Thermodynamics: Steam Powered Carnot Engine Question

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Homework Statement

A steam engine has a cylinder 80cm in diameter with a piston that has a total travel length of 1.8m. Assume that there is no "dead space" between the cylinder head and the piston when the piston is farthest in the cylinder. Steam is piped into the cylinder head from an external boiler at 130psi above atmospheric pressure. The water in the boiler is boiling at the 130psi above atmospheric pressure. There is a "quick release" valve system that can shut off the flow of high pressure steam (at 130psi above atmosphere) at any point in the piston's 1.8m displacement down the cylinder. The displacement until the valve cuts off the flow of steam from the boiler is done at 130psi above atmosphere but after the cut off the steam already in the cylinder expands until the total 1.8m displacement is finished and another valve opens to allow the steam to be exhausted into the atmosphere. Assume the engine is running at 0.60 piston cycles per second.

What should be the piston position at steam cutoff (measured from the cylinder head) for the engine to operate at maximum efficiency?

NOTE: if you cool the steam such that it condenses into liquid water the pressure will decrease drastically, possibly below atmospheric!

Homework Equations

1 (dP / dT) = L / (T * dV)
2 ln(P2/P1)= -(ΔH/R) * ((1/T2) - (1/T1))
3 PV=nRT
4 V=pi*r^2h
5 PV^ɣ=PV^ɣ
6 P^(ɣ-1) * T^(-ɣ) = P^(ɣ-1) * T^(-ɣ)
7 VT^α=VT^α

The Attempt at a Solution

My understanding of this question is that I must determine the largest possible adiabatic expansion, following the initial addition of steam, possible without allowing the steam to condense to a liquid. So the largest ratio of Expansion/Addition possible will be the answer. Using the second variation of the Clapeyron Equation I found that the temperature of the steam is 452K. I started to guess and check after that. So if I allow steam to be added to half the container I'd have 0.9 m^3 of steam at 452K with 0.9 m^3 of expansion. Using a steam table and a little estimation I found that under 144psi steam has a density of 3.11 ft^3/lb which is 285 mol/m^3. So if I allow the engines piston to expand to 0.9 m (1.8 m^3 volume) from the start I have added 513 moles of gas. I looked up the value of ɣ on an adiabatic index that I found and it said that at 200C ɣ=1.310. I did this because I was having trouble find the degrees of freedom for water gas. Using equation 5 I found the new pressure of the system after expansion 400931Pa. Using equation 3 I find the temperature of my final system to be T= 338.7K. I wanted to use equation 1 to determine if the steam had condensed so I looked up the heat of vaporization at the pressure of the fully expanded piston (400931Pa). With a little estimation and some conversion factors I found that the ΔH of vaporization was 38667 J/mol. I used the steam table again to estimate the volume change between liquid and steam water 8.1*10^-3 m^3/mol. Plugging everything into equation 1 I get that I need to drop the pressure below 13935Pa to get a phase change. From there I don't know how to minimize the initial addition of steam and maximize the adiabatic expansion of steam without condensing to a liquid and losing all pressure. I also don't know if I've made numerous errors throughout my calculations.

 

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My understanding of this problem is that I must find the volume of initial steam to add so that the adiabat is as large as possible without condensing the steam. Point B to Point C on the following image is that adiabatic transition that I am talking about.

http://img694.imageshack.us/img694/2987/carnotcycle.png
Using a steam table from http://www.energysolutionscenter.org/BoilerBurner/Eff_Improve/Primer/SteamTables.pdf" I got the following values. The ɣ is from the adiabatic index on Wikipedia.

Vgas=3.5*10^-3 m^3/mol
Vliq=2.0*10^-5 m^3/mol
Volume of full cylinder = 0.90735 m^3
P = 992736Pa
ɣ=1.13
T=453K
P = 992736Pa
ΔHvap=36260J/mol

The following is my attempt to solve for the initial amount of steam added:
http://img190.imageshack.us/img190/9198/mapleg.png
Equation 4 is the pressure and temperature of steam at which a phase change to liquid water will begin. Equation 5 is a rearranged form of equation 3 giving the final pressure after expansion. Equation 6 Is Equation 2 with pressure from equation 5 and the number of moles based on the amount of volume added. Equation 7 is equation 4 with equation 5 and equation 6 plugged into it. Volume is the only variable and I have attempted to solve for it, but Maple 11 will not solve for me. Here is a picture of what happens when I attempt to solve in Maple:
http://img19.imageshack.us/img19/3853/losti.png

Anyone know how to solve for V or whether or not I'm doing this right?

 

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I noticed I clearly made a mistake on my last post with my construction of equation 5. The denominator should bethe total volume of the cylinder fully expanded (0.90477m^3). The corrected formula for pressure is now found in equation 1 of this post. Using equation 2 I have the highest ratio of dP/dT possible without a phase change 23001.19256 Pa/K. The value I got from Maple when solving for the initial volume of steam added was 0.9073500003 m^3.
http://img44.imageshack.us/img44/5019/thermosolved.png
That's bigger than the volume of the cylinder! It seemed crazy so I plotted dP/dT for a volume change from 0 to 0.9 m^3 and sure enough it seems that it is impossible to have a conversion in this cylinder. This seems to mean that I would get the highest efficiency from the smallest addition of steam possible. I thought I might have screwed up the Clapeyron equation so I plotted T vs V and P vs V for initial volumes of steam ranging from 0 to 0.9 m^3.

From the looks of the PV plot an initial volume of steam of 0.1 m^3 would result in a final pressure of 100,000 Pa (roughly atmospheric). Looking at the temperature for this point I find around 50 K. Steam has definitely condensed by this point.

http://img521.imageshack.us/img521/9647/pvtv.png
What am I doing wrong?