VM160

VM160
Solid Cylinder with Harmonic Temperature Load

Overview

Reference:	F. B. Hildebrand, Advanced Calculus for Applications, 2nd Edition, Prentice-Hall, Inc., Englewood, NJ, 1976, pg. 447, equations 38-44.
Analysis Type(s):	Thermal Analysis (ANTYPE = 0)
Element Type(s):	Axisymmetric-Harmonic 8-Node Thermal Solid Elements (PLANE78)
Input Listing:	vm160.dat

Test Case

A long solid cylinder has a harmonically-varying temperature load along its circumference represented by a cosine function with positive peaks at Θ = 0° and 180° and negative peaks at Θ = 90° and 270°. Determine the temperature distribution along the radius at Θ = 0 and Θ = 90°.

Figure 225: 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 symmetric harmonic function (Mode 2) 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 solution: T(r, Θ) = T_o * (r/r_o)² * cos (2 Θ).

Results Comparison

		Target	Mechanical APDL	Ratio
Mode = 2 Angle = 0°	T, °F (Node 1)	0.0	0.0	-
	T, °F (Node 3)	5.0	5.0	1.00
	T, °F (Node 5)	20.0	20.0	1.00
	T, °F (Node 7)	45.0	45.0	1.00
Mode = 2 Angle = 90°	T, °F (Node 1)	0.0	0.0	-
	T, °F (Node 3)	-5.0	-5.0	1.00
	T, °F (Node 5)	-20.0	-20.0	1.00
	T, °F (Node 7)	-45.0	-45.0	1.00