VM111

VM111
Cooling of a Spherical Body

Overview

Reference:	F. Kreith, Principles of Heat Transfer, 2nd Printing, International Textbook Co., Scranton, PA, 1959, pg. 143, ex. 4-5.
Analysis Type(s):	Thermal Analysis (ANTYPE = 4)
Element Type(s):	2D Thermal Solid Elements (PLANE55)
Input Listing:	vm111.dat

Test Case

Determine the temperature at the center of a spherical body, initially at a temperature T_o, when exposed to an environment having a temperature T_e for a period of 6 hours. The surface convection coefficient is h.

Figure 158: Spherical Body Problem Sketch

Material Properties

Geometric Properties

K=(1/3) BTU/hr-ft-°F

γ = 62 lb/ft³

c = 1.075 Btu/lb-°F

h = 2 Btu/hr-ft²-°F

r_o = 2 in = (1/6) ft

T_o = 65°F

T_e = 25°F

Analysis Assumptions and Modeling Notes

Since the problem is axisymmetric, only a one-element sector is needed. A small angle Θ = 15° is used for approximating the circular boundary with a straight-side element. Nodal coupling is used to ensure circumferential symmetry. Automatic time stepping is used. The initial integration time step (6/40 = 0.15 hr) is based on δ²/4 α, where δ is the element characteristic length (0.0555 ft) and α is the thermal diffusivity (k/γc = 0.005 ft²/hr). POST1 is used to extract results from the solution phase.

Results Comparison

Time = 6 hr	Target[1]	Mechanical APDL	Ratio
T, °F	28.0	28.96	1.034

Based on graphical estimates.