On the positivity of entropy productionin multiphase systems

Aaron Romo Hernandez,Denis Dochain, B. Erik Ydstie, Nicolas Hudon

Universite catholique de Louvain

Institute of Information and Communication Technologies,Electronics and Applied Mathematics.

July 3, 2019

1

Multiphase chemical systems are everywhere

Liquid-gas reactions Crystal growth processes The atmosphere

We call a system where matter exists simultaneouslyin two or more states of aggregation a multiphase system

2

Multiphase operations consume 10% of the world’s energy

Challenges in process systems:

I Energy demands could beincremented by 48% in 2040

I The dynamical properties ofmultiphase systems are stillnot fully understood

2010 2020 2030 2040

Time (years)

200

250

300

Energy Consumed (Quadrillion BTU)

Industrial Energy

Consumption Projection

Source: eia.gov

2

Multiphase operations consume 10% of the world’s energy

Challenges in process systems:

I Energy demands could beincremented by 48% in 2040

I The dynamical properties ofmultiphase systems are stillnot fully understood

Limit cycles

Steady state multiplicity

3

Can we use thermodynamics to establishthe stability properties of multiphase systems?

4

Outline

Convexity properties of thermodynamic systemsLimitations in multiphase systems

Internal entropy production in multiphase systemsCompartmental modeling

On the positivity of internal entropy productionInternal entropy production as a Lyapunov function candidate

Phenomenological interpretation of the obtained resultsStable temperature profilesStable mass concentration profiles

Conclusions and future work perspectives

5

Thermodynamics allows for a better understanding of chemical processes

V (m 3)

0 0.01 0.02 0.03 0.04 0.05

S (

J/K

)

3100

3200

3300

3400

3500

3600

3700

Entropy for a single phase system

V (m 3)

0 0.01 0.02 0.03 0.04 0.05

S (

J/K

)

1650

1700

1750

1800

1850

1900

×10-3

0 2

1680

1700

1720

Entropy for a multiphase system

6

How can we characterize the stability properties of multiphase systems?

Every physicochemical systemproduces entropy at a rate

Σ = Σe + Σi

where

I Σe is the environmentalentropy production, and

I Σi is the internal entropyproduction.

Fl,out

Fg,out

Fg,in

Fl,in

Liquid-gas reactor

Second Law of Thermodynamics

Σi ≥ 0

7

A compartmental approach to modeling multiphase systems

Liquid-Gas Reactor Abstract Multiphase System

8

Internal flows increase the amount of entropy in multiphase systems

Spatial inhomogeneities[Tl , Pl , µl

]6=[Tg , Pg , µg

]are known to cause:

I Mass and

I Energy

to move between subsystems at arates Jnj and JQ (J/sec).

Internal flows (Jnj , JQ) are known toproduce entropy at a rate

Σı =c∑

=1

Jnj Xnj + JQ XQ ,

where (Xnj ,XQ) are known to be thedriving forces behind flows (Jnj , JQ).

8

Internal flows increase the amount of entropy in multiphase systems

Spatial inhomogeneities[Tl , Pl , µl

]6=[Tg , Pg , µg

]are known to cause:

I Mass and

I Energy

to move between subsystems at arates Jnj and JQ (J/sec).

Internal flows (Jnj , JQ) are known toproduce entropy at a rate

Σı =c∑

=1

Jnj Xnj + JQ XQ ,

where (Xnj ,XQ) are known to be thedriving forces behind flows (Jnj , JQ).

9

Internal entropy production model

Energy, mass and momentum balances can berepresented as a non-linear system of differentialalgebraic equations

d

dt

[ζlζg

]=

[fl (ζl , ζg ,w)fg (ζl , ζg ,w)

]0 = g(ζl , ζg ,w)

where the Jacobian ∂g/∂w is non singular.

Given state functions

ζ 7→ (Jnj , Xnj , JQ , XQ)

the system produces entropy as described by

Σı =c∑

=1

Jnj Xnj + JQ XQ ,

[Romo-Hernandez et al. 2018]

10

Entropy production behaves numerically as a Lyapunov function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

time (s)

0

0.02

0.04

0.06

0.08

0.1

0.12

i (J/m

ol.K

)

Tl(0) = 75.8792

Tl(0) = 77.4594

Tl(0) = 78.0804

Tl(0) = 82.1978

Figure: Σı vs t, for a methanol-water liquid-vapor system initialized

far from thermodynamic equilibrium (Teq = 78.09oC)

Result

Numerical evidence shows that

Σı =c∑

=1

Jnj Xnj + JQ XQ ,

behaves like a Lyapunov function for the equilibrium state ζ?.

11

Following the numerical results, we investigate if Σı is a Lyapunov function

To show that

Σı(ζ) =c∑

=1

Jnj Xnj + JQ XQ (1)

is a Lyapunov function candidate to characterize the stability of equi-librium ζ? in multiphase systems, we demonstrate that:

1. The internal entropy production (1) is zero at ζ?.

2. The internal entropy production (1) is positive when ζ 6= ζ?.

12

Writing the internal entropy production using alternative flows and forces

Internal entropy production rate

Σı = JQ XQ +c∑

=1

Jnj Xnj ,

where JQ :=∑c

j=1 Jnj hlj + λliδTl .

Note that the previous can be rewritten as

Σı = λli δTl Xe +c∑

=1

Jnj(Xnj + hlj Xe

)where

Xnj + hlj Xe = − 1

Tl(µgj − µlj)

∣∣Tl

= −R ln

(yjxjγj

)+ R ln

(fljPl

),

and flj and γj represent the fugacity and activity coefficient of component j .

12

Writing the internal entropy production using alternative flows and forces

Internal entropy production rate

Σı = JQ XQ +c∑

=1

Jnj Xnj ,

where JQ :=∑c

j=1 Jnj hlj + λliδTl . Note that the previous can be rewritten as

Σı = λli δTl Xe +c∑

=1

Jnj(Xnj + hlj Xe

)

where

Xnj + hlj Xe = − 1

Tl(µgj − µlj)

∣∣Tl

= −R ln

(yjxjγj

)+ R ln

(fljPl

),

and flj and γj represent the fugacity and activity coefficient of component j .

12

Writing the internal entropy production using alternative flows and forces

Internal entropy production rate

Σı = JQ XQ +c∑

=1

Jnj Xnj ,

where JQ :=∑c

j=1 Jnj hlj + λliδTl . Note that the previous can be rewritten as

Σı = λli δTl Xe +c∑

=1

Jnj(Xnj + hlj Xe

)where

Xnj + hlj Xe = − 1

Tl(µgj − µlj)

∣∣Tl

= −R ln

(yjxjγj

)+ R ln

(fljPl

),

and flj and γj represent the fugacity and activity coefficient of component j .

13

Alternative flows and forces

Alternative Flows and Forces

Internal entropy production for a two phase system

Σı = J ′Q X ′

Q +c∑

=1

J ′nj X

′nj . (2)

where

X ′Q = Tl − Tg , J ′

Q = λliδTl

TgTlJ ′nj = RJnj , X ′

nj = ln

(y?jyj

).

Note that

y?j = Klj xj :=γj(x)P sat

j (Tl)

Plxj ,

represents the gas composition in the system at equilibrium.

14

Internal entropy production is zero at equilibrium I

Equilibrium

The thermodynamic equilibrium stateis defined as a state ζ? such that

0 = µ? − µgj (P,T , y1, . . . , yc )∣∣ζ?

0 = µ? − µlj (P,T , x1, . . . , xc )∣∣ζ?

1 =c∑

k=1

xk∣∣ζ?

=c∑

k=1

yk∣∣ζ?

where j = 1, . . . , c.

Remark

Assuming that solutions of the equilibriumstate are unique, pressure, temperature andcomposition in a multiphase system at equi-librium satisfy

Pi = Pl = Pg

Ti = Tl = Tg

xij = xjyij = yj

Spatial Homogeneity

15

Internal entropy production is zero at equilibrium II

Theorem 1:Let ζ? stand for the equilibrium state for a multiphase whereconcentration in the gas yj and concentration in the liquid xj fulfill:

yj − xj 6= 0 ∀j .

Let

Σı(ζ) =c∑

=1

Jnj (ζ)Xnj (ζ) + JQ(ζ)XQ(ζ)

represent the internal entropy production. Then Σı(ζ?) = 0.

Proof.The proof follows from showing that interface flows (Jnj , Je) and con-

jugated driving forces (Xnj ,Xe) are zero at ζ? as a consequence of

spatial homogeneity at equilibrium.

16

Theorem: Positivity of internal entropy production I

DefinitionLet f = [f1, f2]t depend on the deviation from equilibrium

f1(ζ) = (Tl − Ti ) · (Ti − Tg )

f2(ζ) =(y?j − yj

)· φ(ζ)

where y?j represents the gas equilibrium concentration

y?j = Klj xj ,

and where φ(ζ) is a function that depends on the molar transport

Jn(ζ) =βl(xj − xij

)− βg

(yj − yij

)yj − xj

.

Letg(ζ) = Klj − yij/xij ,

where Klj = Kj (ζl ) stands for the multiphase equilibrium

partition ratio.

17

Theorem: Positivity of internal entropy production II

Theorem 2:Let ζ represent the state for a multiphase system. Let thethermodynamic variables be defined as a function

ζ 7→ (Tα,Pα, x1, y1, . . . , xc , yc , xi1, yi1, . . . , xic , yic , Jn),

where the output represents temperature, pressure, molarcomposition, and interface transport, respectively.

There exists a domain of thermodynamic consistency

Θ0 :={ζ∣∣f (ζ) ≥ 0, and g(ζ) = 0

},

such thatΣı(ζ) > 0 ζ ∈ Θ0

where

Σı(ζ) =c∑

=1

Jnj Xnj + JQ XQ .

18

Theorem 2 has clear phenomenological implications

Multiphase Temperature Profile Multiphase Mass Profile

Results from Theorem 2:

I Non-equilibrium systems where temperature profiles changemonotonically are consistent with the second law.

I There is a finite number of molar concentration profileconfigurations in multiphase systems far from thermodynamicequilibrium.

19

Conclusions and future work I

On the positivity of entropy production

We have shown that it is possible for a multiphase systemto operate far from equilibrium while having a positiveinternal entropy production.

Future Work

I What are the conditions such that entropy production decreases in timealong the dynamics of multiphase systems?

I Can our methodology be extended to azeotropic systems and toliquid-vapor processes with multiple equilibria?

20

Appendix: 12 feasible concentration profiles I

21

Appendix: 12 feasible concentration profiles II

22

Appendix: 12 feasible concentration profiles III