# On the positivity of entropy production in multiphase systems Multiphase Temperature Pro le...

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On the positivity of entropy production in multiphase systems

Aarón Romo Hernández, Denis Dochain, B. Erik Ydstie, Nicolas Hudon

Université 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 simultaneously in 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 be incremented by 48% in 2040

I The dynamical properties of multiphase systems are still not fully understood

2010 2020 2030 2040

Time (years)

200

250

300

E n e r g y C o n s u m e d ( Q u a d r i l l i o n B T U )

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 be incremented by 48% in 2040

I The dynamical properties of multiphase systems are still not fully understood

Limit cycles

Steady state multiplicity

3

Can we use thermodynamics to establish the stability properties of multiphase systems?

4

Outline

Convexity properties of thermodynamic systems Limitations in multiphase systems

Internal entropy production in multiphase systems Compartmental modeling

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

Phenomenological interpretation of the obtained results Stable temperature profiles Stable 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 system produces entropy at a rate

Σ̇ = Σ̇e + Σ̇i

where

I Σ̇e is the environmental entropy production, and

I Σ̇i is the internal entropy production.

F l,out

Fg,out

Fg,in

F l,in

Liquid-gas reactor

Second Law of Thermodynamics

Σ̇i ≥ 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 a rates Jnj and JQ (J/sec).

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

Σı = c∑

=1

Jnj Xnj + JQ XQ ,

where (Xnj ,XQ) are known to be the driving 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 a rates Jnj and JQ (J/sec).

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

Σı = c∑

=1

Jnj Xnj + JQ XQ ,

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

9

Internal entropy production model

Energy, mass and momentum balances can be represented as a non-linear system of differential algebraic equations

d

dt

[ ζl ζg

] =

[ 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 o

l. K

)

T l (0) = 75.8792

T l (0) = 77.4594

T l (0) = 78.0804

T l (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 h̄lj + λliδTl .

Note that the previous can be rewritten as

Σı = λli δTl Xe + c∑

=1

Jnj ( Xnj + h̄lj Xe

) where

Xnj + h̄lj Xe = − 1

Tl (µgj − µlj)

∣∣ Tl

= −R ln (

yj xjγj

) + R ln

( flj Pl

) ,

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 h̄lj + λliδTl . Note that the previous can be rewritten as

Σı = λli δTl Xe + c∑

=1

Jnj ( Xnj + h̄lj Xe

)

where

Xnj + h̄lj Xe = − 1

Tl (µgj − µlj)

∣∣ Tl

= −R ln (

yj xjγj

) + R ln

( flj Pl

) ,

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 h̄lj + λliδTl . Note that the previous can be rewritten as

Σı = λli δTl Xe + c∑

=1

Jnj ( Xnj + h̄lj Xe

) where

Xnj + h̄lj Xe = − 1

Tl (µgj − µlj)

∣∣ Tl

= −R ln (

yj xjγj

) + R ln

( flj Pl

) ,

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 = λli δTl TgTl

J ′nj = RJnj , X ′ nj = ln

( y?j yj

) .

Note that

y?j = Klj xj := γj(x)P

sat j (Tl)

Pl xj ,

represents the gas composition in the system at equilibrium.

14

Internal entropy production is zero at equilibrium I

Equilibrium

The thermodynamic equilibrium state is 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 equilibrium state are unique, pressure, temperature and composition in a multiphase system at equi- librium satisfy

Pi = Pl = Pg Ti = Tl = Tg xij = xj yij = yj

Spatial Homogeneity

15

Internal entropy production is zero at equilibrium II

Theorem 1: Let ζ? stand for the equilibrium state for a multiphase where concentration 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 homoge