I’m Gaylon Campbell, a senior research scientist here at METER Group. Today, we’ll be talking about how to measure and understand thermal properties of materials. Now, a number of different transient methods have been proposed for measuring thermal properties.

In this lesson, we’ll look at just the lined heat source or the heated needle measurements. All of them follow the general pattern outlined here. We use a small needle with the heater running its length and at least one temperature sensor. If they’re combined in one needle, the sensor is placed in the center.

If there are several temperature sensors, they’ll be placed at fixed points along the needle. It’s also possible to have two needles with a heater in one and a temperature sensor in the other. The needle is placed in the sample and a known amount of heat is applied for a known amount of time. The temperature the sensors recorded and compared to the temperature from a model of a line heat source.

The thermal properties of the model are adjusted until the measured and model data match. When they do, the thermal properties of the material are the same as those from the model.

If we use a single needle, such as the one shown here on the left (100 millimeters long and 2.4 millimeters in diameter), and we place that in something with the thermal conductivity of point five watts per meter Kelvin, then we’d measure temperatures like those you see in the graph. In the top line here, we plot the difference between the temperature before the heat was applied and time, but time is shown on the log scale.

The two other lines are for 1 watt per meter Kelvin and 2 watts per meter Kelvin. You can see the slope of the line is as inversely proportional to the thermal conductivity of the material. Now, we can ask why we get a straight line when we do a semi log plot of the data.

Here’s a solution of the model for a line heat source heated as we did in the previous slide:

• q is the heat input per unit length
• k is the thermal conductivity
• r is the needle radius
• D is thermal diffusivity
• t is time
• Ei is the exponential integral, and that can be approximated by a series that’s shown on the bottom.
• Gamma is a constant.

So for the times that we typically use for these measurements, the first two terms in this series are a good approximation of the exponential integral. The first two terms are shown in the top equation.

We rearrange those to get the second equation to show that if we plot temperature versus logarithm of time, we should get a straight line with the conductivity equal to the power per unit length divided by four pi times the slope. Now the model we use for temperature response is for a massless, infinitely small, infinitely long line heat source. That’s a best crude approximation of the behavior of a heated needle. We could derive a better model that more closely approximates a needle, but such a model would be a lot more complicated.

It turns out that simply adding a time offset to the model we’ve been using does a lot to improve its performance. Without a time offset, we normally fit our straight line after skipping the first third of the data we collect. In order to get a good fit, if we put a time offset in, we can get our measurements in a much shorter time, and we can use all of the data. The early data are the best, since that’s when the temperature changes the most.

The drawback is the equation now is a non-linear equation. We have to use a non-linear solver to get a solution, but that’s a small price to pay for the improved performance that we get.

We could do the experiment differently, as shown here. We could have two needles. In this example, each of these needles is 30 millimeters long and 1.3 millimeters in diameter, with heaters in one needle and a temperature sensor in the center of the other needle. On the right, we plot the change in temperature due to heating as before, but here we use a linear time scale. Now the height of the peak is inversely proportional to the volumetric specific heat you can see for the three graphs here. The model we use for this is two equations: one for heating and one after the heat has been shut off. We have a computer program that optimizes the model parameters, k and D, until the predicted temperature rise best matches the measured temperature rise.

Since the diffusivity D is the ratio of conductivity to capacity, we can use the conductivity and diffusivity to calculate the capacity.