VM147

VM147
Gray-Body Radiation within a Frustum of a Cone

Overview

Reference:R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd Edition, Hemisphere Publishing Corporation, 1981, pg. 277, prob. 9.
Analysis Type(s):
Thermal Analysis (ANTYPE = 0)
AUX12 (Radiation View Factor Utility)
Element Type(s):
3D Conduction Bar Elements (LINK33)
2D Thermal Surface Effect Elements (SURF151)
Superelement (or Substructure) Elements (MATRIX50)
Input Listing:vm147.dat

Test Case

A frustum of a cone has its base heated (q1) as shown. The top is held at temperature T3, while the side is perfectly insulated. All the surfaces are diffuse-gray (with emissivities ε1, ε2, ε3, respectively). Determine the temperature T1, achieved by surface 1 as a result of radiation exchange within the enclosure.

Figure 203: Gray-Body Radiation Problem Sketch

Gray-Body Radiation Problem Sketch

Material PropertiesGeometric PropertiesLoading
ε1 = .06
ε2 = 0.8
ε3 = 0.5
ri = 0.050 m
r2 = 0.075 m
h = 0.075 m
T3 = 550 K
q1 = 6000 W/m2

Analysis Assumptions and Modeling Notes

An axisymmetric model is used for the cone. The radiating surfaces are modeled using three LINK33 elements. The non-hidden method (VTYPE) is used since there are no blocking or obscuring surfaces within the enclosure (i.e. all radiating surfaces fully "see" each other). The radiation matrix is written using 50 circumferential divisions (GEOM). Since all the radiating surfaces form an enclosure, no space node is specified. Heat flux on the bottom surface is applied using SURF151 (surface effect element). The value of Stefan-Boltzmann constant is specified in consistent units as 5.6696E-8 W/m2-K.

Results Comparison

Non-hidden methodTargetMechanical APDLRatio
T1 , K9049071.003