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 B
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
To provide a physical interpretation of Eq. (B.1), which is Eq. (2.4.9) with γv = 1, we consider a dispersed system and an averaging volume in the shape of a rectangular parallelepiped ΔxΔyΔz with its centroid located at (x, y, z), as illustrated in a. Its top view is shown b.
Clearly, for those elements of the dispersed phase k that are completely inside the averaging volume, where δAk is the closed surface of the element. Such an element, labeled ⓐ in b, may be a bubble or a droplet, spherical or nonspherical. Next, we consider those elements of the dispersed phase that are intersected by the boundary surface ΔAx + (Δx/2). One such element is labeled ⓑ in b. The unit outdrawn normal vector nk can be represented by where i, j, and k are unit vectors pointing in the positive directions of x, y, and z-axis, respectively, and e1, e2, and e3 are the direction cosines of nk. If we denote the portion of the interfacial area of element ⓑ that is inside the averaging volume v by δAk, [x + (Δx/2)], and its area of intersection with the surface ΔAx + (Δx/2) by δAk, x + (Δx/2), then
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- Publisher: Cambridge University PressPrint publication year: 2011