VM113

VM113
Transient Temperature Distribution in an Orthotropic Metal Bar

Overview

Reference:	P. J. Schneider, Conduction Heat Transfer, 2nd Printing, Addison-Wesley Publishing Co., Inc., Reading, MA, 1957, pg. 261, ex. 10-7.
Analysis Type(s):	Thermal Analysis (ANTYPE = 4)
Element Type(s):	2D Thermal Solid Elements (PLANE55)
Input Listing:	vm113.dat

Test Case

A long metal bar of rectangular cross-section is initially at a temperature T_o and is then suddenly quenched in a large volume of fluid at temperature T_f. The material conductivity is orthotropic, having different X and Y directional properties. If the surface convection coefficient is h, determine the temperature distribution in the slab after 3 seconds in the following locations of the bar:

center
corner edge
face centers of the bar

Figure 160: Orthotropic Metal Bar Problem Sketch

Material Properties	Geometric Properties	Loading
k_x = 20 Btu/hr-ft-°F k_y = 3.6036 Btu/hr-ft-°F γ = 400 lb/ft³ c = 0.009009 Btu/lb-°F h = 240 Btu/hr-ft²-°F	a = 2 in = (2/12) ft b = 1 in = (1/12) ft	T_o = 500°F T_f = 100°F

Material Properties

Geometric Properties

k_x = 20 Btu/hr-ft-°F

k_y = 3.6036 Btu/hr-ft-°F

γ = 400 lb/ft³

c = 0.009009 Btu/lb-°F

h = 240 Btu/hr-ft²-°F

a = 2 in = (2/12) ft

b = 1 in = (1/12) ft

T_o = 500°F

T_f = 100°F

Analysis Assumptions and Modeling Notes

A nonuniform grid (based on a geometric progression) is used in both X and Y directions to model a quarter of the bar cross-section. Automatic time stepping is used. The initial integration time step = (3/3600)(1/40) is greater than (δ²/4α), where δ is the shortest element length (0.0089 ft) and α is the thermal diffusivity (k_y/γc = 1.0 ft²/hr).

Results Comparison

Time = 3 sec. (=0.0008333 hr.)	Target[1]	Mechanical APDL	Ratio
T, °F (Node 1)	459.	457.	1.00
T, °F (Node 7)	151.	158.	1.05
T, °F (Node 13)	279.	288.	1.03
T, °F (Node 2)	202.	204.	1.01

Based on graphical estimates.