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

Question

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

Answer 1

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.