Axial conduction effect in flow through circular porous passages with prescribed wall heat flux

Abstract

This study presents a series solution for computation of the steady state temperature field in circular ducts with prescribed wall heat flux. The ducts are filled with fluid saturated porous materials. The developed methodology includes a simple transformation that improves the convergence of this series solution. The acquired solution includes the contribution of axial conduction that leads to a modified Graetz-type solution for these fluid passages. Finally, this solution is augmented by the contribution of frictional heating.

Appendix: Application of orthogonality condition

The objective of this analysis is to use the orthogonality condition for determination of A _m in Eq. 8 and B _m for insertion in Eq. 9. Replacing the parameter −λ ² _m with β ² _m and R _m (r) with Φ _m (r) in Eq. 8, it becomes

$$ \theta_{1} (\bar{x},r) = \sum\limits_{m = 1}^{\infty } {A_{m} \Upphi_{m} (r)e^{{\beta_{m}^{2} \bar{x}}} } \quad {\text{when}}\;\bar{x} < 0 $$

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Also, replacing the parameter R _m (r) with Ψ _m (r) in Eq. 9, it becomes

$$ \theta_{2} (\bar{x},r) = \sum\limits_{m = 1}^{\infty } {B_{m} \Uppsi_{m} (r)e^{{ - \lambda_{m}^{2} \bar{x}}} } \quad {\text{when}}\;\bar{x} \ge 0 $$

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Then, the compatibility conditions as given by Eqs. 13a, 13b leads to the relations

$$ \sum\limits_{m = 1}^{\infty } {A_{m} \Upphi_{m} (r)} = \sum\limits_{m = 1}^{\infty } {B_{m} \Uppsi_{m} (r)} + 2F(r) $$

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and

$$ \sum\limits_{m = 1}^{\infty } {A_{m} \beta_{m}^{2} \Upphi_{m} (r)} = - \sum\limits_{m = 1}^{\infty } {B_{m} \lambda_{m}^{2} \Uppsi_{m} (r)} + 2 $$

(29)

Next, one can multiply both sides of Eq. 28 by β ² _n /Pe ² − u/U, both sides of Eq. 29 by 1/Pe ², and add the resulting relations to get

$$ \sum\limits_{m = 1}^{\infty } {A_{m} \left( {{\frac{{\beta_{m}^{2} + \beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \Upphi_{m} (r) = \sum\limits_{m = 1}^{\infty } {B_{m} \left( {{\frac{{ - \lambda_{m}^{2} + \beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)\Uppsi_{m} (r)} + {\frac{2}{{Pe^{2} }}} + 2F(r)\left( {{\frac{{\beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right) $$

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To apply the orthogonality condition, one can multiply both sides of Eq. 30 by Φ _n (r)rdr for any given n and integrate over r to produce

$$ \begin{aligned} \sum\limits_{m = 1}^{\infty } {A_{m} } \int\limits_{0}^{1} {\left( {{\frac{{\beta_{m}^{2} + \beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \Upphi_{m} (r)\Upphi_{n} (r)rdr & = \sum\limits_{m = 1}^{\infty } {B_{m} } \int\limits_{0}^{1} {\left( {{\frac{{ - \lambda_{m}^{2} + \beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \\ & \times \Uppsi_{m} (r)\Upphi_{n} (r)rdr + 2\int\limits_{0}^{1} {\left[ {{\frac{1}{{Pe^{2} }}} + F(r)\left( {{\frac{{\beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \right]} \Upphi_{n} (r)rdr \\ \end{aligned} $$

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It is to be noted from the orthogonality condition in [8] that the first term on the right side of Eq. 31 would vanish since $- \beta_{n}^{2} \ne \lambda_{m}^{2}$ as they are in different domains for the same eigenfunctions. Therefore, the non-zero terms when $\beta_{n}^{2} = \beta_{m}^{2}$ for m = n lead to the value of

$$ A_{n} = {\frac{{2\int_{0}^{1} {\left[ {{\frac{1}{{Pe^{2} }}} + F(r)\left( {{\frac{{\beta_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \right]} \Upphi_{n} (r)rdr}}{{\int_{0}^{1} {\left( {{\frac{{2\beta_{n}^{2} }}{{{\text{Pe}}^{2} }}} - \frac{u}{U}} \right)} \Upphi_{n}^{2} (r)rdr}}}\;{\text{for}}\;n = 1,2,3, \ldots $$

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The aforementioned procedure can be modified for determination of coefficient B _m for insertion in Eq. 9. Multiplying both sides of Eq. 28 by $- \lambda_{n}^{2} /Pe^{2} - u/U$, both sides of Eq. 29 by 1/Pe ², and then adding the resulting relations yields

$$ \begin{aligned} \sum\limits_{m = 1}^{\infty } {A_{m} \left( {{\frac{{\beta_{m}^{2} - \lambda_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)\Upphi_{m} (r)} & = \sum\limits_{m = 1}^{\infty } {B_{m} \left( {{\frac{{ - \lambda_{m}^{2} - \lambda_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)\Uppsi_{m} (r)} \\ & + {\frac{2}{{{\text{Pe}}^{2} }}} + 2F(r)\left( {{\frac{{ - \lambda_{n}^{2} }}{{{\text{Pe}}^{2} }}} - \frac{u}{U}} \right) \\ \end{aligned} $$

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Finally, multiplying both sides of Eq. 33 by Ψ _n (r)rdr for a given n and integrating over r produces

$$ \begin{aligned} \sum\limits_{m = 1}^{\infty } {A_{m} } \int\limits_{0}^{1} {\left( {{\frac{{\beta_{m}^{2} - \lambda_{n}^{2} }}{{Pe^{2} }}} - \frac{u}{U}} \right)} \Upphi_{m} (r)\Uppsi_{n} (r)rdr & = - \sum\limits_{m = 1}^{\infty } {B_{m} } \int\limits_{0}^{1} {\left( {{\frac{{\lambda_{m}^{2} + \lambda_{n}^{2} }}{{Pe^{2} }}} + \frac{u}{U}} \right)} \\ & \times \Uppsi_{m} (r)\Uppsi_{n} (r)rdr + 2\int\limits_{0}^{1} {\left[ {{\frac{1}{{Pe^{2} }}} - F(r)\left( {{\frac{{\lambda_{n}^{2} }}{{Pe^{2} }}} + \frac{u}{U}} \right)} \right]} \Uppsi_{n} (r)rdr \\ \end{aligned} $$

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In this case, the left side of Eq. 34 vanishes due to the orthogonality condition [8] since $- \beta_{n}^{2} \ne \lambda_{m}^{2}$; therefore, the non-zero terms when $\lambda_{n}^{2} = \lambda_{m}^{2}$ for m = n lead to the value of

$$ B_{n} = {\frac{{\int\limits_{0}^{1} {\left[ {{\frac{1}{{Pe^{2} }}} - F(r)\left( {{\frac{{\lambda_{n}^{2} }}{{Pe^{2} }}} + \frac{u}{U}} \right)} \right]} \Uppsi_{n} (r)rdr}}{{\int\limits_{0}^{1} {\left( {{\frac{{2\lambda_{n}^{2} }}{{Pe^{2} }}} + \frac{u}{U}} \right)} \Uppsi_{n}^{2} (r)rdr}}}\;{\text{for}}\;n = 1,2,3, \ldots $$

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