You received the following letter as a heat transfer expert

You received the following letter as a heat transfer expert. You need to answer the letter taking into consideration that your consulting fees depend on your answer. We have been testing long, cylindrical samples of stainless steel in our laboratory under constant heat flux conditions at the outer radius. We are attempting to determine the length of time it takes the surface to reach its melting temperature of 1480 degree C. Two tests on 10 cm diameter cylinders have given the following results The initial temperature in each test was 27 degree C. The laboratory assistant feels that the data should be fitted with the following hyperbola: q\"t = 1.5 times 10s where q\" is in W/m^2, and t in seconds. I\'m hesitant to rely on the hyperbola even though it closely fits the data and has the right qualitative behavior. I would appreciate learning the answers to the following questions: Is the hyperbola correct in predicting no influence in diameter Is the hyperbola correct in predicting that the thermal properties of the material have no effect Does heat theory support the hyperbola conclusion If the hyperbola is not valid, please provide a better relation

Solution

This is clearly a transient heat transfer problem for semi infinite cylinder. It depends on Biot and Fourier numbers where the characteristic dimension of radius is required. So hyperbola is not correct in predicting no influence in diameter. It is also wrong to conclude that thermal properties of the material like conductivity and diffusivity will have no effect. Also, the heat transfer theory does not support the hyperbola conclusion. - ans (a, b and c).