on the positivity of entropy production in multiphase systems · multiphase temperature pro le...
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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