To determine the thermal conductivity of a structural material, a large 15 cm-thick slab of the material was subjected to a uniform heat flux of 2500 W/m2 , while thermocouples embedded in the wall 2.5 cm. intervals are read over a period of time. After the system had reached equilibrium, an operator recorded the thermocouple readings shown below for two different environmental conditions
From these data, determine an approximate expression for the thermal conductivity as a
function of temperature between 40 and 208°C.
GIVEN
Thermal conductivity test on a large, 6-in.-thick slab
Thermocouples are embedded in the wall 2 in. apart
Heat flux (q/A) = 2500 W/m2
Two equilibrium conditions were recorded (shown above)
FIND
An approximate expression for thermal conductivity as a function of temperature between 40 and
2080C
ASSUMPTIONS
One dimensional conduction
SKETCH
The thermal conductivity can be calculated for each pair of adjacent thermocouples using the equation
for one dimensional conduction,
Solving for thermal conductivity
This will yield a thermal conductivity for each pair of adjacent thermocouples which will then be
assigned to the average temperature for that pair of thermocouples. As an example, for the first pair of
thermocouples in Test 1, the thermal conductivity (ko) is
The average temperature for this pair of thermocouples is
Thermal conductivities and average temperatures for the rest of the data can be calculated in a similar
manner
These points are displayed graphically on the following page.
We will use the best fit quadratic function to represent the relationship between thermal conductivity
and temperature
The constants a, b, and c can be found using a least squares fit.
Let the experimental thermal conductivity at data point n be designated as kn. A least squares fit of the
data can be obtained as follows
The sum of the squares of the errors is
By setting the derivatives of S (with respect to a, b, and c) equal to zero, the following equations
result
For this problem
Solving for a, b, and c
Therefore the expression for thermal conductivity as a function of temperature between 40 and
200°C is
This empirical expression for the thermal conductivity as a function of temperature is plotted with the
thermal conductivities derived from the experimental data in the above graph.
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