absorption simulation

8/10/2019 Absorption simulation

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Gas absorption with rst order chemical reaction in alaminar falling lm over a reacting solid wall

M. Danish * , R.K. Sharma, S. AliDepartment of Chemical Engineering, Aligarh Muslim University, Aligarh, UP, India

Received 1 December 2005; received in revised form 1 November 2006; accepted 16 February 2007Available online 12 March 2007

Abstract

In the present work, a general case of gas absorption with rst order irreversible chemical reaction in a liquid lm, forlaminar ow over a solid wall, has been analyzed theoretically. First order chemical reaction between the diffused soluteand the wall is also considered. Laplace transform followed by power series method has been applied to solve the govern-ing equations. Thereafter, the obtained analytical solution of the developed general model has been successfully veried byan explicit numerical scheme. The general model has also been reduced to six simplied cases, tackled by previous workersand an excellent agreement in the solutions is observed. Moreover, the results are validated by the experimental data avail-able in the literature. The obtained concentration proles in both the phases have been used to nd the absorption ratesand enhancement factor.

Further, theeffectof various parameters on concentration proleof solute, enhancement factorand eigen-valueshave beenanalyzed.Existenceof positive eigen-values for thecounter-currentabsorption is also describedwhich wasnotpointedoutbyanyof thepreviousresearchers. It is concludedthat in a co-current system, forHattanumbergreater than 7 approximately, thevariation in enhancement factor follows a linear trend on a loglog plot and that there is no effect of rate constants for highervalues of Hatta number. It is further shown that the reaction at solid wall has appreciable effect on the absorption rate.

2007 Published by Elsevier Inc.

Keywords: Absorption; Falling lm; Chemical reaction; Mass transfer; Enhancement factor; Laplace transform

1. Introduction

Physical and chemical gas absorption is one of the most important separation processes and is widelyemployed in chemical and allied industries either for separating undesirable components from a gas or forthe manufacturing purposes of some important chemicals. Generally, the former one is due to the stringentenvironment norms, set nowadays and the latter one is the requirement of industry either for quality or designpurposes. Because of these vital reasons, a vast amount of theoretical as well as experimental research materialis available and still increasing.

0307-904X/$ - see front matter 2007 Published by Elsevier Inc.doi:10.1016/j.apm.2007.02.019

* Corresponding author. Tel.: +91 5712721152; fax: +91 5712700042.E-mail address: [email protected] (M. Danish).

Available online at www.sciencedirect.com

Applied Mathematical Modelling 32 (2008) 901929www.elsevier.com/locate/apm
mailto:[email protected]:[email protected]


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Nomenclature

A i series coefficients for the solution of dimensionless concentration in the liquid lma n nth coefficient of the series solutionB dimensionless gas phase or surface resistance parameter equivalent to Biot number for mass

transferB i series coefficients for the solution of dimensionless concentration in the gas phaseC a concentration of the solute A in the liquid phase, g/cm

3

C A dimensionless concentration of the solute A in the liquid phaseC A Laplace transform of C Ac0 1st term in series solution of C A corresponding to k0 0C 0 dimensionless solute concentration in the entering liquidC a0 solute concentration at gas liquid interface, g/cm

3

C A0 dimensionless solute concentration at gasliquid interfaceC g concentration of the solute A in the gas phase, g/cm

3

C g 0 initial gas phase concentration of the solute A, g/cm3

C G dimensionless gas phase concentration of the solute AC G Laplace transform of C G D AB diffusion coefficient of solute A in the liquid phase, cm

2 /s D AG diffusion coefficient of solute A in the gas phase, cm

2 /sE enhancement factor R R01 F i series coefficients for the solution of absorption rate f series function of gk G mass transfer coefficient for innitesimally small gas liquid interface, cm/sk 1 rate constant for 1st order chemical reaction in liquid lm, 1/sk 2 rate constant for 1st order chemical reaction at solid wall, cm/sK 1 dimensionless rate constant for rst order chemical reaction in the liquid lm k 1d

2

D AB

K 2 dimensionless rate constant for rst order chemical reaction between solute and dissolving solid

wall k 2d D AB log logarithm of a quantity on base10m the amount of gas absorbed per unit area after a contact time s, g/m 2

ffiffiffiffiffiffi M 0p a form of Hatta number ffiffiffiffiffiffiffiffiffiffiffiffip4 K 1Z q ffiffiffiffiffiffiffi M 00p a form of Hatta number independent from rate constant ffiffiffiffiffiffiffip4 Z q R total mass rate of absorption, g/s R01 total mass rate of absorption without reaction in an innitely deep, stagnant liquid, g/ss Laplace transform variableV g gas phase constant velocity, cm/sV y velocity in y-direction, cm/s

V y

0

V z average linear velocity of the liquid lm in the direction of ow, cm/sV z linear velocity of the falling lm in the direction of ow, cm/sV max maximum linear velocity of the liquid lm in the direction of ow, cm/s y coordinate along the thickness of the liquid lm from interface, cmY dimensionless coordinate along the thickness of the liquid lm from wallz coordinate along the direction of ow of the liquid lm, cmZ dimensionless coordinate along direction of ow of the liquid lm

Greek symbolsd thickness of the liquid lm, cmd1 width of the gas ow channel, cm

902 M. Danish et al. / Applied Mathematical Modelling 32 (2008) 901929


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The process of isothermal gas absorption with and without chemical reaction in the liquid lm for laminarow over a solid surface has been analyzed by many researchers. Higbie in 1935 [1] carried out the analysis of gas absorption process by assuming penetration theory model for short time of contact and Pigford in 1941 [2]attempted the same by using parabolic velocity prole for long contact time. Later on, Pigford extended thiswork for various other situations [38]. Nysing and Kramers [9] carried out an experimental study of CO 2absorption in solutions of Na 2 CO 3 /NaHCO 3 and K 2 CO 3 /KHCO 3 accompanied by 1st order chemical reac-tion in a wetted wall column, to verify the penetration theory for short contact time. Olbrich and Wild [10]and Walker and Davis [11] extended the work of physical gas absorption for several simple geometries.Absorption with nite gas phase resistance was further demonstrated by Tamir and Taitel [12] and Guptaet al. [13]. The same problem was studied by Tamir and Taitel [14], Stepanek and Achwal [15] and Bestand Horner [16], but with the addition of rst order chemical reaction in the liquid lm. Continuing withthe above problem, Datta and Rinker [17] provided solution of gas absorption for short time of contact, tak-ing into account the nite gas phase resistance and axial decrease of the solute concentration in gas phase withrst order chemical reaction in the liquid lm. Later, Riazi and Faghri [18] and Riazi [19] stretched the abovework for zero order chemical reaction in the liquid lm. Elperin and Fominykh [20] in 2003, derived the coef-

cients of mass transfer during chemical absorption from a single Taylor bubble in the approximation of thethin concentration boundary layer in liquid phase and veried the results with those of Nysing and Kramer [9].Recently, a comprehensive model of gas absorption with chemical reaction has been developed by Sharma [21]in 2004 in which most of the above cases have been included.Sharma, Danish and Ali [22] in 2006, solved theoriginal penetration theory model of Higbie [1] by using Similarity transform, Laplace transform and Sinetransform methods. Various parameters considered by different authors for the isothermal gas absorptionproblem along with chemical reaction are listed in Table 1 .

The case of reaction at the solid wall has not been considered so far. In the present work, an attempt hasbeen made to obtain an analytical solution for a more general case of absorption with nite resistance andaxial decrease of the solute concentration in gas phase and irreversible rst order chemical reaction in liquidlm in laminar ow over a reacting plane wall. A combined method of Laplace transform and power series hasbeen applied to arrive at the concentration proles of both the phases, which are eventually used to evaluateabsorption rate and enhancement factor. Moreover, the effects of gas ow rate, gas phase resistance and reac-tion rate constants on concentration proles and enhancement factor have been shown and discussed in detail.Subsequently, various simplied models, studied by different workers have been derived. To get more con-dence over the derived general analytical solution, a comparison with the results obtained by an explicitnumerical scheme has been carried out. By reducing the general analytical solution for a simplied case of CO 2 absorption in Na-buffer solution accompanied by 1st order chemical reaction, the expression for absorp-tion rate has been derived. Thereafter, the absorption rate is compared successfully with the available exper-imental data of Nysing and Kramer [9] and Elperin and Fominykh [20]. This substantiates the developedmodel and its analytical solution.

The rest of the paper is organized as follows: In Section 2, the partial differential equations are formulatedfor the general gas absorption model. Section 3 describes the solution for general case using Laplace transformand power series methods; this section also illustrates the derivation of the expression for enhancement factor

g dimensionless distance along the thickness of the liquid lmki i th eigen-value for the solution of dimensionless liquid phase concentrationq density of the owing liquid, g/cm 3

x ratio of liquid to gas volumetric ow ratess contact time ( z =V max

, s

/ m absorption rate, g/cm 2 s.

Subscriptsi; j index variablesn nth value of a parameter

M. Danish et al. / Applied Mathematical Modelling 32 (2008) 901929 903


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and reduction of the general model and its solution to various simplied cases. In the same section, a compar-ison between the analytical and numerical solutions has been included and the results obtained from analyticalsolution are veried with the available experimental data. Section 4 presents a detailed discussion on theobtained results and is followed by Section 5, which concludes the whole study.

2. Physical model and model equations

A thin liquid lm, in fully developed laminar ow, is falling down over a dissolving solid wall and isexposed to a gas, which ows either co-currently or counter-currently with respect to the lm; the schematicview is shown in Fig. 1. Mass transfer of solute takes place from gas to liquid phase. Time of contact betweensolute and liquid is long enough to presume a parabolic velocity prole. Steady state conditions are assumed to

prevail. A constant gas phase resistance is present and the concentration of solute in gas phase varies axiallydue to the absorption by liquid. Irreversible rst order chemical reaction takes place between the solute andliquid. It is further assumed that the diffused solute in liquid phase also reacts with the wall with the sameorder of reaction; physical properties of both the uids remain constant.

The law of conservation of mass is applied on an innitesimally small element in the liquid lm as well as inthe gas phase, as shown in Fig. 1, and the governing relations [23,24] are obtained as below.

2.1. Steady state modeling equations for liquid phase

The steady state modeling equation for liquid phase is obtained by applying the species balance over aninnitesimally small element in liquid phase as shown in Fig. 1. In doing so, it is assumed that there is novelocity component in y-direction and the axial diffusive transport is negligible in comparison to convectivetransport. The ow is fully developed and the density of the liquid and diffusivity of the solute are constant.Incorporating these assumptions and putting the well-known parabolic velocity prole in the thus obtainedmass balance equation, we get the following linear partial differential equation along with the relevant bound-ary conditions:

V max 1 y 2

d2 o C ao z D AB o2C a

o y 2 k 1C a C 0: 1

B.C.1: C a C 0 at z 0.B.C.2: D ABo C ao y k G C g C a at y 0.

B.C.3: D ABo C

ao y k 2C a C 0 at y d.

Table 1Various parameters considered by different workers

No. Authors Years d1 d f K 1 K 2 B x n1 Higbie 1935 X NA 2 Pigford 1941 NA X 3 Nysing and Kramers 1958 X NA X 14 Olbrich and Wild 1969 NA X 5 Tamir and Taitel 1971 NA X X 6 Tamir and Taitel 1975 NA X X X 17 Stepnek and Achwal 1976 NA X X X 18 Best and Horner 1979 NA X X 19 Riazi and Fagri 1985 NA X X 010 Datta and Rinker 1984 X NA X X X 111 Gupta et al. 1986 NA X X X 112 Elperin and Fominykh 2003 X NA X 0, 113 Present analysis 2006 NA X X X X X 1



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2.2. Steady state modeling equations for gas phase

The species balance over an innitesimally small element in gas phase, shown in Fig. 1, yields Eq. (2), withthe assumptions that gas ows with uniform velocity and the contribution of axial diffusion is insignicant incomparison to convective transport. After some simple mathematical manipulations one easily arrives at thefollowing 1st order linear ordinary differential equation:

dC g d z

k G d1V g C g C a: 2

The associated B.C. is:

B.C.1: C g C g 0 at z 0.The preceding equations and the respective B.Cs are non-dimensionalized, using the following denitions

for dimensionless variables:

g d y

d ; C A

C a C 0C g 0 C 0

; C G C g C g 0C g 0 C 0

; Z D ABV max d

2 z ;

B

k G d

D AB; K 2

k 2d

D AB; K 1

k 1d

2

D ABand x

V max d

V g d1:

Gas LiquidC go C o

V g = constant, Plug Flow

V z (y)

Y ,

z, Z

Wall

C g (z) = constant C g (z)

C a (z,0)

Reaction atwall =

k 2.(C a - C o)

1 Interface

C a (z,y)g

g g AG z z

dC V C D

dz

gg g AG z z

z z

dC V C D

dz+ +

( )G g ak C C

P1P3

P4

P2

( ) a z a AB z z

dC V y C D

dz

( ) a z a AB z z

dC V y C Ddz + z

a

y a AB y y y y

C V C D

y+ +

1( )a ok C C

y

z

1 2

3

4

:

: ( )

: ( )

g g

aG g a AB

a AB a o

At P C C At P

C At P k C C D

y

C At P D k C C

y

= =

=

a y a AB y

y

C V C D

y

y

2

Fig. 1. Schematic diagram of gas absorption process over a dissolving solid wall with axial variation of gas phase concentration.



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The governing equations thus take the following form:

For liquid phase :

g2 go C Ao Z

o 2C Ao g2 K 1C A: 3

B.C.1: C A 0 at Z 0 8g.B.C.2: o C Ao g B1 C G C A at g 1 8Z .B.C.3: o C Ao g K 2C A at g 0 8Z .

For gas phase :o C G o Z x BC G C A 1jg1 at g 1 8Z :

Using B.C.2 for liquid phase, the above equation can be rewritten as:o C G o Z x

oC Ao g g1 at g 1 8Z : 4

B.C.1: C G 0 at Z 0.Eqs. (3) and (4) including all the B.Cs constitute the general model. The parameter x can be regarded as the

ratio of the maximum possible absorption capacity to the rate at which gas enters into the channel [13]. It maybe noted that the gas phase concentration C g 0, at the top of the column, is the inlet gas concentration for co-current ow and the exit concentration when the gas ows counter-currently. The positive sign in Eq. (4) refersto the counter-current ow, while the negative sign refers to the co-current ow. C 0 is the initial concentrationin the liquid lm i.e. concentration of solute in the liquid lm before absorption starts.

3. Solution by Laplace transform with power series method

After taking the Laplace transform of model Eqs. (3) and (4) along with their B.Cs, using the followingdenition [25,26]

C A Z 10 C Ae sZ dZ 5one obtains the equations, in transformed form as:

For liquid phase :

g2 g sC A d2C Adg2 K 1C A: 6

B.C.2: dC Adg B 1 s C G C A at g 1 8Z .B.C.3: dC Adg

K 2C A at g

0

8Z .

For gas phase :

sC G x dC A

dg g1: 73.1. Evaluation of liquid phase concentration prole

The transformed ordinary differential equation (6) is solved by using a power series solution of the form:

C A

X1

n0an gn:

8



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Substituting the above in Eq. (6) leads to the following recurrence relation:

an s

nn 12an 3 an 4

K 1 an 2nn 1

; 9where, a1 K 2a0, a2 K 12 a0 and a3 s3 K 1 K 26

a 0.

Hence, one nds an independent series solution of C A as:

C A a01 K 2:g K 12 g2 s3 K 1 K 26 g3 or

C A a0 f ; where; f is a function of s and g: 10

The coefficient a0 is obtained from B.C.2 as:

a0 B

s Bx s B s Bx

f o f o g g1

and the solution in transformed form becomes:

C A 1 s f s Bx

B

o f o g g1 f , : 11

After applying the residue theorem [26,27], the following expression for inverse Laplace transform of C A isobtained:

C A X1

i0lim s! ki s ki

s Bx B

o f o g s f g1 f e sz ; 12

where, ki denote the eigen-values which are the zeros of the following relation:

s Bx B

o f o g s: f g1 0: 13

Applying LHospital rule to evaluate the limit in Eq. (12), the required solution for concentration prole in theliquid lm is found to be:

C A X1

i0 Ai f eki Z ; where; Ai 1

1 B

o f o g f

s Bx B

o 2 f o so g s

o f o s g1 ski : 14

Fig. 2 demonstrates the liquid phase concentration proles C A for x 1:0, K 1 0:1, K 2 0:1 and B 106 atdifferent Z .3.2. Evaluation of gas phase concentration prole

To nd the gas phase concentration prole, the above expression for liquid phase concentration i.e. Eq.(14), is substituted in Eq. (4) to get

o C G o Z x X1i0 Ai

o f o g g1 ekiZ 15

and is integrated. Alternatively, one can substitute Eq. (11) in Eq. (7) and obtain the gas phase concentrationby taking its inverse Laplace transform.

The subsequent steps are not as straight forward as were in the case of liquid phase and depend upon themagnitude of rst eigen-value. The two cases, covering all the intricacies that arise in such conditions, aredescribed below:

3.2.1. Case (a): k0 60These types of cases arise, when velocities of both phases have non-zero but nite values (i.e. x6

0) andnone of the two reaction rates are zero ( K 1 60; K 2 60).



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On integration with respect to Z , with limits from 0 to Z , Eq. (15) leads to:

C G x X1

i0 Aiki

o f o g g1eki Z X

1i0

Aiki

o f o g g1" #:

This may be rearranged as:

C G X1

i0 Bi1 eki : z ; where; Bi

x Aiki

o f o g g1

; i 0; 1; 2 . . . 16Eq. (16) represents the concentration prole for solute in gas phase for those cases where magnitude of rsteigen-value is non-zero. Fig. 3 portrays the concentration prole of solute in gas phase for x 1:0, K 1 0:1; K 2 0:1 and B 106. Table 2 presents the eigen-values and series coefficients of the same, for dif-ferent values of x with K 1 0:1; K 2 0:1 and B 106.3.2.2. Case (b): k0 0Following two sub-cases generate zero as their rst eigen-value:

(i) All the absorption models with stationary liquid lm and/or very high gas velocity (i.e. x 0), with/without any of the two reactions and for any value of gas phase resistance.(ii) All the absorption models with no reactions ( K 1 K 2 0) and with any values of x and gas phaseresistance.

Here, only the second sub-case is described and the same approach can be adopted for the rst one.

3.2.2.1. Sub-case: k0 0.For such systems, the liquid phase concentration prole C A may be expressed as:C A c0 X

1i1

Ai f ekiZ ; where; c0 A0 f eki Z j sk00: 17

Also, A0 is the series coefficient evaluated for the rst eigen-value k0 0at g 1 or c0 1= 1 x o 2 f o so g

h ig1; s0.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DISTANCE FROM SOLID WALL,

D I M E N S I O N L E S S L I Q U I D P H A S E

C O N C E N T R A T I O

N ,

C A

z=0.01 z=0.05 z=.1 z=.2 z=.3 z=.4 z=.5 z=.6 z=.7 z=1

B = 10 6 , = 1.0K 1 = 0.1, K 2 = 0.1

Fig. 2. Development of concentration prole in the liquid lm, for nite gas phase resistance and axial variation of gas phaseconcentration for counter-current system over a dissolving wall under rst order chemical reaction.



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Consequently, the required liquid phase concentration prole is expressed as:

C A 1

1 x o2 f

o so gh ig1; s0 X1

i1 Ai f eki z 18

and from Eq. (4), the gas phase concentration prole is obtained as:

C G X1

i1 Bi1 eki : z ; where; Bi

x Aiki

o f o g g1; i 1; 2; 3 . . . 19

The above mentioned Eq. (19) represents the concentration prole of solute in gas phase for those cases wheremagnitude of rst eigen-value is zero and the gas possesses an axial variation in the solute concentration.Fig. 4 shows the concentration prole of solute in gas phase for x 1:0, K 1 K 2 0 and B 106.

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DIMENSIONLESS AXIAL DISTANCE, Z

D I M E N S I O N L E S S G A S P

H A S E

C O N C E N T R A T I O N , C

G

= 0.4

= 1.0

K1 = 0.1K2 = 0.1

B = 10 6

Fig. 3. Gas phase concentration prole for counter-current ow.

Table 2Eigen-values and series coefficients for counter-current gas absorption system at various range of axial variation parameter x , with K 1 0:1, K 2 0:1 and B 106n x 0:0 x 0:4 x 1:0

ki Ai Bi ki Ai Bi x F i ki Ai Bi x F i0 0.0000 0.8679 0.0937 1.1048 1.3005 0.4090 1.6913 2.13651 5.6465 1.1944 0.0000 4.5732 1.4408 0.24415 2.8532 2.0393 0.99242 40.311 0.5297 0.0000 39.285 0.5414 0.0262 37.739 0.5558 0.06813 106.97 0.3528 0.0000 105.95 0.3557 0.0096 104.42 0.3590 0.02444 205.64 0.2687 0.0000 204.62 0.2698 0.0049 203.09 0.2711 0.01255 336.31 0.2189 0.0000 335.29 0.2195 0.0030 333.76 0.2201 0.00766 498.97 0.1857 0.0000 497.95 0.1860 0.0020 496.42 0.1864 0.00517 693.64 0.1619 0.0000 692.62 0.1621 0.0014 691.09 0.1623 0.00368 920.30 0.1439 0.0000 919.28 0.1440 0.0011 917.76 0.1442 0.00279 1178.9 0.1298 0.0000 1177.9 0.1299 0.0008 1176.4 0.1300 0.002110 1469.4 0.1178 0.0000 1468.3 0.1187 0.0006 1466.7 0.1187 0.0017



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3.3. Enhancement factor

The inuence of chemical reaction and nite lm thickness on the absorption is best summarized by com-paring the actual absorption rate R , with that of an innitely deep stagnant liquid with no chemical reaction R01 , as used in [16]. The ratio R= R

0

1 is equal to the well-known enhancement factor E . The absorption ratefrom penetration model, for physical absorption in an innitely deep stagnant liquid exposed to a gas for atime period z =V max , is given by:

R01 C g 0 C 0dV max W ffiffiffiffiffiffi4Z pr : 20In the present condition, the total amount of gas absorbed per unit elapsed time is evaluated from the diffusiveux at interface and is given as:

R D ABW Z z

0

o C ao Y Y d d z : 21

Yet again, to determine the derivative term in the above expression, we consider two different cases dependingupon the magnitude of rst eigen-value.

3.3.1. Case (a): k0 60As explained earlier, these types of cases arise when both the uids have non-zero but nite velocities(x 60) and none of the reaction rates are zero ( K 1 60; K 2 60). For such conditions, we directly use Eq.(14) for liquid phase concentration prole and substitute it in Eq. (21) to get the following expression forabsorption rate:

R

C a0 C 0dV max W X1

i0 F i1 eki Z ; 22

where, F i Aikio f o g g1 Bix ; i 0; 1; 2 . . .A little use of mathematics leads to the following expression for enhancement factor E

E R R01

P1i0 F i1 eki Z

ffiffiffiffiffiffi4Z

pr : 23

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


D I M E N S I O N L E S S G A S P H A S E

C O N C E N T R A T I O N

, C

G

= 1.0

Fig. 4. Development of gas phase concentration prole for counter-current ow for x 1:0, K 1 0:0, K 2 0:0 and B 106 at Z 1:0.



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In Fig. 5, the enhancement factor E has been plotted on a loglog graph as a function of the modulus ffiffi M 00p (a form of Hatta number independent of rate constant), for various values of x with K 1 K 2 0:1 and B 106. The modulus is dened as: ffiffiffiffiffiffiffi M 00p ffiffiffiffiffiffiffiffiffiffiffiffi

p4 Z r : 24

3.3.2. Case (b): k0 0As argued earlier, two situations yield zero as their rst eigen-values i.e.(i) x 0,(ii) K 1 K 2 0.We now proceed to nd the expression for enhancement factor E for the rst case only, since for the second

situation, it does not have any meaning. By rearranging the liquid phase concentration prole, as given in Eq.(17) and following the same procedure, as adopted in Section 3.2.2.1 the expression for c0 is found to be:

c0 cosh ffiffiffiffiffiffi K 1p g

K 2 sinh ffiffiffiffiffiffi K 1p g ffiffiffiffiffiffi K 1p

ffiffiffiffiffiffi K 1p B

sinh

ffiffiffiffiffiffi K 1p

K 2 cosh


B cosh


K 2 sinh



:

Finally, one obtains the required concentration prole for liquid phase as:

C A cosh ffiffiffiffiffiffi K 1p g

K 2 sinh ffiffiffiffiffiffi K 1p g ffiffiffiffiffiffi K 1p ffiffiffiffiffiffi K 1p B sinh ffiffiffiffiffiffi K 1p K 2 cosh ffiffiffiffiffiffi K 1p B cosh ffiffiffiffiffiffi K 1p

K 2 sinh ffiffiffiffiffiffi K 1p ffiffiffiffiffiffi K 1p X

1

i1 Ai f eki Z : 25

With the substitution of C A from the above equation, we can evaluate the gas absorption rate as: R

C a0 C 0dV max W Z

K 1 sinh ffiffiffiffiffiffi K 1p K 2 cosh ffiffiffiffiffiffi K 1p ffiffiffiffi K 1p

B ffiffiffiffiffiffi K 1p sinh ffiffiffiffiffiffi K 1p K 2 cosh ffiffiffiffiffiffi K 1p ffiffiffiffiffiffi K 1p cosh ffiffiffiffiffiffi K 1p K 2 sinh ffiffiffiffi K 1p X

1

i1 F i

1 eki Z

where; F i

Ai

ki

o f o

g g1

; i

1; 2; 3 . . .

26

0.001

0.01

0.1

1

10

100

1000

10000

0.01 0.1 1 10 100

''

E N H A N C E M E N T F A C T O R

, E

= 1.0 = 0.4

= 0.0

= -0.4

= -1.0 = -2.0

Fig. 5. Loglog graph showing effect of x on variation of E with p M 00 for cocurrent and counter-current ow systems at K 1 0:1, K 2 0:1 and B 106.



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For gas phase :o C G o Z 0 at g 1 8Z : 31

B.C.1: C G 0 at Z 0.The solution methodology for the above equations is as follows:

For liquid phase:Taking Laplace transform of Eq. (30) and using B.C.1, one gets:

g2 g sC A d2C Adg2

: 32The other two boundary conditions now become:

B.C.2: C A 1 s and B.C.3: dC Adg 0.Substituting, C A

P1n

0angn in Eq. (32), and using B.C.3, the following recurrence relation is

obtained:

an s

nn 12a n 3 a n 4; where a1 0; and a2 0 and a3

s3 a0:

Thus, the transformed concentration prole turns out to be:

C A a 0 1 s3

g3 s12

g4 s4

45g6 a0 f s; g: 33

B.C.2 gives a0 1

sf jg1and C A

f s; g sf jg1

. After taking its inverse Laplace transform one nds:

C A X10

lim s! ki s k

i sf jg1 f s; ge

sZ ; where; ki are roots of sf jg1 0:Using LHospital rule for evaluating the limits in the above equation, the following expression for C A isobtained:

C A X10

lim s! ki

1

f sd f d s g1

f s; ge sZ X10

Ai f ki; geki Z ;

where Ai lim s! ki1

f sd f

d s g1or

C A A0 X11

Ai f ki; geki Z 1 X11

Ai f ki; gekiZ ;

where; A0 lim s! k001

f sd f d s g1

1:

34

For gas phase :Now, the gas phase Eq. (31) is integrated which gives the gas phase concentration C G

0 throu-

ghout.



14/29


15/29

Following the earlier solution methodology one nds the liquid phase concentration as:

C A A0 X11

Ai f ki; geki Z cosh ffiffiffiffiffiffi K 1p gcosh ffiffiffiffiffiffi K 1p X

11

Ai f ki; g eki Z ; 40

where, A0 lim s! k001

f so f o s jg1 cosh

ffiffiffiffi K 1p g

cosh

ffiffiffiffi K 1p and ki are the roots of sf jg1 0.Again for gas phase, we have concentration C G 0 throughout.

3.4.4. Reduced model 4 ( x 0, K 1 60; K 2 0 and B 6 1)The process of gas absorption with rst order chemical reaction and with nite gas phase resistance and nogas ow was worked out by [14,15]. For this situation the general model Eqs. (3) and (4) reduce to:

For liquid phase :

g2 go C Ao Z

o 2C Ao g2 K 1C A: 41

B.C.1: C A 0 at Z 0 8g.B.C.2: o C Ao g B1 C A at g 1 8Z .B.C.3: o C Ao g 0 at g 0 8Z .For gas phase :

o C G o Z 0 at g 1 8Z : 42

B.C.1: C G 0 at Z 0.

For this case we get the following expression for liquid phase concentration:

C A A0 X11

Ai f ki; geki Z 1

cosh ffiffiffiffiffiffi K 1p ffiffiffiffi K 1p B sinh ffiffiffiffiffiffi K 1p X

11

Ai f ki; geki Z ; 43where,

A0 lim s! k001

1 B

d f dg f s 1 B

o 2 f o so g

o f o s g1

1cosh ffiffiffiffiffiffi K 1p ffiffiffiffi

K 1p B sinh ffiffiffiffiffiffi K 1p

and ki are the roots of s 1 Bd f dg f

g

1 0.

Similarly, for gas phase we have concentration C G 0 throughout.

3.4.5. Reduced model 5 ( x 60, K 1 0; K 2 0 and B 6 1)Physical gas absorption with nite gas phase resistance and gas ow was theoretically analyzed by [13]. Thepresence of non-zero x couples equations of both phases. The gas phase equation is solved after the liquidphase concentration prole is found. For this case, we have a non-zero gas phase dimensionless concentrationunlike all the previous cases derived. The governing equations become:

For liquid phase :

g2 go C Ao Z

o 2C Ao g2 : 44



16/29

B.C.1: C A 0 at Z 0 8g.B.C.2: o C Ao g B1 C G C A at g 1 8Z .B.C.3: o C Ao g 0 at g 0 8Z .

For gas phase :

o C G o Z x

o C Ao g g1

at g 1 8Z : 45B.C.1: C G 0 at Z = 0.

Again, going through the same steps, we have the liquid phase concentration prole as given below:

C A A0 X11

Ai f ki; geki Z 1

1 x o2 f

o so g g1 X11

Ai f ki; g eki Z ; 46where, A0 lim s! k00 11 x o 2 f o so g

g

1

and ki are the roots of o f

o g s x B

B

sf

g1 0.

After putting this in Eq. (45), we get the gas phase concentration C G as:

o C G o Z x

o C Ao g g1

x o

o g X10

Ai f ki; gekiZ !g1

x X10

Aio f ki; g

o g eki Z !

g1:

Integration of the above expression and using B.C.1 for gas phase gives:

C G x X10

Aiki

o f ki; go g g1

1 eki Z !X10

Bi 1 eki Z ; 47where, Bi

x Ai

ki

o f ki ;gog g1

.

3.4.6. Reduced model 6 ( x 0, K 1 0; K 2 0, B 1 and y d 1)This is the simplest case rst considered by [1], while founding the penetration theory for gas absorption forshort time of contact. Recently, Sharma et al. [22] have obtained the solution for this case by three differenttechniques i.e. Laplace transform, Similarity transform and Sine transform. We describe below the solution byLaplace transform only. The governing equations are as follows:

For liquid phase :

V maxo C ao z D AB

o 2C ao y 2

: 48B.C.1: C a C 0 at z 0.B.C.2: C a C g 0 at y 0.B.C.3: C a C 0 at y d.

This system is made non-dimensionalized by using the same denitions except g y d (used for this caseonly). Hence, the following parabolic partial differential equation is obtained:o C Ao Z

o 2C Ao g2

: 49B.C.1: C A 0 at Z 0.B.C.2: C A

1 at g

0.

B.C.3: C A 0 at g 1.



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Taking the Laplace transform with respect to Z on both sides of diffusion equation (49) and incorporatingthe boundary condition at Z 0 gives:

sC A d2C Adg2

: 50The solution of the ordinary differential Eq. (50) is given [57] as:

C A Ae g ffiffi sp Beg ffiffi

sp ; 51where, C A is a function of s and g. The constants A and B are found with the help of the boundary conditions 2and 3, and the solution in transformed form becomes:

C A e g ffiffi

sp

s : 52

The inverse transform of this expression is obtained by using FourierMellin inversion theorem [26] as:

C Ag; Z 12p i

limb!1

Z

cib

c ibC Ae sZ d s: 53

This is approximated by the contour integral [26,28] which is given by:

I e sZ C A d s: 54Putting C A from Eq. (52) in Eq. (54) and after doing some mathematical manipulations, the following liquidphase concentration prole of the absorbed solute, for short contact time, is obtained

C A erfc g

ffiffiffiffiffiffi4Z p : 55

3.5. Numerical solution of the general model

The developed analytical solution has been veried using a simple explicit numerical scheme for many setsof parameter values. The scheme utilizes nite difference approach and the discretized model domain is dem-onstrated in Fig. 13. The adopted nite difference method has been programmed in MATLAB 6 for ease andthe results are saved as an EXCEL data le. Only the liquid and gas phase concentration proles are shown inFigs. 14 and 15 respectively, for one set of parameter values pertaining to the general case. From the two plots,it is clear that the results show a good agreement with the analytical solutions presented in Figs. 2 and 3 .

3.5.1. Numerical schemeThe whole problem domain of interest was divided into two parts: one for the liquid lm and the other for

gas phase, as depicted in Fig. 13. The liquid part was discretized along two directions; while gas phase was dis-

cretized along axial coordinate only. Subsequently, at their respective border points, B.Cs have been specied.

Discretization in liquid phasesThe dimensionless liquid lm thickness g and the dimensionless axial distance Z are divided into N and M

equal parts, respectively. The general model equation in liquid lm at a point i; j 1 is expressed as:g2 g

o C Ao Z i; j1

o 2C Ao g2 i; j1

K 1C Ai; j1

; or

g2 go C Ao Z i; j1

o 2C Ao g2 i; j

K 1C Ai; j

56

and the associated B.Cs are:



18/29

B:C:1 : C Ai; 0 0; 8i 2 0; N 57B:C:2 :

o C Ao g N ; j1

B1 C G C Aj N ; j1; 8 j 2 0; M 58B:C:3 :

o C Ao g

0; j1 K 2C Aj0; j1; 8 j 2 0; M : 59

Discretization in gas phaseOnly axial discretization is needed in gas phase and is given below:

o C G o Z j1

xo C Ao g N ; j1

; 8 j 2 0; M : 60The B.C. for gas phase is introduced as:

B:C:1 C G 0 0: 61Solution algorithm :

1. Concentrations C A and C G of liquid and gas phases at the starting of the boundary are specied using Eqs.(57) and (61).

2. Using the discretized form of Eq. (56), the liquid phase concentrations C Ai; j 1at the next axial positionare calculated for all i 2 1; N 1 .3. Thereafter, from Eq. (59), the liquid phase concentration at wall is obtained.4. Next, C A N ; j 1 and C G j 1 are evaluated from Eqs. (58) and (60) simultaneously.5. The procedure is repeated for the next axial step.

3.6. Verication of the mathematical model with the experimental results

Though the problem of gas absorption with and without chemical reaction has been tackled by many work-ers and a vast amount of material, both theoretical and experimental, is available. Mostly, the experimentalstudies have been carried out to nd the related unknown parameters such as rate constant, diffusivity andsaturation concentration of solute in liquid etc. or the derived quantities like average or total absorption rate,rather than to nd the concentration proles of both the phases. For the purpose of verication, the experi-mental data are chosen from [9], in which an experimental study of CO 2 absorption in the Na buffer solutionsof various strengths was carried out for short time of contact. The whole process of absorption was accom-panied by 1st order homogeneous chemical reaction and the rate of absorption of CO 2 was measured in a wet-ted wall column.

The equations used in [9] for concentration prole, mass transfer rate per unit area and the total amount of

gas absorbed per unit area, in a certain time interval, have been reproduced below in terms of the symbols usedin the present paper:

C aC g 0

12

exp y ffiffiffiffiffiffiffiffik 1 D ABr erfc y 2 ffiffiffiffiffiffiffi zD ABV maxq ffiffiffiffiffiffiffiffiffiffik 1 z

V maxs 0B@1CA

12

exp y ffiffiffiffiffiffiffik 1 D ABr erfc y 2 ffiffiffiffiffi zD ABV maxq ffiffiffiffik 1 z

V maxs 0B@1CA

: 62

The rate of mass transfer through a unit area of the gas liquid interface is given [9] as:

/ m D ABo C ao y y

0 C g 0

ffiffiffiffiffiffiffiffiffiffiffiffi D ABk 1

p erf ffiffiffiffiffiffiffiffiffiffik 1 z V maxs e

k 1 z V max

ffiffiffiffiffiffiffiffiffip k 1 z

V max

q 0B@

1CA

: 63

http://-/?-http://-/?-http://-/?-http://-/?-


19/29

The total amount of gas absorbed per unit area, after a contact time s z V max is given [9] as:

ms Z s

0/ m dt C g 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi D AB z V maxr ffiffiffiffiffiffiffiffiffiffik 1 z V maxs 12

ffiffiffiffiffiffiffik 1 z

V max

q

0B@

1CA

erf ffiffiffiffiffiffiffiffik 1 z V maxs e k 1 z V max

ffiffiffipp 26

4

375

: 64

In order to compare our results with the experimental data [9], the above quantities are now evaluated fromthe present model equations for the case when x 0, K 1 60; K 2 0 and B 1.The rate of mass transfer through a unit area of the gas liquid interface is given by:

/ m D ABo C ao y y 0

D ABo C ao Y Y d

D ABC g 0

d

o C Ao g g1

: 65The total amount of gas absorbed per unit area after a contact time s z

D z 0V max

d2Z D Z 0 D AB

is given (D z 0 is theheight of end effects [9]) as:

ms Z s

0 / m dt 1V max Z

z

0 / m d z C g 0dZ Z

0

o C Ao g g1 dZ ; 66

where

o C Ao g g1

ffiffiffiffiffiffi K 1p tanh ffiffiffiffiffiffi K 1p X1

i1 Ai

o f o g g1eki Z 67

or

ms C g 0d Z

ffiffiffiffiffiffi K 1p tanh

ffiffiffiffiffiffi K 1p X

1i1

F i1 eki Z !; and F i Aiki o f o g g1C g 0d

D ABsd2 ffiffiffiffiffiffi K 1p tanh ffiffiffiffiffiffi K 1p X

1i1

F i1 eki

D ABs

d2 !: 68The values of chosen parameters are same as those used for the runs 5 and 6 of [9] and are listed below:

Run No. 5 of [9]: C g 0 2:31 104 g cm 3; k 1 0:56 s 1; d 0:04 cm; D AB 1:17 10 5 cm2 s 1; V max 32 cm s 1; D z 2 cm.

Table 3Eigen-values and series coefficients for experimental data of gas absorption with rst order chemical reaction, x 0, K 1 76:5812 and144.957265, K 2

0 and B

1No. of eigen-values For K 1 76:5812 For K 1 144:957265ki Ai F i ki Ai F i

0 0.000000 0.0003165843 0.000000 0.0000118087 1 107.6611314 0.0099496611 0.00757368037899 185.8518694 0.0015920387 0.003694235286062 166.3823198 0.0348871199 0.00762832934056 257.1461211 0.0074742510 0.004195374151633 247.9379144 0.0656352839 0.00637605737965 350.5565811 0.0183812004 0.003919938655344 354.9497511 0.0911870510 0.00508712756082 468.4541818 0.0324252865 0.003414529633975 489.75184370 0.1068187151 0.00403212778099 612.5116393 0.0468084563 0.002893324525546 654.1933116 0.1134069768 0.00321390217994 784.0615829 0.0592897557 0.002434522556747 849.4538967 0.1139208620 0.00259132581195 984.2776336 0.0687116330 0.001979177408118 1076.1708969 0.1110551986 0.00198260994417 1214.1646337 0.0748928678 0.002628075045929 1334.5388988 0.1062974349 0.00842749504505 1474.1656341 0.0330288412 0.01797114826218

10 1625.5418988 0.0011747340 0.00518678921917 1726.177 1.358284e-05 0.00013246396498



20/29

Run No. 6 of [9]: C g 0 2:25 104 g cm 3; k 1 1:06 s 1; d 0:04 cm; D AB 1:17 10 5 cm2 s 1; V max 32 cm s 1; D z 2 cm.

The obtained results have been shown in Fig. 16 and an excellent agreement is found between the analyticalsolution and the experimental data. A detailed discussion of the results can be found in the next section. Eigen-

values and other coefficients for these runs have been shown in Table 3 .4. Results and discussion

The concentration proles of solute, in both phases depend upon four parameters namely x , K 1 , K 2 and B .The liquid phase concentration shows variation in both y- and z-directions while, the gas phase concentrationvaries with z only. The parameters also affect absorption rate in the liquid phase. An assessment of the plots of various quantities for a range of parameter values reveals these variations.

4.1. Parameter x

Figs. 3 and 6 show that as gas phase velocity increases, both the liquid and gas phase concentrationsincrease in counter-current systems. This is also true for co-current systems. When x is small, the axial var-iation in gas phase concentration is negligible and the error introduced by assuming a constant axial concen-tration is also small. However, as x increases, the variation in gas phase concentration becomes large.

x depends on the relative liquid to gas volumetric ow rates. For the gases of larger solubility, large axialvariation in the gas composition can be expected.

In the light of the above discussion, Figs. 3 and 15 may cause some confusion in the sense that the concen-tration of solute in gas phase is large for lower gas velocity (i.e. x 1:0) as compared to the one for higher gasvelocity (i.e. x 0:4) for counter-current ow, for all values of Z . This is in sharp contrast with the abovediscussion. This issue is resolved below.

A close look on Eq. (2) or (4) reveals that the slope o C G o Z decreases with increase in gas phase velocity. Thismeans that for higher gas phase velocities, the outlet and inlet concentrations are nearly same. Whereas, forlower gas velocities the slope is positive which indicates that inlet concentration is greater than the outlet con-centration C g 0. Since the outlet concentration C g 0 has been used in all our calculations, and because the slope

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DIMENSIONLESS DISTANCE FROM SOLID WALL,

D I M E N S I O N L E S L I Q U I D P H A S E


, C

A

= 1.0 = 0.4 = 0.0

Fig. 6. Development of liquid phase concentration prole for counter-current ow, x varies from 0.0 to 1.0 for, K 1

0:1, K 2

0:1 and

B 106 at Z 1:0.



21/29

tends to zero for higher gas velocities, this C g 0 will be available through out the axial direction for x 0.However, for non-zero slopes, a concentration greater than C g 0, but lesser than inlet concentration is observedfor all axial positions and hence, a higher gas phase concentration at a particular Z is seen for lower gas veloc-ities and vice-versa. To understand these arguments clearly, one can also study the plots by shifting the con-centration proles such that the inlet concentrations become same for all gas velocities, which accordingly

makes the outlet concentrations C g 0 different.Fig. 5 shows the variation of enhancement factor with Hatta number on a loglog plot for various values of x . As is evident from the plot, the absorption rate for co-current system is much lower than that of constant gasphase concentration. This is due to the gradual decline of gas phase concentration in a co-current ow resulting in a nite amount of solute absorption. In counter-current or co-current systems, increasein gas phase velocity leads to an increase in enhancement factor E . In co-current systems, it is also observed,that enhancement factor E decreases linearly on a loglog plot for Hatta number greater than 7.

4.2. Parameter K 1

As liquid lm reaction rate constant K 1 increases, the exit liquid phase concentration decreases which isportrayed in Fig. 7. This is due to the enhanced rate of reaction which increases the absorption rate; it canalso be seen from the increasing slope of liquid phase concentration proles at interface. Higher the absorptionrate, higher is the axial variation in gas phase concentration as shown in Fig. 8.

Fig. 9 shows the variation of E with ffiffiffiffiffiffiffi M 00p (Hatta number) and K 1 for co-current as well as counter-currentsystems. Initially, for any of the systems, the effect of K 1 is negligible since the rate of reaction is small enough(due to very low concentration in the liquid lm) to impress any change in enhancement factor E . In co-cur-rent systems, for Hatta number greater than 10, the absorption rate overcomes the reaction rate and satura-tion starts. This behavior is shown by a single line with constant negative slope in spite of different values of K 1 , while, in the case of counter-current system no such relation is found. The enhancement factor E increasessteadily with ffiffiffiffiffiffiffi M 00p as the value of parameter K 1 increases. The rate of absorption for counter-current ow ishigher than for co-current ow for xed values of all process parameters. The reaction parameter K 1 , indicatesthe ratio of the rate of reaction to the rate of diffusion in relation to the thickness of the lm. Thus, a small

value of K 1 may be due to a very slow reaction, a very thin lm or a high rate of diffusion. The effect of lmthickness on the rate of absorption is seen only when K 1 is less than 5. At higher values of K 1 , the rate of

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1




, C

A

K1 = 0.1

K1 = 2.0

K1 = 0.4

K1 = 0.0

Fig. 7. Development of liquid phase concentration prole for counter-current ow over a dissolving solid wall, K 1 varies from 0 to 2 forx 0:4, K 2 0:1 and B 106 at Z 1:0.



22/29

reaction is sufficiently high to prevent the absorbed gas molecules from reaching the solid wall for a particularZ .

4.3. Parameter K 2

The effect of K 2 on concentrations of both the phases and on enhancement factor E follows approximatelythe same pattern as that of K 1 , while keeping other parameters constant. But these effects are slightly less due

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1



C O N C E N T R A T I O N ,

C G

K1 = 0.0

K1 = 0.4

K1 = 2.0

K1 = 0.1

Fig. 8. Gas phase concentration prole for counter-current ow over a dissolving solid wall, K 1 varies from 0 to 2 for x 0:4, K 2 0:1and B 106.

0.01

0.1

1

10

100

0.01 0.1 1 10 100

''


, E

0.0

0.0

0.1

0.1

0.4

0.4

K1 =2.0

2.0

COUNTER-CURRENT

CO-CURRENT

Fig. 9. Loglog graph showing effect of K 1 on variation of E with p M 00 for co-current and for counter-current ows in gas phase at K 2 0:1 and B 106.



23/29

to the more pronounced effect of liquid phase reaction as compared to the reaction at the wall. With the helpof Fig. 10, the combined as well as independent effects of parameters K 1 and K 2 on the enhancement factor E can be easily appreciated.

4.4. Parameter B

Figs. 11 and 12 show liquid and gas phase concentration proles for a counter-current system at steadystate. It can easily be noticed that as the surface resistance increases i.e. parameter B decreases, the exit liquid

0.1

1

10

100

0.1 1 10 100''


, E

0.1 &0.1

K1 = 0.2 &K2 = 0.0

0.1 &0.0

K1 = 0.0 &K2 = 0.1

Fig. 10. Loglog graph showing effect of K 1 and K 2 on variation of E with p M 00.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

DISTANCE FROM SOLID WALL,



, C

A

B = 10 6

B = 10

at Z = 1.0

at Z = 0.01

K1 = 0.1K2 = 0.1w = 0.4

Fig. 11. Development of liquid phase concentration prole for counter-current ow, B varies from 10 to 106 for x

0:4 at Z

1:0 and

Z 0:01.



24/29

phase concentration decreases and a similar pattern is observed in the gas phase concentration at the interface.It is also worth mentioning that in a counter-current system, the liquid phase concentration at interface maybe greater than the outlet gas phase concentration C g 0 after some Z . This is because, the gas phase boundarycondition at Z 0, for the two systems i.e. co-current as well as counter-current, uses C g 0 which is the inletconcentration for co-current ow and the outlet concentration for counter-current ow. In case of co-currentow, the inlet concentration C g 0 is the highest concentration at interface among all axial positions whereas, forcounter-current ow it is the lowest. Therefore, in counter-current system the liquid lm after some Z , encoun-ters a gas phase concentration which is greater than C g 0. While, in co-current gas ow, the dimensionless liquidphase concentration C A will be less than 1 for all values of Z .

4.5. Reduction of general model into simple cases

In Section 3.5, six simple cases have been successfully derived by reducing the general model for differentsituations. All these cases were earlier researched by [1,2,4,10,1216,22].

4.6. Numerical scheme

The numerical technique chosen is of explicit type, thus it is simple to implement. But, while discretizing theproblem, care should be taken in selecting a proper step size. At times, larger increments may destabilize themethod or may even give poor results. Though, smaller steps do give accurate results but the computationaltime increases manifolds due to increase in the number of machine calculations.

For the selected parameters, we have taken N 20 50 and M 10 6 10 5. More powerful numericalapproaches will not only yield good results but also in a reasonable time. The employed scheme is pictoriallydepicted in Fig. 13.

4.7. Validation by experimental data

The present general model and its solutions are compared with the available simplied experimental case of gas absorption accompanied by 1st order chemical reaction in lm only, with no gas phase resistance and no

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1



C O N C E N T R A T I O N C

G

B = 10 6

B = 10

K1

= 0.1

K2

= 0.1

= 0.4

Fig. 12. Gas phase concentration prole for counter-current ow.



25/29

gas side ow. A close look on this model reveals that it can be considered as a practical example of reducedmodel 3, described in Section 3. The amount of gas absorbed per unit area after a contact time s, calculatedpresently and the one found by the other two groups [9,20], has been shown in Fig. 16 on a loglog scale. It iseasily understood that the outcome from our model and that of Nysing and Kramers [9] match outstandinglyand is even better than those discussed by Elperin and Fomynikh [20]. In initial stages or for short contacttime, our model deviates slightly from those of [9,20]. This is due to the limited terms considered in Eq.(40), which makes the solution incompatible with B.C.1

C AjZ ! 0 0 cosh ffiffiffiffiffiffi K 1p gcosh ffiffiffiffiffiffi K 1p X

11

Ai f ki; g: 69Finite terms in the above equation do not satisfy it for small Z , while for larger Z , only few terms are sufficient.That is why the present results deviate to a small extent only for initial values of Z and this point can also beveried by varying the number of terms in Eq. (69).

4.8. Miscellaneous

Laplace transform technique is found to be superior to separation of variables method as it does not requirehomogeneous boundary conditions and can easily be applied to systems of linear differential equations. Eigen-values are found by the hybrid scheme of bisection and NewtonRaphson methods. Accuracy in eigen-values

Gas LiquidCgo Co

, i

Z, j

Wall

1 Interface

N, 0 N-1, 0 i+1, 0 i, 0 i-1, 0 3, 0 2, 0 1, 0

0, 1

0, 2

0, 3

0, 4

0, 5

0, 6

0, j-1

0, j

0, j+1

0, M-1

0, M

N, 1

N, 2

N, 3

N, 4

N, 5

N, 6

N, j-1

N, j

N, j+1

N, M-1

N, M

i, ji+1, j i-1, j

i, j+1 Z

Fig. 13. Grid points in numerical scheme.



26/29

decreases with their numbers and therefore, only eleven eigen-values are presented. However, 11th eigen-valuewas determined with some error. Higher eigen-values could not be found due to the divergence of the series inEq. (13). It is also observed that the 11th eigen-value cannot be found if we express Eq. (3) in terms of dimen-sionless variable g y d (where, y coordinate starts from the interface). The nth eigen-value can be found from aquadratic polynomial of n, which helped in predicting the initial guess for all the cases. This point was alsomentioned by [10]. It is also observed that the magnitude of eigen-values increases with an increase in anyof the rate constants while in case of increased surface resistance, it decreases. Further, the magnitudes of eigen-values for co-current ow are greater than the corresponding eigen-values for counter-current owfor the same absolute values of the parameters.

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

D I M E N S I O N L E S S C O N C E N T R A T I O N

, C

A

Z=0.01 Z=0.05 Z=0.1 Z=0.2 0.3 Z=0.4 0.5 Z=0.6 0.7 Z=1.0

B = 10 6, = 1.0K1 = 0.1, K

2 = 0.1


Fig. 14. Liquid phase concentration prole evaluated from numerical scheme.

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1




, C

G = 1.0

= 0.

4

K1 = 0.1

K2 = 0.1

B = 10 6

Fig. 15. Gas phase concentration prole evaluated from numerical scheme.



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An important point regarding eigen-values is also discovered, that the rst eigen-value becomes positive forcounter-current ow with axial variation in the gas phase concentration which was not mentioned by any of the previous authors. The magnitude of this value increases with the increase in gas ow rate. Moreover, forthe two situations i.e. (i) x 0 and (ii) K 1 K 2 0, we get zero as the rst eigen-value. Table 4 shows the rsteigen-value for different values of parameters.

For special cases, our results show excellent agreement with those obtained by [1] for simple physical

absorption for small contact time; [4,10] for simple physical absorption for large time of contact; [9,16,20]for gas absorption with 1st order chemical reaction; [12] for gas absorption with surface resistance; [14] forgas absorption accompanying 1st order chemical reaction with nite gas phase resistance and [13] for gasabsorption with surface resistance and with gas side ow.

Table 4First eigen-value for various sets of parameters values

S. no. x K 1 K 2 B k01 0.0 0.1 0.1 106 0.00002 0.4 0.1 0.1 106 0.09373 1.0 0.1 0.1 106 0.40904 0.0 0.0 0.0 106 0.00005 0.4 0.0 0.0 106 0.00006 1.0 0.0 0.0 106 0.00007 0.4 0.0 0.1 106 0.04838 0.4 2.0 0.1 106 0.58369 0.4 0.1 0.0 106 0.0517

10 0.4 0.1 2.0 106 0.335611 0.0 0.0 0.0 1 0.000012 0.0 0.0 0.0 104 0.000013 0.0 0.1 0.0 1 0.000014 0.0 0.1 0.0 104 0.000015 0.4 0.0 0.0 10

6 0.0000

16 0.4 0.0 0.0 10 0.0000

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

-1 -0.5 0 0.5 1 1.5 2 2.5 3

Run 5 Run 6

log

l o g

( m ( ) /

) +

7 . 0

Ref. [20]Ref. [20]

Ref. [9] &

Present work

Ref. [9] &Present work

Fig. 16. Comparison with experimental data.



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5. Conclusions

A more general situation of gas absorption with nite gas phase resistance, axial variation in the gas phaseconcentration and irreversible 1st order chemical reaction in the liquid lm and at the wall is analyzed.Eigen-values and series coefficients for various cases are also calculated and some of them are shown in

the Tables 2 and 3 . Effects of various parameters on the liquid and gas phase concentrations as well as on the enhancement fac-tor E are shown and it is found that a strong surface resistance leads to a negligible rate of gas absorption,while a higher reaction rate increases the absorption rate for both the co-current and counter-current sys-tems. Though, for counter-current system, the effect of K 1 (or K 2 ) on enhancement factor E may be observedover the whole axial distance but for co-current system there is a limited region where this effect is visible,beyond which K 1 (or K 2 ) does not have any effect and hence absorption rate for simple physical absorptioncan be used. Changes due to reaction at wall follow a similar pattern as those exhibited by the reaction inliquid lm but these are less pronounced. For both the systems, the increase in gas phase velocity increasesthe enhancement factor E . The total mass absorbed by a co-current system is constant and does not varywith Z after attaining the steady state, while in case of counter-current system, it increases with the axial dis-tance Z , which can be veried by evaluating the enhancement factor in terms of Hatta number

ffiffiffi M 00p .

It is also concluded that the exit liquid phase concentration for B 1000 is nearly equal to the value for B 1. It implies that for B 1000 and above, the gas lm resistance is almost negligible. These resultsare expected to be useful in measuring diffusivities, mass transfer coefficients, solubilities and reaction rateconstants for the experiments of gaseous mixtures. The present analysis is a step forward, as it is usefulwhen gas is not necessarily pure and it is applicable for the condition of low as well as high gas ow rates.

The reaction parameter K 1 , indicates the ratio of the rate of reaction to the rate of diffusion in relation tothe thickness of the lm. Thus, a small value of K 1 could be due to a very slow reaction or a very thin lm ora high rate of diffusion. For K 1 greater than 5, the lm thickness does not have any effect on the rate of absorption. The higher values of K 1 prevent the penetration of the absorbed gas molecules towards the solidwall. This is the reason that the results predicted from our analysis match very well with the experimentallyreported results of [9], where K 1 5 and the parabolic velocity prole was not taken into account and thus,the lm thickness did not come into picture. When x is small, the axial variation in the gas phase is negligible and hence the error introduced by assum-ing a constant axial concentration is small. However, as x increases, the error becomes large.

The present general model has been successfully reduced to some of the previously studied simple models.Results obtained from an explicit numerical technique, programmed in MATLAB 6, have been presentedand compared with those of analytical solutions and an excellent agreement between the two has beenfound.

Absorption rates found from the analytical solution, for a simplied case, are veried with the experimentaldata of [9]. The conformity between the two substantiates the present work.

Acknowledgements

The authors are gratefully indebted to Dr. Surendra Kumar, Professor, Department of Chemical Engineer-ing, I.I.T. Roorkee, (Roorkee-247667, U.A., India.) for his invaluable guidance in the preparation of manu-script. We are also thankful to Professor S. M. Yusuf, (Retd.), Department of Mechanical Engineering,A.M.U., (Aligarh-202002, U.P., India.) for his helpful suggestions and to Ms. Arees Qamareen, Lecturer,Department of Mechanical Engineering, A.M.U., (Aligarh-202002, U.P., India.) for her support in formattingthe document.

References

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