VVT32 - Radiation heat transfer in a conical furnace

Description

The purpose of this case is to model a furnace that has the form of a truncated cone and determine the temperature of the lateral wall using Monte Carlo, GPU computed view factors, and deterministic methods.

Geometry

The furnace is modeled as a solid truncated cone with the two circular bases. The top diameter (D1) of the cone is 600 mm, the bottom diameter (D2) is 1200 mm, and the cone height (H) is 1200 mm.

Simulation model

This model uses the Thermal solution type.

The top and the bottom cone surfaces are meshed with a 2D triangular null shell elements with an element size of 100 mm. The lateral cone surface is meshed with a quadrilateral null shell mesh with an element size of 100 mm.

The following thermo-optical properties of the elements were defined:

• Top surface emissivity: ε₁ = 0.70
• Bottom surface emissivity: ε₂ = 0.40
• Lateral surface emissivity: ε₃ = 1

All surfaces radiate from the bottom.

The mesh has no material or physical properties assigned to it.

The following boundary conditions are applied:

• Temperature constraint on the top surface mesh with a value of T = 500 K.
• Temperature constraint on the bottom surface mesh with a value of T = 650 K.
• Thermal Loads: Heat Load on the bottom surface mesh with a value of q = 1400 W.
• Radiation Enclosure on the bottom selection using different calculation methods:
  • Deterministic with element subdivision equal to 3.
  • Monte Carlo with a defined Monte Carlo Setting modeling object and the Calculate View Factors only check box selected with 15 000 number of rays.
  • GPU Computed View Factors with 15 000 number of rays. This model is computed using a graphics processing unit (GPU) with an NVIDIA Quadro P2000 graphics card enabled, and the View Factors GPU check box selected in the Parallel Configuration File in the Solution dialog box.

The following solution options are set:

• Solution Type = Steady State

This model is computed using a graphics processing unit (GPU) with an NVIDIA Quadro P2000 graphics card.

The default solver parameters are selected.

Theory

The following thermal circuit diagram shows the radiation heat transfer of three surfaces: top surface (1), bottom surface (2) and lateral surface (3).

For parallel coaxial disks of unequal radius, the shape factor is computed as follows:

With:

Where i and j are indices designating the coaxial disks, r is the radius of the disk and H is the cone height.

The shape factor from the lateral surface (3) to the base surface (i=1,2):

With:

Where i,j = 1,2 and i≠j.

Since the top and bottom surfaces are flat, F₁₁ = F₂₂ = 0, and based on the enclosure identity:

The radiative resistances can be determined as follows:

Using the Stephan-Boltzmann law, the radiosity on bottom surface can be computed:

The energy balance can be written as:

The radiosity on lateral surface is then:

With:

The temperature on the lateral surface can be calculated with the Stephan-Boltzmann law:

Results

The following table compares the average temperature and the radiosity results at the bottom lateral surface, predicted by the thermal solver, with the calculated theoretical results.