Flow Characterization in Triply Periodic Minimal Surface (TPMS)-Based Porous Geometries: Part 1—Hydrodynamics

Abstract

The modeling of flow and heat transfer in porous media systems has always been a challenge, and the extended Darcy transport models are used for macro-level analysis. However, these models are subjected to the limitations depending upon the porous geometry such as pore size, pore type, effective porosity, tortuosity, permeability, and the flow characteristics. The forced convective flow of an incompressible viscous fluid through a channel filled with four different types of porous geometries constructed using the Triply Periodic Minimal Surface (or TPMS) model is presented in this study. Four TPMS lattice shapes, namely Diamond, I-WP, Primitive, and Gyroid, are created with same volume fraction of solid subdomain as 0.68 (or void fraction as 0.32). Using different configurations for the solid subdomain by treating it as (a) solid, (b) fluid, and (c) porous zone, three different classes of porous structures are further generated for each TPMS lattice. The present study is executed with the objective to investigate the effect of shape–morphology, tortuosity, microporosity, and effective porosity on permeability and inertial drag factor. A pore-scale direct numerical simulation approach is performed for the first two types of porous media by solving the Navier–Stokes equations. The specific microporosity is quantitatively induced in the solid subdomain where Darcy–Forchheimer equation is solved, whereas the Navier–Stokes equations are solved for the void subdomain in the third type of porous media. The results reveal that Darcy flow regime exists up to the mean velocity value of U < 0.0025 m/s (Re < 10) for all the cases discussed here, and it deviates at the higher mean velocity. The conductance to the flow shown by Darcy number has the maximum and minimum values for the Primitive Type 2 and I-WP Type 1 cases. The inertial drag coefficient is minimum in Diamond lattice and maximum in Primitive lattice at lower porosity (0.32), while Primitive lattice has minimum and I-WP lattice has maximum value of inertial drag coefficient for higher porosity (~ 1).

Appendix

See Appendix Tables

Table 10 Different correlations for effective viscosity (Almalki and Hamdan 2016)

10 and

Table 11 Properties of porous medium used in Type 3 treatment (Hetsroni et al. 2006)

11.

1.1 Equation for Momentum Conservation in porous media

• a. Darcy model (for low flow rates, ${\mathrm{Re}}_{p}\ll 1$)

  $$\nabla p = - \frac{\mu }{K}\vec{V}$$

  (14)

• b. Extended Darcy model (for intermediate to high flow rates)

  $$\nabla p = - \frac{\mu }{K}\vec{V} - \frac{\rho \beta }{{\sqrt K }}\vec{V}\left| {\vec{V}} \right| - \frac{{\alpha \rho^{2} }}{\mu }\vec{V}\left| {\vec{V}} \right|^{2} \,\,\,\,\,\,\,\,(R{\text{e}}_{{\text{p}}} { > 1)}$$

  (15)

• c. Brinkman extended Darcy model (for higher flow rates and high porosity, $\varepsilon \to 1$)

  $$\nabla p = - \frac{\mu }{K}\vec{V} - \frac{\rho \beta }{{\sqrt K }}\vec{V}\left| {\vec{V}} \right| - \frac{{\alpha \rho^{2} }}{\mu }\vec{V}\left| {\vec{V}} \right|^{2} + \mu^{*} \nabla^{2} \vec{V}$$

  (16)

In Eq. (16), ${\mu }^{*}$ is the effective viscosity of the porous medium and some important relations between the porosity and effective viscosity are mentioned in Table 10.

1.2 Calculation of Tortuosity

Tortuosity is calculated by taking two points in the direction of the flow and measuring actual length of flow path (length of fluid trajectory) and minimum length (length of straight line) between these two points. It is then defined as below:

$$\tau = \frac{{{\text{length}}\,{\text{of}}\,{\text{actual}}\,{\text{path}}}}{{{\text{length}}\,{\text{of}}\,{\text{minimum}}\,{\text{path}}}}$$

(17)