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

IN H E A T A N D M A S S 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 VAPORCONDENSATION IN THE PRESENCEOF 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

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R. Taylor and M.K. Noah

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equations, all of the foregoing being based on a film model of steady state mass t r a n s f e r .

The experimental results that are available [ 8 , 9 ] confirm the

general conclusion that the film models which take i n t e r a c t i o n e f f e c t s into account are of comparable accuracy and are far superior to methods of the effective diffusivity

type.

the i t e r a t i v e calculations convergence d i f f i c u l t i e s

Their drawback is the length and complexity of that are required.

The methods may also e x h i b i t

[10-12] and c e r t a i n l y take much more computer time

than the e f f e c t i v e d i f f u s i v i t y

methods.

In 1975 Burghardt and Krupiczka [13] published an approximate solution of the Maxwell-Stefan equations for mass t r a n s f e r in multicomponent mixtures containing one or more i n e r t species. mass t r a n s f e r involves no i t e r a t i o n .

Their method of c a l c u l a t i n g rates of Bandrowski and Kubaczka [9] included

t h i s method in t h e i r comparison of theory and experiment and found i t to be only s l i g h t l y tially

less accurate than the more rigorous methods but needed substan-

less computer time.

Krishna [14,15] has also presented an e x p l i c i t

method of c a l c u l a t i n g rates of mass t r a n s f e r in multicomponent mixtures.

More

r e c e n t l y , Taylor and Smith [16] extended and s i m p l i f i e d the e x p l i c i t method of Burghardt and Krupiczka [13] to many of the other determinacy conditions that are encountered in other s i t u a t i o n s .

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 solution of the Maxwell-Stefan equations showed the new method [16] to be of good accuracy over the e n t i r e ternary composition t r i a n g l e ;

i t s poorest r e l a t i v e performance coming in mixtures

containing i n e r t species.

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

i f the mass t r a n s f e r rates are low (small d r i v i n g forces). From the designer's point of view a b e t t e r test of the accuracy of the e x p l i c i t methods would be a comparison of the composition p r o f i l e s predicted by the various mass t r a n s f e r models for a p a r t i c u l a r process.

The objective of

the study described here was to use the e x p l i c i t methods to determine the composition p r o f i l e s in a v e r t i c a l

tube in which a binary vapor is condensing

in the presence of a non-condensing gas.

In essence, we are repeating the

design calculations 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 r a n s f e r . The notation used here corresponds, wherever possible, to that of Krishna and coworkers [ I - 3 ] ;

the few newly introduced symbols are defined in the t e x t

at the point of t h e i r f i r s t

use.

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SIMULATION OF BINARY VAPOR COND~'~SATIGN

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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 condensate.

The differential material balances for each species of the mixture

are [2] dGi d~

-

dLi N.xDZ = 1 d(

i = 1,2..n

(1)

where Gi and Li are the molar flow rates of species i in the vapor and liquid phases respectively.

The Ni are the constituent molar fluxes, positive i f

condensation actually occurs, zero for inert species and negative i f evaporation takes place.

{ is a dimensionless distance measured down from the top of

the tube. An energy balance for the vapor leads to Gt%

dTb d~

-

(2)

qb xDZ

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. It 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, Ni and q, at each step.

The explicit

methods of mass transfer described above lead to the following equations for the fluxes, Nl and N2 in a ternary gas/vapor mixture. N1 :

Kll---(Ylb-Yii)+

KI2-=(Y2b-Y21)

(4)

N2 :

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

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of Taylor and Smith [16] and of Krishna [14,15] lead to the following expressions for the Kij. (6)

Kll = kl3(Ylk23 + Y3kl2)/Y3 S Kl2 =

Ylk13k23/Y3 S

(7)

K21 =

Y2kl3k23/Y3 S

(8)

K22 =

k23(Y2k13 + Y3kl2)/Y3 S

(9)

S

= Ylk23 + Y2k13 + Y3k12

The Yi in equations (6-I0) are the arithmetric averages of YiI and Yib" k i j are the mass transfer coefficients of the binary i - j pairs. These

(I0) The

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 profile in the vapor "film". The e x p l i c i t method of Krishna [14,15] leads to linear concentration profiles and, hence, ~ is just unity. be

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

: -

- ¢ 2

e~+l e¢_l

;

~

= In ~Y31~ \ Y 3b}

(II)

Continuity of the mass transfer rates across the vapor-liquid interface requires that the Ni calculated from equations (4,5) be equal to the same fluxes in the l i q u i d phase. In many condensing systems the principal resistance to mass transfer resides in the vapor phase. A description of mass transfer in the l iq u i d phase is not, therefore, required and one of two extreme conditions is assumed to hold [1,2,18]. ( i)

The l i q u i d phase is completely mixed l a t e r a l l y with regard to composition.

This corresponds to i n f i n i t e liquid phase mass transfer coefficients and the l i q u i d composition calculated from a material balance along the flow path n Lk i = 1,2..n (12) Xil = Li/k

~l

VOI. 9, NO. 6

(ii)

SIMULATION OF BINARY VAPOR COND~SATICN

467

The liquid phase is completely unmixed, corresponding to zero liquid

phase mass transfer coefficients.

In this case the interracial composition of

the condensate is given by the relative rates of condensation Xil

=

n N~/ S Nk

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 Yil

= 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] qw

: ho(TI-Tc)

:

qb

: hy ~ee-1 (Tb-T I)

n }~ Ni~i i:l

qb + ;

E :

+ hy~(Tb-Tl)

n ~ i=l

NiCpi/hy

(15) (16)

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] d i f f e r in their respective methods of estimating the Kij. Taylor and Smith [16] estimate the Kij from binary kij 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 all 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 flux, ZNi , 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 Ni and q from equations (4-16) are described by Krishna and Standart [7] and by Webb and Taylor [19].

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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 noncondensing species, a i r or helium (3); the l a t t e r 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 differ, 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 II.

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

transfer rates, method II uses the e x p l i c i t method of Taylor and Smith [16] and method I l l uses the e x p l i c 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 slightly 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 k i 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 d i f f u s i v i t y 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 I I .

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 the top of the Condenser Tube Example l Inert Gas

Air

Example 2

Example 3 1.841 x lO-4

l .841 x lO-4

1.841 x lO "5

0.921 x I0-5

Air

Example 4

Gt(kmol/s)

1.841 x lO-4

G3(kmol/s)

1.197 x lO-4

Air 1.841 x 10-4 7.364 x lO-5

Air

Yl

0.2129

0.4500

0.7000

0.4500 O.5000

Y2

0.1369

0.1500

0.2000

Tb(K)

344.20

360.0

360.0

365.0

Tc(K)

308.15

308.15

308.15

308.15

Lc(kg/s)

0.03

0.04

0.06

0.06

Example 5

Example 6

Example 7

Example 8

Gt(kmol/s)

Helium 1.841 x lO"4

Helium 1.841 x lO"4

1.841 x lO -4

G3(kmol/s)

1.197 x lO-4

Helium 1.841 x lO-4 4.603 x i0-5

1.841 x lO-5

5.233 x lO-5

Yl

0.2129

0.5000

0.7000

0.3500

Y2 Tb(K)

0.1369

0.2500

0.2000

0.3500

344.20

350.00

360.00

355.00

308.15

308.15

308.15

308.15

0.03

0.05

0.06

0.05

Inert Gas

Tc(K) Lc(kg/s)

Condensate Mixing: Unmixed

Helium

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R. Taylor and M.K. Noah

Vol. 9, No. 6

TABLE I I Outlet Conditions Predicted Using Three Different Mass Transfer Models Example Method Gt x lO-4 (kmol/s)

Yl

Y2

Tb (K)

T (~)

CPU Time

(sec)

1

I II Ill

1.516 1.516 1.516

0.1703 0.1703 0.1704

0.0402 0.0402 0.0402

309.33 309.33 309.33

296,05 296.05 296,05

2,36 1,68 1,61

2

I II Ill

1.034 1.034 1.036

0.2532 0.2533 0.2542

0.0348 0.0347 0.0350

315.12 315.12 315.15

287,46 287.46 287.50

1,80 l.lO 1.06

3

I II Ill

0.307 0.307 0.317

0.3605 0.3618 0.3760

0.0408 0.0394 0.0429

324.12 324.12 324.53

283.80 283,80 283.92

4.04 1,15 1.09

4

I II III

0.138 0.138 0.147

0.2481 0.2492 0.2689

0.0888 0.0851 0.1036

320.33 320.35 322.42

280.42 280.42 280.46

3.25 1.50 1.34

5

I II Ill

1.481 1.481 1.481

0,1565 0,1565 0.1565

0.0351 0.0351 0.0351

302.91 302.91 302.91

294.76 294.76 294.76

4.51 3.27 3.17

6

I II Ill

0.579 0.579 0.580

0,1747 0,1743 0.1750

0.0309 0.0313 0.0317

299.96 299.98 300.05

283.52 283.52 283.53

4.28 2.60 2.36

7

I II Ill

0.233 0.233 0.234

0.1897 0.1890 0.1917

0.0213 0.0220 0.0227

301.20 301.24 301.50

282.92 282.92 282.93

29.12 14.83 13.87

8

I II Ill

0.681 0.681 0.683

0.1466 0.1460 0.1464

0.0433 0.0440 0.0444

299.67 299.71 299.77

284.67 284.67 284.70

4.53 2.77 2.59

Concluding Remarks Currently available experimental evidence [8,9] point to the a b 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 e x p l i c i t method of Taylor and Smith [16] has a sounder theoretical basis than the e x p l i c i t method of Krishna [14,15] and is, therefore, tentatively recommended for incorporation into condenser design procedures.

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471

We would emphasize that the e x p l i c i t methods are just as easy to use as the effective d i f f u s 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 KrishnaStandart models. The e x p l i c i t methods demand s i g n i f i c a n t l y 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).

[II]

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).

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R. Taylor and M.K. Noah

Vol. 9, No. 6

[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).