Kinetic Modeling of Isothermal or Non-isothermal Adsorption in a Pellet: Application to Adsorption Heat Pumps

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Montastruc, Ludovic and Floquet, Pascal and Mayer, Volker and

Nikov, Iordan and Domenech, Serge ( 2010) Kinetic Modeling of

Isothermal or Non-isothermal Adsorption in a Pellet: Application

to Adsorption Heat Pumps. Chinese Journal of Chemical

Engineering, vol. 18 (n°4). pp. 544-553. ISSN 1004-9541

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Kinetic Modeling of Isothermal or Non-isothermal Adsorption in a

Pellet: Application to Adsorption Heat Pumps

Ludovic Montastruc1,2,*_{, Pascal Floquet}1

, Volker Mayer3, Iordan Nikov2 and Serge Domenech 1 1

Université de Toulouse, Laboratoire de Génie Chimique, U.M.R. 5503 CNRS/INP/UPS, 31432 Toulouse, France

2

ProBioGEM Laboratory, Polytech’lille, 59665, Villeneuve d’Ascq, France

3

Laboratoire Paul Painlevé, U.M.R. CNRS 8524, U.F.R. de Mathématiques, 59 655 Villeneuve d’Ascq Cedex, France Abstract Understanding the interaction between a fluid and a solid phase is of fundamental importance to the de-sign of an adsorption process. Because the heat effects associated with adsorption are comparatively large, the as-sumption of isothermal behavior is a valid approximation only when uptake rates are relatively slow. In this article, we propose to determine when it is needed to choose the isothermal or non-isothermal assumption according to two physical parameters  (ratio convection/capacity) and  (quantity of energy/capacity). The proposed problem is solved by a mathematical method in the Laplace domain. When o (infinitely high heat transfer coefficient) or

o0 (infinitely large heat capacity), the limiting case is isothermal. When the diffusion is rapid (<10) the kinetics

of sorption is controlled entirely by heat transfer. If the adsorption process is to be used as a heat pump, it shall be represented by an isotherm model with  and  as high as possible.

Keywords adsorption, non-isothermal, energy, pellet

1 INTRODUCTION

Since the early 1960s, separation processes by adsorption, such as pressure swing adsorption or thermal swing adsorption, have become important industrial operations [1, 2]. At present, an application in the field of adsorption heat pumps (AHPs) repre-sents one of the most promising technical applications of the adsorption process. Understanding the interac-tion between a fluid and a solid phase is of fundamen-tal importance to the design of an adsorption process. These cyclic processes are based on a parallel ar-rangement of columns packed with adsorbent pellets. Because the heat effects associated with adsorption are comparatively great, the assumption of isothermal behavior is a valid approximation only when uptake rates are relatively slow.

A basic adsorption heat pump cycle consists of four main parts: an adsorber, which is a container filled with an adsorbent (such as zeolite, active carbon, silica gel), a condenser, an evaporator, and an expan-sion valve. Basically, an adsorption heat pump oper-ates by cycling adsorbate between adsorber, condenser, and evaporator. In the adsorption heat pump cycle, adsorption phenomena play the same role of me-chanical power, so that the working fluid can be

cir-culated in the cycle without any mechanical power [3].

An adsorption heat pump cycle consists of four steps,

which are isosteric heating (ab), isobaric desorption

(bc), isosteric cooling (cd) and isobaric adsorption

(da) as shown in Fig. 1.

In recent years, the importance of adsorption heat pumps and adsorption refrigeration systems has in-creased since these systems can directly utilize the

* To whom correspondence should be addressed. E-mail: ludovic.montastruc@ensiacet.fr Figure 1 Thermodynamic cycle of a basic adsorption heat pump

primary thermal energy sources and additionally the waste heat generated in various industrial processes. The important advantages of the adsorption heat pumps can be described as follows [3]:

• they can operate with thermal driving energy sources such as waste heat, solar, and geothermal en-ergies;

• they can work with low temperature driving en-ergy sources;

• they do not require moving parts for circulation of working fluid;

• they have long life time;

• they operate without noise and vibration; • they have simple principle of working; • they do not require frequent maintenance; • they are environmental friendly since they do not contain any hazardous materials for environment; and

• they can be employed as thermal energy storage device.

On the contrary, the main disadvantages of the adsorption heat pump systems are as follows:

• they have low COP values;

• they are intermittently working principles; • they require high technology and special design to maintain high vacuum with working pairs used such as zeolite-water, silica gel-water, and activated carbon-methanol;

• they have large volume and mass relative to traditional mechanical heat pump systems.

AHPs are thermally driven and can, therefore, make use of waste heat as well as thermal renewable energy resources. Moreover, AHPs can utilize high energy fuels more effectively than modern condensing boiler heating systems and, consequently, a reasonable amount of the resulting environmental pollution could be avoided. One of the most interesting working pairs for AHPs is zeolite-water, which is non-poisonous, non-flammable and, moreover, has no ozone depletion potential. The adsorption of water vapor into a porous solid adsorbent, such as zeolite, results in the release of the heat of adsorption. Accordingly, the temperature of the adsorbent particle rises, which reduces its water uptake capacity. Therefore, the heat of adsorption has to be removed before further mass transfer of vapor into the adsorbent particle takes place. The evaluation of the heat and mass transfer rates accompanying such an adsorption process is therefore a very important aspect in designing and optimizing the operation of an adsorption heat pump.

Many experimental studies were realized during the last twenty years to analyze the influence of

vari-ous parameters in transient period [46]. Bonnissel et

al. [7] studied a rapid thermal swing adsorption

proc-ess based on a composite adsorbent bed. The high ef-fective conductivity of the bed and the fast dynamics of the thermoelectric elements allow the cycles (cool-ing and adsorption/heat(cool-ing and desorption) to be run in 1020 min. A model based on experimental results is developed that accounts for the usual mechanisms of equilibrium and heat- and mass-transfer kinetics in adsorption, coupled with the specific dynamic

behav-ior of the thermoelectric elements.

Moreover, many studied were realized in the thermal effects in a spherical adsorbent pellet [815]. For each model, some hypotheses were assumed.

During adsorption, the heat generated is partly transferred to the surrounding fluid and is partly accu-mulated in the particle, leading to a temperature increase. The calculation of the temperature uptake curves is based on mathematical models of different complexity depending on the simplifying assumptions being made. The most widely applied simplifying assumptions are as follows [14].

(1) The isotherm is linear. This assumption is par-ticularly useful in the derivation of analytical solutions

of the model equations [8]. In most practical systems,

however, the isotherms are nonlinear and then these models are inaccurate. The linearization of the iso-therm leads to the underestimation of the intraparticle mass-transfer rate and consequently the predicted maxi-mum particle temperature is lower than that observed.

(2) The external mass-transfer resistances are negligible. This assumption is rather controversial be-cause at the initial stages of the process when the heat effects are most significant, the external mass-transfer resistances are usually rate controlling. As shown by Hills

[12], this assumption leads to greater errors than the as-sumption that the isotherm is temperature independent.

(3) The adsorption isotherm parameters are

in-dependent of temperature. As shown by Hills [12], in

the case of small temperature rises (less than 10 K), the assumption does not lead to significant errors.

(4) The diffusion coefficient, heat capacity and heat of sorption are independent of temperature.

(5) The particle temperature gradients are negligi-ble and the heat transfer resistance is contributed by the film surrounding the particle. As shown by Brunovska et al. [16] the lumped thermal model of nonisothermal adsorption is a very good approximation to the com-plete model for many practical systems.

(6) The intraparticle mass-transfer rate can be approximated using driving force models. Bowen and

Rimmer [17] applied Vermeulen’s quadratic driving

force model [18] in the analysis of experimental results for the system of water vapor-alumina. Bhaskar and

Do [19] developed approximate models for

noniso-thermal adsorption with a linear isotherm using the parabolic (LDF model) and the nth order approxima-tion of the intraparticle concentraapproxima-tion profile

In this study, a methodology to choose a good adsorbent in particular according to its thermodynamic properties is presented. We focus our attention on the interaction fluid/solid. The simplest case to consider is a single microporous adsorbent particle exposed to a step change in sorbate concentration at the external surface of the particle at time zero. Heat transfer is assumed to be sufficiently rapid relative to the sorp-tion rate, so that temperature gradients both through the particle and between particle and surrounding fluid are negligible. Two cases are considered, isothermal condition and adsorption behavior with temperature effect. A mathematical development is presented for

each case. According to two developed parameters  (ratio convection/capacity) and  (quantity of en-ergy/capacity), the difference between isothermal and non-isothermal is analyzed.

2 ISOTHERMAL ASSUMPTIONS

During adsorption, heat is generated, which is partly transferred to the surrounding fluid and partly accumulated in the particle leading to an increase of temperature. The calculation of the temperature up-take curves is based on mathematical model of differ-ent complexity depending on the simplifying assump-tions being made. Since adsorption is exothermic and the heat of sorption must be dissipated by the heat transfer, there is, in general, a difference in tempera-ture between the adsorbent particle and the ambient fluid when sorption is taking place. Whether or not this temperature difference is significant depends on the relative rates of mass and heat transfer. By the simple theoretical analysis it may be shown that in a batch adsorption experiment it is the dissipation of heat from the external surface of the adsorbent sample, rather than the conduction of heat within the adsorbent, that is generally the rate limiting heat transfer process. When the heat transfer resistance may be neglected, the system is treated as isothermal case.

First, we consider some isothermal conditions in the pellet and the isotherm is q . The isotherm is q₀ supposed to relate the concentration q inside the pellet (or at the external boundary) to the concentration of the external fluid. Another basic assumption is that the pellet is treated as a homogenous medium (contact with the fluid occurs only at the external boundary). Thus, the diffusivity Dc is constant in the pellet.

The system is modeled by the following equations:

2 c ₂ c 0 c c 0 0 2 0, (0 ) ( , 0) (0 ) ( , ) 0 0 0 r q q q D t r r r t r r r q r t q r r r q r t q t q t r    ­w §w _ w · _{ ! } °_w ¨_{©w w} ¸_¹ ° ° _c °    ®  ! ° ° _w ° _{ !} w °¯ (1)

Using the method based on a separation of vari-ables with ( , )q r t T t R r( ) ( ), and using superposition of the elementary solutions, we obtain





 

2 c 1 0 0 0 1 ( 1) 2 sin e n n D t n n n q q q q r r O O O  f   c  

¦

(2)

If u



qq₀

 

/ q₀c q₀



represents the reduced

concentration in the pellet, K r r/ _c the reduced

ra-dius and W D t r_c / _c2 the reduced time,





2 2 1 1 ( 1) ( , ) 2 sin e n n n u n n W K W K K  f  S  _S S

¦

(3)

Figure 2 represents the 3D evolution of u versus the radius and reduced time (step in time 0.01; step in radius 0.05).

Figure 2 3D representation of reduced concentration ac-cording to radius and time

The value q r t is equal to ( , )_c q₀ z , for solving 0 the system analytically.

With W D t r_c / _c, the analytic solution is given by the familiar expression

2 2 2 2 1 6 e 1 n t n m m n W  S f f  S

¦

(4)

where m represents the quantity adsorbed in the pel-_t

let at time t and m_f the quantity adsorbed in the pellet at final time. We denote ( )QW m m_t/ _f.

In practice m m_t/ _f will be approximated by [8]

2 2 2 2 1 6 e ( ) 1 n N N n Q n W W   S S

¦

(5)

Figure 3 shows different approximations and it seems that the convergence is quite fast. Starting from

5

N , we cannot see any difference in Q_N( )W from

the graph. Let us check it by the following simple er-ror estimation:

Figure 3 Numerical approximation on m m_t/ _f values

ver-sus the reduced time 

2 2 2 2 1 2 2 2 2 2 1 6 e ( ) ( ) 6 1 6 d 6 n N n N N n N Q Q n x n x N W W W f  S  f _f   S S S S

¦

¦

³

İ İ (6)

This gives an approximate absolute error of 0.12

for N . In reality, the error is much smaller if  5

stays away from zero. Indeed, if W Wı ₀ı , the up-0

per estimation can be improved to the following ex-ponential error bound

2 2 2 2 0 0 2 2 4 3 0 6 e 3 e ( ) ( ) d n N N _N Q Q t x x N W W W W  S  S f  S

³

İS (7)

For example, if the reduced time is at least W₀ 0.01

and N , we get for the absolute error a much better 5

approximation of 0.000418.

3 NON-ISOTHERMAL ASSUMPTIONS In this case, we assume that intracrystalline dif-fusion is the only significant resistance to the mass transfer. The sorbate concentration at the surface of each particle in the sample is therefore always in equi-librium, at the temperature of the sample, with the sorbate concentration in the ambient fluid. In addition, the thermal conductivity is assumed to maintain an essentially uniform temperature throughout the entire adsorbent particle. The rate of heat transfer between the adsorbent and the ambient fluid, which is assumed to obey Newton’s law, is finite, so that during the transient adsorption there is a time-dependent tem-perature difference between the adsorbent and ambient fluid. The system is subjected, at time zero, to a small difference step change in ambient sorbate concentra-tion from a previously established equilibrium condi-tion. We assume that the concentration equilibrium between fluid phase and adsorbed phase is established quickly at the free surface of the pellet and the equi-librium adsorbed phase concentration is assumed to vary linearly with both temperature and fluid phase con-centration. Subject to these approximations, the system may be described by the following equation in re-duced values Q q t( ) /q_f, K r r/ _c and W D t r_c / _c2. In opposite to the previous part, the reduced val-ues permit to find an analytic solution in the Laplace domain. The mathematical model is

2 2 1 Q Q K W K K W w w § w · ¨ ¸ w w © w ¹ (8a) ( , ) 0, ( 0) ( , 0 ) 0, (if 0 1) QK W W  QK  İK (8b) s (1, ) ( ) Q W Q T (8c) 0 0 Q K K w w (8d)



0



0 0 c d d ( ) d d / p c T T Q T ha H q q q q D r U U W f W f  '    (8e)

The equilibrium relationship at the crystal surface is assumed to be linear with respect to both concentra-tion and temperature:

* 0 s 0 1 q T T Q T q_f q  w § ·  _{¨ ¸} w _©  _¹ (8f) Equation (8a) is written in the Laplace domain, and ( , )Q K p is a root of 2 2 2 2 1 2 ( , 0) Q Q Q pQ QK K pQ K K K K K K § · w w w w  _{¨ ¸}  w _© w _¹ w w     _ because ( , 0 )QK  (9) 0 The determination of ( , )Q K p is presented in Appen-dix A, in which we obtain





0 0 sh : ( , ) p ; p Q p a a p K K K   R (10) Thus,





 

s sh : ( , ) ( ) sh p p Q p Q p p K K K    (11)

In the Laplace domain, ( )Q p represents the av-erage concentration Q in the pellet

 





1 _{2 s} 1 ₂ 0 0 sh ( ) ( ) 3 ( , ) d 3 d sh p Q p Q p Q p p K K K K K K K

³



³

 _ (12) so Q p( ) 3Q ps( ) pcoth

 

p 1 p ª¬  º¼   ₍₁₃₎

With Eqs. (8e) and (8f), we have the following equation



s



* 0 c 1 d d ( ) d d / p c Q Q T ha H q q D r _q T U U W f W  '   §w · ¨_w ¸ © ¹ (14) With Eq. (8f), s



0



* d d d d q q Q T q T W W f  §w · ¨_w ¸ © ¹ , Eq. (14) becomes





* s s c 1 d d ( ) d d / p p q Q Q Q ha T H c W W D r Uc §w · ¨_w ¸ _ © ¹ '  (15) With c p r ha c D D U and * p H q c T E ' §¨_©w_w ·¸_¹, Eq. (15) becomes

s s d d (1 ) d d Q Q Q E D W  W  (16)

In the Laplace domain, Eq. (16) is

s ( ) ( ) ( ) 1 pQ p p Q p p D E   D   (17)

Equations (13) and (17) allow us to write

 

^

`

s ( ) ( ) 3 coth 1 ( ) p Q p p p p p D E D  ª _{˜  }º _ ¬ ¼  ₍₁₈₎

Finally, transferring Q p_s( ) in Eq. (17), we obtain

 

 

^

`

2 3( ) coth 1 ( ) 3 coth 1 ( ) p p p Q p p p p p D E D ª º  _¬ ˜  _¼ ª _{˜  }º _ ¬ ¼  ₍₁₉₎

The method to obtain the original ( )QW from

( )

Q p is presented in Appendix B, which shows

 

 

 

2 1 ( ) 1 coth 1 9 e 1 coth 1 3 coth 1 2 n n n p n n _{n n} n n n Q p p p p p p p p W W E f  ª _ º « »  « » ¬ ¼ ª _ º « »  ˜  « » ¬ ¼

¦

p_n (20) C The restriction in the real domain is presented in Appendix C. The final result is [1]

 

 

 

2 2 2 1 2 cot 1 9 e ( ) 1 cot 1 1 3 cot 1 2 n n n q n n n n n n n q q q Q q q q q q W W E  f ª  º  « » ¬ ¼  ª  º  _« ˜  _» ¬ ¼

¦

n q R (21) where q_nare the roots of the following equation [1]:



2



 

 

3 sin 3 cos 0

n n n n

q q q q

D  E  E (22)

The parameters  and  are variables in a first approach used to solve the PDE system. The two pa-rameters represent respectively the ratio of conductiv-ity to heat capacconductiv-ity and the ratio of the quantconductiv-ity of en-ergy and heat capacity.

4 LOCATION OF THE ROOTS q AND DIS-_n

CUSSION OF THE SOLUTION Q

In order to understand the solution Q we first discuss the roots of the transcendental Eq. (17). Then we give an error estimation of the Nth approximation of Q similar to the one obtained in the isothermal case.

4.1 Location of the roots

Let us have a look on the particular situation 0

E , which implies infinitely large heat capacity

and corresponds to the isothermal case. Then Eq. (22) becomes simply



Dq_n2



sin

 

q_n and the roots 0 are q_n S , n Nn  plus a special root qc D .

The general case is not so much different.

Sup-pose 0E! , then it is easy to conclude from the

pe-riodicity of the sine and cosine functions that every interval of the form [nS, (m S contains exactly 1) ] one root qn of Eq. (22). In addition, there is one more special root, say qc , coming from the factor



2 ₃



n

q

D  E in Eq. (22). The location of qc depends

on the sign of (D3 )E .

If 3D E! , then 0 qc | D3E . More

pre-cisely, we have that qc  D3E İ . S

If 3D Eİ , then 0 qc(0, / 2)S .

In particular, one must take care of this root in

the numerical approximations of Q if (D3 )E is

small or negative. 4.2 Error bound

We first have to find a more convenient

expres-sion of Q since the one given in Eq. (17) is not

suit-able. Consider again the case E . Then cot( )0 q_n is not defined since q_n S . In order to do this, let us n start with Eq. (26), from which follows that

 

2 3Eq_ncot q_n q_n  D 3E and

 

2 cot 1 3 n n n q q q D E  

We, therefore, can reformulate the coefficients of the series in Eq. (17)



















2 2 2 2 2 2 2 2 2 2 2 2 2 3 9 2 3 9 2 3 3 3 1 n n n n n n n n n n q q c q q q q q q q D E D D E E D E E D D §  · ¨ ¸ ¨ ¸ © ¹  ª _ º    « » « » ¬ ¼  ª   º  ¬ ¼ (23)

which finally leads to the expression

2 2 1 e ( ) 1 6 n q n n n Q C q W W 

¦

f  with













2 2 2 2 2 3 3 1 n n n n q C q q D E E D D  ª   º  ¬ ¼ (24)

Notice that, from this expression, one can see very well that this solution converges to the isothermal

situation when Eo (and that the coefficient, say 0

( )

Cc Cc E , corresponding to the special roots qc

disappears at the limit).

For the numerical approximation of the solution

we have again a good estimate for QQ_N , where



2



2 1 ( ) 1 6 e n / N q N n n n

Q W 

¦

C  W q . Since we have precise

information on the locations of qn, we can proceed like

in the isothermal case. Let us first focus on the sign of





2

3 3 1 _n

B Eª_¬ E q Dº_{¼ . There are two cases.}

Case 1: If Bı then we have immediately 0

1 n C İ . Case 2: B . If 0 1



2



2 0 2 qn D B   İ  , then,

similar to case 1, we get













2 2 2 2 2 2 2 1 2 n n n n q C q q D D D     İ .

The remaining situation 1



2



2

2 n B  q D is equivalent to









2 2 4 2 1 1 3 9 1 2D D Eqn  2qn  E qn

The right hand term of this inequality is negative and

the left hand term is strictly positive while D !

2

2q_n 6E. However, since B , which implies that 0









2 2 2 2

3(1 )q_n 2q_n 6 q_n 3 q_n 2

D! E  E    , we see

that this last situation never occurs while q_n! 2. Finally we get C_n İ for every 2 nı and for 2 every  and .

We can now proceed like in the isothermal case. Let us simply indicate how to get the estimation cor-responding to the first differential Eq. (1). In order to do so, we make the convention that the summation of

the approximation Q is taken over the first N roots _N

plus the special one q if qİN. If q >N then we have the following estimation (otherwise the error is a

little smaller): 2 2 2 2 2 2 1 ( ) ( ) 2 2 1 2 2 e e ( ) ( ) 12 e e 12 ( ) ( ) 12 1 1 n q q N n N n N n n N Q Q q q N n N N W W W W  c f    S f  S  § ·  ¨_¨  ¸_¸ c © ¹ ª º  « » S S « » ¬ ¼ § _ · ¨ ¸ S © ¹

¦

¦

İ İ İ for every Nı (25) 2 which is about two times the error estimation obtained in the isothermal case. If the reduced time W Wı ₀!0, then one has again the exponential decay of the error like in Eq. (6) due to the factor _e SN2 2W0_.

5 DIFFERENCE BETWEEN NON-ISOTHERMAL AND ISOTHERMAL CASES

The adsorption characteristics of water vapor on zeolite are essential data in determining the energetic performance of adsorption chillers. From the view-point of the fundamental design of adsorption chillers, the investigation of the adsorption isotherm is perti-nent for the purpose of system modeling.

In terms of the adsorption chiller performance, the adsorbent must have good thermal conductivity as well as a large specific surface area. To estimate dynamic adsorption capacity using the parameter, we compare the two assumptions, isothermal or non-isothermal. In

Table 1, we summarize the data used [2023]. In this

case, we found D 16.3 and E 2.07.

Figure 4 shows the difference between the two models. In the non-isothermal model, the uptake curve commonly shows a rapid initial uptake followed by a slow approach to equilibrium because the external

Table 1 The used values found in the literature

'H/kJ·mol1 _(wq*_/wT)/mol·kg1_·K1 _c

p/J·kg1·K1 h/m2·K·W1 a/m1 rc/mm U/kg·m3 D/m2·s1

36 0.053 920 10 2.48 1.1 1100 2×109

Figure 4 Comparison between isothermal and non-isothermal adsorption kinetics for water on zeolite

heat transfer controlled the adsorption. Let us recall that the isothermal model represents the ideal condi-tion in the heat adsorpcondi-tion process.

In the isothermal case, the equilibrium is reached very quickly. The maximum difference between the two models is obtained in the first phase of the batch adsorption. This phenomenon is due to the fast tem-perature rising when the adsorption starts. The heat of adsorption produced in the pellet is not transferred into the fluid due to the external heat transfer. In Fig. 5, different graphs represent the square distance between the adsorbed quantity with an isothermal model and a non-isothermal assumption at different reduced time. For the calculation, the number of terms N is 20, and the  value is between 0 and 40 and  between 0 and 10. If  is close to zero and  is large, we observe that when the reduced time is lower than 0.25, the calcu-lated errors increases with the reduced time. And when the reduced time is higher than 0.25, the calcu-lated errors decreases.

To increase the process efficiency, it is necessary to raise the external heat transfer. It can be modified by the hydrodynamics around the pellet. The values of  and  can determine the “efficiency” of the heat ad-sorption process. If the  value is small and/or close to

zero, the adsorption curve can be analyzed as in the isothermal case. However, this means that the heat pump process cannot be operated because it will not produce enough heat. The  value is more significant, because it represents the ratio of the convection to the thermal capacity. To reach the ideal case,  should be very large. The external heat transfer will then not be a limiting phenomenon, and the adsorption is close to the isothermal case.

6 CONCLUSIONS

In this article, we show the importance of the pellet adsorption study before designing a heat ad-sorption process. The difference between the two models, isothermal or non-isothermal, could be meas-ured by the values of two parameters  and . When

o (infinitely high heat transfer coefficient) or o0

(infinitely large heat capacity), the limiting case is isothermal. When the diffusion is rapid (small ) the kinetics of sorption is controlled entirely by heat transfer. If the adsorption process is to be used as a heat pump, it shall be represented by an isotherm model with  and  as high as possible. Then, these

(a) Reduced time 0.01 (b) Reduced time 0.05

(c) Reduced time 0.1 (d) Reduced time 0.25

(e) Reduced time 0.5 (f) Reduced time 0.75

Figure 5 Square distance between the isothermal and non-isothermal model for different  value between 0 to 40 and for different  value between 0 to 10

two parameters could determine if the couple adsorb-ent/ adsorbate can be used in the case of an adsorption heat pump process.

NOMENCLATURE

a external surface area per unit particle volume cp heat capacity

D diffusivity h heat transfer coefficient H adsorption enthalpy

mt mass adsorbed per unit of adsorbent during time t( q q0c)

m mass adsorbed per unit of adsorbent during the equilibrium

time ( q_fq₀c)

Q reduced mass adsorbed ( q q/ _f) Qs reduced mass adsorbed for r rc

Q reduced average mass adsorbed q mass adsorbed per unit of adsorbent

q mass adsorbed per unit of adsorbent at equilibrium time

q0 initial mass adsorbed per unit of adsorbent at r rc 0

qc initial mass adsorbed per unit of adsorbent q average mass adsorbed per unit of adsorbent r radius coordinate of pellet

rc pellet radius

t time

u reduced concentration in the pellet [ ( q q₀) /(q₀cq₀)]  parameter [ harc/(Uc Dp )]

 parameter [ ' wH( q*/wT) /cp]

 reduced radius coordinate  density, kg·m3  reduced time ( D t rc / c2)

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of solute adsorption at heterogeneous solid/solution interfaces: On the possibility of distinguishing between the diffusional and the sur-face reaction kinetics models”, Appl. Surf. Sci., 253, 58275840 (2007).

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703711(2001).

APPENDIX A Determination of ( ,QK p)

First, we search a solution like an integer serial function using Q( , )K p

¦

akKk. We obtain 1 1 1 1 0 1 ( 1) _k k 2 ( 1) _k k _k k 0 k k k k k a K k a K pa K f f f        

¦

¦

¦

(A1)

And then, we transform the following system

1 1 1 1 1 1 0, 2 0 1, ( 1) 2( 1) 0 ( 1)( 2) 0 k k k k k k a k k k a k a pa k k a pa      ½ °     ¾ ° Ÿ    _¿ ı (A2)

From 1  odd,k ak and 0   k 1 2l the recursive formula leads to 0 2 2 3 pa a u , 2 2 0 0 2 4 4 5 2 3 4 5 5! p a p a pa a u u u u so _{2 0} (2 1)! l l p a a l (A3) Finally ₀ 2 0 ( ) ( , ) (2 1)! l l l p Q p a l K f K 

¦

 , so that we can write

 

2( 1) 2( 1) 0 0 ( , ) (2 1)! l l l p a Q p l p K K K  f  

¦

 (A4) Thus,





0 0 sh : ( , ) p ; p Q p a a p K K K   R (A5)

APPENDIX B Determination of ( )QW original of ( )

Q p in the Laplace domain with p C

To obtain ( )QW original of Q p( ), it is necessary to use the residue theorem with just the simple pole-zero.

1. Beginning behavior

We use the development of coth

 

p in the vicinity of

zero, coth

 

1 1 1

 

2 3 p p O p p ª _{ } º « » ¬ ¼ . It results in

 

1

 

2 coth 1 3 p p  pO p . While deferring in Eq. (19), it comes

 





 

 

2 2 2 2 2 1 3( ) ( ) 3 ( ) 1 3 3 ( ) 1 1 ( ) p p O p Q p p p O p p p O p p p p O p D E D D D E ª º  _«  _» ¬ ¼ ­ ª _ º_{ } ½ ® _{«¬ »¼} ¾ ¯ ¿ ª º  _¬  _¼     (B1) ( )

Q p has a simple pole and the residue Res

 

Q, 0 1. 2. Calculation of the residues for the solutions p from n

the equation g p( ) 3E_» pcoth

 

p    1º_¼ D p 0 In the case of a simple pole Ÿg p( )



pp_n



h p( )

( ) ( ) ( n) ( ) g pc h p  pp h pc , from which g pc( n) h p( n) because (g pn) 0

 

 

3 0, coth 3 n n n n p g p p p D E E    (B2)

 

2

 

coth 1 coth ( ) 3 1 2 2 p p g p p Eª  º c «  » « » ¬ ¼ (B3) Thus,





 

 

 

2 2 Res ( )e , 3 coth 1 e 2 3 coth 1 coth 3 2 n p n p n n n n n n n n n Q p p p p p p p p p p p W W D E E ª _º ¬ ¼ ª º  _¬  _¼ ­ ½ ° _{ª º} °    ® _{¬ ¼} ¾ ° ° ¯ ¿  (B4) According to Eq. (B2) Dp_n 3E_¬ª p_ncoth

 

p_n 1º_{¼ .} Therefore,

 

^

`

 

 

 

2 2 Res ( )e , 3 3 coth 1 coth 1 e 2 3 coth 1 coth 3 2 n p n p n n n n n n n n n n Q p p p p p p p p p p p p W W E E E ª _º ¬ ¼ ª º ª º  _¬  _{¼ ¬}  _¼ ­ ½ ° _{ ª  º} ° ® _{¬ ¼} ¾ ° ° ¯ ¿  (B5)

 

 

 

2 2 2 Res ( )e , 6 coth 1 e 2 coth 1 coth 3 n p n p n n n n n n n n Q p p p p p p p p p p W W E ª _º ¬ ¼ ª º  _¬  _¼ ­ ½ ° _{ª º} °    ® _{¬ ¼} ¾ ° ° ¯ ¿  (B6)

 

 

 

2 2 n Res ( )e , coth 1 9 e 2 3 coth 1 coth 3 2 p n p n n n p n n n n n Q p p p p p p p p p W W E ª _º ¬ ¼ § _ · ¨ ¸  ¨_© ¸_¹ ­ ½ ° _{ ª  º} ° ® _{¬ ¼} ¾ ° ° ¯ ¿  (B7)

 

 

 

2 2 Res ( )e , coth 1 9 e coth 1 coth 3 1 2 n p n n n p n n n n n Q p p p p p p p p p W W E ª _º ¬ ¼ § _ · ¨ ¸  ¨_© ¸_¹ ­ _ ª _ º½ ° _{¬ ¼}° ® ¾  ° ° ¯ ¿  (B8)

 





 

 

2 Res ( )e , 9 coth 1 e coth 1 coth 3 1 1 2 n p n p n n n n n n n Q p p p p p p p p p W W E ª _º ¬ ¼   ­ ª _ º ½ ° _{¬ ¼} ° ®  ¾ ° ° ¯ ¿   (B9)

When we sum the terms, we obtain

 

 

 

1 coth 1 9 e ( ) 1 1 coth 1 3 coth 1 2 n n n p n n _{n n} n n n p p p Q p p p p p W W E f § _ · ¨ ¸  ¨_© ¸_¹  ª _ º « »   « » ¬ ¼

¦

(B10) where p are the roots of the following equation n

 

3E_¬ª p_ncoth p_n   1_¼º D p_n 0 (B11)

APPENDIX C Restriction to the real domain

From Eq. (20) or (B10), the restriction on the real domain is carried out as follows.

By considering Eq. (B11), for a particular value p _n

 

_n 3 _ncoth

 

_n 1 _n 0 g p Eª_¬ p p   º_¼ D p (C1) where

 

n 3 nch

   

n sh n



n



sh

 

n 0 g p Eª_¬ p p  p º_¼ Dp p (C2) If p_n iq_nwhere qnR from which Eq. (C1) is written,

while noting (M iq_n) g p( _n)

 

 

 

 

2

 

3 ch sh sh n n n n n n iq iq iq iq iq iq O M Eª_¬  º_{¼ ¬}ªD º_¼ (C3)

 









 



2 e e 3 e e e e 0 n n n n n n iq iq n n iq iq iq iq n iq q iq M D E       ª _{  } º ¬ ¼ (C4)

 





 

 

 

2 2 sin 3 2 cos 2 sin 0 n n n n n n iq q i q iq q i q M D E  ˜  ª ˜  ˜ º ¬ ¼ (C5)

 

^



2



 

 

 

`

2 sin 3 cos sin 0

n n n n n n

iq i q q q q q

M D  Eª_¬  º_¼

(C6) Let us assume that

 

 

2 n n iq q i M \ R , it comes

 



2



 

 

3 sin 3 cos 0 n n n n n q q q q q M D  E  E (C7) According to Eq. (C1),

 

2 coth 1 1 3 3 n n n n p q p p D D E E   _  _{ because} 2 n n p  (C8) q From Eq. (B7)

 

2 2 cot 1 1 3 3 n n n n q p q q D D E E   _   _{ (C9)}

Thus p_ncoth

 

p_n q_ncot

 

q_n . While replacing

 

coth

n n

p p by q_ncot

 

q_n where q_nR , we obtain the

following result

 

 

 

2 2 2 1 2 cot 1 9 e ( ) 1 cot 1 1 3 cot 1 2 n q n n n n n n n n n q q q Q q q q q q W W E  f §  ·  ¨ ¸ © ¹  ª  º  _«  _» ¬ ¼

¦

(C10)

Let us recall that q are solutions of Eq. (C7), n qnR ,