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. Outline. Definition Thermodynamics of nucleation Kinetics of Nucleation Versions of nucleation

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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

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


Outline

Outline

  • Definition

  • Thermodynamics of nucleation

  • Kinetics of Nucleation

  • Versions of nucleation

  • Results of equations of state & subroutine

  • Results of Nucleation rate & subroutine

  • Conclusions


Definition of nucleation and nucleation rate

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.


Thermodynamics of nucleation

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


Metastable and unstable regions for the van der waals fluid

Metastable and unstable regions for the van der Waals fluid

  • 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


Schematic pressure temperature phase diagram for a pure substance

Schematic pressure – temperature phase diagram for a pure substance

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.


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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

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


The maximum work of formation composite by two terms

The Maximum work of formation composite by two terms.


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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)


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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


Nucleation rates of ethanol and methanol using saft and pc saft eoss

-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


Nucleation rates of ethanol and methanol using saft and pc saft eoss

The Concentration Of n-size Droplet At

Equilibrium Equals

And

Then, we can reach to


Three versions of classical theory

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 =Pv /Pve (Supersaturation Ratio)


Pressure vs density isotherm

Pressure vs density isotherm


Saft pc saft eoss

SAFT & PC-SAFT EOSs

●SAFT…Statistical Associating Fluid Theory

● PC-SAFT….Perturbed-Chain Statistical

Associating Fluid Theory


Saft and pc saft eos

SAFT and PC-SAFT EOS

●A is the reduced free helmholtz energy

●The association term is just for polar fluids


Equations for equilibrium

Equations for equilibrium

and


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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


Nucleation rates of ethanol and methanol using saft and pc saft eoss

Binodal points of ethanol And

methanol using SAFT and PC-SAFT


Nucleation rate is given by

Nucleation rate is given by

where


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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


Nucleation rates of ethanol and methanol using saft and pc saft eoss

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


Gibbs s formula improves classical nucleation rates for methanol based on saft eos

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


Gibbs s formula improves classical nucleation rates for ethanol based on pc saft eos

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


Conclusions

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 .


Thank you

Thank You


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