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Prediction of mass transfer from spheres and cylinders in forced convection
Chemical Engineering Science, 1963, Vol. 18, pp. 457466.
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Prediction of mass transfer from spheres and cylinders in forced convection R. W. GRAFTON Boots Pure Drug Co. Ltd., Station Street, Nottingham (Receiued in revised form 9 January 1963) Abstract--An expression is obtained relating mass transfer, due to forced convection, to hydrodynamic conditions over the entire surface of a sphere. In the case of the forward stagnation point no experimental data are required for the hydrodynamic conditions. The relationship may also be applied to circular cylinders in cross flow. Up to the point of separation of forward flow anestablishedmethod,employingpressuredistribution data, is suggested for the estimation of hydrodynamic conditions, whilst a method is developed for estimating the conditions over the area contined by the wake. By simple analogy predictions may be made for heat transfer which are in good agreement with experimental data. In the case of the forward stagnation point of a sphere(orcircularcylinderincrossflow) thenormal dimensionless forced convection transfer groups are shown to be independent of Reynolds number but the groups for the over-all or partial surfaces are shown to be dependent on Reynolds number, contrary to much popular opinion. No form of this dependency is suggested on account of its complexity but the observation explains the reason for various investigators obtaining differentconstants, for different ranges of Reynolds number, in the usual dimensionless expressions.
THE idea that heat and mass transfer from interfaces are partly or wholly influenced by fluid-dynamic conditions is a common one and several workers have sought means of predicting transfer rates from a knowledge of such data, but many of the analyses are limited in application and often rather tedious to apply. The success of such a proposition depends on obtaining a suitable description of the hydrodynamic conditions at the interface and using the data in relating expressions for transfer rates. The problem is a difficult one, for the major portions of the transfer and fluid resistances occur very close to the interfaces-these zones commonly being referred to as resistance films. Apart from providing means of prediction, solutions of the problem can indicate controlling factors and assist in interpreting experimental data. For chemical engineering unit operations, a case of particular interest is that of transfer from a spherical surface to a continuous fluid phase. The particular problem discussed in this analysis is mass transfer from solid spheres due to forced convection in a continuous incompressible fluid, where the
effect of forced convection is such that there is no disturbance as a result of natural convection and the magnitude of molecular diffusion outside the boundary layer is negligible. Comparisons are also made for circular cylinders in cross flow, particularly for cases of undetached wakes i.e. Reynolds numbers of <30, and interpretations are also made for heat transfer. In the analysis of hydrodynamic conditions it is assumed that boundary layers are laminar-i.e. boundary layer Reynolds numbers of < 6 x lo6 [I] ; this limitation is satisfied by most, if not all, conditions met in chemical engineering. HYDRODYNAMIC CONDITIONS OVER SURFACE UP TO SEPARATION
Several methods of analysis are available for conditions up to separation of forward flow [2-61. More recent publications on the solution of boundary layer equations are principally concerned with critical and supersonic velocities, especially for two dimensional aerofoils. The above cited solutions are either of limited scope or laborious in application. The method chosen is that of TOMOTIKA [7]. TOMOTIKA’S solution is obtained by assuming a 457
R.W.
GRAETON
fourth degree polynomial velocity distribution within the laminar boundary layer, as suggested by POHLHAUSEN[8], and substituting this in the momentum integral equation for a volume of revolution obtained by MILIKAN [9] from the Navier-Stokes and continuity equations. The resulting differential equation relates the boundary layer conditions of a sphere by two dimensionless parameters : z -
Ud2 and UV
I=:
%Z 0
For a sphere the value of 1 varies from 4.716 from the forward stagnation point to - 12 at separation. TOMOTIKAexplains how this relation may be used to obtain the boundary layer conditions from a knowledge of variation of the pressure normal to the surface. There are not many published data for the variation of the surface pressure of a sphere in forced convection except for Reynolds numbers of the order of 105--the critical range where the total drag coefficient suffers a sudden reduction owing to a change in the hydrodynamic conditions, e.g. FAGE [lo]. FLOCHSBART [l I] gives an expression for the variation of U/U, for the upstream half of the surface, viz : -
u
varies from 7.052 at the forward stagnation point to - 12 at separation. There are several published data describing experimental surface pressure distributions for circular cylinders in cross flow, e.g. THOM [14] gives data for a wide range of Reynolds numbers. In the same way as for a sphere, potential flow theory may be used to indicate conditions at the forward stagnation point ; this gives a general value of Z = 353 for all values of Reynolds number. HYDRODYNAMIC CONDITIONS OVER SURFACEDOWNSTREAM OF SEPARATION Whereas a number of solutions of boundary layer equations are applicable up to the point of separation of forward flow, no similar procedure available for the area contained by the wake has been found in the literature. The method of TOMOTIKAis unsuitable because the negligible variation of the surface pressure indicates by BERNOULLI’S theorem that 1 dU --= u,
0
de
Thus it is necessary to reconsider the whole analysis. The boundary conditions for the Pohlhausen laminar boundary layer velocity distribution,
;=‘o+‘(;)+“~(~)2+~~(~)3+~4(~
= 1.56 - 0.36402t13 - 0~02466W5
(1)
UO
where 8 is the angle in radians which a point on the surface subtends with the forward stagnation point, but this expression does not extend as far as separation and is obviously incomplete since it is independent of Reynolds number. Potential flow theory (see MILNE-THOMPSON [12]) gives rise to a distribution of pressure normal to the surface which approximates experimental data for the surface near the forward stagnation point; the approximation is such that it may be considered to depict actual conditions at this forward point. This indicates a general value of Z = 3-14 for all Reynolds numbers at the forward stagnation point. In the case of a circular cylinder in cross flow HOWARTH [13] obtained a solution for boundary layer hydrodynamic conditions similar to that for spheres given by TOMOTIKA, where for the cylinder A
for the case of no variation in the surface pressure, are y=o u = 0,
a% ayz-
-0
y=6
au a% ay=ay2=
u = u,
These give
458
a, = 0
ur =2 a, =o a3 =
2
u4 = 1
o
Prediction of mass transfer from spheres and cylinders in forced convection
Substituting equation (1) with these boundary conditions in MILIKAN'Smomentum integral equation for a volume of revolution
s
a *
ax, ru’
dy - U &
s
ru dy 0
au ) 6 y=o
=U~~~rdy-ro~(
(2)
FIG.
and noting that for a sphere of radius a, dx = a de and r. = a sin 8, a differential equation for the boundary layer thickness 6 is obtained: da2 de + 262 cot
1260 av 37 u
e = -.-
On integration, and for convenience writing in terms of the already mentioned dimensionless boundary layer parameter, Z = Uoa2/av equation (3) gives z = 34.054 $4(e,)
(4)
where
4m =
&
(se, - * sin 2e,) ,
and 8, is the angle position on surface subtends with the rear stagnation point. For small values of 8,, 4(e,) = 46,. Equation (4) does not fully describe the conditions over the whole of the surface confined by the wake since U, the velocity parallel to the surface at the edge of the boundary layer, must vanish at the rear stagnation and separation points, i.e. there is some variation in the pressure normal to the surface for which no allowance is made. However, U may be expressed in terms of U,, the free velocity within the wake, viz. U = U, cos $, where ~+9is the angle between the surface and the contour of circulation near the surface. Such data may be obtained from wake traces e.g. GARNER and GRAFTON [151; these may also be used to indicate values of U,. Thus in its complete form, equation (4) may be written as z = 34.054 u, 90 u, cos l/G
1. Contour of wake of sphere at
Re = 500.
U,/U, are not normally readily available but it has been found that the following suggestion, at least for a sphere Reynolds number of 500, affords a good approximation over the major portion of the surface in question except for near the rear stagnation and separation points. In the case of the rear stagnation point of a sphere the error, although apparently great, is not serious since only a relatively small area is involved. The method consists of ignoring the variation in +, i.e. assuming cos $ = 1.0, and choosing a value of U,/U, such that 34*054(U,/U,) +(e,) at separation is equal to the value of 2 at separation obtained for the forward flow; this is equivalent to noting that the boundary layer thickness has a unique value at separation. A known, or assumed, flow contour may then be used to give the variation in I,+,or the error due to neglecting the variation in +. For a circular cylinder in cross flow, equation (4a) should be replaced by
z = 34.054 3 8, u, cos + where 8, is measured in radians, noting that it is only applicable for case of undetached wakes, i.e. at low Reynolds numbers, < 30. MA,SSTRANSFERTHROUGH BOUNDARY LAYER (i) D@m’on equation FICK’S
law [ 161 gives, for constant diffusivity,
a? ad DBx"=x
(9
A typical wake contour, for use in obtaining $ for equation (4a), is shown in Fig. 1. Data for $ and 459
where c’ is concentration expressed as mass per unit volume. Putting pc = c’, where c is concentration expressed as mass per unit mass, the diffusion
R. W. GRAFTON
equation may be written, for negligible variation in density, 2 ac PD$=Pz For convenience, this expression is rearranged as follows:
Aty=O
C=C#-J, Aty = 6,
2 axatac PD&f=P~a,t=P'g
where u is the linear velocity of solution in the x direction. For an element of boundary layer, taking y normal to the surface and x parallel to the surface, such that &/az = 0 where z is a mutually perpendicular axis, the general transfer equation is pD[$+$]
constants depending on x. The condition at the limits are
a2c--m ac ay2-PD ay
(5)
ac a2C o c= Cd,, -=-F= ay ay
These limits give - 12a a; =6+g Where CI= c0 - c,, (i.e. the concentration difference across the boundary layer) and C = G,m/pD. (iii) Rate equation
=pu;+pv;
From equation (6)
At the surface, i.e. y = 0, noting that there is no slip along the surface which indicates that
(i3yzo =(~),=, =O
-12a
=-
(6 + 06, Since’the rate of transfer from the surface, m, is
the rate equation becomes
-pD
(!fay1y=. pD
(pu),,=,
is the mass of solution passing over unit area of surface in unit time, i.e. the rate of solution from the surface. Denoting this rate of mass transfer per unit area by m, the equation may be written as
i.e.
(5)
Or
12a
m=6,‘6+ 12a 5’6,5 r2 + 6c - 12a = 0
giving
(ii) Bowtdary layer concentration gradient In the same way as a fourth-degree polynomial velocity distribution has been assumed for the hydrodynamic boundary layer, a quartic concentration distribution is assumed :
where 6, is the mass-transfer boundary layer thickness, i.e. the value ofy at ac/ay = 0, and a’,,, etc., are 460
$+=-3+4(9+12ar) i*e* m = $
C-3 + J(9 + 12a)] ,
(7)
root of the solution of the quadratic is extraneous. Normally a is very small, i.e. 12~1$ 9, and the
Prediction of mass transfer from spheres and cylinders in forced convection
relationship given by equation (7) is not, in a convenient form, but may be simplified as am 1 pD __=-.-. aa
It should be noted that the average value of M for a portion of surface of a sphere is given by j’M sin 0 de/j sin 8 de
12
2 6,
J(9
+ 12a)
Thus for 12~1< 9 2pDa m=T
(8)
(iv) Mass transfer prediction relationship For the rate equation to be of any assistance a relation between mass and hydrodynamic boundary layer thickness is required. LEVICH [17] states that for the case of systems where the Prandtl number is much greater than 1.0 (i.e. liquids), the mass transfer boundary layer is equal to the hydrodynamic boundary layer thickness times the reciprocal of the cube root of the Schmidt number. This fact is used to write equation (8) in terms of the hydrodynamic boundary layer thickness, 2pDa m= 6 sc0.33
(8a)
Noting that m/pa is the mass transfer film coefficient expressed as rate of mass transfer per unit area per unit concentration difference (in mass per unit volume units), equation (8a) may be written in terms of the Sherwood number Sh/Sc”.33 = 2d/6 This may be rearranged so as to express the hydrodynamic boundary layer thickness in terms of the dimensionless parameter, 2 = U,S’/av viz: Sh
2d
m=-jj-
=
8Re -. z
Jc)
Denoting the dimensionless mass transfer group Sh/Sc”*33 Reoe5 by M, the general mass transfer prediction relationship may be expressed simply as M = J(8lz)
(9)
For the case of the forward stagnation point of a sphere it has already been pointed out that the value of Z is independent of Reynolds number and is equal to 3.14. Thus for this point M has the general value of 1.60. Similarly, in the case of a circular cylinder in cross flow the general value of M at the forward stagnation point is 1.51. In the case of other parts of the surface the value of M will depend on the form of 2 appertaining to the particular flow characteristics and will approach the value at the forward stagnation point for Re + 0. DISZUSSION
Equation (9) enables a prediction to be made of mass transfer from the entire surface of a sphere from a knowledge of the hydrodynamic conditions. These conditions for the area up to separation may be estimated solely from the distribution of the pressure normal to the surface and for the area confined by the wake, an approximate method of determination is given which requires a knowledge of the shape of the contour of circulation within the wake and of the velocity of circulation just outside the boundary layer. Where this latter information is not available, an approximate value for the circulation velocity may be obtained as described in the section dealing with the hydrodynamic conditions downstream of separation. In the case of the forward stagnation point no experimental data are necessary since the hydrodynamic conditions are adequately described by the potential flow theory. As an illustration, the prediction calculation is given for a sphere at a Reynolds number of 500 (see Tables 1 and 2). The wake circulation contour used is illustrated in Fig. 1 and the variations of 2 and M, i.e. the hydrodynamic and the forced convection transfer conditions, over the surface are shown in Figs. 2 and 3. Fig. 4 permits the graphical integration necessary in determining average values of M. The value of the circulation velocity within the wake was obtained as described above, see p. 3. The average calculated values of M and the
461
R. W. GRAFTON
Table 1. Sphere-Re = 500. Forward jlow area. (Solution of hydrodynamic conditions according to method of TOMOTIKA [7] from variation of surface pressure obtained by the writer)
0 10 20
3.15 3.20 3.4
1.60 1.58 1.54
o*OOO 0.174 0.342
0000 0.275 0.527
o*OOO 0.175 0.349
2 50 60 70 80 90 100 110 114
3.8 4.3 5.1 2:;
1.45 1.36 1.25 1.17 1.08 1.04 0.88 0.744 0.556 0.480
0.500 0.643 0.766 0.866 0.940 0.985 1.000 0.985 0.940 0.914
0.125 0.875 0.958 1.015 1.015 1.025 0.880 0.732 0.522 0.438
0.524 0.698 0,873 1.047 1.222 1.396 1.571 1.745 1.920 1.990
7.4 10.2 14.5 26.0 34.8
Reverse flow
&disionce
FIG. 2.
calculated values at the stagnation points are given in Table 3. It is interesting to note that prediction of minimum transfer occurs downstream of the prediction of separation; this agrees with experimental observations of GARNER and SUCKLING [18].
In
comparing the calculated values of M with experimental data, it must be noted that the prediction confines itself to the transfer associated with forced convection. This implies that the flow is sufficiently great to nullify any natural convection
from
SphereRe
forward
stagnotion
point,
= 500. Hydrodynamic over surface.
degrees
conditions
effects.
The usual form of experimental transfer relationships is
Sh =A +.MRe> where fl is independent of Reynolds number. Thus the mass transfer effected by forced convection is that indicated by fi and it is this that should be compared with calculated values of M provided the value off1 is negligible. The transfer indicated byf,, which is negligible when compared with fi at
Table 2. Sphere-Re = 500. Reversesow area (wake). (Solution according to method described)
eo
40
114 120 130 140 150 155 160 165 170 175 180
66 60 E 30 25 20 15 10 5 0
16O 31 22 0 0 z 57 68 !z
ws* 086 0.93 1.00 1.00 0.87 0.72 0.55 0.38 0.17 0.00
t Obtained by extrapolation. For reverse flow Z = 34.054 Uo/Ul+(0l)/cos
4(el) 0,410 0.391 0.250 0.187 0.150 0.118 0.088 0.058 0.029 0.000
z - lJ@ au 3;2 28.4 16.9 12.6 11.6 11.1 10.9 10.5 11.3 10*5t
M= 4/(8/z) 0.500 0.532 0.690 0.798 0.832 0.850 0.858 0.875 0.843 0.875
$I, where for Re = 500, UO/UI= l-98.
462
sine
Msin e
e=
0.866 0.766 0.643 0.500 0.423. 0.342 0.259 0.174 0.087 O+IOO
0.433 0408 0444 0.399 0.352 0291 0.222 0.152 0.073 0.000
1.990 2.094 2.269 2443 2.618 2.705 2.793 2.880 2.967 3.054 3.142
Prediction of mass transfer from spheres and cylinders in forced convection
8 distance
from
forwrd
stagnation
win?,
rod.
4. Sphere-Re = 500. Prediction of forced convection mass transfer from hydrodynamic conditions over FIG.
0
20 6’-distance
40
60 from
80
forward
100 stagnation
Surface. point,
degrees
Table 3. Sphere-Re = 500. Calculated forced convection transfer group
FIG. 3. SphereRe = 500. Prediction of forced convection mass transfer from hydrodynamic conditions over surface.
Reynolds numbers, is commonly thought to be that associated with molecular diffusion and to be dependent on the Grasshof and Schmidt numbers. GARNERand SUCKLING[ 181have reported experimentally determined mass transfer observations for the slow dissolution of solid spheres in water under forced convection conditions. These experimental and the calculated values of A4 are compared below in Table 4, together with the data of GARNERand GRAFTON[15], recalculated for dilIusity data predicted by OTHMERand THAKAR [19] instead of normally
Average for whole surface Average for forward flow area Average for reverse flow area (wake) Forward stagnation point Rear stagnation point
encountered
Table 4. Comparison of experimental
M=
Whole surface Forward flow area Reverse flow area Forward stagnation point Rear stagnation point
0.95 1.01 0.63 1.60 0*815
determined data by HIXSONand BAUM[20] on the suggestion of GARNERand SUCKLING[18]. It has already been pointed out that the calculated value of A4at the forward stagnation point of a sphere for forced convection is independent of Reynolds number and is equal to 160. FROSSLING[21] gives
and calculated values of A4 (Sphere-Re = 500) Sh/&O.SSReO*S
Experimental values of GARNERand SUCKLINQ[18]
Experimental values of GARNERand GRAFFON[15] (recalculated)
Calculated values
0.95 1.08 0.67 1.68 0.87
0.91 I.04 0.71 1-39 0.56
0.90 1.01 0.63 160 0.875
463
R. W. GRAFT~N
some data for the sublimation of naphthalene in air and his experimental values of M for forced convection are 1.87 at Re = 48 and I.90 at Re = 1,060, whilst VENEZIANet al. [22] obtained a value of l-2 for cooling spheres. The average calculated value of M for forced convection for the whole surface of a sphere of 0.90 at Re = 500 is compared with experimentally determined values in Table 5 below :
Table 6. Mass transfer from the forward stagnation point of Q in. D sphere of benzoic acid in water M = Sh/Sc”.s3Reo.5
Table 5. Over-all mass transfer for sphere
Experimenter GARNERand GIZAFTON [15]t GARNERand KEEY[23] GARNERand SUCKLING[18] FROSSLING [21] MCCIJNEand WEHELM[24] Calculated
Re range
Experimental value of M = Sh/ScO’a8ReO’5
500
0.91
250-750
0.94
500 2-1000
0.95 0.55
30-130 500
1.38 0.90
t Values recalculated for revised ditTusivity data (as above). The close agreement between the calculated values and the experimental values of GARNERand KEEYand GARNERand SUCWNG is accounted for by the relevance of the Reynolds number, the value of M being dependent on flow conditions, having a value approaching that of the forward stagnation point (i.e. M -+ 160) for Re -+ 0 (i.e. 2 --t 3.14 for all the surface).
The above discussion assumes that all effects due to natural convection are absent which is so at reasonably high Reynolds number, the actual value depending on the system. GARNER and HOFFMAN [25] carried out dissolution experiments with 3 in. diameter compressed benzoic acid spheres in water at low Reynolds numbers and the disturbance due to natural convection can be seen from Table 6 based on their reported data (they do not include for the value of the Schmidt number and a likely value for the system of Sc-“r3 = 0.09 has been included). The fact that the “down flow” values are lower than for “up flow” is rather surprising since it appears more likely that less disturbance at the forward stagnation point due to natural convection would accrue during “down flow” than during “up flow”. Similar comparisons may be made of calculated and experimental values for circular cylinders in
Remarks
Re UP flow
Down flow
3 10 :
3.1 2.1 1.4 1.8
3.2 1.8 1.1 1.3
70 100
1.4 1.4
1.2 1.3
Calculated value
1.6
1.6
Data of GARNERand HOFFMAN[25]
Unique value for forced convection effects for all Re
cross flow. In the case of the forward stagnation point the calculated value of M for forced convection is independent of Reynolds number and is equal to 1.5. Much of the data for circular cylinders are for heat transfer and for these cases M is taken as Nu/Pr o.33Re0.5. Table 7 gives some comparisons of calculated and experimental values. A calculation for the whole surface of a circular cylinder in cross flow, based on pressure data of THOM [6,14], has been made for a Reynolds number of 500. The average calculated value of M for forced convection is 0.9, whereas the heat transfer value given by HILPERT [32] for Reynolds numbers of O-l-4000 is 0.48 and that by MCADAMS [33], for Reynolds numbers of 50-10,000 is O-60, but the Table 7. Circular cyhnders in cross flow (forward stagnation point)
Experimenter
Re
Experimental value qf M = Sh/Sc”‘aaReO’5 = ~u/~O~S3&-$5
KRUB-IILIN[26] 39,500
1.4
SMALL[27]
39,600
1.45
DXrRLSCH [28] 39,600 WINDMGand 32,800
1.05 1.2
CHEYNEY
Heat transfer Heat transfer Absorption Sublimation
1291
DREWand RYAN 1301 KLEM [31] Calculated value
464
Remarks
32,800
1.0
39,600
0.95 1.5
Heat transfer Heat transfer For all Revalues
Prediction of mass transfer from spheresand cylindersin forced convection significance of Reynolds number range is stressed, as for the sphere ; this indicates a fall in the value of M for increasing Reynolds number.
k K m M
CONCLUSION
Nu
It is seen that there is remarkably good agreement between calculated and experimental values of the transfer group, M, associated with forced convection, even when applied to heat transfer. This indicates that forced convection heat and mass transfer are analogous when expressed in this way and that they are both directly dependent on hydrodynamic conditions. The agreement between the predicted and experimental values further indicates the applicability of the assumptions made to describe the hydrodynamic conditions. The arrangement of influencing factors obtained in this analysis of the problem suggests that experimental data, not subject to disturbing influences, should be related by such expressions. The analysis indicates that the standard practice of relating transfer rate to the square root of the Reynolds number is not fully comprehensive.
Pr r
rleo Scs Sh 2 U
u UO u, V X
Y Z
Acknowledgement-This
discussion is based on work carried out in the Department of Chemical Engineering of the University of Birmingham under the direction of Professor F. H. GARNER whose interest and encouragement is gratefully acknowledged.
NOTATION Radius of sphere or circular cylinder Dimensionless constants in boundary layer velocity profile equation a’s, a’4 Dimensionless constants in boundary layer concentration profile equation Concentration, mass of solute per unit c mass of solution Concentration, mass of solute per unit volume of solution co Concentration at surface car Concentration at outer limit of boundary layer d Diameter of sphere or circular cylinder D Diffusivity of solute in solution fi,fz Molecular and forced convection functions describing mass transfer h Heat transfer coefficient a
ao, al, m, mr a4 do,
B’l,
U’8,
465
z
Thermal conductivity Mass transfer fihu coefficient (dimensions of length/tune) Rate of mass transfer per unit area of surface Forced convection transfer group, = Sh/ScO.ssReO.5 = Nu/prO.ssReO’5 Nusselt number (dimensionless) = hd/k Prandtl number (dimensionless) = sp/k Distance co-ordinate, measured from axis of symmetry Value of r at surface Reynolds number (dimensionless) = dUoplr Specific heat Schmidt number (dimensionless) = v/D Sherwood number (dimensionless) = Kd/D Interval of time Velocity, parallel to surface, within boundary layer Value of u at outer limit of boundary layer Undisturbed velocity of fluid Free velocity within wake Velocity perpendicular from surface Distance co-ordinate, measured along surface Distance co-ordinate, measured perpendicular from surface Distance co-ordinate, mutually perpendicular to x and y Dimensionless parameter describing hydrodynamic boundary layer = UoP/av Concentration difference across mass transfer boundary layer = Co - Cd, Hydrodynamic boundary layer thickness = Q(aular) = 0 Mass transfer boundary layer thickness = QrVad-0 Angle point on sphere, or circular cylinder, subtends with forward stagnation point Angle point on sphere, or circular cylinder, subtends with rear stagnation point Absolute viscosity of fluid Kinematic viscositv of fluid Dimensionless parameter relating to mass transfer boundary layer = GmlpD Density of fluid Trigonometric function = (l/sin2 e,) (if?, - a sin 28,) Angle between wake contour near surface and the surface
R. W. GRAFTON REFERENCES
[l] TINMANR., Zng. Grau. 195264 L.6. 121 BLA~IUSH.. Z. Math. Phvs. 190856 1. i3j GREEN G., Phil. Msg. (skr. 7) 193112 1. [4] VON MIS= R., Z.A.M.M. 19297 425. [5] FALKNERV. M. and SKAN S. W., A.R.C.R. & M. 1930No. 1314. [6] THOM A., A.R.C.R. & M. 1928No. 1176. [7] TOMOTIKA S., A.R.C.R. & M. 1936No. 1678. [8] POHLIUUSENE., Z. Math. Mech. 19211252. [9] MJUKAN C. B., Amer. Sot. Mech. Engrs. Trans. 1932 54 A.P.M. 29. WI FAGE A., A.R.C.R. & M. 1937No. 1766.’ FLOCHSBARTO., Phys. Z. 1927 28 461. 5::; MILNE-THOMPSONI;. M., Theoretical Hydrodynamics (2nd Ed.) Macmillan, London (1949). u31 HOWARTHL., A.R.C.R. &M. 1934 No. 1632. u41 THOM A., A.R.C.R. & M. 1928 No. 1194. u51 GARNERF. H. and GRAPTONR. W., Proc. Roy. Sac. 1954 A224 64. WI FICK A., Phil. Msg. 1855 4 (10) 30. u71 LEVICH V. G., Dokl. Akad. Nauk 1951 78 1105. R. D., Amer. Inst. Chem. Engrs. J. 19584 114. WI GARNER F. H. and SUCKLINGS OTHMERD. C. and THAKARM. S., Zndastr. Engng. Chem. 1953 45 589. Hrxso~ A. W. and BALM S. J.. Zndustr. Elwns. Chem. 1942 34 120. Ki i21] FROSSLINON., Ger. Beit. Geophysik. 1938 g2170. [22] VENEZIANE., CRESPOM. J. and SAGE B. H., Amer. Inst. Chem. Engrs. J. 1962 8 383. [23] GARNER F. H. and KEEY R., Chem. Engng. Sci. 1958 9 119. [24] MCCUNE C. K. and WILHELMR. H., Zndustr. Engng. Chem. 1949 41 1124. 1251 F. H. HOFFMAN J. M.. Amer. Inst. Chem. Enars. J. 1960 6 (4). i26j GARNER K.RUZHILIN G. and H., J. Tech. Phys. 8’ 123 1938. . ,, 579. [U] [28] [29] [30] [31] [32] [33]
SMALL S., J. Phil. Msg. 1935 19251. tiI-IRIXH W., Forschungsarbeiten 1929 322 1. WINDING C. C. and CHENEYA. J., Zndustr. Engng. Chem. 1948 40 1087. DREW T. B. and RYAN P. W., Trans. Amer. Zndust. Chem. Engrs. 1931 29 174. KLEIN V., Arch. Warmwirt, 1934 15 150. HILPERTR., Forsch. Zng. 1933 4 215. MCADAMS W. H., Heat Transmission (2nd Ed.), McGraw-Hill, New York 1942.
- L’auteur obtient une expression reliant le transfer? de masse, du a la convection for&e, ReSUlU~ a l’etat hydrodynamique de la surface dune sphere. Dans le MS du point d’equilibre extreme aucune don&e n’est necessaire pour les conditions hydrodynamiques. Les relations peuvent &e egalement utilisees pour des cylindres pla&s dans un koulement transversal. Au-dessus du point de separation de l%coulement anterieur, l’autreu preconise une methode utilisant la distribution des pressions pour l’evaluation des conditions hydrodynamiques. Des evaluations par analogies dormant des r&hats conformes aux experiences peuvent &re effectut%s pour le transfert de chaleur. Dans le cas du point d’@rilibre extrgme pour une sphere (ou un cylindre place transversalement a l’&coulement), on montre que le groupe sans dimension relatif au transfert en convection for&e est independant du nombre de Reynolds; par contre les groupes relatifs aux surfaces partielles ou globales en dependent contrairement a une opinion repandue. Ces relations ne sont pas analysees en raison de leur complexite mais l’experience explique les raisons qui ont amen& de nombreux chercheurs a trouver des constantes differentes pour des nombres de Reynolds varies.
466
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Prediction of mass transfer from spheres and cylinders in forced convection R. W. GRAFTON Boots Pure Drug Co. Ltd., Station Street, Nottingham (Receiued in revised form 9 January 1963) Abstract--An expression is obtained relating mass transfer, due to forced convection, to hydrodynamic conditions over the entire surface of a sphere. In the case of the forward stagnation point no experimental data are required for the hydrodynamic conditions. The relationship may also be applied to circular cylinders in cross flow. Up to the point of separation of forward flow anestablishedmethod,employingpressuredistribution data, is suggested for the estimation of hydrodynamic conditions, whilst a method is developed for estimating the conditions over the area contined by the wake. By simple analogy predictions may be made for heat transfer which are in good agreement with experimental data. In the case of the forward stagnation point of a sphere(orcircularcylinderincrossflow) thenormal dimensionless forced convection transfer groups are shown to be independent of Reynolds number but the groups for the over-all or partial surfaces are shown to be dependent on Reynolds number, contrary to much popular opinion. No form of this dependency is suggested on account of its complexity but the observation explains the reason for various investigators obtaining differentconstants, for different ranges of Reynolds number, in the usual dimensionless expressions.
THE idea that heat and mass transfer from interfaces are partly or wholly influenced by fluid-dynamic conditions is a common one and several workers have sought means of predicting transfer rates from a knowledge of such data, but many of the analyses are limited in application and often rather tedious to apply. The success of such a proposition depends on obtaining a suitable description of the hydrodynamic conditions at the interface and using the data in relating expressions for transfer rates. The problem is a difficult one, for the major portions of the transfer and fluid resistances occur very close to the interfaces-these zones commonly being referred to as resistance films. Apart from providing means of prediction, solutions of the problem can indicate controlling factors and assist in interpreting experimental data. For chemical engineering unit operations, a case of particular interest is that of transfer from a spherical surface to a continuous fluid phase. The particular problem discussed in this analysis is mass transfer from solid spheres due to forced convection in a continuous incompressible fluid, where the
effect of forced convection is such that there is no disturbance as a result of natural convection and the magnitude of molecular diffusion outside the boundary layer is negligible. Comparisons are also made for circular cylinders in cross flow, particularly for cases of undetached wakes i.e. Reynolds numbers of <30, and interpretations are also made for heat transfer. In the analysis of hydrodynamic conditions it is assumed that boundary layers are laminar-i.e. boundary layer Reynolds numbers of < 6 x lo6 [I] ; this limitation is satisfied by most, if not all, conditions met in chemical engineering. HYDRODYNAMIC CONDITIONS OVER SURFACE UP TO SEPARATION
Several methods of analysis are available for conditions up to separation of forward flow [2-61. More recent publications on the solution of boundary layer equations are principally concerned with critical and supersonic velocities, especially for two dimensional aerofoils. The above cited solutions are either of limited scope or laborious in application. The method chosen is that of TOMOTIKA [7]. TOMOTIKA’S solution is obtained by assuming a 457
R.W.
GRAETON
fourth degree polynomial velocity distribution within the laminar boundary layer, as suggested by POHLHAUSEN[8], and substituting this in the momentum integral equation for a volume of revolution obtained by MILIKAN [9] from the Navier-Stokes and continuity equations. The resulting differential equation relates the boundary layer conditions of a sphere by two dimensionless parameters : z -
Ud2 and UV
I=:
%Z 0
For a sphere the value of 1 varies from 4.716 from the forward stagnation point to - 12 at separation. TOMOTIKAexplains how this relation may be used to obtain the boundary layer conditions from a knowledge of variation of the pressure normal to the surface. There are not many published data for the variation of the surface pressure of a sphere in forced convection except for Reynolds numbers of the order of 105--the critical range where the total drag coefficient suffers a sudden reduction owing to a change in the hydrodynamic conditions, e.g. FAGE [lo]. FLOCHSBART [l I] gives an expression for the variation of U/U, for the upstream half of the surface, viz : -
u
varies from 7.052 at the forward stagnation point to - 12 at separation. There are several published data describing experimental surface pressure distributions for circular cylinders in cross flow, e.g. THOM [14] gives data for a wide range of Reynolds numbers. In the same way as for a sphere, potential flow theory may be used to indicate conditions at the forward stagnation point ; this gives a general value of Z = 353 for all values of Reynolds number. HYDRODYNAMIC CONDITIONS OVER SURFACEDOWNSTREAM OF SEPARATION Whereas a number of solutions of boundary layer equations are applicable up to the point of separation of forward flow, no similar procedure available for the area contained by the wake has been found in the literature. The method of TOMOTIKAis unsuitable because the negligible variation of the surface pressure indicates by BERNOULLI’S theorem that 1 dU --= u,
0
de
Thus it is necessary to reconsider the whole analysis. The boundary conditions for the Pohlhausen laminar boundary layer velocity distribution,
;=‘o+‘(;)+“~(~)2+~~(~)3+~4(~
= 1.56 - 0.36402t13 - 0~02466W5
(1)
UO
where 8 is the angle in radians which a point on the surface subtends with the forward stagnation point, but this expression does not extend as far as separation and is obviously incomplete since it is independent of Reynolds number. Potential flow theory (see MILNE-THOMPSON [12]) gives rise to a distribution of pressure normal to the surface which approximates experimental data for the surface near the forward stagnation point; the approximation is such that it may be considered to depict actual conditions at this forward point. This indicates a general value of Z = 3-14 for all Reynolds numbers at the forward stagnation point. In the case of a circular cylinder in cross flow HOWARTH [13] obtained a solution for boundary layer hydrodynamic conditions similar to that for spheres given by TOMOTIKA, where for the cylinder A
for the case of no variation in the surface pressure, are y=o u = 0,
a% ayz-
-0
y=6
au a% ay=ay2=
u = u,
These give
458
a, = 0
ur =2 a, =o a3 =
2
u4 = 1
o
Prediction of mass transfer from spheres and cylinders in forced convection
Substituting equation (1) with these boundary conditions in MILIKAN'Smomentum integral equation for a volume of revolution
s
a *
ax, ru’
dy - U &
s
ru dy 0
au ) 6 y=o
=U~~~rdy-ro~(
(2)
FIG.
and noting that for a sphere of radius a, dx = a de and r. = a sin 8, a differential equation for the boundary layer thickness 6 is obtained: da2 de + 262 cot
1260 av 37 u
e = -.-
On integration, and for convenience writing in terms of the already mentioned dimensionless boundary layer parameter, Z = Uoa2/av equation (3) gives z = 34.054 $4(e,)
(4)
where
4m =
&
(se, - * sin 2e,) ,
and 8, is the angle position on surface subtends with the rear stagnation point. For small values of 8,, 4(e,) = 46,. Equation (4) does not fully describe the conditions over the whole of the surface confined by the wake since U, the velocity parallel to the surface at the edge of the boundary layer, must vanish at the rear stagnation and separation points, i.e. there is some variation in the pressure normal to the surface for which no allowance is made. However, U may be expressed in terms of U,, the free velocity within the wake, viz. U = U, cos $, where ~+9is the angle between the surface and the contour of circulation near the surface. Such data may be obtained from wake traces e.g. GARNER and GRAFTON [151; these may also be used to indicate values of U,. Thus in its complete form, equation (4) may be written as z = 34.054 u, 90 u, cos l/G
1. Contour of wake of sphere at
Re = 500.
U,/U, are not normally readily available but it has been found that the following suggestion, at least for a sphere Reynolds number of 500, affords a good approximation over the major portion of the surface in question except for near the rear stagnation and separation points. In the case of the rear stagnation point of a sphere the error, although apparently great, is not serious since only a relatively small area is involved. The method consists of ignoring the variation in +, i.e. assuming cos $ = 1.0, and choosing a value of U,/U, such that 34*054(U,/U,) +(e,) at separation is equal to the value of 2 at separation obtained for the forward flow; this is equivalent to noting that the boundary layer thickness has a unique value at separation. A known, or assumed, flow contour may then be used to give the variation in I,+,or the error due to neglecting the variation in +. For a circular cylinder in cross flow, equation (4a) should be replaced by
z = 34.054 3 8, u, cos + where 8, is measured in radians, noting that it is only applicable for case of undetached wakes, i.e. at low Reynolds numbers, < 30. MA,SSTRANSFERTHROUGH BOUNDARY LAYER (i) D@m’on equation FICK’S
law [ 161 gives, for constant diffusivity,
a? ad DBx"=x
(9
A typical wake contour, for use in obtaining $ for equation (4a), is shown in Fig. 1. Data for $ and 459
where c’ is concentration expressed as mass per unit volume. Putting pc = c’, where c is concentration expressed as mass per unit mass, the diffusion
R. W. GRAFTON
equation may be written, for negligible variation in density, 2 ac PD$=Pz For convenience, this expression is rearranged as follows:
Aty=O
C=C#-J, Aty = 6,
2 axatac PD&f=P~a,t=P'g
where u is the linear velocity of solution in the x direction. For an element of boundary layer, taking y normal to the surface and x parallel to the surface, such that &/az = 0 where z is a mutually perpendicular axis, the general transfer equation is pD[$+$]
constants depending on x. The condition at the limits are
a2c--m ac ay2-PD ay
(5)
ac a2C o c= Cd,, -=-F= ay ay
These limits give - 12a a; =6+g Where CI= c0 - c,, (i.e. the concentration difference across the boundary layer) and C = G,m/pD. (iii) Rate equation
=pu;+pv;
From equation (6)
At the surface, i.e. y = 0, noting that there is no slip along the surface which indicates that
(i3yzo =(~),=, =O
-12a
=-
(6 + 06, Since’the rate of transfer from the surface, m, is
the rate equation becomes
-pD
(!fay1y=. pD
(pu),,=,
is the mass of solution passing over unit area of surface in unit time, i.e. the rate of solution from the surface. Denoting this rate of mass transfer per unit area by m, the equation may be written as
i.e.
(5)
Or
12a
m=6,‘6+ 12a 5’6,5 r2 + 6c - 12a = 0
giving
(ii) Bowtdary layer concentration gradient In the same way as a fourth-degree polynomial velocity distribution has been assumed for the hydrodynamic boundary layer, a quartic concentration distribution is assumed :
where 6, is the mass-transfer boundary layer thickness, i.e. the value ofy at ac/ay = 0, and a’,,, etc., are 460
$+=-3+4(9+12ar) i*e* m = $
C-3 + J(9 + 12a)] ,
(7)
root of the solution of the quadratic is extraneous. Normally a is very small, i.e. 12~1$ 9, and the
Prediction of mass transfer from spheres and cylinders in forced convection
relationship given by equation (7) is not, in a convenient form, but may be simplified as am 1 pD __=-.-. aa
It should be noted that the average value of M for a portion of surface of a sphere is given by j’M sin 0 de/j sin 8 de
12
2 6,
J(9
+ 12a)
Thus for 12~1< 9 2pDa m=T
(8)
(iv) Mass transfer prediction relationship For the rate equation to be of any assistance a relation between mass and hydrodynamic boundary layer thickness is required. LEVICH [17] states that for the case of systems where the Prandtl number is much greater than 1.0 (i.e. liquids), the mass transfer boundary layer is equal to the hydrodynamic boundary layer thickness times the reciprocal of the cube root of the Schmidt number. This fact is used to write equation (8) in terms of the hydrodynamic boundary layer thickness, 2pDa m= 6 sc0.33
(8a)
Noting that m/pa is the mass transfer film coefficient expressed as rate of mass transfer per unit area per unit concentration difference (in mass per unit volume units), equation (8a) may be written in terms of the Sherwood number Sh/Sc”.33 = 2d/6 This may be rearranged so as to express the hydrodynamic boundary layer thickness in terms of the dimensionless parameter, 2 = U,S’/av viz: Sh
2d
m=-jj-
=
8Re -. z
Jc)
Denoting the dimensionless mass transfer group Sh/Sc”*33 Reoe5 by M, the general mass transfer prediction relationship may be expressed simply as M = J(8lz)
(9)
For the case of the forward stagnation point of a sphere it has already been pointed out that the value of Z is independent of Reynolds number and is equal to 3.14. Thus for this point M has the general value of 1.60. Similarly, in the case of a circular cylinder in cross flow the general value of M at the forward stagnation point is 1.51. In the case of other parts of the surface the value of M will depend on the form of 2 appertaining to the particular flow characteristics and will approach the value at the forward stagnation point for Re + 0. DISZUSSION
Equation (9) enables a prediction to be made of mass transfer from the entire surface of a sphere from a knowledge of the hydrodynamic conditions. These conditions for the area up to separation may be estimated solely from the distribution of the pressure normal to the surface and for the area confined by the wake, an approximate method of determination is given which requires a knowledge of the shape of the contour of circulation within the wake and of the velocity of circulation just outside the boundary layer. Where this latter information is not available, an approximate value for the circulation velocity may be obtained as described in the section dealing with the hydrodynamic conditions downstream of separation. In the case of the forward stagnation point no experimental data are necessary since the hydrodynamic conditions are adequately described by the potential flow theory. As an illustration, the prediction calculation is given for a sphere at a Reynolds number of 500 (see Tables 1 and 2). The wake circulation contour used is illustrated in Fig. 1 and the variations of 2 and M, i.e. the hydrodynamic and the forced convection transfer conditions, over the surface are shown in Figs. 2 and 3. Fig. 4 permits the graphical integration necessary in determining average values of M. The value of the circulation velocity within the wake was obtained as described above, see p. 3. The average calculated values of M and the
461
R. W. GRAFTON
Table 1. Sphere-Re = 500. Forward jlow area. (Solution of hydrodynamic conditions according to method of TOMOTIKA [7] from variation of surface pressure obtained by the writer)
0 10 20
3.15 3.20 3.4
1.60 1.58 1.54
o*OOO 0.174 0.342
0000 0.275 0.527
o*OOO 0.175 0.349
2 50 60 70 80 90 100 110 114
3.8 4.3 5.1 2:;
1.45 1.36 1.25 1.17 1.08 1.04 0.88 0.744 0.556 0.480
0.500 0.643 0.766 0.866 0.940 0.985 1.000 0.985 0.940 0.914
0.125 0.875 0.958 1.015 1.015 1.025 0.880 0.732 0.522 0.438
0.524 0.698 0,873 1.047 1.222 1.396 1.571 1.745 1.920 1.990
7.4 10.2 14.5 26.0 34.8
Reverse flow
&disionce
FIG. 2.
calculated values at the stagnation points are given in Table 3. It is interesting to note that prediction of minimum transfer occurs downstream of the prediction of separation; this agrees with experimental observations of GARNER and SUCKLING [18].
In
comparing the calculated values of M with experimental data, it must be noted that the prediction confines itself to the transfer associated with forced convection. This implies that the flow is sufficiently great to nullify any natural convection
from
SphereRe
forward
stagnotion
point,
= 500. Hydrodynamic over surface.
degrees
conditions
effects.
The usual form of experimental transfer relationships is
Sh =A +.MRe> where fl is independent of Reynolds number. Thus the mass transfer effected by forced convection is that indicated by fi and it is this that should be compared with calculated values of M provided the value off1 is negligible. The transfer indicated byf,, which is negligible when compared with fi at
Table 2. Sphere-Re = 500. Reversesow area (wake). (Solution according to method described)
eo
40
114 120 130 140 150 155 160 165 170 175 180
66 60 E 30 25 20 15 10 5 0
16O 31 22 0 0 z 57 68 !z
ws* 086 0.93 1.00 1.00 0.87 0.72 0.55 0.38 0.17 0.00
t Obtained by extrapolation. For reverse flow Z = 34.054 Uo/Ul+(0l)/cos
4(el) 0,410 0.391 0.250 0.187 0.150 0.118 0.088 0.058 0.029 0.000
z - lJ@ au 3;2 28.4 16.9 12.6 11.6 11.1 10.9 10.5 11.3 10*5t
M= 4/(8/z) 0.500 0.532 0.690 0.798 0.832 0.850 0.858 0.875 0.843 0.875
$I, where for Re = 500, UO/UI= l-98.
462
sine
Msin e
e=
0.866 0.766 0.643 0.500 0.423. 0.342 0.259 0.174 0.087 O+IOO
0.433 0408 0444 0.399 0.352 0291 0.222 0.152 0.073 0.000
1.990 2.094 2.269 2443 2.618 2.705 2.793 2.880 2.967 3.054 3.142
Prediction of mass transfer from spheres and cylinders in forced convection
8 distance
from
forwrd
stagnation
win?,
rod.
4. Sphere-Re = 500. Prediction of forced convection mass transfer from hydrodynamic conditions over FIG.
0
20 6’-distance
40
60 from
80
forward
100 stagnation
Surface. point,
degrees
Table 3. Sphere-Re = 500. Calculated forced convection transfer group
FIG. 3. SphereRe = 500. Prediction of forced convection mass transfer from hydrodynamic conditions over surface.
Reynolds numbers, is commonly thought to be that associated with molecular diffusion and to be dependent on the Grasshof and Schmidt numbers. GARNERand SUCKLING[ 181have reported experimentally determined mass transfer observations for the slow dissolution of solid spheres in water under forced convection conditions. These experimental and the calculated values of A4 are compared below in Table 4, together with the data of GARNERand GRAFTON[15], recalculated for dilIusity data predicted by OTHMERand THAKAR [19] instead of normally
Average for whole surface Average for forward flow area Average for reverse flow area (wake) Forward stagnation point Rear stagnation point
encountered
Table 4. Comparison of experimental
M=
Whole surface Forward flow area Reverse flow area Forward stagnation point Rear stagnation point
0.95 1.01 0.63 1.60 0*815
determined data by HIXSONand BAUM[20] on the suggestion of GARNERand SUCKLING[18]. It has already been pointed out that the calculated value of A4at the forward stagnation point of a sphere for forced convection is independent of Reynolds number and is equal to 160. FROSSLING[21] gives
and calculated values of A4 (Sphere-Re = 500) Sh/&O.SSReO*S
Experimental values of GARNERand SUCKLINQ[18]
Experimental values of GARNERand GRAFFON[15] (recalculated)
Calculated values
0.95 1.08 0.67 1.68 0.87
0.91 I.04 0.71 1-39 0.56
0.90 1.01 0.63 160 0.875
463
R. W. GRAFT~N
some data for the sublimation of naphthalene in air and his experimental values of M for forced convection are 1.87 at Re = 48 and I.90 at Re = 1,060, whilst VENEZIANet al. [22] obtained a value of l-2 for cooling spheres. The average calculated value of M for forced convection for the whole surface of a sphere of 0.90 at Re = 500 is compared with experimentally determined values in Table 5 below :
Table 6. Mass transfer from the forward stagnation point of Q in. D sphere of benzoic acid in water M = Sh/Sc”.s3Reo.5
Table 5. Over-all mass transfer for sphere
Experimenter GARNERand GIZAFTON [15]t GARNERand KEEY[23] GARNERand SUCKLING[18] FROSSLING [21] MCCIJNEand WEHELM[24] Calculated
Re range
Experimental value of M = Sh/ScO’a8ReO’5
500
0.91
250-750
0.94
500 2-1000
0.95 0.55
30-130 500
1.38 0.90
t Values recalculated for revised ditTusivity data (as above). The close agreement between the calculated values and the experimental values of GARNERand KEEYand GARNERand SUCWNG is accounted for by the relevance of the Reynolds number, the value of M being dependent on flow conditions, having a value approaching that of the forward stagnation point (i.e. M -+ 160) for Re -+ 0 (i.e. 2 --t 3.14 for all the surface).
The above discussion assumes that all effects due to natural convection are absent which is so at reasonably high Reynolds number, the actual value depending on the system. GARNER and HOFFMAN [25] carried out dissolution experiments with 3 in. diameter compressed benzoic acid spheres in water at low Reynolds numbers and the disturbance due to natural convection can be seen from Table 6 based on their reported data (they do not include for the value of the Schmidt number and a likely value for the system of Sc-“r3 = 0.09 has been included). The fact that the “down flow” values are lower than for “up flow” is rather surprising since it appears more likely that less disturbance at the forward stagnation point due to natural convection would accrue during “down flow” than during “up flow”. Similar comparisons may be made of calculated and experimental values for circular cylinders in
Remarks
Re UP flow
Down flow
3 10 :
3.1 2.1 1.4 1.8
3.2 1.8 1.1 1.3
70 100
1.4 1.4
1.2 1.3
Calculated value
1.6
1.6
Data of GARNERand HOFFMAN[25]
Unique value for forced convection effects for all Re
cross flow. In the case of the forward stagnation point the calculated value of M for forced convection is independent of Reynolds number and is equal to 1.5. Much of the data for circular cylinders are for heat transfer and for these cases M is taken as Nu/Pr o.33Re0.5. Table 7 gives some comparisons of calculated and experimental values. A calculation for the whole surface of a circular cylinder in cross flow, based on pressure data of THOM [6,14], has been made for a Reynolds number of 500. The average calculated value of M for forced convection is 0.9, whereas the heat transfer value given by HILPERT [32] for Reynolds numbers of O-l-4000 is 0.48 and that by MCADAMS [33], for Reynolds numbers of 50-10,000 is O-60, but the Table 7. Circular cyhnders in cross flow (forward stagnation point)
Experimenter
Re
Experimental value qf M = Sh/Sc”‘aaReO’5 = ~u/~O~S3&-$5
KRUB-IILIN[26] 39,500
1.4
SMALL[27]
39,600
1.45
DXrRLSCH [28] 39,600 WINDMGand 32,800
1.05 1.2
CHEYNEY
Heat transfer Heat transfer Absorption Sublimation
1291
DREWand RYAN 1301 KLEM [31] Calculated value
464
Remarks
32,800
1.0
39,600
0.95 1.5
Heat transfer Heat transfer For all Revalues
Prediction of mass transfer from spheresand cylindersin forced convection significance of Reynolds number range is stressed, as for the sphere ; this indicates a fall in the value of M for increasing Reynolds number.
k K m M
CONCLUSION
Nu
It is seen that there is remarkably good agreement between calculated and experimental values of the transfer group, M, associated with forced convection, even when applied to heat transfer. This indicates that forced convection heat and mass transfer are analogous when expressed in this way and that they are both directly dependent on hydrodynamic conditions. The agreement between the predicted and experimental values further indicates the applicability of the assumptions made to describe the hydrodynamic conditions. The arrangement of influencing factors obtained in this analysis of the problem suggests that experimental data, not subject to disturbing influences, should be related by such expressions. The analysis indicates that the standard practice of relating transfer rate to the square root of the Reynolds number is not fully comprehensive.
Pr r
rleo Scs Sh 2 U
u UO u, V X
Y Z
Acknowledgement-This
discussion is based on work carried out in the Department of Chemical Engineering of the University of Birmingham under the direction of Professor F. H. GARNER whose interest and encouragement is gratefully acknowledged.
NOTATION Radius of sphere or circular cylinder Dimensionless constants in boundary layer velocity profile equation a’s, a’4 Dimensionless constants in boundary layer concentration profile equation Concentration, mass of solute per unit c mass of solution Concentration, mass of solute per unit volume of solution co Concentration at surface car Concentration at outer limit of boundary layer d Diameter of sphere or circular cylinder D Diffusivity of solute in solution fi,fz Molecular and forced convection functions describing mass transfer h Heat transfer coefficient a
ao, al, m, mr a4 do,
B’l,
U’8,
465
z
Thermal conductivity Mass transfer fihu coefficient (dimensions of length/tune) Rate of mass transfer per unit area of surface Forced convection transfer group, = Sh/ScO.ssReO.5 = Nu/prO.ssReO’5 Nusselt number (dimensionless) = hd/k Prandtl number (dimensionless) = sp/k Distance co-ordinate, measured from axis of symmetry Value of r at surface Reynolds number (dimensionless) = dUoplr Specific heat Schmidt number (dimensionless) = v/D Sherwood number (dimensionless) = Kd/D Interval of time Velocity, parallel to surface, within boundary layer Value of u at outer limit of boundary layer Undisturbed velocity of fluid Free velocity within wake Velocity perpendicular from surface Distance co-ordinate, measured along surface Distance co-ordinate, measured perpendicular from surface Distance co-ordinate, mutually perpendicular to x and y Dimensionless parameter describing hydrodynamic boundary layer = UoP/av Concentration difference across mass transfer boundary layer = Co - Cd, Hydrodynamic boundary layer thickness = Q(aular) = 0 Mass transfer boundary layer thickness = QrVad-0 Angle point on sphere, or circular cylinder, subtends with forward stagnation point Angle point on sphere, or circular cylinder, subtends with rear stagnation point Absolute viscosity of fluid Kinematic viscositv of fluid Dimensionless parameter relating to mass transfer boundary layer = GmlpD Density of fluid Trigonometric function = (l/sin2 e,) (if?, - a sin 28,) Angle between wake contour near surface and the surface
R. W. GRAFTON REFERENCES
[l] TINMANR., Zng. Grau. 195264 L.6. 121 BLA~IUSH.. Z. Math. Phvs. 190856 1. i3j GREEN G., Phil. Msg. (skr. 7) 193112 1. [4] VON MIS= R., Z.A.M.M. 19297 425. [5] FALKNERV. M. and SKAN S. W., A.R.C.R. & M. 1930No. 1314. [6] THOM A., A.R.C.R. & M. 1928No. 1176. [7] TOMOTIKA S., A.R.C.R. & M. 1936No. 1678. [8] POHLIUUSENE., Z. Math. Mech. 19211252. [9] MJUKAN C. B., Amer. Sot. Mech. Engrs. Trans. 1932 54 A.P.M. 29. WI FAGE A., A.R.C.R. & M. 1937No. 1766.’ FLOCHSBARTO., Phys. Z. 1927 28 461. 5::; MILNE-THOMPSONI;. M., Theoretical Hydrodynamics (2nd Ed.) Macmillan, London (1949). u31 HOWARTHL., A.R.C.R. &M. 1934 No. 1632. u41 THOM A., A.R.C.R. & M. 1928 No. 1194. u51 GARNERF. H. and GRAPTONR. W., Proc. Roy. Sac. 1954 A224 64. WI FICK A., Phil. Msg. 1855 4 (10) 30. u71 LEVICH V. G., Dokl. Akad. Nauk 1951 78 1105. R. D., Amer. Inst. Chem. Engrs. J. 19584 114. WI GARNER F. H. and SUCKLINGS OTHMERD. C. and THAKARM. S., Zndastr. Engng. Chem. 1953 45 589. Hrxso~ A. W. and BALM S. J.. Zndustr. Elwns. Chem. 1942 34 120. Ki i21] FROSSLINON., Ger. Beit. Geophysik. 1938 g2170. [22] VENEZIANE., CRESPOM. J. and SAGE B. H., Amer. Inst. Chem. Engrs. J. 1962 8 383. [23] GARNER F. H. and KEEY R., Chem. Engng. Sci. 1958 9 119. [24] MCCUNE C. K. and WILHELMR. H., Zndustr. Engng. Chem. 1949 41 1124. 1251 F. H. HOFFMAN J. M.. Amer. Inst. Chem. Enars. J. 1960 6 (4). i26j GARNER K.RUZHILIN G. and H., J. Tech. Phys. 8’ 123 1938. . ,, 579. [U] [28] [29] [30] [31] [32] [33]
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- L’auteur obtient une expression reliant le transfer? de masse, du a la convection for&e, ReSUlU~ a l’etat hydrodynamique de la surface dune sphere. Dans le MS du point d’equilibre extreme aucune don&e n’est necessaire pour les conditions hydrodynamiques. Les relations peuvent &e egalement utilisees pour des cylindres pla&s dans un koulement transversal. Au-dessus du point de separation de l%coulement anterieur, l’autreu preconise une methode utilisant la distribution des pressions pour l’evaluation des conditions hydrodynamiques. Des evaluations par analogies dormant des r&hats conformes aux experiences peuvent &re effectut%s pour le transfert de chaleur. Dans le cas du point d’@rilibre extrgme pour une sphere (ou un cylindre place transversalement a l’&coulement), on montre que le groupe sans dimension relatif au transfert en convection for&e est independant du nombre de Reynolds; par contre les groupes relatifs aux surfaces partielles ou globales en dependent contrairement a une opinion repandue. Ces relations ne sont pas analysees en raison de leur complexite mais l’experience explique les raisons qui ont amen& de nombreux chercheurs a trouver des constantes differentes pour des nombres de Reynolds varies.
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