Cubic equations of state -- rearranged / solved forms

[JDB]

Does anyone have a reference in English that shows pressure-explicit
cubic equations of state (e.g., the Soave-Redlich-Kwong EOS, or the
Peng-Robinson model thereof) rearranged and solved to be
VOLUME-EXPLICIT (i.e., to yield volume as a direct function of
temperature, pressure, and substance-specific constants/parameters)?

I'd also be interested in seeing these same pressure-explicit cubic
equations of state in a rearranged/solved form to be EXPLICIT IN THE
COMPRESSIBILITY FACTOR Z.

Thanks for your assistance.

[Mark Tarka]

jdrb...@comcast.net (JDB) wrote in message news:<617fdd1.03040...@posting.google.com>...

I haven't touched this stuff in quite some time,
but would guess that it is a "do-it-yourself"
sort of situation. OTOH, check-in with the
industrial nerds...they do neat stuff that never
sees the light of day :-)

If you come up empty handed, then feel free to
massage the math in any reasonable way that gets
you to a reasonable answer. Iterative solutions
count :-)

Mark (Helps, if you can program in Basic, Fortran, ...)

[Terry Wilder]

Doesn't the Virial carry to Z on expansion to volume?
"JDB" <jdrb...@comcast.net> wrote in message
news:617fdd1.03040...@posting.google.com...

[Allan Harvey]

There's a book by S.M. Walas called "Phase Equilibria in Chemical Engineering"
that has a lot of things like this (how to do calculations with cubic equations
of state). I don't remember whether it has exactly what you are asking for,
but it is probably worth finding it in a library and taking a look.

Your other course would be to simply rearrange it in the form of a standard
cubic equation in volume (like a quadratic equation but with one higher term)
and then find the "cubic formula" (analogous to the quadratic formula) in a
math book (I think it is also in the CRC Handbook, and it must be on the Web
someplace).

--------------------------------------------------------------
Dr. Allan H. Harvey, Boulder, CO
"Any opinions expressed are mine, and should not be attributed to
my employer, my wife, or my cats."

[Josh Halpern]

Allan Harvey wrote:

Does anyone have a reference in English that shows pressure-explicit
cubic equations of state (e.g., the Soave-Redlich-Kwong EOS, or the
Peng-Robinson model thereof) rearranged and solved to be
VOLUME-EXPLICIT (i.e., to yield volume as a direct function of
temperature, pressure, and substance-specific constants/parameters)?

I'd also be interested in seeing these same pressure-explicit cubic
equations of state in a rearranged/solved form to be EXPLICIT IN THE
COMPRESSIBILITY FACTOR Z.
    

There's a book by S.M. Walas called "Phase Equilibria in Chemical Engineering"
that has a lot of things like this (how to do calculations with cubic equations
of state).  I don't remember whether it has exactly what you are asking for,
but it is probably worth finding it in a library and taking a look.

Your other course would be to simply rearrange it in the form of a standard
cubic equation in volume (like a quadratic equation but with one higher term)
and then find the "cubic formula" (analogous to the quadratic formula) in a
math book (I think it is also in the CRC Handbook, and it must be on the Web
someplace).

It is a horror.  As I remember it is a page of tight type as
compared to the single equation for the quadratic. You don't
get much insight out of it either. There is a real reason
why people solve these equations numerically.   What I
would suggest is using a symbolic algebra packsage (Maple,
Mathematica for sure, I'm not sure if MathCad would be up
to it) to solve the cubic equation for V.  You have the added
advantage that you can do all graphing of the solutions
directly in the software.

josh halpern

[Martin Brown]

Josh Halpern wrote:

> Allan Harvey wrote:
>
>> > Does anyone have a reference in English that shows
>> > pressure-explicit
>> > cubic equations of state (e.g., the Soave-Redlich-Kwong EOS, or the
>> > Peng-Robinson model thereof) rearranged and solved to be
>> > VOLUME-EXPLICIT (i.e., to yield volume as a direct function of
>> > temperature, pressure, and substance-specific
>> > constants/parameters)?
>> >

>> > Your other course would be to simply rearrange it in the form of a
>> > standard
>> > cubic equation in volume (like a quadratic equation but with one
>> > higher term)
>> > and then find the "cubic formula" (analogous to the quadratic
>> > formula) in a
>> > math book (I think it is also in the CRC Handbook, and it must be
>> > on the Web
>> > someplace).
>> >
> It is a horror. As I remember it is a page of tight type as
> compared to the single equation for the quadratic. You don't
> get much insight out of it either. There is a real reason
> why people solve these equations numerically. What I
> would suggest is using a symbolic algebra packsage (Maple,
> Mathematica for sure, I'm not sure if MathCad would be up
> to it) to solve the cubic equation for V.

It isn't all that bad. Worse than the quadratic form certainly, but
tolerable with care.

Closed form solution is online at the numerical recipes sample pages

http://www.ulib.org/webRoot/Books/Numerical_Recipes/bookfpdf/f5-6.pdf

For some physical problems if you can be sure there is only one real
root then the closed form becomes an attractive proposition even if the
terms in the expression are unwieldly. (beware of numerical
cancellation)

Regards,
Martin Brown

[Oscar Lanzi III]

There are two ways to solve cubic equations of state for volume. One
approach is to use a standard solution method for cubic equations which
is widely available in math references. The other is to use one root to
chase another. Suppose you know or specify the vapor phase (molar)
volume and want to know the liquid phase volume. First calculate the
pressure, then form the cubic equaiotn, then synthetically divide out
the known root that is the vapor volume. This leaves a quadratic
equation whosesmaller root is the liquid phase volume. It's a neat way
to simplify VLE calculations.

--OL