I fixed different thermocouples in the agarose gel bar to monitor temperature increase with time, upon exposure to an external heating source. I used the attached equation but found an opposite trend.
Good day, Mai. With the measurement of the coefficient of thermal conductivity, you need to clearly understand which technique you are using. First, the measurement results depend on the geometry of the sample. You write that you are using a bar, but this formula is for a hollow cylinder. Secondly, you must measure the temperatures T1 and T2 in a stationary mode, when they are established and remain constant over time. What are t1 and t2 in a formula? Usually the symbol t denotes time. There should be no time in this equation. These should be radii: r1 inner and r2 outer radius of the hollow cylinder. According to the theory of dimensions, the formula should contain one more quantity of the dimension of length. In this case, the length of the cylinder L. It should be a factor for (T2-T1) . There is also an error in the formula. There you need to use for the cylinder 2Pi instead of 4Pi. Well, and most importantly, the sign is confused. Should be (T1-T2). (see equation (20.8) here https://lfirmal.com/polyj-cilindr-mnogoslojnyj-cilindr, in this case small t denotes temperature ).
If you want to evaluate thermal conductivity before measurements, then try using simple models. You can read about them in this paper Huang Y. et al. Tailoring the electrical and thermal conductivity of multi-component and multi-phase polymer composites // International Materials Reviews. 2020. V. 65. Pp. 129-163 (see Table 8).
Do you know the right equation because my bar is considered more rectangular? I took my measurements at different times ranging from 0 min. up to 10 min.
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