Calculate the compressibility factor and the molar volume of methanol steam at 200°C and 10 bar
a) using the ideal gas law
b) with the virial equation truncated after the 3rd virial coefficient
(using R instead of Rgas as the symbol for the ideal gas constant reduces the size and readability of more complex expressions but overwrites the unit definition of R (Rankine))
For solving the equation the root-function is used. Finding the root requires a starting value for the volume:.
For not too high pressures where the effect of the 3rd virial coefficient is not
predominant, fast convergence can be achived by iterating the value of v_it:
The solution of interest for v (and at the same time v_it for the next iteration) of this quadratic equation in v can be found via "Symbolics, Variable, Solve". The cursor has to be on one of the "v" variable symbols in the equation above:
As for every physically realistic solution, R*T*v2 is not equal to zero, it is sufficient to search for the solution of
Place the cursor on one of the "v" varable symbols and use "Symbolics, Variable, Solve". As the solution is too complex to display for MathCAD, the symbolic processor Maple places the following string into the clipboard:
MATRIX([[1/6*(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)-6*(-1/3*A1-1/9*A2^2)/(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+1/3*A2], [-1/12*(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+3*(-1/3*A1-1/9*A2^2)/(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+1/3*A2+1/2*I*3^(1/2)*(1/6*(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+6*(-1/3*A1-1/9*A2^2)/(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3))], [-1/12*(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+3*(-1/3*A1-1/9*A2^2)/(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+1/3*A2-1/2*I*3^(1/2)*(1/6*(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3)+6*(-1/3*A1-1/9*A2^2)/(36*A1*A2+108*A0+8*A2^3+12*(-12*A1^3-3*A1^2*A2^2+54*A1*A2*A0+81*A0^2+12*A0*A2^3)^(1/2))^(1/3))]])