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## Contents

1 Motivation

2 Multi-component gas flow

2.1 Introduction

2.2 Approaches to Multicomponent modelling

2.2.1 Volume Of Fluid approach

2.2.2 Mixture approach

2.2.3 Eulerian approach

2.2.4 Model comparisons

2.3 Closure

3 Mathematical Modelling

3.1 Drag modelling

3.1.1 Effects of Diffusion

3.1.2 Maxwell-Stefan relations for modelling drag

3.2 Porosity Modelling

3.3 Energy transport

3.4 Closure

4 Numerical Simulation

4.1 Open Source CFD: OpenFOAM

4.2 Solver: multiphaseEulerFoam

4.2.1 Solution Method: Phase fraction

4.2.2 Solution Method: Pressure-Velocity coupling

4.2.3 Semi-implicit treatment of Drag term

4.3 Modified solver: multicomponentPorousFoam

4.3.1 Source term: Maxwell-Stefan drag

4.3.2 Source term: Porous drag

4.3.3 Energy transport

4.3.4 Solution algorithm

4.4 Closure

5 Validation

5.1 Case1: Loschmidt tube

5.1.1 Case Setup

5.1.2 Results

5.2 Case2: Flow through a channel

5.2.1 Case Setup

5.2.2 Validation of temperature transport

5.2.3 Validation of porosity effects

5.3 Closure

6 Conclusions

A Loschmidt tube

A.1 Scilab Program to calculate 1-D analytical solution

A.2 Animation of initial diffusion of gases

B Scilab program to calculate Nusselt number

Bibliography

## List of Figures

2.1 Examples of multiphase flows

2.2 Multicomponent flow in porous media

2.3 Volume of Fluid approach

2.4 Solution procedure of mixture modelling approach

3.1 Fundamental types of pore diffusion

3.2 Momentum interaction in multicomponent systems

3.3 Collision Scenario of two molecules

3.4 Interaction of molecules in a control volume

3.5 Movement of molecules of a single phase in porous region

4.1 Flow chart of multiphaseEulerFoam solver

4.2 Schematic to demonstrate MULES convective-only transport solution

4.3 Procedure to solve phase volume fraction

4.4 Solution procedure of phase continuity equation

5.1 Experimetal set up of loschmidt tube

5.2 Simulation model of loschmidt tube

5.3 Phase fraction ofArgon in left tube: Analytical vs. Calculated

5.4 Phase fraction of CH4 in left tube : Analytical vs. Calculated

5.5 Phase fraction of H2 in left tube: Analytical vs. Calculated

5.6 Calculation of Nusselt number

5.7 Case set up for a channel flow

5.8 Velocity profile of a non porous zone 2

5.9 Temperature profile of a non porous zone 2

5.10 Temperature and velocity variation (non porous zone 2) at x= 0.1m

5.11 Velocity profile for a non porous zone 2

5.12 Temperature profile for a porous zone 2

5.13 Temperature and velocity variation (porous zone 2) at x= 0.1m

A.1 Initial diffusion of Argon

A.2 Initial diffusion of Methane

A.3 Initial diffusion of Hydrogen

## List of Tables

2.1 Comparison of mixture model and Eulerian model

4.1 Phase properties and functions to access them

4.2 Member function in multiphaseSystem

4.3 Computing inter-component drag force by maxwellStefan() function

4.4 Computing porous drag multiplier by porousSource.H

4.5 Variables and functions required for energy transport

4.6 Solution procedure for temperature of the mixture

4.7 multicomponentPorousFoam algorithm

5.1 Initial composition

## Nomenclature

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## 1. Motivation

Impact of soot is quite adverse on the environment. There is a need for the reduction of emission which contributes to soot . A recent study by the American Geophysical Union in their article “Bounding the role of black Carbon in the climate system: A scientific assessment” [4] gives an objective point of view and sheds light on the importance of soot reduction. It indicates that the effect of soot on the change in climate is twice as much as the previous estimates. It further states that fully mitigating soot can save 1 — 2 million lives by avoiding diseases caused from soot pollution. Complete statistics and details about the hazardous effects of soot can be viewed in the cited article.

The use of Diesel Particulate Filters (DPF) to reduce emission is widespread in automotive and related industries. A first step towards soot reduction is to thorougly understand various phenomena occuring in a DPF. Diesel filter involves a multicomponent flow consisting of Particulate matter, Carbon-Di-Oxide, Oxygen, Nitrogen etc. The mixture encounters the filter which is a porous region. The presence of porous region further influences the motion of the components in the filter. Hence it is important to model the presence of the porous media with the mixture and capture the interaction effects among individual components and the components with the porous matrix.

Approaches used to calculate gas transport is generally based upon parameterized and simplified models like the dusty gas model, mean transport model or binary friction model which predominantly depend on emperical parameters to model the effects of porosity. Another approach is to resolve the porous media and to consider the porosity effects. Though this is quite accurate in predicting actual flow phenomena, it is not generic in nature and is a trade off between having a generic model to represent porous media and computation time. Hence a need for an independant model is imminent which does not base itself upon emperical parameters to a large extent and is generic without compromising the accuracy of the solution.

Simulating the behaviour of a multicomponent gas mixture in porous domains which is an abstract form of flow phenomena occuring in DPF, using Computational Fluid Dynamics (CFD) is undertaken in this thesis. The momentum exchange between individual components is modelled. Momentum source due to the presence of porous structure in the domain is also accounted for. A semi-heuristic term for porous drag force which considers both Darcy effect and Knudsen diffusion in micro porous domains is used.

Volume-averaging method of Slattery is used to model porous media. The article by Piesche and Goll [27] gives a direction to the modelling of the required terms. In this work, the development of the terms are explained and modelled. The modelled terms are then implemented into the open source field manipulation software OpenFOAM and a solver multicomponentPorousFoam is developed.

Finally, the implementation of modelled terms and capability of the solver to simulate the flow scenario is substantiated.

## 2. Multi-component gas flow

### 2.1 Introduction

Transport of more than one fluid is imperative in fluid mechanics given the burgeoning applications involving many fluids interacting in a domain. Multiphase flows do not restrict the presence of different phases only in terms of solids, liquids and gases but is percieved in a broader sense. Eventually all the different phases are expressed in numbers in terms of density, viscosity, molecular weight, molecular diameter and other properties of individual phases. Hence, it is quite obvious that a group of particles having the same properties is represented as one phase rather than abiding by the conventional definition. This means that this group has similar dynamic responses. Some examples of multiphase flows are smoke from a chimney dispersing into the atmosphere, bubble flow due to aeration and blood flow. Further examples are shown in the schematic 2.1.

Figure 2.1 gives an inital perspective of multiphase flows. In bubble flow, the presence of bubbles induce drag into the flow domain. In a slurry, the behavior of the dispersed phase is influenced by the continuous medium. Further, in this kind of flow, if the dispersed particles can be divided into two groups which largely differ in their respective diameters, then the number of phases must be treated as three instead of two since the change in diameter means a change in dynamic response by a group of identical particles. The spray dryer involves a fluid stream comprising of a solute and a solvent which is sent into a chamber containing hot vapour. This results in the vaporization of the fluid resulting in the separation of solute particles which is collected at the bottom of the chamber. These examples give an insight and emphasizes the importance of understanding multiphase flow phenomenon.

Multiphase flows and multicomponent flows are two terminologies that varies in a trifle manner but demands to be distinguished. Multiphase flows involve fluids mixed above molecular level whereas multicomponent flows comprises fluids mixed at the molecular level. Multiphase flows require interface capturing. Multicomponent flows do not require interface capturing since various components are mixed at the molecular level. Exhaust gas treatment which involves the presence of diesel filter defined as a porous matrix defines the scope of this work.

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Figure 2.1: Examples of multiphase flows

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Figure 2.2: Multicomponent flow in porous media

The figure 2.2 abstracts the whole process that occur in the treatment of exhaust gases. A multitude of gases and particulate matter mixed at the molecular level enters a channel. This multicomponent mixture encounters the porous media. The characterisitc property of a porous media is a randomised distribution of pores along space and varying pore diameters. At this juncture, it is important to note all the components flow through as one mixture and there is only a small flux due to molecular diffusion that drifts from the main mixture. “ This is because of Knudsen numbers(kn) being greater than 1 x 10-3 because of elevated temperatures, low operating pressures and pore sizes on the microscale”[27]. The methods used to model multiphase or multicomponent flow is addressed in the following section.

### 2.2 Approaches to Multicomponent modelling

The two distinct treatment of multiphase flows are Lagrangian and Euler-Euler. The former is a particle tracking technique used when there is a dispersed phase interacting with a continuous media. It can largely be remarked that this method works well when there is a one way coupling between the interacting phases. Consider n particles with negligible mass being carried on by a fluid stream. This results in a one way coupling since the flow of fluid influences the particles but not the vice versa. We can go ahead and state that there is a momentum lost from the fluid through drag to the particles but not the other way around. Thus the whole process is controlled by the dominant continuous phase. But in flow domains where no phase is dominant, a need arises to formulate a method where all the phases involved are equally weighed.

The Euler-Euler method models multiple phases with this approach. As formerly mentioned, the interest of this work deals with multicomponent modelling, we can directly decide to model it with the Euler-Euler method since there is no phase which is dominant when the components are mixed at the molecular level. The coupling between the phases is achieved by volume fraction. The Volume fraction of phases varies across space and over time. Simultaneous occupation of a phase by two fluids is not possible. Hence closure is obtained by a condition that the sum of volume fractions of all the phases to be unity.

#### 2.2.1 Volume Of Fluid approach

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Figure 2.3: Volume of Fluid approach

Volume of Fluid (VOF) model is a surface-tracking technique applied to a fixed Eulerian mesh. “It is designed for two or more immiscible fluids where the position of the interface between the fluids is of interest”[1]. Surface tracking is done by evaluating the values at individual contol volumes. Consider two phases interacting as shown in the figure 2.3. A cell is considered to be completely filled by air when the phase fraction is one and completely empty when the phase fraction is zero.

When the volume fraction is between unity and zero in a particular cell, then it is marked with the presence of interface. This demands an equation in order to solve the phase fraction of individual phases which is given by:

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The above equation 2.1 contains the substantive derivative of mass of individual phases. mpq is the mass transfer from phase p to phase q and mqp is the mass transfer from phase q to phase p. Saq is source terms for mass of individual phases. Only volume averaged field variables are solved. Hence the individual phases share this common field variables proportionate to their volume fraction in all the cells. The volume averaged momentum equation is:

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The energy equation shared by the phases is:

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Energy and temperature are treated as mass averaged variables in the above equation 2.3. Sh is the heat source term. Energy shared by individual phases is proportionate to its mass fraction in the whole mixture. Surface tension can also be modelled by adding a source term to the momentum equation. Thus the VOF approach models the required effects in case of immiscible fluids where the need of interface tracking exists.

#### 2.2.2 Mixture approach

The mixture modelling is one of the methods to model multiphase systems. In this approach, a single continuity equation and momentum equation is solved for the mixture. The mixture is assumed to be the averaged field properties of the field variables. Though the averaging is quite similar to the VOF approach, it has to be noticed that there was no equation solved for the conservation of mass in VOF approach since the interest was mainly in the interface.

Contrastingly, in the mixture approach, the continuity equation for the mixture is solved. Hence it can model phases moving with different velocities. It is assumed that the individual phases with different velocities diffuse away from the whole mixture with a drift flux. This drift is represented as the relative velocity between the mixture and the individual phases. The generic equations for the mixture approach ([1],[18]) is explained subsequently .

The continuity and momentum equations for a single phase q can be written as:

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where rq represents the rate of mass generation of phase q at the interface. Momentum exchange between the interface is represented by the source term Mq. Tq is the average viscous stress tensor and TTq is the turbulent stress tensor. To obtain the equations for the mixture as a whole, all the equations of individual phases are added. The continuity equation for the mixture is:

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The mass in the whole system is conserved. Hence, the sum of rq over individual phases must be zero;

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The continuity equation thus becomes:

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where the mixture density is:

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and mixture velocity is:

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The drift flux needs to have a reference velocity which is represented by the mixture velocity at the mass centre. The momentum equation for the mixture is obtained by adding the momentum equation of all phases:

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The second term on the left hand side equation 2.11 can be rewritten in terms of mixture density and mixture velocity:

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The momentum equation in terms of mixture variables is then:

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where Tm, rTm represent the average viscous and turbulent stress because of the slip of a phase. The pressure of the mixture is:

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But more often the partial pressure and the mixture pressure is taken as the same barring a few cases. The term Mm is the contribution of the sum of surface tension from individual phases.

The formulation for relative(slip) velocity is needed since the continuity and momentum equations are expressed in terms of it following which the continuity equation for phase fraction can be solved. In this approach, an algebraic formulation for slip velocity is constructed. Density differences of individual phases has a major influence on particles with different physical properties resulting in a varied dynamic response. The relative velocity between two phases q and j is further coupled to the drift velocity by:

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The algebraic formulation for the relative velocity(slip) between two phases generally considers that the mass of a particle in the mixture is very small and that the particle reaches a terminal velocity over a short distance. In short, a local equilibrium is assumed. The additional force due to the relative velocity of the particle with respect to a fluid is represented by the drag force on the particle by the fluid. Thus, the relative velocities of a particular phase is obtained which can be used in the phase continuity and momentum equations subsequently.

The expression of Manninen et al.[18] is generally used:

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where a is the secondary phase particle acceleration:

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and Tq is the particle relaxation time

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where dq is the particle diameter of phase q.

The continuity equation for a phase which is incompressible and for a process where no phase changes occur is:

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An overview of the complete solution procedure is shown in figure 2.4.

Thus, the mixture approach sums up the continuity and momentum equations of individual phases and circumvents solving for the velocity of phases by algebraic formulation for the relative velocity between phases and coupling drift velocity to relative velocity. It assumes local equilibrium over short spatial scales.

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Figure 2.4: Solution procedure of mixture modelling approach

#### 2.2.3 Eulerian approach

Eulerian approach relies on solving the momentum and continuity equations for all the phases. “In the Euler-Euler approach, the different phases are treated mathematically as inter-penetrating continua”[1]. The coupling between the phases is established by an interaction term in the momentum equation of each phases which contains the relative velocity of a phase with respect to all the other phases. Unlike the former approaches, a number of phases can be modelled which can move with a particular velocity. The concept of volume fraction is considered and a particular space cannot be occupied by two phases. The volume occupied by a phase is proportionate to its phase fraction similar to the mixture theory approach.

Though continuity and momentum equations are solved for individual phases, the effect of all these phases has to be considered in the multiphase framework. Hence, characteristic properties of viscosity and density of a particular phase have to be modified to fit the multiphase framework. It is straightforward to predict that the effect of physical property from a phase in the multiphase domain is proportional to its phase fraction. Hence the effective density used in the equations are:

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and viscosity

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If all terms related to mass transfer across interfaces are set to zero, the continuity equation for a phase is

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The momentum balance equation is:

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The interaction term for Rpq is modelled as an interphase drag term which is a function of relative velocity of the phases:

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Kpq here is the interphase momentum exchange coefficient. Lift and virtual mass forces are not in the scope of this work and will not be discussed exhaustively. Various models of drag, heat transfer or molecular diffusion between the phases can be modelled using the generic interphase momentum transfer term Rpq. The phase fraction is solved from the continuity equation and the effective densities of phases are updated.

Eulerian modelling is used in complicated flow phenomenon where the formerly stated approaches do not work. The complexity involved is the courtesy of allowing the phases to have its individual velocities and maintaining the boundedness of phase fractions. Additionally, it is computationally expensive since n continuity and momentum equations are solved in every iteration and demands larger memory requirements. The equations in this section are referred from [27] and [1].

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Table 2.1: Comparison of mixture model and Eulerian model

#### 2.2.4 Model comparisons

A suitable multiphase modelling approach has to be selected which suffices and represents all the processes that occur in case of flow through the porous media. It can directly be analysed that VOF approach is out of the purview of our work since interface tracking in not an important requirement. Now we are left with two approaches which needs diligent comparison. Table 2.1 shows the capabilities of the two modelling approaches.

Though the mixture approach seems an attractive option for our flow scenario, it assumes local equilibrium between the phases and the relaxation time of a particle must be lesser than the timescale of the problem. This means that a strong coupling between the phases must be present. A sudden acceleration of a few phases may occur as the flow encounters porous media which might weaken the coupling.

Another drawback of using mixture approach for our flow scenario is the requirement of the slip boundary condition for velocity at the wall. “ The usual boundary condition, that the velocity of a gas bounded by a wall should be equal to that of the wall at every point of its surface, is not exactly fulfilled when there exists a velocity gradient perpendicular to the wall, a temperature gradient parallel to the wall or-in case of a mixture- a concentration gradient parallel to the wall”[17].

Both temperature and concentration gradients exist parallel to the wall as a result of porous media in the flow domain. The porous media has its characteristic thermal conductivity which is different than the multiphase mixture. Hence there is a temperature gradient over the porous domain parallel to the wall. Existence of concentration gradient parallel to the wall can be attributed to the permeability of the porous media. Because of the permeability and a knudsen number greater than 0.01 in the porous domain amounts to the presence of varied mass across its sides. There is relatively more mass towards the inlet than the outlet.

The aforementioned reasons establishes the fact that the mixture theory though with its lower computation time fails to fulfill the present requirements. Though with its considerably higher computation time, we proceed with the Eulerian multiphase model as it is capable of modelling the desired effects.

### 2.3 Closure

The three main approaches of multiphase modelling; VOF, mixture theory and Eule- rian were introduced in this chapter. A broader perspective of considering multiphase media and the slight distinction between multiphase and multicomponent flows were discussed. Though this work concerns multicomponent flows, the term multiphase has been used to explain the approaches in order to retain the semantics of prevalent literatures on the same. Further, the characteristics of all the approaches were evaluated and compared. The Eulerian multiphase modelling was selected for modelling our flow problem with reasons stated.

## 3. Mathematical Modelling

Mathematical modelling is imperative in order to express a physical system in a materialized form. A mathematical representation of systems or processes generates a possibility to study and analyse the influence of different components in the system. It starts with transforming the various processes and interactions into governing equations. The governing equations have to be modelled in such a way that, it must posses the capability to express all the important phenomena that may occur. The equations must also convene the application of boundary conditions to realistically simulate the physics of the system.

In fluid systems, the continuity or the mass conservation, momentum conservation and the energy conservation are the three fundamental forms of equations that require mathematical modelling. Differential equations are used quite often in order to express such processes where there is a continuous variation of a quantity over space and time. A solution of the differential equation in terms of time and domain thus results in a possibility of monitoring the behaviour of a quantity over the whole domain and its variation with time. Most of the mathematical systems results in non-linear equations. The non-linearities pose some difficulties in finding an analytical solution to the modelling systems.

The Navier-Stokes equation models the mass and momentum conservation of a fluid system and will be used in this work. The momentum interaction and porous drag terms are developed and are considered as source terms in the momentum equation. Energy equation for the pseudo homogeneous medium is modelled where the mixture and the porous matrix is assumed to have the same temperature and a local thermodynamic equilibrium exists between the individual phases. The subsequent sections deals with the development of mathematical models to express the flow phenomena of our system.

### 3.1 Drag modelling

#### 3.1.1 Effects of Diffusion

Diffusion is a prime mechanism responsible for a variety of processes involving mass transfer. This broad spectrum involves bulk gas or liquid diffusion, knudsen diffusion

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Figure 3.1: Fundamental types of pore diffusion

in pores and molecular diffusion inside micro pores. The driving force for diffusion is ascribed to the concentration gradients that exist in space. The tendency is always to flow from a region of higher concentration to a region of lower concentration. The subject of interest here is porous diffusion. To distinguish between different kind of porous diffusion mechanisms, we need to first distinguish between the different kinds of pores which are macro pores(size > 50nm), micro pores(size < 2nm) and meso pores(size between 2nm-50nm). The three types of diffusion mechanisms are shown in the figure 3.1 referred from [24].

In high pressure systems the number of molecules are so high that collision occurs more frequently amongst the molecules than with the wall. Large pore sizes also accommodate for such free molecular collisions. This kind of diffusion is termed as Bulk diffusion. Knudsen diffusion arises when the mean free path of a molecule is lesser than the charateristic length of the domain. Hence, the molecule travels lesser than the diameter in case of a circular tube before collision which results in a high probability of molecule-wall collisions. Surface diffusion occurs primarily in micro pores when the molecular species are adsorbed on the surface of porous media. “Bulk and Knudsen diffusion occur together and it is prudent to take both mechanisms into account rather than assume that one or other mechanism is ‘controlling’ ”[24].

The physical meaning of interaction between molecules of different components in a multicomponent system can be related to drag effects. The interpretation is that, molecules travelling with different velocities in a system exchange momentum among them due to relative velocity. It is visualised in the figure 3.2. The inter component

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Figure 3.2: Momentum interaction in multicomponent systems

momentum exchange is directly proportional to the relative velocity. A suitable model which considers the stated effects of diffusion and replicates the appropriate interaction effects between components must be selected and integrated to the multicomponent framework. Diffusion of ideal gas mixtures is considered in this work.

#### 3.1.2 Maxwell-Stefan relations for modelling drag

Consider a collision scenario of two molecules with mass m1 and m2 travelling with velocities u1 and u2 respectively. The assumption of collision is completely elastic. There is no loss of energy during the process. The other important assumption is that the molecules are spherical in shape. An example of such a process is collision between two billiard balls.

The two molecules carry a momentum of m1v1 and m2v2. If u1' and v2' are velocities after the collision and the path traced by the particles before and after collision is assumed to be the same, then according to the law of conservation of momentum:

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- Quote paper
- Chethan Mohan Kumar (Author), 2013, CFD simulation of multicomponent gas flow through porous media, Munich, GRIN Verlag, https://www.grin.com/document/211786

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