# What the rate of heat transfer between my thumb and a surface of known dimensions and properties?

I am trying to calibrate a high speed thermocouple for measuring heat transfer rates. One of the calibration techniques I am using is sticking my thumb to the thermocouple (which is embedded flush into a solid steel surface onto which I am pressing my finger) and measuring the result.

The time taken for my thumb to fully compress against the surface appears to be trivial judging from the temperature trace we are recording; let's assume this is the case anyway.

We have used the measurements to derive time-accurate heat transfer rates (peaking around 22kW/m^2 for anyone interested - seems awfully high), but now I wish to compare that peak starting value (before my thumb cools and the surface heats up) to the expected result based on what I think should be simple theory.

How would I go about predicting this? Specifically - what is a good analytical approximation of the heat transfer over a short but finite period of time immediately following contact between my thumb and the metallic surface?

(By short I mean a period sufficiently short that a) temperature changes do not propagate to the end of either the thermocouple or my finger and, hopefully b) a numerical solution is not necessary to provide a solution).

Some background information:

The thermocouple is constantan/chromel with known (and assumed to be uniform) thermal properties. The thermocouple is insulated from the rest of the metal surface and any cross-conduction can be ignored. The temperature of both my thumb and the surface prior to contact are known.

Let's assume my body temperature is also uniform throughout my thumb, and that the thermal properties can be approximated by those of water (is this a good assumption?). My thumb and the thermocouple are semi-infinite over the time period.

• I am completely out of my field in this but why are you not using an infrared source shining on the steel? Or shining on an intermediate material with known infrared absorption properties? Oct 24 '13 at 4:16
• @annav I am using an infrared sensor to measure the temperature of both my thumb skin surface and the metal - what I need to know is what the heat exchange rate should be between them given those known conditions. Oct 24 '13 at 22:17

What you are attempting seems in no way empirical as it depends on many factors (how hard you press, contact area, skin properties) all of which can vary and make it difficult to predict. If you insist, I would recommend that you use the solution for the 1D semi-infinite transient heat equation with constant temperature boundary condition $T_s$ (the temperature of your finger). The solution for the heat transfer rate is: $$\dot q_s = \frac{k (T_s - T_i)}{\sqrt{\pi \alpha t}}$$
where $k$ is the thermal conductivity, $T_s$ is the surface temperature BC, $T_i$ is the initial temperature and $\alpha$ is the thermal diffisuvity.