VVT42 - Non-gray radiation exchange between two parallel plates

Description

The purpose of this validation test is to determine the equilibrium temperature of a thermally free plate and compare the result with a theoretical prediction for non-gray radiative exchange between two parallel plates.

Geometry

The geometry consists of two centered parallel square plates, each with an area of 1 m². The plates are separated by a distance of 0.1 m, such that see each other with a view factor of 0.826997.

Simulation model

Both plates are meshed with quadrilateral thin shell element. One plate is held at a prescribed temperature, while the second plate is allowed to reach radiative equilibrium. The top surface emissivity is defined as a function of wavelength in the field table:

The following boundary conditions are applied:

• Temperature constraint applied to the element 1 with a value of T₁ = 60 K.
• Radiation: All Radiation with the Include Radiative Environment option selected using:
  • Deterministic with element subdivision equal to 5
  • Monte Carlo with 15 000 number of rays and selected Calculate View Factors only
  • Hemicube Rendering

This model uses the Simcenter 3D Space Systems Thermal solver.

The following solution options are set:

• Solution Type = Steady State
• Radiative Environment Temperature: T_env = 0 K

The following radiation solver parameters are set:

• Bands = Specify Wavelength
• Wavelength Breakpoint Table = [0, 40, 80, 120, 1200]

Theory

At equilibrium, the net heat transfer rate to the free plate is zero. Therefore, the radiative heat transfer rate emitted by plate 2 is equal to the radiative heat transfer rate absorbed by plate 2:

The emitted heat rate from plate 2 is:

The absorbed heat rate by plate 2 is:

where:

• σ = 5.670374419 × 10^-8 W/m²·K⁴ is the Stefan-Boltzmann constant.
• A₁ and A₂ are the areas of plates 1 and 2, respectively.
• T₁ is the prescribed temperature of plate 1.
• T₂ is the equilibrium temperature of plate 2.
• VF₁₂ is the view factor from plate 1 to plate 2.
• VF₂₁ is the view factor from plate 2 to plate 1.
• ε_1g is the emissivity of plate 1 in radiation band g.
• ε_2g is the emissivity of plate 2 in radiation band g.
• p_g(T) is the blackbody radiation fraction in band g at temperature T.

The equilibrium temperature T₂=46.04 K is obtained by iteratively solving the nonlinear equation Q_2,emit​(T₂​)−Q_2,abs​(T₂​)=0, because the blackbody radiation fractions p_g​(T₂​) depend on the unknown temperature T₂.

The blackbody radiation fractions for the four bands at T₁=60 K and T₂=46.04 K are found using the radiation function table [20]:

Results

The following table compares the temperature computed by the thermal solver with the theoretical temperature obtained by iteratively solving the nonlinear equation. Simulation results using deterministic and Monte Carlo methods are in agreement with theoretical values. The Hemicube method shows a larger discrepancy, which is likely due to view-factor discretization and resolution effects associated with closely spaced parallel plates.