**The DQMoM for the adaptive characterization method of continuous components in OpenFOAM® **

[conference paper: 9th OpenFOAM® Workshop, Zagreb, Croatia, 06/2014] [download]

**ABSTRACT SHOW / HIDE**

Continuous mixtures are characterized by a large amount of components with similar properties such that it is not feasible to determine each one of them. These mixtures can be represented by a continuous component using a distribution function to characterize its composition. Examples of continuous mixtures are oil fractions and polymer solutions. The semi-continuous mixture is defined as a multicomponent mixture with continuous component and discrete components with known compositions. In order to solve the mass transport equation of a continuous component using the conventional discrete component approach, the distribution function has to be described by pseudo-components which are usually determined by quadrature based methods. However, the elevated number of pseudo-components required to capture the composition changes during mass transfer process still represents a drawback for Computational Fluid Dynamics (CFD) simulations. The present work consists on the implementation of the DQMoM for the solution of mass transport equation for the continuous component. It is based on the extension of the adaptive Quadrature Method of Moments (QMoM) for continuous thermodynamics to field problems. It was implemented in the OpenFOAM-1.6-ext as a compressible ideal gas flow solver. The DQMoM for continuous component were compared with the conventional discrete component model (DCM) in a mixing flow of a mixture of 57 hydrocarbons with nitrogen. It was shown that 4 adaptive DQMoM pseudo-components were enough to reproduce the DCM mixture properties with a 3% accuracy. Furthermore, the DQMoM CFD solution was approximately two times faster than the conventional solution for the mixture flow.

**Multicomponent mass transfer in the compressible flow of semi-continuous mixtures using adaptive characterization method**

[conference paper: AIChE Annual Meeting, San Francisco, USA, 11/2013] [ download ]

**ABSTRACT SHOW / HIDE**

Continuous mixtures are characterized by a large amount of components with similar properties which makes the determination of its exact composition unfeasible. These mixtures can be represented by a continuous component using a distribution function to characterize its composition. The distribution variable can be the molar mass or any other convenient property. Examples of continuous mixtures are oil fractions and polymer solutions. If the mixture has some known components, it is treated as a semi-continuous mixture. In order to solve the mass transport equation of a continuous component using the conventional discrete component approach, the distribution function has to be described by pseudo-components which are usually determined by quadrature based methods. However, the elevated number of pseudo-components required to capture the composition changes during mass transfer process still represents a drawback for CFD simulations. The present work developed the mathematical model and numerical solution of the multicomponent mass transport equations for the flow of semi-continuous mixtures. The model is an extension of the adaptive Quadrature Method of Moments (QMoM) for continuous thermodynamics to field problems. The method is called Direct QMoM for continuous thermodynamics. It was implemented in a CFD open-source package (OpenFOAM) as a compressible ideal gas flow solver. The DQMoM for continuous thermodynamics was compared with the conventional discrete component model (DCM) in a mixing flow of two mixtures consisting of 57 hydrocarbons and nitrogen. It was shown that 4 adaptive DQMoM pseudo-components were enough to reproduce the properties of the DCM mixture with a 2% accuracy. Furthermore, the DQMoM CFD solution was approximately two times faster than the DCM solution for the mixture flow.

**Comparison of different techniques for the characterization of complex mixtures**

[conference paper: International Conference on Chemical Thermodynamics, Buzios, Brazil, 08/2012]

**ABSTRACT SHOW / HIDE**

Complex mixtures pose a challenge for process simulation because of the high number of components and the problematic determination of each single composition. Examples of such mixtures include polymer systems, petroleum, shale and coal oil, among others, which are of course very important in industrial practice and therefore of great interest. This fact implies great difficulties for the use of rigorous process simulation tools, which are generally based on the concept of chemical compounds. Moreover, even in the case that the individual composition of each component could be determined, the number of species would be prohibitive for a computer simulation in reasonable time. The traditional approach in this case is to characterize the mixture in terms of global properties (distillation temperatures, vapor pressure, API gravities and so on), which can be related to a mixture of pseudo-components by means of empirical correlations.

In past years, several methodologies have been proposed in order to characterize complex mixtures in terms of components, either real or pseudo, and to represent the mixture by means of a smaller number of components, without losing excessively precision in phase equilibrium calculations. This reduction aims at decreasing the computational burden for process simulation. In this work, two different methods with this regard are compared: the quadrature method of moments for continuous thermodynamics (QMoM) and the Method of the Substitute Mixture (MSM). QMoM is based in the representation of the mixture molar fraction as a continuous or discrete distribution in terms of a continuous variable (e.g., the molar mass). This distribution is discretized using a Gauss-Christoffel quadrature, whose abscissas are the pseudo-components and the weights are their molar fractions. The Gauss-Christoffel quadrature is calculated from the moments of the molar fraction distribution by efficient numerical procedures. The other approach (MSM) replaces the original mixture, whose components may be unknown, by a mixture of real components from a databank, in order to match, in the best possible way, a set of given global properties of the original mixture - basically, the true boiling point (TBP) curve. MSM also permits to choose the number of components to be used for the characterization.

In order to compare both strategies, three benchmark mixtures with distinct composition are considered, constituted by a homologous series of linear alkanes from C7 to C50. QMoM and MSM are then employed to characterize, that is, to represent the mixture in terms of new components, and the resulting representations are used in PT-flash calculations. In order to test the accuracy of the methods for the prediction of stream properties, and also to verify the reduction in the computation burden, the results are compared with the flash solution considering the original mixture. Moreover, a simple heuristic approach, i. e., the selection of the most abundant components in the mixture, was also tested. Aspects such as the ease of implementation and the rate of convergence in terms of the number of pseudo-components are also analyzed.

The results show that, whereas MSM was unable to represent adequately the mixtures for flash calculations, QMoM needed only 6 pseudo-components in order to achieve a mean 1.4% accuracy in the vaporization fractions in the flash of these three mixtures (maximum error of 1.8%). The heuristic approach reached acceptable results if about 80% of the original composition is accounted for. Improvements in computational speed with MSM and the heuristic method are modest, in the range of 15%, mainly due to the fact that the property calculation method in the example was simple. On the other hand, the flash calculation using QMoM with 6 pseudo-components was 6.4 times faster than the computation using the original mixture characterization.

**Simulating mass transfer in the flow of a continuous mixture using adaptive characterization**

[conference paper: 4th Latin American CFD Workshop Applied to the Oil and Gas Industry, Rio de Janeiro, Brazil, 06/2010] [ download ]

**ABSTRACT SHOW / HIDE**

The chemical industry needs solutions for the CFD simulation of separation and mixture processes that deal with continuous mixtures. A continuous mixture is a kind of complex mixture that has so many components that it is impractical to determine each and every one of them [1]. On the other hand, these components have similar properties that can usually be considered to be a function of only one characterization variable, such as the molar mass or normal boiling temperature. Therefore, these mixtures are usually considered to be one component which has a continuous distribution of its properties in terms of the characterization variable, also called the distribution variable. This kind of characterization implies that usual mass conservation equations and phase equilibrium relations become differential equations for the distribution function of composition in terms of density or mass fraction. In order to solve these equations using the conventional discrete component approach, the continuous has to be described by pseudo-components, which is called its characterization.

There are several methods for mixture characterization. In the past, ad-hoc rules were tested and employed [2]. Then, the method that uses the M abscissas of Gauss-Legendre or Gauss-Laguerre quadratures as the values of the characterization variable for the pseudo-components was developed [3]. Later, Lage [4] developed an adaptive characterization method that uses the M abscissas of the Gauss-Christoffel quadrature in which the weight function is the distribution function itself. This method implies in the conservation of the 2M moments of the distribution function and it was shown to be much more accurate than the other quadrature methods [4]. As it can be applied to either continuous or discrete mixtures, it can also be considered a method of order reduction, as it can characterized a mixture using a much smaller number of components.

The present work extends Lage [4] method to field problems by considering the problem of gas flow in a 2D channel with a boundary condition of vaporization/condensation of a hydrocarbon continuous mixture under the hypothesis of low mass transfer rates. This method can be considered the extension of the DQMOM method for continuous mixture characterization and was implemented in OpenFOAM. The simulation results using the new method were compared to those obtained by applying the usual multicomponent approach to the original mixture. As examples of the results, Figure 1 shows the mixture density in the channel flow calculated by using the original 57 pseudo-components of a given mixture and the DQMOM method with 6 pseudo-components. Figure 2 shows how good are the approximations of the bubble point at 422.95K for the mixture along the channel centerline when the DQMOM is used with several values of pseudo-components.