Multicomponent Mass Transfer in the Compressible Flow of ...

OverviewMathematical Model

Results

Multicomponent Mass Transfer in the Compressible Flow ofSemi-Continuous Mixtures Using the Adaptive Characterization

Method2013 AIChE Annual Meeting

Livia Flavia Carletti [email protected]

AdvisorsPaulo L. C. Lage Luiz Fernando Silva

[email protected] [email protected]

Universidade Federal do Rio de JaneiroLTFD/PEQ/COPPE

San Franscico, November 5th , 2013 1 / 19

LTFD/PEQ/COPPE Livia Jatoba - [email protected] 2013 AIChE Annual Meeting


Results

Complex MixturesGoalsContinuous ThermodynamicsQMoM for Continuous Mixtures

Complex Mixtures

large number of components polymer solutionsoil fractionsbiofuels and coal-derived liquids

similar properties

↓

problematic determination

very small concentration

↓

numerical errors

RATZSCH, M. T., KEHLEN, H., Fluid Phase Equilibria, v. 14, pp. 225–234, 1983.

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Results


Goals

Develop a method to solve a compressible flow with mass transfer andadaptive characterization of the multicomponent mixture.

Reduce the number of equations to be solved by using adaptivepseudo-components.

Compare with the conventional solution for the multicomponent transportequations.

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Results


A Modeling Approach

The Continuous Component Model (CCM):· mixture composition is described by a distribution function.· also known as the continuous thermodynamic model.

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50 100 150 200 250 300 350 400

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0.1

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0.2

0.25

0.3

0.35

y

M (kg/kmol)

ρc =

∫Ω

ρc(M )dM

Yc(M ) =ρc(M )

ρ

∂ρc(M )

∂t+∇·[ρc(M )υ]+∇· j c(M ) = 0

transport equation for a distribution function⇓

solution through numerical methods to discretize the distribution function⇓

mixture characterization approximated by pseudo-componentsCOTTERMAN, R. L., PRAUSNITZ, J. M., Industrial & Engineering Chemistry Process Design And Development, v. 24, n. 2, pp. 434–443, 1985.

LIU, J. L., WONG, D. S. H., Fluid Phase Equilibria, v. 129, pp. 113–127, 1997. 4 / 19



Results


QMoM for Continuous Mixtures

adaptive Gaussian quadrature.

each quadrature point represents a discretized pseudo-component.

ρc(M ) ≈Np∑j=1

ρpj δD(M −Mpj )

uses the moments of the continuous component distribution function.

λk =

∫Ω

M k ρc(M )dM

ρc(M ) as continuous distribution

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x

M (kg/kmol)

or

ρc(M ) as discrete distribution

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50 100 150 200 250 300 350 400

x

M (kg/kmol)

LAGE, P. L. C., Computers & Chemical Engineering, v. 31, pp. 782–799, 2007.

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Results

Direct QMoM for Continuous MixturesCFD Implementation

Mathematical Model I

Semi-continuous mixture.

ρ =

Nesp∑A=1

ρA +

∫Ω

ρc(M )dM

Isothermal compressible flow.

Ideal gas.

PISO couppling.

Mass transport equations - Fick Model.Species - Discrete Component Model (DCM):

∂ρA

∂t+∇ · [ρAυ]−∇ · [DAm∇ρA] +∇ ·

[DAm

∇ρρρA

]= 0

Continuous component - Continuous Component Model (CCM):

∂ρc(M )

∂t+∇·[ρc(M )υ]−∇·[Dm (M )∇ρc(M )]+∇·

[Dm (M )

∇ρρρc(M )

]= 0

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Results


Direct QMoM for Continuous Mixtures I

Numerical method to solve the continuous component transport equation.

QMoM quadrature approximation for the continuous component.

ρc(M ) ≈Np∑j=1

ρpj δD(M −Mpj )

⇓

∂ρc(M )

∂t+∇·[ρc(M )υ]−∇·[Dm(M )∇ρc(M )]+∇·

[Dm(M )

∇ρρρc(M )

]= 0

⇓∫Ω

M k (·)dM

Weighted-abscissa definition: ηj = ρpjMpj

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Results


Direct QMoM for Continuous Mixtures II

Solution for 2Np transport equations: ρpj and ηj .

∂ρpj∂t

+∇ ·(ρpjυ

)−∇ ·

[Dm(Mpj )∇ρpj

]+∇ ·

[Dm(Mpj )ρpj

∇ρρ

]= aj

∂ηj∂t

+∇ · (ηjυ)−∇ ·[Dm(Mpj )∇ηj

]+∇ ·

[Dm(Mpj )ηj

∇ρρ

]= bj

aj and bj determined through the solution of the linear system:

Np∑j=1

(1− k)M kpj aj +

Np∑j=1

kM k−1pj bj =

Np∑j=1

Sj

where Sj depends on the diffusive model and properties gradients.

MCGRAW, R., Aerosol Science and Technology, v. 27, n. 2, pp. 255–265, 1997.

MARCHISIO, D., FOX, R., Journal of Aerosol Science, v. 36, n. 1, pp. 43–73, jan. 2005.

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Results


CFD Implementation

2 solvers implemented in OpenFOAM®.

DCM

ρ =

N∑A=1

ρA

∂ρA∂t

+∇ · (ρA υ) +∇ · jA = 0

DCM + CCM(DQMoM)

ρ =

Nesp∑A=1

ρA +

Np∑j=1

ρpj

Calculate aj and bj

∂ρA

∂t+∇ · (ρA υ) +∇ · jA = 0

∂ρpj

∂t+∇ ·

(ρpj υ

)+∇ · j pj = aj

∂ηj

∂t+∇ · (ηjυ) +∇ · Γ pj = bj

Calculate new Mpj

where N >> Np + Nsp

DCM - Discrete Component Method, CCM - Continuous Component Method, DQMoM - Direct Quadrature Method of Moments 9 / 19



Results


CFD Convergence Control

Mixed tolerance criteria used to control the convergence at each time step.

max

[|ϕk − ϕk−1|εabs + εrel |ϕk |

]< 1.0

where εabs = 10−7 and εrel = 10−5 forϕ = ρA, ρpj , ηj

DCM

ρ =

N∑A=1

ρA

∂ρA∂t

+∇ · (ρA υ) +∇ · jA = 0

DCM + CCM(DQMoM)

ρ =

Nesp∑A=1

ρA +

Np∑j=1

ρpj

Calculate aj and bj

∂ρA

∂t+∇ · (ρA υ) +∇ · jA = 0

∂ρpj

∂t+∇ ·

(ρpj υ

)+∇ · j pj = aj

∂ηj

∂t+∇ · (ηjυ) +∇ · Γ pj = bj

Calculate new Mpj

DCM - Discrete Component Method, CCM - Continuous Component Method, DQMoM - Direct Quadrature Method of Moments10 / 19



Results

The Comparison Case-Study: 2D Mixing Flow

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YA

MA (kg/kmol)

N = 57YN2

= 0.028

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0.015

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0.035

0.04

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YA

MA (kg/kmol)

N = 57YN2

= 0.16

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YA

MA (kg/kmol)

N = 57

YN2 = 0.063

inlet 1

inlet 2

υ1 = 0.12m/s

υ2 = 0.1m/s

21 cm x 50 cm x 1 cm N = 57 –> Np = 4, 5, 6 and 8 650K 11 / 19



Results

Accuracy of the Transient Solution

Results for DCM vs DQMoM at the outlet

Area mean value Characterization error

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8 9 10

ρ_ (kg

/m3 )

t(s)

N = 57

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7 8 9 10

δ(ρ_ )%

t(s)

Np = 4 Np = 5 Np = 6 Np = 8

ρ =

∑Nf

f =1(ρ)f |Sf |∑Nf

f =1 |Sf |

Nf : number of volume facesf : interpolations of ρ to each face centersSf : normal surface area vector of each face.

δ(ρ)% =|ρDQMoM − ρDCM |

|ρDCM |100%

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Results



Characterization error

At steady state, δ(ρ) decreases as Np

increases;

The mesh convergence error at the outletfor ρ were below 0.02%;

δ(ρ) reaches peak values at t ≈ 3s;

δ(ρ) increase with Np for t < 4s;

Convergence control of ρpj :

max

[|ρkpj − ρ

k−1pj |

εabs + εrel |ϕk |

]< 1.0

The same value for the εabs = 10−7 wasused ∀Np .

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6 7 8 9 10δ(

ρ_ )%t(s)

Np = 4 Np = 5 Np = 6 Np = 8

δ(ρ)% =|ρDQMoM − ρDCM |

|ρDCM |100%

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Results



ρpj at the outlet at t = 2.5s

As Np increases, there are somepseudo-components with very smallconcentrations.

εabs dominates the convergence control.

The numerical errors during the transientsolution were more strictly controlled forsmaller Np values.

The smallest ρpj for Np = 4 is ≈ 130 times greaterthan the smallest ρpj for Np = 8.

10-4

10-3

10-2

10-1

100

50 100 150 200 250 300 350 400

ρ_ pj (

kg/m

3 )

M_

pj (kg/kmol)

Np = 4Np = 8

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Results


Results for DCM vs DQMoM at the outletCharacterization error for Pbub Characterization error for Pdew

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1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

δ(P_

bub)%

t(s)

Np = 4 Np = 5 Np = 6 Np = 8

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10δ(P_

dew)%

t(s)

Np = 4 Np = 5 Np = 6 Np = 8

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Results

DQMoM Evolution for Np = 4

Area mean value at the outlet

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0.8

1.0

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

ρ_

pj (

kg/m

3 )

t (s)

ρ1ρ2ρ3ρ4

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10

M_

pj (

kg/k

mol

)t (s)

Mp1

Mp2

Mp3

Mp4

ϕ =

∑Nff=1

(ϕ)f |Sf |∑Nff=1|Sf |

Nf : number of volume facesf : interpolations of ϕ to each face centersSf : normal surface area vector of each face. 16 / 19



Results

Computational Costs: DCM vs DQMoM

Mesh size: 100,000 hexahedral volumes.

8 Xeon X5675 cores.

t interval ∆t Simulation time (hours) Speedup(s) (s) DCM DQMoM (Np = 4)

[0, 3.5] 10−5 174.55 89.35 1.95[3.5, 10] variable (Co=0.6) 78.07 35.97 2.17

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Results

Conclusions

The QMoM for continuous thermodynamics was extended to fieldproblems (called DQMoM).

DQMoM was applied to the flow of a semi-continuous compressiblemixture.

The continuous component is adaptively characterized at each point in thedomain.

Good accuracy of DQMoM was obtained with just a fewpseudo-components.

Fewer pseudo-components has larger concentrations, which allows a bettercontrol of the numerical error.

DQMoM solution was two times faster than that using the conventionalmethod.

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Results

Acknowledgments: Petrobras and CNPq for the financial support.

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