University, UK ABSTRACT - WIT Press Free and Moving Boundary Problems INTRODUCTION The effect of radial turbulent mixing in absorption films has not been comprehensively treated

Mass transfer in turbulent flow

A. Jahanmiri", E.T. Woodbind

^ Chemical Engineering Department, Shiraz

University, Shiraz, Iran

^Department of Chemical Engineering, UMIST

University, UK

ABSTRACT

In order to assess the effect of turbulence on therate of mass transfer in the liquid phase, work hasbeen carried out on gas absorption by liquid flowingover spheres.

The hydrodynamics of the liquid flow over sphereswas modelled incorporating eddy viscosity. Thecoefficients of the eddy viscosity expression wereevaluated by measuring the liquid hold-up over astring of twenty spheres of 3.7 cm diameter. The rangeof the liquid Reynolds numbers was between 30-600. Theform of eddy viscosity was found to be of the form:

Mass transfer in the liquid film flowing over asphere was modelled based on eddy diffusivity[1] incorporating the hydrodynamic model results forphysical or chemical absorption accompanied by ainstantaneous reaction and a pure or mixed gas phase.The system of partial differential equations of masstransfer with moving boundaries was solved numericallyin order to obtain the concentration profiles of thereactants, position of the plane of reaction, and therate of absorption. The predicted results agreed wellwith the experimental results. The mass transferanalysis revealed that the rate of gas absorptionincreases with increasing the liquid flow rate and theliquid reactant concentration in the chemicalabsorption processes.

Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

472 Free and Moving Boundary Problems

INTRODUCTION

The effect of radial turbulent mixing inabsorption films has not been comprehensively treatedyet if early turbulent mixing could be induced in filmof liquid over packing then liquid film resistancecould be significantly reduced if not eliminated.

Turbulence could be induced either by wave motionon the surface or through eddies generated as aconsequence of high velocity gradients in thin films.Different shapes of packing have been used for theexperimental study. Stephen and Morris [2] usedcircular disks,with the vertical plane of each diskbeing at right angles to the one above. Althoughthis may well develop turbulence it has somedifficulties [3]. Spherical packing has been used asa simpler and an alternative packing piece by manyinvestigations [4-9] in gas absorption processes. Inthis system liquid flows over each sphere and, onpassing through the meniscus between the packingelements cause an interruption similar to that betweenpacking pieces in packed tower. The existingmathematical analysis for this system has beendeveloped based on laminar flow and the liquid surfacehas been assumed to be ripple free. Comparisonsbetween the theoretical and the experimental resultsshow that, when the liquid flow rates are low ,agreement was achieved based on the assumption oflaminar ripple-free flow (Figure 1).

Lynn et al. [5] used a simple method of solvingthe hydrodynamics of liquid flow over a sphere in thelaminar flow region. The thickness and velocityprofile of liquid film were found as:

(1)

Mass transfer in laminar flow over sphere has beenanalyzed both experimentally and theoretically[4,5,7,8]. Davidson et al. [7] derived a mass transferequation in the liquid film by considering an elementbounded by two stream lines and the two radial lines.If there is no convection across the stream lines andif diffusion in the flow direction is neglected thenthe following partial differential equation is


Free and Moving Boundary Problems 473

deduced:

For small depth of penetration solution of theabove equation gives [7]:

(C-CQ) /(Cj-Co) =erfc(P/2/Z%pT (5)

The experimental and the theoretical rate of CCâbsorption with pure water show an agrement only forthe low liquid flow rate [4] (Figure 2).

The divergence in both hydrodynamics and masstransfer results may be due to the fact that as thefree surface of the liquid film flows down by gravityeffect, it is disturbed by wave motion and ripples. Itwas found theoretically that laminar motion isunstable for any value of Reynolds number, whileexperiments lead to the conclusion that there exits acritical value of Reynolds number for the transitionfrom laminar state to the wave-motion (N̂ =7) [10].However the analysis of the liquid film flowaccompanied with wave-motion is valid up to Reynoldsnumber of about 120-130 , [11,12]. At these Reynoldsnumber the dominating mechanism of momentum and masstransfer is probably due to unsteady moleculardiffusion into a largely laminar fluid which isintermittently mixed by eddies associated with thesurface wave structure. At higher liquid flow ratesthe laminar and wave treatment does not satisfy theexperimental results, and also does not predict theexistence of wave-motion on the free surface of theliquid.

In turbulent flow the liquid moves in a veryirregular manner causing an exchange of momentum fromone portion of the liquid to another in a mannersimilar to the molecular momentum transfer but on amuch larger scale. In the vicinity of the wall themotion of the eddies is damped because they cannot move radially, and the transfer process depends onmolecular motion. Near the wall the controllingmolecular transfer is very slow, but the distances aresmall while in the main turbulent stream the transferby eddies is rapid.

The eddies are characterized by their velocitiesand by the distance over which these velocities changesignificantly. By analogy with Newton's law forlaminar flow one of the earliest proposal forturbulent shear stress is [13]:



If it is assumed that both the molecular and eddytransport take place by parallel processes, then twocoefficients may be added together. Therefore thetotal shear stress, (T̂ ) is:

=-(H + - (7)

The main difficulties in analyzing the turbulentflow is associated with the turbulent viscosity.

HYDRODYNAMICS

Levich [14] has analyzed the general nature ofturbulent motion qualitatively at developed turbulentstate. The scale of motion in turbulent flow is one ofthe characteristic of the eddies over which thevelocities of eddies are changed. His results showthat, the scale of motion decreases from turbulentzone to the solid surface, and the rate of reductionincreases. In general the variation of eddy viscosityin the turbulent film has been show to be described bya single expression as:

(8)

where the power n is varied from 1 to 4 , and a is afunction of physical and flow characteristics, and yîs the distance measured from the solid wall. Byanalogy with the results of analysis of mass transferstudy in turbulent flow over the flat surface in termsof eddy diffusion [1], the coefficient a is assumedas:

Then the eddy viscosity in term of y, thedistance from free surface is considered as:

Vc=A.£e*(y<,-y)" (10)

where :

_ 4* (11)

A differential element of liquid flow as shown inFigure (3) is considered to obtain flow equation. The



assumptions made for analysis are:1-The thickness of the liquid film is very smallcompared to the radius of sphere (R » VQ)2-Steady state condition3-Newtonian flow4-Neglect changes in the rate of momentum in the flowdirection5-Neglect acceleration effect in the flow direction.

w

Figure 3. Liquid film over sphere and force acting onelement.

Force balance on the differential element andtaking differential gives:

(12)

The boundary conditions are:

u=o at y=y. (13)

du =Tj at y=0 (14)

where, r^ is the shear stress on the liquid surfacecaused by the gas flow.

Depending on the value of n in Equation (10)different forms of velocity distribution are obtained.The values 2 and 3 were used in this study, if n isequal to 2 the velocity distribution is obtained as:

To assess the velocity of the hydrodynamics



analysis in terms of the undetermined coefficients ofthe eddy viscosity, it is to predict the hold up ofthe liquid over a sphere, and then to check theprediction against experiment.

^ ,/a+yy A/a ' (0

On the other hand the average velocity in thefilm is equal to:

(18)

where Q is the liquid flow rate.

The thickness of the liquid film is determined bysolving the above equations for yo implicity.

The contact angle for the top of the first sphereis found by determining the thickness of the liquidfilm over the rod.

MASS TRANSFER

Mass transfer in turbulent flow over the flatsurface in terms of eddy diffusion has been studied inthe past [12-20]. Levich [14] studied the diffusionalflux in turbulent flow by considering the nature ofthe turbulent motion near the free surface of theliquid. He quoted that at Pr»l, we can assume themajor diffusional resistance is offered bydiffusion sublayer. Then the mass-flux could writtenas DI 3c/ 3y where, D< is the turbulent diffusioncoefficient. He found the turbulent coefficientdecreases as the liquid surface is approached as:

D,ay2 (19)

where y is the distance measured from the freesurface.

Later on King [15] analyzed the liquid phase masstransfer processes to and from a free liquid interfaceinvolving a damped eddy diffusivity in the vicinity ofthe surface. He used a general form, ay" for eddydiffusivity, and analyzed the mass transfercoefficient as a function of age of element of surfacefor different values of power n from zero to infinity.

Lamourelle and Sandal [1] followed the methods



adapted by Levich [14] and King [15] and tented masstransfer near a free surface in terms of eddydiffusivity by measuring the mass transfer coefficientof the liquid phase for gas absorption into aturbulent liquid flowing down a long wetted-wallcolumn. They used helium, hydrogen, oxygen and carbondioxide absorption into distilled water over aReynolds number range between 1300-8300. The resultswere interpreted in terms of an eddy diffusivity withthe general form of, ay". It was shown that the eddydiffusivity increases as the square of the distancefrom the interface and confirmed the conclusion ofprevious investigators such as Levich [14] and Davies[20]. By comparing with the experimental results itwas found that the eddy diffusivity in the vicinity ofthe free surface is:

Dc=7 . 9x10-5. Rei-*™.y* (20)

Later on, the above form for eddy diffusivity wasconfirmed in gas absorption accompanied by first orderand instantaneous chemical reaction [16-19].

Equation (18) is valid for the regions adjacentto the free surface. However as a valid form it wasused in this study over the whole film thickness,obviously in a real film the turbulence will decreasenear the support wall but this not modelled because,of the large schmidt number as usually encountered ingas absorption, the major resistance to mass transferoccurs close to the free surface. So it is importantto know the eddy diffusivity is only accurate in thisregion. The magnitude of the error, involved in thisassumption was estimated by using a more realisticeddy diffusivity distribution in the bulk of liquidand it was found that, the mass transfer coefficientdiffered by less than 0.05% [18].

By considering a differential ring of liquid filmwrite down a solute mass balance and takingdifferential gives:

U——=R-——(D —— ) (21)

Where D? is the total diffusivity which is consideredas:

Dj=D+£y* (22)

where:

ci=7.9x10-5.aei'^s (23)



The similar assumptions as considered before,are madehere.

In physical absorption if the gas phase is purethen the concentration is constant over the whole ofthe liquid film is:

at 6=61 , O^y^y^ (24)

at y=y* ,8̂ 8̂ % (25)

Cj=A^ at y=0 , 0^<0<02 (26)

In chemical Absorption if an instantaneousreaction takes place in the liquid film between thedissolved gas (A) and the reagent (B) , a hypotheticalsurface where the concentration of both component arezero exist (plane of reaction). Concentrationdistribution in the liquid phase are described by asystem of two partial differential equations as:

dA dy * dy

If the soluble gas is absorbed from a gas mixturethen the boundary condition becomes:

CB=CBI ^ 8=61 (29)

at y=y, (30)dy

C,=0 at 6=6, (32)

(34)



where:

(35)

At the plane of reaction, whose location is afunction of y and 6, the concentration of reactantsis zero and the motion of the plane of reaction isdescried as [21]:

d8(36)

The motion of the plane of reaction and theconcentration profiles qualitatively sketched inFigure (4).

Q, cB,

gas phase

liquid film interface

plane of reaction

Figure 4. Motion of plane of reaction in liquid film.

Numerical method was used for solving the systemof partial differential equations. Since theboundaries of the system (ie. the thickness of thefilm and the position of the plane of reaction) arevarying a dimensionless variable was used as:

(37)

By using finite difference, the partialdifferential equation (20) is changed to:

Where the coefficient A, B, C and the right handside are known. These set of equations may be written



(38)

in matrix form as:

W V = X (39)

The first matrix is a three diagonal and theothers are column matrix. The answer is:

V = X (40)

EXPERIMENTAL PROCEDURE

The apparatus used consists of a cylindricalcolumn of 70 cm height contain a string of twentypolypropylene spheres each of 37 mm diameter, whichmounted on a rod Figure (5) . The liquid flow rate wasin the range of (0-17 m/s) . The carrier gas wassupplied form cylinder B and the soluble gas fromcylinder A. In order to measure the variation of gasconcentration, side stream sample points wereconsidered.

Inlet Liquid

Solvent Tank

ffiifl r-)0

C

0,

— fel

Thermostat ic Tank

i Humidification/Tl Bottle

Inlet Gas

ReserviourITank

Figure 5. Schematic diagram of experimental apparatus.

In hydrodynamic studies, the liquid hold-up wasmeasured for different flow rates. Hold-up consist oftwo parts, dynamic and static hold-up. The effect ofvarious surfactant was tested by using three different



surface agents.

Mass transfer experiments were carried out bymeasuring the rate of gas absorption and the gasconcentration profile in the column for both physicaland chemical systems. Different gas-liquid systemswere used. Ammonia-water,carbon dioxide-water forphysical absorption and carbon dioxide-sodiumhydroxide solution, hydrogen sulphide-NTA.Fe^+ forchemical absorption were used.

RESULTS AND DISCUSSION

The hold-up prediction obtained from thehydrodynamic model for two value 2 and 3 for power nin Equation (10). The experimental and predictedresults are shown in Figure (6) . The experimentalresults show that the measured hold-up increasesrapidly at lower liquid flow rate and dose not show asubstantial deviation with different surface activeagents. The solid line shows the hydrodynamic modelresult for the value of n = 2 which is the best curvesthrough the experimental results after varying theconstants A and B. The dotted curves show thepredicted hold-up for laminar flow. The laminar andturbulent flow predictions have similar shapes andthey start to diverge from each other at a Reynoldsnumber of about 50 and the divergence increases withliquid flow rate. A possible reason for theexperimental hold-up results, below the Reynoldsnumber of 140, lying below the predicted curves,couldbe that at low liquid flow rated, the wetting ofsphere is not complete.

By comparing the predicted results for two values2 and 3 for power n and the experimental results,there appears to be little difference between the twopredictions. However n = 2 seems to give slightlybetter agreement so it was the basis for masstransfer prediction. Then the Equation (10) change to:

v,=7 .SxlO-s.#e2-2i(y^_y)2 (41)

A typical velocity profile in the liquid film atthe equator if we assume no shear on the liquid surfaceis shown in Figure (7). The effect of shear surfacecaused by increasing the upward gas flow isillustrated in Figure (8).

Analysis of mass transfer in the turbulent regainswas in terms of eddy diffusivity base on Equation(19). A typical concentration of reactants in the



liquid film at different angles is shown in Figure(9) . To examine the validity of the assumption madethe form of eddy diffusivity, arrange of experimentswere carried out.

The experimental and predicted results for CCâbsorption by water in terms of the liquid side masstransfer coefficient (K,) are shown in Figure (10) .The predicted results, by turbulent assumption matchwell with those of the experimental results forReynolds number higher than 250. In the lower rangethe predicted results are higher than the experimentaland as mentioned before this may be due toincomplete wetted surface of the sphere. The laminarflow results, shown by dotted line is located underthe experimental and predicted turbulent results.Their divergence increases by increasing the liquidflow rate, to which is attributed the effect ofturbulence in the liquid film.

In the second part, the concentration profile,using two systems, in the column were examined. Theseincluded the physical absorption of ammonia from airby pure water, and hydrogen sulphide from air by Fe*+chelated by NTA solution. The experimental results areshown with the predicted results in Figures (11) and(12) . The results show that the model predictions arein agreement with those of the experimental ones whichlead to the conclusion the reliability of the masstransfer analysis.

NOMENCLATURE

A Eddy viscosity coefficienta Eddy viscosity coefficienta' Eddy diffusivity coefficientB Reynolds Power in eddy viscosityC Concentration in liquid film (mol/lit)CQ Concentration at the point of entry (mol/lit)D Molecular diffusivity (m^/s)g Gravity acceleration (m/s^)n Power in eddy viscosity or eddy diffusivity

expressionP Dimensionless liquid film thicknessQ Liquid flow rate (m^/s)R Radius of sphere (m)u Velocity (m/s)y Distance measured from liquid surface (m)YQ Liquid film thickness on sphere (cm)y% Distance from wall (m)6 Angle measured from top of sphere (rad)X Position of plane of reaction measured from

interface (m)



p Liquid densityr Shear stress (Pa)v Kinematic viscosity (nf/s)A0 Thickness of liquid film (m)

SUBSCRIPTSA Solute gasB Liquid reactanti Condition at liquidt TurbulentT Totali Grid number in radial directionj Grid number in flow directioi

REFERENCES

1 Lamourelle,A.P. and O.C. Sandall,Chem.Eng.Sci.,27,1035,1972

2 Stephen,E.J. and G.A.Morris,Chem. Eng. Progr.,47, 232, 1951

3 Taylor,R.F.& Roberts,Chem. Eng. Sci.,5,168,19514 Davidson,J.R. et al.,Trans. Instn. Chem. Eng.,

37,122,19595 Lynn,S.,J.R. Straatemerier and H. Kramers,Chem.

Eng. Sci.,4,63,19556 Yoshida,F. and T.Koyanagi Ind. Eng. Chem.,50,

365,19587 Davidson,J.F. and E.J. Cullen,Trans. Instn.

Chem.,35,150,19688 Wild,J.D.,and O.E. Potter,!. Chem. E. Symposium

Seris,28,30,19689 Tamir,A., J.C. Merchak and P.D. Virkar, Chem.

Eng.,35,1393 198810 Ruckenstein,E. and C.Berbente,Chem. Eng. Sci.,

20,795,196511 Banerjee,S.,E.Rhodes and D.S.Scott,Chem.Eng.

Sci.,22,43,196712 Bunch,D.W. and M.R.Strunk, AIChE.J. 11,1108,196513 Bird,R.B.,W.E.Stewart and E.N.Lightfoot,

"Transport Phenomena", Wiley, New York, 196014 Levich,V.G.,"Physicochemical Hydrodynamics",

Pretice-Hall, 196215 King,C. J., I&EC. Fundamentals, 5, 1, 196616 Kayihan,R. and O.C.Sandall, AIChE J.,20,402,197417 Sandall,O.C.,Int. J. Heat Mass Transfer,17,459,

197418 Menez,G.D. and O.C.Sandall,Ind. Eng. Chem.

Fundam.,13,72,197419 Mendez,F. and O.C.Sandall, AIChE J.,21,534,197520 Davies,J.T., Proc. R. Soc., A290, 515, 196621 Astarita,G.,"Mass Transfer with Chemical

Reaction", Elsevier Publishing Company,p.55,1967



Figure 1: Experiment and Predicted data of the hold-un for one sphere byDavidson et at [ 4 ]

o Experimental results for pure waterA Liesepol solution

2 4 6 8 10 12 14

Liquid rate (ml/s)

Figure 2: Carbon dioxide absorption by two sphere [4 )



70

60-.

50--

r_ 40--

20-.

Turbulent ModelLaminar Model

A0V*+

Surfactant

T.X-100S.L.SC.T.M.A.BC.T.M.A.B

Cone.gr/lit

0.0050.0050.0050.01

Solvent

Water

n=2A=7.9E-5B=2.21

50 JOO 150 200 250 300 350 400 450 500 550 600 650Re

Figure 6: Exoerimental and Predicted data of hold-uo as a function ofReynolds number for different surfactant

0.00 -r

0. 05

0.15-.o

* r <u

0.25(Jo^ 0.30 - •

0.35-'.

0.40

Q=14.00ml/s\Re=490DP=0.037mDP/Z=0.0 Pa/m

0.36 0.32 0.28 0.24 0.20 0.16 0.12 0.08 0.04 0.00distance in liquid film (mm)

Figure 7: Velocity profile in liquid film at equator


Solute Concentration (mol/lit)

Velocity (m/s)

-4- -f- -H§

-1

2= -4^ 5> -H

Surface of sphere

_,. _.. o

Q-

GdocCL



15.0

IS. 5-

12. 0-

10.5-

9.0 -

7.5

^601=^ 4.5mujs *•*

1.5

0.0

PHYSICAL ABSORPTIONCO^ (PURE) -WATER

A RUN NO.2V RUN NO. 5

Laminar Model DTurbulent Model

50 WO ISO 200 250 200 350 400 450 500 550 600 650 700Re

Figure 10: Experimental and Predicted Liquid Side Mass Transfer Coefficients

PHYSICAL ABSORPTIONAMMONIA & AIR WATERC 1 CIOD =

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0Height of bed (m)

Figure 11: Experimental and Predicted profile of gas concentration inabsorption column


Gas Concentration (ppm)oo00

CD(D

CL

O

OQ

CL

Oo;nT3


Documents

University, UK ABSTRACT - WIT Press Free and Moving Boundary Problems INTRODUCTION The effect of radial turbulent mixing in absorption films has not been comprehensively treated