Solving an implicit equation in matlab

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Yogan
Yogan 2022-11-13
编辑： Bruno Luong 2022-11-13
Dear all,
I need to solve the following equation:
and the equation that coorelates by the following equation( the term 5 was ignored):
and the variables b, A, B, C, k, Tc can be founded in this table:
As the variavle V can not be isolated, we should solve it implicit. I am trying witht he following code:
Tc = 367.85;
Pc = 3374.84;
R = 7.290839*10.^-2;
b = 3.973395*10.^-4;
k = 5.033;
A2 = -0.1398755;
A3 = 4.031561*10^-4;
A4 = -1.508838*10.^-7;
B2 = 1.677524*10.^-4;
B3 = -6.739375*10.^-7;
B4 = 0;
C2 = -1.518125;
C3 = 5.521381*10^-5;
C4 = 0;
Vapor_pressure_ref = 100
syms F P V T
F = -P + ((R*T)/(V-b)) + (((A2) + (B2*T) + (C2*exp(-k*T/Tc)))/(V-b)^2) + (((A3) + (B3*T) + (C3*exp(-k*T/Tc)))/(V-b).^3) + (((A4) + (B4*T) + (C4*exp(-k*T/Tc)))/(V-b).^4);
dV_dT = - diff(F,T)/diff(F,V)
Eq = (V - (dV_dT*T))
int(Eq,P,Vapor_pressure_ref,1)
However,the output is:
-(17211847986098922057056496272843303410021431145154622943954950247403281191336156397430653844910382510721336111875*exp(3956657/1051000) + 171221942040754850037313460159985281792124894590638846549063088630653073472919905693201072792645916265515881332736)/(1065840624671895280778428055080140800000*(64450689896085148577946341147025193190720307559496600271150138269013626089*exp(3956657/1051000) + 236272089264744816453435291457063368284234553216075863541003706881470889984))
How to simplify it?
  6 个评论
显示 4更早的评论
Torsten
Torsten 2022-11-13
编辑：Torsten 2022-11-13
Once I have the DV/DT done, theoreticaly I need just to multiply by -T, add V and integrate from Pressure 1 to pressure 2.
Yes, but what are these T and V in the integrand ? Both dF/dT and dF/dV depend on P,T and V. And T and V appear directly in the expression of the integrand. So you must know which T and V values are to be inserted in the integrand for the pressures between 1 and 100 in order to evaluate the integral.
Answer:
As I wrote, T must be prescribed and V must be derived as the solution of your equation-of-state for given P within the integration process and the specified T.
Bruno Luong
Bruno Luong 2022-11-13
编辑：Bruno Luong 2022-11-13
"T must be prescribed"
Correct. OP forgets to tell us.

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回答（2 个）

Bruno Luong
Bruno Luong 2022-11-13
编辑：Bruno Luong 2022-11-13
Ran in:
The numerical answer is -0.284179329144572
format long
-(17211847986098922057056496272843303410021431145154622943954950247403281191336156397430653844910382510721336111875*exp(3956657/1051000) + 171221942040754850037313460159985281792124894590638846549063088630653073472919905693201072792645916265515881332736)/(1065840624671895280778428055080140800000*(64450689896085148577946341147025193190720307559496600271150138269013626089*exp(3956657/1051000) + 236272089264744816453435291457063368284234553216075863541003706881470889984))
ans =
-0.284179329144572
Bruno Luong
Bruno Luong 2022-11-13
编辑：Bruno Luong 2022-11-13
D=1/(V-b) is a root of fifth order polynomial, where the coefficients depend on P, T, etc...
It looks like it is small non-linear correction of the linear equation.
P = RT*D
I assume the equation has single real root in the range you consider, so actually it is not really an implicit equation, it is explicit equation if you write it as root of the polynomial.
It seems int is smart enough to figure this fact out, and able to gie back the expression you look for.

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