THE DRYING OF SOLIDS Prediction of Critical Moisture Content D. B. BROUGHTON Massachusetts Institute of Technology, Cambridge, Mass.
The average moisture concentration at the critical point in the drying of a solid is expressed in terms of the drying conditions and the nature of the material. This relation is developed on the assumption that the surface moisture concentration at the critical point is a function only of the nature of the material. The relation proposed here is shown to be supported by several sets of data on kaolin and clays.
I
N T H E drying of slabs with air, the process divides itself into
two distinct stages. I n the first, the rate remains constant at a rate approximating that of evaporation from a free liquid surface. The water distribution becomes nonuniform, with maximum concentration at the center and minimum at the faces. When the integrated average water concentration falls to a certain value, known as the critical average moisture content, the second period is initiated, with a decrease in drying rate. During this period the rate is roughly proportional, at each point in the process, to the average free water content of the solid. These facts allow the complete drying schedule of a material to be approximated, if the critical average moisture content is known. Consequently, it is of considerable importance to be able to predict the critical average moisture content for a particular material being dried under given conditions. Sherwood and Gilliland (6) presented a method of predicting a critical average moisture content from the rate of drying in the constantrate period and the diffusion constant of liquid through the solid. This method assumes that water migrates within the solid in accordance with the diffusion law, and that the constantrate period ends‘when the free water concentration at the surface falls to zero. Use of this method requires one drying test to obtain the diffusion constant and involves a graphical solution of the diffusion equations. More recent work by Ceaglske and Hougen (1) showed that in coarse granular solids, such as sand, water movement does not follow the diffusion equations, which call for development of a parabolic distribution aa the constant-rate period continues. The actual distribution curves determined were S-shaped, and were correlated by the assumption that water moves through the solid in response to capillary forces instead of to concentration gradients. However, the data of Sherwood (4) show that for a number of materials, including wood, brick clay, and soap, the water distribution follows that predicted from the diffusion equations fairly closely, The method of correlation presented in this paper depends on the validity of the diffusion equations in predicting water distribution and, consequently, would not be expected to apply to materials of the type investigated by Ceaglske and Hougen. By making use of the fact that in the constant-rate period the diffusion equations indicate rapid approach to a parabolic mois-
ture distribution in a slab, it is possible to simplify the method of Sherwood and Gilliland (6) to a point where the relation between critical average moisture content and drying conditions for a given material can be represented graphically by a straight line. This method has added flexibility, since it is not necessary to assume that the free water concentration a t the surface has fallen to zero at the critical point. I n fact, the data indicate that this does not occur. The derivation which follows is based on the assumption that the surface free-moisture concentration at the critical point is dependent on the nature of the material but not on the drying conditions. If the constant-rate period (Figure 1) is long enough to establish a parabolic water distribution across the slab,
Differentiating and substituting z = 0 and z = 2L gives gradients at the surface,
I
At any time,
X I
FL-i FIGURE 1
Substituting for C from Equation 1 and integrating,
TABLE I. DATAON DRYING OF KAOLIN AND CHINACLAY SLABS Cia,,
Florida knolin
E n liah
ekns clay
Air Velocity,
Ft./Sec. 10.4 9.6 10.4 10.4 7.5 10.8 10.8 10.2 10.4 10.4 10.4 10.6 10.4 9.4 7.9
II
1.
156 132 172 110 142 166 107 167 167 140 166 117 167 167 167
tw
... ‘95 75 92 106 128 135 132 108 128 93 121 138 103
n
0.346 0.250 0.377 0.197 0.228 0.368 0.252 0.169 0.222 0.242 0.341 0.168 0.236 0.134 0.233
Wc 0.396 0.372 0.360 0,388 0.350 0.369 0.337 0,258 0,297 0.168 0,197 0.205 0.132 0,120 0.165
1.52.-2.3 1.50 1.28 1.47 1.68 1.08 1.50 1.02 0.67 0.90 1.16 1.38 0.97 0.95 C.52 0.94
combined with Equations 3 and 4 gives: 3O(Ca,,
L
- C*)
)
Applying Equation 5 at the critical point and noting that C =
pW and CY
= I’p(dW/d8)c,
CYL w. = w*c+ T =j 1184
(6)
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1945
Dece&r,
FIGURE L CR/T/CAL MOISTURIC CONT€NTS
,
.
from viscosity data on water in their temperature range. Considering the fact that the reproducibility of their experimental critical moisture contents was not better than 2% moisture, the points fall fairly close t o the predicted straight line, Figure 3 shows results of a similar test of data obtained by the author on two other types of clay. The slabs were 17.5 X 17.5 X 2.0 om., and were dried from one face a t a variety of air temperatures, humidities, and velocities. All samples were prepared similarly by hand-kneading the clay with enough water to give an initial water content, W, of about 0.55. The clay was dried in a thin sheet metal pan, and the sides and bottom of the slab were covered with tin foil to prevent drying from those surfaces. The data (Table I) fall reasonably close t o a straight line in Figure 3. None of these curves extrapolate t o an intercept, W,, of zero, as was assumed in the method of Sherwood and Gilliland. Comings and Sherwood (8) observed in some cases a recovery in drying rate following the initial fall, and attribute this to breaking away of the stock from the pan because of shrinkage during drying. This effect was not observed during these tests, presumably because of the ability of the tin foil to shrink with the stock. Having once established the position of the line on a plot of the type of Figures 2 and 3,the critical average moisture content is readily predicted graphically or by direct use of Equation 8. It would be expected that data for slabs of other thicknesses would fall on lines having the same intercept but slopes directly proportional to the thickness. No suitable data have been found to test this effect. For other regular shapes, similar relations c m be derived from the diffusion equations. Thus, for long cylinders, the analog of Equation 8 is:
,
20
1s
& 0
e
10
s --AIR
A /R V€L OC1TY FROM Z 0 TO 13. 1 F Z /SEC. TEMPERATURE FROM 59% TOI1S%
I
0
I
I
I
1185
I
48
QO
32
24 \
16
(9) 8
For spheres the corresponding relation is:
'
kR/T/CAL MO/STUR€ CONTENTS OF FLORlDA KAOL1U AN0 CUGL/SN CHINA CLAY
E 0
4
8
12
a
).SO t
