Posted by Edwin on .
Two spheres S1 and S2 of the same metal and having the same type of surface, have radii 20cm and 10cm respectively. Compare their rates of fall of temperature as they cool (a) when S1 is at 50 degree centigrade, S2 is at 40 degree centigrade, the surroundings are at 20 degree centigrade and Newton's law applies (b) when S1 is at 400 degree centigrade, S2 at 300 degree centigrade the surroundings are at 200 degree centigrade and stefan's law applies.
For the Newton's Law case,
the heat loss rate rate is h*A(T - 20). and this equals M C dT/dt. M is the mass, C the specific heat and
A is the area of the sphere(s). The differential equation can be solved for T(t) in terms of A, h, M and C. It results in an exponential approach to T = 20 C.
You have not been told the value of the "film heat transfer coefficient", h, although one could calculate it for natural of forced convection using heat transfer engineering formulas that are known. You may have to answer this question using equations rather than graphs with real numbers. Try to use dimensionless groups wherever you can.
In the "Stefan", more propertly called the Stefan-Boltzmann case,
e*sigma*A*(T^4 - 473^4) = M C dT/dt,
where T must be expressed in Kelvin.
e is the emissivity, which they may want you to assume is 1, and sigma is the Stefan Boltzmann constant.
I am going to have to leave the solving of the differential equations, and the "comparing" up to you. They are not difficult.
In a real situation, both types of heat transfer will usually be present, with the radiative "Stefan" type dominating at higher tempertures.