## 1. Introduction

Cooling demand has continued to increase in recent years, due to the need to achieve high thermal comfort in residential buildings. This has led to the wide application of vapor compression cooling systems, resulting in a significant increase in electricity peak loads in summer [

1]. Therefore, researchers have put a lot of efforts into developing efficient, environmentally friendly air-conditioning systems driven by thermal energy sources such as solar, geothermal and industrial waste thermal energy. Recently, solar-powered adsorption chillers have attracted increasing attention because of the use of environmentally benign refrigerants such as water with zero ozone depletion potential, and ability to be driven by any type of low-grade thermal energy source. Silica gel-water adsorption chillers are more favorited for low-temperature solar energy under 80 °C which can be obtained using flat-plate solar collectors (FPC), evacuated tube solar collectors (ETC) or compound parabolic collectors (CPC) [

2,

3,

4]. Although solar-powered adsorption chillers have a very promising future, this technology is not yet widely commercialized due to low system coefficient of performance (COP) and high manufacturing cost of the chiller [

5,

6].

Many researchers have put a lot of effort into improving the system COP and specific cooling capacity (SCP) using different technologies such as heat and mass recovery, multi-bed and multi-stage technologies, and improvement of the structure of sorption beds [

7,

8,

9]. These technologies substantially improved the performance of adsorption systems and hence a lot of studies have been conducted to investigate and improve the system performance of adsorption cooling systems. Chang et al. [

10] experimentally investigated the cooling power and COP of a closed-type silica gel-water adsorption chiller under different operating conditions. A COP of 0.45, cooling power of 4.3 kW and SCP of 176 W·kg

^{−1} was achieved at hot water temperature, 80 °C, cooling water temperature, 30 °C and chilled water temperature, 14 °C. Wang and Chua [

11] studied a two-bed silica gel-water adsorption chiller with a water-circulation heat recovery scheme using a lumped parameter method. The results showed that the lumped parameter model was reliable to evaluate the chiller performance and optimize the design of adsorption chillers. Rezk and Al-Dadah [

12] numerically evaluated the performance of a two-bed silica gel-water adsorption chiller with heat and mass recovery. This study showed that the chiller cooling capacity increased by 3% and COP reduced by 2.3%. Deshmukh et al. [

13] proposed a continuous three-bed silica gel water solar-adsorption cooling system with a liquid refrigerant storage tank. This system showed the ability to produce a cooling capacity of 0.8 kW for 24 h. The average system COP and SCP reached 0.63 and 337.5 kJ·kg

^{−1}, respectively. Alsaqoor et al. [

14] experimentally studied the effect of the switching time on a three-bed dual-evaporator silica gel-water adsorption chiller driven by waste heat from a cogeneration plant. The results showed that the highest cooling capacity was 90.5 kW and COP was 0.645 at a switching time of 900 s. Chang et al. [

15] studied a two-bed, closed-type, solar-powered, silica gel-water adsorption chiller. The chiller achieved a cooling capacity (CC) of 9 kW, COP of 0.37 and SCP of 72 W·kg

^{−1} adsorbent at a cooling water temperature of 30 °C, a hot water temperature of 80 °C, and chilled water inlet temperature of 14 °C.

The intermittent nature of solar radiation is one of the challenges for solar-driven adsorption refrigeration (SAR) systems. Hassan et al. [

16] proposed continuous cold production in SAR systems. A dynamic analysis of the CTAR cycle (constant temperature adsorption refrigeration) was performed to compare with the theoretical operation. The results demonstrated that the constant temperature adsorption process could be achieved in the theoretical CTAR cycle. The isothermal process could be approached through enhancing heat transfer between the adsorption bed and the ambient environment during the pre-cooling process. The proposed solar-adsorption chiller achieved a COP of 0.402, cooling capacity of 363.8 W and SCP of 1.82 W·kg

^{−1} under continuous operating conditions. In a similar effort, modeling of a solar-adsorption chiller using activated carbon and methanol as the working pair was performed by Hassan et al. [

17]. The chiller could produce 2.63 kg cold-water per day at 0 °C from water at 25 °C per kg of adsorbent and achieved a COP of 0.66. Habib et al. [

18] evaluated the effect of adsorbent-adsorbate pairs such as silica gel-water, activated carbon fiber (ACF)-ethanol and activated carbon-methanol on the adsorption chiller performance under Malaysia weather conditions. The results showed that the highest COP value was achieved in the silica gel-water cycle at the regeneration temperature below 70 °C while the COP was found to be highest in an ACF-ethanol adsorption cycle at the regeneration temperature above 70 °C. Some authors numerically studied a solar thermal driven dual-mode, four-bed silica gel–water adsorption chiller under weather conditions typical of Durgapur, India [

19]. At a source temperature above 60 °C, the chiller worked as a single-stage four-bed mode; but functioned as a two-stage adsorption mode at the driving source temperature below 60 °C. At a cooling water temperature of 30 °C, the two-stage mode chiller produced a cooling effect at a source temperature as low as 40 °C. Zhai et al. [

20] investigated a solar-powered adsorption air-conditioning system that consisted of a 150 m

^{2} solar collector and two adsorption chillers with a capacity of 8.5 kW each under weather conditions typical of Shanghai, China. The system achieved a daily average solar COP of 0.15 and a system COP of 0.35, respectively. Sumathy et al. [

21] optimized crucial parameter effects on an adsorption cooling system driven by a flat-plate solar collector. The results showed a maximum SCP at a collector area of 150 W·m

^{−2}.

As detailed in the above literature, the solar-adsorption chiller system has been widely studied in general. However, there is a lack of research in the literature about the economics of solar-driven adsorption chillers. This remains a very complex issue because weather conditions vary from one location to another. Tsoutsos et al. [

22] conducted an economic analysis of absorption versus adsorption solar thermal cooling systems. The results showed that adsorption systems were more expensive than absorption systems by 50% in terms of capital cost. However, the adsorption chiller could be operated under low solar energy temperatures which implied that the adsorption chiller had greater potential. Ugla et al. [

23] compared the performance of conventional vapor-compression, solar LiBr–H

_{2}O absorption, and solar photovoltaic (PV) vapor-compression. Payback Period (PbP) and the Net Present Value (NPV) were used to investigate a commercial building in Khobar City, Saudi Arabia. The results showed that solar absorption systems were more economically feasible in comparison to solar PV-vapour-compression systems. Furthermore, the feasibility of both solar-powered systems improved as the size of the commercial building and the electricity consumption rate increased. This indicated an economic potential for solar-adsorption systems due to the similarity between the adsorption and absorption systems. Alahmer et al. [

24] evaluated a CPC solar-adsorption cooling system using Transient System Simulation (TRNSYS) software (Version 17). The study indicated that this form of adsorption cooling could potentially be economically used in cities with high solar irradiation.

This paper presents dynamic modeling of a solar thermally driven two-bed silica gel–water continuous adsorption cooling system which is driven by a compound parabolic concentrator (CPC). Perth city in Western Australia (latitude of 32 °N, longitude of 115.8 °E and an elevation of 0.0 m) is selected for the case study to evaluate the economic feasibility of adsorption air-conditioning systems. In Perth, the daily maximum ambient temperature is around 30 to 40 °C for three to four months in summer which substantially increases the electricity consumption for air-conditioning systems. Meanwhile, Perth city has one of the world’s highest radiation intensities which averages over 5.4 peak sun hours each day up to over 8 peak sun hours [

25]. This provides excellent potential for solar cooling. A mathematic model, Fourier series is proposed to simulate solar radiation data according to the actual solar radiation obtained from Meteonorm software version 7.0. An economic analysis using the PbP and Life-Cycle Saving (LCS) methods is conducted to study the economic potential of applying solar-adsorption cooling technologies for weather conditions like Perth city. The optimum solar collect area is studied. The research work provides useful information for decision-makers and engineers in the adoption and design of solar-adsorption cooling technologies.

## 3. Mathematical Modelling

#### 3.1. Correlation Equation of Solar Radiation Data

The radiation database consists of hourly measurements; however, continuous data is required for the simulation procedure. Therefore, Fourier series are used to simulate the solar radiation data according to the actual hourly-based solar radiation. The objective of using the Fourier series expansion is to provide suitable solar radiation predictions to a large ecosystem model. Then, the actual radiation intensity pattern of Perth city can be emulated. The outputs are expected to be similar to reality, in a statistical manner, so that appropriate amounts of energy are allocated in the virtual ecosystem.

In general, the solar radiation can be described as a function of the Sun–Earth distance, the Earth’s inclination and the Sun’s zenith angle. The proposed model, instead of studying the atmosphere and the clouds, studies the historical on-surface radiation measurements. The variable of interest is the daily overall solar energy, (Wh·m^{−2}). One year’s daily overall solar energy is considered as a signal, and one signal is analyzed at a time. Multiple years’ signals of one city bear the same statistical characters. The statistical characters are summarized with a set of models which would be used to synthesize new annual daily overall solar energy on demand. The synthesized data should be statistically similar to the original set. The output of this model represents an averaged, typical trend of the history, rather than forecast any future extreme situations.

The historical solar radiation records for Perth city were acquired from Perth Meteorology. The data was recorded as hourly overall energy, in kJ·m^{−2}. For each day, hourly values were summed up to obtain the daily overall solar energy (Wh·m^{−2}). The identical analysis procedure was performed for each day of Perth city, resulting in one individual model for Perth city depicting the statistical characteristics of the local solar radiation pattern.

To simplify the calculation, only a simple sine function is considered in the present analysis. The solar radiation,

I can be expressed by:

The objective function is:

A derivative of the objective function with the coefficient

a_{k} is:

where the coefficients

${a}_{k}$ are improved by iterative operation until the desired level of the fitting is obtained.

A is the parameter to control the convergence and is set at 0.01 in this analysis. The iterative equation for the coefficient

${a}_{k}$ is shown below:

Initial values for all

${a}_{k}$ are set at 0. A convergence criterion to stop the program is set as:

where

tol is set at

${10}^{-8}$ in the present study.

#### 3.2. Empirical Equation for Day Temperature Data

The day temperature function is calculated based on the following equation [

27]:

where

i is the time difference between the times of maximum radiation and maximum temperature. In the present case, it is selected as 1 hour.

#### 3.3. Energy Balance for Adsorption Chiller

In the bed-energy balance analysis, the assumptions include uniform temperatures, pressures, and concentrations throughout the bed, thermal equilibrium in the adsorber and constant thermal properties such as density and specific heat for the adsorbent and water vapor. Therefore, the energy balance equation for the sorption beds is expressed by:

where

$\delta $ equals 0 or 1 refers to the desorption and adsorption process. It depends on whether the sorption bed works as a desorber or adsorber. The left-hand side in Equation (7) indicates the internal energy change of the sorption bed including adsorbent material, vapor within the adsorbent bed and metal used in bed. While the right side consists of desorption/adsorption heat, energy transfers from the evaporator to bed or from the bed to the condenser, and energy transfers between the bed and cooling or heating fluid during the sorption process.

The energy balance equation for the condenser is expressed by:

Equation (9) states that the internal energy change of the condenser material and the refrigerant condensation within the condenser is equal to the energy released during the condensation process and the energy transfers from the bed to the condenser, with the addition of energy rejected from the condenser.

The energy balance equation for the evaporator is expressed by:

Equation (11) indicates the change of internal energy in the evaporator process is equal to the energy transfer from the condenser to the evaporator, the energy transferred to the adsorber and energy supplied from the chilled water.

The mass balance equation of the refrigerant inside the evaporator is expressed by:

The vapor adsorption rate in the sorption bed is calculated using the linear driving force kinetic equation:

The numerical values of

D_{so} and

E_{a} for silica gel water are evaluated experimentally in the literature [

22]. Adsorption equilibrium is expressed by a modified Freundlich equation that has been proven in the literature [

25]:

The constants of A

_{0}$~$A

_{3} and B

_{0}$~$B

_{3} are adapted from the literature [

26] and listed below:

${A}_{0}$= −6.5314 kg × kg^{−1} of dry adsorbent

${A}_{1}$= 0.72452 × 10^{−1} kg×kg^{−1} of dry adsorbent K^{−1}

${A}_{2}$= −0.23951 × 10^{−3} kg×kg^{−1} of dry adsorbent K^{−2}

${A}_{3}$ = 0.25493 × 10^{−6} kg×kg^{−1} of dry adsorbent K^{−3}

${B}_{0}$= −15.587

${B}_{1}$= 015915 K^{−1}

${B}_{2}$= −0.50612 ×10^{−3} K^{−2}

${B}_{3}$= 0.5329 ×10^{−6} K^{−3}

The energy balance equation for each solar collector is expressed by:

Due to a small temperature difference between the heating and cooling fluid, the log mean temperature difference (LMTD) method is used to model the outlet temperature of the heat source or heat sink, and the outlet temperature is given by:

where,

i = 1, number of pipes in the solar collector.

#### 3.4. Performance Analysis

Collector efficiency is calculated using the equation reported by Clausse et al. [

28]:

where;

${\overline{T}}_{HW}$ is the mean temperature for hot fluid and it is evaluated by:

${\overline{T}}_{HW}=\left(\frac{{T}_{HW,in}+{T}_{HW,out}}{2}\right)$;

I: solar radiation.

The cyclic average cooling capacity (CACC) is determined by:

The cycle COP is defined as a ratio of cooling capacity to total required thermal energy and is calculated for each single cycle; while the solar system COP in a cycle (COP

_{SC}_{, cycle}) is defined as a ratio of cooling capacity to the net solar energy input; and COP

_{SC}_{, net} is calculated on the period of cold production. These COPs are calculated as [

24,

27]:

After sunset, the

COP_{SC}_{, net} equation becomes

where

I is the solar irradiation intensity,

A_{cr} is one collector area, and

n is the number of collectors.

The above differential equations are solved using implicit finite difference approximation with a tolerance of 10

^{−4} for the convergence calculation. Details of the simulation procedure were discussed in our previous publication [

7]. The flow chart of a solar-driven two-bed adsorption chiller model was demonstrated in

Figure 2.

Table 1 lists the operating conditions and design parameters used in the present simulation. The assumptions include: (i) the temperature and pressure are uniform in each adsorber; (ii) no heat loss to the environment as the system is well insulated; (iii) the thermal energy transfer along the pipes connecting the system components is neglected; and (iv) the pressure differences between the sorption bed and the condenser or between the sorption bed and the evaporator are neglected [

29].

#### 3.5. Economic Analysis

Life-Cycle Saving (LCS) is employed for economic analysis to evaluate economic potential of the solar thermal driven adsorption cooling system and to find the optimum solar collector area. The LCS of the solar system over a conventional system is defined as the difference between the reduction in fuel costs and the increase in expenses resulting from the additional investment for the solar system. It is calculated by the following equation [

30,

31]:

where

P_{1} refers to the factor of the life-cycle fuel cost over the first year fuel cost savings;

P_{2} is a factor of life cycle of the additional capital investment over the initial investment;

C_{A} is the solar energy investment cost that is directly proportional to the collector area;

C_{E} is the solar energy investment cost that is independent of the collector area;

C_{F} is the unit cost of delivered conventional energy for the first year of analysis;

L is the total load, and

Ft is the total solar fraction of the solar system. More details to evaluate P

_{1} and P

_{2} can be found in the literature [

31].

The following assumptions are taken to estimate the costs of using a solar adsorption chiller [

32]:

The initial investment cost of the solar-adsorption cooling system is estimated and listed in

Table 2 [

22,

33,

34,

35]

The life span of the cooling system is set as 20 years according to available products in the market

The period of the loan is 15 years

In Australian markets, the inflation rate in fuel price and the interest rate is around 2.8% and 12 %, respectively

The salvage factor for the auxiliary cooling system is assumed to be 20% of the capital cost

The maintenance factor is assumed to be 1.1% of the capital cost for the auxiliary cooling system

The electricity price is 0.33 USD kWh^{−1} in the Perth market

The simple Payback Period (PbP) method is used to analyze the feasibility of the solar thermal adsorption cooling system. It considers the time required to recover initial investment costs from the savings in electricity consumption by using this proposed solar cooling system. PbP is calculated by [

30,

31]:

where

i is the change in energy prices relative to general inflation in the country or to energy inflation alone,

C is the initial investment cost of installing the solar cooling equipment (USD) and

E is the annual energy saving in USD which represents the difference between the cost of the solar cooling and conventional electric cooling systems.