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Model Discussion of Transpiration Cooling with Boiling

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Abstract

The two-phase mixture model has been widely used to describe the performances of fluid flow and heat transfer within porous media with liquid phase change. However, this model was based on two important assumptions: the temperature is constant (T f = const) in two phase region, while fluid temperature is locally equal to the solid matrix temperature (T s = T f). These assumptions result in an inveracious numerical phenomenon, i.e.,: a thermal insulating layer within the porous matrix in numerical simulations. This numerical phenomenon is not real, because the solid matrix is made of thermal conductive material. To modify the mathematical model of the transpiration cooling problem with boiling, this paper presents an improved model, which is based on that the Gibbs free energy of liquid phase and vapor phase are equal in two-phase region. Temperature variation in two-phase region is considered, and fluid temperature is locally different from the solid matrix temperature (T s ≠ T f), therefore the local heat transfer through the convection between solid and fluid is considered as well. Numerical calculations of the transpiration cooling problem with boiling are carried out with the improved model. The numerical results such as the variations of temperatures of fluid and solid, the saturation and pressure of fluid within porous media, are reasonable, and the inveracious issue of the thermal insulating layer is successfully resolved.

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Abbreviations

z :

Coordinate

L :

Length

T :

Temperature

h :

Specific enthalpy

k :

Thermal conductivity

H :

Specific revised enthalpy

c p :

Specific heat capacity

s :

Saturation

t :

Time

\({\vec {u}}\) :

Velocity

j :

Diffusive mass flux

\({\langle \dot {n} \rangle}\) :

Volumetric mass rate of evaporation

K :

Absolute permeability

k r :

Relative permeability

p :

Pressure

\({\vec {g}}\) :

Gravitational constant

d p :

Averaged particle diameter of porous media

p c :

Capillary pressure

q :

Volumetric heat generation rate induced by convection

h sl :

Convective heat transfer coefficient between solid and liquid

h sv :

Convective heat transfer coefficient between solid and vapor

R :

Mole gas constant

M :

Mole mass

g :

Specific Gibbs free energy

B :

Saturation equation associated with the phase state

ṁ :

Mass flow rate of coolant

Q :

Heat flux on the hot side surface

D :

Discrete system of the equations

N :

Nodes number

Γ:

Effective diffusion coefficient

ρ :

Density

ɛ :

Porosity

λ:

Relative mobility

μ :

Kinetic viscosity

σ :

Surface tension coefficient

α :

Specific surface

η :

Entropy

ξ :

Intermediate variable

ζ :

Intermediate variable

χ :

Phase state

s:

Solid

l:

Liquid

v:

Vapor

f:

Fluid

sat:

Saturated state

eff:

Effective

sf:

Solid–fluid

0:

Reference state

in:

Inlet of porous media

c:

Coolant tank

References

  • Bau H.H., Torrance K.E.: Boiling in low-permeability porous materials. Int. J. Heat Mass Transf. 25, 45–55 (1982)

    Article  Google Scholar 

  • Benard J., Eymard R., Nicolas X. et al.: Boiling in porous media: model and simulations. Transp. Porous Media 60, 1–31 (2005)

    Article  Google Scholar 

  • Bridge L.J., Wetton B.R.: A mixture formulation for numerical capturing of a two-phase/vapor interface in a porous medium. J. Comput. Phys. 225, 2043–2068 (2007)

    Article  Google Scholar 

  • Duval F., Fichot F., Quintard M.: A local thermal non-equilibrium model for two-phase flows with phase-change in porous media. Int. J. Heat Mass Transf. 47, 613–639 (2004)

    Article  Google Scholar 

  • Fichot F., Duval F., Tregoure N.: The impact of thermal non-equilibrium and large-scale 2D/3D effects on debris bed reflooding and coolability. Nucl. Eng. Des. 236, 2144–2163 (2006)

    Article  Google Scholar 

  • Landis, J.A., Bowman, W.J.: Numerical study of a transpiration cooled rocket nozzle. In: 32nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, July 1–3, Lake Buena Vista, FL, pp. 1996–2580 (1996)

  • Leontiev A.I.: Heat and mass transfer problems for film cooling. ASME J. Heat Transf. 121, 509–527 (1999)

    Article  Google Scholar 

  • Li H.Y., Leong K.C., Jin L.W. et al.: Transient two-phase flow and heat transfer with localized heating in porous media. Int. J. Therm. Sci. 49, 1115–1127 (2010)

    Article  Google Scholar 

  • Likhanskii V.V., Loboiko A.I., Khoruzhii O.V.: Two phase convection in a porous medium with nonuniform volume heat release. Atomic Energy 82, 180–186 (1997)

    Article  Google Scholar 

  • Peterson G.P., Chang C.S.: Heat transfer analysis and evaluation for two-phase flow in porous-channel heat sinks. Numer. Heat Transf. A 31, 113–130 (1997)

    Article  Google Scholar 

  • Rahli F., Topin F., Tadrist L. et al.: Analysis of heat transfer with liquid–vapor phase change in a forced-flow fluid moving through porous media. Int. J. Heat Mass Transf. 39, 3959–3975 (1996)

    Article  Google Scholar 

  • Shi J.X., Wang J.H.: A numerical investigation of transpiration cooling with liquid coolant phase change. Transp. Porous Media 87, 703–716 (2011)

    Article  Google Scholar 

  • Soller, S., Kirchberger, C., Kuhn, M.: Experimental investigation of cooling techniques and materials for highspeed flight propulsion systems. In: 16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference, Oct. 19–22, Bremen, Germany, AIAA-2009-7374 (2009)

  • Sozen M., Vafai K.: Analysis of the non-thermal equilibrium condensing flow of a gas through a porous bed. Int. J. Heat Mass Transf. 33, 1247–1261 (1990)

    Article  Google Scholar 

  • Vafai K., Sozen M.: Analysis of energy and momentum transport for fluid flow through a porous bed. ASME J. Heat Transf. 112, 690–699 (1990)

    Article  Google Scholar 

  • van Foreest A., Sippel M., Gulhan A. et al.: Transpiration cooling using liquid water. J. Thermophys. Heat Transf. 23, 693–702 (2009)

    Article  Google Scholar 

  • Wang C.Y.: A fixed-grid numerical algorithm for two-phase flow and heat transfer in porous media. Numer. Heat Transf. B 32, 85–105 (1997)

    Article  Google Scholar 

  • Wang C.Y., Beckermann C.: A two-phase mixture model of liquid–gas flow and heat transfer in capillary porous media I. Formulation. Int. J. Heat Mass Transf. 36, 2747–2758 (1993)

    Article  Google Scholar 

  • Wang J.H., Wang H.N.: A discussion of transpiration cooling problems through an analytical solution of local thermal nonequilibrium model. J. Heat Transf. Trans. ASME 128, 1093–1098 (2006)

    Article  Google Scholar 

  • Yuki K., Abei J., Hashizume H. et al.: Numerical investigation of thermofluid flow characteristics with phase change against high heat flux in porous media. ASME J. Heat Transf. 130, 012602 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Jinzhai Road No. 96, Hefei, 230027, Anhui, People’s Republic of China

    Kuan Wei, Jianhua Wang & Mao Mao

Authors
  1. Kuan Wei
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  2. Jianhua Wang
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  3. Mao Mao
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Correspondence to Kuan Wei.

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Wei, K., Wang, J. & Mao, M. Model Discussion of Transpiration Cooling with Boiling. Transp Porous Med 94, 303–318 (2012). https://doi.org/10.1007/s11242-012-0006-0

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  • DOI: https://doi.org/10.1007/s11242-012-0006-0

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