MECH 395 2020 Midterm Exam Solution Notes
I(a)_{3}marks
The thermal diffusivity α = k / (ρ c) of a material is a ratio of its heat conducting to heat storing
ability, whereare k, ρ and c the termal conductivity, density and specific heat capacity. Units for
α are (m2/s). It can be used to find how quickly heat will diffuse through a given length of the
material. With 1D heat conduction across a length L, the change in temperature at one face
would be felt at the opposite face in a time t that can be estimated from L2 ≈ 2αt. (An answer
without “2” also earns full credit).
We could use this for estimate of heat transfer time through (1) one end of a metal rod of length
L, with cylindrical surface insulated, or (2) one face of a large plane wall, facing a sudden change
in temperature, so that 1D heat conduction is a reasonable approximation.
Answers I(d) _{8}marks. 2 marks each for Parts 1 through 4.
(i) Given steady-state and linear temperature profiles in walls A(1-2) and B(2-3), there is no heat
generation in either segment, so conservation of energy requires heat flux values 𝑞𝑞2" and 𝑞𝑞3" to be
equal.
(ii) From the conclusion in (i), -kA(dt/dx)A = -kB(dt/dx)B at the interface 2 between walls A and B.
Since slope (dt/dx)A > (dt/dx)B, it follows that kA < kB.
(iii) The local slope is shown to be increasing as we move from face 4 to face 3 of C. Hence the
magnitude of the heat transfer rate is increasing as we move from wall 4 to wall 3, which implies
that volumetric heat generation 𝑞𝑞̇ should be occuring in region C. In this case, one cannot
represent the heat conduction in region C by a thermal resistance. Such a representation would
require a constant thermal current (heat transfer rate) flowing through wall C.
(iv) The temperature profile has zero slope at wall 4, which implies zero heat transfer into wall C
at this boundary. As seen from the local slopes of the profiles at interface 3, there is heat transfer
occuring out of region C at this interface. Conservation of energy dictates that there must
volumetric energy generation within wall C.