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Nucleation Rates Of Ethanol And Methanol Using SAFT And PC-SAFT EOSs

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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

- Definition
- Thermodynamics of nucleation
- Kinetics of Nucleation
- Versions of nucleation
- Results of equations of state & subroutine
- Results of Nucleation rate & subroutine
- Conclusions

- 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.

- 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

- The binodal curve (solid dome)separates one-phase and two-phase states
- TC = critical temperature
- The spinodal curve (dashed dome) separates metastable and unstable states
- One true horizontal isotherm is shown

SPINODAL

LINE

- 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.

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:

- 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

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)

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

-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

The Concentration Of n-size Droplet At

Equilibrium Equals

And

Then, we can reach to

●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 =Pv /Pve (Supersaturation Ratio)

●SAFT…Statistical Associating Fluid Theory

● PC-SAFT….Perturbed-Chain Statistical

Associating Fluid Theory

●A is the reduced free helmholtz energy

●The association term is just for polar fluids

and

Subroutine Of Equilibrium Vapor-liquid pressure

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

end do

Binodal points of ethanol And

methanol using SAFT and PC-SAFT

where

Subroutine Of P-form

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 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

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

- 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 .

Thank You