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Multi-Scale approach for conduction heat transfer: one and two-equation models

Part 2: results for a stratified medium

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The averaged temperature fields calculated with the one and two-equation models, obtained in the Part 1 of this work, are compared with the analytical exact solution in the particular case of a stratified medium. Although the results obtained by the one-equation (without assuming the thermal equilibrium between the phases) and two-equation models are generally in good agreement with the exact solution, the two-equation model is more appropriate to deal with high contrast values for the thermophysical properties.

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Correspondence to H. P. Amaral Souto.

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Communicated by Cristina Turner.

This work was supported in part by the National Council for Scientific and Technological Development (CNPq-Brazil) through Grant 305958/2012-7.


Appendix A

The heat conduction problem (2.1)–(2.6) for the medium shown in Fig. 1 has to be solved. To construct the integral transform pair for the solution of this problem, a representation of an arbitrary function in terms of the eigenfunctions of the associated eigenvalue problem is needed. This eigenvalue problem is obtained by the separation of variables technique applied to the homogeneous version of the heat conduction problem (2.1)–(2.6). Then, the integral transform pair for the temperature functions with respect to the \(x\) variable is readily obtained from the eigenfunctions and the representation is split into two parts. We find for the \(\beta \)-phase

$$\begin{aligned} {T}_{\beta }(x, y, t)=\sum ^{\infty }_{m=0}\frac{\cos (\omega _{m}x)}{L/2}\, \theta ^{m}_{\beta }(\omega _{m}, y, t), \end{aligned}$$
$$\begin{aligned} \theta ^{m}_{\beta }(\omega _{m}, y, t)=\int \limits _{0}^{L}\cos (\omega _{m}x')\, {T}_{\beta }(x', y, t)\, {\mathrm{d}}x' \end{aligned}$$

and for the \(\sigma \)-phase

$$\begin{aligned} {T}_{\sigma }(x, y, t)=\sum ^{\infty }_{m=0}\frac{\cos (\omega _{m}x)}{L/2}\, \theta ^{m}_{\sigma }(\omega _{m}, y, t), \end{aligned}$$
$$\begin{aligned} \theta ^{m}_{\sigma }(\omega _{m}, y, t)=\int \limits _{0}^{L}\cos (\omega _{m}x')\, {T}_{\sigma }(x',y,t)\, {\mathrm{d}}x', \end{aligned}$$

where \(\omega _{m}=(2m+1)\pi /2L\).

Taking now the integral transform of Eqs. (2.1) and (2.2), multiplying both sides of these equations by \(\cos (\omega _{m}x)\) and integrating over the region \(0\le x\le L\) leads to

$$\begin{aligned} \int \limits _{0}^{L}\cos (\omega _{m}x)\frac{\partial ^{2}{T_{i}}}{\partial {x}^{2}}\, {\mathrm{d}}x \ , \end{aligned}$$

where the subscript \(i\) is defined as \(\beta \) or \(\sigma \). This term is evaluated integrating twice by part and taking into account the boundary conditions (symmetry) of the problem:

$$\begin{aligned} \int \limits _{0}^{L}\cos (\omega _{m}x)\frac{\partial ^{2}{T_{i}}}{\partial {x}^{2}}\, {\mathrm{d}}x&= \left[ \cos (\omega _{m}x)\frac{\partial {T_{i}}}{\partial {x}}\right] ^{L}_{0}+\omega _{m}\int \limits _{0}^{L} \frac{\partial {T_{i}}}{\partial {x}}\sin (\omega _{m}x)\, {\mathrm{d}}x \nonumber \\&= \Big [\sin (\omega _{m}x)T_{i}\Big ]^{L}_{0}-\omega ^{2}_{m}\int \limits _{0}^{L} T_{i}\cos (\omega _{m}x)\, {\mathrm{d}}x \nonumber \\&= -\omega ^{2}_{m}\theta ^{m}_{i} \, . \end{aligned}$$

The integral transform leads to the following ordinary differential equations for the \(\theta ^{m}_{\beta }\) and \(\theta ^{m}_{\sigma }\) transformed temperatures

$$\begin{aligned} \frac{1}{{\alpha }_{\beta }}\frac{\partial {\theta ^{m}_{\beta }}}{\partial {t}}= \frac{\partial ^{2}{\theta ^{m}_{\beta }}}{\partial {y}^{2}}-\omega ^{2}_{m}\theta ^{m}_{\beta }+ \frac{(-1)^{m}}{\omega _{m}}\frac{{q}_{\beta }}{{k}_{\beta }} \end{aligned}$$


$$\begin{aligned} \frac{1}{{\alpha }_{\sigma }}\frac{\partial {\theta ^{m}_{\sigma }}}{\partial {t}}= \frac{\partial ^{2}{\theta ^{m}_{\sigma }}}{\partial {y}^{2}}-\omega ^{2}_{m}\theta ^{m}_{\sigma }+ \frac{(-1)^{m}}{\omega _{m}}\frac{{q}_{\sigma }}{{k}_{\sigma }} \, . \end{aligned}$$

In the integral transformation, Eqs. (2.4)–(2.6) are used. Therefore, the boundary conditions for the heat conduction problem are embedded in the result.

Taking the Laplace transform of Eqs. (5.7) and (5.8), we obtain

$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\overline{\theta }^{\,m}_{\beta }-\left( \frac{p}{{\alpha }_{\beta }}+\omega ^{2}_{m} \right) \overline{\theta }^{\,m}_{\beta }=-\frac{(-1)^{m}}{\omega _{m}} \frac{{q}_{\beta }}{{k}_{\beta }}\frac{1}{p} \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\overline{\theta }^{\,m}_{\sigma }-\left( \frac{p}{{\alpha }_{\sigma }}+\omega ^{2}_{m} \right) \overline{\theta }^{\,m}_{\sigma }=-\frac{(-1)^{m}}{\omega _{m}} \frac{{q}_{\sigma }}{{k}_{\sigma }}\frac{1}{p} \end{aligned}$$

where \(\overline{\theta }\) is the transformed temperature and \(p\) is the Laplace variable.

The solutions \(\overline{\theta }^{\,m}_{\beta }\) and \(\overline{\theta }^{\,m}_{\sigma }\) of Eqs. (5.9) and (5.10) are given by

$$\begin{aligned} \overline{\theta }^{\,m}_{\beta }=\frac{(-1)^{m}}{\omega _{m}\, p} \frac{1}{\left( \displaystyle \frac{p}{{\alpha }_{\beta }}+\omega ^{2}_{m}\right) } \frac{{q}_{\beta }}{{k}_{\beta }}+A_{m}\cosh \left[ \sqrt{\frac{p}{{\alpha }_{\beta }} +\omega ^{2}_{m}}\,(y + e_{\beta })\right] \end{aligned}$$


$$\begin{aligned} \overline{\theta }^{\,m}_{\sigma }=\frac{(-1)^{m}}{\omega _{m}\, p} \frac{1}{\left( \displaystyle \frac{p}{{\alpha }_{\sigma }}+\omega ^{2}_{m}\right) } \frac{{q}_{\sigma }}{{k}_{\sigma }}+B_{m}\cosh \left[ \sqrt{\frac{p}{{\alpha }_{\sigma }} +\omega ^{2}_{m}}\,(e_{\sigma }-y)\right] . \end{aligned}$$

Applying the boundary conditions at \(y=0\)

$$\begin{aligned} {\overline{T}}_{\beta }&= {\overline{T}}_{\sigma } \end{aligned}$$
$$\begin{aligned} {k}_{\beta }\frac{\partial {{\overline{T}}_{\beta }}}{\partial {y}}&= {k}_{\sigma }\frac{\partial {{\overline{T}}_{\sigma }}}{\partial {y}} \end{aligned}$$

leads to

$$\begin{aligned} A_{m}\cosh \left( \sqrt{{\zeta }_{\beta }}\,e_{\beta }\right) +\frac{(-1)^{m}}{{\zeta }_{\beta }} \frac{1}{\omega _{m} p}\frac{{q}_{\beta }}{{k}_{\beta }}=B_{m}\cosh \left( \sqrt{{\zeta }_{\sigma }} \,e_{\sigma }\right) +\frac{(-1)^{m}}{{\zeta }_{\sigma }}\frac{1}{\omega _{m} p} \frac{{q}_{\sigma }}{{k}_{\sigma }},\qquad \end{aligned}$$
$$\begin{aligned} {k}_{\beta }A_{m}\sqrt{{\zeta }_{\beta }}\sinh \left( \sqrt{{\zeta }_{\beta }}\,e_{\beta }\right) =-{k}_{\sigma }B_{m} \sqrt{{\zeta }_{\sigma }}\sinh \left( \sqrt{{\zeta }_{\sigma }}\,e_{\sigma }\right) , \end{aligned}$$


$$\begin{aligned} {\zeta }_{\beta }=\frac{p}{ {\alpha }_{\beta } }+\omega ^{2}_{m}, \end{aligned}$$
$$\begin{aligned} {\zeta }_{\sigma }=\frac{p}{ {\alpha }_{\sigma } }+\omega ^{2}_{m}. \end{aligned}$$

Equations (5.15) and (5.16) are used to find the \(A_{m}\) and \(B_{m}\) values.

Finally, the temperature fields \({\overline{T}}_{\beta }\) and \({\overline{T}}_{\sigma }\) can be determined from Eqs. (5.1), (5.11), (5.3) and (5.12):

$$\begin{aligned} {\overline{T}}_{\beta }&= \frac{2}{L}\sum ^{\infty }_{m=0}\left\{ A_{m}\cosh \left[ \sqrt{{\zeta }_{\beta }}\,\left( y+e_{\beta }\right) \right] +\frac{(-1)^{m}}{{\zeta }_{\beta }}\frac{1}{\omega _{m} p} \frac{{q}_{\beta }}{{k}_{\beta }}\right\} \cos (\omega _{m}x)\end{aligned}$$
$$\begin{aligned} {\overline{T}}_{\sigma }&= \frac{2}{L}\sum ^{\infty }_{m=0}\left\{ B_{m}\cosh \left[ \sqrt{{\zeta }_{\sigma }}\,\left( e_{\sigma }-y\right) \right] +\frac{(-1)^{m}}{{\zeta }_{\sigma }}\frac{1}{\omega _{m} p} \frac{{q}_{\sigma }}{{k}_{\sigma }}\right\} \cos (\omega _{m}x) \end{aligned}$$

and the temperatures \({T}_{\beta }\) and \({T}_{\sigma }\) are obtained by numerical inversion of Laplace transformed temperatures.

Appendix B

For a unit cell of the medium shown in Fig. 1, the closure equations for the one-equation model are given by

$$\begin{aligned} \frac{\mathrm{d }^{2}{\chi ^{y}_{\beta }}}{\mathrm{d }y^{2}}=0 \qquad \qquad \mathrm{on} \ Y_{\beta } \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d }^{2}{\chi ^{y}_{\sigma }}}{\mathrm{d }y^{2}}=0 \qquad \qquad \mathrm{on} \ Y_{\sigma } \end{aligned}$$

along with the boundary conditions at the \(\partial Y_{\beta \sigma }\) interface

$$\begin{aligned} \chi ^{y}_{\beta }&= \chi ^{y}_{\sigma } \end{aligned}$$
$$\begin{aligned} -{k}_{\beta }\frac{\mathrm{d }{\chi ^{y}_{\beta }}}{\mathrm{d }y}&= -{k}_{\sigma }\frac{\mathrm{d }{\chi ^{y}_{\sigma }}}{\mathrm{d }y} +\left( {k}_{\beta }-{k}_{\sigma }\right) \end{aligned}$$

Here, we have taken \(\chi ^{x}_{\beta }=\chi ^{x}_{\sigma }=0\) as the solution of the associated homogeneous problems.

Integration leads to:

$$\begin{aligned} \chi ^{y}_{\beta }={A}_{\beta } \, y+{B}_{\beta } \end{aligned}$$
$$\begin{aligned} \chi ^{y}_{\sigma }={A}_{\sigma } \, y+{B}_{\sigma } \end{aligned}$$

and from the boundary conditions at the \(\beta \)\(\sigma \) interface for \(y=0\), we obtain

$$\begin{aligned} {B}_{\beta }&= {B}_{\sigma } \end{aligned}$$
$$\begin{aligned} -{k}_{\beta }{A}_{\beta }&= -{k}_{\sigma }{A}_{\sigma }+\left( {k}_{\beta }-{k}_{\sigma }\right) \end{aligned}$$

Now taking the intrinsic volume average of \(\chi ^{y}_{\beta }\) and \(\chi ^{y}_{\sigma }\) and imposing \(\big <\chi ^{y}_{\beta }\big >^{\beta }=0\) and \(\big <\chi ^{y}_{\sigma }\big >^{\sigma }=0\), we obtain

$$\begin{aligned} \left<\chi ^{y}_{\beta }\right>^{\beta }={B}_{\beta }-{A}_{\beta } \, e_{\beta } = 0 \end{aligned}$$
$$\begin{aligned} \left<\chi ^{y}_{\sigma }\right>^{\beta }={B}_{\sigma }+{A}_{\sigma } \, e_{\sigma }= 0 \, . \end{aligned}$$

Thus, after substitution of \({A}_{\beta }, \, {A}_{\sigma }, \, {B}_{\beta }\) and \({B}_{\sigma }\) into Eqs. (6.5) and (6.6) the following final result is obtained

$$\begin{aligned} \chi ^{y}_{\beta }=\frac{{k}_{\sigma }-{k}_{\beta }}{{k}_{\beta }+\displaystyle \frac{e_{\beta }}{e_{\sigma }}{k}_{\sigma }}\left( y + e_{\beta } \right) \end{aligned}$$
$$\begin{aligned} \chi ^{y}_{\sigma }=\frac{{k}_{\beta }-{k}_{\sigma }}{{k}_{\sigma }+\displaystyle \frac{e_{\sigma }}{e_{\beta }}{k}_{\beta }}\left( y - e_{\sigma } \right) \, . \end{aligned}$$

Appendix C

Taking the Laplace transform of Eqs. (2.16) and (2.17) gives

$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\hat{\overline{T}}^{2}_{\beta }-\frac{p}{{\alpha }_{\beta }} \hat{\overline{T}}^{2}_{\beta } =-\frac{1}{p}\frac{{\hat{q}}_{\beta }}{{k}_{\beta }}-\frac{1}{{k}_{\beta }}\left[ {k}_{\beta }- {(\rho c_{p})}_{\beta }\left<\alpha \right>\right] \frac{\mathrm{d }^{2}{}}{\mathrm{d }x^{2}}\left<\overline{T}\right> \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\hat{\overline{T}}^{2}_{\sigma }-\frac{p}{{\alpha }_{\sigma }} \hat{\overline{T}}^{2}_{\sigma }=-\frac{1}{p}\frac{{\hat{q}}_{\sigma }}{{k}_{\sigma }} -\frac{1}{{k}_{\sigma }}\left[ {k}_{\sigma }-{(\rho c_{p})}_{\sigma }\left<\alpha \right>\right] \frac{\mathrm{d }^{2}{}}{\mathrm{d }x^{2}}\left<\overline{T}\right> \end{aligned}$$

where \(\hat{\overline{T}}\) is the transformed temperature fluctuation and \(p\) is the Laplace variable. Now taking into account the relation (2.24)

$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\hat{\overline{T}}^{2}_{\beta }-\frac{p}{{\alpha }_{\beta }} \hat{\overline{T}}^{2}_{\beta } =-\frac{1}{p}\frac{{\hat{Q}}_{\beta }}{{k}_{\beta }} \end{aligned}$$
$$\begin{aligned} \frac{\mathrm{d }^{2}{}}{\mathrm{d }y^{2}}\hat{\overline{T}}^{2}_{\sigma }-\frac{p}{{\alpha }_{\sigma }} \hat{\overline{T}}^{2}_{\sigma }=-\frac{1}{p}\frac{{\hat{Q}}_{\sigma }}{{k}_{\sigma }} \end{aligned}$$


$$\begin{aligned} {\hat{Q}}_{\beta }={\hat{q}}_{\beta }+{k}_{\beta }\left( \frac{\left<\alpha \right>}{{\alpha }_{\beta }}-1 \right) \frac{\left<q\right>}{\left<k\right>}\frac{\cosh \left[ \sqrt{\displaystyle \frac{p}{\left<\alpha \right>}}x\right] }{\cosh \left[ \sqrt{\displaystyle \frac{p}{\left<\alpha \right>}}L\right] } \end{aligned}$$
$$\begin{aligned} {\hat{Q}}_{\sigma }={\hat{q}}_{\sigma }+{k}_{\sigma }\left( \frac{\left<\alpha \right>}{{\alpha }_{\sigma }}-1 \right) \frac{\left<q\right>}{\left<k\right>}\frac{\cosh \left[ \sqrt{\displaystyle \frac{p}{\left<\alpha \right>}}x\right] }{\cosh \left[ \sqrt{\displaystyle \frac{p}{\left<\alpha \right>}}L\right] } \, , \end{aligned}$$

the solutions for \(\hat{\overline{T}}^{2}_{\beta }\) and \(\hat{\overline{T}}^{2}_{\sigma }\) are given by

$$\begin{aligned} \hat{\overline{T}}^{2}_{\beta }=\frac{{\alpha }_{\beta }}{p^{2}} \frac{{\hat{Q}}_{\beta }}{{k}_{\beta }} +A\cosh \left[ \sqrt{\frac{p}{{\alpha }_{\beta }}}\,(y+e_{\beta })\right] \end{aligned}$$


$$\begin{aligned} \hat{\overline{T}}^{2}_{\sigma }=\frac{{\alpha }_{\sigma }}{p^{2}}\frac{{\hat{Q}}_{\sigma }}{{k}_{\sigma }}+B\cosh \left[ \sqrt{\frac{p}{{\alpha }_{\sigma }}}\,(e_{\sigma }-y)\right] \, . \end{aligned}$$

Applying the boundary conditions at \(y=0\)

$$\begin{aligned} \hat{\overline{T}}^{2}_{\beta }&= \hat{\overline{T}}^{2}_{\sigma } \end{aligned}$$
$$\begin{aligned} {k}_{\beta }\frac{\mathrm{d }{}}{\mathrm{d }y}\hat{\overline{T}}^{2}_{\beta }&= {k}_{\sigma }\frac{\mathrm{d }{}}{\mathrm{d }y}\hat{\overline{T}}^{2}_{\sigma } \end{aligned}$$

leads to

$$\begin{aligned} A\cosh \left( \sqrt{\frac{p}{{\alpha }_{\beta }}}\,e_{\beta }\right) +\frac{{\alpha }_{\beta }}{p^{2}}\frac{{\hat{Q}}_{\beta }}{{k}_{\beta }}=B\cosh \left( \sqrt{\frac{p}{{\alpha }_{\sigma }}}\,e_{\sigma }\right) +\frac{{\alpha }_{\sigma }}{p^{2}}\frac{{\hat{Q}}_{\sigma }}{{k}_{\sigma }} \end{aligned}$$
$$\begin{aligned} {k}_{\beta }A\sqrt{\frac{p}{{\alpha }_{\beta }}}\sinh \left( \sqrt{\frac{p}{{\alpha }_{\beta }}}\, e_{\beta }\right) =-{k}_{\sigma }B\sqrt{\frac{p}{{\alpha }_{\sigma }}}\sinh \left( \sqrt{\frac{p}{{\alpha }_{\sigma }}}\,e_{\sigma }\right) \, . \end{aligned}$$

From Eqs. (7.11) and (7.12) the \(A\) and \(B\) values are found and the temperatures \(\hat{T}^{2}_{\beta }\) and \(\hat{T}^{2}_{\sigma }\) are obtained by means of a numerical inversion (Frigo and Johnson 1998) of Laplace transformed temperatures.

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Moyne, C., Souto, H.P.A. Multi-Scale approach for conduction heat transfer: one and two-equation models. Comp. Appl. Math. 33, 433–449 (2014).

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Mathematics Subject Classification (2000)