A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State

Handle URI:
http://hdl.handle.net/10754/623029
Title:
A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State
Authors:
Fan, Xiaolin; Kou, Jisheng; Qiao, Zhonghua; Sun, Shuyu ( 0000-0002-3078-864X )
Abstract:
This paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase fluid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng-Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng-Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is also very efficient. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EoS, it is unconditionally stable through introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EoS. An efficient numerical algorithm is provided to compute the coefficient in the correction term. Finally, some numerical examples are illustrated to verify the theoretical results and efficiency of the established algorithms. The numerical results match well with laboratory data.
KAUST Department:
Physical Sciences and Engineering (PSE) Division
Citation:
Fan X, Kou J, Qiao Z, Sun S (2017) A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State. SIAM Journal on Scientific Computing 39: B1–B28. Available: http://dx.doi.org/10.1137/16M1061552.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
Issue Date:
19-Jan-2017
DOI:
10.1137/16M1061552
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
The work of the first and fourth authors was supported by funding from King Abdullah University of Science and Technology (KAUST). The work of the second author was partially supported by National Natural Science Foundation of China (11301163). The work of the third author was partially supported by Hong Kong Research Grant Council GRF grants 509213, 15302214 and NSFC/RGC joint research scheme N HKBU204/12.
Additional Links:
http://epubs.siam.org/doi/10.1137/16M1061552
Appears in Collections:
Articles; Physical Sciences and Engineering (PSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorFan, Xiaolinen
dc.contributor.authorKou, Jishengen
dc.contributor.authorQiao, Zhonghuaen
dc.contributor.authorSun, Shuyuen
dc.date.accessioned2017-03-20T08:46:08Z-
dc.date.available2017-03-20T08:46:08Z-
dc.date.issued2017-01-19en
dc.identifier.citationFan X, Kou J, Qiao Z, Sun S (2017) A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State. SIAM Journal on Scientific Computing 39: B1–B28. Available: http://dx.doi.org/10.1137/16M1061552.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/16M1061552en
dc.identifier.urihttp://hdl.handle.net/10754/623029-
dc.description.abstractThis paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase fluid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng-Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng-Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is also very efficient. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EoS, it is unconditionally stable through introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EoS. An efficient numerical algorithm is provided to compute the coefficient in the correction term. Finally, some numerical examples are illustrated to verify the theoretical results and efficiency of the established algorithms. The numerical results match well with laboratory data.en
dc.description.sponsorshipThe work of the first and fourth authors was supported by funding from King Abdullah University of Science and Technology (KAUST). The work of the second author was partially supported by National Natural Science Foundation of China (11301163). The work of the third author was partially supported by Hong Kong Research Grant Council GRF grants 509213, 15302214 and NSFC/RGC joint research scheme N HKBU204/12.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/16M1061552en
dc.rightsArchived with thanks to SIAM Journal on Scientific Computingen
dc.subjectConvex splittingen
dc.subjectEquation of stateen
dc.subjectMulticomponent two-phase systemsen
dc.subjectPeng-Robinson EoSen
dc.subjectSequential splittingen
dc.subjectVan der Waals EoSen
dc.titleA Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of Stateen
dc.typeArticleen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionSchool of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei, 432000, , Chinaen
dc.contributor.institutionDepartment of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, , Hong Kongen
kaust.authorFan, Xiaolinen
kaust.authorSun, Shuyuen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.