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

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

of 27

  • date post

    24-Aug-2020
  • Category

    Documents

  • view

    1
  • download

    0

Embed Size (px)

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

  • 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