MEASURING THERMAL CONDUCTIVITY USING A DIVIDED BAR: NEW EQUATIONS FOR IRREGULAR SAMPLES

Divided bar apparatus have been used to measure thermal conductivity of rock samples from the 1950s to the present. It has been accepted practice to only measure circular samples with the same surface dimensions as the circular heat source & sink surfaces in the divided bar. The divided bar apparatus measures thermal conductivity (λ) by subjecting standards and samples to a constant heat flow (q), in a straight line from the heat source to the heat sink, and applying the basic heat flow equation: q = λ (dT/dZ). Since the heat flow is constant across both standard layers and the sample and we measure the thermal gradient (dT/dZ) across the standards and the sample, we can set the two equations equal and solve for λ_sample, giving the following relationship:

λ_sample = λ_standard * ((ΔT_st1 + ΔT_st2) / 2) * (ΔZ_sample / (ΔT_sample * ΔZ_standard))

When a sample is obtained that is of adequate size to match the heat source & sink dimensions, thermal conductivity measurements can be made without any corrections. Samples are often not large enough or of the correct proportions to match the surface of the heat source & sink. The work presented here has been focused on corrections to modify the standard equation to be used on samples of irregular size and testing the limitations of the corrections. The new equation for handling samples of irregular sizes and shapes is:

λ_sample = (λ_standard * A_standard * ΔZ_sample * (ΔT_st1 + ΔT_st2)) / (2 * ΔT_sample * A_sample * ΔZ_standard)

Testing has shown that this equation can be used for any shape sample that does not extend past the edges of the heat source and sink and has a surface area at least 50% as large as the heat source & sink. For smaller samples the thermal conductivity value must be divided by an additional correction factor, where x is the surface area of the sample as a percentage of the surface area of the heat sink & source:

Correction Factor = 1.34x² - 1.74x + 1.57