Use of a Downscaling Procedure for Macroscopic Heat Source Modeling in Porous Media

Abstract

This work deals with the challenges brought by nonlinear heat sources when upscaling heat transfer in porous media. These difficulties are exemplified through applying the volume averaging method to a simple convective heat transfer problem in a porous medium featuring a nonlinear heterogeneous heat source. The most general solution proposed requires the availability of an estimated value of the heat source in the averaging volume, which can be obtained through a multiscale approach making use of a downscaling methodology. The downscaling methodology yields pore-scale governing equations in a sub-domain of the porous medium allowing to deal with the lack of information about the thermal behavior of the sub-domain’s vicinity. Solving the downscaled equations allows to reconstruct the temperature field and the heat source in the sub-domain with a good accuracy. This approximated reconstruction of the temperature field in sub-domain makes it possible to compute accurate estimates of the macroscopic heat source. In practice, the computational cost of the multiscale approach can be reduced by storing the results of the downscaling procedure in a table which takes as entries the limited number of macro-scale dependencies of the downscaled problem’s solution. In an example, the resulting heat source table is used as an input in a heuristic macro-scale transport model and compared to classic approaches. The use of the heat source reconstructed by downscaling results in a significant improvement of the accuracy of the macro-scale solution when the temperature driving the heat source significantly deviates from the macro-scale temperature.

Appendices

A Comments on Length-Scale Constraints

This study, being based on the volume averaging method, requires the analysis of terms in local equations at multiple stages of development. Terms in equations are estimated and compared with the others to decide on which ones are dominant. The estimation of terms is based on the analysis of their order of magnitude. This is a very common practice in the volume averaging literature.

It is important to firstly distinguish the purpose of estimates. When performing upscaling with the volume averaging method for macroscopically uniform media, simplifications are done at both the pore scale and the macroscopic scale:

• at the macroscopic scale, it is important to ensure that insignificant terms are not kept, as they would introduce unnecessary complexity;

• at the pore scale, the goal is to ensure that all the fields involved are either constant or periodic over the averaging volume (or can be forcibly considered so), in order to set ground for the usage of periodic boundary conditions.

The principles of this analysis are presented by Whitaker (1999), and thorough details are given by Quintard and Whitaker (1994a, b, c, d, e).

The length-scale analysis required to perform the upscaling of the problem presented in Sect. 2 is performed in the more general case presented by Whitaker (1998) and can be summarized by:

• the deviation equation can be written in a local form if

  $$\begin{aligned} \frac{l_{T_0}}{r_0}&\ll 1 \end{aligned}$$

  (44a)
  $$\begin{aligned} \frac{r_0^2}{L_\varepsilon L_{T,1}}&\ll 1 \end{aligned}$$

  (44b)

• the average of deviations can be neglected and the average of a quantity can be considered constant over the averaging volume if

  $$\begin{aligned} \frac{l_{T_0}}{L_{T,0}}&\ll 1 \end{aligned}$$

  (44c)
  $$\begin{aligned} \frac{r_0^2}{L_{T,0} L_{T,1}}&\ll 1 \end{aligned}$$

  (44d)

In Eq. (44), $l_{T,0}$ is the typical variation length scale of T, $r_0$ is the size of the averaging volume, $L_\varepsilon$ is the typical variation length scale of porosity and $L_{T,n}$ is the typical variation length scale for the n-th derivative of $\langle {T}\rangle ^\mathrm {f}$ (e.g. $L_{T,1}$ is the typical variation length scale of $\nabla \langle {T}\rangle ^\mathrm {f}$). It is worth mentioning that to our knowledge, terms with an order higher than 2 in the Taylor series developments used for length-scale analysis is always omitted; repeating the analysis for orders higher than 2 yields constraints of the form of Eq. (54). In the case considered, to the classic set of constraints (44) must be added constraint (55), which ensures that a local approximation of the source term is appropriate.

The derivation process for the downscaled equations presented in Sect. (3.1) requires calculus close to that performed for the upscaling procedure, also requiring to provide a local form for the cell problem (i.e. approximating the macro-scale varying quantities with constant values). This leads to length-scale constraints identical to those appearing during the volume averaging process. It follows that the length-scale constraints of both methodologies are compatible: that using the downscaling methodology is relevant in the same conditions as using the upscaling methodology.

B Development of the Flux Boundary Condition for the Energy Cell Problem

This appendix contains the developments leading to Eqs. (35b) and (35c). We first insert decomposition (24) into Eq. (2c) and get:

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\widetilde{T}^* = - \mathbf {n} \cdot k \mathbf {\nabla }\left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \frac{1}{2} \mathbf {y} \mathbf {y} : \mathbf {\nabla }\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \cdots \right) \,\, \text {at } \mathscr {A}^* \backslash \mathscr {R}^* \end{aligned}$$

(45)

where $\mathbf {r} = \mathbf {x} + \mathbf {y}$.

$\mathbf {n} \cdot k \mathbf {\nabla }\left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \right)$ is equal to zero, and we trivially show that

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\left( \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \right) = \mathbf {n} \cdot k \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \end{aligned}$$

(46)

The remaining terms are non-periodic and can be neglected if

$$\begin{aligned} \forall n \in \llbracket 1, +\infty \llbracket , \frac{r_0}{L_{T,\,n}} \ll 1 \end{aligned}$$

(47)

where $L_{T,\,n}$ is the typical variation length scale of the n-th order derivative of $\langle {T}\rangle ^\mathrm {f}$ (e.g. $L_{T,\,1}$ is the typical variation length scale of $\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f}$). We therefore write:

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\widetilde{T}^* = - \mathbf {n} \cdot k \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \,\, \text {at } \mathscr {A}^* \backslash \mathscr {R}^* \end{aligned}$$

(48)

Following a similar process, we insert decomposition (24) into Eq. (2b):

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\widetilde{T}^*= & {} - \mathbf {n} \cdot k \mathbf {\nabla }\left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \frac{1}{2} \mathbf {y} \mathbf {y} : \mathbf {\nabla }\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \cdots \right) \nonumber \\&+ S \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \frac{1}{2} \mathbf {y} \mathbf {y} : \mathbf {\nabla }\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \cdots + \widetilde{T}^* \right) \,\, \text {at } \mathscr {R}^*\nonumber \\ \end{aligned}$$

(49)

The gradient term is expanded and processed as before. To process the source term, we define the non-periodic part of $T^*$

$$\begin{aligned} \varTheta ^*&= T^* - \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \nonumber \\&= \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \frac{1}{2} \mathbf {y} \mathbf {y} : \mathbf {\nabla }\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \cdots \end{aligned}$$

(50)

The heterogeneous source term has a non-periodic part due to the presence of $\varTheta ^*$. We will establish the negligibility condition for the non-periodic terms. We first use a Taylor series expansion to write

$$\begin{aligned} S \left( T^* \right) = \underbrace{ S \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) }_{\text {periodic}} + \underbrace{ \varTheta ^* \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) + \frac{{\varTheta ^*}^2}{2} \frac{\mathrm {d}^{2}S}{\mathrm {d}T^2} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) + \cdots }_{\text {non-periodic}} \end{aligned}$$

(51)

The first term in the non-periodic part can be expanded:

$$\begin{aligned} \varTheta ^* \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) = \left( \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \frac{1}{2} \mathbf {y} \mathbf {y} : \mathbf {\nabla }\mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \cdots \right) \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \end{aligned}$$

(52)

The first term is estimated

$$\begin{aligned} \mathbf {y} \cdot \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) = \mathscr {O}\left( \frac{r_0 \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) }{L_{T,\,0}} \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \right) \end{aligned}$$

(53)

and can be neglected compared to the periodic part if the following constraint is valid:

$$\begin{aligned} \frac{r_0}{L_{T,\,0}} \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \frac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \ll S \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \end{aligned}$$

(54)

Similar estimates can be calculated for all terms of Eq. (51), leading to the following constraints:

$$\begin{aligned} \forall n \in \llbracket 1, + \infty \llbracket , \frac{{r_0}^n}{\prod _{k = 0}^{n-1} L_{T,\,k}} \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) \frac{\dfrac{\mathrm {d}S}{\mathrm {d}T} \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) }{S \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) } \ll 1 \end{aligned}$$

(55)

Finally, the heterogeneous source boundary condition becomes:

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\widetilde{T} = - \mathbf {n} \cdot k \mathbf {\nabla }\langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + S \left( \langle {T}\rangle ^\mathrm {f} \left( \mathbf {x} \right) + \widetilde{T}^* \right) \,\, \text {at } \mathscr {R}^* \end{aligned}$$

(56)

“Appendix A” contains comments on length-scale constraints.

C Closure Problems for the Volume Averaging Procedure

The closure problems are derived by inserting Eq. (17) into Eq. (16). This results in two independent closure problems whose solutions depend on the cell Péclet number. The formulations below include the substitution of boundary conditions in the integral terms and take into account the medium homogeneity hypothesis.

Fig. 14

Effective conductivity used in Sect. 4 (versus $\mathrm{Pe}^\mathrm {cell} = \overline{v} / \left( a A \right)$)

1.1 Problem I

$$\begin{aligned} \rho c_p \widetilde{\mathbf {v}} \cdot \mathbf {\nabla }\mathbf {b} + \rho c_p \mathbf {v}&= \mathbf {\nabla } \cdot \left( k \mathbf {\nabla }\mathbf {b} \right)&\text {in } \mathscr {V}^*_\mathrm {f} \end{aligned}$$

(57a)

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }\mathbf {b}&= - \mathbf {n} k&\text {at } \mathscr {A}^* \end{aligned}$$

(57b)

$$\begin{aligned} \mathbf {b} \left( \mathbf {r} + \mathbf {l}_i \right)&= \mathbf {b} \left( \mathbf {r} \right) \end{aligned}$$

(57c)

$$\begin{aligned} \langle {\mathbf {b}}\rangle&= \mathbf {0} \end{aligned}$$

(57d)

1.2 Problem II

$$\begin{aligned} \rho c_p \widetilde{\mathbf {v}} \cdot \mathbf {\nabla }r&= \mathbf {\nabla } \cdot \left( k \mathbf {\nabla }r \right) - \frac{R^*}{\varepsilon }&\text {in } \mathscr {V}^*_\mathrm {f} \end{aligned}$$

(58a)

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }r&= 1 + \alpha&\text {at } \mathscr {R}^* \end{aligned}$$

(58b)

$$\begin{aligned} \mathbf {n} \cdot k \mathbf {\nabla }r&= 0&\text {at } \mathscr {A}^* \backslash \mathscr {R}^* \end{aligned}$$

(58c)

$$\begin{aligned} r \left( \mathbf {r} + \mathbf {l}_i \right)&= r \left( \mathbf {r} \right) \end{aligned}$$

(58d)

$$\begin{aligned} \langle {r}\rangle&= 0 \end{aligned}$$

(58e)

Solving the closure problems is beyond the scope of this study. However, the computations performed in Sect. 4 require values for the $K_ xx$ conductivity tensor coefficient. Quintard et al. (1997) provide suitable information and Fig. 14 plots the values used.