VM108

VM108
Temperature Gradient Across a Solid Cylinder

Overview

Reference:	F. B. Hildebrand, Advanced Calculus for Applications, 2nd Edition, Prentice-Hall, Inc., Englewood, NJ, 1976, pg. 447, eqs. 38-44.
Analysis Type(s):	Thermal Analysis (ANTYPE = 0)
Element Type(s):	Axisymmetric-Harmonic 4-Node Thermal Solid Elements (PLANE75)
Input Listing:	vm108.dat

Test Case

Heat is conducted across the diameter of a long solid cylinder. The temperature loading along the circumference is antisymmetric about the Y-Z plane and varies sinusoidally with peaks occurring at Θ = 0° and Θ = 180°. Determine the temperature distribution along the radius at Θ = 0°.

Figure 154: Solid Cylinder Problem Sketch

Material Properties

Geometric Properties

k = 1 Btu/hr-ft-°F

r_o = 20 ft

T_o = 80°F

Analysis Assumptions and Modeling Notes

The axial length of the model is arbitrarily chosen to be 5 ft. The temperature loading is applied as a harmonic function (mode = 1) around the periphery of the cylinder. To obtain the theoretical solution, equations 43 and 44 in F. B. Hildebrand, Advanced Calculus for Applications are used. Applying the temperature boundary condition and the requirement that T(r,Θ) should be finite and single-valued leads to the following solution: T(r,Θ) = T₀ * (r/r₀) * cosΘ.

Results Comparison

Mode = 1 (angle =0°)	Target	Mechanical APDL	Ratio
Node 1 T, °F	0.0	0.0	-
Node 2 T, °F	20.0	20.0	1.00
Node 3 T, °F	40.0	40.0	1.00
Node 4 T, °F	60.0	60.0	1.00