by

computed by using formula (6).

(7) and (8) correspondently.

(10) correspondently.

the still water, (W/m2

correspondently.

37

through the glass, (W/m<sup>2</sup>

qew: heat transfer by evaporation (W/m<sup>2</sup>

qrw: heat transfer by radiation (W/m2

qcfg: heat transfer by convection (W/m2

qw-b and qb are the heat transfer (W/m2

). mf: the airflow mass rate, in kg/s.

).

with Pr <sup>¼</sup> <sup>ν</sup>

given by [10]

computed as

(W/m<sup>2</sup>

for Tg:

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:075ð Þ Gr:Pr <sup>1</sup>=<sup>3</sup> (25)

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:<sup>664</sup> � Re<sup>1</sup>=<sup>2</sup> � Pr<sup>1</sup>=<sup>3</sup> (27)

� � (28)

) from the still water to the flow, which

) from the still water to the glass, which is

� � (29)

) from the still water to the basin and

).

) from the air to the glass, which is

TW � Tf

Tf � Tg

ð Þ Tw � Tf (26)

$$a\_w \mathbf{Q}'\_T = q\_{cwf} + q\_{ew} + q\_{rw} + q\_{w-b} + M\_w \frac{dT\_w}{dt} \tag{21}$$

$$a\_b Q\_T^"+q\_{w-b} = q\_b + M\_b \frac{dT\_b}{dt} \tag{22}$$

qcwf: heat transfer by convection from the still water to the air (W/m<sup>2</sup> ). In theory, the blower flowing the air must use energy as low as possible, or it should be powered by solar PV system. Depending on the flow velocity, the process of heat transfer in the still may be in natural or forced mode. Therefore, in this mathematical model, the coefficient of heat transfer in the still is computed by using both Reynolds and Grashof numbers for the forced and natural convection relations separately; then the larger one is chosen [10]:

$$Gr = \frac{\text{g}\beta^{\prime}\Delta TL^{3}}{\nu^{2}}\tag{23}$$

$$Re = \frac{\text{VD}\_h}{\nu} \tag{24}$$

where L is the distance between the water surface and the glass, in m; g = 9.81 m/s<sup>2</sup> is the gravity constant; β' is the volumetric expansion coefficient, in K�<sup>1</sup> ; for air β<sup>0</sup> ¼ 1=T; ΔT is the difference between the water and the glass temperatures, in <sup>0</sup> C; ν is the kinematic viscosity, in m/s<sup>2</sup> ; V is the airflow velocity, in m/s; Dh <sup>¼</sup> <sup>4</sup>ð Þ flow area wetted perimeter is the hydraulic diameter of the solar still.

If the natural mode dominates, the heat transfer by convection from the still water to the airflow can be calculated from

The Mathematical Model of Basin-Type Solar Distillation Systems DOI: http://dx.doi.org/10.5772/intechopen.83228

$$Nu = \frac{h\_{cuf}L}{k} = 0.075(Gr.Pr)^{1/3} \tag{25}$$

with Pr <sup>¼</sup> <sup>ν</sup> <sup>α</sup> being the Prandtl number.

In order to have the same format with Dunkle's expression [11], Tf is replaced for Tg:

$$\mathbf{q\_{cwf}} = \mathbf{0.884} \left[ (\mathbf{T\_w} - \mathbf{T\_f}) + \frac{(\mathbf{T\_w} - \mathbf{T\_f})(\mathbf{T\_w} + 273.15)}{(268 \times 10^3 - \mathbf{p\_w})} \right]^{1/3} (\mathbf{T\_w} - \mathbf{T\_f}) \tag{26}$$

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

If the forced convection dominates, the relation between Nu and Re is given by [10]

$$Nu = \frac{h\_{cwf} D\_h}{k} = 0.664 \times Re^{1/2} \times Pr^{1/3} \tag{27}$$

Considering Tw = 50°C and Tf = 40°C and introducing the corresponding air properties into Eq. (27), the convective heat transfer rate between the basin water and the flow can be computed by

$$q\_{cwf} = 3.91 \left(\frac{V}{D\_h}\right)^{1/2} \left(T\_W - T\_f\right) \tag{28}$$

qew: heat transfer by evaporation (W/m<sup>2</sup> ) from the still water to the flow, which is calculated using formula (5) with pf and Tf are replaced for pg and Tg.

qrw: heat transfer by radiation (W/m2 ) from the still water to the glass, which is computed by using formula (6).

qcfg: heat transfer by convection (W/m2 ) from the air to the glass, which is computed as

$$q\_{\rm cfg} = 2.8 \left( \frac{V^{4/5}}{L\_s^{1/5}} \right) \left( T\_f - T\_\text{g} \right) \tag{29}$$

with V the airflow velocity (m/s) and Ls are the still length (m).

qca and qra are, respectively, the heat transfer rates by convection and radiation (W/m<sup>2</sup> ) from the glass and the ambient around the still, calculated from formulae (7) and (8) correspondently.

qw-b and qb are the heat transfer (W/m2 ) from the still water to the basin and from the basin to the ambient around the still and computed from formulae (9) and (10) correspondently.

QT: the global solar irradiation dropping on the still (W/m2 ).

Q'T: the global solar irradiation dropping on the water, after transmitting through the glass, (W/m<sup>2</sup> ).

Q"T: global solar irradiation dropping on the basin, after transmitting through the still water, (W/m2 ).

mf: the airflow mass rate, in kg/s.

mew: the mass rate evaporating from the basin water to the airflow, in kg/s. αb, αw,and α<sup>g</sup> : solar absorption ratios of the basin, of the water and of the glass correspondently.

qew þ qcwf ¼ qcfg þ mfð Þþ hout � h Mf

<sup>T</sup> <sup>¼</sup> qcwf <sup>þ</sup> qew <sup>þ</sup> qrw <sup>þ</sup> qw�<sup>b</sup> <sup>þ</sup> Mw

<sup>T</sup> <sup>þ</sup> qw�<sup>b</sup> <sup>¼</sup> qb <sup>þ</sup> Mb

theory, the blower flowing the air must use energy as low as possible, or it should be powered by solar PV system. Depending on the flow velocity, the process of heat transfer in the still may be in natural or forced mode. Therefore, in this mathematical model, the coefficient of heat transfer in the still is computed by using both Reynolds and Grashof numbers for the forced and natural convection relations

qcwf: heat transfer by convection from the still water to the air (W/m<sup>2</sup>

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

where L is the distance between the water surface and the glass, in m; g = 9.81 m/s<sup>2</sup> is the gravity constant; β' is the volumetric expansion coefficient, in

C; ν is the kinematic viscosity, in m/s<sup>2</sup>

wetted perimeter is the hydraulic diameter of the solar still.

Re <sup>¼</sup> VDh ν

; for air β<sup>0</sup> ¼ 1=T; ΔT is the difference between the water and the glass temper-

If the natural mode dominates, the heat transfer by convection from the still

ΔTL<sup>3</sup>

mew <sup>¼</sup> qew hfg

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

Distillation - Modelling, Simulation and Optimization

αbQ″

αwQ<sup>0</sup>

separately; then the larger one is chosen [10]:

water to the airflow can be calculated from

K�<sup>1</sup>

36

Figure 3.

atures, in <sup>0</sup>

Dh <sup>¼</sup> <sup>4</sup>ð Þ flow area

dTf

dTw

<sup>ν</sup><sup>2</sup> (23)

; V is the airflow velocity, in m/s;

¼ mfð Þþ wout � w mew�<sup>g</sup> (20)

dTb

dt (19)

dt (21)

). In

(24)

dt (22)

Mb, Mf, Mw, and Mg: heat mass are unit area of the basin, the air, the water in the still, and the glass (J/m<sup>2</sup> .°C).

Tb, Tf, Tw, and Tg: respectively, the basin, the airflow, the still water, and the glass temperatures (°C).

hfg: latent heat of vaporization of water at Tf (J/kg).

wout and win: the air-vapor mixture's moisture contents exit and enter the still (kg/kg).

hout and hin: respectively, the enthalpies of air exiting and entering the still (J/kg). The air enthalpy exiting the still hout can be computed as the temperature Tf function as follows:

$$h\_{out} = \left(T\_f + w\_{out} \left(2500 + 1.81T\_f\right)\right) \times 10^3\tag{30}$$

The yield of the distillate in the solar still depends on the air and the glass temperatures. Water will condense on the glass surface only when the airflow dew point temperature Tfd is higher than the glass temperature Tg. In this case, the amount of the distillate produced from the glass mew-g can be computed from (kg/s m<sup>2</sup> ):

$$
\dot{m}\_{ew-\text{g}} = \frac{q\_{con-\text{g}}}{h\_{\text{fg}}} \tag{31}
$$

The simulation of the preheater includes (i) computing the heat transfer coefficient for the coil, (ii) computing the coil effectiveness, and then (iii) calculating the

The simulation of the dehumidifying includes finding consistent values of tem-

Figures 4 and 5 show the computed distillate yields and still water temperatures from the mathematical model compared to those from the experiments. As shown in these figures, the simulation model developed in this study gave very accurate calculated results. Hence, one can confidently use this program to simulate solar

air and cooling water temperatures leaving the coil.

DOI: http://dx.doi.org/10.5772/intechopen.83228

The Mathematical Model of Basin-Type Solar Distillation Systems

perature and humidity by using an iterative process.

The water calculated and measured temperatures in a passive solar still.

The measured and predicted distillate outputs of a conventional solar still.

experimental results

4.1 The passive solar still

passive stills.

Figure 4.

Figure 5.

39

4. The comparison of results from numerical modeling and

hfg: latent heat of vaporization of water at Tf, (J/kg).

qcon-g = hcon-g(Tf – Tg): heat transfer by condensation from the airflow to the glass. Using the Nusselt to calculate

$$Nu = \frac{h\_{con-g}L\_c}{k} = 0.943 \left(\frac{\text{g}^2 \sin \beta h\_{\text{fg}}L\_c^3}{\mu k \Delta T}\right)^{1/4} \tag{32}$$

where Lc is the length of the glass, in m; Lc = Ls; k is the thermal conductivity, in W/m K; g = 9.81 m/s<sup>2</sup> gravity constant; β is the slope of the glass, in degree; ρ is the air density, in kg/m<sup>3</sup> ; ΔT is the dew point temperature difference between the airflow and the glass, in °K; μ is absolute viscosity, in Pa s.

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

$$q\_{con-g} = 70.93 \left(\frac{\sin \beta}{\Delta T L\_c}\right)^{0.25} \tag{33}$$

Hence, with five formulae from (18) to (19), five parameters Tg, Tw, Tf, wout, and Tb, can be found.
