In this example we are going to look at adiabatic compression of a non-ideal gas.
It's carbon dioxide we are given feed conditions we have the outlet pressure.
So we are compressing from 1 bar to 10 bar and the question is what is the outlet temperature
and how much work do we have to put in per time if we are compressing 2.5 moles per second
of carbon dioxide.
And we are going to use the Peng-Robinson EOS to model this non-ideal gas.
We are also going to use the heat capacity as a function of temperature so we are going
to use this spreadsheet that has this programmed into it.
So first thing we want to do is set up a diagram that represents the process.
So we put the inlet conditions to our compressor, 290 K, 1 bar, H1 and S1 are the enthalpy per
mole coming in and the entropy per mole coming in.
And we want to determine the enthalpy leaving per mole and the entropy.
So we can determine the temperature.
Now we are going to assume it's reversible otherwise we need more information about the
compressor.
We know it's adiabatic.
So if we write down the first law for this system and we are running this at steady state
as we like to run a compression.
General first law says change in internal energy with respect to time is equal to n
dot would be a molar flow rate in times enthalpy per mole minus molar flow rate out which in
this case is the same because we are at steady state times the enthalpy per mole out plus
Q dot plus W dot is the general energy balance that we will use.
And then of course this is zero because it's steady state and this is 0 because it's adiabatic.
So we have an equation for the rate that we are adding work to the system.
This is going to be the total work added per time, and it's molar flow rate, enthalpy out
minus enthalpy in.
So I'm going to use the EOS spreadsheet for Peng-Robinson Equation.
What we are going to do for these initial conditions of 290 K and 1 bar is read off
these equations for H1 and S1.
And I'll show you a little later in this screencast the output from this spreadsheet.
So these are the input values.
You have to keep in mind that these are for a specific reference conditions.
The spreadsheet allows us to change the reference conditions but we have to use the same reference
condition for the inlet and the outlet streams.
So we we want to know H2 and we know S2 is 90.99.
We do not know T2.
We know P2 is 10 bar.
So in the spreadsheet we can put in 10 bar and then we are going to adjust T2 to get
the correct value for the entropy.
Once we do that we can determine the enthalpy.
Turns out we are 469.5 K that gives us this entropy at this pressure.
So temperature increases significantly due to the compression we are putting energy into
the gas, that raises the temperature of the gas.
And then we can read off the spreadsheet the value for H2.
So we can now calculate the work.
N dot( Hout- Hin) which is H2 - H1.
So 2.5 moles/sec is the feed rate, 24342 J/ mol is the outlet value for the enthalpy and
the inlet value is 17224.
So we do the calculation, convert it to KJ and then the correct number of significant
figures is justified, at most, is two in this problem.
So we calculate the work input per time and we've calculated the outlet temperature.
So now let's look at the spreadsheet to see what this looks like.
So the inlet conditions in the spreadsheet you have to input the critical temperature,
critical pressure and the eccentric factor.
And then the heat capacity terms as a function of temperature, a+b*t etc... that is described
in detail in this spreadsheet and we show in another screencast how to use this spreadsheet.
We also input the inlet temperature and the inlet pressure.
The enthalpy is here and the entropy.
And then what we are going to do is for the outlet conditions what we've changed is the
pressure.
And then we adjusted this, we could certainly use excel solver to do this, but we adjusted
this temperature to make this essentially the same value, presented in terms of significant
figures.
And then we can read off this exit enthalpy and calculate then the work.
So for this particular example it may not differ dramatically from an ideal gas as long
as we use the heat capacities as a function of temperature.
But this is a more general approach as we went to higher pressures, gases are going
to be less, less ideal.