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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
)(
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
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
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
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
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 .
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