The Drying of Solids. V I Diffusion Equations for the Period of Constant Drying Rate E. R. GILLILANDAKD T. K. SHERWOOD, Massachusetts Institute of Technology, Cambridge, Mass.
I
S PART I of this series
(?)I Thus for any initial m o i s t u r e Equations are presented f o r fhe moisture dist h e r e w a s presented t h e distribution f(x'), the moisture tribution in a solid during tileperiod of drytheoretical e q u a t i o n for concentration, may lie c a l c u ing at a constant rate. T h e y are shown to be usethe drying of a solid slab under lated for any point in slab fui in the length Of the constant-rate conditions where the resistance after any time, 0, of drying a t period and fhe crifical moisture content, thus t o vapor removal from the solid the constant rate, a. For the supplying a n important missing /ink in the case of a n i n i t i a l u n i f o r m surface is negligible compared moisture d i s t r ill u t i o n (i. e., with the resistance to internal procedure of the conlplete drying f(z') = A ) Equation 2 reduce? diffusion of liquid through the curve. to: solid. This equation has been found to be applicable to the drying of various slow-drying solids in cases where internal diffu(X - R)' KO sion controls from the start of the drying process, and no initial v = A - K cy ___ [ 2R constant rate is obtained. Part IV (4) pointed out that a parabolic moisture distribution through the slab is approached --Ka2mW m,(z - R ) as drying a t a constant rate progresses, and presented the equaR 2 cos R (3) tion for drying with internal liquid diffusion controlling for 1 the case of a n initial parabolic moisture distribution. These conditions correspond to the common case where the drying It is clear from Equation 3 that as 0 becomes large, L' approcess includes an initial constant-rate period of sufficient proaches a parabolic function of 2, thus substantiating the length for the parabolic gradient t o be set up, followed by a previous conclusions. Figure 1 represents Equation 3 falling-rate period in which internal liquid diffusion controls. graphically, in terms of the dimensionless groups K 0 / R 2 Sewman (1) has also published diffusion equations for drying, and (A - v ) K / a R . Where the diffusion constant, K , is based on the assumption of a parabolic moisture distribution known or can be estimated from drying data, it is a relatively a t the end of the constant-rate period. The time required for the parabolic gradient t o be set u p clearly depends on the nature of the material, the thickness of the slab, and the rate of drying in the constant-rate period. The true moisture distribution, after different intervals of drying a t a constant rate, may be obtained by integration, for the proper boundary conditions, of the fundamental diffusion equation:
]
6V -
= : K 62V -
6e
(1)
6x2
For the constant-rate period, the conditions are: 6v CY 6V - - Ea:a t x = 2 R - =Katx=i) 62
62
v = f(z') at 0 = 0 constant rate of drying, as weight per unit time 0,
where LY = per unit face area
*
As a solution of Equation 1 for these conditions: u
=
lR f(x')dx'
+ R1
f3.
K+n%
-4
E
R2
REPRESEVT4TIOU FIGCRE 1. GRAPHICAL
2R
cos m 2R x ' dx' R 1 ~ ~ ~ f(d)
OF
EQUATIOV 3
simple matter t o determine the moisture distribution in a slab after any period of drying a t a constant rate. The plot may be used, for example, t o determine the maximum drying rate corresponding to the maximum allowable moisture gradient in the drying of a ceramic piece sensitive to warping or cracking.
1
E s T I a f . u r I o s O F THE CRITICAL LIOISTURE C O S T E K T
K+mW
-- E' 1
co6 mn(s - R )
R
I
(2)
21, 976 (1929); part 111, 22, 132 (1930); part 11, INDENa. cHEV, Part IV. 24, 307 (1932); Part V, 25, 311 (1933)
Previous articles of this series (3, 4) have postulated a period in the drying process in which the drying is controlled by the rate of internal liquid diffusion, and during which the resistance to vapor removal from the surface is negligible. This mechanism of drying is encountered in the drying of practically all classes of materials a t low moisture contents and is of major importance when drying materials have a low internal diffusion constant or a large thick-
1134
October, 1933
IXDUSTRIAL AND ENGINEERING CHEhfISTRY
ness, or under conditions of high air velocity and low air humidity. Such periods frequently follow immediately after the constant-rate period, and the average moisture content a t the junction of the two periods is termed the critical moisture content. If the resistance to vapor removal from the surface is negligible, the moisture concentration on the wrface may presumably approach closely tlica equilibrium or hygroscopic moisture corresponding t o the humidity of the surrounding air. Drying may be expected to proceed a t a constant rate until the surface concentration has dropped to eisentially this equilibrium moisture. The critical moisture content should, thc>refore, be predictable from the diffusion constant and the known rate of drying in the constant-rate period, for case; where internal diffusion hecomes controlling a t the critical point.
1135
the surface in a much shorter time, and the constant-rate period would hare been negligible.
TABLEI. D a ~ ao s TESTS RTN i?,
RICIC6 Tlmp
c L 4 Y >1IX
K a t era
.ll7nutca 0 10 24
41 .5i
no
103
I15 125 1:35 145
I55
170 180 190 200
%E --
l i
a
% 26.3 24.5 22.8 20.6 18.8 1.5 .5 13.8 13.3 12.8 12.1 11.4
R u n 73, BRICKCL.AYMIS Time Water" .Minutes % n 27.3
11.3 10.6 10.2 10.1 9.7 9.5 9-. 2
(3)
4i
20.4
51 67 87 102 119 138 162 183 205 216
18.8 16.5
14.5
13.4 12.4 11.4 10.6 9.9 9.5
RCN 11, HEMLOCK WOOD Time Watera €lours % n 127 1 112 > 96.8 3 83.5 73.6 4 64.9 5 57.2 6 7 51.7 8 46.1
m
...
41.8 38.5
9 10
30.8
12 14 16 18 20 21
9.0 (3)
...
26.4
20.9 16.5 14.3 12.1 6.6
r
...
. .
..
Dry basis.
Material Face dimensions, cm. Slab thickness, cm. Net bane-dry weight, grams Exposed area
c,
Air temg:, C . , , Wet-bul temp h i r velocity, meters/sec. R a t e of drying in constant rate period, grams/hour/sq. cm. Critical moisture content, % water, dry basis
72 Brick clay mix 7.0 X 7.0
2.54
196.2
Two large faces 24.5 16.4 10.6
0.157 16.5
2
$
The critical moisture contents for these tests have been calculated, making the assumption that the constant-rate period endq when the surface moisture concentration has dropped to the equilibrium value. The time of drying a t a constant rate is obtained from Figure 1, using the curve for r = 0. An example of this calculation is giren for the test on wood, for which the dry density is 0 54: 0 0312 X 0 75
C Y R
11 Hemlock mix wood 7 . 0 X 7 . 0 15.15 X 14.8 0.75 2.54 91 193.9 Two large I'IY o large faces faces 25.0 25.0 17.0 ..... 15.2 3.7
0.174
'%$'1
FIGURE3. C A L C U L A T E D FIGURE 2. C A L C C L A T E D CUR\-ESOF MOISTUREDISCURIES O F h1OISTURE DISTRIBUTION I N THE C O \ TRIBUTIOS IN THE CORSTANTSTAKT-RATE DRYINGOF 4 RATED R Y I ~OG F 4 SLAB OF SL4B O F HEMLOCK\ f O O U BRICKCr AI h11x (RUT 72) (RUN 11)
=
73 Brirk clay
16.5
j R
( A - v)K - (1 27 - 0 066) X 0 54 X 1 7 X 10-8 X 3600 X 2
TABLE11. COSDITIOKS OF TEST Run
I
SURFACE 0
0 34
From Figure 1, K8/R2 = 0 090, m-hence
o=
0.09 X 0 . 7 5 * 4 X 1 . 7 X 10-6 x 3600 = 2 ' 1 hours
The corresponding critical moisture content is 127 - ~
0.0312
:
~
~ X 100 ~ = 2 95% ~ moisture, 0 2 dry ~ basis ~
93
In order to test this theory, the calculated and experimentally determined critical points have been compared for two tests on drying a brick clay mix, and one test on the drying of a hemlock wood slab. In all three tests constant-rate periods were observed, and sharp critical points obtained. The data and conditions of the three tests are given in Tables I and 11. The data on clay are the same as those analyzed in a previous article (I),for which an approximate value of K of 0 7 2 x sq. em. per second was obtained. A similar analysis of the data on wood for the falling rate period gives a n approximate value of K of 1 7 x sq. em. per second for diffusion in hemlock in a direction tangential to the annual rings. Using these values of K , Figure 1 has been used to calculate the moisture distribution through slabs a t short time intervals for each test. The resulting curves for runs 72 and 11are shown in Figures 2 and 3. It is apparent from these curves that by the time the surface moisture concentration has dropped to the equilibrium value, the distribution is very nearly parabolic, even for the slow-drying wood. However, with a higher air velocity or temperature, or 'with a smaller value of the diffusion constant, the equilibrium moisture concentration would have been reached a t
These values may he compared with the experimental values of 2 . 2 hours and 93 per cent, respectively. The calculated and experimental values of these quantities for the three tests are as follows: CONSTANT-RATE
Rrs
hf4TERIkL
72 73 11
Clay Clay Wood
PERIOD
Obsvd Hours 1.22 1.12 2 2
Calcd. Hours 1 08
0.93
2 1
CRITICAL ~IOISTURE CONTENT Obsvd. Calcd
%
%
16.5 16.5 93
17 8
19 1 95
The agreement between the calculated and observed values is seen t o be quite good, in spite of the fact that the values of K used were obtained from data on the falling-rate period, over a range of moisture concentrations considerably lower than in the constant-rate periods, to which the calculations apply. The results substantiate the concept of a fallingrate period with internal liquid diffusion controlling, during which the moisture concentration a t the solid surface is very nearly the equilibrium value correqponding t o the surrounding air. SOMENCLATURE A
moisture concentration for the case of an initial uniform moisture distrihution, grams/cc.
= 'initial . '
INDUSTRIAL AND EKGINEERING CHEMISTRY
1136
diffusion constant, sq. cm./sec. m = term number of series in Equations 2 and 3 n = term number of series in Equation 2 R = half slab thickness, cm. v = moisture concentration, grams/cc. x = distance from one face of the slab, in a direction normal to the surface, cm. f(x') initial moisture distribution, as a function of distance x' from one face a = rate of drying, grams/sec./sq. cm. K
=
0 ?r
1
= = =
Vol. 25, No. 10
time, sec. 3.142 KB/Rz LITERATURE CITED
(1) Newman, Trans. Am. Inst. Chem. Eng., 27, 203 (1931). (2) Sherwood, T.K., IND.ENG.CHEW,21, 12 (1929). (3) Ibid., 21, 976 (1929). (4) Ibid., 24, 307 (1932).
RECEIVED May 4, 1933.
Vapor-Liquid Equilibria of Hydrocarbon Mixtures E. C. BROMILEY AND D. QUIGGLE,Pennsylvania State College, State College, Pa.
F
EW vapor-liquid equilibrium data are available for hydrocarbon mixtures, so this work was undertaken to
find binary mixtures more suitable than benzene-toluene or benzene-carbon tetrachloride for testing the efficiency of fractionating columns. Four binary and two ternary mixtures were investigated, as follows: Binary: N-HeDtane-toluene Tolugne-N-octane 2,2,4Trimethylpentane (iso&tane)-N-octane Normal heptane-methylcyclohexane Ternary: N-Heptane-methylcyclohexane-toluene N-Heptane-methylcyclohexane-acetone
I n the ternary mixtures the concentrations of toluene and acetone were kept reasonably constant. PROPERTIES OF hI.4TERIALS USED The S-heptane was that approved by the Bureau of Standards for knock rating purposes. It was obtained from the California Chemical Company and had the following properties : Boiling point a t 760 mm., Freezing point, C . Densitv. - . d?O Refractive index, n y
C.
98.4 -90.8 0,6839 1,3878
The 2,2,4-trimethylpentane was also the st'andard fuel approved by the Bureau of Standards for knock ratings. It was obtained from the Rohm and Haas Company of Philadelphia. Its properties were: Boiling point a t 760 mm., Freezing point, O C. Density, d i 0 Refractive index, n y
C.
99.2 -107.5 0.6919 1.3916
The N-octane was prepared in the course of some research in this laboratory, and had the following physical propert'ies : Boiling point a t 760 mm., Density, dzo Refractive index, n 2
C.
125.4 0.7026 1.3970
The toluene was a fractionated sample of Baker's analyzed material with the following properties : Boiling point a t 760 mm., Density, d i 0 Refractive index, ny
C.
C. P.
110.4 0.8659 1.4966
The methylcyclohexane was Eastman's technical grade which was fractionated and then washed with concentrated sulfuric acid, neutralized with sodium carbonate, washed with water, and dried. Its properties were:
Boiling point a t 760 mm., Density, d i 0 Refractive index, ng
C.
100.8 0.7693 1.4232
APPARATUS The vapor-liquid equilibrium was obtained in a simple and compact equilibrium still reported by Othmer (3, 4). I n addition to the electric heating of the still a winding of asbestoscovered chrome1 wire was placed around the still within about 0.5 inch (1.27 cm.) of the surface of the liquid. By adjusting the heat in this winding, condensation of vapors was avoided until they reached the condenser. Condensation of vapors before the condenser would have given erroneous results while a slight superheating of the vapors would not. Since the entire apparatus was of glass, condensation of vapors in the still could be readily seen and the necessary heat applied to avoid it. This method of obtaining vapor-liquid equilibria is very satisfactory and is better than the methods used by Evans ( I ) , Zawidski ( 7 ) , or Rosanoff, Lamb, and Breithut (Or). It is simple in operation, is subjected t o few errors if condensation in the still is avoided, and gives reproducible results. The still was designed to operate with about 200 cc. of material, but variations in quantities from 75 t o 250 cc. gave equally satisfactory results. AXALYTICAL PROCEDURE To make the analyses of the equilibrium mixtures as simple as possible, density and refractive index were used whenever there was sufficient difference in these properties. Mixtures of known composition were prepared and measurements made of the refractive index with an Abb6 refractometer, of density with a Nicol pycnometer, and of boiling point with a specially constructed Cottrell apparatus capable of being operated a t any desired pressure. From these data suitable curves were drawn and used as a basis for analysis. These data are given in Tables I-IV under Known Mixtures. Figure 1 shows a convenient method of plotting density or refractive index against composition. I n the cases of the X-heptane-toluene and N-octanetoluene mixtures both density and refractive index were used for analysis, for with an accuracy of 2 in the fourth decimal place for either density or refractive index, these analyses are correct within 0.1 to 0.2 per cent. For the 2,2,4-trimethylpentane (isooctane)-N-octane mixtures, boiling points a t 760 mm. were used for analysis. Boiling points could be read to 0.1' C. and this permitted greater accuracy than when using density or refractive index for analysis. The boiling point was found to vary linearly
