Introduction to Thermodynamics Models for Process Engineering How to Choose a Thermodynamic Model

Introduction to Thermodynamics models for process engineering

How to choose a thermodynamic model - Simulation

Baptiste Bouillot

École Nationale Supérieure des Mines de Saint-Etienne - Centre SPIN

2019-2020 Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models Why study Thermodynamics ?

Thermodynamic properties calculation (Hv, Hl, Cp,...) Phase equilibrium prediction

2/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models Why study Thermodynamics ?

Thermodynamic properties calculation (Hv, Hl, Cp,...) Phase equilibrium prediction

2/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models How does it work ?

P (bar)

220 C (Point critique)

Liquide Solide

(s) (T) P 1 point triple

0,006 Gaz 273,15

273,16 373,15 673,16 T (K) Data base Correlations Thermodynamic models (6= correlations)

3/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models How does it work ?

60 (S) T=50°C P1

50 Liquide

40 courbe de bulle

P (kPa) Liquide + Vapeur 30

courbe de rosée 20 Vapeur

(S) P2 10 0 x1, y1 1 Data base Correlations Thermodynamic models (6= correlations)

3/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models How does it work ?

A 0 1

xB miscibilité

M(x ,x ,x ) α A B C xA x B

α α x A D β

1 immiscibilité 0 B 0 α β 1 x C xC x C C Data base Correlations Thermodynamic models (6= correlations)

3/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models Issues : Thermodynamics and Chemical Engineering

Express magnitudes that characterizes phase properties and phase equilibria from measurable quantities (T, P...) !

Thermodynamic Equilibria Pure phases and mixtures properties LV Pure phase : single Thermodynamic LL component potentials : LLV Mixtures : more than 1 H(T, P, ...), component G(T, P), A(T, V) LS Mixing properties S ... Chemical potential Cp calculation Cv Fugacities ......

4/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models Objectives

Sensitize the engineer to be to Thermodynamics and its importance, Identify issues and choose the apropriate thermodynamic model, Know the theory and how to use a complex thermodynamic model in a process

5/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Phase equilibrium theory Equilibrium equationss, issues of phase equilibrium Thermodynamic models Outline

1 Thermodynamics reminders

2 Phase equilibrium theory

3 Equilibrium equationss, issues of phase equilibrium

4 Thermodynamic models

6/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Outline

1 Thermodynamics reminders

2 Phase equilibrium theory

3 Equilibrium equationss, issues of phase equilibrium

4 Thermodynamic models

7/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Rappels

Magnitude symbol unit Temperature TK Pressure P Pa Volume V m3 Molar composition x - Internal Energy UJ Enthalpy HJ Entropy SJ/K Gibbs “Free Energy” (Potential) GJ Helmoltz “Free Energy” (Potential) AJ Chemical Potential µ J/mol

8/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Potentials and Fundamental relations

Fundamental relation A thermodynamic system can be perfectly deﬁned by its fundamental relation : - S = f (U, V, N) - U = f (S, V, N)

How to do without it ? 2 “equations of state”, OR : 0 Mechanic equation under the form f (P, Vm, T) = 0

Thermal relation under the form Cpm = T/N.(∂S/∂T)p = Cp(T)

These equations are usually unknown !

9/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Usuall Thermodynamic quantities

Quantity symbol differential expression P ∗ Internal Energy U dU = TdS − PdV (+ i µidNi ) P ∗ Enthalpy H dH = TdS + VdP (+ i µidNi ) P ∗ Gibbs “Free Energy” (Potential) G dG = −SdT + VdP (+ i µidNi ) P ∗ Helmoltz “Free Energy” (Potential) A dA = −SdT − PdV (+ i µidNi )

→ Which one to use ? ? ?

→“Best” differential expression ? ? ?

10/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Usuall Thermodynamic quantities

Temperature and Pressure are measurable quantities, therefore the “bests”.

How many degree of freedom ? :

F = N − Φ + 2 − R (1)

avec : F degree of freedom, N number of components (molecules), Φ number of phases, R constraints (additional relations such as, constant composition, chemical reaction, azeotropy...).

11/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Gibbs Phase Rule

How many degree of freedom ? :

F = N − Φ + 2 − R (1)

avec : F degree of freedom, N number of components (molecules), Φ number of phases, R constraints (additional relations such as, constant composition, chemical reaction, azeotropy...).

P (bar)

220 C (Point critique)

Example : Liquide - Single component system/single phase : Solide (s) (T) F = 2 P 1 - Single component / two phases : F = 1 point triple - Singel component / three phases : F = 0 0,006 Gaz 273,15

273,16 373,15 673,16 T (K) 11/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Gibbs Phase Rule

How many degree of freedom ? :

F = N − Φ + 2 − R (1)

avec : F degree of freedom, N number of components (molecules), Φ number of phases, R constraints (additional relations such as, constant composition, chemical reaction, azeotropy...).

60 (S) T=50°C P1

50 Liquide

40 courbe de bulle

Example : Binary system ? P (kPa) Liquide + Vapeur 30

courbe de rosée 20 Vapeur

(S) P2 10 0 x1, y1 1 11/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Gibbs Phase Rule

How many degree of freedom ? :

F = N − Φ + 2 − R (1)

avec : F degree of freedom, N number of components (molecules), Φ number of phases, R constraints (additional relations such as, constant composition, chemical reaction, azeotropy...).

60 (S) T=50°C P1

50 Example : Liquide - 1 phase : F = 3 40 courbe de bulle

( T-P-x for instance), P (kPa) Liquide + Vapeur - two phases : F = 2 30

courbe de rosée ( T-P ou T-x ou P-x for instance) 20 Vapeur

Remember that : P dU = TdS − PdV + i µdNi

et que : U = f (S, V, Ni)

These two equations lead to the standard Gibbs-Duhem equation : l’Équation de Gibbs-Duhem : X SdT − VdP + Nidµi = 0 (2) i Why is this equation essential ?

12/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Fundamental properties Phase equilibrium theory Phase rule Equilibrium equationss, issues of phase equilibrium Gibbs-Duhem Thermodynamic models Gibbs-Duhem equation

Remember that : P dU = TdS − PdV + i µdNi

et que : U = f (S, V, Ni)

These two equations lead to the standard Gibbs-Duhem equation : l’Équation de Gibbs-Duhem : X SdT − VdP + Nidµi = 0 (2) i Why is this equation essential ?

It provides the chemical potential variation as a function of measurable quantities, temperature adn pressure

12/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Outline

1 Thermodynamics reminders

2 Phase equilibrium theory

3 Equilibrium equationss, issues of phase equilibrium

4 Thermodynamic models

13/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Deﬁnition

Different kind of equilibrium...

Phase α Phase β T, P T, P α β (µ1, µ2) (µ1, µ2)

Gibbs (1875) : Chemical potential, chemical equilibrium :  µα = µβ  1 1 (3)  α β µ2 = µ2 µ NEEDS to be calculated ! ! ! unless other functions are introduced, such as : α β f1 = f1 α β (4) f2 = f2 14/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Chemical potential

Gibbs-Duhem

dµi = −SmidT + VmidP (5) integrating :

Z T Z P 0 0 µi(T, P) = µi(T , P ) − SmidT + VmidP (6) T0 P0

Integrals Reference state dP integration : - Major issue of applied - “Mechanical” equation of state :’ thermodynamics f o(P, V, T) = 0 dT integration : - Usually no absolute value ! - Thermal relation (Cvm, Cpm)

Reference state choice ? ? ? 15/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Fugacity

Lewis (1908) → reference state = ideal gas :

Z P 0 µi(T, P) = µi(T, P ) + VmidP (7) P0

16/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Fugacity

Lewis (1908) → reference state = ideal gas :

Z P 0 µi(T, P) = µi(T, P ) + VmidP (8) P0 With perfect gas law : P µ (T, P) − µ (T, P0) = RT ln (9) i i P0

Lewis (1908) → reference state = ideal gas :

Z P 0 µi(T, P) = µi(T, P ) + VmidP (10) P0 With perfect gas law : P µ (T, P) − µ (T, P0) = RT ln (11) i i P0 Lewis suggested :

0 fi µi(T, P) − µi(T, P ) = RT ln 0 (12) fi

Deﬁnition : f i → 1 lorsque P → 0 (13) xiP Fugacity can be considered as a corrected pressure

By choosing the right reference state (µ0, or f 0), it can be demonstrated that :

α β α β µi = µi ⇐⇒ fi = fi (14)

Short exercise : Check equation 14

17/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Fugacity, activity and equilibrium

By choosing the right reference state (µ0, or f 0), it can be demonstrated that :

α β α β µi = µi ⇐⇒ fi = fi (14)

α α α 0 fi µi = µi + RT ln α 0 (15) fi If same reference state (T0, P0):

α 0 β 0 fi = fi α 0 β 0 (16) µi = µi hence : f αf β 0 µα − µβ = µα 0 − µβ 0 + RT ln i i (17) i i i i α 0 β fi fi α β α β therefore µi = µi ⇐⇒ fi = fi → OK ! ! ! (N.B. : also works with reference state at (T, P0)

By choosing the right reference state (µ0, or f 0), it can be demonstrated that :

α β α β µi = µi ⇐⇒ fi = fi (14)

Lewis then introduced the activity :

f (T,P,x) a = f 0(T,P0,x0) (18)

hence the standard formula :

µ = µ0 + RT ln a (= µ0 + RT ln γx) (19)

18/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Introduction

Idéalité Non idéalité

id reference fi = xifi (20)

Issue : Means : Choose the right reference state to Choose the right ideal state describe deviation from ideality

Two approaches :

Residual approach : Excess approach : ideal state = Perfect gas ideal state = Ideal solution

means : means : Equation of state Excess functions Fugacity coefﬁcient φ Acitvity coefﬁcient γ

19/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach

Every thermodynamic function can be expressed as follow :

Xreal(T, P) = Xidealgas(T0, P0) +[ Xidealgas(T0, P) − Xidealgas(T0, P0)] +[ Xidealgas(T, P) − Xidealgas(T0, P)] +[ Xreal(T, P) − Xidealgas(T, P)] (21)

gaz idéal à gaz idéal à T0 et P T et P

gaz idéal à gaz réel à T0 et P0 T et P

Deﬁnition : On appelle Residual quantity :

Xres(T, P) = [Xreal(T, P) − Xidealgas(T, P)] (22)

20/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach

Thanks to Maxwell equation for perfect (ideal) gas (GP) : from zero pressure, from inﬁnite molar volume,

res R P ∂Xreal ∂Xidealgas X (T, P) = |T − |T dP 0 ∂P ∂P (23) res R Vm ∂Xreal ∂Xidealgas X (T, P) = |T − |T dVm ∞ ∂Vm ∂Vm

21/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach

Thanks to Maxwell equation for perfect (ideal) gas (GP) :

res R P ∂Xreal ∂Xidealgas X (T, P) = |T − |T dP 0 ∂P ∂P (24) res R Vm ∂Xreal ∂Xidealgas X (T, P) = |T − |T dV ∞ ∂Vm ∂Vm

ideal gas ideal change ideal change property (P0, T0) at (T0, P) at (T, P) residual ideal R T ideal res U = U (T0, P0, N) + 0 + C (T, N)dT + U (T, P, N) T0 v ideal R T ideal res H = H (T0, P0, N) + 0 + C (T, N)dT + H (T, P, N) T0 p ideal P R T ideal res A = A (T0, P0, N) + NRT0 ln - (S (T, P, N) + NR)dT + A (T, P, N) P0 T0 ideal P R T ideal res G = G (T0, P0, N) + NRT0 ln - S (T, P, N)dT + A (T, P, N) P0 T0 Cideal(T,N) ideal P R T p res S = S (T0, P0, N) - NR ln + dT + S (T, P, N) P0 T0 T

Thanks to Maxwell equation for perfect (ideal) gas (GP) :

real idealgas res R P ∂Xi ∂Xi X (T, P) = |T − |T dP 0 ∂P ∂P real idealgas (25) res R V ∂Xi ∂Xi X (T, P) = |T − |T dV ∞ ∂V ∂V

quantity residual V = f (P, T, Ni) P = f (V, T, Ni) res R P ∂v R V ∂P U (T, P) = V − T |P dP + RT − Pv = T |V − P dV 0 ∂T ∞ ∂T res R P ∂V R V ∂P H (T, P) = V − T |P dP = T |V − P dV + PV − RT 0 ∂T ∞ ∂T res R P RT R V RT PV A (T, P) = 0 V − P dP + RT − PV = ∞ −P + V dV − RT ln RT res R P RT R V RT PV G (T, P) = 0 V − P dP = ∞ −P + V dV − RT ln RT + RTPV res R P ∂V R R V ∂P R PV S (T, P) = − |P + dP = |V − dV + R ln 0 ∂T P ∞ ∂T V RT

Thanks to Maxwell equation for perfect (ideal) gas (GP) :

real idealgas res R P ∂Xi ∂Xi X (T, P) = |T − |T dP 0 ∂P ∂P real idealgas (26) res R V ∂Xi ∂Xi X (T, P) = |T − |T dV ∞ ∂V ∂V

Chemical potential :

real res real idealgas fi µi = µi − µi = RT ln idealgas (27) fi

Definition The fugacity coefficient is defined as follow :

real φ = fi = fi i idealgas xV P (28) fi i

Thanks to the previous expression of residual quantities :

( R P v real − v idealgas dP = R P v − RT dP res 0 i i 0 i P µi = RT ln φi = R V ∂P RT (29) − |T,V,Nj − dV − RT ln Z ∞ ∂Ni V

Deﬁnition Compressibility factor Z :

PVm Z = RT (30) Quantiﬁes the deviation from perfect gas behaviour (0, 2 < Z < 1, 2) (mechanic behavior) 22/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach - fugacity coefﬁcient

Deﬁnition :

PVm Compressibility factor Z = RT Quantiﬁes the deviation from perfect gas behaviour (0, 2 < Z < 1, 2) (mechanic behavior)

Z can also be expressed as :

PVm Vm Z = = GP (31) RT Vm Usually, Z < 1 → P ↑⇒ Z ↓ (increase of molecular interactions)

23/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach - fugacity coefﬁcient

For a pure gas, or a gas mixture (other expression) :

Z P res Zmix − 1 µi = RT ln φi = dP (32) 0 P With Gibbs-Duhem equation for a mixture :

res P Gm (T, P) = RT i xi ln φi (33)

24/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Residual approach - fugacity coefﬁcient

For a pure gas, or a gas mixture (other expression) :

Z P res Zmix − 1 µi = RT ln φi = dP (32) 0 P With Gibbs-Duhem equation for a mixture :

res P Gm (T, P) = RT i xi ln φi (33)

To calculate φ, an “Equation of State” (EoS) is necessary :

- Van der Waals, - Peng-Robinson (PR), - Soave-Redlich-Kwong (SRK)..

Advantages and drawbacks of previous approach : Allows the integration of volumetric quantities → H, U,... BUT less recommended for the liquid phase

25/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Introduction to Excess approach

Advantages and drawbacks of previous approach Allows the integration of volumetric quantities → H, U,... BUT less recommended for the liquid phase

Deﬁnition : Excess quantity XE :

XE(T, P, x) = Xrealsolution(T, P, x) − Xidealsolution(T, P, x) (34)

Quantiﬁes the deviation from ideal state (ideal solution) Reserved more speciﬁcally for dense phases

Advantages and drawbacks of previous approach Allows the integration of volumetric quantities → H, U,... BUT less recommended for the liquid phase

Deﬁnition : Grandeur d’excès XE :

XE(T, P, x) = Xrealsolution(T, P, x) − Xidealsolution(T, P, x) (35)

Quantiﬁes the deviation from ideal state (ideal solution) Reserved more speciﬁcally for dense phases

Case of equilibrium :

E E real ref fi gi = µi = µi − µi = RT ln reference (36) xifi 25/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Excess approach : fundamental difference with residual approach

solution idéale à gaz idéal à gaz idéal à T0 et P T et P T et P solution réelle à gaz idéal à gaz réel à T et P T0 et P0 T et P

No thermodynamic quantity integration

→ need to know the properties of the “ideal” reference solution

26/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Activity coefﬁcient

Deﬁnition : activity coefﬁcient

0 γi = fi/xifi (37)

Brings three main equations :

E P Gm(T, P, x) = RT i xi ln γi (38)

and : E µi = RT ln γi (39) or : 0 0 µi(T, P) = µi (T, P) + RT ln xiγi = µi (T, P) + RT ln ai (40)

27/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Activity coefﬁcient

Deﬁnition : activity coefﬁcient

0 γi = fi/xifi (37)

Brings three main equations :

E P Gm(T, P, x) = RT i xi ln γi (41)

and : E µi = RT ln γi (42) or : 0 0 µi(T, P) = µi (T, P) + RT ln xiγi = µi (T, P) + RT ln ai (43) To calculate γ, an activity coefﬁcient model is needed : NRTL, UNIQUAC, UNIFAC...

Deﬁnition : activity coefﬁcient

0 γi = fi/xifi (37)

Symmetrical convention Asymmetrical convention Most common case Ideality Less common case : Ideality according to Raoult : according to Henry

f 0(T, P) = f pure(T, P) f 0(T, P) = f inﬁnitedillution(T, P) i i (44) i i γi → 1 when xi → 1 γ1 → 1 when x1 → 1 (solvent) γ2 → 1 when x2 → 0 (solute) (45) choice of convention according to γHenry = f2 (46) the situation, the data held... 2 x2H2,1

Deﬁnition : activity coefﬁcient

0 γi = fi/xifi (37)

x x

Gibbs-Duhem P SdT − VdP + i Nidµi = 0 Thermodynamic equilibrium equation α β α β µi = µi ⇐⇒ fi = fi Chemical potential µ = µ0 + RT ln f /f 0 µ = µ0 + RT ln a Residual approach Excess approach 0 fi = xiφiP fi = xiγif res P E P Gm (T, P) = RT i xi ln φi Gm(T, P, x) = RT i xi ln γi R P Z−1 E RT ln φ = 0 P dP µi = RT ln γi real real fi fi fi ai φi = = γi = = idealgas xiP ref xi fi fi Compressibility factor convention Z = PVm/RT Raoult (sym.) Henry (asym.) f 0 = f pur f 0 = f inf .dillution

28/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Objectives

Next objectives :

To know and be able to write mathematically phases equilibrium To know and be able to apply the different approaches To choose a thermodynamic model judiciously

29/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Exercises

in this class, you will handle models and solve phase equilibrium problems

"Coding" and formalization of balance Use of professional software Pro/II problems "by hand" (Matlab, Excel...)

30/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Exercice 1 : Découverte de l’outil Pro/II

Découverte de l’outil Pro/II

31/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Exercice 1 : Découverte de l’outil Pro/II

Exercice : Créer un ﬂux de matière d’un molécule au choix Simuler ensuite le ﬂux, et ouvrir le rapport pour observer l’état de la matière. P.S. : Les cases rouges indique un manque de Paramétrisation !

1/ À partir d’une équation d’état de type Van der Waals, exprimer l’enthalpie H d’un corps pur en phase gaz

2/ Expliciter la procédure de calcul numérique.

On rappelle les expressions d’une équation d’état cubique et de l’enthalpie résiduelle : RT a P = − (47) (Vm − b) (Vm − br1)(Vm − br2) avec a = a(T), et :

Z P Z Vm res ∂Vm ∂P Hm (T, P) = Vm − T |P dP = T |Vm − P dVm + PVm − RT 0 ∂T ∞ ∂T (48)

32/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Chemical equilibrium ? Phase equilibrium theory Residual approach Equilibrium equationss, issues of phase equilibrium Excess approach Thermodynamic models Conclusion - Objectives - Exercise Exercice 2 : Calcul d’enthalpie - cas du corps pur

À partir d’une équation d’état de type Van der Waals, exprimer l’enthalpie H d’un corps pur en phase gaz RT a P = − (49) (Vm − b) (Vm − br1)(Vm − br2) L’équation précédente permet d’exprimer ∂P/∂T ainsi : ∂P R da/dT = − (50) ∂T |Vm (Vm − b) (Vm − br1)(Vm − br2) soit :

res R Vm h RT T (da/dT) RT a i H (T, P) = − − − dVm m ∞ (Vm−b) (Vm−br1)(Vm−br2) (Vm−b) (Vm−br1)(Vm−br2) +PVm − RT (51) donc : Z Vm res da dVm Hm (T, P) = a − T + PVm − RT (52) dT ∞ (Vm − br1)(Vm − br2) Remarque : il est possible d’exprimer analytiquement l’intégrale, faisant intervenir une fonction logarithme...

Ensuite :

Z P Z T 0 0 0 0 res H(T, P) = H (T , P ) + dP + Cp(T)dT + H (T, P) (53) P0 T0

H0 est l’origine (enthalpie comportement Gaz Parfait) 0 o Cp la capacité caloriﬁque à l’état de Gaz Parfait (f molécule) 0 2 souvent écrit : Cp = A + B × T + C × T ... Hres peut se calculer par l’équation d’état ! → da/dT, v = f (T, P)

0 → se référer à des tables pour Cp → calculer Hres

32/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Outline

1 Thermodynamics reminders

2 Phase equilibrium theory

3 Equilibrium equationss, issues of phase equilibrium

4 Thermodynamic models

33/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Vapor phase

Gas fugacity always expressed with the residual approach :

V V V fi = Pxi φi (54)

For an ideal (perfect) gas : V φi = 1 (55)

34/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Phase liquide

Liquid fugacity : 2 approaches :

Residual approach

L L L fi = Pxi φi (56)

Excess approach

L ref L L fi = fi xi γi (57)

symmetrical : asymmetrical :

L sat sat L L L ∗ fi = Pi φi xiγi Poynt(P) fi = xi γi Hi (59) (58)

35/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Solid Phase

Fugacity of solids generally represented by an excess approach :

S S S ref S fi = xi fi γi (60)

and more generally S S pur L fusion fi = fi = fi (T , P) (61)

36/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Équilibres LV

Very common in Process Engineering : condensation (heat exchanger, heat pump...), boiling (heat exchanger, refrigeration...), gas expansion (turbines), distillation (separation of components...), absorption (gas washing, puriﬁcation...), ... Separation of constituents from their volatility

37/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Équilibres LV

V L V L L V Gm = Gm ⇔ µ = µ (62) µi = µi L V (64) fi = fi

Clapeyron’s equation Several forms (approach ?) dPsat ∆HLV = (63) dT T∆VLV

Correlation, or EoS (PR, SRK...)

38/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 3a : PV diagram calculation “by hand”

Molecule : ethane (Pc = 4.88MPa, Tc = 305.4K, ω = 0.099) Cubic EoS (VdW kind) : Peng-Robinson (PR) (p.63 of poly)

LIQUIDE - Procedure ? Algorithm ? TA - PR Programming VAPEUR

- Resolution

- Comparison of experimental values (RMSE), CAMPUS data

LIQUIDE

TA N.B. : We give as part of PR : √ VAPEUR Pb a Vm+(1+ 2)b racines de l'équation cubique ln φ = Z − 1 − ln Z − − √ ln √ RT 2 2bRT Vm+(1− 2)b

39/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 3a : PV diagram calculation “by hand”

Correction : equilibrium condition : µV = µL, ou f V = f L soit xV φV P = xLφLP → φV = φL

' ''

' '''''''

ϕ ϕ ϕ '' ϕ

'' q 1 Pn pred exp 2 RMSE = n i=1 (x − x ) 39/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 3b : PV diagram calculation with Pro/II

LIQUIDE TA

VAPEUR Molecule : ethane use of Pro/II’s case study Comparison equation of cubic state equation Van der Waals LIQUIDE PR

TA SRK... VAPEUR

racines de l'équation cubique

40/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 3b : PV diagram calculation with Pro/II

liquid/gas approach gas approach liquid approach φ/φ (homogeneous) Residual γ/φ (heterogeneous)Residual Excess (symmetrical) γ/φ (heterogeneous) Excess (asymmetrical)

Phase 120 Vapeur

100

60 Température (°C) 40 liq xi x vap 20 Phase i liquide

0 0,20 0,40 0,60 0,80 1 fraction molaire de l'espèce i

Partition coefﬁcient

V xi Ki = L (65) xi

41/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium LV equilibrium - Mixtures

liquid/gas approach gas approach liquid approach φ/φ (homogeneous) Residual γ/φ (heterogeneous)Residual Excess (symmetrical) γ/φ (heterogeneous) Excess (asymmetrical)

Partition coefﬁcient

V xi Ki = L (65) xi

liquid/gas approach gas approach liquid approach φ/φ (homogeneous) Residual γ/φ (heterogeneous)Residual Excess (symmetrical) γ/φ (heterogeneous) Excess (asymmetrical)

P P

P P P

PPPjPi PPPjPi

Partition coefﬁcient

V xi Ki = L (65) xi

water/ethanol mixture :

It can be seen that from a water/ethanol solution with a low concentration of ethanol, it is impossible to obtain very pure ethanol.

limit concentration at column head = azeotrope composition ≈ 0.90

Obligation to go through the azeotropic distillation

42/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Simple case : Raoult’s law

Simple case of gas solubility : Raoult’s law

Assumptions : V V V L L L f1 = x1 f1 pur et f1 = x1 f1 pur : Raoult’s ideality V f1 pur = P : ideal gas L sat f1 pur = P1 : no inﬂuence of pressure on the fugacity of the liquid, and the vapour at equilibrium with the pure liquid is an ideal gas

V L sat x1 P = x1 P1 (66) Good approximation in simple cases (ideal gas mixtures)

43/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium LL equilibrium

Common in Process Engineering : extraction (solvent, solute extraction), distillation (heteroazeotropic distillation...) Separation of constituents from their deviation from ideality

α β µi = µi α β (67) fi = fi or (usually γ/γ): n α α β β xi γi = xi γi (68)

Partition coefﬁcient xα γβ Kα/β = i = i (69) i β γα xi i

44/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Illustration of a liquid-liquid equilibrium (LLE)

A 0 1

xB miscibilité

M(x ,x ,x ) α A B C xA x B

α α x A D β

1 immiscibilité 0 B 0 α β 1 x C xC x C C

45/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Illustration of a liquid-liquid equilibrium (LLE)

α β

α α β β

Generic case

α β α/β xi γi ∀i∈(•,•,•)Ki = β = γα xi i

46/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium LL equilibrium- Focus on extraction

Generic case

α β α/β xi γi ∀i∈(•,•,•)Ki = β = γα xi i

Generic case

α α

β β

α β α/β xi γi ∀i∈(•,•,•)Ki = β = γα xi i

Let’s complicate things...

α β V µi = µi = µi α β V (70) fi = fi = fi

T T

Téb A Vapeur

T B L1+V éb L2+V

Liquide 1 Liquide 2

Liquide 1 + liquide 2

A composition B

47/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 4 : LV equilibrium for binary mixture

mixture : propane, n-butane cubic EoS SRK (p. 62 poly)

Phase Vapeur 120 1- Procedure ? Algorithm ?

100 2- recovery on CAMPUS of the SRK equation and the incomplete code 80 → Exo4 propane/n-butane VLE (ressources élèves)

60 3- Coding of missing parts (iterative loops) Température (°C) 40 liq xi 4- Comparison of experimental values (RMSE), x vap 20 Phase i CAMPUS data liquide 5- Using the VLE function on Pro/II 0 0,20 0,40 0,60 0,80 1 fraction molaire de l'espèce i

48/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 4 : LV equilibrium for binary mixture

Correction :

V L V L V V L L equilibrium condition : ∀i, µi = µi , or fi = fi hence xi φi P = xi φi P We obtain the following equation :

V V L  φprop(T,P,xprop) xprop  L L = V = Kpropane φprop(T,P,xprop) xprop V V L (71) φbut(T,P,xprop) (1−xprop)  L L = V = Kn−butane φprop(T,P,xprop) (1−xprop) System with 2 equations, 2 unknowns (optional). Two variables can be set, for example T and P.

V L So you need two loops : One on xpropane and One on xpropane for example.

The convergence criterion is compliance with the system of equations (71)

Crystallization is very common (once again...) : the puriﬁcation of a solute, the separation of species in solution, the generation of an active principle ingredient, ... 25% to 30% of the turnover of the chemical industry, 80% of the pharmaceutical industry.

49/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Équilibres LS

Deﬁnition : Solubility We call “solubility” the LSE : Concentration

courbe de solubilité

zone de sursaturation

zone de sous-saturation

Température 49/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 5 : LS equilibrium

Propose a method/algorithm to calculate an LS equilibrium For a solution in a solvent By using the chemical potential Using the Gibbs-Helmoltz equation

50/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Exercise 5 : LS equilibrium

Propose a method/algorithm to calculate an LS equilibrium

Thermodynamic equilibrium : solide liquide µi = µi (72) let’s take as a reference state : pure liquid at T and P → symmetrical excess approach : solide liquide pur µi = µi + RT ln (xiγi) (73) therefore : solide liquide pur solide liquide pur SL µi − µi Gmi − Gmi ∆Gmi ln (xiγi) = = = (74) RT RT RT + Gibbs-Helmoltz : ∂G/T H = − (75) ∂T T2 from where (assuming H independent T) : ∆Hmfus Tfus ln x2γ2 = 1 − (76) RTfus T and more rigorously : ∆Hmfus Tfus ∆CPm Tfus Tfus ln x2γ2 = 1 − − ln − + 1 (77) RTfus T R T T 50/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Thermodynamic Flash

Deﬁnition : Thermodynamic Flash A Thermodynamic Flash is a unit operation for phase separation, usually at given temperature and pressure. It is based on both thermodynamic equilibrium equations and mass balance calculations.

FIGURE – Thermodynamic Flash (“Techniques de l’ingénieur (be8031)”) 51/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Thermodynamic Flash

Flowrate vapor Note that : θ = Flowrate feed . Flash PT :

FIGURE – Flash algorithm (“Techniques de l’ingénieur (be8031)”) 51/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Pure phases Phase equilibrium theory LV equilibrium Equilibrium equationss, issues of phase equilibrium LL(V) equilibrium Thermodynamic models LS equilibrium Thermodynamic Flash

Flowrate vapor Note that : θ = Flowrate feed . Flash QP :

FIGURE – Flash algorithm (“Techniques de l’ingénieur (be8031)”) 51/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Outline

1 Thermodynamics reminders

2 Phase equilibrium theory

3 Equilibrium equationss, issues of phase equilibrium

4 Thermodynamic models

52/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Molecular interactions

53/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Molecular interactions

Molecular interactions (attractive and repulsive forces).

δ+ δ+ δ- (1) four different attractive δ- interactions :

dispersion forces (or London δ+ δ- δ+ (2) δ- forces),

δ+ Debye forces, δ- (3) δ+ Keesom forces, δ- chemical forces (such as (4) hydrogen bonds). O H O H H H

+ repulsive short distance → hard sphere

Ideal model no deviation from ideal gas (→ perfect gas), or ideal solution :

α γi = 1 α (78) φi = 1

a few approaches : small pressures (< 2bar) and high temperatures, “weak” interactions (simple (light) hydrocarbons, same chain length...), similar interactions, or that cancel each other out (example : eau/acetone mixture), simple gas phase compared to a complex co-existing liquid phase, ideal solid phases.

54/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Introduction to the Theorem of corresponding states

55/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Introduction to the Theorem of corresponding states

Lenard-Jones interaction potential h i r0 12 r0 6 (r) = 40 r − r (79)

= 0.F(r, r0) quantiﬁes the dispersive forces, and (0, r0) = f (substance).

Lenard-Jones interaction potential h i r0 12 r0 6 (r) = 40 r − r (79)

= 0.F(r, r0) quantiﬁes the dispersive forces, and (0, r0) = f (substance).

55/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Theorem of corresponding states : F an universal function

”Theoreme” Three is an equation, for each “simple” ﬂuid, that describes its state thanks to reduced variables. This equation is the same for each ﬂuid presenting the same reduced variables : V T P F , , = 0 (80) Vc Tc Pc

In practice, this means : res res X (Tr, Pr) = X0 (Tr, Pr) PVm (81) Z = Z(Tr, Pr) = RT However, molecules are rarely “simple”...

56/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Theorem of corresponding states : F an universal function

In practice, this means : res res X (Tr, Pr) = X0 (Tr, Pr) PVm (81) Z = Z(Tr, Pr) = RT However, molecules are rarely “simple”...

Acentric factor ω deviation from the sphericity of a molecule. for example : methane : ω = 0 butane : ω = 0, 20

⇒ Z = Z(Tr, Pr, ω) OR Z = Z(Tr, Pr, Zc) 56/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Relation Zc ↔ ω

Deﬁnition ω an arbitrary deﬁnition of the acentric factor ω is :

sat ω ≡ − log P − 1 (82) Pc Tc=0.7

Relation ω ↔ Zc

A correlation between Zc and ω can be written (Pitzer):

Zc = 0.291 − 0.08ω (83)

Warning ! ! ! this is only a correlation ! ! !

57/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software introduction on the equations of state

58/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software introduction on the equations of state

Deﬁnition Equation under the form f (P, T, V, N) = 0 ↔ inheritance of the corresponding states

Usefulness Allows the calculation of thermodynamic properties (residual function)

PV = NRT (84) et les autres : Virial equation, cubic equations, SAFT/CPA kind equations.

Deﬁnition : Virial equation Development from the perfect gas : PV B C D Z = m = + + + + ... 1 2 3 (85) RT Vm Vm Vm s’écrit aussi : PV Z = m = 1 + B0P + C0P2 + D0P3 + ... (86) RT adapts to mixtures and liquid/gas phases (from 2nd order)

Advantages : All coefﬁcients have a physical meaning More precise Drawbacks : Potentially requires many coefﬁcients

59/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Virial equation

1er ordre PV B Z = m = 1 + (87) RT Vm only vapor phase at low pressure

2nd ordre PV B C Z = m = + + 1 2 (88) RT Vm Vm only vapor phase at higher pressure

other types Benedict Webb Rubin (BWR) Starling-BWR Soave-BWR Lee-Kesler... 60/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Virial equation

1er ordre PV B Z = m = 1 + (87) RT Vm only vapor phase at low pressure

2nd ordre PV B C Z = m = + + 1 2 (88) RT Vm Vm only vapor phase at higher pressure

Fugacity coefﬁcient ?

m m m 2 X 3 X X ln φ = x B + x x C − ln Z i i ij 2 i k ijk mix (89) Vm 2V j=1 m j=1 k=1

60/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Cubic EoS

Equation of state ?

f (T, P, V, N) = 0 (90)

Cubic EoS Deﬁnition : V3 + AV2 + BV + C = 0 m m m (91) Z3 + A0Z2 + B0Z+C0 = 0 General shape : RT a(T, x) P = − (92) Vm − b(x) (Vm − c1b(x))(Vm − c2b(x))

61/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Cubic EoS

a is the attractive parameter, a molecule/molecule interaction coefﬁcient called binary interaction parameter

b is the repulsion parameter or the effective molecular volume. a) b) v b

v-b

Cas des mélanges :

Pm Pm a = i=1 j=1 xixjaij Pm (93) b = i=1 xibi Note that other rules exist...

62/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Équations d’état classiques

Van der Waals, Nobel prize (1910) : a P + 2 (Vm − b) = RT (94) Vm Redlich-Kwong : RT a/T0,5 P = − (95) Vm − b Vm(Vm + b) Soave-Redlich-Kwong (SRK) : RT a α(T) P = − c (96) Vm − b Vm(Vm + b) Peng-Robinson (PR) : RT a(T) P = − 2 2 (97) Vm − b Vm + 2bVm − b

63/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Équations d’état classiques

 P = RT − a(T,x)  Vm−b (Vm−c1b(x))(Vm−c2b(x))  2 2  ∗ R Tc  a = a α(Tr)  Pc b = b∗ RTc (98) Pc √  2  α(Tr) = [1 + m(1 − Tr]  2 m = M0 + M1ω + M2ω

Équation d’état Paramètre VdW SRK PR√ c1 0 0 −1 − √2 c2 0 -1 −1 + 2 a∗ 27/64 1/[9(21/3 − 1)] 0, 45724 b∗ 0 (21/3 − 1)/3 0, 07780 M0 0, 500 0, 48 0, 37464 M1 1, 588 1, 574 1, 54226 M2 −0, 1757 −0, 176 −0, 26992

Équation SAFT res res 1 ∂A /kT µi (T, V)/kT = (99) V ∂Ni V,T,Ni6=k objective : represent highly non-ideal mixtures

Basics : interaction term for “segments” Aseg, Chain formation term of identical spheres Achain, associative term between sites Aassoc,

Ares = Aseg + Achaine + Aassoc (100)

64/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software SAFT equation and Helmoltz function form

Équation SAFT res res 1 ∂A /kT µi (T, V)/kT = (99) V ∂Ni V,T,Ni6=k objective : represent highly non-ideal mixtures

Ares = Aseg + Achaine + Aassoc (101)

assoc assoc interaction entre segments (k , ε )

formation de chaîne

site d'association

(σ, ε) Variants : PC-SAFT, PPC-SAFT, SAFT-VR...,

65/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software γ models presentation

objectives of these models : calculation of excess quantities, to take into account the various interactions and different "forces" (hydrogen bonds, dipole-dipole, Van der Waals, ionic...).

Moyen : Activity coefﬁcient calculation

Reminders the free enthalpy of two-component mixture is written :

pur pur G(T, P, NA, NB) = NAGmA + NBGmB + ∆Gmix (102) avec

∆Gmix = ∆Hmix − T∆Smix (103)

γ is involved in both contributions !

Standard approach : res comb ∀i, ln γi = ln γi + ln γi (104)

semi-predictifs prédictifs Van Laar contribution de groupe : Margules UNIFAC Wilson UNIFAC (modiﬁé) Hildebrand Scatchard Hildebrand Polynomial γ Flory-Huggins NRTL modèles COSMO : eNRTL COSMO-RS NRTL-SAC ADF-COSMO UNIQUAC COSMO-SAC

semi-predictive models :

"simple" models : More complex models : Van Laar, Wilson, Margules, NRTL (+ variants), UNIQUAC.

66/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Simple equations

Van Laar equation GE ABx x m = 1 2 (105) RT Ax1 + Bx2 or : 2 Ax1 ln γ2 = B × (106) Ax1 + Bx2

Margules equation G E m = Ax x (107) RT 2 1 or (2 parameters version) :

2 3 lnγ2 = Ax1 + Bx1 (108)

Very simple models that cannot model equilibria involving highly non-ideal mixtures 67/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Semi-predictive models - local composition

Local composition the local composition around a molecule i is independent of the local composition around a different molecule j

68/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Semi-predictive models - local composition

Wilson’s model GE X X m = − x ln(1 − x A ) (109) RT i j ji i j good representation of the enthalpy free of excess (miscible solvents)

NRTL model E Gm τ21G21 τ12G12 = x1x2 + (110) RT x1 + x2G21 x2 + x1G12 excels in all highly non-ideal, miscible or immiscible systems

Modèle UNIQUAC

combinatorial residual ln γi = ln γi + ln γi (111)

69/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Predictive models - group contribution

Group contribution :

Need to know the binary interaction parameters ! ! ! Most famous : UNIFAC, UNIFAC mod. Dortmund. etc... ASOG... 69/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software UNIFAC model

70/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software UNIFAC model

UNIFAC : Group contribution method

ln γ = ln γC + ln γR γC = entropic contribution (combinatorial)

C Φi z θi Φi X ln γ = ln + qi ln + li − xjlj xi 2 Φi xi j

γR = enthalpic contribution (residual)

R X (i) h (i)i ln γ = νk ln Γk − ln Γk k interaction coefﬁcient (ex : OH ↔ COOH) a Ψ = exp − mn mn T

NRTL-SAC NRTL-SAC (NonRandom Two-Liquids Segment Activity Coefﬁcient) model is a NRTL-based approach adapted to liquid/solid equilibria Division of molecules into conceptual segments Hydrophobic (X) Polar attractive (Y+) and repulsive (Y−) Hydrophilic (Z) → four parameters (XY+ Y− Z)

Polaire attracteur

Hydrophobe Hydrophile

Polaire répulsif

combinatorial residual − + ln γi = ln γi + ln γi (composition, T, [XY Y Z]) (112)

71/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Predictive models - Quantum chemistry

Quantum chemistry approach :

Need to do quantum calculations beforehand ! ! !

COSMO-RS, Most famous models : COSMO-SAC ADF-COSMO...

72/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software COSMO-based models

COSMO COSMO = COductor-like Screening MOdel Different thermodynamic models derived from the COSMO theory : COSMO-RS, COSMO-SAC, ADF-COSMO...

73/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Basic principle (COSMO-SAC 2002)

+ + + + + + ε≠∞ + + + + + S + S + + -- + S S - + - - + - - - + + - + - - + - + + + S + + + + + + S - S - + + - - + - - S - - + - - S + + + - + - - - - + - S + - + + + - S - + - + - S - + - + - +

c) Charges are a) Charges are turned oﬀ turned on

ε=∞

S S S S

S b) cavitation S S S S energy S

S S

a) the surface charges of the molecule are “turned off”, b) The molecule is inserted into a perfect conductor, c) Surface charges are returned.

74/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software COSMO-SAC equation

∆G∗rest − ∆G∗rest ln γ = ln γSG + i/S i/i (113) i/S i/S RT SG Res with ln γi/S the entropic contribution, and ln γi the enthalpic contribution :

Res X ln γi = ni pi (σm)[ln ΓS (σs) − ln Γi (σm)] (114) σm X −∆W(σm, σn) ln Γ (σ )= − ln{ p (σ )Γ (σ ) exp } (115) S m S n S n kT σn

∆W(σm, σn) ≡ interactions between surfaces σm et σn :

α0 ∆W(σ , σ )= (σ + σ )2 + c max[0, σ − σ ]min[0, σ + σ ] (116) m n 2 m n hb acc hb don hb

75/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Autres modèles

Many other models exist.. Regular solution (prédictif) Method sets (Grayson-Streed, Chao et Seader) Models for electrolytes (Pitzer model)

Regular solutions

Gaz à très basse pression Mélange des gaz idéaux

Évaporation de Liquéfaction du chaque espèce mélange

P Liquides purs solution

2 RT ln γ1 = v1Φ1(δ1 − δ2) (117)

Solubility parameters are used for molecules 76/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Autres modèles

Many other models exist.. Regular solution (prédictif) Method sets (Grayson-Streed, Chao et Seader) Models for electrolytes (Pitzer model)

CS et GS - Chao-Seader : Redlich-Kwong EoS + Regular solutions

- Grayson-Streed : CS + correlation

Liquid-Vapor equilibria for heavy and hydrogenated hydrocarbons, P < 200 bars

76/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Autres modèles

Many other models exist.. Regular solution (prédictif) Method sets (Grayson-Streed, Chao et Seader) Models for electrolytes (Pitzer model)

Models for electrolytes

ln γ = ln γcomb + ln γres + ln γLR (118)

eNRTL eUNIQUAC eUNIFAC ePC-SAFT...

pseudo-components : T

Téb. moyenne 4 pseudocomposant 4 (Téb. moyenne 4) M < M4 T4

Téb. moyenne 3 pseudocomposant 3 (Téb. moyenne 3) M4 < M < M3 T3

Téb. moyenne 2 pseudocomposant 2 (Téb. moyenne 2) M3 < M < M2 T2

M2 < M < M1 Téb. moyenne 1 pseudocomposant 1 (Téb.moyenne 1)

M1 < M

77/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software How to select a model ?

Editors recommendation (cf. users manual)

non Hydrocarbures oui (uniquement ?)

oui non polarité non supercritique ? oui

oui eau + non azéotrope hydrocarbures non

oui oui electrolytes non proche Pcritique non oui P > 10 bars P > 10 bars P > 1 bar non non oui non oui oui pseudo composant non oui

e-NRTL, CPA UNIQUAC solution UNIQUAC SAFT SRK SRK SRK e-UNIQUAC, NRTL régulières NRTL Wagner (surtout), PC-SAFT PSRK PR Idéal GS e-UNIFAC UNIFAC UNIQUAC UNIFAC PPR PR SRK, PSRK SAFT (et variantes) BK10 SAFT Flory PPR (+ variantes) e-PC-SAFT, PR, ... Pitzer PC-SAFT

78/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Process Simulation and Thermodynamics Software

There are several types of software in process simulation :

“Flowsheeting” software Software to draw process ﬂow diagrams

79/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Process Simulation and Thermodynamics Software

There are several types of software in process simulation :

Thermodynamics softwares Allows thermodynamic calculations to be performed, for example : Simulis R

Thermodynamics

There are several types of software in process simulation :

Process simulation software Combine ﬂowsheet with thermodynamics and material/energy balance .

Three “leaders” : Aspen One, Pro/II, ProSim.

FIGURE – Aspen HYSYS R

80/85 Baptiste Bouillot Thermodynamics Thermodynamics reminders Molecular interactions Phase equilibrium theory Equation of State EoS Equilibrium equationss, issues of phase equilibrium Activity coefﬁcient models Thermodynamic models Process Simulation and Thermodynamics Software Three leaders in process simulation

FIGURE – Pro/II R

FIGURE – ProSimPlus

CAPE-OPEN Communication standard between software.

FIGURE – Illustration of the interface between Aspen Plus and ChemSep to replace a distillation column 81/85 Baptiste Bouillot Thermodynamics Conclusion - Exercices

Exercice 6 : LSE with NRTL-SAC

An R&D Process Engineer at Sanoﬁ-Aventis is looking for a good crystallization solvent for his molecule AB10C20

He already has some data on the solubility of his molecules in textitsome solvents.

Help him use the thermodynamic model NRTL-SAC for textitpredict solubility (orders of magnitude) in other solvents to save time (and money !). For that : Develop a algorithm of calculation solubility (SL balance). Use the data already acquired for determine the parameters of the AB10C20 molecule. Apply the method, and predict solubility in isopropanol. Solvent parameters and model code (CAMPUS) are given :

Solvent X Y- Y+ Z solubility Temperature (K)

acetone 0.131 0.109 0.513 0. 0.19992 293.15 ethanol 0.256 0.081 0. 0.507 0.1671 293.15 chloroforme 0.278 0. 0.039 0. 0.2718 293.15 cyclohexane 0.892 0. 0. 0. 0.1122 298.15 octanol 0.766 0.032 0.624 0.335 0.343 298.15 ethyl acetate 0.322 0.049 0.421 0. 0.1850 293.15

82/85 Baptiste Bouillot Thermodynamics Conclusion - Exercices

Exercice 7 : Adapter des paramètres, “Créer” une molécule We are interested in the VL balances of propylene glycol mixtures (Mono-,di-,tri-,tetra-)

Only the ﬁrst three are present in the Pro/II database....

Create the tetrapropylene glycol molecule in pro/II (from the tri-prop...) and perform DPG/TPG TPG/TePG equilibrium calculations (CAMPUS data) with : An equation of state (PR or SRK) NRTL UNIFAC the ideal model ( ? ! !)

83/85 Baptiste Bouillot Thermodynamics Conclusion - Exercices

Exercice 7 : Adapter des paramètres, “Créer” une molécule

83/85 Baptiste Bouillot Thermodynamics Conclusion - Exercices

Exercice 8 : Choose a thermodynamic model

An engineer is working on an ethanol production process. To size his process, he uses the Pro/II tool.

Help him to choose a thermodynamic model relevant for his simulation (NB : some data are given in the appendix on CAMPUS). À the view of the molecules present, what is the potentially suitable model(s) Quickly check on Pro/II using some data (see on CAMPUS)

Main reaction : C H + H O = CH CH OH 2 4 2 3 2 Secondary reactions : 2(CH CH OH) = (CH CH ) O + H O (Ethyl ether) 3 2 3 2 2 2 C2H2 + H2O = CH3CHO (acétaldéhyde) 2(CH CHO) = CH CHCHCHO + H O (crotonaldehyde) 3 3 2 These molecules need to be separated...

84/85 Baptiste Bouillot Thermodynamics Conclusion - Exercices

The End

85/85 Baptiste Bouillot Thermodynamics