**1. Introduction**

The continuous increase in energy demand raises the need for alternative energy sources. Energy consumption from air conditioning will continue to increase which will raise the need for innovative solutions in many industrial sectors. One of the significant sources alternative widely and increasingly studied in the industry is solar-based technologies. Solar energy is used for a wide range of applications such as electricity production, desalination, cooling, heating, etc. Solar ponds are relatively simple and yet effective thermal storage [1].

A solar pond consists of a body of salty water which collects solar energy and converts it for thermal storage. A solar pond can be either convective or nonconvective. The principle of convective solar ponds largely depends on the water's surface being covered by an insulating material to store the system's collected heat; the most commonly used solar pond of this type is the shallow solar pond. Nonconvective solar ponds, however, operate by limiting the process of natural convection by the use of a water collector or storage medium.

in essence, consisted of two homogeneous layers: an upper convecting layer (a lower salt content) and a lower non-convecting layer which acts as the thermal

prospects of solar ponds in many regions and climates [8].

equations [10] used for steady-state flat plate solar energy collectors.

ration would be the key for an economical solar pond system.

observed to affect solar radiation penetration with increasing depths.

using solar ponds for seawater desalination in Qatar.

The authors considered the varying temperature ranges that a standard-sized dwelling would encounter seasonally in varying locations and climates. They subsequently reported that the use of a solar pond would undoubtedly provide adequate heating at prices, competitive with those of conventional heating solutions, for the various climates and geographic locations they considered; including the Arctic Circle. Due to its simplified nature, the model developed by Rabl and Nielsen [4] revived solar pond research and had subsequently been used to study the

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source*

Kooi [9] developed an analytical model to study operating characteristics, such as its temperature distributions and energy fluxes, on the assumption that the nonconvective zone temperature was equal to the ambient wet-bulb temperature. The model would allow the analysis of the performance of three-layer solar pond systems to be conducted in a similar calculation model to the Hottel-Whillier-Bliss

With validation, established models such as that of Rabl and Nielsen [4] deemed the steady-state salt gradient of a solar pond system to be similar to that of a flat plate collector. Kooi [11] also modified the formulae developed by Rabl and Nielsen [4] to account for reflected radiation. He assumed that, since most solar ponds would operate near the solubility limit, this would naturally increase the reflectivity of the solar pond floor. Kooi concluded that the increase of reflectivity, in fact, reduced the efficiency of the steady-state system and that avoidance of supersatu-

Other authors also studied the influence of a solar pond's physical properties on its thermal storage efficiency. Wang and Seyed-Yagoobi [12] developed the equations reported by Kooi [11] to investigate the influence of the water's clarity and salt concentrations on the penetration of solar radiation underwater. The authors used turbidity as a parameter for the solar pond's water clarity. The authors reported that solar radiation did not affect the penetration of solar radiation underwater. However, the clarity of the water was found to be imperative, as the turbidity was

Karakilcik et al. [13] assessed the performance effect of the presence of shade on each of the solar pond zones. The authors reported a major influence on the solar pond's efficiency caused by the solar pond's shading effect. Another potential performance effect was proposed by Jaefarzadeh and Akbarzadeh [14]. The authors suggested that wind-induced mixing could affect the salinity gradient required for an effectively operating solar pond. The use of floating rings on the surface of the pond would help to mitigate such effects and thus improve performance

With a great amount of progress focussed on the analysis of the efficiency, performance and adverse effect mitigation, the salt gradient solar pond has been shown by many accounts to be a very promising technology for energy storage that can be adapted to many climates and geographical locations. Elsarrag et al. [15] reviewed the possibility of supplying the necessary energy required for the regeneration stage of a liquid desiccant cooling system using a salt gradient solar pond. The authors considered different solar pond system configurations, designs and solute materials which would be suitable for implementations in a potential solarpowered desiccant cooling system. Sayer et al. [16] researched the feasibility and performance gel pond and compared with the salinity gradient solar pond for low temperature applications. Amro and Yusuf [17] conducted a theoretical study of

storage part of the pond.

*DOI: http://dx.doi.org/10.5772/intechopen.89632*

year-round.

**41**

In this study, a salinity gradient solar pond (SGSP) is designed and constructed. The SGSP is a large, low-cost solar-thermal energy collection and storage system which consists of a large body of saltwater (with salinity gradient) such that solar energy incident on the pond is partially transmitted to the bottom of the pond where a portion (20–30%) of it is absorbed.

A typical SGSP consists of three regions: the upper-convective zone (UCZ), the non-convective zone (UCZ) and the lower-convective zone (LCZ). The UCZ is the topmost layer of the solar pond and is a relatively thin layer (usually 0.1–0.5 m) which contains almost no salinity (about 0–5% concentration). The NCZ is the middle region (of about 0.7–1.5 m thickness) and has an increasing concentration (salinity gradient) relative to the UCZ, and it also acts as insulation on the LCZ, because convection motion in the NCZ is ideally suppressed if the concentration gradient is sufficiently large. The LCZ is the layer in which the salt concentration is the greatest (about 26%), and there is no concentration gradient in it, as depicted in **Figure 1** [3].

Resultantly, large amounts of heat can potentially be stored in these systems [4]. The utilisation of solar ponds as energy carriers was first conceived by Tabor [5, 6] when observing the natural phenomenon in a Hungarian lake at the turn of the twentieth century. The studies proposed the possibility of simulating the natural phenomenon experimentally for energy production. The authors laid the allimportant foundations of current solar pond research conducted to date. However, the work of Tabor [6] was not developed as far as its potential would allow. Nevertheless, in recent decades, with the current global search for alternative energy, as aforementioned, solar ponds have gained a substantial increase in interest in academia and industry [1, 3, 7].

A wide variety of models have been developed to investigate the energy properties and potential capabilities of solar ponds. Rabl and Nielsen [4] first examined the possibility of utilising the thermal energy from solar ponds for space heating, by deriving a set of formulae specific to a certain type of salt gradient pond. The pond,

**Figure 1.** *SGSP solar radiation distribution [2].*

#### *Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source DOI: http://dx.doi.org/10.5772/intechopen.89632*

in essence, consisted of two homogeneous layers: an upper convecting layer (a lower salt content) and a lower non-convecting layer which acts as the thermal storage part of the pond.

The authors considered the varying temperature ranges that a standard-sized dwelling would encounter seasonally in varying locations and climates. They subsequently reported that the use of a solar pond would undoubtedly provide adequate heating at prices, competitive with those of conventional heating solutions, for the various climates and geographic locations they considered; including the Arctic Circle. Due to its simplified nature, the model developed by Rabl and Nielsen [4] revived solar pond research and had subsequently been used to study the prospects of solar ponds in many regions and climates [8].

Kooi [9] developed an analytical model to study operating characteristics, such as its temperature distributions and energy fluxes, on the assumption that the nonconvective zone temperature was equal to the ambient wet-bulb temperature. The model would allow the analysis of the performance of three-layer solar pond systems to be conducted in a similar calculation model to the Hottel-Whillier-Bliss equations [10] used for steady-state flat plate solar energy collectors.

With validation, established models such as that of Rabl and Nielsen [4] deemed the steady-state salt gradient of a solar pond system to be similar to that of a flat plate collector. Kooi [11] also modified the formulae developed by Rabl and Nielsen [4] to account for reflected radiation. He assumed that, since most solar ponds would operate near the solubility limit, this would naturally increase the reflectivity of the solar pond floor. Kooi concluded that the increase of reflectivity, in fact, reduced the efficiency of the steady-state system and that avoidance of supersaturation would be the key for an economical solar pond system.

Other authors also studied the influence of a solar pond's physical properties on its thermal storage efficiency. Wang and Seyed-Yagoobi [12] developed the equations reported by Kooi [11] to investigate the influence of the water's clarity and salt concentrations on the penetration of solar radiation underwater. The authors used turbidity as a parameter for the solar pond's water clarity. The authors reported that solar radiation did not affect the penetration of solar radiation underwater. However, the clarity of the water was found to be imperative, as the turbidity was observed to affect solar radiation penetration with increasing depths.

Karakilcik et al. [13] assessed the performance effect of the presence of shade on each of the solar pond zones. The authors reported a major influence on the solar pond's efficiency caused by the solar pond's shading effect. Another potential performance effect was proposed by Jaefarzadeh and Akbarzadeh [14]. The authors suggested that wind-induced mixing could affect the salinity gradient required for an effectively operating solar pond. The use of floating rings on the surface of the pond would help to mitigate such effects and thus improve performance year-round.

With a great amount of progress focussed on the analysis of the efficiency, performance and adverse effect mitigation, the salt gradient solar pond has been shown by many accounts to be a very promising technology for energy storage that can be adapted to many climates and geographical locations. Elsarrag et al. [15] reviewed the possibility of supplying the necessary energy required for the regeneration stage of a liquid desiccant cooling system using a salt gradient solar pond. The authors considered different solar pond system configurations, designs and solute materials which would be suitable for implementations in a potential solarpowered desiccant cooling system. Sayer et al. [16] researched the feasibility and performance gel pond and compared with the salinity gradient solar pond for low temperature applications. Amro and Yusuf [17] conducted a theoretical study of using solar ponds for seawater desalination in Qatar.

surface being covered by an insulating material to store the system's collected heat; the most commonly used solar pond of this type is the shallow solar pond. Nonconvective solar ponds, however, operate by limiting the process of natural

In this study, a salinity gradient solar pond (SGSP) is designed and constructed. The SGSP is a large, low-cost solar-thermal energy collection and storage system which consists of a large body of saltwater (with salinity gradient) such that solar energy incident on the pond is partially transmitted to the bottom of the pond

A typical SGSP consists of three regions: the upper-convective zone (UCZ), the non-convective zone (UCZ) and the lower-convective zone (LCZ). The UCZ is the topmost layer of the solar pond and is a relatively thin layer (usually 0.1–0.5 m) which contains almost no salinity (about 0–5% concentration). The NCZ is the middle region (of about 0.7–1.5 m thickness) and has an increasing concentration (salinity gradient) relative to the UCZ, and it also acts as insulation on the LCZ, because convection motion in the NCZ is ideally suppressed if the concentration gradient is sufficiently large. The LCZ is the layer in which the salt concentration is the greatest (about 26%), and there is no concentration gradient in it, as depicted

Resultantly, large amounts of heat can potentially be stored in these systems [4]. The utilisation of solar ponds as energy carriers was first conceived by Tabor [5, 6] when observing the natural phenomenon in a Hungarian lake at the turn of the twentieth century. The studies proposed the possibility of simulating the natural phenomenon experimentally for energy production. The authors laid the allimportant foundations of current solar pond research conducted to date. However, the work of Tabor [6] was not developed as far as its potential would allow. Nevertheless, in recent decades, with the current global search for alternative energy, as aforementioned, solar ponds have gained a substantial increase in

A wide variety of models have been developed to investigate the energy properties and potential capabilities of solar ponds. Rabl and Nielsen [4] first examined the possibility of utilising the thermal energy from solar ponds for space heating, by deriving a set of formulae specific to a certain type of salt gradient pond. The pond,

convection by the use of a water collector or storage medium.

where a portion (20–30%) of it is absorbed.

*Low-temperature Technologies*

interest in academia and industry [1, 3, 7].

in **Figure 1** [3].

**Figure 1.**

**40**

*SGSP solar radiation distribution [2].*

This study aims to investigate theoretically and validate a salt gradient solar pond experimentally as a desiccant and energy source in a hot-humid climate. The model is comprised of energy balances of the pond (including each salt gradient, pond wall, and surface area), saltwater thermo-physical properties and soil temperature.

Assuming an initial pre-stable ideal state, convection is ideally suppressed in the pond, thus heat flow is primarily by conduction. Thus, the energy balance can be expressed in terms of the one-dimensional heat conduction equation in differential

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source*

where the thermo-physical properties (density, thermal conductivity and specific heat capacity) of the saltwater vary with temperature and concentration.

The temperature profile of the solar pond can be obtained from an energy balance of the different zones of the solar pond. With the assumptions that [2]:

• The temperature variation in the horizontal direction is assumed negligible. Thus the temperature and concentration distribution can be considered one-

• The three zones of the pond (UCZ, NCZ and LCZ) are considered distinct

• The bottom surface of the pond is assumed appropriately blackened; as such the radiation reaching the LCZ is completely absorbed by the saltwater and the

• Due to the presence of convection in the UCZ and the LCZ, the temperature and concentration in these zones are considered uniformly constant; such that they can be treated as single cells with a thickness *zu* and *zl*, respectively.

• The temperature varies with depth in the NCZ, and as such, in applying the energy balance, this part of the pond can be divided into several imaginary

• The pond is considered very large. Thus, the side effects such as convection

Due to convection in the UCZ, it can be treated as having a uniform tempera-

*QUCZ* ¼ *QNU* þ *Qsolar* � *QU* (5)

*k* ¼ 0*:*5553 � 0*:*0000813*S* þ 0*:*0008ð Þ *T* � 20 (2)

*ρ* ¼ 998 þ 0*:*65*C* � 0*:*4ð Þ *T* � 20 (3)

*Cp* <sup>¼</sup> <sup>4180</sup> <sup>þ</sup> <sup>4</sup>*:*396*<sup>C</sup>* <sup>þ</sup> <sup>0</sup>*:*0048*S*<sup>2</sup> (4)

For example, for NaCl pond, the following correlations are widely

þ *g z* \_ð Þ� , *t L z*ð Þ , *t* (1)

*ρCp ∂T <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂z k ∂T ∂z* 

*DOI: http://dx.doi.org/10.5772/intechopen.89632*

**3. Energy analysis of the solar pond**

enough to have a clear fixed boundary.

layers, *i* of thickness Δ*z* each.

**3.1 Upper-convective zone (UCZ)**

**43**

current at the wall can be ignored [19].

ture. The heat balance equation for the UCZ can be given as:

form as:

employed [18]:

dimensional.

pond's bottom.

## **2. Modelling of a salinity gradient solar pond**

The behaviour of a solar pond, like any other solar-thermal collector, is majorly influenced by its geographical location. The pond of interest for the title-study has been constructed and has been analysed for more than 10 months in a hot-humid climate. Initially, the weather data for the location, as obtained from the NASA Atmospheric Science Data Centre at 25.2867°N and 51.5333°E (the geographical location of the salinity gradient solar pond of interest) is given in **Table 1**.

The temperature profile needs to be determined to characterise the thermal behaviour of the pond. The temperature varies with depth (and time). The temperature profile of the solar pond can be obtained from an energy balance of the solar pond. The general energy balance equation is in the form:



**Table 1.** *Metrological data for Doha.* *Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source DOI: http://dx.doi.org/10.5772/intechopen.89632*

Assuming an initial pre-stable ideal state, convection is ideally suppressed in the pond, thus heat flow is primarily by conduction. Thus, the energy balance can be expressed in terms of the one-dimensional heat conduction equation in differential form as:

$$
\rho C\_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial \mathbf{z}} \left( k \frac{\partial T}{\partial \mathbf{z}} \right) + \dot{\mathbf{g}}(\mathbf{z}, t) - L(\mathbf{z}, t) \tag{1}
$$

where the thermo-physical properties (density, thermal conductivity and specific heat capacity) of the saltwater vary with temperature and concentration.

For example, for NaCl pond, the following correlations are widely employed [18]:

$$k = 0.5553 - 0.0000813 \text{S} + 0.0008(T - 20) \tag{2}$$

$$
\rho = \text{998} + \text{0.65C} - \text{0.4} (T - \text{20}) \tag{3}
$$

$$\text{Cp} = 4\text{180} + 4.396\text{C} + 0.0048\text{S}^2\tag{4}$$

## **3. Energy analysis of the solar pond**

The temperature profile of the solar pond can be obtained from an energy balance of the different zones of the solar pond. With the assumptions that [2]:


#### **3.1 Upper-convective zone (UCZ)**

Due to convection in the UCZ, it can be treated as having a uniform temperature. The heat balance equation for the UCZ can be given as:

$$Q\_{UCZ} = Q\_{NU} + Q\_{solar} - Q\_U \tag{5}$$

This study aims to investigate theoretically and validate a salt gradient solar pond experimentally as a desiccant and energy source in a hot-humid climate. The model is comprised of energy balances of the pond (including each salt gradient, pond wall, and surface area), saltwater thermo-physical properties and soil

The behaviour of a solar pond, like any other solar-thermal collector, is majorly influenced by its geographical location. The pond of interest for the title-study has been constructed and has been analysed for more than 10 months in a hot-humid climate. Initially, the weather data for the location, as obtained from the NASA Atmospheric Science Data Centre at 25.2867°N and 51.5333°E (the geographical location of the salinity gradient solar pond of interest) is given in **Table 1**.

The temperature profile needs to be determined to characterise the thermal behaviour of the pond. The temperature varies with depth (and time). The temperature profile of the solar pond can be obtained from an energy balance of the

*rate ofheat*

*into the element*

*flow*

1

0

BBBBB@

*rate of heat generation in the elemental*

*layer*

1

CCCCCA

CCCCCA þ

1

CCCCCA

**/day) Tamb, av (°C) V (m/s) RH (%)**

*layer*

*flow*

*rate of heat*

*out from the elemental layer*

solar pond. The general energy balance equation is in the form:

0

BBBBB@

0

BBBBB@

Jan 3.42 19.5 4.11 52.5 Feb 4.25 20.1 4.71 51.8 Mar 4.88 22.5 4.44 51.4 Apr 5.84 26.7 4.12 47.3 May 6.92 31.4 4.52 42.2 Jun 7.4 33.7 4.75 41.7 Jul 7.01 35.2 4.35 42 Aug 6.57 35.4 4.28 43.5 Sep 5.84 33.4 3.83 44.8 Oct 4.84 30 3.5 47.8 Nov 3.78 25.9 3.52 50.2 Dec 3.2 21.9 4.05 52.9

�

1

CCA ¼

*rate ofchange ofenergy content of elemental layer*

**Month Insolation (kWh/m<sup>2</sup>**

**2. Modelling of a salinity gradient solar pond**

temperature.

*Low-temperature Technologies*

0

BB@

**Table 1.**

**42**

*Metrological data for Doha.*

$$Q\_U = Q\_{U\epsilon} + Q\_{Ur} + Q\_{U\epsilon} + Q\_{U\epsilon} \tag{6}$$

The solar radiation intensity, I, at a given layer (depth) in the pond can be obtained as a fraction of the radiation that penetrates the pond's surface.

The solar radiation in the pond decays exponentially with depth

$$I\_x = \text{\textquotedblleft}I\_o \tag{7}$$

The evaporative heat loss rate *Q*\_

*DOI: http://dx.doi.org/10.5772/intechopen.89632*

The heat loss through the sidewall *Q*\_

differential form as:

*z*1*AρuCpu*

*TU t*ð Þ <sup>þ</sup><sup>1</sup> ¼ *TU t*ð Þ þ

last layer as Tf.

**45**

*Ue* is obtained as:

*Tu* <sup>þ</sup> <sup>230</sup> � � (21)

*Ta* <sup>þ</sup> <sup>230</sup> � � (22)

*Cpa* ¼ 1*:*005 þ 1*:*82*Rs* (23)

*Us* ¼ *CwAwU*ð Þ *Tu* � *TGU* (24)

*Us* can be obtained as:

<sup>¼</sup> *kpkc Spkc* þ *Sckp*

The solar energy absorbed by the zone can be obtained as the difference between

With the foregoing, the energy balance equation for the UCZ can be written in

þ *<sup>∂</sup>I*ð Þ *<sup>z</sup>*,*<sup>t</sup> <sup>∂</sup><sup>z</sup>* � *<sup>q</sup>*\_

þ *βA*ð Þ *<sup>z</sup>*<sup>1</sup> *I τ*ð Þ 0,*<sup>t</sup>* � *τ*ð Þ *<sup>z</sup>*1,*<sup>t</sup>*

� � � *<sup>Q</sup>*\_

þ *βAe u*ð Þ*I τ*ð Þ 0,*<sup>t</sup>* � *τ*ð Þ *<sup>z</sup>*1,*<sup>t</sup>*

þ *βAe u*ð Þ*I τ*ð Þ 0,*<sup>t</sup>* � *τ*ð Þ *<sup>z</sup>*1,*<sup>t</sup>*

*u t*ð Þ ( )

(20)

(25)

*<sup>u</sup>* (26)

� � � *<sup>Q</sup>*\_

� � � *<sup>Q</sup>*\_

*<sup>u</sup>* (27)

*u t*ð Þ (28)

(29)

*Ue* <sup>¼</sup> *<sup>λ</sup>hc*ð Þ *Pu* � *Pw <sup>A</sup>* 1*:*6*CpaPatm*

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source*

*Pu* <sup>¼</sup> exp 18*:*<sup>403</sup> � <sup>3885</sup>

*Pw* <sup>¼</sup> *RH* exp 18*:*<sup>403</sup> � <sup>3885</sup>

*Q*\_

*Q*\_

*ρCp ∂TU <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂z k ∂T ∂z* � �

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *kA <sup>∂</sup><sup>T</sup>*

*∂z* � � � � *z*¼*z*<sup>1</sup>

2 *A*1

*k*1 2 *A*1

Thus, the UCZ layer temperature can be obtained as:

Here, the heat balance for the NCZ can be given as:

Thus, the heat balance in non-differential form can be written as:

<sup>2</sup> *<sup>T</sup>*ð Þ 1,*<sup>t</sup>* � *TU t*ð Þ � � *Δz*<sup>1</sup> 2

<sup>2</sup> *<sup>T</sup>*ð Þ 1,*<sup>t</sup>* � *TU t*ð Þ � � *Δz*<sup>1</sup> 2

The NCZ is assumed to be divided into several imaginary layers i, with the first

*Q*ð Þ *<sup>i</sup>*,*t*þ<sup>1</sup> ¼ *Q*ð Þ *<sup>i</sup>*þ1,*<sup>t</sup>* þ *Q*ð Þ *solar* � *Q*ð Þ *<sup>i</sup>*�1,*<sup>t</sup>* � *Qs i*ð Þ ,*<sup>t</sup>* (30)

and last layers having a boundary with the UCZ and LCZ denoted as 1 and f, respectively; thus, the temperature in the first layer can be denoted as T1 and in the

*∂TU*

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup>

*Δt zuAuρuCpu*

**3.2 Non-convective zone (NCZ)**

*z*1*AρCp*

*TU t*ð Þ <sup>þ</sup><sup>1</sup> � *TU t*ð Þ

*Cw* <sup>¼</sup> <sup>1</sup> *Rw*

the radiation entering the zone and the radiation leaving the zone.

The fraction (*τ*) varies with the depth (z), and can be expressed as:

$$
\pi = 0.36 - 0.08 \ln \left( z \right) \tag{8}
$$

While, the solar radiation that penetrates the pond's surface can be expressed in terms of the incident radiation on the pond's surface; taking into consideration that not all the incident rays penetrate (refracted) at the surface, as some are reflected back.

$$I\_{\sigma} = \beta I \tag{9}$$

$$\beta = 1 - \frac{1}{2} \left[ \frac{\sin^2(\theta\_i - \theta\_r)}{\sin^2(\theta\_i + \theta\_r)} + \frac{\tan^2(\theta\_i - \theta\_r)}{\tan^2(\theta\_i + \theta\_r)} \right] \tag{10}$$

are related to the refractive index, *n* (=1.33 for water) as:

$$n\sin\theta\_i = n\sin\theta\_r\tag{11}$$

The angle of incidence can be obtained from:

$$
\cos\theta\_i = \cos\delta\cos\theta\cos\alpha + \sin\delta\sin\theta \tag{12}
$$

$$\delta = 23.45 \sin \left( \frac{360(284 + N)}{365.25} \right) = 23.45 \sin \left( \frac{360(N - 80)}{370} \right) \tag{13}$$

$$
\omega = \frac{2\pi(h - 12)}{24} \text{ (in rad)}\tag{14}
$$

$$
\omega = \frac{\text{360} (h - 12)}{24} \text{ (in degree)} \tag{15}
$$

The convective heat loss rate from the pond's surface, *Q*\_ *Uc* is:

$$
\dot{Q}\_{L\&} = A h\_c (T\_u - T\_a) \tag{16}
$$

where the convective heat transfer coefficient can be obtained as:

$$h\_c = \text{5.7} + \text{3.8V} \tag{17}$$

The heat loss rate due to radiation *Q*\_ *Ur* is

$$
\dot{Q}\_{Ur} = \varepsilon \sigma A \left( T\_u{}^4 - T\_{sky}{}^4 \right) \tag{18}
$$

The sky temperature *Tsky* can be determined as:

$$T\_{sky} = T\_a - \left(\mathbf{0.55} + \mathbf{0.061}\sqrt{P\_w}\right)^{0.25} \tag{19}$$

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source DOI: http://dx.doi.org/10.5772/intechopen.89632*

The evaporative heat loss rate *Q*\_ *Ue* is obtained as:

*QU* ¼ *QUc* þ *QUr* þ *QUe* þ *QUs* (6)

*Iz* ¼ *τIo* (7)

*Io* ¼ *βI* (9)

sin *θ<sup>i</sup>* ¼ *nsinθ<sup>r</sup>* (11)

370 � �

<sup>24</sup> ð Þ in rad (14)

<sup>24</sup> ð Þ in degree (15)

*Uc* ¼ *Ahc*ð Þ *Tu* � *Ta* (16)

*hc* ¼ 5*:*7 þ 3*:*8*V* (17)

<sup>4</sup> � � (18)

*Uc* is:

(10)

(13)

(19)

tan <sup>2</sup>ð Þ *<sup>θ</sup><sup>i</sup>* � *<sup>θ</sup><sup>r</sup>* tan <sup>2</sup>ð Þ *θ<sup>i</sup>* þ *θ<sup>r</sup>*

cos *θ<sup>i</sup>* ¼ *cosδcosθcosω* þ *sinδsinθ* (12)

<sup>¼</sup> <sup>23</sup>*:*45 sin <sup>360</sup>ð Þ *<sup>N</sup>* � <sup>80</sup>

*τ* ¼ 0*:*36 � 0*:*08 ln ð Þ*z* (8)

The solar radiation intensity, I, at a given layer (depth) in the pond can be

While, the solar radiation that penetrates the pond's surface can be expressed in terms of the incident radiation on the pond's surface; taking into consideration that not all the incident rays penetrate (refracted) at the surface, as some are reflected

ð Þ *θ<sup>i</sup>* � *θ<sup>r</sup>*

ð Þ *θ<sup>i</sup>* þ *θ<sup>r</sup>*

þ

� �

obtained as a fraction of the radiation that penetrates the pond's surface. The solar radiation in the pond decays exponentially with depth

The fraction (*τ*) varies with the depth (z), and can be expressed as:

sin <sup>2</sup>

sin <sup>2</sup>

are related to the refractive index, *n* (=1.33 for water) as:

365*:*25 � �

The convective heat loss rate from the pond's surface, *Q*\_

*Q*\_

*Q*\_

The sky temperature *Tsky* can be determined as:

*<sup>ω</sup>* <sup>¼</sup> <sup>2</sup>*π*ð Þ *<sup>h</sup>* � <sup>12</sup>

*<sup>ω</sup>* <sup>¼</sup> <sup>360</sup>ð Þ *<sup>h</sup>* � <sup>12</sup>

where the convective heat transfer coefficient can be obtained as:

*Ur* ¼ *εσA Tu*

*Tsky* <sup>¼</sup> *Ta* � <sup>0</sup>*:*<sup>55</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>061</sup> ffiffiffiffiffiffi

*Ur* is

<sup>4</sup> � *Tsky*

� � p <sup>0</sup>*:*<sup>25</sup>

*Pw*

*<sup>β</sup>* <sup>¼</sup> <sup>1</sup> � <sup>1</sup> 2

The angle of incidence can be obtained from:

*<sup>δ</sup>* <sup>¼</sup> <sup>23</sup>*:*45 sin 360 284 ð Þ <sup>þ</sup> *<sup>N</sup>*

The heat loss rate due to radiation *Q*\_

back.

*Low-temperature Technologies*

**44**

$$
\dot{Q}\_{U\epsilon} = \frac{\lambda h\_c (P\_u - P\_w) A}{1.6 C\_{p\_d} P\_{atm}} \tag{20}
$$

$$P\_u = \exp\left(18.403 - \frac{3885}{T\_u + 230}\right) \tag{21}$$

$$P\_w = R\_H \exp\left(18.403 - \frac{3885}{T\_a + 230}\right) \tag{22}$$

$$C\_{p\_d} = \mathbf{1.005} + \mathbf{1.82Rs} \tag{23}$$

The heat loss through the sidewall *Q*\_ *Us* can be obtained as:

$$
\dot{Q}\_{Us} = C\_w A\_{wU} (T\_u - T\_{GU}) \tag{24}
$$

$$C\_w = \frac{1}{R\_w} = \frac{k\_p k\_c}{S\_p k\_c + S\_c k\_p} \tag{25}$$

The solar energy absorbed by the zone can be obtained as the difference between the radiation entering the zone and the radiation leaving the zone.

With the foregoing, the energy balance equation for the UCZ can be written in differential form as:

$$
\rho \mathbf{C}\_p \frac{\partial T\_U}{\partial t} = \frac{\partial}{\partial \mathbf{z}} \left( k \frac{\partial T}{\partial \mathbf{z}} \right) + \frac{\partial I\_{(x,t)}}{\partial \mathbf{z}} - \dot{q}\_u \tag{26}
$$

$$k z\_1 A \rho \mathcal{C}\_p \frac{\partial T\_U}{\partial t} = k A \frac{\partial T}{\partial x} \bigg|\_{x=x\_1} + \beta A\_{(x\_1)} I \left( \tau\_{(0,t)} - \tau\_{(x\_1,t)} \right) - \dot{Q}\_u \tag{27}$$

Thus, the heat balance in non-differential form can be written as:

$$\begin{aligned} \, \_{L^{2}A} \boldsymbol{\rho}\_{u} \mathbf{C}\_{p\_{u}} \frac{T\_{U(t+1)} - T\_{U(t)}}{\Delta t} &= \frac{k\_{\frac{\Delta t}{2}} (T\_{(1,t)} - T\_{U(t)})}{\frac{\Delta t}{2}} + \beta \mathbf{A}\_{\epsilon(u)} I \left(\boldsymbol{\tau}\_{(0,t)} - \boldsymbol{\tau}\_{(x\_{1},t)}\right) - \dot{Q}\_{u(t)} \end{aligned} \tag{28}$$

Thus, the UCZ layer temperature can be obtained as:

$$T\_{U(t+1)} = T\_{U(t)} + \frac{\Delta t}{\mathbf{z}\_u \mathbf{A}\_u \rho\_u \mathbf{C}\_{p\_u}} \left\{ \frac{\mathbf{k}\_\sharp \mathbf{A}\_\ddagger (T\_{(1,t)} - T\_{U(t)})}{\frac{\Delta \mathbf{z}\_1}{2}} + \beta \mathbf{A}\_{\epsilon(u)} I(\mathbf{z}\_{(0,t)} - \boldsymbol{\tau}\_{(x\_1, t)}) - \dot{\mathbf{Q}}\_{\omega(t)} \right\} \tag{29}$$

#### **3.2 Non-convective zone (NCZ)**

The NCZ is assumed to be divided into several imaginary layers i, with the first and last layers having a boundary with the UCZ and LCZ denoted as 1 and f, respectively; thus, the temperature in the first layer can be denoted as T1 and in the last layer as Tf.

Here, the heat balance for the NCZ can be given as:

$$Q\_{(i,t+1)} = Q\_{(i+1,t)} + Q\_{(solar)} - Q\_{(i-1,t)} - Q\_{s(i,t)} \tag{30}$$

Thus, the energy balance can be written in differential form as:

$$\begin{split} A\rho\_i \mathbf{C}\_{p\_i} \frac{\partial T}{\partial \mathbf{t}} &= \frac{\partial}{\partial \mathbf{z}} \left( k A \frac{\partial T}{\partial \mathbf{z}} \right)\_{x\_{i+1}} + \beta I \frac{\partial \left( A\_{\epsilon(x)} \tau\_{(x)} \right)}{\partial \mathbf{z}} - \frac{\partial}{\partial \mathbf{z}} \left( k A \frac{\partial T}{\partial \mathbf{z}} \right)\_{x\_{i-1}} \\ &- C\_w \frac{\partial A\_{w(x)}}{\partial \mathbf{z}} \left( T\_{(i,t)} - T\_G \right) \end{split} \tag{31}$$

**3.4 Dimensions of the solar pond**

*DOI: http://dx.doi.org/10.5772/intechopen.89632*

*3.4.1 Surface areas of the LCZ*

zone (as shown in **Figure 2**). The average area is:

Thus, the area is:

Similarly,

**Figure 2**

**47**

*The dimensional characteristics of the solar pond.*

relation to the dimensional references of the zones.

The dimensional characteristics of each zone of the solar pond would need to be

The cross-sectional area of the LCZ, *Al* may be calculated at mid-plane of the

where the average length of the pond at the level (measured at mid-level):

*<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *zl* 2 *tanϑ*

> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *zl* 2 *tanφ*

� *W* � 2

*xl* ¼ *X* � 2

*wl* ¼ *W* � 2

*z*<sup>2</sup> þ *<sup>z</sup>* 2 *tanϑ* 

*tanϑ*

� *<sup>W</sup>* � <sup>2</sup> *<sup>z</sup>*<sup>2</sup>

Similarly, the average width of the pond can be given as:

*Al* ¼ *X* � 2

*Az*<sup>2</sup> <sup>¼</sup> *<sup>X</sup>* � <sup>2</sup> *<sup>z</sup>*<sup>2</sup>

*Al* ¼ *xl* � *wl* (39)

(40)

(41)

(42)

(43)

*<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *zl* 2 *tanφ*

*tanφ*

initially ascertained to obtain the temperature predictions of each of the pond's zones. The geometry of the pond could be characterised as shown in **Figure 2**. Here, the relevant cross-sectional surface areas (required for heat transfer) are deduce in

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source*

Or,

$$\begin{aligned} \Delta \mathbf{z}\_i \mathbf{A}\_i \rho\_i \mathbf{C}\_{p\_i} \frac{\partial T}{\partial t} &= kA \frac{\partial T}{\partial \mathbf{z}} \Big|\_{x = x\_{i+1}} + \beta \mathbf{A}\_{\varepsilon(i)} I \Big( \mathbf{r}\_{\left(i - \frac{1}{2}t\right)} - \mathbf{r}\_{\left(i + \frac{1}{2}t\right)} \Big) - kA \frac{\partial T}{\partial \mathbf{z}} \Big|\_{x = x\_{i-1}} \\ &- C\_w \mathbf{A}\_{w(i)} \Big( T\_{\left(i, t\right)} - T\_G \Big) \end{aligned} \tag{32}$$

And in non-differential form:

$$
\Delta\mathbf{z}\_{i}\mathbf{A}\_{i}\rho\_{i}\mathbf{C}\_{p\_{i}}\frac{T\_{(i,t+1)}-T\_{(i,t)}}{\Delta t} = \frac{\mathbf{k}\_{\left(i+\frac{1}{2}\right)}\mathbf{A}\_{\left(i+\frac{1}{2}\right)}\left(T\_{(i+1,t)}-T\_{(i,t)}\right)}{\Delta\mathbf{z}\_{i}} + \beta\mathbf{A}\_{\epsilon(i)}I\left(\mathbf{r}\_{\left(i-\frac{1}{2}\right)} - \mathbf{r}\_{\left(i+\frac{1}{2}\right)}\right)}{\Delta\mathbf{z}\_{i}}\tag{33}
$$

$$
$$

Hence, the temperature of a layer of the NCZ can be expressed as:

$$\begin{split} T\_{(i,t+1)} &= T\_{(i,t)} + \frac{\Delta t}{\Delta z\_i A\_i \rho\_i C\_{p\_i}} \left\{ \frac{k\_{\{i+\frac{1}{2}\}} A\_{\{i+\frac{1}{2}\}} \left( T\_{(i+1,t)} - T\_{(i,t)} \right)}{\Delta z} \\ &+ \beta A\_{\varepsilon(i)} I \left( \tau\_{\{i-\frac{1}{2}t\}} - \tau\_{\{i+\frac{1}{2}t\}} \right) - \frac{k\_{\{i-\frac{1}{2}\}} A\_{\{i-\frac{1}{2}\}} \left( T\_{(i,t)} - T\_{(i-1,t)} \right)}{\Delta z} \\ &- C\_w A\_{w(i)} \left( T\_{(i,t)} - T\_{G(i)} \right) \right\} \end{split} \tag{34}$$

#### **3.3 Lower-convective zone (LCZ)**

Using the same procedure that outlined previously, the heat balance equation for the LCZ can be expressed as:

$$Q\_L = Q\_{(solar)} - Q\_{L \to N} - Q\_{s(i,t)} - Q\_G - Q\_{ext} \tag{35}$$

$$\begin{aligned} \varepsilon\_l A\_l \rho\_l \mathbf{C}\_{p\_l} \frac{\partial T\_l}{\partial t} &= \beta A\_{\epsilon(x\_l)} I \left( \tau\_{(x\_l, t)} \right) - k A \frac{\partial T}{\partial x} \Big|\_{x = x\_2} - \mathbf{C}\_w A\_{w(x\_l)} \left( T\_{(l, t)} - T\_G \right) \\ &- \mathbf{C}\_{\text{gw}} A\_{x\_3} \left( T\_{(l, t)} - T\_G \right) - \mathbf{Q}\_{\text{ext}} \end{aligned} \tag{36}$$

$$\begin{aligned} \varepsilon\_l \mathbf{z}\_l \mathbf{A}\_l \rho\_l \mathbf{C}\_{p\_l} \frac{T\_{(l,t+1)} - T\_{(l,t)}}{\Delta t} &= \beta \mathbf{A}\_{\mathbf{r}(x\_3)} \mathbf{I} \left( \tau\_{(x\_3,t)} \right) - \frac{k\_l \mathbf{A}\_{x\_2} \left( T\_{(l,t)} - T\_{(f,t)} \right)}{\frac{\Delta x}{2}} \\ &- \mathbf{C}\_w \mathbf{A}\_{w(x\_l)} \left( T\_{(l,t)} - T\_G \right) - \mathbf{C}\_{\mathbf{g}w} \mathbf{A}\_{x\_3} \left( T\_{(l,t)} - T\_G \right) - \mathbf{Q}\_{\text{ext}} \tag{37} \end{aligned} \tag{37}$$

Thus, the temperature of the LCZ can be given as:

$$\begin{split} T\_{(l,t+1)} &= T\_{(l,t)} + \frac{\Delta t}{z\_l A\_l \rho\_l \mathbf{C}\_{pl}} \left\{ \beta \mathbf{A}\_{\mathbf{e}(x\_3)} I \left( \tau\_{(x\_3,t)} \right) - \frac{k\_l \mathbf{A}\_{\mathbf{z}\_2} \left( T\_{(l,t)} - T\_{(f,t)} \right)}{\frac{\Delta \mathbf{z}}{2}} \\ &- \mathbf{C}\_w \mathbf{A}\_{w(\mathbf{z}\_l)} \left( T\_{(l,t)} - T\_G \right) - \mathbf{C}\_{\mathbf{u}\mathbf{g}} \mathbf{A}\_{\mathbf{z}\mathbf{g}} \left( T\_{(l,t)} - T\_G \right) - \mathbf{Q}\_{\text{ext}} \right\} \end{split} \tag{38}$$

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source DOI: http://dx.doi.org/10.5772/intechopen.89632*

## **3.4 Dimensions of the solar pond**

Thus, the energy balance can be written in differential form as:

*zi*þ<sup>1</sup>

þ *βI*

*T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *TG*

<sup>þ</sup> *<sup>β</sup>Ae i*ð Þ*<sup>I</sup> <sup>τ</sup> <sup>i</sup>*�<sup>1</sup>

*<sup>A</sup> <sup>i</sup>*þ<sup>1</sup> ð Þ<sup>2</sup>

Hence, the temperature of a layer of the NCZ can be expressed as:

(

<sup>2</sup> ð Þ,*<sup>t</sup>* � *<sup>τ</sup> <sup>i</sup>*þ<sup>1</sup>

� �

� �)

*<sup>k</sup> <sup>i</sup>*þ<sup>1</sup> ð Þ<sup>2</sup>

<sup>2</sup> ð Þ,*<sup>t</sup>*

� � � *kA <sup>∂</sup><sup>T</sup>*

� *CgwAz*<sup>3</sup> *T*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> *<sup>β</sup>Ae z*ð Þ<sup>3</sup> *<sup>I</sup> <sup>τ</sup>*ð Þ *<sup>z</sup>*3,*<sup>t</sup>*

Thus, the temperature of the LCZ can be given as:

(

*Δt zlAlρlCpl*

� *CwAw z*ð Þ*<sup>l</sup> T*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

*Δzi*

*Δzi*

*<sup>A</sup> <sup>i</sup>*þ<sup>1</sup> ð Þ<sup>2</sup>

�

Using the same procedure that outlined previously, the heat balance equation for

*∂z* � � � � *z*¼*z*<sup>2</sup>

� *CwAw z*ð Þ*<sup>l</sup> T*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

*βAe z*ð Þ<sup>3</sup> *I τ*ð Þ *<sup>z</sup>*3,*<sup>t</sup>*

� � � *CwgAz*<sup>3</sup> *<sup>T</sup>*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

*<sup>A</sup> <sup>i</sup>*�<sup>1</sup> ð Þ<sup>2</sup>

*<sup>∂</sup> Ae z*ð Þ*τ*ð Þ*<sup>z</sup>* � � *<sup>∂</sup><sup>z</sup>* � *<sup>∂</sup>*

<sup>2</sup> ð Þ,*<sup>t</sup>* � *<sup>τ</sup> <sup>i</sup>*þ<sup>1</sup>

*T*ð Þ *<sup>i</sup>*þ1,*<sup>t</sup>* � *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � �

> *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *T*ð Þ *<sup>i</sup>*�1,*<sup>t</sup>* � �

> > *Δz*

*<sup>k</sup> <sup>i</sup>*�<sup>1</sup> ð Þ<sup>2</sup>

*T*ð Þ *<sup>i</sup>*þ1,*<sup>t</sup>* � *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � �

*<sup>A</sup> <sup>i</sup>*�<sup>1</sup> ð Þ<sup>2</sup>

*QL* ¼ *Q*ð Þ *solar* � *QL*!*<sup>N</sup>* � *Qs i*ð Þ ,*<sup>t</sup>* � *QG* � *Qext* (35)

� � � *klAz*<sup>2</sup> *<sup>T</sup>*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *<sup>T</sup>*ð Þ *<sup>f</sup>*,*<sup>t</sup>*

*Δz*

� *CwAw z*ð Þ*<sup>l</sup> T*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

� � � *Qext* (36)

� � � *CgwAz*<sup>3</sup> *<sup>T</sup>*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *TG*

� � � *klAz*<sup>2</sup> *<sup>T</sup>*ð Þ *<sup>l</sup>*,*<sup>t</sup>* � *<sup>T</sup>*ð Þ *<sup>f</sup>*,*<sup>t</sup>*

� � � *Qext*

� � *Δz* 2

� �

� � *Δz* 2

� � � *Qext*

) (38)

(37)

� �

*∂z*

<sup>2</sup> ð Þ,*<sup>t</sup>*

� � (32)

� � (31)

*kA <sup>∂</sup><sup>T</sup> ∂z* � �

> � *kA <sup>∂</sup><sup>T</sup> ∂z* � � � � *z*¼*zi*�<sup>1</sup>

<sup>þ</sup> *<sup>β</sup>Ae i*ð Þ*<sup>I</sup> <sup>τ</sup> <sup>i</sup>*�<sup>1</sup>

*T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *T*ð Þ *<sup>i</sup>*�1,*<sup>t</sup>* � �

<sup>2</sup> ð Þ,*<sup>t</sup>* � *<sup>τ</sup> <sup>i</sup>*þ<sup>1</sup>

� *CwAw i*ð Þ *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *TG*

� �

� �

<sup>2</sup> ð Þ,*<sup>t</sup>*

(33)

(34)

*zi*�<sup>1</sup>

*kA <sup>∂</sup><sup>T</sup> ∂z* � �

*<sup>∂</sup>Aw z*ð Þ *∂z*

� *CwAw i*ð Þ *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *TG*

*<sup>k</sup> <sup>i</sup>*þ<sup>1</sup> ð Þ<sup>2</sup>

*<sup>k</sup> <sup>i</sup>*�<sup>1</sup> ð Þ<sup>2</sup>

�

*Δt ΔziAiρiCpi*

� *CwAw i*ð Þ *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* � *TG i*ð Þ

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>β</sup>Ae z*ð Þ<sup>3</sup> *<sup>I</sup> <sup>τ</sup>*ð Þ *<sup>z</sup>*3,*<sup>t</sup>*

� *Cw*

*AρiCpi*

*Low-temperature Technologies*

Or,

*ΔziAiρiCpi*

*ΔziAiρiCpi*

*∂T <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂z*

*∂T*

And in non-differential form:

*T*ð Þ *<sup>i</sup>*,*t*þ<sup>1</sup> � *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>*

*T*ð Þ *<sup>i</sup>*,*t*þ<sup>1</sup> ¼ *T*ð Þ *<sup>i</sup>*,*<sup>t</sup>* þ

*Δt* ¼

<sup>þ</sup> *<sup>β</sup>Ae i*ð Þ*<sup>I</sup> <sup>τ</sup> <sup>i</sup>*�<sup>1</sup>

**3.3 Lower-convective zone (LCZ)**

the LCZ can be expressed as:

*∂Tl*

*T*ð Þ *<sup>l</sup>*,*t*þ<sup>1</sup> ¼ *T*ð Þ *<sup>l</sup>*,*<sup>t</sup>* þ

*T*ð Þ *<sup>l</sup>*,*t*þ<sup>1</sup> � *T*ð Þ *<sup>l</sup>*,*<sup>t</sup>*

*zlAlρlCpl*

*zlAlρlCpl*

**46**

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *kA <sup>∂</sup><sup>T</sup> ∂z* � � � � *z*¼*zi*þ<sup>1</sup>

The dimensional characteristics of each zone of the solar pond would need to be initially ascertained to obtain the temperature predictions of each of the pond's zones. The geometry of the pond could be characterised as shown in **Figure 2**. Here, the relevant cross-sectional surface areas (required for heat transfer) are deduce in relation to the dimensional references of the zones.

#### *3.4.1 Surface areas of the LCZ*

The cross-sectional area of the LCZ, *Al* may be calculated at mid-plane of the zone (as shown in **Figure 2**).

The average area is:

$$A\_l = \mathbf{x}\_l \cdot \mathbf{w}\_l \tag{39}$$

where the average length of the pond at the level (measured at mid-level):

$$\mathbf{x}\_{l} = \mathbf{X} - \mathbf{2}\left(\frac{\mathbf{z}\_{2} + \frac{\mathbf{z}\_{l}}{2}}{\tan \theta}\right) \tag{40}$$

Similarly, the average width of the pond can be given as:

$$w\_l = W - 2\left(\frac{x\_2 + \frac{x\_l}{2}}{\tan \rho}\right) \tag{41}$$

Thus, the area is:

$$A\_l = \left(X - 2\left(\frac{z\_2 + \frac{z}{2}}{\tan \theta}\right)\right) \cdot \left(W - 2\left(\frac{z\_2 + \frac{z\_1}{2}}{\tan \phi}\right)\right) \tag{42}$$

Similarly,

$$A\_{z\_1} = \left(X - 2\left(\frac{x\_2}{\tan \theta}\right)\right) \cdot \left(W - 2\left(\frac{x\_2}{\tan \rho}\right)\right) \tag{43}$$

**Figure 2** *The dimensional characteristics of the solar pond.*

Side wall area, *Aw z*ð Þ*<sup>l</sup>* ,

$$A\_{w(x\_l)} = \mathfrak{A}A\_{w(x\_l), W} + \mathfrak{A}A\_{w(x\_l), X} \tag{44}$$

*Aw*ð Þ *<sup>Δ</sup>zi* ¼ 2

**3.5. Soil temperature**

obtained as [20]:

**4. Calculation procedure**

respectively.

**49**

**5. Modelling results**

where

*Δzi*

*DOI: http://dx.doi.org/10.5772/intechopen.89632*

*sin<sup>ϑ</sup> <sup>W</sup>* � *zi*�<sup>1</sup>

2 *tan<sup>φ</sup>* � *zi*þ<sup>1</sup>

*zi*�<sup>1</sup>

*zi*þ<sup>1</sup>

such requires to be calculated separately as a function of depth and time.

*Tg*ð Þ¼ *z*, *t Tg* þ *A*0*e*

� �

2 *tanφ*

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source*

*zi* ¼ *z*<sup>1</sup> þ ð Þ *i* � 1 *Δz* þ

Although the earth surface temperature at the location can easily be obtained alongside the metrological data; the temperature of the soil varies with depth, as

The annual variation of the average soil temperature at different depths can be

�*z*

*<sup>d</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffi 2*Dh=ω*

*<sup>ω</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

The solar radiation intensity and ambient temperature data were obtained from the meteorological data for Qatar. The calculation was initialised (at time t = 0) by setting the temperature of the various layers of the pond to be equal to the ambient temperature; while setting the initial salinities of the UCZ and LCZ as equal to 2 and 26%, respectively (i.e. assuming that the pond was initially stabilised artificially). In the sequence of calculation, the parameters—heat transfer coefficients and the properties of the liquid (in the different layers)—are first determined by the initial (ambient) temperature. Then, the obtained liquid properties are employed together with the solar radiation to determine the temperatures of the different layers of the pond at time interval *Δt*. The temperature of any layer at a time interval is determined with the liquid properties previously obtained with the preceding temperature of that layer. Using the same procedure, the temperatures of the layers for any selected time interval or time of the day can be calculated. In the simulation, the thickness of UCZ, NCZ and LCZ were taken to be 0.2, 1.3 and 0.5 m,

Results were computed by performing energy balances throughout each layer and sub-layer of the pond to achieve accurate temperatures. As the middle NCZ layer has the greatest depth and in turn has the most changes in salinity and density, it had to be divided into more layers. Each layer, as they have different densities,

*<sup>d</sup>* sin *ω*ð Þ� *t* � *t*<sup>0</sup>

*Ao* <sup>¼</sup> *Tg*ð Þ <sup>0</sup> max � *Tg*ð Þ <sup>0</sup> min � �*=*<sup>2</sup> (60)

þ 2 *Δzi sinφ* *<sup>X</sup>* � *zi*�<sup>1</sup> 2 *tan<sup>ϑ</sup>* � *zi*þ<sup>1</sup>

<sup>2</sup> ¼ *z*<sup>1</sup> þ ð Þ *i* � 1 *Δz* (57)

<sup>2</sup> ¼ *z*<sup>1</sup> þ *iΔz* (58)

*z d*

p (61)

<sup>365</sup> (62)

h i (59)

*Δz*

2 *tanϑ* � � (55)

<sup>2</sup> (56)

The area of one side (along the width) is:

$$A\_{w(z\_l),W} = \frac{1}{2} \left[ \left( W - 2 \left( \frac{z\_2}{\tan \rho} \right) \right) + \left( W - 2 \left( \frac{z\_3}{\tan \rho} \right) \right) \right] \frac{z\_l}{\sin \theta} \tag{45}$$

Hence,

$$A\_{w(x\_l),W} = \frac{\mathbf{z}\_l}{\sin \theta} \left[ W - \frac{\mathbf{z}\_2}{\tan \rho} - \frac{\mathbf{z}\_3}{\tan \rho} \right] \tag{46}$$

Similarly, the area of an adjacent side along the length is:

$$A\_{w(x\_l),X} = \frac{z\_l}{\sin \rho} \left[ X - \frac{z\_2}{\tan \theta} - \frac{z\_3}{\tan \theta} \right] \tag{47}$$

Thus, the total area of the four side walls of the LCZ is:

$$A\_{w(z\_l)} = 2\frac{\mathbf{z\_l}}{\sin\theta} \left[ W - \frac{\mathbf{z\_2}}{\tan\rho} - \frac{\mathbf{z\_3}}{\tan\rho} \right] + 2\frac{\mathbf{z\_l}}{\sin\rho} \left[ X - \frac{\mathbf{z\_2}}{\tan\theta} - \frac{\mathbf{z\_3}}{\tan\theta} \right] \tag{48}$$

### *3.4.2 Surface areas of the UCZ*

Following the same procedure, the area for the UCZ can be deduced to be:

$$A\_{u} = \left(X - 2\left(\frac{0 + \frac{z\_{u}}{2}}{\tan \theta}\right)\right) \cdot \left(W - 2\left(\frac{0 + \frac{z\_{u}}{2}}{\tan \phi}\right)\right) \tag{49}$$

$$A\_{u} = \left(X - \frac{z\_{u}}{\tan \theta}\right) \cdot \left(W - \frac{z\_{u}}{\tan \phi}\right)$$

Similarly, the total area of the four side walls for the UCZ can be obtained as:

$$A\_{w(x\_u)} = 2\frac{z\_u}{\sin\theta} \left[ W - \frac{0}{\tan\rho} - \frac{z\_1}{\tan\rho} \right] + 2\frac{z\_u}{\sin\rho} \left[ X - \frac{0}{\tan\theta} - \frac{z\_1}{\tan\theta} \right] \tag{50}$$

$$A\_{w(x\_u)} = 2\frac{z\_u}{\sin\theta} \left[ W - \frac{z\_1}{\tan\rho} \right] + 2\frac{z\_u}{\sin\rho} \left[ X - \frac{z\_1}{\tan\theta} \right] \tag{51}$$

#### *3.4.3 Surface areas of the NCZ*

$$A\_i = \left(X - 2\frac{z\_i}{\tan\theta}\right) \cdot \left(W - 2\frac{z\_i}{\tan\rho}\right) \tag{52}$$

$$A\_{i+\frac{1}{2}} = \left(X - 2\frac{z\_{i+\frac{1}{2}}}{\tan\theta}\right) \cdot \left(W - 2\frac{z\_{i+\frac{1}{2}}}{\tan\phi}\right) \tag{53}$$

$$A\_{i-\frac{1}{2}} = \left(X - 2\frac{z\_{i-\frac{1}{2}}}{\tan\theta}\right) \cdot \left(W - 2\frac{z\_{i-\frac{1}{2}}}{\tan\phi}\right) \tag{54}$$

The total surface area of the four side walls of an elemental layer in the NCZ is:

*Solar Pond Driven Air Conditioning Using Seawater Bitterns and MgCl2 as the Desiccant Source DOI: http://dx.doi.org/10.5772/intechopen.89632*

$$A\_{w(\Delta \mathbf{z}\_i)} = 2 \frac{\Delta \mathbf{z}\_i}{\sin \theta} \left[ W - \frac{\mathbf{z}\_{i-\frac{1}{2}}}{\tan \rho} - \frac{\mathbf{z}\_{i+\frac{1}{2}}}{\tan \rho} \right] + 2 \frac{\Delta \mathbf{z}\_i}{\sin \rho} \left[ X - \frac{\mathbf{z}\_{i-\frac{1}{2}}}{\tan \theta} - \frac{\mathbf{z}\_{i+\frac{1}{2}}}{\tan \theta} \right] \tag{55}$$

where

Side wall area, *Aw z*ð Þ*<sup>l</sup>* ,

*Low-temperature Technologies*

*Aw z*ð Þ*<sup>l</sup>* ,*<sup>W</sup>* <sup>¼</sup> <sup>1</sup>

*Aw z*ð Þ*<sup>l</sup>* <sup>¼</sup> <sup>2</sup> *zl*

*3.4.2 Surface areas of the UCZ*

*Aw z*ð Þ*<sup>u</sup>* <sup>¼</sup> <sup>2</sup> *zu*

*3.4.3 Surface areas of the NCZ*

**48**

Hence,

The area of one side (along the width) is:

*<sup>W</sup>* � <sup>2</sup> *<sup>z</sup>*<sup>2</sup>

*Aw z*ð Þ*<sup>l</sup>* ,*<sup>W</sup>* <sup>¼</sup> *zl*

Similarly, the area of an adjacent side along the length is:

*Aw z*ð Þ*<sup>l</sup>* ,*<sup>X</sup>* <sup>¼</sup> *zl*

Thus, the total area of the four side walls of the LCZ is:

� �

*tan<sup>φ</sup>* � *<sup>z</sup>*<sup>3</sup>

*sin<sup>ϑ</sup> <sup>W</sup>* � *<sup>z</sup>*<sup>2</sup>

*Au* ¼ *X* � 2

*sin<sup>ϑ</sup> <sup>W</sup>* � <sup>0</sup>

*Aw z*ð Þ*<sup>u</sup>* <sup>¼</sup> <sup>2</sup> *zu*

*Ai*þ<sup>1</sup>

*Ai*�<sup>1</sup>

� � � �

*tanφ*

*sinϑ*

*sinφ*

*tanφ*

Following the same procedure, the area for the UCZ can be deduced to be:

*tanϑ* � �

*tanφ*

*<sup>W</sup>* � *<sup>z</sup>*<sup>1</sup> *tanφ* � �

*tanϑ* � �

*zi*þ<sup>1</sup> 2 *tanϑ* � �

*zi*�<sup>1</sup> 2 *tanϑ* � �

The total surface area of the four side walls of an elemental layer in the NCZ is:

Similarly, the total area of the four side walls for the UCZ can be obtained as:

<sup>þ</sup> <sup>2</sup> *zu sinφ*

<sup>0</sup> <sup>þ</sup> *zu* 2 *tanϑ* � � � �

*Au* <sup>¼</sup> *<sup>X</sup>* � *zu*

*tan<sup>φ</sup>* � *<sup>z</sup>*<sup>1</sup>

� �

*sinϑ*

*Ai* <sup>¼</sup> *<sup>X</sup>* � <sup>2</sup> *zi*

<sup>2</sup> ¼ *X* � 2

<sup>2</sup> ¼ *X* � 2

2

*Aw z*ð Þ*<sup>l</sup>* ¼ 2*Aw z*ð Þ*<sup>l</sup>* ,*<sup>W</sup>* þ 2*Aw z*ð Þ*<sup>l</sup>* ,*<sup>X</sup>* (44)

*tanφ*

*tanφ*

*<sup>X</sup>* � *<sup>z</sup>*<sup>2</sup>

<sup>0</sup> <sup>þ</sup> *zu* 2 *tanφ*

� � � �

*tanφ* � �

*<sup>X</sup>* � <sup>0</sup>

*tanφ*

*zi*þ<sup>1</sup> 2 *tanφ*

*zi*�<sup>1</sup> 2 *tanφ*

*tan<sup>ϑ</sup>* � *<sup>z</sup>*<sup>1</sup> *tanϑ*

� �

*<sup>X</sup>* � *<sup>z</sup>*<sup>1</sup> *tanϑ* h i

*tan<sup>ϑ</sup>* � *<sup>z</sup>*<sup>3</sup> *tanϑ*

h i

*sin<sup>ϑ</sup>* (45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

<sup>þ</sup> *<sup>W</sup>* � <sup>2</sup> *<sup>z</sup>*<sup>3</sup>

*tan<sup>φ</sup>* � *<sup>z</sup>*<sup>3</sup>

*tan<sup>ϑ</sup>* � *<sup>z</sup>*<sup>3</sup> *tanϑ*

h i

� �

� � � � � � *zl*

*<sup>W</sup>* � *<sup>z</sup>*<sup>2</sup>

*<sup>X</sup>* � *<sup>z</sup>*<sup>2</sup>

<sup>þ</sup> <sup>2</sup> *zl sinφ*

� *W* � 2

� *<sup>W</sup>* � *zu*

<sup>þ</sup> <sup>2</sup> *zu sinφ*

� *<sup>W</sup>* � <sup>2</sup> *zi*

� *W* � 2

� *W* � 2

� �

� �

� �

$$z\_i = z\_1 + (i - 1)\Delta z + \frac{\Delta z}{2} \tag{56}$$

$$z\_{i - \frac{1}{2}} = z\_1 + (i - 1)\Delta z \tag{57}$$

$$z\_{i+\frac{1}{2}} = z\_1 + i\Delta z \tag{58}$$

#### **3.5. Soil temperature**

Although the earth surface temperature at the location can easily be obtained alongside the metrological data; the temperature of the soil varies with depth, as such requires to be calculated separately as a function of depth and time.

The annual variation of the average soil temperature at different depths can be obtained as [20]:

$$T\_{\mathcal{g}}(\mathbf{z},t) = \overline{T}\_{\mathcal{g}} + A\_0 e^{\frac{-\overline{\mathbf{z}}}{d}} \sin\left[o(t - t\_0) - \frac{\mathbf{z}}{d}\right] \tag{59}$$

$$A\_o = \left( T\_{\mathcal{g}(0)\max} - T\_{\mathcal{g}(0)\min} \right) / 2 \tag{60}$$

$$d = \sqrt{{}^{2D\_{b}}\!\!\!\!\!\!\!\!\!\/}\_{a}\tag{61}$$

$$
\omega = \frac{2\pi}{365} \tag{62}
$$
