VM115

VM115
Thermal Response of a Heat-generating Slab

Overview

Reference:	P. J. Schneider, Conduction Heat Transfer, 2nd Printing, Addison-Wesley Publishing Co., Inc., Reading, MA, 1957, pg. 283, article 11-4, eq. (11-21) and pg. 309, article 12-8.
Analysis Type(s):	Thermal Analysis (ANTYPE = 4)
Element Type(s):	3D Conduction Bar Elements (LINK33)
Input Listing:	vm115.dat

Test Case

An infinite plate of thickness , initially at a uniform temperature T_o, is subjected to a sudden uniformly distributed heat generation rate and a surface temperature T_s. Determine the temperature distribution in the plate after 12 minutes.

Figure 163: Heat-Generating Slab Problem Sketch

Material Properties

Geometric Properties

k = 20 Btu/hr-ft-°F

γ = 500 lb/ft³

c= 0.2 Btu/lb-°F

= 1 ft

T_o = 60°F

T_s = 32°F (t>0)

= 4 x 10⁴ Btu/hr-ft³

Analysis Assumptions and Modeling Notes

A 1 ft² area is used for the conduction elements. Due to symmetry only half of the plate is modeled. Automatic time stepping is used. The initial integration time step (0.01 hr) is based on δ²/4α, where δ is the element length (0.1 ft) and α is the thermal diffusivity (k/γc = 0.2 ft²/hr).

Results Comparison

Time = 0.2 hr	Target	Mechanical APDL	Ratio
T, °F (Node 1)	32.00	32.00	1.000
T, °F (Node 3)	75.75	75.37	0.995
T, °F (Node 4)	103.99	103.26	0.993
T, °F (Node 5)	120.80	119.79	0.992
T, °F (Node 6)	129.46	128.27	0.991
T, °F (Node 2)	132.10	130.85	0.991

Using up to three terms of the infinite series solution in equation 11-21 of P. J. Schneider, Conduction Heat Transfer.