KMM 342E Mathematical Modeling in Chemical Engineering
Spring 2023
Homework 8
1) The fuel oil pipe that supplies the heating system of a house is laid 1 m below the ground. Around
a temperature of 2oC, the viscosity of the fuel oil increases to a point at which pumping becomes
almost impossible. Assume that the initial ground temperature is 10oC.
Physical properties: k = 0.38 W/m·K and α = 4×10−7 m2/s.
∂T
∂2 𝑇
=α 2
∂t
∂𝑧
a) Specify the initial and boundary conditions.
b) Obtain an expression for the unsteady-state temperature profile by introducing a dimensionless
temperature variable:
θ=
T − 𝑇0
𝑇∞ − 𝑇0
c) When the air temperature drops to −15oC, how long is it before there are problems in the heating
system?
2) A spherical fuel element of radius R is initially at a uniform temperature of T0. At t = 0, energy
is generated within the sphere at a uniform rate of Q (W/m3). The outside surface temperature is
kept constant at T∞ by a coolant. The governing differential equation for temperature distribution
within the sphere is given by
a) Write the appropriate initial and boundary conditions.
b) Rewrite the model equation and conditions using the following dimensionless
quantities:
d) A solution is proposed in the following form:
Rewrite the model equation and conditions for 𝑢𝑇 (𝜏, 𝜁) as the dependent variable.
Explain briefly the reason of using this variable transformation.
e) Obtain the solution for 𝑢𝑇 (𝜏, 𝜁) using separation of variables method.
Due date: May 22, 2023