SAMPLE PROBLEMS Question 1. A composite cylindrical wall is composed of two materials of thermal conductivity kA and kB, which are separated by a very thin, electrical resistance heater for which interfacial contact resistances are negligible. Liquid pumped through the tube is at a temperature T∞,i and provides a convection coefficient hi at the iner surface of the composite. The outer surface is exposed to ambient air, which is at T∞,o and provides a convection coefficient of ho. Under steady-state conditions, a uniform heat flux of qh” is dissipated by the heater. a) sketch the equivalent thermal circuit of the system and express all resistances in terms of

relevant variables. b) Obtain an expression that may be used to determine the heater temperature, Th. c) Obtain an expression for the ratio of heat flows to the outer and iner fluids, qo´/ qi´. How

might the variables of the problem be adjusted to minimize this ratio? Question 2. A steady-state, finite difference analysis has been performed on a cylindrical fin with a diameter of 12 mm and a thermal conductivity of 15 W/m.K. The convection process is characterized by a fluid temperature of 25oC and a heat transfer coefficient of 25 W/m2.K.

+r1

r2

r3

Internal flowT8 ,i, hi

Resistance heaterqh”, Th

Ambient airT8 ,o, ho

+r1

r2

r3

Internal flowT8 ,i, hi

Resistance heaterqh”, Th

Ambient airT8 ,o, ho

To T1 T2 T3

T8 , h

Δx

DTo=100o CT1=93.4o CT2=89.5o C

To T1 T2 T3

T8 , h

Δx

DTo=100o CT1=93.4o CT2=89.5o C

a) The temperatures for the first three nodes, separated by a spatial increment of of x=10 mm, are given in the sketch. Determine the fin heat rate.

b) Determine the temperature at node 3, T3. Question 4. The wall of a spherical tank of 1-m diameter contains an exothermic chemical reaction and is at 200 oC when the ambient air temperature is 25 oC . What thickness of urethane foam is required to reduce the exterior temperature to 40 oC , assuming the convection coefficient is 20 2W/m K ⋅ for both situations? What is the percentage reduction in heat rate achieved by using the insulation?

Question 5. A nuclear fuel element of thickness 2L is covered with a steel cladding of thickness b. Heat generated within the nuclear fuel at a rate !q is removed by a fluid at T∞ , which adjoins one surface and is charecterized by a convection coefficient h. The other surface is well insulated, and the fuel and the steel have thermal conductivities of fk and sk , respectively. a) Obtain the equation for the temperature distribution T(x) in the nuclear fuel. Express your results in terms of !q ,

k f L, b, sk , h, and T∞ . b) Sketch the temperature distribution for the entire system.

Question 6. Annular aluminum fins that are 2 mm thick and 15 mm long are installed on an aluminum tube of 30 mm diameter. The thermal contact resistance between a fin and the tube is known to be

2 x 10-4 2m K/W⋅ . If the tube wall is at 100 oC and the adjoining fluid is at 25 oC , with a convection coefficient of 75 2W/m K⋅ , what is the rate of heat transfer from a single fin? What would be the rate of heat transfer if the contact resistance could be eliminated?

Question 7. Steam flowing through a long, thin walled pipe maintains the pipe wall at a uniform temperaure of 500 K. The pipe is covered with an insulation blanket comprised of two different materials, A and B. The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which T∞=300K and h= 25 W/m2.K. For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures Ts,2(A) and Ts,2(B)?