Nucleation Rates Of Ethanol And Methanol Using SAFT And

PC-SAFT EOSs

Fawaz HrahshehAdvisor ::Dr. Abdalla ObeidatCo-advisor ::Dr. H. Al-Ghanem

Department of PhysicsJUST

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

• 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

• 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

Metastable and unstable regions for the van der

Waals fluid

• 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.

SPINODALLINE

Schematic pressure – temperature phase diagram

for a pure substance

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:

VPNFvvb

-The Helmholtz free energy after the formation equals:

APVPVVnnNFllvllva )()(

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

APPVWvl

)(

The Maximum work of formation composite by two terms.

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:

)()()()1()1()1()( nnAnfnnAnfnJ

At steady state

])(

)(

)1(

)1()[1()1()(

nD

nf

nD

nfnDnAnJ

)()(

)1()1(

nAnD

nAnD

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

)1()(),(

nJnJ

t

tnf

At equilibrium

)1()( nJnJAnd

q

nDnAJ

1

)]()(/1[

1

)exp()1()(Tk

WDnD

B

The Concentration Of n-size Droplet AtEquilibrium Equals

And

KTm

Pv

2

Then, we can reach to

]/exp[)/(/2 *2 TkWTkPvmJBBvl

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)

W p ( / ) / ( )1 6 3 3 2 p p pre f vap

p l

kT Sln

2

2)/()3/16(

lw

)ln/()3/16(3

skTwl

Pressure vs density isotherm

SAFT & PC-SAFT EOSs

● SAFT…Statistical Associating Fluid Theory

● PC-SAFT….Perturbed-Chain Statistical

Associating Fluid Theory

SAFT and PC-SAFT EOS

RT

A

RT

A

RT

A

RT

A

RT

A

RT

A assocdischainhsideal

●A is the reduced free helmholtz energy

●The association term is just for polar fluids

Equations for equilibrium

and

levePP

)()(leve

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

Subroutine Of Equilibrium Vapor-liquid pressure

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

Nucleation rate is given by

/KTWCL eJJ

*

0

2

0

2

KT

P

mJ

l

where

gama=(24.23d0-0.09254d0*(T-273.15d0))*1.0d-3Vl=(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-12Jp=(Jo*dexp(-W2))

Subroutine Of P-form

Subroutine Of S-form

gama=(24.23d0-0.09254d0*(T-273.15d0))*1.0d-3W3=((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-12Js=Jo*dexp(-W3/(Kl*T)) !Nucleation rate

Subroutine Of Actual Pressure Pg=satu*p(equg,T)guess3=1.20d0*equgfun1=p(guess3,t)-Pgdfun1=dp(guess3,t)do while(dabs(fun1/dfun1)>error)fun1=p(guess3,t)-Pgdfun1=dp(guess3,t)root1=guess3-fun1/dfun1guess3=rowg

Subroutine Of Internal Pressuremewg=mew(rowg,t)guess=1.2d0*equlfun=mew(guess,t)-mewgdfun=dmew(guess,t)do while(dabs(fun/dfun)>error)fun=mew(guess,t)-mewgdfun=dmew(guess,t)root=guess-fun/dfunguess=rowlend do

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.

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 .

710810

Thank You