Finite strip method applied to steady heat conduction and thermal radiation in a planar slab: absorbing ̶ emitting gray material and parallel diffuse surfaces

Abstract

This paper addresses a new finite strip method for the analysis of simultaneous heat conduction and thermal radiation in a planar slab with diffuse surfaces and filled with an absorbing and emitting material considered as a gray medium. The gray material is discretized into a finite number of strips where the temperature is approximated with quadratic expansions in local coordinates whose coefficients are unknowns inside each strip. The finite strip method consists in a set of discrete equations corresponding to energy balance equations united to the compatibility conditions of both temperature and heat flux between consecutive strips. The gray material is articulated with different combinations of thermal and optical properties. Numerical results for the temperature fields and the conductive, radiative and total heat fluxes are presented in graphical and tabular forms and they compared favorably with equivalent results employing standard calculation techniques. The three main features attributable to the finite strip method are simplicity, quick calculation, good convergence and quality results.

1. Introduction

The combination of heat conduction and thermal radiation mechanisms is normally encountered in semitransparent materials as cited in the textbooks by Howell et al. (2011) and Modest (2013). Practical engineering applications of semitransparent solids, such as industrial furnaces, glass manufacturing, fiber and foam insulations, high performance windows, solar collectors have been reported in publications by Viskanta and Anderson (1975) and Campo et al. (1986,1987).

Several theoretical and numerical studies exist for the prediction of the total heat transfer by simultaneous conduction and radiation in gray materials; that is materials with extinction coefficient independent of wavelength. As noted by Mishra et al. (2006), theoretical models for the quantification of radiation heat transfer have been recognized as computationally intensive and time consuming. These features are attributed to the difficulty and high processing cost involving the radiation mechanism, which are due to the integro-differential nature of the governing energy equations (Howell et al. (2011) and Modest (2013). However, reasonable efforts have been directed in the past to reduce the processing time by developing new models and enhancing the computational procedures of the existing models. Representative publications on these efforts are those by Mishra et al. (2004), Anteby et al. (2000), Ratzell III and Howell (1982). A pioneering theoretical formulation for radiation heat transfer in a planar slab was developed by Viskanta and Grosh (1962a). These authors generated a rigorous solution for the case of one-dimensional gray medium by way of a complex procedure that transforms the integro-differential energy equation into a nonlinear integral energy equation. The nonlinear integral equation was solved by an iterative procedure. The authors also extended their formulation to investigate the effects of boundary emissivities on radiation heat transfer in a gray medium (Viskanta and Grosh, 1962b). More recently, the discrete transfer method (Shah, 1979) and the collapsed dimension method (Mishra, 1997) have been implemented to address the same problem. Afterwards, a comparative study between the two methods was carried out in Talukdar and Mishra (2002). The emerging results are in very good agreement with those published by Viskanta and Grosh (1962a,b). However, an inevitable drawback was that the number of iterations required for sub-problems dominated by radiation is very large. According to Talukdar and Mishra (2002), for sub-problems dictated by conduction-radiation parameters N smaller or equal to 0.01 an under relaxation technique is mandatory to attain satisfactory convergence.

For a particular case of transparent materials, Dai and Fang (2014) used the thermal response factor method developed by Mitalas (1968) to estimate the heat transmission in bodies. In fact, the peculiarity of this work is that the absorbed solar radiation was treated as an internal source term in the descriptive energy equation.

The objective of the present work revolves around the application of a new finite strip method for the heat transfer analysis by simultaneous heat conduction and thermal radiation in an absorbing and emitting gray material forming a planar slab. Further, convection heat transfer is ignored. The method in question consists of an iterative tangent nonlinear formulation in which the gray medium is discretized into a finite number of strips. Inside each finite strip, the temperature variation is approximated using quadratic expansions in local coordinates whose coefficients are the primary unknowns in the problem. The resulting discrete equations correspond to a collection of balance energy equations and compatibility conditions of temperature and heat flux between successive strips. Unquestionably, the main features inherent to the finite strip method are simplicity and quick convergence. For verification, the finite strip method is applied to a planar slab problem with different combinations of thermal and optical properties. Numerical results in terms of temperature fields as well as conductive, radiative and total heat fluxes are presented in graphical and tabulated forms. Besides, the relative importance of each heat transfer mode is elucidated. A detailed comparison of the numerical results with others obtained by well established theoretical models demonstrates a good balance between accuracy and the number of iterations needed for convergence.

2. Steady Conservation Equation of Energy

A planar slab of thickness D made from a homogeneous, isotropic material is sketched in Figure 1. The material filling the planar slab is assumed to be gray, absorbing, emitting and nonscattering. For simplicity, the material properties are considered independent of temperature. The bounding parallel surfaces 1 and 2 in the planar slab are idealized as opaque and diffuse and are maintained at temperatures T₁ and T₂ . It is envisioned that heat transfer in the planar slab occurs by simultaneous heat conduction and thermal radiation. In addition, to avoid convection heat transfer, the planar slab is situated in a perfect vacuum.

The one dimensional geometry is a good approximation for the energy transport in many physical situations, such as insulation, atmospheres and furnaces. Besides, the one dimensional geometry is also a building block for the preliminary analysis of more involved geometries dealing with two and three dimensions.

Under steady-state conditions, the total heat flux q_t through the planar slab is constant. Thereby, the total heat flux q_t is quantified by the additive relation

where q_c(x) is the conductive heat flux and q_r(x)is the radiative heat flux occurring at the point with coordinate x as illustrated in Figure 1. The conductive heat flux q_c(x) is obtained from Fourier’s law (Arpaci, 1966):

where β_c is the thermal conductivity. Upon introducing the optical coordinate κ = Kx, where K is the material extinction coefficient, the radiative heat flux q_r(x) as given by Howell et al. (2011) and Modest (2013) is:

In this equation, i⁺(0) and i⁻(κ_D) represent the forward and backward total intensities at the bounding surfaces 1 and 2, respectively, κ_Dis the optical thickness and η is the refractive index of the gray material. Furthermore, μ = cosΘ, wherein Θ is the angle between the radiation direction and the x-axis as indicated in Figure 1. The blackbody total intensity i_b′ depends on the temperature T with the formula (Howell et al., 2011 and Modest, 2013).

where the Stefan-Boltzmann constant σ = 5.67 × 10⁻⁸ W/(m²K⁴). The quantities E₂( ) and E₃( ) in Eq. (3) emerge from the exponential integral function (Abramowitz and Stegun, 1965):

for n = 2 and n = 3.

The total intensities i⁺(0)and i⁻(κ_D) at the two bounding surfaces 1 and 2 can be evaluated by the pair of expressions

and

where ε₁ and ε₂ denote the respective surface emissivity of bounding surfaces 1 and 2.

Within the platform of simultaneous conduction-radiation heat transfer, the governing conservation equation of energy is taken from Howell et al. (2011) and Modest (2013):

Using Eqs. (2) and (3), allows us to write Eq. (7) in the form of a nonlinear integro-differential equation

which is subject to the prescribed temperature boundary conditions

From a fundamental framework, the pair of Eqs. (8) and (9) constitutes a nonlinear and nonlocal problem. On one hand, the nonlinear part presents difficulties because the blackbody total intensity i_b′ depends on the temperature field T (x), which is not known a priori. On the other hand, the non-local part implies that the total heat flux q_t at a point x depends on both the temperature T (x) and the temperature gradient dT/dx.

3. Numerical Computational Procedure: The New Finite Strip Method

Owing that the central objective in the study is to produce manageable discrete equations, the planar slab is divided into a finite number of L strips, each with variable thickness d_α (1 ≤ α ≤ L) as shown in Figure 2. Accordingly, a local coordinate ${\stackrel{-}{x}}^{\left(\alpha \right)}$ is assigned at the mid-plane of each strip, so that

The temperature T (x) at each strip is approximated by means of a quadratic expansion in the corresponding local coordinate $\stackrel{-}{x}$. Correspondingly, for a typical α–strip, the temperature is given by the equation

where T_o^(α) , T₁^(α) and T₂^(α) are the unknown temperature coefficients. It can be shown that: a) T_o^(α) represents the mean temperature in the α–strip and b) β_cT₁^(α) is the conduction heat flux at the mid-plane of the α–strip. Using this approximate temperature field, the energy equation (8) specialized at the mid-plane ${\stackrel{-}{x}}^{\left(\alpha \right)}$ = 0 of the α–strip can be channeled through the following expression

where κ_c^(α) is the optical coordinate at the mid-plane of the strip. The partial derivatives of Ψ_α with respect to the unknown temperature coefficients T_o^(γ), T₁^(γ) andT₂^(γ) associated with the γ−strip are represented by the following system of integro-partial differential equations

where δ_αγ stands for the Kronecker delta function.

Under the premises that the finite strips are thin, the blackbody total intensities i_b′ along with their derivatives in Eqs. (12)-(14) can be approximately evaluated in terms of the mean temperature T_o in each strip. By way of this simplification, the exponential integral functions E_n over the optical coordinate can be evaluated explicitly. The integrals can be treated with different quadrature schemes, such as Gaussian, Lobatto, Chebyshev, and Newton-Cotes (Press et al., 1986). The integrals in question are of the form

where w_j are the weight coefficients corresponding to the n discrete points μ_j. The differences between the various quadrature schemes lie in the values of w_j and μ_j. Consider f(μ) to be a polynomial of degree m, having m +1 coefficients. For Gaussian quadrature, which contains the 2n weights and points to be arbitrary, the maximum value of m for which the summation is exact is m = 2n – 1. In this work, the precise four-point Gaussian quadrature scheme (Press et al., 1986) was implemented.

Actually, the present model generates a system of L energy equations, wherein each equation is connected to a particular finite strip. It is important to add that the energy equations depend on the unknown temperature coefficients. As expected, additional equations have to be written to comply with the compatibility conditions of temperature and heat flux at the interfaces of adjacent finite strips. In equation form, these compatibility conditions are equivalent to:

a) The temperature compatibility at the interface between two adjacent strips is

b) The conduction heat flux compatibility at the interface between two adjacent strips is

Besides, the temperature boundary conditions at the bounding surfaces are

Overall, a nonlinear system of algebraic equations consisting of 3L equations and 3L unknown temperature coefficients is formed by Eqs. (11), (16), (17) and (18). In general, the nonlinear system of algebraic equations can be written compactly as follows

where is the vector of temperature coefficients

An iterative solution procedure is implemented to solve the above system of algebraic equations. Applying the Taylor’s series expansion of Ψ_i to the vector ${\overrightarrow{T}}_m$ corresponding to a certain iteration m, a linearization technique leads to the equation

After neglecting higher order terms. In this equation, represents the Jacobian matrix of the function Ψ_i and δ${\overrightarrow{T}}_m$ indicates the incremental vector of temperature coefficients defined by

Next, combining Eqs. (21) and (22), the unknown temperature coefficients T_m+1 are obtained iteratively using the equation

Here, the iterative process begins by guessing an initial temperature field in the gray material occupying the planar slab, which is associated with the limiting condition of heat conduction.

Convergence of the nonlinear system of algebraic equations in Eq. (19) is achieved when the Euclidean norm of the n-th increment of temperature normalized by the temperature is less than a pre-set tolerance. Then, the convergence criterion is given by the ratio

where ‖ ‖represents the Euclidian norm and ∣T_max∣ signifies the absolute value of the larger temperature between T₁ and T₂. The pre-set tolerance is usually set at 10⁻³. Excellent convergence patterns are obtained employing a relatively small number of iterations (normally, from two to six) in the algorithm.

4. Validation of the New Finite Strip Method

The conduction-radiation parameter is defined by N = β_cK/4σT_ref³, where β_c is the thermal conductivity, K is the extinction coefficient and T_ref is a reference temperature. In this work, T_ref has been taken as the largest temperature at the bounding surfaces, i.e., either T₁ or T₂. For high values of the conduction-radiation parameter N, heat conduction is the dominant mechanism, whereas for small values of N, thermal radiation is the dominant mechanism.

For gray materials, the accuracy of the algorithm has to be demonstrated by comparing the computed results provided by the new finite strip method in terms of temperature and heat fluxes with comparable published results that are available in the heat transfer literature.

For the computational domain, uniform meshes with 100, 150 and 200 finite strips are constructed. The number of iterations required for convergence usually varied between n = 2 and 6, being the maximum value (n = 6) associated with the limiting case of pure radiation N = 0. An adequate tolerance with an error equal to 0.001 is imposed. As it can be seen in Figure 3, the non-dimensional temperature distributions are in excellent agreement with those published by Talukdar and Mishra (2002). It should be added that in the paper by these authors, the number of iterations required for gray materials having various conduction-radiation parameters N = 0.1, 0.01, 0.001 and 0.0001 amounts to 80, 120, 600 and 650, respectively. For those particular cases connected to very small conduction-radiation parameter, such as N ≤ 0.01. under relaxation was necessary.

5. Presentation and Discussion of the Numerical Results

Figure 3 displays the non-dimensional temperature distributions Θ = T/T₁ varying with the relative distance x/D for the case of a gray material owing an optical thickness κ_D =1, temperature ratio T₂/T₁ = 0.5 coupled with four conduction-radiation parameter N = 0, 0.01, 0.1 and 10 together with a high surface emissivityε =1.0. With the exception of a high N = 10, the three non-dimensional temperature distributions exhibit a characteristic S-shape. For limiting pure radiation with N = 0 and weak radiation with N = 0.01 the two non-dimensional temperature distributions coincided with those obtained by Talukdar and Mishra (2002). For limiting pure radiation with N = 0, the non-dimensional temperature Θ at x/D = 1 attains a relatively high value of 0.75. This behavior demonstrates a non-dimensional temperature jump from 0.5 to 0.75. For weak radiation with N = 0.01 the non-dimensional temperature Θ at x/D = 1 attains the value of Θ = 0.50, that is the non-dimensional temperature of the right bounding surface. For limiting pure conduction with a high N = 10, the non-dimensional temperature distribution shows the negative sloped straight line ending with Θ = 0.5 at x/D = 1.

For the limiting case of pure radiation with N = 0, the non-dimensional temperature distributions in gray materials with temperature ratio T₁/T₂ = 0.5, different surface emissivity and optical thickness, are plotted in Figure. 4. Notice that a different temperature ratio T₂/T₁ = 0.5 was used in Figure 3. To assess the goodness of the new strip model in describing the heat transfer phenomena for pure radiation with N = 0, the respective non-dimensionless temperature distributions are compared with those reported in the seminal work by Viskanta and Grosh (1962b). It is observable that for all optical thicknesses κ_D the results are of excellent quality. Specifically, the largest difference of about 2.6% corresponds to a high optical thickness κ_D =10.

In Figure 4(a) for a small optical thickness K_D = 0.1 and surface emissivities ε = 0.1, 0.5 and 1, the non-dimensional temperatures in the gray material remain almost constant from the left to the right bounding surfaces.

In Figure 4(b) for a moderate optical thickness K_D = 1, the non-dimensional temperature curve remains constant for a surface emissivities ε = 0.1, and switches to monotonic sloped for higher surface emissivities of ε = 0.5 and 1. The curve slopes increase gradually with increments in the surface emissivity ε.

In Figure 4(c) for a large optical thickness K_D = 10, the non-dimensional temperature curves exhibit a parabolic behavior, which is accentuated with increments in the surface emissivity going from ε = 0.1 to 1.

The non-dimensional heat flux is conveniently defined by the relation ζ = q/(σT_ref⁴), where q denotes heat flux. Non-dimensional heat fluxes computed by the present model for gray materials characterized with optical thickness κ_D =10, temperature ratio T₂/T₁ = 0.5 and different values of the conduction-radiation parameter N are portrayed in Figure. 5. Here again, the surface emissivity takes two extreme values ε =1.0 and ε = 0.1. In Figure 5, the conductive, radiative and total non-dimensional fluxes are represented by ζ_c, ζ_r and ζ_t, respectively. A variety of straight lines, concave lines and convex lines are observed in the figure. For the extreme case dealing with pure radiation (N = 0), the maximum number of iterations needed for convergence was 5. On the contrary, for dominant conduction (N = 10), only 2 iterations were needed to achieve satisfactory convergence. Again, to maintain uniformity, a tolerance of 0.001 was pre-set in advance.

The numerically-obtained non-dimensional total fluxes ζ_t are listed in Tables 1 and 2. It is observable that the numbers compared favorably with those reported in the seminal publication by Viskanta and Grosh (1962b).

6. Conclusions

A new computational method, called the finite strip method is developed in this work dealing with the heat transfer analysis of simultaneous heat conduction and thermal radiation in an absorbing and emitting gray material filling a planar slab. The finite strip method seeks to transform the integro partial differential equation of energy into a nonlinear system of algebraic equations. The computational efficiency of the finite strip method is verified through a series of critical tests utilizing gray materials accounting for a wide variety of thermal properties and optical properties, such as the conduction-radiation parameter N, the surface emissivity ε and the optical thickness κ_D.

Comparisons of the computed non-dimensional temperatures Θ and the non-dimensional heat fluxes ζ_t with their counterparts from reliable publications in the heat transfer literature demonstrate that the finite strip method is capable of delivering results of superb accuracy for engineering applications. Of equal importance, the finite strip method necessitates a lesser number of iterations, small CPU times and remarkable convergence when compared against the standard numerical methods, such as finite differences and finite elements. Surely, these are unique features attributed to the finite strip method.