Chemical Reaction Engineering Model Library Manual

8/11/2019 Chemical Reaction Engineering Model Library Manual

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Part number: CM021606

C h e m i c a l R e a c t i o n E n g i n e e r i n g M o d e l L i b r a r y M a n u a l 19982013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474;7,623,991; 8,219,373; and 8,457,932. Patents pending.This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement ( www.comsol.com/sla) and may be used or copied only under the terms of the licenseagreement.COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are eitherregistered trademarks or trademarks of COMSOL AB. All other trademarks are the property of theirrespective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by,sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm. Version: November 2013 COMSOL 4.4
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1 | D E T E R M I N I N G A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N

Determining Arrhenius Parametersus ing Parameter Es t imat ion

Introduction

This model shows how to use the Parameter Estimation feature in the ReactionEngineering physics interface to find the Arrhenius parameters of a first order reaction.

Note: This model requires the Optimization Module.

Inspiration for this example is taken fromRef. 1.

Model Definition

Benzene diazonium chloride in the gas phase decomposes to benzene chloride andnitrogen according to:

(1)

The reaction is first order with the rate:

(2)

where the temperature dependent rate constant given by:

(3)

Above, A is the frequency factor (1/s) and E is the activation energy (J/mol).

In order to evaluate the Arrhenius parameters, A and E , a set of experiments wereconducted using a perfectly mixed isothermal batch system. The concentration ofbenzene diazonium chloride was monitored as function of time for the temperatures;T = 313 K, 319 K, 323 K, 328 K, and 333 K.

N

N2+N

ClCl

k

r kc PhN2Cl=

k A E

R g T

----------- exp=


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The five experimental data sets are available ascomma separated value files (csv -files)together with the model file for this example.

Results and Discussion

Parameter estimation calculations give the values A = 1.11016 (1/s) and E = 116 (kJ/mol) for the frequency factor and activation energy, respectively.

Plots of the model results and the associated experimental data points are shown below.

Figure 1: Model results and experimental data for PhN 2 Cl concentration as function oftime.


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Notes About the COMSOL Implementation

The parameter estimation solver will be more efficient in finding an optimal parameterset if the model experiences similar sensitivity with respect to changes in parameter values. In this problem we therefore define a parameter A ex that is to be estimatedtogether with the activation energy E , such that the rate constant is written as

(4)

The frequency factor A is then evaluated as: (5)

The data indicates that the rate constant is of the order ~110-3 (1/s) at T = 323 K.Taking this into account and using an initial guess for the activation energy of 150 kJ/mol, an initial guess is set for A ex = 49.

Reference 1. H.S. Fogler,Elements of Chemical Reaction Engineering 4th ed. , p. 95, PrenticeHall, 2005.

Model Library path: Chemical_Reaction_Engineering_Module/Optimization/activation_energy

Modeling Instructions

From theFile menu, chooseNew.

N E W

1 In the New window, click theModel Wizard button.

M O D E L W I Z A R D

1 In the Model Wizard window, click the0D button.2 In the Select physics tree, selectChemical Species Transport>Reaction Engineering (re) .3 Click theAdd button.4 Click theStudy button.

k A ex E

R g T -----------

expexp=

A A ex ln=


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5 In the tree, selectPreset Studies>Time Dependent .6 Click theDone button.

G L O B A L D E F I N I T I O N S

Parameters1 On the Home toolbar, clickParameters .2 In the Parameters settings window, locate theParameters section.3 In the table, enter the following settings:

R E A C T I O N E N G I N E E R I N G

Reaction 11 On the Physics toolbar, clickGlobal and chooseReaction .

2 In the Reaction settings window, locate theReaction Formula section.3 In the Formula edit field, typePhN2Cl=>PhCl+N2 .

Species: PhN2Cl 1 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

PhN2Cl.2 In the Species settings window, locate theGeneral Expressions section.

3 In the c0 edit field, type1000 .Now, add a Parameter Estimation feature, define parameters and set initial values.

Parameter Estimation 11 On the Physics toolbar, clickGlobal and chooseParameter Estimation .2 In the Parameter Estimation settings window, locate theControl Variables section.3 In the Control variables table, enter the following settings:

Create separate Experiment features for the data collected at different temperatures(T_iso).

Name Expression Value Description

T_iso 313[K] 313.00 K temerature

Variable Initial value Lower bound Upper bound

Aex 49

E 150e3


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Experiment 11 Right-clickComponent 1>Reaction Engineering>Parameter Estimation 1 and choose

Experiment .2 In the Experiment settings window, locate theExperimental Data section.3 Click theBrowse button.4 Browse to the models Model Library folder and double-click the file

activation_energy_experiment313K.csv .5 Click theImport button.

6 In the table, enter the following settings:

7 Locate theExperimental Parameters section. ClickAdd.8 In the table, enter the following settings:

Experiment 21 Right-clickParameter Estimation 1 and chooseExperiment .2 In the Experiment settings window, locate theExperimental Data section.

3 Click theBrowse button.4 Browse to the models Model Library folder and double-click the file

activation_energy_experiment319K.csv .5 Click theImport button.6 In the table, enter the following settings:

7 Locate theExperimental Parameters section. ClickAdd.

Data column Use Model variables

time tconc PhN2Cl (313K) c_PhN2Cl

Parameter names Parameter expressions

T_iso 313




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Experiment 31 Right-clickParameter Estimation 1 and chooseExperiment .2 In the Experiment settings window, locate theExperimental Data section.3 Click theBrowse button.

4 Browse to the models Model Library folder and double-click the fileactivation_energy_experiment323K.csv .

5 Click theImport button.6 In the table, enter the following settings:


Experiment 41 Right-clickParameter Estimation 1 and chooseExperiment .2 In the Experiment settings window, locate theExperimental Data section.3 Click theBrowse button.4 Browse to the models Model Library folder and double-click the file

activation_energy_experiment328K.csv .

5 Click theImport button.6 In the table, enter the following settings:

Parameter names Parameter expressionsT_iso 319


time t

conc PhN2Cl (323K) c_PhN2Cl


T_iso 323




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Experiment 51 Right-clickParameter Estimation 1 and chooseExperiment .2 In the Experiment settings window, locate theExperimental Data section.

3 Click theBrowse button.4 Browse to the models Model Library folder and double-click the file

activation_energy_experiment333K.csv .5 Click theImport button.6 In the table, enter the following settings:


Next, return to the Reaction Engineering feature and introduce the parameters tothe reaction model.

9 In the Model Builder window, clickReaction Engineering .10 In the Reaction Engineering settings window, locate theGeneral section.11 In the T edit field, typeT_iso .

1: PhN2Cl=>PhCl+N21 In the Model Builder window, underComponent 1>Reaction Engineering click1:

PhN2Cl=>PhCl+N2 .2 In the Reaction settings window, locate theRate Constants section.3 Select theUse Arrhenius expressions check box.4 In the A f edit field, typeexp(Aex) .


T_iso 328




T_iso 333


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5 In the E f edit field, typeE.

S T U D Y 1

Step 1: Time Dependent1 In the Model Builder window, expand theStudy 1 node, then clickStep 1: Time

Dependent .2 In the Time Dependent settings window, locate theStudy Settings section.3 In the Times edit field, typerange(0,50,5000) .

Optimization1 On the Study toolbar, clickOptimization .2 In the Optimization settings window, locate theOptimization Solver section.3 From theMethod list, chooseLevenberg-Marquardt .

Solver 11 On the Study toolbar, clickShow Default Solver .

2 In the Model Builder window, expand theStudy 1>Solver Configurations>Solver1>Optimization Solver 1 node, then clickTime-Dependent Solver 1 .

3 In the Time-Dependent Solver settings window, click to expand theOutput section.4 From theTimes to store list, chooseSpecified values .5 Click to expand theAbsolute tolerance section. Locate theAbsolute Tolerance section.

In the Tolerance edit field, type1e-5 .6 From theMethod list, chooseUnscaled .7 In the Tolerance edit field, type1e-5 .8 On the Home toolbar, clickCompute .

R E S U LT S

Experiment 1 Group1 In the Model Builder window, expand theResults>Experiment 1 Group node, then

clickGlobal 1 .2 In the Global settings window, locate theData section.3 From theParameter selection (T_iso) list, chooseFrom list .4 In the Parameter values (T_iso) list, choose313 , 319 , 323 , 328 , and333 .

This restricts the plot to the results associated with T_iso = 313 K. Alternatively, youcould have chosen From list and then selected the proper entries.


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5 Locate they-axis data section. ClickConcentration (comp1.re.c_PhN2Cl) in theupper-right corner of the section. Locate thex-axis data section. ClickTime (t) in

the upper-right corner of the section. On the1D plot group toolbar, clickPlot .Experiment 2 Group1 In the Model Builder window, expand theResults>Experiment 2 Group node, then

clickGlobal 1 .2 In the Global settings window, locate theData section.3 From theParameter selection (T_iso) list, chooseFrom list .

4 In the Parameter values (T_iso) list, select319 .5 Locate they-axis data section. ClickConcentration (comp1.re.c_PhN2Cl) in the

upper-right corner of the section. Locate thex-axis data section. ClickTime (t) inthe upper-right corner of the section. On the1D plot group toolbar, clickPlot .


clickGlobal 1 .2 In the Global settings window, locate theData section.3 From theParameter selection (T_iso) list, chooseFrom list .4 In the Parameter values (T_iso) list, select323 .5 Locate they-axis data section. ClickConcentration (comp1.re.c_PhN2Cl) in the



clickGlobal 1 .2 In the Global settings window, locate theData section.3 From theParameter selection (T_iso) list, chooseFrom list .4 In the Parameter values (T_iso) list, select328 .

5 Locate they-axis data section. ClickConcentration (comp1.re.c_PhN2Cl) in theupper-right corner of the section. Locate thex-axis data section. ClickTime (t) inthe upper-right corner of the section. On the1D plot group toolbar, clickPlot .


clickGlobal 1 .


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2 In the Global settings window, locate theData section.3 From theParameter selection (T_iso) list, chooseFrom list .4 In the Parameter values (T_iso) list, select333 .5 Locate they-axis data section. ClickConcentration (comp1.re.c_PhN2Cl) in the


In the last step, output the estimated parameter to a table.

Derived Values

On the Results toolbar, clickEvaluate All .Taking scaling into account E is found to be 1.16e5 and Aex is evaluated to 36.94.


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1 | B O A T R E A C T O R F O R L O W P R E S S U R E C H E M I C A L VA P O R D E P O S I T I O N

Boat Reactor for Low PressureChemical Vapor Deposi t ion

Introduction

Chemical vapor deposition (CVD) is an important step in the process ofmanufacturing microchips. A common application is the deposition of silicon on

wafers at low pressure. Low-pressure reactors are used to get a high diffusivity of thegaseous species, which results in a uniform deposition thickness, because the processbecomes limited by the deposition kinetics (Ref. 1 andRef. 2).

Figure 1: Schematic of a boat reactor and the principle of the deposition process.

This example models the momentum and mass transport equations coupled to thereaction kinetics for the deposition process. It treats a low-pressure boat reactor, wherethe goal of the simulation is to describe the rate of deposition as a function of the fluidmechanics and kinetics in the system.

The gas, in this case silane (SiH4), enters the reactor from the left and reacts on the

wafers to form hydrogen and silicon. The remaining mixture leaves the reactorthrough the outlet on the right. The deposition of silicon on the surface of the wafersdepends on the local concentration of silane, which is determined by the operatingconditions for the reactor.

More details about this example can be found inElements of Chemical ReactionEngineering by H. Scott Fogler (Ref. 1).


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Model Definition

First assume that the density of the gas is constant throughout the reactor. This impliesthat the reacting gas is either diluted in a carrier gas or that the conversion in thereactor is small. Moreover, only account for the mass transport of the reactant gas, inthis case silane, and assume constant temperature in the reactor.

In the wafer bundle convection transport can be neglected, so that the reacting gas canonly be transported through diffusion. To save time and computational memory, alsosimplify the geometrical description of the wafer bundle by modeling it as an

anisotropic medium. To this end, because silane cannot diffuse through the physical wafers, assume that the axial diffusivity in the wafer bundle is zero. Furthermore,correct the diffusivity in the radial direction according to the degree of packing in thebundle. Finally, neglect the influence of the support boat on the transport process thatholds the wafer bundle in place. The structure of the boat reactor means that the 3Dgeometry can be reduced to a 2D axisymmetric model. The modeling domain is shownin Figure 2.

Figure 2: The model geometry showing the domain and boundary labels.

The chemical reaction accounted for in this example is:

Axial symmetry

Wafer bundle

domain

Free flow domain

Walls

Inlet

Outlet


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(1)

The rate of this reaction depends on the partial pressure of silane and the temperaturein the reactor.

The assumptions mentioned above in combination with the chemical reaction for thedeposition process make it possible to define an equation system. The momentumequations and the continuity equations for laminar flow in cylindrical coordinates read

(2)

Here (SI unit: kg/(ms)) denotes the viscosity; (SI unit: kg/m3) is the density ofthe gas;u andv (SI unit: m/s) refer to the velocity vectorsr- and z-components,respectively; and p (SI unit: Pa) is the pressure.

The mass transport in the free-fluid domain is given by the following equation,expressed in cylindrical coordinates:

(3)

Here D denotes the diffusivity (SI unit: m2/s) and c is the concentration of silane(SI unit: mol/m3). You obtain the corresponding mass transport equation for the

wafer bundle domain by neglecting transport by convection and adding a reaction-rateterm for the dissociation of silane:

(4)

Because you neglect diffusion in the axial direction, the effective diffusivity tensor, D eff , only has anrr -component. In the equation above,k (SI unit: m/s) denotes therate constant for the reaction, and S a (SI unit: m2/m 3) refers to the specific surfacearea.

You solve the system of equations defined above by using the proper boundaryconditions. For laminar flow, no-slip conditions apply at the reactor-wall surface andbetween the free channel and the wafer bundle:

SiH4 g Si s 2H2 g +

rr u

r

------

z

r u

z

------

r u u

r

------ v u

z

------+

r p

r

------+ + + 0= in ff

rr v

r------

z

r v z

------ r u v

r------ v v

z------+

r p z

------+ + + 0= in ff

r ur

------ v z

------+ v+ 0= in ff

r D r c

r-----

z

D r c z

----- r u c

r----- r v c

z-----+ + + 0 in ff =

r D eff rr, r

cr

----- r kS a c in wb=


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(5)

At the symmetry axis, the radial velocity component vanishes:

(6)

The last three conditions for the momentum equations and continuity equation are

(7)

For the mass transportEquation 3 andEquation 4, the boundary conditions are

(8)

where D i represents the diffusivity inff or wb depending on to which boundarysegment you apply the equation. This equation implies that there is no fluxperpendicular to these boundaries. At the inlet, the composition of the gas is known, which yields:

(9)

At the outlet, assume that the transport of species takes place mainly by convection andneglect the concentration gradients perpendicular to this boundary:

(10)

It remains to discuss the material parameters appearing inEquation 4: D eff, rr , S a , andk. First, calculate the specific surface area (that is, the area per unit volume) of the waferbundle, S a , by assuming a certain pitch between the wafers; seeFigure 3.

u v 0 0 = at wall , iw ff , , and wb ff ,

u 0 at sym=

u 0= at inv v 0= at in

p p 0= at out u u T + n 0= at out

D i rrcr

----- D i zzc

z-----

n 0= at wall , sym , and iw

c c 0 at in=

D cr

----- D c z

----- n at 0 out=


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Figure 3: Calculation of the specific surface area.

Furthermore, to estimate the effective diffusivity in the radial direction inside the waferbundle, multiply the diffusivity in the free-fluid domain by the ratio of the contact areabetween the free gas and the wafer bundle to the total lateral surface area of the wafer-bundle domain:

(11)

The rate constant,k (SI unit: m/s), is a function of the partial pressure of silane. At600 C and a total system pressure of 25 Pa,Ref. 2 provides the valuek = 8.0610 3 m/s.

A crucial characteristic of the reactors performance is the silicon deposition rate,Si

, which expresses the growth rate of the silicon layer on the wafers. The amount ofsilicon deposited on the wafers, expressed in mass per unit area per unit time, is theproduct of the rate constant,k, the silane concentration,c, and the molar mass ofsilicon, M Si (SI unit: kg/mol). Dividing the so obtained quantity by the density ofsilicon, Si (SI unit: kg/m3), gives the deposition rate:

(12)

In this model, you study the radial and axial distribution ofSi inside the wafer bundle.

D ef f rr, 1d wd cc--------

D in wb=

Sik cM Si

Si

----------------- (nm/min)=


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Figure 4 shows the concentration distribution in the boat reactor, indicating that theconversion is quite small.


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Figure 4: Concentration distribution in the reactor. Figure 5 shows the flow distributionin the reactor.

Figure 5: Flow distribution in the reactor. The surface color and the arrows both representthe velocity.


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The plots inFigure 6 display the deposition rate for the inlet velocities1 m/s (toppanel) and2 m/s (bottom panel).

Figure 6: The deposition rate in the wafer bundle for the inlet velocities 1 m/s (top) and2 m/s (bottom).


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In both cases, the highest deposition rate is obtained near the reactor inlet and closeto the free-fluid channel. The difference in deposition rate between the center and

periphery of the wafers is approximately 0.1 nm/min (or roughly 2.5%), and thatalong the length of the reactor approximately 0.5 nm/min (roughly 12.5%). Thus, asdesired, the variations in the deposition rate inside the reactor are rather small.

Moreover, comparing the plots it is evident that the deposition rate changes onlymarginally when the gas inlet velocity is doubled, showing that convection does nothave a major influence on reactors of this type.

References 1. H. Scott Fogler,Elements of Chemical Reaction Engineering , 3rd ed., PrenticeHall, 1999.

2. A.T. Voutsas and M.K. Hatalis, Structure of As-Deposited LPCVD Silicon Filmsat Low Deposition Temperatures and Pressures, J. Electrochem. Soc., vol. 139, no. 9,pp. 26592665, 1992.

Model Library path: Chemical_Reaction_Engineering_Module/Surface_Reactions_and_Deposition_Processes/boat_reactor



1 Go to theModel Wizard window.2 Click the2D axisymmetric button.3 ClickNext .4 In the Add physics tree, selectFluid Flow>Single-Phase Flow>Laminar Flow (spf) .

5 ClickAdd Selected .6 In the Add physics tree, selectChemical Species Transport>Transport of Diluted Species

(chds) .7 ClickAdd Selected .8 ClickNext .


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9 Find theStudies subsection. In the tree, selectPreset Studies for SelectedPhysics>Stationary .

10 ClickFinish.


Parameters1 On the Home toolbar,Model Builder window, clickParameters .2 In the Parameters settings window, locate theParameters section. ClickLoad from

File, and double click the fileboat_reactor_parameters_1.txt in the modelsmodel library folder.

D E F I N I T I O N S

Variables 11 In the Model Builder window, underModel 1 right-clickDefinitions and choose

Variables .2

In theVariables

settings window, locate theVariables

section.3 In the table, enter the following settings:

G E O M E T R Y 1

Create the geometry. To simplify this step, insert a prepared geometry sequence:

1 On the Geometry toolbar, click Import/Export and chooseInsert Sequence .2 Browse to the models Model Library folder and double-click the file

boat_reactor.mph . Then clickBuild all on the Geometry toolbar.


Explicit 11 In the Model Builder window, underModel 1 right-clickDefinitions and choose

Selections>Explicit .2 Right-clickExplicit 1 and chooseRename .3 Go to theRename Explicit dialog box and typewafers in theNew name edit field.4 ClickOK.5 Select Domain 2 only.

Name Expression Description

Delta_Si c*k*M_Si/rho_Si Silicon deposition rate

S l d i h CO SO l i h i 4 4


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Explicit 21 In the Model Builder window, right-clickDefinitions and chooseSelections>Explicit .

2 Right-clickExplicit 2 and chooseRename .3 Go to theRename Explicit dialog box and typereactor in theNew name edit field.4 ClickOK.5 Select Domain 1 only.


Parameters1 On the Hone toolbar,Model Builder window, clickParameters .2 In the Parameters settings window, locate theParameters section. ClickLoad from

File, and double click the fileboat_reactor_parameters_2.txt in the modelsmodel library folder.

L A M I N A R F L O W

1 In the Laminar Flow settings window, locate theDomain Selection section.2 From theSelection list, choosereactor .

Fluid Properties 11 In the Model Builder window, expand theLaminar Flow node, then clickFluid

Properties 1 .2 In the Fluid Properties settings window, locate theFluid Properties section.3 From the list, chooseUser defined . In the associated edit field, typerho .4 From the list, chooseUser defined . In the associated edit field, typeeta .

Inlet 11 In the Model Builder window, right-clickLaminar Flow and chooseInlet .2 Select Boundary 2 only.3 In the Inlet settings window, locate theVelocity section.

4 In the U 0 edit field, typev0 .Outlet 11 Right-clickLaminar Flow and chooseOutlet .2 Select Boundary 9 only.

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T R A N S P O R T O F D I L U T E D S P E C I E S

Convection and Diffusion 11 In the Model Builder window, expand theModel 1>Transport of Diluted Species node,

then clickConvection and Diffusion 1 .2 In the Convection and Diffusion settings window, locate theDiffusion section.3 In the D c edit field, typeD.4 Locate theModel Inputs section. From theu list, chooseVelocity field (spf/fp1) .

Convection and Diffusion 21 In the Model Builder window, right-clickTransport of Diluted Species and choose

Convection and Diffusion .2 In the Convection and Diffusion settings window, locate theDomain Selection section.3 From theSelection list, choosewafers .4 Locate theDiffusion section. From the symmetry property list, chooseDiagonal .5 In the D c table, enter the following settings:

Reactions 11 Right-clickTransport of Diluted Species and chooseReactions .2 In the Reactions settings window, locate theDomain Selection section.3 From theSelection list, choosewafers .4 Locate theReactions section. In the R c edit field, type-(k*S_a)*c .

Inflow 11 Right-clickTransport of Diluted Species and chooseInflow.2 Select Boundary 2 only.3 In the Inflow settings window, locate theConcentration section.

4 In the c0,c edit field, typec0 .Outflow 11 Right-clickTransport of Diluted Species and chooseOutflow.2 Select Boundary 9 only.

D_eff 0

0 0

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M E S H 1

1 In the Model Builder window, underModel 1 clickMesh 1 .

2 In the Mesh settings window, locate theMesh Settings section.3 From theElement size list, chooseExtra fine .4 Click theBuild All button.

S T U D Y 1

Step 1: Stationary 1 In the Model Builder window, underStudy 1 clickStep 1: Stationary .2 In the Stationary settings window, click to expand theStudy Extensions section.3 Select theContinuation check box.4 ClickAdd.5 In the table, enter the following settings:

6 In the Model Builder window, right-clickStudy 1 and chooseCompute .

R E S U LT S

Velocity (spf)1 In the Model Builder window, underResults clickVelocity (spf) .

2 In the 2D Plot Group settings window, locate theData section.3 From theParameter value (v0) list, choose1.4 In the Model Builder window, underResults>Velocity (spf) clickSurface 1 .5 In the Surface settings window, clickReplace Expression in the upper-right corner of

the Expression section. From the menu, chooseTransport of Diluted Species>Speciesc>Concentration (c) .

6 Click thePlot button.7 Click theZoom Extents button on the Graphics toolbar.8 ClickReplace Expression in the upper-right corner of theExpression section. From

the menu, chooseLaminar Flow>Velocity magnitude (spf.U) .9 In the Model Builder window, right-clickVelocity (spf) and chooseArrow Surface .10 In the Arrow Surface settings window, locate theArrow Positioning section.

Continuation parameter Parameter value list

v0 1 1.5 2



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p y


11 Find thez grid points subsection. In thePoints edit field, type25 .12 Click thePlot button.

13 Click theZoom Extents button on the Graphics toolbar.Data Sets1 In the Model Builder window, underResults right-clickData Sets and chooseSolution .2 In the Solution settings window, locate theSolution section.3 From theSolution list, chooseSolver 1 .4 Right-clickResults>Data Sets>Solution 2 and chooseAdd Selection .5 In the Selection settings window, locate theGeometric Entity Selection section.6 From theGeometric entity level list, chooseDomain .7 Select Domain 2 only.

2D Plot Group 61 In the Model Builder window, right-clickResults and choose2D Plot Group .2 Right-click2D Plot Group 6 and chooseSurface .3 In the Surface settings window, locate theData section.4 From theData set list, chooseSolution 2 .5 From theParameter value (v0) list, choose1.6 ClickReplace Expression in the upper-right corner of theExpression section. From

the menu, chooseDefinitions>Silicon deposition rate (Delta_Si) .7 Click thePlot button.8 Click theZoom In button on the Graphics toolbar.9 Locate theData section. From theParameter value (v0) list, choose2.10 Click thePlot button.



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p y

1 | C A R B O N D E P O S I T I O N I N H E T E R O G E N E O U S C A T A L Y S I S

Carbon Deposi t ion in HeterogeneousCata lys is

Introduction

Carbon deposition onto the surface of solid catalysts is commonly observed inhydrocarbon processing. Carbon deposits can affect both the activity of catalysts as well as the flow of gas through a catalyst bed.

This example investigates the thermal decomposition of methane into hydrogen andsolid carbon. In the first model you look at the isothermal process occurring in an idealreactor, simulated with the 0D Reaction Engineering interface. The influence ofcarbon deposition on catalyst activity is also considered. In the second model, youstudy the effect that the carbon deposits have on the fluid flow. The second simulationtakes both time and space dependencies into account.

Model Definition

C H E M I S T R Y

Methane decomposes over a Ni/Al2O3 catalyst according to the overall chemicalreaction:

(1)

Under atmospheric pressure, temperate ranging from 490 to 590C and volumefraction of hydrogen between 0 and 40%, the following reaction rate expression hasbeen reported in the literature (Ref. 1):

(2)

where

(3)

CH4

C 2 H2

+

r k P CH 4

P H 22

K P----------

1 k H P H 2+ 2

---------------------------------------=

k k 0 20.492104200 R g T

------------------- - exp=



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and

k0 inEquation 3 is2.3110 5(mol/(m3s)).The unit for pressure inEquation 2 is bar.

I D E A L R E A C T O R M O D E L

This model treats the isothermal decomposition of methane (Figure 2) in a perfectlymixed reactor with constant volume. The species mass balances are summarized by

The rate term, R i (mol/(m3s)), takes into account the reaction stoichiometry, v i, thereaction rate, r (mol/(m3s)), and the catalyst activity, a:

The mass balances of the reacting species are then

The time dependence of the catalytic activity is expressed by

(4)

where

k H 163200 R g T

-------------------- 22,426 exp=

K p 5.088 105 91200

R g T ----------------

exp =

dc idt

-------- R i=

R i v i ra=

dc CH 4dt

---------------- ra=

dcCdt---------- ra

=

dc H 2dt

------------ 2 ra=

da

dt------- k

ar 2 c

Cc

a=

k a k a 0135600 R g T

-------------------- 32.007 exp=



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Whereka0 is8.32410 6 ((m3/mol)3s) . Solving the mass balances provides theevolution of the species concentrations over time. The fact that carbon is in the solidphase is taken into account by removing its effect on gas phase physical properties. Thepressure in the reactor is a function of only the methane and hydrogen concentrations:

S PA C E - A N D T I M E - D E P E N D E N T M O D E L

The second model takes both fluid flow and the chemical reaction into account.It iscreated by Generate Space-Dependent Model submenu under Reaction Engineering

interface and solved in COMSOL Multiphysics.The flow reactor is set up in 2D, as illustrated below:

Figure 1: A flow reactor is set up in 2D. Methane enters from the left and reacts in the porous catalytic bed in the mid-section of the geometry.

M O M E N T U M B A L A N C E S

The flow in the free channel section is described by the Navier-Stokes equations:

(5)

wherer denotes density (kg/m3), u represents the velocity (m/s), is the dynamic viscosity (Pas)), and p refers to the pressure (Pa). In the porous domain, the Brinkmanequations govern the flow:

(6)

Here p is the porosity andk denotes permeability (m2) of the porous medium. As youcan see inEquation 4 andEquation 5, the momentum-balance equations are closelyrelated. The term on the right-hand side of the Navier-Stokes formulation corresponds

P R g T c CH 4 c H 2+ =

wall

wall

OutletInlet Porous catalytic bed

tu u u T + p I+ + u u=

u 0=

P----- t

u P

----- u u T

+ p I+ +

--- u=

u 0=



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to momentum transported by convection in free flow. In the Brinkman formulation,this term is replaced by a contribution associated with the drag force experienced bythe fluid as it flows through a porous medium. COMSOL Multiphysics automaticallycombines free and porous-media flow to solve the equations simultaneously.

The boundary conditions for the flow are:

inlet

walls

outletMass transport in the reactor is described by the diffusion-convection equations:

where D i denotes the diffusion coefficient (m2/s) and ci is the species concentration(mol/m3). The term R

i (mol/(m3s)) corresponds to the species net reaction rate.

In the free channel, the inlet condition is set as (concentrations)

At the outlet, use the convective flux condition

All other boundaries, use the insulating or symmetry condition

B A L A N C E F O R V O I D F R A C T I O N

The void fraction of the catalytic bed decreases as carbon is deposited. This, in turn,affects the flow through the reactor. A balance for the void fraction, or porosity, of thebed is given by:

Wherekpor is constant, M C (kg/mol) is carbon molar weight andsoot (kg/m3) isdeposited carbon density. This equation can be implemented in the Domain ODEs andDAEs physics interface of COMSOL Multiphysics, resulting in porosity distribution

u n u 0=

u 0=

p 0=

tci D i ci c i u+ + R i=

c c in=

n D c 0=

n D c c u+ 0=

ddt------ k pos

rM Csoot

---------------=



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across the catalytic bed as a function of time. The initial porosity of the bed is assumedto be = 0.4.

The porosity is related to the permeability of the porous domain by the expression(Ref. 2):

(7)

In this way, the porosity balance couples the mass and momentum balances describingthe reacting system.


I D E A L R E A C T O R M O D E L

Figure 2 shows the concentration transients of methane, hydrogen and depositedcarbon as methane decomposes over a Ni/Al2O3 catalyst.

Figure 2: Concentration transients of methane decomposition over a Ni/Al 2O 3 catalyst.

00

----- 3.55=



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Figure 3 shows the deactivation of catalyst during methane decomposition. Theactivity of catalyst decreases rapidly at the early stage of reaction, then decreases slowly.

Figure 3: Change of activity of catalyst with reacting time.

The reactor pressure increases (seeFigure 4) with the proceeding of decompositiondue to the gas expanding according to the reaction (Equation 1).

The effect of activity of catalyst on decomposition is shown inFigure 5. The reactionrate with activity of catalyst constant is obvious larger than that with deactivation ofcatalyst.



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Figure 4: Reactor pressure during methane decomposition.

Figure 5: Comparison of concentration transients under two conditions of catalyst: 1)deactivation; 2) constant activity (1).



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S PA C E - D E P E N D E N T M O D E L

This model concerns the stationary space-dependent model in which the Free andPorous Media Flow physics interface is used.

Figure 6 shows the parabolic velocity profile (2D) in the reactor. The flow velocity ofreacting gas is reduced from 0.45 mm/s to about 0.30 mm/s due to the limitedpermeability in the porous domain while it is almost 0.45 mm/s in other two freechannel sections (Figure 7). As the same reason as that for velocity, there is moderatepressure drop along the catalytic bed (seeFigure 8 andFigure 9).

Figure 6: Velocity flow field in the 2D reactor under stationary state.


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Figure 9: Pressure drop along the packed catalyst bed.

S PA C E - A N D T I M E - D E P E N D E N T M O D E L

The following results concern a space- and time-dependent model simulated in a fullycoupled model consisting of Reaction Engineering, Transport of Diluted Species, Freeand Porous Media Flow, and Domain ODEs and DAEs physics interfaces.

The reacting gas passes through the catalytic bed whose permeability is correlated toits porosity as inFigure 6. The fully coupled model is solved by using the result fromstationary Free and Porous Media Flow as the initial values.

Figure 10 shows the 2D profile for the concentration of methane at reacting time4000 s.Figure 11 shows the comparison of concentration distributions for methaneand hydrogen at different reacting times. At reacting time 4000 s, the methane and

hydrogen concentrations are 12.5 and 5 mol/m3 at the end of catalytic bed,respectively. The average residence time in the catalytic bed is 400 mm/0.3 (mm/s) which equals 1333 s. InFigure 2, the methane and hydrogen concentrations are alsoabout 12.5 and 5 mol/m3 at the reacting time 1333 s. This means the contribution ofdiffusion to the mass transfer is negligible compared to convection.



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Figure 12 shows the velocity field. It is very similar to the velocity field understationary state (Figure 6). Figure 13 shows the pressure distribution in reactor. Thepressure drop is obvious larger than that under stationary state (Figure 8). In thesimulation condition, the effect of carbon deposition on porosity is significant(Figure 14). Figure 15 shows the permeability distribution along the catalytic bed atdifferent reacting times.

Figure 10: Concentration distribution of methane as a function of the bed position atreacting time 4000 s.



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Figure 11: Concentration distribution of CH 4 and H 2 along the center line of reactorunder fully coupled physics interfaces at different reacting times.

Figure 12: Velocity flow field in the 2D reactor at reacting time = 4000 s.



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Figure 13: Distribution of pressure in the reactor for reacting gas passing through a cleancatalyst under transient state at reacting time 4000 s.

Figure 14: Porosity distribution in the 2D reactor after reacting time 4000 s.



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Figure 15: Permeability distribution along the center line of reactor at different reactingtimes

References

1. S.G. Zavarukhin and G.G. Kuvshinov, The kinetic model of formation of

nanofibrous carbon from CH4H2 mixture over a high-loaded nickel catalyst withconsideration for the catalyst deactivation, J. Appl. Catal. A , vol. 272, pp. 219227,2004.

2. E.A. Borisova and P.M. Adler, Deposition in porous media and clogging on thefield scale,Phys. Rev. E , vol. 71, p. 016311-1, 2005.

Model Library path: Chemical_Reaction_Engineering_Module/Heterogeneous_Catalysis/carbon_deposition


From theFile menu, chooseNew.



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N E W

1 In the New window, click theModel Wizard button.


1 In the Model Wizard window, click the0D button.2 In the Select physics tree, selectChemical Species Transport>Reaction Engineering (re) .3 Click theAdd button.4 Click theStudy button.5 In the tree, selectPreset Studies>Time Dependent .

6 Click theDone button.


Reaction 11 On the Physics toolbar, clickGlobal and chooseReaction .2 In the Reaction settings window, locate theReaction Formula section.

3 In the Formula edit field, typeCH4=>C+2H2 .Species 1 Add a special species representing the catalyst activity.

1 On the Physics toolbar, clickGlobal and chooseSpecies .2 In the Species settings window, locate theSpecies Formula section.3 In the edit field, typea .

When a new species is created a mass balance equation is set up along with it. In thiscase: the left-hand side is defined internally in the software and the right-hand sidecorresponds to the expression given in the R edit field. Note also that you can removethe effect of catalyst activity from your model by selecting the Lock concentration/activity check box.This removes the species mass balance and sets the concentration ofthe species to the value entered in the Initial concentration edit field.


Load the model parameters from a text file.

Parameters1 On the Home toolbar, clickParameters .2 In the Parameters settings window, locate theParameters section.3 ClickLoad from File .



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4 Browse to the models Model Library folder and double-click the filecarbon_deposition_parameters.txt .


Load the model variables from a text file.

Variables 11 In the Model Builder window, underComponent 1 right-clickDefinitions and choose

Variables .2 In the Variables settings window, locate theVariables section.

3 ClickLoad from File .4 Browse to the models Model Library folder and double-click the file

carbon_deposition_variables.txt .


1: CH4=>C+2H21 In the Model Builder window, underComponent 1>Reaction Engineering click1:

CH4=>C+2H2 .2 In the Reaction settings window, locate theReaction Rate section.3 From theReaction rate list, chooseUser Defined .4 In the r edit field, typec_a*k*(p_CH4-p_H2^2/Kp)/(1+kH*sqrt(p_H2))^2 .

Species: CH41 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

CH4.2 In the Species settings window, locate theGeneral Expressions section.3 In the c0 edit field, typec_CH4in .

Species: H21 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

H2.

2 In the Species settings window, locate theGeneral Expressions section.3 In the c0 edit field, typec_H2in .

Species: a1 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

a.


I th tti i d l t th ti


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2 In the Species settings window, locate theGeneral Expressions section.3 In the c0 edit field, type1 .

4 From theRate expression list, chooseUser Defined .5 In the R edit field, type-ka*r_1^2*c_C*c_a .6 In the Model Builder window, clickReaction Engineering .7 In the Reaction Engineering settings window, locate theGeneral section.8 In the T edit field, type850[K] .9 In the p edit field, typeR_const*T*(c_CH4+c_H2) .

S T U D Y 1

Step 1: Time Dependent1 In the Model Builder window, underStudy 1 clickStep 1: Time Dependent .2 In the Time Dependent settings window, locate theStudy Settings section.3 In the Times edit field, typerange(0,400,4000) .4 On the Home toolbar, clickCompute .

R E S U LT S

Concentration (re)1 In theModel Builder window, underResults right-clickConcentration (re) and choose

Rename .2 Go to theRename 1D Plot Group dialog box and typeConcentration (CH4, C,

H2, re) in theNew name edit field.3 ClickOK.

Concentration (CH4, C, H2, re)1 In theModel Builder window, expand theResults>Concentration (CH4, C, H2, re) node,

then clickGlobal 1 .2 In the Global settings window, clickReplace Expression in the upper-right corner of

the y-axis data section. From the menu, chooseReaction Engineering>Concentration(comp1.re.c_CH4) .

3 ClickAdd Expression in the upper-right corner of they-axis data section. From themenu, chooseReaction Engineering>Concentration (comp1.re.c_C) .

4 ClickAdd Expression in the upper-right corner of they-axis data section. From themenu, chooseReaction Engineering>Concentration (comp1.re.c_H2) .


5 On the 1D l t toolbar clickPl t


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5 On the 1D plot group toolbar, clickPlot .

1D Plot Group 2

1 On the Home toolbar, clickAdd Plot Group and choose1D Plot Group .2 In the Model Builder window, underResults right-click1D Plot Group 2 and choose

Rename .3 Go to theRename 1D Plot Group dialog box and type Activity(a, re) in theNew

name edit field.4 ClickOK.

Act ivity(a, re)1 On the 1D plot group toolbar, clickGlobal.2 In the Global settings window, clickReplace Expression in the upper-right corner of

the y-axis data section. From the menu, chooseReaction Engineering>Concentration(comp1.re.c_a) .

3 Locate they-Axis Data section. In the table, enter the following settings:

4 Click to expand theLegends section. Clear theShow legends check box.5 On the 1D plot group toolbar, clickPlot .Plot the pressure in the reactor versus reacting time.

1D Plot Group 3


Rename .3 Go to theRename 1D Plot Group dialog box and typePressure(re) in theNew

name edit field.4 ClickOK.

Pressure(re)1 On the 1D plot group toolbar, clickGlobal.2 In the Global settings window, clickReplace Expression in the upper-right corner of

the y-axis data section. From the menu, chooseReaction Engineering>Pressure(comp1.re.p) .

3 Click to expand theLegends section. Clear theShow legends check box.

Expression Unit Descriptioncomp1.re.c_a mol/m^3 Activity


4 On the 1D plot group toolbar clickPlot


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Now study the reaction when the catalyst activity is held constant (initial value).

Species: a1 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

a.2 In the Species settings window, locate theGeneral Expressions section.3 Select theLock concentration/activity check box.

R O O T

On the Home toolbar, clickAdd Study .

A D D S T U D Y

1 Go to theAdd Study window.2 Find theStudies subsection. In the tree, selectPreset Studies>Time Dependent .

3 In the Add study window, clickAdd Study .S T U D Y 2

Step 1: Time Dependent1 In the Model Builder window, underStudy 2 clickStep 1: Time Dependent .2 In the Time Dependent settings window, locate theStudy Settings section.3 In the Times edit field, typerange(0,400,4000) .4 On the Home toolbar, clickCompute .

R E S U LT S

Concentration (re)Compare the concentrations between locked and unlocked species a.

1 In theModel Builder window, underResults right-clickConcentration (re) and chooseRename .

2 Go to theRename 1D Plot Group dialog box and typeConcentration comparison(re) in theNew name edit field.

3 ClickOK.


Concentration comparison (re)


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Concentration comparison (re)1 In the Model Builder window, expand theResults>Concentration comparison (re)

node, then clickGlobal 1 .2 In the Global settings window, clickReplace Expression in the upper-right corner of

the y-axis data section. From the menu, chooseConcentration (comp1.re.c_CH4) .3 ClickAdd Expression in the upper-right corner of they-axis data section. From the

menu, chooseConcentration (comp1.re.c_C) .4 ClickAdd Expression in the upper-right corner of they-axis data section. From the

menu, chooseConcentration (comp1.re.c_H2) .

5 Locate theLegends section. From theLegends list, chooseManual.6 In the table, enter the following settings:

7 Right-clickResults>Concentration comparison (re)>Global 1 and chooseDuplicate .8 In the Global settings window, locate theData section.9 From theData set list, chooseSolution 1 .10 Click to expand theTitle section. From theTitle type list, chooseManual.11 In the Title text area, typeConcentration comparison .12 Locate theLegends section. In the table, enter the following settings:



Synchronize the Reaction Engineering physics with mass and momentum transportphysics interfaces to generate a time- and space-dependent model.

Legends

CH4: locked

C: locked

H2: locked

Legends

CH4: unlocked

C: unlocked

H2: unlocked


Species: a


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Spec es: a1 In the Model Builder window, underComponent 1>Reaction Engineering clickSpecies:

a.2 In the Species settings window, locate theGeneral Expressions section.3 Clear theLock concentration/activity check box.

Generate Space-Dependent Model 11 In the Model Builder window, right-clickReaction Engineering and chooseGenerate

Space-Dependent Model .2 In theGenerate Space-Dependent Model settings window, locate theGeometry Settings

section.3 From theGeometry to use list, choose2D: New.4 Locate thePhysics Interfaces section. From theMomentum balance list, chooseFree

and Porous Media Flow: New .5 Select theCreate inflow and outflow features check box.6 Locate theStudy Type section. From theStudy type list, chooseTime dependent .

7 Locate theSpace-Dependent Model Generation section. Click theCreate/Refresh button.

C O M P O N E N T 2

Set up the geometry as a union of two rectangles.

G E O M E T R Y 1

In the Model Builder window, expand theComponent 2 node.

Rectangle 11 Right-clickGeometry 1 and chooseRectangle .2 In the Rectangle settings window, locate theSize section.3 In the Height edit field, type0.1 .

Rectangle 2

1 In the Model Builder window, right-clickGeometry 1 and chooseRectangle .2 In the Rectangle settings window, locate theSize section.3 In the Width edit field, type0.4 .4 In the Height edit field, type0.1 .5 Locate thePosition section. In thex edit field, type0.4 .


6 Click theBuild All Objects button.


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C O M P O N E N T 2

Add the Domain ODE and DAE interface for modeling the porosity change in theporous domain.

On the Home toolbar, clickAdd Physics .

A D D P H Y S I C S

1 Go to theAdd Physics window.2 In theAdd physics tree, selectMathematics>ODE and DAE Interfaces>Domain ODEs and

DAEs (dode) .3 In the Add physics window, clickAdd to Component .

D O M A I N O D E S A N D D A E S

1 In the Model Builder window, underComponent 2 clickDomain ODEs and DAEs .2 Select Domain 2 only.

3 In the Domain ODEs and DAEs settings window, click to expand theDependentvariables section.4 Locate theDependent Variables section. In theField name edit field, typepor .5 In the Dependent variables table, enter the following settings:

Distributed ODE 11 In the Model Builder window, underComponent 2>Domain ODEs and DAEs click

Distributed ODE 1 .2 In the Distributed ODE settings window, locate theSource Term section.3 In the f edit field, type-k_por*por*root.comp1.re.r_1*M_C/rho_soot .

Initial Values 11 In the Model Builder window, underComponent 2>Domain ODEs and DAEs clickInitial

Values 1 .2 In the Initial Values settings window, locate theInitial Values section.3 In the por edit field, typepor0 .

por


T R A N S P O R T O F D I L U T E D S P E C I E S 1


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Convection and Diffusion 1

1 In the Model Builder window, expand theComponent 2>Transport of Diluted Species1 node, then clickConvection and Diffusion 1 .2 In the Convection and Diffusion settings window, locate theDiffusion section.3 In the D cCH4 edit field, typeD_CH4.4 In the D cH2 edit field, typeD_H2.

Reactions

1 In the Model Builder window, underComponent 2>Transport of Diluted Species 1 clickReactions .2 Select Domain 2 only.

Inflow 11 In the Model Builder window, underComponent 2>Transport of Diluted Species 1 click

Inflow 1 .2 Select Boundary 1 only.

Outflow 11 In the Model Builder window, underComponent 2>Transport of Diluted Species 1 click

Outflow 1 .2 Select Boundary 10 only.

Convection and Diffusion 21 On the Physics toolbar, clickDomains and chooseConvection and Diffusion .2 Select Domain 2 only.3 In the Convection and Diffusion settings window, locate theModel Inputs section.4 From theu list, chooseVelocity field (fp/fp1) .5 Locate theDiffusion section. In the D cCH4 edit field, typek_eff*D_CH4 .6 In the D cH2 edit field, typek_eff*D_H2 .

F R E E A N D P O R O U S M E D I A F L O W 11 In the Model Builder window, underComponent 2 clickFree and Porous Media Flow 1 .2 In the Free and Porous Media Flow settings window, locate thePhysical Model section.3 From theCompressibility list, chooseCompressible flow (Ma


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1 In the Model Builder window, expand theFree and Porous Media Flow 1 node, thenclickFluid Properties 1 .

2 In the Fluid Properties settings window, locate theFluid Properties section.3 From the list, chooseUser defined . In the associated edit field, typerho .4 From the list, chooseUser defined . In the associated edit field, typeeta .

Inlet 11 In the Model Builder window, underComponent 2>Free and Porous Media Flow 1 click

Inlet 1 .2 Select Boundary 1 only.3 In the Inlet settings window, locate theBoundary Condition section.4 From theBoundary condition list, chooseLaminar inflow .5 Locate theLaminar Inflow section. In theU av edit field, typeu_in .

Outlet 11 In the Model Builder window, underComponent 2>Free and Porous Media Flow 1 click

Outlet 1 .2 Select Boundary 10 only.

Porous Matrix Properties 11 On the Physics toolbar, clickDomains and choosePorous Matrix Properties .2 Select Domain 2 only.3 In the Porous Matrix Properties settings window, locate thePorous Matrix Properties

section.4 From the p list, chooseUser defined . In the associated edit field, typepor .5 From the list, chooseUser defined . In the associated edit field, typekappa .


Variables 2

1 In the Model Builder window, underComponent 2 right-clickDefinitions and chooseVariables .

2 In the Variables settings window, locate theVariables section.




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M E S H 1

Build a mesh with Boundary Layers.

Size1 In the Model Builder window, underComponent 2 right-clickMesh 1 and choose

Boundary Layers .2 In the Size settings window, locate theElement Size section.3 From theCalibrate for list, chooseFluid dynamics .

Boundary Layer Properties1 In the Model Builder window, underComponent 2>Mesh 1>Boundary Layers 1 click

Boundary Layer Properties .

2 Select Boundaries 26, 8, and 9 only.3 In the Boundary Layer Properties settings window, locate theBoundary Layer

Properties section.4 In the Number of boundary layers edit field, type4 .5 Click theBuild All button.

S T U D Y 3

Set up a stationary study for the Free and Porous Media Flow physics. The result is alsoused later as initial values for the fully coupled model.

Step 2: Stationary 1 On the Study toolbar, clickStudy Steps and chooseStationary>Stationary .2 In the Stationary settings window, locate thePhysics and Variables Selection section.3 In the table, enter the following settings:

Name Expression Unit Description

kappa kappa0*(por/por0)^3.55

m permeability

Physics Solve for Discretization

Reaction Engineering physics

Transport of Diluted Species 1 physics

Domain ODEs and DAEs physics


Step 2: Time Dependent1 Right clickSt d 3>St 2 St ti and chooseM U


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1 Right-clickStudy 3>Step 2: Stationary and chooseMove Up.2 In the Model Builder window, underStudy 3 right-clickStep 2: Time Dependent and

chooseDisable.3 On the Home toolbar, clickCompute .

R E S U LT S

Velocity (fp1)1 In the Model Builder window, underResults right-clickVelocity (fp1) and choose

Rename .2 Go to theRename 2D Plot Group dialog box and typeStationary velocity (fp1)

in theNew name edit field.3 ClickOK.

Stationary velocity (fp1)Plot an arrow surface of the velocity field.

1 Right-clickResults>Velocity (fp1) and chooseArrow Surface .2 In the Arrow Surface settings window, clickReplace Expression in the upper-right

corner of theExpression section. From the menu, chooseFree and Porous Media Flow1>Velocity field (u,v) .

3 Click to expand theTitle section. From theTitle type list, chooseNone.4 Locate theArrow Positioning section. In thePoints edit field, type10 .5 Locate theColoring and Style section. Select theScale factor check box.6 In the associated edit field, type100 .7 On the 2D plot group toolbar, clickPlot .

Data SetsCreate a dataset for 1D plotting.

1 On the Results toolbar, clickCut Line 2D .

2 In the Cut Line 2D settings window, locate theLine Data section.3 In rowPoint 1 , sety to 0.05 .4 In rowPoint 2 , sety to 0.05 .

1D Plot Group 7 1 On the Home toolbar, clickAdd Plot Group and choose1D Plot Group .


2 In the Model Builder window, underResults right-click1D Plot Group 7 and chooseRename


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Rename .3 Go to theRename 1D Plot Group dialog box and typeVelocity(center line,

fp1) in theNew name edit field.4 ClickOK.5 In the 1D Plot Group settings window, locate theData section.6 From theData set list, chooseCut Line 2D 1 .

Velocity(center line, fp1)1 On the 1D plot group toolbar, clickLine Graph .2 In the Line Graph settings window, clickReplace Expression in the upper-right corner

of they-axis data section. From the menu, chooseFree and Porous Media Flow1>Velocity magnitude (fp.U) .


1D Plot Group 81 On the Home toolbar, clickAdd Plot Group and choose1D Plot Group .2 In the Model Builder window, underResults right-click1D Plot Group 8 and choose

Rename .3 Go to theRename 1D Plot Group dialog box and typePressure(center line,

fp1) in theNew name edit field.4 ClickOK.5 In the 1D Plot Group settings window, locate theData section.

6 From theData set list, chooseCut Line 2D 1 .7 Click to expand theAxis section. Select theManual axis limits check box.8 In the x minimum edit field, type0.4 .9 In the x maximum edit field, type0.8 .10 In the y minimum edit field, type0 .11 In the y maximum edit field, type1.2 .

Pressure(center line, fp1)1 On the 1D plot group toolbar, clickLine Graph .2 In the Line Graph settings window, clickReplace Expression in the upper-right corner

of they-axis data section. From the menu, chooseFree and Porous Media Flow1>Pressure (p) .



A D D S T U D Y

Now study the fully coupled model consisting of the Reaction Engineering Transport


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Now study the fully coupled model consisting of the Reaction Engineering, Transportof Diluted Species, Free and Porous Media Flow, Domain ODEs and DAEs interfaces.

1 Go to theAdd Study window.2 Find theStudies subsection. In the tree, selectPreset Studies>Time Dependent .3 In the Add study window, clickAdd Study .

S T U D Y 4

Step 2: Stationary 1 On the Study toolbar, clickStudy Steps and chooseStationary>Stationary .2 In the Model Builder window, underStudy 4 right-clickStep 2: Stationary and choose

Move Up.3 In the Stationary settings window, locate thePhysics and Variables Selection section.4 In the table, enter the following settings:

Step 2: Time Dependent1 In the Model Builder window, underStudy 4 clickStep 2: Time Dependent .2 In the Time Dependent settings window, locate theStudy Settings section.3 In the Times edit field, typerange(0,400,4000) .

Solver 41 On the Study toolbar, clickShow Default Solver .2 In the Model Builder window, expand theSolver 4 node, then clickDependent

Variables 2 .3 In the Dependent Variables settings window, locate theScaling section.4 From theMethod list, chooseManual.5 In the Scale edit field, type10 .6 On the Home toolbar, clickCompute .

Physics Solve for Discretization

Reaction Engineering physics

Transport of Diluted Species 1 physics

Domain ODEs and DAEs physics


R E S U LT S


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Data Sets

1 On the Results toolbar, clickCut Line 2D .2 In the Cut Line 2D settings window, locate theData section.3 From theData set list, chooseSolution 4 .4 Locate theLine Data section. In rowPoint 1 , sety to 0.05 .5 In rowPoint 2 , sety to 0.05 .

1D Plot Group 13


Rename .3 Go to theRename 1D Plot Group dialog box and typeConcentration 1D (chds1)

in theNew name edit field.4 ClickOK.5 In the 1D Plot Group settings window, locate theData section.6 From theData set list, chooseCut Line 2D 2 .7 From theTime selection list, chooseManual.8 In the Time indices (1-11) edit field, type2,3,11 .9 Locate theAxis section. Select theManual axis limits check box.10 In the x minimum edit field, type0.4 .11 In the x maximum edit field, type0.8 .12 In the y minimum edit field, type0 .13 In the y maximum edit field, type15 .

Concentration 1D (chds1)Plot the concentration distributions for CH4 and H2 at different reacting times.

1 On the 1D plot group toolbar, clickLine Graph .

2 In the Line Graph settings window, clickReplace Expression in the upper-right cornerof they-axis data section. From the menu, chooseTransport of Diluted Species1>Species cCH4>Concentration (cCH4) .

3 Click to expand theLegends section. Select theShow legends check box.4 From theLegends list, chooseManual.




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6 On the 1D plot group toolbar, clickLine Graph .7 In the Line Graph settings window, locate theData section.8 From theData set list, chooseCut Line 2D 2 .

9 From theTime selection list, chooseManual.10 In the Time indices (1-11) edit field, type2,3,11 .11 ClickReplace Expression in the upper-right corner of they-axis data section. From

the menu, chooseTransport of Diluted Species 1>Species cH2>Concentration (cH2) .12 Locate theLegends section. Select theShow legends check box.13 From theLegends list, chooseManual.



Velocity (fp1)1 In the Model Builder window, underResults right-clickVelocity (fp1) and choose

Arrow Surface .2 In the Arrow Surface settings window, clickReplace Expression in the upper-right

corner of theExpression section. From the menu, chooseFree and Porous Media Flow1>Velocity field (u,v) .

3 Locate theTitle section. From theTitle type list, chooseNone.4 Locate theArrow Positioning section. In thePoints edit field, type10 .5 Locate theColoring and Style section. Select theScale factor check box.6 In the associated edit field, type100 .7 On the 2D plot group toolbar, clickPlot .

Legends

400 CH4800 CH4

4000 CH4

Legends

400 H2

800 H2

4000 H2


Pressure (fp1) 1This is the distribution of pressure drop in porous media section of reactor at reacting


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time = 4000 s.

1 In the Model Builder window, expand theResults>Pressure (fp1) 1 node.

2D Plot Group 12This is the porosity distribution in porous media section of reactor at reacting time =4000 s.

1D Plot Group 141 On the Home toolbar, clickAdd Plot Group and choose1D Plot Group .2 In the Model Builder window, underResults right-click1D Plot Group 14 and choose

Rename .3 Go to theRename 1D Plot Group dialog box and typePermeability (center

line, ODE and DAE) in theNew name edit field.4 ClickOK.5 In the 1D Plot Group settings window, locate theData section.

6 From theData set list, chooseCut Line 2D 2 .7 From theTime selection list, chooseManual.8 In the Time indices (1-11) edit field, type2,3,5,6,11 .

Permeability (center line, ODE and DAE)1 On the 1D plot group toolbar, clickLine Graph .2 In the Line Graph settings window, clickReplace Expression in the upper-right corner

of they-axis data section. From the menu, chooseDefinitions>permeability (kappa) .3 Locate theLegends section. Select theShow legends check box.4 From theLegends list, chooseManual.5 In the table, enter the following settings:


Legends

400 s

800 s

1600 s

2000 s

4000 s



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Homogeneous Charge Compress ion


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1 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E

Igni t ion of MethaneIntroduction

Homogeneous Charge Compression Ignition (HCCI) engines are being considered asan alternative to traditional spark- and compression-ignition engines. As the nameimplies, a homogeneous fuel/oxidant mixture is autoignited by compression with

simultaneous combustion occurring throughout the cylinder volume. Combustiontemperatures under lean burn operation are relatively low, resulting in low levels ofNOx emission. Furthermore, the fuels homogeneous nature as well as the combustionprocess itself lead to low levels of particulate matter being produced.

Although HCCI combustion shows much promise, the method also suffers from anumber of recurring problems, one of the more important being ignition timing. Thefollowing model examines the HCCI of methane, investigating ignition trends as afunction of initial temperature, initial pressure, and fuel additives.This model solves the mass and energy balances describing the detailed combustion ofmethane in a variable-volume system. The large amount of kinetic and thermodynamicdata required to set up the problem is readily made available by importing relevant filesinto the Reaction Engineering interface.

Model Definition

It is difficult to form the uniform mixtures required for HCCI with conventional dieselfuel. Natural-gas fuels, on the other hand, readily produce homogeneous mixtures andhave the potential to serve as HCCI fuels. This example considers the combustion ofmethane, as described by the GRI-3.0 mechanism, incorporating a detailed reactionmechanism of 53 species taking part in 325 reactions. The files describing the reactionkinetics and thermodynamics of the GRI-3.0 mechanism are available on the Internet

(Ref. 1), and you can import these files directly into the Reaction Engineering interfaceVA R I A B L E V O L U M E R E A C T O R

This model represents the combustion cylinder with a perfectly mixed batch system of variable volume, a reactor type that is predefined in the Reaction Engineeringinterface.Figure 1 shows a drawing of an engine cylinder, and it points out parameters


relevant for calculating the instantaneous cylinder volume.


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Figure 1: The volume of a combustion cylinder can be expressed as a function of time withthe slider-crank relationship. This diagram shows the key geometric parameters. La is thelength of the crank arm, Lc gives the length of the connecting rod, D equals the cylinderdiameter, and is the crank angle.

The volume change as a function of time is described by the slider-crank equation:

(1)

where,V is the cylinder volume (m3), V c gives the clearance volume (m3), CR equalsthe compression ratio, and R denotes the ratio of the connecting rod to the crank arm( L c/ L a). Further, is the crank angle (rad), which is also a function of time

(2)

where N is the engine speed in rpm, and t is the time (s).

The engine specifications used in the model are:

ENGINE SPECIFICATION VARIABLE NAME VALUE

Bore D 13 cm

Stroke S 16 cm

Connecting rod Lc 26.93 cm

Crank arm La 8 cm

Engine speed N 1500 rpm

Compression ratio CR 15

La

Lc

D

V V c------ 1 CR 1

2

----------------------- R 1 R 2 sin 2cos+ +=

2 N

60

------------ t=


Equation 3 includes the clearance volume,V c, which is calculated from

(3)VV s


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(3)

V s is the volume swept by the piston during a cycle from the equation

(4)

Figure 2 shows the calculated cylinder volume as a function of the crank angle. Thepiston is initially at bottom dead center (BDC), corresponding to a crank angle of180 degrees.

Figure 2: Cylinder volume as function of crank angle. The crank angle is defined as beingzero at top dead center (TDC).

M A S S A N D E N E R G Y B A L A N C E SThe mass balances describing a perfectly mixed reactor with variable volume aresummarized by

(5)

V cs

CR 1

-----------------------=

V s D 2

4----------- S=

d Vc i dt

----------------- VR i=


whereci represents the species concentration (mol/m3), and R i denotes the speciesrate expression (mol/(m3s)).

F id l i h b l i


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For an ideal gas mixture, the reactor energy balance is

(6)

whereC p,i is the species molar heat capacity (J/(molK)),T is the temperature (K),and p gives the pressure (Pa). In this equation,Q is the heat due to chemical reaction(J/s)

(7)

where H j is the enthalpy of reaction (J/(molK)), andr j equals the reaction rate (mol/(m3s)). Qext denotes heat added to the system (J/s). The model being describedassumes adiabatic conditions, that is,Qext = 0.

The kinetic and thermodynamic data for methane combustion is available in the form

of data input files. Once the files are imported into the Reaction Engineering interface,the software automatically sets up the mass and energy balances detailed inEquation 5 andEquation 6.

To complete the model setup, all that remains is to define the initial conditions. In thismodel, methane is combusted under lean conditions, that is, supplying more than thestoichiometric amount of oxidizer. The stoichiometric requirement of the oxidizer(air) to combust methane is found from the overall reaction:

(8)

Assuming that the composition of air is 21% oxygen and 79% nitrogen, thestoichiometric air-fuel ratio is

(9)

The equivalence ratio relates the actual air-fuel ratio to the stoichiometric requirements

(10)

This model sets the equivalence ratio to = 0.5 .

V r c i C p idT dt--------

i

Q Q+ ex t V rdpdt-------+=

Q V r H j r j=

CH4 + O2 N2 CO2 H2O N2+( )2 2 7.523.76 + +

A F stoicm ai rm fuel--------------

stoic

4.76 2 M ai r1 M fuel

------------------------------------= =

A F stoic A F

----------------------------=


FromEquation 9 andEquation 10 it is possible to calculate the molar fraction of fuelin the reacting mixture as


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(11)

and subsequently the initial concentration is

(12)

The initial pressure and the initial temperature are variable model parameters.


Figure 3 shows the cylinder pressure as a function of time when a methane-air mixtureis compressed and ignites. The piston starts at bottom dead center (BDC) and reachestop dead center (TDC) after 0.02 s. At BDC the pressure is set to1.510 5 Pa, is0.5 ,and the compression ratio isCR = 15. The initial temperature is varied from400 K to800 K.

Figure 3: Pressure traces illustrating the compression and ignition of fuel in an enginecylinder. The initial temperature varies between 400 K and 800 K.

x fuel1

4.76 2 1+ -------------------------------------=

cfuel x fuel p init

Rg T init-------------------------=


Consistent with literature results, methane does not ignite at an initial temperature of400 K (Ref. 2). Furthermore, the induction delay decreases with increasing initialtemperature. The induction delay time can be evaluated from t

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Chemical Reaction Engineering Model Library Manual