2021  OriginalPaper  Chapter Open Access
A Coupled TwoDomain Approach for Transpiration Cooling
Authors: Valentina König, Michael Rom, Siegfried Müller
Publisher: Springer International Publishing
1 Motivation
2 Mathematical Modeling
2.1 Hot Gas Domain
2.2 Porous Medium Domain
2.3 Coupling Conditions
3 Numerical Methods

a linear elliptic system for the two heat equations to determine the temperatures for the coolant and the solid;

a nonlinear transport system consisting of the continuity and the DarcyForchheimer equation to determine the density of the coolant and the Darcy velocity.
4 Numerical Results
4.1 Nonuniform Injection into a Subsonic Hot Gas Channel Flow
Flow

Porous material



Mach number
\(M_{\infty }\)

0.144

Throughflow direction

Parallel

Inflow temperature
\(T_{\infty }\)

375.05 K

Porosity
\(\varphi \)

\(12.36~\%\)

Inflow pressure
\(p_{\infty }\)

88, 570 Pa

Solid heat conductivity
\(\kappa _{s,\text {par}}\)

15.19 W/(m K)

Isothermal channel wall temp.
\(T_{\text {W}}\)

362.6 K

Darcy coefficient
\(K_D\)

\(5.98 \cdot 10^{13}\) m
\(^2\)

Integral coolant mass flow rate
\(\dot{m}_c\)

1.14 g/s

Forchheimer coefficient
\(K_F\)

\(7.86 \cdot 10^{8}\) m

Reservoir pressure
\(p_{\text {R,num}}\)

216, 900 Pa

Volumetric heat transfer coef.
\(h_v\)

\(10^6\) W/(m
\(^3\)K)

Coolant reservoir temperature
\(T_{c}\)

300.15 K


Back side temperature
\(T_{b}\)

321.45 K

4.2 Uniform Injection into a Supersonic Nozzle Flow
Nozzle flow

Porous medium flow



Mach number
\(M_{\infty }\)

1.0

Integral coolant mass flow rate
\(\dot{m}_c\)

0.58 g/s

Inflow temperature
\(T_{\infty }\)

3, 288 K

Coolant temperature
\(T_{c}\)

333.15 K

Inflow pressure
\(p_{\infty }\)

1.698 MPa

Specific gas constant helium
\(R_c\)

2, 077 J/(kg K)

Outflow pressure
\(p_{\text {out}}\)

1, 000 Pa


Isothermal wall temperature
\(T_{\text {W}}\)

333.15 K
