TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY

LAYER FLOW IN A CATALYTIC REACTOR

By

PATRICK D. GRIFFIN

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2006

Copyright 2006

by

Patrick D. Griffin

iii

ACKNOWLEDGMENTS

Siemens and the National Aeronautics and Space Administration supported this

research. I thank Dr. David Mikolaitis, Dr. David Hahn, and Dr. Corin Segal for their

assistance. I also thank my parents for their continued support and involvement in and

out of my educational development.

iv

TABLE OF CONTENTS page

ACKNOWLEDGMENTS ................................................................................................. iii

LIST OF TABLES............................................................................................................. vi

LIST OF FIGURES .......................................................................................................... vii

ABSTRACT..................................................................................................................... viii

CHAPTER

1 INTRODUCTION ........................................................................................................1

2 REDUCTION OF CONSERVATION EQUATIONS .................................................7

Applying Assumptions .................................................................................................8 Continuity Equation...............................................................................................9 Species Continuity Equations..............................................................................10 Momentum Equations .........................................................................................12 Energy Equation ..................................................................................................15

Order of Magnitude Analysis .....................................................................................21 Continuity Equation.............................................................................................22 Species Continuity Equations..............................................................................23 Axial Momentum Equation .................................................................................25 Vertical Momentum Equation .............................................................................27 Energy Equation ..................................................................................................30

Unit Analysis ..............................................................................................................34 Continuity Equation.............................................................................................34 Species Continuity Equations..............................................................................35 Momentum Equation ...........................................................................................36 Energy Equation ..................................................................................................37

Summary of Governing Equations .............................................................................40

3 PROGRAM METHODOLOGY ................................................................................41

Discretization..............................................................................................................43 Parameters and Conditions .........................................................................................45 Input and Output Files ................................................................................................47 Initial Conditions of a Stage .......................................................................................49

v

Stage One.............................................................................................................50 Blasius Solution...................................................................................................51 Subsequent Stages ...............................................................................................53

Solving Governing Equations.....................................................................................53 Momentum Equation ...........................................................................................57 Continuity Equation.............................................................................................59 Species Continuity Equations..............................................................................61 Energy Equation ..................................................................................................63 Species/Energy System of Equations ..................................................................67

4 TESTING....................................................................................................................71

Case One.....................................................................................................................72 Results of Case One....................................................................................................73 Case Two ....................................................................................................................75 Results of Case Two ...................................................................................................76 Case Three ..................................................................................................................79 Results of Case Three .................................................................................................80 Case Four ....................................................................................................................83 Results of Case Four...................................................................................................83

5 PROGRAM LIMITATIONS AND IMPROVEMENTS............................................90

6 CONCLUSION...........................................................................................................91

LIST OF REFERENCES...................................................................................................93

BIOGRAPHICAL SKETCH .............................................................................................95

vi

LIST OF TABLES

Table page 2-1 Equations modeling the flow....................................................................................40

2-2 Units of the governing equations. ............................................................................40

4-1 Parameters and conditions of case one.....................................................................73

4-2 Parameters and conditions of case two.....................................................................76

4-3 Parameters and conditions of case three...................................................................79

4-4 Parameters and conditions of case four....................................................................83

vii

LIST OF FIGURES

Figure page 2-1 Dimensionless variables. ..........................................................................................21

3-1 Flow chart for single stage modeling. ......................................................................55

3-2 Flux components in the species/energy system........................................................68

3-3 Source components in the species/energy system....................................................69

3-4 Boundary conditions of the species/energy system..................................................70

4-1 Axial velocity profiles of case one. ..........................................................................74

4-2 Pressure plot of case one. .........................................................................................75

4-3 Axial velocity profiles of case two...........................................................................77

4-4 Pressure plot of case two. .........................................................................................78

4-5 Reduction in methane concentrations of case two. ..................................................78

4-6 Axial velocity profiles of case three.........................................................................81

4-7 Pressure plot of case three. .......................................................................................82

4-8 Reduction in methane concentrations of case three. ................................................82

4-9 Axial velocity profiles of case four. .........................................................................84

4-10 Pressure plot of case four. ........................................................................................85

4-11 Temperature profiles of case four. ...........................................................................86

4-12 Methane concentrations of case four........................................................................86

4-13 Hydrogen concentrations of case four. A) Mass fractions of atomic hydrogen. B) Mass fractions of diatomic hydrogen. .................................................................87

viii

Abstract of Thesis Presented to the Graduate School

of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY LAYER FLOW IN A CATALYTIC REACTOR

By

Patrick D. Griffin

August 2006

Chair: David Mikolaitis Major Department: Mechanical and Aerospace Engineering

Problems associated with fossil fuels are increasing interest in alternative forms of

energy production. Hydrogen is quickly becoming a popular option, but the efficient,

affordable production of hydrogen is needed for it to become a viable source of energy.

Catalytic reformation of hydrocarbons and alcohols appears to be a promising means of

hydrogen production, but little is known about the surface chemistry. Research on

heterogeneous catalyst and their reaction mechanisms is growing. A greater

understanding of the surface chemistry could yield cheaper, more effective catalysts. The

evolving chemistry of the surface catalyst is in need of a flexible software program to test

new surface mechanisms.

A program is developed to model chemically reacting flow through a catalytic

reactor. The reactor is represented in two-dimensional Cartesian coordinates with

negligible body forces acting on the fluid. The flow is characterized as a steady, low

Mach number, boundary layer flow of a Newtonian fluid. Basic principles of mass,

ix

species mass, momentum, and energy conservation are expressed mathematically and

simplified. These principles are transformed into the equations controlling the behavior

of the fluid and its motion through a process of applying assumptions, an order

magnitude analysis, and a unit analysis. A code is written to numerically solve the

resulting system of coupled governing equations. The methodology of constructing the

program is decomposed into developing an orthogonal computational mesh,

quantitatively defining the reactor and flow, locating chemical data and solutions,

establishing initial boundary conditions, and solving the governing equations. The

program is used to model four different flows: one with no chemistry, the second with

only gas chemistry, and the third and fourth with gas and surface chemistry. Calculated

solutions from each case are examined to confirm that the software produces reasonable

results and is operational. The software is found to predict the point of ignition when the

initial temperature is great enough to cause catalytic combustion.

1

CHAPTER 1 INTRODUCTION

The world is becoming increasingly aware of its dependence on fossil fuels. This

fuel is meeting over eighty-five percent of our country’s energy demands, which includes

everything from electricity to transportation [1]. The power of fossil fuels lies in the

atomic bonds of the hydrocarbons that make up these fuels. Energy is released by

breaking these bonds in the process of combustion. The burning of fossil fuels also

releases harmful byproducts that include: carbon monoxide, carbon dioxide, and nitrogen

oxides. The carbon released into the atmosphere is originally trapped underneath the

earth’s surface, leading to an overall increase of carbon oxides in the atmosphere. Some

believe these byproducts are leading to weather changes and health problems around the

world.

Energy extraction from fossil fuels is a relatively easy process and the fuel is

readily available in deposits beneath the earth’s surface. For these reasons, fossil fuels

have become the main source of the world’s energy production. The finite source is

nonrenewable and will eventually run out. Decreasing supplies will lead to a rise in fuel

cost and alternative forms of energy will become cheaper than fossil fuels. Economics

involved with the decrease in fuel supplies will dictate that the world turn to alternative

forms of energy.

Whether for ecological or economical reasons, the world will need to find

alternative forms of energy. Some look to the most abundant element in the universe,

hydrogen. Hydrogen is a clean, renewable source of energy that can be used in

2

combustion engines and fuel cells. Fuel cells are very efficient at producing electricity

from hydrogen with the byproduct being water. A major obstacle in this alternative fuel

is the affordable production of the energy carrier. Hydrogen rarely stands alone in its

pure form. Most of the earth’s hydrogen is bonded to oxygen and carbon, in the form of

water, alcohols, and hydrocarbons. Water is an extremely stable molecule and takes a

great deal of energy to extract hydrogen atoms. This energy must come from renewable

sources if we wish to address the problems associated with fossil fuels. Hydrogen

extraction from alcohols and hydrocarbons is much easier. However, fossil fuels are

currently the main source of hydrocarbons. About ninety-five percent of the hydrogen

supply comes from the catalytic steam reforming of natural gas according to the US

Department of Energy [2]. Natural gas is a relatively clean fossil fuel consisting mostly

of methane. But natural gas is still a finite resource that will eventually run out. A very

promising renewable source of hydrogen comes from ethanol. Ethanol is an alcohol that

can be derived from biomass such as corn. Fuels produced from biomass release carbon

into the atmosphere that is originally in the atmosphere leading to zero net-production of

carbon oxides [3]. There are many promising energy alternatives to fossil fuels.

However, fossil fuels are so entrenched in our way of life, economically and politically,

that few expect a quick transition away from fossil fuels. Most believe that hydrogen

production will initially come from fossil fuels, with a gradual transition to renewable

sources of hydrogen production.

The world’s attraction to the hydrogen economy is leading to an increased interest

in heterogeneous catalyst for converting hydrocarbons and alcohols into the energy

carrier. Surface catalysts are useful in increasing the reaction rates in combustors and

3

reformers. Catalytic combustors burn the fuel over a catalyst. This burns fuel at a lower

temperature, which decreases the amount of nitrogen oxides produced in the exhaust [4].

Catalytic reformers transform complex hydrocarbons and alcohols into hydrogen by

stripping the fuel of their hydrogen atoms. In either case, the fuel molecule is adsorbed

by the catalytic surface. The molecule forms a bond with the surface, usually through an

oxygen or carbon atom. This weakens the adjacent bonds between the oxygen or carbon

atom and the hydrogen atoms. The hydrogen atoms now begin to break off the molecule.

The product molecule will detach from the surface once it is finished reacting with the

catalyst. This leaves the surface free to adsorb a new reactant molecule. The catalyst

provides reaction pathways with lower activation energies. In effect, the catalyst lowers

the energy needed to break a molecule apart [5].

The efficient, affordable production of hydrogen is needed for this alternative fuel

to become a viable source of energy. The efficiency of a metal to catalyze a given

molecule is defined by how well the catalyst adsorbs the reactants and desorbs the

products.

Silver, for example, isn't a good catalyst because it doesn't form strong enough attachments with reactant molecules. Tungsten, on the other hand, isn't a good catalyst because it adsorbs too strongly. Metals like platinum and nickel make good catalysts because they adsorb strongly enough to hold and activate the reactants, but not so strongly that the products can't break away. [6]

The efficiency of the catalyst has no affect on the metal’s price. The price is

dependant on the demand and rarity of the metal. As mentioned above, platinum and

nickel are two common metals used in catalyst. Platinum cost approximately $1000 per

ounce, where nickel cost around $0.4 per ounce [7]. With such a large disparity between

efficiency and price, a greater understanding of the surface chemistry could lead to

cheaper, more effective catalyst.

4

Catalytic reformation of hydrocarbons and alcohols appears to be a promising

means of hydrogen production, but little is know about the surface reactions. Surface

catalysts are not fully understood because the chemistry around the surface is difficult to

measure, especially in normal operating conditions. In the past, catalysts were treated as

a black box. The black box representation of the catalyst usually consists of one global

surface reaction or a small series of reduced mechanisms. Modifications to the black box

can be made until the model accurately reproduces the experimental data. While this

method is adequate for engineering applications, it does not accurately represent the

chemistry involved [4]. Many studies have recently taken place in attempts to understand

the surface reaction mechanisms of the heterogeneous catalyst [8-11]. The studies are

mostly concerned with determining the reaction pathways and the step-by-step chemical

degradation process of molecules. This is leading to new chemical reactions being added

to the surface chemistry. The evolving chemistry of the surface catalyst is in need of a

flexible software program to test the new mechanisms being added.

A program adaptable to the changing surface chemistry is developed in this study.

The program models a two-dimensional, chemically reacting flow though a catalytic

reactor. The fluid motion is characterized as a steady, low Mach number, boundary layer

flow. The catalytic reactor consists of a heterogeneous catalyst covering the inside

surface of a pipe or channel. The fluid motion is modeled as a flow through two flat

plates with a pressure gradient. The two flat plates are modeled as catalytic surfaces and

are identical. Basic principles of mass, species mass, momentum, and energy

conservation are employed to generate the model. These principles are expressed

mathematically and simplified for this specific problem. The process of reducing the

5

principles into the governing equations consists of applying assumptions, an order

magnitude analysis, and a unit analysis.

A software code is written to numerically solve the resulting governing equations.

The calculated solutions thermodynamically and kinetically define the fluid and its

motion. The code is written in MATLAB, a programming language created by

MathWorks that is used in many chemical flow simulations. MATLAB provides several

built-in capabilities that make the software well suited for this problem. It’s

compatibility with Cantera being one such capability. Cantera is a free software package

developed by Professor David Goodwin at the California Institute of Technology to solve

problems concerning chemical reactions [12]. The program utilizes Cantera software to

manage the chemistry. The methodology of the program’s development includes the

creation of an orthogonal computational mesh to resolve the equations. Then the

establishment of parameters and conditions that quantify the reactor and fluid flow is

performed. The location of input and output data is defined and initial boundary

conditions are set. Finally the governing equations are solved and the flow in the

catalytic reactor is modeled. The program is tested with four different cases: one with no

chemistry, another with only gas chemistry, and two with gas and surface chemistry.

Each case is modeled and the results are examined to confirm that the software produces

reasonable results and is operational. In the future, calculated solutions can be compared

to experimental measurements.

New surface mechanisms can be tested with the program resulting from this study.

Improved chemical kinetic data are updated in Cantera. The new chemistry is processed

by Cantera and incorporated into the program. Any change in the chemistry being used

6

to model the flow is done so inside the separate software of Cantera and not the main

program. Because the data is stored separately, the program is able to remain flexible

with the type of catalyst and fuel being used. This also allows the type of reaction

pathways to change as our understanding of catalyst grows without altering the code. As

a result the program is adaptable to the varying surface reaction pathways. Comparing

the two-dimensional model to experimental data provides a means of validating the

accuracy of the new chemistry. With a better understanding, heterogeneous catalyst

might be the key to the clean, renewable source of energy.

7

CHAPTER 2 REDUCTION OF CONSERVATION EQUATIONS

The chemically reacting flow through the reactor is modeled by numerically

solving the governing equations. Four of the equations are derived on the principles of

mass, momentum, and energy conservation [13]. The velocity field, pressure, and

temperature in the reactor are determined with these four equations. Knowing two

independent thermodynamic properties would adequately model the flow if it were not

for the chemistry taking place. The catalytic surface is expected to induce chemical

activity changing the fluid composition. A set of equations is needed to determine this

changing chemical composition. The species continuity equations satisfy this need and

are called upon to calculate the composition of the flow. One equation is needed to

determine the mass fraction of a single atom or molecule. As a result, the number of

equations inside this set is equal to the number of species used to model the flow, denoted

as N.

The mass, momentum, species and energy equations along with an equation of state

are all that is needed to determine flow properties through out the reactor. These

governing equations are coupled to one another in several different ways. All of them

contain properties dependent on the flow variables. For example, the momentum,

species, and energy equations contain transport properties such as viscosity, diffusion

coefficients, and thermal conductivity. Most of these properties are dependent on the

pressure, temperature, and composition of the flow. Properties dependent on the flow

variables indirectly couple the equations to one another. The equations are also directly

8

coupled to one another. Vertical gradients of the species mass fraction not only appear in

the set of N species continuity equations, but also the final form of the energy equation.

In addition to this, the velocity and velocity gradients can be found in all of the equations.

This makes for a group of highly coupled equations that control the behavior of the flow.

Equations of mass, momentum, species, and energy conservation are broken down

and modified to reflect this specific model while Cantera processes the equation of state

for an ideal gas. Conservation equations are transformed into the governing equations by

applying assumptions characterizing the reactor and flow. Governing equations are

individually examined in an order magnitude analysis after the assumptions are made.

Dominant terms in a given equation are found by comparing their magnitude to the

magnitude of other terms in the equation. Neglecting the weak terms and retaining the

strong terms further reduce the equations. A unit analysis or unit check is applied to the

resulting system of equations to ensure the validity of the equations. The process also

establishes the units of each variable, property, and solution.

Applying Assumptions

Turns goes through a similar process of simplifying the conservation equations for

a steady one-dimensional flow [14]. Instead of one dimension, the computational space

of the catalytic reactor is modeled in two-dimensional orthogonal space. These two

dimensions are the rectangular coordinates x and y, which represent the axial direction

and vertical direction respectfully. Once the governing equations are reduced to their

two-dimensional form, they are simplified by making assumptions about the fluid and its

motion. The chemically changing fluid is always considered a Newtonian fluid, which

carries many assumptions with it. Most importantly of which is that the shear stress is

linearly proportional to the rate of deformation. Another important assumption concerns

9

the fluid’s motion. The flow is modeled as a steady-state flow, meaning all fluid

properties are independent of time. As a result, a partial derivative of any quantity with

respect to time is zero. More assumptions are made in order to reduce the governing

equations and are discussed during that process below. For the most part, the analysis

mirrors that of a boundary layer flow. However, the catalytic surface creates density

variations in the flow and compressibility must not be ignored.

Continuity Equation

The analysis begins with the reduction of the continuity or mass equation. The

continuity equation is a mathematical representation of the conservation of mass that

states that mass cannot be created or destroyed. In a Eulerian method of description, the

conservation of mass is described as the time rate of change of mass in a control volume

being equal to the net flux of mass through the control surface. Equation 2-1 is the vector

form of the continuity equation.

( ) 0=⋅∇+∂∂ V

tρρ (Equation 2-1)

The steady flow assumption leads to the partial derivative with respect to time

being zero. The first term in Equation 2-1 is dropped as a result, leaving only the mass

flux in vector notation. The catalytic reactor is being modeled in a two-dimensional

space. Therefore, the mass flux is written out into its two-dimensional form with u and v

representing the x and y component of the velocity, respectfully [15].

( ) ( ) 0u v

x yρ ρ∂ ∂

+ =∂ ∂

(Equation 2-2)

Further simplification is restricted due to the fact that density (defined as ρ)

variations occur in the flow. Equation 2-2 represents the continuity equation for the flow

10

field being modeled. This equation proves to be very important in the reduction of the

other governing equations. However, it is not the form used by the program. The

computer code uses the mass equation to determine the vertical velocity and some

mathematical manipulation is needed before the equation reaches its final form below.

( )uv vy x y

ρ ρρ∂∂ ∂

= − −∂ ∂ ∂

(Equation 2-3)

Species Continuity Equations

Much like the continuity equation, the species continuity equation requires that the

rate of gain of a single species mass in a control volume equals the net flux of the species

mass in through the control surface. Dissimilarity in the two equations arises due to the

chemistry. Instead of equaling zero, it equals the net chemical production of that species

in the control volume. The continuity equation of species i is shown as Equation 2-4 and

the set consists of one equation for each species. The time rate of change of the species

mass is zero because the flow is steady. The species mass flux is expanded out into its

two-dimensional Cartesian coordinate form and the result is Equation 2-5.

( )ii

i mmtY ′′′=′′⋅∇+

∂∂

&&ρ

(Equation 2-4)

( ) ( ),, i yi xi

mmm

x y

′′′′ ∂∂′′′+ =

∂ ∂

&&& (Equation 2-5)

On the right hand side of the species continuity equation lies the net chemical

production of species i in the control volume ( im′′′& ). This is determined using Cantera,

which gives the chemical production in moles. Therefore, the species mass chemical

production is replaced with the molar chemical production times the molecular weight of

11

the species. The species mass entering the control volume, know as the species mass

flux, transpire as a result of two modes, bulk flow and diffusion [14].

( ) ( ),, i yi xi i

mmMW

x yω

′′′′ ∂∂+ =

∂ ∂

&&& (Equation 2-6)

, , ,i x i i x Diffm Yu mρ′′ ′′= +& & (Equation 2-7a)

, , ,i y i i y Diffm Y v mρ′′ ′′= +& & (Equation 2-7b)

The first term in Equations 2-7a and 2-7b is the mass flux due to the bulk flow. It

is equal to the product of the density, species mass fraction, and fluid velocity component

corresponding to the direction of the mass flux. The second term is the mass flux due to

diffusion. The species mass flux can now be placed in the reduced species continuity

equation and the two modes separated from each other.

( ) ( ) ( ) ( ), , , ,i i i x Diff i y Diff i iYu Y v m m MWx y x yρ ρ ω∂ ∂ ∂ ∂′′ ′′+ + + =

∂ ∂ ∂ ∂&& & (Equation 2-8)

The chain rule is applied to the bulk flow terms and the process is shown in

Equation 2-9. This leaves the continuity equation being multiplied by the mass fraction

plus two mass fraction gradient terms being multiplied by the density and velocity. The

continuity equation is equal to zero via Equation 2-2. After dropping the mass equation

and replacing the two bulk flow terms with Equation 2-9, the species continuity equation

reduces to Equation 2-10.

( ) ( ) ( ) ( )

_

i ii i i

Continuity Equation

u v Y YYu Y v Y u vx y x y x y

ρ ρρ ρ ρ ρ

∂ ∂⎡ ⎤ ∂ ∂∂ ∂+ = + + +⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦144424443

(Equation 2-9)

( ) ( ), , , ,i i

i x Diff i y Diff i iY Yu v m m MWx y x y

ρ ρ ω∂ ∂ ∂ ∂′′ ′′+ + + =∂ ∂ ∂ ∂

&& & (Equation 2-10)

12

Diffusion is a result of concentration gradients, temperature gradients, pressure

gradients, and uneven body forces. Ordinary diffusion from concentration gradients is

the only mode of diffusion considered in this model. The species mass diffusion is

approximated using a mixture-averaged diffusion coefficient [14]. The mass diffusion

terms are replaced with Equations 2-11a and 2-11b inside the governing equation.

Equation 2-12 is the species continuity equation after all the assumptions are applied.

Further simplification is possible with an order magnitude analysis.

, ,i

i x Diff imYm Dx

ρ ∂′′ = −∂

& (Equation 2-11a)

, ,i

i y Diff imYm Dy

ρ ∂′′ = −∂

& (Equation 2-11b)

i i i iim im i i

Y Y Y Yu v D D MWx y x x y y

ρ ρ ρ ρ ω⎛ ⎞∂ ∂ ∂ ∂∂ ∂⎛ ⎞+ + − + − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

& (Equation 2-12)

Momentum Equations

The momentum or Navier-Stokes equation is analogous to Newton’s law of

momentum conservation. The momentum equation states that the rate of change of linear

momentum per unit volume equals the net momentum flux through that volume plus the

sum of forces acting on the volume. This is mathematically written in vector form as

Equation 2-13. Forces acting on the control volume are broken up into the surface forces

and body forces. Surface forces are defined as the divergence of the stress tensor. The

stress tensor is shown below in its rectangular, two-dimension form as Equation 2-14.

( ) ( ) BFVVtV ρσρρ

+⋅∇=⋅∇+∂

∂ (Equation 2-13)

xx xy xx xy

yx yy yx yy

pp

σ σ τ τσ

σ σ τ τ−⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

sr (Equation 2-14)

13

The stress tensor is made up of the shear stress acting on the surface plus the

pressure acting normal to the volume. Again, the flow is considered steady with

negligible body forces. Therefore, the time derivative and body force terms are dropped.

The vector form of the momentum equation is separated into its x-component and y-

component equations. Equation 2-15a represents the two-dimensional Cartesian

momentum equations in the x-direction, while Equation 2-15b is in the y-direction.

( ) ( ) yxxxuu uvx y x y

σρ ρ σ ∂∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ (Equation 2-15a)

( ) ( ) xy yyvu vvx y x y

σ σρ ρ ∂ ∂∂ ∂+ = +

∂ ∂ ∂ ∂ (Equation 2-15b)

The x-momentum equation is used as one of the governing equations in the

computer program. Equation 2-15b, on the other hand, is not explicitly used in the

computer code. It is used to gain some insight into the behavior of the pressure. Both

equations simplify in a similar manner so both are reduced collectively. In this process,

the x-component equation is given first followed by the y-component equation. The

chain rule is applied to the momentum convection on the left hand side of the two Navier-

Stokes equations.

( ) ( )

_

yxxx

Continuity Equation

u v u uu u vx y x y x y

σρ ρ σρ ρ∂∂ ∂⎡ ⎤ ∂∂ ∂

+ + + = +⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦144424443

(Equation 2-16a)

( ) ( )

_

xy yy

Continuity Equation

u v v vv u vx y x y x y

σ σρ ρρ ρ

∂ ∂∂ ∂⎡ ⎤ ∂ ∂+ + + = +⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦144424443

(Equation 2-16b)

This results in the continuity equation being multiplied by the x-velocity in

Equation 2-16a and by the y-velocity in Equation 2-16b. The continuity equation is equal

14

to zero and the first term in these equations is dropped accordingly. The two momentum

equations are reduced to Equations 2-17a and 2-17b.

yxxxu uu vx y x y

σσρ ρ∂∂∂ ∂

+ = +∂ ∂ ∂ ∂

(Equation 2-17a)

xy yyv vu vx y x y

σ σρ ρ

∂ ∂∂ ∂+ = +

∂ ∂ ∂ ∂ (Equation 2-17b)

The flow through the reactor consists of a Newtonian fluid. The stress acting on a

Newtonian fluid has no preferred direction, meaning the stress tensor matrix is symmetric

and its components are defined below. By definition the shear stress of a Newtonian

fluid is proportional to the rate of deformation. Shear stresses of a Newtonian fluid are

written below as functions of the velocity gradients [15].

xx xx pσ τ= − (Equation 2-18a)

yy yy pσ τ= − (Equation 2-18b)

xy yx xy yxσ σ τ τ= = = (Equation 2-18c)

223xx

u u vx x y

τ µ µ⎛ ⎞∂ ∂ ∂

= − +⎜ ⎟∂ ∂ ∂⎝ ⎠ (Equation 2-19a)

223yy

v u vy x y

τ µ µ⎛ ⎞∂ ∂ ∂

= − +⎜ ⎟∂ ∂ ∂⎝ ⎠ (Equation 2-19b)

xyu vy x

τ µ⎛ ⎞∂ ∂

= +⎜ ⎟∂ ∂⎝ ⎠ (Equation 2-19c)

Replacing the stress components with their definitions above and separating the

pressure gradient, the two momentum equations become,

15

223

u u pu vx y x

u u v u vx x x x y y y x

ρ ρ

µ µ µ

∂ ∂ ∂+ = − +

∂ ∂ ∂

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ − + + +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(Equation 2-20a)

223

v v pu vx y y

u v v u vx y x y y y x y

ρ ρ

µ µ µ

∂ ∂ ∂+ = − +

∂ ∂ ∂

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(Equation 2-20b)

Equations 2-20a and 2-20b represent the momentum equation in the x-direction and

y-direction respectfully. Both are reduced to their final form via an order of magnitude

comparison.

Energy Equation

The Energy Equation requires that the rate of change per unit volume is equal to the

net energy flux into that volume due to convection, heat, and work [13].

( ) BFVVqVeVVet

⋅+⋅∇+⋅∇−=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅∇+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂ ρσρρ

22

22

(Equation 2-21)

From left to right in Equation 2-21, the first term is the time rate of change of

energy, which is zero because of the steady flow assumption. The second term represents

the flux of energy due to convection and equals the heat transferred into the control

volume plus the work done by the surface forces and the body forces. The work done by

the surface forces is determined using the stress tensor of Equation 2-14. The body force

is assumed to be negligible; therefore, the work done by the body force is neglected.

Equation 2-21 is written out in its two-dimensional Cartesian form with the assumed

simplifications.

16

( ) ( )

2 2

2 2

yxxx xy yx yy

V Vu e v ex y

qq u v u vx y x y

ρ ρ

σ σ σ σ

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂+ + + =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∂∂ ∂ ∂− − + + + +∂ ∂ ∂ ∂

(Equation 2-22)

The stress components are replaced with the shear stress and pressure. The

pressure is separated from the shear stress terms, leaving two pressure work terms at the

end of the energy equation.

( ) ( ) ( ) ( )

2 2

2 2

yxxx xy xy yy

V Vu e v ex y

qq u v u v up vpx y x y x y

ρ ρ

τ τ τ τ

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂+ + + =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∂∂ ∂ ∂ ∂ ∂

− − + + + + − −∂ ∂ ∂ ∂ ∂ ∂

(Equation 2-23)

Placing the x and y pressure work terms on the far right of Equation 2-23 into the

corresponding x and y energy convection terms on the left, Equation 2-23 becomes

Equation 2-24. The internal energy and pressure is replaced by the enthalpy, defined in

Equation 2-25 as the internal energy plus the product of the pressure and the specific

volume. The energy transfer due to the shear stress work is replaced by a variable called

(τ_work) to save space. Simplification of this term is possible via an order of magnitude

analysis of the governing equation, but first the energy convection and heat flux terms are

reduced. As of now the energy equation can be written out as Equation 2-26.

( )2 2

_2 2

yx qqp V p Vu e v e workx y x yρ ρ τ

ρ ρ∂⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ∂∂ ∂

+ + + + + =− − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ (Equation 2-24)

ph eρ

= + (Equation 2-25)

( )2 2

_2 2

yx qqV Vu h v h workx y x y

ρ ρ τ∂⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ∂∂ ∂

+ + + = − − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ (Equation 2-26)

17

First the two energy convection terms on the left hand side of Equation 2-26 are

simplified. These expressions consist of an enthalpy flux and kinetic energy flux, both

due to bulk flow. The two convection terms are separated into enthalpy convection and

kinetic energy convection. Performing the chain rule on the two kinetic energy

convection terms produces four separate terms. Equation 2-28 illustrates the process.

Two of these terms are combined to form the kinetic energy multiplying the continuity

equation, which equals zero. After dropping this term, the kinetic energy flux is replaced

with the last two terms of the equation above and the energy equation now takes the form

of Equation 2-29.

( ) ( )

( )

2 2

2 2

_yx

uh vh V Vu vx y x y

qq workx y

ρ ρρ ρ

τ

⎡ ⎤ ⎡ ⎤∂ ∂ ⎛ ⎞ ⎛ ⎞∂ ∂+ + + =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∂∂− − +∂ ∂

(Equation 2-27)

( ) ( ) ( ) ( )

2 2

22 2

_

2 2

2 2 2Continuity Equation

V Vu vx y

u vV u vV Vx y x y

ρ ρ

ρ ρ ρ ρ

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂+ =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∂ ∂⎧ ⎫ ∂ ∂+ + +⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭144424443

(Equation 2-28)

( ) ( ) ( ) ( )

( )

2 2

2 2

_yx

uh vh u vV Vx y x y

qq workx y

ρ ρ ρ ρ

τ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂∂∂

− − +∂ ∂

(Equation 2-29)

Moving over to the right hand side of the energy equation, the heat flux terms are

now simplified. The heat flux is determined using Fourier’s Law of Heat conduction plus

the flux of enthalpy [14]. The enthalpy flux here is due only to diffusion. The flux of

enthalpy from the bulk flow is already accounted for in the convection term. The vector

equation of the heat flux is broken up into the two Cartesian coordinate components, x

18

and y. Equations 2-7a and 2-7b are used to replace the species diffusion mass flux inside

the sum of the heat flux.

( ),1

N

i iDiffi

q k T m h=

′′= − ∇ +∑v uv uuv

& (Equation 2-30)

( ) ( ) ( ), , ,1 1 1

N N N

x i x Diff i i x i i ii i i

T Tq k m h k m h uY hx x

ρ= = =

∂ ∂′′ ′′= − + = − + −∂ ∂∑ ∑ ∑& & (Equation 2-31a)

( ) ( ) ( ), , ,1 1 1

N N N

y i y Diff i i y i i ii i i

T Tq k m h k m h vY hy y

ρ= = =

∂ ∂′′ ′′= − + = − + −∂ ∂∑ ∑ ∑& & (Equation 2-31b)

Taking the partial derivative of the two equations above produces Equation 2-32a

and 2-32b. The partial derivative is not affected by the species sum and therefore can be

moved inside the sum. Similarly, the mass flux due to the bulk flow is not affected by the

sum and can be moved outside of the sum. The sum located inside the partial derivative

of the last term contains the product of the species mass fraction and species enthalpy.

Since the species enthalpy is given on a mass basis, the sum is equal to the specific

enthalpy of the flow. The last term can be rewritten as the partial derivative of the

density, velocity component, and enthalpy product.

( ),1 1

N Nx

i x i i ii i

q Tk m h u Y hx x x x x

ρ= =

∂ ∂ ∂ ∂ ∂ ⎡ ⎤⎡ ⎤ ⎛ ⎞′′⎡ ⎤= − + −⎜ ⎟ ⎢ ⎥⎣ ⎦⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎣ ⎦∑ ∑& (Equation 2-32a)

( ),1 1

N Ny

i y i i ii i

q Tk m h v Y hy y y y y

ρ= =

∂ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂ ⎡ ⎤′′⎡ ⎤= − + −⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦∂ ∂ ∂ ∂ ∂ ⎣ ⎦⎣ ⎦ ⎝ ⎠∑ ∑& (Equation 2-32b)

[ ],1

Nx

i x ii

q Tk m h uhx x x x x

ρ=

∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎛ ⎞′′⎡ ⎤= − + −⎜ ⎟⎣ ⎦⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠∑ & (Equation 2-33a)

[ ],1

Ny

i y ii

q Tk m h vhy y y y y

ρ=

∂ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂′′⎡ ⎤= − + −⎜ ⎟⎢ ⎥ ⎣ ⎦∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠∑ & (Equation 2-33b)

19

The two heat flux terms of the energy equation are replaced with Equations 2-33a

and 2-33b above. Once this is complete the energy equation takes the form of Equation

2-34 shown below.

( )uhxρ∂∂

( )vhyρ∂

+∂

( ) ( )

[ ]

2 2

, ,1 1

2 2N N

i x i i y ii i

V Vu vx y

T Tk k m h m hx x y y x y

uhx

ρ ρ

ρ

= =

∂ ∂+ + =

∂ ∂

⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎛ ⎞′′ ′′⎡ ⎤⎡ ⎤+ − − +⎜ ⎟⎜ ⎟⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎝ ⎠∂∂

∑ ∑& &

[ ]vhyρ∂

+∂

( )_ workτ+

(Equation 2-34)

The enthalpy convection cancels with the modified enthalpy diffusion of the heat

flux. The chain rule is performed on the expressions inside the two sums. The process,

shown in Equations 2-35a and 2-35b, leaves the species enthalpy times the partial

derivative of the species mass flux plus the species mass flux times the enthalpy gradient.

Equations 2-7a and 2-7b are used again to replace this species mass flux. This procedure

takes the original partial derivatives and splits it into three terms each.

, ,, , , ,

i x i xi i ii x i i i x i i i x Diff

m mh h hm h h m h uY mx x x x x x

ρ′′ ′′∂ ∂∂ ∂ ∂∂ ′′ ′′ ′′⎡ ⎤ = + = + +⎣ ⎦∂ ∂ ∂ ∂ ∂ ∂

& && & & (Equation 2-35a)

, ,, , , ,

i y i yi i ii y i i i y i i i y Diff

m mh h hm h h m h vY my y y y y y

ρ′′ ′′∂ ∂∂ ∂ ∂∂ ′′ ′′ ′′⎡ ⎤ = + = + +⎣ ⎦∂ ∂ ∂ ∂ ∂ ∂

& && & & (Equation 2-35b)

Replacing the two partial derivatives with their expanded expressions above, the

process of simplifying the sums can begin. For the last two expressions in Equations 2-

36a and 2-36b, the gradient of the species enthalpy is equal to the product of the species

specific heat and the species temperature gradient. Every species comprising the fluid at

a given point in the flow is assumed to have the same temperature, which means the

temperature gradient can be moved outside of the sums. This process is done for all four

of the terms containing enthalpy gradients.

20

,, , ,

1 1 1 1

N N N Ni x i i

i x i i i i x Diffi i i i

m h hm h h uY mx x x x

ρ= = = =

′′∂⎛ ⎞ ∂ ∂∂ ⎛ ⎞ ⎛ ⎞⎛ ⎞′′ ′′⎡ ⎤ = + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠∑ ∑ ∑ ∑

&& & (Equation 2-36a)

,, ,

1 1 1 1y

N N N Ni y i i

i y i i i i Diffi i i i

m h hm h h vY my y y y

ρ= = = =

′′∂⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂∂ ′′ ′′⎡ ⎤ = + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠∑ ∑ ∑ ∑

&& & (Equation 2-36b)

( )1 1

i

N Ni

i i p pi i

h T TuY u Y c ucx x x

ρ ρ ρ= =

∂ ∂ ∂⎛ ⎞ = =⎜ ⎟∂ ∂ ∂⎝ ⎠∑ ∑ (Equation 2-37a)

( )1 1

i

N Ni

i i p pi i

h T TvY v Y c vcy y y

ρ ρ ρ= =

⎛ ⎞∂ ∂ ∂= =⎜ ⎟∂ ∂ ∂⎝ ⎠

∑ ∑ (Equation 2-37b)

( ), , , ,1 1 1

i i

N N Ni i

i x Diff i x Diff p im pi i i

h YT Tm m c D cx x x x

ρ= = =

∂ ∂∂ ∂⎛ ⎞ ⎛ ⎞′′ ′′= = −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∑ ∑ ∑& & (Equation 2-38a)

( ), , , ,1 1 1

i i

N N Ni i

i y Diff i y Diff p im pi i i

h YT Tm m c D cy y y y

ρ= = =

⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂′′ ′′= = −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∑ ∑ ∑& & (Equation 2-38b)

In Equations 2-37a and 2-37b, the product of the species mass fractions and

specific heats summed over every species equals the specific heat of the flow. This is

multiplied by the mass flux, which equals the flow density times the proper velocity

component. The diffusion mass flux inside the sum of the last two expressions is

approximated using the mixture-averaged diffusion equation, Equation 2-11. After

replacing the four terms with four equations above, Equations 2-36a and 2-36b are added

together and rearranged before being placed into the energy equation.

, ,1 1

,,

1 1

_

i

N N

i x i i y i p pi i

N Ni yi x i i

i im pi i

Species Continuity

T Tm h m h uc vcx y x y

mm Y YT Th D cx y x x y y

ρ ρ

ρ

= =

= =

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞′′ ′′⎡ ⎤⎡ ⎤ + = + +⎜ ⎟⎜ ⎟⎣ ⎦ ⎣ ⎦∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞⎜ ⎟′′′′ ∂ ⎛ ⎞∂⎡ ⎤ ⎡ ⎤∂ ∂∂ ∂⎜ ⎟+ − +⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝ ⎠⎜ ⎟⎝ ⎠

∑ ∑

∑ ∑

& &

&&

1442443

(Equation 2-39)

The two dimensional gradient of the species mass flux is replaced with the species

chemical production via the species continuity equation, Equation 2-6. The energy

21

equation is reduced to the Equation 2-40. Further simplification is performed with an

order magnitude analysis in the next section.

( ) ( )

( )

( )

2 2

1

1

2 2

_i

N

p p i i ii

Ni i

im pi

V Vu v T Tk kx y x x y y

T Tuc vc h MWx y

Y YT TD c workx x y y

ρ ρ

ρ ρ ω

ρ τ

=

=

∂ ∂ ⎡ ⎤∂ ∂ ∂ ∂⎡ ⎤+ = + −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦∂ ∂

− − +∂ ∂

⎛ ⎞⎡ ⎤∂ ∂∂ ∂+ +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎝ ⎠

∑

∑

(Equation 2-40)

Order of Magnitude Analysis

An order of magnitude comparison between terms in a given equation determines

which terms must be reserved and which terms can be neglected. Governing equations

that are modified for this specific model are simplified further by eliminating the

insignificant terms. It is necessary to nondimensionalize the equation prior to comparing

terms. Variables are nondimensionalized with the uniform properties of the flow entering

the reactor. Most of the properties are chosen such that the resulting magnitudes are on

the order of one. The dimensionless variables and their magnitude are shown below in

Figure 2-1.

( )

( )

( )

* 1

* 1

** ?

uuUvvU

ρρρ∞

= = Ο

= = Ο

= = Ο

( )

( )

( )

* 1

* 1

* 1

TTT

ppUρ

µµµ

∞

∞

∞

= = Ο

= = Ο

= = Ο

( )

( )

( )

* 1

* 1

xxLyyH

HL HL

δ

= = Ο

= = Ο

→ = Ο

( )

( )

( )

*

*

* 1

1

1i

imim

pp

p

kkkDDD

cc

c

∞

= = Ο

= = Ο

= = Ο

Figure 2-1. Dimensionless variables.

All but two of the dimensionless parameters have a magnitude on the order of one.

The unknown magnitude of the vertical velocity is found with the continuity equation.

The characteristic distance in the axial direction, L, is much greater than the characteristic

22

distance in the vertical direction, H. A dimensionless parameter with a very small

magnitude, denoted by О(δ), is produced when the characteristic height is divided by the

characteristic length.

Continuity Equation

Equation 2-2 is the two-dimensional continuity equation that is reduced based on

the steady flow assumption. Flow properties are replaced with their appropriate

dimensionless variables. After some algebraic rearranging, the mass equation is rewritten

in its dimensionless form.

( ) ( ) 0u v

x yρ ρ∂ ∂

+ =∂ ∂

(Equation 2-2)

( ) ( )* * * *0

* *u vL

x H yρ ρ∂ ∂

+ =∂ ∂

(Equation 2-41)

All of the known dimensionless variables have an order magnitude of one. It has

already been noted that the characteristic length is much larger than the characteristic

height. This produces a relatively small value that divides the vertical mass flux term. In

order to balance the mass equation, the dimensionless y-velocity must have the same

order magnitude as the division of the height by the length.

( )( ) ( ) ( )1 1 * 01 1

vδ

Ο+ =

Ο Ο Ο (Equation 2-42)

( )*v δ= Ο (Equation 2-43)

While the continuity equation remains unchanged, the comparison of terms reveals

that the vertical velocity of the flow is small compared to axial velocity. This is a

common result in boundary layer flow analysis. Growth of the boundary layer is dictated

by the viscosity, or momentum transfer, and does not affect the entire flow until farther

23

downstream. The flow does not consist entirely of a boundary layer flow. However, a

vertical velocity does not exist at the entrance of the reactor, on the surface, or at the

centerline of the pipe or channel. The vertical velocity remains much smaller than the

axial velocity through out the reactor because of these boundary conditions.

Species Continuity Equations

The species continuity equation is reduced based on the assumptions of a steady,

two-dimensional flow, with ordinary diffusion being the only mode of diffusion. This

equation is shown below.

i i i iim im i i

Y Y Y Yu v D D MWx y x x y y

ρ ρ ρ ρ ω⎛ ⎞∂ ∂ ∂ ∂∂ ∂⎛ ⎞+ + − + − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

& (Equation 2-12)

The dimensionless form of the species mass equation is obtained by replacing the

flow variables with their proper dimensionless counterpart. The species mass fraction is

exempt from this part of the process because it is already a dimensionless quantity that

varies between zero and one. The unknown magnitude of the mass fraction does not pose

a problem since it is found in every term on the left hand side of the equation. As a result

it affects the magnitude of each term equally. After some algebraic manipulation, the left

hand side is rewritten in its dimensionless form as Equation 2-44. The right side of the

species equation, the species chemical production, is not compared to the rest of the

equation. Neglecting this term would result in the modeling of a non-reacting flow.

Therefore, the convection and diffusion terms are the only terms considered.

2* *

2

* * * ** *

* ** * * *

i i

i iim im

Y YLu vx H y

Y YD LD DUL x x H y y

ρ ρ

ρ ρ

⎛ ⎞∂ ∂⎛ ⎞ + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎡ ⎤⎛ ⎞∂ ∂∂ ∂⎛ ⎞− + −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

(Equation 2-44)

24

Note that the mass-averaged diffusion coefficients are nondimensionalized by an

arbitrary value. This value is chosen such that the quantity of the dimensionless property

is roughly one. This ensures that the dimensionless diffusion coefficients have an order

magnitude of one, but the size of the value relative to the product of the incoming

velocity and characteristic length is unknown. The species diffusion terms cannot be

compared to the species bulk flow terms as a result. However, the comparison between

the diffusion terms inside the brackets is still possible.

( ) ( ) ( ) ( ) ( )

21 11 1D

ULδ

δ δ

⎡ ⎤⎛ ⎞⎢ ⎥Ο + Ο + Ο + ⎜ ⎟⎜ ⎟Ο Ο⎢ ⎥⎝ ⎠⎣ ⎦

(Equation 2-45)

Both of the bulk flow terms have a magnitude on the order of one. The first term

inside the brackets, corresponding to diffusion in the axial direction, also has a magnitude

of one. The second term corresponds to the diffusion in the vertical direction and has a

magnitude much greater than one. The order magnitude comparison of the species

continuity equation shows that the x-component of the species diffusion is much smaller

than the vertical diffusion and can be neglected. Information about the size of the

characteristic diffusion coefficient relative to the product of the characteristic velocity

and length is needed to determine the parameter multiplying the vertical diffusion inside

the brackets. The parameter must be very small, on the order of О(δ)2, in order for the

vertical diffusion to be of the same magnitude as the two bulk flow terms. This means

that the species mass transfer from the bulk flow is much greater than the species mass

transfer due to diffusion. Though this is most likely the case for the flow being modeled,

further restricting the flow to this assumption does not simplify the equation. The species

25

continuity equation for the flow through the reactor is now reduced down to Equation 2-

46 after dropping the axial diffusion term.

i i iim i i

Y Y Yu v D MWx y y y

ρ ρ ρ ω⎛ ⎞∂ ∂ ∂∂

+ + − =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠& (Equation 2-46)

Axial Momentum Equation

Analysis of the axial momentum equation is performed in the same manner as the

other equations. Flow properties are nondimensionalized by their characteristic variables.

The comparison begins with the momentum equation governing a steady, two-

dimensional flow of a Newtonian fluid in the x-direction. Characteristic variables are

rearranged, and the dimensionless form of the x-momentum equation is shown as

Equation 2-47. The viscous term inside the brackets is compared separately from the

momentum flux and pressure gradient terms due to the length of the expression. The

comparison of the dimensionless momentum flux and pressure gradient terms is now

possible. Excluding the vertical velocity, all of the dimensionless variables have a

magnitude on the order of one. The division of the characteristic length by the

characteristic height produces a relatively large value. This value is multiplied by the

dimensionless vertical velocity, which is a small quantity. The overall effect produces a

momentum flux and pressure gradient terms that all have the same order magnitude of

one. As a result, none of these terms is less important than the other and none of the

three can be ignored.

223

u u pu vx y x

u u v u vx x x x y y y x

ρ ρ

µ µ µ

∂ ∂ ∂+ = − +

∂ ∂ ∂

⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎡ ⎤ − + + +⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

(Equation 2-20a)

26

{ }2

* * ** * * * _* * * x

u L u p Lu v termx H y x U

ρ ρ τρ∞

∂ ∂ ∂+ = − +

∂ ∂ ∂ (Equation 2-47)

( ) ( )( ) ( ) ( ) { }21 1 1 _

x

L termU

δτ

δ ρ∞

ΟΟ + Ο = −Ο +

Ο (Equation 2-48)

Viscous terms inside the brackets are transformed into dimensionless variables and

compared to each other. Equation 2-49 represents the dimensionless form of the viscous

term. The characteristic properties are reorganized and the viscous term is now

multiplied by the inverse of the Reynolds number. The Reynolds number is a common

dimensionless parameter used to compare inertial forces to viscous forces. The Reynolds

number in Equation 2-50 is based on the length of the reactor and therefore is a

comparison of these two forces in the axial direction.

{ }2

2

_

223

x

L termU

L u u v u vU x x x x y y y x

τρ

µ µ µρ

∞

∞

=

⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎡ ⎤ − + + +⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

(Equation 2-49)

2

2

* 2 * *2 * ** * * 3 * *

* *** * *

u u L vx x x x H y

UL L u L vy H y H x

µ µµ

ρµ

∞

∞

⎧ ⎫⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂ ∂⎡ ⎤ − + +⎪ ⎪⎢ ⎥⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎣ ⎦⎪ ⎪⎨ ⎬

⎡ ⎤⎛ ⎞∂ ∂ ∂⎪ ⎪+⎢ ⎥⎜ ⎟⎪ ⎪∂ ∂ ∂⎝ ⎠⎣ ⎦⎩ ⎭

(Equation 2-50)

Magnitudes of each term that comprise the viscous momentum transfer expression

can now be compared to one another. Every expression inside the brackets is of the order

of one, except for a single term. This term is underlined twice in Equation 2-51 and has a

magnitude much greater than one. The result is a significant reduction of the viscous

term. With the exception of the highlighted term, every expression is neglected and the

complex viscous expression is simplified to just one term. Information about the

27

magnitude of the Reynolds number is needed to compare the viscous term with the rest of

the momentum equation. The inverse of the Reynolds number must have a magnitude of

О(δ)2 for the remaining viscous term to be of a similar size as the momentum flux and

pressure gradient. A large Reynolds number assumption forces the viscous term to

balance with the other terms in the equation. It also forces the inertial forces of the flow

to be more significant than the viscous forces. This is a reasonable assumption because

the Reynolds number is based on the axial direction, where inertial forces are expected to

be greater than the viscous forces [16].

( ) ( ) ( )( ) ( )

( )( )

21 11 1

Reδ δδ δ δ

⎧ ⎫⎡ ⎤⎡ ⎤ ⎛ ⎞Ο Ο⎪ ⎪⎢ ⎥Ο − Ο + + +⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎢ ⎥Ο Ο Ο⎣ ⎦ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

(Equation 2-51)

The order of magnitude comparison of the Navier-Stokes equation in the axial

direction produces a couple of useful results. The complex momentum transfer due to

viscosity is simplified to a single term. This reduces the x-momentum equation to its

final form used in the computer code. In addition, the Reynolds number must be large for

the viscous momentum transfer to be of a size comparable to the rest of the momentum

equation. The large Reynolds number result is used later in the magnitude comparison of

the y-momentum equation.

u u p uu vx y x y y

ρ ρ µ⎛ ⎞∂ ∂ ∂ ∂ ∂

+ = − + ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ (Equation 2-52)

( )21Reδ

=Ο

(Equation 2-53)

Vertical Momentum Equation

The momentum equation in the vertical direction is not used directly in the

computer program. It is used to gain some insight into the behavior of the pressure

28

through out the flow. The process of comparing terms in the y-momentum equation is the

same as the process for the x-momentum equation. The Navier-Stokes equation for a

steady, two-dimensional flow of a Newtonian fluid is nondimensionalized by the

characteristic scales. The resulting dimensionless equation takes the form of Equation 2-

54 after some algebraic manipulation. Again, the viscous term is broken down separately

because of the length of the expression.

223

v v pu vx y y

u v v u vx y x y y y x y

ρ ρ

µ µ µ

∂ ∂ ∂+ = − +

∂ ∂ ∂

⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪+ + − +⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

(Equation 2-20b)

{ }2

* * ** * * * _* * * y

v L v L p Lu v termx H y H y U

ρ ρ τρ∞

∂ ∂ ∂+ = − +

∂ ∂ ∂ (Equation 2-54)

Comparison of the momentum flux and pressure terms is postponed until the

magnitude of the viscous momentum transfer is known. After the flow properties are

converted to their dimensionless form, the viscous term becomes Equation 2-55. Similar

to the viscous term in the axial Navier-Stokes equation, the parameter multiplying the

viscous term is the Reynolds number. Analysis of the axial momentum equation

determined that the Reynolds number is on the order of 1/О(δ)2. Knowing the magnitude

of the Reynolds number allows the viscous term to be compared to the momentum flux

and pressure gradient terms. But first an order magnitude comparison must be performed

on all the terms inside the brackets.

2

2

* * ** 2 ** * * * *

2 * *** 3 * *

L u v L vx H y x H y y

UL L u L vH y x H y

µ µµ

ρµ

∞

∞

⎧ ⎫⎡ ⎤⎛ ⎞ ⎡ ⎤∂ ∂ ∂ ∂ ∂+ + −⎪ ⎪⎢ ⎥⎜ ⎟ ⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦⎪ ⎣ ⎦ ⎪

⎨ ⎬⎡ ⎤⎛ ⎞∂ ∂ ∂⎪ ⎪+⎢ ⎥⎜ ⎟⎪ ⎪∂ ∂ ∂⎝ ⎠⎣ ⎦⎩ ⎭

(Equation 2-55)

29

( ) ( ) ( ) ( ) ( ) ( )( )

1 1 1 1 1Re

δδ

δ δ δ δ

⎧ ⎫⎡ ⎤ ⎛ ⎞ ⎡ ⎤Ο⎪ ⎪+Ο + − Ο +⎜ ⎟⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟Ο Ο Ο Ο⎪ ⎪⎣ ⎦ ⎝ ⎠ ⎣ ⎦⎩ ⎭ (Equation 2-56)

As a reminder, all of the dimensionless properties, excluding the vertical velocity,

have a magnitude of one. The dimensionless vertical velocity and the division of the

characteristic height by the length have a magnitude much less than one. The result is the

magnitude of all but a single term inside the brackets of Equation 2-55 reducing to an

order of 1/О(δ). The single remaining term, underlined twice in Equation 2-56, has a

magnitude of О(δ). This is much smaller than the other terms and can be neglected. As a

result, the dimensionless viscous term in the vertical momentum equation has an order

magnitude of 1/О(δ). The process of nondimensionalizing the equation is complete. An

analysis of the vertical momentum equation is now possible with the knowledge of the

viscous term’s magnitude.

{ }* * * 1* * * * _* * * Re y

v L v L pu v termx H y H y

ρ ρ τ∂ ∂ ∂+ = − +

∂ ∂ ∂ (Equation 2-57)

( ) ( )( ) ( ) ( ) ( ) ( )

21 1δδ δ δ

δ δ δ⎧ ⎫Ο ⎪ ⎪Ο + Ο = − +Ο ⎨ ⎬Ο Ο Ο⎪ ⎪⎩ ⎭

(Equation 2-58)

The magnitude of every term, with the exception of one, is a very small quantity,

О(δ). The exception is underscored in Equation 2-58 and is relatively large compared to

the rest of the equation. The highlighted value corresponds to the pressure gradient in the

y-direction. The pressure gradient is considerably larger than the other terms and

dominates this equation. Neglecting all of the irrelevant terms, the dimensionless

pressure gradient is the only term remaining. This results in the pressure gradient in the

y-direction being essentially zero.

30

0py∂

≈∂

(Equation 2-59)

The order of magnitude comparison of the vertical momentum equation reveals that

pressure through out the flow is a weak function of the vertical position. Pressure can be

treated as strictly a function of the axial direction, x.

Energy Equation

The magnitude comparison of the energy equation begins with Equation 2-40. The

equation is simplified based on the assumptions of a steady, two-dimensional flow with

negligible work done on the fluid by the body forces. Magnitudes of each similar

expression are compared to one another separately. First, the two kinetic energy

convection terms are compared to each other. Then the two heat conduction terms are

compared and so on. The work done by the shear stress is evaluated last.

( ) ( )

( )

( )

2 2

1

1

2 2

_i

N

p p i i ii

Ni i

im pi

V Vu v T Tk kx y x x y y

T Tuc vc h MWx y

Y YT TD c workx x y y

ρ ρ

ρ ρ ω

ρ τ

=

=

∂ ∂ ⎡ ⎤∂ ∂ ∂ ∂⎡ ⎤+ = + −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦∂ ∂

− − +∂ ∂

⎛ ⎞⎡ ⎤∂ ∂∂ ∂+ +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎝ ⎠

∑

∑

(Equation 2-40)

Analysis of the energy convection on the left hand side of Equation 2-40 is a

qualitative process. The couple of convection terms are not nondimensionalized, but a

simplification is possible with the understanding of the behavior of the velocity field.

The energy being transferred consists of kinetic energy, which is proportional to the

magnitude of the flow velocity squared. Balancing the continuity equation proved that

the vertical component of the velocity is much smaller than the axial component.

Squaring both components only make this difference more pronounce.

31

( ) ( )2 22 2 2 2 2 2 2 2 2* * 1V u v U u v U V uδ⎡ ⎤⎡ ⎤= + = + = Ο +Ο → =⎣ ⎦ ⎣ ⎦ (Equation 2-60)

The contribution of the vertical component to the magnitude of the flow velocity is

neglected. The kinetic energy is calculated using only the axial component of the flow

velocity and the kinetic energy convection is rewritten as Equation 2-61. The two partial

derivatives of the x-velocity squared are performed to obtain the final form of the kinetic

energy transfer.

( ) ( )2 22

2 2u uu v u uu uvx y x y

ρ ρ ρ ρ∂ ∂ ∂ ∂

+ = +∂ ∂ ∂ ∂

(Equation 2-61)

Moving over to the right hand side of Equation 2-40, the two heat conduction terms

are now compared in a much more quantitative procedure. Variables are replaced with

their dimensionless counterparts in Equation 2-62 so a dominant term can be found.

These terms cannot be compared to the rest of the energy equation because the magnitude

of the parameter outside of the brackets of Equation 2-63 is unknown. Comparing the

magnitudes of the two expressions reveals the dominant term. It is underlined twice and

corresponds to the heat conduction in the vertical direction. Being much smaller than

vertical conduction, the heat conduction in the axial direction is neglected in the final

form of the energy equation.

2

2 2

* ** ** * * *

T Tk kx x y y

k T T L Tk kL x x H y y∞ ∞

⎡ ⎤∂ ∂ ∂ ∂⎡ ⎤ + =⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂⎡ ⎤ +⎨ ⎬⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎩ ⎭

(Equation 2-62)

( )( )22

11k TL δ∞ ∞

⎧ ⎫⎛ ⎞⎪ ⎪Ο + ⎜ ⎟⎨ ⎬⎜ ⎟Ο⎪ ⎪⎝ ⎠⎩ ⎭

(Equation 2-63)

32

The energy transfer due to the diffusion of enthalpy is the next term to be reduced.

Every variable and property is replaced with its dimensionless representation in Equation

2-64. The order magnitude analysis reveals that the second term, underscored in

Equation 2-65, is much larger than the other. The lesser of the two terms is the enthalpy

diffusion in the axial direction and is neglected in the energy equation.

1

2* *

2 21

* *** * * *

i

i

Ni i

im pi

Np i i

im pi

Y YT TD cx x y y

Y YD c T T L TD cL x x H y y

ρ

ρ ρ

=

∞ ∞

=

⎛ ⎞⎡ ⎤∂ ∂∂ ∂+ =⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎝ ⎠

⎛ ⎞⎡ ⎤∂ ∂∂ ∂+⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎝ ⎠

∑

∑ (Equation 2-64)

( ) ( )( )22

1

11 1N

p

i

D c TL

ρδ

∞ ∞

=

⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥Ο Ο + ⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟Ο⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠

∑ (Equation 2-65)

The last expression to be analyzed in the energy equation is the energy transfer due

to work done by the shear stress. It is greatly simplified by comparing the magnitude of

each term that comprises the work. Shear stress is defined as Equations 2-19a-c for a

Newtonian fluid. The energy transfer expression becomes Equation 2-67 after the shear

stress definitions are substituted. The flow properties are replaced with their proper

dimensionless variables and characteristic scales.

( ) ( ) ( )_ xx xy xy yywork u v u vx y

τ τ τ τ τ∂ ∂= + + +∂ ∂

(Equation 2-66)

223

223

u u v u vu u vx x x y y x

v u v u vv v uy y x y y x

µ µ µ

µ µ µ

⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂− + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂− + + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

(Equation 2-67)

33

2

2

2 2

2 2

* 2 * *2 * * * ** 3 * *

* * ** ** *

* 2 * *2 * * * ** 3 * *

* * ** ** *

u u L vu ux x H yU

L x L u vvH y x

v H u vv vy L x yU L

L H y u H vuy L x

µ µµ

µ

µ µµ

µ

∞

∞

⎡ ⎤⎛ ⎞∂ ∂ ∂− + +⎢ ⎥⎜ ⎟∂ ∂ ∂∂ ⎝ ⎠⎢ ⎥ +⎢ ⎥∂ ⎛ ⎞∂ ∂⎢ ⎥+⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞∂ ∂ ∂− + +⎢ ⎥⎜ ⎟∂ ∂ ∂∂ ⎝ ⎠⎢ ⎥

⎢ ⎥∂ ⎛ ⎞∂ ∂⎢ ⎥+⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

(Equation 2-68)

The dimensionless form of the viscous work can now be used to determine which

terms dominate the energy transfer expression. Most of the terms have an order

magnitude of one. There are two terms that do not have this magnitude. One is very

small with a magnitude on the order of О(δ)2, while the other has a large magnitude and

is underlined twice. The highlighted term is substantially greater than the other terms

inside the brackets and can be considered the only dominant term. Neglecting all the

other weak terms, the energy transfer due to the shear stress work is reduced to Equation

2-70.

( ) ( ) ( )( )

( )( ) ( )

( )( )

( )( )

( )( ) ( )

( )( )

2

2

2 2 2 2 2

2 2 2 2 2

1 1

1

UL

δ δδ

δ δµ

δ δ δ δ

δ δ δ δ δ

∞

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞Ο ΟΟ − Ο + + +Ο +⎪ ⎪⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Ο Ο⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪

⎨ ⎬⎡ ⎤⎛ ⎞⎛ ⎞Ο Ο Ο Ο⎪ ⎪⎢ ⎥⎜ ⎟− + + +⎜ ⎟⎪ ⎪⎢ ⎥⎜ ⎟⎜ ⎟Ο Ο Ο Ο Ο⎜ ⎟⎝ ⎠⎪ ⎪⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭

(Equation 2-69)

uuy y

µ⎡ ⎤∂ ∂⎢ ⎥∂ ∂⎣ ⎦

(Equation 2-70)

The order magnitude analysis of the energy equation has greatly simplified the

governing equation. The kinetic energy convection, heat conduction, enthalpy diffusion,

and shear stress work are analyzed individually. These four modes of energy transfer are

not compared to one another because no assumptions are made about which mode is

34

more important. Analysis of the velocity field has revealed that the axial component can

be used to determine the magnitude of the velocity at any point of the flow. The heat

conduction in the x-direction and many of the terms that comprise the shear stress work

are neglected due to the analysis. As a result of all this simplification, the energy

equation becomes Equation 2-71.

( )

2

1 1i

p p

N Ni

i i i im pi i

u u T T Tu uv k uc vcx y y y x y

Y T uh MW D c uy y y y

ρ ρ ρ ρ

ω ρ µ= =

⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂+ = − − −⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

⎛ ⎞⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂+ +⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎝ ⎠

∑ ∑ (Equation 2-71)

Unit Analysis

Units are substituted into the governing equations to ensure each expression in an

equation balances with the other expressions. Replacing variables and properties with

their units reveals several important aspects of these quantities. The inspection validates

that the equations were reduced without misplacing any variables or properties. It also

locates properties that require a unit conversion and determines units of the calculated

solutions. Note that Cantera calculates the properties with the International System (SI)

of measurement [17]. In order to keep unit conversions to a minimum, the variables also

use this system of measurement.

Continuity Equation

The main program calculates the vertical velocity component with the mass

equation. Solving the mass equation for the y-velocity produces Equation 2-3. Each

variable and property is replaced with its units. The density is given in kilograms per

meter cubed by Cantera. Therefore, the velocity and differential distances are measured

in meters per second and meters, respectfully. From Equation 2-72, it is clear that the

mass equation balances with equivalent units on both side of the equation.

35

( )uv vy x y

ρ ρρ∂∂ ∂

= − −∂ ∂ ∂

(Equation 2-3)

3

kg mm

1s m 3

kg mm

= −1

s mm

− 3

1kgs m m 3 3

kg kgm s m s

→ =⋅ ⋅

(Equation 2-72)

Unit analysis of the mass equation reveals that no unit conversion of the density is

necessary and the two differential step sizes should be given in similar units. Units of the

calculated vertical velocity depend on the units of the axial velocity, which is defined by

the initial condition. Although units cancel each other out in the mass equation, the other

governing equations prove that SI units should be used for the variables.

Species Continuity Equations

The reduced species continuity equation is algebraically reorganized in a form the

program can solve. This form is discussed more in section Solving Governing Equations.

Equation 2-46 is transformed into Equation 2-73. International System of measurement

is used for the velocity and differential distances, and their units are meters per second

and meters, respectfully. The density is still kilograms per meter cubed and the mass

fraction is dimensionless. Cantera gives a mixture-averaged diffusion coefficient in

meters squared per second. The unit of the net production rate is kilomoles per second

meter cubed, and the molecular weight is given in kilograms per kilomole. All of these

units are placed into Equation 2-73 to complete the analysis.

i i iim i i

Y Y Yu D v MWx y y y

ρ ρ ρ ω⎛ ⎞∂ ∂ ∂∂

= − +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠& (Equation 2-73)

3

kg mm

1s m

1m

=2

3

kg mm

1s m 3

kg mm

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

1s m

kmol+ 3

kgm s kmol⋅

3 3

kg kgm s m s

=⋅ ⋅

(Equation 2-74)

36

The unit check of the species continuity equation shows that converting the

molecular production rate into a mass production rate is indeed necessary. No other unit

conversion is required if SI units are used for the velocity components and differential

step sizes. The process proves the equation is reduced correctly from a unit analysis

point of view, and its solution is dimensionless.

Momentum Equation

The reduced momentum equation is reorganized into a form similar to Equation 2-

73. Equation 2-75 is the form of the momentum equation solved by the program. Units

of the velocity components, differential distances, and density remain unchanged. The

pressure and dynamic viscosity is also given in SI units. The SI unit of measurement for

pressure is the Pascal, which equals a kilogram per meter per second squared. Cantera

reports the dynamic viscosity in units of Pascal-second. A Pascal-second is equivalent to

a kilogram per meter-second. Variables and properties are replaced with these units and

result in balanced Equation 2-76.

u u p uu vx y y x y

ρ µ ρ⎛ ⎞∂ ∂ ∂ ∂ ∂

= − −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ (Equation 2-75)

3

kgm 2

m ms

1s m

1 kg mm m s

=⋅

1s m 2 3

1kg kgm s m m

⎛ ⎞− −⎜ ⎟ ⋅⎝ ⎠

2

m ms

1s m

2 2 2 2

kg kgm s m s

=⋅ ⋅

(Equation 2-76)

The analysis finds that none of the properties determined by Cantera necessitate a

unit conversion. Units of the simplified momentum equation balance accurately and its

solution, the axial velocity, is calculated in meters per second.

37

Energy Equation

Like the last two governing equations, the reduced energy equation is organized

into the form of Equation 2-77.

( ) 2

1 1i

p p

N Ni

im p i i ii i

T T u Tuc k u vcx y y y y

Y T u uD c h MW u uvy y x y

ρ µ ρ

ρ ω ρ ρ= =

⎡ ⎤∂ ∂ ∂ ∂ ∂= + − +⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦

⎛ ⎞⎡ ⎤∂ ∂ ∂ ∂− − −⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎝ ⎠

∑ ∑ (Equation 2-77)

Units of the flow properties already discussed in the previous governing equations

remain the same. Several new properties are encountered in the energy equation. The

temperature, thermal conductivity, species enthalpy, and specific heat of the fluid and

species i are used exclusively by this equation. The temperature is measured in degrees

Kelvin, and the thermal conductivity is given in units of watts per meter-Kelvin. A watt

per meter-Kelvin is equivalent to a Joule per meter-Kelvin-second. Specific heat of the

fluid can be determined on a mass basis in Cantera. The unit of the fluid’s specific heat

is Joules per kilogram-Kelvin. Each expression is analyzed individually with the units

established. Moving left to right in Equation 2-77, the axial energy convection is

analyzed first. Substituting units into the axial energy convection shows that the term has

units of Joules per cubic meter second. Units of the other expressions must reduce to this

unit to balance the energy equation.

p

kgTucx

ρ ∂=

∂ 3

mm

Js kg K⋅

Km 3

Jm s

=⋅

(Equation 2-78)

The next expression evaluated is the energy conduction along with the viscous

term. Units of the viscous term are equivalent to the energy conduction at a Joule per

cubic meter second and Equation 2-79 balances with Equation 2-78.

38

1T u Jk uy y y m m K

µ⎡ ⎤∂ ∂ ∂

+ =⎢ ⎥∂ ∂ ∂ ⋅⎣ ⎦

Ks⋅

mm

+kg

s m s⋅ms

1m 3

Jm s

⎡ ⎤=⎢ ⎥

⋅⎢ ⎥⎣ ⎦ (Equation 2-79)

Units of the second expression reduce to the same units of the axial energy

convection. The vertical energy convection also reduces to these units and is shown

below in Equation 2-80.

p

kgTvcy

ρ ∂=

∂ 3

mm

Js kg K⋅

Km 3

Jm s

=⋅

(Equation 2-80)

Cantera reports the species specific heat in a column vector that has been

nondimensionalized by the universal gas constant. Multiplying the vector by the

universal gas constant produces specific heats with the units of Joules per kilomole-

Kelvin. The sum of the energy diffusion expression includes the species specific heat.

The term highlighted is added to convert the species specific heat from a molar basis to a

mass basis. It is the inverse of the species molecular weight and must be added to the

enthalpy diffusion expression for the units to conform to the rest of the energy equation.

Equation 2-82 illustrates the modification.

2

1i

Ni

im pi

kg mY TD cy y

ρ=

⋅⎛ ⎞⎡ ⎤∂ ∂=⎜ ⎟⎢ ⎥∂ ∂⎣ ⎦⎝ ⎠

∑ 3

J Km s

⋅⋅ kmol 2K m⋅ ⋅

kmolkg 3

Jm s

⎛ ⎞=⎜ ⎟⎜ ⎟ ⋅⎝ ⎠

(Equation 2-81)

1 1

i

i

N Npi i

im p imi i i

cY YT TD c Dy y MW y y

ρ ρ= =

⎛ ⎞ ⎛ ⎞⎡ ⎤∂ ∂∂ ∂⇒⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎝ ⎠

∑ ∑ (Equation 2-82)

Much like the species specific heat, Cantera reports the enthalpy of each species in

a dimensionless column vector. The vector is nondimensionalized by the universal gas

constant and the temperature of the fluid. After multiplying the vector by these two

properties, the resulting enthalpy has the units of Joules per kilomole. Unit analysis of

39

the enthalpy production expression is done in Equation 2-83. Because the species

enthalpy is reported on a molar basis, the chemical production rate does not need to be

converted to a mass basis. The molecular weight term is underscored and is dropped

such that the unit of this expression is consistent with the other terms of the energy

equation.

( )1

N

i i ii

Jh MWkmol

ω=

=∑ kmol3 3

kg J kgm s kmol m s kmol

=⋅ ⋅

(Equation 2-83)

( ) ( )1 1

N N

i i i i ii i

h MW hω ω= =

⇒∑ ∑ (Equation 2-84)

The last expression in the unit check is the kinetic energy convection. Equation 2-

85 shows that the expression needs no modification.

2 kgu uu uvx y

ρ ρ∂ ∂− =

∂ ∂

2

3

mm 2s

m 1s m

2J s⋅kg 2m⋅ 3

Jm s

⎛ ⎞⎜ ⎟ =⎜ ⎟ ⋅⎝ ⎠

(Equation 2-85)

Analysis of the energy equation reveals that energy diffusion and enthalpy

production expressions required modification. Molecular weights are added to the energy

diffusion term and removed from the enthalpy production term. After these

modifications, the energy equation becomes Equation 2-86 where the units of each

expression are equal and the equation balances. Temperature being the dependent

variable of the energy equation is calculated in degrees Kelvin.

( ) 2

1 1

i

p p

N Np i

im i ii ii

T T u Tuc k u vcx y y y y

c Y T u uD h u uvMW y y x y

ρ µ ρ

ρ ω ρ ρ= =

⎡ ⎤∂ ∂ ∂ ∂ ∂= + − +⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦⎛ ⎞∂ ∂ ∂ ∂

− − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠∑ ∑

(Equation 2-86)

40

Summary of Governing Equations

In conclusion the governing equations are applied to the two-dimensional modeling

of the catalytic reactor. Each equation is reduced based on assumptions describing the

fluid and its motion. Through an order magnitude analysis these equations are simplified

further. The units of each term are verified and the equations are balanced. The resulting

equations solved by the program are summarized in Table 2-1.

Table 2-1. Equations modeling the flow.

Principle Equation Equation number

Mass Conservation

( )uv vy x y

ρ ρρ∂∂ ∂

= − −∂ ∂ ∂

2-3

Species Mass Conservation

i i iim i i

Y Y Yu D v MWx y y y

ρ ρ ρ ω⎛ ⎞∂ ∂ ∂∂

= − +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠& 1, 2,3,...i N= 2-73

Momentum Conservation

u u p uu vx y y x y

ρ µ ρ⎛ ⎞∂ ∂ ∂ ∂ ∂

= − −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ 2-75

Energy Conservation

( ) 2

1 1

i

p p

N Np i

im i ii ii

T T u Tuc k u vcx y y y y

c Y T u uD h u uvMW y y x y

ρ µ ρ

ρ ω ρ ρ= =

⎡ ⎤∂ ∂ ∂ ∂ ∂= + − +⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦⎛ ⎞∂ ∂ ∂ ∂

− − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠∑ ∑

2-86

The unit analysis also determines the units of properties and variables found in the

governing equations. Units of each fluid property are compiled in Table 2-2.

Table 2-2. Units of the governing equations. Property Variable Units Property Variable Units

Differential step sizes dx, dy m Production

rates ωi kmol/m3·s

Diffusion coefficients Dim m2/s Thermal

conductivity k W/m2·K

Density ρ kg/m3 Specific heats cp J/kg·KEnthalpies hi J/kmol Temperature T KMass fractions Yi dimensionless Velocity

components U, v m/s

Viscosity µ Pa·sMolecular weights MWi kg/kmol Pressure p Pa

41

CHAPTER 3 PROGRAM METHODOLOGY

Flow variables are calculated throughout the catalytic reactor via a step-by-step

process of solving the governing equation. The process begins by creating a discrete

mesh of points to numerically solve the equations. Solutions to the flow variables are

recorded at these points. The next step involves establishing parameters and conditions

of the reactor and fluid. These values characterize the reactor and initial conditions of the

flow. Folders to import and export data must also be defined. Once the first three steps

are completed, the code can begin to find the solutions. The program sets the initial

conditions and solves the simplified governing equations in Table 2-1.

The code is written in MATLAB and consists of a main program with three

subprograms. One of the subprograms finds the initial velocity components. The main

program creates the mesh, finds fluid properties, and sets conditions needed to solve the

equations. Information from the main program is sent to the other two subprograms.

Solutions are found by the subprograms and sent back to the main program where it is

saved in the solution variables. MATLAB is chosen above other programming languages

because of its built-in ability to handle vectors, vector operations, and partial differential

equations. MATLAB incorporates several computational tools capable of solving partial

differential equations. A function called pdepe is used to solve the momentum equation

as a single equation. It is also used to solve the energy equation and species continuity

equations as a set of coupled partial differential equations. This makes MATLAB well

suited for modeling chemically reacting flows. Another useful property of MATLAB is

42

its compatibility with Cantera. Cantera is a free software package developed by

Professor David Goodwin at the California Institute of Technology to solve problems

concerning chemical reactions. The main MATLAB program calls upon this software to

determine the thermodynamic, transport and chemical kinetic properties of the flow and

catalytic surface. Cantera is able to construct objects of different phases and tie the

phases together through an interface. This allows for the chemical interaction between

the gas and surface [17].

Several studies attempt to model catalytic combustion similar to this model. A

study by the National Institute for Advanced Transportation Technology at the University

of Idaho modifies an existing code. Lawrence Livermore National Laboratory provides

the existing Hydrodynamics, Combustion, and Transport (HCT) code. The finite-

difference code, HCT, utilizes the same principles of conservation for its calculations.

Dissimilarity occurs in the application of the governing equations to the one-dimensional

time-dependent catalytic combustion, opposed to the two-dimensional steady-state

catalytic reactor modeled by this program. Still, the study offers some insight into the

chemistry and equations involved with modeling a catalytic combustor [5]. In a second

study, Chou et al. [4] uses CURRENT with CHEMKIN and SURFACE CHEMKIN

software to model a two-dimensional monolith catalytic combustor. CURRENT is a

code developed by Winters et al. [18] for low Mach number chemically reacting flows.

The study discusses the chemistry and boundary conditions of the model and compares

the calculations to experimental data. This program uses a similar symmetric boundary

condition at the centerline.

43

Discretization

The two-dimensional computational space of the reactor is broken down and

discretized before the equations can be numerically solved. The mesh is that of a planar

geometry with the height determined by the reactor’s radius. The upper boundary is

moved to the centerline and the lower boundary is still the catalytic surface. This reduces

the height of the computational space and in turn reduces computation time and memory

used by the computer. Now is a good time to mention that the centerline is assumed to be

a streamline and symmetric conditions are assumed to exist at this boundary due to the

two identical plates modeling the surface of the pipe or channel. This assumption affects

the boundary conditions discussed in the section Solving Governing Equations. The

length of the reactor is broken down into stages, the first stage being the entrance. This is

also intended to reduce the time needed to calculate a solution. It is expected that the

flow changes relatively fast in the beginning of the reactor when the catalyst is first

encountered. This corresponds to the first few stages of the computational space. To

help resolve the solution in these stages a smaller differential step size in the x-direction

is chosen. Once the properties reach a quasi-steady state, the step size can be increased to

help lower the computation time.

An orthogonal mesh is created for every stage. Each stage has its axial direction

discretized in a linear manner, where every point is an equal distance apart. The distance

is set for a given stage but can change from stage to stage. This allows the user to adjust

the axial step size of a stage if the program cannot converge on a solution. A possible

source of this problem is a significant change of flow properties in the x-direction. Recall

that the governing equations are simplified based on the assumption that the characteristic

length is much larger than the characteristic height. In other words, the vertical gradients

44

are much larger than the axial gradients. While this assumption is still applicable, there

may be areas where a change in step size is needed, such as the reactor’s entrance.

The point separation in the y-direction, in contrast to the axial point placement, is

the same throughout the reactor. Although the vertical point placement must remain the

same for every stage, it is not restricted to only a linear displacement. The point

displacement is set as a power of the point location. For example, setting the power to

one would position the points linearly. Setting the power to two creates a quadratic point

displacement, leading to more points near the surface. A larger power places more points

near the surface. Varying the power allows the user to control the location of the points

in the vertical direction. This aids the program in resolving the varying chemical

composition near the surface. The catalytic surface serves as the main source of the

chemical reaction in the flow. Therefore, it is expected that most of the chemical change

will occur near the surface. More points are needed near the surface to determine the

change in the chemical composition in the vicinity of the catalyst. A tight mesh near the

surface also helps resolve the fluid velocity boundary layer.

Velocity, temperature, and composition variables are not found for the entire stage

at once. Instead the stage’s mesh is broken up further into mini-meshes. A mini-mesh

contains all the vertical points for a group of three axial locations. Governing equations

are solved one mini-mesh at a time due to the coupling of the equations. The pressure,

temperature and mass fraction of a mini-mesh must be approximated prior to solving the

equations. Jumping ahead might seem premature because the governing equations meant

to calculate the variables have not been solved yet. However, fluid properties dependent

on the solution are imbedded inside the equations. These properties must be established

45

in order to solve the equations. One can now being to appreciate the complex coupling of

the governing equations. Once the equations are solved, the program updates the

variables and moves downstream to the next mini-mesh.

Parameters and Conditions

Parameters and conditions of the catalytic reactor and incoming flow are set inside

the code of the main program. All of the values, composition being the only exception,

must equal a real scalar. The computer code begins by setting parameters of the catalytic

reactor, such as the radius, stage length, stage number, surface temperature, and the

distance of the non-reactive surface. Dimensions of the computational space are

constructed with the height and length of the stage. The stage number is simply the

sequential numbering of each stage for which a solution is calculated. The value of this

number determines the data used to set the incoming conditions and the output folder in

which the export files are stored. This is discussed further in the sections Initial

Conditions of a Stage and Input and Output Files. In the energy equation, the

temperature at the wall or surface boundary is held constant at the value entered. The

distance of the non-reactive surface refers to the entrance of the reactor where there is no

catalyst on the surface. This is only important for the first stage and can be ignored for

any other stage. The differential step sizes in the vertical and axial direction are also set

at this point, along with the power used to discretize the vertical direction. These values

are used to construct the two-dimensional mesh described in the section Discretization.

After the parameters of the reactor are entered, the conditions of the incoming flow

are defined. The speed, temperature, pressure, and composition of the flow entering the

reactor are established. The incoming flow is assumed to be a uniform flow where the

velocity is purely in the axial direction. Initially there is no vertical component to the

46

fluid’s velocity and the speed of the x-velocity is the same at every point. Therefore, only

a single quantity is needed to define the velocity vector entering the reactor. The flow’s

chemical composition is initially modeled as a well-mixed fluid. This simply means that

the species mass fractions are also the same at every point entering the reactor. The

composition is the only value entered as a string variable. This string contains the name

and mass faction of the species present in the incoming flow. Cantera reads the string to

set the composition of the gas. The program uses these values to set the initial conditions

of the variables.

The last parameter to set is the PC variable, also a scalar. This variable controls

whether the program iterates on a solution and if so, how many times the iteration takes

place. An inherent delay in the solution process exists because the governing equations

are decoupled. The delay is exaggerated by properties that are dependent on the solution

inside the equation. Some of these properties include density, viscosity, and diffusion

coefficients. With no iteration (PC equal to one), the program numerically solves the

governing equations for one mini-mesh. Then the program updates the variables and

properties and moves one differential step downstream to solve the equations at the next

three axial locations. The program is continually updating the properties prior to moving

downstream; therefore the delay is expected to be small. To improve the calculation one

may choose to iterate on a solution using a predictor/corrector type method. To iterate on

a solution the PC variable is set to quantity greater than one. For example, the program

iterates once on a calculation if the variable is equal to two. Iteration occurs by solving

the equations and updating the properties with the known solution. This could be seen as

a predictor step, now to correct the calculation. Instead of moving one differential step

47

downstream the program recalculates the solution for the same three axial locations based

on the updated properties. If the PC variable is three, the iteration occurs twice, and so

on.

Input and Output Files

Input and Output file names are given prior to operating the program to direct

import and export data. The input text file contains the chemical data Cantera require to

model the gas and solid of the catalytic reactor. Naming the input file informs the

program where the chemical data are located to import. Data determine properties found

within the governing equations. Only the filename of the input file is needed if it is

located in Cantera’s current working directory. This directory is initially set as the data

folder inside Cantera’s main folder, which is installed with the free software. The

pathname of the output folder provides the program the location of the export folders.

Export folders must be created inside the output folder and given the name Stage1,

Stage2, Stage3… etc. The solutions of a stage are recorded in the folder with the

corresponding number. Therefore, the export folder of a stage must exist before seeking

the solution of that stage. The entire pathname is stored in the string variable saveFile.

Not only is this string variable used to export solutions of a stage; it is also used to import

initial conditions for most of the stages. This is discussed further in the section Initial

Conditions of a Stage.

Considerable amounts of data are required to model the gas and solid of the

catalytic reactor. Cantera accesses this data via the input text file specified. These files

contain information on the chemical kinetics, thermodynamics, and transport properties

of many different species. Data consistent with the modified Arrhenius function

determines the chemical kinetic properties of the gas phase reactions. This data include

48

activation energy, pre-exponential coefficients and temperature exponent. In addition to

this, surface reactions apply reactive sticking probability. The thermodynamic properties

are determined using a NASA polynomial parameterization or Shomate parameterization.

Coefficients of either parameterization are incorporated in the data of the input file.

Information needed to calculate transport properties based on either a multi-component or

mixture-averaged transport model is also included. The multi-component transport

model provides a more accurate solution than the mixture-averaged model. However, the

multi-component model requires more data and computation time than its counterpart

[17].

The program saves several variables to the output folder for every stage. The value

of each variable is saved as a double precision scalar, vector, or matrix in an ASCII file.

The axial location, axial velocity, pressure, temperature, mass fraction of each species,

pressure gradient, and vertical velocity are all stored in the export folder. The axial

location is saved in order to keep track of which discretized points in the mesh the

various solutions correspond. The x-location is saved as a vector that begins at zero and

ends at the length of the stage. The axial velocity is recorded at every point in the stage

and the variable is saved as a matrix. It is possible to record the vertical velocity as a

matrix in a similar manner with little addition to computation time. This is due to the fact

that the variable is already determined to solve the governing equations. However, the y-

component of the velocity is so small when compared to the x-component that it is not

recorded as part of the solution. This will help retain memory space for the other

properties. Two independent thermodynamic properties are recorded to

thermodynamically define the fluid. One of these properties is the pressure, which does

49

not vary in the vertical direction. Being only one-dimensional, the pressure at each axial

location is recorded and the variable is saved as a vector. The other thermodynamic

property is temperature and it remains a function of both dimensions, x and y.

Temperature at every discretized point is calculated and saved as a matrix in the stage’s

export folder. The composition of the fluid is recorded as mass fractions of each species.

Like the temperature, the mass fractions are a function of both dimensions. The mass

fraction of each species is saved as a matrix into its own file. As a result, the number of

mass fraction files saved in the export folder is equal to the number of species, N. All the

properties needed to kinetically and thermodynamically define the flow are recorded.

The only other variables saved are the pressure gradient and vertical velocity. The

pressure gradient is recorded as a scalar and the vertical velocity is saved as a vector.

Both variables correspond to the last axial position of the stage and are used as initial

conditions of the next stage.

Initial Conditions of a Stage

The program can operate once all of the parameters, conditions, and file names are

designated. Initial boundary conditions of the stage’s first mini-mesh are established

before solving the governing equations. Velocity components at all vertical points in the

first axial location are required to define the momentum equation and its initial boundary

condition. The pressure, temperature, and composition in the first mini-mesh must also

be defined to estimate the properties inside the governing equations. The model requires

only one pressure value per x-location, because the pressure is independent of the vertical

direction. The result is only three scalars being required to define the pressure in the

mesh. Temperature and species mass fractions are two-dimensional and must be set for

every point in the stage’s first mini-mesh. The process used to define these variables

50

depends on the stage number. Conditions of the initial stage or stage one are based on the

values discussed in the section Parameters and Conditions. Every other stage uses the

solution of the previous stage to set these initial boundary conditions. The program can

begin to solve the governing equations once the initial conditions are set.

Cantera creates a gas object and surface object prior to defining initial conditions.

The gas is adjusted to the pressure, temperature, and composition entering the reactor and

the two objects are connected through an interface. The gas object is created at this time

because Cantera provides a simple means to set the composition variable of the first

stage. Only the composition’s string variable is needed to establish the mass fraction of

all the species initially present. Cantera can take the composition of the gas object and

return the mass fraction of every species. This is much easier than searching for the

species not present and setting their mass fraction to zero. Cantera also ensures that the

sum of the mass fractions equals one.

Stage One

The reactor is characterized by the absence of a catalytic surface at its entrance.

The catalyst does not begin until further downstream. This is where stage one begins and

the initial boundary conditions of the first mini-mesh are determined. Minimal change in

the conditions should occur over the non-reactive surface with the exception of the two

velocity components. Therefore, the initial conditions of the temperature, composition,

and pressure remain the flow conditions entering the reactor. Temperature and mass

fractions at the first three axial locations are approximated by the values entered as initial

conditions. The surface temperature is set to the value entered as a parameter. The initial

pressure is equal to the pressure of the incoming flow, and the pressure at the next two

differential steps is calculated with the pressure gradient. In contrast to the other

51

variables, the surface affects the velocity vector. A boundary layer develops changing

the profile of the axial velocity, which produces a vertical velocity. Blasius solution is

used to model the boundary layer and determine the two velocity components.

Blasius Solution

Axial velocity is quantified by two values at a point in the beginning of the reactor.

The singularity point is located on the front edge of the reactor, where the incoming flow

first encounters the surface. A finite value is given to the uniform velocity entering the

reactor. The velocity at this point must also equal zero due to the boundary conditions of

the velocity. To overcome the singularity point, Blasius solution is used to calculate the

two velocity components at the end of the non-reactive surface, where the first stage

begins.

H. Blasius is well known for obtaining an exact solution to a laminar boundary

layer flow over a flat plate. Blasius is able to find a similarity solution to the continuity

and momentum equations through proper scaling and nondimensionalization of the two

equations. In his solution, the dimensionless stream function replaces the two velocity

components as the dependent variable. The two coordinates, x and y, are also combined

into one dimensionless independent variable. Blasius transforms the two partial

differential equations into one ordinary differential equation. A power series expansion

or numerical methods can then be used to solve the third order, nonlinear equation. The

dimensionless stream function and its derivative are used to calculate the axial and

vertical velocity [15].

Blasius solution describes a two-dimensional, steady, incompressible flow with no

pressure gradient. Recall that the assumption of constant density is not applicable to this

model due to the chemistry involved. A pressure gradient equal to zero is also not

52

accurate because the flow is assumed to have pressure changes in the axial direction.

However, Blasius solution is used as a reasonable estimate to the velocity profile over the

non-reactive surface. The change in density is caused mostly by the catalytic surface, and

the catalyst is not present in the region that Blasius solution is employed. A change in

density from species diffusing upstream is possible, but the effect should be negligible.

The production of a new species with a diffusion velocity great enough to overcome the

axial velocity is needed for this to occur. The behavior of the pressure gradient also

permits the use of Blasius solution. The pressure slowly decreases as the flow moves

downstream. This produces a decreasing pressure gradient that has an initial value of

zero. The change in pressure is small and a zero pressure gradient should be a reasonable

approximation at the entrance of the reactor.

Fortunately a non-reactive surface is located in the region of the singularity point.

Blasius solution can be used to generate the velocity profile at the end of the non-reactive

surface. The main program calls on one of the subprograms, a function called Blasius.

The function imports the axial differential step size, stage length, y-coordinate vector,

location where the catalytic surface begins, viscosity, and initial speed of the flow. A

shooting method determines the dimensionless stream function and its derivative.

Because the equation developed by Blasius is dimensionless, the calculated values of the

stream function are independent of the values imported. The axial velocity vector and

vertical velocity vector are determined using the dimensionless variables with the

imported variables. The axial velocity is treated as the initial condition of stage one. The

vertical velocity is used in the momentum equation. Note that the function Blasius is

53

only used in the first stage, where the non-reactive surface is located. Other stages use

the solutions of the previous stage to set the velocity profile.

Subsequent Stages

If the stage number is greater than one, initial boundary conditions are taken from

the export folder of the previous stage. The program locates and loads initial quantities

with the saveFile variable. The last value of the preceding stage’s pressure vector

becomes the initial pressure of the current stage. The vertical velocity vector and

pressure gradient (scalar) are loaded from the preceding stage’s output files. Initial

values of the current stage’s axial velocity are set to equal the x-velocity at the end of the

last stage. Temperature and composition variables are all that remain to import. Recall

that the temperature and mass fraction must be set for all three axial locations. The first

axial location is equal to the last axial location in the previous stage’s matrix, similar to

the axial velocity variable. For the next two differential steps downstream, the x-location

vector of the last stage is imported. This vector along with the temperature and mass

fractions of the previous stage determine the variable’s gradient at the end of the last

stage. A second-order backward-difference formula is used to estimate this gradient [19].

Values of the next two axial locations in the variable of the current stage are linearly

extrapolated using the gradient. With the initial conditions of the stage set, the program

is prepared to solve the equations controlling the behavior of the flow.

Solving Governing Equations

Governing equations are solved for three axial locations at a time. Remember that

the equations contain properties dependent on the solution. Solving one mini-mesh at a

time allows the program to update properties inside the equations prior to calculating the

solution one step downstream. For the same reason, the pressure, temperature, and

54

composition at these three places must be approximated before solving the equations. For

the stage’s first mini-mesh, the method for these approximations is described in the

section Initial Conditions of a Stage. Approximations of the other mini-meshes are

calculated with the solution of the previous mini-mesh. Note that two of the axial

locations of the next mini-mesh are repeated locations of the previous mini-mesh because

the program moves only one differential step.

Two MATLAB functions, or subprograms, are written to solve the momentum

equation and the species/energy equations separately. These two functions are named

Momentum and Species_Energy. The process of the solving the governing equations for

a stage is illustrated in the flow chart of Figure 3-1. Once the initial approximations for a

single mini-mesh are established, the program begins with the momentum equation in the

x-direction. The main program calls on Cantera to find properties embedded in the

momentum equation. Information is exported to the subprogram Momentum, which is

then used to solve the equation. The subprogram sends the solution, the axial velocity,

back to the main program. The mass or continuity equation calculates the vertical

velocity and confirms that the solution of the momentum equation is accurate. The

pressure gradient is updated and the momentum equation is solved again if the boundary

condition of the vertical velocity at the centerline is not reacted. If the y-velocity equals

zero, the species continuity and energy equations are solved together with the

Species_Energy function. Again, the data needed to solve the system of equations are

provided by Cantera and sent to the subprogram. This time the composition and

temperature are sent back as the solution. Once the solutions are calculated and any

iteration correction is performed, the program updates the variables, saves the data in the

55

export folder, and moves one differential step downstream. The temperature and

composition of the next axial location are predicted using a linear extrapolation from the

previous two x-locations. The program loops back and solves the equations for the next

mini-mesh. This continues until the end of the stage is reached or the program cannot

resolve the solutions.

Figure 3-1. Flow chart for single stage modeling.

The subprograms Momentum and Species_Energy use a partial differential

equation (PDE) solver provided by MATLAB to numerically solve the equations. The

solver numerically computes the momentum equation as a single equation. It is also used

to solve the energy equation and species continuity equations as a set of coupled partial

differential equations. The solver, named pdepe, calculates the solution of partial

differential equations with the form shown below in Equation 3-1. The gradient of the

dependent variable with respect to x is multiplied by a coupling term, c. This term along

with the flux term, f, and the source term, s, are functions of the two independent

variables, the dependent variable and its vertical gradient.

56

, , , , , , , , ,m mz z z zc x y z x x f x y z s x y zy x y y y

− ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂= +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

(Equation 3-1)

The symmetry parameter, m, along with the coupling, flux, and source term define

the PDE. The mesh spacing of the two independent variables, one initial condition, and

two boundary conditions are all that remain to solve the equation. The mesh spacing is

simply the mini-mesh discussed in the section Discretization. Function pdepe selects the

x-mesh dynamically to resolve the solution, but only reports the answer at the mesh

points specified. Strictly speaking the initial condition is a boundary condition. It is the

value of the dependent variable at the first of the three axial locations. The initial

boundary condition of the dependent variable needs to be given as a function of y. The

other two boundaries are found at the catalytic surface and centerline. Both must fit the

form shown in Equation 3-2. Boundary conditions are expressed in terms of p, q, and f.

The flux term f is already defined in the PDE above, so only p and q are needed to

establish the boundary conditions.

( ) ( ), , , , , , 0zp x y z q x y f x y zy

⎛ ⎞∂+ =⎜ ⎟∂⎝ ⎠

(Equation 3-2)

Some fluid properties, such as density, are converted into functions of y to conform

to Equation 3-1 above. Most of these properties are functions of both dimensions.

However, properties dependent only on the vertical direction should be an acceptable

representation for several reasons. Fluid properties vary more in the vertical direction

than the axial direction and are a stronger function of y. In addition to this, the functions

only need to represent the properties for the three x-locations of the mini-mesh. Initial

boundary conditions also need to be turned into functions of y. Built-in MATLAB

functions spline and unmkpp generate the function representations. Properties are found

57

at every vertical point in the middle of the three x-locations and saved in a vector. In the

case of the initial condition, the vector contains the initial values of the dependent

variable. The function spline uses the vector to create twenty separate piecewise

polynomials of the form of the cubic spline. Function unmkpp extracts the four

coefficients of the each polynomial and saves it into a four-by-twenty matrix for each

representation. The matrices are exported to either the Momentum function or

Species_Energy subprogram to reconstruct the piecewise polynomial. The coefficients

and a heavyside step function connect the piecewise polynomials inside the subprogram.

The result is a smooth function representation of the initial conditions or fluid properties

embedded in the governing equation.

Momentum Equation

Solving the momentum equation begins by guessing the pressure gradient.

Pressure at the three axial locations is determined with the guessed pressure gradient.

The pressure along with the approximated temperature and composition are used to

determine the density and viscosity. These are the properties found in the simplified

momentum equation.

u u p uu vx y y x y

ρ µ ρ⎛ ⎞∂ ∂ ∂ ∂ ∂

= − −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ (Equation 2-75)

Properties are determined at every vertical point in the middle of the three x-

locations and transformed into functions of y for the coupling, flux, and source terms.

Comparing the momentum equation to the form used by the pdepe function, it is evident

that the axial velocity replaces the dependent variable, z. The symmetry parameter, m, is

zero. The coupling, flux, and source terms equal Equation 3-3, Equation 3-4, Equation 3-

5, respectfully.

58

, , , uc x y u uy

ρ⎛ ⎞∂

=⎜ ⎟∂⎝ ⎠ (Equation 3-3)

, , , u uf x y uy y

µ⎛ ⎞∂ ∂

=⎜ ⎟∂ ∂⎝ ⎠ (Equation 3-4)

, , , u p us x y u vy x y

ρ⎛ ⎞∂ ∂ ∂

= − −⎜ ⎟∂ ∂ ∂⎝ ⎠ (Equation 3-5)

The coupling term, c, contains the density and axial velocity. This term is allowed

to be a function of the dependent variable. As a result, only the density must be

transformed into a function. The flux term, f, equals the viscosity multiplied by the

vertical gradient of the axial velocity. To fit Equation 3-1, the gradient will remain but

the viscosity is represented by a function of the vertical direction. The last two terms in

the momentum equation are combined into the source term. These two terms consist of

the pressure gradient and the product of the density, y-velocity, and y-gradient of the

dependent variable. Recall that the pressure is only a function of the axial location.

Therefore, the pressure gradient remains constant at a given x-location and does not need

to be transformed into a function of y. The vertical gradient of the x-velocity is allowed

inside the source term. The density and y-velocity product is the only element of the

source term transformed into a function. The axial velocity’s initial boundary condition

is also transformed into a function of y.

Boundary conditions at the surface and centerline are all that remain to solve the

momentum equation. Appling the no-slip assumption, the axial velocity is zero on the

catalytic surface. The centerline of the reactor is assumed to be a streamline with a

vertical gradient of the x-velocity equal to zero. Equation 3-6a and 3-6b show these two

conditions in a form recognized by the pdepe function.

59

( )0 0 0u u f= → + ⋅ = (Equation 3-6a)

( ) ( )0 0 1 0u uy y

µ∂ ∂= → + ⋅ =

∂ ∂ (Equation 3-6b)

Boundary conditions of the momentum equation are defined inside the subprogram

Momentum. At the surface, or y equal to zero, p equals the dependent variable and q is

zero. The centerline condition dictates that p equals zero and q equals one. This sets the

flux term to zero at the boundary. The flux term is the product of the viscosity and

vertical gradient of the axial velocity. Since the viscosity is finite, the gradient must

equal zero, which is the condition sought.

The Momentum subprogram can now be used to solve the momentum equation.

The function imports the discretized mesh and guessed pressure gradient. Coefficients of

the initial boundary condition, coupling, flux, and source terms are also imported. These

are the coefficients of the piecewise cubic polynomial. The second-order, nonlinear PDE

is solved and the axial velocity at the three axial locations is returned to the main

program.

Continuity Equation

The axial velocity solution must be verified because the value of the pressure

gradient is assumed. This value directly affects the momentum equation by being part of

the source term. It also indirectly affects the solution by changing the properties

dependent on the pressure. Equation 2-3 is the mass or continuity equation that

calculates the vertical velocity. At the same time, the solution of the mass equation acts

as a check to the momentum equation.

First, the gradient of the density and x-velocity product is determined at every

vertical point at the end of the mini-mesh. This partial derivative is calculated with a

60

second-order backward-difference formula. The y-gradient of the density is found with a

second-order central difference formula with varying spacing. Once these two gradients

are found, the partial derivative of the vertical velocity is approximated with another

second-order central difference formula with varying spacing in Equation 3-7 [19].

Solving for the velocity at the next mesh point produces Equation 3-8.

( )uv vy x y

ρ ρρ∂∂ ∂

= − −∂ ∂ ∂

(Equation 2-3)

( )( )( )

( )2 21 1

1

1

1j j j

j jj j jj

v a v a v uv

x ya a y yρ ρρ + −

−

+ − − ∂ ∂= − −

∂ ∂+ − (Equation 3-7)

where ( )( )

1

1

j j

j j

y ya

y y+

−

−=

−

( )

( )( )( )

( )( )11 2 2

1

1

1

1

1

jjj j j

jjj j j j

j j

uv

x y a a y yv

a v a v

a a y y

ρ ρ

ρρ ρ−

+−

−

⎡ ⎤∂ ∂− − −⎢ ⎥

∂ ∂ + −⎢ ⎥= ⋅⎢ ⎥

− −⎢ ⎥⎢ ⎥+ −⎣ ⎦

(Equation 3-8)

Equation 3-8 is used to find the vertical velocity component at the last of the three

x-locations. This new vertical velocity becomes the variable used by the next mini-mesh

downstream. The species and energy equations still use the original y-velocity for their

calculations. Once the y-velocity is found at every vertical point, its value at the

centerline is checked. Being a streamline boundary condition, there should be no flow

across the boundary and the y-velocity should roughly equal zero. If the velocity does

not meet this requirement, the pressure gradient is adjusted and the program loops back to

the momentum equation. The amount of the adjustment is proportional to the size of the

y-velocity at the centerline. A weighted correction modifies the pressure gradient. This

61

continues until the centerline y-velocity is less than one ten-thousandths. At which point

the axial velocity of the mini-mesh is saved or spliced to the axial velocity variable of the

entire stage. The program then moves on to the remaining two governing equations.

Now is an excellent moment to discuss the reasoning behind breaking apart the

momentum and mass equations from the other governing equations. It has already been

shown that all of the equations are highly coupled and should be solved as such.

However, that approach leads to a very problematic and time-consuming calculation due

to the unknown pressure gradient. Solving the entire group of equations until the correct

pressure gradient is found would take a great amount of computing time. Decoupling the

momentum and mass equations significantly reduces the time of the calculation. This

method does not come without its disadvantages. Separating the governing equations

creates a delay in the solution. This delay can be overcome with the iteration process

already discussed in the section Parameters and Conditions.

Species Continuity Equations

The remaining equations are not decoupled, but instead are solved simultaneously

by the function pdepe. The set of species equations are solved for the mass fraction of

each atom or molecule. The number of equations in this set is equal to the total number

of species in the model, defined as N. Equation 2-46 shows the simplified species

equation.

1, 2,3,...i i iim i i

Y Y Yu D v MW i Nx y y y

ρ ρ ρ ω⎛ ⎞∂ ∂ ∂∂

= − + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠& (Equation 2-46)

Mass fraction of species i is the dependent variable of the species equations. Again,

the symmetry parameter, m, is zero. Cantera determines the density, diffusion

coefficients, net gas production rates, and molecular weights found in the equation.

62

These quantities and the two velocity components are used to create the coupling, flux,

and source terms shown in the three equations below.

, , , ii

Yc x y Y uy

ρ⎛ ⎞∂

=⎜ ⎟∂⎝ ⎠ (Equation 3-9)

, , , i ii im

Y Yf x y Y Dy y

ρ⎛ ⎞∂ ∂

=⎜ ⎟∂ ∂⎝ ⎠ (Equation 3-10)

, , , i ii i i

Y Ys x y Y v MWy y

ρ ω⎛ ⎞∂ ∂

= − +⎜ ⎟∂ ∂⎝ ⎠& (Equation 3-11)

The density and axial velocity make up the coupling term. Axial velocity is no

longer the dependent variable as it is in the momentum equation. This means that the

multiple of the density and axial velocity must be transformed into a function of y. The

flux term, f, can be found inside the parenthesis of Equation 2-46. It equals the density

multiplied by the species mixture-averaged diffusion coefficient and the vertical gradient

of the mass fraction. The flux term is allowed to be a function of dependent variable’s

vertical gradient. Therefore, only the density and diffusion coefficient product is

represented by a function. The source term becomes the combination of the last two

terms of the species equation. This term equals the species mass production minus the

product of the density, y-velocity, and vertical gradient of the dependent variable. The

product of the density and y-velocity is transformed into a function representation, while

the vertical gradient is left unaltered. This y-velocity is the original vector and not the

velocity found from the mass equation. The species mass production and initial boundary

condition are also changed to a function of the y-direction.

Two boundary conditions of the species equations can be connected to the species

mass flux. At the surface boundary is a heterogeneous catalyst where species can be

63

created or destroyed. Assuming a steady-state model, the species flux can be equated to

the production rate on the catalyst. Species mass flux into the surface equals the

destruction rate and the flux away from the surface is the creation rate. Mathematically

written in Equation 3-12 and reorganized into Equation 3-13a to fit the form defined by

the PDE solver. Cantera determines these production rates for each species. A

symmetric boundary condition is applied to the upper boundary. Resulting in the vertical

gradient of a species mass fraction approximately equaling zero at the centerline.

Equation 3-13a and 3-13b show these two conditions in a form recognized by the pdepe

function.

,i

i surface i imYMW Dy

ω ρ ∂= −

∂& (Equation 3-12)

( ) ( ), 1 0ii surface i im

YMW Dy

ω ρ⎛ ⎞∂

+ ⋅ =⎜ ⎟∂⎝ ⎠& (Equation 3-13a)

( ) ( )0 0 1 0i iim

Y YDy y

ρ⎛ ⎞∂ ∂

= → + ⋅ =⎜ ⎟∂ ∂⎝ ⎠ (Equation 3-13b)

Parameter p equals the mass production rate and q is one for the lower boundary at

y equal zero. The upper boundary condition has p equal to zero while q equals one. This

sets the gradient equal to zero because neither the density nor the diffusion coefficient of

the flux term equal zero. The system of partial differential equation is defined along with

their initial and boundary conditions. Before the species equations are solved, the energy

equation is added to the group.

Energy Equation

The energy equation contains kinetic energy terms defined by the velocity field.

Kinetic energy terms are in the form of x and y gradients of the axial velocity. The

64

solution of the momentum equation is used for the x-velocity. Both partial derivatives

are calculated at every vertical point in the mini-mesh. A simple second-order central-

difference formula is applied to estimate the axial gradient. The vertical gradient is a

little more complicated because the spacing in the y-direction may vary. A second-order

central-difference formula that is modified for varying point spacing is used for the core

of the calculations. The y-gradient at the surface is found with either a first-order or

second-order forward-difference formula for equal spacing. If the y spacing is linear,

then the second-order formula is used. The first-order equation is used if the spacing is

non-linear [19]. Although it is first order, the error should be small because the spacing

near the surface is tight. The gradient at the centerline equals zero due to the boundary

condition of the momentum equation. Equation 2-86 is the simplified energy equation

with the two kinetic energy terms at the end of the equation.

( ) 2

1 1

i

p p

N Np i

im i ii ii

T T u Tuc k u vcx y y y y

c Y T u uD h u uvMW y y x y

ρ µ ρ

ρ ω ρ ρ= =

⎡ ⎤∂ ∂ ∂ ∂ ∂= + − +⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦⎛ ⎞∂ ∂ ∂ ∂

− − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠∑ ∑

(Equation 2-86)

Equation 2-86 contains many thermodynamic and transport properties that need to

establish. Cantera retrieves the properties at every vertical point in the middle of the

three x-locations and saves them to vectors. The main program uses the vectors to create

the function representations of the coupling, flux, and source terms. It is apparent that the

temperature is now the dependent variable of the PDE. The symmetry parameter is zero

and the coupling, flux, and source term are listed below.

, , , pTc x y T ucy

ρ⎛ ⎞∂

=⎜ ⎟∂⎝ ⎠ (Equation 3-14)

65

, , , T T uf x y T k uy y y

µ⎛ ⎞∂ ∂ ∂

= +⎜ ⎟∂ ∂ ∂⎝ ⎠ (Equation 3-15)

( )

1

2

1

, , , iN

p ip im

i i

N

i ii

c YT T Ts x y T vc Dy y MW y y

u uh u uvx y

ρ ρ

ω ρ ρ

=

=

⎛ ⎞⎛ ⎞ ∂∂ ∂ ∂= − + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂− −

∂ ∂

∑

∑ (Equation 3-16)

The coupling term is the multiple of the density, axial velocity, and specific heat.

The entire expression is transformed into a function y. The flux term, found inside the

brackets, is made up of two parts. The first equals the thermal conductivity times the

temperature’s axial gradient. Second is the combination of the viscosity, x-velocity and

its y-gradient. The two parts must be kept separate for the flux term to remain a function

of the temperature gradient. Thermal conductivity is turn into one function, while the

second part is turned into another. The remaining five terms are grouped into the source

term. For the first representation, the product of the density, y-velocity, and specific heat

are changed to a function of y, and the temperature gradient remains a variable. The

second term consists of a complicated sum containing the species mass fraction gradient.

This is where the coupling between the governing equations directly takes effect.

Calculation of the sum is addressed in the section Species/Energy System. The third

expression is the other sum in the equation. However, it is not nearly as difficult as the

last because it does not contain any of the system’s dependent variables. This sum is

simply calculated in the main program and added with the last two kinetic energy terms.

The last three terms in the energy equation are combined into a function representation.

Initial boundary condition of the temperature is transformed into a function for the

subprogram.

66

Boundary conditions are established for the temperature at the surface and

centerline. The temperature at the catalytic surface (y=0) is held constant. This lower

boundary condition is shown in Equation 3-17a. The upper boundary condition is

characterized by no heat flux. Temperature of the flow is uniform when it enters the

reactor. At which point, the catalyst induces chemistry in the flow and heat production

occurs at the surface. The temperature begins to increase at the surface and slowly

expand up to the centerline. A thermal boundary layer is created and heat flux across the

streamline is zero until the layer reaches the centerline. A long distance is needed for this

to occur and the heat flux remains zero at the streamline for the short distance of the

reactor. Equations 3-17a and 3-17b show the conditions in a form recognized by the

pdepe function.

( ) ( )0 0surface surfaceT T T T f= → − + ⋅ = (Equation 3-17a)

( ) ( )0 0 1 0T T uk uy y y

µ⎛ ⎞∂ ∂ ∂

= → + ⋅ + =⎜ ⎟∂ ∂ ∂⎝ ⎠ (Equation 3-17b)

In Equation 3-17a, p is the dependent variable minus the temperature at the surface

and q equals zero. This produces the constant value at the surface. The upper condition

is created with parameter p equal to zero and q equal to one. This sets the flux term,

which consists of two parts, to zero at the boundary. The first part is the heat flux and the

second contains the gradient of the axial velocity. A problem arises because only the heat

flux should be zero. However, the second term vanishes at the centerline due to the

boundary condition of the momentum equation. The result is the proper symmetry

condition at the centerline boundary.

67

Species/Energy System of Equations

The species and energy equations are combined for the pdepe function to solve.

The resulting system of (N+1) equations is shown below. The flux and source terms are

split into three parts to accommodate the various expressions in each equation.

[ ] [ ] [ ] [ ] [ ] [ ] [ ]1 2 3 1 2 3Z Z Z Zc f f f s s sx y y y y

⎛ ⎞∂ ∂ ∂ ∂ ∂= + + − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(Equation 3-18)

where ( )1 2, , , ,NZ Y Y Y T= K

Components of the dependent variable vector, Z, consist of each species mass

fractions and the temperature. Coefficients of the cubic spline generated for the coupling,

flux, and source terms are grouped together. Each component of the vectors c, f1, f3, s1,

and s3 represents the cubic spline function of that component. The other two expressions

(f2 and s2) generate the sums involving the system’s dependent variables. The coupling

term of each equation is combined into one group, c. Separating this term is not

necessary because it does not contain any mass fractions or temperature variables.

[ ]

p

uu

cu

uc

ρρ

ρρ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M (Equation 3-19)

The flux term is broken up into three separate collections. The first group, f1, is the

combination of the species equations’ flux terms and the energy equation’s thermal

conductivity expression. These terms are multiplied by the axial gradient of the

dependent variable. The Nth component of f1 is set to zero and replaced with a sum in f2.

Group f3 is a result of the additional flux term in the energy equation. The first N

68

components of this group are zero because none of the species equations contain an

additional term.

[ ]

1

2

1,

1

0

m

m

N m

DD

fD

k

ρρ

ρ −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M [ ]

00

2 1,01

0

Zf dot Fny

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎛ ⎞∂

= ⎢ ⎥ ⎜ ⎟∂⎝ ⎠⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

M [ ]

00

30

f

uuy

µ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥∂⎢ ⎥

∂⎣ ⎦

M

Figure 3-2. Flux components in the species/energy system.

The group, f2, is a result of “the fact that the sum of all the species diffusion fluxes

must be zero” [13:227]. This fact is rearranged and shown as the sum in Equation 3-18.

It suggested that this equation “be applied to the species in excess, which in many

combustion systems is N2” [13:227]. Diatomic nitrogen is the last species listed in the

input file used in the tests. This corresponds to the Nth component in the dependent

variable vector. Note that the Nth component in f1 is zero and negative one in f2. The

result is this sum replacing the mass-averaged diffusion flux in the last species equation.

MATLAB’s built-in dot product is utilized to create the sum. The last two values of Fn1

are zero because the sum does not include the Nth specie (diatomic nitrogen) or

temperature.

1

1

1,N

iim

i

Y ZD dot Fny y

ρ−

=

⎛ ⎞ ⎛ ⎞∂ ∂=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

∑ (Equation 3-20)

where ( )1 2 1,1 , , ,0,0m m N mFn D D Dρ ρ ρ −= L

The source term is also grouped into three expressions. Group s1 is a combination

of the terms multiplying the gradient of the dependent variable. The second group, s2

69

produces the complicated sum discussed in the section Energy Equation of this chapter.

All the remaining terms are compiled into s3.

1

p

vv

sv

vc

ρρ

ρρ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

M [ ]

00

2 2,01

Zs dot Fny

⎡ ⎤⎢ ⎥⎢ ⎥ ⎛ ⎞∂⎢ ⎥= ⎜ ⎟∂⎢ ⎥ ⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦

M

( )

1 1

2 2

2

1

3N N

N

i ii

MWMW

sMW

u uh u uvx y

ωω

ω

ω ρ ρ=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂− − −⎢ ⎥

∂ ∂⎣ ⎦∑

&

&

M

&

Figure 3-3. Source components in the species/energy system.

The sum in the energy equation, shown in Equation 3-21, is a function of the

species mass fractions. The energy equation is not decoupled from the species equation

and the mass fractions are dependent variables in the system. To reproduce this sum, all

of the properties (this excludes the species mass fractions) are transformed into functions

of y for each species and stored in a vector. The dot product of this vector and the

dependent variable vector gradient recreates the sum inside the subprogram

Species_Energy. The sum is then multiplied by the temperature gradient. Note that the

temperature gradient is not part of the sum, consequently the last value in Fn2 is zero.

12,

Npi im i

i i

c D Y Zdot FnMW y y

ρ

=

⎛ ⎞ ⎛ ⎞∂ ∂=⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

∑ (Equation 3-21)

where 1 1 2 2

1 2

2 , , ,0p m p m pN Nm

N

c D c D c DFn

MW MW MWρ ρ ρ⎛ ⎞

= ⎜ ⎟⎝ ⎠

L

The boundary conditions of both equations are also grouped together. Figure 3-4

shows the conditions for the system of equations. For the lower boundary condition,

parameter p is broken up into two groups, because the dependent variable is part of the

temperature’s lower boundary condition.

70

1, 1

2, 2

,

0 10 1

00 11 0

surface

surface

N surface N

surface

MWMW

Z fMW

T

ωω

ω

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦

&

&

M M M

&

0 10 1

00 10 1

f

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

M M

Figure 3-4. Boundary conditions of the species/energy system.

Function Species_Energy imports information to solve this system of governing

equations. The subprogram imports the discretized mini-mesh and all the polynomial

coefficients. Parameters p and q of the system’s boundary conditions, which are

compiled in the main program, are also sent to the subprogram. The function then

calculates the solution of each component in the dependent variable vector and returns it

to the main program. A mass fraction less than 1E-20 is treated as error and the value is

set to zero. Solutions for the mini-mesh are spliced to their corresponding variables for

the entire stage. The program moves one differential step downstream and loops back to

the momentum equation. Recall that before solving the equations for a given mini-mesh

the temperature and composition must be defined at all three axial locations. The

program linearly extrapolates these values from the previous two x-locations. The same

process solves the governing equations for the next mini-mesh. Its solution is spliced to

the stage’s solution variable and the program moves on. This continues until the end of

the stage is reached or the differential x-step is not small enough to resolve the solution.

71

CHAPTER 4 TESTING

A process of running the code for several cases and examining the solutions is

performed in order to test the program. Four cases are used to test the software.

Beginning with no chemistry in the first case, chemistry is slowly introduced to the other

cases. The second case involves only gas chemistry while the last two tests include both

gas and surface chemistry. Gas and surface chemical reactions are modeled at five

hundred and seven hundred degrees Kelvin. Slowly introducing chemistry to the model

will aid in locating errors during the debugging process should any problem arise. Each

case uses the input file named ptcombust.cti, which is provided by Cantera as part of the

software package. This file contains data for the methane/oxygen surface mechanism on

platinum developed by O. Deutschmann. The input file ptcombust.cti calls on the file

gri30.cti, also part of the Cantera package, to manage the gas reactions. The file gri30.cti

contains data for the optimized GRI-Mech mechanism and for this program calculates

transport properties based on a mixture-averaged transport model. Once the program

finds the solution for a given case, the results are examined. No experimental data is

available at this time to compare to the program’s solutions. However the different tests

can confirm that the software produces reasonable results and is operational.

Several parameters and conditions that characterize the reactor and incoming flow

are similar for the four cases. The reactor has a radius or thickness of two centimeters

and a length of thirty centimeters. Therefore, the height of the mesh is two centimeters

and the sum of stage lengths equals thirty centimeters. The distance of the non-reactive

72

surface at the entrance of the reactor is given the variable name Lnocat and is equal to

one centimeter. In case one, a catalytic surface is not present and the one-centimeter

value only determines the location of the initial velocity condition found with Blasius

solution. The differential step size in the vertical direction is set at four-hundredths of a

centimeter. This mesh spacing in the y-direction is not linear. In order to place more

points near the surface, the power discussed in the section Discretization is set to four.

The PC variable is equal to one so no iteration occurs. Temperature of the surface is set

to four hundred degrees Kelvin for case one and two. The initial temperature of the flow

is also four hundred degrees Kelvin for the first two cases. A mixture of air and methane

at one atmosphere of pressure comprise the fluid entering the reactor of every test.

Case One

Case one models a chemically inactive gas passing through a reactor with no

catalytic surface. The flow is essentially a non-reactive flow through a pipe or channel

with a pressure gradient. With no chemical reactions taking place, density remains the

same and the entire system of governing equations is altered. All of the equations could

be simplified for an incompressible flow and the set of species equations could be

removed all together. Although modifying the code in this way would defeat the purpose

of the test. To test the program only fluid properties are changed, while the code remains

unaltered. Turning the gas chemistry off is achieved by equating the species mass

production rate to zero. This only affects the source term, s3 in Figure 3-3, in the species

continuity equations. Creating a surface with no catalyst also exclusively affects the

species continuity equations. Production rates at the surface are forced to zero changing

the lower boundary conditions in Figure 3-4.

73

One stage is used to model the flow of the first case. Recall that the computational

space of the reactor can be broken up into stages. Being able to change the axial

differential step size of each stage allows the program to resolve the changing

composition. However there is no varying composition because the chemistry is

removed in this test. The reactor does not need to be split into stages for this reason. The

fluid mixture, given in mass fractions in Table 4-1, enters with a velocity of one meter

per second. Other parameters and conditions of this test run are listed in Table 4-1.

Table 4-1. Parameters and conditions of case one. Reactor Parameters Initial Flow Conditions

Radius 0.02 m Velocity 1 m/s Lnocat 0.01 m Temperature 400 K Stage length 0.30 m Pressure 101325 Pa Surface temp 400 Kdy 0.0004 m

power 4

Composition (mass fractions)

CH4:0.004, O2:0.23,

N2:0.752, AR:0.014

No. of Stages 1 PC 1 dx (Stage1) 0.01 m Input file ptcombust.cti

Results of Case One

The solution should mirror that of a viscid two-dimensional laminar flow through

two flat plates with the pressure slowly decreasing. Figure 4-1 illustrates the axial

velocity profile at four different locations. As expected the axial velocity solution

resembles a boundary layer flow increasing from zero at the surface to the centerline

velocity. The centerline velocity increases to compensate for the loss of mass flux near

the surface. This is shown in Figure 4-1, where the centerline velocity increasing

downstream as the boundary layer grows. Note the overshoot in the velocity profile at x

equal zero. This is due to the program creating a function representation of the initial

74

velocity condition. Other than the overshoot the velocity solution is a smooth continuous

model of what is expected for flow over a flat plate with changing pressure.

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.0 0.2 0.4 0.6 0.8 1.0 1.2

x-velocity (m/s)

y (m)

x=0x=0.1x=0.2x=0.3

Figure 4-1. Axial velocity profiles of case one.

Increase in the centerline velocity should lead to a decrease in pressure. The

pressure change of Figure 4-2 shows this to be the condition. The pressure slowly

decreases downstream from its initial value of one atmosphere. The incompressibility of

case one allows the use of Bernoulli’s Equation to calculate the change in pressure. This

provides an alternate means of finding the pressure with the axial velocity and ensures

that the solutions of the program are consistent. The velocity at the streamline or

centerline, where viscous effects are not present, is used in Bernoulli’s Equation. The

pressure difference calculated from both the program and Bernoulli’s Equation is graphed

in Figure 4-2. The change in pressure predicted by the program and Bernoulli’s Equation

is very similar and the behavior is typically found in the beginning stages of a pipe or

channel flow.

75

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0 0.05 0.1 0.15 0.2 0.25 0.3

x (m)

P-Pi

(Pa)

programBernoulli

Figure 4-2. Pressure plot of case one.

Variations in the species mass fractions should not exist because all chemistry is

neglected. Temperature should also have minor changes for the same reason in addition

to the low Mach number of the flow. This is the case for the first test of the program. As

expected, the calculated composition and temperature remain constant. The software

produces the expected solutions for all variables in case one.

Case Two

A flow characterized by gas reactions and no surface reactions is modeled in case

two. The flow is that of a chemically reacting fluid passing over a non-catalytic surface.

The only difference between case two and case one is the presence of gas chemistry in

the flow. Cantera determines the value of the species mass production rates. Unlike case

one, these values are not forced to equal zero. Removing the surface chemistry is

achieved by altering the lower boundary conditions of the species continuity equations.

Surface production rates are set to zero just as they are in case one.

76

Three stages are used to model the flow in case two. Because the temperature of

the flow is relatively low, little change in the composition is expected. However, using

more than one stage will test the process involved with multiple stages. This includes the

saving and loading of variables and the smooth connection of the stages. The length of

each stage is one centimeter. The same fluid composition enters the reactor, but the

initial velocity is now half a meter per second. Parameters and conditions of this test run

are listed in Table 4-2.

Table 4-2. Parameters and conditions of case two. Reactor Parameters Initial Flow Conditions

Radius 0.02 m Velocity 0.5 m/s Lnocat 0.01 m Temperature 400 K Stage length 0.30 m Pressure 101325 Pa Surface temp 400 Kdy 0.0004 mpower 4No. of Stages 3dx (Stage1) 0.01 m

Composition (mass fractions)

CH4:0.004, O2:0.23,

N2:0.752, AR:0.014

dx (Stage2) 0.01 m PC 1 dx (Stage3) 0.01 m Input file ptcombust.cti

Results of Case Two

The axial velocity calculated in the second test is graphed in Figure 4-3 for three x-

locations. Similar to the first test, the velocity is recognized as a typical pipe or channel

flow solution. The initial velocity is half a meter per second and the centerline velocity

increases from this value as the flow becomes fully developed. The presence of gas

chemistry does not appear to affect the solution of the momentum equation. Variation in

the composition is not anticipated and the velocity profile is comparable to that in case

one. The reduction in the initial velocity does remove the overshoot found in Figure 4-1.

77

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.0 0.1 0.2 0.3 0.4 0.5 0.6

x-velocity (m/s)

y (m)

x=0x=0.1x=0.2x=0.3

Figure 4-3. Axial velocity profiles of case two.

Minor variations in the density do not change the velocity, meaning the pressure

should also behave the same. Figure 4-4 illustrates that the pressure decreases

downstream much like the pressure in case one. A difference in the program’s solution

and Bernoulli’s solution is noticeable and there is almost a twenty-two percent difference

between the two. The general behavior of the pressure is consistent with expectations;

however, the software produces values that are not validated by Bernoulli’s Equation.

As expected, the temperature remains constant at four hundred degrees Kelvin.

Some changes in the temperature do occur but are very small and can be considered

numerical error. Some of the mass fractions also contain small fluctuations. Figure 4-5

shows the change in the mass fraction of the species methane. While some of the species

mass fractions behave oddly, it is most likely a product of numerical error. The program

has a second-order accuracy and the largest step size equals one centimeter. The

78

resulting error has the size of one ten-thousandths, which is greater than the error seen in

Figure 4-5.

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 0.05 0.1 0.15 0.2 0.25 0.3

x (m)

P-Pi (Pa)

program

Bernoulli

Figure 4-4. Pressure plot of case two.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-3E-12 -2E-12 -1E-12 0 1E-12 2E-12

reduction of CH4 mass fraction

y (m)

x=0

x=0.1

x=0.2

x=0.3

Figure 4-5. Reduction in methane concentrations of case two.

79

The software produces reasonable solutions for a chemically reacting flow without

a catalyst. The low temperature in this test results in little gas reactions and the mass

fractions remain nearly constant. A large incoming temperature will lead to combustion

of the fuel/air mixture. Care is taken to avoid combustion because the governing

equations are reduced based on the expectation that characteristic length scales in the

axial direction are large. The smooth connection of multiple stages is also confirmed by

the second test. This can be seen in Figure 4-4 where the pressure is a continuous

function of x. At this point the code calculates expected values for a flow with and

without chemical gas reactions.

Case Three

A complete test of the software is performed in case three where gas and surface

chemistry both exist. As it is originally intended, the model is that of a chemically

reacting fluid flow over a catalytic surface. Cantera finds the gas and surface production

rates used by the set of species continuity equations. Unlike the previous two tests, these

values are not forced to equal zero.

Table 4-3. Parameters and conditions of case three. Reactor Parameters Initial Flow Conditions

Radius 0.02 m Velocity 0.5 m/s Lnocat 0.01 m Temperature 500 K Stage length 0.30 m Pressure 101325 Pa Surface temp 500 Kdy 0.0004 mpower 4No. of Stages 2

Composition (mass fractions)

CH4:0.004, O2:0.23,

N2:0.752, AR:0.014

dx (Stage1) 0.001 m PC 1 dx (Stage2) 0.001 m Input file ptcombust.cti

Minimal change in the composition is encountered due to the low temperature and

only two stages are applied. The temperature of the gas and catalytic surface is increased

80

to five hundred degrees Kelvin and the differential step size in the axial direction

decreases to one millimeter. The other parameters and conditions are similar to the

second test and all are found in Table 4-3.

Results of Case Three

At first the original software does not obtain a solution for the entire flow in case

three and changes are made accordingly. Two centimeters into the reactor the program is

unable to resolve the changing composition and the code prematurely terminates. This

problem is found to be associated with the application of the dot products in the

subprogram Species_Energy. Once these dot products are removed, the program is able

to solve the entire computational space. The dot product of Equation 3-20 is replaced

with the mixture-averaged diffusion coefficient. The Nth species equation is now similar

to the rest of the species equations. The other sum, Equation 3-21, is no longer

performed by the dot product but is found by adding the function representations of each

species. These two dot products create the sums involving the gradients of the species

mass fractions. The problem is not noticeable in case one because the change in the mass

fractions is zero. This problem might be the source of the odd behavior seen in some of

the mass fractions and pressure difference of the second test.

The software is able to model the entire reactor after the corrections are made. The

velocity profiles graphed in Figure 4-6 are typical of boundary-layer growth in the

presence of a pressure gradient and are consistent with the models of the first two tests.

Again, the velocity increases at the centerline and the boundary layer grows as the fluid

moves downstream.

81

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

x-velocity (m/s)

y (m)

x=0

x=0.05

x=0.1

x=0.2

x=0.3

Figure 4-6. Axial velocity profiles of case three.

Figure 4-7 shows the pressure change calculated from the program and Bernoulli’s

Equation. Little difference is seen between the two solutions and both agree favorably.

It is evident that after the corrections are made, the program produces reasonable values

for the pressure in case three. The general behavior is also consistent with that of the

other two tests.

Temperature of the computational space remains constant at five hundred degrees

Kelvin. Like case two the low temperature means gas reactions are at a minimum, but

the presence of the catalytic surface generates chemical reactions. The reactions produce

a slight increase in the temperature just above the surface but the change is minimal.

The chemical decomposition of methane illustrated in Figure 4-8 also seems

logical, but the values are small enough to be considered numerical error. The species

mass fraction decreases from its initial value at the catalyst and the effect diffuses to the

82

centerline as the flow moves downstream. The decomposition is not sufficient to

generate any significant chemical activity.

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.00 0.05 0.10 0.15 0.20 0.25 0.30

x (m)

P-Pi(Pa)

programBernoulli

Figure 4-7. Pressure plot of case three.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-3.5E-11 -3E-11 -2.5E-11 -2E-11 -1.5E-11 -1E-11 -5E-12 0 5E-12

reduction of CH4 mass fraction

y (m)

x=0

x=0.1

x=0.2

x=0.3

Figure 4-8. Reduction in methane concentrations of case three.

83

The third case test reveals that the use of dot products in the subprogram,

Species_Energy, leads to resolution problems in the code. After removing the dot

products, the software produces good results. However, case three does not produce a

considerable amount of chemical activity and a higher temperature is use in case four.

Case Four

Similar to the third test, case four is another complete test of the software where

gas and surface chemistry both exist. The incoming gas temperature and surface

temperature is increased to seven hundred degrees Kelvin in an attempt to generate

chemical reactions. Three stages are applied in an attempt to resolve the changing gas

composition. Parameters and conditions are listed in Table 4-4.

Table 4-4. Parameters and conditions of case four. Reactor Parameters Initial Flow Conditions

Radius 0.02 m Velocity 0.5 m/s Lnocat 0.01 m Temperature 700 K Stage length 0.30 m Pressure 101325 Pa Surface temp 700 Kdy 0.0004 mpower 4No. of Stages 3dx (Stage1) 0.001 m

Composition (mass fractions)

CH4:0.004, O2:0.23,

N2:0.752, AR:0.014

dx (Stage2) 0.00001 m PC 1 dx (Stage3) 0.000001 m Input file ptcombust.cti

Results of Case Four

The program is not able to obtain a solution for the entire flow in case four. Nearly

five centimeters into the catalytic reactor, rapid change in the fluid’s composition is

followed by a large increase in temperature. It appears that an initial temperature of

seven hundred degrees Kelvin is sufficient to cause ignition of the air/fuel mixture over

the catalytic surface. The software is unable to resolve the rapidly changing flow

variables after this point. This is due to the fact that the code being tested is not designed

84

to model a combustion process. Governing equations are reduced based on the

assumption of a relatively large characteristic length. Large axial gradients involved with

the ignition of the fuel will cause the code to terminate at the point of ignition.

The velocity profiles, Figure 4-9, behave similarly to the other test and do not show

any error prior to ignition. Inaccuracy in the axial velocity at the point of combustion is

visible just above the surface in the boundary layer. The combustion of the fuel leads to a

temperature increase in this same region. The large temperature change causes the

density, found in the momentum equation, to change rapidly leading to error in the

velocity solution.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.00 0.10 0.20 0.30 0.40 0.50 0.60

x-velocity (m/s)

y (m)

x=0

x=0.025

ignition

Figure 4-9. Axial velocity profiles of case four.

The change in pressure is graphed in Figure 4-10. This plot shows that the pressure

decreases from its initial value of one atmosphere and appears more linear than the other

pressure graphs. Although combustion occurs, significant change in density is not

present until ignition and Bernoulli’s Equation calculates a pressure difference nearly

85

identical to the pressure change found by the program. Ignition is predicted just before

five centimeters into the reactor where the pressure gradient becomes very steep. The

pressure’s behavior at this point is unexpected and is attributed to the resolution problems

associated with combustion.

-0.009

-0.008

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 0.01 0.02 0.03 0.04 0.05

x (m)

P-Pi(Pa)

programBernoulli

Figure 4-10. Pressure plot of case four.

Temperature is graphed at four axial locations in Figure 4-11. The temperature

continues to increase just above the catalytic surface as the flow moves downstream. The

exothermic reactions induced by the catalyst lead to the temperature increase in the

boundary layer. This variable becomes large and unstable just before the code

terminates, which is visible in Figure 4-11. At the point of ignition, the temperature

increases to over eight thousand degrees Kelvin. This value cannot be viewed as an

accurate representation of the temperature. However, it appears that the ignition of the

fuel is occurring just above the catalytic surface.

86

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

650 700 750 800 850 900 950 1000 1050 1100

Temp (K)

y (m)

x=0

x=0.02

x=0.04

ignition

Figure 4-11. Temperature profiles of case four.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

CH4 mass fraction

y (m)

x=0x=0.025ignition

Figure 4-12. Methane concentrations of case four.

The chemical decomposition of methane in case four, before resolution problems

arise, behaves much like the mass fraction reduction graphed in Figure 4-8. The mass

87

fraction slowly decreases at the catalyst and the reduction effect diffuses up, away from

the surface. Significant methane decomposition occurs right before combustion ends the

program. This can be seen in the plot of the methane mass fraction in Figure 4-12.

Many species are produced once significant amounts of methane are broken down.

Two such species are atomic and diatomic hydrogen and their mass fractions are graphed

in Figure 4-13. It seems that the greater initial temperature (700K) produces the desired

effect of chemical activity. However, the large temperature also produces other species

such as OH radicals, and temperature continually grows to the point of ignition. The

code is not designed to process combustion and consequently ends at this point. As

expected the catalyst aids in the production of hydrogen and the mass fraction of both the

hydrogen atom and molecule increase at the surface. The “wiggle” found in the graph of

Figure 4-13 at the last axial position is a result of the absolute value of an overshoot.

Cantera cannot process negative mass fractions and any overshoot into the negative must

be adjusted.

Figure 4-13. Hydrogen concentrations of case four. A) Mass fractions of atomic

hydrogen. B) Mass fractions of diatomic hydrogen.

88

The fourth test predicts ignition of the fuel just above the catalytic surface nearly

five centimeters into reactor. The greater initial temperature reveals that the program is

unable to model catalytic combustion, but can forecast the point of ignition. At this point

the software is unable to resolve the rapidly changing flow variables. However, solutions

past this point are no longer physically realistic because the assumptions made to

simplify the governing equations are not valid. Characteristic length scales in the axial

direction become much shorter in the combustion process, which result in very large

gradients. The program’s resolution problems can be attributed to these large gradients

found in some of the variables being determined. The incoming temperature of five

hundred degrees Kelvin in case three is too low to produce any significant chemical

activity. While the initial temperature of case four is too great and causes combustion.

Two additional tests are preformed to better understand the temperature

dependence of chemical activity in the reactor. Both tests are similar to case three and

four, but use an initial temperature of five hundred fifty and six hundred degrees Kelvin

respectfully. The solution of the case using a temperature of five hundred fifty is very

similar to the solution of case three. The temperature and composition of the flow remain

nearly constant. The other solution, using an initial temperature of six hundred degrees

Kelvin, is similar to case four. The temperature continues to increases as the flow moves

downstream until ignition is reacted. Comparable to case four, the composition begins to

change at this point with the decomposition of methane and the production OH radicals

and other species. The point of ignition is further downstream than case four due to the

lower initial temperature. It is clear that the chemical activity is highly dependent on the

initial temperature. An initial temperature in the range of five hundred fifty to six

89

hundred degrees Kelvin is the temperature needed to cause ignition in the reactor being

modeled.

90

CHAPTER 5 PROGRAM LIMITATIONS AND IMPROVEMENTS

The program possesses several limiting characteristics when modeling a reacting

flow. Calculated solutions are second-order estimates due to the finite difference

equations. Error from these estimates could propagate into the governing equations

causing inaccuracies in the calculated solutions. The software uses a mixture-averaged

transport model in order to minimize the time needed to solve the system of equations.

The temperature at the catalytic surface is held constant. Initial conditions of a mini-

mesh and fluid properties embedded in the equations are transformed into smooth cubic

spline functions. Errors are undoubtedly produced in this process and rapid changes are

not converted to smooth functions very well. Consequently, this program cannot model

past the point of combustion and is only physically accurate for a relatively slow

reformation process.

To improve the program, the mini-mesh could be enlarged to include more than

three axial locations and the use of higher-order finite difference equations would be

possible. Increasing the size of the mini-mesh worsens the effect of the delay discussed

in the section Parameters and Conditions and iteration would probably be needed. The

code could also be modified to support a multi-component transport model. Both

changes would improve the accuracy of the solution but greatly increase the computation

time. The lower boundary condition of the temperature could also be modified to

represent a more realistic adiabatic surface or a surface with heat transfer.

91

CHAPTER 6 CONCLUSION

A program is created to validate new surface mechanisms of heterogeneous

catalysts. The adaptable program models a chemically reacting flow over a catalytic

surface. The catalytic reactor is represented in two-dimensional Cartesian coordinate

form with negligible body forces acting on the fluid. The flow is characterized as a

steady, low Mach number, boundary layer flow of a Newtonian fluid. The principles of

mass, species mass, momentum, and energy conservation are expressed mathematically

and simplified into the governing equations. The model is constructed by numerically

solving the system of coupled partial differential equations. The code, which consists of

a main program with three subprograms, is written in MATLAB and uses Cantera to

calculate chemical properties based on a mixture-averaged transport model. Allowing

Cantera to manage the chemistry independent of the main code allows the program to

remain flexible with the varying reaction pathways. Four different cases are utilized to

test the program. Calculated solutions from each case are examined to confirm that the

software produces reasonable results and is operational. The software is found to predict

the point of ignition in the fourth test where the initial temperature is great enough to

cause catalytic combustion.

Calculated values need to be compared to experimental data to truly determine the

accuracy of the program. If the comparison between experimental data and the model

reveals error in the program, improvements could be made to the code. Sacrificing

computation time for accuracy might be necessary. Once the solutions of the program

92

are proven acceptable, the program can begin to test surface mechanisms of catalyst. The

program could aid in the development of cheaper, more efficient heterogeneous catalyst.

93

LIST OF REFERENCES

1. U.S. Department of Energy. Fossil Fuels. Retrieved February 2006, from http://www.energy.gov/energysources/fossilfuels.htm

2. U.S. Department of Energy. (2004, June 1). Nuclear Plants May Be Clean Hydrogen Source. Retrieved February 2006, from http://www.eurekalert.org/features/doe/2004-06/dnl-npm061404.php

3. Fatsikostas, A., Kondarides, D., & Verykios, X. (2001). Steam Reforming of Biomass-derived Ethanol for the Production of Hydrogen for Fuel Cell Applications [Electronic version]. Chem. Commun., 2001, 851-852.

4. Chou, C., Chen, J., Evans, G., & Winters, W. (2000). Numerical Studies of Methane Catalytic Combustion inside a Monolith Honeycomb Reactor Using Multi-Step Surface Reactions. Combustion Science and Technology, 150, 27-58.

5. Steciak, J., Beyerlein, S., Jones, H., Klein, M., Kramer, S. and Wang, X. National Institute for Advanced Transportation Technology University of Idaho. (2001, September). Catalytic Reactor Studies. Retrieved November 2005, from http://www.webs1.uidaho.edu/niatt/publications/Reports/KLK317_files/KLK317.htm

6. Clark, J. (2002). Types of Catalysis. Retrieved February 2006, from http://www.chemguide.co.uk/physical/catalysis/introduction.html#top

7. KITCO. (2002, March). New York Spot Price. Retrieved March 2006, from http://www.kitco.com/market/

8. Aghalayam, P., Park, Y., Fernandes, N., Papavassiliou, V., Mhadeshwar, A., & Vlachos, D. (2003). A C1 Mechanism for Methane Oxidation on Platinum [Electronic version]. Journal of Catalysis, 213, 23-38.

9. Di Cosimo, J., Apesteguia, C., Gines, M., & Iglesia, E. (2000). Structural Requirements and Reaction Pathways in Condensation Reactions of Alcohols on MgyAlOx Catalysts. Journal of Catalysis, 190, 261-275.

10. Deutschmann, O., Schwiedernoch, R., Maier, L., & Chatterjee, D. (2001). Natural Gas Conversion in Monolithic Catalysts: Interaction of Chemical Reactions and Transport Phenomena [Electronic version]. Natural Gas Conversion VI, Studies in Surface Science and Catalysis, 136, 251-258.

94

11. Hirschl, R., Eichler, A., & Hafner, J. (2004). Hydrogenation of Ethylene and Formaldehyde on Pt (111) and Pt80Fe20 (111): A Density-functional Study. Journal of Catalysis, 226, 273-282.

12. CANTERA. (2006, April 29). CANTERA, Object-Oriented Software for Reacting Flows. Retrived May 1, 2006, from http://www.cantera.org

13. Aeronautics Learning Laboratory for Science Technology and Research. (2004 March 12). Aeronautics – Fluid Dynamics – Level 3, Flow Equations. Retrieved February 28, 2006, from http://www.allstar.fiu.edu/aero/Flow2.htm

14. Turns, Stephen R. (2000). An Introduction to Combustion: Concepts and Applications, second edition. Boston: McGraw Hill.

15. Fox, R., & McDonald, A. (1998). Introduction to Fluid Mechanics, fifth edition. New York: John Wiley & Sons, Inc.

16. Panton, R. (1996). Incompressible Flow, second edition. New York: John Wiley & Sons, Inc.

17. Goodwin, D. (2003). Defining Phases and Interfaces, Cantera 1.5. California Institute of Technology, Pasadena, CA.

18. Winters, W., Evans, G., & Moen, C. (1996). CURRENT - A Computer Code for Modeling Two-Dimensional, Chemically Reacting, Low Mach Number Flows. Sandia Report SAND97-8202, Sandia National Laboratories, Livermore, CA.

19. Tannehill, J., Anderson, D., & Pletcher, R. (1997). Computational Fluid Mechanics and Heat Transfer, second edition. Philadelphia: Taylor & Francis.

95

BIOGRAPHICAL SKETCH

Patrick D. Griffin is a graduate student at the University of Florida, Department of

Mechanical and Aerospace Engineering, where he is studying fluid mechanics. He was

accepted to the University of Florida in 2000 where he received his Bachelor of Science

degree in aerospace engineering in 2003 with the honor of summa cum laude. He has

tutored a variety of engineering courses, including the Thermodynamics and Fluid

Mechanics Lab, as a teaching assistant from 2002 to 2004. He is a member of Tau Beta

Pi Engineering Honor Society, Phi Kappa Phi and Phi Theta Kappa Honor Society. As a

graduate research assistant, he has studied the fluid mechanics and chemistry involved

with catalytic reformation and combustion.