Tuesday, September 30, 2008
TE 303 - HW #4, P1 - Compression of Cooling Air by a Linear Spring - 15 pts
Warm air is conatined in a piston-and-cylinder device oriented horizontally, as shown below. The air cools slowly from an initial volume of 0.003 m3 to a final volume of 0.002 m3. During this process, the spring exerts a force that varies linearly from 900 N to a final value of zero N. The atmospheric pressure is 100 kPa and the area of the piston face is 0.018 m2. Friction between the piston and cylinder wall can be neglected because the process occurs so slowly. For the air, determine the initial and final pressures in kPa and the boundary work for the process in kJ.
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11 comments:
do the pressures need to be converted to absolute? and if so, what gage pressure do we use?
Anonymous 12:53 PM:
Good question. I think the easiest way to understand this problem is to think in terms of absolute P and to use Pabs. The catch is that we are NOT given either the absolute or the gauge pressure inside the cylinder. That is OK. The key is that the boundary work, Wb, is the force that is overcome times the distance moved. In this case, the force that is overcome consists of the combination of Patm on the back of the piston AND Fspring / Area. The SUM of these two pressures is the P that the expansion must overcome. The tricky part is determining Fspring. It is a linear spring so the force it exerts can be written in terms of a slope and an intercept. Use the information in the problem statement to determine the appropriate slope * Vcyl + intercept. Best of luck !
ANSWERS AND HINTS:
P1 = 150 kPa
W = +/- 125 kJ (YOU determine the sign)
See Dr. B's comment above for a good hint. Be sure to notice that the force of the spring is LINEAR with position (and thus with volume).
I was just wondering if the answer was supposed to be .125KJ rather than 125 KJ...
Anonymous 4:28 PM:
Yes, the answer is 0.125 kJ or 125 J. Good eye for units !
The piston is contracting from State 1 to State 2 (volume decreases). Does that mean that the spring is pushing inwards (along with the contracting)?
Dear anonymous #2,
Your system for this problem is the fluid within the piston. The pressure acting upon it for compression is the sum of the atomospheric pressure and the pressure due to the spring (spread over the area of the piston). Therefore, both forces are acting UPON the system.
For work, must one split the path up into an isochoric path and an isobaric path and use the equation for work if it's isobaric-- W=P(v1-v2) ?
Dear Sydney,
In general, if you can break up your path into isochoric vs. isobaric, then yes, you can do that to solve the problem. That was the point of the hypothetical process paths! I'm glad that you saw that connection.
However, for this problem, you do NOT need to do that. Instead, you should get a functional form for Pressure (see earlier comments) and integrate that.
Remember, the spring is acting on the system by moving the piston.
I worked this out with an integral for each thing. I got the force of pressure for the cylinder and atmosphere in terms of x (y=mx+b for the cylinder, and a constant for atmospheric) and got an equation in terms of x, then integrated the whole thing over displacement. I got 127 joules of work, is my method accurate? Thank you
Dear Kris,
I'm not sure I follow your logic. Can we please discuss it before or after class tomorrow?
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