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Transfer from a sphere into a fluid in laminar flow
Chemical
Engineering Science, lOsO,Vol. 12,pp.215to80% l'ergomon Press Ltd., London.
Transfer
Printed in Great Witsin
from a sphere into a fluid in laminar flow M.
LINTON*
and
Division of Physical Chemistry, C.S.I.R.O.
K.
L. SUTHERLAND*
Chemical Research Laborstories,
Melbourne
(Received 28 November 1252) Abstract-Boundary layer theory and experimental results for the transfer of heat and mass from spheres in forced flow are compared over the range of conditions: 1 <
Re < 1.8 x 106, 0.5 < Pr < 106, 2 < Nu < 540.
The overall and local solution rates of 2/S in. diameter spheres of benzoic acid were measured in uniform ilow in a water tunnel at 490 < Re < 7580. The overall transfer was correlated by Nu = 0.582Relia Prlis The relative distribution of local transfer rates over the front half of the sphere was in fair agreement with theory and other workers but the absolute local values at the front stagnation point were about 44) per cent lower than theory and differed considerably among other workers. The discrepancies may in part be due to a gradual transition from viscous flow to potential flow outside the boundary layer as Re increases from 1 to 10 5. The relative transfer from the rear half of the sphere was observed to increase with Ii%. The angle of separation 0, found experimentally varies approximately according to 0 s = 83 + 101 Re-li3 The average transfer rate over the front of the sphere is, theoretically proportional to Re@ Pr1/3. Over the whole sphere 0.5 < Nu Re-li2 Pr-lls < 1.0 for IO2 < Re < 105. The approximation is only rough because of the factors dependent on Re mentioned above, and the following secondary factors : a lower limit to transfer rate set by diffusion or conduction, of the free stream, density convection unless Gr Q Re 2, turbulence and/or non-uniformity interference from sphere supports, influence of duct walls, uncertainty in transfer properties of fluids, variation of fluid properties with concentration or temperature, and dust accumulation on the sphere. R&sum&-La couche limite theorique et les r6sultats experimentaux pour le transfert de chaleur et le transfer% de masse a partie de spheres dans un t!coulement force, sont compares dans l’intervalle : 1 < Re < 1,s x lo6 + 0,5 < Pr < log + 2 < Nu < 544.3. Les vitesses de dissolution locale et globale de spheres d’acide benzoique de 3/8in. de diambtre ont et6 mesurdes dans un tunnel d’eau ii 490 < Re < 7580. Le transfer% global est. doMe par Nu = 0,532 Rel/* Prlls La distribution des vitesses de transfer% sur la moitie avant de la sphere s’accorde bien avec la theorie et les autres expt%mentateurs, mais les valeurs locales absolues au point de stagnation avant, sont environ 40% inf&ieures A la theorie et varient avec les exp6rimentateurs. Les differences sont dues en partie B la transition graduelle de l’dcoulement visqueux en Bcoulement potentiel en &hors de la couche limite quand Re augmente de 1 B 106. Le transfer% relatif sur la moitie arriere de la sphere augmente avec Re. Le point de separation 0, trouve expdrimentafement varie comme 0, = 83 + 191 Re-l/s. La vitesse de tranafert moyenne sur le front de la sphere est thkoriquement proportionnelfe a Rer/s pyr/s. Sur la sphere entiere 0,5 c Nu Re-l/s Pr-l13 < 1,0 pour lo2 < Re < 10s. L’approximation est un peu gross&e ir cause des facteurs dependants de Re deja. mentiormes et des facteurs secondaires suivants : limite inferieure de vitesse de transfert par diffusion ou 214
Transfer
from 8 sphere into a fluid in lsminar
flow
conduction ; importance de la convection (sauf pour Gr Q Res) turbulence et non uniformiti du courant libre ; interf&ence des supports de spheres ; influence des parois de la conduite ; incertitude des propriet& de transfer+. des fluides, variation des propri&.es du fluide avec la concentration ou la temperature, et accumulation des poussieres sur la sphere. Zusammenfassung-Fur den W&me- und Stoffiibergang an Kugeln in enwungener StrGmung wurden die Ergebnisse der Grenzschichttheorie und der Versuche miteinander verglichen und zw8r hn Bereich von 1 < Re < 1,8 x 10 5;0,5
,
Nu = 0,582 Re112 Pr1j3 Die relative Verteilung der ijrtlichen Ubergangskoelhzienten auf der Stirmseite der Kugel war ln guter ubereinstimmung mit der Theorie und anderen Versuchsergebnissen, aber die absoluten Brtllchen Werte am vorderen Staupunkt waren etwa 4Q% unter denen der Theorie und differierten betr%chtlich mit fremden Versuchsergebnissen. Die Unstimmigkeiten m%en zum Teil durch einen schrittweisen Ubergang vom z&hen Fliessen zur Potentialisttimung ansserbalb der Grenzscblcht beim Anwachsen von Re von 1 bis lo5 verursacht sein. Die relative ubertragung an der Riickseite der Kugel stieg mit Re an. Der Winkel des Ablijsungspunktes e, 5ndert.e sich nsch den Versuchsergebnlssen ungef5hr nach der Gleichung 8 8 = 83 + 191 Re-l13 Die mlttlere vbertr8gung an der Vorderseite der Kugel ist nach der Theorie proportional zu Re112 Prl’s. Uber die ganze Kugel ist 0,5 < Nu Re-lls Pr-l13 cc 1,0 fur lo2 < Re < 106. Es handelt sich dabei urn grobe Merung wegen der oben erwiihnten von Re abhihqigen Faktoren und wegen folgender sekund%rer Faktoren: Eine untere Grenze fur die Ubertagung lnfolge Diffusion oder wiirmeleitung, Konvektion durch Dichteunterschied, sofern nicht Gr < Re2, turbulenter und/oder nicht einheitlicher Freistrom, Einwirkung der K~gelhalterung, Ein6uss der Kanalwiinde, Unsicherheit in den Stoffgr&sen der Fliissigkeit, Abhitngigkeit der Stoffgr&ssen von der Konzentration oder der Temperstur, Staubsns8mmbmg auf der Kugel. 1.
INTRODUCTION
THIS paper compares the transfer predicted from the theory of a laminar boundary layer, with experimental results, for local heat and mass transfer around the front of a sphere which is held in a moving fluid. The theory suggests also an approximate but very general expression for overall transfer, and experimental deviations are shown to be due to the influence of many secondary factors. The transfer of mass or heat from a solid sphere into a fluid flowing at moderate velocities has been treated theoretically by FR~~SSLING [6] and AKSEL’RUD [ 11. FR~~SSLING [6] calculated the velocity distribution and mass transfer* in the laminar boundary layer using power series. By a method outlined in Appendix 1 below, he calculated numerically the *For convenience we will refer mninly to mass transfer although analogous statements apply to heat transfer.
coefficients in the series for the particular case of Prandtl numberl_ = 2.532 (corresponding to naphthalene evaporating in air). We will express the local rate of mass transfer as the dimensionless local Nusselt number Nu (8) defined in terms of experimental quantities by
0
where B is the polar angle, d the diameter of the sphere, D the diffusion coefficient, m the mass of the sphere, A the area of the sphere, t the time, C, is the concentration at saturation and C, that in the free stream. FR~~SSLING [6] was able to show that quite generally Nu (19) co Re1j2 and that for the particular case of Pr + co (when the transfer boundary layer is thin compared to the flow boundary layer) Nu (0) oc Pr”“. Consequently the quantity Nu (8) Re-‘Is Pr-“’ which tFor
215
(1)
definitions see the end of this paper.
M. LINTON and K. L. SUTHERLAND
we shall designate as the transfer number Tr (8) approaches a constant value for any given value of 8. Following the method of FR~~SSLING[6] we obtain in the manner shown in Appendix 1 for the sphere at Pr = 2.532
for Pr + w using the approximate polynomial method. The following equation is derived from AKSEL’RUD’S Tr (8) =
1.523 (1 -
Tr (8) = Nu (8) Re-‘12 Pr-‘/s = 1.366 (1 -
published
graphs
0.1728
0.0114
Q2 -
8*) =
1.523fs
(8)
(7)
I.0
0.1837 82 +
+ 0.00696 8* + . . .) = 1.366f, (8)
(2)
Because FR~~SSLING was unable to eliminate the Prandtl number from his equations generally the coefficients in equation (2) are functions of Pr. From the particular solution, Pr + co, one finds at the front stagnation point (B = 0) Tr (0) = Nu (0) Re-‘12 Pr-‘/a = 1.472
0.8
(3)
From coefficients tabulated by FR&SLING [6] we calculated the transfer at the front stagnation point for the range 0.5 < Pr < 100 and can represent the variation with Prandtl number to f 0.5 per cent by 0.22 0
Tr (0) = 1.478 -
30
0.158 PrM1/afor 0.5 < Pr < co (4)
FIG. 1. The theotetical distribution of transfer rates with polar angle, taking that at the front stagnation point as unity.
Other approximate solutions to the transfer number at the stagnation point have been derived by FR~~SSLING [6] and SIBULKIN[25]. FR&SLING [6] following POLHAUSEN[22] and KROUJILINE [15], used polynomials to express the velocity and concentration distributions and from his solutions we find that, to f 1 per cent, Tr (0) = lGi3 -
0.190 PT’/s
0.1 < Pr < co (5)
whilst SILBULKIN using similar expressions but with different integral equations gave values which we can represent to f 0.5 per cent by Tr (0) = 1.50 -
0.182 Pr+s
0.6 -C Pr < 10 (6)
Thus the theoretical transfer numbers at the front stagnation point are in quite reasonable agreement when calculated by three different methods. Equations (P)-(6) indicate ‘that Tr (0) only varies about 15 per cent as Pr varies from co down to 0.5. AKSEL’RUD[l] has also calculated the variation in mass transfer around the sphere from the front stagnation point to the separation point
60 e*
: theory of FIGSSLING at Pr = 2.532, Equation (2). fi (0) Full curve : theory of AKSEL’RUD at Pr = CO,f2 (0)
Broken curve
Equation
(7).
The distribution of transfer rates c&lculated by the exact method by FR~~SSLING [6] for Pr = 2.532 -fi (19) in equation (2)-and by the approximate method by AKSEL’RUD [l] for Pr = w - fa (8) in equation (7) - are plotted in Fig. 1. The agreement between the two curves is good up to 60”. Beyond this point fi(0)will not be accurate due to the absence of terms higher than 8* which have not been calculated so far. Thus fi (0) and fa (0) are independent of Re and Pr. Hence Tr (0) is independent of Re and varies only slightly with Pr due to the variation in Tr (0) discussed above. The above theories do not, so far, predict mass transfer at the rear of the sphere, or consequently
216
Transfer from a sphere into a fluid in laminar flow
the total mass transfer, because of separation of the flow with its profound effect on the boundary layer. We have confined a comparison of experiment and theory (Section 4b) to the leading surface of a sphere. AKSEL’RUD [l] calculated the total transfer over the sphere by graphical integration and obtained From pump
14in. squore ‘low diffuser
.l5in.
Tr = 0.80
He assumed the transfer to be constant for 8, Q B < 180’ and the separation angle 0, to be llO“, but the experiments presented below show that neither of these assumptions is valid. However, if we postulate that in the wake a velocity and diffusion boundary layer develop from the rear stagnation point and obey the same form of relations as those at the front but differing in the values of the constants, then the overall transfer number could again be constant except insofar as the separation point varies and Tr (0) varies with PT. The available experimental data discussed in Section 4, show that Tr does lie within fairly narrow limits.
square constonl head inlet section
2.
25:l
‘2
-1,-
3 in. square
tronsporent working section with
c’
‘5
Groduoted velocity control mechanism i
Contraction
screw adopter
opening
f-
EXPERIMENTAL
METnoD
Wder tumel
(a) _ Square grid of &in. oluminium sheet 01 ‘4 in. cenlres
(8)
Uniform flow for velocities of 3-50 cm set-l was achieved over a considerable area of the working section of 3 in. x 3 in. of a vertical water tunnel by the design shown in Fig. 2 (a). The water was fed from a constant head, the level being maintained by a centrifugal pump and overflow weir. An essential feature is the diffuser, shown in Fig. 2 (b), the remaining vorticity after passage through this being removed by an aluminium grid at the lower end of the inlet section. The design of the contraction section was supplied by the Aeronautical Research
3in. square to 3in. diom.
I
round section 2in.2 k-
feed pipe
3in. pt round section
&-.--~7
Brass control valve and seat
ISin.
flange,
_ 30in.squore discharge receiver box To Row diffuser !$in. O.D. pwcelain
beods
.3in. porcelain insulating eods in co, Iin. layer Choin drive’
I%&=
14in.;quore golvonized diffuser box
:ible hose connex
iron
‘Perfomted
(b) The diffuser
(a) Main section FIG. 2.
The
water tunnel
217
zinc screen
M. LINTON and K. L. SIJTHERLANII %&I. Wall lin. LD. rubber tube
Hose clip (a) Adapter
7/8in.O.D.Xfin.Sross tube with end plates
he1nZS.S.
Spot weld qround rmooth
FIG. 3. (a)
Unit for insertinp.sphere
Mounting
for sphere
(b) Support cylinder
into the tunnel
Laboratory, Dept. of Supply, and was smoothly connected into the polymethylmethacrylate working section which had 1.5 in. diameter screw ports at the sides. These ports carried the mounts for test objects. A square to round conversion section led to the circular outlet where the flow was controlled by a blunt-nosed conical valve. Water collected in the receiver box was fed to the pump by flexible hose to prevent transmission of vibration. The tunnel was supported by a suitably braced steel frame. Most of the construction was in wood, internal surfaces being coated with water-resistant resins. It would be more desirable-but more costly-to
(c)
Sphere support
construct the tunnel in metal which would avoid The tunnel required swelling and distortion. regular cleaning, algal growth being controlled by small additions of copper sulphate. The raw water supply contained sufficient suspended matter to cause difficulties and it was treated with an alum-lime mixture*, the floe being allowed to settle for 24 hr when the supernatant was found to be sufficiently clear that it did not appreciably scatter a light beam. The water temperature was controlled by heating or cooling coils to 20.0 f 0*3’C.
218
*135galum
and 45gslaked
lime in 3Wgal
water.
Transfer from a sphere into a fluid in laminarflow
(b)
Spheres and th+Grsuppvrt
Each test sphere was moulded on a fine stainless wire attached to a heavier support wire (Fig. 3c) which fitted into a cylinder (Fig. 9b) carried, as a slide fit, by the adaptor (Fig. 3a) which screwed into the port. The cylinder acted as a water seal and when being removed escape of water was prevented by using a clamp closing the rubber tube. The adaptor and cylinder were fitted flush with the internal wall. Solution in the time (15 see) required to insert or remove a sphere was negligible. The sphere was centred in the tunnel by projecting an image of it and the working section on a ground glass screen. To reduce any effect which dust may have on transfer, the spheres were rotated rapidly for a few seconds every 10 min. Where there was any sign of dust this treatment led to its removal. The spheres were of benzoic acid, pressed in a mild steel mould heated to 110 “C. A mark on the support wire served to orient the sphere. The sphere profiles were measured before and after solution either with a co-ordinate travelling microscope or from tracings done by hand at 30 magnifications using a profile projector. The spheres before solution were actually prolate spheroids the major axis being 0-005cm longer than the minor. The mean diameter was 0.958 f 0.003cm and the mean density 1.286g cm-3. The rate of solution of the spheres was a maximum at the front, a minimum near the point of separation and rose to a smaller maximum at the rear. From the resulting change in shape measured as changes in radius, the local and overall solution rates were calculated as indicated in Sections 3 (b) and 8 (c) below. The spheres and their support wire were also weighed before and after solution and the overall solution rate calculated as shown in Section 3 (b) below. The experiments lasted between one and four hours, in which time the average diameter was reduced to about 90 per cent of the initial value. (c)
pressure mercury arc emitting 100 flashes/set illuminated the working section through a tall beaker filled with water acting as a cylindrical lens. The traces were then photographed using a 6 in. lens. The magnification was determined using a known object in the image plane.
3. (a)
EXPERIMENTAL
RESULTS
Unifbrmity of jZow in the wder tunnel
The flow in at least the central 4 cm x 4 cm of the working section was uniform to within the accuracy of measurement of velocity, viz. f 1 per cent at 3 cm set-1 and f O-1 per cent at 54 cm set-l. The turbulence, as indicated by the maximum observed lateral displacement from the vertical path was less than 0.5 per cent in the same zone. At 54 cm set-’ up to 3 per cent departure from vertical path could be observed at 1 cm from the wall but the internal core of the working section was still satisfactory. (b)
Over-all solution rates
The sphere changed both its size and shape during the several hours in which it was being dissolved. The theory presented above applied to the i~nitial solution rate. Fig. 4 shows that the average solution rate (m,, - m,)/t during a period of solution t, varies linearly with the square root of the flow velocity, (or Re”‘). As
Velocity distribution
The velocity distribution in the working section was determined from photographs of the movement of aluminium flakes. A 250 W a.c. high 219
FIG. 4. The solution rate (averaged over the period t) of a benzoic acid sphere (d = 0.958 cm) as a function of water flow velocity at 20 “C. The straight line is (mO - ml)/ = 0.439 x 10-6 u,l/z.
hf. LINTON
shown in Appendix 2 the initial solution rate can hence be calculated, assuming the sphere remains spherical, as
m. -
ml
2rnbi2
’ mo1/2 +
t
m11/2
(9)
where m, is the initial mass and m1 that after a time t. The observed change in shape was less than 0.1 diameters and the linearity of the points in Fig. 4 indicates that its effect on the solution rate was negligible. The overall Nusselt number for the sphere was obtained by substituting the initial surface area and the initial solution rate derived from equation (9) in equation (1) in The diffusion coefficient of integrated form. benzoic acid in water at 20 “C! was taken as 0.882 x 1O-5cm2 see-l (WILKE et al. [28]) and the saturation solubility as 3.11 g 1-l (KREVELEN and KREKELS [14]).These give Pr = 1210. The mean density of the pellets, 1.286 g cm-s, was used to calculate the local and integral rates of solution from the change in size. The integral solution rates are determined by substituting equations (10) and (11) in (1) and integrating, where,
Nu
pr-‘/’
=
0.669
Re”‘481
(12)
with a coefficient of correlation of 0.984. The best fit assuming that n = l/2 is Nu Pr-“a (c)
O-582 Re112
(18)
Local solution rates
The local values of the Nusselt number, Nu (e), were obtained from J2 m
3A=
Ar (8)
Pm At
(14)
and equation (1) where Ar (8) is the change in radius at angle 8. The results are plotted in Fig. 6.
(10)
rno1i2+ m11i2
At
at positions Bi (75” + i 15",i = 0, 1, 2, . . . ll), and ABi = 15”. Slight irregularities in the final sphere shape caused the calculated solution rates to be less accurate than those obtained by direct weighing. The variation of Nusselt number with Reynolds number obtained by both methods is shown in Fig. 5. The results by direct weighing were fitted by least mean squares to an equation of the form Nu Pr-‘Is = a Ren. The best fit is
I.2
2rn,‘/’
~rnzA&,
L. SUTHERLAND
and K.
0”
F
where
I.0 ‘0.6
‘;
AC = (7r/2) Aei(ro- T,)(r.+ pi)2 sin Si
(11)
h N c 3
is the loss in volume from a zone of the sphere, the ri are the radii after an interval of time At
s
0.6 0.4 0.2
PIG. 6. The local solution rate of benzoic acid spheres in water in forced convection. + Separation of laminar layer, Reynolds numbers are as shown. The broken curve is Nu (0) Pr-r/s Re-ri2 = 0.94 (1~00-6050 B2-O*069 e4) and is the best fit to the data.
1
00
Re
FIG. 5. The overall solution rate of benzoic acid spheres in water in forced convection. The linear relation is given by Nu = 0.582 Re1i2 P&s. l By direct weighing A By integration over the surface
The best fit of an equation of the form of equation (2) to the mean values of Tr (8) at O’, 30°,60' and 90” is Tr
220
(e) =
0.94(1.00 -
0.05082 - 0.06984) (15)
Transfer from a sphere into a fluid in laminar flow
(d)
0, = 83’ for large Reynolds numbers ; the value of 83” was found by FAGE [3] at Re = 1.57 x 106. Relation (17) is in good agreement with observations in the range 15 < Re < 1000, made by
Angle of separation
After a sphere had been dissolving for some time, it developed a marked ridge where the solution rate was a minimum (0 N 90°-100°) and which was close to the zone of separation of the boundary layer. This separation was detected by illuminating the system with accurately parallel light and studying the stream lines visible due to differences in refractive index. The separation as observed by us from the position of the ridge is fitted by the relation 0, -
GARNER from
Expressions
data
GRAFTON
obtained
evaporation Fig.
7.
Nusselt over
of
83 = 660 Re-‘/’
(13)
83 = 191 Re-‘/’
numbers
the
ference
by
by
of
are
at each 8.
the
are
to note
the overall
on
the
shown
that
to
and
the
their
circumcontribu-
and as separation
4.
DISCUSSION
Requirements
of Theory
The theories of transfer considered introduction to this paper assume
in the
(1) potential
Id
IO'
flow outside the boundary layer, uniform far from the sphere. (2) the velocity and transfer boundary layers are thin compared to the radius of the sphere. This ensures negligible contribution by molecular diffusion. (3) negligible natural convection. (4) freedom from turbulence or other interference with flow. (5) constant fluid properties.
IO'
Re
FIG. 7. The angle of separation, B,, as a function of Reynolds number. 0 FR&SLIN~ [5] ; l GARNER and GRAFTON [7] ; x LINTON and SUTHERLAND(this paper) ; - - - TANEDA [26].
Table 1. Summary
of experiments -
Re
Transfer
Ekperimen
__
on the distribution
1
--
of transfer rates on. the sphere
References
Theory
Nu (W-Son) Nu (W-180*)
(Eqn. (4))
Mass
136724
1.23
1.37
FR&.SLINO[5]
3.9
Mass
200-I 766
1.11
1.46
GARNER~~~&JCKLING[~]
1.6
Heat
1530-4206
1.16
1.30
Hsu andS~~~[9]
2.6
Mass
1490-7580
0.94
1.46
This paper
1.9
Heat
44-15
x 104
0.77
1.30
CARY [2]
1.0
Heat
87-18
x IO4
1.46
1.30
XENAKIS
-
is
influence on
rate.
(a)
Kl
in
the overall
values
close to this zone it has a considerable
to
[5]
also
the maximum
at the equator
and
from the summation
local
proportional Thus
GARNER
Values derived
FR~SSLING
are obtained
surface
tion is made
(17)
(16) and (17) are asymptotic
[7],
naphthalene
It is important
contributions
as shown in Fig. 7 but nearly as good a fit is 0, -
and
SUCKLING [8] and by TANEDA [26].
221
etal.[30]
1.02-0.66
nitrate
Water
Benzene
Mercury
M
M
M
H
M
H
H and M
Benzene
Aniline, water,
Bensoic acid
Benaoic acid
1%
M
Methanol
n-Heptane
Adipic acid
Bensoic acid,
Bensoic acid,
Naphthalene,
wate
Aniline, benzene
Benaoic acid
Potassium
Solute
H H H
_M M
M M H M
Tratlsj-.?rN
Air Air Air
Air .4ir Air Water Oil Water Air Air Air Water Air
0.7 0.7 0.7 73-10.7 2ocb380 1210 0*8-0~7 0.6-0.7 0.5 1210 0.60.7
1210 120&1525
1560 2.3 x lo6 0.7 0.6-2.53
_-
Pr
. _
-
I
-
_
7
-
-
300-480
1.4-11 x 10’
d
I
-_
Y
_,“._
0.42 + 2f 0.43 Re0.06 0.37 ReO’l
0.61 1.18 0.66 + 3.2f 0.56 + 3.7f 0.56 + 5-Oj 0.43 Re0’06 0.43 Re0.06 0.65 0.55 + 2j 0.582 o%o + 2f
0.55 + 54 f 0.63 + 2 f
J
[7]
[lo]
[29][
_L
..--
XENAKIS et al. 301
WILLIAMS
TANG et al. [27]
_
HANZ and MARSHALL [23]
This paper
MAXWELL and STORROW [I91
MAISEL and SHERWOOD [la]
LINTON and SHERWOOD [Ia]
KRAMERs [13]
I~oEn0
Hsu and SAGE [B]
GARNER and SUCKLING [8]
. > ‘\
j
7
[5] GARNER and GRAFION
FR&SLING
GARY [2]
AKSEL’RUD [l]
AKSEL’RUD [l]
References
convection
T
in forced
0.82 1.1 Re-l/e 0.59 0.55 + 2f
pr-l/3
0.32-1.6 0-033-15 15.3-30.5
0.95 0.064.11
1.2 O-69 0.7-1.26 0.7-1.26 0.7-l -26 1.27 2.6-3.5 2.6-3.5 -
1.27 0.95-1.9
0.4-o.7 12.7 o*olLo~o9
-
(cm)
tf = &-l/z
622 -
25-35 39-87 1l-30 11-45 lo-30 31-65 4O-160 40-160 P-30 120-510 2-10
70-240 loo-280
120-540 60-200 loo-200 2-20
Nu
20-l 500 -
c 10
104 1.7 x 103 220-6300 350-1900 1.5-10 x 104 1.3-10 x 104 106 10s
M = mass
,~. _ ,,, ..- _
*H = heat
-
1.3 x 104 N 104
10s
Gr
on over-all transfer of heat and mass from spheres -
SO-1OOO 0.67 0.7 2-50 x lo3 8.7-18 x 104 0.7
15OO-42OO 16OO-5720 10&2ooo 60-1OOO l-40 2ooo-12,600 2oo&42,ooo 3ooo-42,ooo 2-1500 300-7600 2.1-200
2O-850 loo-700
lOO-3500 0.1-2.5 4-15 x 104 2.3-800
Water Oil Air Air Water Water
Re
Fluid
Table 2. Summary of experiments
.
.
Transfer from a sphere into a fluid in laminar flow
mentally for natural convection alone from flat plates that
Of the many observations (Tables 1 and 2) reported on transfer from a sphere to a flowing fluid few satisfy all these assumptions. To compare theory with both our own data and those of others these conditions are examined in detail. (1)
Nu = 2 + 0.5 (Gr Pr)“‘25
Potential flow (fluid of xero viscosity)
Diffusion,
and boundary layer thickness
For the theory to apply the velocity and transfer boundary layers must be thin compared to the radius of the sphere. The latter ensures that transfer by diffusion is negligible and requires that Re’j2 Pr’/s > 2. This condition often fails to apply, particularly to the evaporation of liquid drops (see Table 2). Thus the temperature profile determined by RANZ and MARSHALL[23] for a typical evaporating drop shows that the thickness of the temperature boundary layer at the equator was 8.0 drop radii. (8)
1Oa < Gr Pr < lo6 where & is the Grashof number. The first term is for diffusion and the second term arises from density convection and will only be much less than forced convection if Gr < Re’j2 Pr1/12 (19) in the range
The theory of FR~SSLING can be applied to potential flow with any arbitrary velocity distribution outside the boundary layer. The potential flow around a sphere immersed in an infinite fluid is assumed in the analysis given in Section 1 and Appendix 1. Real flow around the front of the sphere changes gradually from viscous flow at Re < 1 to approximately potential flow at Re > lo6 (as can be seen by comparing theoretical experimental pressure distributions and (SCHLICHTING[24] Fig. l-9). Thus at Re = 20 the pressure distribution calculated by KAWAGUTI [ll] is intermediate between those for viscous flow and potential flow. We achieved an initial uniform velocity profile in the water tunnel close to the throat of the contraction section and worked at Reynolds numbers of up to several thousand to approximate to potential flow. Experiments in which the flow was parabolic e.g. GARNER and SUCKLING [8], would not be expected to agree accurately with the theory. (2)
(13)
Density convection
When a temperature or concentration gradient is high, natural convection currents are supefimposed on the forced flow altt?ring the rate of transfer. PIRET et al. [20], [21] found experi-
Although based on a flat plate, condition (19) should also apply to the sphere. Table z shows that Tr increases with decreasing Re and this is consistent with the increasing importance of the density convection or diffusion. GARNER and SUCKLING [8] observed density convection in horizontal flow at Re = 200 but the effect would be present to the limit given by equation (19). Such an effect is clear also in KRAMERS’ results [IS]. We made observations on potassium nitrate crystals and spheres of sodium hydroxide which showed considerable density convection in horizontal flow up to the highest Re used by AKSEL’RUD [l] when he studied the former material. Our tests are qualitatively in agreement with equation (19). (4)
Turbulence and other interference
(i) Turbulence. This may be present in the stream or arise from the supports for the sphere. Turbulence also arises in the boundary layer of the sphere, but only above Re = lo6 which is outside the range considered by us. MAISEL and SHERWOOD [18] show that in the range 2000 < Re < 20,000 turbulence in the free stream of intensity greater than 5 per cent increases mass transfer. Some of the data in Table 2 show an increase in Tr at Re > 5000 consistent with increasing turbulence. For Re < lo4 a stream is probably laminar in effect if the turbulence is less than 5 per cent. Turbulence of 04 to 15 per cent (which we estimate from their range of critical Reynolds numbers) may partly account for the high values of Tr obtained by XENAKIS et al. [SO] at Re 2: 106. In those experiments of GARNER and GRAFTON [7] and GARNER and SUCKLING[8] in which screens were close used to
223
M.
LINTON
and K. L. SUTHERLAND
the test sphere excessive turbulence was probably present. In some of the experiments of Hsu and SAGE [2], GARNER and GRAFTON [7], CARY [2] and XENAKIS et al. (301, the sphere supports seemed excessively large or the sphere either was supported at the equator or at the front. All these arrangements lead to turbulence. We determined the effect which a single side support had at Re between 490 and 1120. The transfer rates behind the support were considerably increased and the wake on that side oscillated because vortices were shed periodically. The wake on the opposite side of the support was steady but even midway the influence of the support could be observed. With the support shown in Fig. 3 (c) the refractive index streamers, which could be readily observed in parallel light, trailed away from the separation point on both sides showing no visible asymmetry or instability due to the support. No asymmetry in the solution of the sphere due to the presence of the support was evident. (ii) Duct walls. The ratio of duct diameter to sphere diameter must always be sufficiently large to avoid interaction of their boundary layers. (iii) Dust. We have observed that the solution rate is decreased particularly at the front stagnation point by deposition of dust. Consequently care was taken to eliminate this error, but it seems to have escaped the attention of other investigators. Unfortunately’ there is no simple way of assessing its importance in measurements other than our own. (5)
Variation of &id
properties
The properties of the fluid vary with concentration or temperature and hence between the sphere and the free stream. Where these variations are appreciable average values are often used. In most of the work considered here errors due to this variation were negligible (e.g. this paper). (b) (1)
Comparison of theory and experiment Local transfer
r&s
The curves showing the variation of transfer rates with angle are shown in Fig. 8, and these
80
40
0
120
160
6PIG. 8.
The local
heat and mass transfer around
A CARY [2]
191;
A
; 0
SUTHERLAND (present
FR~SLING
XENAKIS data)
(this
;
et al. paper)
broken
numbers
spheres. [5] ; 0 NW and [39] ; l LINTON ; + Separation
lines,
separation
SAGE and point
points
of
other investigators.
are obtained by taking the average transfer, at a given angle, over the ranges of Reynolds numbers stated in Table 1. As we have already pointed out, many data do not satisfy the assumptions of the theory. Those of FR&SLING and ourselves appear to do so, and the data by Hsu and SAGE on evaporation at the 5 per cent turbulence level are possibly satisfactory. On the other hand, XENAKIS worked so close to the critical Reynolds number for a sphere that free stream turbulence was probably excessive. Likewise CARY’S work has been criticized on the instrumental side by KOROBKIN [12] and his ratio of duct to sphere diameter was too small. There is a wide variation in these experimental results particularly in Tr (0), the transfer number at the front stagnation point. In general the value of Tr (0) lies below the theoretical value (compare equations (4), (5) or (6) with Fig. 8 and Table 1). Part of the difference may be due to the scatter in results and for any investigator this is of the same order as those shown in our Fig. 6 and seems to be due to the difficulty of the measurements. The angular distributions are more easily
224
Transfer
from a sphere into a fluid in laminar flow
compared by considering the ratios Tr (Q)/Tr (0) as in Fig. 9. The theoretical curve derived by AKSEL’RUD [l] is also shown and it is seen that there is substantial agreement between all investigators and theory.
No closer correlation than this seems possible because the ratio of the integral transfer rate of the front half to the rear half decreases with Re (shown in Table 1) due to relatively increased transfer at the rear. This greatly influences the transfer. If Tr (0) varies with Pr as given in equation (4) then presumably Tr will vary in a like fashion ; however, as Table 1 shows the variation is only 10 per cent. AKSEL’RUD* [l] derived the over-all transfer for the particular case of Re < 1 and Pr > 1. The agreement is good even for KRAMER? data where 1 < Re < 40 and 200 < Pr < SS0.f
5.
20
0
40
60
80
100
CONCLUSIONS
We may conclude that, despite the many secondary factors influencing the experimental results, the boundary layer theory describes reasonably the distribution of the rate of transfer of either heat or mass around the leading half of a sphere. However further investigation is needed on the factors which cause theory and observation to deviate considerably in transfer at the front stagnation point. The form of the relation for the overall transfer rates connecting Nu, Re and Pr is suggested by the theory, and various experimental results lead to a useful set of limits, i.e. 0.5 < Nu Re-‘l2 Pr-‘/3 < 1.0 for the range lo2 < Re < 106, for a variety of systems.
120
60
FIG.9. Relative distribution of local heat and mass transfer rates on the front portion of a sphere. The curve is that given by AKSEL’RUD’S theory (see Fig. 1).
Aclcnor&dgements-We wish to acknowledge helpful suggestions on the manuscript by Dr. H. R. C. PRATT, Chemical Engineering Section, C.S.I.R.O.
Symbols as in Fig. 8, except, 0 GARNER and SUCKLINP, [8]
*FRIEDLANDER [4] has given (2)
Over-all
iransfer
analysis
rates
The over-all rate of transfer has been studied by many investigators (Table 2). The ranges of Re, Pr, Gr, Nu and d are wide. The value of Tr was calculated from these data where possible, average duct velocities were replaced by sphere approach velocities and the majority of results can be represented by 0.5 <
Tr < 1.0 for 100 < Re < lo5 0.5 <
and
Pr < lo8
(20) 225
but
it seems AKSEL’RUD’S analysis.
an
a inferior
similar theoretical approximation to
tTwo recent papers, (GARNER F. H. and KEEYR. B., Chem. Engng. Sn’. 1958 9 119 and 208) on mass transfer in forced convection at low Reynolds numbers and in natural convection respectively, only became available to us after the present paper had been written. The experimental results are similar to those presented in Fig. 6 and Fig. 8 of this paper. The work particularly confirms the important effect of density convection on both local and overall transfer rates in forced convection. The present authors do not agree with the derivation of the theory given in the first paper.
M. LINTON and K. L. SUTHERLAND
APPENDIX
Laminar
1
so that rr = 1, rs = -
The velocity and concentration distributions in the boundary layer are determined by the modified Navier Stokes, continuity, and mass-transfer equations. These for a body of revolution (about the axis of flow) are respectively (SCHLICHTING[24]) = U(&~
3 (ur) + 3 (*) &r 3Y
+ “g
u=--, JS JY
_ 0
(1.2)
U;;+“+$
$ldr
ha
r&
7=Y
X* = x/d,
2u, ( Y1
y* = y Rel/2/d,
1b*
=
“/urn,
1b,*
q* = y* (2ur*)r/*
Expressing
$
U,sinz
(1.4) I.0
(1.5)
U(+)=u,B+Us83+Ug~5+... us = -
U,J@,,, = U,/80rSo
c (Cs -
us = etc.
(1.6)
Similarly the surface of the sphere can be given in series form r = r,sin(r/r,)
...,
= rro + rszm + r5z5 +
...
U* (x) =
=
ll,,d”/U,,
,...,
Re112 and v* = v u co
(1.12)
us*) h3* (?*) etc.
(1.13
)
The concentration C in the boundary layer will also be a function of x and Y which may be expressed in series form
this in series form
we have u1 = 3U,/2ro,
(1.11)
We will designate the equations thus modified by (l.l*)-(l.ll*). Equations (l.l*) and (1.2*) can be solved by substituting equations (1.4*)-(1.9*) together with the appropriate boundary conditions and equating like coefficients of z*. One obtains a series of ordinary non-linear differential equations which can be solved numerically for the coefficient functions Jr* (‘I*), g,* (T+), h3* (~2) etc. where
on the sphere.
0
0
r/s
wrthn=l,a,5
U,sinB=
~1.10)
U (4/U,,
fs* (v*) = gs* (v*) + @a* ur*/rr* :
. .]
r* = r/d,
T*n = d”-’ rn with n = 1, 3, 5,
X
U(z)=
+
FR&SLING [6] modified equations (l.l)-(1.11) to make them dimensionless and independent of Y by, in effect, substituting
Y
of co-ordinates
(1.9)
where Jr (v), fs (v), etc. are coefficient functions expressing the dependence on y and
where (see Fig. 10) y is the distance normal to the surface of a point in the fluid, B is the distance on the sphere surface of the normal from the front stagnation point’ r is the distance from the axis of a point on the sphere surface, a, I) are the velocity components in the boundary layer parallel and normal to the surface respectively and To solve the above equations, C is the concentration. U(z), the velocity distribution outside the boundary layer (i.e. for y + co), must be known and assuming this to be the theoretical non-viscous “potential” flow we have for the sphere of radius r,,
Definition
b$
by introducing
+ 2usfs (7))zs + 326615(7+5
(1.3)
FIG. 10.
v=-----
(1.8)
which satisfies equation (1.2). Within the boundary layer the velocity distribution and hence I# is a function of both a and y and FR~SSLING [6] showed that in analogy to the series (1.5) # can be expressed
(1.1)
2
ld,, etc.
Equations (1.1) and 1.2) can be combined a stream function $ (z, y) defined by
boundary layer calculations of the local trarhsfer rates on the sphere
US+“:
l/6 rzo, r5 = l/l20
(1.7) 226
C,)
= c* = cs + c20*a
+ c*o*s
+ .
..
(1.142)
where the coefficients Cs, C,, C4, etc. are functions of y* (or v*). When substituting equation (1.14*) in equation (1.3*) a series of ordinary differential equations can be obtained for a new set of coefficient functions F,, (v*), G, (v*), H, (v*), etc. when one also substitutes (1 .I I*) and
Transfer
from a sphere into a iiuld m laminar
c, = Fo(11*), c, = 2% *
= -..% G, (‘I*) +
‘s*
k = 2 (rngljB - m11i2) 2 mo - ml ??I = ~ m,1/2 + mIl/2 t t
(1.15f)
II, (‘I*) etc.
71* us*
u1*
ilow
and the initial solution rate is
The equations involve coefficients fi*, fs*, etc. and were solved numerically by FIX&SLING up to terms involving x4. Now the local Nusselt number is given theoretically in terms of the concentration gradient in the boundary layer by
k,,, mo1i2 =
3
?? ~ ( 2Y 1 y=o C, -
d
mo1j2 +
zlc*
w
NOTATION
(1.16)
1 +=l-J 3y*
n n C c* d D
Substituting for (X*/J?*) in equation (1.26) from equation (1.14*) with appropriate substitution of the coefficient functions from equation (1.X*) we obtain F,’ (2ul*)l/2
= [-
+ Hi,‘) ,x*~ -
3 (2/u1*)‘/2
-
2 (2/u1*)li2
ug* F4’x4
-
us*
f1 (q), f3 (TJ)=
(1.17)
where dashes denote differentiation with respect to q*. For the particular value of Pr = 2.532, corresponding to nnphthalene vaporization into air, and T* = 0, FR~SSLINC [6] calculated F,’ = - 0760, F4’ = - 0.212, G,’ = - 0.319, and Ii,’ = - 0.101 so that = I.862
-
I.369 x*~ + + 0.2705 x*4 +
...
(1.18)
which on dividing both sides by Prl/s (remembering Pr = 2.532) and inserting x* = e/2 finally becomes equation (2) of the text.
APPENDIX
Initial
2
solution
rate
The over-all rate of solution for a sphere is dm
= kA (C, -
C,)
(2.1)
-dt where k = “D R&a d
Prl/s
(2.2)
as is borne out by the experiments. Expressing A and d throughout in terms of m and the density, P,,, we have dm
k,,, rn112
(2.3)
-z= where Ii2 (C, Integrating
C,)
= = = = = =
fi (O), f2 (0) =
. . .]
‘)* = 0
Nu (0) Ile-li2
(2.6)
=
(G
(G,’
m11/2
Cc0
= Re1i2
Nu (/3) Re-l12
2m01/2
- ml 1
Nu (0) = -
(2.5)
(2.4)
(2.3) 227
Fo (v*)* F, (v*), g k, m m0 m, r; ri rs, r6’ r,, * m
= = = = = = = = =
constant area of sphere cm2 concentration in solution g cm-s C/(C, - C,) diameter of sphere cm diffusion coefficient cm2 see-l functions of 0 defined in equations (2) and (7) respectively functions of 7 defined in Appendix 1
G,
(?*I,
4
(‘I*)
=
definedinAppendixl
gravitational acceleration cm set-2 constant defined in Appendix 2 mass of sphere g initial mass of sphere g mass of sphere at a time, t P -. radius of a sphere at zero time cm radius of a sphere zone after a time 1 cm constants defined in Appendix 1 rl’ d=-1 r n, 12 = 1, 3, 5, . . . r = polar radius, defined in Fig. 10 Ar (/3) = change in radius at angle 0 cm q = quantity of heat transferred cal 1=time set T = temperature OC T, = temperature on sphere surface “C u = velocity of fluid in the boundary layer parallel to the sphere surface cm se+ u* = u/u, 841, Us, 146, zr, = constants defined in Appendix 1 u,* = u, dn/U,, n = 1,3,5,. .. 72,U(z) = velocity of fluid external to the boundary layer cm se& U, = velocity of fluid at infinity (free stream) cm se+ o = velocity of fluid in the boundary layer normal to the sphere surface cm see-l v* = v Re112/U, 8 = distance on the sphere, from the front stagnation point, defined in Fig. IO cm ++ =x/d
M. LINTON and K. L. SUTHERLAND y = distance into the fluid, normal to the sphere surface, defined in Fig. 10 cm y* = y Re1/2/d OL= thermal diffusivity of fluid cm2 se& ,8 = volume coefficient of thermal expansion of fluid per ‘C ‘1 = y (226r/“)1/2 ?)* = y* (2u,*)1/2 X = thermal conductivity of fluid cal cm-l set-l “C-l 0 = polar angle radian @i = angular positions of zones on the sphere Atli = angle subtended by a zone of the sphere 0, = angle of separation degree Y = kinematic viscosity cm2 see-l p = density of fluid g cm-s g cm-3 Pm = density of sphere # = a stream function defined by equation (1.9) Appendix 1 * = Asterisk denotes dimensionless quantities and equations Subscripts Unless otherwise 0 s 03 B
= = = =
refers refers refers refers
specified to to to to
1 k=
1 k(O)= NUA
Nu (8) =
Over-all mass transfe coefficien
dm
Y4(C,-CC,)dt b2m
-(C,-C&
(
h4iJt ) B
Local mass transfer coefficieni
Over-all Nusselt number
D
k (0) d
Local Nusselt number
D
Pr = v/D
Prandtl number
Re = Ud/v
Reynolds
Cr I = @* B(T, vs h=
-
T,)
Grashof number
1 A (T, -
T,)
Over-all heat transfer coefficient
!!! dt
Local heat transfer coefficient Nu = hd/X Nu (0) = h (0) d/X Pr = v/a
&er-all Nusselt number Local Nusselt number Prandtl number
For heat and mass transfer
initial values (t = 0) saturated solution values in the free stream (at infinity) local values at 0
Tr = Nu Re-r/s
Pr-11s
Tr (0) = Nu (8) Re-li2
Pr-ll*
Over-all transfer number Local tmnsfer number
Tr (0) = Nu (6) Re-l12 Pr-ri3
Transfer number at front stagnation point
Nu (O” -
Average Nusselt numbers fGO” < 0 < ‘90, and 90” < f? < 180” respectively
Definitions For mass transfer Grashof number
OO”), Nu (90’ -
180’)
REFERENCES PI
AKSEL’RUD G. A. Zh.&.
PI
CARY
r31
FACE A. Aeronaut.
khim. 1953 27 1445.
J. R. Trans. Amer. Sot. Mech. Engrs.
number
For heat transfer
1953 75 483.
Research. Conun. Rept. and Mem. No. 1766 1937.
[41
FRIEDLANDER S. K. Amer. Inst. Chem. Engrs. J. 1957 3 43.
[51
FRBSSLINGN. Gerlands Beitr. Geophys.
PI
FR~SSLIN~ N. Lunds Univ. Arsskr. N.F. AVD. 36 No. 4 1940.
PI
GARNER F. H. and GRAFTONR. W. PTOC.Roy. Sot. 1954 A 224 64.
1938 52 170.
PI
GARNER F. H. and SUCKLINGR. D. Amer. Inst. Chem. Engrs. J. 1958 4 114.
[91
Hsu N. T. and SAGE B. H. Amer. Inst.
Chem. Engrs. J. 195’7 3 405.
[191
INGEBO R. D. Natl. Advisory
Comm. Aeronaut. Tech. Notes No. 2368 1951.
PI
KAWAGUTI M. Rept. Inst. Sci. and Technol. Univ. Tokyo
WI
KOROBKIN I. Amer. Sot. Mech. Engrs.
1131
KRAMERS H. Physica 1946 12 61.
P41
KREVELEN D. W. VAN and KREKELS
P51
KROUJILINE G. Zh. tekh. fiz. 1938 5 289.
WI
LINTON W. H. JR. and SHERWOOD
1950 4, 154. (C.S.I.R.O.
Translation
Paper 54-F-18 Milwaukee Meeting, September
J. T. C. RecueiZ1956
69 1519.
T. K. Chum. Engng. PTO~T.1950 46 258. 228
1954.
No, 2466).
Transfer from a sphere into a fluid in laminar flow [17]
MCADAMS W. H. Heal Transmission (3rd Ed.) McGraw-Hill
[18]
MAISEL D. S. and SHERWOODT. K. Che9n.Engng. Progr. 1950 46 131, 172.
New York 1054.
[lQ]
MAXWELL
[20]
PIRET E. L., KYTE J. R. and MADDEN A. J. Chum. Engng.
[21]
PIRET E., MATHERSW. G. and MADDEN A. J.
[22]
POHLHAVSEN E. Z. angers. Math.
[23]
RANZ W. E. and MARSHALLW. R. JR. Chem. Engng.
[24]
SCHLICHTINCH. GrenzschichtTheotie G. Braun, Karlsruhe 1951. Engl. Transl. by Kestin, J., Bownduy Theory. McGraw-Hill, New York 1955.
[25]
SIBVLKIN M. J. Aero. Sci. 1952 19 5’70.
[26]
TANEDA S. J. Phys. Sot. Japan 1956 11 1104.
[27]
TANG Y. S., DUNCAN J. M. and SCHWEYER H. E. Natl. Advisory Comm. Aeronaut. Tech. Notes No. 2367 1953.
R. W. and STORROWJ. A. Chem. Engng. Sci. 1957 6 204.
I&us~T.
PTogr. 1953 49 653.
Engng.
Chem. 1957 49 Qtil.
Mech. 1921 1 115. Progr.
1952 48 141, 173, 247. LU@T
[28]
WILKE C. R., TOBIAS C. W. and EISENBERG M. Chenz.Engng. Progr.
[29]
WILLIAMS G. C., Sc.D. Thesis in Chemical Engineering, M.I.T. 1942 ref. McAdams [17].
1953 49 663.
[30]
XENAKIS G., AMERMAN A. E. and MICHELSONR. W., Wright Air Development Center, Tech. Rept. 53-117, Ohio, April 1953.
Engineering Science, lOsO,Vol. 12,pp.215to80% l'ergomon Press Ltd., London.
Transfer
Printed in Great Witsin
from a sphere into a fluid in laminar flow M.
LINTON*
and
Division of Physical Chemistry, C.S.I.R.O.
K.
L. SUTHERLAND*
Chemical Research Laborstories,
Melbourne
(Received 28 November 1252) Abstract-Boundary layer theory and experimental results for the transfer of heat and mass from spheres in forced flow are compared over the range of conditions: 1 <
Re < 1.8 x 106, 0.5 < Pr < 106, 2 < Nu < 540.
The overall and local solution rates of 2/S in. diameter spheres of benzoic acid were measured in uniform ilow in a water tunnel at 490 < Re < 7580. The overall transfer was correlated by Nu = 0.582Relia Prlis The relative distribution of local transfer rates over the front half of the sphere was in fair agreement with theory and other workers but the absolute local values at the front stagnation point were about 44) per cent lower than theory and differed considerably among other workers. The discrepancies may in part be due to a gradual transition from viscous flow to potential flow outside the boundary layer as Re increases from 1 to 10 5. The relative transfer from the rear half of the sphere was observed to increase with Ii%. The angle of separation 0, found experimentally varies approximately according to 0 s = 83 + 101 Re-li3 The average transfer rate over the front of the sphere is, theoretically proportional to Re@ Pr1/3. Over the whole sphere 0.5 < Nu Re-li2 Pr-lls < 1.0 for IO2 < Re < 105. The approximation is only rough because of the factors dependent on Re mentioned above, and the following secondary factors : a lower limit to transfer rate set by diffusion or conduction, of the free stream, density convection unless Gr Q Re 2, turbulence and/or non-uniformity interference from sphere supports, influence of duct walls, uncertainty in transfer properties of fluids, variation of fluid properties with concentration or temperature, and dust accumulation on the sphere. R&sum&-La couche limite theorique et les r6sultats experimentaux pour le transfert de chaleur et le transfer% de masse a partie de spheres dans un t!coulement force, sont compares dans l’intervalle : 1 < Re < 1,s x lo6 + 0,5 < Pr < log + 2 < Nu < 544.3. Les vitesses de dissolution locale et globale de spheres d’acide benzoique de 3/8in. de diambtre ont et6 mesurdes dans un tunnel d’eau ii 490 < Re < 7580. Le transfer% global est. doMe par Nu = 0,532 Rel/* Prlls La distribution des vitesses de transfer% sur la moitie avant de la sphere s’accorde bien avec la theorie et les autres expt%mentateurs, mais les valeurs locales absolues au point de stagnation avant, sont environ 40% inf&ieures A la theorie et varient avec les exp6rimentateurs. Les differences sont dues en partie B la transition graduelle de l’dcoulement visqueux en Bcoulement potentiel en &hors de la couche limite quand Re augmente de 1 B 106. Le transfer% relatif sur la moitie arriere de la sphere augmente avec Re. Le point de separation 0, trouve expdrimentafement varie comme 0, = 83 + 191 Re-l/s. La vitesse de tranafert moyenne sur le front de la sphere est thkoriquement proportionnelfe a Rer/s pyr/s. Sur la sphere entiere 0,5 c Nu Re-l/s Pr-l13 < 1,0 pour lo2 < Re < 10s. L’approximation est un peu gross&e ir cause des facteurs dependants de Re deja. mentiormes et des facteurs secondaires suivants : limite inferieure de vitesse de transfert par diffusion ou 214
Transfer
from 8 sphere into a fluid in lsminar
flow
conduction ; importance de la convection (sauf pour Gr Q Res) turbulence et non uniformiti du courant libre ; interf&ence des supports de spheres ; influence des parois de la conduite ; incertitude des propriet& de transfer+. des fluides, variation des propri&.es du fluide avec la concentration ou la temperature, et accumulation des poussieres sur la sphere. Zusammenfassung-Fur den W&me- und Stoffiibergang an Kugeln in enwungener StrGmung wurden die Ergebnisse der Grenzschichttheorie und der Versuche miteinander verglichen und zw8r hn Bereich von 1 < Re < 1,8 x 10 5;0,5
,
Nu = 0,582 Re112 Pr1j3 Die relative Verteilung der ijrtlichen Ubergangskoelhzienten auf der Stirmseite der Kugel war ln guter ubereinstimmung mit der Theorie und anderen Versuchsergebnissen, aber die absoluten Brtllchen Werte am vorderen Staupunkt waren etwa 4Q% unter denen der Theorie und differierten betr%chtlich mit fremden Versuchsergebnissen. Die Unstimmigkeiten m%en zum Teil durch einen schrittweisen Ubergang vom z&hen Fliessen zur Potentialisttimung ansserbalb der Grenzscblcht beim Anwachsen von Re von 1 bis lo5 verursacht sein. Die relative ubertragung an der Riickseite der Kugel stieg mit Re an. Der Winkel des Ablijsungspunktes e, 5ndert.e sich nsch den Versuchsergebnlssen ungef5hr nach der Gleichung 8 8 = 83 + 191 Re-l13 Die mlttlere vbertr8gung an der Vorderseite der Kugel ist nach der Theorie proportional zu Re112 Prl’s. Uber die ganze Kugel ist 0,5 < Nu Re-lls Pr-l13 cc 1,0 fur lo2 < Re < 106. Es handelt sich dabei urn grobe Merung wegen der oben erwiihnten von Re abhihqigen Faktoren und wegen folgender sekund%rer Faktoren: Eine untere Grenze fur die Ubertagung lnfolge Diffusion oder wiirmeleitung, Konvektion durch Dichteunterschied, sofern nicht Gr < Re2, turbulenter und/oder nicht einheitlicher Freistrom, Einwirkung der K~gelhalterung, Ein6uss der Kanalwiinde, Unsicherheit in den Stoffgr&sen der Fliissigkeit, Abhitngigkeit der Stoffgr&ssen von der Konzentration oder der Temperstur, Staubsns8mmbmg auf der Kugel. 1.
INTRODUCTION
THIS paper compares the transfer predicted from the theory of a laminar boundary layer, with experimental results, for local heat and mass transfer around the front of a sphere which is held in a moving fluid. The theory suggests also an approximate but very general expression for overall transfer, and experimental deviations are shown to be due to the influence of many secondary factors. The transfer of mass or heat from a solid sphere into a fluid flowing at moderate velocities has been treated theoretically by FR~~SSLING [6] and AKSEL’RUD [ 11. FR~~SSLING [6] calculated the velocity distribution and mass transfer* in the laminar boundary layer using power series. By a method outlined in Appendix 1 below, he calculated numerically the *For convenience we will refer mninly to mass transfer although analogous statements apply to heat transfer.
coefficients in the series for the particular case of Prandtl numberl_ = 2.532 (corresponding to naphthalene evaporating in air). We will express the local rate of mass transfer as the dimensionless local Nusselt number Nu (8) defined in terms of experimental quantities by
0
where B is the polar angle, d the diameter of the sphere, D the diffusion coefficient, m the mass of the sphere, A the area of the sphere, t the time, C, is the concentration at saturation and C, that in the free stream. FR~~SSLING [6] was able to show that quite generally Nu (19) co Re1j2 and that for the particular case of Pr + co (when the transfer boundary layer is thin compared to the flow boundary layer) Nu (0) oc Pr”“. Consequently the quantity Nu (8) Re-‘Is Pr-“’ which tFor
215
(1)
definitions see the end of this paper.
M. LINTON and K. L. SUTHERLAND
we shall designate as the transfer number Tr (8) approaches a constant value for any given value of 8. Following the method of FR~~SSLING[6] we obtain in the manner shown in Appendix 1 for the sphere at Pr = 2.532
for Pr + w using the approximate polynomial method. The following equation is derived from AKSEL’RUD’S Tr (8) =
1.523 (1 -
Tr (8) = Nu (8) Re-‘12 Pr-‘/s = 1.366 (1 -
published
graphs
0.1728
0.0114
Q2 -
8*) =
1.523fs
(8)
(7)
I.0
0.1837 82 +
+ 0.00696 8* + . . .) = 1.366f, (8)
(2)
Because FR~~SSLING was unable to eliminate the Prandtl number from his equations generally the coefficients in equation (2) are functions of Pr. From the particular solution, Pr + co, one finds at the front stagnation point (B = 0) Tr (0) = Nu (0) Re-‘12 Pr-‘/a = 1.472
0.8
(3)
From coefficients tabulated by FR&SLING [6] we calculated the transfer at the front stagnation point for the range 0.5 < Pr < 100 and can represent the variation with Prandtl number to f 0.5 per cent by 0.22 0
Tr (0) = 1.478 -
30
0.158 PrM1/afor 0.5 < Pr < co (4)
FIG. 1. The theotetical distribution of transfer rates with polar angle, taking that at the front stagnation point as unity.
Other approximate solutions to the transfer number at the stagnation point have been derived by FR~~SSLING [6] and SIBULKIN[25]. FR&SLING [6] following POLHAUSEN[22] and KROUJILINE [15], used polynomials to express the velocity and concentration distributions and from his solutions we find that, to f 1 per cent, Tr (0) = lGi3 -
0.190 PT’/s
0.1 < Pr < co (5)
whilst SILBULKIN using similar expressions but with different integral equations gave values which we can represent to f 0.5 per cent by Tr (0) = 1.50 -
0.182 Pr+s
0.6 -C Pr < 10 (6)
Thus the theoretical transfer numbers at the front stagnation point are in quite reasonable agreement when calculated by three different methods. Equations (P)-(6) indicate ‘that Tr (0) only varies about 15 per cent as Pr varies from co down to 0.5. AKSEL’RUD[l] has also calculated the variation in mass transfer around the sphere from the front stagnation point to the separation point
60 e*
: theory of FIGSSLING at Pr = 2.532, Equation (2). fi (0) Full curve : theory of AKSEL’RUD at Pr = CO,f2 (0)
Broken curve
Equation
(7).
The distribution of transfer rates c&lculated by the exact method by FR~~SSLING [6] for Pr = 2.532 -fi (19) in equation (2)-and by the approximate method by AKSEL’RUD [l] for Pr = w - fa (8) in equation (7) - are plotted in Fig. 1. The agreement between the two curves is good up to 60”. Beyond this point fi(0)will not be accurate due to the absence of terms higher than 8* which have not been calculated so far. Thus fi (0) and fa (0) are independent of Re and Pr. Hence Tr (0) is independent of Re and varies only slightly with Pr due to the variation in Tr (0) discussed above. The above theories do not, so far, predict mass transfer at the rear of the sphere, or consequently
216
Transfer from a sphere into a fluid in laminar flow
the total mass transfer, because of separation of the flow with its profound effect on the boundary layer. We have confined a comparison of experiment and theory (Section 4b) to the leading surface of a sphere. AKSEL’RUD [l] calculated the total transfer over the sphere by graphical integration and obtained From pump
14in. squore ‘low diffuser
.l5in.
Tr = 0.80
He assumed the transfer to be constant for 8, Q B < 180’ and the separation angle 0, to be llO“, but the experiments presented below show that neither of these assumptions is valid. However, if we postulate that in the wake a velocity and diffusion boundary layer develop from the rear stagnation point and obey the same form of relations as those at the front but differing in the values of the constants, then the overall transfer number could again be constant except insofar as the separation point varies and Tr (0) varies with PT. The available experimental data discussed in Section 4, show that Tr does lie within fairly narrow limits.
square constonl head inlet section
2.
25:l
‘2
-1,-
3 in. square
tronsporent working section with
c’
‘5
Groduoted velocity control mechanism i
Contraction
screw adopter
opening
f-
EXPERIMENTAL
METnoD
Wder tumel
(a) _ Square grid of &in. oluminium sheet 01 ‘4 in. cenlres
(8)
Uniform flow for velocities of 3-50 cm set-l was achieved over a considerable area of the working section of 3 in. x 3 in. of a vertical water tunnel by the design shown in Fig. 2 (a). The water was fed from a constant head, the level being maintained by a centrifugal pump and overflow weir. An essential feature is the diffuser, shown in Fig. 2 (b), the remaining vorticity after passage through this being removed by an aluminium grid at the lower end of the inlet section. The design of the contraction section was supplied by the Aeronautical Research
3in. square to 3in. diom.
I
round section 2in.2 k-
feed pipe
3in. pt round section
&-.--~7
Brass control valve and seat
ISin.
flange,
_ 30in.squore discharge receiver box To Row diffuser !$in. O.D. pwcelain
beods
.3in. porcelain insulating eods in co, Iin. layer Choin drive’
I%&=
14in.;quore golvonized diffuser box
:ible hose connex
iron
‘Perfomted
(b) The diffuser
(a) Main section FIG. 2.
The
water tunnel
217
zinc screen
M. LINTON and K. L. SIJTHERLANII %&I. Wall lin. LD. rubber tube
Hose clip (a) Adapter
7/8in.O.D.Xfin.Sross tube with end plates
he1nZS.S.
Spot weld qround rmooth
FIG. 3. (a)
Unit for insertinp.sphere
Mounting
for sphere
(b) Support cylinder
into the tunnel
Laboratory, Dept. of Supply, and was smoothly connected into the polymethylmethacrylate working section which had 1.5 in. diameter screw ports at the sides. These ports carried the mounts for test objects. A square to round conversion section led to the circular outlet where the flow was controlled by a blunt-nosed conical valve. Water collected in the receiver box was fed to the pump by flexible hose to prevent transmission of vibration. The tunnel was supported by a suitably braced steel frame. Most of the construction was in wood, internal surfaces being coated with water-resistant resins. It would be more desirable-but more costly-to
(c)
Sphere support
construct the tunnel in metal which would avoid The tunnel required swelling and distortion. regular cleaning, algal growth being controlled by small additions of copper sulphate. The raw water supply contained sufficient suspended matter to cause difficulties and it was treated with an alum-lime mixture*, the floe being allowed to settle for 24 hr when the supernatant was found to be sufficiently clear that it did not appreciably scatter a light beam. The water temperature was controlled by heating or cooling coils to 20.0 f 0*3’C.
218
*135galum
and 45gslaked
lime in 3Wgal
water.
Transfer from a sphere into a fluid in laminarflow
(b)
Spheres and th+Grsuppvrt
Each test sphere was moulded on a fine stainless wire attached to a heavier support wire (Fig. 3c) which fitted into a cylinder (Fig. 9b) carried, as a slide fit, by the adaptor (Fig. 3a) which screwed into the port. The cylinder acted as a water seal and when being removed escape of water was prevented by using a clamp closing the rubber tube. The adaptor and cylinder were fitted flush with the internal wall. Solution in the time (15 see) required to insert or remove a sphere was negligible. The sphere was centred in the tunnel by projecting an image of it and the working section on a ground glass screen. To reduce any effect which dust may have on transfer, the spheres were rotated rapidly for a few seconds every 10 min. Where there was any sign of dust this treatment led to its removal. The spheres were of benzoic acid, pressed in a mild steel mould heated to 110 “C. A mark on the support wire served to orient the sphere. The sphere profiles were measured before and after solution either with a co-ordinate travelling microscope or from tracings done by hand at 30 magnifications using a profile projector. The spheres before solution were actually prolate spheroids the major axis being 0-005cm longer than the minor. The mean diameter was 0.958 f 0.003cm and the mean density 1.286g cm-3. The rate of solution of the spheres was a maximum at the front, a minimum near the point of separation and rose to a smaller maximum at the rear. From the resulting change in shape measured as changes in radius, the local and overall solution rates were calculated as indicated in Sections 3 (b) and 8 (c) below. The spheres and their support wire were also weighed before and after solution and the overall solution rate calculated as shown in Section 3 (b) below. The experiments lasted between one and four hours, in which time the average diameter was reduced to about 90 per cent of the initial value. (c)
pressure mercury arc emitting 100 flashes/set illuminated the working section through a tall beaker filled with water acting as a cylindrical lens. The traces were then photographed using a 6 in. lens. The magnification was determined using a known object in the image plane.
3. (a)
EXPERIMENTAL
RESULTS
Unifbrmity of jZow in the wder tunnel
The flow in at least the central 4 cm x 4 cm of the working section was uniform to within the accuracy of measurement of velocity, viz. f 1 per cent at 3 cm set-1 and f O-1 per cent at 54 cm set-l. The turbulence, as indicated by the maximum observed lateral displacement from the vertical path was less than 0.5 per cent in the same zone. At 54 cm set-’ up to 3 per cent departure from vertical path could be observed at 1 cm from the wall but the internal core of the working section was still satisfactory. (b)
Over-all solution rates
The sphere changed both its size and shape during the several hours in which it was being dissolved. The theory presented above applied to the i~nitial solution rate. Fig. 4 shows that the average solution rate (m,, - m,)/t during a period of solution t, varies linearly with the square root of the flow velocity, (or Re”‘). As
Velocity distribution
The velocity distribution in the working section was determined from photographs of the movement of aluminium flakes. A 250 W a.c. high 219
FIG. 4. The solution rate (averaged over the period t) of a benzoic acid sphere (d = 0.958 cm) as a function of water flow velocity at 20 “C. The straight line is (mO - ml)/ = 0.439 x 10-6 u,l/z.
hf. LINTON
shown in Appendix 2 the initial solution rate can hence be calculated, assuming the sphere remains spherical, as
m. -
ml
2rnbi2
’ mo1/2 +
t
m11/2
(9)
where m, is the initial mass and m1 that after a time t. The observed change in shape was less than 0.1 diameters and the linearity of the points in Fig. 4 indicates that its effect on the solution rate was negligible. The overall Nusselt number for the sphere was obtained by substituting the initial surface area and the initial solution rate derived from equation (9) in equation (1) in The diffusion coefficient of integrated form. benzoic acid in water at 20 “C! was taken as 0.882 x 1O-5cm2 see-l (WILKE et al. [28]) and the saturation solubility as 3.11 g 1-l (KREVELEN and KREKELS [14]).These give Pr = 1210. The mean density of the pellets, 1.286 g cm-s, was used to calculate the local and integral rates of solution from the change in size. The integral solution rates are determined by substituting equations (10) and (11) in (1) and integrating, where,
Nu
pr-‘/’
=
0.669
Re”‘481
(12)
with a coefficient of correlation of 0.984. The best fit assuming that n = l/2 is Nu Pr-“a (c)
O-582 Re112
(18)
Local solution rates
The local values of the Nusselt number, Nu (e), were obtained from J2 m
3A=
Ar (8)
Pm At
(14)
and equation (1) where Ar (8) is the change in radius at angle 8. The results are plotted in Fig. 6.
(10)
rno1i2+ m11i2
At
at positions Bi (75” + i 15",i = 0, 1, 2, . . . ll), and ABi = 15”. Slight irregularities in the final sphere shape caused the calculated solution rates to be less accurate than those obtained by direct weighing. The variation of Nusselt number with Reynolds number obtained by both methods is shown in Fig. 5. The results by direct weighing were fitted by least mean squares to an equation of the form Nu Pr-‘Is = a Ren. The best fit is
I.2
2rn,‘/’
~rnzA&,
L. SUTHERLAND
and K.
0”
F
where
I.0 ‘0.6
‘;
AC = (7r/2) Aei(ro- T,)(r.+ pi)2 sin Si
(11)
h N c 3
is the loss in volume from a zone of the sphere, the ri are the radii after an interval of time At
s
0.6 0.4 0.2
PIG. 6. The local solution rate of benzoic acid spheres in water in forced convection. + Separation of laminar layer, Reynolds numbers are as shown. The broken curve is Nu (0) Pr-r/s Re-ri2 = 0.94 (1~00-6050 B2-O*069 e4) and is the best fit to the data.
1
00
Re
FIG. 5. The overall solution rate of benzoic acid spheres in water in forced convection. The linear relation is given by Nu = 0.582 Re1i2 P&s. l By direct weighing A By integration over the surface
The best fit of an equation of the form of equation (2) to the mean values of Tr (8) at O’, 30°,60' and 90” is Tr
220
(e) =
0.94(1.00 -
0.05082 - 0.06984) (15)
Transfer from a sphere into a fluid in laminar flow
(d)
0, = 83’ for large Reynolds numbers ; the value of 83” was found by FAGE [3] at Re = 1.57 x 106. Relation (17) is in good agreement with observations in the range 15 < Re < 1000, made by
Angle of separation
After a sphere had been dissolving for some time, it developed a marked ridge where the solution rate was a minimum (0 N 90°-100°) and which was close to the zone of separation of the boundary layer. This separation was detected by illuminating the system with accurately parallel light and studying the stream lines visible due to differences in refractive index. The separation as observed by us from the position of the ridge is fitted by the relation 0, -
GARNER from
Expressions
data
GRAFTON
obtained
evaporation Fig.
7.
Nusselt over
of
83 = 660 Re-‘/’
(13)
83 = 191 Re-‘/’
numbers
the
ference
by
by
of
are
at each 8.
the
are
to note
the overall
on
the
shown
that
to
and
the
their
circumcontribu-
and as separation
4.
DISCUSSION
Requirements
of Theory
The theories of transfer considered introduction to this paper assume
in the
(1) potential
Id
IO'
flow outside the boundary layer, uniform far from the sphere. (2) the velocity and transfer boundary layers are thin compared to the radius of the sphere. This ensures negligible contribution by molecular diffusion. (3) negligible natural convection. (4) freedom from turbulence or other interference with flow. (5) constant fluid properties.
IO'
Re
FIG. 7. The angle of separation, B,, as a function of Reynolds number. 0 FR&SLIN~ [5] ; l GARNER and GRAFTON [7] ; x LINTON and SUTHERLAND(this paper) ; - - - TANEDA [26].
Table 1. Summary
of experiments -
Re
Transfer
Ekperimen
__
on the distribution
1
--
of transfer rates on. the sphere
References
Theory
Nu (W-Son) Nu (W-180*)
(Eqn. (4))
Mass
136724
1.23
1.37
FR&.SLINO[5]
3.9
Mass
200-I 766
1.11
1.46
GARNER~~~&JCKLING[~]
1.6
Heat
1530-4206
1.16
1.30
Hsu andS~~~[9]
2.6
Mass
1490-7580
0.94
1.46
This paper
1.9
Heat
44-15
x 104
0.77
1.30
CARY [2]
1.0
Heat
87-18
x IO4
1.46
1.30
XENAKIS
-
is
influence on
rate.
(a)
Kl
in
the overall
values
close to this zone it has a considerable
to
[5]
also
the maximum
at the equator
and
from the summation
local
proportional Thus
GARNER
Values derived
FR~SSLING
are obtained
surface
tion is made
(17)
(16) and (17) are asymptotic
[7],
naphthalene
It is important
contributions
as shown in Fig. 7 but nearly as good a fit is 0, -
and
SUCKLING [8] and by TANEDA [26].
221
etal.[30]
1.02-0.66
nitrate
Water
Benzene
Mercury
M
M
M
H
M
H
H and M
Benzene
Aniline, water,
Bensoic acid
Benaoic acid
1%
M
Methanol
n-Heptane
Adipic acid
Bensoic acid,
Bensoic acid,
Naphthalene,
wate
Aniline, benzene
Benaoic acid
Potassium
Solute
H H H
_M M
M M H M
Tratlsj-.?rN
Air Air Air
Air .4ir Air Water Oil Water Air Air Air Water Air
0.7 0.7 0.7 73-10.7 2ocb380 1210 0*8-0~7 0.6-0.7 0.5 1210 0.60.7
1210 120&1525
1560 2.3 x lo6 0.7 0.6-2.53
_-
Pr
. _
-
I
-
_
7
-
-
300-480
1.4-11 x 10’
d
I
-_
Y
_,“._
0.42 + 2f 0.43 Re0.06 0.37 ReO’l
0.61 1.18 0.66 + 3.2f 0.56 + 3.7f 0.56 + 5-Oj 0.43 Re0’06 0.43 Re0.06 0.65 0.55 + 2j 0.582 o%o + 2f
0.55 + 54 f 0.63 + 2 f
J
[7]
[lo]
[29][
_L
..--
XENAKIS et al. 301
WILLIAMS
TANG et al. [27]
_
HANZ and MARSHALL [23]
This paper
MAXWELL and STORROW [I91
MAISEL and SHERWOOD [la]
LINTON and SHERWOOD [Ia]
KRAMERs [13]
I~oEn0
Hsu and SAGE [B]
GARNER and SUCKLING [8]
. > ‘\
j
7
[5] GARNER and GRAFION
FR&SLING
GARY [2]
AKSEL’RUD [l]
AKSEL’RUD [l]
References
convection
T
in forced
0.82 1.1 Re-l/e 0.59 0.55 + 2f
pr-l/3
0.32-1.6 0-033-15 15.3-30.5
0.95 0.064.11
1.2 O-69 0.7-1.26 0.7-1.26 0.7-l -26 1.27 2.6-3.5 2.6-3.5 -
1.27 0.95-1.9
0.4-o.7 12.7 o*olLo~o9
-
(cm)
tf = &-l/z
622 -
25-35 39-87 1l-30 11-45 lo-30 31-65 4O-160 40-160 P-30 120-510 2-10
70-240 loo-280
120-540 60-200 loo-200 2-20
Nu
20-l 500 -
c 10
104 1.7 x 103 220-6300 350-1900 1.5-10 x 104 1.3-10 x 104 106 10s
M = mass
,~. _ ,,, ..- _
*H = heat
-
1.3 x 104 N 104
10s
Gr
on over-all transfer of heat and mass from spheres -
SO-1OOO 0.67 0.7 2-50 x lo3 8.7-18 x 104 0.7
15OO-42OO 16OO-5720 10&2ooo 60-1OOO l-40 2ooo-12,600 2oo&42,ooo 3ooo-42,ooo 2-1500 300-7600 2.1-200
2O-850 loo-700
lOO-3500 0.1-2.5 4-15 x 104 2.3-800
Water Oil Air Air Water Water
Re
Fluid
Table 2. Summary of experiments
.
.
Transfer from a sphere into a fluid in laminar flow
mentally for natural convection alone from flat plates that
Of the many observations (Tables 1 and 2) reported on transfer from a sphere to a flowing fluid few satisfy all these assumptions. To compare theory with both our own data and those of others these conditions are examined in detail. (1)
Nu = 2 + 0.5 (Gr Pr)“‘25
Potential flow (fluid of xero viscosity)
Diffusion,
and boundary layer thickness
For the theory to apply the velocity and transfer boundary layers must be thin compared to the radius of the sphere. The latter ensures that transfer by diffusion is negligible and requires that Re’j2 Pr’/s > 2. This condition often fails to apply, particularly to the evaporation of liquid drops (see Table 2). Thus the temperature profile determined by RANZ and MARSHALL[23] for a typical evaporating drop shows that the thickness of the temperature boundary layer at the equator was 8.0 drop radii. (8)
1Oa < Gr Pr < lo6 where & is the Grashof number. The first term is for diffusion and the second term arises from density convection and will only be much less than forced convection if Gr < Re’j2 Pr1/12 (19) in the range
The theory of FR~SSLING can be applied to potential flow with any arbitrary velocity distribution outside the boundary layer. The potential flow around a sphere immersed in an infinite fluid is assumed in the analysis given in Section 1 and Appendix 1. Real flow around the front of the sphere changes gradually from viscous flow at Re < 1 to approximately potential flow at Re > lo6 (as can be seen by comparing theoretical experimental pressure distributions and (SCHLICHTING[24] Fig. l-9). Thus at Re = 20 the pressure distribution calculated by KAWAGUTI [ll] is intermediate between those for viscous flow and potential flow. We achieved an initial uniform velocity profile in the water tunnel close to the throat of the contraction section and worked at Reynolds numbers of up to several thousand to approximate to potential flow. Experiments in which the flow was parabolic e.g. GARNER and SUCKLING [8], would not be expected to agree accurately with the theory. (2)
(13)
Density convection
When a temperature or concentration gradient is high, natural convection currents are supefimposed on the forced flow altt?ring the rate of transfer. PIRET et al. [20], [21] found experi-
Although based on a flat plate, condition (19) should also apply to the sphere. Table z shows that Tr increases with decreasing Re and this is consistent with the increasing importance of the density convection or diffusion. GARNER and SUCKLING [8] observed density convection in horizontal flow at Re = 200 but the effect would be present to the limit given by equation (19). Such an effect is clear also in KRAMERS’ results [IS]. We made observations on potassium nitrate crystals and spheres of sodium hydroxide which showed considerable density convection in horizontal flow up to the highest Re used by AKSEL’RUD [l] when he studied the former material. Our tests are qualitatively in agreement with equation (19). (4)
Turbulence and other interference
(i) Turbulence. This may be present in the stream or arise from the supports for the sphere. Turbulence also arises in the boundary layer of the sphere, but only above Re = lo6 which is outside the range considered by us. MAISEL and SHERWOOD [18] show that in the range 2000 < Re < 20,000 turbulence in the free stream of intensity greater than 5 per cent increases mass transfer. Some of the data in Table 2 show an increase in Tr at Re > 5000 consistent with increasing turbulence. For Re < lo4 a stream is probably laminar in effect if the turbulence is less than 5 per cent. Turbulence of 04 to 15 per cent (which we estimate from their range of critical Reynolds numbers) may partly account for the high values of Tr obtained by XENAKIS et al. [SO] at Re 2: 106. In those experiments of GARNER and GRAFTON [7] and GARNER and SUCKLING[8] in which screens were close used to
223
M.
LINTON
and K. L. SUTHERLAND
the test sphere excessive turbulence was probably present. In some of the experiments of Hsu and SAGE [2], GARNER and GRAFTON [7], CARY [2] and XENAKIS et al. (301, the sphere supports seemed excessively large or the sphere either was supported at the equator or at the front. All these arrangements lead to turbulence. We determined the effect which a single side support had at Re between 490 and 1120. The transfer rates behind the support were considerably increased and the wake on that side oscillated because vortices were shed periodically. The wake on the opposite side of the support was steady but even midway the influence of the support could be observed. With the support shown in Fig. 3 (c) the refractive index streamers, which could be readily observed in parallel light, trailed away from the separation point on both sides showing no visible asymmetry or instability due to the support. No asymmetry in the solution of the sphere due to the presence of the support was evident. (ii) Duct walls. The ratio of duct diameter to sphere diameter must always be sufficiently large to avoid interaction of their boundary layers. (iii) Dust. We have observed that the solution rate is decreased particularly at the front stagnation point by deposition of dust. Consequently care was taken to eliminate this error, but it seems to have escaped the attention of other investigators. Unfortunately’ there is no simple way of assessing its importance in measurements other than our own. (5)
Variation of &id
properties
The properties of the fluid vary with concentration or temperature and hence between the sphere and the free stream. Where these variations are appreciable average values are often used. In most of the work considered here errors due to this variation were negligible (e.g. this paper). (b) (1)
Comparison of theory and experiment Local transfer
r&s
The curves showing the variation of transfer rates with angle are shown in Fig. 8, and these
80
40
0
120
160
6PIG. 8.
The local
heat and mass transfer around
A CARY [2]
191;
A
; 0
SUTHERLAND (present
FR~SLING
XENAKIS data)
(this
;
et al. paper)
broken
numbers
spheres. [5] ; 0 NW and [39] ; l LINTON ; + Separation
lines,
separation
SAGE and point
points
of
other investigators.
are obtained by taking the average transfer, at a given angle, over the ranges of Reynolds numbers stated in Table 1. As we have already pointed out, many data do not satisfy the assumptions of the theory. Those of FR&SLING and ourselves appear to do so, and the data by Hsu and SAGE on evaporation at the 5 per cent turbulence level are possibly satisfactory. On the other hand, XENAKIS worked so close to the critical Reynolds number for a sphere that free stream turbulence was probably excessive. Likewise CARY’S work has been criticized on the instrumental side by KOROBKIN [12] and his ratio of duct to sphere diameter was too small. There is a wide variation in these experimental results particularly in Tr (0), the transfer number at the front stagnation point. In general the value of Tr (0) lies below the theoretical value (compare equations (4), (5) or (6) with Fig. 8 and Table 1). Part of the difference may be due to the scatter in results and for any investigator this is of the same order as those shown in our Fig. 6 and seems to be due to the difficulty of the measurements. The angular distributions are more easily
224
Transfer
from a sphere into a fluid in laminar flow
compared by considering the ratios Tr (Q)/Tr (0) as in Fig. 9. The theoretical curve derived by AKSEL’RUD [l] is also shown and it is seen that there is substantial agreement between all investigators and theory.
No closer correlation than this seems possible because the ratio of the integral transfer rate of the front half to the rear half decreases with Re (shown in Table 1) due to relatively increased transfer at the rear. This greatly influences the transfer. If Tr (0) varies with Pr as given in equation (4) then presumably Tr will vary in a like fashion ; however, as Table 1 shows the variation is only 10 per cent. AKSEL’RUD* [l] derived the over-all transfer for the particular case of Re < 1 and Pr > 1. The agreement is good even for KRAMER? data where 1 < Re < 40 and 200 < Pr < SS0.f
5.
20
0
40
60
80
100
CONCLUSIONS
We may conclude that, despite the many secondary factors influencing the experimental results, the boundary layer theory describes reasonably the distribution of the rate of transfer of either heat or mass around the leading half of a sphere. However further investigation is needed on the factors which cause theory and observation to deviate considerably in transfer at the front stagnation point. The form of the relation for the overall transfer rates connecting Nu, Re and Pr is suggested by the theory, and various experimental results lead to a useful set of limits, i.e. 0.5 < Nu Re-‘l2 Pr-‘/3 < 1.0 for the range lo2 < Re < 106, for a variety of systems.
120
60
FIG.9. Relative distribution of local heat and mass transfer rates on the front portion of a sphere. The curve is that given by AKSEL’RUD’S theory (see Fig. 1).
Aclcnor&dgements-We wish to acknowledge helpful suggestions on the manuscript by Dr. H. R. C. PRATT, Chemical Engineering Section, C.S.I.R.O.
Symbols as in Fig. 8, except, 0 GARNER and SUCKLINP, [8]
*FRIEDLANDER [4] has given (2)
Over-all
iransfer
analysis
rates
The over-all rate of transfer has been studied by many investigators (Table 2). The ranges of Re, Pr, Gr, Nu and d are wide. The value of Tr was calculated from these data where possible, average duct velocities were replaced by sphere approach velocities and the majority of results can be represented by 0.5 <
Tr < 1.0 for 100 < Re < lo5 0.5 <
and
Pr < lo8
(20) 225
but
it seems AKSEL’RUD’S analysis.
an
a inferior
similar theoretical approximation to
tTwo recent papers, (GARNER F. H. and KEEYR. B., Chem. Engng. Sn’. 1958 9 119 and 208) on mass transfer in forced convection at low Reynolds numbers and in natural convection respectively, only became available to us after the present paper had been written. The experimental results are similar to those presented in Fig. 6 and Fig. 8 of this paper. The work particularly confirms the important effect of density convection on both local and overall transfer rates in forced convection. The present authors do not agree with the derivation of the theory given in the first paper.
M. LINTON and K. L. SUTHERLAND
APPENDIX
Laminar
1
so that rr = 1, rs = -
The velocity and concentration distributions in the boundary layer are determined by the modified Navier Stokes, continuity, and mass-transfer equations. These for a body of revolution (about the axis of flow) are respectively (SCHLICHTING[24]) = U(&~
3 (ur) + 3 (*) &r 3Y
+ “g
u=--, JS JY
_ 0
(1.2)
U;;+“+$
$ldr
ha
r&
7=Y
X* = x/d,
2u, ( Y1
y* = y Rel/2/d,
1b*
=
“/urn,
1b,*
q* = y* (2ur*)r/*
Expressing
$
U,sinz
(1.4) I.0
(1.5)
U(+)=u,B+Us83+Ug~5+... us = -
U,J@,,, = U,/80rSo
c (Cs -
us = etc.
(1.6)
Similarly the surface of the sphere can be given in series form r = r,sin(r/r,)
...,
= rro + rszm + r5z5 +
...
U* (x) =
=
ll,,d”/U,,
,...,
Re112 and v* = v u co
(1.12)
us*) h3* (?*) etc.
(1.13
)
The concentration C in the boundary layer will also be a function of x and Y which may be expressed in series form
this in series form
we have u1 = 3U,/2ro,
(1.11)
We will designate the equations thus modified by (l.l*)-(l.ll*). Equations (l.l*) and (1.2*) can be solved by substituting equations (1.4*)-(1.9*) together with the appropriate boundary conditions and equating like coefficients of z*. One obtains a series of ordinary non-linear differential equations which can be solved numerically for the coefficient functions Jr* (‘I*), g,* (T+), h3* (~2) etc. where
on the sphere.
0
0
r/s
wrthn=l,a,5
U,sinB=
~1.10)
U (4/U,,
fs* (v*) = gs* (v*) + @a* ur*/rr* :
. .]
r* = r/d,
T*n = d”-’ rn with n = 1, 3, 5,
X
U(z)=
+
FR&SLING [6] modified equations (l.l)-(1.11) to make them dimensionless and independent of Y by, in effect, substituting
Y
of co-ordinates
(1.9)
where Jr (v), fs (v), etc. are coefficient functions expressing the dependence on y and
where (see Fig. 10) y is the distance normal to the surface of a point in the fluid, B is the distance on the sphere surface of the normal from the front stagnation point’ r is the distance from the axis of a point on the sphere surface, a, I) are the velocity components in the boundary layer parallel and normal to the surface respectively and To solve the above equations, C is the concentration. U(z), the velocity distribution outside the boundary layer (i.e. for y + co), must be known and assuming this to be the theoretical non-viscous “potential” flow we have for the sphere of radius r,,
Definition
b$
by introducing
+ 2usfs (7))zs + 326615(7+5
(1.3)
FIG. 10.
v=-----
(1.8)
which satisfies equation (1.2). Within the boundary layer the velocity distribution and hence I# is a function of both a and y and FR~SSLING [6] showed that in analogy to the series (1.5) # can be expressed
(1.1)
2
ld,, etc.
Equations (1.1) and 1.2) can be combined a stream function $ (z, y) defined by
boundary layer calculations of the local trarhsfer rates on the sphere
US+“:
l/6 rzo, r5 = l/l20
(1.7) 226
C,)
= c* = cs + c20*a
+ c*o*s
+ .
..
(1.142)
where the coefficients Cs, C,, C4, etc. are functions of y* (or v*). When substituting equation (1.14*) in equation (1.3*) a series of ordinary differential equations can be obtained for a new set of coefficient functions F,, (v*), G, (v*), H, (v*), etc. when one also substitutes (1 .I I*) and
Transfer
from a sphere into a iiuld m laminar
c, = Fo(11*), c, = 2% *
= -..% G, (‘I*) +
‘s*
k = 2 (rngljB - m11i2) 2 mo - ml ??I = ~ m,1/2 + mIl/2 t t
(1.15f)
II, (‘I*) etc.
71* us*
u1*
ilow
and the initial solution rate is
The equations involve coefficients fi*, fs*, etc. and were solved numerically by FIX&SLING up to terms involving x4. Now the local Nusselt number is given theoretically in terms of the concentration gradient in the boundary layer by
k,,, mo1i2 =
3
?? ~ ( 2Y 1 y=o C, -
d
mo1j2 +
zlc*
w
NOTATION
(1.16)
1 +=l-J 3y*
n n C c* d D
Substituting for (X*/J?*) in equation (1.26) from equation (1.14*) with appropriate substitution of the coefficient functions from equation (1.X*) we obtain F,’ (2ul*)l/2
= [-
+ Hi,‘) ,x*~ -
3 (2/u1*)‘/2
-
2 (2/u1*)li2
ug* F4’x4
-
us*
f1 (q), f3 (TJ)=
(1.17)
where dashes denote differentiation with respect to q*. For the particular value of Pr = 2.532, corresponding to nnphthalene vaporization into air, and T* = 0, FR~SSLINC [6] calculated F,’ = - 0760, F4’ = - 0.212, G,’ = - 0.319, and Ii,’ = - 0.101 so that = I.862
-
I.369 x*~ + + 0.2705 x*4 +
...
(1.18)
which on dividing both sides by Prl/s (remembering Pr = 2.532) and inserting x* = e/2 finally becomes equation (2) of the text.
APPENDIX
Initial
2
solution
rate
The over-all rate of solution for a sphere is dm
= kA (C, -
C,)
(2.1)
-dt where k = “D R&a d
Prl/s
(2.2)
as is borne out by the experiments. Expressing A and d throughout in terms of m and the density, P,,, we have dm
k,,, rn112
(2.3)
-z= where Ii2 (C, Integrating
C,)
= = = = = =
fi (O), f2 (0) =
. . .]
‘)* = 0
Nu (0) Ile-li2
(2.6)
=
(G
(G,’
m11/2
Cc0
= Re1i2
Nu (/3) Re-l12
2m01/2
- ml 1
Nu (0) = -
(2.5)
(2.4)
(2.3) 227
Fo (v*)* F, (v*), g k, m m0 m, r; ri rs, r6’ r,, * m
= = = = = = = = =
constant area of sphere cm2 concentration in solution g cm-s C/(C, - C,) diameter of sphere cm diffusion coefficient cm2 see-l functions of 0 defined in equations (2) and (7) respectively functions of 7 defined in Appendix 1
G,
(?*I,
4
(‘I*)
=
definedinAppendixl
gravitational acceleration cm set-2 constant defined in Appendix 2 mass of sphere g initial mass of sphere g mass of sphere at a time, t P -. radius of a sphere at zero time cm radius of a sphere zone after a time 1 cm constants defined in Appendix 1 rl’ d=-1 r n, 12 = 1, 3, 5, . . . r = polar radius, defined in Fig. 10 Ar (/3) = change in radius at angle 0 cm q = quantity of heat transferred cal 1=time set T = temperature OC T, = temperature on sphere surface “C u = velocity of fluid in the boundary layer parallel to the sphere surface cm se+ u* = u/u, 841, Us, 146, zr, = constants defined in Appendix 1 u,* = u, dn/U,, n = 1,3,5,. .. 72,U(z) = velocity of fluid external to the boundary layer cm se& U, = velocity of fluid at infinity (free stream) cm se+ o = velocity of fluid in the boundary layer normal to the sphere surface cm see-l v* = v Re112/U, 8 = distance on the sphere, from the front stagnation point, defined in Fig. IO cm ++ =x/d
M. LINTON and K. L. SUTHERLAND y = distance into the fluid, normal to the sphere surface, defined in Fig. 10 cm y* = y Re1/2/d OL= thermal diffusivity of fluid cm2 se& ,8 = volume coefficient of thermal expansion of fluid per ‘C ‘1 = y (226r/“)1/2 ?)* = y* (2u,*)1/2 X = thermal conductivity of fluid cal cm-l set-l “C-l 0 = polar angle radian @i = angular positions of zones on the sphere Atli = angle subtended by a zone of the sphere 0, = angle of separation degree Y = kinematic viscosity cm2 see-l p = density of fluid g cm-s g cm-3 Pm = density of sphere # = a stream function defined by equation (1.9) Appendix 1 * = Asterisk denotes dimensionless quantities and equations Subscripts Unless otherwise 0 s 03 B
= = = =
refers refers refers refers
specified to to to to
1 k=
1 k(O)= NUA
Nu (8) =
Over-all mass transfe coefficien
dm
Y4(C,-CC,)dt b2m
-(C,-C&
(
h4iJt ) B
Local mass transfer coefficieni
Over-all Nusselt number
D
k (0) d
Local Nusselt number
D
Pr = v/D
Prandtl number
Re = Ud/v
Reynolds
Cr I = @* B(T, vs h=
-
T,)
Grashof number
1 A (T, -
T,)
Over-all heat transfer coefficient
!!! dt
Local heat transfer coefficient Nu = hd/X Nu (0) = h (0) d/X Pr = v/a
&er-all Nusselt number Local Nusselt number Prandtl number
For heat and mass transfer
initial values (t = 0) saturated solution values in the free stream (at infinity) local values at 0
Tr = Nu Re-r/s
Pr-11s
Tr (0) = Nu (8) Re-li2
Pr-ll*
Over-all transfer number Local tmnsfer number
Tr (0) = Nu (6) Re-l12 Pr-ri3
Transfer number at front stagnation point
Nu (O” -
Average Nusselt numbers fGO” < 0 < ‘90, and 90” < f? < 180” respectively
Definitions For mass transfer Grashof number
OO”), Nu (90’ -
180’)
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AKSEL’RUD G. A. Zh.&.
PI
CARY
r31
FACE A. Aeronaut.
khim. 1953 27 1445.
J. R. Trans. Amer. Sot. Mech. Engrs.
number
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1953 75 483.
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[41
FRIEDLANDER S. K. Amer. Inst. Chem. Engrs. J. 1957 3 43.
[51
FRBSSLINGN. Gerlands Beitr. Geophys.
PI
FR~SSLIN~ N. Lunds Univ. Arsskr. N.F. AVD. 36 No. 4 1940.
PI
GARNER F. H. and GRAFTONR. W. PTOC.Roy. Sot. 1954 A 224 64.
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PI
GARNER F. H. and SUCKLINGR. D. Amer. Inst. Chem. Engrs. J. 1958 4 114.
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PI
KAWAGUTI M. Rept. Inst. Sci. and Technol. Univ. Tokyo
WI
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P51
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WI
LINTON W. H. JR. and SHERWOOD
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Paper 54-F-18 Milwaukee Meeting, September
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[18]
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