Consider a finite solid circular cylinder of diameter D and length L with temperature-dependent thermal conductivity k(T). The external surface of the cylinder is cooled by a fluid at temperature T through a uniform heat transfer coefficient h (Robin bc). It is assumed that the fluid has infinite thermal capacity so that its temperature remains constant at T.
The initial and boundary conditions are:
T(r, z, t) = T at r = 0; aT/aR = 0 or aT/aZ = 0
T(r, z, t) = T at t = 0
h[T(r, z, t) - T] = k(T)[a^2T/aR^2 + a^2T/aZ^2] at the external surface of the cylinder
Fig.1: Geometry of a solid cylinder
The variation in thermal conductivity with temperature in the temperature range of interest can be approximated as a linear function and expressed as:
k(T) = ko[1 + (T - T0)]
where ko is the thermal conductivity of the material at the reference temperature and α is the temperature coefficient of thermal conductivity.
Obtain the temperature distribution and the heat transfer rate.