1. Nucleation Rates Of Ethanol And Methanol Using SAFT And PC-SAFT EOSs Fawaz Hrahsheh Advisor ::Dr. Abdalla Obeidat Co-advisor ::Dr. H. Al-Ghanem Department of Physics JUST

2. Outline Definition Thermodynamics of nucleation Kinetics of Nucleation Versions of nucleation Results of equations of state & subroutine Results of Nucleation rate & subroutine Conclusions

3. Definition of Nucleation And Nucleation Rate The nucleation is the process of formation of the critical size droplet (embryo) which has the ability to grow spontaneously till the phase transition. The nucleation rate is the rate of formation of the critical size droplets per unit volume per unit time.

4. Thermodynamics of nucleation The vapor which can nucleate is the supersaturated vapor The work of formation consists from two term The critical size droplets have the ability to grow spontaneously The maximum work of formation equals the difference in the Helmholts free energy

5. The binodal curve (solid dome)separates one-phase and two- phase states T C = critical temperature The spinodal curve (dashed dome) separates metastable and unstable states One true horizontal isotherm is shown Metastable and unstable regions for the van der Waals fluid

6. Solid black lines represent points of equilibrium two- phase coexistence c is the critical point b is the triple point When, say, gas is forced to cross lines ab or bc it is no longer the thermodynamically stable phase. The transition to the new stable phase is not instantaneous. SPINODAL LINE Schematic pressure – temperature phase diagram for a pure substance

7. The work of formation -The work of Formation is the work which is needed To form the critical size droplet - The Helmholtz free energy before the formation equals: -The Helmholtz free energy after the formation equals:

8. The Maximum Work Of Formation The maximum work of formation consists from two terms: the bulk (volumetric) term and the surface term At critical size, There is an thermodynamic equilibrium

9. The Maximum work of formation composite by two terms.

10. Kinetics Of Nucleation Becker and Döring assumed that the clusters change its size by absorbing single molecule (1- cluster) or by emitting single molecule (reversible process)

11. The difference between the formation of n-size cluster by absorbing single molecule into (n-1)- size cluster and its destruction by emitting single molecule equals: At steady state Then

12. -The n-size cluster can be formed by emitting single molecule from (n+1)-cluster and it can be destroyed by absorbing single molecule -the total time-variation of concentration of –size droplet is the difference between the two methods At equilibrium And

13. The Concentration Of n-size Droplet At Equilibrium Equals And Then, we can reach to

14. Three versions of classical theory ● Gibbs’s exact formula: ● version 1: use bulk surface tension for ● Version 2: liquid droplet is incompressible, – and ● Version 3: the vapor is an ideal gas – and –S =P v /P ve (Supersaturation Ratio)

15. Pressure vs density isotherm

16. SAFT & PC-SAFT EOSs ● SAFT …Statistical Associating Fluid Theory ● PC-SAFT …. Perturbed-Chain Statistical Associating Fluid Theory

17. SAFT and PC-SAFT EOS ● A is the reduced free helmholtz energy ● The association term is just for polar fluids

18. Equations for equilibrium and

19. Do rho(1)=guess1 rho(2)=guess2 k(1,1)=dp(rho(1),T) k(1,2)=-dp(rho(2),T) k(2,1)=dmew(rho(1),T) k(2,2)=-dmew(rho(2),T) f(1)=p(rho(2),T)-p(rho(1),T) f(2)=mew(rho(2),T)-mew(rho(1),T) z=k(2,1)/k(1,1) k(2,1)=0.0d0 f(2)=f(2)-(z*f(1)) k(2,2)=k(2,2)-(z*k(1,2)) u(2)=f(2)/k(2,2) u(1)=(f(1)-k(1,2)*u(2))/k(1,1) rho(1)=rho(1)+u(1) rho(2)=rho(2)+u(2) error1=0.0d0 do i=1,2 error1=error1+f(i)**2 end do error1=dsqrt(error1) if (error1<error) exit guess1=rho(1) guess2=rho(2) end do Subroutine Of Equilibrium Vapor-liquid pressure

20. Binodal points of ethanol And methanol using SAFT and PC-SAFT

21. Nucleation rate is given by where

22. gama=(24.23d0-0.09254d0*(T-273.15d0))*1.0d-3 Vl=(1.0d0/(rowl*N)) W1=((16.0d0/3.0d0)*b*(gama**3)/((Pl-Pv))**2) W2=(W1/(Kl*T)) Jo=(dsqrt((2.0d0*gama)/(b*(MM/N)))*Vl*(Pv/(Kl*T))**2)*1.0d-12 Jp=(Jo*dexp(-W2)) Subroutine Of P-form Subroutine Of S-form gama=(24.23d0-0.09254d0*(T-273.15d0))*1.0d-3 W3=((16.0d0/3.0d0)*b*(Vl**2)*(gama**3)/& &((Kl*T*dlog(satu))**2))*1.0d-12 !(n.m) W4=(W3/(Kl*T)) Jo=(dsqrt((2.0d0*gama)/(b*(MM/N)))*Vl*(Pv/(Kl*T))**2)*1.0d-12 Js=Jo*dexp(-W3/(Kl*T)) !Nucleation rate

23. Subroutine Of Actual Pressure Pg=satu*p(equg,T) guess3=1.20d0*equg fun1=p(guess3,t)-Pg dfun1=dp(guess3,t) do while(dabs(fun1/dfun1)>error) fun1=p(guess3,t)-Pg dfun1=dp(guess3,t) root1=guess3-fun1/dfun1 guess3=rowg Subroutine Of Internal Pressure mewg=mew(rowg,t) guess=1.2d0*equl fun=mew(guess,t)-mewg dfun=dmew(guess,t) do while(dabs(fun/dfun)>error) fun=mew(guess,t)-mewg dfun=dmew(guess,t) root=guess-fun/dfun guess=rowl end do

24. Gibbs’s formula improves classical nucleation rates for METHANOL based on SAFT EOS.

25. Gibbs’s formula improves classical nucleation rates for ETHANOL based on PC-SAFT EOS.

26. Conclusions the methanol and ethanol gases are not ideal. SAFT and PC-SAFT EOSs improve the binodals for methanol and ethanol at low temperature where the deviation from the experimental values approach zero. SAFT EOS gives better for the nucleation rates by one order of magnitude when compare with PC-SAFT EOS for methanol, that was clear in fitting value of nucleation rates for SAFT EOS was and PC-SAFT EOS was.

27. Thank You