Modeling heat and fluid flow in porous media

David J. Lopez Penha∗, Lilya Ghazaryan∗,

Bernard J. Geurts∗, Steffen Stolz†,∗ & Markus Nordlund†

∗Dept. of Applied Mathematics †Philip Morris International R&D

University of Twente, Enschede Philip Morris Products S.A., Neuchatel

The Netherlands Switzerland

ECCOMAS CFD 2010, Lisbon, Portugal

June 14–17, 2010

D.J. Lopez Penha et al.

Porous media

Amorphous nano-porous material

Source: http://gubbins.ncsu.edu/research.html

• Porous media may have complicated pore geometries

• Difficult to build practicable body-fitted grids

D.J. Lopez Penha et al.

Goals

1. Develop “gridding-free” method for computing heat & fluidflow in porous media

2. Method allows arbitrary pore geometries

D.J. Lopez Penha et al.

Goals

1. Develop “gridding-free” method for computing heat & fluidflow in porous media

2. Method allows arbitrary pore geometries

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Representing complex geometries

• Cartesian grid representation of fluid & solid domains

• Trade-off: large spatial resolutions — efficient numericalalgorithms

D.J. Lopez Penha et al.

Representing complex geometries

• Cartesian grid representation of fluid & solid domains

• Trade-off: large spatial resolutions — efficient numericalalgorithms

D.J. Lopez Penha et al.

Fluid dynamics

• Incompressible Navier-Stokes equations for fluid & soliddomains:

∇ · u = 0,∂u

∂t+ u · ∇u = −∇p +

1

Re∇2u+ f

• Methodology: immersed boundary method

• Force f: approximates no-slip condition (volume-penalization)

f = −1

ǫΓ(x) · u, ǫ ≪ 1

• Γ(x): phase-indicator function (Γ = 1 in solid; Γ = 0 in fluid)

D.J. Lopez Penha et al.

Fluid dynamics

• Incompressible Navier-Stokes equations for fluid & soliddomains:

∇ · u = 0,∂u

∂t+ u · ∇u = −∇p +

1

Re∇2u+ f

• Methodology: immersed boundary method

• Force f: approximates no-slip condition (volume-penalization)

f = −1

ǫΓ(x) · u, ǫ ≪ 1

• Γ(x): phase-indicator function (Γ = 1 in solid; Γ = 0 in fluid)

D.J. Lopez Penha et al.

Discretization

• Symmetry-preserving finite-volume method

• Staggered grid (uniform Cartesian)

• Implicit time-integration of force f

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Conjugate heat transfer

• Single temperature equation for fluid & solid domains:

∂T

∂t+ u · ∇T =

1

RePr∇ · (α∇T )

• Thermal diffusivity α: discontinuous if αf 6= αs

α(x) = (1− Γ)αf + Γαs

• Solid domains: convective term vanishes =⇒ diffusion only

D.J. Lopez Penha et al.

Conjugate heat transfer

• Single temperature equation for fluid & solid domains:

∂T

∂t+ u · ∇T =

1

RePr∇ · (α∇T )

• Thermal diffusivity α: discontinuous if αf 6= αs

α(x) = (1− Γ)αf + Γαs

• Solid domains: convective term vanishes =⇒ diffusion only

D.J. Lopez Penha et al.

Discretization

• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces

• Auxiliary temperatures {T xi ,j ,T

yi ,j} on cell surfaces:

T xi ,j =

αi ,jTi ,j + αi+1,jTi+1,j

αi ,j + αi+1,j

Tyi ,j =

αi ,jTi ,j + αi ,j+1Ti ,j+1

αi ,j + αi ,j+1

D.J. Lopez Penha et al.

Discretization

• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces

• Auxiliary temperatures {T xi ,j ,T

yi ,j} on cell surfaces:

T xi ,j =

αi ,jTi ,j + αi+1,jTi+1,j

αi ,j + αi+1,j

Tyi ,j =

αi ,jTi ,j + αi ,j+1Ti ,j+1

αi ,j + αi ,j+1

D.J. Lopez Penha et al.

Discretization

• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces

• Auxiliary temperatures {T xi ,j ,T

yi ,j} on cell surfaces:

T xi ,j =

αi ,jTi ,j + αi+1,jTi+1,j

αi ,j + αi+1,j

Tyi ,j =

αi ,jTi ,j + αi ,j+1Ti ,j+1

αi ,j + αi ,j+1

D.J. Lopez Penha et al.

Discretization

• Physics: heat flux α∇T continuous =⇒ ∇T discontinuous atjumps in α =⇒ special care discretizing ∇T on jumpinterfaces

• Auxiliary temperatures {T xi ,j ,T

yi ,j} on cell surfaces:

T xi ,j =

αi ,jTi ,j + αi+1,jTi+1,j

αi ,j + αi+1,j

Tyi ,j =

αi ,jTi ,j + αi ,j+1Ti ,j+1

αi ,j + αi ,j+1

αi ,j ,Ti ,j αi+1,j ,Ti+1,j

αi ,j+1,Ti ,j+1

T xi ,j

Tyi ,j

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Plane-Poisuielle flow with isothermal walls

101

102

10−1

ny

‖uh−

u‖ ℓ

p

101

102

10−4

10−3

10−2

10−1

ny

‖Th−

T‖ ℓ

p• ℓp-norm of error in velocity & temperature (p = {2,∞})

• Velocity: first-order in ℓ∞

• Temperature: second-order in ℓ∞

D.J. Lopez Penha et al.

Porous medium: inline arrangement of squares

x

y

D/2

H

H

Q

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y• Porosity: φ = 0.75 =⇒ 25% solid grid cells

• Literature: Nakayama et al., J. Heat Transfer, 124, 746–753(2002)

D.J. Lopez Penha et al.

Porous medium: inline arrangement of squares

• Gradient average pressure ∂〈p〉f /∂x vs. grid resolution:

(nx × ny )∖

Re 10 100 600

32× 32 6.65 0.711 0.12464× 64 7.16 0.768 0.135128× 128 7.45 0.800 0.143

181× 181 (Nakayama et al.) 7.82 0.835 0.154

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Realistic porous medium

• Left: solid data from µCT-imaging (black: solid material)

• Right: Cartesian grid representation using Γ(x) [grid: (32)2]

D.J. Lopez Penha et al.

Realistic porous medium

• Left: solid data from µCT-imaging (black: solid material)

• Right: Cartesian grid representation using Γ(x) [grid: (64)2]

D.J. Lopez Penha et al.

Realistic porous medium

• Left: solid data from µCT-imaging (black: solid material)

• Right: Cartesian grid representation using Γ(x) [grid: (128)2]

D.J. Lopez Penha et al.

Simulated field variables

0 0.5 1 1.5 20

0.5

1

1.5

2

z

y0 0.5 1 1.5 2

0

0.5

1

1.5

2

zy

• Left: contours out-of-plane velocity [grid: (256)2]

• Right: fluid-solid temperature [grid: (256)2]

D.J. Lopez Penha et al.

Outline

1 Modeling fluid flow in porous media

2 Modeling heat flow in porous media

3 Validation tests

4 Application to realistic porous medium

5 Conclusions

D.J. Lopez Penha et al.

Conclusions

Immersed boundary method:

• Reliably capture laminar fluid dynamics with adequate gridresolution

• Include arbitrarily shaped solid objects using Cartesian gridrepresentation

• First-order accurate in fluid dynamics & second-order accuratein heat transfer

D.J. Lopez Penha et al.