On the Uncertainty of a Mathematical Model for Drying of a Wood

Article pubs.acs.org/EF

On the Uncertainty of a Mathematical Model for Drying of a Wood Particle Mehran Jalili,*,† Andrés Anca-Couce,†,‡ and Nico Zobel†,§ †

Institute of Energy Engineering, FG EVUR, Technische Universität Berlin, Sekr. RDH 9, Fasanenstrasse 89, 10623 Berlin, Germany Institute for Process and Particle Engineering, Graz University of Technology, Infeldgasse 21 b, 8010 Graz, Austria § Business Unit of Process and Plant Engineering, Fraunhofer Institute for Factory Operation and Automation IFF, Joseph-von-Fraunhofer-Strasse 1, 39106 Magdeburg, Germany ‡

ABSTRACT: The reliability of the predictions of a mathematical model is a prerequisite to employ it. The drying process of a pine wood particle is modeled using a quasi-continuous multiphase porous media model. Different values and expressions from the literature of nine physical properties and transport coefficients are listed in this study. Lower and upper bounds of these model variables are determined and applied to the model, and a bounded output of the model is calculated. This study shows that taking arbitrary values of the model variables from the literature may lead to approximately 16 times differences in the calculated drying time of a single wood particle. By global sensitivity analysis, the most effective variables are determined. The variations of the coefficients of water vapor transport, except gas relative permeability, have a high impact on the presented drying model. For reliability of the predictions of the drying model, at least the values of the gas intrinsic permeability and water vapor diffusivity as well as capillary pressure have to be determined by own measurements in the studied conditions. One can use any of the reported values or expressions of effective thermal conductivity and specific heat capacity of wood as well as gas relative permeability and bound water diffusivity for modeling drying processes of wood particles, because all of them produce similar model predictions.

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INTRODUCTION Drying is a very important stage in the processes involving energy utilization of biomass materials. An accurate description of the drying process is required for optimal design of dryers and reactors for pyrolysis, gasification, or incineration. Therefore, models describing the drying of wood particles are of high interest for industry and research. An appropriate drying model must be capable of providing reliable predictions for different operating conditions under which measurements are very complex and challenging with currently available techniques. Every drying model must be able to describe all relevant and effective physical and chemical phenomena occurring during the studied process. However, for the applicability of the model predictions in practical cases, reliable information regarding material properties and transport coefficients of governing equations is required. The advances in the field of transport models for drying of porous media and also their restrictions and applications have been reviewed by Vu.1 Fernandez and Howell2 and Jianmin and Fangtian3 surveyed and discussed various models dealing with the simulation of moisture migration during drying of wood. Perre4 showed how to define a relevant model for wood drying simulations and explained the validity of the different models in different applications, including low- and high-temperature drying. The drying models established most recently can accurately describe all of the physical and chemical phenomena taking place during wood drying. However, as Koumoutsakos,5 Plumb et al.,6 and Kang and Chung7 stated, the determination of model variables have received little attention thus far. In some studies,8−12 own experimental data of a few of the parameters of the drying model have been experimentally determined. In some other studies, however, most model © 2013 American Chemical Society

parameters are chosen from the literature without any experimental investigation if these selected values are really appropriate for the studied case.7 A method for quantifying the impact of model parameters on the model predictions is the sensitivity analysis. The methods of sensitivity analysis can be categorized into local and global. Local sensitivity methods based on model derivatives with respect to the parameters are commonly employed to study the role of model parameters in the model. However, when the model variables are very uncertain and cover a large range, the techniques of local sensitivity analysis are not very useful and global sensitivity analysis is used instead. In the global sensitivity analysis, the output variability is evaluated when the model variables can vary in their whole uncertainty ranges.13 Sensitivity analysis has been applied to wood-drying models in several previous works: Plumb et al.6 showed that their model for low-temperature drying of wood (i.e., the drying takes place below the normal boiling point of water) is very sensitive with respect to the intrinsic gas permeability. Perre et al.14 showed that the latter is also true for high-temperature drying of wood (which occurs at temperatures above the normal boiling point of water). Di Blasi15 and Fyhr and Rasmuson16 have analyzed the sensitivity of their models for high-temperature drying of woody biomass particles for a limited number of physical properties. A comprehensive sensitivity analysis has been conducted by Nassrallah and Perre17 on low-temperature drying of wood material. Most of these sensitivity analyses of wood drying models have been conducted by arbitrarily changing a model parameter, Received: June 21, 2013 Revised: October 18, 2013 Published: November 4, 2013 6705

dx.doi.org/10.1021/ef401156s | Energy Fuels 2013, 27, 6705−6717

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Whitaker’s theory, which was further developed by Perre and his coworkers.17,25 The macroscopic differential equations have been obtained by averaging the conservation laws through a volumeaveraging method. The resulting equations are in terms of average field quantities, and they are presented in a general form. The sum of volume fractions for all of the phases is equal to 1

irrespective of the relevant uncertainty of this parameter. Fyhr and Rasmuson16 reported that they considered the relevant uncertainties for the input parameters, but they did not state how they determined those relevant uncertainties. Gronli18 has presented an excellent review of empirical correlations and values for physical properties of wood, but he conducted the sensitivity analysis regarding the pyrolysis model and not the drying model. Couture et al.19 investigated the impact of different correlations for the relative permeabilities of gas and liquid on their model for lowtemperature drying of wood material. Thus far, no comprehensive global sensitivity analysis has been conducted that takes into account the whole range of uncertainty reported in the literature for the parameters of a wood drying model. The objectives of this study are 3-fold: (1) to review the different correlations that can be used for the physical properties of a species of softwood, e.g., pine wood, as well as for the transport coefficients, (2) to determine the range of predicted drying times obtained with different model variables, and (3) to conduct a global sensitivity analysis of the drying model in which the model parameters are changed not by an arbitrary value but by values according to the range of established correlations for the respective parameter. In this study, the impact of nine model parameters (effective thermal conductivity, specific heat capacity, gas and liquid intrinsic permeabilities, gas and liquid relative permeabilities, capillary pressure, bound water diffusivity, and effective diffusivity of water vapor) on the predictions of a model for high-temperature drying of wood are investigated. To achieve it, first the mathematical model for drying of a woody biomass particle is introduced. Then, the numerical solution procedure is outlined and validated. After that, the results are presented and discussed. Finally, the conclusions of this study are presented.

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εg + εs + εfw + εbw = 1

(1)

where the volume fraction of each phase is obtained by εγ = ⟨ργ⟩/ (⟨ργ⟩γ) and the averaged quantities are given by

⟨ϕ⟩ =

1 Λ

∫Λ

⟨ϕ⟩γ =

ϕ dΛ γ

1 Λγ

∫Λ

ϕ dΛ γ

(2)

The moisture content (dry basis), MC, neglecting the mass of water vapor, is given by

MC = M fw + Mbw =

⟨ρbw ⟩ + ⟨ρfw ⟩ ⟨ρSD ⟩

(3)

Mass conservation of the mixture of water vapor and air is given as

∂(εg⟨ρg ⟩g ) ∂t

+ ∇(⟨ρg ⟩g ⟨Vg⟩) = ⟨ωv̇ ⟩

(4)

where the density of the mixture is equal to the sum of densities of air and water vapor

⟨ρg ⟩g =

⟨ρi ⟩g

∑

(5)

i = v,a

The total gas pressure, using the state equation, is written as

⟨Pg⟩g =

⟨ρg ⟩g R⟨T ⟩ Mg

(6)

The molecular weight of the gas mixture, Mg, is given by −1 ⎛ ⟨ρi ⟩g ⎞ ⎜ ⎟ Mg = ⎜ ∑ g ⎟ ⎝ i = v,a ⟨ρg ⟩ Mi ⎠

MATHEMATICAL MODEL

During the drying process of a porous wood particle, heat and mass transfer as well as phase change (evaporation and recondensation) may occur simultaneously. This complex interplay of different phenomena has been modeled using a continuum approach based on Whitaker’s theory20 by several authors (for instance, see refs 4, 11, 17−19, 21, and 22). Nowadays, Whitaker’s theory is a well-known model to describe drying processes in porous media. Because this model is the most rigorously formulated and comprehensive model in this field,23 it has been selected in this paper. Therefore, in this study, a one-dimensional drying process inside a single wood particle has been modeled on the basis of the multiphase transport theory of Whitaker. The model considers water in three forms: free and bound water as well as water vapor. All mechanisms of heat and mass transfer including free water flow because of capillary forces, diffusion of bound water, water vapor, and air flow because of convection and diffusion, and heat transfer by convection, diffusion, and conduction are incorporated in the model. This current model is an extension of a previous particle model developed to model pyrolysis of a dry biomass particle.24 The main simplifying assumptions of this mathematical model are as follows: (1) Porous medium is considered to be onedimensional, homogeneous, and rigid. (2) Local thermal equilibrium exists between all of the phases (all of the phases are at the same temperature at each location). (3) Convection of free water and water vapor follows Darcy’s law. (4) The bound water phase moves through the solid phase by molecular diffusion. (5) All gas species including water vapor and air obey the ideal gas law. (6) There is binary diffusion in the mixture of water vapor and air. (7) The enthalpy for all phases is a linear function of the temperature. (8) No degradation of the solid phase occurs; i.e., the solid density is constant. Governing Equations. In this section, a full set of governing equations in drying of wood material is presented on the basis of

(7)

The mass conservation of water vapor is written as

∂(εg⟨ρv ⟩g ) ∂t

+ ∇⟨ρv Vv⟩ = ⟨ωv̇ ⟩

(8)

where

⎛ ⟨ρ ⟩g ⎞ ⟨ρv Vv⟩ = ⟨ρg ⟩g ⟨Vg⟩ − ⟨ρg ⟩g Dveff ∇⎜⎜ v g ⎟⎟ ⎝ ⟨ρg ⟩ ⎠

(9)

The mass conservation equation of free and bound water is written as ∂(⟨ρfw ⟩ + ⟨ρbw ⟩)

+ ∇(⟨ρfw Vfw⟩ + ⟨ρbw Vbw⟩) = −⟨ωv̇ ⟩

∂t

(10)

where ⟨ρfw Vfw⟩ = ρw ⟨Vfw⟩

(11)

⎛ ⟨ρ ⟩ ⎞ ⟨ρbw Vbw⟩ = −⟨ρSD ⟩D bw ∇⎜⎜ bw ⎟⎟ ⎝ ⟨ρSD ⟩ ⎠

(12)

The superficial gas-phase velocity and the superficial free-water velocity are calculated using Darcy’s law

⟨Vg⟩ = −

K gK rg

⟨Vfw⟩ = − 6706

μg

∇⟨Pg⟩g (13)

KlK rl ∇⟨Pw⟩w μw

(14)


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where w

g

⟨Pw⟩ = ⟨Pg⟩ − Pc μg ⟨ρg ⟩g =

∑

particle; that is, the velocities and the gradients of the dependent variables are equal to 0 at this point. The boundary conditions at the surface of the particle are written as follows:26

(15)

μi ⟨ρi ⟩g

(⟨ρfw Vfw⟩ + ⟨ρbw Vbw⟩ + ⟨ρv Vv⟩)n|surf

(16)

i = v,a

g g = (εg + εfw + εbw )β(⟨ρv ⟩surf − ⟨ρv ⟩∞ )

The energy equation in porous media, with the local thermal equilibrium assumption, is written as

(keff Δ⟨T ⟩) + ⟨ρfw Vfw⟩Δh v + ⟨ρbw Vbw ⟩Δhsorp)n|surf

∂⟨T ⟩ (C ps⟨ρSD ⟩ + C pw⟨ρfw ⟩ + C pw⟨ρbw ⟩ + C pgεg⟨ρg ⟩ ) ∂t g

= − α(⟨Tsurf ⟩ − ⟨T∞⟩) g g ⟨Pg⟩surf = ⟨Pg⟩∞ = Patm

+ (C pw⟨ρfw Vfw⟩ + C pw⟨ρbw Vbw⟩ + C pg⟨ρg Vg⟩ +

∑

(17)

where

Δhv = Δhw + Δhsorp Δhv = Δhw

(MC < M fsp)

(MC ≥ M fsp)

■

(18)

and the diffusion flux in the gas phase is given by

⎛ ⟨ρ ⟩g ⎞ ⟨ρi Ui⟩ = −⟨ρg ⟩g Dieff ∇⎜⎜ i g ⎟⎟ ⎝ ⟨ρg ⟩ ⎠

∑ i = v,a

C pi

(19)

⟨ρi ⟩g ⟨ρg ⟩g

(20)

The phase change, including evaporation and recondensation, can be formulated on the basis of the hypothesis that, inside the pores of the particle, water vapor is in phase equilibrium with free and bound water. The partial pressure of water vapor in the equilibrium state is proportional to the saturated vapor pressure, which can be calculated by

⟨Pveq⟩g = Pvsat(T ) ⟨Pveq⟩g

=

(MC > M fsp)

Pvsat(T )h(Mbw ,

T)

(MC ≤ M fsp)

(21) Psat v ,

is obtained by where the equilibrium or saturated vapor pressure, an empirical equation and the relative humidity, h(Mbw,T), is given by the sorption isotherm of wood. The phase change rate can be calculated by two methods based on the equilibrium assumption. In one method, it is assumed that the water vapor density in each point in the particle is equal to the equilibrium water vapor density of that point, which can be calculated using the state equation and eq 21. Knowing the water vapor density, the phase change rate is calculated using the mass conservation equation of water vapor. In the second method, the phase change rate is given by a mass transfer expression, with the difference between the equilibrium vapor density and the local vapor density.

⟨ωv̇ ⟩ = Koεg(⟨ρveq ⟩g − ⟨ρv ⟩g )

(25)

NUMERICAL SOLUTION PROCEDURE

The resulting system of equations, which has to be solved, includes partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations (AEs). When all of the AEs and the ODEs are substituted into the PDEs, a system consisting of PDEs is obtained. This system of PDEs is transformed into a system of ODEs using the method of lines.24,28 The spatial derivatives of PDEs are discretized by the finite volume method. In this work, the LIMEX solver29 has been used for solving the resulting system of ODEs (relative and absolute error tolerances are set to 10−6). The model has been implemented in a FORTRAN-based computer code. The formulation, presented in this study, has been validated successfully with experimental data over different ranges of conditions in drying of various porous materials, such as wood,6,30 brick,23 food,26 and lignite.31 These works highlight the acceptance and validity of this model in drying of porous media. A comparison to the numerical simulation results of Gronli18 is made to verify that (1) the model equations are implemented correctly and that (2) the numerical solution procedure used in this study is appropriate. Gronli’s numerical results have been chosen as a case study because he used the complete set of Whitaker’s formulation, including all phenomena taking place during drying of a wood particle, as in this paper. He presented the model predictions for a one-dimensional simulation of pyrolysis of a wet wood pellet with an initial moisture content of 0.3 kg/kg (db) and a length of 0.06 m, which has been exposed to a heat flux of 130 kW/m2 for 10 min. In this paper, 152 grid points have been used for the simulation along the half length of the particle (0.03 m), as Gronli did. All model parameters were the same as reported in Gronli’s thesis (page 209), with one exception regarding the calculation of the evaporation rate. While in Gronli’s model, the evaporation rate equation has been formulated implicitly by setting the partial water vapor pressure equal to the saturated vapor pressure; in this model, eq 22 gives the phase change rate of moisture explicitly. Figure 1 shows the comparison between the results obtained in this study and the results of Gronli for the spatial profiles of moisture content, temperature, and evaporation rate inside the wet wood particle under intensive heating conditions after 600 s. The results are in very good agreement.

The specific heat capacity for the gaseous mixture, Cpg, is defined by

C pg =

(24)

The total vapor flux in the boundary condition has contributions from evaporation of free water, bound water, and water vapor existing at the surface. When the volume fraction of free water, bound water, and water vapor are multiplied to the vapor flux at the particle surface, the total vapor flux at the boundary is determined.26 α and β are heat and mass transfer coefficients at the surface, respectively, and the effect of Stefan flow is neglected in this paper.27

C pi⟨ρi Ui⟩)∇⟨T ⟩

i = v,a

= ∇(keff ∇⟨T ⟩) − ⟨ωv̇ ⟩Δhv + ⟨ρbw Vbw ⟩∇(Δhsorp)

(23)

(22)

Here, Ko is the reciprocal of the equilibration time. Considering the diameter of wood pores, the equilibration time is very small; therefore, Ko must be chosen big enough (>105 s−1) to satisfy the equilibrium condition. In this paper, the latter method has been used to determine the phase change rate as a source term in the equations with an explicit expression. Initial and Boundary Conditions. Initially, the biomass particle is at ambient pressure and temperature conditions, and the initial moisture content is introduced in the model. Because the free and bound water are assumed to be in equilibrium with water vapor inside the particle, the pore spaces are initially assumed to be filled by humid air. Symmetry is assumed for the boundary condition at the center of

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RESULTS AND DISCUSSION This study focuses on drying of a long cylindrical pine wood particle. The particle is considered to be one-dimensional in radial direction. A staggered grid was used, and a gridindependence study has been performed, as it is later explained. Physical properties of the wood particle as well as initial and boundary values have been chosen, so that they are representing typical values of pine wood and typical conditions 6707


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Table 1. Model Parameters parameter

correlation/value

⟨ρSD⟩ Tbc ε Patm, Pbc MCinit α β rp Mfsp h(Mbw,T) Psat v (T) Δhw Δhsorp

450 423 0.7 101325 0.8 5.69 + 0.0098Tbc α/950 0.0125 0.598−0.001 T −3 1 − (1 − (Mbw/Mfsp))6.453 × 10 T exp(24.1201 − 4671.3545T−1) 3.1749 × 106 −2460T 400Δhw(1 − (Mbw/Mfw))2

unit kg m−3 K

reference 33 33

N m−2 kg kg−1 W m−2 K−1 m s−1 m kg kg−1 N m−2 J kg−1 J kg−1

33 9 1 12 18 34 34 8

model based on Whitaker’s theory, determination of heat and mass transfer coefficients of this model is very difficult. These coefficients depend upon grain orientation and physicochemical properties of the wood species. Furthermore, these coefficients depend upon the moisture content and temperature of the particle, which are variable during the drying process and may affect these coefficients significantly. Therefore, the determination of these coefficients for different species of wood over different conditions is very time-consuming. The information on the transport coefficients of this model is scarce and limited to a few species of wood over a rather limited range of conditions. This fact increases the uncertainty degree of these coefficients in modeling. On the other hand, the accuracy of the predictions of a mathematical model depends upon the accuracy of these coefficients. Available values and empirical correlations of the transport coefficients in previous works, which can be used for modeling of the drying process in pine wood, are listed in the Appendix. It is obvious that their variation is remarkable, although these correlations have been used only for a single species of wood: pine. Because the drying time is the most important output variable in the drying process, this quantity is considered as the model output. The drying time is calculated according to the approximation that the particle is supposed to be dried when its average moisture content reaches 0.001. This value (0.001) has been selected, as in ref 15, to avoid wasting computational time and numerical problems around zero moisture content. When different empirical expressions are plotted and different values are compared, maximum and minimum values of each transport coefficient are determined over the conditions of this study. Thanks to sensitivity analyses conducted previously,15−17 the qualitative effect of variations of effective thermal conductivity, capillary pressure, bound water diffusivity, effective diffusivity of water vapor, and permeability on the drying time is known. On the basis of this information and reasoning, two extreme scenarios according to two sets of nine empirical correlations and values for model parameters, presented in Table 2, are defined. In one scenario, the maximum drying time is obtained, and in the other scenario, the minimum drying time is obtained. If the final average moisture content decreases 10 times, from 0.001 to 0.0001, the drying time for the cases of maximum drying time, reference (later defined), and minimum drying time based on 120 grid points will increase 15.27, 6.78, and 5.47%, respectively. The grid-independence study was performed by increasing the number of grid points from 30 to 60, 120, and 240. When 120 grid points is chosen, the numerical solution is almost

Figure 1. Spatial profiles of moisture content, temperature, and evaporation rate inside the particle after 600 s.

of drying of wood chips in industrial dryers. They are listed in Table 1. For the nine model parameters, different correlations and values from the literature are investigated and listed in the Appendix. All other coefficients are taken from ref 32. Extreme Scenarios. Despite the comprehensive descriptions of transport phenomena in drying of wood material by the 6708


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Table 2. Model Variables for Two Extreme Scenarios of Modelinga scenario 1 (maximum drying time)

model variable keff (W m−1 K−1)

Cp (J kg−1 K−1) Dbw (m2 s−1) 2 −1 Deff v (m s )

Pc (N m−2)

Kg (m2) Kl (m2) Krg

Krl

a

scenario 2 (minimum drying time)

6 × 104⟨ρSD⟩((1 + MC)/(1 + 0.15)) − 0.166, if (MC > Mfsp) 6 × 104⟨ρSD⟩((1 + MC)/(1 + 0.44MC)) − 0.166, if (MC ≤ Mfsp) ref 11 (1113 + 4.85T + 4185MC)/(1 + MC) refs 14, 25, and 36 exp(−12.8183933 + 10.8951601Mbw − (4300/T) ref 38 10−3[1.192 × 10−4(T1.75/P)] ref 14 (77.5−0.185(T−273))×10−3[(3150/(S+10−4))−((1047+3.368⟨ρSD⟩)/(1.02−S)) + 149.8⟨ρSD⟩(1 − S) + 52350 + 168.4⟨ρSD⟩ − (3150/(1 + 10−4))] ref 40 5 × 10−18 ref 39 0.1Kg ref 8 0.95(1 − (Mfw/Mcr))2 + 0.05, if (0 < Mfw < Mcr) 0.05((Msat − Mfw)/(Msat − Mcr)), if (Mcr < Mfw < Msat) ref 39 S0.5[1 − (1 − S1/m)m]2 m = (1 − (1/1.921)) ref 7

1.5(0.142 + 0.46MC) ref 25 (CpwMC + 1357)/(1 + MC) ref 37 exp(−9.9 + 9.8Mbw − (4300/T)) ref 39 ε6[1.192 × 10−4(T1.75/P)] ref 11 exp(16.38 − 0.3909MC − 17.761MC2 + 21.228MC3− 7.0784MC4) refs 7 and 41 5 × 10−15 ref 8 10Kg ref 39 1 + (2S − 3)S2 ref 14 0.95(Mfw/Mcr)2 ref 39

Sirr = 0.07; Msat = 1.33; and Mcr = 0.8.

independent of the mesh. The order of reduction of the error was calculated on the basis of the Richardson extrapolation method35 for two extreme scenarios as well as the reference case. For the cases of maximum drying time, reference, and minimum drying time based on 120 grid points, the order of reduction of the error is 0.955, 0.914, and 0.973, respectively, and the total discretization error is 1.35, 1.62, and 1.95%, respectively. The errors obtained with 120 grid points are considered small enough in comparison to the differences obtained in total drying time with the different inputs employed in this work. The results are presented on the basis of 120 grid points. The temporal resolution is determined by the LIMEX solver, as explained in ref 29. In Figure 2, numerical simulation results of moisture loss dynamics during drying of a pine wood particle are shown.

They have been calculated on the basis of the two extreme scenarios. The maximum drying time is 16.45 times bigger than the minimum drying time. As seen, using the model parameters arbitrarily from the literature can lead to a significant inaccuracy of the predicted drying time. This big difference between maximum and minimum drying times is the response of the studied model against these two extreme scenarios. This shows that two researchers can obtain numerical simulation results with up to a 16 times difference, although both of them can correctly argue that they have used properties of pine wood that are reported in the literature. It should be emphasized that all model parameters considered in this study have been established for only one type of wood: pine. It can be expected that the range of predicted drying times is even bigger if parameters of different types of wood are used. Despite this significant uncertainty inherent in wood-drying models, it was shown in previous studies that modeling results and experimental results agree very well, even if the parameters that have been used were determined for different types of wood. This may in some cases be attributed to the fact that the errors because of simplifications of the physical model on one hand and the errors related to model parameters on the other hand can cancel each other. Considering the findings of this work, it may be supposed that the predictions of these models for other conditions than the conditions for which they have been evaluated can be rather inaccurate. Global Sensitivity Analysis. Global sensitivity analysis is used to identify the contribution of each model parameter to the variation of the model output, i.e., the drying time. In this study, the one-at-a-time (OAT) approach is applied to perform the global sensitivity analysis.13,42 The effect of variation in each model variable on the model output is evaluated, while the other model variables are held constant at their base values. Hence, a set of base or reference values of each model parameter is needed. To this end, an intermediate correlation

Figure 2. Evolution of the average moisture content based on two extreme scenarios of modeling. 6709


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Table 3. Reference Case of Model Variables for Drying of Pine Wood on Transverse Directiona model variable

unit

reference

Scots pine

37

Scots pine

37

m2 s−1

pine

11

⎡ ⎛ T1.75 ⎞⎤ Dveff = 0.05ε⎢1.192 × 10−4⎜ ⎟⎥ ⎝ P ⎠⎦ ⎣

m2 s−1

southern pine

8

Pc = 1.364 × 105σ(T )(M fw + 1.2 × 10−4)−0.63

N m−2

keff = 0.17681 + 0.83535 × 10 (T − 273) + 0.2765MC Cp =

J kg

Cpw MC + 1710

D bw =

−1

K

−1

1 + MC

⎡⎛ − 2590.1 ⎤ ⎞ 1 1046.63 exp⎢⎜ + 11.954⎟MC − − 12.35⎥ ⎝ ⎠ ⎣ ⎦ T T ⟨ρSD ⟩

K g = 4 × 10−16

Kl = K g

⎛ 1 − S ⎞3 K rg = ⎜ ⎟ ⎝ 1 − Sirr ⎠ K rl = S 3 a

species

W m−1 K−1

−4

softwood

39

m

2

pine (sapwood)

19

m

2

pine

19

pine

19

pine

19

−3

σ(T) = (1.28 × 10 − 0.185T)10 ; Sirr = 0.07. 2

Figure 3. (a) Effective thermal conductivity versus moisture content and (b) evolution of the average moisture content based on minimum and maximum effective thermal conductivities.

Figure 4. (a) Specific heat capacity versus moisture content and (b) evolution of the average moisture content based on minimum and maximum specific heat capacities.

or value of each model parameter is defined as the reference case and shown in Table 3. The global sensitivity analysis is

performed by varying each model parameter (OAT) from its minimum value to its maximum value. Although some 6710


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Figure 5. (a) Bound water diffusivity versus moisture content and (b) evolution of the average moisture content based on minimum and maximum bound water diffusivities.

Figure 6. (a) Effective diffusivity of water vapor versus temperature and (b) evolution of the average moisture content based on minimum and maximum effective diffusivities of water vapor.

parameters, such as relative permeability of gas and liquid, may be related to each other, in this work, all model parameters are considered to be independent from each other. The model parameters are categorized into thermal parameters, diffusive transport coefficients, and convective transport coefficients. Thermal Parameters. Thermal parameters include the effective thermal conductivity and the specific heat capacity of a wet wood particle. Different correlations and values for the effective thermal conductivity with respect to different grain directions have been reported for dry as well as wet pine wood in the literature. Also, for the specific heat capacity of dry and wet pine wood particles, several correlations and values are available. For the sensitivity analysis, the effective thermal conductivity of wet pine wood in transverse direction is considered to be the arithmetic mean of the values in radial and tangential directions. For a case that only thermal conductivity of dry wood is available, the effective thermal conductivity of wet wood is obtained by adding the contribution to thermal conductivity of the moisture content. The contribution to thermal conductivity of the moisture content is calculated by multiplying the volume fraction of the moisture with the thermal conductivity of water, according to ref 15. In the same way, by adding the specific heat capacity of water, the specific heat capacity of wet wood is calculated. The contribution of the

gas phase is neglected in the calculation of the specific heat capacity of wet wood. The specific heat capacity and thermal conductivity of water are considered to be constant and equal to 4200 J kg−1 K−1 and 0.615 W m−1 K−1, respectively.15 Effective Thermal Conductivity. Figure 3 shows the results of mass loss during drying of a single wood particle for minimum and maximum effective thermal conductivity of wet wood. Although the two correlations differ by a factor of approximately 3, the drying time varies only by 14%. This finding is in agreement with results obtained by Ben Nassrallah and Perre,17 who also concluded that their drying model is not very sensitive to the changes in the effective thermal conductivity parameter. Specific Heat Capacity. Figure 4a shows that the correlations for the specific heat capacity of a wet wood particle differ by a factor of approximately 1.5, but nevertheless, the difference in the results is negligible, as shown in Figure 4b. Therefore, one can use all of the expressions for specific heat capacity of wet wood listed here for the simulation of the drying of a pine wood particle without a significant error. Diffusive Transport Coefficients. The bound water diffusivity and the effective diffusivity of water vapor are two model variables that characterize diffusive flow inside the 6711


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Figure 7. (a) Capillary pressure versus saturation and (b) evolution of the average moisture content based on minimum and maximum capillary pressures.

Figure 9. (a) Gas relative permeability versus saturation and (b) evolution of the average moisture content based on minimum and maximum gas relative permeabilities.

Figure 8. Evolution of the average moisture content based on minimum and maximum gas intrinsic permeabilities.

Figure 10. Evolution of the average moisture content based on minimum and maximum liquid intrinsic permeabilities.

particle. In the literature, there are few studies dealing with the determination of mass transport coefficients in wood. The separation of the effects of the different modes of mass transfer

during the drying process is experimentally extremely challenging. In light of these experimental difficulties, it is not 6712


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parameter causes a significant uncertainty on the predicted drying time, of a factor of approximately 2, as shown in Figure 6b. This 97% increase of the drying time is actually not in agreement with the results reported by Bonneau and Puiggali,11 who stated that the vapor diffusion has a minor effect on the moisture transport during wood drying. However, Ben Nassarrah and Perre17 reported that, in the case of wood drying, the resistance of the porous medium to gaseous migration is very important and showed that their model is sensitive to the changes in effective diffusivity of water vapor. The latter statement is approved with the results obtained in this study. Convective Transport Coefficients. Capillary pressure and intrinsic and relative permeabilities of liquid are the model variables influencing the convection of free water in the pores of a particle. The convective flow of gas, including water vapor and air, is dependent upon the intrinsic and relative permeabilities of the gas. Capillary Pressure. Presented values of minimum and maximum capillary pressure in the pores of wood, under studied conditions, range from 103 to 106 Pa over different saturations, as shown in Figure 7a. This 3 orders of magnitude difference between capillary pressures affects the drying time considerably. The drying time associated with the maximum value of capillary pressure is 47% bigger than the minimum drying time, as shown in Figure 7b. Fyhr and Rasmuson16 have reported a similar remarkable effect of changes in the capillary pressure on their model. Gas Intrinsic Permeability. The intrinsic permeability refers to the permeability in a single-phase saturated medium. The reported values for this model parameter differ by 3−4 orders of magnitude, even for one type of wood.43 Figure 8 shows that the ratio of maximum to minimum drying time, associated with the minimum and maximum values of gas intrinsic permeability, respectively, is 2.93. This 193% discrepancy between the predictions of a model are not acceptable. Therefore, for reasonable accurate predictions of the drying time, it is recommended that at least the order of magnitude of gas intrinsic permeability is experimentally determined for the wood under examination. Gas Relative Permeability. The relative permeability is defined as the permeability of a fluid phase in an unsaturated medium. As Di Blasi15 reported, under the conditions examined in this work, the convective liquid-phase flow is 2 or 3 orders of magnitude larger than the vapor flow. For this reason, a 3 times difference between maximum and minimum values of the relative permeability of gas (shown in Figure 9a) is not enough to make a considerable change in the results of the model, as shown in Figure 9b. Under these conditions, the model is not sensitive to the different expressions of the relative permeability of gas listed here, and one can use each of these expressions freely. In the extreme conditions of high-temperature drying, which are encountered in initial stages of thermochemical conversion of biomass, the gas-phase convective transport is prominent in the drying process and it is expected that the model would be more sensitive to the changes in the value of relative permeability of gas. Liquid Intrinsic Permeability. Perre and Karimi44 reported that entrapped air and particulate matter may appear in the pathway of the liquid water flow during the measurement of the intrinsic permeability of liquid water in wood. They stated that, for these reasons, the measured value of the gas intrinsic permeability is much more accurate than the liquid intrinsic

Figure 11. (a) Liquid relative permeability versus saturation and (b) evolution of the average moisture content based on minimum and maximum liquid relative permeabilities.

surprising that there is a very broad range of values and correlations for these diffusion coefficients reported by different authors, even for a certain type of wood. Diffusion Coefficient of Bound Water. As shown in Figure 5b, the effect of the variation of bound water diffusivity on the drying time over the studied conditions is negligible. The ratio of maximum to minimum drying time resulting, respectively, from minimum and maximum values of the diffusion coefficient of bound water is 1.065. The variation between maximum and minimum values of the diffusion coefficient of bound water reaches up to 3 orders of magnitude for some temperatures and moisture contents, as shown in Figure 5a, while the maximum drying time is only 6.5% bigger than the minimum drying time. Effective Diffusivity of Water Vapor. Because several parameters affect the effective diffusivity of water vapor in porous media (molecular diffusivity, porosity, and tortuosity), this coefficient is very variable from one kind of material to another. Recently, Kang7 pointed out that the effective diffusivity of water vapor in wood is actually still unknown. Figure 6a shows that a wide range of values have been used for the effective diffusivity of water vapor in wood by researchers in the literature. The wide range of variation of this model 6713


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which could be used for modeling the drying of pine wood, have been collected from the literature. When lower and upper bounds of the values of these variables are found and applied to the model, a bounded model output is calculated. When the variation of each model variable is applied, from its minimum to maximum value, to the model, the sensitivity analysis has been performed. The following points are concluded from the results: (1) The range of values of the model variables that are reported in the literature for a certain species of wood (pine) is considerably broad. For some properties or transport coefficients, 3−4 orders of magnitude differences in the values have been reported. This shows the variety of properties in wood, even in the same species. Therefore, taking an arbitrary value of a model variable from the broad range of reported values, without enough justifications, may lead to inaccuracy. (2) The impact of the big variations in the values of the model variables on the model output is surprising; there is a 16 times difference between the predicted maximum and minimum drying times. It means that the selection of the model variables from the literature can lead to totally wrong predictions of the model output, the total drying time. (3) (i) The global sensitivity analysis shows that the impact of the variation of thermal parameters, including effective thermal conductivity and specific heat capacity, on the model output is almost negligible. Therefore, each correlation listed in this paper for the above-mentioned variables can be adopted for modeling drying of pine wood under the studied conditions. (ii) The model output changed slightly with the different correlations of bound water diffusivity. However, the effect of the variation of the effective diffusivity of water vapor on the model output was significant. Therefore, the adoption of a correlation for bound water diffusivity is not critical for drying modeling, but a wrong selection of the value of effective diffusivity of water vapor may lead to 100% error in the predictions of drying

permeability. By considering the phenomena of pit aspiration during the drying process, Perre et al.14 assumed the liquid intrinsic permeability being 5 times greater than the gas intrinsic permeability. Stanish et al.8 have taken into account the value of the liquid intrinsic permeability being 1 order smaller than the gas intrinsic permeability. The effect of maximum and minimum values of the liquid intrinsic permeability, listed in Table 2, is significant on the model results, as shown in Figure 10, and the maximum drying time is 25% bigger than the minimum drying time. This considerable sensitivity to not big changes in the value of the parameter can be related to the prominent contribution of the convective liquid-phase flow in the drying process under the studied conditions.15 Di Blasi concluded in her paper15 that, for fast drying under extreme conditions, the convective liquid-phase transport does not significantly affect the drying process; therefore, under those conditions, it is expected that the sensitivity of the model would be much smaller to the changes in the value of liquid intrinsic permeability. Liquid Relative Permeability. Over the studied range of saturation, there are 3−6 orders of magnitude differences between the maximum and minimum value of the relative permeability of liquid water, as shown in Figure 11a. The significant effect of this wide range of changes in the value of the liquid relative permeability on the model results is shown in Figure 11b; the maximum drying time is 35% bigger than the minimum drying time.

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CONCLUSION A one-dimensional drying model based on the Whitaker model has been implemented for drying of an infinite cylindrical wood particle. Nine physical properties of wood as well as transport coefficients of the model have been investigated as model variables. Different values and correlations of these parameters,

Table 4. Effective Thermal Conductivity of Pinewood in Transverse Direction empirical correlation

species

k T = 0.142 + 0.46MC

reference

softwood

25

southern pine

8

Scots pine

37

Scots pine

37

softwood

39

pine

11

kR = 2k T keff = 1.5(0.142 + 0.46MC) keff =

⟨ρSD ⟩ 1000

(0.4 + 0.5MC) + 0.024

k T = 0.1989 + 0.8314 × 10−4(T − 273) kR = 0.1990 + 0.8393 × 10−4(T − 273) keff = 0.17681 + 0.83535 × 10−4(T − 273) + 0.2765MC k T = 0.15894 + 0.47048 × 10−3(T − 273) kR = 0.14749 + 0.43657 × 10−3(T − 273) keff = 0.131075 + 0.453525 × 10−3(T − 273) + 0.2765MC

keff = (0.129 − 0.049MC)(0.986 + 2.695MC) [1 + 0.001(2.05 + 4MC)(T − 273)] if (MC > 0.4) keff = (0.0932 − 0.0065MC)(0.986 + 2.695MC)[1 + 0.00365(T − 273)] if (MC ≤ 0.4)

1 + MC − 0.166 if (MC > M fsp) 1 + 0.15 1 + MC = 6 × 104⟨ρSD ⟩ − 0.166 if (MC ≤ M fsp) 1 + 0.44MC

keff = 6 × 104⟨ρSD ⟩ keff

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Table 5. Specific Heat Capacity of Pine Wood empirical correlation

species

Cp = Cp =

14, 25, and 36

0.0022 3.32MC + 2.95 4057MC + 526 (T )2 + (T ) + 1 + MC 1 + MC 1 + MC

Scots pine

37

Cpw MC + 1710

Scots pine

37

spruce pine

45

Scots pine

37

1 + MC

Cp = − 9.12 × 10−2 + 4.4 × 10−3T Cp =

reference

softwood

4185MC + 4.85T + 1113 Cp = 1 + MC

Cpw MC + 1357 1 + MC

Table 6. Bound Water Diffusivity in Pinewood in Transverse Direction empirical correlation

D bw

⎛ 4300 ⎞⎟ = exp⎜− 9.9 + 9.8Mbw − ⎝ T ⎠

D bw = exp(− 30.39 + 5.46MC + 2.54 × 10−2T ) D bw =

⎡⎛ − 2590.1 ⎤ ⎞ 1 1046.63 exp⎢⎜ + 11.954⎟MC − − 12.35⎥ ⎠ ⎣⎝ T ⎦ T ⟨ρSD ⟩

⎛ 4300 ⎞⎟ D bw = exp⎜− 12.81 + 10.89Mbw − ⎝ T ⎠

reference 11

0.05ε Dv − a 1.37

southern pine

2

Dveff = 0.05ε 2Dv − a

southern pine

8

Dveff = 10−3Dv − a

softwood

14

6

= ε Dv − a

Dveff =

a

species pine

Dveff

reference 39

Scots pine

46

pine

11

softwood

38

predictions of the model that are 3 times higher (or lower) than the accurate results. Despite the unpromising view on modeling drying of a single wood particle, because of the errors that may arise from wrongly adopting the model variables from the literature, it should also be stated that, if only some of the most effective parameters are experimentally determined in the studied case, then the model output would be in a band with acceptable broadness. To obtain reliable predictions of a model of drying of a single wood particle, the gas intrinsic permeability, effective diffusivity of water vapor, and capillary pressure, in order of their importance, must be experimentally determined for the studied conditions. Finally, it should be pointed out that the presented results have been obtained for typical conditions in industrial dryers, i.e., moderate heating conditions. For different conditions, as low or extreme heating conditions, the results may be different.

Table 7. Effective Diffusivity of Water Vapor in Pine Wood in Transverse Directiona empirical correlation

species softwood

Dv−a = [1.192 × 10−4(T1.75/P)].

time. (iii) The variation of all investigated convective transport coefficients, except gas relative permeability, showed a remarkable impact on the predicted drying time. Among all model variables, the gas intrinsic permeability is the most significant parameter affecting the model output. Wrong adoption of the value of gas intrinsic permeability may lead to Table 8. Capillary Pressure in Pinewood in Transverse Direction empirical correlation

species

reference

Pc = exp(16.348 − 0.3909MC − 17.761MC + 21.228MC − 7.0789MC )

red pine

41

Pc = 8.4 × 104S −0.63

southern pine

47

Pc = 101325[1.937(S)exp(− 3.785S) + (0.093(1 − S)S −1.4] × 1

pine (sapwood)

11

Pc = 1.364 × 105σ(T )(M fw + 1.2 × 10−4)−0.63

softwood

39

⎡ 3150 1047 + 3.368⟨ρSD ⟩ Pc = σ(T )⎢ + 149.8⟨ρSD ⟩(1 − S) + 52350 −4 − 1.02 − S ⎣ S + 10

softwood

40

2

3

4

− 2.79 × 10−3(T − 273))

+ 168.4⟨ρSD ⟩ − a

3150 ⎤ ⎥ 1 + 10−4 ⎦

σ(T) = 1.28 × 102 − 0.185T)10−3; S = Mfw/(Mmax − Mfsp); and Mmax = ⟨ρl⟩l((1/⟨ρSD⟩) − (1/⟨ρs⟩s)). 6715


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APPENDIX Available values and empirical correlations of the transport coefficients in previous works, which can be used for modeling of the drying process in pine wood (Tables 4−12).

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Table 9. Gas Intrinsic Permeability in Pinewood in Transverse Direction empirical correlation

species

reference

K g = 5 × 10

southern pine

8

K g = 4 × 10−15

softwood

36

K g = 1.096 × 10−15

maritime pine

48

K g = 4 × 10−16

pine (sapwood)

19

K g = 1 × 10−16

pine

11

K g = 5 × 10−18

softwood

39

−15

Table 10. Liquid Intrinsic Permeability in Pinewood in Transverse Direction empirical correlation

species

reference

Kl = 10K g

southern pine

8

Kl = 5K g

softwood

36

Kl = K g

pine

19

Kl = 0.1K g

softwood

39

NOMENCLATURE Cp = specific heat capacity at constant pressure (J kg−1 K−1) Dbw = bound water diffusivity (m2 s−1) Deff i = effective diffusivity of component i in the gas mixture (m2 s−1) h(Mbw,T) = relative humidity k = thermal conductivity (W m−1 K−1) Ko = phase change rate constant (s−1) K = intrinsic permeability (m2) Kr = relative permeability M = molecular weight (kg mol−1) MC = dry-basis moisture content (kg kg−1) Mfw = dry-basis moisture content of free water (kg kg−1) Mbw = dry-basis moisture content of bound water (kg kg−1) Mcr = critical moisture content (kg kg−1) Mirr = irreducible moisture content (kg kg−1) Mfsp = moisture content at the fiber saturation point (kg kg−1) Msat = saturated moisture content (kg kg−1) Mmax = maximum possible value for the moisture content (kg kg−1) n = unit normal vector with direction out of surface P = pressure (N m−2) Pc = capillary pressure (N m−2) ⟨Pg⟩g = pressure in the gas phase (N m−2)

Table 11. Gas Relative Permeability in Pinewood in Transverse Directiona empirical correlation

species

reference

softwood

14

⎛ 1 − S ⎞3 K rg = ⎜ ⎟ ⎝ 1 − Sirr ⎠

pine

19

K rg = (1 − S)3

pine

19

softwood

39

K rg = 1 + (2S − 3)S

2

2 ⎛ M ⎞ K rg = 0.95⎜1 − fw ⎟ + 0.05 Mcr ⎠ ⎝

K rg = 0.05 a

Msat − M fw Msat − Mcr

if (0 < M fw < Mcr)

if (Mcr < M fw < Msat)

Sirr = 0.07; Msat = 1.33; and Mcr = 0.8.

Table 12. Liquid Relative Permeability in Pinewood in Transverse Directiona empirical correlation

species

reference

softwood

39

softwood

14

⎛ S − Sirr ⎞ K rl = ⎜ ⎟ ⎝ 1 − Sirr ⎠

pine

19

K rl = S 0.5[1 − (1 − S1/ m)m ]2

pine

7

⎛ M ⎞2 K rl = 0.95⎜ fw ⎟ ⎝ Mcr ⎠ K rl = 0.05

if (0 < M fw < Mcr)

M fw − Mcr + 0.95 Msat − Mcr

if (Mcr < M fw < Msat)

K rl = S 3 3

a

Sirr = 0.07; Msat = 1.33; Mcr = 0.8; and m = (1 − (1/1.921)). 6716


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⟨Pw⟩w = pressure in free water (N m−2) rp = radius of the particle (m) R = universal gas constant (8.3144 J mol−1 K−1) S = saturation (%) t = time (s) T = temperature (K) ⟨Vi⟩ = superficial velocity vector of component i (m s−1) Δh = latent heat of vaporization (J kg−1) Δhsorp = differential heat of sorption (J kg−1)

(10) Perre, P. Int. Commun. Heat Mass Transfer 1987, 14, 519−529. (11) Bonneau, P.; Puiggali, J.-R. Wood Sci. Technol. 1993, 28, 67−85. (12) Siau, J. Transport Processes in Wood; Springer-Verlag: New York, 1984. (13) Brun, F.; Wallach, D.; Makowski, D.; Jones, J. W. Working with Dynamic Crop Models: Evaluation, Analysis, Parameterization, and Applications, 1st ed.; Elsevier Science: Amsterdam, Netherlands, 2006. (14) Perre, P.; Moser, M.; Martin, M. Int. J. Heat Mass Transfer 1993, 36, 2725−2746. (15) Blasi, C. D. Chem. Eng. Sci. 1998, 53, 353−366. (16) Fyhr, C.; Rasmuson, A. AIChE J. 1996, 42, 2491−2502. (17) Nasrallah, S. B.; Perre, P. Int. J. Heat Mass Transfer 1988, 31, 957−967. (18) Gronli, M. G. A theoretical and experimental study of the thermal degradation of biomass. Ph.D. Thesis, The Norwegian University of Science and Technology, Trondheim, Norway, 1996. (19) Couture, F.; Jomaa, W.; Puiggali, J.-R. Transp. Porous Media 1996, 23, 303−335. (20) Whitaker, S. Adv. Heat Transfer 1977, 13, 119−203. (21) Ouelhazi, N.; Arnaud, G.; Fohr, J. Transp. Porous Media 1992, 7, 39−61. (22) Melaaen, M. C. Numer. Heat Transfer, Part A 1996, 29, 331− 355. (23) Zhang, Z.; Yang, S.; Liu, D. Heat TransferAsian Res. 1999, 28, 337−351. (24) Anca-Couce, A.; Zobel, N. Fuel 2012, 97, 80−88. (25) Perre, P.; Turner, I. W. Int. J. Heat Mass Transfer 1999, 42, 4501−4521. (26) Halder, A.; Dhall, A.; Datta, A. K. J. Heat Transfer 2011, 133. (27) Blasi, C. D.; Branca, C.; Sparano, S.; Mantia, B. L. Biomass Bioenergy 2003, 25, 45−58. (28) Ascher, U. M.; Petzold, L. R. Computer Methods for Ordinary Differential Equations and Differential−Algebraic Equations; Society for Industrial and Applied Mathematics: Philadelphia, PA, 1998. (29) Deuflhard, P.; Hairer, E.; Zugck, J. Numer. Math. 1987, 51, 501−516. (30) Bellais, M. Modelling of the pyrolysis of large wood particles. Ph.D. Thesis, Royal Institute of Technology (KTH), Stockholm, Sweden, 2007. (31) Zhang, K.; You, C. Energy Fuels 2010, 24, 6428−6436. (32) Gronli, M. G.; Melaaen, M. C. Energy Fuels 2000, 14, 791−800. (33) Forest Products Society. Wood Handbook: Wood as an Engineering Material; Forest Products Society: Madison, WI, 2010. (34) Raznjevic, K. Handbook of Thermodynamic Tables and Charts; Hemisphere Publishing Corporation: Washington, D.C., 1976. (35) Ferziger, J. H.; Peric, M. Computational Methods for Fluid Dynamics, 3rd ed.; Springer: New York, 2001. (36) Perre, P.; Turner, I. W. AIChE J. 2006, 52, 3109−3117. (37) Olek, W.; Weres, J.; Guzenda, R. Holzforschung 2003, 57, 317− 325. (38) Perre, P.; Turner, I. W. Holzforschung 2001, 55, 417−425. (39) Perre, P.; Degiovanni, A. Int. J. Heat Mass Transfer 1990, 33, 2463−2478. (40) Perre, P.; Turner, I. W. Holzforschung 2001, 55, 318−323. (41) Tremblay, C.; Cloutier, A.; Fortin, Y. Wood Sci. Technol. 1996, 30, 361−371. (42) Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. Global Sensitivity Analysis: The Primer, 1st ed.; Wiley-Interscience: Hoboken, NJ, 2008. (43) Bao Fucheng, L. J.; Stavros, A. Holzforschung 1999, 53, 350− 354. (44) Perre, P.; Karimi, A. Maderas: Cienc. Tecnol. 2002, 4, 50−68. (45) Koch, P. Wood Sci. 1969, 1, 203−214. (46) Hukka, A. Holzforschung 1999, 53, 534−540. (47) Spolek, G.; Plumb, O. Wood Sci. Technol. 1981, 15, 189−199. (48) Perre, P.; Agoua, E. J. Porous Media 2010, 13, 1017−1024.

Greek Letters

α = heat transfer coefficient (W m−2 K−1) β = mass transfer coefficient (m s−1) ε = volume fraction Λ = volume (m3) Λγ = volume of phase γ (m3) μ = dynamic viscosity (kg m−1 s−1) ⟨ργ⟩ = phase-averaged density of phase γ (kg m−3) ⟨ρi⟩γ = intrinsic density of component or phase i in phase γ (kg m−3) ϕ = general variable ⟨ω̇ i⟩ = phase change rate of component i in phase γ (kg m−3 s−1)

Subscripts

a = air atm = atmosphere bc = boundary condition bw = bound water cr = critical eff = effective equ = equilibrium fw = free water fsp = fiber saturation point g = gas mixture phase i = component or phase i init = initial irr = irreducible l = liquid free-water phase s = solid sat = saturated state SD = solid dry wood surf = external surface of particle v = water vapor w = water γ = phase γ ∞ = ambient

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REFERENCES

(1) Vu, T. H. Influence of pore size distribution on drying behaviour of porous media by a continuous model. Ph.D. Thesis, Otto von Guericke University of Magdeburg, Magdeburg, Germany, 2006. (2) Fernandeza, M. L.; Howella, J. R. Drying Technol. 1997, 15, 2343−2375. (3) Jianmin, C.; Fangtian, D. J. Northeast For. Univ. 1993, 4, 112−117. (4) Perré, P. How to get a relevant material model for wood drying simulation? Proceedings of the 1st Workshop “State of the Art for Kiln Drying”; Edinburgh, U.K., Oct 13−14, 1999. (5) Koumoutsakos, N. Modelling radio frequency/vacuum drying of wood. Ph.D. Thesis, The University of British Columbia, Vancouver, British Columbia, Canada, 2001. (6) Plumb, O.; Spolek, G.; Olmstead, B. Int. J. Heat Mass Transfer 1985, 28, 1669−1678. (7) Kang, W.; Chung, W. J. Wood Sci. 2009, 55, 91−99. (8) Stanish, M. A.; Schajer, G. S.; Kayihan, F. AIChE J. 1986, 32, 1301−1311. (9) Alves, S.; Figueiredo, J. Chem. Eng. Sci. 1989, 44, 2861−2869. 6717


On the Uncertainty of a Mathematical Model for Drying of a Wood

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