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In this screencast we are going to go through an example using a cubic equation of state,
and take the pressure inside of a gas tank. Specifically with this example we are told
that a propane tank is charged with 10 pounds of propane gas at 25 degree Celsius. We are
given the dimensions of the tank, and we are asked to calculate the tank pressure using
the SRK equation of state and compare that to what pressure we would get if we where
using the ideal gas law. So the SRK and EOS equations are cubic equations of state. Meaning
that there are going to have a term for the specific volume in the third power. Again
these help characterize deviation from ideal behavior taking into consideration in things
like the molecular size interactions, shapes, and ect. The SRK equation states the following
form. Where we have pressure related to the gas constant R times the system temperature
and that is going to be divided by the specific molar volume Vhat. Minus B, which is an empirical
function, has to do with critical temperature and pressure. We will get to that in a second,
and the second term has this alpha and a, also empirical functions, and that is going
to be divided by specific molar volume, times the specific molar volume plus b. So you can
see if we rewrote this we would have the specific molar volume at a third order. Hence the equation.
So it is important that we basically go through and look at some of these empirical functions.
We have alpha, a, and b. We need to figure out how we calculate these. So that we can
solve for the pressure. So I will write down what each of those empirical functions are.
Basically these have been calculated through a number of empirical correlations. Modeling
the different gasses at different temperatures, and pressures. So I have written down all
of these empirical functions. As you can see a is a function of the critical temperature,
and critical pressure. B is also a function of the critical temperature and critical pressure.
We also have alpha, which is a function of the reduced temperature Tr in this other variable
m, m is a function of the acentric factor omega, and the reduced temperature is the
system temperature divided by the critical temperature. We have looked these up and we
can figure out for what ever has we want to look up the critical temperature and pressure.
So we can figure out what these values are. As well as the eccentric factor, and these
can be looked up in say the chemical hand book, or also in most of your textbooks. So
you could calculated these by hand. Assume that you carry all of the units our correctly.
I have gone ahead and plugged all of this into an excel spread sheet. So that we can
easily make sure that we have all our units in check, and then again calculated all these
very simply. So lets go through that. So on the left side of this spread sheet I have
typed all the variables we can look up or that we know from the problem statement. Such
as the diameter and height the tank. It is easy to calculate what the radius will then
be. You can see that it is just half of the diameter, and then we are given the mass of
the tank, and we know the system temperature in Celsius. We can convert that to kelvin
pretty easily. So all my numbers in the light blue are numbers that were given in the problem
statement and the yellow numbers are numbers that we needed to look up. Such as the molecular
weight of propane, and I have that in kg/kmol. The critical temperature and critical pressure.
Are given in kelvin and atm. I have converted atm to Pa. Just so that we can keep our units
a little easier to work with, and we have also looked up the eccentric factor 0.152
for propane, and our gas constant R. We are going to need that for both our a and b values.
The first thing we need to do is figure out what our specific molar volume of propane
and the tank is. Now to do this we are going to have to figure out what the volume of the
tank is and how much propane we have in the tank. So the volume of the tank if we assume
a cylinder, which is pretty good asumption in this case. It is just pi R squared H. We
can calculate the amount of moles of propane in the tank, by the mass divided by the molecular
weight of propane. The specific molar volume is then is our volume of the gas divided by
the moles of the gas. This should give us something in the order of meters cubed per
mole. You can see here I have my volume as Pi R squared, times the height and this is
all in inches cubed. I have converted that to meters cubed. I have calculated the mass
in kg and have divided by the molecular weight so that we have it in moles, and just converted
to molar basis of a kmol. That gives use the specific volume of 4.7 times ten to the minus
4th meters cubed per mole. At this point it is just a matter of plugging in the values.
As you can see our a value here is the 0.4247 times the R TC square both of these values
divided by PC. We have the units here. We do the same thing for b, except this time
it is 0.08664 times RTC over PC. You solve your value for m. You can check this. Should
match up we have our eccentric factor here, and then squared, and our alpha is going to
be based on both the reduced temperature, which you can see I have B7, which is just
some temperature. Right here divided by the critical temperature. Which is cell B6 take
the square root of this term. Subtract it from 1, and multiply this by our or m term,
and we take the square of this whole term, and we should get 1.15. Then it is just a
matter of plugging this all in to 1 equation to get a pressure. So then I break this down
into 2 terms. The first term solves for pressure was that RT over molar volume minus P=b. So
you can see that I have those terms highlighted here, and the second term I am subtracting,
and this time it is going to be the alpha, a, divided by the molar volume times the molar
volume plus the term b. So when you you take that second term and subtract it from the
first you are going to get a value in Pa. As long as your units are checked out, and
I can convert that to atm. So based on this we are seeing that 10 pound of propane given
the tank that we have is about 17 atm. Now if we did this by using the ideal gas equation
state. Where P equals RT over the molar volume. Specific molar volume. I also did that in
excel just to keep it simple, and you can see that we get a pressure of about 52 atm.
So in this case we could calculate the percent error. If we use the ideal gas law as 52 minus
17 over 17 times 100. This is about 200 percent. So 200 percent error if we use ideal gas law
instead of using the cubic equation state. In this case the SRK equation state. So hopefully
this gives you an idea on how to use a cubic state and why it is important to choose different
equations of states to model gas behavior.