Finding Isentropic Enthelpy, knowing Isentropic Entropy

[WhiteWolf98]

Homework Statement

The source question is very long, and most likely unneeded. This question is about an actual refrigeration cycle. For the first part of the question, I'm to use the given isentropic efficiency to calculate $h_{2s}$.
$P_1=140~kPa$
$T_1=-10°C$
$P_2=1~MPa$
$\eta = 0.78$

Relevant Equations

$\eta=\frac {h_{2s}-h_1} {h_2 - h_1}$

A short background: My question focuses solely on the part of the refrigeration cycle to do with the compressor, where the cycle begins. The first state is before the refrigerant enters the compressor, and the second state is after the refrigerant leaves the compressor. My goal is to obtain $h_2$; but for that, I need $h_{2s}$.

From the Thermodynamic Tables:

$h_1=h(140~kPa,~-10°C)=246.37~kJ/kg$

Easy enough to obtain. All that's left is $h_{2s}$. From the T-s diagram of the refrigeration cycle, it can be seen that:

$s_{2s}=s_1$

$s_1=s(140~kPa,~-10°C)=0.9724~kJ/kg\cdot K$

So I know that the entropy at state $2s$ is $0.9724~kJ/kg\cdot K$

Now this is where I'm stuck. I don't know how to get $h_{2s}$.

State 1 I know for sure the refrigerant is superheated. And state 2, I'm near to certain it's still superheated.

In other questions, I've been able to work out $h_{2s}$ when state 2 is a mixture. I use the entropies to work out quality,

$x=\frac {s-s_f} {s_{fg}}$

And then knowing the quality, work out $h_{2s}$:

$h_{2s}=x(h_{fg})+h_f$

I can't do that though if they're both superheated. There's no, 'quality' or, 'x', nor any saturated liquid values. This has come up once before this time, and I was unable to answer it then too. Any help would be appreciated.

 

[dRic2]

If it is superheated it's easier. You just need a table for the superheated vapour properties or a graph and you read $h_2$ knowing both $P_2$ and $s_2s$. A priori I'm not sure how you can deduce if you have a mixture or a superheated gas. But if you have a superheated gas and you try to work out the quality $x$ you should get an absurd result (bigger than 1 for example...) so that might be a way to check things out if you lack experimental data. But pressure is usually plotted in thermodynamics diagrams for water (or other coolants) so if you locate the right point after the compression took place you should be able to see wether you have a mixture or not. Hope this helps.

 

[Chestermiller]

Let me guess. Your refrigerant is 134a. Look in your superheated vapor tables at 10 bars and about 55 C.

 

[WhiteWolf98]

Greetings to you both. I arrived at my solution, acting on your advice. It was achieved through interpolation.

Using interpolation, I found the temperature to be around $56.1°C$. Using interpolation again, and that value for temperature, I found the enthalpy ($h_{2s}$) at that temperature to be around $289~kJ/kg$. I only had table values for $50$ and $60$ degrees, which is why I had to interpolate.

Thanks for the help! You have my gratitude.