Book contents
- Frontmatter
- Contents
- Figures and Table
- Foreword
- Foreword
- Foreword
- Nomenclature
- Preface
- Acknowledgments
- 1 Introduction
- 2 Averaging relations
- 3 Phasic conservation equations and interfacial balance equations
- 4 Local volume-averaged conservation equations and interfacial balance equations
- 5 Time averaging of local volume-averaged conservation equations or time-volume-averaged conservation equations and interfacial balance equations
- 6 Time averaging in relation to local volume averaging and time-volume averaging versus volume-time averaging
- 7 Novel porous media formulation for single phase and single phase with multicomponent applications
- 8 Discussion and concluding remarks
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- References
- Index
Appendix D
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Figures and Table
- Foreword
- Foreword
- Foreword
- Nomenclature
- Preface
- Acknowledgments
- 1 Introduction
- 2 Averaging relations
- 3 Phasic conservation equations and interfacial balance equations
- 4 Local volume-averaged conservation equations and interfacial balance equations
- 5 Time averaging of local volume-averaged conservation equations or time-volume-averaged conservation equations and interfacial balance equations
- 6 Time averaging in relation to local volume averaging and time-volume averaging versus volume-time averaging
- 7 Novel porous media formulation for single phase and single phase with multicomponent applications
- 8 Discussion and concluding remarks
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- References
- Index
Summary
The Fourier law of isotropic conduction for fluid phase k is which is valid for variable conductivity. Because ∇uk = cvk∇Tk, Eq. () can be written in a form relating the heat flux vector and the gradient of internal energy. Thus, where $\beta _k = \frac{{\kappa _k }}{{c_{vk} }}$. When κk or cvk, or both, vary with temperature, we write Accordingly, In deriving Eq. (), the relation ${}^{2i} \langle {\nabla {}^{2i} \langle {u_k } \rangle _{LF} } \rangle = \nabla {}^{2i} \langle {u_k } \rangle _{LF} $ has been used. Subsequently, time averaging leads to When βk is a constant, ${}^{2i} \langle {\beta _k } \rangle _{LF} = \beta _k $, and β̃kLF = β′k = 0. In addition, ${}^{2i} \langle {\nabla \tilde u_{kLF} } \rangle = 0$. Consequently, Eq. () simplifies to which is precisely the result given by Eq. ().
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- Publisher: Cambridge University PressPrint publication year: 2011