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Gus. Srp. Purif: Vol. IO. No. 1, pp. 41-46, 1996Copyright :(: 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0950-4214/96 $15.00 + 0.00

ELSEVlER 0950-4214(95)00024-O

Modeling of gas absorption into turbulent filmswith chemical reaction

Mohammad R. Riazi

Chemical Engineering Department, Kuwait University, PO Box 5969, Safat 13060, Kuwait

A mathematical model has been developed to predict the rates of gas absorption in turbu-lent falling liquid films with and without the first order homogeneous reaction and externalgas phase mass transfer resistance. The eddy viscosity model used to describe the flowdistribution is the van Driest model, modified in the outer region of the film by the use ofan eddy diffusivity deduced from gas absorption measurements. The results are given forspecial cases to illustrate the effects of turbulence, reaction rate and gas phase resistanceson the concentration profiles and the rates of gas absorption. Copyright @ 1996 ElsevierScience Ltd

Keywords: mathematical model; gas absorption; turbulent films

Nomenclature

Damping length constant (26v(p/~o))C Concentration of dissolved gas in the liquidC Equilibrium concentration of gas in liquidC.5 Concentration of gas at the free surface

D Molecular diffusion coefficientg Acceleration due to gravityg, Gravity constant (= 1 in SI unit system)k First-order reaction rate constantk, Mass transfer coefficient in liquid phasekG Mass transfer coefficient in gas phaseK Constant (= 0.4)K ConstantL Prandtl mixing lengthm Dimensionless rate of absorptionN Dimensionless parameter (= kc/D[u2/g]/3)RC? Reynolds numberSC Schmidt number (= v/D)SC, Turbulent Schmidt number (= EM/ED)

Sh Sherwood number (= k,.z/D)u Liquid velocity in axial directionI Distance normal to the surfaceZ Axial distance

Greek symbols

:Reaction rate constant (= k(v/g2)13)Liquid loading

s Liquid film thicknessED Eddy diffusivityEM Kinematic eddy viscosity0 Surface tensionP Liquid density/l Liquid viscosityV Liquid kinematic viscosityTO Shear stress at the free surface

Superscript

Non-dimensionalized quantity

Introduction laminar falling liquid films has been studied in some

Development of mathematical models to predict the rateof gas absorption into liquid films is important in thedesign and operation of falling film reactors, which arewidely used for gas-liquid reactions such as sulfonationor chlorination, as well as in gas separation and purifica-tion units. From a review of the published literature onwetted-wall columns, it is evident that while the problemof gas absorption with or without chemical reaction in

detail, the c&e bf turbulent flow, especially whenchemical reaction is involved, has received less attention.The problem of gas absorption without chemicalreaction in laminar films was solved by Olbricht andWild2 using the series expansion method. For the case ofgas absorption into laminar films with chemical reaction,the governing differential equations were solved by bothanalytical and numerical techniques as shown by

Riazi3,4. The effect of gas phase mass transfer resistanceson the rate of gas absorption for laminar liquid films waspreviously discussed by Riazi3. The effect of turbulenceFax: (+965) 4839498; email: riazi@kucOl .kuniv.edu.kw

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42 Modeling of gas absorption: M.R. Riazi

in the gas phase as well as the effects of interfacial dragat the gas-liquid interface on the rate of gas absorptionwere shown by Riazi and Fagri4. The problem ofphysical gas absorption into a turbulent liquid film wastreated to some extent by Lamourelle and Sandall.Based on the gas absorption measurements, theyobtained an expression for the liquid phase eddy diffu-sivity in the region near the free surface. Menez andSandall studied the problem of gas absorption accom-panied by first-order chemical reaction in a liquidflowing in fully developed turbulent flow. They obtainedasymptotic solutions for which solute concentrates onlya short distance into the liquid film, because of a slowrate of diffusion or very high rate of reaction where onlyeddy diffusivity in the region near the free surface wasused to describe the turbulence in the liquid film.

The main objective of this work is to show the effectsof turbulence in liquid falling films on the rate of gas

absorption when combined with chemical reaction andgas phase mass transfer resistances. For our work, theeddy diffusivity given by Lamourelle and Sandall isused for the region near the free surface, while the vanDriest viscosity model is used for the region near thewall.

Formulation

Let us consider the system shown in Figure 1. A liquidinitially free of the absorbing species at z = 0 flows downthe surface of a vertical and impermeable wall under theinfluence of gravity. The absorbing species are absorbedby the liquid where it undergoes a (pseudo) first-orderirreversible chemical reaction. It is assumed that the gasphase concentration of absorbing species is constant andthe interfacial shear stress at the gas-liquid interface isneglected. Furthermore, it is assumed that the diffusion

QIEI1i q u i d

Figure 1 Schematic of a falling liquid film showing the coord-inate system

in the axial direction is negligible. Under these condi-tions, the steady state mass balance on the absorbingspecies in the liquid phase for turbulent flow is:

(1)

The coordinate system used and the physical descriptionof the absorbing film are shown in Figure 1. The term u isthe axial velocity of liquid film and can be found fromthe momentum equation after neglecting the pressuregradient and axial terms:

g[(YfEdg g=oThe solution of the above equations for momentum andmass transfer requires the specifications of the boundaryand initial conditions:

at the inlet, z = 0

c=o

at the wall, y = 0

(3)

u= 0

aC 0_=ay(4)

at the interface, y = 6

a u

5=Op(c*_c)

where 6 is the film thickness, C is the concentration inequilibrium with gas phase, and ko is the mass transfercoefficient in the gas phase. Upon integration of Equa-tion (2) with the corresponding boundary conditionsgiven in Equations (4) and (5), the velocity profile can beobtained:

L!=

In the above equations, if flow is laminar thenEM=&Eg=o.

Introducing the following non-dimensional variables

will transform Equations (l), (3), (4), (5) and (6) to thefollowing form:

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Modeling of gas absorption: M.R. Riazi 43

C=O

atJ=O

(8)

ac3-O

at J = S

ac NT&y=zp(-c) (10)

and

U=./

v (1 -y/J) dq(11)

0 EM

where a = k(v/g2)/, N = kC/D(v2/g)i3 and cM = 1+EMfu.

Note that when the mass transfer resistance in the gas

phase is negligible, N = 00, the boundary conditiongiven in Equation (10) reduces to c = 1 at 7 = &.

In order to solve the equations for the turbulent case,it is necessary to introduce some empirical profiles forthe eddy diffusivity. Some typical models for the fallingfilm are introduced by Gutierrez-Gonzalez et al..Accurate specifications of the eddy diffusivity close tothe wall and also close to the free surface are much moreimportant than in the middle of the film, due to lowresistances in the central region. It is customary that formodelling Ed the flow is divided into two regions, aninner region where the turbulent transport is dominatedby the presence of the wall and an outer wall-like region.The model proposed here is described by use of the vanDriest model, modified in the outer region of the film byuse of an eddy diffusivity deduced by Lamourelle andSandall from gas absorption measurements.

Various assumptions have been made in order todescribe the mean velocity distribution near the wall. Apopular kinematic eddy viscosity model for this region,as mentioned before, is provided by van Driest whoassumed the following modified expression for thePrandtl mixing length theory:

L = K!y[l - exp( i)]

where A is a damping length constant defined as26vm and K = 0.4. For the falling film van Driestviscosity is used in the inner layer of film liquid. There-fore for the inner region:

du1 -_v + Jz/ + 4L2 + 4L(S - y)g&J= 2L2EM =

-_v + J$ + 4L2 + 4L2(6 - y)g2

Non-dimensionalization transfers the above equation inthe following form:

1 + 1 + 0.64~*[1 - exp(-J/26)]*(1 -p/6)EM =

2 (12)

Lamourelle and Sandal15, by measuring the masstransfer coefficient for the liquid phase for gas absorp-tion into a turbulent liquid Aow down a wetted column,obtained the following pattern distribution for sn at atemperature of 25C

Ed = 0.284Rei.678(~ - y)2 (13)

in which Ed is in ft2 h-. In order to generalize the aboveequation to temperatures other than 25C and to liquidsother than water, it must be rendered dimensionless in a

manner which Levich8 has indicated:

where k is a constant and 0 is the surface tension. Theabove constant can be calculated from Equation (13),which yields the following result:

ED- = 6.4 x lo-: Rc.~*(~ - y)*u