Transport in Porous Media

, Volume 122, Issue 3, pp 575–593 | Cite as

A New Model and its Application to Investigate Transpiration Cooling with Liquid Coolant Phase Change

  • Wenjie Dong
  • Jianhua Wang


This paper presents a new mathematical model, semi-mixing model (SMM), to describe transpiration cooling with coolant phase change from liquid into vapor through two-phase process. The local heat exchange of fluid-solid within pores is considered in this model, and therefore it is closer to real transpiration cooling condition. The differences from the separated phase model and two-phase mixture model are that SMM can overcome the trouble of tracking phase change interface and avoid the inveracious numerical phenomenon, i.e., a thermal insulating layer occurs within the porous matrix. Using SMM, the corresponding numerical method is realized to simulate the processes of coolant moving, absorbing heat and evaporating within porous matrix. To validate SMM and the numerical strategy, an experiment is conducted. Using the validated SMM and numerical strategy, the effects of two-dimensional coolant injection rate and two-dimensional heat flux on transpiration cooling characteristics are simulated and analyzed. The simulations and analysis discover several interesting and valuable phenomena.


Modeling Two-dimensional Transpiration cooling Phase change Porous media 

List of symbols


Thickness of porous matrix, m


Length of porous matrix, m


Temperature, K

x, y

Cartesian coordinate, m


Heat flux, W m\(^{-2}\)


Fluid-to-solid heat transfer rate, W m\(^{-3}\)


Mass flow rate per unit area, kg m\(^{-2}\) s\(^{-1}\)


Interfacial mass transfer rate, kg m\(^{-3}\) s\(^{-1}\)


Phase saturation


Velocity vector

v\(^{\prime }\)

Superficial or Darcian velocity vector


Velocity components along x and y axes, m/s


Pressure, Pa


Gravity vector


Permeability, m\(^{2}\)


Specific enthalpy, J kg\(^{-1}\)


Latent heat of evaporation, J kg\(^{-1}\)


Gas constant of air, J kg\(^{-1}\) K\(^{-1}\)


Heat transfer coefficient, W m\(^{-2}\) K\(^{-1}\)


Fluid-to-solid heat transfer coefficient, W m\(^{-2}\) K\(^{-1}\)


Thermal conductivity, W m\(^{-1}\) K\(^{-1}\)


Particle diameter, m


Prandtl number


Reynolds number


Specific heat, J kg\(^{-1}\,\)K\(^{-1}\)

Greek Symbols

\(\rho \)

Density, kg m\(^{-3}\)

\(\phi \)


\(\mu \)

Dynamic viscosity, N s m\(^{-2}\)


\(\infty \)

Hot air



l, v

Liquid, vapor

i, f

Fluid in different region








Saturated state



This work is supported by the Natural Science Foundation of China (Contract No. 51376168). We also thank Prof. Bangcheng Ai, Dr. Nan Wu, Dr. Siyuan Chen and Dr. Xiaoguang Luo for providing professional opinions.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Thermal Science and Energy EngineeringUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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