1. No category

Physics 285
Orange Coast College
Arnold Guerra III
Extra Examples – Chapter 17
1. A brass ring of diameter 10.00 cm at 20.00C is heated and slipped over an aluminum rod of
diameter 10.01 cm at 20.00C. Assuming the average coefficients of thermal expansion are
constant for brass and aluminum,
(a) to what temperature must this combination be cooled to separate them? Is this attainable?
(b) What if the aluminum rod is 10.02 cm in diameter?
2. Calculate the final equilibrium temperature when 150 grams of steam initially at 100oC is
passed into 200 grams of liquid water and 25 grams of ice both initially at 0 oC in a calorimeter
container. That is, the liquid water AND the ice are initially at 0 oC. Ignore any heat energy
exchanges to the calorimeter and the surroundings. {For water: the latent heat of fusion is
3.33x105 J/kg, the heat of vaporization is 2.26x106 J/kg, and the specific heat capacity is 4186
J/kg oC}. If the final temperature of the system is 0 oC, what mass of ice remain, but if the final
temperature of the system is 100 oC, what mass of steam remains?
3. Heat Conduction
The heat (energy) flow down a temperature gradient through stationary matter is given by
Fourier’s law of heat conduction as
! = −! ∇!
where
! ≡ heat current density = heat flowing per unit time, per unit area
∇! ≡ temperature gradient
! ≡ thermal conductivity of the material
We can think of −∇! as a thermal “driving force” which drives thermal energy. The total heat
current
I
flowing
across
an
area
A
is
!=
! ∙ !!
where !! = ! !" and ! is a unit vector along the normal to the surface area. “Steady State” heat
flow occurs when the heat flowing across an area is the same at all times, i.e., I = constant.
Derive expressions for the steady state heat current I for the following geometries:
(a) heat flow in one-dimension across an area A, with no heat source
(b) radial heat flow in a cylinder, with no heat source
(c) radial heat flow in a sphere, with no heat source
(d) radial heat flow in a cylinder, with a heat source S = heat generated per unit time, per unit
volume
4. Starting with Fourier’s law of heat conduction, ! = −! ∇! , and the fact that ∇ ∙ ! !" is the
net outward heat current across the boundary of the volume element dV, show that the
temperature satisties the equation
!"
!
= !∇! ! + !
!"
! !!
!
where D = ! ! ! is the thermal diffusivity of the medium, and s is the heat source density (heat
!
generated per unit volume per unit time).
5. Greenhouse Effect
Earth absorbs solar energy and radiates infrared. The solar energy incident on earth is π R2 J
where R is earth radius and J = 1350 W m-2 is the solar constant. Also, earth radiates from its
entire surface area 4 π R2 . In the following questions, assume earth’s surface temperature to be
uniform at Ts .
(a) Find the steady state value of Ts assuming the earth absorbs ALL the incident solar
energy, and that the earth is black (emissivity = 1) for infrared. (numerical answer
required).
(b) Find the steady state value of Ts taking into account that earth absorbs only 0.65 of the
incident solar energy. (The remaining 35% is reflected from clouds and ice). You’ll
discover that at this temperature earth would not support life.
(c) Now find the steady state value of Ts taking into account the “greenhouse effect” of
atmospheric infrared absorption and emission. The diagram below shows the ground at
the surface temperature Ts and the atmosphere, represented as a thin black layer, at
temperature Ta . The atmosphere absorbs 95% of the infrared radiation emitted by the
earth, and can therefore be approximated as a single black layer. The diagram shows the
infrared radiation emitted upward by the earth, and both upward and downward radiation
emitted by the atmosphere. Assume that the ground absorbs 0.475 of the incident solar
energy and the atmosphere absorbs 0.175 of the incident solar energy (for a total of 0.65).
Write the heat energy balance (heat gained = heat lost) for the ground and for the
atmospheric layer to obtain two equations for Ts and Ta . Solve for these temperatures and
evaluate them numerically. Then look up on WolframAlpha the average surface
temperature for the earth and compare this value with your answer obtained for the
ground temperature using the above greenhouse effect model.