Benchmark C7

VMC7
One-Dimensional Transient Heat Transfer

Overview

Reference:

J. P. Holman, Heat Transfer, 4th Edition, McGraw-Hill Book Co., Inc., New York, NY, 1976, pg. 106.

Analysis Type(s):

Thermal Analysis (ANTYPE = 4)

Element Type(s):

2D 6-Node Triangular Thermal Solid Elements (PLANE35)

2D Thermal Solid Elements (PLANE55)

2D 8-Node Thermal Solid Elements (PLANE77)

Input Listing:

vmc7.dat

Test Case

A semi-infinite solid, initially of temperature T_o, is suddenly subjected to a convection environment with convection coefficient h and ambient temperature T₁. Determine the surface temperature after 2 seconds for a series of test cases with increasing mesh refinement for each element type.

Figure 600: Semi-Infinite Solid Problem Sketch

Material Properties

Geometric Properties

k = 54 W/m-°C

ρ = 7833 kg/m³

c = 0.465 J/kg-°C

a = 1.0 m

See Comments below

b = a/2 Nm

Parameter Definitions

N = No. of elements in longitudinal direction

N3 = Initial Δt increment

h = 50 W/m²-°C

T_o = 0°C

T₁ = 1000°C

Representative Mesh Options

Results Comparison

Target Solution: T = 157.25

ETYP	N	Deflection Δt_min	Cumulative Iterations	Surface Temperature	Temperature Ratio
55	6	.5	4	142.953	.909
55	8	.25	8	150.472	.957
55	16	.0667	30	155.655	.990
77	6	.5	4	151.993	.967
77	8	.25	8	154.573	.983
77	16	.0667	30	156.523	.995
35	6	.5	4	151.934	.966
35	8	.25	8	154.560	.983
35	16	.0667	30	156.524	.995

Assumptions, Modeling Notes, and Solution Comments

One-dimensional heat transfer is assumed along the model length, a, which is chosen such that no significant change in temperature occurs at the end region. This is done to ensure an infinite length approximation. The model width, b, is chosen such that the element aspect ratio remain constant for all test cases.
Time step optimization was activated to automatically increment the time step during the solution. The minimum Δt increments chosen were arbitrary.
Results tabulation includes the cumulative number of iterations per run. From these results it is shown that the time step opened up from the initial value, for the Δt = 0.5 and 0.0667 cases.
All three elements are shown to converge to the correct solution. The quadratic elements (PLANE35, PLANE77) provide a more accurate solution than the linear PLANE55 element for a similar mesh configuration.