MODEL OF A PROCESS FOR DRYING Eucalyptus spp AT HIGH TEMPERATURES

MODEL OF A PROCESS FOR DRYING Eucalyptus spp AT HIGH TEMPERATURES

P. C. C. PINHEIRO¹, T. J. RAAD¹ and M. I. YOSHIDA²

¹Departamento de Engenharia Mecânica, UFMG, 31270-901 Belo Horizonte, MG, Brazil

e-mail: [e-mail], e-mail: [e-mail]

²Departamento de Química, Icex, UFMG - Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazil

(Received: May 14, 1998; Accepted: October 13, 1998)

Abstract - A mathematical model of a process for drying of Eucalyptus spp is presented. This model was based on fundamental heat and mass transfer equations and it was numerically solved using a segregated finite volume method. Software in the FORTRAN language was developed to solve the mathematical model. The kinetic parameters of drying for Eucalyptus spp were experimentally obtained by isothermal thermogravimetry (TG). The theoretical results generated using the mathematical model were validated by experimental data.

Keywords: Diffusion mechanism, heat transfer, kinetic, mathematical model, numerical method, porous media, thermogravimetry, wood.

INTRODUCTION

The process of drying wood has been studied by several researchers to explain the behaviour and nature of free water, bound water and water vapor movement through porous structure of the wood. These studies have provided information about the behaviour of physical properties of the material such as: cell-wall structure, thermal conductivity, permeability, the diffusion coefficient and others that depend on moisture content. However, considering that there are several species of wood, data on specific woods are scarce.

In the present work, a two-dimensional mathematical model was developed to simulate an isothermal process for drying of wood. It was validated by experimental data obtained from isothermal thermogravimetric tests of Eucalyptus grandis, aimed at studying changes in wood properties during drying and also at providing a correlation between drying temperature and the effective diffusion coefficient of the species analysed by studying the influence of the original wood and the barked wood samples undergoing the drying process.

Because of the complexity of the model and its set of non-linear equations, the solution must be based on numerical techniques, which have become more and more popular due to the high processing velocity of computers. Several mathematical models of the heat and mass transfer process presented in the literature [Alves and Figueiredo (1989), Souza and Nebra (1996), Stanish et al (1986)] use numerical methods to solve the set of complex equations and most of them have employed the finite volume method for one-dimensional coordinates.

The studies concerned with biomass combustion and carbonization are always conducted at high temperatures, and they naturally include the use of a drying process prior to the other process. Thus, the present model was purposefully developed for an extensive temperature range (40 to 1000⁰C) and its intend is also to provide support to studies of these biomass processes.

MATHEMATICAL MODEL

Several simultaneous events occur during the drying of the wood at high temperatures [Alves and Figueiredo (1989)]: heat is conducted into the center of wood; the sample starts to dry, especially at the outer edges, where the temperature is higher; bound and free water moves outwards by capillarity and diffusion; and water vapor moves outwards by convection and diffusion.

The theoretical model proposed in this work is based on fundamental equations, which describe the simultaneous heat and mass transfer in porous media. Physical considerations of the model are: I - the solid phase is inert and rigid, i.e., sample volume is constant during the process; II - the liquid phase is regarded as continuous, i.e., the free water is not separated from the bound water; III - considering that the specific heat transfer area between phases is considerably large due to the structural characteristics of the lignocellulosic matrix, it is possible to assume the existence of a thermodynamic quasi-equilibrium state, where the temperatures of gaseous, liquid and solid phases are equal, i.e., T_S = T_L = T_G = T, [Berger and Pei (1973)]; IV - the properties of each phase taken as constant during drying are: solid thermal conductivity, solid specific heat and directional permeabilities; V - the heat and mass convective transport which develop due to the increase in internal pressure induced by drying are described by the Darcy equation for porous medium flow; VI - viscous effects can be neglected due to the small magnitude of the flows involved [Min and Emmonds (1972)]; VII - considering that the wood is composed of a fine porous structure, mainly of unconnected pores, migration of moisture occurs by liquid diffusion and capillary liquid flow may be neglected.

The control volume and the boundary conditions where the balance equations will be applied are shown in Figure 1.

Liquid Mass Balance

(1)

where is the liquid to total volume ratio (m³ liq/m³ sol), r _L is the density of the liquid (kg/m³), t is time (s), is the velocity vector of the liquid (m/s) and is the rate of mass evaporation per unit volume (kg/m³.s); and the terms of equation 1 are: storage mass of liquid in the control volume, flow of liquid mass through the system and rate of volumetric evaporation of the liquid/vapor phase change.

Water Vapor Mass Balance

(2)

(3)

where is the sample porosity (m³ of empty pores)/(m³ solid), X_Vis the concentration of vapor in the air/vapor mixture (kg vapor/kg air-vapor mixture), is the density of the gas mixture (kg/m³), is the velocity vector of the gas mixture (m/s), is the diffusive term of the vapor concentration (kg/m².s) and D_EFF is the effective diffusion coefficient of gas (m²/s); and the terms of equation 2 are: storage of vapor mass in the control volume, flow of vapor mass through the system by convection and diffusion and rate of volumetric evaporation of liquid/vapor phase change.

Air Mass Balance

(4)

where X_A is the concentration of air in the air/vapor mixture (kg air/kg air-vapor mixture); and the terms of equation 4 are: storage of air mass in the control volume and flow of air mass through the system by convection and diffusion.

Figure 1:
Physical Domain - Control volume and boundary conditions.

Liquid Momentum Equation (Darcy’s Law)

(5)

where is the directional permeability of the liquid (m²), is the dynamic viscosity of the liquid (N.s/m²), P_G is the pressure of the gas phase (Pa) and P_C is the capillary pressure of the liquid (Pa). The pressure gradient was neglected, since no liquid movement was regarded in the present model.

Water Vapor-Air Mixture Momentum Equation (Darcy’s Law)

(6)

where is the directional permeability of the gas phase (m²) and is the dynamic viscosity of the gas phase (N.s/m²).

Thermal Balance

The thermal balance is governed by equation 7, where T is the temperature of the system (K), is the density of the solids (kg/m³), Cp_S is the specific heat of the solids (kJ/kg.K), Cp_L is the specific heat of the liquid (kJ/kg.K), Cp_G is the specific heat of the gas phase (kJ/kg.K), k_E is the equivalent thermal conductivity (kJ/m.s.K) and is the vaporization enthalpy of the liquid (kJ/kg); and the terms of equation 7 are: energy storage in the control volume, energy that moves through the system by diffusion, energy that moves through the system by convection, energy dissipation due to pressure change and heat necessary for liquid/vapor phase change.

(7)

Thermodynamic Equilibrium - Vapor Mass Fraction

In order to attain thermal equilibrium between the liquid and vapor phases, the vapor mass fraction should be such that the partial pressure of the vapor (P’_V) should be equal to its saturation pressure (P_VS) at temperature of the mixture. Therefore, in wood drying, the concentration of vapor in the air/vapor mixture inside the pores can be obtained by thermodynamic relations. According to Dalton’s Law of Additive Pressure applied to the air/vapor mixture, one can show that:

(8)

(9)

(10)

(11)

Combining equations 8 to 11, one can obtain:

(12)

where is the density of the vapor (kg/m³), is the density of the air (kg/m³), P’_V is the partial pressure of the vapor (Pa), P_VS is the saturation pressure of the vapor (Pa), R_V is the vapor equivalent constant (kJ/kg.K) and R_A is the air equivalent constant (kJ/kg.K).

Mass Rate of Evaporation

The mass rate of evaporation was obtained in two different ways, as follows:

Firstly, the mass rate of evaporation was expressed explicitly by taking it from the water vapor mass balance (equation 2), since vapor concentration is given by equation 12.

(13)

Secondly, an equation to compute the mass rate of evaporation can be derived with a combination of the liquid mass balance (equation 1) with a first-order-Arrhenius type equation. From the general kinetic equation:

(14)

(15)

(16)

where a is the evaporated mass fraction (kg/kg), k is the drying rate (s^-1), f(a ) is a kinetic mechanism function (kg/kg), e ₀ is the initial liquid to total volume ratio (m³ liq/m³ sol), T_SUR is the temperature of evaporation (K), E is the activation energy (kJ/kmol), A is the Arrhenius equation pre-exponential factor (s^-1) and R is the universal gas constant (8.314kJ/kg.K).

The kinetics of drying process were determined by a mathematical correlation between the kinetic equations and the isothermal TG data analysis. Software was developed in to analyse TG experimental data in order to find the most probable kinetic mechanism for the drying of wood. For all tests, the kinetic mechanism most likely to occur is F1, which is a first-order mechanism described by the unimolecular decay law. This expression gives the kinetic function as f(a ) = 1-a . Combining this function and equations 14 to 16 and comparing the result with equation 1 and neglecting the convective term, one can arrive at:

(17)

Drying Kinetic Mechanism Coupling

Hann [1964] has shown that the drying of porous material occurs in two regimes: (1) an initial regime during which the drying rate is constant and evaporation occurs at the surface and in a manner similar to that of a free liquid surface and (2) a falling rate regime, where the porous body offers additional resistance to moisture movement. The model simulates these two mechanisms by coupling the mass rate of evaporation , computed by mass balance equations 1 to 4 and vapor mass fraction (equation 12), with the mass rate of evaporation computed by equation 17, which represents these two regimes.

To obtain the necessary kinetic parameters to compute the drying rate, which represents the falling rate period, the following procedure was used:

(18)

Using experimental data from two thermogravimetric curves for wood samples and their respective evaporation temperatures, one can obtain kinetic parameters E and A. Then, any other evaporation temperature needed to simulate the wood drying process can be calculated by the energy balance (equation 7).

The coupling between the two drying mechanisms must be done in a manner that assures process continuity. Thus, drying rate k was used as a continuity parameter at the change in mechanism. During the simulation process a parameter k₁ is calculated using the following relationship:

(19)

where is the mass rate of evaporation computed by equation 13, (kg/m³.s). The change in mechanism will occur when the two parameter values, k and k₁, become equal. At this moment, the mass rate of evaporation can no longer be computed by balance equation 13, and it is now computed by equation 17. Thereafter, vapor concentration X_V can no longer be evaluated by equation 12 and it is now calculated by vapor balance equation 2.

In the same way, the density of the gas mixture has to follow a continuity process. So, for the first drying mechanism, it will be obtained by equation 8 and for the second drying mechanism as follows:

Using thermodynamic relations, according to Amagat’s law of additive volumes, under the same absolute pressure,

(20)

(21)

(22)

(23)

(24)

(25)

where m_V is the water vapor mass (kg), m_A is the air mass (kg), m_T is the total vapor-air mixture mass (kg), V_V is the volume filled by vapor (m³), V_A is the volume filled by air (m³), V_G is the volume filled by the vapor-air mixture (m³) and V_S is the total volume of the solids.

Solving the set of algebraic equations 20 to 25, one can obtain the vapor-air mixture density:

(26)

(27)

(28)

Equivalent Thermal Conductivity

It is necessary to determine the equivalent value of the thermal conductivity of the material as a whole, since no phase separation was considered in the overall energy equation. The equation proposed by Brailsford and Major [1963] was used to achieve the equivalent thermal conductivity of materials k_E, composed of a continuous medium with a uniform disperse phase. It is expressed as follows in equation 29.

(29)

(30)

where k_S is the thermal conductivity of the solid (kW/m.K), k_L is the thermal conductivity of the liquid (kW/m.K) and k_G is the equivalent thermal conductivity of the gas (kW/m.K).

Effective Diffusion Coefficient Equation

The binary bulk diffusivity D_AV of air-water vapor mixtures is given by [Stanish et al (1986)]:

(31)

where T_REFis the oven drying temperature (K) and P_ATM is the atmospheric pressure (1x10⁵ Pa).

In wood, factor a _F is used to account for closed pores resulting from the cellular nature of the solid, which would increase gas outflow resistance, so the equation of effective diffusion coefficient D_EFF for wood drying is:

(32)

Convective Heat Transfer Coefficient Equation

Since the experimental tests were done at a very low air flow velocity (3.33x10^-6 m³/s), in the present model the free convection mechanism was used to evaluate the convective heat transfer coefficient. In natural convection inside an annular free space between concentric vertical cylinders, where viscous strengths overcome buoyant forces, maintaining the flow practically stationary, the Nusselt number Nud is equal to one. Therefore, the convective heat transfer coefficient equation can be expressed as [Özisik (1985)]:

(33)

where h is the heat transfer coefficient (W/m².K), d is the annular free space between concentric cylinders (m) and k is the thermal conductivity of the air (W/m.K).

Convective Mass Transfer Coefficient Equation

The Chilton-Colburn analogy equation was used to compute the convective mass transfer coefficient [Bird et al (1960)]:

(34)

where h_M is the convective mass transfer coefficient (kg/m².s), Pr is the Prandtl number and Sc is the Schmidt number.

(35)

(36)

Absolute Pressure

The absolute pressure of the gas phase is achieved by the Simplec method of velocity-density-pressure coupling. This method consists in coupling the continuity equation (equations 2 and 4), gas momentum equation 6 and ideal gas equation of state 8, converted into their discrete counterparts. The coupling algorithm is detailed by Maliska [1995].

Thermodynamic and Transport Property Equations

The physical properties of air, water and vapor that depend on temperature are represented in the present model by polynomial equations (Table 1) [Pakowski et al (1991)]:

Table 1:
Transport property equations [Pakowski et al (1991)]

Equation	Coefficient						Temp rang	Eq N0.
	a0	a1	a2	a3	a4	a5
	2.8223	1.1828 x10-2	-3.5047 x10-5	3.6010 x10-8			0 to 3500C	(37)
	1.8830	-0.1674 x10-3	0.8439 x10-6	-0.2697 x10-9			20 to 12000C	(38)
	1.0093	-4.0403 x10-5	6.1760 x10-7	-4.0972 x10-10			-40 to 10000C	(39)
	0.3838	5.2533 x10-3	-6.3689 x10-6				0.01 to 3500C	(40)
	1.7153 x10-2	1.9569 x10-4	-3.3839 x10-6	3.1203 x10-8	-1.1539 x10-10	1.6104 x10-13	0.01 to 3000C	(41)
	2.4250 x10-2	7.8891 x10-5	-1.7903 x10-8	-8.5705 x10-12			-40 to 10000C	(42)
	267.155	374.2	0.38				0.01 to 3500C	(43)
	133.322	18.3036	-3816.4	-46.13			0.01 to 3500C	(44)

The physical properties are defined as follows: Cp_L is the specific heat of the liquid (kJ/kg.K), Cp_V is the specific heat of the vapor (kJ/kg.K), Cp_A is the specific heat of the air (kJ/kg.K), k_Lis the thermal conductivity of the liquid (W/m.K), k_V is the thermal conductivity of the vapor (W/m.K), k_A is the thermal conductivity of the air (W/m.K), DH_Vis the latent heat of evaporation (kJ/kg) and P_VS is the saturation pressure of the vapor (Pa).

Initial and Boundary Conditions

The following equations are used in a two-dimensional model applicable to cylindrical coordinates and axis-symmetric geometry (Table 2).

Table 2:
Boundary conditions equations

For t = 0, " r, " x	For r = 0, " t	For r= R, x = 0, x = L , t > 0
	-	-

where r is the radial linear dimension (m), x is the axial linear dimension (m), R is the control volume ray, L is the sample height, is the unit

vector normal to the surface, T¥ is the ambient temperature (300K), U₀ is the initial moisture (kg liq/kg sol), X_REF is the vapor mass fraction of the drying agent (0.03 kg vapor/kg mixture) and X_V0 is the initial vapor concentration.

Model Constant Parameters

(Table 3)

Table 3:
Model constant parameters

Notation	Parameter	Unit	Value	Reference
a G	gas permeability - x direction	(m2)	1.0x10-10	Alves and Figueiredo (1989)
	gas permeability - r direction	(m2)	1.0x10-13
m G	dynamic viscosity of the gas	(N.s/m2)	2.81x10-4	Gieck and Gieck (1990)
kS	thermal conductivity of the solids	(kW/m.K)	1.59x10-4	Gieck and Gieck (1990)
RV	vapor equivalent constant	(kJ/kg.K)	0.462	Gieck and Gieck (1990)
RA	air equivalent constant	(kJ/kg.K)	0.287	Gieck and Gieck (1990)
e	solids porosity	(m3 air/m3sol)	0.64	Gieck and Gieck (1990)
l	solids emissivity	-	0.95	Gieck and Gieck (1990)
s	Stefan-Boltzman constant	(kW/m2.K4)	5.673x10-11	Perry and Green (1984)

NUMERICAL METHOD

The two-dimensional model described was numerically solved, in an iterative way, due to high non-linearities, by the Finite Volume Method. The solution domain used is defined by axis-symmetric geometry (Figure 2).

Figure 2: Numerical domain and elementary volume.

The balance equations (1, 2, 4 and 7) can be written in a general form for variable f , in cylindrical coordinates. All terms used in the mathematical model’s differential equations are represented by the following general transport equation:

(45)

where f is the transport property, r is the mass density associated with the transport property, l is the transient term parameter, G is the diffusive coefficient of the transport property, u and v are axial and radial velocity associated with the transport property, t is time, x is the axial linear dimension, r is the radial linear dimension and S is the source term associated with the transport property.

The general equation terms (45) are compared with the respective parameters of physical equations 1, 2, 4 and 7 (Table 4).

Integrating each term of equation 45 into the 2p r perimeter, in space and time for the elementary volume shown in Figure 2, where the transport property does not vary in q direction (D q = 1 radian), one can arrive in equation 46.

Table 4:
General equation parameters

Eq n0	Mathematical Model Physical Equations	General Equation Parameters
		f	l	G	Sf
1
2
4
7

where ; ; represents elementary volume (m³), r_P is the radial linear distance between r = 0 and point P (m), is the transport property calculated in prior time t; ;; ; ; represents the control volume interface area in the axial direction (m²); is the control volume interface area n (m²); is the control volume interface area s (m²); r_n is the radial linear distance between r = 0 and side n (m); r_s is the radial linear distance between r = 0 and side s (m); and D_i represents the boundary diffusive terms where: , , and . The source term has the linear form , where S_P represents the coefficient of f _P and S_C is the term of which does not depend explicitly on f _P. To guarantee positive coefficients and to avoid numerical solution divergence, the value of S_P must always be negative [Patankar (1980)].

Using the interpolation upwind scheme to evaluate the convective flow and the central difference to evaluate the diffusive flow at the elementary volume boundaries, one can arrive in equation 47.

where coefficient a _i is defined by a _i = 0.5 for and a _i = -0.5 for .

Rearranging the terms of equation 47, one can obtain the discretizated general equation

(48)

where the coefficients are given by:

(46)

(47)

; ;

;

;

;

The boundary conditions used are given in Table 2. A software in the FORTRAN language was specially developed to solve the set of equations. To determine the numerical mesh and to evaluate the differential equations, Conduct’s [Patankar (1991)] software subroutines were used. After a convergence test in time and space of a very refined mesh, a 99-volume mesh was chosen. Temperature, density and absolute pressure are located at the middle of the control volume and the variable velocity are located at the control volume interface (Figure 3). A PC-Pentium-133 MHz computer was used to run the software and it took about ten minutes to obtain the desired results for each sample analysed.

Figure 3:
Numerical mesh and variable locations.

EXPERIMENTAL TESTS

Recently cut, cylindrical green wood samples of Eucalyptus spp were submitted to isothermal thermogravimetry tests. The samples were divided in three groups: the first and third groups were kept as cut and the second was carefully barked. This procedure was followed to observe the wood bark influence on the drying process. All groups of samples were dried in a METTLER TA4000 thermobalance. The maximum capacity of the oven’s inner dimensions was used as a geometric parameter to determine the diameter and height of each sample. The drying tests were carried out in a dry air atmosphere, at a flow of 3.33x10^-6 m³/s and at oven isothermal temperatures in the range of 75 to 225⁰C. Wood density was measured according to the MB-1269/79-ABNT norm. A thermocouple set inside the wood was used to measure the temperature of the third group samples. The experimental and calculated data are listed on Tables 5 to 7.

TREF D(mm) L(mm) mi(g) mf(g) U0(%) r S TSUR h DAV.PG a F[x] a F[r] 75 8.65 13.30 0.6881 0.5795 18.74 742.4 50.4 14.0 3.36 0.80 0.20 100 8.80 13.30 0.7177 0.6000 19.62 742.4 60.2 15.2 3.80 0.80 0.20 125 8.93 13.70 0.7604 0.6365 19.46 742.4 68.0 16.4 4.25 0.80 0.20 150 8.40 12.93 0.6342 0.5318 19.26 742.4 76.0 15.2 4.73 0.80 0.20 175 8.70 13.00 0.6903 0.5743 20.20 742.4 78.4 17.0 5.23 0.80 0.20 200 8.52 13.45 0.6910 0.5688 21.48 742.4 81.9 16.9 5.75 0.80 0.20 225 8.60 13.05 0.6848 0.5636 21.50 742.4 84.9 17.4 6.30 0.80 0.20

TREF	D(mm)	L(mm)	mi(g)	mf(g)	U0(%)	r S	TSUR	h	DAV.PG	a F[x]	a F[r]
75	8.65	13.30	0.6881	0.5795	18.74	742.4	50.4	14.0	3.36	0.80	0.20
100	8.80	13.30	0.7177	0.6000	19.62	742.4	60.2	15.2	3.80	0.80	0.20
125	8.93	13.70	0.7604	0.6365	19.46	742.4	68.0	16.4	4.25	0.80	0.20
150	8.40	12.93	0.6342	0.5318	19.26	742.4	76.0	15.2	4.73	0.80	0.20
175	8.70	13.00	0.6903	0.5743	20.20	742.4	78.4	17.0	5.23	0.80	0.20
200	8.52	13.45	0.6910	0.5688	21.48	742.4	81.9	16.9	5.75	0.80	0.20
225	8.60	13.05	0.6848	0.5636	21.50	742.4	84.9	17.4	6.30	0.80	0.20

Table 5: Experimental and calculated data on original samples - 1^st group

TREF D(mm) L(mm) mi(g) mf(g) U0(%) r S TSUR h DAV.PG a F[x] a F[r] 75 8.98 12.30 0.9526 0.5417 75.86 694.7 49.0 15.1 3.36 0.85 0.85 100 9.33 12.75 1.0739 0.6051 77.47 694.7 54.9 17.4 3.80 0.85 0.85 125 9.02 12.50 0.9947 0.5545 79.38 694.7 60.8 16.8 4.25 0.85 0.85 150 9.42 12.60 1.0992 0.6100 80.21 694.7 64.9 19.5 4.73 0.85 0.85 175 9.19 12.43 1.0463 0.5727 82.69 694.7 68.5 19.1 5.23 0.85 0.85 200 9.53 12.65 1.1312 0.6264 80.59 694.7 71.2 21.8 5.75 0.85 0.85 225 8.89 13.34 1.0131 0.5756 82.05 694.7 73.3 19.1 6.30 0.85 0.85

TREF	D(mm)	L(mm)	mi(g)	mf(g)	U0(%)	r S	TSUR	h	DAV.PG	a F[x]	a F[r]
75	8.98	12.30	0.9526	0.5417	75.86	694.7	49.0	15.1	3.36	0.85	0.85
100	9.33	12.75	1.0739	0.6051	77.47	694.7	54.9	17.4	3.80	0.85	0.85
125	9.02	12.50	0.9947	0.5545	79.38	694.7	60.8	16.8	4.25	0.85	0.85
150	9.42	12.60	1.0992	0.6100	80.21	694.7	64.9	19.5	4.73	0.85	0.85
175	9.19	12.43	1.0463	0.5727	82.69	694.7	68.5	19.1	5.23	0.85	0.85
200	9.53	12.65	1.1312	0.6264	80.59	694.7	71.2	21.8	5.75	0.85	0.85
225	8.89	13.34	1.0131	0.5756	82.05	694.7	73.3	19.1	6.30	0.85	0.85

Table 6: Experimental and calculated data on barked samples - 2^nd group

TREF D(mm) L(mm) mi(g) mf(g) U0(%) r S TSUR h DAV.PG a F[x] a F[r] *115 10.82 9.63 0.9768 0.4697 108.0 530.9 73.0 - - - - *140 10.32 8.30 0.7768 0.3687 110.7 530.9 80.0 - - - - 115 10.46 9.20 0.8629 0.4197 105.6 530.9 72.5 25.8 4.07 0.75 0.10 140 10.44 7.55 0.6791 0.3430 98.0 530.9 79.2 26.7 4.54 0.75 0.10

TREF	D(mm)	L(mm)	mi(g)	mf(g)	U0(%)	r S	TSUR	h	DAV.PG	a F[x]	a F[r]
*115	10.82	9.63	0.9768	0.4697	108.0	530.9	73.0	-	-	-	-
*140	10.32	8.30	0.7768	0.3687	110.7	530.9	80.0	-	-	-	-
115	10.46	9.20	0.8629	0.4197	105.6	530.9	72.5	25.8	4.07	0.75	0.10
140	10.44	7.55	0.6791	0.3430	98.0	530.9	79.2	26.7	4.54	0.75	0.10

Table 7: Experimental and calculated data on wood samples - 3^rd group

(*) The drying temperatures

T_REF of these tests were submitted to evaporation temperature measurements by a thermocouple set inside the samples, and the last two were submitted to thermogravimetric tests.

Table 8: Kinetic parameters of 1

Sample Group	Drying Temperature (0C)	Evaporation Temperature (0C)	Drying Rate k (s-1) x 103	Activation Energy E (kJ/mol)	Pre-exponential factor A (s-1)
1	100	60.2	1.412	53.78	3.776 x 105
	125	68.0	2.205
2	100	55.0	1.786	56.44	1.742 x 106
	125	60.7	2.805
3	115	72.5	1.748	61.70	3.485 x 106
	140	79.2	2.672

RESULTS AND DISCUSSION

Wood Drying Kinetics

The following parameters are necessary to simulate the wood drying process described by this model:

- to obtain the kinetic parameters Activation Energy E and pre-exponential factor A by solving the linear equation 18, using two thermogravimetric curves of cylindrical wood samples to obtain drying rate k by mathematical correlation and their respective evaporation temperatures T_SUR (see Figure 4). The results for the three sample groups are shown in Table 8;

- to obtain the effective diffusion coefficient (correction factor a ) of wood analysed by adjustment of theoretical and experimental kinetic curves by mathematical correlation;

- once have been obtained correction factor a and kinetic parameters E and A, the software can provide the evaporation temperature by the iterative method, using balance energy equation 7 for each simulation of the wood drying process;

- to provide the other input data: drying temperature desired for simulation T_REF (K), vapor concentration in the flow of the drying agent X_REF (kg vapor/kg mixture), heat transfer coefficient h (kW/m².K), sample geometry ray r (m), height L (m), dry basis initial moisture U₀ (kg water/kg solids), solid density r _S (kg/m³), permeability in both the axial and the radial directions a _G (m²), the solids porosity e and ambient temperature T¥ (K).

The kinetic diffusion mechanism F1 means that, in spite of the fact that the drying process takes place at high isothermal oven temperatures, on the moving drying surfaces of the samples, the water vaporises at temperatures below the boiling point of water, at atmosphere pressure, in the temperature range and under the experimental condition studied. Otherwise, the driving force of drying of the effective diffusion coefficient is directly affected by the isothermal oven temperature.

Simulation of Wood Drying

The theoretical drying curves obtained by the model and the experimental curves obtained by isothermal thermogravimetry (TG) are shown in Figures 5 and 6.

Figure 4: Experimental temperature of evaporation of the 3^rd sample group

Figure 5: Kinetic curves of experimental and model data for original wood - 1^st sample group

Figure 6: Kinetic curves of experimental and model data for barked wood - 2^nd sample group

Figure 7: Kinetic curve and internal temperature profile for the 3^rd sample group.

The two kinetic mechanisms, constant drying rate and first-order decay drying rate, can be easily observed in the set of curves. The fit between theoretical data and experimental data is very good, which validates the model and its physical assumptions.

Wood is an anisotropic material with much varied physical properties. As a consequence, even though a lot of research on simulation of drying of porous media has been carried out [Bramhall (1977), Spolek and Plumb (1980)], the complete validation of these models is very difficult. By the model proposed and validated by experimental tests, one can see that the drying mechanism is strongly influenced by parameters such as permeability and effective diffusion coefficients.

The unknown effective diffusion coefficient of vapor for Eucalyptus spp under different drying temperatures was determined by adjustment of the model’s theoretical alpha correction factor and experimental data. The values of the a factor for the two experiments, with original wood samples and barked wood samples, showed that wood bark has a big influence on the drying process. The water vapor gas flow encountered heavy resistance by the wood bark due to its compact cell structure and most probably to the much larger number of blocked pores present in the bark than in the core and alburnum of the wood.

Internal Temperature Profile for the Wood

The theoretical and experimental internal temperature profile and respective kinetic curve of one of the third sample group is shown in Figure 7. The main characteristics of the temperature profile are a flat temperature profile due to a constant drying rate representing the first kinetic mechanism and a continuous and increasing period where the temperature tends to reach the oven drying temperature, representing the second kinetic mechanism. The main experimental parameters that influence the evaporation temperature profile during the drying process are temperature, flow velocity and relative humidity of the drying agent on the sample. The evaporation temperature is directly proportional to parameter changes in the drying agent. Thus, the greater the drying flow velocity, the greater the heat change by convection between the drying agent and the sample and, thereafter, the flatter the temperature profile.

CONCLUSION

A model of a process for drying Eucalyptus spp was introduced with the goal of attaining a better understanding of the mechanism and its influence on several parameters that control the process. This model was validated by isothermal thermogravimetric tests, and a very good agreement was observed.

The simulation of wood drying in a wide range of temperatures can be done with essentially two experimental thermogravimetric tests and their respective evaporation temperatures and wood properties as input data. The tests must be done at temperatures below 200⁰C where the model gives good precise output data since wood combustion and carbonization do not occur. However, the simulation of drying can be applied at very high temperatures such as 1000⁰C, since a coupling between drying and wood carbonization models can be done. In addition, a complete wood drying/carbonization model can improve studies concerned with quality and energy optimization in charcoal formation.

ACKNOWLEDGEMENT

The authors would like to acknowledge the financial support received from CAPES and FAPEMIG.

• Alves, S.S., Figueiredo, J.L., A Model for Pyrolysis of Wet Wood, Chemical Engineering Science, 44(12), pp. 2861-2869 (1989).
• Berger, D., Pei, D.C.T., Drying of Hygroscopic Capillary Porous Solids - A Theoretical Approach, International Journal of Heat and Mass Transfer, 16, pp. 293-302 (1973).
• Bird, R.B., Stewart, W.E., Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, New York, NY, 780pp (1960).
• Brailsford, A.D., Major, K.G., The Effect of Irradiation on the Electric Resistivity and Thermal Conductivity of Uranium. Journal Nucl. Mater, 8(2), pp. 241-247 (1963).
• Bramhall, G., Mathematical Model for Lumber Drying, I. Principles, II. The Model, Wood Science and Technology, 12, pp. 14-34 (1977).
• Gieck, K., Gieck, R., Engineering Formulas, 6th Ed. McGraw-Hill, New York (1990).
• Hann, R.A., Drying of Yellow-Poplar at Temperatures above 100^oC, Forest Prod. Journal, 6, pp.215-220 (1964).
• Maliska, C.R., Transferęncia de Calor e Mecânica dos Fluidos Computacional, LTC, Rio de Janeiro, RJ, Brazil, 424pp (1995).
• Min, K., Emmonds, K.W., Drying of Porous Media, Proc. Heat Transfer and Fluid Mechanics Inst., Stanford Univ. Press, pp. 2-18 (1972).
• Özisik, M.N., Heat Transfer, A Basic Approach, McGraw-Hill, Inc., New York, 661pp (1985).
• Pakowski, Z., Bartczak, Z., Strumillo, C., Stenström, S., Evaluation of Equations Approximating Thermodynamic and Transport Properties of Water, Steam and Air for Use in Cad of Drying Process. Drying Technology, 9(3), pp. 753-773 (1991).
• Patankar, J.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C., 197pp (1980).
• Patankar, J.V., Computation of Conduction and Duct Flow Heat Transfer, Innovative Research Inc., Maple Grove, MN, USA, 354pp (1991).
• Perry, R.H., Green, D., Perrys Chemical Engineers Handbook, 6th ed. McGraw-Hill, New York (1984).
• Souza, M.E.P., Nebra, S.A., Heat and Mass Transfer in the Drying of Wood, Proceedings of the 10^th International Drying Symposium IDC96, A, pp. 695-702 (1996).
• Spolek, G.A., Plumb, O.A., A Numerical Model of Heat and Mass Transport in Wood During Drying, Drying80, Proceedings of the 2^nd International Symposium, 2, pp. 84-92 (1980).
• Stanish, M.A., Schajer, G.S., Kayihan, F., Mathematical Model of Drying for Hygroscopic Porous Media, AICHE Journal, 32(8), pp. 1301-1311 (1986).

Publication Dates

• Publication in this collection
  07 Dec 1998
• Date of issue
  Dec 1998

History

• Received
  14 May 1998
• Accepted
  13 Oct 1998