VVT3 - Radiation between two perpendicular surfaces

Description

The purpose of this test case is to determine the temperature of an insulated surface that exchanges radiant heat with an isothermal surface and the surrounding environment.

Geometry

The model is sufficiently simple, no part geometry is necessary.

Simulation model

The mesh of elements (1) and (2) is made of quadrilateral null shell element with dimensions of 0.5 m by 0.5 m. The top surface emissivity of the element (1) is 0.6 and element (2) is 1.

The two-element mesh has no material and physical properties assigned to it. Consequently, the thermal conductivity is zero, which is of importance for this activity in order for radiant heat exchange to be the main heat transfer mechanism.

The following boundary conditions are applied:

• Temperature constraint on the element (1) with a value of T =1000 K.
• Radiation: Radiation Enclosure on the top-side region with selected Include Radiative Environment.

The following solution options are set:

• Solution Type=Steady State
• Results Options: Select view factor sums, net radiative fluxes, radiosity fluxes and irradiance fluxes. The radiative environment temperature is set to T₃ = 300 K.

The default solver parameters are selected.

Theory

From [1], the view factor between surfaces (1) and (2) is F₁₂=0.2. Since surfaces elements (1) and (2) are convex, F₁₁=F₁₂=0 and by the enclosure identity, F₁₃=F₂₃=0.8. Radiant heat exchange for this system can be modeled using a resistive network. The surface resistance of surface (1) is defined as:

The remaining resistances are given by:

The equivalent resistance of the network is:

The blackbody emissive power of surface (1) is given by the Stefan-Boltzmann law:

while that of the surroundings is E_b3 = 459.2 W/m². Then, the overall heat transfer rate through the network is defined as:

while the radiosity of surface (2) is obtained realizing that there is zero net heat flow at that surface:

Finally, because surface (2) is an insulator, J₂ = E_b3 and by the Stefan-Boltzmann law again, T₂ = 599.8 K.

Results

The following table compares the thermal solver results calculated using both finite volume and finite element discretization methods with the theoretical results. A good agreement is found.