Heat Transfer

Homework #2
Professor: Albio Gutiérrez – albio.gutierrez@correounivalle.edu.co – Office E33-2005
Instructions:

• This homework must be submitted in groups of two students.

• This homework may be submitted using handwriting if desired as well as utilizing a word processor
such as MS Office or OpenOffice. If handwriting is utilized, no cross-outs or illegible words or
expressions will be allowed. In this case, the point will not be graded.
• The professor will select one exercise from numerals 1 to 8. This represents 50% of the total
homework’s grade.
• The professor will also select one exercise from numerals 9 to 12. This represents the remaining 50% of
the total homework’s grade.
• The file must be saved as a pdf file and uploaded to the folder named HMW#2.
• The file should be named with the last names of the students (both last names) followed by HM2. For
instance: Ocampo-Charria_Ortiz-Diaz_HM2.pdf.
• The submission deadline for this homework is April 8th.

Advice:
• Carefully follow the previous instructions. A good engineer, a professional one, should know how to
follow instructions for submission of his/her duties in both, academia and industry.
• Use computer programing to automate the solution process whenever you find it convenient.

Problems:

1. Derive the general heat conduction equation in a) rectangular coordinates, b) cylindrical coordinates and, C)
spherical coordinates.

2. Select ten technological solid materials commonly utilized in heat transfer applications and fill a table with
their following properties in SI units: density, specific heat, thermal conductivity and thermal diffusivity.
Organize the selected materials according to their thermal diffusivity value. Repeat the table for the imperial
unit system.

3. A furnace wall is constructed of 0.2 m thick fireclay brick having ka = 1.0 W/mK. This is covered on the
outer surface with a 0.03 m thick layer of insulating material having kb = 0.07 W/mK. The furnace inner
brick surface is at 1250 K and the outer surface of the insulation material is at 310 K. Calculate the steady
state heat transfer rate through the wall in W/m2, and determine the interfacial temperature between the
brick and the insulation (neglect the thermal contact resistance).

4. The composite wall of an oven consists of three materials, two of which are of known thermal conductivity,
kA = 20 W/mK and kC = 50 W/mK, and known thickness, LA = 0.30 m and LC = 0.15 m. The third material,
 B, which is sandwiched between materials A and C, is of known thickness, LB = 0.15 m, but unknown
thermal conductivity kB.

Under steady-state operating conditions, measurements reveal an outer surface temperature of Ts,o = 20°C,
an inner surface temperature of Ts,i = 600°C, and an oven air temperature of T∞ = 800°C. The inside
convection coefficient h is known to be 25 W/m2K. What is the value of kB?

5. A house has a composite wall of plywood, fiberglass insulation, and plaster board, as indicated in the
sketch. On a cold winter day, the convection heat transfer coefficients are ho = 60 W/m2K and hi =30
W/m2K. The total wall surface area is 350 m2.

Consider the use of more realistic conditions for which the outside air is characterized by a daily basis
varying temperature of the form:

Estimate the daily heat loss through the wall assuming quasi-steady conditions for which changes in energy
storage within the wall may be neglected.

6. The performance of gas turbine engines may be improved by increasing the tolerance of the turbine blades
to hot gases emerging from the combustor. One approach to achieving high operating temperatures involves
application of a thermal barrier coating (TBC) to the exterior surface of the blade, while passing cooling air
through the blade. Typically, the blade is made from a high-temperature superalloy, such as Inconel (k = 25
W/mK), while a ceramic, such as zirconia (k = 1.3 W/mK), is used as a TBC.
 Problem Sketch

Additional Drawings

Consider conditions for which hot gases at T∞,o = 1700 K and cooling air at T∞,i = 400 K provide outer and
inner surface convection coefficients of ho = 1000 W/m2K and hi = 500 W/m2K, respectively. If a 0.5-mm
thick zirconia TBC is attached to a 5-mm-thick Inconel blade wall by means of a metallic bonding agent,
which provides an interfacial thermal resistance of Rt,c = 10e-04 m2K/W, can the Inconel be maintained at
a temperature that is below its maximum allowable value of 1250 K? Radiation effects may be neglected,
and the turbine blade may be approximated as a plane wall. Plot the temperature distribution with and
without the TBC. Are there any limits to the thickness of the TBC?

7. Consider one-dimensional steady state heat conduction in the radial direction with a constant volumetric
heat generation rate and derive the expressions for the temperature distribution along the domain of a) a
cylindrical wall and, b) a spherical wall. Both domains having the BCs: T(r1) = Ts1 and T(r2) = Ts2.

8. Steam at 250 °F flows in an insulted pipe. The pipe is made of stainless steel and has an inside radius of 2.0
in and an outside radius of 2.25in. The pipe is covered with one-inch layer of 85% magnesia insulation. The
inside heat transfer coefficient is, hi = 15 Btu/hft2°F, and the outside heat transfer coefficient is, ho = 2.2
Btu/hft2°F. Determine the overall heat transfer coefficient and the heat transfer rate from the steam per foot
of pipe length if the surrounding air temperature is 65 °F.

9. Following a similar procedure that the one shown in class, derive the solution for the temperature
distribution and the rate of heat transfer for fins of uniform cross section under the following BCs at the tip:
a) Adiabatic, b) prescribed temperature and, c) infinite length.

10. An aluminum cylindrical rod (k = 132 Btu/hft°F), having a diameter of 0.375 in and a length of 4 in, is
attached to a surface having a temperature of 200 °F. The rod is exposed to ambient air at 70 °F, and the
heat transfer coefficient along the length and the end is 1.5 Btu/hft2°F. Determine the temperature
distribution and the heat flux a) neglecting the heat transfer at the end and, b) accounting for the heat
transfer at the end.
11. Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular
profile to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a distance of 4 mm
(250 fins/m). The convection coefficient associated with the bare wall is 40 W/m2K, while that resulting
from attachment of the fins is 30 W/m2K.

12. A long rod of 20-mm diameter and a thermal conductivity of 1.5 W/mK has a uniform internal volumetric
thermal energy generation of 106 W/m3. The rod is covered with an electrically insulating sleeve of 2-mm
thickness and thermal conductivity of 0.5 W/mK. A spider with 12 ribs and dimensions as shown in the
sketch has a thermal conductivity of 175 W/mK, and is used to support the rod and to maintain
concentricity with an 80-mm-diameter tube. Air at T∞ = 25°C passes over the spider surface, and the
convection coefficient is 20 W/m2K. The outer surface of the tube is well insulated. We wish to increase
volumetric heating within the rod, while not allowing its centerline temperature to exceed 100°C. Determine
the impact of the following changes, which may be effected independently or concurrently: (i) increasing
the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and
(iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous
carbon or quartz). Recommend a realistic configuration that yields a significant increase in 𝑞̇ .