Modern Engineering Thermodynamics

16.7 Reynolds Transport Theorem 669
Then,
dm T /dt = −0:532 A pffiffiffiffiffiffi
m T RT os / V T T os
= −0:532 A p
R ffiffiffiffiffiffi
T os/VT
mT
p ffiffiffi
= − 0:532 lbm. R / ðlbf⋅sÞ
4:80 × 10 −5 ft 2 ½53:34 ft.lbf/ ðlbm.RÞŠ
h
i
× ð70:0 + 459:67 RÞ 1/2 /ð1:00 ft 3 Þ
m T
= − ð0:0313 s −1 Þm T
or
dm T
= −ð0:0313Þdt
m T
Integrating this result from the initial mass in the tire m Ti to the mass in the tire m Tτ when the valve stem unchokes at time τ
gives
or
Now, m Ti /m Tτ = ½p os V T /ðRT os ÞŠ i
/ ½p os V T /ðRT os ÞŠ τ
= p osi /p osτ ,so
ln ðm Tτ /m Ti Þ = −ð0:0313Þτ
τ = 31:95½ln ðm Ti /m Tτ ÞŠseconds
τ = 31:95 ln ðp osi /p osτ Þ = 31:95 ln ð50:0/27:8Þ = 18:7s
Exercises
19. Determine the time required for the valve stem in Example 16.8 to unchoke when the tire pressure is increased from
50.0 to 70.0 psia. Assume all the other variables remain unchanged. Answer: τ = 29.5 s.
20. If the internal temperature of the tire in Example 16.8 is maintained at 0.00°F rather than 70.0°F during the deflation
process, determine the time required for the valve stem to unchoke. Assume all the other variables remain unchanged.
Answer: τ = 20.1 s.
21. Suppose the valve core is not removed in Example 16.8 but instead the tire has a slow leak through a hole 1.00 × 10 –4 inches
in diameter. Determine how long it would take for the leak hole to unchoke when all the other variables remain unchanged.
Answer: τ = 1.64 × 10 7 s = 190. days.
16.7 REYNOLDS TRANSPORT THEOREM
In Chapter 2, we define a closed system as any system in which mass does not cross the system boundary, but
energy (heat and work) may cross the boundary. An open system then is defined to be any system in which
both mass and energy may cross the system boundary. In classical mechanics, closed and open systems are
called Lagrangian and Eulerian systems, respectively; and their use in problem solving is referred to as Lagrangian
analysis and Eulerian analysis.
The Lagrangian analysis technique is named after the French mathematician Joseph Louis Lagrange (1736–
1813). Basically, it involves solving the equations of energy and motion for a fixed mass (or closed) system. The
Eulerian analysis technique is named after the Swiss mathematician Leonhard Euler (1707–1783). It involves
solving the equations of energy and motion for a nonconstant spatial volume (or open system). For a given
situation, usually one or the other of these techniques is easier to use, but regardless of their ease of application,
both must give the same results. Therefore, we must be able to mathematically transform our governing equations
back and forth between these two analysis frames. The Reynolds transport theorem is a method of carrying
out this transformation.
In Chapter 2, we define a simple balance equation for any extensive property X as
and on a rate basis as
X G = X T + X P (2.11)
_X G = _X T + _X P (2.12)