PolymathPlus Report
📈   Nonlinear Equation 2022-04-01 12:31 

# Example 7 - Single NLE
# Molar Volume: Van Der Waals equation.
# Verified Solution: V=0.574892, Z=0.871827
# Ref.: Comput. Appl. Eng. Educ. 6: 171-172, 1998

f(V)=(P+a/(V^2))*(V-b)-R*T

P=56
R=0.08206
T=450
Tc=405.5
Pc=111.3
Pr=P/Pc
a=27*(R^2*Tc^2/Pc)/64
b=R*Tc/(8*Pc)
Z=P*V/(R*T)

V(min)=0.4
V(max)=1


Calculated values

    Variable Value f(x) Initial Guess Initial f(x)
1 V 0.57489191 3.2E-10 0.7 5.9E00

    Variable Value Initial Value
1 a 4.1969459 4.1969459
2 b 0.03737122 0.03737122
3 P 56 56
4 Pc 111.3 111.3
5 Pr 0.50314465 0.50314465
6 R 0.08206 0.08206
7 T 450 450
8 Tc 405.5 405.5
9 Z 0.87182677 1.0615539

Root Finding Plot

 -10 0 10 20 0.2 0.4 0.6 0.8 1 1.2 V Zero Line f(V)



Formatted equations

  f V ( ) = + P a V 2 V b ( ) R T = 0
  P = 5 6
  R = 0 . 0 8 2 0 6
  T = 4 5 0
  T c = 4 0 5 . 5
  P c = 1 1 1 . 3
  P r = P P c
  a = 2 7 R 2 T c ( ) 2 P c 6 4
  b = R T c 8 P c
  Z = P V R T
  0 . 4 < V < 1

Nonlinear equations

1 f(V) = (P+a/(V^2))*(V-b)-R*T = 0

Explicit equations

1 P = 56
2 R = 0.08206
3 T = 450
4 Tc = 405.5
5 Pc = 111.3
6 Pr = P/Pc
7 a = 27*(R^2*Tc^2/Pc)/64
8 b = R*Tc/(8*Pc)
9 Z = P*V/(R*T)

General Settings

Total number of equations 10
Number of implicit equations 1
Number of explicit equations 9
Elapsed time 0.00 sec
Reporting digits 10
Solution method safenewt
Max iterations 150
Tolerance F 1E-07
Tolerance X 1E-07
Tolerance min 1E-07