18-20 June, 2003, Budapest, Hungary OSSKI Center (Törley Palace)

with Exhibition and Pre-Session on Thermal Energy in Hungarian
"THERMO-BRIDGE"
between East and West for technology transfer and information exchange

P01

Drying is a thermal process in which heat and mass transfer occur simultaneously. Heat is transferred by convection from heated air to the moist products pellet to raise the temperatures of both the solid and moisture that is present. Mass transfer occurs as the moisture travels to the evaporative surface of the pellet and then into the circulating air as water vapor. The heat and mass transfer rates are related to the velocity and temperature of the circulating air.

The driving force for evaporation is the difference between the vapor pressure of the water in the product and the partial pressure of the water vapor in the surrounding atmosphere. Increasing the temperature of a wet solid increases the vapor pressure of the water in the solid, while increasing the temperature of air decreases the partial pressure of water vapor in the air. Moisture will migrate from an area of high vapor pressure to one of low pressure. Recently, Dincer et al. (2000) has illustrated the moisture gradient resulting from this migration of moisture from within the heated product, where it has high vapor pressure, through micropores to the wet surface, where it is evaporated into the surrounding atmosphere as water vapor with a lower partial pressure.

The mathematical modeling of solids drying processes is of great importance, and the knowledge of temperature and moisture distributions within the product and understanding the fundamental mechanisms during the drying process are essential for process design, handling practices, quality control, and energy savings. The principle of modeling is based on having a set of mathematical equations, which can adequately characterize the process. The solution of these equations must allow for the prediction of the process parameters as a function of time. In the literature, a number of investigations have been undertaken by many researchers (e.g, Dincer et al., 2000; Ruiz-Cabrera et al., 1997; Akiyama et al., 1996; Rogers and Kaviany, 1990; Perre et al., 1993; Ssu-Hsueh and Marrero, 1996; Dincer and Dost, 1996) on heat and moisture transfer analysis during drying of various solid products.

The main goal of this study is to model the drying process, which can adequately predict the temperature and moisture distributions inside a cylindrical moist solid subjected to air drying, and to validate the model with experimental data obtained from the literature.

Drying of moist products is a complicated process involving simultaneous, coupled heat and mass transfer phenomena, which occur inside the moist solid being dried. These coupled phenomena make the analysis of the drying mechanism so complicated that extensive efforts have been devoted to the development of theoretical models of heat and mass transfer to describe the drying phenomena in moist products. The theory was initially proposed by Philip et al. (1957) and Luikov (1973), noting that it was not just the moisture content but also the moisture transfer potential, which was the more effective parameter in predicting moisture movement in moist products. Temperature gradient was considered to be one of the driving forces for moisture flow in moist products (Mukherjee et al., 1997).

The mathematical modeling pertinent to the drying process is based on the Fourier law of heat conduction and Fick’s law of mass diffusion. The governing Fickian equation is exactly in the form of the Fourier equation of heat transfer, in which temperature and thermal diffusivity are replaced with concentration and moisture diffusivity, respectively.

• Thermophysical properties of the moist material and drying air are constant.

• There is negligible shrinkage or deformation of material during drying.

• No heat generation inside the material takes place.

• The temperature of the drying air is constant.

• Two-dimensional temperature and moisture distributions are considered in a cylindrical moist material (i.e., variations occur in the r and z directions)

The mathematical equations governing the drying process in a two-dimensional rectangular product with the appropriate boundary conditions are given as:

• For heat transfer:

where the initial and boundary conditions are:

at r = 0;  at r = R;  at z = 0;  at z = L;

• For moisture transfer:

where initial and boundary conditions are:

at r = 0;  at r = R;  at z = 0;  at z = L;

and D is the moisture diffusivity, whose dependence on temperature is of the form of Arrhenius expression (for details, see Ruiz-Cabrera et al., 1997):

(3)

where T is the absolute temperature in K.
The dimensionless temperature and moisture content can be written as follows:

(4)

and

(5)

The solution of the above governing equations is difficult to obtain using analytical methods. Moreover, considerable assumptions have to be considered in order to obtain a closed form solution with many inadequacies. Therefore, approximate methods of solution are used to solve the governing equations.

The method used in the present study is the explicit finite difference approximation where the governing equations are first transformed into difference equations by dividing the domain of the solution to a grid of points in the form of mesh and the derivatives are expressed along each mesh point referred to as a node. The numerical grid of an axisymmetric cylindrical product is shown in Figure 1. The index i represents the mesh points in the z-direction starting with i = 0 being one boundary and ending at i = m, the other boundary, while index j represents the mesh points in the R-direction starting from j = 0. Thus, the finite difference representation of the mesh points is as follows:

z_i = iz for i = 0,1,2,…m and r_j = jr for j = 0,1,2,…n

where ?z and ?r represents the grid sizes in the z and r-directions respectively and the subscripts denote the location of the dependent variable under consideration, i.e., T_i,j means the temperature at the i’th z-location and j’th r-location. Knowing the value of a dependent variable at the initial time step, unknown values at next time steps are calculated using the finite difference equations. The finite difference representations of the governing equations can be written in the following form:

• For heat transfer:
   

  (6)

where

and

The initial and boundary conditions are as follows:

	at r = 0;		at r = R;		at z = 0;
	at z = L;		where	;

• For moisture transfer:

where

and

The initial and boundary conditions become:

	at r = 0;		at r = R;		at z = 0;
	at z = L;		where		and

The above finite difference equations are used to obtain temperature and moisture distributions inside the cylindrical product at different time periods. The grid independent tests are conducted to ensure the presence of grid independent results in the simulation. Stability analysis is performed in order to investigate the boundedness of the exact solution of the finite-difference equations using von Neumann’s method. The method introduces an initial line of errors as represented by a finite Fourier series and applies in a theoretical sense to initial value problems. The stability criterions obtained for the above finite difference equations are:

Thus the above conditions have to be satisfied in order to have a converged solution.

In this section temperature and moisture profiles obtained for the cylindrical product at different time periods are presented. The product considered in the simulation was a date of diameter 0.0084 m and length 0.02 m. Thermophysical properties and the drying conditions used in the simulation are listed in Table 1.

Figures 2-4 exhibit the temperature distribution in the cylindrical product for different time periods. Temperature in the cylinder increases as the time period progresses. This is because of the higher ambient temperature than the temperature of the product. Moreover, temperature distribution inside the product is non-uniform, giving an indication that the temperature dependent moisture diffusivity varies in the product, which in turn affects the rate of moisture diffusion in the product.

The moisture distributions inside a cylindrical product for different time periods are shown in Figures 5-7. The moisture content inside the cylinder reduces as the time period increases. The reduction rate of moisture content is higher in the surface region compared to the interior of the product. Moreover, in the early heating period moisture content reduces rapidly and as the heating period progresses the rate of reduction of moisture content becomes less, i.e., it reduces almost steadily with progressing heating period. This effect is more pronounced in the surface vicinity. The rapid drop of moisture content in the early heating period occurs because of the high moisture gradient in this region, which in turn derives considerable diffusion rates from the inside to the surface.

The predicted temperature and moisture distribution in a cylindrical product are compared with the experimental data taken from the literature and are shown in Figures 8 and 9. Experimental drying conditions and product properties are listed in Table 2. A high agreement is found between the numerical results and the experimental data taken from Simal et al. (1998).

Figure 2: 3-D plot of temperature distribution inside a cylindrical date after 60 s.

Figure 3: 3-D plot of temperature distribution inside a cylindrical date after 180 s.

Figure 4: 3-D plot of temperature distribution inside a cylindrical date after 300 s.

Figure 5: 3-D plot of moisture distribution inside a cylindrical date after 60 s.

Figure 6: 3-D plot of moisture distribution inside a cylindrical date after 180 s.

Figure 7: 3-D plot of moisture distribution inside a cylindrical date after 300 s.

Figure 8: Measured and calculated center temperature distributions in a cylindrical broccoli.

Figure 9: Measured and calculated centerline moisture distributions in a cylindrical broccoli.

Table 1: Thermophysical properties and drying conditions used in the simulation.

k	0.337 W/mK from Dincer (1997)
	1319 kg/m3 from Dincer (1997)
cp	(0.837+1.256M)1000J/kgK from Dincer (1997)
h	25-250 W/m2K
hm	0.0001 m/s
Ti	298 K
Td	323 K
Mi	3 kg/kg (db)
RH	0.42 kg/kg (db)

Table 2: Experimental drying conditions and product properties.

Product	Broccoli cylinder
Size	Diameter 0.007 m and length 0.02 m
Ti	298 K
Td	333 K
Mi	9.57 kg/kg (db)
RH	1.18 kg/kg (db)
k	0.148+0.493Mi (W/mK) from Dincer (1997)
	2195.27 kg/m3 from Dincer (1997)
cp	(0.837+1.256Mi) 1000 (J/kgK) from Dincer (1997)
Reference	Simal et al. (1998)

In this paper, the numerical solution for the temperature and moisture distribution inside a two-dimensional cylindrical product subject to drying under convective boundary conditions at the surface is presented. It is found that the temperature rises rapidly in the early heating period and as the heating period progresses, the rise of temperature becomes almost steady with advancing heating period. The moisture gradient is higher in the early heating period and as heating progresses, the moisture gradient remains almost steady. In addition, validation of the results obtained from the present analysis is performed with experimental data available in the literature. A considerably good agreement is found between the model results and measured values for the temperature and moisture distributions inside the product.

The authors acknowledge the support provided by King Fahd University of Petroleum and Minerals for this work under the KFUPM research grant # ME/ENERGY/203.

A constant		M dry-base moisture content (kg/kg)
B constant		M* reduced (dimensionless) moisture content
cp specific heat capacity at constant pressure (J/kg K)		n number of mesh points of the numerical grid in the r-direction
D moisture diffusivity (m2/s)		r radial coordinate
Do pre-exponential factor of Arrhenius equation (m2/s)		R radius of cylindrical product (m)
h heat transfer coefficient (W/m2K)		RH relative humidity (kg/kg)
hm moisture transfer coefficient (m/s)		t time (s)
k thermal conductivity (W/mK)		T temperature (K)
L length, m		T* dimensionless temperature
m number of mesh points of the numerical grid in the z- direction		z coordinate
Greek Symbols a thermal diffusivity (m2/s) density (kg/m3)		Subscripts d drying air i initial

[1] Akiyama, T., Liu, H. and Hayakawa, K., (1997). Hygrostress multi-crack formation and propagation in cylindrical viscoelastic food undergoing heat and moisture transfer processes, Int. J. Heat & Mass Transfer 40(7), 1601-1609.
[2] Dincer, I., Sahin, A.Z., Yilbas, B.S., Al- Farayedhi, A.A. and Hussain, M.M (2000). Exergy and energy analysis of food drying systems. Progress Report 2, KFUPM Project # ME/ENERGY/203.
[3] Dincer, I. (1997). Heat Transfer in Food Cooling Applications, Taylor & Francis, Washington, DC.
[4] Dincer, I. and Dost, S. (1996). A modeling study for moisture diffusivities and moisture transfer coefficients in drying of solid products, International Journal of Energy Research 20, 531-539.
[5] Luikov, A. V. (1973), Systems of differential equation of heat and mass transfer in capillary porous, Int. Journal of Heat and Mass Transfer, 18, 1-14.
[6] Mukherjee, D., Puri, V. M., and Anantheswaran, R. C., (1997), Measurement of coupled heat and moisture transfer coefficients for selected vegetables, Drying Technology, 15(1), 71-94.
[7] Perre, P., Moser, M. and Martin, M. (1993). Advances in transport phenomena during convective drying with superheated steam and moist air, Int. J. Heat & Mass Transfer 36(11), 2725-2746.
[8] Philip, J. R. and De Vries, D. A. (1957), Moisture movement in porous materials under temperature gradients, Trans. Amer. Geophys. Unin., 222-232.
[9] Rogers, J.A and Kaviany, M. (1990). Variation of heat and mass transfer coefficients during drying of granular beds, Transactions of the ASME 112, 668-674.
[10] Ruiz-Cabrera, M.A., Salgado-Cervantes, M.A., Waliszewski-Kubiak, K.N. and Garcia Alvarado, M.A. (1997). The effect of path diffusion on the effective moisture diffusitivity in carrot slabs, Drying Technology 15(1), 169-181.
[11] Simal, S., Rosello, C., Berna, A., and Mulet, A., (1998), Drying of shrinking cylinder-shaped bodies, Journal of Food Engineering, 37, 423-435
[12] Ssu-Hsueh, S. and Marrero, T.R. (1996). Experimental study of simultaneous heat and moisture transfer around single short porous cylinders using convection drying by a psychrometry method, Int. J. Heat & Mass Transfer 39(17), 3559-3565.

Contact details: Ibrahim Dincer, Ph.D.
Professor
School of Manufacturing Engineering
University of Ontario Institute of Technology
2000 Simcoe Street North
Oshawa, Ontario L1H 7K4, Canada
www.uoit.ca
Phone: 905-721-3209 or 721-3111 (Ex: 2573)
Fax: 905-721-3140
E-mail: Ibrahim.Dincer@uoit.ca
Web Site: www.geocities.com/ibrahimdincer

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