**Heat Conduction Assessment**

Problem 1 (30 pts)

A long semi-infinite cylinder of unit radius has zero temperature imposed on its surfaces (r = 1)

and T = 100 on its base (z = 0). Plot the steady temperature distribution along the axis for a)

constant thermal conductivity k = 10, b) variable thermal conductivity, k = 10(1 + 0.01T)/2, c)

compare the heat flows through the bottom surface for these two cases.

Problem 2 (35 pts)

A case of periodic heat transfer is exhibited in the cylinder of a reciprocating internal combustion

engine. Determine a) the depth of penetration of the temperature oscillations into the cylinder

wall, b) the fluctuation of the cylinder surface temperature, c) the maximum and minimum surface

heat flux into the surface. Assume that the engine is an engine operating at 2000 rpm, the thermal

diffusivity = 1.7×10-5 m2 /s, and the thermal conductivity *k *= 50 W/mK, the gas temperature

fluctuates with a peak-to-peak amplitude of 1000oC, and the heat transfer coefficient between

the gas and the cylinder surface, *h *= 500 W/m2K. Neglect the effect of the cylinder wall curvature

and simplify to a semi-infinite solid.

As part of the problem, derive then the steady periodic temperature field in a semi-infinite solid

subject to convection at the surface from a 1-D semi-infinite medium of periodic free stream

temperature through a constant heat transfer coefficient.

Problem 3 (30 pts)

A solid sphere of radius b, thermal conductivity k, is initially at temperature T(r,0)=F(r). At times

t>0, heat is generated in the sphere at the volumetric time rate g(r,t). Heat is lost from the surface

r=b into a medium of zero temperature through a heat transfer coefficient h. Using the Green’s

function method, obtain the transient temperature distribution in the sphere.

Problem 4 (30 pts)

A semi-infinite solid is initially at zero temperature. Suddenly, a constant heat flux of magnitude

q is supplied over a surface strip of width W. Find the transient temperature field in the solid.

Problem 5 (35 pts)

A solid is confined in a half-space x>0 is initially at the melting temperature Tm. For times t>0, the

boundary surface at x=0 is subjected to a constant heat flux q”. Using the integral method of

solution and a second-degree polynomial approximation for the temperature, obtain an

expression for the location of the solid-liquid interface as a function of time. Problem 6 (20 pts)

Consider two semi-infinite solids, whose densities, thermal conductivities and diffusivities are *1*,

k1, 1 and 2, k2, 2. The solids are initially at the temperatures *T**1 *and *T**2 *and at *t *= 0 they are

brought into perfect contact. Solve for the transient temperature fields in the two materials using

the Laplace Transform method and find the interface temperature.

Problem 7 (20 pts)

Consider the steady temperature field in the vicinity of a corner of a very large and very long prism

(schematic shown below). The horizontal side (along x) is kept at a temperature of 0oC and the

vertical side (along y) at a temperature of 100oC.

- a) sketch the isotherms and the adiabats in the vicinity of the corner
- b) give an expression for the temperature distribution in the vicinity of the corner.
- c) Give an expression for the y-variation of the heat flux near the corner and across the side

at x=0.

x

y

100

0