Applied Energy 90 (2012) 280–287

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Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

A non-uniform pressure and transient boundary condition based dynamicmodeling of the adsorption process of an adsorption refrigeration tube

Y.L. Zhao, Eric Hu ⇑, Antoni BlazewiczSchool of Mechanical Engineering, The University of Adelaide, SA 5005, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 September 2010Received in revised form 18 December 2010Accepted 20 December 2010Available online 14 January 2011

Keywords:Non-uniform pressureTransient boundary conditionDriving forceART

0306-2619/$ - see front matter Crown Copyright � 2doi:10.1016/j.apenergy.2010.12.062

⇑ Corresponding author. Tel.: +61 83130545.E-mail address: eric.hu@adelaide.edu.au (E. Hu).

The adsorption refrigeration tube (ART) has drawn increasing attention because of its advantages of nomoving parts, compact structure and use of low grade heat as the heat source. A new design of ARTwas developed, in which activated carbon–methanol was selected as the working pair for either refriger-ation or air-conditioning purposes. Based on the typical configuration of the generator of the ART, adynamic mathematical model was developed under the non-uniform pressure assumption and theintroduction of a transient boundary condition of diffusion equation. Moreover, experiments were con-ducted for validating the model. The experimental data and numerical results were compared in termsof temperature development inside the carbon bed, with the transient and two popular simplified bound-ary conditions of vapor density respectively. The comparison shows that the transient boundary condi-tion improves accuracy of the model and more importantly it is capable of reflecting the dynamic shiftof dominant driving forces of the adsorption process inside the generator, i.e. shifting to temperaturedriving gradually from diffusion driving.

Crown Copyright � 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Adsorption refrigeration has been studied extensively since thenineteen eighties for its low-grade energy requirement andenvironmentally-friendly refrigerant [1–6]. The coefficient of per-formance (COP) and specific cooling performance (SCP) of adsorp-tion refrigeration systems have been improved in recent years dueto system optimization with advanced thermal cycles [4,5,7,8].However, the optimization increases the complexity of the system,which in turn leads to a decrease in performance reliability of theadsorption refrigeration systems.

Over the last ten years, some researchers studied ART that onlyconsists of integrated adsorber, condenser and evaporator, whichwas aimed at improving system reliability through multi-ARTintegration to achieve continuous and large scale refrigerationcapacity. Tamaninot-Telto and Critoph [9], in 1993, initially pro-posed a cooling tube with monolithic carbon–ammonia as workingpair, in which the monolithic carbon was made in situ within thegenerator in order to reduce contact heat transfer resistance be-tween the generator wall and the adsorbent. Zhao et al. [10] pro-posed a solar powered ART, in which vacuum glass tube wasemployed to contain the adsorbent bed and adsorb solar energyefficiently. Wang and Zhang [11] developed a fin-type adsorption

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cooling tube that finned tube was adopted to improve heat transferwithin the adsorbent bed.

On the other hand, various mathematical models describing theadsorption process were proposed. Some mathematical modelsused lumped parameter method based on the uniform pressureassumption to study the thermal performance [11–13]. However,this kind of model may not represent the accurate system perfor-mance, especially when the thickness of the adsorbent bed exceedsa certain value which results in the uneven temperature profilebecoming significant. Moreover, the assumption of uniform pres-sure may be not accurate as the pressure gradient within theadsorbent bed would act as the principal driving force of adsorp-tion at the very beginning of an adsorption process. For the sideof non-uniform pressure based model, although these kinds ofmodels have been proposed and studied, one important boundarycondition of the diffusion equation (or mass transfer equation), i.e.gas density from the evaporator, all applied simplified constantvalues [14,15]. This simplification is not accurate to some extentsince the dynamic shift of adsorption driving forces cannot be pre-sented under the circumstances. In fact, the shift of driving forcesat the very beginning of the adsorption process, i.e. from diffusiondriving (gradient of gas density) to temperature driving, plays animportant role in following adsorbent temperature developmentand adsorption performance.

The aim of this study is to establish a non-uniform pressure andtransient vapor density based model to investigate the adsorptionprocess of a real adsorption refrigeration cycle. Experiments were

ights reserved.

Nomenclature

x concentration of adsorbed methanol, kg kg�1

t time, sr radial coordinateT temperature, KP pressure, PaM mass, kg or molar mass, kg mol�1

H heat of adsorption, J kg�1

R0 specific gas constant, J mol�1 K�1

Ri inside radius of carbon column, mRo outside radius of carbon column, mI serial number of carbon layersN number of total layers dividedC gas density, mol m�3

cp specific heat, J kg�1 K�1

_Q internal heat source, W m�3

Ded effective diffusion coefficient of porous media, m2 s�1

Dk–m diffusivity of Knudsen and molecular diffusion, m2 s�1

Dk Knudsen diffusivity, m2 s�1

Dm molecular diffusivity, m2 s�1

D coefficient of D–A equationn coefficient of D–A equation

Greek symbolsq density, kg m�3

k thermal conductivity, W m�1 K�1

e total porositya coefficient of the transient boundary condition

Subscriptss solid adsorbentl liquid phaseg gaseous phasec condensinge evaporating or gas from evaporatort transientam ambientcb activated carboneb equivalent property of the adsorbent bedsat saturation statusini initial conditionmax maximum value in studied cyclemin minimum value in studied cycle1 2 3 4 200 400 30 40 status of relevant processes

Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287 281

conducted for validating the model. Moreover, numerical studies ofthe model under two constant vapor densities were conducted andcompared with the experimental data.

2. Configuration description

The ART proposed is comprised of a generator, a condenser, anevaporator, insulation sectors, a circulation pipe, a water jacket

Fig. 1. Cross-section of the ART configuration.

and a control valve, as shown in Fig. 1. Activated carbon (Calgonactivated carbon WS-480) and methanol are selected as the work-ing pair for either refrigeration or air-conditioning purposes. In thegenerator, a porous tube is placed centrally in the carbon as themass transfer channel for the methanol vapor.

This ART has two major advantages, compared to Critoph’s andother researchers’ models [16,17]. First, the condenser and evapo-rator are separated and located at two ends, which thoroughlyhelps to avoid heat losses and heat disturbances. Second, the valveon the external circulation pipe provides convenience to controlthe evaporating pressure, which could enhance the production ofuseful refrigeration capacity.

3. Working principle

A non-valve controlled adsorption refrigeration cycle normallyincludes two processes, which are heating-desorption processand cooling-adsorption process (purposes 1–200–3 and 3–400–1,shown in Fig. 2). However, a valve is usually added to control theeffective evaporating temperature by connecting the evaporator

Fig. 2. The Clapeyron diagram of the valve controlled and non-valve controlledadsorption refrigeration cycles.

282 Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287

to the generator until the pressure in the generator has dropped tothe required pressure. In such valve controlled cycle, the entire cy-cle has four processes, which are isosteric heating, isobaric desorp-tion, isosteric cooling and isobaric adsorption (i.e. processes 1–2,2–3, 3–4, 4–1).

The overall working processes of the ART are next described. Inthe first half cycle (i.e. process 1–2–3), the generator is heated upby circulating heat transfer fluid in the water jacket of the ART.At this moment, the refrigerant (methanol) is desorbed from thecarbon bed and the gaseous methanol via the porous tube risesto the condenser and then flows to the evaporator via the circula-tion pipe after condensing. When this process completes, heattransfer fluid is switched to cooling fluid to start the second halfcycle (i.e. 3–4–1). When the pressure in the generator drops tothe designed evaporating pressure, valve is turned on to connectthe evaporator to the generator to generate the refrigeration pro-cess (4–1) in which evaporating pressure drops to and keeps atthe designed evaporating pressure gradually (process 30–40). Theevaporated gaseous refrigerant then goes back to the adsorbentbed and the adsorption heat is removed continuously by the out-side cooling fluid.

In this study, the model is established for the isobaric adsorp-tion process (process 4–1), which is a key process that refrigerationeffect is generated.

4. Modeling

4.1. Model assumptions

In this study, the heat and mass transfer processes in the gener-ator is considered to be one dimensional (radial). Fig. 3 shows thesimplified one dimensional scheme, in which uniform grid (layer)is adopted. Besides the one dimensional assumption, the othermodeling assumptions are made as follows:

(1) The temperature of heat transfer fluid is considered asconstant.

(2) The particles of activated carbon are uniformly distributedand have identical adsorptive property.

Fig. 3. The grid scheme of the one-dimensional model.

(3) Thermal equilibrium between adsorbed liquid methanol andsolid carbon in the generator is reached locally.

(4) The methanol vapor behaves like the ideal gas.(5) Adsorption equilibrium is reached locally between liquid

methanol and carbon.

4.2. Governing equations

4.2.1. Adsorption state equationThe Dubinin–Astakov equation (D–A) [16] is used as the state

equation, which can be rewritten as:

x ¼ x0 exp �D T lnPsat

P

� �n� �ð1Þ

where x0 is the maximum possible concentration of adsorbate; T isthe local temperature of solid adsorbent; Psat is the saturation pres-sure at local temperature T; P is the local adsorption pressure; D andn are the constants determined experimentally by the adsorbate/adsorbent pair involved.

4.2.2. Energy governing equationAdsorption process is considered as coupled heat transfer and

gas diffusion processes within the adsorbent bed. The energy con-servation equation in cylindrical coordinates is:

ðqcpÞeb@T@t¼ keb

1r@

@rr@T@r

� �þ _Q ð2Þ

where (qcp)eb denotes the total heat capacity of the control volumeof the adsorbent bed; keb refers equivalent thermal conductivity ofthe adsorbent bed and _Q represents internal heat source. Total heatcapacity (qcp)eb and internal heat source can be calculated as fol-lows [18]:

ðqcpÞeb ¼ qscps þ eqgcpg þ xqscpl ð3Þ

_Q ¼ ð1� eÞ � H � qs �@x@t

ð4Þ

4.2.3. Diffusion equationDuring the adsorption process, adsorbates (i.e. methanol vapor)

from the evaporator travel towards the inner positions of the car-bon layers resulted from diffusion as well as adsorption effects. Thedriving forces of the methanol vapor transfer, including the innermass transfer within micropores of carbon particles and externalmass transfer within the void space of the bed, are the local pres-sure gradient and the local methanol density difference. When tak-ing porosity of the carbon bed into account, the equation ofmethanol diffusion can be written as [19]:

e@C@tþ ð1� eÞ � qs

@ðx=MÞ@t

¼ Ded � e1r@

@rr@C@r

� �ð5Þ

where C is the local density of gaseous methanol; x is the concentra-tion of adsorbed methanol (in micropores of the carbon); M is themolar mass of methanol; e is the total porosity of the carbon bedand Ded is the effective diffusion coefficient.

The local density of gaseous methanol is evaluated with theideal gas law, Eq. (6), in which R0 is the gas constant. The temper-ature of gaseous methanol in Eq. (6) is approximately evaluatedwith the correlation of methanol saturation vapor pressure andcorresponding temperature, Eq. (7) [12].

C ¼ P=R0T ð6Þln P ¼ 18:587� 3626:55=ðT � 34:29Þ ð7Þ

For porous media with random twisted and interconnected pores,Ded can be calculated approximately using Eq. (8), in which e isthe total porosity of the bed and Dk–m is the diffusion coefficient

Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287 283

that depends on double effects resulted from Knudsen diffusion(Dk) and molecular diffusion (Dm). Expressions for calculatingthe diffusion coefficient Dk–m and corresponding Knudsen diffu-sion and molecular diffusion are given in Eqs. (9)–(11) respec-tively [20].

Ded ¼ e2Dk—m ð8Þ1

Dk—m¼ 1

Dkþ 1

Dmð9Þ

Dk ¼ 4850dporeðTMÞ1=2 ð10Þ

Dm ¼2T3=2

3p3=2r2Pðj

3Na

MÞ1=2 ð11Þ

In above two equations, i.e. Eqs. (10) and (11), dpore is the porediameter of activated carbon; T is the temperature of gas methanol;M is the molar mass of methanol; r is the Lennard-Jones diameterof the spherical molecules of gas methanol; P is the local adsorptionpressure; j is the Boltzmann constant; Na is the Avogadro’s number[20].

4.3. Initial and boundary conditions

4.3.1. Initial conditionsThe adsorption process numerically studied is assumed start-

ing from point 4 (see Fig. 2), which could be practically achievedby holding system at the point 4 status long enough, with circu-lating cooling water of temperature T4. The temperature of theactivated carbon and concentration of methanol at point 4 areevenly profiled cross the generator, whose values can be calcu-lated using Eq. (1) associated with Pe and xmin. Therefore, theinitial conditions of the above governing equations (Eqs. (2)and (5)) are:

T ¼ 0; Tini ¼ T4 ð12ÞT ¼ 0; Cini ¼ C4 ð13Þ

Table 1Parameters for the simulation.

Symbol Value Unit Symbol Value Unit

qs 420 kg m�3 R0 8314 J mol�1 K�1

qg 0.8 kg m�3 e 0.72C30 6.6 mol m�3 Cps 930 J kg�1 K�1

C40 3 mol m�3 Cpl 2530 J kg�1 K�1

C4 2.5 mol m�3 Cpg 1370 J kg�1 K�1

Tam 298 K L 1,102,000 J kg�1

T4 338 K Ro 0.105 mPc 16,360 Pa Ri 0.055 mPe 7140 Pa keb 0.175

(exper.)W m�1 K�1

x0 0.269 (exper.) kg kg�1 dpore 10(assumed)

Å

n 1.781 (exper.) N 500D 9.08e�6

(exper.)a 0.005

(exper.)

4.3.2. Boundary conditionsDirichlet and Neumann boundary conditions [21] are used

respectively for the heat transfer equation (Eq. (2)), which are:

Tjr¼Ro ¼ Tc ð14Þ@T@r

����r¼Ri

¼ 0 ð15Þ

For the diffusion equation, Eq. (5), boundary condition at Ro appliesmass insulation while that at Ri equals to the vapor density from theevaporator. Therefore two boundary conditions can be written as:

@C@r

����r¼Ro

¼ 0 ð16Þ

Cjr¼Ri ¼ Ce ð17Þ

The value of Ce in Eq. (17) depends on the vapor pressure and itstemperature as shown in Eqs. (6) and (7).

In previous models developed by other researchers [13–15], Ce

(or system pressure Pe) was assumed to be a constant that equalsto C40 (or P4) in the process 4–1. However, the system pressureand the density of methanol vapor from the evaporator are notconstants, especially when the evaporator is just connected tothe generator (i.e. status at point 4 in Fig. 2). There is a (quick) pro-cess for the pressure and corresponding methanol vapor density tostabilize from Pc and C30 to Pe and C40 , which results in a transientcondition of vapor density for the diffusion effect. After this begin-ning stage, the pressure and the density of methanol vapor in theevaporator reach and remain at the stable values. To present this

process, a concept of the transient system pressure (Pt) is intro-duced to reflect the corresponding transient vapor density (Ct).The transient pressure (Pt) is defined as Eq. (18):

Pt ¼Pc � Pe

1þ atþ Pe ð18Þ

where Pc and Pe denote the pressure in the evaporator at statuspoint 30 and 40 (see Fig. 2); t is time; a is a coefficient that reflectsthe time taken by the pressure dropping to Pe from Pc.

Since the pressure falling off process is relatively quick com-pared to the time needed for the whole adsorption process, a is inthe range 0 < a 6 1, which means that the bigger value a is, theshorter time needs for the evaporating pressure dropping. Whena = 1, the evaporating pressure falls off to the designed evaporatingpressure and is dominated by the generator in dozens of seconds.The value of a depends on the practical operating situation and sys-tem inner structure, which can be an empirical value and obtainedexperimentally. Substituting the defined transient pressure into Eq.(6), the transient density of methanol vapor (Ct) is then:

Ct ¼ Pt=R0Tt ð19Þ

4.4. Numerical solutions

The operating conditions, Tc = 30 �C, T40 = Te = 10 �C,T30 = Tam = 25 �C and T4 = 65 �C were chosen for the numericalstudy. The data used in numerical calculations is given in Table1. The thermal properties of methanol and physical property ofcarbon were obtained from literature [22,23]. The adsorptiveproperties of the carbon-methanol working pair were obtainedexperimentally in our lab.

The set of governing equations (i.e., Eqs. (1), (2), and (5)) asso-ciated with corresponding initial and boundary conditions weresolved by finite-volume technique [24] in Matlab environment.For the spatial grids, as schematic diagram shown in Fig. 3, theadsorbent bed was divided into a number of elementary layers/rings in radial direction and each elementary layer was smallenough to be considered having the uniform temperature andpressure within each layer. In the grid scheme, Ri is the inside ra-dius of the carbon column while Ro is the outside radius of the car-bon column; Dr represents the uniform thickness of each layer inspatial discretization. Thus, each carbon layer has an even thick-ness of (Ro � Ri)/N (N denotes the number of total layers).

The central difference and forward difference approaches wereadopted to convert the differential equations into a set of linearequations. The linear equations were solved by Thomas algorithmand Gauss–Seidel iteration technique [24].

284 Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287

5. Experiments

The schematic diagram and photo of the test rig are shown inFig. 4a and b. The configuration of the experimental prototype ofthe ART was modified for the testing purpose, which includes spin-dled evaporator, enlarged generator and non-compact connection.Water baths were used as the simulated heating source and cool-ing source. Temperature was measured by the directly-insertedthermocouple (T type). The thermocouples and the data loggerwere calibrated separately by Fluke calibrators. The volume ofthe evaporator was calibrated using a volume calibration syringe.

The main procedures of the experiments conducted include thefollowing steps:

(1) Heat and degas the carbon bed simultaneously for twelvehours, and then isolate the generator by closing valve V2.

(2) Fill methanol into the evaporator and maintain evaporator atT30 . The amount filled in the evaporator is slightly higherthan the amount xmax �Mcb for compensating the vacuum-ing process caused evaporation.

(3) Charge methanol into the carbon bed. The amount charged isM4 = xmin �Mcb.

(4) Set the status of the generator to point 4 (see Fig. 2) via cir-culating water (at T4) in water jacket until the carbon bedtemperature reaches T4 evenly.

(5) Connect the generator and evaporator and in the meantimecirculate cooling water (at Tc) in water jacket to start theadsorption process.

(6) Record temperatures and pressures.

6. Results and discussion

Fig. 5 shows the comparison of pressure development amongexperimental data and results calculated using the transient pres-sure Pt with a = 0.001, a = 0.005 and a = 0.009 respectively.

Fig. 4. Schematic diagram and photo of the experime

The experimental data shows that evaporating pressure in theevaporator takes about 3000 s to reach the designed evaporatingpressure. In other words, system pressure is dominated by the gen-erator after 3000 s. Moreover, this experimental data also indicatesthat the principal driving force of adsorption is shifted to temper-ature driving over this period (about 3000 s). Because in the fol-lowing process only the decrease of adsorbent temperature couldlead to concentration variation (i.e. adsorption) once the pressurethroughout the adsorbent bed is almost stabilized, according toD–A equation (Eq. (1)).

For three values of a, a = 0.005 gives the best agreement of cal-culated data and experimental data. Therefore, a = 0.005 is selectedfor this particular experimental setting up, for further simulation.

Fig. 6 presents three typical methanol vapor density changingtrends assumed in following simulation models. They are (1) tran-sient Ct with a = 0.005 proposed in current model, (2) setting Ce asa constant that equals to C30, and 3) setting Ce as a constant thatequals to C40 which was used in almost all previous models[14,15,23]. Fig. 6 indicates that with the transient vapor densityassumption, after some time (about 3000 s), value of Ce would bestable towards Ce ¼ C40 .

Fig. 7 shows the experimental data and simulated results oftemperature development under three Ce assumptions i.e. (a)Ce = C30, (b) Ce = C40 and (c) Ce = Ct. It can be seen from the group fig-ures of Fig. 7 that the calculated temperature values under theassumption Ce = Ct (with a = 0.005) has the best agreement withthe experimental data. In other words, introduction of transientboundary condition of vapor density (Ct) makes the simulation ofthe process more accurate, especially at the beginning stage, com-pared to that simulated with Ce = C30 and Ce = C40.

In Fig. 7a, when Ce ¼ C30 was chosen as the boundary conditionfor diffusion equation, the simulated temperature of the relativeinside positions (layers) jumps dramatically, up to 25 �C, at thevery beginning of the process, before starting to decrease. This re-sult indicates there is a strong adsorption process happened in the

ntal setup: (a) schematic diagram and (b) photo.

Fig. 6. Transient vapor density Ct and static densities C30 and C40 .

Fig. 5. Comparison of pressure development among experimental data and results calculated with the transient pressure Pt with a = 0.001 and a = 0.005, a = 0.009respectively.

Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287 285

inner layers, i.e. a lot of adsorption heat generated. However, thesimulated values are much higher than the experimental data,reaching about 10 �C difference in the inner positions. This devia-tion generated is due to the inappropriate boundary conditionCe = C30 because this high vapor density cannot exist over the entireadsorption process, but reaching the low level in about 1500 s,shown in Fig. 6.

In Fig. 7b, when using the popular boundary condition Ce = C40,the simulated temperature of inner layers rises slightly, about4 �C, and then starts to decrease. But for the outer position (e.g.r = 98 mm), the temperature decreases directly without a quicktemperature jump. Comparing with the case at Ce = C30, usingboundary condition Ce = C40 may bring a better agreement withexperimental data. However, the temperature of outer layer andthe temperature jump of the inner layers are lower than the exper-imental values, which resulted from missing the strong diffusionforce several hundred seconds before the vapor density reachingand keeping at the stable value. In other words, if the process of

transient pressure was neglected, the phenomenon of the temper-ature jump could not be well predicted, such as the numericalstudy in literature [14].

Fig. 7c shows the experimental and simulated results of tem-perature development under the introduced transient boundarycondition Ce = Ct with a = 0.005. There is a very good agreement be-tween the calculated data and the experimental data. Therefore themodel developed is valid in terms of predicting the temperaturedevelopment. From Fig. 7, it can also be seen that the simulatedtemperature increases before starting to decrease, which agreeswith the experimental data well. It indicates that the transientboundary condition introduced is able to present the shift of thedominant factors of the adsorption driving force, which is shiftingto temperature driving from vapor density gradient caused diffu-sion driving with the diminishing of the gradient of vapor density.

Fig. 8 shows the dynamic adsorption amount calculated withCe = C40 and Ce = Ct. The adsorption amount calculated with Ce = C40

is about 20% lower than that calculated with the transient boundary

Fig. 7. Comparison of experimental data and modeling results of temperature variation at three positions with static and transient boundary conditions Ce: (a) Ce ¼ C30 , (b)Ce ¼ C40 and (c) Ce = Ct. (Makers represent experimental data and solid lines denote numerical results.)

286 Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287

Fig. 8. Dynamic adsorbed amount of methanol per meter ART of the studied base-case.

Y.L. Zhao et al. / Applied Energy 90 (2012) 280–287 287

condition (Ce = Ct) in the first 2000 s, which is resulted from the ne-glect of the strong diffusion effect at the very beginning moment.However, with time goes on, the gap tends to reduce because thevalue of Ct approaches the constant boundary condition C30 as well.

7. Conclusions

A dynamic model describing the adsorption process in the car-bon/methanol refrigeration tube was established based on coupleddiffusion driving and temperature driving adsorption mechanism.A concept of transient boundary i.e. transient pressure (and tran-sient vapor density) was introduced for the first time into the mod-el. The model was preliminary validated in terms of adsorbenttemperature development after the determination of a value usingexperimental data.

The introduced transient boundary condition provides an effec-tive way to describe the shift of dominant driving forces of theadsorption process, which is very helpful for improving the accu-racy of the model.

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