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Title: What is the largest adiabatic expansion of steam without a phase change?Post by: spinverted on December 06, 2009, 04:29:45 am
I'm having trouble working this knowing that if I allow too much adiabatic expansion, the water vapor will condense to a liquid. I believe that the Clapeyron Equation is used to keep the steam in its phase. (dP / dT) = L / (T * dV)
Steam at 180 psi is piped into an 80cm cylinder with a piston that has a total travel length of 1.8 m. At what distance along the cylinder should I stop the addition of steam and allow the adiabatic expansion to take place if I want to have the Carnot Engine operating at it's highest efficiency? By highest efficiency I mean the largest adiabatic expansion possible without converting the steam into liquid. This is of course balanced against the initial volume of steam added. Could someone point me in the direction of solving this problem? Title: Re: What is the largest adiabatic expansion of steam without a phase change?Post by: Fu-Kwun Hwang on December 06, 2009, 11:51:37 am
Please check out http://en.wikipedia.org/wiki/Triple_point
And look for phase diagram. You will find out the pressure-temperature region for gaseous state, i.e. below the blue line. (http://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Phase-diag.svg/350px-Phase-diag.svg.png) Title: Re: What is the largest adiabatic expansion of steam without a phase change?Post by: spinverted on December 06, 2009, 11:05:15 pm
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 (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/img19/4810/volume.png) Is this the right approach? Title: Re: What is the largest adiabatic expansion of steam without a phase change?Post by: Fu-Kwun Hwang on December 07, 2009, 08:23:15 pm
The equation 4 you found did not match the P-T curve in the previous post.
I would suggest you find P-T curve, and draw new curves for carnot cycle (usually it was dwan on P-V diagram). The two constant temperature curves will become two vertical lines in P-T curve. With ideal gas law: $PV=nRT$, the adiabatic curve $PV^\gamma=C $ (constant can be reduced to $P (\frac{nPT}{P})^\gamma=C$ or $P=C' T^{\gamma/(\gamma-1)}$. Draw starting point with your initial condition: $P_i,T_i$, then follow the above new PT curve before it touch the PT condensation curve. |