VM102

VM102
Cylinder with Temperature Dependent Conductivity

Overview

Reference:	P. J. Schneider, Conduction Heat Transfer, 2nd Printing, Addison-Wesley Publishing Co., Inc., Reading, MA, 1957, pg. 166, article 7-9.
Analysis Type(s):	Thermal Analysis (ANTYPE = 0)
Element Type(s):	2D Thermal Solid Elements (PLANE55)
Input Listing:	vm102.dat

Test Case

A long hollow cylinder is maintained at temperature T_i along its inner surface and T_o along its outer surface. The thermal conductivity of the cylinder material is known to vary with temperature according to the linear function k(T) = C₀ + C₁ T. Determine the temperature distribution in the cylinder for two cases:

k = constant, (i.e. C₁ = 0)
k = k (T).

Figure 144: Cylinder Problem Sketch

Material Properties

Geometric Properties

C₀ = 50 Btu/hr-ft-°F

C₁ = 0.5 Btu/hr-ft-°F²

r_i = 1/2 in = (1/24) ft

r_o = 1 in = (1/12) ft

T_i = 100°F

T_o = 0°F

Analysis Assumptions and Modeling Notes

The axial length of the model is arbitrarily chosen to be 0.01 ft. Note that axial symmetry is automatically ensured by the adiabatic radial boundaries. The problem is solved in two load steps. The first load step uses the constant k. The MP command is reissued in the second load step to specify a temperature-dependent k.

Results Comparison

		Target[1]	Mechanical APDL	Ratio
T, °F (k = constant); first load step	Node 2	73.8	73.7	1.000
	Node 3	51.5	51.5	1.000
	Node 4	32.2	32.2	1.000
	Node 5	15.3	15.2	0.99
T, °F (k = k(T)); second load step	Node 2	79.2	79.2	1.000
	Node 3	59.6	59.5	1.000
	Node 4	40.2	40.2	1.000
	Node 5	20.8	20.7	0.99

Based on a numerical relaxation method.