Simulation of binary vapor condensation in the presence of an inert gas

IN HEATANDMASS TRANSFER 0094-4548/82/060463-I0503.00/0 Vol. 9, pp. 463-472, 1982 ©Pergamon Press Inc. Printed in the United States

SIMULATION OF BINARY VAPOR CONDENSATION IN THE PRESENCE OF AN INERT GAS

Ross Taylor and Matthew K. Noah Department of Chemical Engineering

Clarkson College of Technology Potsdam, New York 13676, USA

(C~,[,~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Two recently developed simplified models of multicomponent mass transfer are employed to determine the composition profiles during the condensation of a binary vapor (methanol-water) from an inert gas (air or helium). I t is found that the profiles and the predicted outlet conditions are in excellent agreement with the results obtained using an exact solution of the Maxwell-Stefan equations to predict the mass transfer rates. I t is concluded that the simplified approach to multicomponent mass transfer calculations will probably be adequate for many design problems, thereby realising important savings in computational time and computer program complexity.

Introduction

The design of heat exchangers to condense multicomponent vapor mixtures

requires the numerical integration of a set of differential material'and

energy balances combined with the solution of the equations giving the rates

of mass and energy transfer at a particular location in the condenser.

Multicomponent mixtures display many interesting interaction effects, such as

reverse or osmotic diffusion, that are not possible in binary mixtures and

which can significantly affect condensation rates.

Krishna and co-workers [I-3] have compared several methods of calculating

the mass transfer rates in a variety of multicomponent condensation design

problems; an effective diffusivity method (which ignores interaction effects),

a solution of the linearised equations of multicomponent mass transfer

developed by Toot [4] and by Stewart and Prober [5] and a method due to

Krishna and Standart [6,7] based on an exact solution of the Maxwell-Stefan

463

464 R. Taylor and M.K. Noah Vol. 9, No. 6

equations, al l of the foregoing being based on a f i lm model of steady state

mass t ransfer . The experimental resul ts that are avai lable [8 ,9] confirm the

general conclusion that the f i lm models which take in teract ion ef fects into

account are of comparable accuracy and are far superior to methods of the

e f fec t i ve d i f f u s i v i t y type. Their drawback is the length and complexity of

the i t e ra t i ve calculat ions that are required. The methods may also exh ib i t

convergence d i f f i c u l t i e s [10-12] and cer ta in ly take much more computer time

than the e f fec t i ve d i f f u s i v i t y methods.

In 1975 Burghardt and Krupiczka [13] published an approximate solut ion of

the Maxwell-Stefan equations for mass t ransfer in multicomponent mixtures

containing one or more iner t species. Their method of ca lcu lat ing rates of

mass t rans fer involves no i t e ra t i on . Bandrowski and Kubaczka [9] included

th is method in t he i r comparison of theory and experiment and found i t to be

only s l i g h t l y less accurate than the more rigorous methods but needed substan-

t i a l l y less computer time. Krishna [14,15] has also presented an e x p l i c i t

method of ca lcu lat ing rates of mass t ransfer in multicomponent mixtures. More

recent ly , Taylor and Smith [16] extended and s imp l i f ied the e x p l i c i t method of

Burghardt and Krupiczka [13] to many of the other determinacy condit ions that

are encountered in other s i tuat ions. A thorough s t a t i s t i c a l comparison [17]

of the e x p l i c i t methods with an exact solut ion of the Maxwell-Stefan equations

showed the new method [16] to be of good accuracy over the ent i re ternary

composition t r i ang le ; i t s poorest re la t i ve performance coming in mixtures

containing iner t species. The e x p l i c i t method of Krishna [14,15] performs well

i f the mass t ransfer rates are low (small dr iv ing forces).

From the designer's point of view a bet ter test of the accuracy of the

e x p l i c i t methods would be a comparison of the composition pro f i les predicted by

the various mass t ransfer models for a par t i cu la r process. The object ive of

the study described here was to use the e x p l i c i t methods to determine the

composition p ro f i l es in a ver t ica l tube in which a binary vapor is condensing

in the presence of a non-condensing gas. In essence, we are repeating the

design calculat ions of Krishna [2 ,3] but use the e x p l i c i t methods [14-16] to

calculate the rates of mass t ransfer .

The notat ion used here corresponds, wherever possible, to that of Krishna

and coworkers [ I - 3 ] ; the few newly introduced symbols are defined in the text

at the point of t he i r f i r s t use.

Vol. 9, No. 6 SIMULATION OF BINARY VAPOR COND~'~SATIGN 465

Mass and Ener9~ Balance Equations

Consider a multicomponent vapor mixture condensing inside a vertical tube.

The vapor enters at the top of the tube and flows co-currently with the con-

densate. The differential material balances for each species of the mixture

are [2]

dG i dL i - N.xDZ = - i = 1,2..n (1) d~ 1 d(

where G i and L i are the molar flow rates of species i in the vapor and liquid

phases respectively. The N i are the constituent molar fluxes, positive i f

condensation actually occurs, zero for inert species and negative i f evapora-

tion takes place. { is a dimensionless distance measured down from the top of

the tube.

An energy balance for the vapor leads to

dT b Gt% d~ - qb xDZ (2)

where qb is the conductive heat flux in the vapor phase. Coolant, in the

annulus surrounding the tube flows countercurrent to the vapor-liquid flow.

An energy balance on the coolant yields [2]

dT c

LcCpc d~ - - qw ~DZ (3)

where qw is the heat flux through the condenser tube wall into the coolant.

I t is not necessary to augment these relations with an energy balance for the

condensate.

Mass and Energy Transfer Rate Equations

The (numerical) integration of equations (I-3) ~equires the calculation

of the mass and energy transfer rates, N i and q, at each step. The expl ic i t

methods of mass transfer described above lead to the following equations for

the fluxes, N l and N 2 in a ternary gas/vapor mixture.

N 1 : Kll---(Ylb-Yii)+ KI2-=(Y2b-Y21 ) (4)

N 2 : K21Z(Ylb-YlI) + K22z(Y2b-Y21) (5)

where the Kij are the multicomponent overall mass transfer coefficients. For

the special case of interest here (ternary mass transfer with N3=O) the methods


of Taylor and Smith [16] and of Krishna [14,15] lead to the following

expressions for the Kij.

Kll = kl3(Ylk23 + Y3kl2)/Y3 S (6)

Kl2 = Ylk13k23/Y 3 S (7)

K21 = Y2kl3k23/Y 3 S (8)

K22 = k23(Y2k13 + Y3k l2 ) /Y3 S (9)

S = Y lk23 + Y2k13 + Y3k12 ( I0 )

The Yi in equations (6-I0) are the arithmetric averages of YiI and Yib" The

k i j are the mass transfer coefficients of the binary i - j pairs. These

coefficients are estimated from an appropriate correlation of, say, the

Chilton-Colburn JD = f(Re) type (see [7,18] for further discussion on this

point).

in equations (4,5) is a correction factor that accounts for the shape

of the concentration prof i le in the vapor " f i lm". The exp l i c i t method of

Krishna [14,15] leads to l inear concentration profi les and, hence, ~ is just

unity. In the method of Taylor and Smith [16], for n =3, N3=0,~ is found to

be

: - ¢ e~+ l ; ~ = In ~Y31~ ( I I ) - 2 e¢_ l \Y3b}

Continuity of the mass transfer rates across the vapor-liquid interface

requires that the N i calculated from equations (4,5) be equal to the same fluxes

in the l iquid phase. In many condensing systems the principal resistance to

mass transfer resides in the vapor phase. A description of mass transfer in

the l iquid phase is not, therefore, required and one of two extreme conditions

is assumed to hold [1,2,18].

( i ) The l iquid phase is completely mixed la te ra l l y with regard to composition.

This corresponds to i n f i n i te l iquid phase mass transfer coefficients and the

l iquid composition calculated from a material balance along the flow path

n = Li/k L k i = 1,2..n (12) Xil ~l

VOI. 9, NO. 6 SIMULATION OF BINARY VAPOR COND~SATICN 467

( i i ) The l iquid phase is completely unmixed, corresponding to zero l iquid

phase mass transfer coefficients. In this case the interracial composition of

the condensate is given by the relative rates of condensation

n X i l = N~/ S N k i = 1 , 2 . . n (13)

k:l

The conditions at the interface (Xil , Y i I ' TI) become completely specified by

assuming equilibrium to exist there

Yi l = Kie Xil i = 1,2..n (14)

(where the Ki e are the equilibrium ratios) and by the requirement that the

energy flux be continuous across i t [2]

n qw : ho (T I -Tc ) : qb + }~ Ni~i + hy~(Tb-T l ) (15)

i : l

n qb : hy ~ (Tb-T I ) ; E : ~ /hy (16) ee-1 i=l NiCpi

h accounts for the resistance to heat transfer in the coolant, the tube wall, o the condensate and in any d i r t films, hy is the vapor phase heat transfer

coefficient, estimated from the JH half of the JH = JD = f(Re) analogy [18].

The ~i are the constituent latent heats of vaporisation.

I t is important to note that even for the special case of mass transfer through a stagnant gas the method of Taylor and Smith [16] and the method of

Burghardt and Krupiczka [13] di f fer in their respective methods of estimating

the Kij. Taylor and Smith [16] estimate the Kij from binary ki j which, in

turn, depend on binary diffusion coefficients. Burghardt and Krupiczka [13]

estimate the Kij as a matrix function of their matrix of multicomponent

"diffusion" coefficients. The method of Taylor and Smith [16] is in exact

agreement with the methods of Toor [4], Stewart and Prober [5] and Krishna and

Standart [6,7] i f al l the binary k i j are equal. The method of Krishna [14,15]

is appended to this l i s t i f , further, the total molar f lux, ZN i , vanishes.

The method of Burghardt and Krupiczka [13] is not in exact agreement with the

more rigorous models even i f all the binary coefficients are equal.

Algorithms for the calculation of the N i and q from equations (4-16) are

described by Krishna and Standart [7] and by Webb and Taylor [19].


Computational Results and Discussion

To test the accuracy of the simplified methods in design applications we

have re-run the examples considered by Krishna [3]. Methanol (1) and water (2)

are condensing in a vertical tube of 2.12 m length, inside diameter, D, 0.0254 m.

The vapors enter the top of the tube at a pressure of 1.0135 bar with the non-

condensing species, air or helium (3); the lat ter giving a mixture which displays

large interaction effects. The coolant, water, leaves the surrounding annulus at

a temperature of 308.15K. All other conditions at the top of the tube are given

in Table I for each of the example problems reported here (the specifications

dif fer, in a few cases, from those given by Krishna [3] which sometimes lead to

unrealistic coolant entrance temperatures). Physical properties and heat and

binary mass transfer coefficients are calculated in the same way that Krishna

and Panchal [2] computed these quantities. In fact, we have used the computer

program given by Krishna [20] with appropriate modifications to include the

simplified methods and to permit the program to be executed on an IBM 4341

computer. Single precision was used in all calculations.

The outlet conditions, predicted using three different methods of estimating

fluxes corresponding to the inlet conditions in Table I are summarised in Table

I I . Method I uses the method of Krishna and Standart [6,7] to predict the mass

transfer rates, method II uses the expl ic i t method of Taylor and Smith [16] and

method I l l uses the expl ic i t method of Krishna [14,15].

I t is immediately clear that the discrepancies between the predicted outlet

conditions are low and that no errors of any consequence whatsoever are

introduced by the simplified method of Taylor and Smith [16]. The method of

Krishna [14,15] does sl ight ly less well, but only in mixtures of high concentration

of condensing vapors. This would be expected on purely theoretical grounds since

method I l l is not an exact calculation of the mass transfer rates i f all the

binary ki j are equal. In an attempt to uncover situations leading to larger

discrepancies between models I and II we have considered many other problems with

specifications differing from those given in Table I. We must admit failure in

this task, the examples reported here are typical of all the problems considered

to date.

Temperature and composition profiles for some of these examples are given

by Krishna [3] for method I and a model based on an effective di f fusiv i ty method

af calculating the mass transfer rates. The profiles computed using an

Vol. 9, NO. 6 SIMULATION OF BINARY VAPOR CONDENSATIQN 469

effective diffusivity method are very different from those calculated using

method I;the temperature and composition profiles computed using the explicit

method [16] were virtually indistinguishable from the profiles for method I and,

for this reason, are not given here. There was l i t t l e difference between the

outlet conditions predicted by the two extremes of condensate mixing,

equations (12,13)(see, also, [8]).

In view of the excellent performance of the explicit methods i t is relevant

to consider the computational time requirements of the various methods. These

results also are reported in Table II . I t is immediately clear that the

advantage lies with the explicit methods which require about half the time

taken by method I.

TABLE I

Specification of Conditions at

Example l Example

Inert Gas Air Air

Gt(kmol/s) 1.841 x lO -4 1.841 x

G3(kmol/s) 1.197 x lO -4 7.364 x

Yl 0.2129 0.4500

Y2 0.1369 0.1500

Tb(K ) 344.20 360.0

Tc(K) 308.15 308.15

Lc(kg/s) 0.03 0.04

Inert Gas

Gt(kmol/s)

G3(kmol/s)

Yl

Y2 Tb(K)

Tc(K)

Lc(kg/s)

the top of the Condenser Tube

2

10 -4

lO -5

Example 5 Example 6

Helium Helium

1.841 x lO "4 1.841 x lO -4

1.197 x lO -4 4.603 x i0-5

0.2129 0.5000

0.1369 0.2500

344.20 350.00

308.15 308.15

0.03 0.05

Example 3

Air

1.841 x lO -4

1.841 x lO "5

0.7000

0.2000

360.0

308.15

0.06

Example 4

Air

l .841 x lO -4

0.921 x I0-5

0.4500

O. 5000

365.0

308.15

0.06

Example 7

Helium

1.841 x lO "4

1.841 x lO -5

0.7000

0.2000

360.00

308.15

0.06

Example 8

Helium

1.841 x lO -4

5.233 x lO -5

0.3500

0.3500

355.00

308.15

0.05

Condensate Mixing: Unmixed


TABLE I I

Outlet Conditions Predicted Using Three Different Mass Transfer Models

Example Method G t x lO -4 Yl Y2 Tb T CPU Time (kmol/s) (K) (~) (sec)

I 1.516 0 . 1 7 0 3 0 .0402 309.33 296,05 2,36 1 I I 1.516 0 . 1 7 0 3 0 .0402 309.33 296.05 1,68

I l l 1.516 0 .1704 0 .0402 309.33 296,05 1,61

I 1.034 0 .2532 0 .0348 315.12 287,46 1,80 2 I I 1.034 0 .2533 0 .0347 315.12 287.46 l. lO

I l l 1.036 0 .2542 0 .0350 315.15 287.50 1.06

I 0.307 0 . 3 6 0 5 0 .0408 324.12 283.80 4.04 3 I I 0.307 0 .3618 0 .0394 324.12 283,80 1,15

I l l 0.317 0 . 3 7 6 0 0 .0429 324.53 283.92 1.09

I 0.138 0 .2481 0 .0888 320.33 280.42 3.25 4 I I 0.138 0 .2492 0.0851 320.35 280.42 1.50

I I I 0.147 0 . 2 6 8 9 0 .1036 322.42 280.46 1.34

I 1.481 0 , 1 5 6 5 0.0351 302.91 294.76 4.51 5 I I 1.481 0 , 1 5 6 5 0.0351 302.91 294.76 3.27

I l l 1.481 0 . 1 5 6 5 0.0351 302.91 294.76 3.17

I 0.579 0 , 1 7 4 7 0 .0309 299.96 283.52 4.28 6 I I 0.579 0 , 1 7 4 3 0 .0313 299.98 283.52 2.60

I l l 0.580 0 .1750 0 .0317 300.05 283.53 2.36

I 0.233 0 .1897 0 .0213 301.20 282.92 29.12 7 I I 0.233 0 .1890 0 .0220 301.24 282.92 14.83

I l l 0.234 0 .1917 0 .0227 301.50 282.93 13.87

I 0.681 0 . 1 4 6 6 0 .0433 299.67 284.67 4.53 8 I I 0.681 0 . 1 4 6 0 0 .0440 299.71 284.67 2.77

I l l 0.683 0 .1464 0 .0444 299.77 284.70 2.59

Concluding Remarks

Currently available experimental evidence [8,9] point to the ab i l i t y of

"film" models to predict, successfully, heat and mass transfer rates during

the condensation of multicomponent vapors. The results reported here suggest

that there is a good case for using one of two recently developed methods of

estimatinq mass transfer rates which eliminate the iteration that is required

in the "film" models of Toot [4], Stewart and Prober [5] and of Krishna and

Standart [6,7]. The expl ic i t method of Taylor and Smith [16] has a sounder

theoretical basis than the expl ic i t method of Krishna [14,15] and is, therefore,

tentatively recommended for incorporation into condenser design procedures.

Vol. 9, No. 6 SIMULATION OF BINARY VAPOR COND~SATICN 471

We would emphasize that the exp l i c i t methods are just as easy to use as the

effect ive d i f f us i v i t y methods and, at least for the systems described here,

the method [16] is just as accurate as the Toor-Stewart-Prober and Krishna-

Standart models. The exp l i c i t methods demand s ign i f icant ly less computer time

than these more rigorous methods. Further testing of the methods with other

systems would appear to be indicated including mixtures with more than three

components and against available experimental data. In connection with this

last point we note that the method of Krishna and Standart [6,7] has already

been found to be in good agreement with experimental data [8,9].

Ackpgwled9ement

This material is based upon work supported by the National Science Foundation under Grant No. CPE8105516.

References

[ I ] R. Krishna, C.B. Panchal, D.R. Webb and I . Coward, Letts. Heat and Mass Transfer, 3, 163 (1976).

[2] R. Krishna and C.B. Panchal, Chem. Eng. Sci . , 3_22, 741 (1977).

[3] R. Krishna, Letts. Heat and Mass Transfer, 6, 137 (1979).

[4] H.L. Toor, A.I .Ch.E.J., I0, 448, 460 (1964).

[5] W.E. Stewart and R. Prober, Ind. Eng. Chem. Fundam., 3, 224 (1964).

[6] R. Krishna and G.L. Standart, A.I .Ch.E.J., 2__22, 383 (1976).

[7] R. Krishna and G.L. Standart, Chem. Eng. Commun., 3, 201 (1979).

[8] D.R. Webb and R.G. Sardesai, Int . J. Multiphase Flow, ~, 507 (1981).

[9] J. Bandrowski and A. Kubaczka, Int. J. Heat Mass Transfer, 2_4_4, 147 (1981).

[ I0 ] R. Taylor and D.R. Webb, Chem. Eng. Coramun., 2, 287 (1980).

[ I I ] R. Taylor and D.R. Webb, Comput. Chem. Eng., 5, 61 (1981).

[12] R. Taylor, Comput. Chem. Eng., 6, 69 (1982).

[13] A. Burghardt and R. Krupiczka, Inz. Chem., 5, 487, 717 (1975).

[14] R. Krishna, Letts. Heat and Mass Transfer, 6, 439 (1979).

[15] R. Krishna, Chem. Eng. Sci. , 36, 219 (1981).

[16] R. Taylor and L.W. Smith, Chem. Eng. Commun., I_4_4, 361 (1982).

[17] L.W. Smith and R. Taylor, Ind. Eng. Chem. Fundam., in press (1982).


[18] D.R. Webb and J.M. McNaught, Developments in Heat Exchanger Technology - I , (D. Chisholm, Ed.), Applied Science Publishers, Barking, Essex, England, p. 71 (1981).

[19] D.R. Webb and R. Taylor, Chem. Eng. Sci., 3__7_7, If7 (1982).

[20] R. Krishna, Ph.D. Thesis, UMIST, England (1975).

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Simulation of binary vapor condensation in the presence of an inert gas