EVAPORATION

OF DROPS

IN AIR

FREE-CONVECTION

V.

UNDER

I. B i i n o v

and

OF

G . V.

BINARY

LIQUID

MIXTURES

CONDITIONS

Plokhikh

UDC 536.423.1:536.25

The e v a p o r a t i o n of d r o p s of m i x t u r e s of t e t r a f l u o r o d i b r o m e t h a n e with benzene, toluene, and xylene is studied at r o o m t e m p e r a t u r e in a i r at r e s t . Sprayed liquid m i x t u r e s a r e widely used in e n e r g e t i c s , the food industry, a g r i c u l t u r e , etc., so that a study of the e v a p o r a t i o n of d r o p s of liquid m i x t u r e s is of p r a c t i c a l as well as t h e o r e t i c a l importance. This r e s e a r c h began with a study of binary m i x t u r e s of liquids having a p p r o x i m a t e l y equal boiling points [1-3]. A n a t u r a l second step is to study binary m i x t u r e s in which the boiling points a r e very different. We r e p o r t h e r e a study of the e v a p o r a t i o n of d r o p s of m i x t u r e s of t e t r a f l u o r o d i b r o m e t h a n e (Freon 114-C2), which boils at 46.2~ with benzene, toluene, and xylene, whose boiling points a r e 80, 111, and 141~ r e s p e c t i v e l y . P a r t of the motivation f o r choosing these m i x t u r e s is that it thus b e c o m e s possible to study s e v e r a l questions involving the s u p p r e s s i o n of the combustion of liquid fuels and questions which a r e of e c o n o m ic i m p o r t a n c e . In the e x p e r i m e n t s we m e a s u r e d the m a s s m, d i a m e t e r d, and t e m p e r a t u r e d of the e v a p o r a t i n g drops. We s e p a r a t e l y d e t e r m i n e d the c o m p o s i t i o n of the v a p o r phase. The m a s s m e a s u r e m e n t s w e r e c a r r i e d out with a m i c r o b a l a n c e [4] having a sensitivity of 1.8- 10 -~ g p e r scale division. The d r o p s w e r e suspended on a g l a s s filament at whose end t h e r e was a ball 1 m m in d i a m e t e r . The d i a m e t e r of the e v a p o r a t i n g drop w a s m e a s u r e d with a m i c r o s c o p e with an ocular. The drop t e m p e r a t u r e was d e t e r m i n e d with a m a n g a n i n - C o n s t a n t a n t h e r m o c o u p l e , with w i r e d i a m e t e r s of 30 and 100 ~. The initial drop d i a m e t e r varied f r o m 1.4 to 1.8 m m . The c o m p o s i t i o n of the v a p o r phase was d e t e r m i n e d in the following m a n n e r : a w e a k n i t r o g e n s t r e a m was p a s s e d through a D r e c h s e i bottle containing the test solution. The s t r e a m , s a t u r a t e d with the v a p o r of the m i x t u r e , then entered a U-shaped g l a s s tube in a D e w a r with liquid nitrogen, f r o m which it e s c a p e d into the a t m o sphere. The v a p o r condensed in the U-shaped tube. A r e f r a c t o m e t e r was used to d e t e r m i n e the r e f r a c t i v e index of the condensate, and then the c o m p o s i t i o n of the condensed liquid was found f r o m a calibration c u r v e . The flow r a t e of the nitrogen in the d e t e r m i n a t i o n of the composition of the v a p o r phase did not exceed 1 l i t e r / m i n and was chosen e m p i r i c a l l y , such that a f u r t h e r reduction of the flow rate did not affect the condensate c o m position. The p r e s s u r e in the s y s t e m w a s a t m o s p h e r i c . We f i r s t c a r r i e d out e x p e r i m e n t s with the liquids used as components in the m i x t u r e s . The change in the drop m d s s during the e v a p o r a t i o n can be d e s c r i b e d well by the equation f r o m [4, 5]: (1)

m = m o - - a t + b t 2,

w h e r e a and b a r e coefficients which depend on the nature of the liquid. The a g r e e m e n t of Eq. (1) with e x p e r i m e n t can be judged on the b a s i s of Fig. 1, which shows e x p e r i m e n t a l data f o r F r e o n and toluene; the so'lid c u r v e s a r e drawn f r o m Eq. (1). The drop d i a m e t e r falls off e s s e n t i a l l y linearly o v e r the range studied. Some e x p e r i m e n t a l data obtained in a study of the evaporation of the m i x t u r e s a r e shown in Fig. 1. We see that the m(t) c u r v e s f o r the mixture differ f r o m the c o r r e s p o n d i n g c u r v e s for the pure liquids. Analysis of the e x p e r i m e n t a l data r e v e a l s that when the initial weight fraction of the F r e o n in the m i x t u r e , x0, is no V. I. Ul'yanov Leningrad E l e c t r o t e c h n i c a l Institute. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 30, No. 5, pp. 848-854, May, 1956. Original a r t i c l e submitted J a n u a r y 7, 1974. !

This material is protected b y copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York, N.Y. !0011. N o part | o f this publication may be reproduced, stored in a retrieval system, or ~ransmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, w i t h o u t written petTnission o f the publisher. A copy o f this article is available from the publisher f o r $ 7. 50.

1

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Fig. 1. Time dependence of the drop mass during evaporation. 1) Toluene; 2) F r e o n ; 3-5) m i x t u r e s of F r e o n with toluene. The weight fraction of F r e o n is: 3) 0.38; 4) 0.72; 5) 0.86. Here t is in seconds. Fig. 2. a: Time.dependence of the drop evaporation r a t e : 1) F r e o n ; 2) toluene; 3-6) mixtures of F r e o n with toluene. The weight fraction of the F r e o n is: 3) 0.91; 4) 0.86; 5) 0.72; 6) 0.38. b: Time depen_ dence of the rate at which the F r e o n evaporates f r o m the drop, PF. 1) P u r e F r e o n ; 2, 3) mixtures of F r e o n with toluene [the weight fraction of the F r e o n in the mixture is: 2) 0.91; 3) 0.83]; 4, 5) mixtures of F r e o n with benzene [the F r e o n fraction is: 4) 0.91; 5) 0.79]; 6) mixture of F r e o n with xyiene with a F r e o n weight fraction of 0.72. Here t is in seconds. higher than 0.2 the m(t) c u r v e s are described satisfactorily by (1) if the coefficients a and b a r e approximately equal to the values of these coefficients for the second component. At x 0 > 0.3 the c u r v e s for the mixtures are not described completely by Eq, (1). N e v e r t h e l e s s , Eq. (1-) does give a s a t i s f a c t o r y description of the first, rapidly falling, and second, slowly falling, parts of the c u r v e s for various values of a and b, which, of c o u r s e , depend on x 0 and the nature of the second component. This c i r c u m s t a n c e can be seen from Table 1, which shows the ratios of the coefficients o for the f i r s t parts of the c u r v e s for the mixtures to the coefficients for drops of pure F r e o n , benzene, toluene, and xylene (aiF, alb, a!t, and alx, respectively). We see from this table that the values of alF increase with increasing x 0, tending toward a value of unity; at a given value of x 0, they are nearly the same for all the mixtures studied. The ratio of a 1 to the coefficient a for the second component is l a r g e r than one and i n c r e a s e s rapidly in the s e r i e s from benzene to toluene to xylene. Analysis of the experimental data shows that the transition from the f i r s t to the second parts of the m(t) c u r v e s occurs a f t e r the evaporation of that fraction of the drop for which the ratio of the m a s s m to the initial m a s s m 0 is essentially equal to the weight fraction of the Yreon in the drop, x 0. Using these e x p e r i m e n t a l data, we can find the rate of evaporation of the drops. F o r this purpose it is convenient to use the quantity p, which is the ratio of the rate of change of the m a s s , m. to the drop d i a m e t e r at the given instant. The results calculated for mixtures of F r e o n and toluene a r e shown in Fig. 2a. We see that t" ~ evaporation rate ~ of the mixtures is initially high; it subsequently d e c r e a s e s , eventually becoming a p p r o x -

TABLE 1. Relative Values of the Coefficients a Second

component B~nzel'l~

Toluene Xylene

.v,~

aiF

aab a~F al t alF OiX

0.38

0,4 2,2 0,3 2,6

0,71

0,75 3,8 0,5 4,2 0,61 14,5

0,79

O,72 3,7

0,85

O, r .

1,0 5,4 O,93 7,8

I

O,86 7,2 0,8 18,0

563

| o-0--

-/aO

O25

0,5O

z?ZY

2 3

xM

~

st

m

o--2

0 yt

Iio-- ~ a

-I~

N

{'--5

/o0

200

9 0

25

Fig. 3

50 D

25

50 t

Fig. 4

Fig. 3. a: Time dependence of drop t e m p e r a t u r e . 1) F r e o n ; 2) toluene; 3-5) m i x t u r e s of F r e o n with toluene [the weight f r a c t i o n of the F r e o n is: 3) 0.38; 4) 0.63; 5) 0.79]. b: Minimum t e m p e r a ture ~*min as a function of the mole f r a c t i o n of F r e o n in the m i x t u r e s . The second component is: 1) xylene; 2) toluene; 3) b e n zene. H e r e ~ is in Celsius d e g r e e s , and t is in seconds. Fig. 4. T i m e dependence of x during the drop evaporation, a) Mixtures of F r e o n with benzene with a F r e o n weight f r a c t i o n of: 1) 0.91; 2) 0.79; 3) 0.38. b : The s a m e , f o r m i x t u r e s of F r e o n with toluene with a F r e o n weight f r a c t i o n of: 4) 0.91; 5) 0.86; 6) 0.71; 7) 0.38. H e r e t is in seconds, and x is e x p r e s s e d as a weight fraction. imately equal to the e v a p o r a t i o n r a t e of the second component. The initial r a t e , ~0, i n c r e a s e s with increasing x0; at x 0 = 0.9 it is a p p r o x i m a t e l y equal to the value of p for the pure F r e o n . The shape of the re(t) c u r v e s f o r the m i x t u r e s of F r e o n with benzene a n d o f F r e o n w i t h x y l e n e is the s a m e . F i g u r e 3a shows s o m e of the m e a s u r e d t e m p e r a t u r e s of the evaporating drops. We see that during the e v a p o r a t i o n of the F r e o n - t o i u e n e d r o p s the t e m p e r a t u r e ,9 initially d e c r e a s e s rapidly and then r e m a i n s e s s e n tially constant throughout the e x p e r i m e n t . A different situation is observed in the e v a p o r a t i o n of d r o p s of m i x tures. In this c a s e ~ initially d e c r e a s e s rapidly, but a f t e r reaching s o m e m i n i m u m value ~ m i n it begins to r i s e again, tending toward the t e m p e r a t u r e of a drop of the second component (which is r e l a t i v e l y involatile). The m i n i m u m t e m p e r a t u r e is lower, the higher the F r e o n concentration in the m i x t u r e , x 0. F i g u r e 3b shows 9 ~ m i n as a function of the mole f r a c t i o n of F r e o n in the drop, x M. T h e s e r e s u l t s c l e a r l y imply that the composition of the d r o p s of these m i x t u r e s changes during the e v a p oration. At x 0 > 0.3 the F r e o n e v a p o r a t e s f r o m the drop rapidly; the F r e o n concentration d e c r e a s e s rapidly, and in the late stage of the e v a p o r a t i o n the e v a p o r a t i o n rate is e s s e n t i a l l y equal to that of the second component. T h e d r o p s of m i x t u r e s with a low initial F r e o n concentration e v a p o r a t e at a rate nearly equal to that of the less volatile component. Using these e x p e r i m e n t a l r e s u l t s we can easily d e t e r m i n e the e v a p o r a t i o n r a t e s of the components. The equations r e q u i r e d f o r these calculations can be e a s i l y derived in the following m a n n e r , a f t e r [4]. We c l e a r l y have

'~d= ~nv § k~"

(2)

If the p r o c e s s is a s s u m e d q u a s i s t e a d y , we can w r i t e mFLF + / n ~ L ~ a (~)m-- ~) S = ~d)~Nu (~)m - - {))"

H e r e m d, m F, and 1:/12a r e the e v a p o r a t i o n r a t e s of the drop, the F r e o n , and the second component, r e s p e c tively; L F and Ir a r e the heats of v a p o r i z a t i o n of the F r e o n and the second component; and ~ and ~m a r e the t e m p e r a t u r e s of the drop and the medium. 564

(3)

T A B L E 2. V a l u e s of A / x

, e, T, and ~ - / t 0 f o r C e r t a i n M i x t u r e s

I Componont Benzene Xylone

xo

Atxo

0,22 [ 0,67 0,39 0,90 0,79 1,0 0,91 1,0 0,91 1,0

~

9

0,08 0,12 0,13 0,17 0,12

~,'to

0, I7 t0,06 0,40 27 0,78 36 0, 19

r

Component xo

Alxo

To1none

0,89 10,08 28 i0,13 0,90 t0,07 31 !0 I ' 19 0,95 I0, I0[ 29 0,29

1

26

0,38 0,72 0,86 0,91

I

~

r i~lto

1,o Io,151 32 !o,43

F r o m Eqs. (2) and (3) we find F

d r = ~ N u (~m-- ~) - - L~[ d L F- - L2 "

(4)

According to the data of Banz and Marshall [6], we have Nu --- 2 + 0.6 Gr'/4Pr '/s .

(5)

Using Eqs. (4) and (5), we can easily find/~F, the evaporation rate of the F r e o n f r o m the drop; f r o m Eq. (2) we can d e t e r m i n e the evaporation rate of the second component. Some of the r e s u l t s obtained in this m a n n e r a r e shown in Fig. 2b; we see that the evaporation rate of the F r e o n f r o m the drop of mixture, /)F, is a strong function of x 0 and falls off rapidly as time elapses. The initial value of PF for the solution with a F r e o n concentration of 0.9 is nearly the same as the evaporation rate of a drop of pure Freon. Interestingly, the/)F(t) c u r v e s c o r r e s p o n d i n g to solutions with the same values of x 0 and different second components essentially coincide. The evaporation rate of the second component f r o m drops in which there is a high initial F r e o n c o n c e n tration is initially small; it i n c r e a s e s as time etapscs, approaching the values of/) for the drop of the pure second component. During the evaporation of mixtures with a low F r e o n concentration, the rate/~ changes little over time and is approximately equal to the evaporation rate of a drop of the pure second component. Since the m a s s and d i a m e t e r of the drops are m e a s u r e d simultaneously in the e x p e r i m e n t s , it is p o s sible to determine the drop density p; using this value and a calibration curve, we can find the weight fraction of the F r e o n in the drop, x, during the evaporation 9 Since the drop d i a m e t e r changes by no more than I mm during the experiment, we can use only the r e s u l t s of experiments which were c a r r i e d out p a r t i c u l a r l y c a r e fully for this purpose. The r e s u l t s calculated for c e r t a i n mixtures of F r e o n and toluene a r e shown by the t r i angles in Fig. 4b. We see that the composition of the drops of mixtures changes during the evaporation, and we can determine the time dependence of x. Dobrynina [5] gave the following equation for the complete mixing of evaporating binary solutions: lnn!:-: ~ dx / mo .j y - - x

(6)

.

Xo

As we mentioned e a r l i e r , in these experiments we studied the composition of the vapor f r o m these m i x tures, tt turns out that x and y can be related satisfactorily by g~ = 6 x 1--y 1--x

or g =

6x 1 - ? ( 5 - - 1)x

(7)

Here 5 is a coefficient. F o r the F r e o n - b e n z e n e mixtures it turns out to be 6; for the F r e o n - t o l u e n e mixtures it is 16. and for the F r e o n - x y l e n e mixtures it is 91. Using e m p i r i c a l equation (7) we can easily calculate the integral on the right side of Eq. (6), finding

/'/l rn-~ =

( ~X_o)[~' t 1 -- XO~ 1-715' \ 1--x ]

(8)

w h e r e t3 = 1 / O . If we d e t e r m i n e the times c o r r e s p o n d i n g to various v a l u e s of the ratio m / m 0 from the experimental c u r v e , we can use Eq. (8) to find the values for the c o r r e s p o n d i n g times if there is complete mixing in the drop. Corresponding calculations were c a r r i e d out for mixtures of F r e o n with benzene, toluene, and xylene. Some of the r e s u l t s a r e shown in Fig. 4a and 4b (circles). We see that the values of x determined f r o m the 565

density and f r o m Eq. (8) a g r e e s a t i s f a c t o r i l y . The c l e a r implication of this a g r e e m e n t is that total mixing o c c u r s in the e v a p o r a t i n g drop of solution* and that Eq. (8), along with the e x p e r i m e n t a l data, can be used f o r r e l i a b l e calculations of the quantity x, which is a m e a s u r e of the c o m p o s i t i o n of the e v a p o r a t i n g drops. T h e r e is a n o t h e r way to d e t e r m i n e x: we c a n use Eqs. (2) and (4) to find the e v a p o r a t i o n r a t e of the F r e o n , ~F. and that of the second, ~2, f r o m the drop of m i x t u r e , and we can use the J(t) c u r v e to find the drop t e m p e r a t u r e ~. Then, noting that the e v a p o r a t i o n r a t e of pure F r e o n is p r o p o r t i o n a l to the s a t u r a t i o n v a p o r density of the drop, and knowing the dependence of the e v a p o r a t i o n rate on the d i a m e t e r d, we c a n e a s i l y calculate the e v a p o r a t i o n r a t e of a drop of p u r e F r e o n , ~pF at a t e m p e r a t u r e ~ and at a d i a m e t e r equal to the d i a m e t e r of the m i x t u r e drop. A s s u m i n g gF :/~pF = CalF :CF, w h e r e CdF and c F a r e the m e a s u r e d concentrations of F r e o n in the v a p o r phase n e a r the s u r f a c e of the m i x t u r e d r o p and n e a r the s u r f a c e of a drop of p u r e F r e o n (expressed in g r a m s p e r cubic c e n t i m e t e r ) , we can find CdF. In the s a m e m a n n e r we can calculate the v a p o r density of the second component, and then it is a s i m p l e m a t t e r to find y and then x, f r o m the y(x) curve. The r e s u l t s calculated in this m a n n e r a g r e e s a t i s f a c t o r i l y with the values of x found f r o m Eq. (8). This a g r e e m e n t again c o n f i r m s that t h e r e is a total mixing in the drop. The dependence of x on the t i m e t can be d e s c r i b e d well by the e m p i r i c a l equation A x ----exp (z) -P I ' z = e (t - - T).

(9)

H e r e A is a quantity a p p r o x i m a t e l y equal to x 0, T is the t i m e at which we have x = 0.5A, and e is a coefficient. F i g u r e 4 c o m p a r e s Eq. (9) with the e x p e r i m e n t ; the solid c u r v e s a r e drawn f r o m this equation, with the a p p r o p r i a t e values of A and e. Some data on A / x 0 , e, T, and v / t 0, w h e r e t o is the t i m e required f o r c o m p l e t e e v a p o r a t i o n of the drop, a r e shown in T a b l e 2. T h i s study has thus made it possible to d e t e r m i n e the e v a p o r a t i o n kinetics of d r o p s of solutions in which the c o m p o n e n t s have v e r y different boiling points. It t u r n s out that t h e r e is an initial rapid e v a p o r a t i o n of the d r o p s ; then the e v a p o r a t i o n r a t e p d e c r e a s e s and tends toward the e v a p o r a t i o n r a t e of the d r o p s of the pure second components. Using the laws of e n e r g y and m a s s c o n s e r v a t i o n , we can find the r a t e s at which the c o m ponents e v a p o r a t e f r o m the m i x t u r e drops. We have found that t h e r e is a complete mixing in the d r o p s and that Eq. (8) can be used to calculate x in an e v a p o r a t i n g drop. It h a s been e s t a b l i s h e d that e m p i r i c a l equation (9) gives a good d e s c r i p t i o n of the time dependence of x. NOTATION m0, m, initial m a s s and instantaneous m a s s of e v a p o r a t i n g d r o p (g); d, drop d i a m e t e r (cm); ~ , ~ m i n , ~ m , t e m p e r a t u r e s of drop, m i n i m u m t e m p e r a t u r e , and t e m p e r a t u r e of medium; x0, initial weight fraction of F r e o n in m i x t u r e ; x, weight f r a c t i o n of F r e o n in the e v a p o r a t i o n d r o p ; XM, initial mole f r a c t i o n of F r e o n ; y, weight f r a c t i o n of F r e o n in v a p o r phase; ~ = m / d { g / s e c - cm); m , rate of change of drop m a s s ; L, heat of v a p o r i z a tion; )~, t h e r m a l conductivity of a i r ; S, s u r f a c e a r e a of drop; Nu, G r , P r , Nusselt, G r a s h o f , and P r a n d t l n u m b e r s ; a , h e a t - t r a n s f e r coefficient; t 0, time required f o r c o m p l e t e e v a p o r a t i o n of drop; c, v a p o r concentration (g/cm3). LITERATURE

i. 2. 3. 4. 5. 6.

A. V. V. V. V. W.

A. V. E. D. V. E.

CITED

R a v d e I ' , V. V. Danilov, and T. N. Cherepnina, Zh. P r i k L Khim., 4.1, 79 (1968). Danilov, A u t h o r ' s A b s t r a c t of C a n d i d a t e ' s D i s s e r t a t i o n , Leningrad (1968). Glushkov, A u t h o r ' s A b s t r a c t of Candidate,s D i s s e r t a t i o n , Odessa (1968). BUnov and V. V. Dobrynina, I n z h . - F i z . Zh., 2._1.1, 2 (1971). Dobrynina, A u t h o r ' s A b s t r a c t of C a n d i d a t e ' s D i s s e r t a t i o n , Leningrad (1972). Ranz and W. R. M a r s h a l l , Chem. Eng. P r o g r . , 4.~8, 141, 173 (1952).

*This conclusion was a l s o verified by d i r e c t o b s e r v a t i o n s : In the e v a p o r a t i n g d r o p s we c l e a r l y observed an i n tense, o r d e r e d motion of dust p a r t i c l e s which had entered the drops; this motion could o c c u r only if there w e r e a convective flow of the liquid, which would tend to equalize the concentration and t e m p e r a t u r e in the drop.

566