calculating enthalpy of mixture from Peng-Robinson EOS

[MSU05]

I know the equation for calculating the enthalpy of a mixture using Peng-Robinson. However, there is a derivative, d(am(T))/dT,
where am(T) = xi*xj*(a(T,i)*a(T,j))^(1/2)
a(T) = ac*(1+k*(1-(T/Tc)^(1/2))^2
k = 0.37464+1.54226*om - 0.26992*om^2
that I am having a difficult time solving. I used my TI-89 and it gave me an extremely long result. Does any one know the solution to this derivative or know where I may find it to verify my solution (which is too long to reproduce here)
Thanks!

 

[sshep]

Can you not make life easy for yourself by just using a numerical derivative? Just a thought, sshep

 

[ColourfulFigsnDiags]

I haven't bothere to expand the equation, but it shouldn't be too hard to solve using a combination of chain and product rules .... to verify you have the correct solution, as sshep said, do it numerically. You could always use Simpson's Rule to solve the derivative. Set it up in excel, pretty simple to do!

Have a read of this

http://www.mactech.com/articles/mactech/Vol.09/09.10/SimpsonsRule/

or do a search on the net

 

[MSU05]

How do I use Simpson's rule to calculate a derivative? I thought it was for integrals.

 

[ColourfulFigsnDiags]

Sorry, you are right ... was having a blonde moment!!!! For a derivative, just do it from first principles using df/dx = lim (DeltaX ---> 0) (f(x+DeltaX)-f(x))/DeltaX ...

ut I still reckon you should solve the derivative! maybe two pages of work max?

 

[ColourfulFigsnDiags]

Ok ... I must be bored today, or you are lazy

d(am)/dT = (xi.xj/2)(a(T,i).d(a(T,j))/dT + a(T,j).d(a(T,i))/dT)^(-0.5)

Now (and I did this very quickly so it is probably wrong) ...
da/dT = ac(k^2/Tc-ac.(T.Tc)^(-0.5).(1-k^2))

Also where is the dependance on i or j in the equation for a?

 

[MSU05]

am(T) = (sumproduct of) xi*xj*(a(T,i)*a(T,j))^(1/2)
a(T) = ac(i)*(1+k*(1-(T/Tc(i))^(1/2))^2
k = 0.37464+1.54226*om(i) - 0.26992*(om(i))^2

ac, om, and Tc are component specific, therefore a(T) is different for each component and am(T) is the sumproduct of the following equation, where i, j are different components:

am(T) = (sumproduct of) xi*xj*(a(T,i)*a(T,j))^(1/2)

 

[ColourfulFigsnDiags]

Ahh ok, well my equation still works (assuming it is right, I did do it very quickly!). you just need to solve da/dT for each component and shove it into the equation.

Read the FAQ ...

http://www.eng-tips.com/faqs.cfm?pid=731&fid=376