March, 1932
INDUSTRIAL AND ENGINEERING
CHEMISTRY
307
ACKNOWLEDGMENT
increasing concentrations of salt, the color changed markedly from a dark to a light blue color,.and that high concentrations of salt resulted in increasing amount of drawn grain. This may be attributed to the shrinkage of the stock as shown in Figure 8. Not only does the salt content affect the shrinkage, but it also affects the concentration of chrome as shown in Figure 9. The maximum and minimum of swelling are due t o the hydrogen-ion content of the liquors, as demonstrated by the authors ( 2 ) in a previous paper on pickling, and to the acid removed from the stock by diffusion, as mentioned in the earlier part of this paper. Strips of the leather treated with 12 per cent Tan R and varying amounts of salt were fat-liquored and analyzed for fat content, and it was found that there was no appreciable difference in the fat content of the leathers. These same strips were given a tear test, and it was found that the strength of the leather increased from 42 to 63 pounds up to 10 per cent salt, and then decreased to 52 pounds for 20 per cent salt.
The writers wish to acknowledge the generous financial support given this investigation by the Hunt-Rankin Leather Company of Peabody, Mass.
LITERATURE CITED (1) Gustavson, K. H., IND.EXG.CHEM.,17, 945 (1925). (2) Theis, E. R., and Goetz, A. W., J. Am. Leather Chem. -4ssocri., 26, 507 (1931). (3) Thomas, A. W., and Foster, S B., J. ISD ENO. C m x , 13, 31 (1921). (4) Wilson, J. A., and Gsllun, E. A , J.A m Leather Chena. Assocn , 15, 273 (1920). RECEIVEDSeptember 14, 1931. Presented before the Division of Leather and Gelatin Chemistry a t the 82nd Meeting of the Amerkan Chemioai Sooiety, Buffalo, N. Y . , August 31 t o September 4, 1931. A. W. Goetz is the Hunt-Rankin fellow in leather technology at Lehigh University.
The Dry:ing of Solids-IV Application of Diffusion Equations T. K. SHERW OOD, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
I
I n the drying of a solid slab, the moisture distribution approached during the constant rateof-drying period is represented by a parabolic function O f the distance .from surface. The equation is given expressing the relation between moisture content and time in the drying of a with internal liquid diffusion controlsolid ling, for the case of a n initial parabolic moisture distribution. Data ape Presented supporting the applicability of this relation to the drying with internat liquid diffusion controlling, subsequenf to a preliminary constant-rate period.
solid a t the start, it cannot be expected to apply to the fallingr a t e p e r i o d in such a case, e v e n t h o u g h internal liquid ing of solids has been outlined, diffusion is controlling. T h e with particular reference to the moisture gradients in the confactors governing the diffusion s t a n t - r a t e period are deterof moisture to the solid surface mined by the nature of the and thence out into the air. solid, its dimensions, and the The drying process was shown rate of drying. They are of in general to be divisible into a constant-rate and a fallinggreat interest in the drying of materials which tend to warp or rate period, the j u n c t i o n of crack, since the difference bethese two periods being termed t w e e n s u r f a c e a n d internal t h e " c r i t i c a l p o i n t * " The m o i s t u r e contents determines falling-rate period was in turn the differential shrinkage, and shown to be frequently in two parts: in the period immediately following the critical point consequently the tendency to warp or crack. In most cases the rate of drying decreases because of a decrease in wetted the shrinkage of clay on drying is completed before the critical surface; and in the last stage of the drying process the rate of point is reached, and drying with air of high humidity is diffusion of water through the solid controls the rate of resorted to in order that the rate of drying, and consequently drying. In the drying of certain slow-drying materials, the differential shrinkage between surface and interior, may however, no constant-rate period occurs, and internal liquid k not be excessive. diffusion is controlling from the first. An equation ( 3 ) was The equation is derived below for the moisture gradient presented for this case, giving the relation between the mois- approached in a slab drying a t a constant rate. Assuming I ' the moisture gradient so derived t o apply a t the critical ture content and the time of drying for an infinite slab: 4 point, the relation between water content and time is given Ke for the subsequent falling-rate period in which internal liquid diffusion controls throughout. K THREE previous articles (3, 4, 5 ) the general mechanism of the air-dry-
4
I
f
2% e-
25
(')'
'
+
'1 9 1
' '
(l)
In the drying of many common materials, a short constantrate period is followed by a falling-rate period in which internal liquid diffusion controls throughout. During the constant-rate period there may be set up an appreciable difference between the moisture contents at the surfme and in the interior of the solid. Since Equation 1 was derived on the assumption of a uniform distribution of moisture in the
*
hIOISTURE
_ PT _ - D- 8T
,1 ,
I
GRADIENTS1N SLAB DURING PERIOD
6x2
K ' 68
COXSTAXT-RATE
(2)
which may, presumably, be employed in an analysis of the conditions in the solid during the .constant rate period. ks long as the rate of drying is constant, the moisture gradient a t the solid surface will remain constant, as given by
INDUSTRIAL AND ENGINEERING CHEMISTRY
308
(3)
and for the case of an infinite slab or sheet, the gradient a t each surface will be the same.
curves will be parallel, and Equation 2, therefore
6T
Vol. 24. No. 3
will be a constant.
9 = constant 6x2
(4)
6T
Integrating and substituting the conditions that when x = R; T = T, when x =R; and T = T, when 0:
On differentiating and substituting x the surface is found:
=
=
0
2 =
0, the gradient a t
From Equation 5 it follows that, after drying an infinite slab for an appreciable length of time at a constant rate, the moisture-gradient curve approaches a parabolic form, no matter what the initial moisture gradient map have been.
I -2
From
R-
Figure 1represents the cross section of a slab, with ordinates representing the moisture concentrations a t different times during the constant-rate period. ABC represents the gradient a t any instant, and EDF the new gradient after a finite length of time. Since no moisture crosses the center line, the slope of the concentration curve is zero a t that point. Furthermore, since the rate of drying is constant, then from Equation 3 the slopes a t A , E , C, and F are equal. Now the area ABCFDE represents the moisture lost; and, since the rate of drying is constant, the areas between successive concentration curves, corresponding to equal time intervals, will be the same. Humps or high points in the
Rd
7-
FIGURE3. PLOTOF E' VS. K8'/R2
CM.
FROM
EXPOSED
SURFACE
FIGURE2. CONCENTRATION CURVES BY TROOP AND WHEELER OBTAINED IN DYING CLAY PT
gradient curves will tend to disappear rapidly because 62' will be numerically large, and from Equation 2 it follows that the rate of decrease of moisture concentrations a t such points will be correspondingly large. The gradient curves will clearly approach a definite shape, after which successive
As a test of the correctness of the above derivation there are available the data of Troop and Wheeler (6),who obtained moisture gradients in clay samples dried a t various temperatures and humidities. Small clay cylinders, 12.7 cm. long and 2.54 cm. in diameter, were held in copper tubes and placed in a thermostat maintained a t a constant temperature. One end face mas exposed to air a t a controlled temperature and humidity, and, as the drying took place wholly from the single exposed end face, the conditions were similar to the drying of a 25.4-em. infinite slab and clay. After various time intervals, identical samples were taken a t several points along the length of the cylinder, and the moisture concentrations obtained. Figure 2 shom concentration curves obtained with air a t 70" C. and 90 per cent relative humidity, these data being typical of the results obtained. The parabolic moisture gradients are clearly indicated, and it is of special interest t o note how quickly the initial flat moisture-distribution curve disappears and the parabolic distribution is approximated. I n the test described, the rate of drying of each sample during the constant-rate period was 2.86 grams in 24 hours. From Figure 2 the average value of (T, - T,)may be found to be 0.053. Substituting these values together with a value
March, 1932
INDUSTRIAL AND ENGINEERING
of R of 12.7 in Equation 6, the value of K’ is found to be 7.8 X lo-‘ grams per second per square centimeter per centimeter thickness per unit dT. It may be noted that K’= KD. Repeating the calculation of K’, using the data of Troop and Wheeler for other tests, values of K’ were obtained varying roughly with the fluidity of water a t the temperature of the thermostat. For example, a t 30” C. the value of K’, calculated from the ralue of T , - TI after drying for 24 hours, is 1.3 X
Where shrinkage is negligible, T D can replace v . Substituting t’he parabola ( 5 ) for f (d), and integrating,
309
pears and the result checks exactly with (l),the corresponding equation for the case where the initial moisture distribution is uniform. Equation 10 represents the theoretical relation between water content and time, for the drying of an infinite slab with liquid diffusion controlling, for the case of an initial parabolic moisture distribution. as a t the end of a constant-rate period.
DRYING AT CONSTANT RATEFOLLOWED BY LIQCIDDIFFUSION CONTROLLINQ The theoretical relation for the drying of an infinite slab, when internal liquid diffusion is controlling, is given above (Equation 1). The equation is applicable, however, only when the moisture distribution is uniform throughout the *lab a t the beginning of the period when internal liquid diffusion controls. Frequently, as in the drying of certain shapes of wood and clay, the period of internal liquid diffusion controlling follows immediately after the constant-rate period, and the moisture distribution a t the start of this period is consequently more nearly parabolic than uniform. It is wident then that Equation 1 does not hold for this case. I t ivould appear that an equation similar to (l),but based on the assumption of an initial parabolic moisture distribution, would be useful in the study of the drying of such materials. Carslaw (1) gives the solution of the basic diffusion equation for an infinite slab where the initial concentration gradient i s j (x’)as
CHEMISTRY
0.15
d
.
%
E 0.10 Y)
U J
w1
c ( I
MI
%
10 20 F R C C WATER, DRY B A S I S .
FIGURE4. RESULTS OF DRYING CWYSLABSIN HIGH-VELOCITY AIR STREAM The coefficients of the two series in Equation 10 are both functions of the ratio, T8/TflA.E’ is consequently a function of the two dimensionless ratios, p and Ta/Tm. This fact suggests a method of plotting Equation 10, as shown in Figure 3, where E’ is plotted vs. K8/R2,for values of Ta/T,,, of 1.0 (Equation 1) to 0 (Equation 11). EXPERIMENTAL DATAON DRYING OF BRICKCLAY I n an experimental study of liquid diffusion in solids, it is found advantageous to exaggerate the importance of the period where liquid diffusion controls, by drying in a high-
It may be noted that when 8’ = 0, and z = R, this reduces t.0 T = T,. More important than the concentration a t any point, however, is the total water content, obtained by integrating the concentration across the slab thickness. Defining E’ as the ratio of the free-water content to the free-water content a t the end of the constant-rate period (the critical point), then
r2“
T dx
Substituting the two series of Equation 8 for T, and the parabola (Equation 5 ) for f (d), and integrating, E’ = -
+se -25p +
,
.
.J
A F T E R S T A R T O F F A L L I N G - R A T E PERIOD
FIGURE 5 . COMPARISON
OF DATA OY DRYINGC L ~ Y SLABSWITH E Q U ~ TION
For the case of 8’ = 0, Equation 10 may be shown to reduce to E’ = 1. Furthermore, when T,,, = T,, the second series disaD-
.
1
velocity air stream, thus vaporizing the water as fast as it diffuses to the surface. The drying of a clay slab 2.54 cm. thick under such conditions %as described in a previous
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ENGlNEERING
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article ( 3 ) . The results of two such tests are shown plotted in Figure 4. The rates of drying obtained by measuring the slopes of the curves of weight us. time are shown plotted against the free-moisture content. Each sample dried at a constant rate down to about 16.5 per cent water, after which
I I I I I 40 60 80 100 120 T I M E A F T E R START OF F A L L I N G R A T E P E R I O D
I 20
-
I
I
140 160 MINUTES.
I
the rate curves are concave upwards and are typical of drying with internal liquid diffusion controlling. The edges of the slabs were covered with tin foil, and consequently diffusion was possible only in a direction normal to the faces, as in an infinite slab. The data are first compared with the theoretical relation (I) by means of Figure 5 . A special coordinate plot is constructed, the ordinate or E' scale being modified t o force the K8' theoretical relation (1) to be linear in E' and ~ 2 Thus, , the data of any single experiment following Equation 1 should fall on a straight line, when plotted as E', the fraction of the K8' critical free moisture vs. F ,or V S . ~ ' ,the time after the end of the constant-rate period. Figure 5 shows lines having a marked curvature a t high values of E', which is to be expected since the derivation of Equation 1 assumes a uniform moisture distribution a t the start, i. e., a t the critical point. I n order to compare the data with Equation 10 which allows for an initial parabolic moisture distribution in the slab, it is necessary to determine T , and T. a t the critical point. These quantities can ordinarily be found using the fact that the average moisture T , a t the critical is Ta plus two-thirds of T , - T,, and by calculating T, - T , from Equation G. I n order to do the latter, K' may be estimated by comparing the data with Equation 1. I n the case of the present data on clay, T,' - T, so calculated is considerably greater than the initial free-moisture content, which was approximately 27 per cent for both samples. This means that the clay was dried a t such a high rate that the parabolic moisture distribution could not be reached in the short constant-rate period, even with the surface concentration, T,, a t zero. I n the extreme case, T,may be assumed to have reached zero at the critical point. Substituting T, = 0 in Equation 10, the result is
Table I shows values of E' calculated from this series, as well as from Equation 1. I n order to compare the data with Equation 11,Figure 6 is shown with the ordinate scale modified to make this relation linear. In this case the data fall on lines
-ALUES OF
Ke
E Eauation 1
E' Eauation 11
0 0.02 0.05 0.10 0.15 0.20 0.30 0.50 1.00
180
FIGURE6 . COMPARISON OF DATAON DRYINGCLAYSLABS WITH EQUATION 11
E AND E' CALCUL4TED FROM EQUATIONS 1 AND 11
I.
KW RP Or
I
Vol. 24, No. 3
reasonably straight a t high values of E', but slightly concave upwards a t the lower ends. Going from a comparison with (1) to a comparison with (ll), the lines representing the data have changed from convex to concave upwards, whereas in each case the lines representing the theoretical equations are straight. It may be concluded: that either the data would compare better with the more rigorous solution (lo), using suitable values of T, and Ta than the extreme cases (1) and (11); or the series (lo), using T, = 0, is approximated closely, and that the curvature of the lines of Figure 6 is due to a decrease of the diffusion constant, K , witli decrease in moisture content. TABLE
0
CHEMISTRY
An approximate value of K' for this clay may be obtained by noting from Figure 6 that E' = 0.605 a t about 75 minutes. Since this E' corresponds to a value of K9'/Rz of 0.20,it follows that K and K'
= =
0.20 X 1.272/75 X 60 = 0.72 X 10-4 K D = 1.6 X 0.72 X lo-' = 1.15 X lo-'
This value, for clay-drying a t room temperature, agrees a t 70" and 1.3 X reasonably well with the values of 7.8 X 10-4 a t 30" C., calculated from the data of Troop and Wheeler for a different clay. NOMENCLATURE A D
E
=
face area
=
ratio of total free-water content to initial free-water content ratio of total free-water content to free-water content at end of the constant-rate period, i. e., at critical point diffusion constant of moisture through solid, with concentrations as weight per unit volume same with concentrations as weight of water per unit weight of dry solid = K D
= density of dry solid
E'
=
K
=
K'
=
P
=-
KT%' 482
half slab thickness = free-moisture concentration, weight of water per unit weight of dry solid T , = same at center line of slab at critical point T. = same at faces of slab at critical point = free-water concentration at anypoint D W = weight of water 2 = distance from slab face 8 = time 8' = time after critical point R T
=
LITERATURE CITED (1) Carslaw, H. S . , "Introduction to the Mathematical Theory of Conduction of Heat in Solids," MacMillan, 1921.
(2) Newman, A. B., Interim publication of Am. Inst. Chem. Eng., August, 1931. (3) Sherwood, T. K., IND.E K G . CHEM., 21, 12 (1929). (4) Sherwood, T. K., I b i d . , 21, 976 (1929). (5) Sherwood, T. K., I b i d . , 22, 132 (1930). ( 6 ) TrooD and Wheeler, Trans. Ceram. Soc., 26, 231, 239, 261 (1926-27); 27, 303 (1927-28).
RECEIVBD December 9, 1931. Presented before the meeting of the American Institute of Chemical Engineers, Atlantic City, N. J., December 9 to 1 1 , 1931.
