Hot (or Cold) to the Touch

At one time or another we’ve all experienced the sensation of feeling a different temperature depending on the material that we are touching.  Try it now—touch a nearby piece of metal with your fingertip, then, with another finger, touch a piece of wood or cloth.  Most likely the metal felt cooler to the touch than the wood did.  

How could this be if both objects have been sitting in the room long enough to come to thermal equilibrium with the surroundings? 

The answer is readily explained by considering contact temperatures and thermal resistance.  Your finger is most likely warmer than both of the objects that you touched if you are sitting in a climate-controlled room, say around 20°C.  As a first approximation, let’s neglect the distance between the surface of your skin and your nerve endings and assume that you are feeling the temperature at the interface between your finger and the object that you were touching.  

 The temperature that you feel will be some kind of average between your warm finger and the cooler object.  Since your finger is warmer, heat will have to flow from your finger into the object.  By considering conservation of energy (this is ignoring the heat generation in your finger and ignoring the temperature changes which occur as the heat flows, both reasonable approximations for the first few instants of contact) it can be shown that the appropriate way of averaging the two temperatures to calculate the interface temperature, T_i, is:

 where the subscript “f”  refers to your finger and the subscript “s” refers to the surface of the object you touch. I am using the symbol “m” to represent a group of material properties:

 

 where “k” is the thermal conductivity, “ρ” is the density, and “c” is the specific heat capacity.  Thermal conductivity is a material property that characterizes how easily heat flows—good conductors like metals have a higher thermal conductivity than thermal insulators like wood.

It is easy to see from these equations that the interface temperature (sensed by your nerve endings) is going to be much closer to the object temperature if the object has a high value for “m” than if that value is low.  In fact, if you touched a perfect thermal insulator (k=0) the interface temperature would just be the temperature of your finger.

Here are some values for “m” for a few substances:

copper: 110;  human skin: 4;  hardwood: 1;  asbestos: 0.8;    

[units are BTU/(ft²*°F*hr^0.5)]

It is also possible, (though quite a bit of work) to derive an interface temperature taking into account the heat generation in your finger.  The average interface temperature for that case turns out to not have an explicit solution, but with a computer, it can be readily solved anyway.