Energy Analysis and Performance Evaluation of the Adsorption Refrigeration System

A complete steady state thermodynamic diﬀerential analysis is developed for the adsorption-based refrigeration systems. e introduced thermodynamic model accurately represents the behaviour of adsorption cooling systems, based on a precise, reasonable, and clear fundamental approach. Based on the energy conservation principle, all components and processes in the system are analyzed. e dynamics of adsorption is expressed by the Dubinin-Astakhov adsorption equilibrium model. All types of energy interactions are evaluated in order to determine the theoretical performance and the operating parameters of the system. Moreover, the actual thermodynamic properties of the refrigerant are considered in developing the model. e case studied is an ice maker which uses activated carbon-methanol as the working pair.


Introduction
e traditional vapor compression refrigeration (VCR) machines are dominating electricity consumers and their operation causes high electricity peak loads [1]. Providing cooling by using a low quality source of energy is a key solution to reduce electrical energy consumption. Low quality energy sources are those sources with low temperature and include solar thermal energy, geothermal energy, and waste heat. Recently, thermally driven cold production (TDCP) systems attract many researchers [2]. e booming progress in the TDCP technology offers a considerable number of systems as alternatives to the VCR machines. ese systems include adsorption, absorption, and ejector systems. An advantage of the TDCP systems is the environmentally benign and natural refrigerants which have zero ozone depleting as well as zero global warming potentials. Moreover, the low heat source driving temperature and its wide range make TDCP systems more attractive.
Adsorption cold production (ACP) systems can be powered by heat sources with temperature which is nearenvironmental temperatures and as low as 50 ∘ C, depending on the adsorbent-refrigerant working pair [3]. Moreover, ACP machines are suitable for working in hot and dry areas because they do not require any extra cooling equipment for high ambient temperatures. Due to the fact that ACP systems use solid absorbent, they are suitable for conditions with serious vibration, such as in �shing boats and locomotives. e most widely used working pairs are activated carbonmethanol [4,5], activated carbon �bers-methanol [6], activated carbon-ethanol [7], activated carbon-ammonia [8], silica gel-water [9], and zeolite-water [10]. Many researchers have been interested with the study of the ACP system both numerically and experimentally [3,11,12]. Moreover, the literature is abundant with numerical studies accomplished on different adsorbent-adsorbate pairs, various ACP system designs, and different system con�gurations [13][14][15][16][17].
Most of the published studies on the ACP systems were dynamic models to simulate the system and to study its performance [4,8,[18][19][20][21]. Although it is an interesting point to investigate the dynamic behaviour of the ACP system, it is most important to understand its theoretical steady state behaviour and characteristics. However, an obvious and detailed information on the fundamental thermodynamic analysis of the ACP system is limited. Moreover, the introduced work in [22,23] proposed many assumptions and simpli�cations to the analysis, without enough reasons and accuracy. Although some of the assumptions in [24,25]  be accepted to some extent, other assumptions appear to be far from the reality of the system behaviour. Although Teng et al. [26] and more recently Hassan et al. [27] presented much more better thermodynamic models, they used an integral analysis which cannot provide precise details for the system behaviour.
It is difficult to �nd any work in the literature which accurately study the steady state thermodynamic behaviour of the ACP system, based on a precise, reasonable, and clear fundamental analysis. erefore, the major motivation behind the current study is to introduce a complete differential thermodynamic analysis of the ACP system.

Thermodynamic Cycle of the ACP System
e ACP system is similar to the basic VCR machine except that the power compressor is replaced with a thermally driven compressor which is the adsorption bed. e main components of the basic system are shown in Figure 1. e system consists of an adsorption reactor, a condenser, an evaporator, a refrigerant storage tank, 3 one-way control valves, and a throttling device. e adsorption bed is composed of a type of solid medium that has affinity to physically adsorb and desorb the refrigerant vapor. e basic adsorption refrigeration thermodynamic cycle is illustrated on the conceptual Clapeyron diagram, Figure 2, and its operation time schedule is represented in Figure 3.
Since refrigerant phase change is monovariant, its equilibrium state is de�ned only by one variable, either by the pressure or the temperature. e dynamic equilibrium condition of the refrigerant is represented by the line on the le in Figure 2. e condition of the gas-solid adsorption equilibrium in the reactor is bivariant and requires two variables to be speci�ed, the concentration ratio and either the temperature or the pressure. erefore, the temperaturepressure dynamic equilibrium of the gas-solid reactor is represented in Clapeyron diagram by the line in the middle for high concentration and the line in the right for the lower concentration ratios. e working region of the adsorption reactor spans from the lower concentration adsorption equilibrium line to the higher concentration adsorption equilibrium line. e adsorbent-adsorbate reactor cycle consists mainly of four processes; pressurization and preheating process at the constant higher concentration (isosteric heating process 1 → 2), desorption-condensation process at constant condenser pressure (isobaric heating process 2 → 3), depressurization and precooling process at the constant lower concentration (isosteric cooling process 3 → 4), and adsorption-evaporation process at constant evaporator pressure (isobaric cooling process 4 → 1) [1,4].
At the beginning of the cycle, state 1, the reactor is isolated from both the condenser and the evaporator by valves and and is completely charged and saturated with the refrigerant. e pressure inside the reactor initially equals the evaporator pressure ev and its temperature is uniform and equals the ambient temperature amb , state 1. When the reactor is heated, both pressure and temperature inside the adsorption bed are elevated. is constant concentration heating phase continues till state 2 where the pressure reaches a value that equals the condenser pressure con . At state 2, valve opens and the refrigerant vapor starts to desorb from the bed and �ows towards the condenser. In parallel with the desorption process, the adsorption reactor is still being heated and the pressure inside the bed is �xed at the condenser pressure. e temperature continues to increase above the generation temperature gen . e refrigerant content inside the reactor continues to decrease as more adsorbate is being freed from the reactor. In the condenser, the refrigerant vapor losses both sensible heat and latent heat and condenses at the preset condensation temperature con corresponding to the condenser pressure con . e condensed refrigerant is then collected and stored in the refrigerant storage tank, Figure  1. It is worth noting that process 1 → 2 → 3 is equivalent to the compression process in case of the VCR system. e desorption process continues while the bed is being heated and its temperature is continuously increasing whereas the pressure keeps its value at the constant condenser pressure. is operation continues for a certain period of time until one of the following stopping criteria is reached.
(1) e temperature of the adsorption bed reaches the maximum allowable value at which the refrigerant becomes chemically unstable.
(2) e reactor temperature reaches the heat source temperature and no more adsorbate can be generated from the bed.
Whichever stopping condition is approached, valve is closed in order to disconnect the adsorption bed from the condenser and start the bed cooling process, 3 → 4. e reactor cooling process takes place at the constant low concentration and both of the temperature and pressure decrease, till its pressure reaches the evaporator pressure ev at state 4 on Clapeyron diagram, Figure 2. At state 4, valve is opened to allow the liquid refrigerant, which is previously stored in the storage tank, to pass through the expansion device. Due to the throttling process, the liquid refrigerant is self-cooled by evaporating some liquid before it enters the evaporator. At the same instant of opening  valve t, valve opens and the refrigerant vapor from the evaporator �ows towards the reactor. is process consists of the refrigerant adsorption within the reactor in parallel with the production of cooling effect inside the evaporator. e reactor is being cooled during this period in order to remove the generated heat from the exothermic adsorption process. e evaporation-adsorption process takes place at the constant evaporator pressure and continues till the adsorption bed retains its higher cycle isostere, at point 1.
e solid adsorbent is a type of porous media that has cavities, channels, or interstices, which are deeper than being wide. e large surface area of the adsorption media is mainly due to the micropore and mesopore internal structure. Porous materials may be one of two types: agglomerates or aggregates. Agglomerates are rigid and consolidated porous material which consists of the assemblage of macroscopic particles whose dimensions exceed those of the pores by many orders of magnitude, like silica gel. Aggregates are loose and unconsolidated assemblage of individual particles, like powders, [28,29]. As illustrated in Figure 4, the total void space, viod , within an element of the adsorbent consists of three components: the interparticle voids, the open pore voids, and the closed pore voids. e open pore type of void spaces is due to the contribution of both the intercrystalline macropores and the intracrystalline micropores void spaces as well as the mesopore spaces. e total porosity of any porous material is de�ned as the ratio between the total void spaces within the solid porous material and the material bulk volume. e bulk volume of a material includes the contribution of skeletal volume, pore void volume, and interparticle space volume [30]. erefore, an adsorption bed with a bulk volume of has its total porosity, , calculated from where is the packed bed porosity which represents the interparticle void spaces within the bed per unit bulk volume of the bed. e particle porosity, , is the the total pore volume, open and closed, per unit bulk volume of the particle and is given in terms of the particle density, , and true density, , as follows; e true density (also, absolute density, helium density, or skeletal density) of the material is calculated by excluding all pores (open and closed) and interparticle voids. erefore it is comparable to the theoretical density of this type of solid in the absence of pores, whereas, particle density (also apparent or mercury density) is the density of the material including all pore system, open and closed pores. If both apparent and real densities are known, the difference of their inverses will provide the total pore volume per unit mass of the material. A third type of density of porous media is the bulk density. e bulk density includes all the pore system as well as interparticle voids between loose particles [28][29][30].
An important parameter in dealing with adsorption in porous media is the adsorbed phase volume fraction, . e adsorbed phase volume fraction is de�ned as the volume of the adsorbate phase, , per unit bulk volume of the adsorption bed, : . ( erefore, the mass of the solid media, sm , the mass of gas phase, , and the mass of the adsorbate phase, , can be expressed in terms of the total bed porosity, , and the adsorbate volume fraction, , as follows: where ( , ) is the density of the gas phase at the temperature and pressure of the adsorption reactor. e density of the adsorbate phase, ( ), is considered equal to the density of the liquid refrigerant at the temperature of the bed. Another important parameter is the adsorbate concentration ratio which is the mass of the adsorbate phase per unit mass of the solid adsorbent porous media. From (4), we can express as follows:

The Adsorption Equilibrium Model
e adsorption characteristics of a certain pair depend on the nature of the adsorbate, the nature of the adsorbent, the reactivity of the surface, the surface area, and the temperature and the pressure at which adsorption process takes place. When a solid surface is exposed to a gas, the molecules of the gas strike the surface of the solid. Some of the striking molecules attach and stick to the solid surface and as a consequence become adsorbed, while others rebound back. e rate of adsorption is large at the beginning of the process because the whole surface is uncovered. e adsorption rate continues to decrease since more and more of the solid surface is being covered by the adsorbate phase molecules. Meanwhile, the rate of desorption increases due to the escaping of the gas molecules from the covered surface, Figure 5. An equilibrium is approached when the rate of adsorption is equal to the rate of desorption. At this point of time, the gas-solid system is said to be in adsorption dynamic equilibrium because the number of molecules sticking to the surface is equal to the number of molecules rebounding from the surface [31]. e amount of adsorbate phase per unit mass of the solid adsorbent at the equilibrium condition is known as the adsorbate equilibrium concentration ratio and is denoted by * . It is noteworthy that the adsorbent concentration ratio for a certain adsorption pair at the equilibrium is a bivariant property which depends upon the pressure of the gas as well as the temperature at which the adsorption process takes place, that is * ( , ). e adsorption equilibrium theories are widely introduced in the literature [31][32][33]. One of the mostly used adsorption equilibrium models is the Dubinin-Astakhov (D-A) model [34][35][36]. e D-A equation has been developed mainly to describe the adsorption of gases by microporus adsorbents especially in case of activated carbon and is better than other models due to its thermodynamic character. According to Dubinin-Astakhov model, the adsorbate concentration ratio at equilibrium * is related to the adsorption process pressure and temperature by the following equation: where is the maximum adsorption capacity, and are constants which depend on the adsorbate-adsorbent pair, and ( ) and ( ) are the adsorbate density and the saturation pressure corresponding to the bed temperature.
Equation (6) can be rearranged in order to give the pressure in an explicit formula. Aer doing this arrangement, Also, the adsorbate volume fraction at the equilibrium condition, * , is found from the following relation:

Isosteric Heat of Adsorption and Desorption
When the gas or the vapor is physically adsorbed in the adsorbent solid media, two types of thermal energy are liberated: the latent heat of the evaporation of the adsorbed liquid and the additional heat of wetting which is due to the electrostatic forces involved in the physical adsorption process. e combined latent heat and the heat of wetting are transformed into sensible heat and dissipated into the adsorbent causing an increase in its temperature [37]. erefore, the process of adsorption is exothermic whereas the process of desorption is endothermic. From the thermodynamics point of view, the adsorption process of a gas or a vapor on a solid adsorbent is a spontaneous process during which the free energy of the system decreases. Moreover, when the gas molecules are adsorbed, the system entropy is decreased since the thermodynamic probability of identifying the microscopic state increases. erefore, it can be shown that the heat of adsorption should be negative and therefore the process is exothermic. e amount of heat required to adsorb or desorb a unit mass of the adsorbate is known as the isosteric heat of adsorption or desorption, respectively. It has been common in the literature to use the Clapeyron-Clausius equation to calculate the heat of adsorption. Despite the simplicity of Clapeyron-Clausius equation, it is applicable only for the ideal gases [38,39]. In the present analysis, the more general Clapeyron equation is used to determine the heat of adsorption or desorption. is isosteric heat per unit mass of adsorbate, sh , is calculated from the Clapeyron equation: By performing a partial differentiation for the pressure equation, (7), with respect to the temperature and substituting it to (9), a formula for the adsorption heat can be obtained in terms of pressure and temperature as follows: It is noteworthy that the difference between the gas phase and the adsorbate phase speci�c enthalpies is equal to the adsorption isosteric heat. us, the speci�c enthalpy of the adsorbed refrigerant can be found from the following equation: Moreover, the speci�c internal energy of the adsorbed phase is expressed by

First Law Analysis of the ACP System
In this section, the mathematical model representing the ACP system operation is developed. is model is based on a steady state �rst law differential analysis in order to calculate all types of energy interactions which take place in all of the system components and all of the processes of the theoretical cycle. e derivation of the mathematical model is based on the actual thermodynamic properties of the refrigerant rather than considering it to behave like a perfect gas in the vapor phase. erefore, either the refrigerant equation of state or the tabulated thermodynamic properties could be used in the numerical solution of the model. In the following subsections, the four process in the theoretical ACP cycle are analyzed �rst. Following that, the condenser, the throttling device, and the evaporator are analyzed.
6.1. Process 1 → 2. In this process, the adsorption bed is preheated at the constant highest isostere line, * max . For a given evaporator temperature, ev , the corresponding evaporator pressure, ev , is the saturation pressure of the refrigerant vapor corresponding to ev . Furthermore, the maximum adsorbate concentration ratio * max is determined from (6) corresponding to the evaporator pressure ev and the ambient temperature amb . e relation between the pressure and temperature along the path 1 → 2 is found from (7) at the constant concentration ratio which equals * max . Furthermore, the variation of the adsorbate volume fraction, * , along the highest concentration path can be determined from (8). During the present process, there is no desorption taking place and therefore, the mass of the gas phase as well as the mass of the adsorbate phase remains constant through the process path 1 → 2. e added energy during this process causes sensible heating of the adsorption bed constituents. A differential amount of energy, (1 → 2) , which is introduced to the adsorption reactor, causes a step change in the adsorption bed state. is in�nitesimal added energy can be represented as illustrated in Figure 6 and is given by where , (1 → 2) , and (1 → 2) are the differential sensible energies absorbed by the adsorption reactor metallic shell, the porous adsorbent media, the adsorbate phase, and the refrigerant vapor phase, respectively, as seen in Figure 6. e energy absorbed by the metallic cover of the adsorption bed causes a step temperature rise of and is given by Also, e thermal energy consumed in sensible heating of the adsorption bed porous media is calculated from (15) e energy required by the adsorbate phase in order to cause an in�nitesimal step change of its state is divided in two parts which has two contributions, as illustrated in Figure 6. e �rst contribution appears in the increase of the adsorbate internal energy. e second part of the energy absorbed by the adsorbate phase contributes in moving the boundary between the gas and the adsorbate phases and causes compression or expansion for the adsorbate phase. erefore, the differential amount of energy input to the adsorbate phase can be expressed as follows: where is given as e adsorbate mass, (1 → 2) , is constant during this process and is given by e thermal energy required by the refrigerant vapor phase is calculated from the following equation where (1 → 2) is the mass of the vapor phase which is constant during this process and is given by 6.2. Process 2 → 3. During this process the adsorption reactor acts as a generator and undergoes the desorption phase at the constant condenser pressure. For a given value of the condensation temperature, con , the condenser pressure is equal to the refrigerant saturation pressure corresponding to con . Moreover, the adsorbate concentration ratio decreases during this process from its maximum value, * max , at state 2 to its minimum value, * min , at state 3. Equation (6) is used to describe this variation along the process path 2 → 3 as a function of the temperature and with the pressure value set to con . Also, the variation of the adsorbate volume fraction, * , along path 2 → 3, is calculated from (8).
It is noteworthy that the total energy required to drive this process has two main effects. e �rst effect causes a sensible heating of all the adsorption bed constituents and a change in their internal energy. e second effect activates the desorption of the refrigerant from the adsorbent media and generates the gas phase. e in�nitesimal amount of energy which is required to cause a step change in the system state along the path 2 → 3 can be expressed as follows: e energies required for sensibly heating the reactor metallic casing and the adsorbent porous solid media are given by the following two equations, respectively: (22) During this isobaric desorption process, the mass of the adsorbate phase, (2 → 3) , is continuously decreasing since the refrigerant is being freed from the adsorption bed and �ows towards the condenser. �ence, the concentration ratio is continuously decreasing along the process path 2 → 3 as well. In this case and according to the conservation of energy principle, the differential amount of energy (2 → 3) which is added to the adsorbate phase, as illustrated in Figure 7, is expressed by where (2 → 3) is the step change in the total internal energy of the adsorbate refrigerant phase and is given by where (2 → 3) is the adsorbate phase mass in this process and is given by e second term in (23) is the in�nitesimal amount of energy which is carried by the desorbed vapor when its phase changes from the adsorbate phase to the gas phase. When a differential mass des changes from the adsorbed phase to the gas phase, the corresponding required energy is calculated from des = des sh .
e change in the mass of the adsorbate phase is due to the releasing of a differential amount from this mass into the gas phase. erefore, we can write des = − 2 → 3 .
Regarding the refrigerant vapor phase, its mass (2 → 3) is varying during the desorption process. By applying the conservation of energy principle on the refrigerant vapor phase, as shown in Figure 7, we get e �rst term in (28), (2 → 3) , is the change in the total internal energy of the vapor refrigerant which is calculated from e differential amount of energy which is transported from the adsorption reactor to the condenser is given by con = ℎ con , con , where con is the differential mass that leaves the adsorption bed and �ows towards the condenser. is in�nitesimal amount of mass equals the negative change of the total refrigerant mass content inside the adsorption reactor. Mathematically, it can be written as follows:

Process 3 → 4.
During this cooling process, the adsorbate concentration ratio remains constant at its minimum cyclic value, * min . e value of * min is calculated from (6) corresponding to the condenser pressure, con , and the highest cycle temperature reached, max . Additionally, the variation of the reactor pressure with the temperature along the path 3 → 4 is found from (7) at the constant concentration ratio, * min . Also, the variation of the adsorbate volume fraction, * , along this minimum concentration path can be determined from (8).
e energy which is lost from the adsorption bed to the ambient heat sink at amb is in a sensible form that results in the reduction of the bed temperature. In this process, there is no adsorption taking place and therefore, the mass of the gas phase as well as the adsorbate phase remains constant through this path.
e differential amount of energy lost from the bed to the ambient surroundings, (3 → 4) , during this stage, as represented in Figure 8, is given by where (3 → 4) , and (3 → 4) are the energies lost from the adsorption bed cover, the porous media, the adsorbate phase, and the gas phase, respectively. e energy lost from the adsorption bed metallic cover and the adsorption bed solid state are calculated from the following equations: By the same methodology which has been followed in process 1 → 2, the adsorbate phase loss in energy is due to the decrease of the adsorbate internal energy and the amount 8 ISRN Mechanical Engineering of energy extracted in the expansion or compression of the boundary. As a consequence, the total amount of energy output from the adsorbate refrigerant to cause a step change in its state is express as (3 → 4) = − * − (3 → 4) , (34) where (3 → 4) is the constant adsorbate phase mass in the present process and is calculated from Also, the thermal energy losses from the refrigerant vapor phase during the current process are calculated by the following expression: where (3 → 4) is the mass of the gas phase which is determined from the following equation: 6.4. Process 4 → 1. In this process, the adsorption bed is being used as an adsorber while it is being cooled at the constant evaporator pressure. e mass of the adsorbate phase inside the reactor (4 → 1) is continuously increasing as more refrigerant vapor leaves the evaporator and enters the adsorption bed. As a consequence, the concentration ratio increases also during this process from its minimum value * min to its maximum value * max . e bed temperature decreases from the adsorption temperature ads to the ambient temperature at the end of the process path 4 → 1. For a given value of ev the evaporator pressure equals the refrigerant saturation pressure corresponding to ev . Equation (6) is used to describe the variation of * as a function of the bed temperature and ev . Also, the variation of * along the path 4 → 1 is calculated from (8).
Energy removed from the bed during the current process consists of the reactor sensible cooling energy in addition to the adsorption energy. As seen in Figure 9, the in�nitesimal total amount of energy removed from the reactor to cause a step change in the system state along the path 4 → 1 is given in a differential form as follows: Energies required for sensibly cooling the bed metallic cover and the adsorbent porous solid media are given by the following two equations, respectively: According to the �rst law of thermodynamics, the differential amount of energy, (4 → 1) , which is taken from the refrigerant adsorbate phase in order to cause a step change of its state along the process path is expressed by where ℎ ev is the refrigerant vapor saturation enthalpy corresponding to the evaporator pressure and (4 → 1) is given by e differential mass ads which is transformed from the vapor phase to the adsorbate phase is given by e corresponding in�nitesimal amount of the adsorption heat ads is calculated from ads = ads sh .
e change in the adsorbed mass 4 → 1 is given as a function of the concentration ratio by Regarding the refrigerant gas phase, from the conservation of energy principle and energy �ow diagram which is illustrated in Figure 9 we get where (4 → 1) is the change in the total internal energy of the vapor refrigerant and is given by where (4 → 1) is expressed as e differential mass that leaves the evaporator and enters the adsorption bed, ev , equals the change of the total refrigerant mass content inside the adsorption reactor. Mathematically, it can be written as follows: 6.5. e Condenser. e refrigerant vapor enters the condenser as soon as it desorbs from the adsorption bed. In the condenser, the refrigerant gas is being desuperheated �rst by removing sensible energy from the superheated vapor across a driving temperature difference of − amb . When the refrigerant vapor reaches the saturated vapor state, it starts to condense and the latent energy of condensation is rejected to the ambient heat sink across a difference of con − amb . Aer the vapor condenses to liquid at a temperature of con , it is being subcooled to the ambient temperature by rejecting energy to the ambient surroundings. e differential condensation energy of con is then given by con = con ℎ con , − ℎ amb , where ℎ ( amb ) is the refrigerant liquid speci�c enthalpy at the ambient temperature. F 9: Energy �ow diagram for the adsorption reactor during process 4 → 1.
6.6. e rottling Process. In the throttling process, the refrigerant conserves its total enthalpy. erefore, the wetness fraction of the refrigerant aer the throttling process is given by where ℎ ( ev ) is the refrigerant saturated liquid speci�c enthalpy corresponding to the evaporator temperatures. e term ( ev ) is the latent energy of the vaporization of the refrigerant at the evaporator temperature. e fraction of liquid that evaporates in this process, , causes a selfcooling of the remaining liquid fraction, , and consequently reduces its temperature from the ambient temperature to the evaporator temperature. 6.7. e Evaporator. In the evaporator, cold production takes place by the evaporation of the saturated liquid refrigerant. e differential cooling effect, ev , associated with a differential amount of refrigerant liquid, ev , is calculated from ev ev ev , where ev is the mass of the refrigerant vapor which is coming from the evaporator and is being adsorbed in the adsorption reactor, (48).

e System Performance.
In general, the efficiency of any cooling system is measured by the coefficient of performance, (COP) , which is a measure of how efficiently the input energy is transformed to a useful output cooling effect. e (COP) is given by In this equation, the refrigeration effect ev is given by and the required input energy to drive the system in is given by e effective refrigerant mass, eff , is the total mass of the desorbed refrigerant from the adsorption reactor which is circulated through the condenser and the evaporator. is refrigerant mass is responsible for providing the cooling effect inside the evaporator and is given by the following equation: Another important parameter which is used in the present study is the refrigerant mass utilization efficiency, rmu . is parameter is de�ned as the ratio between the effective refrigerant mass, eff , to the total refrigerant mass which is required to charge the system. e refrigerant mass utilization efficiency is de�ned by the following equation: When the cooling system is used for freezing applications, the total amount of the ice produced from the ACP system per cycle is given by where is the initial water temperature and , ice , and are the water speci�c heat, the ice speci�c heat, and the latent heat of ice fusion, respectively. If the cooling system is used as a water chiller, the daily production of chilled water is given by ev ev , ev ≥ 273 K.

The Case Study
e case investigated in the present study is an ice maker and produces ice at a temperature of −5 ∘ C. e system operates with activated carbon-methanol as the working pair. e liquid and vapor thermodynamic properties of methanol have been calculated from the methanol equation of state reported by the IUPAC [40]. e base line parameters used in this case study are summarized in Table 1. Moreover, the effect of metallic casing of the adsorption bed is not taken into account in this studied case.

Results and Discussions of the Case Study
Based on the introduced thermodynamic analysis in Section 6 and the case study described in Section 7, a computer program is developed to solve the mathematical model of the ACP system. An energy balance for the whole system including the bed, the evaporator, and the condenser is performed as a numerical experimentation of the developed computer code. It is found that the total energy input to the cycle, add = 1 → 2 + 2 → 3 + ev , equals 318.634 MJ, whereas, the total energy output from the system, rej = 3 → 4 + 4 → 1 + con , equals 318.615 MJ. erefore, the relative error, 100 × ( add − rej )/ add , in the calculations is nearly 0.006% which is an acceptable numerical error and validates the calculation accuracy.
Generally, it is found that the system attains a refrigeration COP of 0.616 with a refrigeration effect produced inside the evaporator estimated to 12.15 MJ per cycle. is cooling effect corresponds to ice production of 27 kg per cycle at −5 ∘ C from water at a source temperature of 25 ∘ C. erefore, every 1 kg of activated carbon inside the adsorption reactor produces ice mass of 0.471 kg per cycle at a temperature of −5 ∘ C. Figure 10 plots the relation between the adsorption bed pressure and temperature for all of the four cycle processes. e adsorption bed pressure rises from the evaporator pressure at 2.91 kPa to the highest cycle pressure at the condenser, 27.79 kPa, during the preheating process. It can be also seen that the adsorbent starts the generation process at a temperature of about 74.65 ∘ C. en the bed pressure remains steady during the generation process till the maximum cycle temperature, 120 ∘ C, is approached. Figure 11 illustrates the variation of the adsorbate phase concentration ratio with the adsorption bed temperature along the cycle. From this plot, the adsorbate concentration ratio varies between its maximum value of 0.24 kg/kg to its minimum value of 0.051 kg/kg during the cycle. is corresponds to a maximum and a minimum contents of an adsorbed refrigerant phase 30  mass of 13.76 kg and 2.97 kg, respectively. As a result, the total refrigerant effective mass which is circulated through the system is 10.79 kg. In other words, the refrigerant mass utilization efficiency is estimated to be 78.43%. e development of the adsorbate volume fraction with the bed temperature is demonstrated in Figure 12. As noticed from this �gure, the adsorbate volume fraction is continuously varying throughout the cycle operation. Slight changes of the adsorbate refrigerant volume fraction are noticed during both of the two isosteric processes. In the isosteric preheating phase, the volume fraction increases with temperature due to the decrease in adsorbed refrigerant density with increasing temperature. It attains its highest cyclic value at the end of this phase, about 0.186, whereas, the decreasing trend in the adsorbed phase volume fraction during the isosteric cooling process is due to the increase in its density with the decreasing bed temperature. It attains its minimum value at the end of isosteric cooling process, about 0.04. Figure 13 presents the evolution of the refrigerant desorbed mass from the adsorption reactor during the isobaric process 2 → 3. It is noticed that the rate of the generated refrigerant has its maximum value at the beginning of the desorption process, when the bed temperature reaches the value of the generation temperature. is rate, as depicted in the plot, is continuously decreasing with increasing the bed temperature and it approaches the minimum value at the end of the desorption phase, at the maximum cycle temperature and the minimum concentration. e total desorbed refrigerant vapor mass at the end of this process equals 10.79 kg. Furthermore, the adsorbed refrigerant mass evolution as a function of the adsorption reactor pressure during process 4 → 1 is demonstrated in Figure 14.
e cumulative sum of the energy added to the adsorption reactor per unit mass of the solid adsorbent is demonstrated in Figure 15 versus the bed temperature. e total amount of the speci�c energy required to activate the adsorption bed is estimated to 343.33 kJ/kg per cycle. A fraction    F 16: �e cumulative sum of speci�c energy added to the bed and the speci�c isosteric energy during the desorption process. from this energy, about 16.15%, is needed to bring the reactor up to the generation temperature, whereas, the constant pressure desorption process requires the remaining part of the total energy added, 83.84%. As noticed from the plot in Figure 15, there is a more rapid increase in the temperature during the preheating process than in the desorption process with respect to the speci�c driving energy. �e reason is that all of the energy input to the adsorption bed during the process 1 → 2 is consumed only in the sensible heating of the bed, whereas, in process 2 → 3, a large portion of energy is consumed in the refrigerant generation process. Figure 16 illustrates a comparison of the speci�c total energy added to the adsorption bed and the speci�c isosteric desorption energy during the desorption process. As provided in the �gure, most of the input energy is being used as activating energy of desorption, about 92.56%. Similarly, the cumulative sum of the speci�c energy removed from the adsorption reactor during both the isosteric cooling as well as the adsorption process is demonstrated in Figure 17. e total energy removed from the bed during this cooling phase is found to be about 315.97 kJ/kg per cycle. About 13.21% of this energy is lost to the ambient during the constant concentration cooling process in a form of sensible cooling. e remaining larger fraction of this energy is then rejected from the reactor to the surroundings during the adsorption process.

Conclusion
A simple thermodynamic differential analysis for adsorption refrigeration systems is introduced in the present work. e analysis is based on the energy conservation principle and the D-A adsorption equilibrium model. e case studied is an ice machine working with activated carbon-methanol as the working pair.