# Oneeq19b

 Name: Oneeq19b - Compressibility factor from the RK equation - one solution Source: Cutlip, M. B. and Shacham, M, Problem Solving in Chemical Engineering with Numerical Methods (2nd Ed), Prentice Hall Inc., 1999 Reference/s Model: 1 implicit equation, indep. variable name: z Average difficulty level Constraints: z>0 Discontinuities: none in the range of interest Initial range: zmin=-0.5, zmax=1.2 Solved by Shacham, M., POLYMATH 5.1, build 19, April 1, 2001 Model Eqs. Compressibility factor from the RK equation - one solution |POLVER05_1 EXCEL FILE f(z)=z^3-z^2-Q*z-r # TEXT FILE P = 200 # POLYMATH FILE Pc=33.5 # T= 631 # Tc=126.2 # Pr=P/Pc # Tr=T/Tc # Asqr=0.4278*Pr/(Tr^2.5) # B=0.0867*Pr/Tr # Q=B^2+B-Asqr # r=Asqr*B # p=(-3*Q-1)/3 # q=(-27*r-9*Q-2)/27 # R=(p/3)^3+(q/2)^2 # V=if(R>0)then((-q/2+sqrt(R))^(1/3))else(0) # WW=if(R>0)then(-q/2-sqrt(R))else(0) # psi1 = if(R<0)then(arccos(sqrt((q^2/4)/(-p^3/27))))else(0) # W = if(R>0)then(sign(WW)*(abs(WW))^(1/3))else(0) # z1 = if(R<0)then(2*sqrt(-p/3)*cos((psi1/3)+2*3.1416*0/3)+1/3)else(0) # z2 = if(R<0)then(2*sqrt(-p/3)*cos((psi1/3)+2*3.1416*1/3)+1/3)else(0) # z3 = if(R<0)then(2*sqrt(-p/3)*cos((psi1/3)+2*3.1416*2/3)+1/3)else(0) # z0=if(R>0)then(V+W+1/3)else(0) # z(min)=-.5, z(max)=1.2 Variable/function values Variable Value f(x) z 0.35 -1.0835E-01 P 200 Pc 33.5 T 631 Tc 126.2 Pr 5.970149254 Tr 5 Asqr 0.045687875 B 0.103522388 Q 0.068551398 r 0.004729718 p -0.401884731 q -0.101654258 R 0.000179362 V 0.400457283 WW 0.037434511 psi1 0 W 0.334521515 z1 0 z2 0 z3 0 z0 1.068312132 Root Variable Value f(x) z 1.06831213384200 2.89E-16 P 200 Pc 33.5 T 631 Tc 126.2 Pr 5.970149254 Tr 5 Asqr 0.04568787490270 B 0.10352238805970 Q 0.06855139798660 r 0.00472971791530 p -0.40188473132000 q -0.10165425798490 R 0.00017936221260 V 0.40045728293280 WW 0.03743451115230 psi1 0 W 0.33452151531760 z1 0 z2 0 z3 0 z0 1.06831213158400 Additional Information The plot can be interpreted as there is a double root in the vicinity of z =0. The analytical solution shows however that there is only a single root. The absolute function value is minimal at z=-0.03267 where f(z) = -0.0035 Function plot