Direct and indirect methods of calculating entropy generation rates in turbulent convective heat transfer problems

Abstract

Computational fluid dynamics (CFD) solutions of turbulent convective heat transfer problems based on the mass, momentum and energy conservation principle provide all information to calculate the entropy production rate in such a transfer process. It can be determined in the post processing phase of a CFD calculation. Two methods are discussed in detail which can provide the information about the entropy production with different degrees of accuracy.

Appendix

1.1 Wall functions for $S^{+}_{{\rm PRO}, {\bar D}} \hbox{ and } S^{+}_{{\rm PRO},{\bar C}}$

The general form we assume for these two wall functions with respect to the mean profiles is

$$S^{+}_{{\rm PRO},i} = {A_i}\,\exp\,\left[-b_i \left(y^{+}- a_{i}\right)^{2}\right]; \quad i={\bar D}, {\bar C}.$$

(16)

In Kock (2003) the constants are determined from asymptotic considerations (y ⁺ → 0) and DNS data. They are listed Table 1.

Table 1 Constants in the wall functions for $S_{{\rm PRO},\overline D} \hbox{ and }S_{{\rm PRO},\overline C}.\;T_{\tau} = -q_w/\varrho c_p u^2_{\tau};\; \hbox{Ec}_{\tau}=u^2_{\tau}/c_pT_{\tau};\; u_{\tau}=\sqrt{\tau_w {\text{/}}\varrho}$

The wall distances$y^{+}_{\ln {\bar D}} \hbox{ and } y^{+}_{\ln {\bar C}}$ correspond to the intersection of the asymptotic representation of the velocity and temperature profiles, respectively, for y ⁺ → 0 and y ⁺ → ∞. They are $y^{+}_{\ln {\bar D}}= 11.6 \hbox{ and } y^{+}_{\ln {\bar C}} = 12.1$ for Pr = 0.71 and $y^{+}_{\ln {\bar C}} = 7.3$ for Pr = 5, respectively.

In order to find a representative value of the entropy production in the wall adjacent volume of finite volume approach, (16) is integrated over this volume. At a distance y ⁺ _mp (mp: midpoint, centre of the volume) we thus get:

$$S^{+}_{{\rm PRO},{\bar D} mp} = \frac{1}{2y^{+}_{mp}} \left( \frac{A_{\overline D}}{2} \sqrt{\frac{\pi}{b_{\overline D}}} \times \left[ \hbox{erf} \left( \sqrt{b_{\bar D}} 2y^{+}_{mp} - \sqrt{b_{\overline D}}{a_{\overline D}}\right) - \hbox{erf} \left(-\sqrt{b_{\overline D}}{a_{\overline D}}\right)\right]\right),$$

(17)

$$S^{+}_{{\rm PRO},{\bar C} mp} = \frac{1}{2y^{+}_{mp}} \left( \frac{A_{\bar C}} {2} \sqrt{\frac{\pi}{b_{\bar C}}} \times \left[ \hbox{erf} \left(\sqrt{b_{\bar C}} 2y^{+}_{mp} - \sqrt{b_{\overline C}}{a_{\overline C}}\right) - \hbox{erf} \left(-\sqrt{b_{\overline C}}{a_{\overline C}}\right)\right]\right).$$

(18)

1.2 Wall functions for $S^{+}_{{\rm PRO},{\ D}^{\prime}} \hbox{ and } S^{+}_{{\rm PRO},{\ C}^{\prime}}$

These wall functions are found by patching the asymptotic representations at their intersection points $y^{+}_{\ln {\bar D}} \hbox{ and } y^{+}_{\ln {\bar C}},$ respectively.

After an integration over the wall adjacent finite volume the midpoint values for the entropy production are

$$\begin{aligned} S^+_{{\rm PRO},D^{\prime} mp} &= \frac{1}{2y^+_{mp}} \left[0.15 Ec_{\tau} \frac{T_w}{T_{\tau}} y^+_{\ln \overline D} \right.\\ &\quad\left.+ Ec_{\tau} \frac{T_w^2}{T_{\tau}^2} \frac{1}{\kappa} \times \left[ \log \left\{ 1 + \frac{T_{\tau}}{T_w} \left( \log (2 y^+_{mp}) + C^+_{\bar D}\right)\right.\right.\right.\\ &\left.\left.\left.\quad- \log \left(1 + \frac{T_{\tau}}{T_w} \left( \log (y^+_{\ln \overline D}) + C^+_{\bar D}\right)\right )\right\} \right ] \right], \end{aligned}$$

(19)

$$\begin{aligned}S^+_{{\rm PRO},C^{\prime} mp} &= \frac{1}{2y^+_{mp}} \left[\vphantom{\frac{1}{T_{\tau}/{T_w} + \log \left(y^+_{\ln \overline C}\right) + C^+_{\overline C}}} 0.15\,Pr y^+_{\ln \overline C}\right.\\&\left.\quad + \frac{1}{T_{\tau}/{T_w} + \log \left(y^+_{\ln \overline C}\right) + C^+_{\overline C}} - \frac{1}{T_{\tau}/{T_w} + \log \left({2 y^+_{mp}}\right) + C^+_{\overline C}}\right].\end{aligned}$$

(20)

with $C^+_{\overline D} = 5.0 \hbox{ and } C^+_{\overline C} = 13.7\,Pr^{2/3}-7.5.$