Help regarding calculating roots.

"Cubic equations of state can be used to establish the pressure(P)-volume(V)-temperature (T) relationship of fluids. A typical one is called Redlich/Kwong equation of state and is in the following form: 
P = (R*T)/(V-b) - a/(V*(V+b)*sqrt(T))

where a and b are fluid specific parameters and R is the gas constant 83.14 cm3.bar/mol.K. When this equation is solved for volume (V) the largest root yield the vapor-phase volume, whereas the smallest root yields the liquid-phase volume.
Use the bisection method calculate the smallest root of the equation, in order to obtain the liquid-phase volume of butane at 350 K and 9.46 bar for which a= 2.9*10^8 and b = 807,"

Hello, can someone help me to understand this problem i am trying to use bisection method but i am not sure about how to do it. I've been trying to reorganize the equation above but to no avail. also can i somehow calculate the smallest root of the equation without an initial temperature to find the liquid-phase volume ?

Thank you for your help

Answer 1

[Walter Roberson]

To solve P = (R*T)/(V-b) - a/(V*(V+b)*sqrt(T)) for V when everything other than V is known, you can subtract the left hand side from both sides, giving

P-P = (R*T)/(V-b) - a/(V*(V+b)*sqrt(T)) - P

or

0 = (R*T)/(V-b) - a/(V*(V+b)*sqrt(T)) - P

now you have an equation that should be equal to 0, and you can do a bisection search for the V that make (R*T)/(V-b) - a/(V*(V+b)*sqrt(T)) - P equal to 0. You don't need to do much rearranging at all.

Hint: consider using anonymous functions.

If you have the symbolic toolbox then you could solve the equation with V and T symbolic in order to get a formula for V in terms of T, but at some point you would still need to substitute in a T in order to get the answer to the assignment.