Drying 2004 – Proceedings of the 14th International Drying Symposium (IDS 2004) São Paulo, Brazil, 22-25 August 2004, vol. A, pp. 501-509
SIMULTANEOUS HEAT AND MOISTURE TRANSFER AND SHRINKAGE DURING DRYING OF CERAMIC MATERIALS
José J. S. Nascimento ¹, Gelmires A. Neves2, Francisco A. Belo3 and Antonio G. B. Lima¹, 1. Department of Mechanical Engineering, CCT, Federal University of Campina Grande, P.O. Box 10069, 58109-970 – Campina Grande, PB, Brazil, E-mail:
[email protected] 2. Department of Material Engineering, CCT, Federal University of Campina Grande, 58109-970 – Campina Grande, PB, Brazil, E-mail:
[email protected] 3. Department of Mechanical Technology, CT, Federal University of Paraíba, 58059-970 – João Pessoa, PB, Brazil, E-mail:
[email protected] Keywords: drying, numerical, experimental, ceramics bricks ABSTRACT This work reports a transient three-dimensional mathematical model to describe the simultaneous heat and mass transport and shrinkage. The model assumes constant thermo-physical properties and convective boundary condition at the surface of the solid. The governing equations were solved using finite-volume method and fully implicit formulation. The mathematical formulation was used to describe the drying of ceramic bricks, in the following air drying conditions: temperatures T=60, 80 and 110ºC and relative humidities RH=10.07, 4.66, 4.96 and 2.30, respectively. The experimental tests were made using two clay materials for production of red ceramic and white ceramic (ball-clay). Several results of the mean moisture content and temperature along the process are shown and analyzed. Numerical results were compared with experimental data and the transport coefficients were determined by the least square error technique. A good agreement was obtained in all experiments. INTRODUCTION The drying is a simultaneous phenomenon of heat and mass transfer and shrinkage. Many researchers has reported drying of clay, such like Norton (1975), Elias (1995), Fricke (1981), Ketelaars et al. (1992), and Hasatani & Itaya (1992), van der Zanden et al. (1996); van der Zanden et al. (1997) and Reeds (1991). During the drying of solid, the shrinkage phenomenon exists, and it alters the drying kinetics and the dimensions of the solid. This phenomenon happens simultaneously with the moisture transport and it
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is more intense in ceramic materials with high initial moisture content, mainly in products of fine granulation. Depending on the drying conditions, structures of the material and geometry of the product, the shrinkage phenomenon can cause cranks, deformations and even fracture inside the solid. Then it is very important to study drying including shrinkage. In the mathematical modeling of shrinkage, the properties relative to the phenomenon will must be incorporated to the model. However, in the literature, there is scarce information on the shrinkage coefficients, as well as of mathematical relationships among the mass diffusivity, shrinkage and density. However, we found in the literature, curves of linear shrinkage versus moisture content (Reeds, 1991, Itaya & Hasatani, 1997; Keteraals et al., 1992; Hasatani & Itaya, 1992; Itaya & Hasatani, 1996, Nascimento et al. 2001). In this sense, this work has as objectives to present a three-dimensional mathematical modeling of the heat and mass transfer in parallelepiped solids including shrinkage and to apply the methodology to the drying of ceramic materials. MATHEMATICAL MODELING Mass transfer model To simplify the model, the following considerations are adopted: • the thermo-physical properties are constant, during the whole diffusion process; • the solid is homogeneous and isotropic; • the moisture content and temperature distributions are uniform in the beginning of the process; • the phenomenon happens under convective and evaporative conditions including heating of vapor produced in the surface of the body. • the heat and mass transfer coefficients are assumed to be constant in the whole process; • the shrinkage of the solid is equal to the volume of water evaporated; • the shrinkage is three-dimensional and symmetry exists in the center of the solid; • the solid is composed of water in liquid phase and solid material. Figure 1 illustrates a solid parallelepiped of dimensions 2R1x 2R2x2R3. For this case, the general differential equation that describes the diffusion phenomenon is: ∂M (1) = ∇.(D∇M ) ∂t Due to the symmetry in the solid, in particular in planes (x=0,y,z), (x,y=0,z), (x,y,z=0) we consider 1/8 of the volume of the solid. The initial, symmetry and boundary conditions are as follows:
Initial condition: Symmetry condition:
x
O
R2
R3
R2
z
R3 R1
R1
Figure 1- Geometrical configuration of the physical problem
M ( x , y, z, t = 0) = M o
(1a)
∂M(x = 0, y, z, t) ∂M(x, y = 0, z, t) ∂M(x, y, z = 0, t) = = = 0 , t>0 ∂x ∂y ∂z
(1b)
Boundary condition: −D
502
y
∂M ( x , y, z, t ) = h m (M( x , y, z, t ) − M e ) for t>0 and z= R1 ∂x
(1c)
∂M ( x , y, z, t ) = h m (M( x , y, z, t ) − M e ) for t>0 and z= R2 ∂y ∂M ( x , y, z, t ) −D == h m (M ( x , y, z, t ) − M e ) in t>0 e z= R3 ∂z
−D
(1d) (1e)
The average moisture content was calculated by: M=
1 M.dV V
(2)
V
Heat transfer model
For heat transfer process, the transient heat conduction equation is given by: ∂(ρC p ) = ∇.(k∇θ) ∂t The boundary conditions are: • Free surface: ρ V ∂M ∂θ( x , y, z, t ) k = h c [(θ ∞ − θ( x = R 1 , y, z, t )] + s h fg + c v ∂x S ∂t ρ V ∂M ∂θ( x , y, z, t ) h fg + c v k = h c [(θ ∞ − θ( x , y = R 2 , z, t )] + s ∂y S ∂t
(3)
[
(θ ∞ − θ x =R )]
(3a)
[
(θ ∞ − θ y=R )]
(3b)
[
(
)]
(3.c)
1
2
ρ V ∂M ∂θ( x , y, z, t ) = h c [(θ ∞ − θ( x , y, z = R 3 , t )] + s h fg + c v θ ∞ − θ z = R 3 ∂z S ∂t • Planes of symmetry: ∂θ( x , y, z, t ) ∂θ( x , y, z, t ) ∂θ( x, y, z, t ) | x =0 = | y =0 = | z =0 = 0 ∂x ∂y ∂z • Initial condition: θ( x , y, z, t = 0) = θ o The average temperature of the body during diffusion phenomenon was calculated as follows: k
θ=
1 θdV V
(4) (5) (6)
V
Shrinkage model
According to Lima (1999), the following equation was utilized to obtain the changes of volume in each time during the drying process: (7) (V )t = Vo β1 + β 2 M
(
)
(
)
Since in t = 0, M = M o and (V)t = Vo (see Figure 2), we have that β = 1 − β 2 M o . Then the equation (7) can be written as follows: (V )t ∗ = β3 + β 4 M (8) Vo
where β 3 = β1 + β 2 M e , β 4 = β 2 (M o − M e ) and M* = (M − Me) /(Mo − Me) . The volume of the parallelepiped is given by: ( V )t = ( R1 )t ( R 2 )t ( R3 )t
The surface area of mass transfer of the solid during the drying process was obtained by:
(10)
503
( Sˆ )t = [( R1 )t ( R3 )t + ( R1 )t ( R3 )t + ( R )t ( R3 )t ]
(11)
y DC
x
α1 (R2) t (R2) t = 0
0
α2
(R3) t
(R3) t = 0
(R1)tt z (R1) t = 0
Figure 2 – Shrinkage of a solid parallelepiped during the drying process
EXPERIMENTAL METHODOLOGY The ceramics brick (ball clay type) were made press-molded of the parallelepiped shape under pressing pressure of 2.5 MPa. The brick was dried by placing it in oven in specified temperature and relative humidity. The first step of experiment was to weight the sample and to measure the length one in intervals of 10 min during drying process. The mass of the sample was obtained by an electronic balance with accuracy of ±0.01g and the dimensions of the sample were measured using a digital calipers. Table 1 presents all air and ceramic bricks drying conditions used in this work. Details of the experimental procedure and drying equipment can be found in Nascimento (2002). Table 1. Air and ceramics bricks experimental conditions used in this work. Sample E60R1 E80BA3 E80R1 E110R1
Air T (ºC) 60 80 80 110
RH (%) 10.07 4.66 4.96 2.30
v (m/s) 0.1 0.1 0.1 ≈0.0
Ceramic brick Mo Me (d.b.) (d.b.) 0.1005 0.00173 0.0765 0.00084 0.2139 0.00158 0.2270 0.00902
2R3 (mm) 20.54 20.49 20.55 20.31
2R1 (mm) 60.45 60.81 60.26 60.25
2R2 (mm) 7.06 5.39 6.55 9.56
t (min) 270 220 270 390
NUMERICAL SOLUTION In this work we use the finite-volume method (Maliska, 1995; Patankar, 1980) to discretize the governing equations. The Fig. 3 represents the differential volume of the physical domain (Fig. 1), where the nodal points (W, E, N, S, F, T), dimensions and length of the control volume are presented. As results the equations (1-3) can be written in the linear form as: (12) A P Φ P = A E Φ E + A W Φ W + A N Φ N + ASΦS + A FΦ F + A T Φ T + A 0P Φ 0P + B
where Φ can be M or θ . The set of equations is solved iteratively using the Gauss-Seidel method. The following convergence criterion was used: Φ
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m +1
−Φ
m
≤ 10−8
(13)
where m represents the m-th iteration in each time. More details can be encountered in Nascimento et al. (2001) and Nascimento (2002). N T
δy n n W
δz t
f δz f
∆z
s ∆x
δy s
δx w
F
E
e
P
w ∆y
t
δx e S
Figure 3 – Control-volume used in this work
The diffusion and mass transfer coefficients were found by varying the D and hm to minimize the sum of squared deviations between the actual and predicted data. The relative deviation between experimental and calculated values (relative residuals, ERMQ) and the variance (S2) are defined as follows:) ˆ m 2 2 ERMQ (14a-b) ERMQ = Mi, Num − Mi,Exp ; S = (mˆ − nˆ ) i =1
(
)
ˆ is the number of experimental points and n where m is the number of parameters fitted (Figliola and Beasley, 1995).
The heat of vaporization of pure water (Table 2) was used instead of the heat of vaporization of the moisture from the brick, at particular moisture content, in the same air drying condition. Table 2 – Heat of vaporization of pure water and specific heat of the water vapor for each drying condition Test 1 3 4 5
Sample E60R1 E80BA3 E80R1 E110R1
hfg (kJ/kg) 1.91093 1.91726 1.91726 1.92759
cv (kJ/kgK) 2358.34 2307.62 2307.62 2227.01
In this work ρs = 1920kg / m3 and c p = 1673.51J / kgK (Incropera & DeWitt, 2002) and k=1 W/m.K (Hasatani & Itaya, 1992) at 100ºC, were used. The convective heat transfer coefficient was obtained as follows (Incropera & DeWitt, 2002): k (15) h c = 0.664 Re1x/ 2 Pr 1 / 3 f 0.6 ≤ Pr ≤ 50 x for test with temperature of 60 and 80oC, and 0.670Ra1x/ 4 kf 0 < Ra x < 10 9 h c = 0.68 + (16) 9 / 16 4 / 9 x [1 + (0.492 / Pr) ] for test with temperature 110oC. RESULTS AND DISCUSSIONS In order to test the formulation presented in this work, different numerical results of the average moisture content of a ceramics brick along the drying process are compared with experimental results. To obtain the numerical results a computational code was implemented using the grid 20 x 20 x 20 points and ∆t = 1s . These conditions were obtained by successive grid and time refinements. Other information about this procedure can be found in Nascimento et al (2001) and Nascimento (2002). Shrinkage coefficient The estimated values of the shrinkage coefficients in the equation (8) applied to ceramics brick are illustrated in Fig. 4 that shows the changes of the volume as a function of the average moisture content during
505
drying process. In this figures we can see the existence of two periods of volume changes. This behavior is in agreement with the results of Hasatani & Itaya (1992) and Ketelaars et al. (1992). According to Elias (1995), the point where the curves intercept is called a critical moisture content, however in this study no constant drying period was encountered because the initial moisture content was low. In the beginning of drying a large moisture removal takes place so the dimensions of solid changes with high velocity. Further, the shrinkage velocity approaches zero. 1.000
The point where the shrinkage shows the new behavior is different depending of the __ material composition. In the brick made of 0.995 V/Vo=0.984676+0.0162859 M* ball clay this point occurred at the lowest __ V/Vo=0.986874+0.0037114 M* dimensionless moisture content. But this 0.990 behavior is not completely explained and requires more studies. It is possible that at too low moisture content, the volume shrinkage is 0.985 not equal to volume of water evaporated and 0.00 0.20 0.40 0.60 0.80 1.00 vapor appears in the pores of solid. Table 3 __ M* presents all the values of the shrinkage and Figure 4 - Comparison between the experimental (o) e predict volumes correlation coefficients and variance for all of red ceramics brick obtained during the drying in oven to the T=80 oC. drying experiments. V/Vo
E80BA3 Experimental
Table 3. Shrinkage and correlation coefficients and variance for all drying experiments Sample
E60R1 E80BA3 E80R1 E110R1
1st segment of shrinkage curve
β3 0.9373134 0.9846760 0.7886403 0.7562295
β4
2nd segment of shrinkage curve
R
0.0678610 0.0162859 0.1854330 0.2424924
0.973 0.930 0.924 0.992
S2 0.947 0.864 0.853 0.983
β3 0.9611617 0.9868740 0.7991150 0.8248540
β4
R
0.0199140 0.0037114 0.2814140 0.0819587
0.954 0.932 0.972 0.920
S2 0.910 0.868 0.945 0.838
Moisture content of the ceramics brick Figures 5-8 illustrate the comparison between numerical and experimental data of the average moisture content obtained during drying. It can be seen in these figures that a good agreement was obtained. Some discrepancies appear at the lowest moisture content due to the fact that for longer drying times the assumption of linear shrinkage is probably not valid, as assumed in equation (7). According to Figure 2, the shrinkage is proportional the dimensions of the solid, characterizing a threedimensional shrinkage, however, this shrinkage happens in the direction of DC segment (one dimensional shrinkage). In this sense, different deformations and shrinkage velocity are generated in the directions of the x, y and z axis. Then, the β 3 and β 4 are the effective shrinkage coefficients. 1.00
0.60 0.40
0.40 0.20
0.00
0.00
40.00
80.00
120.00 t(min)
160.00
200.00
240.00
Figure 5 - Comparison between predicted and experimental
506
0.60
0.20
0.00
Sample E80BA3 Numerical Experimental
0.80 (M-Me)/(Mo-Me)
0.80 (M-Me) /(Mo-Me)
1.00
Sample E60R1 Numerical Experimental
0.00
20.00
40.00 t(min)
60.00
Figure 6 - Comparison between predicted and experimental
dimensionless means moisture content during the drying of ceramics brick (test 1). 1.00
1.00
Sample E80R1 Numerical Experimental
Sample E110R1 Numerical Experimental
0.80 (M -M e) / (M o-M e)
0.80 (M-Me)/(Mo-Me )
dimensionless means moisture content during the drying of ceramics brick (test 2).
0.60 0.40 0.20
0.60 0.40 0.20
0.00 0.00
20.00
40.00
60.00 80.00 t(min)
100.00
120.00
0.00
140.00
0.00
Figure 7 - Comparison between predicted and experimental dimensionless means moisture content during the drying of ceramics brick (test 3).
50.00
100.00
150.00
200.00
250.00
t(min)
Figure 8 - Comparison between predicted and experimental dimensionless means moisture content during the drying of ceramics brick (test 4).
Figures 9-12 presents the means moisture content and temperature of the solid for all drying experiments. It is observed that the temperatures reach the equilibrium temperature very quickly mainly at higher air temperature and lower initial moisture content in the same relative humidity. 1.20
1.20
1.00
1.00
1.00
Sample E60R1- Numerical Moisture Content Temperature
0.60
1.00 Sample E80BA3- Numerical Moisture Content Temperature
0.80 M*
0.60
0.80
1 – θ*
0.60
0.80 0.60
0.40
0.40
0.40
0.40
0.20
0.20
0.20
0.20
0.00
0.00
0.00
50.00
100.00
150.00
200.00
250.00
t(min)
Figure 9 - Predicted dimensionless means moisture content and temperature during the drying of ceramics brick (test 1). 1.20 1.00
M*
0.80 0.60
20
40 t(min)
60
80
Figure 10 - Predicted dimensionless means moisture content and temperature during the drying of ceramics brick (test 2).
1.20
1.20
1.20
1.00
1.00
1.00
0.80
0.80
1 – θ∗
Sample E80R1- Numerical Moisture Content Temperature
0.00 0
0.60
Sample E110R1- Numerical Moisture Content Temperature
0.60
0.80 0.60
0.40
0.40
0.40
0.40
0.20
0.20
0.20
0.20
0.00
0.00 0.00
40.00
80.00 t(min)
120.00
160.00
Figure 11 - Predicted dimensionless means moisture content and temperature during the drying of ceramics brick (test 3).
0.00
1 – θ∗
0.00
M*
M*
0.80
1.20
1 – θ*
1.20
0.00 0.00
50.00
100.00
150.00
200.00
250.00
t(min)
Figure 12 - Predicted dimensionless means moisture content and temperature during the drying of ceramics brick (test 4).
The transport coefficients D and hm as well as the variance obtained in the experiments are shown in Table 4. The small variance indicates that the numerical results present good agreement with the experimental data. The model is very versatile and the technique used has great potential, being able to be used to describe diffusion processes such as drying, wetting, heating and cooling of solids with geometry that
507
varies from one-dimensional rod to parallelepiped, besides plate planes, without restrictions to the nature of the material (fruits, cereals, vegetables, minerals, etc.).
Table 4. Convective mass transfer and mass diffusion coefficients, and variance estimated for all drying experiments D. 10+8 (m2/s)
hm.10+6 (m/s)
Bim Initial
1 – E60R1
0.233
1.650
2 – E80BA3
0.357
3 – E80R1
1.296
Final
S2 (10+3)
hc (W/m2 ºC)
22.65
22.51
0.020
4.190
37.78
37.73
0.131
1.350
3.33
3.14
4 – E110R1 1.520 1.820 Bim=hmR/D and Bic=hcR/k
3.85
3.60
Test
Bic Initial
Final
4.92
0.1393
0.1383
0.1382
0.1380
0.210
4.88 4.90
0.1387
0.1310
0.550
1.39
0.0400
0.0370
CONCLUSIONS From the analysis of the results obtained, the conclusions may be summarized as follows: a) the model and the technique used have great potential and are accurate and efficient to simulate many practical problems of diffusion such as heating, cooling, wetting and drying in parallelepiped solids, including one-dimensional rod as a limiting case; b) The linear shrinkage coefficient of the ceramics brick changes with the drying temperature and presents two shrinkage periods; c) the mass diffusion coefficient increase strongly with the increase of the temperature. ACKNOWLEDGMENT The authors would like to express their thanks to CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), grant # 475098/2003-0, for its financial support to this work. NOTATION Bim, Bic cp cv D ERMQ hc hm h fg
Biot number of mass transfer Biot number of heat transfer Specific heat Specific heat of vapor. Diffusion coefficient Least square error heat transfer coefficient
J/kg oC J/kg oC m2/s (kg/kg)2 W/m2 oC
Mass transfer coefficient Latent heat of vaporization
m/s J/kg
k M, M N, S, E, W, P RH Re Ra S2 Sˆ V v x, y, z
Thermal conductivity Local and average moisture content Nodal points Relative humidity Reynolds number Rayleigh number Variance Area Volume Air velocity Cartesian coordinates
W/moC kg/kg (%) (kg/kg)2 m2 m3 (m/s) m
508
t T Greek Symbols δ, ∆ ρ θ β1 , β2 , β3 , β4
∇ Subscripts e e,w,s,n t
Time Air temperature
s (ºC)
Variation Density, Temperature Shrinkage coefficient Gradient
kg/m3 o C -
Equilibrium Boundary of the nodal points Time
Superscripts * o t
Dimensionless Old Time
LITERATURE Elias, X., (1995) The manufacture ceramics materials, Barcelona-Espanha, 205p. (In Portuguese). Figliola, R. S., Beasley, D.E., (1995), Theory and design for mechanical measurements 2 ed., New York: John Wiley & Sons, 607 p. Fricke, (1981), The ceramics, Ed. Presença Ltda., Lisboa, 152p. (In Portuguese). Hasatani, N., Itaya, Y., (1992), Deformation characteristic of ceramics during drying, International Drying Symposium, pp 190-199, Parte A. Incropera , F. P.; DeWitt., (2002), D. P. Fundamentals of heat and mass transfer, New York-U.S.A, John Wiley & Sons. Itaya, Y., Hasatani, M., (1996), R & D needs-drying ceramics, Drying Technology, Vol. 14, no. 6, pp 1301-1313. Itaya, Y., Taniguchi, Hasatani, M., (1997). A numerical study of transient deformation and stress behavior of a clay slab during drying, Drying Technology, Vol 15, no. 1, pp 1-21. Ketelaars, A. A. J., Jomaa, W, Puiggali, J. R., Coumans, W. J., (1992), Drying shrinkage and stress, International Drying Symposium, A, pp 293-303. Lima, A.G.B., (1999), Diffusion phenomenon in prolate spheroidal solids. Case studied: Drying of banana, Ph.D. Thesis, State University of Campinas, Campinas, Brazil, (In Portuguese). Maliska., C.R. (1995), Computational heat transfer and fluid mechanics, Rio de Janeiro, Brasil, Ed. LTC, 424p. (In Portuguese) Nascimento, J. J. S. (2002), Diffusion phenomenon in parallelepiped solids. Case studied: drying of ceramic materials, Ph. D. Thesis, State University of Paraiba, João Pessoa, Brazil, (In Portuguese). Nascimento., J. J. S., Belo, F. A., Lima A, G. B, and Pontes., L. R. A., (2001), Simultaneous moisture transport and shrinkage during drying of parallelepiped solids, Proceedings of the Second InterAmerican Drying Conference, Boca del Rio, Vera Cruz, Mexico, pp 535-543. Norton., F. H. (1975), Elements of Ceramics, Ed. Addison Wesley, Massachusetts, 350p. Patankar, S.V., 1980, Numerical heat transfer and fluid flow, Ed. Hemisphere Publishing Corporation, New York, USA. Reeds, J. S., (1991), Drying, ASM International Handbook Committee, pp 131-134. van der Zanden, A.J.J, Schoenmakers, A.M.E, Kerkof, P.J.A.M., 1996, Isothermal vapour and liquid transport inside clay during drying, Drying Technology, Vol. 14, no. 3 e 4, pp 647-676. van der Zanden, A.J.J., 1997, Mathematical modeling and numerical techniques in dying technology, Chapter: Modelling and simulating simultaneous liquid and vapour transport in partially saturated porous materials, Ed. Marcel Dekker, Inc., New York, USA.
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