Calculating Flow Rates for Cascading Heat Pumps

[cheme2019]

Homework Statement

I am given that a Chemical Company wants to implement a cascading refrideration cycle to provide 1 MM Btu/hr of cooling. We are given the three compounds we are using.
Evaporator Temperature (F) Compresspr Out Pressure(psia)
Cycle 1 Methane -251.6 580.2
Cycle 2 Ethylene -135.7 377.1
Cycle 3 Ammonia -9.67 203.1

I am suppose to use the Peng Robinson EOS for all three fluids.

Calculate the flow rates of cycle 1, 2 and 3
Calculate the work required in each compressor / lbm of fluid
Calculate the condenser duty in cycle 3 per lbm of flow in cycle 1
Create a stream table
Create an Energy Table
Prepare a UniSim simulation to compare.

Assumptions
Sat'd Vapor going into each compressor
Each compressor is 80% efficient
Sat'D Liquid leaving heat exchanger/cooler
delta P = 0 in Heat exchangers/heater/cooler

Homework Equations

Peng Robinson EOS
P= [\frac {RTρ} {1-bρ} - \frac {aρ^2} {1+2bρ-b^2 ρ^2}]
Enthalpy Departure Functions
Entropy Departure Functions
η = W'/W = ΔH'/ΔH

The Attempt at a Solution

So I want to start with the first compressor because I have an initial temperature and pressure (from antoines) and I have an exiting pressure. So with these I should be able to get an exiting pressure and the amount of work that is needed.
I was going to start off using the Enthalpy departure function with respect to density, but density changes with respect to temperature and I don't know how to intergrate T with a changing ρ and P...
Can anyone help point me in the right direction?

 

[cheme2019]

UPDATE:
So I found out that I can use ΔH = Ws to start out with the compressor. I was able to find T2' to get rho and ΔH', then use the efficiency to get ΔH. So now I'm stuck at my departure function, because density is changing with respect to temperature, so I don't know how to integrate my dZ/dT with a changing temperature and density...
Basically I know
H1-H1ig, H2ig - H1ig
But I can't figure out
H2 - H2ig

 

[cheme2019]

Thanks for the help