ORIGINAL

Wen Wang Æ Ruzhu Wang

Investigation of non-equilibrium adsorption characterin solid adsorption refrigeration cycle

Received: 2 July 2002 / Published online: 31 March 2005� Springer-Verlag 2005

Abstract The adsorber is the key element in anadsorption refrigeration system. Its characteristic cou-pled with heat and mass transfer and adsorption do-minats the cycle performance. In general, the velocity ofadsorption in adsorber is not faster than that of heattransfer, that is, the adsorption rate does not reach theequilibrium value at each status point in real operation.The feature of non-equilibrium adsorption results ofapparent different characteristic on adsorption refriger-ation performance comparing with the case of equilib-rium adsorption model. Based on numerical simulation,both models with equilibrium and non-equilibriumadsorption are compared with each other. The simula-tions condition includes various surface diffusing veloc-ity coefficients. This paper shows the characteristic oftemperature and adsorbed mass rate distribution in ad-sorber and some thermodynamic analysis on adsorptionrefrigeration cycle. It could be observed that non-equi-librium adsorption should be taken into account inadsorption refrigeration particularly for short cycle timeand vacuum system.

Keywords Adsorption Æ Refrigeration ÆNon-equilibrium Æ Heat transfer Æ Cycle

1 Introduction

In porous media, adsorption process is controled byboth adsorbate transportation and diffusion in adsor-bent and inner reaction with adsorbent, between the twoitems, the transportation of gaseous adsorbate in thetunnels of micro pores and the diffusion on its surfaceare more dominated than internal reaction in rare

pressure [1]. Therefore, the macro velocity of adsorption(and desorption) actually relies on the mass transportprocess, and mainly on the process in micro-pores,which is roughly determined by the states of adsorptionpairs, such as: the temperature of adsorbent, density ofadsorbate, coverage of adsorbent, and so on. Therefore,actual adsorption process is not the process only domi-nated by the temperature of adsorbent and the pressureof adsorbate, it is involved with the mass transport anddiffusion in micro-pores and adsorption history also [2].Moreover, the adsorption rate does not reach the equi-librium value at each status point in real operation, theactual process of adsorption/desorption is of non-equi-librium.

In adsorption refrigeration system, the adsorber isserved as a compressor to drive the refrigerant cycling,and itself is driving by heat source outside. The perfor-mance of adsorber is affected by three items, heattransfer, mass transfer and diffusion, and inner reaction,its status need turn over and over between adsorptionand desorption frequently. Therefore, the process in anadsorber should be of non-equilibrium. However, mostdata of adsorption character are got on the situation ofequilibrium adsorption. In case of application inadsorption refrigeration, it should be supposed that theadsorption velocity to reach equilibrium on adsorptionis faster than the heat transfer in adsorber. It wouldbring deviation from reality unavoidably.

Sokoda and Suzuki [3] considered the influence ofdiffusion on the micro pores surface, proposed themodel of adsorption velocity on silica gel–water as be-low, this model was accepted by some researchers also[4, 5]. For simplifying analysis, the lagging effect be-tween adsorption and desorption is neglected.

dxds¼ ksapðx� � xÞ ð1Þ

where x* is the equilibrium adsorption rate per adsor-bent (kg/kg), ksap is the coefficient of diffusion velocityon solid surface, which is related to the diffusion char-acter of adsorbate on the micro pores surface of adsor-

W. Wang (&) Æ R. WangInstitute of Refrigeration and Cryogenics,Shanghai Jiao Tong University, Shanghai,200030, P.R. ChinaE-mail: wangwen3250@hotmail.com

Heat Mass Transfer (2005) 41: 680–684DOI 10.1007/s00231-004-0582-9

bent and related to the surface geometry feature ofadsorbent and potential between adsorbent and adsor-bate [3].

ksap ¼15Dso

R2p

expð�Ea=RT Þ ð2Þ

In Eq. 2, Dso is the coefficient related to diffusioncharacter on adsorbent surface, Ea is the activation po-tential of diffusion on surface, and Rp is the averagediameter of adsorbent particles. Therefore, its value ofksap results from not only the thermal state parameters(such as temperature in Eq. 2) but also the geometryfeature and surface diffusion activation potential ofadsorbent.

Therefore, ksap can be approximately considered as aparameter to describe the character of non-equilibriumon adsorption. The greater is the value of ksap, the fasteris the adsorption (or desorption), then the more far awayfrom non-equilibrium adsorption.

Passos et al. [4] generalized their experimental dataon solar adsorption ice making, and proposed a set ofreference values in Eq. 2 for active carbon–methanolas:15Dso / Rp

2 =7.35·10�3 s�1, Ea/R=978 K. Since thesedata come from experiment about solar ice making, thevalues of temperature and adsorption rate should beaverage values over the adsorbent, and the uneven dis-tribution of adsorption rate would be neglected. Besides,during adsorption or desorption, the geometry charactershould be some changed. These factors all properly ef-fect the adsorbent diffusion and adsorption reactionprocess. However, Eq. 2 and the data [4] provided asimple method to describe the adsorption non-equilib-rium involving diffusion and adsorption reaction inadsorption refrigeration.

Based on this non-equilibrium adsorption model,some numerical simulation and discussion had beendone to discover the effect of non-equilibrium adsorp-tion on adsorption refrigeration and how the non-equi-librium adsorption effect on the adsorption process andheat transfer character.

2 Model for numerical simulation

Supposed an adsorber of adsorption refrigerationemploys active carbon–methanol as the working pair.

The type of adsorber is made of cylinder tube, and theadsorbent is filled annularly connecting with the wallof adsorber, the thickness of adsorbent layer is 5 mm.The outer diameter of tube is 51 mm, and its wallthickness is 2 mm. For continuous cooling supply, thissystem employs two adsorbent beds. Therefore, onebed can keep adsorption process for cool supplyingwhile another bed is being regenerated. Figure 1 showsthe structure of adsorbent bed in the simulation.

For simplifying calculation, the heat transfer in theadsorbent is considered as one-dimensional on radiusdirection. Its control equation is written as Eq. 3. Innumerical calculation, the thickness of adsorbent isuniformly divided into six units along radius direction.

qCp@T@s¼ keff

1

r@T@rþ @

2T@r2

� �þ qqsorp ð3Þ

In Eq. 3, the heat source comes from the reactionheat of adsorption or desorption, which is calculated outof Clausius–Clapeyron equation, Eq. 4. In which, Ts issaturated temperature corresponding to the adsorberpressure during adsorption and desorption,s presentstime, R is the gas constant of methanol, A is an empiricalconstant, 4464.05.

qsorp ¼ RATTs

dxds

ð4Þ

The equilibrium adsorption character of active car-bon with methanol is adopted as Dubinin–Astakhov(D–A) equation, and those coefficients are same asprevious reference [4]:

x� ¼ 0:284 exp �10:21 TTs� 1

� �1:39" #

ð5Þ

The velocity of adsorption is described as Eq. 1.In below calculations and discussion, the design and

working parameters are shown in Table 1 unless spe-cially illustrating. In the table, Tl is the temperature ofcooling fluid used for cooling adsorber in adsorptionprocess. a2 is the convective coefficient between coolingfluid and the wall of adsorber. Th is the temperature ofheating fluid used for heating adsorber in desorptionprocess, its convective coefficient is a1. The temperatureof evaporation is Te, and the temperature of condensa-tion is Tc.

Table 1 Parameters used in the calculation

keff,J/m �C 0.4 Tl,�C 30

q, kg/m3 700 a1,W/m2 �C 200hw,W/m2 �C 50 Th,�C 120Te,�C 5 a2,W/m2 �C 250Tc, �C 25

Fig. 1 Scheme of adsorbent bed

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3 Simulation and discussion

3.1 Comparison of non-equilibrium modelwith equilibrium model

During simulation, the initial conditions are a group ofunified temperatures for each discrete units, their tem-perature profiles in each cycle tend to being steadygradually after 10–15 simulated cycles. After that, we getsome data to analyze. The below analysis and discussionare based on the data.

In Fig. 2, the discrete units are distinguished withnumbers of 1, 2, 3, 4, 5, and 6 with radius reducing.Figures 2 and 3 could be used to compare the temper-ature profiles in 20 minutes cycles between equilibriumand non-equilibrium models. Although the environ-ments for the two kinds of adsorption refrigeration aresame, the temperature profiles are different greatly. Thedifference actually comes from adsorption character incycle, in cycles with non-equilibrium model, theadsorption (and desorption) is not sufficient as same aswith equilibrium model, so more heat is added on the

heat capacities of the adsorbent and the tube, less on theadsorption (or desorption) heat.

Figures 4 and 5 show the profiles of adsorption ratein cycles, in these figures, the horizontal lines representthe adsorption rate during isosteric process, which areresulted from the assumption of constant adsorptionrate at each unit in simulation. Because of differentadsorption velocity, the discrepancy of their fluctuationamplitude is very large, this brings about 0.209 ofCOP for equilibrium model and only 0.0257 of COP fornon-equilibrium model. Moreover, in simulation withnon-equilibrium adsorption model, it can be observedtransitory rearrangement of adsorbate among each unitof adsorbent.

3.2 Analysis on elements leading to non-equilibriumcharacteristic

Without doubt, the non-equilibrium model is moreclosed to real processes of adsorption refrigeration.However, its performance is lower than ideal cycles de-scribed with equilibrium models. What are the factors

Fig. 2 Temperature distribution with equilibrium adsorptionmodel

Fig. 3 Temperature distribution in adsorber with non-equilibriumadsorption model

Fig. 4 Adsorption rate distribution in adsorber with equilibriumadsorption model

Fig. 5 Adsorption rate distribution in adsorber with non-equilib-rium adsorption model

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lead to the distance? Based on the theory from Sokodaand Suzuki [2] ksap should be a parameter to indicate thedegree of non-equilibrium. Figures 6, 7, 8, and 9 showthe effect of ksap. In the simulation cases, we fixed itsvalues at some constants in cycles, such as: 0.000235,0.00235, 0.00535, 0.0135, and these just are assumption.In term of Eq. 1, ksap should be related to temperature.It could be observed that the results with higher ksapclose to that on equilibrium model, and the results withlower ksap close to that on non-equilibrium model. Inthe figures, the total values of temperature and adsorp-tion rate are averaged weighted with each unit volume.

Further, in Figs. 3, 6, and 7, ksap affects the adsorp-tion and desorption processes, that is, changes thetemperature profiles obviously at beginning parts ofcurves after phase switching, particularly in shorter cycletime, the adsorption or desorption rate is not able toreach the maximum amplitudes. However, in long cy-cles, the effect of ksap is reduced, and the curves of non-equilibrium models close gradually to the curve ofequilibrium model.

From Fig. 8 and 9, the differences of averageadsorption rate produced by ksap are very apparent,even in long cycle with 120 min of cycle time. It would

lead to different cycle cooling capacity absolutely. Thatmeans that the effect of surface diffusion character couldnot be neglected.

On the other hand, as mentioned ahead, ksap is in-volved with the temperature, the diffusion character onadsorbent surface, the active potential, and the averagediameter of adsorbent particles (such as in piled silicagel, active carbon, etc.) or the holes size and distributionin adsorbent, and so on. Therefore, we could take con-sideration to improve the performance of adsorptionrefrigeration through changing these parameters in someextent. However, these modifications on surface diffu-sion characters, active potential, even geometry charac-ters are limited by present chemical engineeringtechnology. Moreover, Too small diameter of particlesperhaps leads to more resistance of mass transfer, it isnot helpful to accelerate adsorption (or desorption).

Table 2 shows some designed results of adsorber withtwo-bed continuous cycle, their differences between two

Fig. 6 Temperature curves of adsorber with cycle time of 20 min

Fig. 7 Temperature curves of adsorber with cycle time of 120 min

Fig. 8 Adsorption rate curves of adsorber with cycle time of20 min

Fig. 9 Adsorption rate curves of adsorber with cycle time of120 min

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models are serious. Table 3 shows some designed pro-jects with various cycle times, the differences of param-eters between two models tend to reduce with longingcycle time. Although the reference values about non-equilibrium adsorption from reference [3] are not exactlymatched on short cycles, these comparisons still pointout that non-equilibrium adsorption should not be ne-glected on designing short cycle adsorption heat pumpsystem.

4 Conclusion

In actual adsorber of adsorption refrigeration system, allthe adsorption (or desorption) processes are non-equi-librium, particularly in rare pressure system, theadsorption process is far away from equilibriumadsorption model.

Contrasting to with equilibrium adsorption, the cycleadsorption rate is obviously decreased due to non-equilibrium adsorption in adsorption refrigerationsystem, while the temperature fluctuation amplitude incycle is increased. The status in adsorption refrigerationcycle is influenced by adsorption velocity greatly besidesthe characters of heat and mass transfer.

In real short cycle, the cycle adsorbed mass could notbe sufficient because of the non-equilibrium adsorptionprocess. Therefore, improving the diffusion character onthe micro-pore surface (increasing ksap) is helpful toenhancing the velocity of adsorption reaction and thenthe performance of refrigeration (COP). Moreover, evenin long cycle with low ksap, to increase the value of ksapis much effective to promote the performance ofadsorption refrigeration.

Acknowledgements The authors appreciate the support from theNational Natural Science Found (50106006), National Outstand-ing Youth Founding (50225621), and the State FundamentalResearch Program (G2000026309).

References

1. Xin HW (1997) Reaction mechanics on fractal materials.Shanghai Science & Education Press (in Chinese)

2. Jaroniec M, Madey R (1997) Physical adsorptionh on hetero-geneous solids. Chemical Industry Press (in Chinese)

3. Sokoda A., Suzuki M (1984) J Chem Eng Japan 17(1):524. Passos EF, Escobedo JF, Meunier F (1989) Solar Energy

42(2):1035. Bidywt B. Soha, Boelman Elisa C, Kashiwagi T (1995) ASH-

RAE Trans 101:384

Table 2 Required thickness of adsorbent with two adsorption models with 20 min of cycle time

Thicknessof adsorbent (mm)

COP with equilibriumadsorption model

Length for 1 kWcooling power (m)

COP with non-equilibriumadsorption model

Length for 1 kWcooling power (m)s

3 0.221 14.7 0.0264 1444 0.217 14.2 0.0267 1325 0.209 14.4 0.0257 1306 0.195 15.3 0.0236 1347 0.181 16.4 0.0211 146

Table 3 Differences between two adsorption modes with various cycle times

Cycle time(min)

COP with equilibriumadsorption model

Length for 1 kWcooling power (m)

COP with non-equilibriumadsorption model

Length for 1 kWcooling power (m)

20 0.209 14.4 0.026 129.660 0.302 21.5 0.109 78.2120 0.312 38.3 0.188 77.9240 0.312 81.7 0.256 99.3360 0.312 122.5 0.278 131.0480 0.312 163.3 0.285 167.3

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