Modeling Of Gas Absorption Into Turbulent Films

Gus. Srp. Purif: Vol. IO. No. 1, pp. 41-46, 1996 Copyright :(: 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0950-4214/96 $15.00 + 0.00 0950-4214(95)00024-O

ELSEVlER

Modeling of gas absorption with chemical reaction Mohammad Chemical

into turbulent

films

R. Riazi”

Engineering

Department,

Kuwait

University,

PO Box 5969, Safat 13060, Kuwait

A mathematical model has been developed to predict the rates of gas absorption in turbulent falling liquid films with and without the first order homogeneous reaction and external gas phase mass transfer resistance. The eddy viscosity model used to describe the flow distribution is the van Driest model, modified in the outer region of the film by the use of an eddy diffusivity deduced from gas absorption measurements. The results are given for special cases to illustrate the effects of turbulence, reaction rate and gas phase resistances on the concentration profiles and the rates of gas absorption. Copyright @ 1996 Elsevier Science Ltd Keywords:

mathematical

model;

gas absorption;

Sh u

Nomenclature A

C C’ C.5 D

g g, k

k, kG K’ K’ L’

m N RC?

SC SC,

Damping length constant (26v(p/~o)) Concentration of dissolved gas in the liquid Equilibrium concentration of gas in liquid Concentration of gas at the free surface Molecular diffusion coefficient Acceleration due to gravity Gravity constant (= 1 in SI unit system) First-order reaction rate constant Mass transfer coefficient in liquid phase Mass transfer coefficient in gas phase Constant (= 0.4) Constant Prandtl mixing length Dimensionless rate of absorption Dimensionless parameter (= kc/D[u2/g]‘/3) Reynolds number Schmidt number (= v/D) Turbulent Schmidt number (= EM/ED)

I’ Z

films

Sherwood number (= k,.z/D) Liquid velocity in axial direction Distance normal to the surface Axial distance

Greek symbols

: s ED EM

0 P /l V TO

Reaction rate constant (= k(v/g2)‘13) Liquid loading Liquid film thickness Eddy diffusivity Kinematic eddy viscosity Surface tension Liquid density Liquid viscosity Liquid kinematic viscosity Shear stress at the free surface

Superscript

Non-dimensionalized

quantity

laminar falling liquid films has been studied in some detail, the c&e bf turbulent flow, especially when chemical reaction is involved, has received less attention. The problem of gas absorption without chemical reaction in laminar films was solved by Olbricht and Wild2 using the series expansion method. For the case of gas absorption into laminar films with chemical reaction, the governing differential equations were solved by both analytical and numerical techniques as shown by Riazi3,4. The effect of gas phase mass transfer resistances on the rate of gas absorption for laminar liquid films was previously discussed by Riazi3. The effect of turbulence

Introduction Development of mathematical models to predict the rate of gas absorption into liquid films is important in the design and operation of falling film reactors, which are widely used for gas-liquid reactions such as sulfonation or chlorination, as well as in gas separation and purification units’. From a review of the published literature on wetted-wall columns, it is evident that while the problem of gas absorption with or without chemical reaction in *Fax: (+965)

turbulent

4839498; email: riazi@kucOl .kuniv.edu.kw

41

42

Modeling

of gas absorption:

in the gas phase as well as the effects of interfacial drag at the gas-liquid interface on the rate of gas absorption were shown by Riazi and Fagri4. The problem of physical gas absorption into a turbulent liquid film was treated to some extent by Lamourelle and Sandall’. Based on the gas absorption measurements, they obtained an expression for the liquid phase eddy diffusivity in the region near the free surface. Menez and Sandall studied the problem of gas absorption accompanied by first-order chemical reaction in a liquid flowing in fully developed turbulent flow. They obtained asymptotic solutions for which solute concentrates only a short distance into the liquid film, because of a slow rate of diffusion or very high rate of reaction where only eddy diffusivity in the region near the free surface was used to describe the turbulence in the liquid film. The main objective of this work is to show the effects of turbulence in liquid falling films on the rate of gas absorption when combined with chemical reaction and gas phase mass transfer resistances. For our work, the eddy diffusivity given by Lamourelle and Sandall is used for the region near the free surface, while the van Driest viscosity model is used for the region near the wall.

M.R. Riazi

in the axial direction is negligible. Under these conditions, the steady state mass balance on the absorbing species in the liquid phase for turbulent flow is:

(1) The coordinate system used and the physical description of the absorbing film are shown in Figure 1. The term u is the axial velocity of liquid film and can be found from the momentum equation after neglecting the pressure gradient and axial terms:

g[(YfEdg+g=o The solution of the above equations for momentum and mass transfer requires the specifications of the boundary and initial conditions: at the inlet, z = 0

c=o

(3)

at the wall, y = 0 u=

0

-_= aC 0

Formulation Let us consider the system shown in Figure 1. A liquid initially free of the absorbing species at z = 0 flows down the surface of a vertical and impermeable wall under the influence of gravity. The absorbing species are absorbed by the liquid where it undergoes a (pseudo) first-order irreversible chemical reaction. It is assumed that the gas phase concentration of absorbing species is constant and the interfacial shear stress at the gas-liquid interface is neglected. Furthermore, it is assumed that the diffusion

QIEI 1

ay

(4)

at the interface, y = 6

au

5=O p(c*_c) where 6 is the film thickness, C’ is the concentration in equilibrium with gas phase, and ko is the mass transfer coefficient in the gas phase. Upon integration of Equation (2) with the corresponding boundary conditions given in Equations (4) and (5), the velocity profile can be obtained: L!=

Liquid

In the above equations, if flow is laminar then EM=&Eg=o. Introducing the following non-dimensional variables

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

Figure 1 Schematic of a falling liquid film showing the coordinate system

Modeling of gas absorption: M.R. Riazi

C=O

(8)

atJ=O

Non-dimensionalization the following form:

ac

1+

3-O N

T&y=zp(‘-c)

v (1 -y/J) ./0

1 + 0.64~*[1 - exp(-J/26)]*(‘1

dq’

EM

where a = k(v/g2)‘/‘,

-p/6) (12)

2

(10)

Lamourelle and Sandal15, by measuring the mass transfer coefficient for the liquid phase for gas absorption into a turbulent liquid Aow down a wetted column, obtained the following pattern distribution for sn at a temperature of 25°C

(11)

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

and U=

transfers the above equation in

EM =

at J = S

ac

43

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 condition given 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 for the eddy diffusivity. Some typical models for the falling film are introduced by Gutierrez-Gonzalez et al.‘. Accurate specifications of the eddy diffusivity close to the wall and also close to the free surface are much more important than in the middle of the film, due to low resistances in the central region. It is customary that for modelling Ed the flow is divided into two regions, an inner region where the turbulent transport is dominated by the presence of the wall and an outer wall-like region. The model proposed here is described by use of the van Driest model’, modified in the outer region of the film by use of an eddy diffusivity deduced by Lamourelle and Sandall from gas absorption measurements. Various assumptions have been made in order to describe the mean velocity distribution near the wall. A popular kinematic eddy viscosity model for this region, as mentioned before, is provided by van Driest’ who assumed the following modified expression for the Prandtl mixing length theory: L’ = K!y[l - exp( i)]

(13)

in which Ed is in ft2 h-‘. In order to generalize the above equation to temperatures other than 25°C and to liquids other 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. The above constant can be calculated from Equation (13), which yields the following result: -ED = 6.4 x lo-“:

Rc’.~‘*(~ -

u

y)*

<

For computation purposes, upon non-dimensionalization this reduces to: l/3pv314 Re'.678($__ y)2

&=1+6.47~10~~~

(14)

g,a62~3

The Reynolds number is defined as:

Re=~=4tpudy p

I-1

(15)

in dimensionless form, Equation (15) becomes: n

where A is a damping length constant defined as 26vm and K’ = 0.4. For the falling film van Driest viscosity is used in the inner layer of film liquid. Therefore for the inner region:

du &J=

1I EM =

-_v + Jz/’ + 4L’2 + 4L’“(S - y)g 2L’2 -_v + J$

+ 4L’2 + 4L’2(6 - y)g 2

Re = 4

sn

iidj

(16)

Therefore, the feature of the resulting model is the van Driest’ distribution modified in the outer region by the Lamourelle and Sandall’ model. It is a common practice to use a constant turbulent Schmidt number in the boundary layer analysis and for the present analysis SC, = 1 was used. The overall mass transfer coefficient from the interface to the inlet liquid film is defined as: (17)

Modeling of gas absorption: M.R. Riazi

44

where C, is the surface concentration. The Sherwood number can be calculated from the following equation:

(18) The overall dimensionless mass transfer rate, #z, is: fi= s

a$j=idr

(19)

The differential Equation (7) was put into a finite difference form using an implicit scheme. The CrankNicolson method was used for Equation (7) which yielded a three-diagonal matrix. Four hundred increments were chosen in the J direction, and using a stepby-step advancing technique (? was calculated at different values of Z.

at z = lo6 is shown in Figure 6. As N decreases concentration decreases too. A graphical relation between 8 and Re is shown in Figure 7. Transition from laminar to turbulent flow is at Re = 1200- 1300, for laminar flow, Re = ($)6* and for turbulent flow for values of s> 60, Re can be related to 6 by a linear relation of Re = 63.836- 1383 with less than 1% deviation. Mass transfer rates (m) are shown in Figure 8 for special cases. The two lower curves represent rates of absorption for laminar films (6 = 6.66 or Re = 59.1) for a = 0 (physical absorption alone) and o = 1 respectively. For turbulent flow (8 = 100, Re = 5000) the overall mass transfer rates are shown for a = 0, 1 and 10. As can be seen, the effect

0.8 -

Results

and discussion

The numerical values which were used in our calculations were: u = 8.63 x 10e7 m* s-‘, p = 996.3 kg cme3, D= 1.95x lo-9 m2 s-l, L&=442, g=9.81 m s-*, and g = 0.0689 N m-t. In most of our calculations we used S = 100 which is equivalent to the Reynolds number of 5000. For the physical absorption (Q = 0) development of the concentration profile is shown in Figure 2, calculations were carried up to z = lo6 which is equivalent to the column length of z = 7.24 m. For the case of laminar flow, development of the concentration profile for physical absorption for 6 = 6.66 (Re = 59.1) is shown in Figure 3. The relation between two different values of $1 and $2 is -

21 -=

61 _

22

0 62

l/3

0.6 F 0.4 -

0.2 -

5 x 10’

I 0

I

0.2

I

1

10’

0.4

I

0.6 Fla

0.8

I 1

Figure 2 Development of the concentration profile for no resistance in the gas phase and without chemical reaction for s= 100 (Re = 5000)

Thus z(S = 6.66) = 0.4052(6 = 100) In Figure 3 values of z correspond to the same dimensional axial distance z as in Figure 2. Results shown in Figure 3 are in good agreement (less than 0.5% deviation) with the analytical solution of Olbricht and Wild*. Figure 4 shows the same result for turbulent flow when 8 = 200 (Re = 11383). Comparison of Figures 2 and 3 indicates that turbulent flow concentration is less than in laminar flow while the slope of ac/a~ at free surface in turbulent is greater than in laminar flow. Therefore, as is expected the rate of gas absorption in turbulent flow is much higher than that in laminar flow. The effect of chemical reaction on the concentration profile is shown in Figure 5. Comparison of Figures 5 and 2 shows how the presence of a reaction reduces the concentration of the absorbing species. The effect of mass transfer resistance (N) on the concentration profile

z = 20250(5 x 10’) 0.6 -

0.4

0.2

0

Figure 3 Development of the concentration profile for no resistance in the gas phase and without chemical reaction for 6= 6.66 (Re=59.1)

Modeling of gas absorption: M.R. Riazi

45

0.8 T=2

0.8 -

x 106

N=CQ 4

0.6 3

0. 0

'

I 0.2

I

I 0.4

I

I 0.6

I

1 0.8

I 1

513 Figure 4 Development of the concentration ance in the gas phase and without chemical (Re = 11 383)

profile for no resistreaction for 6 = 200

Figure 6 Concentration cr=l for6=100at.T=106

profile

with

gas phase resistance

and

10000 -

8000 0.6

RC.

t 6000 -

I I 1 I

0 0

0.2

0.4

_ Y/6

0.6

Figure 5 Development of the concentration resistance in the gas phase and a = 1 for 6 = 100

0.8 profile

1 for

no

of the chemical reaction on the absorption rate in the laminar film is greater than that in the turbulent flow. The absorption rate for N = 1 and 01 = 1 is also shown in Figure 8. For higher Reynolds number (8= 200 or Re = 11383) the absorption rate for a = 0 is also shown in Figure 8. It should be noted that in Figure 8 for different dimensionless film thickness, 8, corresponding values of z for the same axial distance, z, are different. In fact for b = 6.66, ~(8 = 6.66) = 0.4052(6 = loo), and for 8 = 200, ~(6 = 200) = 1.262(6 = 100). The eddy diffusivity and eddy viscosity models proposed in this study may also be applied to laminar liquid films in order to consider effects of wavy film flow

Figure 7 Relation between and Reynolds number

the dimensionless

as discussed by Gutierrez-Gonzalez be the next phase of this study.

film thickness,

8,

et al.‘. This would

Conclusions The numerical solution of gas absorption into laminar and turbulent falling films with the combined effects of homogeneous, irreversible first-order chemical reaction and gas phase mass transfer resistance has been presented. Results are given in the form of concentration profile development and the rate of gas absorption. It has been shown that the relative effect of the chemical reaction on the absorption rate in the laminar flow is

Modeling

46 106b

’

’ ’ ’ ’ “‘I

I

M.R. Riazi

It has been shown that in turbulent flow, when Reynolds number or the film thickness increase, absorption rates also increase. With the gas phase mass transfer resistance, the absorption rates can be significantly smaller than those for the case of no resistance, particularly when the parameter N is small,

11’1’1’1 Q =

z = 200 a==0

of gas absorption:

10 1 II

-

,A

References Gutierrez-Gonzalez, J., Mans-Teixido, C. and Costa-Lopez, J. Improved mathematical model for a falling film sulfonation reactor Ind Eng Chem Res (1988) 27(9) 1701-1707 Olbricbt, W.E. and Wild, J.D. Diffusion from the free surface into a liquid film in laminar flow over defined shapes Chem Engng Sci (1969) 24 2.5

Figure 8 Gas absorption rates in laminar and turbulent and without chemical reaction

films with

Riazi, M.R. Estimation of rates and enhancement factors in gas absorption with chemical reaction and gas phase mass transfer resistances Chem Eng Sci (1986) 41( 11) 2925 Riazi, M.R. and Fagri, A. Effect of interfacial drag on gas absorption with chemical reactions AIChE J (1986) 32(4) 696 Lamourelle, A.P. and Sandall, O.C. Gas absorption into a turbulent liquid Chem Engng Sci (1972) 27 1035-1043 Menez, G.D. and Sandall, O.C. Gas absorption accompanied by first-order chemical reaction in turbulent liquids Ind Eng Chem Fundum (1974) 13(l) 72-76

greater than that in the turbulent flow, while the absolute mass transfer rates in the turbulent case are significantly larger than those observed in laminar flow.

van Driest, E.R. On turbulent flow near the wall J Aero Sci (1956) 23 1007 Levicb, V.C. Physiochemical Hydrodynamics Prentice Hall, Englewood Cliffs, New Jersey (1962) 691