A Finite Element Method for Nonlinear and Fully Coupled Transient Hygrothermal Transfer

Abstract

A fully coupling nonlinear finite element algorithm for moisture and heat transfer is developed and implemented. The finite element model starts from a suitable formulation of the constitutive and equilibrium equations. The nonlinearity arises due to the strong moisture and temperature dependence of the material properties. Four examples are presented to check the validity and precision of the finite element model. Two of the examples compare the numerical results with simplified, one-dimensional analytical solutions. Perfect agreement is obtained between the analytical and numerical solutions. The results show that the dependency of the material coefficients on the moisture and temperature is important, even for one-dimensional cases. The other two examples, without direct analytical solution due to the material nonlinearity, show that the fully coupling between the moisture and heat conduction is essential. The formulation can be used for the transient analysis of moisture and heat conduction with arbitrary geometrical dimensions and time-dependent loading conditions.

1. Introduction

Nowadays, porous and fiber-reinforced polymeric matrix composite materials are increasingly used, particularly in the aerospace industry and ocean industry, due to their high mechanical performance, light weight, and great flexibility in design. During the operation, the materials are often exposed to diverse conditions of temperature and humidity [1]. Moisture and temperature have an adverse effect on the performance of composites. Stiffness and strength are reduced with the increase in moisture content and temperature. Besides, residual stresses may also be developed due to thermal expansion and moisture expansion [2].

It is well known that the mechanical characteristics of polymeric composites and porous materials are significantly affected by environmental agents, particularly temperature and humidity. Hence, the analysis of the materials under these environments is an area of concern. Marques and Creus [3] employed a three-dimensional degenerated finite element method to analyze the time-dependent response of polymeric matrix laminated composites subject to mechanical and hygrothermal loads. Kerur and Ghosh [4] considered the hygrothermal and electroelastic couplings and developed a finite element formulation to evaluate the stresses due to large deformation bending of piezoelectric structures. Using the variational principles, Altay and Dokmeci [5] investigated the coupling equations with piezoelectric, thermopiezoelectric, and hygrothermopiezoelectric effects. Vinyas et al. [6] used the finite element method to evaluate the displacements, electric and magnetic potentials of a magnetoelectroelastic beam due to hygrothermal effects. Assie and Mahmoud [7] proposed an uncoupled numerical model to study the hygrothermal response of composite laminates under dynamic bending. Naidu and Sinha [8] studied the large deflection bending of a composite cylindrical shell under hygrothermal environments using the finite element formulation. Wu and Ren [9] developed a higher-order transverse shear model to calculate the transverse shear stresses of angle-ply composite plates under hygrothermal environments. Cinefra et al. [10] studied the crossly plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads and under a hygrothermal environment. Vinyas and Kattimani [11] investigated the 3D response of a magnetoelectroelastic plate subjected to hygrothermal loads by a finite element method. Yi and Sze [12] conducted a finite element analysis of moisture absorption and residual stresses in plastic-encapsulated IC packages in various moisture-soaking conditions. They also obtained hygrothermally induced deformations and stresses in plastic integrated circuit (IC) packages. Wang et al. [13] conducted a numerical and experimental research of the failure modes and fatigue life of carbon fiber-reinforced polymer (CFRP) beams under the coupling action of hygrothermal environment and cyclic load.

All of the above research used known distributions of the temperature and moisture. However, determinations of moisture and temperature, in many situations, are difficult. Three major difficulties are (1) the materials are usually discontinuous, (2) the moisture transfer and heat transfer are coupling phenomena, and (3) the moisture and thermal properties of the materials are strongly moisture- and temperature-dependent. Chang and Weng [14] gave the exact and analytical solution to the coupled heat and moisture transfer equations in porous materials. They used the Laplace transform technique to deal with the time-related parts in the partially differential equations. However, only a one-dimensional problem was considered. Mendes and Phlippi [15] took into account the discontinuity between the interface and predicted heat and moisture transfer in multilayered structures. Khoshbakht et al. [16] constructed a generic theoretical formulation and finite element implementation to model moisture and heat transport in a layered structure consisting of distinct materials. Analytical results were then compared with experimental data to validate the model. Their model considers the effect of temperature on the moisture but ignores the influence of the moisture on the temperature. Therefore, it can be regarded as a semi-coupling. Recently, Beni and Alihemmati [17] proposed a finite element analysis for transient coupled heat and moisture transfer in the cylinders and cylindrical panels of the porous medium. The material properties were considered as constants, and the time-dependent responses were solved numerically by using the Runge–Kutta method. Recently, Zhu et al. [18] considered a variety of data analysis tools for hygrothermal transfer features. A novel approach for peak and valley detection was proposed based on the discrete differentiation of the original data. The research is critical to understanding the hygrothermal transfer mechanisms (HTM) between the walls and the layers inside the walls. Very recently, Wu et al. [19] proposed a temperature and humidity regulation wall (THRW) integrating bio-based HM and PCM model for hygrothermal and energy performance improvement. The temperature, vapor pressure, and heat load on the internal surface of the walls were evaluated. The results show that the optimal wall design varies depending on whether prioritizing hygrothermal or energy performance. Dai et al. [20] reviewed the historic evolution of building greenery with integrated photovoltaic (BGIPV) system, tracing its journey from initial conceptualization and technological research to practical application.

As can be seen from the existing works, the heat and moisture transfer have been studied for (1) with coupling between temperature and moisture but without moisture and temperature-dependent material properties and (2) without coupling between temperature and moisture but with temperature and moisture-dependent material properties. In this paper, a fully coupled finite element model of heat and moisture transfer is proposed, taking into consideration the moisture- and temperature-dependent material properties. The finite element space discretization is used to obtain a system of first-order differential equations about time. The differential equations are solved by the finite difference scheme to obtain the time-dependent response. Some case studies are carried out to show the capability of the proposed finite element model.

2. Governing Equations of Moisture and Temperature Coupling

The phenomenon of moisture and heat transport in composite materials and porous media has been studied for many decades in the science and engineering research community (see review papers [1,20]). A comprehensive approach in modeling the moisture mass and heat transport has been developed, and the proposed governing equation was manipulated and separated into two components defining the mechanisms for isothermal moisture diffusion and heat-induced moisture diffusion. This approach yields a system of coupled partial differential equations consisting of a moisture transfer equation and a heat balance equation.

Suppose in a coordinate system x (whose components are x_i, where i = 1, 2, 3), there is a body occupying a space Ω, which is surrounded by a surface S. The moisture and temperature inside the body may change from point to point and vary with time variable t. Assume H(x, t) is the moisture content (which is the mass of water per mass of wet solid, in kg/m³) and T(x, t) is the temperature (K). Both of them are changed continuously with space coordinates x_i and time t. The basic law of the diffusion of moisture and heat conduction may be stated as [21],

$$f_i=-D_m{H}_{,i}-\gamma D_h{T}_{,i}$$

(1a)

$${\displaystyle \frac{1}{\rho c}}{q}_i=-\lambda D_m{H}_{,i}-D_h{T}_{,i}$$

(1b)

where the comma “,” means the partial derivative (e.g., $H_{,i}$ means $\partial H/\partial x_i$), $q_i$ is the components of the heat flux vector q, f_i is the rate of moisture transfer per unit area along the x_i direction, ρc is the volumetric heat capacity (J/(m³K)), and the notations D_m and D_h are the moisture diffusivity (m²/s) and thermal diffusivity ((W/(m·K))/(kg/m³·J/(kg·K)) = m²/s), respectively. The paramaters γ (in kg/(m³·K)) and λ (in m³·K/kg) are coupling coefficients between the temperature and moisture transfer.

The conservation of mass of moisture (neglecting the generation of internal moisture inside the material) requires that [21]

$$-f_{i,i}={\displaystyle \frac{\partial H}{\partial t}}$$

(2a)

while the conservation of energy (neglecting the generation of internal heat) yields [21]

$$-q_{i,i}=\rho c{\displaystyle \frac{\partial T}{\partial t}}$$

(2b)

Substituting Equation (1a,b) into Equation (2a,b), a system of simultaneous equations is obtained

$$\mathsf{\nabla }\cdot (D_m\mathsf{\nabla }H)+\mathsf{\nabla }\cdot (\gamma D_h\mathsf{\nabla }T)={\displaystyle \frac{\partial H}{\partial t}}$$

(3a)

$$\mathsf{\nabla }\cdot (\lambda D_m\mathsf{\nabla }H)+\mathsf{\nabla }\cdot (D_h\mathsf{\nabla }T)={\displaystyle \frac{\partial T}{\partial t}}$$

(3b)

where ∇ is the Nabla operator. On the left-hand side of Equation (3a), the first term represents the moisture movement due to moisture gradient, and the second term describes the moisture movement due to temperature gradient. While on the left-hand side of Equation (3b), the first term reflects the heat conduction due to moisture gradient, the second term is the heat conduction due to temperature gradient.

The coefficient of moisture diffusion D_m and the thermal diffusivity D_h can strongly depend on moisture concentration and temperature [21]. In this situation, Equation (3) is highly nonlinear, and the exact solution is impossible even for the simplest loading and boundary conditions.

Considering the highly nonlinearly coupled nature of the problem, a finite element code will be developed. For a more systemic analysis, below material properties are introduced: $c_0={\displaystyle \frac{\rho c\lambda D_m}{\gamma D_h}}$, $D_{11}=c_0{D}_m$, $D_{12}=D_{21}=\rho c\lambda D_m$, $D_{22}=\rho c{D}_h$, and a generalized moisture transfer vector J = $c_0$f, where $D_{22}$ is the thermal conductivity (W/(mK)) and $D_{11}$, by analogy, can be understood as the moisture conductivity. By such definitions, the constitutive Equation (1a,b) can be rewritten as

$$J_i=-D_{11}{\displaystyle \frac{\partial H}{\partial x_i}}-D_{12}{\displaystyle \frac{\partial T}{\partial x_i}}$$

(4a)

$$q_i=-D_{21}{\displaystyle \frac{\partial H}{\partial x_i}}-D_{22}{\displaystyle \frac{\partial T}{\partial x_i}}$$

(4b)

where J_i are the components of the vector J. By constitutive Equation (4), the global matrices of the finite element model are symmetric. Because the material properties are temperature- and moisture-dependent, the equivalent coefficients D₁₁, D₁₂, and D₂₂ are also temperature- and moisture-dependent. They are calculated at the temperature T and moisture content H. With the substitution of Equation (4), the equilibrium Equation (2a,b) can be written in terms of the moisture and temperature as

$$\mathsf{\nabla }(D_{11}\mathsf{\nabla }H)+\mathsf{\nabla }(D_{12}\mathsf{\nabla }T)=c_0{\displaystyle \frac{\partial H}{\partial t}}$$

(5a)

$$\mathsf{\nabla }(D_{21}\mathsf{\nabla }H)+\mathsf{\nabla }(D_{22}\mathsf{\nabla }T)=\rho c{\displaystyle \frac{\partial T}{\partial t}}$$

(5b)

The conduction Equation (5a,b) should be solved for given boundary conditions and initial conditions. The initial conditions are the moisture content and temperature distributions when time t is 0. These are $H(x_j,0)={\overline{H}}_0(x_j,0)$ and $T(x_j,0)={\overline{T}}_0(x_j,0)$. The boundary conditions are (1) the surface moisture content and temperature are given

$$H=\overline{H}, \mathrm{on} \mathrm{boundary} S_H$$

(6a)

$$T=\overline{T}, \mathrm{on} \mathrm{boundary} S_T$$

(6b)

and (2) the moisture transfer vector and heat flux are given

$$J_i{n}_i=\overline{J}, \mathrm{on} \mathrm{boundary} S_j$$

(7a)

$$q_i{n}_i=\overline{q}, \mathrm{on} \mathrm{boundary} S_q$$

(7b)

where S_j + S_H = S_q + S_T = S, S is the surface surrounding the body, the overbar means the prescribed value, and n_i are the components of the unit vector n that is normal to the exterior surface of the body. On the boundaries S_j and S_q, the prescribed moisture transfer and thermal fluxes are positive if they are directed toward the exterior of the body.

3. Finite Element Scheme

Equation (5a,b) are fully coupled, nonlinear, and transient-governing equations for the moisture and heat transfer. To derive the system of finite element equations, it is assumed that the body is subjected to a virtual change of moisture content δH and a virtual change of temperature δT. By multiplying Equation (5a) by δH and Equation (5b) by δT and integrating the resulting equations in the whole space domain Ω, it can be seen that

$${\int }_{\Omega }\left[\mathsf{\nabla }\left(D_{11}\mathsf{\nabla }H\right)+\mathsf{\nabla }\left(D_{12}\mathsf{\nabla }T\right)-c_0{\displaystyle \frac{\partial H}{\partial t}}\right]\delta H\mathrm{d}\Omega +{\int }_{\Omega }\left[\mathsf{\nabla }(D_{21}\mathsf{\nabla }H)+\mathsf{\nabla }(D_{22}\mathsf{\nabla }T)-\rho c{\displaystyle \frac{\partial T}{\partial t}}\right]\delta T\mathrm{d}\Omega =0$$

(8)

The fundamental idea of the finite element model is to divide the continuum into a finite number of elements. For each element, the moisture content and temperature at any point inside the element can be expressed in terms of their values at their nodal points of the element by

$$H(x_1,x_2,x_3,t)=\left[N_H\right]\left\{\psi \right\}, T(x_1,x_2,x_3,t)=\left[N_T\right]\left\{\psi \right\}$$

(9)

$$\left\{\psi \right\}=(H_1,T_1,H_2,T_2,\dots ,H_n,T_n{)}^T$$

(10)

in which n is the number of nodes of the element and

$$\left[N_H\right]=(N_1,0,N_2,0,\dots ,N_n,0{)}^T$$

(11a)

$$\left[N_T\right]=(0,N_1,0,N_2,\dots ,0,N_n{)}^T$$

(11b)

are so-called shape function matrices and {Ψ} is a vector containing the moisture content and temperature values at the nodal points of the element. The gradients for moisture content and temperature at any point in the element can be written as

$$\{\partial H/\partial x_1,\partial H/\partial x_2,\partial H/\partial x_3{\}}^T=[B_H\left]\right\{\psi \}$$

(12a)

$$\{\partial T/\partial x_1,\partial T/\partial x_2,\partial T/\partial x_3{\}}^T=[B_T\left]\right\{\psi \}$$

(12b)

where $\left[B_H\right]=\left[L\right]\left[N_H\right]$, $\left[B_T\right]=\left[L\right]\left[N_T\right]$, and [L] denotes a differential operator matrix. Substituting Equation (12) into Equation (4), the general moisture transfers J_i and heat fluxes q_i in the element can be obtained as follows:

$$\left\{J\right\}=\{J_1,J_2,J_3{\}}^T=-D_{11}[B_V\left]\right\{\psi \}-D_{12}[B_T\left]\right\{\psi \}$$

(13a)

$$\left\{q\right\}=\{q_1,q_2,q_3{\}}^T=-D_{21}[B_V\left]\right\{\psi \}-D_{22}[B_T\left]\right\{\psi \}$$

(13b)

Finally, by substituting Equation (9) into Equation (8), the finite element formulation of the moisture and temperature conductions can be obtained as (after assemblage)

$$[C(\left\{\psi \right\})]\left\{\dot{\psi }\right\}+[K(\left\{\psi \right\})]\left\{\psi \right\}=\left\{p\right\}$$

(14)

where the dot represents differentiation with respect to time. The element matrices and general load vector $\left\{p\right\}$ are given by

$$[C(\{\psi \})]={\int }_{\Omega }{c}_0[N_H{]}^T[N_H]d\Omega +{\int }_{\Omega }\rho c[N_T{]}^T[N_T]d\Omega$$

(15a)

$$\begin{array}{l}[K(\{\psi \})]={\int }_{\Omega }[B_H{]}^T{D}_{11}[B_H]d\Omega +{\int }_{\Omega }[B_H{]}^T{D}_{12}[B_T]d\Omega \\ +{\int }_{\Omega }[B_T{]}^T{D}_{21}[B_H]d\Omega +{\int }_{\Omega }[B_T{]}^T{D}_{22}[B_T]d\Omega \end{array}$$

(15b)

$$\{p\}=-{\int }_{S_j}[N_H{]}^T{\overline{h}}_{J}ds-{\int }_{S_q}[N_T{]}^T{\overline{h}}_{q}ds$$

(15c)

The integrals involved in Equation (15) can be evaluated by the Gauss–Legendre numerical technique. Since $c_0$, D₁₁, D₁₂, and D₂₂ depend on the moisture content and temperature, Equation (14) is a system of highly nonlinear, time-related, first-order matrix differential equations. Furthermore, in real applications, the external load {p} may also vary with time.

Equation (14) will be solved by using direct numerical difference in time. For this, the value of {Ψ} is calculated at t_m and is denoted as {Ψ}_m. This is {Ψ}_m = {Ψ(t_m)} for some points $t_m$ in the interval [0, t₀], where t_m = m(Δt), m = 0, …, M and the step length Δt is such that Δt = t₀/M for an integer M. Based on previous experience [22,23], the central differential scheme for the time portion has less truncation error, therefore will be used. The central difference scheme of Equation (14) is

$$\left({\displaystyle \frac{[C]}{\mathsf{\Delta }t}}+{\displaystyle \frac{[K]}{2}}\right)\{\psi {\}}_{m+1}=\left({\displaystyle \frac{[C]}{\mathsf{\Delta }t}}-{\displaystyle \frac{[K]}{2}}\right)\{\psi {\}}_m+{\displaystyle \frac{\{p_m\}+\{p_{(m+1)}\}}{2}}$$

(16)

where m = 0~M. Note that the global matrices [C], [K], and {p} are functions of moisture content and temperature. They depend on the unknown vector {Ψ}. Due to this fact, Equation (16) is nonlinear in nature. To avoid iterative operation for the nonlinear equations, we can assume that [C], [K], and {p} at the time interval (t_m, t_m+1) depend only on {Ψ} at time t_m. Therefore, $\left[C\right],$ $\left[K\right],$ and {p} can be calculated based on the already known {Ψ(t_m)}. As a result, the only unknown in Equation (16) is $\{\psi {\}}_{m+1}$. Thus, Equation (16) is a general family of recurrence relations. The length of the time steps in Equation (16) can be smaller enough such that the calculation precision is satisfactory.

4. Steady-State Problem

If the problem is steady that does not depend on time, the matrix [C] vanishes. In this situation, the finite element system, Equation (14) becomes

$$\left[K\right(\left\{\psi \right\}\left)\right]\left\{\psi \right\}=\left\{p\right\{\psi \left\}\right\}$$

(17)

The equations are still nonlinear since the matrix [K] depends on the unknown $\left\{\psi \right\}$. Equation (17) can be solved by the following iteration procedure:

(1)

At the beginning, the stiffness matrix [K] and the load vector {p} are calculated at the initial moisture and temperature (the reference state). Equation (17) is solved, and the moisture and temperature vector {Ψ} is updated.

(2)

The [K] and the load vector {p} are calculated by using the updated {Ψ}. The finite element Equation (17) is solved again, and the moisture and temperature vector {Ψ} is updated.

(3)

Repeat the procedure (2) until the result is converged.

5. Illustrative Examples

Some typical examples are given to demonstrate the applicability of the proposed finite element model. These include the transient and steady moisture and heat transfer in a one-dimensional bar and a disk.

5.1. Transient 1D Moisture and Heat Transfer with Constant Material Properties

In this example, transient moisture and heat conduction in a bar are considered for which the material properties $D_m$, $D_h$, $\gamma$, and $\lambda$ are all constants. The exact solution to this problem is available and will be used to validate the finite element method.

The 1D form of the equations governing the moisture content and temperature, Equation (3), is

$$[D]{\displaystyle \frac{{\partial }^2}{\partial x^2}}\left\{\begin{array}{c}H\\ T\end{array}\right\}=[E]{\displaystyle \frac{\partial }{\partial t}}\left\{\begin{array}{c}H\\ T\end{array}\right\}$$

(18)

where $[D]=\left[\begin{array}{cc}{D}_m& \gamma D_h\\ \lambda D_m& D_h\end{array}\right]$ and $[E]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. Assume the bar is along the x-axis and its length is denoted as b. The left side and the right side of the bar are at x = 0 and x = b, respectively. Initially, the moisture content and the temperature of the bar are 0. At t = 0, the moisture content and temperature on both sides of the strip are suddenly changed to $H_0$ and $T_0$, respectively.

Assume the solution of Equation (18) is

$$\left\{\begin{array}{c}H\\ T\end{array}\right\}=\left\{\begin{array}{c}{H}_0\\ T_0\end{array}\right\}-{\displaystyle \frac{4}{\pi }}\sum _{m=1,3,5\dots }^{\infty }{\displaystyle \frac{1}{m}}\mathit{sin}\left({\displaystyle \frac{m\pi x}{b}}\right)\mathit{exp}\left(-{\left({\displaystyle \frac{m\pi }{b}}\right)}^2\xi t\right)A\{a\}$$

(19)

in which {a} is an unknown column of two elements, A and $\xi$, constants to be determined. Boundary conditions are automatically met by Equation (18). The constants A, $\xi$, and {a} are determined by satisfying the equilibrium equations and initial conditions of the problem.

Putting Equation (19) into Equation (18) yields

$$\left(\left[D\right]-\xi \left[E\right]\right)\{a\}=0$$

(20)

Thus, $\xi$ has two values, ${\xi }_1$ and ${\xi }_2$, which are the eigenvalues of the matrix $[D]-\xi [E]$. The corresponding eigenvectors for {a} are denoted as $\left\{a_1\right\}$ and $\left\{a_2\right\}$, respectively. For real problem, ${\xi }_1$ and ${\xi }_2$ should be positive.

As a result, the solution can be written as

$$\begin{gathered} \left\{\begin{array}{c}H\\ T\end{array}\right\}=\left\{\begin{array}{c}{H}_0\\ T_0\end{array}\right\}-{\displaystyle \frac{4}{\pi }}\sum _{m=\mathrm{1,3},5\dots }^{\infty }{\displaystyle \frac{1}{m}}\mathit{sin}\left({\displaystyle \frac{m\pi x}{b}}\right) \\ \times \left(\mathit{exp}(-{\left({\displaystyle \frac{m\pi }{b}}\right)}^2{\xi }_{1}t)A_1\{a_1\}+\mathit{exp}(-{\left({\displaystyle \frac{m\pi }{b}}\right)}^2{\xi }_{2}t)A_2\{a_2\}\right) \end{gathered}$$

(21)

The initial conditions will be used to obtain $A_1$ and $A_2$. Therefore, at t = 0, one obtains

$${\displaystyle \frac{4}{\pi }}\sum _{m=1,3,5\dots }^{\infty }{\displaystyle \frac{1}{m}}\mathit{sin}\left({\displaystyle \frac{m\pi x}{b}}\right)\left(A_1\left\{a_1\right\}+A_2\left\{a_2\right\}\right)=\left\{\begin{array}{c}{H}_0\\ T_0\end{array}\right\}$$

(22)

Since ${\displaystyle \frac{4}{\pi }}{\sum }_{m=\mathrm{1,3},\dots }^{\mathrm{\infty }}{\displaystyle \frac{1}{m}}\mathit{sin}\left({\displaystyle \frac{m\pi x}{b}}\right)=1$, Equation (22) yields

$$\left\{\begin{array}{c}{A}_1\\ A_2\end{array}\right\}={\left[\begin{array}{cc}\left\{a_1\right\}& \left\{a_2\right\}\end{array}\right]}^{-1}\left\{\begin{array}{c}{H}_0\\ T_0\end{array}\right\}$$

(23)

The following material properties are used for the numerical calculation: $\lambda =0.122$, $\gamma =2.053$, $D_m={\displaystyle \frac{D_0}{1-\lambda \gamma }}$, and $D_h=10D_m$ [21].

In total, 20 quadratic 2D elements and 42 nodal points are used to construct the grid pattern. There are 21 nodal points along the x direction and 2 nodal points normal to the x direction. The values of H and T as a function are found and are given in

Table 1. Comparison of finite element results and the analytical results (Equation (21)) for sudden change of surface moisture content from 0 to H₀.

$\mathit{t}/{\mathit{t}}_0,{\mathit{t}}_0=\frac{{\mathit{b}}^2}{{\mathit{D}}_0}$	$\mathit{x}/\mathit{b}$	$\mathit{T}/\left(\mathit{\lambda }{\mathit{H}}_0\right)$ Finite Element	$\mathit{T}/\left(\mathit{\lambda }{\mathit{H}}_0\right)$ (Analytic)	$\mathit{H}/{\mathit{H}}_0$ Finite Element	$\mathit{H}/{\mathit{H}}_0$ (Analytic)
0.125	0.0	0.000	0.000	1.000	1.000
	0.1	0.01241	0.01241	0.8850	0.8850
	0.2	0.02360	0.02360	0.7812	0.7812
	0.3	0.03249	0.03248	0.6989	0.6989
	0.4	0.03819	0.03818	0.6460	0.6460
	0.5	0.04015	0.04015	0.6278	0.6278
0.25	0.0	0.0000	0.0000	1.0000	1.0000
	0.1	0.003726	0.003732	0.9655	0.9654
	0.2	0.007087	0.007099	0.9343	0.9342
	0.3	0.009754	0.009770	0.9096	0.9094
	0.4	0.01147	0.01149	0.8937	0.8935
	0.5	0.01206	0.01208	0.8882	0.8880
0.375	0.0	0.0000	0.0000	1.0000	1.0000
	0.1	0.001119	0.001123	0.9896	0.9896
	0.2	0.002128	0.002135	0.9803	0.9802
	0.3	0.002929	0.002939	0.9729	0.9728
	0.4	0.003443	0.003455	0.9681	0.9680
	0.5	0.003620	0.003633	0.9664	0.9663

and

Table 2. Comparison of finite element results and the analytical results (Equation (21)) for sudden change of surface temperature from 0 to T₀.

$\mathit{t}/{\mathit{t}}_0,{\mathit{t}}_0=\frac{{\mathit{b}}^2}{{\mathit{D}}_0}$	$\mathit{x}/\mathit{b}$	T Finite Element	T (Analytic)	Relative Difference (%)	$\mathit{H}/\left(\mathit{\gamma }{\mathit{T}}_0\right)$ Finite Element	$\mathit{H}/\left(\mathit{\gamma }{\mathit{T}}_0\right)$ (Analytic)	Relative Difference (%)
0.01	0.0	1.0000	1.0000	0	0.0000	0.0000	0
	0.1	0.8862	0.8862	0	0.4451	0.4440	0.25
	0.2	0.7878	0.7879	0	0.6886	0.6862	0.35
	0.3	0.7136	0.7137	0.01	0.7394	0.7358	0.49
	0.4	0.6675	0.6675	0	0.7172	0.7155	0.24
	0.5	0.6517	0.6517	0	0.7029	0.7023	0.09
0.02	0.0	1.0000	1.0000	0	0.0000	0.0000	0
	0.1	0.9635	0.9634	0.01	0.3787	0.3781	0.16
	0.2	0.9319	0.9318	0.01	0.6701	0.6698	0.04
	0.3	0.9085	0.9084	0.01	0.8427	0.8409	0.21
	0.4	0.8944	0.8942	0.02	0.9187	0.9161	0.29
	0.5	0.8897	0.8896	0.01	0.9380	0.9352	0.30
0.03	0.0	1.0000	1.0000	0	0.0000	0.0000	0
	0.1	0.9843	0.9843	0	0.3291	0.3288	0.09
	0.2	0.9707	0.9707	0	0.6061	0.6060	0.02
	0.3	0.9606	0.9605	0.01	0.8018	0.8007	0.14
	0.4	0.9545	0.9545	0	0.9121	0.9099	0.24
	0.5	0.9525	0.9525	0	0.9468	0.9442	0.28

for various values of time while x varies from 0.0 to 0.5b in increments of 0.1b. Due to symmetry, the left half and the right half of the bar have the same temperature and moisture distributions. It can be seen that the deviations between the finite element and analytical results are very small. In Table 1, the relative difference between the proposed finite element model and the analytical solution is almost 0; therefore, it is not given. In Table 1, the maximum relative difference is about 0.05%, which is quite small.

Khoshbakht et al. [16] argued that the effect of moisture content on the heat conduction is negligible (i.e., λ in Equation (3b) is taken as 0). They ignored the influence of $D_m$ on the temperature and used the following uncoupling system of moisture and temperature relations:

$$\mathsf{\nabla }\cdot (D_m\mathsf{\nabla }H)+\mathsf{\nabla }\cdot (\gamma D_h\mathsf{\nabla }T)={\displaystyle \frac{\partial H}{\partial t}}$$

(24a)

$$\mathsf{\nabla }\cdot (D_h\mathsf{\nabla }T)={\displaystyle \frac{\partial T}{\partial t}}$$

(24b)

In Figure 1, Figure 2, Figure 3 and Figure 4, results for fully coupling (Equation (3)) and uncoupling (Equation (24)) are shown for sudden moisture and sudden temperature changes on the surfaces of the bar. The results of uncoupling and fully coupling are small when time is small or at the steady state. In fact, at the steady state, Equations (3) and (24) show that the moisture and temperature are governed, respectively, by two independent equations, $\mathsf{\nabla }\cdot (D_m\mathsf{\nabla }H)=0$ and $\mathsf{\nabla }\cdot (D_h\mathsf{\nabla }T)=0$. They are, therefore, totally uncoupling at the steady state. Generally, at the transient state, the differences between the uncoupling and fully coupling are very significant. Therefore, it is important that when studying the transient solutions of the moisture and temperature, fully coupling between the moisture and heat conductions should be considered. Otherwise, the results will be considerably different from the exact values or even completely wrong.

5.2. Steady 1D Moisture and Heat Transfer with Temperature-Dependent Material Properties

In this example, moisture and temperature distributions at steady state in a one-dimensional bar is analyzed with the present FEM method. The length of the bar is 20 mm, the material is T300/5208 composite, and its properties are moisture- and temperature-dependent [21].

$$D_m={\displaystyle \frac{D_0}{1-\lambda \gamma \times 0.607}}\mathit{exp}\left(-{\displaystyle \frac{E_d}{R_{g}T}}\right) ({\mathrm{m}}^2/\mathrm{s})$$

(25a)

$$D_h=10D_m$$

(25b)

$$\lambda =0.122, \gamma =2.053$$

(25c)

$$D_0=4.25\times 10^{-5} ({\mathrm{m}}^2/\mathrm{s})$$

(25d)

$$E_d=5.23\times 10^4 (\mathrm{k}\mathrm{J}/(\mathrm{k}\mathrm{g}·\mathrm{mol}\left)\right)$$

(25e)

$$R_g=8.314 (\mathrm{J}/(\mathrm{mol}·\mathrm{K}\left)\right)$$

(25f)

Boundary conditions are such that the left side of the bar has a temperature of 293 K and a moisture content of 28 kg/m³, and the right side of the bar has a temperature of 313 K and a moisture content of 94 kg/m³.

Distributions of the moisture content and temperature for this example are shown in Figure 5 and Figure 6. For the curves with constant properties, the material properties are calculated at the initial temperature T = 293 K. As expected, if the material properties are constant, the moisture content and temperature profiles are linear. If the temperature-dependent properties are considered, the corresponding temperature and moisture content are considerably higher than those with constant properties. More specifically, at the middle of the bar, moisture content without considering the temperature dependence of the material properties is 61 kg/m³, while the moisture content with temperature-dependent material properties is 17% higher and is 71.4 kg/m³.

5.3. Transient 1D Moisture and Heat Transfer with Temperature-Dependent Material Properties

For this example, the material properties are the same as the example of Section 5.2. Initially, the bar is at a moisture of 28 kg/m³ and a temperature of 293 K. The left side of the bar is kept at constant moisture and constant temperature, and the moisture and temperature at the right side of the bar are suddenly increased to 94 kg/m³ and 313 K, respectively.

Distributions of the moisture and temperature are displayed in Figure 7 and Figure 8 for various values of time. When time is infinite, the moisture content and temperature become steady state, and they are the same as those of Figure 5 and Figure 6. It can be seen that the temperature can reach steady state much faster (10 times earlier) than the moisture content. This is due to the fact that the current material has a much higher value of thermal diffusivity than moisture diffusivity. It is interesting to see that there is a change in the concavity in Figure 7, from 10⁸ s to 10¹⁰ s, for moisture. This is due to the fact that the distribution of moisture is affected not only by the temperature but also by the moisture-dependent properties. At different times, the contributions of the moisture and temperature are different. As such, the distribution of moisture is a very complicated function of time and position.

5.4. Transient 1D Moisture and Heat Transfer in a Solid Disk with Constant Material Properties

The exact solution for this problem is also available. Therefore, the developed FEM model will be used to study the problem, and the results will be compared with the exact solutions. In order to obtain an exact solution, it is assumed that the problem is axisymmetric and the moisture content and heat only vary with the radial coordinate r and time. The disk is initially at 0 moisture content and 0 temperature. The boundary condition is such that the moisture content and temperature on the outer surface of the disk (at r = b) are suddenly changed to H₀ and T₀, respectively.

Since $D_m$, $D_h$, $\gamma$, and $\lambda$ are all constants, the 1D form of the equations governing the moisture content and temperature along the radial direction is as follows:

$$[D]\left({\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\right)\left\{\begin{array}{c}H\\ T\end{array}\right\}=[E]{\displaystyle \frac{\partial }{\partial t}}\left\{\begin{array}{c}H\\ T\end{array}\right\}$$

(26)

where $[D]=\left[\begin{array}{cc}{D}_m& \gamma D_h\\ \lambda D_m& D_h\end{array}\right]$ and $[E]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$.

The conduction Equation (26) can be solved by a standard separation-of-variables technique. The result is

$$\left\{\begin{array}{c}V\\ T\end{array}\right\}=\left\{\begin{array}{c}{H}_0\\ T_0\end{array}\right\}-{\sum }_{n=1}^{\infty }{\displaystyle \frac{2J_0\left({\gamma }_n\frac{r}{b}\right)}{{\gamma }_n{J}_1\left({\gamma }_n\right)}}\left(\exp (-{\left({\displaystyle \frac{{\gamma }_n}{b}}\right)}^2{\xi }_{1}t)A_1\{a_1\}+\exp (-{\left({\displaystyle \frac{{\gamma }_n}{b}}\right)}^2{\xi }_{2}t)A_2\{a_2\}\right)$$

(27)

in which J₀ is the zero-order Bessel function of the first kind, J₁ is the first-order Bessel function of the second kind, γ_n are the roots of the equation J₀(γ_n) = 0, ${\xi }_1$ and ${\xi }_2$ are eigenvalues, and $\left\{a_1\right\}$ and $\left\{a_2\right\}$ are corresponding eigenvectors of Equation (20). The constants A₁ and A₂ are determined by the initial conditions of the problem, and they are the same as Equation (23).

Once again, the material properties are taken as $\lambda =0.122$, $\gamma =2.053$, $D_m={\displaystyle \frac{D_0}{1-\lambda \gamma }}$, and $D_h=10D_m$ [21].

Note that the 2D numerical model has been developed. Of course, the 2D numerical model can be used to solve the 1D problem (which has a known analytical solution) and to verify the numerical model. As such, the problem is also studied by the two-dimensional finite element modeling. Due to axisymmetricity, only one-fourth of the disk needs to model. A finite element mesh configuration with 20 divisions along the radial direction and 10 divisions along the circumferential direction is shown in Figure 9.

Analytical solutions and the finite element solutions are given in

Table 3. Comparison of finite element results and the analytical results (Equation (27)) for sudden change of surface moisture content from 0 to H₀.

$\mathit{t}/{\mathit{t}}_0,{\mathit{t}}_0=\frac{{\mathit{b}}^2}{{\mathit{D}}_0}$	$\mathit{r}/\mathit{b}$	$\mathit{T}/\left(\mathit{\lambda }{\mathit{H}}_0\right)$ Finite Element	$\mathit{T}/\left(\mathit{\lambda }{\mathit{H}}_0\right)$ (Analytic)	Relative Difference (%)	$\mathit{H}/{\mathit{H}}_0$ Finite Element	$\mathit{H}/{\mathit{H}}_0$ (Analytic)	Relative Difference (%)
0.1	0.0	0.08991	0.08987	0	0.1663	0.1659	0.24
	0.2	0.08597	0.08601	0	0.2032	0.2021	0.10
	0.4	0.07388	0.07407	0.25	0.3146	0.3129	0.54
	0.6	0.05373	0.05392	0.4	0.5017	0.4999	0.36
	0.8	0.02744	0.02757	0.5	0.7455	0.7443	0.16
	1.0	0.0000	0.0000	0	1.0000	1.0000	0
0.2	0.0	0.05394	0.05415	0.39	0.4999	0.4980	0.34
	0.2	0.05083	0.05111	0.55	0.5288	0.5260	0.53
	0.4	0.04226	0.04253	0.64	0.6083	0.6057	0.43
	0.6	0.02949	0.02969	0.68	0.7266	0.7247	0.26
	0.8	0.01459	0.01469	0.69	0.8648	0.8638	0.12
	1.0	0.000	0.0000	0	1.0000	1.0000	0
0.3	0.0	0.03075	0.03099	0.78	0.7149	0.7127	0.31
	0.2	0.02894	0.02923	1.0	0.7317	0.7291	0.36
	0.4	0.02399	0.02424	1.04	0.7776	0.7753	0.30
	0.6	0.01668	0.01685	1.02	0.8454	0.8438	0.19
	0.8	0.008227	0.008313	1.05	0.9237	0.9229	0.09
	1.00	0.0000	0.0000	0	1.0000	1.0000	0

and

Table 4. Comparison of finite element results and the analytical results (Equation (27)) for sudden change of surface temperature from 0 to T₀.

$\mathit{t}/{\mathit{t}}_0,{\mathit{t}}_0=\frac{{\mathit{b}}^2}{{\mathit{D}}_0}$	$\mathit{r}/\mathit{b}$	T Finite Element	T (Analytic)	Relative Difference (%)	$\mathit{H}/\left(\mathit{\gamma }{\mathit{T}}_0\right)$ Finite Element	$\mathit{H}/\left(\mathit{\gamma }{\mathit{T}}_0\right)$(Analytic)	Relative Difference (%)
0.01	0.0	0.2837	0.2828	0.32	0.3060	0.3051	0.29
	0.2	0.3206	0.3185	0.66	0.3459	0.3436	0.67
	0.4	0.4247	0.4225	0.52	0.4581	0.4557	0.53
	0.6	0.5851	0.5833	0.31	0.6274	0.6234	0.64
	0.8	0.7833	0.7823	0.13	0.6626	0.6604	0.33
	1.0	1.000	1.0000	0	0.0000	0.0000	0
0.02	0.0	0.6550	0.6528	0.34	0.7066	0.7042	0.34
	0.2	0.6736	0.6710	0.39	0.7266	0.7237	0.40
	0.4	0.7247	0.7224	0.32	0.7785	0.7752	0.43
	0.6	0.8014	0.7998	0.20	0.8061	0.8029	0.40
	0.8	0.8968	0.8960	0.09	0.5902	0.5894	0.14
	1.0	1.000	1.0000	0	0.0000	0.0000	0
0.03	0.0	0.8290	0.8272	0.22	0.8940	0.8920	0.22
	0.2	0.8374	0.8355	0.23	0.9014	0.8989	0.28
	0.4	0.8609	0.8593	0.19	0.9071	0.9044	0.27
	0.6	0.8979	0.8968	0.12	0.8309	0.8297	0.14
	0.8	0.9466	0.9460	0.06	0.5260	0.5258	0.04
	1.0	1.000	1.0000	0	0.0000	0.0000	0

. Both solutions agree with each other very well. If finer mesh configurations are used, the finite element results and the analytical solutions are identical. In both cases, the relative difference between the proposed finite element model and the analytical solution is less than 1%.

6. Conclusions

The constitutive equations governing the moisture content and temperature are usually highly nonlinear. For the majority of applications, the coupling diffusion equations are not amenable to analytical solutions. As such, a numerical method is essential. This paper constructs a finite element numerical technique to study the time-dependent equations that govern the coupled phenomenon of heat and moisture diffusion. The following conclusions can be drawn:

(a)

The proposed numerical code includes the nonlinear coupling between the temperature and moisture.

(b)

The code can consider the temperature- and moisture-dependent material properties.

(c)

The code can study the transient heat and moisture conduction. The finite difference is employed to resolve the time-dependent portion of the formulation. Finite element discretization applies only to the space variables.

(d)

Numerical examples are given to validate the proposed computer code. Accuracy of the proposed model is confirmed.

(e)

Taking into account fully coupling of moisture and heat transfer is essential. As such, all existing theoretical solutions neglecting the coupling are not exact and even completely wrong.

(f)

Although numerical results are presented for two-dimensional problems, the extension to three dimensions is obvious and straightforward.