Drying Mechanism of Sintered Spheres of Glass Beads in

Fukuoka, 814-01 Japan. The drying of sintered spheres of coarse glass beads in superheated steam was studied to elucidate the mechanism leading to the...
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Znd. E n g . C h e m . R e s . 1988,27, 2353-2362

2353

Drying Mechanism of Sintered Spheres of Glass Beads in Superheated Steam Hiromichi Shibata,* Jiro Mada, and Hisashi Shinohara Department of Chemical Engineering, Fukuoka University, 8-19-1, N a n a k u m a , Jonan-ku, Fukuoka, 814-01 J a p a n

The drying of sintered spheres of coarse glass beads in superheated steam was studied to elucidate the mechanism leading to the critical moisture content. T h e experiments were performed a t superheated steam temperatures of 110-170 "C under atmospheric pressure, and the porosities of samples were 0.254-0.367 depending on the sintering conditions. A simple model for superheated steam drying in which evaporation takes place not only on the surface of the sample but also in the thin layer below the surface for the constant-rate period is proposed. The critical moisture content in superheated steam drying is lower than that in air drying for similar drying rates during a constant-rate period and can be estimated from the moisture content a t the end of the first falling rate period in air drying. The predicted critical moisture contents and drying rate curves were in good agreement with the experimental results. Recently, the importance of superheated steam drying has been emphasized because of its numerous advantages (Beeby and Potter, 1984). However, only a few papers have been published on the mechanism of superheated stream drying. Wenzel and White (1951) studied the superheated steam drying of granular solids and reported on drying rate curves, heat-transfer coefficients, and critical moisture contents. Chu et al. (1959) reported on superheated steam drying, air drying, and air-stream mixtures drying of granular solids. Trommelen and Crosby (1970) studied the evaporation of pure liquid drops and the drying of drops containing suspended and dissolved solids in superheated steam. Moyers (1978) reported on the drying of solution droplets in superheated vapor and proposed a model in which a rigid porous crust contributed heat-transfer resistance in series with the gas film resistance. Other materials, such as cellulose pellets (Luikov, 1966) and unglazed pottery (Toei et al., 1966b), were studied in connection with superheated steam drying, air drying, or air-steam mixtures drying. The above investigations experimentally revealed that the critical moisture content in superheated steam drying is lower than that in air drying when the respective drying rates during the constant-rate periods are equal. In this paper, the drying mechanism of sintered spheres of coarse glass beads in superheated steam under atmospheric pressure is described to elucidate the process leading to the critical moisture content. It is generally accepted that, during the constant-rate period, evaporation takes place only on the surface of a sample. If evaporation takes place on a receding front in an inner part of the sample, the falling rate drying period is assumed. However, in this work, the authors conclude that, in superheated steam drying, for the constant-rate period, evaporation takes place not only on the surface of the sample but also in the thin layer below the surface. A simple model of superheated steam drying is proposed and critical moisture contents and drying rate curves are calculated. The predicted results are compared with experimental results.

Experimental Section Apparatus. Figure 1shows a schematic diagram of the experimental equipment. The wind tunnel was designed in the same way as described by Toei et al. (1966a). A convergent nozzle in the wind tunnel was used to produce 0888-588518812621-2353$01.50/0

Table I. Properties of the Samples diameter of material of sample sample ( 2 ~ 1m , x io3 (2R), m 0 0.25-0.42" 0.0235 1 0.4b 0.0252 18 0.25442" 0.0236 12 0.25-0.42" 0.0251 19 0.25-0.42" 0.0245 ''

Spherical glass beads.

e

e,,

0.292 0.316 0.254 0.345 0.367

deg 27.0 27.0 25.0 15.0 7.5

Nonspherical glass beads.

a flat velocity profile (Ranz and Marshall, 1952). The sample of radius R = 1 X m was bung by a cotton thread from a balance in the test section (0.1 m in diameter) through which the drying medium flowed upward after the rate of flow was measured by a quadrant-edge orifice meter (Jorissen and Ithaca, 1956). The temperature profiles were uniform (less than f1.5%) in this test section. Materials. Table I shows the physical properties of the samples tested. The sample spheres were made of spherical glass beads, 2p = 3.6 X lo4 m in mean diameter, but sample 1was made of nonspherical glass beads, 2~ = 4X m in mean diameter. The sample spheres were sintered at 690-710 "C. The dimensions indicated in Figure 2 were obtained from the sintering model equations (Kubo et al., 1958)

(r + p ) 2 p cos 8, sin2 %,

(P

(:

+ 2(r + p)p2 sin eP

+ p ) cos %,

=

r -L

- - -sin:8p

(2) in which %, was determined by scanning electron micrographs of the sintered glass beads. The porosity of the sample (0.2544.367) was determined on an Air Comparison Pycnometer (Beckman-Toshiba, Ltd.). The coordination number of the sample was N = 7.62 according to the equation derived by Ridgeway and Tarbuck (1967) when the porosity before sintering was ci = 0.414. Procedures. The samples were prepared by adding distilled water to sintered spheres after they were degassed in a vacuum container for 20 min. (1)The sample in the test section was weighed at regular intervals on a balance until it was completely dried. The experiments for drying rate curves in superheated steam

1988 American Chemical Society

2354

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988

z

Figure 1. Flow diagram of the experimental apparatus: A, boiler; B, quadrant-edge orifice; C, blower; D, heater; E, screens; F, gastemperature controller; G , sample; H, bottle covered with wet cotton (the bottle contains unvulcanized silicone rubber); I, balance; J, fan; K, wall-temperature controller; L, heater and insulation; M, cotton thread; N, wire (stainless steel); ( f ) flow of drying medium.

Figure 2. Details in sintered part of glass beads.

drying were performed at steam temperatures from 110 to 170 "C and flow rates of 0.3 kg/(m2s). The experiments for air drying were performed a t air temperatures from room temperature to 130 "C and flow rates of 1.4 kg/(m2 S).

m in (2) A copper-constantan thermocouple (1 X diameter) was used to measure the temperature of the sample. The thermocouple junction was attached directly to the surface of the sample to measure surface temperature (ASTM, 1982). The output of the thermocouple was electrically biased and was registered in a full-scale range of 300 pV by a microvoltmeter (PMlGA, TOA Electronics Co., Ltd.). (3) In superheated steam drying, it is impossible to take the sample directly from the test section to measure the moisture distribution. In this work, the following method was developed for the measurement of moisture distributions. Before the sample was removed from the wind tunnel, it was placed in highly viscous unvulcanized silicone rubber

I

("j

Figure 3. Equipment for division of a sample: A, pressure vessel; B, silicone oil; C, blades (high-speed steel) to obtain a disk; D, blades to cut the disk into four parts; E, sample disk; F, sample sphere; G, temperature controller; H, heater; I, fan; J, insulation; K, bottle; L, cup; M, unvulcanized silicone rubber. (KE'ifjShin-Etsu Chem. Lo., Ltd.) in a bottle (hung by a wire in the wind tunnel as shown in Figure 1) which was covered with wet cotton to maintain the sample at a boiling point; this unvulcanized silicone rubber shielded only the surface of the sample and did not penetrate inside the sample because of the high viscosity. This assembly was removed from the wind tunnel and then the bottle with the sample was immediately put into the air bath which was maintained at 100 "C as shown in Figure 3. The sample was cut in the bottle to obtain a disk (thickness t d = 6 mm), the new sides of which were shielded by highly viscous silicone oil (KF96H, Shin-Etsu Chem. Co., Ltd., 1.0 mz/s) supplied from the pressure vessel at 10 atm and 100 "C. (Both the unvulcanized silicone rubber and the silicone oil were preheated at 150 "C in a vacuum for 10 h in order to evacuate the air and the moisture in them.) The disk covered with the unvulcanized silicone rubber and the silicone oil was then immediately moved from the bottle into a cup filled with the silicone oil maintained at 100 "C and was cut into four parts (two outer parts and two inner parts), each size of which was t X td X t , = 6 mm X 6 mm X 6 mm as shown in Figure 3. Each part was then taken out of the cup and the moisture content was determined by the Karl-Fisher method (Moisture Meter MKAII&ADP311A, Kyoto Electronics Co., LTD.). Moisture distributions were measured at moisture contents of 0.007 5 C 5 0.038 at the superheated steam temperature of 117 "C and a flow rate of 0.27 kg/(m2 s). This operation was also used for the measurement of moisture distributions in air drying at room temperature and the flow rate of 1.4 kg/(m2 s). Accuracy of Measurements. The precision of the balance was 0.1 mg for a full-scale range of 200 g. This balance gave reproducible weights of the sample during drying within 1 mg. The microvoltmeter registered the output of the thermocouple within an accuracy of 6 pV, which corresponded to 0.13 "C. In the measurement of moisture distributions, from the time that drying of the sample was stopped to the time that the cutting operation was completed, no condensation occurred within the sample which was maintained at boiling point. The configuration of water in such coarse granular solids was stable as determined by a static method

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2355 0

3.01

MOISTURE CONTENT, 0.1

MOISTURE CONTENT, 9

@ 0.2

I

0

-o- 2 4-.-O*-

-

0'

I

I

.

-- v-o.

-O-*

. *

J

B

3 0

r -, d - -

2

0030

O

T

o"/

2.0

i

oooQcao-o--o---o-o----

fi =1.0 CI =0.133

0,l

w

b-

d 01

f

> LI 0

0

1

0

I

1

I

I

0 0.01 002 0.03 MOISTURE CONTENT. C-DRY BASIS

I

001 0.02 0.03 MOISTURE CONTENT, C -DRY BASIS

Figure 4. Comparison of predicted and observed drying rate curves in sample 0 with OP = 27O and c = 0.292; (- - -) predicted drying rate evaporation-zone model; receding evaporation front model); (0) observed drying rate at 117 O C in superheated steam drying; ( 0 ) observed drying rate at 45.5 "C in air drying; (X) end of the first falling rate period in air drying.

Figure 6. Comparison of (- - -) predicted and (0) observed drying rate curves of sample 18 with 8, = 25' and c = 0.254 at 120 "C in superheated steam drying.

(a

MOISTURE CONTENT, $

0

01

0---

a2 0

I

i

_0_ - - o----o-$5, =10 CI ~0.133

L$

1001

I 1

1 I

I

I I

I

1

I

p,

I

0

1

0.01 0.02 Ob3 MOISTURE CONTENT, C-DRY BASIS

i

Figure 7. Comparison of (- - -) predicted and (0) observed drying rate curves of sample 18 with 8, = 25O and t = 0.254 at 170 O C in superheated steam drying.

6

E-

l

V

%

I

11

MOISTURE CONTENT, $6 0

001 002 003 004 005 MOISTURE CONTENT, C-DRY BASIS

Figure 5. Surface temperature (upper graph) and drying rate (lower graph) as a function of the moisture content in sample 1 with 8, = 27' and c = 0.316 at 120 "C in superheated steam drying: (---) calculated from evaporation-zone model; (- - -) calculated from ob(-) measured by a thermocouple. served drying rate (0);

using Haines' suction pressure apparatus (Ceaglske and Hougen, 1937). Error in the moisture distributions was caused by the loss of moisture due to the evaporation from the new sides of the disk during the first cutting operation. The moisture content corresponding to this loss of moisture was CL = 0.001.

Experimental Results The observed drying rate curves for superheated steam temperatures of 110-170 OC for samples with different sintered angles, sample 0 (0, = 27O), 1 (0, = 27O), 18 (0, = 25O), 12 (0, = 15O), and 19 (0, = 7.5'), are shown in Figures 4-9, respectively. For all samples, the drying rate slightly decreased as the moisture content approached the critical moisture content during the constant-rate period, and for the falling rate period the drying rate decreased steeply. Figure 4 also shows the comparison of the drying rate curve in superheated steam drying to that in air drying for sample 0; the critical moisture content in superheated steam drying was lower than that in air drying, and in the falling rate period the shape of the drying rate curve in superheated steam drying was convex, while that for air drying was concave. Figure 5 also shows a typical measured surface temperature of the sample (solid line). It rose slightly as the moisture content approached the

a2

0.1

I

1

;,.op/

0

(3

f

gt: I

7

$5, =1.0 c1=0209 I

I

I

I

Figure 8. Comparison of (- - -) predicted and (0) observed drying rate curves of sample 12 with 8, = 15" and t = 0.345 a t 110 "C in superheated steam drying. MOISTURE CONTENT, 0

0.1

$ 0.2

I

I

\I

a02 1 0.04 I I 0 0' 6 MOISTURE CONTENT, C-DRY BASIS

O ' O '

Figure 9. Comparison of (- - -) predicted and (0) observed drying rate curves of sample 19 with Op = 7.5O and t = 0.367 a t 114 "C in superheated steam drying.

2356

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988

I

5

10 Wconst

Figure 10. Effect of W,,,,, (drying rate during the constant-rate period) on the observed and predicted critical moisture contents in observed critical moisture sample 1with 0, = 27' and = 0.316: (0) contents in superheated steam drying; ( 0 )observed critical moisture contents in air drying; (- - -) evaporation zone model.

(kg/dS)xlO'

Figure 13. Effect of Wcomt(drying rate during the constant-rate period) on the observed and predicted critical moisture contents in observed critical moisture sample 19 with e, = 7.5' and e = 0.367; (0) contents in superheated steam drying; ( 0 )observed critical moisture contents in air drying; (- - -) evaporation zone model. 04r-

I-

Z

_ _ __

.-----

I- -- C =0166

w Z

t

8, a02 Wci

52

W

5

(kg/JS)xlO

Wconst

Figure 11. Effect of Wconst(drying rate during the contant-rate period) on the observed and predicted critical moisture contents in observed critical moisture sample 18 with 8, = 25' and c = 0.254; (0) contents in superheated steam drying; ( 0 )observed critical moisture contents in air drying; (- - -) evaporation zone model. I-

z

.

e

CRITICAL MOISTURE CONTENT----

0

I

1 I ~

3 L-

-

12

1 6

* e

--L---I --J 0

6

0 CENTER

R SURFACE

12 R (m)xld SURFACE

Figure 14. Moisture distributions in a sample in air drying a t room temperature (about 27 O C ) .

*e e

r

5001 !-

e

0041

-

I

1

I

,

F

5

10 Wconst

(k9/dS)X1o4

Figure 12. Effect of W,, (drying rate during the constant-rate period) on the observed and predicted critical moisture contents in observed critical moisture sample 12 with 0, = 15' and c = 0.345; (0) contents in superheated steam drying; ( 0 )observed critical moisture contents in air drying; (- - -) evaporation zone model.

critical moisture content during the constant-rate period and rose steeply for the falling rate period. The observed critical moisture contents in superheated steam drying and in air drying are shown in Figures 10-13. The observed critical moisture contents in superheated steam drying for samples 1 and 18 (Figures 10 and 11) decreased somewhat with decreasing Wconst(drying rate during the constant-rate period) for temperatures of 110-170 O C , but those for sample 1 2 (Figure 12) were constant. Figures 14 and 15 show the moisture distributions of samples sintered under the same conditions as those of sample 0 in air drying and in superheated steam drying, respectively. In air drying, there was no difference between the moisture content, C1, in the outer part of the sample

-

'12 6 R SURFACE

0 0 CENTER

6

12 R (m)xld SURFACE

Figure 15. Moisture distributions in a sample in superheated steam drying a t 117 "C.

and the moisture content, Cz, in the inner part of the sample for moisture contents in the vicinity of the critical moisture content. In superheated steam drying, at moisture contents of more than C = 0.03, there was no significant difference between C1 and Cz, but at moisture contents of less than C = 0.02 in the constant-rate period, Cl was smaller than C2.

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2357 51

s2

Figure 17. Schematic model of the top view of surface of a sample in the case of cubic array during drying when dry spots begin to evolve.

WN SPOT

Fl

p

FI

P

F,

F,

P

P

Fn

Fn

P

Figure 16. Schematic drawing of evaporation zone: (a) C < CI, (b) C = C,, (c) C, < C < C,, (d) C = C,; (i) 6, when W,,,, (drying rate during the constant-rate period) is low; (ii) 6cmbwhen W,,, is high; (F)region in the funicular state; (P)region in the pendular state.

In all the experiments described above, the samples used were sintered spheres of glass beads, the use of which gave good reproducibility of the experimental results; in sintered spheres of glass beads, the structural properties of the sample for each run were identical. Proposed Mechanism for Superheated Steam Drying In air drying, evaporation takes place only on the surface of the sample a t the wet-bulb temperature during the contant-rate period. Heat is homogeneously transferred through the sample, that is, no parts of the sample can be heated locally because of the great thermal conductivity in the wet material. In superheated steam drying, the resistance to vapor transfer is negligible in the sample as well as in the drying medium, and water in the sample is maintained at a boiling point. There are no essential differences between the water transfer in the funicular state in superheated steam drying and in air drying. In air drying, the water transfer in the funicular state is determined by the capillary pressure curve; the water transfer ceases a t the pendular state (Ceaglske and Hougen, 1937; Pearse et al., 1949). It is supposed that the water transfer in superheated stream drying follows the same scheme. On the basis of the above arguments, the following mechanism in superheated steam drying is proposed. A sample is dried in superheated steam as shown in Figure 16 where a-d are schematic drawings of the cross section in a small part of the sample as drying proceeds during a period defined as the evaporation zone by the authors. It is assumed 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. Figure 16a shows that there are regions, F,, ..., F,, ...,F,,, in the funicular state in the small part of the sample when C < CI (C, is the initial moisture content) and that heat transferred through the boundary layer to the

surface of the sample evaporates the water in the funicular state transferred from other parts of the sample in which the pendular state evolves. In superheated steam drying, there are sinks of heat at the positions with water a t a boiling point. Therefore, not only the water in the funicular state on the surface of the sample but also the water in the pendular state a t the region between the adjacent regions in the funicular state are evaporated by heat transferred through the boundary layer to the surface of the sample. Dry spots begin to evolve at the moisture content C, as shown in Figure 16b. The dry spots extend progressively as drying proceeds; the water in the pendular state in the first layer of sintered glass beads in the sample is completely evaporated and then that in the second layer follows as shown in Figure 16c. Drying proceeds further at moisture contents below the critical moisture content, C,, in air drying which is larger than the moisture content, C,, in the pendular state (Ceaglske and Hougen, 1937). Figure 16d shows that the dry spots develop with a decrease in the moisture content. The depth of the dry spots to the thickness, 6, of the evaporation zone and approaches a maximum, 6,, at the moisture content, C,, where the water transfer ceases; namely the funicular state disappears and only the pendular state remains throughout the small sample. At moisture contents less than C,, an evaporation front rapidly retreats inward in the sample because of the lack of the funicular state. Model Constant-Rate Period. In Figure 16a, there are no dry spots because there is only one particle between the adjacent regions (Fl, ..., Fj, ..., F,) in the funicular state. Figure 16b shows that dry spots begin to evolve at the moisture content, C,, where there are two particles between the adjacent regions. There is no moisture distribution in the funicular state of the small sample with coarse glass beads. Therefore, the moisture content, C,, at which dry spots begin to evolve is given by the expression

in which S 2 is the area in one segment and SIis the area in the region in the funicular state. Figure 17 shows the top view of the surface of the sample in the case of cubic array when dry spots begin to evolve. There is a complicated configuration of water in the evaporation zone as shown in Figure 16, but the experimental results suggest that, in the sample with coarse glass beads, the evaporation zone is thin. Therefore, the evaporation zone is modeled as shown in Figure 18; there are three regions, funicular state region I, dry-spot region 111, and region I1 between funicular state region I and dry-spot region 111. The region between the adjacent funicular state regions extends proportionally with a decrease in the moisture

2358 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988

Figure 18. Model of evaporation zone: (I) funicular state region; (11) region between funicular state region and dry-spot region; (111) dry-spot region.

content. Therefore, the thickness, 6, of the evaporation zone is assumed to be proportional to a decrease in the moisture content and is given by the expression 6 = 6 , - 6-,

c - c, co - cc

for C, 5 C 5 C,

R - 26,/3 R

(4)

The radius, a, of the funicular state region I is mr for moisture contents in the vicinity of C, (n is the number of particles). However, at moisture contents of less than C,,, it is assumed that the radius of the funicular state region decreases suddenly. Therefore, the following expressions are given:

mr (Cca 5 C 5 C,) ~PC/C,, (C, < C < C,,)

a = a

in which T is the temperature within the dry spot, Tgis the temperature of superheated stream, Tbis the boiling point temperature, and K , is the effective thermal conductivity of sintered glass beads. The above equations were solved numerically by a finite-difference method (see Appendix A). Critical Moisture Contents. A t the critical moisture content, C, the funicular state is not present in the sample and only the pendular state remains in it. The ratio of the volume of a dry spot to that of one segment of the evaporation zone is nearly 213 at C, as estimated from the shape of the evaporation zone in Figure 18. Therefore, C, is given by the expression

(5) (6)

cc = cp(

Falling Rate Period. At moisture contents of less than C,, water in the sample is in the pendular state. Therefore, in the falling rate period, the heat transfer within the sample becomes the heat conduction with a receding evaporation front. If the quasi steady state is assumed for the formulation of the receding evaporation front, 5; the following expressions are given

The radius, b, of the dry spot at the moisture content,

C, can be obtained from a mass balance over the evaporation zone [(Cp/3)b(b + 3a) + C I a 2 ] [ R-3 ( R - &I3] + [C,(U + b)2 + (C, - C,)a2](R - 6)3 - ( a + b)'R3C = 0 (7) This equation is valid at moisture contents from C, to C , where region I1 consists of only the pendular state as shown in Figure 16b,c. However, this equation is applied to conditions of moisture contents from C, to C,, in spite of overestimation of the ratio of a over b because, as the moisture content approaches C,, the evaporation rate in the evaporation zone is controlled by 6. There are few dry spots at moisture contents of more than C, and the rate of drying is consistent with the usual drying rate during the constant-rate period. However, at moisture contents of less than C,, the rate of drying in the evaporation zone can be expressed as Wconst= (CF/CF,)H(T, - Ts)/h + (1 - CF/CF,)W~, (8) CF = CIa2/(a + b)'

(9)

in which CF is the moisture content in funicular state region I, CFois that at C,, and W, is the rate of evaporation from the border between region I1 and dry-spot region 111. Wconstis calculated from the temperature distribution of the evaporation zone, which can be obtained as described below: If quasi steady state is assumed for the formation of the evaporation from the evaporation zone as shown in Figure 18, the heat transfer through the evaporation zone is given by the expressions d 2 T + -1- d+ T- = od2T -

r dr dz2 with boundary conditions -k, dT/dz = H(Tg- Tz=0), z = 0 , dr2

(11)

a S r S a + b

T = Tb,

z = 6(r - a ) / b ,

-k, d T / & = 0 ,

05216,

a l r i a + b (12)

r =a

+b

(13)

)

'=

k,(R - i-1 (TsR?

q = H(T,

Tb)

+

T,) + 5.672[ Tg + 273 -

(

) ( :00273)'IFAE - Ts

(15)

in which the second term on the right side of the top line of eq 15 is the sensible heat.

Thickness of Evaporation Zone. It is very difficult to theoretically predict 6, in the unsteady state. Therefore, the characteristics of 6, are assumed to be the following. The thickness of the evaporation zone, 6,, has an upper limit and a lower one with regard to Wconst,which are not functions of the frictional resistance to flow of water but functions of the configuration of water in the funicular state and the effective thermal conductivity in the sample. (1) The lower limit of a,, acmin, corresponding to the position of retreated water in Fiin regions F1, ..., Fi,...,F,, in the evaporation zone as shown in Figure 16d, is determined by the following method. The water transfer ceases when 6 = 6, a t the critical moisture content, C,, in superheated steam drying. On the other hand, the water transfer ceases at the depth, i + ~ r ,of a water-transfer front in air drying. The water transfer both in superheated steam drying and in air drying follows capillary pressure curves of the respective temperatures, which are represented by a dimensionless capillary pressure curve (Kubota et al., 1968). Therefore, when the water transfer ceases, Cair in air drying has the same position of retreated water as a lower limit of 6, in superheated steam drying does; that is, 6 cmin . = j - .air (17) In the coarse glass beads sample with a small diameter, the frictional resistance to flow of water during drying arxl

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2359 the effect of the gravitational force are negligible, resulting in a quite small amount of lair, if any, and evaporation is controlled by the water transfer. Therefore, the watertransfer front, {air, corresponds to a evaporation front, that is, lair

+C

(18)

If the quasi steady state is assumed, the rate of evaporation at the evaporation front, in air drying (Nissan et al., 1959) is given by the expressions

c,

Effective Thermal Conductivity. The following equation for the effective thermal conductivity, k,, in sintered glass beads is derived on the basis of t,he method for the estimation of k, in packed beds (Okazaki et al., 1977; Kunii and Smith, 1960) and in view of the flow of heat through sintered parts (F sin 8’ in radius) of glass beads as shown in Figure 2 k , = (3e3c + 2 - 3e3)kg/2 + 3(1 - c)e3k,,/2, (3e3 - 2)/(3e3) < c < (e3 - 0.49)/e3

k , = k,,, = H ( T , - T,)

5.672

+

[(

T g+ 273

)( -

)Ipm

Ts ;73

t

>’

( >’(

k,, = N3kg 1 (19)

< (3e3 - 2)/(3e3) x

K ( 1 - L/P) -

(K

- 1) COS 8,

~ (-1L/F) -

(K

- 1) COS 8’

(e)( &-)(cos

8’ - cos 8,)

4[(P + p ) sin 8, - p I 2 { was calculated from eq 19 and 20 a t the moisture content at the end of the first falling rate period in air drying which is represented by X in Figure 4 (Ceaglske and Hougen, 1937; Yagi et al., 1957). I , and I, were defined as I , = ( C p - Cx)/(Cca - Cx), Cx 5 C 5 Cca (21) Iy

= (Cx - Cycalc)/(Cx - C y ) ,

Cy

5 C 5 Cx

(22)

If I , < 1, it indicates that the water transfer continues and the funicular state is still present in the sample at moisture contents C, to C,. If I, = 1,it indicates that the water transfer ceases. I y has the same meaning as I,. It is confirmed from the values of I , and I y that the water transfer ceases a t the moisture content at the end of the first falling rate period. The moisture content, Cp, in the pendular state is given by the expression =

“(

R - R{dr/2)

3

where it is assumed that the moisture distribution between the surface of the sample and the water-transfer front, k,, is linear. (2) 6, is not calculated directly from the moisture distribution in superheated steam drying but the mean thickness 6’, is calculated from a mass balance at the moisture content, C

(C,

+ C,)(t

- 6’,)/t

=

c1 + CL

(C2 + CL)(R- 6’c)3/R3= C

(24) (25)

in which t is the thickness of the outer part of the sample as shown in Figure 3. If 6’, > 0 at C < C,, then C1 < C,;this indicates that dry spots are present. From the shape of the evaporation zone in Figure 18, the relationship between 6, and 6‘, is given by the expression 6,

=

36’,/2

(26)

(3) The upper limit of 6,, 6,, was not able to be estimated because of the complexity of configuration of water in the funicular state in the sample.

I

+

ks (27) (2P - 2fL)Z

in which (F

8’ = sin-’

(

+ p ) sin 0, - p

e = (F-~L)/F

r

Discussion Evaporation Zone. Dry spots evolved at A = S2/$ = (m + 3)2/m2 = 6.25 (when m = 2 in eq 3 for regular packing), which was close to A = 7.2 obtained from eq 3 and C, = 0.03 which was determined by the results of the observed moisture distributions. Therefore, the radius of the funicular state region was determined as a = 2i; and then b was calculated from eq 7. was calculated from eq 17-22. I, and Iy for all the 6, samples used in this work are shown in Table 11. From these values, it was confirmed that the water transfer ceased and the funicular state was not present in small samples with coarse glass beads a t the moisture content at the end of the first falling rate period in air drying. As shown in Table 11, Cdr was about 5 X lo4 m in all samples. Hence, Scmh was estimated from eq 17. The mean thickness of the evaporation zone, 6’,, obtained from the experimental moisture distribution for sample 0 and eq 24-25 was 1.2 X m a t a superheated steam temperature of 117 “C and W,, = 2.6 X lo4 kg/(m2 s). This same value of 6’, was applied to samples 1and 18 because the sintering conditions for them were very similar to sample 0. The value 6, = 1.8 X m was calculated from 6’, and eq 26. From these values and eq 4-7, the shape of the evaporation zone was calculated as shown in Figure 19. The effective thermal conductivity in the evaporation zone was calculated from eq 27. Figure 20 shows the relationship between k, and 8, in the case of N = 6 and 8. Drying Rate Curves. As shown in Figure 4, at moisture contents from C, to C,, the drying rate curve (dotted line 1) was calculated from eq 8-13 (evaporation zone model). It slightly decreased with a decrease in the moisture content in the same way as in the observed drying rate curve. At moisture contents of less than C,, the drying rate curve during the falling rate drying period was calculated from eq 15 and 16 (receding evaporation front model) and it decreased steeply (dotted line 2). Figures 5-9 show the predicted drying rate curves for samples 1,

2360 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 Table 11. Thickness of Evaporation Zone and Moisture Content in the Pendular State 6, (when W,,, = Li*(=I%&), CP 2.6 x lo4 kg m-* s-l), m sample m x io3 (from I, I Y 0 0.48 0.0096 0.189 2.40 1.8" 1 0.69 0.0115 0.090 2.75 1.8' 18 0.28 0.078 0.83 0.61 0.181 1.76 0.0101 0.45 av 1.8' 12 0.40 0.098 1.06 0.54 0.242 2.32 0.68 0.157 3.10 0.40 2.71 0.308 0.62 0.185 3.64 0.37 0.152 1.74 0.89 2.67 0.275 0.222 1.92 0.55 0.125 1.22 0.37 0.0157 0.54' 0.54 av 19 0.35 0.190 1.18 2.53 0.78 0.345 0.57 av 0.0196 0.57c

X

lo3

"Obtained from moisture distributions. *The value of 6, for sample 0 was used. ,Estimated from 6,

=

CP (from 6,) 0.0068 0.0093

0.0082 0.0104

0.0084

0.0093

0.0065

0.0111

0.0164

0.0180

c,

a"

constant.

Table 111. Temuerature Distribution" in a Dry Soot for Samule 0 100.20

100.58 100.20

a 6 = 1.706 X

100.90 100.57 100.20

101.17 100.85 100.52 100.20

m, b = 1.773 X a +b

&

I

101.38 101.07 100.77 100.48 100.20

b 101.54 101.24 100.96 100.68 100.43 100.20

101.67 101.38 101.10 100.84 100.60 100.39 100.20

m, C = 0.007, T, = 117 "C, Tb = 100.2

101.77 101.48 101.20 100.95 100.72 100.52 100.34 100.20

101.83 101.54 101.27 101.03 100.81 100.61 100.44 100.30 100.20

101.87 101.58 101.31 101.07 100.85 100.66 100.50 100.36 100.26 100.20

101.88 101.59 101.33 101.08 100.87 100.68 100.51 100.38 100.28 100.22 100.20

6

OC.

(m)x103

' \ \

\

"C=OO10

C=0007

1

LOLL

-

I

__

-

i __

Figure 19. Example for evaporation zone of sample 0 a t 117 "C in superheated steam drying.

12, 18, and 19, which were in good agreement with the observed drying rate curves in all samples. As shown in Figure 5, the surface temperature calculated from the observed drying rate curve in superheated steam drying (chain h e ) was nearly equal to that calculated from the evaporation zone model (dotted line), but it was somewhat higher than the observed surface temperature (solid line) for sample 1 in superheated steam a t a temperature of 120 "C. In samples 1and 18, at C,, there was almost no difference (1 f 0.5 "C above a boiling point) between the surface temperature obtained from the observed rate curves in superheated steam temperatures of 110-170 OC, but in sample 12, at C, there was a difference (0.5-2.5 OC above a boiling point corresponding to superheated steam temperatures of 110-160 "C) between the respective surface temperatures. Consequently, in all samples, the surface temperature at moisture contents of more than C, was only slightly higher than a boiling point. From these facts, it was concluded that the drying rate curves at moisture contents from C, to C, in superheated

SINTERED ANGLE

(9

Figure 20. Effective thermal conductivities of sintered glass beads a t 100 "C for coordination number N = 6 and 8.

steam drying appear to be in the constant-rate period. Critical Moisture Contents. The moisture contents, C,, in the pendular state were calculated from eq 23 and {& at the moisture content, C,, at the end of the f i t falling rate period as shown in Table I1 (third column). Sample 19 was weakly sintered (6 = 7.5'); hence, it was comparatively easy to compare of sample 19 with that of the packed bed of the same glass beads. The decrease in C, due to the effect of sintering was E = 4% for sample 19 (see Appendix B). Therefore, C, in the packed bed was estimated from C, of sample 19 and was 0.0204 or = 0.078, which was in good agreement with +p = 0.085 (Kubota et al., 1968) and 4, = 0.075 (Dombrowski and Brownell, 1954). As shown in Table 11, C, = (seventh column) obtained from the observed critical moisture contents in super-

e,

+,

Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2361 heated steam drying in terms of 6, and eq 14 was nearly equal to C (third column) obtained from Cair except for C, of sampfe 12. From these results, it was confirmed that, at the critical moisture content in superheated steam drying, the water transfer ceased and the funicular state was not present in the small samples with coarse glass beads. C,, (last column), namely the average of C, (seventh column) and C, (third column), was used for the calculation of critical moisture contents. In samples 0, 1, and 18, the critical moisture contents were calculated from Cpav,6, = 1.8 X low3m ( Wcomt= 2.6 X kg/(m2 s), 117 " C ) and ljCmin (Wconst2 9.5 X kg/(m2 s), 170 "C).Figures 10 and 11 show that C, (dotted line) for samples 1 and 18 is affected by dry spots developing with a decrease in Wcomt. In samples 12 and 19, the critical moisture contents, C, (dotted line), were calculated from Cpav and 6, = constant a t Wconstcorresponding to superheated steam temperatures of 110-160 "C as shown in Figures 12 and 13. This dependence of 6, on Wconstwas explained as follows: In quasi steady state,

when A = constant, Tgl< Tgh,and Wconstl< WCOnstFIn samples 12 with small effective thermal conductivities as shown in Figure 20, T,1 < T,.at C, (110 < T, < 160 " C ) for sample 12 as described in drying rate curves, whereas in samples 1 and 18 with large effective thermal conductivities, Td = T h at C, (110 "C < T, < 170 "C). Therefore, for samples 1 and 18,

"c

in which subscripts 1and h are corresponding to low temperature and high temperature and A is the thickness of the gas film. The predicted critical moisture contents were in good agreement with the observed ones except for those for sample 12. Wenzel and White (1951) investigated the critical moisture contents in granular solids in superheated steam drying. However, their experimental conditions were those of continuous drying; hence, our results were not able to be compared with theirs directly.

Conclusion In this work, the following findings concerning the drying characteristics of sintered spheres of coarse glass beads with a small diameter were obtained: The drying rate curves and the critical moisture contents in superheated steam drying can be estimated on the basis of an evaporation zone model during the constant-rate period and a receding evaporation front model during the falling rate period. The predicted results are in good agreement with the observed results. The drying rate curve a t moisture contents in the vicinity of the critical moisture content for air drying appears to be in the constant-rate period for superheated steam drying. The critical moisture content is superheated steam drying can be estimated from the moisture content at the end of the first falling rate period in air drying. The use of sintered spheres of glass beads with a wide range of porosities and effective thermal conductivities gives good reproducibility of the experimental results. We believe that the evaporation zone model could be applied to the superheated steam drying of nonhygroscopic material with the first falling rate period.

Acknowledgment We are grateful to Dr. I. Tanaka of the University of Occupational and Environmental Health for permission to use their scanning electron microscope.

Nomenclature A = ratio of Sz over SI B = radius of curvature of pendular ring, m a = radius in funicular state region, m b = radius in dry spot, m C = moisture content (dry dasis) cH = specific heat in glass bead, J kg-l K-l D = diffusion coefficient of water vapor in air, m2 s-l E = parameter defined in eq B-2 e = parameter defined in eq 27 F A E = overall interchange factor f = shrinkage factor H = heat-transfer coefficient, W mb2K-l Z, = parameter defined in eq 21 Zy = parameter defined in eq 22 K = mass-transfer coefficient, m s-l k, = effective thermal conductivity, W m-l K-' k,, = apparent thermal conductivity of solid part, W m-l K-' k, = thermal conductivity of gas, W m-l K-I k, = thermal conductivity of solid, W m-l K-' L = width of sintered part of glass beads, m m = number of particles N = coordination number P, = saturated vapor pressure of water, Pa Pa = partial vapor pressure of water, Pa Q = angle of pendular ring, rad q = heat flux, W m-2 RG = gas constant, J K-' mol-' R = radius of a sample, m r = coordinate in evaporation zone, m P = radius of glass bead, m S1 = area in funicular state region, m2 S2 = area in one segment, m2 T = temperature, "C Tg = temperature of drying medium, "C T , = temperature on surface of sample, "C Tb = boiling point, "c T , = temperature of sample during falling rate period in air drying, "C t = thickness of outer part of sample as shown in Figure 3, m td = thickness of sample disk as shown in Figure 3, m t , = width of each parts of sample as shown in Figure 3, m Wconst= drying rate during constant-rate period, kg m-2 s-l W , = evaporation rate from the border between region I1 and dry-spot region 111, kg m-2 s-l W , = drying rate during falling rate period, kg m-2 s-l z = coordinate in evaporation zone, m Greek Letters A = thickness of gas film, m 6 = thickness of evaporation zone in superheated steam drying, m 6' = mean thickness of evaporation zone in superheated steam drying, m c = porosity ti = porosity before sintering { = depth of evaporation front, m = depth of water-transfer front in air drying when water transfer ceases, m OP = sintered angle, deg or rad Bo = angle corresponding to boundary of heat flow area for one contact point, rad 0' = sintered angle in neck, rad K = k,/k, A = latent heat of water, J kg-'

cair

2362 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 = radius of curvature in neck, m 4 = moisture content (volume of water in sample/volume of

p

w

void space in sample) = relaxation factor

Literature Cited

Subscripts

av = average of C, calculated in terms of { ~ rand that calculated in terms of 6 , c = critical moisture content in superheated steam drying ca = critical moisture content in air drying calc = calculated in terms of { at y point and C, F = funicular state region h = high temperature I = initial state 1 = low temperature L = loss of moisture during division of a sample o = evolution of dry spot p = pendular state x = x point in drying rate curve in air drying y = y point in drying rate curve in air drying 1 = outer part of sample 2 = inner part of sample

Appendix A Equations 10-13 were solved numerically by the successive over-relaxation method (S.O.R.) (Smith, 1975). The conditions for convergence are AT,,,,,