{"title":"Achieving Convergence in Multiphase Multicomponent Density Gradient Theory Calculations through Regularization","authors":"Paul Maidl, Kai Langenbach, Florian Frank","doi":"10.1021/acs.iecr.4c01669","DOIUrl":null,"url":null,"abstract":"We present a solution strategy for computing the equilibrium density profiles of mixtures consisting of an arbitrary number of components and phases in a closed system at constant temperature. Our approach is based on the density gradient formulation of the Helmholtz energy in the canonical ensemble, which is a functional of the component densities. By extending the corresponding Euler–Lagrange equations with an artificial time, we utilize a time-stepping scheme that converges toward the desired equilibrium state, adopting the approach proposed by Qiao and Sun [<contrib-group><span>Qiao, Z.</span>; <span>Sun, S.</span></contrib-group> <cite><i>SIAM J. Sci. Comput.</i></cite> <span>2014</span>, <em>36</em>, B708–B728]. Numerical methods for this approach are intrinsically incapable of preserving positivity in each time step. This is problematic, since the free energy density, determined by an equation of state, often prohibits nonpositive densities. To overcome this issue, we introduce a regularization of the energy densities that permits nonpositive densities during the time-stepping scheme. At the end of the solution process, the artificial time stabilization and the regularization both vanish, allowing us to obtain the density profile of the mixture, including its bulk compositions and interface profile. In this paper, we utilize the Peng–Robinson equation of state, which is popular in petroleum engineering. However, the proposed solution approach is applicable to arbitrary equations of state.","PeriodicalId":39,"journal":{"name":"Industrial & Engineering Chemistry Research","volume":null,"pages":null},"PeriodicalIF":3.8000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Industrial & Engineering Chemistry Research","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1021/acs.iecr.4c01669","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, CHEMICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We present a solution strategy for computing the equilibrium density profiles of mixtures consisting of an arbitrary number of components and phases in a closed system at constant temperature. Our approach is based on the density gradient formulation of the Helmholtz energy in the canonical ensemble, which is a functional of the component densities. By extending the corresponding Euler–Lagrange equations with an artificial time, we utilize a time-stepping scheme that converges toward the desired equilibrium state, adopting the approach proposed by Qiao and Sun [Qiao, Z.; Sun, S.SIAM J. Sci. Comput.2014, 36, B708–B728]. Numerical methods for this approach are intrinsically incapable of preserving positivity in each time step. This is problematic, since the free energy density, determined by an equation of state, often prohibits nonpositive densities. To overcome this issue, we introduce a regularization of the energy densities that permits nonpositive densities during the time-stepping scheme. At the end of the solution process, the artificial time stabilization and the regularization both vanish, allowing us to obtain the density profile of the mixture, including its bulk compositions and interface profile. In this paper, we utilize the Peng–Robinson equation of state, which is popular in petroleum engineering. However, the proposed solution approach is applicable to arbitrary equations of state.
我们提出了一种计算恒温封闭系统中由任意数量的组分和相组成的混合物平衡密度曲线的求解策略。我们的方法基于典型集合中赫尔姆霍兹能量的密度梯度公式,它是各组分密度的函数。通过对相应的欧拉-拉格朗日方程进行人工时间扩展,我们采用乔和孙提出的方法[Qiao, Z.; Sun, S. SIAM J. Sci. Comput.]这种方法的数值方法本质上无法在每个时间步中保持正向性。这是一个问题,因为由状态方程决定的自由能密度通常禁止非正值密度。为了解决这个问题,我们引入了一种能量密度正则化方法,允许在时间步进过程中出现非正值密度。在求解过程结束时,人工时间稳定和正则化都会消失,这样我们就可以得到混合物的密度曲线,包括其体积成分和界面曲线。本文采用的是石油工程中常用的彭-罗宾逊状态方程。不过,本文提出的求解方法也适用于任意状态方程。
期刊介绍:
ndustrial & Engineering Chemistry, with variations in title and format, has been published since 1909 by the American Chemical Society. Industrial & Engineering Chemistry Research is a weekly publication that reports industrial and academic research in the broad fields of applied chemistry and chemical engineering with special focus on fundamentals, processes, and products.