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(CENG131)[2008](s)hw1~jcychan^_10023.pdf
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CENG 131

CHEMICAL ENG GINEERING THERMODYN NAMICS

HONG KONG UNIVERS SITY OF SCIENCE AND TECHNOLOGY, SPRING G 2009
HHOMEWORK #1 SOLUTIONS
PROBLEM
Advertised is a small toy that will send up a signal flare with a parachute. Upon exa amination, we found that it is simply a metal tubee 2.13 m long and 645 mm2 in cross-sectional are rea. A plug, which contains the parachute, fits into th he tube and a trigger holds it in place. At this posit tion, the bottom edge of the plug is 0.61 m from the e bottom. The plug weighs 1.57 kg. To operate th he toy, the air below the plug is pumped up to a ppressure of about 4.05 bar with a hand pump, and d then the trigger is released, allowing the plug to fly y out the top. The ambient temperature is 298 K, and the constant-volume heat capacity of aair is 2.5 R, where R is the gas constant.
(a)
Assuming the entire devicee operates isothermally, what is the maximum he eight that the plug can go? Hint: To get thhe maximum possible height, you may assume thaat the gas expansion is reversible, th hat is, (i) the gas remains in an equilibrium state a at all times, and

(ii) there is no friction any ywhere in the process.

(b)
Now suppose the expansioon is adiabatic, rather than isothermal. What is th he maximum height that the plug can go o?

(c)
As an engineer, you are intterested in
making the toy work even n better.
Suppose that for safety rea asons, the
metal tube cannot withsta and any
pressure higher than 4.05 bar, and
that it is impossible to cha ange the
overall dimension of the tu ube. But you
can easily change where th he trigger is
positioned in the tube. Wh here would
you place the trigger such that the
maximum height attained by the plug
is the highest possible for the
isothermal case?




SOLUTION
SYSTEM

State 1 State 2 State 33
We will use three different sys stems in this problem, the air below the plug in th he piston-cylinder assembly (control mass, subsc cript A), the plug (control mass, subscript P), and t
the atmosphere (control mass, subscript E). AlAll are closed systems.
We define three different state es: before the trigger is released (subscript 1), wh hen the plug is exactly at the outlet of the tubee (subscript 2), and when the plug reaches maxim mum height (subscript 3).
SYMBOL DEFINITIONS AND GIVENN INFORMATION



For all symbols below, the suubscripts 1, 2 and 3 refer to states defined above e.
11K-1
R = gas constant = 8.314 Jmol-
. = gravitational acceleration = 9.8 ms-2
.. = mass of plug = 1.57 kg
. = cross-sectional area of plu ug = 645 mm2
.... = The constant-volume heeat capacity of air = 2.5R


...= Number of moles of air beelow the plug
.. = height of the plug at state j (.. = 0.61 m, .. = 2.13 m)

.
... = 0.61 m

.
... = 2.13 m


.... = magnitude of velocity of plug at state j
.
.... = 0 m/s
.... = Pressure of system i at state j


. .... = 4.05 bar