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Mass transfer in a rotating cylinder cell—II. turbulent flow
Ekctrochimica
Acta.
MASS
1972. Vol.
17, pp.
TRANSFER
1 I29 to 1137.
Persam
cm Prws.
Printed
in No-
Ireland
IN A ROTATING CYLINDER TURBULENT FLOW*
CELL-II.
D. R. GABE and D. J. ROBINS~N~ Department of Metallurgy, University of Sheffield, Sheffield, Engiand Abstract-A model for turbulent fiow in a concentric cylinder cell. in which the inner cylinder rotates and the outer cylinder remains stationary, has been developed, based on the three-zone concept : a laminar boundary layer next to the rotating cylinder surface, a viscous transition layer and a main-stream turbulent flow. The over-a11 mass-transfer equation has been calculated by considering a turbulent diffusion coefficient enhanced by eddying effects and then solving Fick’s law for the concentration distribution in each zone. The mass-transfer equation, in dimensionless form, is St = const . Re-lIa . SC-~/~. R&urn&--Etude d&tai%e d’un schhma relatif au courant turbulent dans une cellule cylindrique concentrique, dans laquelle le cylindre inthrieur tourne et le cylindre extkrieur reste immobile. On suppose trois 26nes: une couche de jonction lamellaire pr& de la surface cylindrique toumante, une couche de transition visqueusc et un courant turbulent principal. L’Bquation globale de transfert de masse a 6tt 6tablie en considbrant un coefficient de diffusion turbulent exalt6 par des efFets de tourbillons et en r6solvant la loi de Fick relative A la r&partition de la concentration dans chaque z&e. L’equation de transfert de masse sous sa forme adimensionnelle est : St = const. Re-lfa - SC+~~ Zusammenfassung-Es wird ein Model1 fiir die turbulente Striimung in einer konzentrischen Zylinderzelle, bei weIcher der innere Zylinder rotiert und der lussere stillsteht, entwickelt. Es basiert auf dem Dreizonen-Model1 : Eine laminare Unterschicht nahe bei der rotierenden Zylinderoberfllche, eine viskose Uebergangsschicht und ein turbulenter Kern. Die Stofftransportgleichung wurde berechnet, indem man einen durch Wirbcleffekte vergrcsscrten turbulenten DiffusionskoefiGenten annahm und dann das Fick’sche Gesetz fir die Konzentrationsverteilung in jeder Zone l&te. Die Stofftransportgleichung in dimensionloser Form lautet dann: St = Konst. Rc”~SC-~~~ INTRODUCTION
the case of laminar flow, at very low relative flow velocities in a concentric cylinder cell, was considered and it was pointed out that the case of turbulent flow has found greatest application. Empirical correlations of mass-transfer data have been rnad$+ which lead to a relationship of the type
IN
PART
I1
St =
const. Re”Sc*
where St is the Stanton number (IcJu), Re is the Reynolds number (z&/y). SC is the Schmidt number (r/o). each being dimensionless. Such a result draws a close analogy with the Chilton-Colborn equation for momentum transfer and in the case of concentric cylinders the constant has been found to be O-0791, a ca --_H and b ca -_8. In fact values of a and b seem to vary in the ranges -0-30 to -0.33 and -0.60 to -0.67 respectively and may be sensitive to geometrical details of the cei1. Arvia and Carrozza3 were able to vary the diameters of the inner and outer cylinders and found a further dependence expressed as St =
0~079Re4’3Sc4~644(Ra/RI)4’70.
SubsequentlqP they compared their approach found comparabIe relationships.
with that for a plane plate surface8 and
* Manuscript received 16 November 1970; as amended, 3 May 1971. 7 Present address: Cominco Ltd., Trail, Canada. 9
1129
1130
D. R. GAEE and D. J. ROBINSON
A satisfactory agreement has been found for tbe complete mass-momentum-heattransfer analogy’-@ although in the case where an inner cylinder rotated in an axial flow of fluid the analogy failed lo~r because, while both surfaces were similar for momentum transfer, they were not for mass transfer. The analogy, however, should improve as the transition from laminar to turbulent flow regimes is completed. An exact mathematical treatment of turbulent transfer has not yet been possible owing to the difficulty of expressing and defining a true model. However, a semirigorous treatment is possible, which can predict the relationships found experimentally, based on the simple model of Prandtl’s “mixing-length” theory. “MIXING-LENGTH”
THEORY
A fluid is said to be stable if its flow behaviour is laminar or streamlined, but it becomes unstable if turbulent flow, in the form of swirling eddies, develops. The transition is defined in terms of Reynolds number Re which is itself dimensionless but depends upon the flow velocity, the density and viscosity of the fluid, and a characteristic length of the Bow channel. The critical value of Re depends upon ffow geometry and can vary from 108 for rotating concentric cylinders to 106 for a rotating disk. The general mathematical nature of flow stability has been found for only three casesI of which that by Taylorr2-x4 is particularly relevant. Taylor showed that when the inner cylinder rotates instability occurs when the Taylor number Tu is given bY Ta=I+-:]
=29-3,
where &
= w,R -
CR2 Y
-
RI) *
was also shown that instability should be marked by a set of three-dimensional laminar vortices Clling the annulus but that the criterion of stability ceased to be fully valid when (R2 - R,)/R, > Q and L < 20R,-this has been confirmed experimentallY*lo*u.l@ N o direct evidence of the behaviour of the vortices under increasing rotation has been reported but it seems likely that they must interact with one another, becoming more random in size until fully developed turbulence exists. Measurements of the turbulent velocity protie in the rotating cylinder cellI indicated that orZZr2was constant over 83-5 per cent of the annular gap indicating that the “Vorticity Transport” theory was correct away from the surfaces but that Prandtl’s “Momentum Transport” theory, which predicted a steady decrease in qRi2, would seem to be valid near the solid walls [over 10 per cent of the gap at the stationary surface and 6.5 per cent of the gap at the rotating surface (see Fig. l)]. In describing the physical behaviour of flowing fluids they are assumed to be incompressible and Newtonian in nature, so that the shear stress at any point in a laminar flow is du
It
7=ccdy’ where p is the coefficient of viscosity and du/dy is the velocity gradient. In turbulent
Mass transfer in a rotating cylinder cell-II.
Turbulent flow
1131
0
3
FIG.
motion
1. Velocity
an “apparent”
Radial
distance
R2
distribution between the inner rotating cylinder and the outer stationary cylinder in turbulent flow.
eddy viscosity
e is felt such that,
The value of E depends upon the fluid velocity and, clearly, implies that the friction drag at the liquid/solid interface is greater in turbulent than in laminar flow. The eddying gives rise to enhanced diffusion effects so the convective diffusion equation for turbulent flow [cfequation (1) of part I] may be written
However, unlike Ox, D turb probably varies with flow velocity, ie the degree of eddying, and so this equation may be valid only with this condition. The “Mixing-Length” assumes that E = Dtnrb and that theory of Prandtl” turbulence is essentially analogous to the random motion of gas molecules as described by the kinetic theory of gases. This predicts a logarithmic velocity distribution, which was found to exist to witbin a tite distance of the wall for flow in a tube. To overcome this deficiency two zones were postulated to exist-one of developed turbulence far from the wall and a second of laminar flow adjacent to the wall-but it is obvious that some transition must occur, and von Karmanr’ proposed the three-zone model in which the transition takes place in a viscous subIayer where molecular viscosity tends to damp out eddying effects. Levichrs has discussed the addition of a turbulent sublayer. The three-zone model has found wide acceptance and DiessIefl* has proposed values for each of the zones. In the laminar (boundary) zone no eddying occurs and E = 0. In the turbulent main stream flow E = nS . yu where n is a constant. In the viscous-damping transition layer
D. R. GABE and D. J.
1132 which,
when y is small,
approximates
to
z=-
n4yaua Y
A
THEORY
AT
ROBINSON
FOR TURBULENT A ROTATING
MASS CYLINDER
TRANSFER
The theory is based on the Prandtl-von Karman model of a three-zone electrolyte (Fig. 2). The method used is to derive an expression for Dturb, solve Fick’s law for the Cont. Boundary tayE’F
I
I
I
I
I ,
I
I
viscous
Fully
Sublayer
Developed Turbulence
I I I I
II
I I
/Fia.
2. Model
4 -
III
I
of turbulence damping near a solid wall showing a concentration distribution over three zones.
concentration distribution in each zone and show that the major resistance to mass transfer occurs in zone I where molecular effects prevail. The eddy viscosity parameter E, and hence Dtiurb, can be expressed as a function of a length and velocity parameter alone, but in the case of concentric cylinders a curvature term is also necessary (R, the inner cylinder radius), D turb
=
f(
v’,
1, RI)-
From Prandtl’s treatment we assume that the scale or size of the eddies is comparable with the annular gap so that I = y. Taylor20 has calculated that v’ =
B’u gz Y
1 (m” + [ O-057
l)m
(m” +
1)2 I
where d = (R, - R,), y is the distance such that y = 3 2 - . .) and B’ is a constant_ Near R,, y -+ d so that V’ zzz B”u g 1 Y(
Therefore,
we may define the function D turb
0 at R,, m is an index (=
21 w Bu. d>
more closely as =f(%y,
Rd.
mrY sin d ’ 1,2,
Mass transfer in a rotating cylinder cell-II.
Zone
Turbulent flow
1133
III
When R1 is large the flow gap is approximately a straight channel or pipe so that influence of R1 is irrelevant. This suggests that we may assume that two functions which may be distinguished contribute to Dturb such that D turb
u, -g(y,
=f(u>
&)-
Since DiessleP showed that E = FZ”~Ufor this zone, and this has the required dimensions (eg cma/s), it follows that the function g must be dimensionless, and it takes the form
( 1 a-.
YZ
Rl
The simplest case is clearly where a is a constant and x = 1; this will be assumed in the absence of more substantial evidence. The correct selection of g, however, must satisfy the two conditions (a) lim g = (b) limg =
1 as R, + 00, since the pipe expression aCy/R,) when R1 is finite. ---
I
FIG. 3.
is then valid,
Transformation Coordinates
Modified arctangent curve as a solution for the diffusion equation of zone III.
These conditions that
may be adequately
satisfied by the arc tangent curve (Fig. 3) such
Dturb = n2yu arc tan This expression is valid for zone III where the turbulent motion is undamped and complete mixing occurs. Thus there is no concentration gradient and clIr = cb_ Zone 11 In the transition zone viscosity tends to damp out the eddies but it is nevertheless assumed that the kinematic viscosity does not influence the term containing R,. The
D. R. GABE and D.
1134
equation for E given by DiessleP &
is
Dbulb
=
J. ROBINSON
=
2yu . F
n"yu
( >’ -
Y
where the function I; has to be dimensionless. The simplest form possible for P is n’yu Y
which when differentiated becomes d[
’
nayu Y 1-
Two boundary conditions are imposed to allow a smooth transition from one layer to another : limP=
(9
I asy +
8,,
limdF=Oasy-+6,.
(ii)
To account for (ii) the factor (1 -
r;) is introduced such that
Y1
dF = d nZyu (1 -I;).
1
Then by separating variables and integrating between appropriate limits the expression becomes
s
= dF 0 u---n
and
=[‘VU”d
[PI+]
F=l-exp[+].
When y is large, the exponential term is small and F--t I. When y is small, the exponential term can be expanded in an infEte series and by retaining the first two terms only n'yu .
F=1__1++=_ Y
Y
The complete solution for a curved surface is therefore D tnrb=
n”y~[l
-
exp (-
y)]
arc tan (CZg).
When RI is Unite and y is smaI1this expression may be reduced to D
-tnrb -
bySua YJG
'
where b is a constant (= Ural). This expression may now be introduced into the Fick’s law equation with the assumption that u = u,, ie the net flow velocity is
Mass transferin a rotatingcylinderceil-II.
Turbulent flow
unaffected by the transition damping effect, which only affects the eddying. grating between the appropriate limits of zone II we have
C
II
1135
Inte-
+E+jj-$],
=cb
where 8, is the width of the Prandtl hydrodynamic boundary layer (see Fig. 2). Zone I In zone I, y < 8 (the Nernst diffusion-layer thickness) and Dtnrb < D, so that from Fick’s law c= dc = L ‘dy s 01) D so and
=c.+L D ”
C=
Wheny
=
8,c1 = P,
we may combine the above equations, whence J _ -
D(c, 8 -A
cJ ’
where
The order of magnitude of each term is 1 = lo”, 4R,=ula = Re’
YB
D/y
=
l/Se
=
a0 % lOma and
IO--=, 6 =
lo-“,
so
[-$-$I = lo*-
lo*%
104.
The value of R, is about unity, so if b < O-1 A will be much smaller than 8 and can be ignored, yielding the simple equation J=
NC,
8
4 .
Since D varies with Dtnrb inversely as y varies with 8, and D = Dtnrb when y = 6,
D. R. GABE and D. J. ROBINSON
1136
the equation
for Dtnrb derived in considering a=
DyRl L bu,=
In terms of a cd (i = J/zF’) we then have i = Making
the substitution
bl/3zFU;/3D2/3
of cu,R, = Sr =
zone II can give a value for 8,
1
‘I3 -
Y-1’3R;1’3(cb
-
c,).
ul and reverting to dimensionless
numbers,
(2b)lis Re-11s SC-~“_ DISCUSSION
The mass-transfer equation derived has a very similar form to that discussed in the introduction and deduced empirically in several investigations. Previous reports have given the index exponent for Re as -0.30,2m3 -0*31,5 -O*333B and -O-4O,4 while that for SC was -O*59,6 -O.6442*3*7 and -O*666.4 Hence the general agreement between theory and experiment is quite good, although the most complete experimental correlations favour -0.30 and -0-644 respectively. Furthermore, there is good agreement with the conclusions of Kambara et aZ,21 who showed that the limiting current was proportional to 0~~‘~and later22 produced a fuller relationship for a wire of radius r and length L,
The constant (2b)l’3 must be determined experimentally and for the concentric cell in which the inner electrode rotates and is under examination; its value is 0.079.2.3*5 The argument used clearly involves several approximations and assumes a relatively simple model amenable to analytical treatment. The treatment necessarily yieIds indices that are ratios of whole numbers but clearly can be affected by modified assumptions. For example, if in deriving the dimensionless form of function g for zone III we take a value of x other than 1, a change in index will be necessary throughout: in fact if x = 1.0185 the indices for Re and SC will be -0-30 and -0.65 respectively. It is clear, therefore, that despite these inherent shortcomings this analysis yields an equation of the correct form and any subsequent treatment must be based on a refined model. authors gratefully acknowledge tiancial support from the Athlone Fellowship Committee, the Science Research Council and the British Steel Corporation (Strip Mills Division), and facilities made available by Professors A. G. Quarrel1 and G. W. Greenwood_
Acknowledgemen&--The
NOMENCLATURE
RI, Rz d(=R2w U Y c cb es
inner
and outer
cylinder
radii
annular gap angular velocity of rotation peripheral velocity co-ordinate distance across gap concentration of ions in soIution bulk concentration electrode surface concentration
Mass
transfer
in a rotating
cylinder
cell-II.
Turbulent
x Re SC
kinematic viscosity coefficient of viscosity fluid density apparent eddy viscosity diffusion coefficient apparent turbulent diffusion coefficient mass transfer flux current density thickness of Nernst diffusion layer thickness of Prandtl hydrodynamic boundary shear stress velocity distribution turbulent velocity component in JJdirection function of. . , constants integer 1,2,3. . . variable Reynolds number, ud/ y Schmidt number, D/y
St
Stanton
Ta
Taylor
Y P P &
D
D turb J
i
a 4J 7 ii
V'
f,g,F a,b,B,B’ m,n
number, number,
k, -
U
Re
=
(ii:
iL zFc,u
flow
1137
layer
D = Bu
_t ::Y2
REFERENCES Acta 17,112l (1972). 1. D. R. GABE and D. J. ROBINSON, Efectrochim. Sot. 101,306 (1954). 2. M. EISENBERG, C. W. TOBIAS and C. R. WILKE, J. efectrochem. 3. A. J. &tVfA and J. S. W. CARROZZA, Electrochim. Acta 7, 65 (1962). 4. A. J. hVfA, J. S. W. CARROZZA and S. L. MARCHIANO, Eiectrochim. Acta 9, 1483 (1964). D. J. ROBINSON and D. R. GABE, Trans. Inst. Met. Finishing 48, 35 (1970); 49, 17 (1971). 2: G. WRANGLEN and 0. NI=N, ECectrochim. Acta 7, 121 (1962). 7. T. K. SHERWOOD and J. M. RYAN, Chem. Engng. Sci. 11, 81 (1959). 8. T. THEODORSEN and A_ REGIER, N.A.C.A. Report 793 (1944). 9. I. CORNET and R. KAPPESSER. Trans. Inst. them. Engng. 47, 194 (1969). 10. J. R. FLOWER, N. MACLEOD and A, P. SHAHBENDARIAN, Chem. Ellgng. Sci. 24,637 (1969). 11. J. R. FLQWER and N. MAC-D, Chem. Engng. Sci. 24,651 (1969). Stability. Cambridge University Press (1955). 12. C. C. LIN, The Theory of Hydrodynamic 13. G. I. TAYUIR, Phi!. Trans R. Sot. A223, 289 (1923). G. I. TAY~R, Proc. R. Sot. A157,546 (1936). ::: G. I. TA~OR. Proc. R. Sot. A151,494 (1935). Theory, ed. W. F. 16. L. PRANDXX, 7&e ll4echanics of Viscous Fluids, Vol. 3 of Aerodynamic DURAND. p. 34. Springer, Berlin (1935). TH.VON KARhlAN, J. aero. Sci. 1, 1 (1934). ::: V. G. LEVICH, Physiochemical Prentice-Hall, Englewood Cliffs, New Jersey Hydrodynamics. (1962). R. G. DESSLER, N.A.C.A. Report 1210 (1955). ::: G. I. TAYLOR, Proc. R. Sac. A135,678 (1932). 21. T. &+MBARA, T. TSUICAMOXI and I. TACHI, J. elecfrochem. Sot., Japan 18,356, 386 (1950). 22. T. KAMBARA, T. TSUKAMO~ and I. TACHI, J. electrochem. Sot., Japan 19,297 (1951).
Acta.
MASS
1972. Vol.
17, pp.
TRANSFER
1 I29 to 1137.
Persam
cm Prws.
Printed
in No-
Ireland
IN A ROTATING CYLINDER TURBULENT FLOW*
CELL-II.
D. R. GABE and D. J. ROBINS~N~ Department of Metallurgy, University of Sheffield, Sheffield, Engiand Abstract-A model for turbulent fiow in a concentric cylinder cell. in which the inner cylinder rotates and the outer cylinder remains stationary, has been developed, based on the three-zone concept : a laminar boundary layer next to the rotating cylinder surface, a viscous transition layer and a main-stream turbulent flow. The over-a11 mass-transfer equation has been calculated by considering a turbulent diffusion coefficient enhanced by eddying effects and then solving Fick’s law for the concentration distribution in each zone. The mass-transfer equation, in dimensionless form, is St = const . Re-lIa . SC-~/~. R&urn&--Etude d&tai%e d’un schhma relatif au courant turbulent dans une cellule cylindrique concentrique, dans laquelle le cylindre inthrieur tourne et le cylindre extkrieur reste immobile. On suppose trois 26nes: une couche de jonction lamellaire pr& de la surface cylindrique toumante, une couche de transition visqueusc et un courant turbulent principal. L’Bquation globale de transfert de masse a 6tt 6tablie en considbrant un coefficient de diffusion turbulent exalt6 par des efFets de tourbillons et en r6solvant la loi de Fick relative A la r&partition de la concentration dans chaque z&e. L’equation de transfert de masse sous sa forme adimensionnelle est : St = const. Re-lfa - SC+~~ Zusammenfassung-Es wird ein Model1 fiir die turbulente Striimung in einer konzentrischen Zylinderzelle, bei weIcher der innere Zylinder rotiert und der lussere stillsteht, entwickelt. Es basiert auf dem Dreizonen-Model1 : Eine laminare Unterschicht nahe bei der rotierenden Zylinderoberfllche, eine viskose Uebergangsschicht und ein turbulenter Kern. Die Stofftransportgleichung wurde berechnet, indem man einen durch Wirbcleffekte vergrcsscrten turbulenten DiffusionskoefiGenten annahm und dann das Fick’sche Gesetz fir die Konzentrationsverteilung in jeder Zone l&te. Die Stofftransportgleichung in dimensionloser Form lautet dann: St = Konst. Rc”~SC-~~~ INTRODUCTION
the case of laminar flow, at very low relative flow velocities in a concentric cylinder cell, was considered and it was pointed out that the case of turbulent flow has found greatest application. Empirical correlations of mass-transfer data have been rnad$+ which lead to a relationship of the type
IN
PART
I1
St =
const. Re”Sc*
where St is the Stanton number (IcJu), Re is the Reynolds number (z&/y). SC is the Schmidt number (r/o). each being dimensionless. Such a result draws a close analogy with the Chilton-Colborn equation for momentum transfer and in the case of concentric cylinders the constant has been found to be O-0791, a ca --_H and b ca -_8. In fact values of a and b seem to vary in the ranges -0-30 to -0.33 and -0.60 to -0.67 respectively and may be sensitive to geometrical details of the cei1. Arvia and Carrozza3 were able to vary the diameters of the inner and outer cylinders and found a further dependence expressed as St =
0~079Re4’3Sc4~644(Ra/RI)4’70.
SubsequentlqP they compared their approach found comparabIe relationships.
with that for a plane plate surface8 and
* Manuscript received 16 November 1970; as amended, 3 May 1971. 7 Present address: Cominco Ltd., Trail, Canada. 9
1129
1130
D. R. GAEE and D. J. ROBINSON
A satisfactory agreement has been found for tbe complete mass-momentum-heattransfer analogy’-@ although in the case where an inner cylinder rotated in an axial flow of fluid the analogy failed lo~r because, while both surfaces were similar for momentum transfer, they were not for mass transfer. The analogy, however, should improve as the transition from laminar to turbulent flow regimes is completed. An exact mathematical treatment of turbulent transfer has not yet been possible owing to the difficulty of expressing and defining a true model. However, a semirigorous treatment is possible, which can predict the relationships found experimentally, based on the simple model of Prandtl’s “mixing-length” theory. “MIXING-LENGTH”
THEORY
A fluid is said to be stable if its flow behaviour is laminar or streamlined, but it becomes unstable if turbulent flow, in the form of swirling eddies, develops. The transition is defined in terms of Reynolds number Re which is itself dimensionless but depends upon the flow velocity, the density and viscosity of the fluid, and a characteristic length of the Bow channel. The critical value of Re depends upon ffow geometry and can vary from 108 for rotating concentric cylinders to 106 for a rotating disk. The general mathematical nature of flow stability has been found for only three casesI of which that by Taylorr2-x4 is particularly relevant. Taylor showed that when the inner cylinder rotates instability occurs when the Taylor number Tu is given bY Ta=I+-:]
=29-3,
where &
= w,R -
CR2 Y
-
RI) *
was also shown that instability should be marked by a set of three-dimensional laminar vortices Clling the annulus but that the criterion of stability ceased to be fully valid when (R2 - R,)/R, > Q and L < 20R,-this has been confirmed experimentallY*lo*u.l@ N o direct evidence of the behaviour of the vortices under increasing rotation has been reported but it seems likely that they must interact with one another, becoming more random in size until fully developed turbulence exists. Measurements of the turbulent velocity protie in the rotating cylinder cellI indicated that orZZr2was constant over 83-5 per cent of the annular gap indicating that the “Vorticity Transport” theory was correct away from the surfaces but that Prandtl’s “Momentum Transport” theory, which predicted a steady decrease in qRi2, would seem to be valid near the solid walls [over 10 per cent of the gap at the stationary surface and 6.5 per cent of the gap at the rotating surface (see Fig. l)]. In describing the physical behaviour of flowing fluids they are assumed to be incompressible and Newtonian in nature, so that the shear stress at any point in a laminar flow is du
It
7=ccdy’ where p is the coefficient of viscosity and du/dy is the velocity gradient. In turbulent
Mass transfer in a rotating cylinder cell-II.
Turbulent flow
1131
0
3
FIG.
motion
1. Velocity
an “apparent”
Radial
distance
R2
distribution between the inner rotating cylinder and the outer stationary cylinder in turbulent flow.
eddy viscosity
e is felt such that,
The value of E depends upon the fluid velocity and, clearly, implies that the friction drag at the liquid/solid interface is greater in turbulent than in laminar flow. The eddying gives rise to enhanced diffusion effects so the convective diffusion equation for turbulent flow [cfequation (1) of part I] may be written
However, unlike Ox, D turb probably varies with flow velocity, ie the degree of eddying, and so this equation may be valid only with this condition. The “Mixing-Length” assumes that E = Dtnrb and that theory of Prandtl” turbulence is essentially analogous to the random motion of gas molecules as described by the kinetic theory of gases. This predicts a logarithmic velocity distribution, which was found to exist to witbin a tite distance of the wall for flow in a tube. To overcome this deficiency two zones were postulated to exist-one of developed turbulence far from the wall and a second of laminar flow adjacent to the wall-but it is obvious that some transition must occur, and von Karmanr’ proposed the three-zone model in which the transition takes place in a viscous subIayer where molecular viscosity tends to damp out eddying effects. Levichrs has discussed the addition of a turbulent sublayer. The three-zone model has found wide acceptance and DiessIefl* has proposed values for each of the zones. In the laminar (boundary) zone no eddying occurs and E = 0. In the turbulent main stream flow E = nS . yu where n is a constant. In the viscous-damping transition layer
D. R. GABE and D. J.
1132 which,
when y is small,
approximates
to
z=-
n4yaua Y
A
THEORY
AT
ROBINSON
FOR TURBULENT A ROTATING
MASS CYLINDER
TRANSFER
The theory is based on the Prandtl-von Karman model of a three-zone electrolyte (Fig. 2). The method used is to derive an expression for Dturb, solve Fick’s law for the Cont. Boundary tayE’F
I
I
I
I
I ,
I
I
viscous
Fully
Sublayer
Developed Turbulence
I I I I
II
I I
/Fia.
2. Model
4 -
III
I
of turbulence damping near a solid wall showing a concentration distribution over three zones.
concentration distribution in each zone and show that the major resistance to mass transfer occurs in zone I where molecular effects prevail. The eddy viscosity parameter E, and hence Dtiurb, can be expressed as a function of a length and velocity parameter alone, but in the case of concentric cylinders a curvature term is also necessary (R, the inner cylinder radius), D turb
=
f(
v’,
1, RI)-
From Prandtl’s treatment we assume that the scale or size of the eddies is comparable with the annular gap so that I = y. Taylor20 has calculated that v’ =
B’u gz Y
1 (m” + [ O-057
l)m
(m” +
1)2 I
where d = (R, - R,), y is the distance such that y = 3 2 - . .) and B’ is a constant_ Near R,, y -+ d so that V’ zzz B”u g 1 Y(
Therefore,
we may define the function D turb
0 at R,, m is an index (=
21 w Bu. d>
more closely as =f(%y,
Rd.
mrY sin d ’ 1,2,
Mass transfer in a rotating cylinder cell-II.
Zone
Turbulent flow
1133
III
When R1 is large the flow gap is approximately a straight channel or pipe so that influence of R1 is irrelevant. This suggests that we may assume that two functions which may be distinguished contribute to Dturb such that D turb
u, -g(y,
=f(u>
&)-
Since DiessleP showed that E = FZ”~Ufor this zone, and this has the required dimensions (eg cma/s), it follows that the function g must be dimensionless, and it takes the form
( 1 a-.
YZ
Rl
The simplest case is clearly where a is a constant and x = 1; this will be assumed in the absence of more substantial evidence. The correct selection of g, however, must satisfy the two conditions (a) lim g = (b) limg =
1 as R, + 00, since the pipe expression aCy/R,) when R1 is finite. ---
I
FIG. 3.
is then valid,
Transformation Coordinates
Modified arctangent curve as a solution for the diffusion equation of zone III.
These conditions that
may be adequately
satisfied by the arc tangent curve (Fig. 3) such
Dturb = n2yu arc tan This expression is valid for zone III where the turbulent motion is undamped and complete mixing occurs. Thus there is no concentration gradient and clIr = cb_ Zone 11 In the transition zone viscosity tends to damp out the eddies but it is nevertheless assumed that the kinematic viscosity does not influence the term containing R,. The
D. R. GABE and D.
1134
equation for E given by DiessleP &
is
Dbulb
=
J. ROBINSON
=
2yu . F
n"yu
( >’ -
Y
where the function I; has to be dimensionless. The simplest form possible for P is n’yu Y
which when differentiated becomes d[
’
nayu Y 1-
Two boundary conditions are imposed to allow a smooth transition from one layer to another : limP=
(9
I asy +
8,,
limdF=Oasy-+6,.
(ii)
To account for (ii) the factor (1 -
r;) is introduced such that
Y1
dF = d nZyu (1 -I;).
1
Then by separating variables and integrating between appropriate limits the expression becomes
s
= dF 0 u---n
and
=[‘VU”d
[PI+]
F=l-exp[+].
When y is large, the exponential term is small and F--t I. When y is small, the exponential term can be expanded in an infEte series and by retaining the first two terms only n'yu .
F=1__1++=_ Y
Y
The complete solution for a curved surface is therefore D tnrb=
n”y~[l
-
exp (-
y)]
arc tan (CZg).
When RI is Unite and y is smaI1this expression may be reduced to D
-tnrb -
bySua YJG
'
where b is a constant (= Ural). This expression may now be introduced into the Fick’s law equation with the assumption that u = u,, ie the net flow velocity is
Mass transferin a rotatingcylinderceil-II.
Turbulent flow
unaffected by the transition damping effect, which only affects the eddying. grating between the appropriate limits of zone II we have
C
II
1135
Inte-
+E+jj-$],
=cb
where 8, is the width of the Prandtl hydrodynamic boundary layer (see Fig. 2). Zone I In zone I, y < 8 (the Nernst diffusion-layer thickness) and Dtnrb < D, so that from Fick’s law c= dc = L ‘dy s 01) D so and
=c.+L D ”
C=
Wheny
=
8,c1 = P,
we may combine the above equations, whence J _ -
D(c, 8 -A
cJ ’
where
The order of magnitude of each term is 1 = lo”, 4R,=ula = Re’
YB
D/y
=
l/Se
=
a0 % lOma and
IO--=, 6 =
lo-“,
so
[-$-$I = lo*-
lo*%
104.
The value of R, is about unity, so if b < O-1 A will be much smaller than 8 and can be ignored, yielding the simple equation J=
NC,
8
4 .
Since D varies with Dtnrb inversely as y varies with 8, and D = Dtnrb when y = 6,
D. R. GABE and D. J. ROBINSON
1136
the equation
for Dtnrb derived in considering a=
DyRl L bu,=
In terms of a cd (i = J/zF’) we then have i = Making
the substitution
bl/3zFU;/3D2/3
of cu,R, = Sr =
zone II can give a value for 8,
1
‘I3 -
Y-1’3R;1’3(cb
-
c,).
ul and reverting to dimensionless
numbers,
(2b)lis Re-11s SC-~“_ DISCUSSION
The mass-transfer equation derived has a very similar form to that discussed in the introduction and deduced empirically in several investigations. Previous reports have given the index exponent for Re as -0.30,2m3 -0*31,5 -O*333B and -O-4O,4 while that for SC was -O*59,6 -O.6442*3*7 and -O*666.4 Hence the general agreement between theory and experiment is quite good, although the most complete experimental correlations favour -0.30 and -0-644 respectively. Furthermore, there is good agreement with the conclusions of Kambara et aZ,21 who showed that the limiting current was proportional to 0~~‘~and later22 produced a fuller relationship for a wire of radius r and length L,
The constant (2b)l’3 must be determined experimentally and for the concentric cell in which the inner electrode rotates and is under examination; its value is 0.079.2.3*5 The argument used clearly involves several approximations and assumes a relatively simple model amenable to analytical treatment. The treatment necessarily yieIds indices that are ratios of whole numbers but clearly can be affected by modified assumptions. For example, if in deriving the dimensionless form of function g for zone III we take a value of x other than 1, a change in index will be necessary throughout: in fact if x = 1.0185 the indices for Re and SC will be -0-30 and -0.65 respectively. It is clear, therefore, that despite these inherent shortcomings this analysis yields an equation of the correct form and any subsequent treatment must be based on a refined model. authors gratefully acknowledge tiancial support from the Athlone Fellowship Committee, the Science Research Council and the British Steel Corporation (Strip Mills Division), and facilities made available by Professors A. G. Quarrel1 and G. W. Greenwood_
Acknowledgemen&--The
NOMENCLATURE
RI, Rz d(=R2w U Y c cb es
inner
and outer
cylinder
radii
annular gap angular velocity of rotation peripheral velocity co-ordinate distance across gap concentration of ions in soIution bulk concentration electrode surface concentration
Mass
transfer
in a rotating
cylinder
cell-II.
Turbulent
x Re SC
kinematic viscosity coefficient of viscosity fluid density apparent eddy viscosity diffusion coefficient apparent turbulent diffusion coefficient mass transfer flux current density thickness of Nernst diffusion layer thickness of Prandtl hydrodynamic boundary shear stress velocity distribution turbulent velocity component in JJdirection function of. . , constants integer 1,2,3. . . variable Reynolds number, ud/ y Schmidt number, D/y
St
Stanton
Ta
Taylor
Y P P &
D
D turb J
i
a 4J 7 ii
V'
f,g,F a,b,B,B’ m,n
number, number,
k, -
U
Re
=
(ii:
iL zFc,u
flow
1137
layer
D = Bu
_t ::Y2
REFERENCES Acta 17,112l (1972). 1. D. R. GABE and D. J. ROBINSON, Efectrochim. Sot. 101,306 (1954). 2. M. EISENBERG, C. W. TOBIAS and C. R. WILKE, J. efectrochem. 3. A. J. &tVfA and J. S. W. CARROZZA, Electrochim. Acta 7, 65 (1962). 4. A. J. hVfA, J. S. W. CARROZZA and S. L. MARCHIANO, Eiectrochim. Acta 9, 1483 (1964). D. J. ROBINSON and D. R. GABE, Trans. Inst. Met. Finishing 48, 35 (1970); 49, 17 (1971). 2: G. WRANGLEN and 0. NI=N, ECectrochim. Acta 7, 121 (1962). 7. T. K. SHERWOOD and J. M. RYAN, Chem. Engng. Sci. 11, 81 (1959). 8. T. THEODORSEN and A_ REGIER, N.A.C.A. Report 793 (1944). 9. I. CORNET and R. KAPPESSER. Trans. Inst. them. Engng. 47, 194 (1969). 10. J. R. FLOWER, N. MACLEOD and A, P. SHAHBENDARIAN, Chem. Ellgng. Sci. 24,637 (1969). 11. J. R. FLQWER and N. MAC-D, Chem. Engng. Sci. 24,651 (1969). Stability. Cambridge University Press (1955). 12. C. C. LIN, The Theory of Hydrodynamic 13. G. I. TAYUIR, Phi!. Trans R. Sot. A223, 289 (1923). G. I. TAY~R, Proc. R. Sot. A157,546 (1936). ::: G. I. TA~OR. Proc. R. Sot. A151,494 (1935). Theory, ed. W. F. 16. L. PRANDXX, 7&e ll4echanics of Viscous Fluids, Vol. 3 of Aerodynamic DURAND. p. 34. Springer, Berlin (1935). TH.VON KARhlAN, J. aero. Sci. 1, 1 (1934). ::: V. G. LEVICH, Physiochemical Prentice-Hall, Englewood Cliffs, New Jersey Hydrodynamics. (1962). R. G. DESSLER, N.A.C.A. Report 1210 (1955). ::: G. I. TAYLOR, Proc. R. Sac. A135,678 (1932). 21. T. &+MBARA, T. TSUICAMOXI and I. TACHI, J. elecfrochem. Sot., Japan 18,356, 386 (1950). 22. T. KAMBARA, T. TSUKAMO~ and I. TACHI, J. electrochem. Sot., Japan 19,297 (1951).