VM-DX-MECH-002

VM-DX-MECH-002
Optimization of Bar with Temperature-Dependent Conductivity

Overview

Reference:	From the Basic Principle
Analysis Type(s):	Goal Driven Optimization
Element Type(s):	3-D Solid

Test Case

A long bar 2 X 2 X 20 m is made up of material having thermal conductivity linearly varying with the temperature K = k₀*(1 + a*T) W/m-°C, k₀ = 0.038, a = 0.00582. The bar is constrained on all faces by frictionless support. A temperature of 100°C is applied at one end of the bar. The reference temperature is 5°C. At the other end, a constant convection coefficient of 0.005 W/m²°C is applied. The ambient temperature is 5°C.

Input Parameters: — Convection coefficient, coefficient of thermal expansion and length

Response Parameters: — Temperature (scoped on end face), thermal strain

Figure 221: Schematic

Material Properties

E = 2e11 Pa

ν = 0

α = 1.5E-05/°C

K = k₀*(1 + a*T) W/m-°C

k₀ = 0.038

a = 0.00582

Geometric Properties

Breadth B = 2 m

Width W = 2 m

Length L = 20 m

Loading

Frictionless Support (on all faces)

Reference temperature = 5°C

Temperature on end face T = 100°C

Convection on other end face

Convection coefficient h = 5e-3 W/m²°C

Ambient temperature Ta = 5°C

Parameter	Type	Limits	Desired Value	Importance
Length (l)	Input	15 m l 25 m	No Preference	Low
Convection coefficient (h)	Input	0.004 W/m²°C h 0.006 W/m²°C	No Preference	Low
Coefficient of temperature expansion (α)	Input	1.4e-5/°C α 1.6e-5/°C	No Preference	Low
Temperature (T)	Output	n/a	Minimum Possible	High
Thermal strain (ε)	Output	n/a	Minimum Possible	High

Analysis

Temperature:

Thermal strain:

Combined objective function becomes,

Minimizing ϕ we get input parameters as:

l = beam length = 25 m

h = convection coefficient = 0.006 W/m²°C

α = coefficient of thermal expansion = 1.4e-5/°C

Results Comparison

Results	Target	DesignXplorer	Error (%)
Length (l)	25 m	25 m	0.0000
Convection coefficient (h)	0.006 W/m²°C	0.006 W/m²°C	0.0000
Coefficient of thermal expansion (α)	1.40E-5/°C	1.40E-5/°C	0.0000
Temperature (T)	29.650°C	29.812°C	0.5477
Thermal strain (ε)	3.4514E-04 m/m	3.448E-04 m/m	-0.0995