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Flow Characterization in Triply-Periodic-Minimal-Surface (TPMS)-Based Porous Geometries: Part 2—Heat Transfer

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Abstract

Complex physical phenomena take place while dealing with the convective heat transfer in porous medium. Due to involved complexities, most of the earlier numerical studies are performed using various porous models compromising the detailed phenomena. Therefore, a pore-scale simulation has been performed for convective heat transfer in triply-periodic-minimal-surface lattices, with identical void fraction and unit-cell size, but different geometrical shapes (tortuosity), namely Diamond, Inverted Weaire–Phelan, Primitive, and Gyroid. Further, each lattice derived into three different types of porous structures by designing second subdomain as solid (in Type 1), fluid (in Type 2), and microporous zones (in Type 3). The convective heat transfer in a square mini-channel filled with the porous structures is investigated for the range of flow Reynolds number \(0.01<\mathrm{Re}<100\) and \(\mathrm{Pr}=7\). The temperature distributions, solid and fluid Nusselt numbers on the external walls and on the internal walls, and quantitative departure from local thermal equilibrium (LTE) assumption are calculated for different porous media. The effect of porous morphology/tortuosity and effective porosity on the heat transfer is examined. The results revealed that the maximum temperature within the domain is found in Type 2 treatment, leading to inferior heat transfer performance compared to Type 1 and Type 3. Among all the lattices, the Diamond lattice provides more uniform temperature distribution over the external walls and within the volume including solid and fluid. The effective and the internal Nusselt numbers increase drastically for Re > 10. For the range of Re considered here, the Primitive lattice shows the maximum deviation from LTE assumption.

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Abbreviations

a :

Unit cell size in x direction (m)

a sf :

Interfacial area density (\(a_{{{\text{sf}}}} = A_{{{\text{sf}}}} /\forall\)) (m−1)

A :

Area (m2)

A sf :

Interfacial area (m2)

b :

Unit cell size in y direction (m)

c :

Unit cell size in z direction (m)

c p :

Specific heat capacity (J/kg-K)

C :

Level-set constant (–)

h :

Heat transfer coefficient (W/m2-K)

k :

Thermal conductivity (W/m-K)

Da:

Darcy number (\({\text{Da}} = K/L^{2}\)) (–)

K :

Permeability (m2)

l :

Unit cell or pore size (\(l = L/4\)) (m)

L :

Channel width or characteristic length (m)

\(\dot{m}\) :

Mass flow rate (kg/s)

n :

Normal distance from the surface (m)

Nu:

Nusselt number (\(hL/k\)) (–)

p :

Pore-scale pressure (Pa)

P :

Average pressure (Pa)

\(q^{\prime\prime}\) :

Heat flux (W/m2)

R k :

Solid-to-fluid thermal conductivity ratio \(\left( {{\raise0.7ex\hbox{${k_{{\text{s}}} }$} \!\mathord{\left/ {\vphantom {{k_{{\text{s}}} } {k_{{\text{f}}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{{\text{f}}} }$}}} \right)\) (–)

Re:

Channel size Reynolds no. (\(\rho UL/\mu\)) (–)

Rep :

Pore-scale Reynolds no. (\(\rho Ul/\mu\)) (–)

u :

Pore level velocity in x direction (m/s)

v :

Pore level velocity in y direction (m/s)

w :

Pore level velocity in z direction (m/s)

\(\vec{v}\) :

Pore level velocity vector (m/s)

T :

Temperature (K)

U :

Average velocity in x direction (m/s)

V :

Average velocity in y direction (m/s)

W :

Average velocity in z direction (m/s)

\(\vec{V}\) :

Macroscale or average velocity vector (m/s)

x :

X-Direction distance (m)

y :

Y-Direction distance (m)

z :

Z-Direction distance (m)

X :

Lattice size in x direction (m)

Y :

Lattice size in y direction (m)

Z :

Lattice size in z direction (m)

α :

Coefficient of cubic term (–)

\(\beta\) :

Coefficient of quadratic term (–)

\(\gamma\) :

Periodic temperature gradient (K/m)

δ :

Periodic pressure gradient (Pa/m)

ɛ :

Porosity (–)

ɛ o :

Porosity of the lattice (–)

ɛ * :

Microporosity (–)

ɛ eff :

Effective porosity of the lattice (–)

µ :

Dynamic viscosity of the fluid (kg/m-s)

\(\rho\) :

Density of the fluid (kg/m3)

\(\tau\) :

Tortuosity (–)

\(\forall\) :

Volume (m3)

act:

Actual

b:

Bulk associated quantity

eff:

Effective property

ext:

External wall associated quantity

f:

Fluid phase property

int:

Internal wall associated quantity

interface1:

Quantity at interface in first subdomain side

interface2:

Quantity at interface in second subdomain side

min:

Minimum

n:

Normal direction quantity

s:

Solid phase property

sf:

Interface associated quantity

t:

Tangential direction quantity

w:

Wall associated quantity

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Acknowledgements

The first two authors Surendra Singh Rathore and Balkrishna Mehta would like to acknowledge the Indian Institute of Technology, Bhilai, for providing the computational and other peripheral resources through the institute research initiation grant.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Bhilai, Raipur, Chhattisgarh, 492015, India

    Surendra Singh Rathore & Balkrishna Mehta

  2. Numerical Experiment Laboratory (Radiation and Fluid Flow Physics), Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, 175075, India

    Pradeep Kumar

  3. Department of Mechanical Engineering, College of Engineering, Dawadmi, Shaqra University, 11911, Shaqra, Saudi Arabia

    Mohammad Asfer

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  1. Surendra Singh Rathore
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Correspondence to Pradeep Kumar.

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Rathore, S.S., Mehta, B., Kumar, P. et al. Flow Characterization in Triply-Periodic-Minimal-Surface (TPMS)-Based Porous Geometries: Part 2—Heat Transfer. Transp Porous Med 151, 141–169 (2024). https://doi.org/10.1007/s11242-023-02036-x

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