Thermodynamics Problem: Heat and Vapor Formation in a Piston-Cylinder Device

[michael3]

A piston-cylinder device initially contains 10 kg. of water at a temperature of 85 C. The cylinder has a section area of 0.12 meter square and the piston has no mass. A linear spring (spring constant 120,000 N/m) is touching the piston as shown but exerts no force on it initially. Then heat is slowly added to the water in the cylinder, determine: (assume atmospheric pressure to be 100 kPa)

a) How much heat is needed to start to form vapor in the cylinder?

b) How much heat is needed to raise the piston 0.2 meter above its original position.

c) Show the process on a T-v diagram with respects to saturation lines.


For answer a I got 614.4 KJ from m ( u2-u1) the change in the internal energies multiplied by the mass using u2 at 100 kPa at the sat temp. and u1 from Tsat @ 85 C since it can be reasonably substituted for a compressed liquid.

For answer b I got 4.8 KJ

V1= 10kg* .001032 (Vf@85 C)
V2= .01032+.12*.2

One way i did it was find the specific volumes then finding the total volumes and drawing a P-V Diagram and looking at the area under the curve, then checking it by calculating the total work done.

300+100 * 1/2 * (.03432-.01032) =4.8kj

checking using work methods ( 1/2kx^2 + pressure*area*distance)
1/2 * 120 * .2^2 + 100*.2*.12=4.8kj

C) did a check of specific volumes to confirm that the final temp was indeed the saturation temperature. then drew the T-V diagram. 133.52 C is my final temp that i calculated.

vf@300kpa vg@300kpa
.001073<.0034232<.60582 --> is the saturated temperature


Could someone please confirm these results? :)

THANK YOU!

 

[vijay123]

yes you r right(to all i know)(to very sure but in he rite trak)...but 1atm is 101.345kpa...doesnt dat cause a significant variation in you ans?

 

[piercas]

Your calculations and process seem correct. To confirm your results for part a, you can also use the Clapeyron equation, which relates pressure, temperature, and specific volume for a substance undergoing a phase change. In this case, we can assume that the water is in the liquid phase at the initial temperature and pressure, and we want to find the temperature at which it will start to boil (i.e. the vapor pressure will equal the atmospheric pressure). So we can rearrange the Clapeyron equation to solve for temperature:

T = (Pvap/R)(dV/dT)

Where Pvap is the vapor pressure at the given temperature and R is the gas constant. For water, Pvap can be approximated using the Antoine equation:

log(Pvap) = A - (B/(T+C))

Where A, B, and C are constants. Using the given values for atmospheric pressure and the section area of the cylinder, we can solve for the temperature at which the water will start to boil. This should give us a similar result to what you calculated using the change in internal energies.

For part b, you can also use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. So we can set up an equation:

ΔU = Q - W

Where ΔU is the change in internal energy, Q is the heat added, and W is the work done. Since we know the initial and final positions of the piston, we can calculate the work done by the spring and the work done by the atmosphere (pressure * area * distance). Setting these equal to each other and solving for Q should give us the same result as you calculated.

For part c, your approach of using specific volumes and drawing a T-V diagram is correct. Another way to approach it is to use the Clausius-Clapeyron equation, which relates the change in temperature and pressure for a substance undergoing a phase change. This can also be plotted on a T-V diagram to show the process with respect to the saturation lines. Overall, your results seem reasonable and your approach is sound. Good job!