Ind. Eng. Chem. Res. 1990, 29, 614-617
614
Prediction of Drying Rate Curves on Sintered Spheres of Glass Beads in Superheated Steam under Vacuum Hiromichi Shibata* and Jiro Mada Department o f Chemical Engineering, Fukuoka University, 8- 19-1, Nanakuma, Jonan-ku, Fukuoka, 814-01 Japan
Kazumori Funatsu Department of Chemical Engineering, Kyushu University, 6-10- 1, Hakozaki, Higashi-ku, Fukuoka, 812 Japan
The superheated steam drying of sintered spheres of coarse glass beads was investigated under vacuum (48-305 mmHg). The drying rate curves for the constant-rate period and the critical moisture contents were estimated on the basis of an evaporation zone model for superheated steam drying at atmospheric pressure, and the drying rate curves for the falling rate period were estimated from an evaporation front model. The predicted results were in good agreement with the experimental data which revealed t h a t dry spots were present in the thin surface layer of the sintered sphere where evaporation took place during the constant-rate period, resulting in lower critical moisture contents in superheated steam under vacuum than those in air under vacuum, and that for the falling rate period the drying rates were higher than those in air under vacuum. These facts indicated t h a t the evaporation zone model was applicable to superheated steam drying over a wide range of pressures (from 48 mmHg t o atmospheric pressure) and that superheated steam drying under vacuum is superior t o air drying under vacuum. It is very important to elucidate the mechanism of superheated steam drying under vacuum because of its many advantages as described previously (Shibata et al., 1988a). In this study, we already developed an experimental apparatus for superheated steam under vacuum and reported on experimental drying rate curves and critical moisture contents of sintered spheres of coarse glass beads for superheated steam drying under vacuum: it was found from experiments with sintered spheres of coarse glass beads with Os = 18' and 27' that the critical moisture content in superheated steam drying under vacuum was independent of the pressure and was lower than in air drying under vacuum and that the drying rate during the falling rate period in superheated steam drying under vacuum was higher than that in air drying under vacuum. In this work, the drying rate curves and the critical moisture contents in superheated steam drying under vacuum (48-305 mmHg) are estimated on the basis of an evaporation zone model which was proposed for superheated steam drying at atmospheric pressure (Shibata et al., 1988b). These predicted results are compared with the data of superheated steam drying under vacuum and air drying under vacuum, which were obtained by Shibata et al. (1988a). Model In superheated steam at pressures of 48-305 mmHg, the drying rate curves and the critical moisture contents of a sintered sphere of coarse glass beads with the sintered angle 6, = 18O, the porosity, t = 0.344, the radius of sintered sphere R = 1.24 cm, and the mean radius of glass beads r = 0.18 mm are calculated from an evaporation zone model and the predicted effective thermal conductivity under vacuum. The Knudsen number in the sample is below 0.01 in this pressure range (see Appendix A). Therefore, the flow of the drying medium is laminar within the sintered sphere. Constant-Rate Period and Critical Moisture Contents. In superheated steam drying in this pressure range, there is no resistance to flow of the drying medium in the sample, and there are sinks of heat at the positions with water at the boiling point that depends on the pressures of the drying medium. It is assumed that there are no 0888-5885/90/ 2629-0614$02.50/0
differences between the water transfer in the funicular state under vacuum (48-305 mmHg) and at atmospheric pressure and that the radius of the sample with coarse glass beads is so small that the frictional resistance to flow of water and the effect of the gravitational force are negligible. According to the evaporation zone model (Shibata et al., 1988b) as shown in Figure 1, during the constant-rate period, not only the water in the funicular state on the surface of the sample but also the water in the pendular state at the region between adjacent regions in the funicular state is evaporated by heat transferred through the boundary layer by radiation and convection in superheated steam under vacuum; then dry spots begin to evolve on the surface of the sample at the moisture content C,. As drying proceeds, dry spots extend to the thin surface layer of the sample (evaporation zone), and, finally, the water transfer ceases at the critical moisture content when the dry region and the region with the water in the pendular state remain in the evaporation zone. The evaporation zone is modeled as shown in Figure 2. The moisture content (C,) and the parameters in the evaporation zone are given by the expressions (Shibata et al., 1988b)
6 = 6,
-
6,-
c - c, c o - c,
for C, 5 C I C,
(2)
a = mi= for C, I CI C, (3) a a mrC/C,, for C, < C < C, (4) b is obtained from a mass balance over the evaporation zone [(cP/3)b(b+ 3a) + cIa21[~3 - ( R - 6131 + [Cp(a + b)2 + (C, - Cp)a2](R- 613 - ( a + b)2R3C = 0 (5) in which 6 is the thickness of the evaporation zone, 6, is that at the critical moisture content C, in superheated steam drying, C,, is the critical moisture content in air drying, C, is the moisture content in the pendular state, CI is the moisture content in the initial state, and the :C: 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 615 state, the drying rate can be calculated from the expression (Shibata et al., 1988b)
P
F'
P'
F
P
F
Effective Thermal Conductivity. The effective thermal conductivity of sintered glass beads is obtained from the following expressions (Shibata et al., 1989):
P
Figure 1. Schematic drawing of a small part of the evaporation zone in a sample sphere: (F) region in the funicular state; (P) region in the pendular state.
?)(
(F - L)2
k, = (1 - ')[(
~ (-1L/P ~ (-1L/P
+ 0.5U(F - L )
(F - fL)2
+ 0.5U/F) + 0.5U/F) -
(K
- 1) COS 0,
(K
- 1) COS 8'
Nk,[(i;+ p ) sin Os - p I 2 Figure 2. Model of a small part of the evaporation zone in a sample sphere: (I) funicular state region; (11) region between funicular state region and dry-spot region; (111) dry-spot region.
values A = 6.25 and m = 2 obtained in superheated steam drying a t atmospheric pressure (Shibata et al., 1988b) are used, because these values are governed by the coordination of the water in the funicular state, which is independent of the pressure of superheated steam in this pressure range from the assumptions as described above. There are few dry spots at moisture contents of more than C,, and the drying rate is determined only by radiation and convection from the surrounding wall and the drying medium, respectively. However, a t moisture contents of less than C,, dry spots govern the drying rate, which is given by the expressions
( ,,,+ ) (
T,) + 5 . 6 7 4 T,
273
-
Ts;;3)]7E)
6(F - fL)2
in which d = 2 ( -~ L ) is assumed
u= Bo = sin-'
Y + l
(6/N)1/2 e3 - r/(3(2)ll2)
at
2
)
in which 26,/3 is the mean thickness of the evaporation zone. Falling Rate Period. The drying rate can be estimated from a receding evaporation front model because water in the sample is in the pendular state. In the quasi steady
e3
(13)
e3 - r / 6
r
Bo=-
e3 - r / 6 -
+
in which CF is the moisture content in the funicular state region I, CFois that a t C,, and w d , is the evaporation rate from the border between regions I1 and 111. Weonstcan be calculated numerically from the temperature distribution in the evaporation zone in the same way as in superheated steam drying a t atmospheric pressure based on the assumption of quasi steady state (Shibata et al., 1988b). The critical moisture content is determined by the thickness of the evaporation zone and the moisture content in the pendular state because the funicular state is not present within the sample and is given by the expression (Shibata e t al., 198813) R - 26,/3 'p( R
);
(y )(")(
at e < -
(14)
e3
3e3 - 2 3e3
e = ( F - L ) / F ef = ( p - f L ) / ~ (17) The dimensions in the sintered part of the glass beads are calculated from the expressions (Shibata et al., 1988b) Os sin 20, (F + p)2p cos 0, sin2 0, + 2(F p)p* sin os( - -4
+
E) 4
+ ( p 3 + P3)( (F
cos 0, -
5
?):2i;3 3
= 0 (18)
+ p ) COS 8, = F - L
(19) The effective thermal conductivity in sintered glass beads can be calculated as follows: The radius ( p ) in the neck of the sintered glass beads and the width ( L )of the sintered part is calculated from eqs 18 and 19 in which 8, is determined by scanning electron micrographs. 0, is calculated from eq 13 or 14 in which the coordination number ( N ) is determined by the equation obtained by Ridgway and Tarbuck (1967) (see Appendix B), and CP is calculated from the porosity (e), eq 17 and eq 15 or 16. Therefore, the effective thermal conductivity is calculated from eqs 10-12. Results and Discussion The thickness of the evaporation zone, 6, = 3 X m, was determined from a location of the water-transfer front
616 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 MOISTURE CONTENT, $
0
0.1
0.2
I
I
Ng4.0t
-E -
s2.0
0
0.1
I
V
I
0.2
I
I
U
E''
l o ,
CO W E NT, #
%? X h
,
00/1
MOISl%RE
/
w
In,'
I
/
I
5 2.0F co
:
garr OO
0.02
0.04
MOISTURE CONTENT, C-DRY BASIS Figure 3. Comparison of (- - -) predicted drying rate curve and (0) experimental data (Shibata et al., 1988a) for sample with 8, = 18O, t = 0.344,and R = 0.0124 m at 135 O C and at 305.0 mmHg in superheated steam. MOISTURE CONTENT,
0
01
1
a
0.06
I
I
I
1
I
1
0.02 0.04 0.06 MOISTURE CONTENT,C-DRY BASIS
-
Figure 5. Comparison of (- -) predicted drying rate curve and (0) experimental data (Shibata et al., 1988a) for sample with 0, = 1 8 O , c = 0.344,and R = 0.0124 m at 120 "C and at 97.0 mmHg in superheated steam.
9
MOISTURE CONTENT,
0.1
0
0.2
9 0.2
I
i'F/l
I
1:.,20j #\=Lo
En
U
OO
0.02
0.04
I
I
I
0.0 2
0.06
I
I
0.04
I
0.06
MOISTURE CONTENT, C-DRY BASIS
MOISTURE CONTENT, C-DRY BASIS
Figure 4. Comparison of (- - -) predicted drying rate curve and (0) experimental data (Shibata et al., 1988a) for sample with 8, = 1 8 O , c = 0.344 and R = 0.0124 m at 120 O C and a t 201.0 mmHg in superheated steam.
Figure 6. Comparison of (- -) predicted drying rate curve and (0) experimental data (Shibata et al., 1988a) for sample with 0, = 18O, t = 0.344,and R = 0.0124 m a t 100 OC and a t 48.0 mmHg in superheated steam.
a t the critical moisture content in superheated steam, which was estimated from a location of the water-transfer front at the end of the first falling rate period in air drying at atmospheric pressure in the same way as in a previous paper (Shibata et al., 1988b). The radius a was calculated from eqs 3 and 4, and then b was calculated from eq 5. Co was calculated from eq 1. From the parameters in the evaporation zone described above and eqs 6 and 7 (evaporation zone model), the drying rate curves for the constant-rate period in superheated steam drying a t pressures of 48-305 mmHg were calculated a t moisture contents from C,to C,. In this pressure range, they slightly decreased with a decrease in the moisture content and were in good agreement with the experimental data of drying rate curves (Shibata et al., 1988a) as shown in Figures 3-6. The drying rate curves for the falling rate period were calculated from eq 9 (receding evaporation front model) a t moisture contents of less than C,, and the predicted as well as the experimental curves were convex. The critical moisture contents were calculated from eq 8 in which the value C, = 0.0154 was estimated from C, = 0.0204 (&, = 0.078) in packed beds and the ratio of the volume of a pendular ring a t the sintered part of glass beads over that at the packed bed (Shibata et al., 198813). The predicted critical moisture contents were somewhat larger than the experimental data (Shibata et al., 1988a): at the critical moisture contents in superheated steam, the location of the water-transfer front predicted a minimum value of the thickness of the evaporation zone (Shibata et
-
I
A//--
-L-a
97 5 ii 5
" 0
90
ma
I
I
0.1 MOISTURE CONTENT,
0.2
15 4 17 5 12 5 i a
-
0.3
9
Figure 7. Comparison of normalized drying rate curves in superheated steam at pressures of 48-305 mmHg and those in air at pressures of 11.5-760 mmHg: (-) predicted drying rate curve a t 48 mmHg in superheated steam; (---) predicted drying rate curve a t 305 mmHg in superheated steam; symbols denote the experimental data that were taken from Shibata et al. (1988a).
al., 1988b), resulting in overestimation of the critical moisture content in eq 8, but the thickness of the evaporation zone could not affect the critical moisture content in eq 8 too much because the thickness of the evaporation zone should be thin t o maintain the drying rate for the constant-rate period. Figure 7 shows normalized drying rate curves in superheated steam under vacuum and in air under vacuum
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 617 in which the experimental data were taken from Shibata et al. (1988a). There was almost no difference among the predicted normalized drying rate curves in superheated steam at pressures of 48-305 mmHg as well as among the experimental ones. T h e normalized drying rates for the falling rate period in superheated steam were higher than those for air, and the critical moisture contents in superheated steam were lower than those in air. Conclusions The critical moisture contents and the drying rate curves during the constant-rate period for sintered spheres of coarse glass beads with a small diameter were predicted in superheated steam drying at pressures of 48-305 mmHg by an evaporation zone model. The drying rate curves for the falling rate period were estimated from a receding evaporation front model. The predicted results were in good agreement with the experimental data which revealed that during the constant-rate period dry spots were present in the thin surface layer of the sintered sphere where evaporation took place, resulting in lower critical moisture contents in superheated steam drying under vacuum than those in air drying under vacuum, and that during the falling rate period the drying rates in superheated steam under vacuum were higher than those in air under vacuum. From the above results, it was concluded that the evaporation zone model was applicable to superheated steam drying over a wide range of pressures (from 48 mmHg to atmospheric pressure) and that superheated steam drying under vacuum is superior to air drying under vacuum. Nomenclature A = ratio of Szover S1 a = radius in funicular state region, m b = radius in dry spot, m C = moisture content (dry basis) d = diameter of void space, m e, ef = parameter defined in eq 17 FAE= overall interchange factor f =: shrinkage factor H = heat-transfer coefficient, W m-2 K-' k , = effective thermal conductivity, W m-l K-' k , = thermal conductivity of gas, W m-l K-' k , = thermal conductivity of solid, W m-l K-' L = width of sintered part of glass beads, m L, = latent heat of water, J kg-l m = number of particles n = effective number of contact points on a hemisphere for conductive heat transfer N = coordination number P = pressure in drying chamber, mmHg P, = Prandtl number R = radius of a sample r = coordinate in evaporation zone ? = radius of glass bead, m SI= area in funicular state region, m2 Sz = area in one segment, m2 Tb = boiling point, "C T , = temperature of drying medium, "C T , = temperature of sample, "C T,= temperature on surface of sample, "C T, = temperature of wall of drying chamber, "C U = parameter defined in eq 12 Wcanst= drying rate during constant-rate period, kg m-2 s-l W,, = evaporation rate from the border between regions I1 and 111, kg m-2 s-l W , = drying rate during falling rate period, kg m-2 s-l z = coordinate in evaporation zone, m Creek L e t t e r s
a = accommodation coefficient y = ratio of heat capacity at constant pressure over heat
capacity at constant volume 6 = thickness of evaporation zone in superheated steam drying,
m t
= porosity
{ = depth of evaporation front, m d, = sintered angle, deg or rad do = angle corresponding to boundary of heat flow area for
one contact point, deg or rad 0' = sintered angle in neck, deg or rad K = k,/k, X = mean free path of molecules, m p = radius of curvature in neck, m 4 = moisture content (volume of water in sample/volume of
void space in sample) fractional area of heat flow
\k =
Subscripts
c = critical moisture content in superheated steam drying ca = critical moisture content in air drying F = funicular state region I = initial state o = evolution of dry spot p = pendular state Appendix A The Knudsen number ( K n ) in packed beds was given by SCEJ (1988) and Asaeda et al. (1974):
x
K n = ___ = 0.0084 < 0.01 2tr 3(1 - e) in which = k ( T , + 273)/(2)1/2xu2P,P = 48 mmHg, T,, = 37.7 "C, r = 0.018 cm, the porosity in the packed bed before sintering tr = 0.414, the diameter of H 2 0 molecule u = 4.6 X lo-* cm (Kennard, 19381, and k is Boltzmann constant. This value of the Knudsen number in the packed bed before sintering was used for the weakly sintered sample because the shrinkage in diameter of sintered glass beads was only 2 YO. Appendix B The coordination number ( N )was experimentally given by the expression (Ridgway and Tarbuck, 1967) t I = 1.072 - 0.1193N + 0.00431N2 in which tering.
tI
is the porosity in the packed bed before sin-
Literature Cited Asaeda, M.; Yoneda, S.; Toei, R. Flow of Rarefied Gases Through Packed Beds of Particles. J . Chem. Eng. Jpn. 1974, 7, 93-98. Kennard. E. H. Kinetic Theory of Gases. 1st ed.: McGraw-Hill New York, '1938. Ridgway, K.; Tarbuck, K. J. The Random Packing of Spheres. Br. Chem. Enp. 1967, 12, 384-388. SCEJ. Kagaku Kogaku Binran, 5th ed.; The Society of Chemical Engineers Japan: Tokyo, 1988; p 160. Shibata, H.; Mada, J.; Shinohara, H. Steam Drying of Sintered Glass Bead Spheres under Vacuum. Ind. Eng. Chem. Res. 1988a, 27, 2385-2387. Shibata, H.; Mada, J.; Shinohara, H. Drying Mechanism of Sintered Spheres of Glass Beads in Superheated Steam. Ind. Eng. Chem. Res. 1988b,27, 2353-2362. Shibata, H.; Mada, J.; Funatsu, K. Prediction of Effective Thermal Conductivities of Sintered Spherical Powders under Vacuum. Fukuoka Uniu. Reu. Technol. Sci. 1989, 44, in press. 1
.
Receiued for reuiew June 1, 1989 Revised manuscript receiued December 4, 1989 Accepted December 15, 1989
