VM116

VM116
Heat Conducting Plate with Sudden Cooling

Overview

Reference:

F. Kreith, Principles of Heat Transfer, 2nd Printing, International Textbook Co., Scranton, PA, 1959, pg. 161, ex. 4-11.

Analysis Type(s):

Thermal Analysis (ANTYPE = 4)

Element Type(s):

Convection Link Elements (LINK34)

3D Conduction Bar Elements (LINK33)

Input Listing:

vm116.dat

Test Case

A large plate of thickness initially has its left surface at temperature T₁ and the other surface at temperature T₂. The left surface is suddenly subjected to an environment temperature of T = T₂. The convection coefficient on this side is given by h = 2.0 + 0.02 (T - T ) where T is the surface temperature (function of time). Determine T after 7 hours. Graphically display the variation of T with time and the temperature distribution across the plate at 7 hours.

Figure 164: Heat Conducting Plate Problem Sketch

Material Properties

Geometric Properties

k = 2 Btu/hr-ft-°F

γ = 800 lb/ft³

c = 0.833 Btu/lb-°F

= 8 in = (8/12) ft

T₁ = 500°F

T₂ = 100°F

Analysis Assumptions and Modeling Notes

A 1 ft² area is assumed for the conduction elements. The length of the convection element is taken as zero (arbitrarily selected). The finite element model is made the same as the theoretical model for a direct comparison. A steady-state solution is done in the first load step. Automatic time stepping is used. The initial integration time step (7/20 = 0.35 hr) is based on δ²/4α, where δ is the nodal distance within the element (1/12 ft), and α is the thermal diffusivity k/γc = 0.003 ft²/hr). POST26 and POST1 are used to obtain the surface temperature history and the temperature distribution at the final time step, respectively.

Results Comparison

Time = 7 hr	Target	Mechanical APDL	Ratio
T, °F (at X = 0.0 in)	285.[1]	293.[2]	1.03

Based on graphical estimates
Temperature at Node 2

Figure 165: Surface Temperature History Plot

Figure 166: Temperature Distribution Across Thickness Plot