- Email: [email protected]
The mass transfer driving force for high mass flux
Chemical Engineering Science, 1969, Vol. 24, pp. 1655-1660.
Pergamon Press.
Printed in Great Britain.
The mass transfer driving force for high mass flux A. W. NIENOW, R. UNAHABHOKHA and J. W. MULLIN Department of Chemical Engineering, University College London, Torrington Place, London W.C. 1,England (First received 30 May 1969; in revisedform 24 June 1969) Abstract - For the dissolution
of single spheres of moderately soluble electrolytes, can be correlated by an equation of the type
mass transfer data
It is shown that the correlation constant #J is approximately the same for the four different electrolytewater systems studied and is approximately equal to the value obtained by Rowe et al. for heat and low mass flux mass transfer, provided the driving force is defined by the relationship
Any other choice of driving force would, for these systems, lead to different dovalues al1 markedly different from that obtained for low mass flux. INTRODUCTION
driving forces have been used in the field of mass transfer. Generally, provided the driving force is clearly defined, the choice is not of great importante unless use is to be made of the analogy between heat and mass transfer. In this case, the driving force definitions for each process must be compatible, and to test for compatability, both heat and mass transfer data must be available for a given geometry. In an experimental study on low heat and low mass flux mass transfer from a single sphere to an extensive flowing liquid, Rowe, Claxton and Lewis [l] found that their data and those of many other workers could be satisfactorily correlated by the equation MANY DIFFERENT
(ShorNu)
= 2+0*79(Re)1/2(SeorPr)1’3
over the range 20 < Re < 2000. In Eq. (l), the mass transfer coefficient is generally defined by m = k,(C,-C,)
= kc(pOwO-pdb)
(2)
or, since for low mass flux, po = pm = p, m = pk,(oo-o,).
Alternatively,
the coefficient
(3) may be defined by
m = k, Co,-om).
(4)
fOr each System,
This wil1 give a different numerical value for the coefficient but since the Sherwood groups corresponding to the coefficients defined by Eqs. (2) and (4) are k,d/D and k,d/pD respectively, it can be seen that the Sherwood groups wil1 be numerically identical. Thus either k, or k, can be used inEq.(l). Equation (1) should describe the dissolution of a solid into water or its own aqueous solution. However, if the solubility of the solid is greater than one or two per cent, certain complications arise. Firstly, the concentration dependence of the physical properties becomes significant and, since po + p,, the two different Sherwood groups associated with k, and k, wil1 not be numerically equal. Secondly, the mass flux from the surface of the sphere alters the concentration gradient at the surface compared to that obtained under otherwise identical conditions of low mass (or heat) flux. Both Spalding[2] and Stewart[3] showed theoretically that by a suitable choice of driving force, the mass flux range over which the analogy between heat and mass transfer was directly apphcable (as in Eq. (1)) could be extended, though for strict accuracy additional correction factors should be included. However, the present authors know of no experimental evidente that
1655
A. W. NIENOW.
R. UNAHABHOKHA
has been reported to substantiate these theoretical predictions. This paper considers the correlation of moderately high mass flux dissolution data, and in particular shows that the selection of the driving force has a very significant effect on any equation derived from these data and also on the relationship of such an equation to that for low mass flux for the same geometry. EXPERIMENTAL
AND
DATA
CORRELATION
The effect of the concentration-dependent physical properties for a hydrodynamic situation basically identical to that used by Rowe et al.[l] has recently been studied and reported in detail [4,5]. A single sphere of a moderately soluble electrolyte was dissolved into different concentrations of its solutions and into water. The data were then plotted as (Sh. SC-‘/~) against Re and lines of slape 4 were obtained. However, the correlation constant 4 (equivalent to 0.79 in Eq. (1)) depended on the physical property values used and in genera1 was different for each solution concentration. However, a single line sufficed for al1 the data when mean physical properties were used for the Sherwood and Schmidt groups and bulk solution properties for the Reynolds group. Therefore, Eq. (1) became (since for the range under consideration the constant 2 was negligible) sh = t#~(Re~)~‘~(%)~‘~
(5)
for 100 < Re, < 1000. In order to determine CJJand to construct the basic Sh-Sc-Re graphs, some value of driving force had to be selected. Since it appeared to have good theoretical justification [6], the driving force of Spalding [2] was chosen. The basic mass transfer equation then is
The systems studied were potassium and ammonium alum and potassium and ammonium chloride. The alum’s mass flux data were obtained by measuring the total weight loss[7] and gave a value of 4 of 0.81 with a standard devia-
and J. W. MULLIN
tion of &0*05[4,7]. The data obtained from the dissolution of the chloride were treated separately since the mass flux was determined by measuring the rate of recession of the sphere’s surface at the front or stagnation point [5]. In this case the correlation for the stagnation point, 4F was found to be 1.56 with standard deviation of 3.16. In order to compare the two above correlation constants, some means must be found of relating 4F to 4. Few experimental studies for solid-liquid systems have been made and these have al1 been confined to the dissolution of benzoic acid into water. Garner and Suckling[8], Garner and Grafton [9] and Linton and Sutherland [ 101 found & was 1.68, 1.39 and 0.94 respectively. However, discrepencies of this order in experimentally obtained 4 and & values are not unusual and the reasons for this have been explained in detail [ 1,7]. Another, probably better procedure therefore is comparison with the predictions of low flux mass transfer boundary layer theory, particularly as theoretically the choice of B should extend the range over which high and low mass flux are directly comparable. From boundary layer theory, both Aksel’rud [ 1 l] and Froessling [ 121 predicted 4F = 1.53 whilst Garner and Keey [ 131 and Grafton [ 141 predicted values of 1.59 and 1.60 respectively. These are al1 in very good agreement with the present experimental value of 1.56 and with each other. Finally, Aksel’rud [ 1 l] showed theoretically that the ratio c#JJ~ was 1.91. Therefore, since the present experimental & = 1.56, this predicts that 4 for the dissolution of single spheres of ammonium and potassium chloride is 0.82. This value is almost identical to that for potassium and ammonium alum and is also in good agreement with that given in Eq. (1) for low mass flux, constant physical properties. DISCUSSION
AND
CONCLUSIONS
As explained earlier in the paper the choice of different driving forces leads to different numerical values for the Sherwood group. Table 1 shows Sh,/Sh, the ratio of the Sherwood num-
1656
z Z
t171
1161
m = k,(w,-om)
-
--
m = kcp(wo-w,)
1151
-
PD
kd
-
P&JO-W~)
md
kcd D
&=F! PD
m=k,(Y,,-Y,)
U,9,131
[18:20]
W
k,d Pg
_--
Of Sherwood number
D
Of driving force
m = k,(C,-C,)
References
Definition
-
1
1-00
1-0,
Sh
Sha
-
.-
0.896
1.00
>( 1_ wo> 0.896
-
0.896
1.05
-
0.915
wo 2
-
_^
0.896
1.00
0.896
O
-
0.896
I
1.05
0.916
wo 1
.- -
0.729
l+M
0.729
0
wo 1
-
0.745
1.00
0.745
0
..
0.255 00 1
0.745
1.14
,_
0.806
Potassium chloride
.
-
1.01
l+IO
1.01
0
oxtO34
Benzoic acid
for various systems at 20°C
_. -. ._
0.729
1.16
0.750
0.271
0.104
0.104
0
Ammonium chloride
Potassium alum
Ammonium alum
Evaluation of the ratio, Sh&h
Table 1. Different definitions compared
., ,.
_
Bulk conc., O_
Surface conc., o0
A. W. NIENOW,
R. UNAHABHOKHA
bers based on B as the driving force to that resulting from other definitions, and quantifies the ratio for the systems studied in the present work and for benzoic acid. The latter solid has been included to show that for low mass flux, low solubility, the choice of driving force is not important. Also, of course, (7) if the limiting Sherwood number of 2 is neglected, which is reasonable for Sh in the range 100-1000 as found in this work. It is apparent that if the variations in physical properties had been allowed for as set out in Eq. (5), the driving forces defined by Mendelsohn and Yerazunis [15], Ranz and Dickson[ 161 and Eckert[l7] (al1 of whom were concerned with high mass flux) would stil1 have enabled a single correlation to account for the data for each salt. However, 4 would be greater than that obtained using B as the driving force by between about 10 and 30 per cent and, in general, different in each case. The same remarks apply to the almost standard definition of (C, - C,), except that the inclusion of the density would affect the amount of scatter of the data for any one system. The mass ratio definition has seldom been used except in certain crystallisation and dissolution studies[l& 201. However, if physical property changes are allowed for as set out in Eq. (5), a different line wil1 exist for each value of urn. Some other combination of physical properties might enable a single line to correlate al1 the data for any one system with this definition of driving force but this is most unlikely since (I/( 1-0,)) always decreases + as urn increases, whilst physical property values, particularly diffusivityr 191, may increase or decrease with concentration. This paper has not set out to show that a value for 4 of 0.79 to 0.81 is correct. Rowe et al.[l] have shown that the exact determination of I#Jis not practical because of the genera1 inaccuracy of physical property data, particularly diffusivity [ 1, 191, and because there is some indication that both 4 and the exponent of the Reynolds group
and J. W. MULLIN
are themselves functions of Re. Also, theory suggests that the exponent on the Schmidt group is itself a function of Sc [7], Cook [20] found that 4 depended on the structure of the solid surface and it is generally recognized that 4 varies with main stream turbulente. However, providing physical property variations are allowed for as set out in Eq. (5) and a driving force, B = (ti,, -co& 1-03, is used to determine the Sherwood number from the experimental mass flux data, 4 is approximately constant for the systems reported both in this work and by Rowe et al.[l] for heat and low mass flux mass transfer. Conversely, if other definitions of driving force are used, the correlation constant do varies from system to system and in certain cases with concentration. This therefore is experimental evidente that the driving force B does indeed extend the range of mass flux over which the analogy between heat and mass transfer applies and further suggest that B is the best choice of driving force for high mass flux studies.
NOTATION
mass transfer driving force, defined by Eq. (6), dimensionless. C concentration of solute in solution, g cmm3 d sphere diameter, cm D diffusivity, cm2 sec-’ ks mass transfer coefficient defined by Eq. (6), g crnm2sec-’ B-’ k, mass transfer coefficient defined by Eq. (2), cm sec-’ ky mass transfer coefficient defined in Table 1, gcm+ sec-’ Y-1 k, mass transfer coefficient defined by Eq. (4), g cm-2 sec-’ o-l m mass flux, g cm+ sec-’ Nu Nusselt number, dimensionless Pr Prandtl number, dimensionless Re Reynolds number, dulv, dimensionless Sc Schmidt number, dimensionless % Y/Drntdimensionless Sh Sherwood number, dimensionless. sh k8D/p&, dimensionless u mainstream velocity, cm sec+
1658
B
The mass transfer driving force for high mass flux
Y
ratio of solute to solvent dimensionless
in a solution,
Greek symbols kinematic viscosity, cm2 sec-’ correlation constant (sec Eq. (5)) dimensionless density of solution, g cmp3 P mass fraction of solute in solution, dimen6J sionless
Subscripts 00 in the bulk solution B based on B as the driving force F the front or stagnation point of a sphere 0 at the surface int integral
Superscript - mean
REFERENCES UI
PI [31 [41 Vl [61 [71 Vl r91 [lol [lil [121 [131 [141 [IS1 V61 [171 [lg1 u91 Lw
COWE P. N., CLAXTON K. T. and LEWIS J. B., Trans. Instn chem. Engrs 1965 43 14. iPALDING D. B., 1nt.J. Heat Mass Transfer 1960 1 192. ITEWART W. E., A.I.Ch.EJll962 8 42 1. qIENOW A. W., UNAHABHOKHA R. and MULLIN J. W., Ind. Engng Chem. Fundls 1966 5 578. VIENOW A. W., UNAHABHOKHA R. and MULLIN J. W., J. uppl. Chem. 1968 18 154. rlIENOW A. W., Er. chem. Engng 1967 12 1737. JNAHABHOKHA R., Ph.D. Thesis, University of London 1968. JARNER F. H. and SUCKLING R. D., A.I.Ch.EJll958 4 114. JARNER F. H. and GRAFTON R. W., Proc. R. Sec. 1954 224A 64. _.INTON M. and SUTHERLAND K. L., Chem. Engng Sci. 1960 12 214. 4KSEL’RUD G. A., Zh. prikf. Khim., Leningrud 1953 27 1446. FROESSLING N., Beitr. Geophys. 1938 52 70. JARNER F. H. and KEEY R. B., Chem. Engng Sci. 1958 9 119. SRAFTON R. W., Chem. Engng Sci. 1963 18 457. MENDELSOHN H. and YERAZUNIS S.,A.I.Ch.E. Jll965 11834. RANZ W. E. and DICKSON P. F.. Ind. Enww Chem. Fundls 1965 4 345. ECKERT E. R. G. and DRAKE R.‘M. Zntroud;ction to the Trnnsfer of Heut und Muss. McGraw-HiII 1950. GASKA C., Ph.D. Thesis, University of London, 1966. NIENOW A. W., Br. chem. Engng 1965 10 327. MULLIN J. W. and COOK T. P., J. uppl. Chem. 1965 15 145. Résumé-Les données de transfert de masse, pour la dissolution de sphères uniques d’électrolytes modérément solubles, peuvent être mises en corrélation par une équation du genre:
sh = $4 Re.#‘2(%)1’3. On montre que la constante de corrélation + est presque identique pour les quatres systèmes variés électrolyte-eau étudiés et sa valeur est presque égale à la valeur obtenue par Rowe et al. pour le transfert de chaleur et le transfert de masse du flux de faible masse, à condition que la force d’entraînement soit definie par la relation
Tout autre choix de Ia force d’entraînement conduirait, pour ces systèmes, à des valeurs de 4 diverses pour chaque système, toutes différentes d’une façon marquée de celles obtenues pour le flux de masse faible. Zusammenfassung- Für die Auflösung von Einzelkugeln Stoffübertragungswerte durch eine Gleichung des Typs
mässig Iöslicher
Elektrolyte
können
in eine Korrelation gebracht werden. Es wird gezeigt, dass die Korrelationskonstante $ für die untersuchten vier verschiedenen Elektrolyt-Wasser Systeme ungefähr gleich ist und circa dem durch Rowe et af. für Warme-und niedirge Stoffströmungübertragung erhahenen Wertentspricht, voraus-
1659
A. W. NIENOW.
R. UNAHABHOKHA
and J. W. MULLIN
gesetzt, dass die Triebkraft durch die Beziehung:
ausgedrückt wird. Jede andere Waal einer Triebkraft würde bei diesen Systemen zu verschiedenen 4 Werten fiir jedes System führen, die alle sehr verschieden von dem für niedrige Stoffströmung erahltenen sind.
1660
Pergamon Press.
Printed in Great Britain.
The mass transfer driving force for high mass flux A. W. NIENOW, R. UNAHABHOKHA and J. W. MULLIN Department of Chemical Engineering, University College London, Torrington Place, London W.C. 1,England (First received 30 May 1969; in revisedform 24 June 1969) Abstract - For the dissolution
of single spheres of moderately soluble electrolytes, can be correlated by an equation of the type
mass transfer data
It is shown that the correlation constant #J is approximately the same for the four different electrolytewater systems studied and is approximately equal to the value obtained by Rowe et al. for heat and low mass flux mass transfer, provided the driving force is defined by the relationship
Any other choice of driving force would, for these systems, lead to different dovalues al1 markedly different from that obtained for low mass flux. INTRODUCTION
driving forces have been used in the field of mass transfer. Generally, provided the driving force is clearly defined, the choice is not of great importante unless use is to be made of the analogy between heat and mass transfer. In this case, the driving force definitions for each process must be compatible, and to test for compatability, both heat and mass transfer data must be available for a given geometry. In an experimental study on low heat and low mass flux mass transfer from a single sphere to an extensive flowing liquid, Rowe, Claxton and Lewis [l] found that their data and those of many other workers could be satisfactorily correlated by the equation MANY DIFFERENT
(ShorNu)
= 2+0*79(Re)1/2(SeorPr)1’3
over the range 20 < Re < 2000. In Eq. (l), the mass transfer coefficient is generally defined by m = k,(C,-C,)
= kc(pOwO-pdb)
(2)
or, since for low mass flux, po = pm = p, m = pk,(oo-o,).
Alternatively,
the coefficient
(3) may be defined by
m = k, Co,-om).
(4)
fOr each System,
This wil1 give a different numerical value for the coefficient but since the Sherwood groups corresponding to the coefficients defined by Eqs. (2) and (4) are k,d/D and k,d/pD respectively, it can be seen that the Sherwood groups wil1 be numerically identical. Thus either k, or k, can be used inEq.(l). Equation (1) should describe the dissolution of a solid into water or its own aqueous solution. However, if the solubility of the solid is greater than one or two per cent, certain complications arise. Firstly, the concentration dependence of the physical properties becomes significant and, since po + p,, the two different Sherwood groups associated with k, and k, wil1 not be numerically equal. Secondly, the mass flux from the surface of the sphere alters the concentration gradient at the surface compared to that obtained under otherwise identical conditions of low mass (or heat) flux. Both Spalding[2] and Stewart[3] showed theoretically that by a suitable choice of driving force, the mass flux range over which the analogy between heat and mass transfer was directly apphcable (as in Eq. (1)) could be extended, though for strict accuracy additional correction factors should be included. However, the present authors know of no experimental evidente that
1655
A. W. NIENOW.
R. UNAHABHOKHA
has been reported to substantiate these theoretical predictions. This paper considers the correlation of moderately high mass flux dissolution data, and in particular shows that the selection of the driving force has a very significant effect on any equation derived from these data and also on the relationship of such an equation to that for low mass flux for the same geometry. EXPERIMENTAL
AND
DATA
CORRELATION
The effect of the concentration-dependent physical properties for a hydrodynamic situation basically identical to that used by Rowe et al.[l] has recently been studied and reported in detail [4,5]. A single sphere of a moderately soluble electrolyte was dissolved into different concentrations of its solutions and into water. The data were then plotted as (Sh. SC-‘/~) against Re and lines of slape 4 were obtained. However, the correlation constant 4 (equivalent to 0.79 in Eq. (1)) depended on the physical property values used and in genera1 was different for each solution concentration. However, a single line sufficed for al1 the data when mean physical properties were used for the Sherwood and Schmidt groups and bulk solution properties for the Reynolds group. Therefore, Eq. (1) became (since for the range under consideration the constant 2 was negligible) sh = t#~(Re~)~‘~(%)~‘~
(5)
for 100 < Re, < 1000. In order to determine CJJand to construct the basic Sh-Sc-Re graphs, some value of driving force had to be selected. Since it appeared to have good theoretical justification [6], the driving force of Spalding [2] was chosen. The basic mass transfer equation then is
The systems studied were potassium and ammonium alum and potassium and ammonium chloride. The alum’s mass flux data were obtained by measuring the total weight loss[7] and gave a value of 4 of 0.81 with a standard devia-
and J. W. MULLIN
tion of &0*05[4,7]. The data obtained from the dissolution of the chloride were treated separately since the mass flux was determined by measuring the rate of recession of the sphere’s surface at the front or stagnation point [5]. In this case the correlation for the stagnation point, 4F was found to be 1.56 with standard deviation of 3.16. In order to compare the two above correlation constants, some means must be found of relating 4F to 4. Few experimental studies for solid-liquid systems have been made and these have al1 been confined to the dissolution of benzoic acid into water. Garner and Suckling[8], Garner and Grafton [9] and Linton and Sutherland [ 101 found & was 1.68, 1.39 and 0.94 respectively. However, discrepencies of this order in experimentally obtained 4 and & values are not unusual and the reasons for this have been explained in detail [ 1,7]. Another, probably better procedure therefore is comparison with the predictions of low flux mass transfer boundary layer theory, particularly as theoretically the choice of B should extend the range over which high and low mass flux are directly comparable. From boundary layer theory, both Aksel’rud [ 1 l] and Froessling [ 121 predicted 4F = 1.53 whilst Garner and Keey [ 131 and Grafton [ 141 predicted values of 1.59 and 1.60 respectively. These are al1 in very good agreement with the present experimental value of 1.56 and with each other. Finally, Aksel’rud [ 1 l] showed theoretically that the ratio c#JJ~ was 1.91. Therefore, since the present experimental & = 1.56, this predicts that 4 for the dissolution of single spheres of ammonium and potassium chloride is 0.82. This value is almost identical to that for potassium and ammonium alum and is also in good agreement with that given in Eq. (1) for low mass flux, constant physical properties. DISCUSSION
AND
CONCLUSIONS
As explained earlier in the paper the choice of different driving forces leads to different numerical values for the Sherwood group. Table 1 shows Sh,/Sh, the ratio of the Sherwood num-
1656
z Z
t171
1161
m = k,(w,-om)
-
--
m = kcp(wo-w,)
1151
-
PD
kd
-
P&JO-W~)
md
kcd D
&=F! PD
m=k,(Y,,-Y,)
U,9,131
[18:20]
W
k,d Pg
_--
Of Sherwood number
D
Of driving force
m = k,(C,-C,)
References
Definition
-
1
1-00
1-0,
Sh
Sha
-
.-
0.896
1.00
>( 1_ wo> 0.896
-
0.896
1.05
-
0.915
wo 2
-
_^
0.896
1.00
0.896
O
-
0.896
I
1.05
0.916
wo 1
.- -
0.729
l+M
0.729
0
wo 1
-
0.745
1.00
0.745
0
..
0.255 00 1
0.745
1.14
,_
0.806
Potassium chloride
.
-
1.01
l+IO
1.01
0
oxtO34
Benzoic acid
for various systems at 20°C
_. -. ._
0.729
1.16
0.750
0.271
0.104
0.104
0
Ammonium chloride
Potassium alum
Ammonium alum
Evaluation of the ratio, Sh&h
Table 1. Different definitions compared
., ,.
_
Bulk conc., O_
Surface conc., o0
A. W. NIENOW,
R. UNAHABHOKHA
bers based on B as the driving force to that resulting from other definitions, and quantifies the ratio for the systems studied in the present work and for benzoic acid. The latter solid has been included to show that for low mass flux, low solubility, the choice of driving force is not important. Also, of course, (7) if the limiting Sherwood number of 2 is neglected, which is reasonable for Sh in the range 100-1000 as found in this work. It is apparent that if the variations in physical properties had been allowed for as set out in Eq. (5), the driving forces defined by Mendelsohn and Yerazunis [15], Ranz and Dickson[ 161 and Eckert[l7] (al1 of whom were concerned with high mass flux) would stil1 have enabled a single correlation to account for the data for each salt. However, 4 would be greater than that obtained using B as the driving force by between about 10 and 30 per cent and, in general, different in each case. The same remarks apply to the almost standard definition of (C, - C,), except that the inclusion of the density would affect the amount of scatter of the data for any one system. The mass ratio definition has seldom been used except in certain crystallisation and dissolution studies[l& 201. However, if physical property changes are allowed for as set out in Eq. (5), a different line wil1 exist for each value of urn. Some other combination of physical properties might enable a single line to correlate al1 the data for any one system with this definition of driving force but this is most unlikely since (I/( 1-0,)) always decreases + as urn increases, whilst physical property values, particularly diffusivityr 191, may increase or decrease with concentration. This paper has not set out to show that a value for 4 of 0.79 to 0.81 is correct. Rowe et al.[l] have shown that the exact determination of I#Jis not practical because of the genera1 inaccuracy of physical property data, particularly diffusivity [ 1, 191, and because there is some indication that both 4 and the exponent of the Reynolds group
and J. W. MULLIN
are themselves functions of Re. Also, theory suggests that the exponent on the Schmidt group is itself a function of Sc [7], Cook [20] found that 4 depended on the structure of the solid surface and it is generally recognized that 4 varies with main stream turbulente. However, providing physical property variations are allowed for as set out in Eq. (5) and a driving force, B = (ti,, -co& 1-03, is used to determine the Sherwood number from the experimental mass flux data, 4 is approximately constant for the systems reported both in this work and by Rowe et al.[l] for heat and low mass flux mass transfer. Conversely, if other definitions of driving force are used, the correlation constant do varies from system to system and in certain cases with concentration. This therefore is experimental evidente that the driving force B does indeed extend the range of mass flux over which the analogy between heat and mass transfer applies and further suggest that B is the best choice of driving force for high mass flux studies.
NOTATION
mass transfer driving force, defined by Eq. (6), dimensionless. C concentration of solute in solution, g cmm3 d sphere diameter, cm D diffusivity, cm2 sec-’ ks mass transfer coefficient defined by Eq. (6), g crnm2sec-’ B-’ k, mass transfer coefficient defined by Eq. (2), cm sec-’ ky mass transfer coefficient defined in Table 1, gcm+ sec-’ Y-1 k, mass transfer coefficient defined by Eq. (4), g cm-2 sec-’ o-l m mass flux, g cm+ sec-’ Nu Nusselt number, dimensionless Pr Prandtl number, dimensionless Re Reynolds number, dulv, dimensionless Sc Schmidt number, dimensionless % Y/Drntdimensionless Sh Sherwood number, dimensionless. sh k8D/p&, dimensionless u mainstream velocity, cm sec+
1658
B
The mass transfer driving force for high mass flux
Y
ratio of solute to solvent dimensionless
in a solution,
Greek symbols kinematic viscosity, cm2 sec-’ correlation constant (sec Eq. (5)) dimensionless density of solution, g cmp3 P mass fraction of solute in solution, dimen6J sionless
Subscripts 00 in the bulk solution B based on B as the driving force F the front or stagnation point of a sphere 0 at the surface int integral
Superscript - mean
REFERENCES UI
PI [31 [41 Vl [61 [71 Vl r91 [lol [lil [121 [131 [141 [IS1 V61 [171 [lg1 u91 Lw
COWE P. N., CLAXTON K. T. and LEWIS J. B., Trans. Instn chem. Engrs 1965 43 14. iPALDING D. B., 1nt.J. Heat Mass Transfer 1960 1 192. ITEWART W. E., A.I.Ch.EJll962 8 42 1. qIENOW A. W., UNAHABHOKHA R. and MULLIN J. W., Ind. Engng Chem. Fundls 1966 5 578. VIENOW A. W., UNAHABHOKHA R. and MULLIN J. W., J. uppl. Chem. 1968 18 154. rlIENOW A. W., Er. chem. Engng 1967 12 1737. JNAHABHOKHA R., Ph.D. Thesis, University of London 1968. JARNER F. H. and SUCKLING R. D., A.I.Ch.EJll958 4 114. JARNER F. H. and GRAFTON R. W., Proc. R. Sec. 1954 224A 64. _.INTON M. and SUTHERLAND K. L., Chem. Engng Sci. 1960 12 214. 4KSEL’RUD G. A., Zh. prikf. Khim., Leningrud 1953 27 1446. FROESSLING N., Beitr. Geophys. 1938 52 70. JARNER F. H. and KEEY R. B., Chem. Engng Sci. 1958 9 119. SRAFTON R. W., Chem. Engng Sci. 1963 18 457. MENDELSOHN H. and YERAZUNIS S.,A.I.Ch.E. Jll965 11834. RANZ W. E. and DICKSON P. F.. Ind. Enww Chem. Fundls 1965 4 345. ECKERT E. R. G. and DRAKE R.‘M. Zntroud;ction to the Trnnsfer of Heut und Muss. McGraw-HiII 1950. GASKA C., Ph.D. Thesis, University of London, 1966. NIENOW A. W., Br. chem. Engng 1965 10 327. MULLIN J. W. and COOK T. P., J. uppl. Chem. 1965 15 145. Résumé-Les données de transfert de masse, pour la dissolution de sphères uniques d’électrolytes modérément solubles, peuvent être mises en corrélation par une équation du genre:
sh = $4 Re.#‘2(%)1’3. On montre que la constante de corrélation + est presque identique pour les quatres systèmes variés électrolyte-eau étudiés et sa valeur est presque égale à la valeur obtenue par Rowe et al. pour le transfert de chaleur et le transfert de masse du flux de faible masse, à condition que la force d’entraînement soit definie par la relation
Tout autre choix de Ia force d’entraînement conduirait, pour ces systèmes, à des valeurs de 4 diverses pour chaque système, toutes différentes d’une façon marquée de celles obtenues pour le flux de masse faible. Zusammenfassung- Für die Auflösung von Einzelkugeln Stoffübertragungswerte durch eine Gleichung des Typs
mässig Iöslicher
Elektrolyte
können
in eine Korrelation gebracht werden. Es wird gezeigt, dass die Korrelationskonstante $ für die untersuchten vier verschiedenen Elektrolyt-Wasser Systeme ungefähr gleich ist und circa dem durch Rowe et af. für Warme-und niedirge Stoffströmungübertragung erhahenen Wertentspricht, voraus-
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A. W. NIENOW.
R. UNAHABHOKHA
and J. W. MULLIN
gesetzt, dass die Triebkraft durch die Beziehung:
ausgedrückt wird. Jede andere Waal einer Triebkraft würde bei diesen Systemen zu verschiedenen 4 Werten fiir jedes System führen, die alle sehr verschieden von dem für niedrige Stoffströmung erahltenen sind.
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