Transport in Porous Media

, Volume 112, Issue 2, pp 313–332

# Multiphase, Multicomponent Flow in Porous Media with Local Thermal Non-equilibrium in a Multiphase Mixture Model

• Franz Lindner
• Michael Pfitzner
• Christian Mundt
Article

## Abstract

For multiphase, multicomponent flow, the multiphase mixture model of Wang and Cheng (Int J Heat Mass Trans 39(17):3607–3618, 1996) has been used in the literature. In our self-written solver, based on this mathematical background, we use a modified multiphase mixture model for a simple transpiration cooling application. In this modeling approach, we do not solve for every phase or component separately. Instead, we solve for mixture variables and thus only use a simple set of conservation equations, which is valid in the entire domain. The information of concentration of components and phase saturations is incorporated within the multiphase mixture variables and can be extracted out of them easily with arithmetic equations. It has been shown that this approach is completely equivalent to using the separate phase models. We implemented an augmentation to local thermal non-equilibrium; i.e., we allow for different fluid and solid temperatures locally. We show solutions for stationary cases of evaporation and discuss the effect of variation of heat input, inlet temperature, inlet mass flux and fraction of components.

## Keywords

Porous medium Multicomponent fluid Evaporation  Local thermal non-equilibrium Mixture model

## Latin Symbols

$$A^\alpha ,B^\alpha ,C^\alpha$$

Coefficients of Antoine equation of component $$\alpha$$

$$a_s$$

Specific surface of pores

$$b_{sf}$$

Material-fluid parameter for boiling heat flux

$$c_{i}$$

Constants

$$c_{p}$$

Isobaric specific heat capacity

$$d_p$$

Characteristic size of porous medium $$({\mathrm {m}})$$

$$D_c$$

Capillary diffusion coefficient

f

Hindrance function

$$\mathbf g$$

Gravity vector

h

Specific enthalpy

H

Volumetric enthalpy

$$h_{fg}$$

Enthalpy of vaporization

$${\bar{h}}_{s\,l}$$

Mean heat transfer coefficient of solid and liquid phase

$${\bar{h}}_{s\,v}$$

Mean heat transfer coefficient of solid and vapor phase

J

Leverett J-function

$$\mathbf j _k$$

Diffusive mass flux of phase k within the multiphase mixture

$$k$$

Heat conductivity

K

Permeability tensor $$({\mathrm {m}}^2)$$

$$k_r$$

Relative permeability

M

Molar mass

m

Mass $$({\mathrm {kg}})$$

$$\dot{m}$$

Mass flux

n

Constant, number of moles $$({\mathrm {mol}})$$, index for iterations

nx

Number of cells

p

Pressure $$({\mathrm {Pa}})$$

pol

Polynomial to approximate H as a function of

$$p^\alpha$$

Partial pressure of component $$\alpha$$ $$({\mathrm {Pa}})$$

$$p^{\alpha ,0}$$

Equilibrium vapor pressure of component $$\alpha \,({\mathrm {Pa}})$$

$$p_c$$

Capillary pressure (phase pressure difference) $$({\mathrm {Pa}})$$

q

Term in Taylor series expansion for temperature $$({\mathrm {K}})$$

$$\dot{q}_{sf}$$

Heat exchange between solid and fluid phase

$$\dot{q}_\text {boil}$$

$$\dot{q}_\text {total}$$

Net heat flux into the fluid

$$\dot{q}$$

Heat flux at outlet boundary

$$\mathbf r$$

Right-hand side of discretization system

$$s_k$$

Saturation of phase k

s

Short for $$s_l$$, saturation of liquid phase

T

Temperature $$(^\circ {\mathrm {C}})$$

$$\mathbf u$$

Velocity vector

V

Volume $$({\mathrm {m}}^3)$$

w

Mass fraction

x

Coordinate $$({\mathrm {m}})$$, molar fraction

Nu

Nusselt number

Pr

Prandtl number

$$\mathcal {L}$$

Characteristic length $$({\mathrm {m}})$$

## Greek Symbols

$$\varXi$$

Constant for volumetric enthalpy

$$\phi$$

Porosity

$$\gamma _h$$

$$\gamma _\rho$$

Density correction coefficient

$$\lambda$$

Relative mobility

$$\mu$$

Dynamic viscosity $$({\mathrm {Pa}}\,{\mathrm {s}})$$

$$\nu$$

Kinematic viscosity

$$\varOmega$$

Cell length, 1d $$({\mathrm {m}})$$

$$\varrho$$

Density

$$\sigma$$

Surface tension

$$\zeta$$

Measure for global energy conservation

## Superscripts

$$\alpha$$

General component

n

Iteration number

(m)

m-th derivation

## Subscripts

$$\text {boil}$$

Boiling

$$\text {bubble}$$

At bubble point

$$\text {dew}$$

At dew point

$$\text {dryout}$$

At dryout state

$$\text {eff}$$

Effective

f

Fluid

i

Discrete value

$$\text {in}$$

At inlet

k

General phase

l

Liquid phase

$$\text {norm}$$

Normalized

$$\text {exit}$$

At exit

s

Solid phase

$$\text {sat}$$

At saturation state

$$\text {total}$$

Total value

v

Vapor phase

## Notes

### Acknowledgments

This work was developed during a research project with Institut für Technik Intelligenter Systeme (ITIS GmbH) and Institute for Thermodynamics at University of the Federal Armed Forces, Munich.

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## Authors and Affiliations

• Franz Lindner
• 1
Email author
• Michael Pfitzner
• 1
• Christian Mundt
• 1
1. 1.Department of Aerospace Engineering, Institute for ThermodynamicsUniversity of the Federal Armed Forces MunichMunichGermany