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dave84
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Homework Statement
We have a horizontal cylinder with an ideal piston, both are made from a heat nonconductive material. There is 10 liters of steam at 3 bars and 200 degrees Celsius in the cylinder. The cylinder is surrounded with atmosfere with pressure of 1 bar and temperature at 20 degrees Celsius.
At first we hold the piston, then we release it and wait for a standstill. What is the final temperature of steam at that state? How did the enthropy change?
In the second part, we put the system in a stationary electric field E. What is the final temperature of steam in the stationary state of piston?
[tex]E = 5\times 10^{9} \frac{V}{m}[/tex]
[tex]a = 4 \times 10^{-13} \frac{Asm^2 K}{V}[/tex]
so we calculate
[tex]c_p = 1850\frac{J}{kg K}[/tex]
[tex]M = 18\frac{kg}{kmol}[/tex]
[tex]p = 3\text{ bar}; V = 10 l;T = 200°C;[/tex]
[tex]p_0 = 1\text{ bar}; T_0 = 20°C;[/tex]
Homework Equations
[tex]p_e = \frac{a E}{T}[/tex]
The Attempt at a Solution
I solved the first part, not sure if it is correct. Could there be a phase change of steam? I don't know how to calculate that.
[tex]dU = dQ + dW[/tex] and [tex]dQ = 0[/tex]
so I integrated
[tex] c_v dT = -\frac{R T}{M V}dV[/tex]
from [itex]p, V, T[/itex] to [itex]p_0, V_1, T_1[/itex].
The solution is
[tex]T_1 = T \left ( \frac{p_0}{p}\right )^{\frac{\beta}{\beta+1}} \approx 360 K[/tex]
where
[tex]\beta = \frac{1}{\frac{M}{R}c_p - 1}[/tex].
I calculated enthropy using
[tex]\delta S = m c_p \ln \left(\frac{T_1}{T}\right) - \frac{m R}{M} \ln\left( \frac{p_0}{p} \right) \approx -0.7 \frac{J}{K}[/tex]
For the second part, when we turn on the electric field [itex]E[/itex], I get a term in [itex]dU = dW = -pdV - E dp_e[/itex], so I need to integrate
[tex]\frac{mR}{MV}dV = \left( \frac{a E^2}{T^3} - \frac{m c_v}{T}\right) dT[/tex]
but this makes little sense as [itex]T_1[/itex] cannot be calculated easily from this?
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