Heat Transfer Between Radiating Concentric Cylinders

In this application, AcuSolve is used to simulate the heat transfer due to radiation between concentric cylinders. The inner and outer cylinders are held at constant temperature and are defined to be radiation surfaces. AcuSolve results are compared with analytical results for temperature as described in Incropera (2006). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with radiation heat transfer requiring view factor computation.

Problem Description

The problem consists of a fluid with arbitrary material properties between two infinitely long concentric cylinders, as shown in the following image, which is not drawn to scale. The radius of the outer cylinder is 0.0468 m and the radius of the inner cylinder is 0.0178 m. The inner cylinder is defined to have a constant wall temperature at 773.0 K (500 °C). The outer cylinder is defined to have a constant wall temperature at 300.0 K (27 °C). Both cylinders are specified to radiate as grey bodies, with emissivity set to 0.03778.

Figure 1. Critical Dimensions and Parameters Used for Simulating Radiation Between Two Concentric Cylinders

Figure 2. Mesh used for Simulating Radiation Between Two Concentric Cylinders

The simulation was performed as a two-dimensional problem by constructing a volume mesh that contains a single layer of elements normal to the radial and circumferential directions.

AcuSolve Results

The AcuSolve solution converges to a steady state and the results reflect the mean temperature distribution. As the inner diameter is considerably hotter than the outer, a heat flux, due to radiation, is present between the cylinders. The temperature within the medium reduces logarithmically as a function of the radius. The following images show the temperature with the medium as a function of radius from the center. The temperature is compared with the analytical solution as described in Incropera (2006). The image below shows black circles representing the analytical solution and a solid red line for the AcuSolve results.

Figure 3. Contours of Temperature Between the Concentric Cylinders

Figure 4. Non Dimensional Temperature Distribution Within the Annulus as a Function of Non Dimensional Radius

Summary

The AcuSolve solution compares well with analytical results for heat transfer due to radiation between two concentric cylinders. In this application, an arbitrary material is subjected to a heat flux that is calculated based on the view factors within the model. As a result of radiative heat flux a temperature distribution between the two cylinders develops. The AcuSolve solution for the temperature as a function of radius matches well compared to the analytical solution.

Heat Transfer Between Radiating Concentric Cylinders

HyperMesh CFD database file: <your working directory>\annulus_radiation\annulus_radiation.hm

Global

Problem Description
- Analysis type - Steady State
- Temperature equation - Advective Diffusive
- Radiation equation- Enclosure
Auto Solution Strategy
- Convergence tolerance - 0.0001
- Relaxation Factor- 0.4
- Flow- off
- Enclosure radiation- on
Material Model
- Material
  - Conductivity
    - Type - Constant
    - Conductivity- 1.0
Emissivity Model
- Grey Body
  - Emissivity- 0.03778
- Black Body
  - Emissivity- 1.0
Model
Volumes
- Medium
  - Element Set
    - Medium - Fluid
    - Material model- Material
Surfaces
- ID
  - Simple Boundary Condition
    - Type- Wall
    - Temperature BC type- Value
    - Temperature- 773.0 K
  - Radiation Surface
    - Type- Wall
    - Emissivity model- Grey Body
- Max_Z
  - Simple Boundary Condition
    - Type- Symmetry
  - Radiation Surface
    - Type- Wall
    - Emissivity model- Black Body
- Min_Z
  - Simple Boundary Condition
    - Type- Symmetry
  - Radiation Surface
    - Type- Wall
    - Emissivity model- Black Body
- OD
  - Simple Boundary Condition
    - Type- Wall
    - Temperature BC type- Value
    - Temperature- 300.0 K
  - Radiation Surface
    - Type- Wall
    - Emissivity model- Grey Body

References

F. P. Incropera and D. P. DeWitt. "Fundamentals of Heat Transfer - Sixth Edition". John Wiley & Sons. New York. 2006.