- Email: [email protected]
Mass transfer for a wide range of driving force Evaporation of pure liquids
Chemical Engineering Science, 1971, Vol. 26,
pp. 1187-I 194. Pergamon Press. Printed in Great Britain.
Mass transfer for a wide range of driving force Evaporation of pure liquids KOICHI ASANO and SHIGEFUMI FUJITA Department of Chemical Engineering, Tokyo Institute of Technology, Tokyo, Japan (Received
3 November 1970)
Abstract- Measurements were made for the rate of evaporation of pure liquids into air and carbondioxide for a wide range of driving force (B = 0.03-36.1) by a specially designed agitated vessel. Taking into account the effect of variable properties on the rate of mass transfer, a new general form for the correlation of high mass transfer data wasproposed and data were well correlated by that correlation. Agreement between the data and the predicted value by laminar boundary layer theory was excellent. Validities of the assumption of negligible effect of variable properties on the rate of mass transfer in high mass transfer and of the use of p&P as correlating variable were also discussed. INTRODUCTION
well-known fact that a mass transfer through a heterogeneous interface is caused by two different phenomena, mass transfer due to diffusion of a component at the interface and mass transfer due to convective flow induced by diffusion. For mass transfer of low concentration gradient at the interface, we may say low mass transfer, such as mass transfer in absorbing and rectifying columns or in extractors, the amount of mass transfer due to convective flow is very small and can be neglected in comparison with that by diffusion, but for mass transfer of high concentration gradient, we may say high mass transfer, the convective term becomes important and can sometimes become of the same order of magnitude as the diffusional term or more. In other words mass transfer is affected by driving force for a moderate and high driving force range and for this reason an analogy between heat and mass transfer as is the case for low mass transfer cannot be applicable for high mass transfer [l, 2, 7, 121. This is one of the characteristics of high mass transfer. Usually high mass transfer is accompanied by large concentration difference between interface and bulk flow so for the systems of concentration dependent properties, the effect of variable properties on the rate of mass transfer needs to be considered [ 1,6,7]. Although in the past few decades considerable efforts have been directed to the study of high IT IS A
mass transfer in the light of laminar boundary layer theory, very few experimental data are known[3, 8-11, 13, 141. The purpose of this paper is to report some aspects of high mass transfer by measuring rate of evaporation of pure liquids. FUNDAMENTAL EQUATIONS MASS TRANSFER
FOR HIGH
For the flow inside a two-dimensional steady state laminar boundary layer along a flat plate with uni-directional diffusion of a component at the interface, the transport equations may be written as follows:
apu+apv=() ax
ay
,au+vau=ra au
( > p$!z+pvaW=a pDaw ax ay ay( ay> * ax
ay paypaY
(2) (3)
Boundary conditions at the interface are aty=O,
w=w,
(4)
u = u, v=v~=l-w,
-D,
(5)
(aw> ay
where Eq. (6), the convective
1187
8
velocity
(6) at the
KOICHI
ASANO and SHIGEFUMI
interface, is derived by the condition of no mass transfer of the inert component at the interface. From Eq. (6) the mass transfer flux at the interface is given by:
JlOC =
Plh
=
Ps%
_-P& -
, _
aw
w,
s
ay
(
>
FUJITA
where, B=
(w,-w,)/(l-ww,)
Re = U,LIv, SC, = v,lD,
(7)
4s =
(1%
CL8P*IkPm
= p&(w,-w,).
Considering variation of fluid properties with variation of compositions, we will introduce the following dimensionless variables: rl =
U, j- p/pm dylc
In the above equation the Sherwood number is defined according to the definition by Ranz and Dickson [ lo]. If we define an average Sherwood number as:
(16) we get from Eqs. (19)-( 16) the following relation: Sh(l-
w,) =f(Re,Sc,,
B, 4,~p,/pm9LI/L).
(17)
SC = vlD
e=
For perfect gas mixtures the equations for density and viscosity known:
(w,-w)I(w,-ww,)
following are well-
4 = PNPmPm P2
where the dimensionless variable 4 was first introduced by Merk[7] but without theoretical discussion. The transport equations can be rewritten as:
P
=
PlIP2
P2 i
(9) + F-8, = 0.
(10)
Boundary conditions at the interface are: F(0) = -BW,(O)/Sc,
(11)
F,(O) = 0
(12)
0(O) = 0
(13)
e,(o) =
vzsh,,, (1 -
w,) ( p,/p,) -l.Rfe’2.(x/L)“~ (14)
(18)
1+@4,/A4,-1)
p=
+
l+1-w -a12
W
1 1 +&a21
(19) 1
*
From Eqs. (15) (18) and (19) we will find that for a given system the three variables, B, +8 and ps/p,, are functions only of concentration at the interface and that in bulk flow. Therefore we can eliminate the variable & or p,/p, by using the following relations: 48 = g(B,
Islam)
(20-a)
or ~sl~co = g’ (Bv $4.
(20-b)
Substituting Eq. (20-a) or (20-b) into Eq. (17), we get the following fundamental relations for
1188
Mass transfer for a wide range of driving force
mass transfer in general: Sh(1 -w,)
=f(Re,Sc,,
B,p,lp,,L,/L)
(21-a)
or Sh(1 -w,)
=f(Re,
SC,, B, c#+,Lx/L).
(21-b)
We would like to point out the fact that in the present work the Sherwood and Schmidt numbers are defined at interface conditions, whereas in previous work no clear definition has ever been made. For the special case of a constant-property systems, Eq. (21-a) or (21-b) is simplified to the following form: .%(1-w,)
=f(Re,Sc,B,L,/L).
Equation (22) is in accordance previous works [ 1,2,7,12].
(22)
under the liquid surface by copper-constantan thermocouples. The measured liquid temperatures agreed within O*OS”C. Inlet gas was dehumidified by a CaCl,-packed column to absolute humidities of less than 0.004 and its temperature was adjusted by a lo-ohm electric resistance heater and a 1 kW slidac so that the temperature difference between gas and liquid was less than 1°C. The rate of agitation was kept at the maximum that did not cause aeration (250300 revlmin) in order to get homogeneous liquid temperature distribution, because no effect of agitation was observed in the preliminary runs [31. Data were taken for five systems; air-water, air-methanol, air-benzene, air-carbon tetrachloride and carbon dioxide-water. Ranges of variables shown below:
with the results of Re:
SC* EXPERIMENTAL APPARATUS PROCEDURES
B:
AND
The apparatus used in the present work, is a brass-made, 252-mm i.d., lOO-mm depth agitated vessel, with lOO-mm turbine impeller and 8-20 X 95 mm baffles and with heating jacket outside. The gas-liquid contacting is done at a portion of liquid surface (20.7 mm in width and 100 mm in the direction of flow) and a special design precautions are taken to prevent variation of gas-liquid contacting area due to variation in liquid level. The passage of inlet gas consists of a calming grid of 15 mm-2 mm o.d. stainless tubes and of a duct of rectangular cross section (20.7 mm in width and 25 mm in height) and the angle between the duct and liquid surface is about 6 degrees. The details of the agitated vessel are shown in Fig. 1. The rate of mass transfer was calculated by the amount of the liquid evaporated. The value of the driving force was adjusted by adjusting the vapour concentration at the interface, which was calculated from the vapour pressure of the pure liquid at the surface temperature. The surface temperature was estimated by taking the average of the temperatures measured at several points
Pmflp: PslPm:
48:
845 -15,100 o-221.21 o-0336.1 0.97 0*40o-754.45 0*632.55.
EXPERIMENTAL RESULTS DISCUSSIONS
Choice
of characteristic
AND
length
In the calculation of Reynolds and Sherwood number a definition of a characteristic length is necessary. In order to select a proper characteristic length for these variables two sets of data of different contact length, L, = 100 mm and L, = 50 mm (by covering a portion of the gasliquid contacting area with a brass covering), were compared for the air-methanol system at 30°C and the air-carbon tetrachloride system at 60°C. The results for the air-methanol system are shown in Figs. 2a and 2b. In the former where contact length was assumed as the characteristic length, two different lines of correlation were observed but in the latter where the height of the duct was assumed as the characteristic length, a single line of correlation was observed for both sets of data of different contact length. Similar results were observed for
1189
KOICHI ASANO and SHIGEFUMI
I
FUJITA
View window
thread
rid mm o.d. tubes Gas
as inlet
9-20
x95mm
1~~mm-.l Fig. 1. Details of the agitated vessel.
I
Air methanol
2x10'
4
at 3O’C
IO4
6
I
2
II
4x104
--lo3
2
4
6
IO'
2X104
Ro
R6X
Fig. 2a. Characteristic length; case-A L = L,.
Fig. 2b. Characteristic length; case-B L = height of duct.
the data of the air-carbon tetrachloride system. For this reason the height of duct (25 mm) was selected as the characteristic length throughout this paper. From Fig. 2b we may conclude that in this apparatus the Sherwood number is independent of contact length:
Sh = Sh,,,.
(23)
From Eqs. (21-a) (21-b) and (23) we can get the following relations:
1190
Sh(l -w,) =f(Re,Sc,,B,p,/p,)
(24-a)
Mass transfer for a wide range of driving force
kh(l-w,)
=f(Re,Sc,,B,&).
(24-b)
Discussions on the assumption of constant properties Although a considerable concentration difference between the interface and the bulk flow can be expected in high mass transfer, most of previous investigators [3, 8- 11, 13, 141 have assumed constant properties and neglected the effect of variable properties on the rate of mass transfer due to variation in composition, without any experimental proof. In order to confirm the validity of this assumption, data of the airmethanol system, a system of relatively constant properties, and of the air-carbon tetrachloride system, a system whose properties vary widely, are shown as a function of transfer number in Fig. 3, where values of the exponent of the Reynolds and of the Schmidt number were from the latter correlation. From the fact that two different lines of correlation were obtained for the two systems, we may conclude that the assumption of negligible effect of variable properties on the rate of mass transfer is invalid. Correlation of data Considering the above result, data were correlated by Eq. (24-a) or (24-b) by assuming power function type correlation. The coefficients
O-02
I
2
4
6 (I
IO
20
of correlation were determined by the least squares method with the HITAC 5020E Computor of Tokyo University Computing Center. The results were: Sh( 1 - w,) = 0.41 .Sc,@5.Reo.s7.(1 + B)-0.7g.
(P,lPmr”.07
(25-a)
and, Sh( 1 - w,) = 0*42.Sc,0.5.Re067. (1 + B)-0.*2. c$s-o.o2. (25-b) The average deviations from the above correlations were 10.4 per cent both for Eq. (25-a) and for Eq. (25-b). An explanation for the same values of exponent for the Reynolds and Schmidt numbers and for the different values for the transfer number may be clear from the discussion on Eqs. (20). Figures 4a,b and c shows the effect of Reynolds number, Schmidt number and (pJp,), respectively; where the solid line shows Eq. (25-a). In Fig. 4c, a correction by the method of Hanna[6], the only one quantitative approach for the effect of variable properties on the rate of mass transfer available at present: In P~IP~
kvar.cJkomt.p =
PslPm - 1
WI61
4060 -6.10’
+ 8)
Fig. 3. Correlation based on the assumption of negligible effect of variable properties.
1191
IO’
2
4
6
104
2x104
R6
Fig. 4a. Final correlation; effect of Reynolds number.
KOICHI ASANO and SHIGEFUMI
G t5 -8 *m
FUJITA
2.0
I.0
$ 6 0 2 g 6 'z .
0.6 0.4
0.2
2 L iFi 0.1 O-I
0.4 0.6
0.2
I.0
2.0
'
SC,
Fig. 4b. Final correlation; effect of Schmidt number.
solutionfor
xact
: 6 s + I
7 : #o LL
4
6
IO
20
40
60
(I+B)
Fig. 5. Effect of driving force on the rate of mass transfer; a comparison with laminar boundary layer theory.
x Iv) G 3 Z $
2
I 0.4
0.6
I
2
to low mass transfer Sherwood numbers were plotted as a function of transfer number to show the effect of driving force on the rate of mass transfer, where low mass transfer Sherwood numbers were estimated by rearranging Eq. (25-a):
III, 4
66
ps 'pm
Fig. 4c. Final correlation; effect of variable density on the rate of mass transfer.
Sh( 1 - w~)~+ = 0*41Sc,0’5~Re0’67.
is also shown in a dotted line and compared with the present data, assuming the same exponent for the Reynolds, Schmidt and transfer numbers as for the present work. The reason for disagreement between the data and the correction by Eq. (26) may be explained due to the fact that in the derivation of Eq. (26) only the effect of variable density is taken into account, whereas in the present work not only the effect of variable density but also the effects of & and of transfer number are taken into account. Efect of driving force on the rate transfer; a comparison with the theory
(27)
Exact solution for the laminar boundary layer on a flat plate is also shown in a solid line for the purpose of comparison. A good agreement between the data and the theory for a wide range of driving force is worthy of remark, in spite of the difference in the exponents of the Reynolds and of Schmidt numbers in the present work from the theoretical values of 0.5 and 0.33, respectively. This may suggest the existence of a universal function of transfer number irrespective of whether the flow is laminar or turbulent or of the geometrical shape of the interface.
of mass
In Fig. 5 the ratios of high mass transfer Sherwood numbers corrected for the density variation
Use ofp&P as correlating variable Most of previous investigators[5] on mass transfer have used molar driving force in their
1192
Mass transfer for a wide range of driving force
correlation instead of baricentric driving force used here and regarded p&P as an important correlating variable characterizing the effect of driving force on the rate of mass transfer, as transfer number is in the case of baricentric driving force. Recently Asano [3,4] discussed the validity of the use of p&P as correlating variable and showed that p&P varied not only with variation in transfer number but also with variation in the ratio of the molecular weight of the diffusing component to that of the inert component. In order to confirm the validity of the use of p&P, data for air-methanol and for air-carbon tetrachloride were plotted against p&P in Fig. 6, assuming the same exponent for
proposed (Eqs. (21-a) and (2 l-b)). Data for the rate of evaporation of water, methanol, benzene and carbon tetrachloride into air, and water into carbon dioxide by a specially designed agitated vessel were well correlated by the equations to get Eqs. (24-a) and (24-b), respectively with average deviation of 10.4 per cent. For the effect of driving force on the rate of mass transfer, a good agreement between the data and values predicted by laminar boundary layer theory for flat plate was obtained. From comparison between the data for the air-methanol system and for the air-carbon tetrachloride system, invalidity of the use of p&P as correlating variable for high mass transfer was also shown. Acknowledgments-One of the authors. K. A., would like to express’his thanks to Professor S. Ito of Tokyo Institute of Technology for his valuable discussions and-to The Asahi Glass Foundation for the Contribution to Industrial Technology for financial support.
In
0
I
‘2
NOTATION 0.6
I’
.
$)
B
0.3 o-2
0.4
0.6
I.0
0’
PqM’/P Fig. 6. Effect ofp,,/P
on the rate of mass transfer.
the Reynolds and Schmidt number as in the case of baricentric driving force, where molar Sherwood number was calculated by: Sh"
=
J
‘*
c,D,(x,-x,)/L’
(28)
From the fact that two different lines of correlation were obtained, we may conclude that the use of peMlP as correlating variable for mass transfer of wide range of driving force, especially for high mass transfer is invalid. CONCLUSIONS
Taking into account the effect of variable properties on the rate of mass transfer, _new general equations for the correlation of data which are applicable not only for low mass transfer but also for high mass transfer have been 1193 CJ?SVd.Z&No.8-D
constant transfer number Eq. (15) concentration . diffisivity F dimensionless stream function .A, baricentric local mass flux at the interface ‘X molar mass flux at the interface J k baricentric mass transfer coefficient by Eq. (7) L characteristic length L contact length in the direction of flow Re Reynolds number by Eq. (15) SC Schmidt number by Eq. (8) Sherwood number by Eq. Sh baricentric (16) Sh, baricentric local Sherwood number by ‘Eq. (15) molar Sherwood number by Eq. (28) * Sh* velocity; velocity of urn characteristic bulk flow in boundary layer in the u velocity direction of flow V velocity in boundary layer normal to the interface
a12, up1
04
KOICHI
ASANO and SHIGEFUMI
distance in the direction of flow; mole fraction in Eq. (28) Y distance normal to the interface W weight fraction x
dimensionless concentration viscosity kinematic viscosity
C#J dimensionless
p
group defined by Eq. (8)
density
Subscripts co bulk flow S at the interface 1 orA diffusing component 2 or B inert component
_ _
Greek symbols 8 p v
FUJITA
Eq. (8)
REFERENCES [II ACRIVOS A., J. Fluid Mech. 1962 12 337. PI ASANO K., Kagaku Kogaku (Chem. Engng, Japan) 1964 28 538. [31 ASANO K., The Asahi Glass Foundation for the Contribution to Industrial Technology Ann. Rep. 1969 15 13. [41 ASANO K., Saikin no Kagaku Kogaku (RecentAdv. Chem. Engng), pp. 89-103. Maruzen 1970. [51 CAIRNS R. C. and ROPER G. H., Chem. Engng Sci. 1954 4 937. t61 HANNAO.T.,A.I.Ch.E.J119628278. [71 MERK H. J., Appl. scient. Res. 1959 A-8 237. H. and YERAZUNIS H.,A.I.Ch.EJI. 1965’11834. VI MENDELSON R. and MULLIN J. W., Chem. Engng Sci. 1969 24 1665. 191 NIENOW A. W., UNAHABHOKHA [lOI RANZ W. E. and DICKSON P., Ind. Enana Chem. Fundls 1965 4 345. T. and HIRATA A., Kagaku Kogaku (Chem. Engng, Japan) 1970 34 904. [Ill SHIROTSUKA iI21 SPALDING D. B., Convective Mass Transfer. Edward Arnold 1963. S., Kagaku Kogaku (Chem. Engng, Japan) 1966 30 616. [I31 WAKAO N. and FUJISHIRO L. E. and WHITE R. R..A.I.Ch.E. Jll957 369. H41 WESTKAEMPER
R&WII& Une cuve agitte de conception spbiale a set-vi ?Ieffectuer des mesures de la vitesse d’tvaporation de liquides purs dam l’air et dam le gaz carbonique, pour une gamme &endue de forces d’entrainement (B = 0,03-36.1). Une nouvelle forme g&&ale pour la correlation des don&es de transfert de masse tleve, tenant compte de l’effet des proprietes variables sur la vitesse de transfert de la masse, a ete proposee et s’est averee tres satisfaisante pour les donntes relevtes. L’accord entre les don&es et les valeurs pr&lites par la theorie des couches limites lame&tires etait excellent. On discute egalement de la validit de l’hypothbse de l’effet nCgligeable des proprietes variables sur la vitesse de transfert de masse dam le cas dun transfert ClevC ainsi que de l’emploi de p&P en tant que varible de correlation. Zusamme&ssung-Es wurden Messungen iiber die Verdampfungsgeschwindigkeit reiner Fliissigkeitin in Luft turd Kohlendioxyd fur einen weiten Treibkraftsbereich (B = 0,03-36.1) mittels eines besonderskonstruierten Riihrkessels durchgeftlhrt. Unter Beriicksichtigung der Wiikung der Eigenschaften von Variablen auf die Stoffautauschgeschwindigkeit wurde eine neue allgemeine Form fur die Wechselbeziehung von hohen Stolfaustauschdaten entwickelt und die Daten konnten durch diese Korrelation gut auf einander abgestimmt werden. Die Ubereinstimmung zwischen den Daten und der durch die laminare Grenzschichttheorie vorausgesagten Werte war ausgezeichnet. Die Gtiltigkeit der AMahme einer vemachhhsigbarer Wiikung derEigenschaften von %riablen auf die Stoffaustauschgeschwindigkeit bei hohem Stoffaustausch und der Verwendung von p&P als Korrelationsvariable wurden ebenfalis eriirtert.
1194
pp. 1187-I 194. Pergamon Press. Printed in Great Britain.
Mass transfer for a wide range of driving force Evaporation of pure liquids KOICHI ASANO and SHIGEFUMI FUJITA Department of Chemical Engineering, Tokyo Institute of Technology, Tokyo, Japan (Received
3 November 1970)
Abstract- Measurements were made for the rate of evaporation of pure liquids into air and carbondioxide for a wide range of driving force (B = 0.03-36.1) by a specially designed agitated vessel. Taking into account the effect of variable properties on the rate of mass transfer, a new general form for the correlation of high mass transfer data wasproposed and data were well correlated by that correlation. Agreement between the data and the predicted value by laminar boundary layer theory was excellent. Validities of the assumption of negligible effect of variable properties on the rate of mass transfer in high mass transfer and of the use of p&P as correlating variable were also discussed. INTRODUCTION
well-known fact that a mass transfer through a heterogeneous interface is caused by two different phenomena, mass transfer due to diffusion of a component at the interface and mass transfer due to convective flow induced by diffusion. For mass transfer of low concentration gradient at the interface, we may say low mass transfer, such as mass transfer in absorbing and rectifying columns or in extractors, the amount of mass transfer due to convective flow is very small and can be neglected in comparison with that by diffusion, but for mass transfer of high concentration gradient, we may say high mass transfer, the convective term becomes important and can sometimes become of the same order of magnitude as the diffusional term or more. In other words mass transfer is affected by driving force for a moderate and high driving force range and for this reason an analogy between heat and mass transfer as is the case for low mass transfer cannot be applicable for high mass transfer [l, 2, 7, 121. This is one of the characteristics of high mass transfer. Usually high mass transfer is accompanied by large concentration difference between interface and bulk flow so for the systems of concentration dependent properties, the effect of variable properties on the rate of mass transfer needs to be considered [ 1,6,7]. Although in the past few decades considerable efforts have been directed to the study of high IT IS A
mass transfer in the light of laminar boundary layer theory, very few experimental data are known[3, 8-11, 13, 141. The purpose of this paper is to report some aspects of high mass transfer by measuring rate of evaporation of pure liquids. FUNDAMENTAL EQUATIONS MASS TRANSFER
FOR HIGH
For the flow inside a two-dimensional steady state laminar boundary layer along a flat plate with uni-directional diffusion of a component at the interface, the transport equations may be written as follows:
apu+apv=() ax
ay
,au+vau=ra au
( > p$!z+pvaW=a pDaw ax ay ay( ay> * ax
ay paypaY
(2) (3)
Boundary conditions at the interface are aty=O,
w=w,
(4)
u = u, v=v~=l-w,
-D,
(5)
(aw> ay
where Eq. (6), the convective
1187
8
velocity
(6) at the
KOICHI
ASANO and SHIGEFUMI
interface, is derived by the condition of no mass transfer of the inert component at the interface. From Eq. (6) the mass transfer flux at the interface is given by:
JlOC =
Plh
=
Ps%
_-P& -
, _
aw
w,
s
ay
(
>
FUJITA
where, B=
(w,-w,)/(l-ww,)
Re = U,LIv, SC, = v,lD,
(7)
4s =
(1%
CL8P*IkPm
= p&(w,-w,).
Considering variation of fluid properties with variation of compositions, we will introduce the following dimensionless variables: rl =
U, j- p/pm dylc
In the above equation the Sherwood number is defined according to the definition by Ranz and Dickson [ lo]. If we define an average Sherwood number as:
(16) we get from Eqs. (19)-( 16) the following relation: Sh(l-
w,) =f(Re,Sc,,
B, 4,~p,/pm9LI/L).
(17)
SC = vlD
e=
For perfect gas mixtures the equations for density and viscosity known:
(w,-w)I(w,-ww,)
following are well-
4 = PNPmPm P2
where the dimensionless variable 4 was first introduced by Merk[7] but without theoretical discussion. The transport equations can be rewritten as:
P
=
PlIP2
P2 i
(9) + F-8, = 0.
(10)
Boundary conditions at the interface are: F(0) = -BW,(O)/Sc,
(11)
F,(O) = 0
(12)
0(O) = 0
(13)
e,(o) =
vzsh,,, (1 -
w,) ( p,/p,) -l.Rfe’2.(x/L)“~ (14)
(18)
1+@4,/A4,-1)
p=
+
l+1-w -a12
W
1 1 +&a21
(19) 1
*
From Eqs. (15) (18) and (19) we will find that for a given system the three variables, B, +8 and ps/p,, are functions only of concentration at the interface and that in bulk flow. Therefore we can eliminate the variable & or p,/p, by using the following relations: 48 = g(B,
Islam)
(20-a)
or ~sl~co = g’ (Bv $4.
(20-b)
Substituting Eq. (20-a) or (20-b) into Eq. (17), we get the following fundamental relations for
1188
Mass transfer for a wide range of driving force
mass transfer in general: Sh(1 -w,)
=f(Re,Sc,,
B,p,lp,,L,/L)
(21-a)
or Sh(1 -w,)
=f(Re,
SC,, B, c#+,Lx/L).
(21-b)
We would like to point out the fact that in the present work the Sherwood and Schmidt numbers are defined at interface conditions, whereas in previous work no clear definition has ever been made. For the special case of a constant-property systems, Eq. (21-a) or (21-b) is simplified to the following form: .%(1-w,)
=f(Re,Sc,B,L,/L).
Equation (22) is in accordance previous works [ 1,2,7,12].
(22)
under the liquid surface by copper-constantan thermocouples. The measured liquid temperatures agreed within O*OS”C. Inlet gas was dehumidified by a CaCl,-packed column to absolute humidities of less than 0.004 and its temperature was adjusted by a lo-ohm electric resistance heater and a 1 kW slidac so that the temperature difference between gas and liquid was less than 1°C. The rate of agitation was kept at the maximum that did not cause aeration (250300 revlmin) in order to get homogeneous liquid temperature distribution, because no effect of agitation was observed in the preliminary runs [31. Data were taken for five systems; air-water, air-methanol, air-benzene, air-carbon tetrachloride and carbon dioxide-water. Ranges of variables shown below:
with the results of Re:
SC* EXPERIMENTAL APPARATUS PROCEDURES
B:
AND
The apparatus used in the present work, is a brass-made, 252-mm i.d., lOO-mm depth agitated vessel, with lOO-mm turbine impeller and 8-20 X 95 mm baffles and with heating jacket outside. The gas-liquid contacting is done at a portion of liquid surface (20.7 mm in width and 100 mm in the direction of flow) and a special design precautions are taken to prevent variation of gas-liquid contacting area due to variation in liquid level. The passage of inlet gas consists of a calming grid of 15 mm-2 mm o.d. stainless tubes and of a duct of rectangular cross section (20.7 mm in width and 25 mm in height) and the angle between the duct and liquid surface is about 6 degrees. The details of the agitated vessel are shown in Fig. 1. The rate of mass transfer was calculated by the amount of the liquid evaporated. The value of the driving force was adjusted by adjusting the vapour concentration at the interface, which was calculated from the vapour pressure of the pure liquid at the surface temperature. The surface temperature was estimated by taking the average of the temperatures measured at several points
Pmflp: PslPm:
48:
845 -15,100 o-221.21 o-0336.1 0.97 0*40o-754.45 0*632.55.
EXPERIMENTAL RESULTS DISCUSSIONS
Choice
of characteristic
AND
length
In the calculation of Reynolds and Sherwood number a definition of a characteristic length is necessary. In order to select a proper characteristic length for these variables two sets of data of different contact length, L, = 100 mm and L, = 50 mm (by covering a portion of the gasliquid contacting area with a brass covering), were compared for the air-methanol system at 30°C and the air-carbon tetrachloride system at 60°C. The results for the air-methanol system are shown in Figs. 2a and 2b. In the former where contact length was assumed as the characteristic length, two different lines of correlation were observed but in the latter where the height of the duct was assumed as the characteristic length, a single line of correlation was observed for both sets of data of different contact length. Similar results were observed for
1189
KOICHI ASANO and SHIGEFUMI
I
FUJITA
View window
thread
rid mm o.d. tubes Gas
as inlet
9-20
x95mm
1~~mm-.l Fig. 1. Details of the agitated vessel.
I
Air methanol
2x10'
4
at 3O’C
IO4
6
I
2
II
4x104
--lo3
2
4
6
IO'
2X104
Ro
R6X
Fig. 2a. Characteristic length; case-A L = L,.
Fig. 2b. Characteristic length; case-B L = height of duct.
the data of the air-carbon tetrachloride system. For this reason the height of duct (25 mm) was selected as the characteristic length throughout this paper. From Fig. 2b we may conclude that in this apparatus the Sherwood number is independent of contact length:
Sh = Sh,,,.
(23)
From Eqs. (21-a) (21-b) and (23) we can get the following relations:
1190
Sh(l -w,) =f(Re,Sc,,B,p,/p,)
(24-a)
Mass transfer for a wide range of driving force
kh(l-w,)
=f(Re,Sc,,B,&).
(24-b)
Discussions on the assumption of constant properties Although a considerable concentration difference between the interface and the bulk flow can be expected in high mass transfer, most of previous investigators [3, 8- 11, 13, 141 have assumed constant properties and neglected the effect of variable properties on the rate of mass transfer due to variation in composition, without any experimental proof. In order to confirm the validity of this assumption, data of the airmethanol system, a system of relatively constant properties, and of the air-carbon tetrachloride system, a system whose properties vary widely, are shown as a function of transfer number in Fig. 3, where values of the exponent of the Reynolds and of the Schmidt number were from the latter correlation. From the fact that two different lines of correlation were obtained for the two systems, we may conclude that the assumption of negligible effect of variable properties on the rate of mass transfer is invalid. Correlation of data Considering the above result, data were correlated by Eq. (24-a) or (24-b) by assuming power function type correlation. The coefficients
O-02
I
2
4
6 (I
IO
20
of correlation were determined by the least squares method with the HITAC 5020E Computor of Tokyo University Computing Center. The results were: Sh( 1 - w,) = 0.41 .Sc,@5.Reo.s7.(1 + B)-0.7g.
(P,lPmr”.07
(25-a)
and, Sh( 1 - w,) = 0*42.Sc,0.5.Re067. (1 + B)-0.*2. c$s-o.o2. (25-b) The average deviations from the above correlations were 10.4 per cent both for Eq. (25-a) and for Eq. (25-b). An explanation for the same values of exponent for the Reynolds and Schmidt numbers and for the different values for the transfer number may be clear from the discussion on Eqs. (20). Figures 4a,b and c shows the effect of Reynolds number, Schmidt number and (pJp,), respectively; where the solid line shows Eq. (25-a). In Fig. 4c, a correction by the method of Hanna[6], the only one quantitative approach for the effect of variable properties on the rate of mass transfer available at present: In P~IP~
kvar.cJkomt.p =
PslPm - 1
WI61
4060 -6.10’
+ 8)
Fig. 3. Correlation based on the assumption of negligible effect of variable properties.
1191
IO’
2
4
6
104
2x104
R6
Fig. 4a. Final correlation; effect of Reynolds number.
KOICHI ASANO and SHIGEFUMI
G t5 -8 *m
FUJITA
2.0
I.0
$ 6 0 2 g 6 'z .
0.6 0.4
0.2
2 L iFi 0.1 O-I
0.4 0.6
0.2
I.0
2.0
'
SC,
Fig. 4b. Final correlation; effect of Schmidt number.
solutionfor
xact
: 6 s + I
7 : #o LL
4
6
IO
20
40
60
(I+B)
Fig. 5. Effect of driving force on the rate of mass transfer; a comparison with laminar boundary layer theory.
x Iv) G 3 Z $
2
I 0.4
0.6
I
2
to low mass transfer Sherwood numbers were plotted as a function of transfer number to show the effect of driving force on the rate of mass transfer, where low mass transfer Sherwood numbers were estimated by rearranging Eq. (25-a):
III, 4
66
ps 'pm
Fig. 4c. Final correlation; effect of variable density on the rate of mass transfer.
Sh( 1 - w~)~+ = 0*41Sc,0’5~Re0’67.
is also shown in a dotted line and compared with the present data, assuming the same exponent for the Reynolds, Schmidt and transfer numbers as for the present work. The reason for disagreement between the data and the correction by Eq. (26) may be explained due to the fact that in the derivation of Eq. (26) only the effect of variable density is taken into account, whereas in the present work not only the effect of variable density but also the effects of & and of transfer number are taken into account. Efect of driving force on the rate transfer; a comparison with the theory
(27)
Exact solution for the laminar boundary layer on a flat plate is also shown in a solid line for the purpose of comparison. A good agreement between the data and the theory for a wide range of driving force is worthy of remark, in spite of the difference in the exponents of the Reynolds and of Schmidt numbers in the present work from the theoretical values of 0.5 and 0.33, respectively. This may suggest the existence of a universal function of transfer number irrespective of whether the flow is laminar or turbulent or of the geometrical shape of the interface.
of mass
In Fig. 5 the ratios of high mass transfer Sherwood numbers corrected for the density variation
Use ofp&P as correlating variable Most of previous investigators[5] on mass transfer have used molar driving force in their
1192
Mass transfer for a wide range of driving force
correlation instead of baricentric driving force used here and regarded p&P as an important correlating variable characterizing the effect of driving force on the rate of mass transfer, as transfer number is in the case of baricentric driving force. Recently Asano [3,4] discussed the validity of the use of p&P as correlating variable and showed that p&P varied not only with variation in transfer number but also with variation in the ratio of the molecular weight of the diffusing component to that of the inert component. In order to confirm the validity of the use of p&P, data for air-methanol and for air-carbon tetrachloride were plotted against p&P in Fig. 6, assuming the same exponent for
proposed (Eqs. (21-a) and (2 l-b)). Data for the rate of evaporation of water, methanol, benzene and carbon tetrachloride into air, and water into carbon dioxide by a specially designed agitated vessel were well correlated by the equations to get Eqs. (24-a) and (24-b), respectively with average deviation of 10.4 per cent. For the effect of driving force on the rate of mass transfer, a good agreement between the data and values predicted by laminar boundary layer theory for flat plate was obtained. From comparison between the data for the air-methanol system and for the air-carbon tetrachloride system, invalidity of the use of p&P as correlating variable for high mass transfer was also shown. Acknowledgments-One of the authors. K. A., would like to express’his thanks to Professor S. Ito of Tokyo Institute of Technology for his valuable discussions and-to The Asahi Glass Foundation for the Contribution to Industrial Technology for financial support.
In
0
I
‘2
NOTATION 0.6
I’
.
$)
B
0.3 o-2
0.4
0.6
I.0
0’
PqM’/P Fig. 6. Effect ofp,,/P
on the rate of mass transfer.
the Reynolds and Schmidt number as in the case of baricentric driving force, where molar Sherwood number was calculated by: Sh"
=
J
‘*
c,D,(x,-x,)/L’
(28)
From the fact that two different lines of correlation were obtained, we may conclude that the use of peMlP as correlating variable for mass transfer of wide range of driving force, especially for high mass transfer is invalid. CONCLUSIONS
Taking into account the effect of variable properties on the rate of mass transfer, _new general equations for the correlation of data which are applicable not only for low mass transfer but also for high mass transfer have been 1193 CJ?SVd.Z&No.8-D
constant transfer number Eq. (15) concentration . diffisivity F dimensionless stream function .A, baricentric local mass flux at the interface ‘X molar mass flux at the interface J k baricentric mass transfer coefficient by Eq. (7) L characteristic length L contact length in the direction of flow Re Reynolds number by Eq. (15) SC Schmidt number by Eq. (8) Sherwood number by Eq. Sh baricentric (16) Sh, baricentric local Sherwood number by ‘Eq. (15) molar Sherwood number by Eq. (28) * Sh* velocity; velocity of urn characteristic bulk flow in boundary layer in the u velocity direction of flow V velocity in boundary layer normal to the interface
a12, up1
04
KOICHI
ASANO and SHIGEFUMI
distance in the direction of flow; mole fraction in Eq. (28) Y distance normal to the interface W weight fraction x
dimensionless concentration viscosity kinematic viscosity
C#J dimensionless
p
group defined by Eq. (8)
density
Subscripts co bulk flow S at the interface 1 orA diffusing component 2 or B inert component
_ _
Greek symbols 8 p v
FUJITA
Eq. (8)
REFERENCES [II ACRIVOS A., J. Fluid Mech. 1962 12 337. PI ASANO K., Kagaku Kogaku (Chem. Engng, Japan) 1964 28 538. [31 ASANO K., The Asahi Glass Foundation for the Contribution to Industrial Technology Ann. Rep. 1969 15 13. [41 ASANO K., Saikin no Kagaku Kogaku (RecentAdv. Chem. Engng), pp. 89-103. Maruzen 1970. [51 CAIRNS R. C. and ROPER G. H., Chem. Engng Sci. 1954 4 937. t61 HANNAO.T.,A.I.Ch.E.J119628278. [71 MERK H. J., Appl. scient. Res. 1959 A-8 237. H. and YERAZUNIS H.,A.I.Ch.EJI. 1965’11834. VI MENDELSON R. and MULLIN J. W., Chem. Engng Sci. 1969 24 1665. 191 NIENOW A. W., UNAHABHOKHA [lOI RANZ W. E. and DICKSON P., Ind. Enana Chem. Fundls 1965 4 345. T. and HIRATA A., Kagaku Kogaku (Chem. Engng, Japan) 1970 34 904. [Ill SHIROTSUKA iI21 SPALDING D. B., Convective Mass Transfer. Edward Arnold 1963. S., Kagaku Kogaku (Chem. Engng, Japan) 1966 30 616. [I31 WAKAO N. and FUJISHIRO L. E. and WHITE R. R..A.I.Ch.E. Jll957 369. H41 WESTKAEMPER
R&WII& Une cuve agitte de conception spbiale a set-vi ?Ieffectuer des mesures de la vitesse d’tvaporation de liquides purs dam l’air et dam le gaz carbonique, pour une gamme &endue de forces d’entrainement (B = 0,03-36.1). Une nouvelle forme g&&ale pour la correlation des don&es de transfert de masse tleve, tenant compte de l’effet des proprietes variables sur la vitesse de transfert de la masse, a ete proposee et s’est averee tres satisfaisante pour les donntes relevtes. L’accord entre les don&es et les valeurs pr&lites par la theorie des couches limites lame&tires etait excellent. On discute egalement de la validit de l’hypothbse de l’effet nCgligeable des proprietes variables sur la vitesse de transfert de masse dam le cas dun transfert ClevC ainsi que de l’emploi de p&P en tant que varible de correlation. Zusamme&ssung-Es wurden Messungen iiber die Verdampfungsgeschwindigkeit reiner Fliissigkeitin in Luft turd Kohlendioxyd fur einen weiten Treibkraftsbereich (B = 0,03-36.1) mittels eines besonderskonstruierten Riihrkessels durchgeftlhrt. Unter Beriicksichtigung der Wiikung der Eigenschaften von Variablen auf die Stoffautauschgeschwindigkeit wurde eine neue allgemeine Form fur die Wechselbeziehung von hohen Stolfaustauschdaten entwickelt und die Daten konnten durch diese Korrelation gut auf einander abgestimmt werden. Die Ubereinstimmung zwischen den Daten und der durch die laminare Grenzschichttheorie vorausgesagten Werte war ausgezeichnet. Die Gtiltigkeit der AMahme einer vemachhhsigbarer Wiikung derEigenschaften von %riablen auf die Stoffaustauschgeschwindigkeit bei hohem Stoffaustausch und der Verwendung von p&P als Korrelationsvariable wurden ebenfalis eriirtert.
1194