Heat Transfer Between Radiating Concentric Spheres

In this application, AcuSolve is used to simulate the heat transfer due to conduction and radiation between concentric spheres. The inside surface of the inner and the outside surface of the outer sphere are both held at constant temperature, while the gap between them radiates the heat from one sphere to the other. AcuSolve results are compared with analytical results for temperature and heat flux as described in Incropera (2006) and Salazar, et al. (2016). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with radiation heat transfer and conjugate heat transfer with enclosure radiation modeling and the P1 radiation model.

Problem Description

The problem consists of a fluid region with arbitrary material properties between two concentric spheres with surfaces held at fixed temperature, as shown in the following image, which is not drawn to scale. The radius of the outer sphere is 0.04 m and the radius of the inner sphere is 0.01 m. The inner surface of the inner sphere is defined to have a constant wall temperature at 300.0 K (27 ºC). The outer surface of the outer sphere is defined to have a constant wall temperature at 1300.0 K (1027 ºC). The fluid within the spheres is defined as a non-conducting material, allowing heat to transfer via radiation only. The solid and fluid regions equilibrate to a steady temperature and can be compared with a semi-analytical solution for the temperature and heat flux at the fluid boundaries.

Figure 1. Critical Dimensions and Parameters used for Simulating Conduction and Radiation Between two Concentric Spheres

The simulation was performed as a three dimensional problem by constructing a volume mesh that contains a layer of boundary elements normal to the radial direction.

AcuSolve Results

The AcuSolve solution converges to a steady state and the results reflect the mean temperature distribution within the spheres. As the outer diameter is considerably hotter than the inner, a heat flux is present between the spheres. The temperature within the fluid medium increases as a function of the radius, with a slope corresponding to the temperature difference and the specified emissivity of the shells. The following image shows the temperature contour within the medium along with the two dimensional plot of the temperature versus radius. The temperature and heat flux at the radiation surfaces are compared with the analytical solution as described in Salazar, et al. (2016).

Table 1. The Analytical Solution for the Temperature and Heat Flux at r1 and r2 are Presented with the Corresponding AcuSolve Results for the Enclosure (Surface to Surface) Model and the P1 Model
	T(r1) (K)	T(r2) (K)	Q(r1) (W)	Q(r2) (W)
Analytical	578.2	1035.0	-27822.5	12365.6
AcuSolve (Enclosure)	578.1	1035.1	-27836.3	12366.6
AcuSolve (P1)	594.7	1019.2	-29504.6	13109.6
Percent Difference (Enclosure)	0.025%	-0.010%	-0.049%	-0.008%
Percent Difference (P1)	-2.883%	1.538%	-5.993%	-6.009%

Summary

The AcuSolve solution compares well with analytical results for heat transfer due to conduction and radiation between two concentric spheres. In this application, an arbitrary material is subjected to a heat flux that is calculated based on the view factors or the P1 model within the simulation. As a result of the conjugate heat transfer and the radiative heat flux a temperature distribution between the two spheres develops. The AcuSolve solution for the temperature and heat flux at the radiation surfaces matches well compared to the analytical solution with only a minor difference for the enclosure radiation model. The enclosure radiation model was found to have a higher accuracy for this application compared to the P1 model, as expected due to the relatively low optical thickness in this simulation.

Heat Transfer Between Radiating Concentric Spheres

AcuConsole database file: <your working directory>\sphere_radiation\sphere_radiation.acs

Global

Problem Description
- Analysis type - Steady State
- Turbulence equation - Advective Diffusive
- Radiation equation - Enclosure
Auto Solution Strategy
- Convergence tolerance - 1.0e-5
- Relaxation Factor - 0.4
- Flow - off
- Enclosure radiation - on
Material Model
- Inner
- Conductivity
  - Type - Constant
  - Conductivity - 2.0
- Outer
- Conductivity
  - Type - Constant
  - Conductivity - 0.35
- Radiating
- Conductivity
  - Type - Constant
  - Conductivity - 1.0e-6
Emissivity Model
Inner
- Emissivity - 0.5
Outer
- Emissivity - 0.8
Model
Volumes
- Inner
  - Element Set
    - Medium- Solid
    - Material model - Inner
  - Outer
    - Element Set
      - Medium - Solid
      - Material model - Outer
  - Radiating
    - Element Set
      - Medium - Fluid
      - Material model - Radiating
Surfaces
- Inner_Inner_r1
  - Simple Boundary Condition - (disabled to allow for heat flux)
- Inner_Inner_ri
  - Simple Boundary Condition
    - Type - Wall
    - Temperature BC type - Value
    - Temperature - 300.0 K
- Inner_Radiating_r1
  - Simple Boundary Condition (disabled to allow for heat flux)
  - Radiation Surface
    - Type - Wall
    - Emissivity model - Inner
- Outer_Outer_r2
  - Simple Boundary Condition (disabled to allow for heat flux)
- Inner_Outer_ro
  - Simple Boundary Condition
    - Type - Wall
    - Temperature BC type - Value
    - Temperature - 1300.0 K
- Outer_Radiating_r2
  - Simple Boundary Condition (disabled to allow for heat flux)
  - Radiation Surface
    - Type - Wall
    - Emissivity model - Outer

References

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

Salazar, G., Droba, J., Oliver, B., & Amar, A. J. (2016). Development and Verification of Enclosure Radiation Capabilities in the CHarring Ablator Response (CHAR) code. 46th AIAA Thermophysics Conference.