AN ANALYSIS SORPtion.pdf

8/14/2019 AN ANALYSIS SORPtion.pdf

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AN ANALYSISSORPTION

OF A NONISOTHERMAL ONE COMPONENTIN A SINGLE ADSORBENT PARTICLE

A SIMPL IFI ED MODEL

A. BRUNOVSKA and 3. ILAVSKkDepartment of Organic Technology, Slovak Technical University, 880 37Czechoslovakia

Bratislava, Jgnska Street I,

andV. HLAVACEK

Department of Chemical Engineering, Institute of Chemical Technology, 166 28 Prague 6, SuchbLtarova Street1903, Czechoslovakia(Received 21 December 1979; n revised orm 24 Apr i l 1980; accepfed 19 M ay 1980)

Abstract h this paper on approximation is described making it possible to calculate temporal temperature andadsorbed amount profiles during an adsorption process in a single pellet. The suggested model includes heattransfer in an external film and mass transfer inside a porous structure. The profiles calculated from theapproximation and rigorous model are compared and it is shown that for a majority of operating systems thesimplified model can be used and approximates very wel l the results obtained from an exact model.

INTRODUCTION 4The model developed by us recently[ 1,21 takes into j nRpC, g = hR=D )g -AH.,)x-Rconsideration both external and internal gradients of -47rPh,(T- To). (2)temperature. A detailed numerical analysis of themode1[1,2] as well as experimental observations[3] in- The initial and boundary conditions are:dicated that the internal temperature gradients are lowand may be neglected, i.e. the pellet is supposed to t=O,O~xaR:a=a~,T=~,operate isothermally. Moreover, the complete modelresults in a very high computer time expenditure. r>O,x=R : a = a* co, T), (31

We wish to present in the following section a practicalsimpler model, which may be used for describing of most x=0 :$o.sorption processes occurring in a single pellet. The sug-gested model takes into consideration the heat transfer For a general equiIibrium isotherm c = c a, T) eqn (1)resistance in the external film and mass transfer resis- may be rewritten:tance in the porous structure. The goal of this paper istwofold (i) to study the applicability of this model and (ii)to demarcate domains of its validity in important prac-tical cases. (4)

2. MATHEMATI CAL MODELFollowing approximations are introduced(1) Intraparticle mass transfer may be described bymeans of the Ficks law, the diffusivity is assumed

concentration and temperature independent.(2) One component system is considered.(3) Heat transfer resistance is concentrated in the

external fihn.(4) Constant average heat capacities characterize thesystem, heat of adsorption is assumed independent ofconcentration and temperature.

(5) There is an equilibrium between the gas phase andsolid phase concentrations everywhere in the particle.(6) Only the gas phase diffusion is considered.For the assumptions mentioned above we can write:

(1)

Equations (2)-(4) may be rendered dimensionless:

6)and since Q = p q)Jl f+ - K*

g= 3&9_, -0e (8)7=o,o


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An analysis of a nonisothermal one-component sorption 125The occurrence matrix can be rearranged to 4. COMPARISONOF RIGOROUS AND APPROXIMATION MODEZ

In Figs. l(a)-3(b) a comparison of temporal tem-xxx perature and adsorbed amount dependences is shown.xxxx The values of the parameters are the same as in our

xxxx former paper [ ]. Since in the approximation model the. . . parameters Le and Bi do not enter in the descriptionXX-XX individually but as a product Z_QX Bi we can easily

xxx follow the effect of both these parameters on the qualityxxx -. of the approximation if the value of the product J+ex Biis constant. Our calculation have been performed for

A special algorithm for band matrices was used to solve cL.e X Bi = 0.05. Based on the numerical calculations andthis set of linear equations[4]. for the temperature of the ambient gas 70C the over-

0.7s

06.e

02

O.l-

0+0 Olr 0.2 03 OL 05 08 r o.7 Od 0e9 11Fig. I(a). Adsorption in a spherical particle (cLe = I, Bi = 0.05, u = 10, p = 0.3, K, = 0.7, K?= 0, 6 = 0.05).Temperature dependence. - approximation model; - - - - -, integral mean value calculated from rigorousmodel:-.- : -, rigorous model 15 = 0); - . - ..., rigorous model (f = I).

040 03 0.2 03 OC 05 0.6 0.7z 0.8 OS 1 1.r

Fig. l(b). Adsorption in a spherical particle. Adsorbed amount dependence. p, approximation model:- -, rigorous model.


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126 A. BRUNOVSK.~ et al0 7

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Fi i . 2(a). Adso rptio n in a spher ical parti cle cLe=O.S, Bi= 0.1, a = 10, @= 0.3, K, = 0.7, K ? = 0, 8 = 0.05).Temperature dependen ce. - - - - -model; - - I approximation model; -, integral mean value calculated from igorous-, rigorous model (6 = 0); - .. - .., rigorous model (5 = 1).

0 0.1 02 03 OA 05 0.6 0.7 0.8 0.9 1.lT 12Fig. 2(b). Adsorption in a spherical particle. Adsorbed amount dependence. -, approximation model;

-----. rigorous model.

heating of the particle surface is -50C. From the figuresmay be inferred that the agreement between the rigorousand approximation model is very good. For a stronglynonisothermal particle (see Fig. 3a) the approximationmodel is capable of describing the temperature depen-dences with accuracy better than 10% with respect to therigorous model. In this particular case the maximumtemperature difference in the particle is (for T, = 70C)

AT = 30C. The agreement between the adsorbed amountdependences calculated from both models is also verygood.

A comparisoi of the computer time expenditure in-dicates that the approximation model results in a lowercomputer time. In the approximation model a longer timestep may be used. In the example discussed the rigorousmodel requires a time step f I x lo- while for the ap-


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OR

An analysis of a nonisothermal one-component sorption 127

Fig. 3(a). Adsorption in a spherical particle (cL0 = 0.05, Bi = 1, a = 10, fi =0.3, K, = 0.7, K* = 0, 6 = 0.05).Temperature dependence. - approximation model; - - - - -, integral mean value calculated from rigorousmodel:-~-~-,rigorousmodel(~=O):-~~~-~~~,rigorousmodel(~=l).

10 0.1 02 0.3 Olr 05 0.6 07 6.6 6.9 1 1. 1I

2Fig. 3(b). Adsorption in a spherical particle. Adsorbed amount dependence. -, approximation model:- .., - .... rigorous model (f = I).

proximation model a time step % I x lo- was sufficient. Moreover the computer time for a simplified model isAs a result the computer time for the approximation lower by a factor ~10.model is lower by a factor ten.

5. CONCLUSIONSFor nonisothermal sorption processes occurring in asingle adsorbent particle a simplified model was

developed which approximates the temperature profileswithin the particle by an integral mean value. This ap-proximation yields very good results and can be used forcalculation of nonisothermal sorption processes.CES Vol. 36. No 1-l

a4a8a.Bic

NOTATIONsorbate concentration in particleinitial sorbate concentration in particleequilibrium sorbate concentration in particlemonolayer capacity in the LongmunequationBiot numbergas phase concentration


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128

kLeQ4RTTitX

A. BRUNOVSK~ et nl.gas phase concentration in the bulk flowspecific heat of sorbentdiffusion coefficientheat of adsorptionspace step length, gas-to-solid heat transfer

coefficienttime step lengthLewis numberdimensionless gas phase concentrationdimensionless sorbate concentration in par-

ticleradius of particletemperatureinitial temperaturetimespace coordinate

reek sym olsa dimensionless adsorption energy

dimensionless adiabatic temperature risedimensionless parameterporosity of particledimensionless temperature risedimensionless parameters in adsorption iso-thermeffective thermal conducting of sorbent

particledensity of sorbent particledimensionless space coordinatedimensionless time

REFERENCES[I] Brunovska A. et al., Chem. Engng Sci. 1978 33 1385.[2] Brunovska A. et at. C/tern. Engng Sci. 1980 35 757.[3J llavskq J., Brunovskb A. and HIavQEek V.. Chem. EngngSci. inpress.[4] NEmec J., Personal communication (Prague 1977).

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AN ANALYSIS SORPtion.pdf