VM103

VM103
Thin Plate with Central Heat Source

Overview

Reference:

P. J. Schneider, Conduction Heat Transfer, 2nd Printing, Addison-Wesley Publishing Co., Inc., Reading, MA, 1957, pg. 173, article 8-1.

Analysis Type(s):

Thermal Analysis (ANTYPE = 0)

Element Type(s):

4-Node Layered Thermal Shell Elements (SHELL131)

8-Node Layered Thermal Shell Elements (SHELL132)

Input Listing:

vm103.dat

Test Case

Determine the temperature distribution in the thin infinite plate with a cylindrical heat source shown in the following table. The plate also gains heat on the top face from an ambient gas at a temperature T_T and loses heat on the bottom face to an ambient gas at temperature T_B. Assume that no temperature gradient exists through the thickness of the plate.

Figure 145: Thin Plate Problem Sketch

Material Properties

Geometric Properties

k = 5 Btu/hr-ft-°F

h_T = 30 Btu/hr-ft²-°F

h_B = 20 Btu/hr-ft²-°F

t = 0.1 ft

r_i = 0.1 ft

T_T = 100°F

T_B = 0°F

= 250,000 Btu/hr-ft³

Analysis Assumptions and Modeling Notes

Since the problem is axisymmetric only a one-element sector is needed. A small angle Θ = 10° is used for approximating the circular boundary with a straight-sided element. The outer radius r_o is arbitrarily selected at a point where the heat source should have negligible effect. Small elements are used adjacent to the heat source where the temperature gradient is the largest. Nodal coupling is used to ensure circumferential symmetry. Triangular and quadrilateral elements are used.

Results Comparison

SHELL131
	Target[1]	Mechanical APDL	Ratio
T, °F (at Node 1)	226.3	227.0	1.00
T, °F (at Node 2)	103.2	102.4	0.99
T, °F (at Node 3)	73.8	73.6	1.00
T, °F (at Node 4)	65.8	64.6	0.98
T, °F (at Node 5)	62.8	61.6	0.98
T, °F (at Node 6)	60.8	60.6	1.00
T, °F (at Node 7)	60.2	60.2	1.00
T, °F (at Node 8)	60.0	60.2	1.00
T, °F (at Node 9)	173.1	164.9	0.95
T, °F (at Node 10)	130.7	126.8	0.97

SHELL132
	Target[1]	Mechanical APDL	Ratio
T, °F (at Node 1)	226.3	234.8	1.04
T, °F (at Node 2)	103.2	107.2	1.04
T, °F (at Node 3)	73.8	74.4	1.01
T, °F (at Node 4)	65.8	64.7	0.98
T, °F (at Node 5)	62.8	61.6	0.98
T, °F (at Node 6)	60.8	60.6	1.00
T, °F (at Node 7)	60.2	60.4	1.00
T, °F (at Node 8)	60.0	60.4	1.01
T, °F (at Node 9)	173.1	170.5	0.98
T, °F (at Node 10)	130.7	131.7	1.01

Based on graphical estimate