Text: Question 3 (12 marks.) (MATH2011/2100/7100)
A textiles company would like to know the thermal properties of a new material. The material is 1 cm thick and is primarily constructed of polyester with an overall thermal diffusivity of c^2 = 0.2 cm^2/s. As part of the testing procedure, the material is wrapped around a drum, placed in an environment at a temperature of 10°C, and the drum's temperature is regulated so that the temperature gradient on the outer surface of the material is 25°C/cm.
(a) Give one reason why a 1-dimensional heat equation is appropriate here, and thus deduce a model of heat transfer through the material could be
∂T/∂t = 0.2∂^2T/∂x^2,
T(x,0) = 10,
T(0,t) = 10,
T(1,t) = 35,
where T is temperature (°C), x is depth into the material (cm), and t is time (seconds).
(b) Write T(x, t) = Tsteady(x) + Ttransient(x, t) where Tsteady(x) is the steady state solution and Ttransient(x, t) is the transient. Show that the steady state solution is Tsteady(x) = 10 + 25x and T satisfies the following equations:
0.2∂^2Tsteady/∂x^2 = 0,
Tsteady(0) = 10,
Tsteady(1) = 35.
(c) Use separation of variables, T(x, t) = F(x)G(t), to show that
T(x, t) = Σ[50(1/n+1)sin(nπx)e^(-0.2(nπ)^2t)],
where Σ denotes the sum over all non-negative integers n and m.
