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

technologies and devices, the solar stills are widely used because they are designed and operated in a manner that is consistent with the technological level and economic conditions of the poor and developing countries and communities.

This is the main aim of this chapter.

Distillation - Modelling, Simulation and Optimization

chapter as well.

simulations, MATLAB.

30

tion solar still with enhanced water recovery.

The most popular solar stills are passive type, in which distillation process occurs within the still through evaporation and condensation [1]. They are simple in design and manufacture, easy to operate, usually small, and reasonably cheap. Passive solar stills only use solar energy to remove the salts or impurities in saline or brackish water; thus, it is environmentally friendly and saving energy. Therefore, it is still of value to study in this type of stills to continue improving its efficiencies and designs.

The main drawback of this type of solar distillation system is low energy efficiency and distillate productivities. Hence, many active distillation systems such as solar still coupled with flat plate or evacuated tube collectors, solar still coupled with parabolic concentrator, solar still coupled with heat pipe, solar still coupled with hybrid PV/T system, multistage active solar distillation system, multi-effect active solar distillation system, etc. have been developed theoretically and experimentally [2]. However, a forced circulation solar still with enhanced water recovery has not been researched and presented. Therefore, this type of solar still has been developed and modeled, both theoretically and experimentally, and will be presented in this

In terms of numerical analyses of passive and active solar distillation systems, there are several models presented in literatures [2–8]. Sampathkumar et al. [2] comprehensively reviewed mathematical models applied to predict the performances of active solar distillation systems and concluded that Kumar and Tiwari's model [3] was most suitable for evaluating the internal heat transfer coefficients and hourly yield accurately except in extreme cases. However, Dwivedi and Tiwari [4] observed from their studies in passive solar still that Dunkle's model [5] gave better agreement between theoretical and experimental results. Madhlopa and Johnstone [6] numerically modeled a passive solar still with separate condenser and claimed that the distillation productivity of their still was 62% higher than that of the conventional passive solar still. Ahsan et al. [7] reviewed a few numerical models of a tubular solar still and compared them with Dunkle and Ueda models. Recently, Edalatpour et al. [8] reviewed the latest developments in numerical simulations for solar stills including the use of computational fluid dynamics (CFD)

Based on the above literature review, it is obvious that although Dunkle's model is one of the oldest thermal model for predicting the internal heat transfers of solar stills, it still can be used to accurately present the performance of heat transfers inside the solar stills. However, there is no research found in the literature review that consistently uses Dunkle's equations to develop the numerical models for both passive and active solar stills. Therefore, this chapter will use this approach to develop the mathematical models for a conventional solar still and a forced circula-

2. The mathematical model of a passive basin-type solar still

The relationships of heat and mass transfer in a solar still under steady-state conditions were first studied in 1961 by Dunkle [5]. Based on this initial work, this research has developed the transient mode of the solar still in which all heat and mass coefficients and still parameters are calculated using the formulae within the model. The weather data used for simulation will be either input from actual

measured data or data generated from a sub-computer program developed by the author and linked to the main program [9].

The processes of heat and mass transfer in a passive solar still are indicated in Figure 1. In order to develop the formulae for the energy and mass balances in the still, the following assumptions are made:


As can be seen in Figure 1, the heat and mass transfer inside the solar still occurs as follows: the solar incidence QT from the sun reaches the glass, part of it will be reflected Qr, part will be absorbed by the glass Qα, and the remaining Q' will transfer through the glass and reach the basin water. Then, Q' absorbed into the basin water will be partially reflected back to the glass under convection qcw, evaporation qew, and radiation qrw, partially transfer to the basin qw-b, and the remaining will increase the temperature of the basin water Mw dTw dt . The basin, in its turn, gains the energy partially from the sun (αQ″ <sup>T</sup>), partially from the water (qw-b).

This gained energy will be partially lost from surroundings qb, and the remaining will increase the temperature of the basin Mb dTb dt . Similarly, the energy going in the glass includes the reflected energy from the basin water through convection qcw, evaporation qew, radiation qrw, and the energy absorbed from the sun αgQT. This gained energy of the glass will partially transfer to the ambient through convection qca and radiation qra and partially increase the temperature of the glass Mg dTg dt .

Based on these assumptions and the heat and mass transfer explained above, the energy balances for the glass, for the basin water and for the basin, are

$$q\_{cw} + q\_{ew} + q\_{rw} + a\_{\text{g}} \mathbf{Q}\_{T} = \left(q\_{ra} + q\_{ca}\right) + \mathbf{M}\_{\text{g}} \frac{dT\_{\text{g}}}{dt} \tag{1}$$

$$a\_w Q'\_T = q\_{cw} + q\_{cw} + q\_{rw} + q\_{w-b} + M\_w \frac{dT\_w}{dt} \tag{2}$$

$$a\_b \mathbf{Q}\_T^"+q\_{w-b} = q\_b + M\_b \frac{dT\_b}{dt} \tag{3}$$

qra: heat transfer by radiation from the glass to the ambient around the still

qw�b: heat transfer by convection from the still water to the absorbing surface of

where hw�<sup>b</sup> is the coefficient of convection from the water to the basin

qb: heat transfer by convection from the basin to the surroundings of the still

where hb is the coefficient of convection from the basin to the ambient around

<sup>¼</sup> <sup>δ</sup>insul kinsul þ 1 hi

kinsul (W/m.°C) and δinsul (m) are the basin thermal conductivity and the

hi: the combination of heat transfer coefficients by convection and radiation from the basin insulation to the ambient surroundings, which can be derived

<sup>T</sup>: global solar irradiation dropping on the still water, after transmitting

<sup>T</sup>: global solar irradiation dropping on the basin, after transmitting

αb, αw, and αg: solar radiation absorption coefficients of the basin, water,

Mb, Mw, and Mg: solar radiation heat capacities per unit area of the basin,

Tb, Tw, and Tg : transient temperatures of the basin, of the water, and of the

dt <sup>¼</sup> <sup>∝</sup>gQT <sup>þ</sup> qcw <sup>þ</sup> qew <sup>þ</sup> qrw � qra <sup>þ</sup> qca � � (12)

� � (13)

<sup>T</sup> <sup>þ</sup> qw�<sup>b</sup> � qb (14)

<sup>T</sup> � qcw <sup>þ</sup> qew <sup>þ</sup> qrw <sup>þ</sup> qw�<sup>b</sup>

1 hb

qra <sup>¼</sup> <sup>ε</sup>g<sup>σ</sup> Tg <sup>þ</sup> <sup>273</sup> � �<sup>4</sup> � ð Þ Ta <sup>þ</sup> <sup>273</sup> <sup>4</sup> h i (8)

qw�<sup>b</sup> <sup>¼</sup> hw�bð Þ Tw � Tb (9)

qb ¼ hbð Þ Tb � Ta (10)

.

(11)

(W/m2 ):

(W/m2

(W/m2 ):

Q0

Q″

33

the basin (W/m2

.°C).

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

where ε<sup>g</sup> is the emission of the glass.

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

The Mathematical Model of Basin-Type Solar Distillation Systems

):

.°C):

from formulae (6) and (7).

through the still water, in W/m<sup>2</sup>

Mg dTg

> Mw dTw

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

and glass, respectively.

water, and glass, in J/m2

glass, respectively, in °C.

thickness of the basin insulation, respectively.

QT: global solar irradiation to the cover, in W/m<sup>2</sup>

.

.

.°C.

Formulae (1), (2), and (3) can be derived as follows:

dt <sup>¼</sup> <sup>∝</sup>wQ<sup>0</sup>

Mb dTb

dt <sup>¼</sup> <sup>∝</sup>bQ″

where

qcw: heat transfer by convection from the still water to the glass (W/m2 ), which is calculated by using Dunkle's equation:

$$q\_{cw} = 0.884 \left[ (T\_w - T\_\text{g}) + \frac{(p\_w - p\_\text{g})(T\_w + 273)}{(268.9 \times 10^3 - p\_w)} \right]^{1/3} \left( T\_w - T\_\text{g} \right) \tag{4}$$

with pw and pg being the partial pressure of water vapor at the temperatures of the basin water and the glass, respectively (in Pa).

qew: heat transfer by evaporation from the still water to the glass (W/m<sup>2</sup> ):

$$q\_{ew} = 16.276 \times 10^{-3} q\_{ew} \frac{\left(p\_w - p\_g\right)}{\left(T\_w - T\_\text{g}\right)}\tag{5}$$

qrw: heat transfer by radiation from the basin water to the glass cover (W/m<sup>2</sup> ), given by

$$q\_{rw} = \varepsilon\_w \sigma \left[ (T\_w + 273)^4 - \left( T\_\text{g} + 273 \right)^4 \right] \tag{6}$$

where ε<sup>w</sup> is the emission of water

$$
\sigma = 5.67 \times 10^{-8} \text{W/m}^2 \text{.K}^4
$$

qca: heat transfer by convection from the glass to the ambient around the still (W/m2 ), calculated as [10]

$$q\_{ca} = (\mathbf{5.7}\_w + \mathbf{3.8})(T\_\mathbf{g} - T\_a) \tag{7}$$

with W being the velocity of wind (m/s) and Ta the temperature of the atmosphere (°C).

qra: heat transfer by radiation from the glass to the ambient around the still (W/m2 ):

$$q\_{\rm nt} = \epsilon\_{\rm g} \sigma \left[ \left( T\_{\rm g} + 273 \right)^{4} - \left( T\_{\rm a} + 273 \right)^{4} \right] \tag{8}$$

where ε<sup>g</sup> is the emission of the glass.

This gained energy will be partially lost from surroundings qb, and the remaining

glass includes the reflected energy from the basin water through convection qcw, evaporation qew, radiation qrw, and the energy absorbed from the sun αgQT. This gained energy of the glass will partially transfer to the ambient through convection

Based on these assumptions and the heat and mass transfer explained above, the

<sup>T</sup> <sup>¼</sup> qcw <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

" #<sup>1</sup>=<sup>3</sup>

with pw and pg being the partial pressure of water vapor at the temperatures of

qrw: heat transfer by radiation from the basin water to the glass cover (W/m<sup>2</sup>

qrw <sup>¼</sup> <sup>ε</sup>w<sup>σ</sup> ð Þ Tw <sup>þ</sup> <sup>273</sup> <sup>4</sup> � Tg <sup>þ</sup> <sup>273</sup> � �<sup>4</sup> h i

<sup>σ</sup> <sup>¼</sup> <sup>5</sup>:<sup>67</sup> � <sup>10</sup>�8W=m<sup>2</sup>

qca: heat transfer by convection from the glass to the ambient around the still

qca ¼ ð Þ 5:7<sup>w</sup> þ 3:8 Tg � Ta

with W being the velocity of wind (m/s) and Ta the temperature of the

qcw

pw � pg � �

Tw � Tg

:K4

qew: heat transfer by evaporation from the still water to the glass (W/m<sup>2</sup>

ð pw � pg Þð Þ Tw þ 273 <sup>268</sup>:<sup>9</sup> � <sup>10</sup><sup>3</sup> � pw � �

qcw: heat transfer by convection from the still water to the glass (W/m2

qca and radiation qra and partially increase the temperature of the glass Mg

energy balances for the glass, for the basin water and for the basin, are

qcw þ qew þ qrw þ αgQT ¼ qra þ qca

dTb

� � <sup>þ</sup> Mg

dTb

dt . Similarly, the energy going in the

dTg

dTw

dTg dt .

), which

):

),

(6)

dt (1)

dt (2)

dt (3)

Tw � Tg � � (4)

� � (5)

� � (7)

will increase the temperature of the basin Mb

Distillation - Modelling, Simulation and Optimization

αwQ<sup>0</sup>

is calculated by using Dunkle's equation:

qcw ¼ 0:884 Tw � Tg

where ε<sup>w</sup> is the emission of water

), calculated as [10]

where

given by

(W/m2

32

atmosphere (°C).

αbQ″

� � <sup>þ</sup>

the basin water and the glass, respectively (in Pa).

qew <sup>¼</sup> <sup>16</sup>:<sup>276</sup> � <sup>10</sup>�<sup>3</sup>

qw�b: heat transfer by convection from the still water to the absorbing surface of the basin (W/m2 ):

$$q\_{w-b} = h\_{w-b}(T\_w - T\_b) \tag{9}$$

where hw�<sup>b</sup> is the coefficient of convection from the water to the basin (W/m2 .°C).

qb: heat transfer by convection from the basin to the surroundings of the still (W/m2 ):

$$q\_b = h\_b (T\_b - T\_a) \tag{10}$$

where hb is the coefficient of convection from the basin to the ambient around the still (W/m<sup>2</sup> .°C):

$$\frac{\mathbf{1}}{h\_b} = \frac{\delta\_{insul}}{k\_{insul}} + \frac{\mathbf{1}}{h\_i} \tag{11}$$

kinsul (W/m.°C) and δinsul (m) are the basin thermal conductivity and the thickness of the basin insulation, respectively.

hi: the combination of heat transfer coefficients by convection and radiation from the basin insulation to the ambient surroundings, which can be derived from formulae (6) and (7).

QT: global solar irradiation to the cover, in W/m<sup>2</sup> .

Q0 <sup>T</sup>: global solar irradiation dropping on the still water, after transmitting through the glass, in W/m<sup>2</sup> .

Q″ <sup>T</sup>: global solar irradiation dropping on the basin, after transmitting through the still water, in W/m<sup>2</sup> .

αb, αw, and αg: solar radiation absorption coefficients of the basin, water, and glass, respectively.

Mb, Mw, and Mg: solar radiation heat capacities per unit area of the basin, water, and glass, in J/m2 .°C.

Tb, Tw, and Tg : transient temperatures of the basin, of the water, and of the glass, respectively, in °C.

Formulae (1), (2), and (3) can be derived as follows:

$$M\_{\rm g} \frac{dT\_{\rm g}}{dt} = \infty\_{\rm g} Q\_T + q\_{cw} + q\_{cw} + q\_{rw} - \left(q\_{ra} + q\_{ca}\right) \tag{12}$$

$$M\_w \frac{dT\_w}{dt} = \mathfrak{\infty}\_w Q\_T' - \left(q\_{cw} + q\_{ew} + q\_{rw} + q\_{w-b}\right) \tag{13}$$

$$M\_b \frac{dT\_b}{dt} = \mathfrak{\infty}\_b \mathcal{Q}\_T^"+ q\_{w-b} - q\_b \tag{14}$$
