ME 223 ASSIGNMENT #2 2013
Assignments handed in by 6 PM on Thursday January 31 will receive a 5 point bonus.
Assignments handed in after that but by 4 PM on Friday February 1 will receive full credit but no
bonus. No assignments will be accepted after 4 PM on Friday February 1.
LECTURE SCHEDULE AND READING
Section in Class Notes
2. BASICS OF HEAT CONDUCTION
2.1 Basic Concepts
2.2 General Conduction Equation
2.3 Examples: Steady Conduction in Walls
Date
Th Jan. 24
F Jan. 25
M Jan. 28
Section in Text
2.1 – 2.2
2.3 - 2.4
2.5
PROBLEMS
All of the problems are from the text. However, some have been modified, so be sure to use the
versions on this sheet rather than the versions in the text.
2-17 (10 points) In a solar pond, the absorption of solar energy can be modeled as heat
generation described by egen = e0 e−bx , where e0 is the rate of heat absorption at the top surface per
unit volume and b is a constant. Obtain a formula for the total rate of heat absorption in a water
layer of surface area A and thickness L at the top of the pond.
2.20E (15 points) The resistance wire of a 800-W iron is 60 in long and has a diameter D.
Assume that the hot wire loses all of its heat by radiation at its surface. Neglect any incoming
radiation to the wire, and assume that the filament is nichrome with an emissivity ε = 0.4. Find
and plot the wire surface temperature in degrees R as a function of diameter in the range
0.02 ≤ D ≤ 0.10 in. (You may use any computational platform you like to produce the graph.
The solutions to be handed out will use Mathematica.) The melting point of Nichrome is 3010
R. What part of your range of D values gives an acceptable design for the filament?
2-25 (15 points) Starting with an energy balance on a volume element in the form of a spherical
shell, derive the one-dimensional transient heat conduction equation for a sphere with constant
thermal conductivity and no heat generation. Assume that the temperature is a function of only
the time t and the radial coordinate r.
ME 223 ASSIGNMENT #2
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2.51 (10 points) A spherical metal ball of radius ro is heated in an oven to a temperature Ti
throughout and is then taken out of the oven and dropped into a large body of water at
temperature T∞ where the sphere is cooled by convection with an average convective heat
transfer coefficient of h. Assume the temperature of the sphere depends only on time t and the
spherical radial coordinate r, and assume that the material properties are all constant. Formulate
the problem mathematically by giving the relevant equation, boundary conditions and initial
condition. Do not solve.
2-53 (15 points) Water at an average temperature of T∞ = 90°C flows through a pipe. The inner
and outer radii of the pipe are r1 = 6 cm and r2 = 6.5 cm, respectively. The outer surface of the
pipe is wrapped with a thin electric heater that provides 400 W per meter length of the pipe. The
exposed surface of the heater is heavily insulated so that the entire heat flow generated by the
heater is transferred inward to the pipe. Heat is then transferred from the inner surface of the pipe
to the water by convection with a heat transfer coefficient h = 85 W/m2·K. Assuming a constant
thermal conductivity for the pipe wall of k = 55 W/m ⋅ K , find the temperature at the outer pipe
wall adjacent to the electric heater. (Answer: 102.6 °C )
2-58 (20 points) The view below shows a section through the front plate of a furnace. The plate
has a thickness L = 20 mm, and the plate material has a thermal conductivity of 25 W/m·K. The
furnace is in a room with surrounding air temperature of 20 °C and an average convective heat
transfer coefficient h = 10 W/m2·K. If the inside surface of the furnace front is subjected to
uniform heat flux of q0 = 5 kW/m2 and the outer surface has an emissivity ε = 0.30, determine
T0 , the inside surface temperature of the furnace front plate. Assume steady state. (Hint: Find
an equation for TL first. You will have to solve that equation numerically. Once you have TL ,
finding T0 is straightforward.)
ME 223 ASSIGNMENT #2
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2-70E (15 points) Consider a steam pipe of length L = 30 ft, inner radius r1 = 2 in, outer radius r2
= 2.4 in, and thermal conductivity k = 7.2 Btu/h·ft·°F. Steam is flowing through the pipe at an
average temperature of 300°F, and the average convective heat transfer coefficient on the inner
pipe surface is given to be h = 1.25 Btu/h·ft2·°F. The average temperature on the outer surface of
the pipe is T2 = 175°F, and you may assume that the temperature in the pipe depends only on the
cylindrical radial coordinate r.
(a) (5 points) Formulate the problem by writing the differential equation and the boundary
conditions for steady one-dimensional heat conduction through the pipe.
(b) (5 points) Solve the problem to obtain the distribution of temperature in the pipe.
(c) (5 points) Find the rate of heat loss from the steam through the pipe. (Answer: 4883 Btu/h)