# Determinig the position of the piston at state 2

1. Nov 7, 2015

### Doubell

1. The problem statement, all variables and given/known data
A piston-cylinder device attached below is shown in Figure 6.A-31where
Patm = 100 kPa
Tamb = 150°C
piston
Ac = 0.01m2
mp = 100 kg
water
T1 = 350°C
P1 = 400 kPa
z1 = 0.5m
Figure 6.A-31: Piston-cylinder device.
The initial position of the piston is z1 = 0.5 m and the piston cross-sectional area is Ac = 0.01 m2.
The mass of the piston is mp = 100 kg. The cylinder contains water that is initially at T1 = 350ºC
and P1 = 400 kPa. The surroundings are at Patm = 100 kPa and Tamb = 150ºC. The piston is
initially held in place by a pin to prevent it from moving due to the internal pressure. At some
time, the pin is removed and the piston quickly and violently shoots upward under the action of
the internal pressure. The piston motion continues for some time until eventually the oscillations
are damped out and the piston obtains a new equilibrium position at state 2 where it is in
mechanical equilibrium with the surroundings (i.e., a force balance on the piston can be used to
provide the internal pressure at state 2). There is no heat transfer between the contents of the
piston and the surroundings during the time required by the equilibration. Note: this is an
irreversible mechanical equilibration process. You do not know, nor is there any way to
determine, the pressure of the water acting on the lower surface of the cylinder during this
process. However, you do know the pressure of the atmosphere acting on the upper surface of the
piston during the process. Your system selection should be informed by these facts.
a.) Determine the position of the piston at state 2, z2.
b.) Determine the temperature of the water at state 2, T2.
c.) What is the entropy generated by the process of moving from state 1 to state 2, Sgen,1-2?
d.) What is the work transfer from the water to the piston during the process of going from state
1 to state 2, Wout,1-2?
After some time has passed, heat transfer between the water to the surroundings causes the water
to come to a final temperature that is equal to the temperature of the surroundings, Tamb. This is
an irreversible thermal equilibration process that must result in entropy generation because heat is
being transferred through a temperature gradient. The piston is allowed to move freely during
this process.
e.) Determine the position of the piston at state 3, z3.
f.) Determine the heat transferred from the water to the surroundings during this process, Qout,2-3.
g.) Determine the entropy generated by the process of moving from state 2 to state 3, Sgen,2-3.
h.) generate a temperature-entropy diagram that shows states 1, 2, and 3.
i.) Plot the entropy generated by the process of moving from state 1 to state 2 as a function of P1
for 100 < P1 < 500 kPa. You should see that there is an optimal pressure at which the entropy
generated by this process is minimized. Explain why this is the case.
j.) What initial pressure and temperature should you use if you want to minimize the total
entropy generated by the equilibration processes (i.e., you want to minimize Sgen = Sgen,12 +
Sgen,23). Why?

2. Relevant equations
P1 = F1/A1 ,

3. The attempt at a solution
my initial approach is as the problem suggests to start with a force balance to find the internal pressure at state two though then determine the temperature with thermodynamic table. is this a possible good start to the problem and also what would be the correct force balance equation? any help would be really appreciated.

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2. Nov 8, 2015

### Staff: Mentor

This is a very meaty problem. The place to start is to determine the amount of water you are dealing with. Show us how you determine that.

Also, let F(t) represent the force that the water exerts on its side of the piston at time t. Write a Newton's 2nd law force balance on the piston, taking into account the weight of the piston, the mass and acceleration of the piston, the force of the air on the other side of the piston, and the force F.

Chet

3. Nov 9, 2015

### Doubell

i wanted to know if it is right to say at state two at mechanical equilibrium the pressure P2 = Patm + (the Weight of the piston/area of the piston)?

4. Nov 9, 2015

### Staff: Mentor

Yes, but to get the work of the gas on the piston, you need to consider the force balance on the piston (because the piston oscillates, as indicated in the problem statement).

Chet