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Mass transfer coefficients on truncated slabs
Pergamon Press.
Chemical Engineering Science, 1969, Vol. 24, pp. 14451460.
Printed in Great Britain.
Mass transfer coefficients on truncated slabs ANSGAR SORENSEN Department of Chemical Engineering, Technical University of Denmark, Lyngby, Denmark (First received 7 February
1969; in revisedform
24 March 1969)
Abstract- Local and average mass transfer coefficients on a sharp-edged
plate and on truncated slabs of various thicknesses were obtained experimentally. The results show that the mass transfer coefficients do not only depend on the air velocity and the distance from the leading edge, but that the thickness of the slabs is also of significance. The local and mean &factors are correlated by two different Reynolds numbers, Re, and Red and from these correlations mass transfer coefficients on truncated slabs can be calculated. The experimental results indicate the existence of a critical Redvalue, Red,Cr. If Red is less than Red,Cr, the mass transfer coefficients do not depend on Red, but follow the expressions that apply to a sharp-edged plate, whereas for Recvalues larger than Red,wr the mass transfer coefficients clearly depend on Red. The results also indicate, that a laminar boundary layer exists behind the stationary eddy which is formed at the leading edge. 1NTRODUCTlON
DESIGN of convective try dryers requires knowledge of mass and heat transfer coefficients on the surface of the material to be dried. The transfer coefficients depend on the geometry of the tray and on the physical properties and velocity of the drying medium. Considerable experimental work has been carried out to obtain quantitative expressions for heat and mass transfer coefficients on plane surfaces [ l-5,8,10-15]. However the majority of the experiments has been performed under conditions that make them inapplicable to prediction of transfer coefficients on trays, and the remaining results show inconsistencies so large, that it so far has not been possible to advance a general expression for this situation. For flow parallel to a sharp-edged flat plate, the length of the plate in the flow direction is sufficient to characterize the geometrical configuration. For truncated slabs of finite thickness, such as trays, this is not the case. Here one also has to specify the thickness of the slab. The purpose of the present work is to investigate how the mass transfer coefficients on the surface of a slab placed in an infinite air-stream depend on the air velocity, the distance from the leading edge of the slab, and the thickness of this slab. On this basis expressions will be set up
THE
from which local and average coefficients can be calculated. EXPERIMENTAL
EQUIPMENT
mass transfer
AND
PROCEDURE
In principle the experimental technique rests on a method employed by Danckwerts and Anolick[ 141. However, in several respects the technique used in the present work differs from that of Danckwerts and Anolick. Thus the preparation of the slabs and the sampling mentioned below were performed in a different way. Rectangular glass slabs of dimensions 0.30 m X O-25 m and of varying thickness were used as test objects. The test slabs were covered with a thin layer of naphthalene of uniform thickness. The weight of naphthalene deposited per unit area was determined, and the slabs were placed in a wind-tunnel where the naphthalene-layer was sublimated under predefined experimental conditions. During the sublimation a front was formed, which divided the slab into two distinct regions, one which was still covered with naphthalene and one where the layer of naphthalene had already sublimated. The movement of the front was photographed, and local mass transfer coefficients were determined by measuring the time used for the sub-
1445
A. S@RENSEN
limation at various distances from the leading edge, assuming that the driving force was held constant through the experiment. The experiments were carried out under such conditions that cAo, = 0, and cAO4 c. The time-averaged value of the local mass transfer coefficient, which is defined as
(1) d
can therefore be obtained from the expression
where rx is the time for the front to move to a point at a distance z from the leading edge.
b
APPARATUS
The experiments were performed in a windtunnel. The section where the test slabs were placed was l-5 m long, 0.5 m wide and O-3 m high. The top side was provided with windows, which made it possible to photograph the slabs. In order to get a, flat velocity profile in the test section, two grids were placed in front of it, and the velocity distribution was checked with a pitot tube. The flow through the test section was measured using a venturimeter. The air was heated by an electrical heater, and the temperature of the air was measured by a calibrated, thermocouple and was radiation-shielded recorded on a potentiometric recorder. The temperature was controlled by a simple on-off controller. The amplitude of the temperature fluctuations was less than 0~4°C and the drift throughout the experimental period did not exceed 0.2°C. EXPERIMENTAL
TECHNIQUE
The coating of the test slabs with naphthalene was performed in an apparatus as shown in Fig. 1. “a” is a water-cooled heat exchanger which kept the temperature of the slabs at a suitable level during the coating. “b” is another heat
Fig. 1. a. Heat exchanger (cool); b. Heat exchanger (warm); c. Naphthalene; d. Test slabs; e. Additional slabs; f. Spacers.
exchanger, the purpose of which was to heat the naphthalene-source used for the coating-process. This exchanger was made out of 12 mm steelplate. On one side of this plate channels were milled for hot water circulation. On the other side, two mm of the plate material was removed except at the edges. In the flat tray formed in this way, a naphthalene-layer was cast, and the layer was rasped until the surface became flush with the edges. Before the coating, a test slab “d” was placed directly upon the cool exchanger. To prevent edge-effects, smaller slabs of the same thickness as the test slab were placed in front of it and behind it. 2 mm spacers “f” were mounted at the comers of the slab, and the hot exchanger was placed on the spacers, the naphthalene-layer facing downwards. The temperature of the hot exchanger was about 50°C. Under these circumstances about 1 mg/cm2 naphthalene was deposited in 10 min. The technique used did not give a quite uniform thickness of the naphthalene-layer, but the
1446
Mass transfer coefficients on truncated slabs
variations could be held within about 10 per cent. Just after the coating, samples were collected from the surface to determine the weight of naphthalene deposited per unit area on the slab. On each slab four samples were collected along a straight line. From this the thickness of the deposit could be calculated by interpolation at any distance from the leading edge. The sampling process was carried out with an apparatus as shown in Fig. 2. It consists of a solid metal cylinder with a sharp-edged nylon ring “a” fitted concentrically to one end. There
PRESENTATION DIRECT
AND
ANALYSIS
EXPERIMENTAL
OF THE
RESULTS
Forty five runs were carried out partly using truncated slabs, 1, 2, 4, 8 and 16 mm thick, and partly using a 2 mm glass plate with a sharp leading edge. The experiments were performed at four different air velocities, as shown in Table 1, where the runs are indicated by their numbers. All experiments except the one marked “*“, which suffered from an experimental error, were used in the subsequent treatment of the data. Table 1 Plate-thickness d@ml
Air-velocity Vmm/sec 1 a
1.25 2.5 5.0 10.0
I Fig. 2. a. Nylon ring; b. Connection to suction flask; c. Rubber plug; d. Ampulla cover.
.
are two borings in the axial direction. One of them, “b”, is connected to a vacuum pump via a suction flask, while the other opens into a larger boring fitted with a rubber plug “c” and an ampulla-cover “d”. The sampling apparatus was placed on the test slab with the nylon ring facing down. When the vacuum pump was started, the apparatus adhered to the slab, demarcating a well defined area coated with naphthalene. Pure ethanol was injected through the ampulla cover and was drawn to the suction flask together with the naphthalene enclosed, which was dissolved in the ethanol. The amount of naphthalene was then determined by ultraviolet spectrophotometry. After the sampling process, the slab was transfered to the wind-tunnel, where it was placed horizontally. The sublimation, which took place at temperatures between 20-3O”C, was recorded by photographing the test slab at suitable time intervals.
Vol. 24 No. 9-C
5 677 1,2 8,9,10,11 3,4 12,13
4
8
16
Sharpedged
14,15 16,17 l&19 20,2 1
22,23 24,* 25,26 27,28
29,30 31,32 33.34 35,36
1s,2s 3s,4s 5s,6s 7s,gs
The time-averaged values [k& for the local mass transfer coefficients during the sublimation period were calculated from Eq. (2) and plotted against the distance from the leading edge, z. In Fig. 3 the mass transfer coefficients on a 2 mm truncated slab have been plotted as an example. 3o01--
50 6. 70 6r
200 L i -3 -
1447 C.ES.
5
9r IO II 12 13
A v 0 0
too --a
-
Fig. 3. Time-averaged values for local mass transfer coefficients on 2 mm truncated slab vs. distance from the leading edge.
A. SBRENSEN
The time-averaged values for the sharp-edged plate all appeared to decrease with increasing distance from the leading edge, while all the experiments on truncated slabs, except for run no. 5, showed a characteristic maximum for the transfer coefficients at some distance from the leading edge. According to Danckwerts and Anolick [ 141, the presence of this maximum is due to the existence of a stationary eddy just behind the leading edge. The eddy, which is shown in Fig. 4, causes large mass transfer coefficients at the point of reattachment “A”, while relatively small coefficients are recognized on the part of the plate, which is covered by the eddy. From point “A”, a boundary layer is formed, the character of which was not investigated by Danckwerts and Anolick, but was supposed to be turbulent.
Fig. 4. Flow past truncated slab.
The local mass transfer coefficients on a flat plate surrounded by a laminar boundary layer are given by the following theoretical equation r91, j, = 0.332 Re,-Q.5 (3) where Re, = :
and j, = 2 &+?‘3
I 2
IO'
2 bq Rs,
5
Id
2
5
I Id
Fig. 5. Time-averaged values for local j,-factors on sharpedges plate vs. Re,.
suggested way. The experimental points group about a straight line with slope -05 in accordance with Eq. (3). However the experimental values are about 65 per cent larger than prescribed by this equation. As a characteristic illustration to the experiments performed on truncated slabs, Fig. 6 shows the results from the runs with the 8 mm slab plotted in a j, vs. Re,-diagram on a logarithmic scale. As it appears from the plot, the mass transfer coefficients in the case of truncated slabs cannot be correlated in this way. However, it is worth noticing that the asymptotic values of the slopes of the curve-branches to the right of the maximum are about-O-5. A few of the runs, especially some of those performed at the largest air velocity used, show deviations from this general pattern as they seem to converge to less negative slopes.
while the transfer coefficients on a flat plate surrounded by a turbulent boundary layer are described by the expression [ 16,171 j, = O-0288 Re,-02.
5
2
.?a Icr2 3
(4)
An attempt was made to correlate the timeaveraged values here found as indicated by Eqs. (3) and (4). Figure 5 shows the experimental results for the sharp-edged plate plotted in a logarithmic [j,] & vs. Re,-diagram. As can be seen, the results correlate well in the 1448
s
5
2
2ro 280 5
1 I d
I 2
:‘.._,, 5
I IO'
2
5
loq R+
Fig. 6. Time-averaged values for local j,-factors on 8 mm truncated slab vs. Re,.
Id
Mass transfer coefficients on truncated slabs
If .the boundary layer as proposed by Danckwerts and Anolick is developed from the point of reattachment, then the characteristic length z used in Figs. 5 and 6 is irrelevant as far as the truncated slabs are concerned and must be replaced by the length z-A, where A is the distance from the leading edge to the point of reattachment. In Fig. 7, the results for the 8 mm slab are replotted with z- zmax. as the characteristic is the distance from the length, where z,,. leading edge to the point where the mass transfer coefficient reaches its maximum value. The temporary assumption is here made, that the position of this maximum is identical with the point of reattachment.
Fig. 7. Time-averaged values for local jr,-factors on 8 mm truncated slab vs. ( u, (z - .zmax.))/v.
The plot shows that now the experimental points for the individual runs can be represented with reasonable accuracy by straight lines having slopes about -0.5. This indicates that a laminar boundary layer is primarily developed from the point of reattachment. On this basis the following working hypothesis is proposed. When air flows tangentially past a truncated slab, a stationary eddy will be formed at the leading edge, and for the experimental conditions considered, a laminar boundary layer will be developed behind this eddy.
than A, then the velocity boundary layer and the concentration boundary layer will not be developed from the same point. This will, however, change the concentration profile and through that affect the mass transfer coefficients. For a flat plate surrounded by a laminar boundary layer with no mass transfer through the front part, x’, the following expression will apply to the local mass transfer coefficient, kCe, at a distance x from the starting point of the boundary layer;[18], k,H
=k,[l-(;~]-“3.
In this equation, kc is the local mass transfer coefficient that would exist if mass transfer did take place all the time right from the starting point of the velocity boundary layer. In the experiments considered, x’ is the distance from the starting point of the velocity boundary layer to the naphthalene front. Thus x’ is a time variable and consequently k,” is also time dependent as already indicated by the superfix, 8. As will now be shown, kc can be computed on the basis of the hypothesis proposed and the experimental time-averaged coefficients. During the experimental work it was observed that the truncated slabs were completely covered with naphthalene during the first part of the sublimation process, and that only after some time, bare patches were formed at some distance from the leading edge. Thus in this first period (0, tI), the velocity boundary layer and the concentration boundary layer start from the same point. That is, x’ = 0. Then from Eqs. (I) and (3, k,B = kc = [kc]:
Using Eqs. (1) and (6) we get,
CONVERSION OF THE DIRECT EXPERlMENTAL RESULTS
When the naphthalene front is situated at a distance from the leading edge, which is larger 1449
.
(6)
A. S@RENSEN
The ratio [k,]&:lk, is called F, and from this definition, Eq. (7) can be expressed in the following way, kl"a:
k, =
(8)
+--F)
when the first bare patch is formed the thickness at point x is (11)
W(x),, = w0-- W(x)KW
where W(X)(~,~~) is the amount of naphthalene sublimated at x in the time interval (0, tl). From Eqs. (2) and (6) we get
+F'
z
(12)
From Eq. (8) it is possible to calculate local mass transfer coefficients, kc, if F is known. On the basis of the working hypothesis, Eq. (5) is assumed to apply to the boundary layer on the slabs. From Eq. (5) and the expression
On the assumption that a laminar boundary layer exists, kc must be expected to follow an expression of the form kc =
Kx-O+
(13)
Combining Eqs. (1 1), (12) and (13), one obtains
P”dM”h
W(x),, = WO--jy-K~+5.
F=
--
kc
(9)
we get
F=--$---,; [l-($)Sr]-1’3d6J.
(10)
From this equation F can be calculated as x’ = x’(0) is known from the experimental results. F was computed in this way for a number of cases. The resulting values are plotted in Fig. 8 against the dimensionless position variable dx/A, which will be defined in the following.
(14)
For the largest part of the boundary layer region, this expression will describe the thickness of the naphthalene-layer when ~9= tl, but on the other hand Eq. (14) cannot be correct near the starting point, as finite values for tl and thus for ka,_o, were observed in the experiments. To obtain a better description of the thickness of the deposit for small x-values, the following empirical equation was used instead of Eq. (14). W(x),, =
Wo( 1-exp
(15)
(-9))
where
(16) This expression satisfies the conditions
_______-.___-.and W(x),, + W,forx + m. Further Eq. ( 15) does not deviate much from Eq. (14) except for small values of 4x/A. In connection with the remaining calculations, Eq. (13) is assummed to apply. Using the Eqs. ( 13) and ( 15) and the expression RTR’(x)t, [kJk = (tt-tl)PAOMA
-A/,
(17)
we get by substitution in Eq. ( 10)
Fig. 8. F vs. l/x/A. In practice it is difficult and in some cases impossible to determine F wi:h reasonable accuracy in this way, partly because a number of (x’, 0) pairs have to be known in front of the point considered and partly because the integrand in Eq. (10) is infinite for x’ = x. For these reasons, F was also computed as follows. At the start of the run, the thickness of the naphthalenelayer is W,, all over the surface of the slab. At the time tl,
~+-exp(-%9P~_ (tt-f,)Prl&f.J
,,i,
I” ,l_(~~]-l/J& 11
X
(18)
which can be rearranged to the following Volterra integralequation of the first kind,
1450
q1/3(1-exp(-qqZ’S))
=&fq
”
[q-q’]-“%(q’)dq’
(19)
Mass transfer coefficients on truncated slabs
calculated from Eq. (22), local mass transfer coefficients kc were computed from Eq. (8). As mentioned above, the time-averaged values indicated that the mass flux through the boundary layer could be described by an expression of the form
where
B =
WoRT PA~MAK
=
const.
The solution to this integral equation is [ 191
$ScW=
-exp(-$u”)]du
y 0
(20)
314
in whichu = 7
.
Combination of Eqs. (10) and (18) gives F~B[l-Elp(-~)]~x
(
y
V
,,a[ I- exp(-$@)]du
LP(1 -u)-
from which F can be computed as a function of d.x/A. For x + 0 and x + m F can be calculated analytically from Eq. (22) rw-(t) lim. F = x-0 ,-I U(1_uU)-2,3dU = ‘%’ and x-m
(24)
and
St=&_ VCO
(22)
0
lim.F=
2[Re,-Re,] >
where Re A =u,d
I
(23)
>
where C is a constant for each run. To examine the validity of Eq. (23), the calculated k,-values for the experiments on truncated slabs were plotted in a ((St)-” vs. Re,)-diagram. Because x = z-A, Eq. (23) can be rearranged to give the following expression St-2 =
When Eq. (20) is inserted in this expression, the result is
t
-0.5
(21)
b--t,
F=
C 7 (
rmw @(
1-
= 1.37. u) -*‘3du
In the general case F was calculated by numerical integration of Eq. (22). The relation between F and the dimensionless position variable dx/A as calculated from Eq. (22) is shown in Fig. 8. As it appears from the diagram, this relation is in close agreement with the values of F determined from the experimental (x’, @-values. This indicates that the expression (15) used to describe the thickness of the naphthalene-layer is satisfactory for the present purpose. The determination of F for a given value of x has to be performed in general by an iterative procedure as A contains the unknown factor K. By starting from a suitable value for K and using in turn Eqs. (16), (22), (8) and (13), still better values for K are obtained.
From the time-averaged values determined experimentally and the corresponding F-values
So if Eq. (23) is valid, the experimental results will follow a straight line with slope (SC~/~/C)~ and crossing the abscissa-axis at the value ReA. For the vast majority of the experiments considered, a straight line could be drawn through the experimental points, but in most cases, as must be expected, positive deviations were experienced near the abscissa axis, corresponding to smaller mass transfer coefficients than described by Eq. (23). The extent of this region clearly depends on ReA, and its order of magnitude is from 4 to 1 ReA. For a few of the experiments, especially some of those performed at air velocities of IO m/set, negative deviations from the linear relationship were experienced at high Re,-values. This effect can be explained as a transition from laminar to turbulent flow conditions in the boundary layer. For run no. 5 the straight line that was drawn through the experimental points passed through the origin, i.e. ReA = 0. The ReA-values obtained in this way were plotted against v,z,,,~~./Y. The plot showed that there was no systematic deviation between
1451
A. SBRENSEN
these two quantities. Thus the assumption zmax.= A which was made by the calculation of F is correct. The identity of z,,,,. and A and the linear dependence between W2 and Re, both support the working hypothesis proposed on the existence of a laminar boundary layer immediately behind the stationary eddy. The slope of the straight lines which is a measure of the value of the const. C in Eq. (23), varies between the runs in such a way that C increases when the air- velocity and the thickness of the slab are increased. This indicates that C depends on the Reynolds number, Red = u,d/u. A relationship between C and Red is easily understood on the basis of observations made by Danckwerts and Anolick[l4], who found that a strongly turbulent area, caused by eddy separation from the stationary eddy, exists above the boundary layer (Fig. 4). The turbulence in this region will influence on the flow conditions in the boundary layer and thereby change the mass transfer coefficient. Following this mechanism, the transfer coefficients will depend on the condition of the stationary eddy, which in turn depends on the Reynolds number, Red. In Fig. 9, the values of C as found from the (St+ vs. Re,)-diagrams are plotted against Red on a logarithmic scale. As may be seen there exists a close correlation between C and Red in the interval considered. The straight line drawn through the points represents the expression C = 0.152 Red0’179,
(25)
(standard deviation 9 per cent) which was found through linear regression analysis. Substitution of Eq. (25) into Eq. (23) gives j, = O-152 Redo.17e[Re, - Realve5.
(26)
In Fig. 10, ReA is plotted against Red in a logarithmic diagram. It appears that ReA with reasonable accuracy can be described by the relationship
I.0
0
g J
0.5 %
0.2
ri 2
log Re, Fig. 9. The constant C in Eq. (23) as a function of Red.
0
0
3.0 2.0
30
I 4.0
109 Red
Fig. 10. Reynolds number ReA vs. Reynolds number Red.
ReA = 49-g RedoaG1,
(27)
(standard deviation 19 per cent). In Fig. 11, the converted experimental results are plotted in a logarithmic diagram, which on the abscissa axis has (Re, - ReA) and on the ordinate axis has (j, X Red-0’179). The experimental points situated in the interval 0 < Re,- ReA 5 4 ReA, where deviations from Eq. (26) must be expected, are marked by small circles, while the rest of the points are marked by larger circles. In Fig. 11 the Re&-values found from the (St-” vs. Re,)-diagrams are used. When Eq. (27) is used to describe Rea, this gives rise to a minor increase in the scattering of the points, but even then the standard deviation from Eq. (26) does not exceed 15 per cent in the region Re,-ReA > )ReA. The time-averaged values obtained from the
1452
Mass transfer coefficients on truncated slabs
Fig. 11. Local mass transfer coefficients on truncated slabs ’ correlated by Eq. (26). The ReA-values used, are obtained from St+ vs. Re, plots.
experiments made on the sharp-edged plate were converted as described for the truncated slabs. For the vast majority of the experimental points, the Fvalues computed from Eq. (22) differed very little from the values one would obtain by assuming [k&+ 33for x -+ 0. In this case, A = 0 because t, = 0 and Eq. (18) takes the form
The solution corresponding with Eq. (20) then becomes u”~( 1 - u)-z’3du
rww JI
= 1.37.
#a( 1 - u) -z’Sdu
In Fig. 12, the local mass transfer coefficients, kc, on the sharp-edged plate are plotted in a logj,
3.0
4.0 log Re,
Fig. 12. Localj,-factors on sharp-edged plate vs. Re,.
j, = O-407 Rezma5,
5.0
(30)
(standard deviation 12.5 per cent). From Eqs. (3) and (30) it appears that the mass transfer coefficients obtained from Eq. (30) are 22.5 per cent in excess of the coefficients given by the theoretical Eq. (3). ON THE EXISTENCE OF A CRITICAL Red-VALUE
(29)
which gives F=
vs. log&,-diagram. The results are correlated by the following equation
Equations (26) and (30) show that if Eq. (26) applies to all Rehvalues, the mass transfer coefficient at an arbitrary distance x from the starting point of the boundary layer will become smaller for a truncated slab than for a sharpedged plate, if Red is less than a certain value. This would not agree with the above mentioned assumption made by Danckwerts and Anolick [14] on the existence of a turbulent region outside the boundary layer. From this model, one should expect the mass transfer coefficients on truncated slabs to approach the transfer coefficients on sharp-edged plates, when the turbulent activity outside the boundary layer approaches zero. If in Eqs. (26) and (30), (Re,ReJ and Re,, respectively, are equated to Re,, andjDCsharp)is set equal to jD(truncated)r then we find that Red equals 245. This is then the value of Red for which the mass transfer coefficients
1453
A. SfBRENSEN
calculated by Eqs. (26) and (30) take on the same values. The only run which was performed at a Reynolds number Red less than 245, was run no. 5 (d= 2 mm, o, = 1.25 mlsec), which in two respects distinguishes itself from all the rest of the experiments performed on truncated slabs. As realized from Fig. 3, this run does not show any maximum for the time-averaged value of the mass transfer coefficient, and the experimental values when plotted in a (Ste2 vs. Re,)-diagram fall on a straight line through the origin, which according to the previous assumptions means that no stationary eddy exists at the leading edge. From this, run no. 5 must be expected to obey Eq. (30), which applies to a sharp-edged plate. This is supported by the experimental results, as run no. 5 can be described by the expression j, = O-396 Re,-0.5
From Fig. 13, local mass transfer coefficients on truncated slabs can easily be calculated if the air velocity, the thickness of the slab, and the distance from the leading edge are known. THE MEAN
MASS TRANSFER
COEFFICIENT
For most practical purposes, it is of more interest to know the mean values of the local mass transfer coefficients for a given surface area rather than to know the local coefficients themselves. A relation will therefore be set up which makes it possible to calculate mean transfer coefficients on the basis of the experimental results obtained. The mean value of the local mass transfer coefficient for the surface area between z and z + 6z is defined by the equation
(31)
which is in quantitative agreement with Eq. (30). The experiments thus indicate that the stationary eddy at the leading edge is only formed if Red exceeds a certain critical value, R&r, and that this critical Reynolds number has a value of about 245. Equation (26) is represented in Fig. 13, where j, is plotted against Re, with Red as a discrete variable. The family of curves are bounded partly by the expression Re, = 1.5 ReA, as Eq. (26) only applies to Re,-values larger than 1.5 ReA, and partly by Eq. (30), which applies to the sharp-edged plate.
This expression can be put on a dimensionless form by introducing the j,-factor. The following equation then applies to the surface area between z=Oandz=z 1 + jDdz. JD(0.Z) = ; J0
(33)
T
From this expresston, J~~~,~,values for the truncated slabs were computed by numerical integration using the local j,-values obtained, from the experiments, whereas the mean values on the sharp-edged plate were found by integrating Eq. (30) to give .&,Co,Z, = O-812 ReFw5.
(34)
This expression must also be expected to apply to truncated slabs if Red is less than about 245. To get to an analytical expression describing for truncated slabs the integral in Eq. (33) &CO,+, is split into two parts
logRe, Fig. 13. Local&factors on truncated slabs, calculated from Eq. (26).
where the contents in the square brackets according to Eq. (32), are identical with jmo.a, 1454
Mass transfer coefficients on truncated slabs
which is the mean value of the coefficient for the surface area between z = 0 and z = A. In Fig. 14, the experimental values ofJ;)Co,L\) are plotted against Red on a logarithmic scale. Even
0.6
F 0.5
Fig. 14. Mean values of the j,-factor for the eddy-area, plotted as a function of Red.
0.4
runs performed under identical experimental conditions show considerable scattering. It appears, however, that j,,,,, can, as a reasonable approach be described by the empirical exnression 2.42 x lo3 htO,A, =
2.02 x 105+Red1'3'
(36)
i
As mentioned above, Eq. (26) describing the local &factors in the boundary layer region is not valid for the front part of the boundary layer just behind the stationary eddy. To extend the validity of Eq. (26) to include this region, a correction factor 7 is introduced. .experimental
7 = &ez_ReA,-0.5'
(37)
Fig. 15,~ is plotted against the dimensionless position-variable A/z in the interval 4 < A/z < 1. As shown, r) can be described by the empirical expression (38) q=l_ 4” z .
In
0
Inserting Eq. (38) into Eq. (35) gives + 0. 152Re~017s&?A-o’5 x[1 - (x+ I0
I)-“IX-WX]
(39)
06
0.7
0.6
0.9
I.0
A/Z Fig. 15. Correction factor r) vs. A/z.
where X=z--A --_=--1 Re, A &A
’
The ratio between the Jbto,,,-factors obtained from the experiments and the values calculated from the correlation (39) was formed. The mean value of this ratio for all the experiments performed was l-004, and the standard deviation was 0.115. In Eq. (39) &,,A, as Well as &?A depend on Red as described by Eqs. (36) and (27), while X depends on both Re, and Red. Thus the equation describesjb,o,z, as a function of the two Reynolds numbers Re, and Red. In Fig. 16, &,to,z, as calculated from Eq. (39) is plotted against Re, with Red as the discrete variable. The left hand curve-branches clearly show the negative influence from the stationary eddy on the &o,z,-factor, while the rather strong dependence of the right hand curve-branches on Red demonstrates the positive effect which the eddy
1455
A. S@RENSEN
vestigation. From these plots local and mean mass transfer coefficients can easily be calculated when the air velocity, the physical properties of the air and the geometrical configuration are known. Especially Fig. 17 is convenient for most practical purposes, as shown by the following example. 2
s
I04
2
5
Id
EXAMPLE
logRe, Fig. 16. Mean values of j,-factor for the region z, = 0 to z = z vs. Re, and Red.
exerts through the formation of a turbulent region outside the boundary layer. In Fig. 17, .J”,,,~, calculated from Eq. (39) is plotted against Re, for different values of z/d. A specific curve in this diagram thus shows how jD(O,rjdepends on the air velocity for a given slab or for geometrically similar slabs.
Moist ceramic slabs 20 cm long and 1 cm thick are dried in an air stream moving parallel to the surface of the slabs. Calculate the mean mass transfer coefficient, when the air velocity is 3 mlsec. The kinematic viscosity of the air is 1.7 X lo-” m%ec and the Schmidt number for the air-water system is 0.60.
zld=$=20.
From Fig. 17:
kxo,z, =
On the whole, the curves form parallel straight lines with slopes of about - O-31, which gives a velocity dependence of the form
Powell[lO] as well as Shepherd, Hadlock and Brewer[ 131 have carried out mass transfer experiments on trays or slabs 1 ft long and 2 in. thick, i.e. L/d = 6. Powell’s results are described by the equation
(40) .&OW
This dependence, however, does not apply when Red is less than about 245. In that case an expression of the form
kcto,,,- tho.50
1O-2m/set.
X
COMPARISON WITH PREVIOUS INVESTIGATIONS
Fig. 17. Mean values of the j,-factor for the region z = 0 to z = z vs. Re, and z/d.
I;c(o.x) - Ym@6S.
6*8X 10-S x 3.0 = 2.9 0.71
0*151Re;030
while the results of Shepherd summarized by the expression
(42) er al. can be
jmo,w = 0.11 2ReLwZ5.
(41)
should apply as demonstrated by run no. 5. Figures 16 and 17 together with Fig. 13 constitute the main results of the present in-
=
These results are compared Fig. 18 where the equation
1456
(43)
with Eq. (39) in
Jmo,Zj= 0-036Re,-0’2
(44)
Mass transfer coefficients on truncated slabs
Q-p ..\‘-
.
-__
Shaphcrd This correlation POWII
Eq.(44)
Fig. 19. Comparison between Eq. (39) and the work of Powell [lo] on a thin truncated slab.
Fig. 18. Comparison between Eq. (39) and the works of Shepherd[ 131and Powell[ lo]. Equation (44) applies to a turbulent boundary layer on a flat plate.
which applies to a turbulent boundary layer on a flat plate is also shown. Sherwood [4] performed drying experiments on a slab made from sulphite pulp, 5 cm long and 1.5 cm thick. The results showed that
5 2
5 &I
IO’
1 2
Rc,
Fig. 20. Comparisonbetween Eq. (39) and the results obtained by Danckwerts[ 141. Raaschou, Becker and Janholm [ 151 investigated the drying of linen, which was stretched on a frame. The results of this work demonstrated that
I@& - U,O’50.
(46)
The velocity dependences obtained by [lo], [4] thus agree quite well with the velocity dependence found in this work for Red > 245, while the dependence obtained by [ 151 agrees with this work for Red < 245. Powell [lo] also carried out experiments on a “thin” truncated slab about 20 cm long, and found that for this situation i&o,Ljat small velocities was proportional to u,O’~,while at larger velocities it was proportional to v,O.~~.So his results support the assumption on the existence of a critical value for Red. In Fig. 19, they are compared with the values obtained from Eq. (39) for a 5 mm slab , of the same length as the one used by Powell. In Fig. 20, the results obtained by Danckwerts and Anolick [ 141 for a a in. truncated slab by v, = [ 131 and
2.15 mlsec are compared present correlation.
with the results of the
SUMMARY AND CONCLUSION Time-averaged values for the local mass transfer coefficients were measured on truncated slabs and on a sharp-edged plate placed in a tangential air stream. The primary results for the slabs indicated the existence of a laminar boundary layer behind the stationary eddy formed at the leading edge. On this assumption, the time-averaged values were converted into “ordinary” local mass transfer coefficients that will exist if mass transfer takes place on the whole surface all the time. The local mass transfer coefficients obtained in this way support the assumption of the existence of a laminar boundary layer. The coefficients found on the sharp-edged plate are about 22 per cent larger than expected
1457
A. SBRENSEN
from the theoretical Eq. (3), which indicates that the experimental results perhaps suffer from a systematic error. This might be due to the effect of the turbulence intensity in the air-stream because, as shown by Thomas [7,20,21], the intensity of turbulence can greatly enhance the mass flux through’ a laminar boundary layer. However, this quantity was not measured in the experiments, and thus it is hard to say if the intensity of turbulence is of any significance in the present case. The results from the experiments performed on truncated slabs indicate the existence of a critical value for the Reynolds number Red. Judging from the present data, no stationary eddy is formed if Red is less than about 245, and this being the case, the local mass transfer coefficients on truncated slabs follow the expression that applies to a sharp-edged plate. Correlations were set up for local and mean mass transfer coefficients on truncated slabs and on sharp-edged plates. From these expressions, mass transfer coefficients can easily be calculated if the velocity and physical properties of the air and the length and thickness of the slab are known. Owing to the analogy existing between convective heat and mass transfer, the correlations can also be applied to predict heat transfer coefficients, when thej,-factors in Figs. 13, 16 and 17 are replaced by the correspondingjH-factors.
NOTATION
defined by Eq. (16) C dimensionless constant defined by Eq. (23) c total concentration in gas phase, kmole/m3 cA concentration of sublimating component, kmole/m3 D diffusion coefficient, m*/sec d thickness of slab, cm or m F = kl:: jD local jD-factor (mass transfer taking place from the whole surface), dimensionless A
1458
time-averaged value of local j,-factors in the time interval t1 to t2, dimensionless mean value of the j,-factor for the area between z = z and z = z +6z, dimensionless K constant defined by Eq. (13) kc local mass transfer coefficient (mass transfer from the whole surface), mlsec or m/hr time-averaged value of the local mass transfer coefficient for the time interval between tI and tz, m/set or n&r local mass transfer coefficient at time 8, mlsec or m/hr mean value of mass transfer coefficient for the area between z = z and z = z + 6z, mlsec or mlhr L length of slab in velocity direction, morcm molarweight of the sublimating comMA ponent, kglkmole P A0 vapour pressure of sublimating component, ton-. R universal gas constant, 62.36 (m3 torr.)/(deg. kmole) Red = v,d/y, dimensionless Re, = v,z/v, dimensionless ReA = vmA/v, dimensionless Rf&i,cr = critical value of Red, dimensionless SC = Dp/p Smith number, dimensionless St = kc/v, Stanton number, dimensionless T absolute temperature, “K front to t, time for the naphthalene reach the position z = z, set time for the first bare patch to be t1 formed on the surface, set in the undisturbed airVCO air velocity stream far from the slab, mlsec wo initial thickness of the naphthalenelayer, kg/m* thickness of the deposit at the W(x),, position x, when t = tl, kg/m* naphthalene sublimated at x in the time interval 0 to tl, kg/m*
Mass transfer coefficients on truncated slabs
X
= (Z - A)/A
= Re,lReA - 1,
dimen-
sionless distance from the starting point of the boundary layer, m x’ distance between the naphthalene front and the starting point of the boundary layer, m Y distance normal to the surface, m Z distance from the leading edge Zmax. distance between the leading edge and the point where the mass transfer coefficient reaches its maximum value, m x
Greek symbols A distance from the leading edge to the starting point of the boundary layer, m kinematic viscosity mYsec V 77 correction factor defined by Eq. (37) P density kgIm3 _ _ Subscripts
0 on the surface ~0 in the undisturbed air-stream
REFERENCES [l] MILLAR F. G., Can. met. Mem. 1937 143. [2] WADES. H., Trans. Znstn them. Engrs 1942 M 1. [3] PASQUILL F., Proc. R. Sot., Land. 1943 A182 75. [4] SHERWOOD T. K., Ind. Engng Chem. 1929 21976. [5] HINCHLEY J. W. and HIMUS G. W., Trans. Instn them. Engrs 1924 2 57. [6] BIRD R. B., STEWART W. E. and LIGHTFOOT E. N., Transport Phenomena. Wiley 1962. [7] THOMAS D. G., A. 1. Ch. E. J/l965 11520. [8] JU CHIN CHU, LANE A. M. andCONKLIN D., Znd. Engng Chem. 1953 45 1586. [9] POHLHAUSEN E.,Z. angew. Math. Mech. 19211115. [lo] POWELL R. W., Trans. Instn chem. Engrs 1940 18 36. [ 1l] MAISEL D. S. and SHERWOOD T. K., Chem. Engng Prog. 1950 46 13 I. [ 121 WENZEL L. and WHITE R. R., Znd. Engng Chem. 195 143 1829. [ 131 SHEPHERD C. B., HADLOCK C. and BREWER R. C., Ind. Engng Chem. 1938 30 388. [14] DANCKWERTS P. V. and ANOLICK C., Trans. Znstn them. Engrs 1962 40 203. [15] RAASCHOU P. E., BECKER F. C. and JANHOLM C., Dunmarks Nururuidensk, No. 9, Samfund 1925. [16] SCHLICHTING H., Grenzschicht Theorie. Braun 1958. [ 171 COLBURN A. P., Trans. Am. Inst. Chem. Engrs 1933 29 174. [18] KROUJILINEG., Tech. Phys. U.S.S.R. 19363 183,311. [19] MARGENAU H. and MURPHY G. M., The Mathematics ofPhysics and Chemistry. Van Nostrand 1956. [20] THOMAS D. G.,A. I. Ch. E. JI 1%5 11848. [21] THOMAS D. G.,A. I. Ch. E. JI 1966 12 124. R&sum& Des coefficients de transfert de masse locaux et moyens ont ttC obtenus experimentalement sur une plaque a bords tranchants et sur des dalles tronquees d’epaisseur variable. Les r&sultats dkmontrent que les coefficients de transfert de masse sont non seulement fonction de la v&cite de lair et de la distance du bord d’attaque, mais 6galement de l’epaisseur des dalles. Les facteurs j, sont en correlation par deux nombres de Reynolds ditferents, Re, et Red et, sur la base de ces correlations, il est possible de calculer les coefficients de transfert de masse sur les dalles. Les rtsultats experimentaux indiquent l’existence dune valeur critique Red, Red,,. Si Red est inferieur a Red,er, les coefficients de transfert de masse ne dependent pas de Red, mais suivent les expressions qui se rapportent B une plaque B bords tranchants, tandis que pour les valeurs Re,, sup&ieures B Re,,,,, les coefficients de transfert de masse sont clairement fonction de Red. Ces resultats indiquent Cgalement qu’il existe une couche limite laminaire derriere le tourbillon stationnaire qui s’est form& sur le bord d’attaque. Zusammenfaasung- &tliche und durchschnittliche Stofflibergangswerte an einer scharfkantigen Platte und an abgestumpften Scheiben verschiedener Dicke konnten auf experimentelle Weise erhalten werden. Die Ergebnisse zeigen, dass die StolBlbergangswerte nicht nur von der Luftgeschwindigkeit und dem Abstand von der Vorderkante abhiingen, sondern daas such die Dicke der Scheiben van Bedeutung ist. Die ortlichen und durchschnittlichen j,-Faktoren werden durch zwei verschiedene Reynolds -Zahlen, Re, und Red, in Wechselbeziehung gebracht, und mit Hilfe dieser Korrelationen kiinnen Stofflibergangswerte an abgestumpften Scheiben berechnet werden. Die experimentellen
1459
A. SORENSEN Etgebnisse weisen auf das Bestehen eines kritischen RehWertes, Ret,, hin. 1st Red niedriger als Red,cr so hiingen die Stoffubergangswerte nicht von Red ab, sondem nchten sich nach den auf eine schartkantige Platte anwendbaren Ausdrilcken, wiihrend bei Re=Werten, die grosser als Re+, sind, die Stoffdbetgangswerte eindeutig von Red abhiingen. Die Ergebnisse deuten femer darauf hut, dass hinter dem stationiiren Wirbel, der sich an der Vorderkante bildet, eine laminare Grenzschicht besteht.
1460
Chemical Engineering Science, 1969, Vol. 24, pp. 14451460.
Printed in Great Britain.
Mass transfer coefficients on truncated slabs ANSGAR SORENSEN Department of Chemical Engineering, Technical University of Denmark, Lyngby, Denmark (First received 7 February
1969; in revisedform
24 March 1969)
Abstract- Local and average mass transfer coefficients on a sharp-edged
plate and on truncated slabs of various thicknesses were obtained experimentally. The results show that the mass transfer coefficients do not only depend on the air velocity and the distance from the leading edge, but that the thickness of the slabs is also of significance. The local and mean &factors are correlated by two different Reynolds numbers, Re, and Red and from these correlations mass transfer coefficients on truncated slabs can be calculated. The experimental results indicate the existence of a critical Redvalue, Red,Cr. If Red is less than Red,Cr, the mass transfer coefficients do not depend on Red, but follow the expressions that apply to a sharp-edged plate, whereas for Recvalues larger than Red,wr the mass transfer coefficients clearly depend on Red. The results also indicate, that a laminar boundary layer exists behind the stationary eddy which is formed at the leading edge. 1NTRODUCTlON
DESIGN of convective try dryers requires knowledge of mass and heat transfer coefficients on the surface of the material to be dried. The transfer coefficients depend on the geometry of the tray and on the physical properties and velocity of the drying medium. Considerable experimental work has been carried out to obtain quantitative expressions for heat and mass transfer coefficients on plane surfaces [ l-5,8,10-15]. However the majority of the experiments has been performed under conditions that make them inapplicable to prediction of transfer coefficients on trays, and the remaining results show inconsistencies so large, that it so far has not been possible to advance a general expression for this situation. For flow parallel to a sharp-edged flat plate, the length of the plate in the flow direction is sufficient to characterize the geometrical configuration. For truncated slabs of finite thickness, such as trays, this is not the case. Here one also has to specify the thickness of the slab. The purpose of the present work is to investigate how the mass transfer coefficients on the surface of a slab placed in an infinite air-stream depend on the air velocity, the distance from the leading edge of the slab, and the thickness of this slab. On this basis expressions will be set up
THE
from which local and average coefficients can be calculated. EXPERIMENTAL
EQUIPMENT
mass transfer
AND
PROCEDURE
In principle the experimental technique rests on a method employed by Danckwerts and Anolick[ 141. However, in several respects the technique used in the present work differs from that of Danckwerts and Anolick. Thus the preparation of the slabs and the sampling mentioned below were performed in a different way. Rectangular glass slabs of dimensions 0.30 m X O-25 m and of varying thickness were used as test objects. The test slabs were covered with a thin layer of naphthalene of uniform thickness. The weight of naphthalene deposited per unit area was determined, and the slabs were placed in a wind-tunnel where the naphthalene-layer was sublimated under predefined experimental conditions. During the sublimation a front was formed, which divided the slab into two distinct regions, one which was still covered with naphthalene and one where the layer of naphthalene had already sublimated. The movement of the front was photographed, and local mass transfer coefficients were determined by measuring the time used for the sub-
1445
A. S@RENSEN
limation at various distances from the leading edge, assuming that the driving force was held constant through the experiment. The experiments were carried out under such conditions that cAo, = 0, and cAO4 c. The time-averaged value of the local mass transfer coefficient, which is defined as
(1) d
can therefore be obtained from the expression
where rx is the time for the front to move to a point at a distance z from the leading edge.
b
APPARATUS
The experiments were performed in a windtunnel. The section where the test slabs were placed was l-5 m long, 0.5 m wide and O-3 m high. The top side was provided with windows, which made it possible to photograph the slabs. In order to get a, flat velocity profile in the test section, two grids were placed in front of it, and the velocity distribution was checked with a pitot tube. The flow through the test section was measured using a venturimeter. The air was heated by an electrical heater, and the temperature of the air was measured by a calibrated, thermocouple and was radiation-shielded recorded on a potentiometric recorder. The temperature was controlled by a simple on-off controller. The amplitude of the temperature fluctuations was less than 0~4°C and the drift throughout the experimental period did not exceed 0.2°C. EXPERIMENTAL
TECHNIQUE
The coating of the test slabs with naphthalene was performed in an apparatus as shown in Fig. 1. “a” is a water-cooled heat exchanger which kept the temperature of the slabs at a suitable level during the coating. “b” is another heat
Fig. 1. a. Heat exchanger (cool); b. Heat exchanger (warm); c. Naphthalene; d. Test slabs; e. Additional slabs; f. Spacers.
exchanger, the purpose of which was to heat the naphthalene-source used for the coating-process. This exchanger was made out of 12 mm steelplate. On one side of this plate channels were milled for hot water circulation. On the other side, two mm of the plate material was removed except at the edges. In the flat tray formed in this way, a naphthalene-layer was cast, and the layer was rasped until the surface became flush with the edges. Before the coating, a test slab “d” was placed directly upon the cool exchanger. To prevent edge-effects, smaller slabs of the same thickness as the test slab were placed in front of it and behind it. 2 mm spacers “f” were mounted at the comers of the slab, and the hot exchanger was placed on the spacers, the naphthalene-layer facing downwards. The temperature of the hot exchanger was about 50°C. Under these circumstances about 1 mg/cm2 naphthalene was deposited in 10 min. The technique used did not give a quite uniform thickness of the naphthalene-layer, but the
1446
Mass transfer coefficients on truncated slabs
variations could be held within about 10 per cent. Just after the coating, samples were collected from the surface to determine the weight of naphthalene deposited per unit area on the slab. On each slab four samples were collected along a straight line. From this the thickness of the deposit could be calculated by interpolation at any distance from the leading edge. The sampling process was carried out with an apparatus as shown in Fig. 2. It consists of a solid metal cylinder with a sharp-edged nylon ring “a” fitted concentrically to one end. There
PRESENTATION DIRECT
AND
ANALYSIS
EXPERIMENTAL
OF THE
RESULTS
Forty five runs were carried out partly using truncated slabs, 1, 2, 4, 8 and 16 mm thick, and partly using a 2 mm glass plate with a sharp leading edge. The experiments were performed at four different air velocities, as shown in Table 1, where the runs are indicated by their numbers. All experiments except the one marked “*“, which suffered from an experimental error, were used in the subsequent treatment of the data. Table 1 Plate-thickness d@ml
Air-velocity Vmm/sec 1 a
1.25 2.5 5.0 10.0
I Fig. 2. a. Nylon ring; b. Connection to suction flask; c. Rubber plug; d. Ampulla cover.
.
are two borings in the axial direction. One of them, “b”, is connected to a vacuum pump via a suction flask, while the other opens into a larger boring fitted with a rubber plug “c” and an ampulla-cover “d”. The sampling apparatus was placed on the test slab with the nylon ring facing down. When the vacuum pump was started, the apparatus adhered to the slab, demarcating a well defined area coated with naphthalene. Pure ethanol was injected through the ampulla cover and was drawn to the suction flask together with the naphthalene enclosed, which was dissolved in the ethanol. The amount of naphthalene was then determined by ultraviolet spectrophotometry. After the sampling process, the slab was transfered to the wind-tunnel, where it was placed horizontally. The sublimation, which took place at temperatures between 20-3O”C, was recorded by photographing the test slab at suitable time intervals.
Vol. 24 No. 9-C
5 677 1,2 8,9,10,11 3,4 12,13
4
8
16
Sharpedged
14,15 16,17 l&19 20,2 1
22,23 24,* 25,26 27,28
29,30 31,32 33.34 35,36
1s,2s 3s,4s 5s,6s 7s,gs
The time-averaged values [k& for the local mass transfer coefficients during the sublimation period were calculated from Eq. (2) and plotted against the distance from the leading edge, z. In Fig. 3 the mass transfer coefficients on a 2 mm truncated slab have been plotted as an example. 3o01--
50 6. 70 6r
200 L i -3 -
1447 C.ES.
5
9r IO II 12 13
A v 0 0
too --a
-
Fig. 3. Time-averaged values for local mass transfer coefficients on 2 mm truncated slab vs. distance from the leading edge.
A. SBRENSEN
The time-averaged values for the sharp-edged plate all appeared to decrease with increasing distance from the leading edge, while all the experiments on truncated slabs, except for run no. 5, showed a characteristic maximum for the transfer coefficients at some distance from the leading edge. According to Danckwerts and Anolick [ 141, the presence of this maximum is due to the existence of a stationary eddy just behind the leading edge. The eddy, which is shown in Fig. 4, causes large mass transfer coefficients at the point of reattachment “A”, while relatively small coefficients are recognized on the part of the plate, which is covered by the eddy. From point “A”, a boundary layer is formed, the character of which was not investigated by Danckwerts and Anolick, but was supposed to be turbulent.
Fig. 4. Flow past truncated slab.
The local mass transfer coefficients on a flat plate surrounded by a laminar boundary layer are given by the following theoretical equation r91, j, = 0.332 Re,-Q.5 (3) where Re, = :
and j, = 2 &+?‘3
I 2
IO'
2 bq Rs,
5
Id
2
5
I Id
Fig. 5. Time-averaged values for local j,-factors on sharpedges plate vs. Re,.
suggested way. The experimental points group about a straight line with slope -05 in accordance with Eq. (3). However the experimental values are about 65 per cent larger than prescribed by this equation. As a characteristic illustration to the experiments performed on truncated slabs, Fig. 6 shows the results from the runs with the 8 mm slab plotted in a j, vs. Re,-diagram on a logarithmic scale. As it appears from the plot, the mass transfer coefficients in the case of truncated slabs cannot be correlated in this way. However, it is worth noticing that the asymptotic values of the slopes of the curve-branches to the right of the maximum are about-O-5. A few of the runs, especially some of those performed at the largest air velocity used, show deviations from this general pattern as they seem to converge to less negative slopes.
while the transfer coefficients on a flat plate surrounded by a turbulent boundary layer are described by the expression [ 16,171 j, = O-0288 Re,-02.
5
2
.?a Icr2 3
(4)
An attempt was made to correlate the timeaveraged values here found as indicated by Eqs. (3) and (4). Figure 5 shows the experimental results for the sharp-edged plate plotted in a logarithmic [j,] & vs. Re,-diagram. As can be seen, the results correlate well in the 1448
s
5
2
2ro 280 5
1 I d
I 2
:‘.._,, 5
I IO'
2
5
loq R+
Fig. 6. Time-averaged values for local j,-factors on 8 mm truncated slab vs. Re,.
Id
Mass transfer coefficients on truncated slabs
If .the boundary layer as proposed by Danckwerts and Anolick is developed from the point of reattachment, then the characteristic length z used in Figs. 5 and 6 is irrelevant as far as the truncated slabs are concerned and must be replaced by the length z-A, where A is the distance from the leading edge to the point of reattachment. In Fig. 7, the results for the 8 mm slab are replotted with z- zmax. as the characteristic is the distance from the length, where z,,. leading edge to the point where the mass transfer coefficient reaches its maximum value. The temporary assumption is here made, that the position of this maximum is identical with the point of reattachment.
Fig. 7. Time-averaged values for local jr,-factors on 8 mm truncated slab vs. ( u, (z - .zmax.))/v.
The plot shows that now the experimental points for the individual runs can be represented with reasonable accuracy by straight lines having slopes about -0.5. This indicates that a laminar boundary layer is primarily developed from the point of reattachment. On this basis the following working hypothesis is proposed. When air flows tangentially past a truncated slab, a stationary eddy will be formed at the leading edge, and for the experimental conditions considered, a laminar boundary layer will be developed behind this eddy.
than A, then the velocity boundary layer and the concentration boundary layer will not be developed from the same point. This will, however, change the concentration profile and through that affect the mass transfer coefficients. For a flat plate surrounded by a laminar boundary layer with no mass transfer through the front part, x’, the following expression will apply to the local mass transfer coefficient, kCe, at a distance x from the starting point of the boundary layer;[18], k,H
=k,[l-(;~]-“3.
In this equation, kc is the local mass transfer coefficient that would exist if mass transfer did take place all the time right from the starting point of the velocity boundary layer. In the experiments considered, x’ is the distance from the starting point of the velocity boundary layer to the naphthalene front. Thus x’ is a time variable and consequently k,” is also time dependent as already indicated by the superfix, 8. As will now be shown, kc can be computed on the basis of the hypothesis proposed and the experimental time-averaged coefficients. During the experimental work it was observed that the truncated slabs were completely covered with naphthalene during the first part of the sublimation process, and that only after some time, bare patches were formed at some distance from the leading edge. Thus in this first period (0, tI), the velocity boundary layer and the concentration boundary layer start from the same point. That is, x’ = 0. Then from Eqs. (I) and (3, k,B = kc = [kc]:
Using Eqs. (1) and (6) we get,
CONVERSION OF THE DIRECT EXPERlMENTAL RESULTS
When the naphthalene front is situated at a distance from the leading edge, which is larger 1449
.
(6)
A. S@RENSEN
The ratio [k,]&:lk, is called F, and from this definition, Eq. (7) can be expressed in the following way, kl"a:
k, =
(8)
+--F)
when the first bare patch is formed the thickness at point x is (11)
W(x),, = w0-- W(x)KW
where W(X)(~,~~) is the amount of naphthalene sublimated at x in the time interval (0, tl). From Eqs. (2) and (6) we get
+F'
z
(12)
From Eq. (8) it is possible to calculate local mass transfer coefficients, kc, if F is known. On the basis of the working hypothesis, Eq. (5) is assumed to apply to the boundary layer on the slabs. From Eq. (5) and the expression
On the assumption that a laminar boundary layer exists, kc must be expected to follow an expression of the form kc =
Kx-O+
(13)
Combining Eqs. (1 1), (12) and (13), one obtains
P”dM”h
W(x),, = WO--jy-K~+5.
F=
--
kc
(9)
we get
F=--$---,; [l-($)Sr]-1’3d6J.
(10)
From this equation F can be calculated as x’ = x’(0) is known from the experimental results. F was computed in this way for a number of cases. The resulting values are plotted in Fig. 8 against the dimensionless position variable dx/A, which will be defined in the following.
(14)
For the largest part of the boundary layer region, this expression will describe the thickness of the naphthalene-layer when ~9= tl, but on the other hand Eq. (14) cannot be correct near the starting point, as finite values for tl and thus for ka,_o, were observed in the experiments. To obtain a better description of the thickness of the deposit for small x-values, the following empirical equation was used instead of Eq. (14). W(x),, =
Wo( 1-exp
(15)
(-9))
where
(16) This expression satisfies the conditions
_______-.___-.and W(x),, + W,forx + m. Further Eq. ( 15) does not deviate much from Eq. (14) except for small values of 4x/A. In connection with the remaining calculations, Eq. (13) is assummed to apply. Using the Eqs. ( 13) and ( 15) and the expression RTR’(x)t, [kJk = (tt-tl)PAOMA
-A/,
(17)
we get by substitution in Eq. ( 10)
Fig. 8. F vs. l/x/A. In practice it is difficult and in some cases impossible to determine F wi:h reasonable accuracy in this way, partly because a number of (x’, 0) pairs have to be known in front of the point considered and partly because the integrand in Eq. (10) is infinite for x’ = x. For these reasons, F was also computed as follows. At the start of the run, the thickness of the naphthalenelayer is W,, all over the surface of the slab. At the time tl,
~+-exp(-%9P~_ (tt-f,)Prl&f.J
,,i,
I” ,l_(~~]-l/J& 11
X
(18)
which can be rearranged to the following Volterra integralequation of the first kind,
1450
q1/3(1-exp(-qqZ’S))
=&fq
”
[q-q’]-“%(q’)dq’
(19)
Mass transfer coefficients on truncated slabs
calculated from Eq. (22), local mass transfer coefficients kc were computed from Eq. (8). As mentioned above, the time-averaged values indicated that the mass flux through the boundary layer could be described by an expression of the form
where
B =
WoRT PA~MAK
=
const.
The solution to this integral equation is [ 191
$ScW=
-exp(-$u”)]du
y 0
(20)
314
in whichu = 7
.
Combination of Eqs. (10) and (18) gives F~B[l-Elp(-~)]~x
(
y
V
,,a[ I- exp(-$@)]du
LP(1 -u)-
from which F can be computed as a function of d.x/A. For x + 0 and x + m F can be calculated analytically from Eq. (22) rw-(t) lim. F = x-0 ,-I U(1_uU)-2,3dU = ‘%’ and x-m
(24)
and
St=&_ VCO
(22)
0
lim.F=
2[Re,-Re,] >
where Re A =u,d
I
(23)
>
where C is a constant for each run. To examine the validity of Eq. (23), the calculated k,-values for the experiments on truncated slabs were plotted in a ((St)-” vs. Re,)-diagram. Because x = z-A, Eq. (23) can be rearranged to give the following expression St-2 =
When Eq. (20) is inserted in this expression, the result is
t
-0.5
(21)
b--t,
F=
C 7 (
rmw @(
1-
= 1.37. u) -*‘3du
In the general case F was calculated by numerical integration of Eq. (22). The relation between F and the dimensionless position variable dx/A as calculated from Eq. (22) is shown in Fig. 8. As it appears from the diagram, this relation is in close agreement with the values of F determined from the experimental (x’, @-values. This indicates that the expression (15) used to describe the thickness of the naphthalene-layer is satisfactory for the present purpose. The determination of F for a given value of x has to be performed in general by an iterative procedure as A contains the unknown factor K. By starting from a suitable value for K and using in turn Eqs. (16), (22), (8) and (13), still better values for K are obtained.
From the time-averaged values determined experimentally and the corresponding F-values
So if Eq. (23) is valid, the experimental results will follow a straight line with slope (SC~/~/C)~ and crossing the abscissa-axis at the value ReA. For the vast majority of the experiments considered, a straight line could be drawn through the experimental points, but in most cases, as must be expected, positive deviations were experienced near the abscissa axis, corresponding to smaller mass transfer coefficients than described by Eq. (23). The extent of this region clearly depends on ReA, and its order of magnitude is from 4 to 1 ReA. For a few of the experiments, especially some of those performed at air velocities of IO m/set, negative deviations from the linear relationship were experienced at high Re,-values. This effect can be explained as a transition from laminar to turbulent flow conditions in the boundary layer. For run no. 5 the straight line that was drawn through the experimental points passed through the origin, i.e. ReA = 0. The ReA-values obtained in this way were plotted against v,z,,,~~./Y. The plot showed that there was no systematic deviation between
1451
A. SBRENSEN
these two quantities. Thus the assumption zmax.= A which was made by the calculation of F is correct. The identity of z,,,,. and A and the linear dependence between W2 and Re, both support the working hypothesis proposed on the existence of a laminar boundary layer immediately behind the stationary eddy. The slope of the straight lines which is a measure of the value of the const. C in Eq. (23), varies between the runs in such a way that C increases when the air- velocity and the thickness of the slab are increased. This indicates that C depends on the Reynolds number, Red = u,d/u. A relationship between C and Red is easily understood on the basis of observations made by Danckwerts and Anolick[l4], who found that a strongly turbulent area, caused by eddy separation from the stationary eddy, exists above the boundary layer (Fig. 4). The turbulence in this region will influence on the flow conditions in the boundary layer and thereby change the mass transfer coefficient. Following this mechanism, the transfer coefficients will depend on the condition of the stationary eddy, which in turn depends on the Reynolds number, Red. In Fig. 9, the values of C as found from the (St+ vs. Re,)-diagrams are plotted against Red on a logarithmic scale. As may be seen there exists a close correlation between C and Red in the interval considered. The straight line drawn through the points represents the expression C = 0.152 Red0’179,
(25)
(standard deviation 9 per cent) which was found through linear regression analysis. Substitution of Eq. (25) into Eq. (23) gives j, = O-152 Redo.17e[Re, - Realve5.
(26)
In Fig. 10, ReA is plotted against Red in a logarithmic diagram. It appears that ReA with reasonable accuracy can be described by the relationship
I.0
0
g J
0.5 %
0.2
ri 2
log Re, Fig. 9. The constant C in Eq. (23) as a function of Red.
0
0
3.0 2.0
30
I 4.0
109 Red
Fig. 10. Reynolds number ReA vs. Reynolds number Red.
ReA = 49-g RedoaG1,
(27)
(standard deviation 19 per cent). In Fig. 11, the converted experimental results are plotted in a logarithmic diagram, which on the abscissa axis has (Re, - ReA) and on the ordinate axis has (j, X Red-0’179). The experimental points situated in the interval 0 < Re,- ReA 5 4 ReA, where deviations from Eq. (26) must be expected, are marked by small circles, while the rest of the points are marked by larger circles. In Fig. 11 the Re&-values found from the (St-” vs. Re,)-diagrams are used. When Eq. (27) is used to describe Rea, this gives rise to a minor increase in the scattering of the points, but even then the standard deviation from Eq. (26) does not exceed 15 per cent in the region Re,-ReA > )ReA. The time-averaged values obtained from the
1452
Mass transfer coefficients on truncated slabs
Fig. 11. Local mass transfer coefficients on truncated slabs ’ correlated by Eq. (26). The ReA-values used, are obtained from St+ vs. Re, plots.
experiments made on the sharp-edged plate were converted as described for the truncated slabs. For the vast majority of the experimental points, the Fvalues computed from Eq. (22) differed very little from the values one would obtain by assuming [k&+ 33for x -+ 0. In this case, A = 0 because t, = 0 and Eq. (18) takes the form
The solution corresponding with Eq. (20) then becomes u”~( 1 - u)-z’3du
rww JI
= 1.37.
#a( 1 - u) -z’Sdu
In Fig. 12, the local mass transfer coefficients, kc, on the sharp-edged plate are plotted in a logj,
3.0
4.0 log Re,
Fig. 12. Localj,-factors on sharp-edged plate vs. Re,.
j, = O-407 Rezma5,
5.0
(30)
(standard deviation 12.5 per cent). From Eqs. (3) and (30) it appears that the mass transfer coefficients obtained from Eq. (30) are 22.5 per cent in excess of the coefficients given by the theoretical Eq. (3). ON THE EXISTENCE OF A CRITICAL Red-VALUE
(29)
which gives F=
vs. log&,-diagram. The results are correlated by the following equation
Equations (26) and (30) show that if Eq. (26) applies to all Rehvalues, the mass transfer coefficient at an arbitrary distance x from the starting point of the boundary layer will become smaller for a truncated slab than for a sharpedged plate, if Red is less than a certain value. This would not agree with the above mentioned assumption made by Danckwerts and Anolick [14] on the existence of a turbulent region outside the boundary layer. From this model, one should expect the mass transfer coefficients on truncated slabs to approach the transfer coefficients on sharp-edged plates, when the turbulent activity outside the boundary layer approaches zero. If in Eqs. (26) and (30), (Re,ReJ and Re,, respectively, are equated to Re,, andjDCsharp)is set equal to jD(truncated)r then we find that Red equals 245. This is then the value of Red for which the mass transfer coefficients
1453
A. SfBRENSEN
calculated by Eqs. (26) and (30) take on the same values. The only run which was performed at a Reynolds number Red less than 245, was run no. 5 (d= 2 mm, o, = 1.25 mlsec), which in two respects distinguishes itself from all the rest of the experiments performed on truncated slabs. As realized from Fig. 3, this run does not show any maximum for the time-averaged value of the mass transfer coefficient, and the experimental values when plotted in a (Ste2 vs. Re,)-diagram fall on a straight line through the origin, which according to the previous assumptions means that no stationary eddy exists at the leading edge. From this, run no. 5 must be expected to obey Eq. (30), which applies to a sharp-edged plate. This is supported by the experimental results, as run no. 5 can be described by the expression j, = O-396 Re,-0.5
From Fig. 13, local mass transfer coefficients on truncated slabs can easily be calculated if the air velocity, the thickness of the slab, and the distance from the leading edge are known. THE MEAN
MASS TRANSFER
COEFFICIENT
For most practical purposes, it is of more interest to know the mean values of the local mass transfer coefficients for a given surface area rather than to know the local coefficients themselves. A relation will therefore be set up which makes it possible to calculate mean transfer coefficients on the basis of the experimental results obtained. The mean value of the local mass transfer coefficient for the surface area between z and z + 6z is defined by the equation
(31)
which is in quantitative agreement with Eq. (30). The experiments thus indicate that the stationary eddy at the leading edge is only formed if Red exceeds a certain critical value, R&r, and that this critical Reynolds number has a value of about 245. Equation (26) is represented in Fig. 13, where j, is plotted against Re, with Red as a discrete variable. The family of curves are bounded partly by the expression Re, = 1.5 ReA, as Eq. (26) only applies to Re,-values larger than 1.5 ReA, and partly by Eq. (30), which applies to the sharp-edged plate.
This expression can be put on a dimensionless form by introducing the j,-factor. The following equation then applies to the surface area between z=Oandz=z 1 + jDdz. JD(0.Z) = ; J0
(33)
T
From this expresston, J~~~,~,values for the truncated slabs were computed by numerical integration using the local j,-values obtained, from the experiments, whereas the mean values on the sharp-edged plate were found by integrating Eq. (30) to give .&,Co,Z, = O-812 ReFw5.
(34)
This expression must also be expected to apply to truncated slabs if Red is less than about 245. To get to an analytical expression describing for truncated slabs the integral in Eq. (33) &CO,+, is split into two parts
logRe, Fig. 13. Local&factors on truncated slabs, calculated from Eq. (26).
where the contents in the square brackets according to Eq. (32), are identical with jmo.a, 1454
Mass transfer coefficients on truncated slabs
which is the mean value of the coefficient for the surface area between z = 0 and z = A. In Fig. 14, the experimental values ofJ;)Co,L\) are plotted against Red on a logarithmic scale. Even
0.6
F 0.5
Fig. 14. Mean values of the j,-factor for the eddy-area, plotted as a function of Red.
0.4
runs performed under identical experimental conditions show considerable scattering. It appears, however, that j,,,,, can, as a reasonable approach be described by the empirical exnression 2.42 x lo3 htO,A, =
2.02 x 105+Red1'3'
(36)
i
As mentioned above, Eq. (26) describing the local &factors in the boundary layer region is not valid for the front part of the boundary layer just behind the stationary eddy. To extend the validity of Eq. (26) to include this region, a correction factor 7 is introduced. .experimental
7 = &ez_ReA,-0.5'
(37)
Fig. 15,~ is plotted against the dimensionless position-variable A/z in the interval 4 < A/z < 1. As shown, r) can be described by the empirical expression (38) q=l_ 4” z .
In
0
Inserting Eq. (38) into Eq. (35) gives + 0. 152Re~017s&?A-o’5 x[1 - (x+ I0
I)-“IX-WX]
(39)
06
0.7
0.6
0.9
I.0
A/Z Fig. 15. Correction factor r) vs. A/z.
where X=z--A --_=--1 Re, A &A
’
The ratio between the Jbto,,,-factors obtained from the experiments and the values calculated from the correlation (39) was formed. The mean value of this ratio for all the experiments performed was l-004, and the standard deviation was 0.115. In Eq. (39) &,,A, as Well as &?A depend on Red as described by Eqs. (36) and (27), while X depends on both Re, and Red. Thus the equation describesjb,o,z, as a function of the two Reynolds numbers Re, and Red. In Fig. 16, &,to,z, as calculated from Eq. (39) is plotted against Re, with Red as the discrete variable. The left hand curve-branches clearly show the negative influence from the stationary eddy on the &o,z,-factor, while the rather strong dependence of the right hand curve-branches on Red demonstrates the positive effect which the eddy
1455
A. S@RENSEN
vestigation. From these plots local and mean mass transfer coefficients can easily be calculated when the air velocity, the physical properties of the air and the geometrical configuration are known. Especially Fig. 17 is convenient for most practical purposes, as shown by the following example. 2
s
I04
2
5
Id
EXAMPLE
logRe, Fig. 16. Mean values of j,-factor for the region z, = 0 to z = z vs. Re, and Red.
exerts through the formation of a turbulent region outside the boundary layer. In Fig. 17, .J”,,,~, calculated from Eq. (39) is plotted against Re, for different values of z/d. A specific curve in this diagram thus shows how jD(O,rjdepends on the air velocity for a given slab or for geometrically similar slabs.
Moist ceramic slabs 20 cm long and 1 cm thick are dried in an air stream moving parallel to the surface of the slabs. Calculate the mean mass transfer coefficient, when the air velocity is 3 mlsec. The kinematic viscosity of the air is 1.7 X lo-” m%ec and the Schmidt number for the air-water system is 0.60.
zld=$=20.
From Fig. 17:
kxo,z, =
On the whole, the curves form parallel straight lines with slopes of about - O-31, which gives a velocity dependence of the form
Powell[lO] as well as Shepherd, Hadlock and Brewer[ 131 have carried out mass transfer experiments on trays or slabs 1 ft long and 2 in. thick, i.e. L/d = 6. Powell’s results are described by the equation
(40) .&OW
This dependence, however, does not apply when Red is less than about 245. In that case an expression of the form
kcto,,,- tho.50
1O-2m/set.
X
COMPARISON WITH PREVIOUS INVESTIGATIONS
Fig. 17. Mean values of the j,-factor for the region z = 0 to z = z vs. Re, and z/d.
I;c(o.x) - Ym@6S.
6*8X 10-S x 3.0 = 2.9 0.71
0*151Re;030
while the results of Shepherd summarized by the expression
(42) er al. can be
jmo,w = 0.11 2ReLwZ5.
(41)
should apply as demonstrated by run no. 5. Figures 16 and 17 together with Fig. 13 constitute the main results of the present in-
=
These results are compared Fig. 18 where the equation
1456
(43)
with Eq. (39) in
Jmo,Zj= 0-036Re,-0’2
(44)
Mass transfer coefficients on truncated slabs
Q-p ..\‘-
.
-__
Shaphcrd This correlation POWII
Eq.(44)
Fig. 19. Comparison between Eq. (39) and the work of Powell [lo] on a thin truncated slab.
Fig. 18. Comparison between Eq. (39) and the works of Shepherd[ 131and Powell[ lo]. Equation (44) applies to a turbulent boundary layer on a flat plate.
which applies to a turbulent boundary layer on a flat plate is also shown. Sherwood [4] performed drying experiments on a slab made from sulphite pulp, 5 cm long and 1.5 cm thick. The results showed that
5 2
5 &I
IO’
1 2
Rc,
Fig. 20. Comparisonbetween Eq. (39) and the results obtained by Danckwerts[ 141. Raaschou, Becker and Janholm [ 151 investigated the drying of linen, which was stretched on a frame. The results of this work demonstrated that
I@& - U,O’50.
(46)
The velocity dependences obtained by [lo], [4] thus agree quite well with the velocity dependence found in this work for Red > 245, while the dependence obtained by [ 151 agrees with this work for Red < 245. Powell [lo] also carried out experiments on a “thin” truncated slab about 20 cm long, and found that for this situation i&o,Ljat small velocities was proportional to u,O’~,while at larger velocities it was proportional to v,O.~~.So his results support the assumption on the existence of a critical value for Red. In Fig. 19, they are compared with the values obtained from Eq. (39) for a 5 mm slab , of the same length as the one used by Powell. In Fig. 20, the results obtained by Danckwerts and Anolick [ 141 for a a in. truncated slab by v, = [ 131 and
2.15 mlsec are compared present correlation.
with the results of the
SUMMARY AND CONCLUSION Time-averaged values for the local mass transfer coefficients were measured on truncated slabs and on a sharp-edged plate placed in a tangential air stream. The primary results for the slabs indicated the existence of a laminar boundary layer behind the stationary eddy formed at the leading edge. On this assumption, the time-averaged values were converted into “ordinary” local mass transfer coefficients that will exist if mass transfer takes place on the whole surface all the time. The local mass transfer coefficients obtained in this way support the assumption of the existence of a laminar boundary layer. The coefficients found on the sharp-edged plate are about 22 per cent larger than expected
1457
A. SBRENSEN
from the theoretical Eq. (3), which indicates that the experimental results perhaps suffer from a systematic error. This might be due to the effect of the turbulence intensity in the air-stream because, as shown by Thomas [7,20,21], the intensity of turbulence can greatly enhance the mass flux through’ a laminar boundary layer. However, this quantity was not measured in the experiments, and thus it is hard to say if the intensity of turbulence is of any significance in the present case. The results from the experiments performed on truncated slabs indicate the existence of a critical value for the Reynolds number Red. Judging from the present data, no stationary eddy is formed if Red is less than about 245, and this being the case, the local mass transfer coefficients on truncated slabs follow the expression that applies to a sharp-edged plate. Correlations were set up for local and mean mass transfer coefficients on truncated slabs and on sharp-edged plates. From these expressions, mass transfer coefficients can easily be calculated if the velocity and physical properties of the air and the length and thickness of the slab are known. Owing to the analogy existing between convective heat and mass transfer, the correlations can also be applied to predict heat transfer coefficients, when thej,-factors in Figs. 13, 16 and 17 are replaced by the correspondingjH-factors.
NOTATION
defined by Eq. (16) C dimensionless constant defined by Eq. (23) c total concentration in gas phase, kmole/m3 cA concentration of sublimating component, kmole/m3 D diffusion coefficient, m*/sec d thickness of slab, cm or m F = kl:: jD local jD-factor (mass transfer taking place from the whole surface), dimensionless A
1458
time-averaged value of local j,-factors in the time interval t1 to t2, dimensionless mean value of the j,-factor for the area between z = z and z = z +6z, dimensionless K constant defined by Eq. (13) kc local mass transfer coefficient (mass transfer from the whole surface), mlsec or m/hr time-averaged value of the local mass transfer coefficient for the time interval between tI and tz, m/set or n&r local mass transfer coefficient at time 8, mlsec or m/hr mean value of mass transfer coefficient for the area between z = z and z = z + 6z, mlsec or mlhr L length of slab in velocity direction, morcm molarweight of the sublimating comMA ponent, kglkmole P A0 vapour pressure of sublimating component, ton-. R universal gas constant, 62.36 (m3 torr.)/(deg. kmole) Red = v,d/y, dimensionless Re, = v,z/v, dimensionless ReA = vmA/v, dimensionless Rf&i,cr = critical value of Red, dimensionless SC = Dp/p Smith number, dimensionless St = kc/v, Stanton number, dimensionless T absolute temperature, “K front to t, time for the naphthalene reach the position z = z, set time for the first bare patch to be t1 formed on the surface, set in the undisturbed airVCO air velocity stream far from the slab, mlsec wo initial thickness of the naphthalenelayer, kg/m* thickness of the deposit at the W(x),, position x, when t = tl, kg/m* naphthalene sublimated at x in the time interval 0 to tl, kg/m*
Mass transfer coefficients on truncated slabs
X
= (Z - A)/A
= Re,lReA - 1,
dimen-
sionless distance from the starting point of the boundary layer, m x’ distance between the naphthalene front and the starting point of the boundary layer, m Y distance normal to the surface, m Z distance from the leading edge Zmax. distance between the leading edge and the point where the mass transfer coefficient reaches its maximum value, m x
Greek symbols A distance from the leading edge to the starting point of the boundary layer, m kinematic viscosity mYsec V 77 correction factor defined by Eq. (37) P density kgIm3 _ _ Subscripts
0 on the surface ~0 in the undisturbed air-stream
REFERENCES [l] MILLAR F. G., Can. met. Mem. 1937 143. [2] WADES. H., Trans. Znstn them. Engrs 1942 M 1. [3] PASQUILL F., Proc. R. Sot., Land. 1943 A182 75. [4] SHERWOOD T. K., Ind. Engng Chem. 1929 21976. [5] HINCHLEY J. W. and HIMUS G. W., Trans. Instn them. Engrs 1924 2 57. [6] BIRD R. B., STEWART W. E. and LIGHTFOOT E. N., Transport Phenomena. Wiley 1962. [7] THOMAS D. G., A. 1. Ch. E. J/l965 11520. [8] JU CHIN CHU, LANE A. M. andCONKLIN D., Znd. Engng Chem. 1953 45 1586. [9] POHLHAUSEN E.,Z. angew. Math. Mech. 19211115. [lo] POWELL R. W., Trans. Instn chem. Engrs 1940 18 36. [ 1l] MAISEL D. S. and SHERWOOD T. K., Chem. Engng Prog. 1950 46 13 I. [ 121 WENZEL L. and WHITE R. R., Znd. Engng Chem. 195 143 1829. [ 131 SHEPHERD C. B., HADLOCK C. and BREWER R. C., Ind. Engng Chem. 1938 30 388. [14] DANCKWERTS P. V. and ANOLICK C., Trans. Znstn them. Engrs 1962 40 203. [15] RAASCHOU P. E., BECKER F. C. and JANHOLM C., Dunmarks Nururuidensk, No. 9, Samfund 1925. [16] SCHLICHTING H., Grenzschicht Theorie. Braun 1958. [ 171 COLBURN A. P., Trans. Am. Inst. Chem. Engrs 1933 29 174. [18] KROUJILINEG., Tech. Phys. U.S.S.R. 19363 183,311. [19] MARGENAU H. and MURPHY G. M., The Mathematics ofPhysics and Chemistry. Van Nostrand 1956. [20] THOMAS D. G.,A. I. Ch. E. JI 1%5 11848. [21] THOMAS D. G.,A. I. Ch. E. JI 1966 12 124. R&sum& Des coefficients de transfert de masse locaux et moyens ont ttC obtenus experimentalement sur une plaque a bords tranchants et sur des dalles tronquees d’epaisseur variable. Les r&sultats dkmontrent que les coefficients de transfert de masse sont non seulement fonction de la v&cite de lair et de la distance du bord d’attaque, mais 6galement de l’epaisseur des dalles. Les facteurs j, sont en correlation par deux nombres de Reynolds ditferents, Re, et Red et, sur la base de ces correlations, il est possible de calculer les coefficients de transfert de masse sur les dalles. Les rtsultats experimentaux indiquent l’existence dune valeur critique Red, Red,,. Si Red est inferieur a Red,er, les coefficients de transfert de masse ne dependent pas de Red, mais suivent les expressions qui se rapportent B une plaque B bords tranchants, tandis que pour les valeurs Re,, sup&ieures B Re,,,,, les coefficients de transfert de masse sont clairement fonction de Red. Ces resultats indiquent Cgalement qu’il existe une couche limite laminaire derriere le tourbillon stationnaire qui s’est form& sur le bord d’attaque. Zusammenfaasung- &tliche und durchschnittliche Stofflibergangswerte an einer scharfkantigen Platte und an abgestumpften Scheiben verschiedener Dicke konnten auf experimentelle Weise erhalten werden. Die Ergebnisse zeigen, dass die StolBlbergangswerte nicht nur von der Luftgeschwindigkeit und dem Abstand von der Vorderkante abhiingen, sondern daas such die Dicke der Scheiben van Bedeutung ist. Die ortlichen und durchschnittlichen j,-Faktoren werden durch zwei verschiedene Reynolds -Zahlen, Re, und Red, in Wechselbeziehung gebracht, und mit Hilfe dieser Korrelationen kiinnen Stofflibergangswerte an abgestumpften Scheiben berechnet werden. Die experimentellen
1459
A. SORENSEN Etgebnisse weisen auf das Bestehen eines kritischen RehWertes, Ret,, hin. 1st Red niedriger als Red,cr so hiingen die Stoffubergangswerte nicht von Red ab, sondem nchten sich nach den auf eine schartkantige Platte anwendbaren Ausdrilcken, wiihrend bei Re=Werten, die grosser als Re+, sind, die Stoffdbetgangswerte eindeutig von Red abhiingen. Die Ergebnisse deuten femer darauf hut, dass hinter dem stationiiren Wirbel, der sich an der Vorderkante bildet, eine laminare Grenzschicht besteht.
1460