Convective Heat Transfer Between a Bead Packing and Its Bounding Wall: Part I—Theory

Abstract

Forced convective heat transfer between fluid-saturated bead packings and the solid containing walls is important phenomena in different engineering fields. In practical applications, the volume-averaged governing equations and the use of Reynolds-Averaged Navier–Stokes (RANS) Computational Fluid Dynamics (CFD) simulations can provide a reasonable description of the averaged transport phenomena in bead packings without too much computational resources. However, it is still a challenge to treat conjugate problems because the presence of a bounding wall causes different modeling issues. Firstly, different exponential functions have been proposed for RANS CFD simulations to model the porosity variation of bead packings near the wall. These functions are usually determined by an inverse approach which solves the governing equations to approximate the physically measured flow fields (e.g., velocity, temperature). Given the variety of existing porosity models, it is difficult to select the most appropriate one in a particular case. Secondly, the volume-averaged quantities of the bead packing are incompatible with the local point quantities of the wall. This largely increases the difficulty in defining the boundary conditions at the wall. In the present work, a modeling framework is presented to simulate the forced convective heat transfer from the bead packing to its containing wall. These two critical issues are addressed by deriving volume-averaged governing equations using a non-constant Representative Elementary Volume (REV). In Part I of the article, the theoretical derivations are presented in detail. A procedure is also developed to select an appropriate porosity model from the packing microstructure. Part II of the article completes the investigation by validating the predictive capability of the derived governing equations by experiments.

Appendices

Appendix 1: Volume-Averaged Momentum Equation

We begin our analysis by averaging Eq. 21:

$$\frac{1}{V}{\int }_{{V}_{f}}\frac{\partial }{\partial t}\left({\rho }_{f }{{\varvec{u}}}_{{\varvec{f}}}\right) \mathrm{d}V+\frac{1}{V}{\int }_{{V}_{f}}\nabla \cdot \left({\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\right) \mathrm{d}V=-\frac{1}{V}{\int }_{{V}_{f}}\nabla p \mathrm{d}V+\frac{1}{V}{\int }_{{V}_{f}}\nabla \cdot \overline{\overline{\tau }} \mathrm{d}V+\frac{1}{V}{\int }_{{V}_{f}}{\rho }_{f}{\varvec{g}} \mathrm{d}V$$

(33)

With Eq. 17, the first term on the left-hand side of Eq. 33 can be expressed as:

$$\frac{1}{V}{\int }_{{V}_{f}}\frac{\partial }{\partial t}\left({\rho }_{f }{{\varvec{u}}}_{{\varvec{f}}}\right) \mathrm{d}V=\frac{\partial }{\partial t}\left({\phi }_{V} {\langle {\rho }_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}\right)$$

(34)

Using Gray’s theorem (1983) and considering no mass transfer between phases, the second term on the left-hand side of Eq. 33 writes:

$$\frac{1}{V} {\int }_{{V}_{f}}\nabla \cdot \left({\rho }_{f} {{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\right) dV=\nabla \cdot \left({\phi }_{V} {\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}\right)+\frac{\nabla V}{V} \left({\phi }_{V} {\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}-{\phi }_{A} {\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}^{f}\right)$$

(35)

We make use of the simplified form of ${\langle {\rho }_{f} {{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}$ developed by Faghri and Zhang (2006):

$${\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}={\langle {\rho }_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}={\langle {\rho }_{f}\rangle }^{f }\left({\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}{\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}+{\langle {{\varvec{u}}\boldsymbol{^{\prime}}}_{{\varvec{f}}}{{\varvec{u}}\boldsymbol{^{\prime}}}_{{\varvec{f}}}\rangle }^{f}\right)$$

(36)

where ${{\varvec{u}}\boldsymbol{^{\prime}}}_{{\varvec{f}}}$ is the fluctuating part of ${{\varvec{u}}}_{{\varvec{f}}}$ defined as ${{\varvec{u}}\boldsymbol{^{\prime}}}_{{\varvec{f}}}={{\varvec{u}}}_{{\varvec{f}}}-{\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}$. Following the development discussed in Eq. 19 and assuming low variations within the REV, ${\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}$ writes:

$${\langle {\rho }_{f}{{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}^{f}={\langle {\rho }_{f}\rangle }_{A}^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}^{f}={\langle {\rho }_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}$$

(37)

Directing the attention to the right-hand side of Eq. 33, with Gray’s theorem (1983), the first term writes:

$$-\frac{1}{V}{\int }_{{V}_{f}}\nabla p \mathrm{d}V=- \nabla \left({\phi }_{V} {\langle p\rangle }^{f}\right)-\frac{\nabla V}{V} {\langle p\rangle }^{f} \left({\phi }_{V}-{\phi }_{A}\right)- \frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}} p{\mathrm{d}}A$$

(38)

The pressure $p$ is then decomposed as an intrinsic averaged term ${\langle p\rangle }^{f}$ and a fluctuating part $p^{\prime}$, i.e., $p={\langle p\rangle }^{f}+p^{\prime}$. Equation 38 is then expressed as:

$$-\frac{1}{V}{\int }_{{V}_{f}}\nabla p \mathrm{d}V=- {\phi }_{V} \nabla {\langle p\rangle }^{f}-{\langle p\rangle }^{f} \nabla {\phi }_{V}- \frac{{\langle p\rangle }^{f}}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}} {\mathrm{d}}A-\frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}} p^{\prime}{\mathrm{d}}A-\frac{\nabla V}{V} {\langle p\rangle }^{f} \left({\phi }_{V}-{\phi }_{A}\right)$$

(39)

We make use of the equation proposed by Whitaker (2013):

$$\frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}} {\mathrm{d}}A=- \nabla { \phi }_{V}$$

(40)

and write Eq. 39 as:

$$-\frac{1}{V}{\int }_{{V}_{f}}\nabla p \mathrm{d}V=- {\phi }_{V} \nabla {\langle p\rangle }^{f}-\frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}} p^{\prime}{\mathrm{d}}A-\frac{\nabla V}{V} {\langle p\rangle }^{f} \left({\phi }_{V}-{\phi }_{A}\right)$$

(41)

The second term on the right-hand side of Eq. 33 can be written as:

$$\frac{1}{V} {\int}_{{V_{f} }} \nabla \cdot \overline{\overline{\tau }} \,{\text{d}}V = \nabla \cdot \left( \phi_{V} {\langle \overline{\overline{\tau }} \rangle}^{f} \right) + \frac{\nabla V}{V} {\langle \overline{\overline{\tau }} \rangle}^{f} \left( {\phi_{V} - \phi_{A} } \right) + \frac{1}{V} {\int}_{{A_{fs} }} {\varvec{n}}_{{{\varvec{fs}}}} \overline{\overline{\tau }} \user2{ }{\text{d}}A$$

(42)

As ${\varvec{g}}$ is constant, the last term on the right-hand side of Eq. 33 can be written explicitly as:

$$\frac{1}{V} {\int }_{{V}_{f}}{\rho }_{f}{\varvec{g}} \mathrm{d}V = \frac{{\varvec{g}}}{V} {\int }_{{V}_{f}}{\rho }_{f} \mathrm{d}V={\phi }_{V} {\langle {\rho }_{f}\rangle }^{f}{\varvec{g}}$$

(43)

Substitution of Eqs. 34 to 37 and Eqs. 41 to 43 into Eq. 33 yields finally:

$$\begin{aligned} & \frac{\partial }{{\partial t}}\left( {\phi _{V} ~\left\langle {\rho _{f} } \right\rangle ^{f} ~\left\langle {\varvec{u}_{\varvec{f}} } \right\rangle ^{f} } \right) + \nabla \cdot \left( {\phi _{V} ~\left\langle {\rho _{f} } \right\rangle ^{f} ~\left\langle {\varvec{u}_{\varvec{f}} } \right\rangle ^{f} \left\langle {\varvec{u}_{\varvec{f}} } \right\rangle ^{f} } \right) \\ & \quad = - \phi _{V} \nabla \left\langle p \right\rangle ^{f} + \nabla \left( {\phi _{V} \left\langle {\overline{\overline{\tau }} } \right\rangle ^{f} } \right) + \phi _{V} ~\left\langle {\rho _{f} } \right\rangle ^{f} \varvec{g} - \nabla \cdot \left( {\phi _{V} ~\left\langle {\rho _{f} } \right\rangle ^{f} ~\left\langle {\varvec{u^{\prime}}_{\varvec{f}} \varvec{u^{\prime}}_{\varvec{f}} } \right\rangle ^{f} } \right) \\ & \quad + ~\frac{1}{V}{\int}_{{A_{{fs}} }} \varvec{n}_{{\varvec{fs}}} ~\left( {\overline{\overline{\tau }} - p^{\prime}} \right)\varvec{~}{\text{d}}A + \frac{{\nabla V}}{V}~\left( {\phi _{V} - \phi _{A} } \right)~\left( {\left\langle {\overline{\overline{\tau }} } \right\rangle ^{f} - \left\langle {\rho _{f} } \right\rangle ^{f} ~\left\langle {\varvec{u}_{\varvec{f}} \varvec{u}_{\varvec{f}} } \right\rangle ^{f} - \left\langle p \right\rangle ^{f} } \right) \\ \end{aligned}$$

(44)

This gives Eq. 23 in the main text.

Appendix 2: Volume-Averaged Thermal Energy Equations

2.1 Fluid Phase Volume-Averaged Thermal Energy Equation

The following equation is obtained by averaging Eq. 24 over a REV of volume $V$:

$$\frac{1}{V}{\int }_{{V}_{f}}\frac{\partial }{\partial t}\left({\rho }_{f }{C}_{f }{T}_{f}\right) \mathrm{d}V+\frac{1}{V}{\int }_{V}\nabla \cdot \left({\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\right) \mathrm{d}V=\frac{1}{V}{\int }_{{V}_{f}}\nabla \cdot \left({\lambda }_{f }\nabla {T}_{f}\right) \mathrm{d}V$$

(45)

The integration and differentiation of the first term on the left-hand side of Eq. 45 can be interchanged, due to the independency in time:

$$\frac{1}{V}{\int }_{{V}_{f}}\frac{\partial }{\partial t}\left({\rho }_{f }{C}_{f }{T}_{f}\right) \mathrm{d}V=\frac{\partial }{\partial t}\left(\frac{1}{V} {\int }_{{V}_{f}}{\rho }_{f }{C}_{f }{T}_{f} dV\right)=\frac{\partial }{\partial t}\left({{\phi }_{V} \langle {\rho }_{f }{C}_{f }{T}_{f}\rangle }^{f}\right)$$

(46)

Based on Gray’s theorem (1983) for non-constant REV and the assumption that no mass transfer occurs between phases, the second term on the left-hand side of Eq. 45 writes:

$$\frac{1}{V} {\int }_{{V}_{f}}\nabla \cdot \left({\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\right) \mathrm{d}V=\nabla \cdot \left({\phi }_{V} {\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}\right)+\frac{\nabla V}{V} \left({\phi }_{V} {\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}-{{\phi }_{A} \langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}^{f}\right)$$

(47)

We make use now of the analysis presented by Whitaker (2013) and Ochoa-Tapia and Whitaker (1993) to simplify the terms $\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle$, ${\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}$ and $\langle {\rho }_{f }{C}_{f }{T}_{f}\rangle$ as follows:

$${\langle {\rho }_{f }{C}_{f }{T}_{f}\rangle }^{f}={\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f}$$

(48)

$${\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }_{A}^{f}={\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}$$

(49)

$${\langle {\rho }_{f }{C}_{f }{T}_{f }{{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}={\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}$$

(50)

Using Gray’s theorem (1983) another time, the term on the right-hand side of Eq. 45 writes as follows:

$$\frac{1}{V}{\int }_{{V}_{f}}\nabla \cdot \left({\lambda }_{f }\nabla {T}_{f}\right) \mathrm{d}V=\nabla \cdot \left({{\phi }_{V} \lambda }_{{f}_{\mathrm{eff}}} \nabla {\langle {T}_{f}\rangle }^{f}\right)+\frac{\nabla V}{V}{ \lambda }_{{f}_{\mathrm{eff} }}\nabla {\langle {T}_{f}\rangle }^{f }\left({\phi }_{V}-{\phi }_{A}\right)+\frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}}\cdot \left({\lambda }_{f }\nabla {T}_{f}\right){\mathrm{d}}A$$

(51)

where ${\lambda }_{{f}_{\mathrm{eff}}}$ is the effective thermal conductivity of the fluid. Considering the turbulence of the flow, ${\lambda }_{{f}_{\mathrm{eff}}}$ is computed as:

$${\lambda }_{{f}_{\mathrm{eff}}}={\lambda }_{f}+\frac{{\langle {C}_{f}\rangle }^{f} {\mu }_{t}}{{\mathrm{Pr}}_{\mathrm{t}}}$$

(52)

where ${\mu }_{t}$ is the eddy viscosity, and ${\mathrm{Pr}}_{\mathrm{t}}$ the turbulent Prandtl number. The effect of turbulence modeling will be discussed in detail in Part II. Substituting Eqs. 46 to 51 into Eq. 45 provides:

$$\frac{\partial }{\partial t}\left({\phi }_{V} {\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f}\right)+\nabla \cdot \left({\phi }_{V} {\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}\right)=\nabla \cdot \left({{\phi }_{V} \lambda }_{{f}_{\mathrm{eff}}}\nabla {\langle {T}_{f}\rangle }^{f}\right)+\frac{1}{V}{\int }_{{A}_{fs}}{{\varvec{n}}}_{{\varvec{f}}{\varvec{s}}}\cdot \left({\lambda }_{f }\nabla {T}_{f}\right){\mathrm{d}}A+\frac{\nabla V}{V}\left({\phi }_{V}-{\phi }_{A}\right)\left({\lambda }_{{f}_{\mathrm{eff} }}\nabla {\langle {T}_{f}\rangle }^{f }-{\langle {\rho }_{f}\rangle }^{f} {\langle {C}_{f}\rangle }^{f} {\langle {T}_{f}\rangle }^{f} {\langle {{\varvec{u}}}_{{\varvec{f}}}\rangle }^{f}\right)$$

(53)

This gives Eq. 26 in the main text.

2.2 Solid Phase Volume-Averaged Thermal Energy Equation

Averaging Eq. 25 over a REV of volume $V$ gives:

$$\frac{1}{V}{\int }_{{V}_{s}}\frac{\partial }{\partial t}\left({\rho }_{s }{C}_{s }{T}_{s}\right) \mathrm{d}V=\frac{1}{V}{\int }_{{V}_{s}}\nabla \cdot \left({\lambda }_{s }\nabla {T}_{s}\right) \mathrm{d}V$$

(54)

where ${V}_{s}$ is the volume of solid beads in the REV. The solid phase intrinsic volume average of a quantity $\psi$ is defined as:

$${\langle \psi \rangle }^{s}=\frac{1}{{V}_{s}} {\int }_{{V}_{s}}\psi \mathrm{d}V=\frac{1}{\left(1-{\phi }_{V}\right) V} {\int }_{{V}_{s}}\psi \mathrm{d}V$$

(55)

The integration and differentiation of the term on the left-hand side of Eq. 54 can be interchanged. Applying the simplification proposed by Whitaker (2013) and Ochoa-Tapia and Whitaker (1993) gives:

$$\frac{1}{V}{\int }_{{V}_{s}}\frac{\partial }{\partial t}\left({\rho }_{s }{C}_{s }{T}_{s}\right) \mathrm{d}V=\frac{\partial }{\partial t}\left[{(1-\phi }_{V}) {\langle {\rho }_{s}\rangle }^{s} {\langle {C}_{s}\rangle }^{s} {\langle {T}_{s}\rangle }^{s}\right]$$

(56)

The term on the right-hand side of Eq. 54 may be expressed in the same way as the term in fluid phase equation:

$$\frac{1}{V}{\int }_{{V}_{s}}\nabla \cdot \left({\lambda }_{s }\nabla {T}_{s}\right) \mathrm{d}V=\nabla \cdot \left[{\left({1-\phi }_{V}\right) \lambda }_{s }\nabla {\langle {T}_{s}\rangle }^{s}\right]+\frac{\nabla V}{V}{ \lambda }_{s }\nabla {\langle {T}_{s}\rangle }^{s }\left({\phi }_{A}-{\phi }_{V}\right)+\frac{1}{V}{\int }_{{A}_{sf}}{{\varvec{n}}}_{{\varvec{s}}{\varvec{f}}}\cdot \left({\lambda }_{s }\nabla {T}_{s}\right){\mathrm{d}}A$$

(57)

Substitution of Eq. 56 and Eq. 57 into Eq. 54 results in:

$$\frac{\partial }{\partial t}\left[{(1-\phi }_{V}) {\langle {\rho }_{s}\rangle }^{s} {\langle {C}_{s}\rangle }^{s} {\langle {T}_{s}\rangle }^{s}\right]=\nabla \cdot \left[{\left({1-\phi }_{V}\right) \lambda }_{s }\nabla {\langle {T}_{s}\rangle }^{s}\right]+\frac{1}{V}{\int }_{{A}_{sf}}{{\varvec{n}}}_{{\varvec{s}}{\varvec{f}}}\cdot \left({\lambda }_{s }\nabla {T}_{s}\right){\mathrm{d}}A+\frac{\nabla V}{V}{ \lambda }_{s }\nabla {\langle {T}_{s}\rangle }^{s }\left({\phi }_{A}-{\phi }_{V}\right)$$

(58)

This gives Eq. 27 in the main text.

Appendix 3: Calculation of the Volume and Area Porosity from LIGGGHTS Data

To determine the value of ${\phi }_{V}$ and ${\phi }_{A}$ for a given REV, we firstly loop through all the beads in the packing and calculate the distance between each bead center and the REV. If a bead is located entirely out of the REV, it will be excluded from the following calculation. For the other beads, there are two possible situations: the bead is located entirely and partially inside the REV. In the first situation, the entire bead’s volume will be used to calculate ${\phi }_{V}$ and ${\phi }_{A}$ without any surface area term. In the second situation, the intersection volume (${V}_{\mathrm{ins}}$) and surface area (${A}_{\mathrm{ins}}$) are calculated by the following equations:

$${V}_{\mathrm{ins}}=\frac{\pi }{12 {d}_{ss}} {\left(R+{r}_{s}-{d}_{ss}\right)}^{2}\left({d}_{ss}^{2}+2{ d}_{ss} {r}_{s}-3 {r}_{s}^{2}+2 {d}_{ss} R+6 {r}_{s} R-3{R}^{2}\right)$$

(59)

$${A}_{\mathrm{ins}}=2 \pi R \frac{{r}_{s}^{2}-{\left(R-{d}_{ss}\right)}^{2}}{2 {d}_{ss}}$$

(60)

where ${d}_{ss}$ is the distance between the two sphere centers, $R$ is the radius of the spherical REV and ${r}_{s}$ is the radius of the bead. A schematic diagram is given in Fig. 

Fig. 6

Schematic of a typical spherical REV that intersects with a bead

6. We calculated and summed up the volume and area of each single bead to obtain the total beads occupied volume and bounding area for the REV. Then, the value of ${\phi }_{V}$ and ${\phi }_{A}$ can be determined.