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

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    Type
    Article
    Authors
    Fan, Xiaolin cc
    Kou, Jisheng cc
    Qiao, Zhonghua
    Sun, Shuyu cc
    KAUST Department
    Computational Transport Phenomena Lab
    Earth Science and Engineering Program
    Physical Science and Engineering (PSE) Division
    Date
    2017-01-19
    Online Publication Date
    2017-01-19
    Print Publication Date
    2017-01
    Permanent link to this record
    http://hdl.handle.net/10754/623029
    
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    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.
    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.
    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.
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Scientific Computing
    DOI
    10.1137/16M1061552
    Additional Links
    http://epubs.siam.org/doi/10.1137/16M1061552
    ae974a485f413a2113503eed53cd6c53
    10.1137/16M1061552
    Scopus Count
    Collections
    Articles; Physical Science and Engineering (PSE) Division; Earth Science and Engineering Program; Computational Transport Phenomena Lab

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