Thermodynamics: Insulated Box Partition Question

[Master1022]

Homework Statement

A thermally insulated, rigid vessel is divided into two equal compartments. One contains steam at 100 bar and 400 degrees celsius, and the other is evacuated. The partition is removed. Calculate the resulting pressure and temperature.

(Please let me know if this is the wrong section)

Homework Equations

1st Law of Thermodynamics: $Q - W = \Delta U$

The Attempt at a Solution

from the 1st law, Q = 0 (insulated) and W = 0, thus internal energy will remain constant.

At the start
Initially, at 100 bar, the saturation temperature is 310.96 deg celsius and thus we need to look in the superheated tables.

we find that the specific internal energy for steam @ 100 bar and 400 deg C is: u = 2835.8 kJ/kg and the specific volume is: v = 26.408 $\times 10^{-3} m^3/kg$

After the partition is removed
we know that the internal energy remains the same and also mass is conserved, thus $u_{f} = 2835.8 kJ/kg$.

Given that the volume doubles, I thought that the specific volume would double, thus $v_{f} = 2 \times v$

For this type of problem, the textbook then calculates the dryness fraction/ quality. However, given that the steam is superheated, I am unsure of what to do. Once the book has a dryness fraction, it goes on to guess pressure and temperature values that will yield the same internal energy.

I have just thought of using the new specific volume to try and pinpoint the conditions, but how would I know where to start interpolating? The v value lies in a number of areas...

Thanks in advance.

 

[Chestermiller]

If you check the saturated steam tables, there are no states of vapor in which the internal energy is as high as 2835. So certainly, all mixtures of saturated liquid and saturated vapor are going to have internal energies less than 2835. Therefore, the final state must be a superheated vapor. If it were an ideal gas, then the final temperature would be 400 C and the final pressure would be 50 bars. So this is the region of the saturated steam tables where you should be looking. You need to interpolate to find the combination of temperature and pressure in this vicinity at which the internal energy is 2835 and the specific volume is 0.0528.

 

[Master1022]

If you check the saturated steam tables, there are no states of vapor in which the internal energy is as high as 2835. So certainly, all mixtures of saturated liquid and saturated vapor are going to have internal energies less than 2835. Therefore, the final state must be a superheated vapor. If it were an ideal gas, then the final temperature would be 400 C and the final pressure would be 50 bars. So this is the region of the saturated steam tables where you should be looking. You need to interpolate to find the combination of temperature and pressure in this vicinity at which the internal energy is 2835 and the specific volume is 0.0528.

Thank you for your response. When looking in the saturated tables, I see that I can interpolate in 2 ways (either by looking at adjacent pressure or adjacent temperature) Is there a 'proper' way of deciding between the two or would they come out the same?

At the moment, I am thinking about doing:
1. Interpolate to find the corresponding temperature for the specific volume at 50 bar
2. Using our temperature to find the u value at that temperature (at 50 bar)
- this turns out to be lower than required

Would the following step be suitable?
3. We need to interpolate between pressure values. Therefore, will need to find the u value at the same temperature for the next pressure step up. Then I will need to interpolate between the 2 u values at the same temperature, but different pressures

Many thanks

 

[Chestermiller]

There's no one right way of doing this. I would try something like this.
1. Interpolate on U to get the corresponding temperatures at the bounding pressures (40 bars and 60 bars).
2. Interpolate to get the specific volumes at the interpolated temperatures
3. Interpolate on specific volume to get the temperature and pressure at the desired specific volume.