Main
Main
Main
Instructions: Open book. Group study is encouraged, but the solutions submitted MUST BE YOUR OWN
WORK. Please show your work steps. State your assumptions and justify your equations. Partial credit
will be awarded on careful and clear arguments. Wrong unsupported numerical answers can only receive
zero credit. Please upload your solutions in the designated Assignment 2 folder in myCourses.
Now consider the spherical blast explosion shown in the figure below. We define the system to be the initial
explosive material and its subsequent products. There is no mass transfer across the system boundary throughout
this problem. To simplify the analysis, we divide the overall explosion process into three steps. Assume that the
temperature and pressure inside our system are always spatially uniform (but of course change in time).
v. (2 points) Step I (0 1): Initially, the system is at ambient temperature (T0) and pressure (P0). The
chemical reactions taking place within the system convert some explosive material into high-temperature,
high-pressure gas products.
vi. (2 points) Step II (1 2): The system expands very rapidly until its pressure is equilibrated with the ambient
pressure P0.
vii. (2 points) Step III (2 3): The system comes to thermal equilibrium with the environment while the
pressure remains constant and equal to P0.
Spherical blast
wave. Image capture
by Prof. Tétreault-
Friend during her
summer SURE
project working for
Prof. Frost many
years ago.
𝑄𝑅 ~𝐴𝑅 𝑇𝑅4
However, in order to minimize the mass of a space vehicle or satellite, it is necessary to minimize the area of the
radiator since its mass (𝑚𝑅 ) is directly proportional to its surface area, such that
𝑚𝑅 ~𝐴𝑅 .
Suppose that a space vehicle or satellite contains a reversible Carnot engine which experiences a positive heat
transfer 𝑄 (not fixed) from an energy source (“hot heat reservoir) whose temperature is fixed at 𝑇 . This engine
delivers a fixed amount of power 𝑊 and ultimately uses the depths of space as an energy sink (“cold heat
reservoir).
a) Show that for fixed 𝑊 and 𝑇 , three local minima and/or maxima of the radiator mass exist for this
system when the temperature of the radiator is such that
b) Show that 𝑇𝑅 0.75𝑇 corresponds to the absolute minimum of the radiator mass by plotting 𝑚𝑅 (or 𝐴𝑅 )
versus 𝑇𝑅 .
Hints: You may wish to find the roots of the first derivative of a given continuous function in order to find its local
minima and maxima, i.e. 𝑓 𝑥 0. You can then determine graphically which root is the global minimum.