Numerical Simulation of the Laminar Diffustion Flame in a Simplified Burner

8/10/2019 Numerical Simulation of the Laminar Diffustion Flame in a Simplified Burner

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LDC-2014-00112/23/2014

NUMERICAL SIMULATION OF THE LAMINAR DIFFUSION

FLAME IN A SIMPLIFIED BURNER

Lawrence D. Cloutman

[email protected]

Abstract

The laminar ethylene-air diffusion flame in a simple laboratory burner was simu-

lated with the COYOTE reactive flow program. This program predicts the flow field,

transport, and chemistry for the purposes of code validation and providing physical

understanding of the processes occurring in the flame. We show the results of numerical

experiments to test the importance of several physical phenomena, including gravity,

radiation, and differential diffusion. The computational results compare favorably with

the experimental measurements, and all three phenomena are important for accuratesimulations.

c2014 by Lawrence D. Cloutman. All rights reserved.

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1 Introduction

Flower and Bowman [1, 2, 3] measured the temperature and velocity fields in a simple

laboratory-scale burner. Although their primary goal was to study the sooting properties of

this burner, these measurements are suitable for validation of computational fluid dynamicsprograms. The geometry is simple, the flow is laminar, and the combustion occurs in an

ethylene-air diffusion flame. The burner is a simple Wolfhard-Parker burner enclosed in a

rectangular chamber as shown in figure 1. Fuel is fed through a rectangular tube 0.8 cm wide

by 8.0 cm long, and coflowing air surrounds the tube. Flame stability requires the initial air

velocity to exceed the initial fuel velocity, and these are 22.0 and 7.0 cm/s, respectively. A

steel screen was placed 1.5-2.5 cm outside the flame to reduce flickering. The results reported

here are for a pressure of one atmosphere.

The experimental measurements of temperature and vertical velocity are the subject

of this study. A small silica-coated thermocouple was use to measure gas temperatures. The

measurements have been corrected for radiative losses and have estimated uncertainties of

4%. Velocity measurements were made with standard LDV techniques.

This report is an updated version of reference [4]. Simulations of the experiment were

performed with the COYOTE computational fluid dynamics program [5], which uses a par-

tially implicit algorithm to solve the full transient Navier-Stokes equations on a nonuniform

grid. The model includes a real-gas caloric equation of state, arbitrary chemical kinetics,

transport coefficients from a Lennard-Jones model, a simple radiative heat loss model, and

mass diffusion based on either Ficks law or the full Stefan-Maxwell equations. The ethy-

lene combustion is modeled by a one-step global Arrhenius kinetics rate. Also included are

three kinetic reactions for thermal NOx production and six molecular dissociation reactions

required to close the thermal NO reaction set and produce accurate temperatures.

Steady-state solutions, when they exist, are found by assuming an arbitrary initial

flow and allowing the transient to decay. While this approach is less efficient computationally

than a direct steady state approach, it has several advantages. First, it does not make an ad

hocassumption that the final flow field is truly steady. That is, it allows the final solution

to contain quasi-periodic features such as flame flickering. If the final flow is truly steady,

then that type of solution will evolve naturally. Second, it automatically performs a flamestability study. If, for example, the flame cannot be sustained, it should be extinguished

in the calculation (at least to the extent that the gas physics included in the calculation is

an accurate representation of the physical system). Third, it allows easy incorporation of

complex, realistic gas physics.

We show the results of numerical experiments that test the importance of several

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physical phenomena, including gravity, radiative heat loss, and differential diffusion based

on the Stefan-Maxwell equations. The base case includes all three of these phenomena.

Several additional runs were made. One used zero gravity. The second neglected radiative

losses. The third used Ficks law and a unit Lewis number for all species instead of the

Stefan-Maxwell equations. The computational results are in excellent agreement with theexperimental measurements of velocity and temperature provided all three phenomena are

included. We find that the solutions are quite sensitive to gravity, NOx predictions will

require the use of the radiation model to get adequate accuracy in the temperature field,

and the solutions are sensitive to the mass transport model.

Section 2 presents the governing equations and describes the gas physics. Section

3 describes the geometry of the experimental apparatus and the problem setup. Section 4

describes several solutions for this burner. Conclusions are presented in Section 5.

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2 Governing Equations

The program is based on the Navier-Stokes equations for a mixture of compressible gases.

We use the single (mass weighted) velocity representation and Eulerian coordinates.

Mass conservation is expressed by the continuity equation for each species :t

+ (u) = J+ R, (1)

where is the density of species , uis the fluid velocity, J is the diffusional mass flux of

species , and R is the rate of change of species by chemical reactions. The diffusional

flux is a complex function of the flow that often is approximated by Ficks law,

J= D(/), (2)

in some of the solutions. D is the species diffusivity (assumed here independent of), andis the total density. We also use the formalism of Ramshaw [6, 7], which is an approximate

treatment of the full Stefan-Maxwell equations. TheR are assumed to be known functions

of the composition and thermodynamical variables. The global rate for ethylene oxidation,

for example, is

RC2H4 = 6.4 1012 WC2H4[O2]

1.65 [C2H4]0.1 exp(15000/T) g/cm3s, (3)

where WC2H4 is the molecular weight of ethylene, and square brackets denote molar concen-

tration (mol/cm3

). This rate predicts the correct laminar flame speed under stoichiometricconditions at one atmosphere.

The momentum equation is

(u )

t + (uu ) =

F P S, (4)

where P is the pressure, and F is the body force per unit mass acting on species , which

in the present application is the gravitational acceleration g. The viscous stress tensor is

S= [u + (u)T] 1( u) U, (5)

where is the coefficient of viscosity, 1 = 2/3 is the second coefficient of viscosity, and

Uis the unit tensor.

We use the thermal internal energy equation to express energy conservation:

(I)

t + (Iu) = P u S: u q +

HR+

F J Lrad, (6)

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where I is the specific thermal internal energy, andH is the heat of formation of species .

Note that for F = g, the next-to-the-last term vanishes. The heat flux q is approximated

by the sum of Fouriers law and enthalpy diffusion:

q= T+ h

J

, (7)

where h is the specific enthalpy of species .

The radiative heat loss term Lrad is described in [8]. A complete treatment of the

radiative transfer would be extremely complex and computationally challenging, so we con-

sider only a highly simplified model that is valid in the optically thin limit, namely a local

radiative heat sink. We use a slight generalization of the approximation used by Chao, Law,

and Tien [9], namely

Lrad = 4 KP (T4 T4

w), (8)

where is the Stefan-Boltzmann constant. The wall temperature Tw is assumed to be the

same for all walls. The functionKP is related to the Planck mean opacity, P, by

KP(T, {})

=P =

T4

0

k(T, {})B(T) d, (9)

where {} is the set of all species densities, k is the monochromatic absorption coefficient,

and B is the Planck function. Planck mean absorption coefficients for three important

radiating species (CO, CO2, and H2O) were taken from [10].

The equation of state is assumed to be given as the sum of the partial pressures of

an ideal gas for each species. Transport coefficients are computed from the Lennard-Jones

model [11]. The JANAF tables [12, 13, 14] provide a homogeneous set of thermochemical

data for a large collection of materials, and these tables are used to supply the specific

enthalpy and heat of formation for each species of interest.

Chemical reactions are divided into two groups as shown in table 1. The first group is

treated kinetically, with the rates assumed to be of generalized Arrhenius form. The second

group, reactions 1e through 6e, is assumed to be in chemical equilibrium.

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Table 1. One-Step Fuel Oxidation and Thermal NOx Mechanisms

Reaction A E/R (K)

1. C2H4 + 3 O2 2 CO2 + 2 H2O 6.4 1012 0.0 15000 [C2H4]

0.1 [O2]1.65

2a. O2 + 2 N2 2 N + 2 NO 1.5587 1014 0.0 67627 [O2]

0.5 [N2]1.0

2b. O2 + 2 N2 2 N + 2 NO 7.5 1012 0.0 0 [N] [NO]

3a. 2 O2 + N2 2 O + 2 NO 2.6484 1010 1.0 59418 [O2] [N2]

0.5

3b. 2 O2 + N2 2 O + 2 NO 1.6 109 1.0 19678 [O] [NO]

4. N2 + 2 OH 2 H + 2 NO 2.123 1014 0.0 57020 [N2]

0.5 [OH]1e. H2 2H2e. N2 2N3e. O2 2O

4e. O2 + H2 2OH5e. O2 + 2H2O 4OH6e. O2 + 2CO 2CO2

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3 Problem Description

The geometry of the experiment is shown in figure 1. The burner is nothing more than a

rectangular metal tube with a 0.8 by 8.0 cm cross section through which ethylene flows. The

tube is surrounded by the axial air flow, which is confined in a rectangular chamber whosewalls are several centimeters from the tube. The high aspect ratio of the open end of the

tube allows us to assume the flame is uniform along most of the length of the opening. The

two dimensional simulations were made in a plane perpendicular to the long direction of the

slot. We also assume bilateral symmetry about a line bisecting the slot the long way, so

we place the lower left hand corner of the grid at the center of the slot, but 0.5 cm below

the opening. The wall of the slot is assumed to be a sheet of metal 0.1 cm thick and is

represented in the two-dimensional plots as a series of xs. The base case uses a uniform grid

of 1 mm square zones and has 30 by 60 zones. One run was made with 0.5 mm square zones.

The fuel, ethylene, is allowed to flow into the bottom of the mesh through horizontal zones

numbers 2 through 5 in the coarse grid (number 1 is the fictitious zone at the left side of the

mesh), and air flows in through zones 6 through 31. An obstacle representing the edge of

the burner occupies the first 6 zones vertically (including the bottom fictitious zone) of the

6th column.

Our inflow boundary condition is the type (ii) of Rudy and Strikwerda [15] with

specified density. We assume that the inflowing gases have a temperature of 300 K and a

pressure of 1.013 106 dynes/cm2. The inflow velocities are 7 cm/s for the ethylene and 22

cm/s for the air velocity. The inflow density is 1.131 103 g/cm3 for the ethylene. The

air was assumed to be a mixture of five species with densities of 2 .688 104 g/cm3 O2,

8.766 104 N2, 5.292 107 CO2, 7.217 10

8 H2O, and 1.489 105 argon. Unless

otherwise noted, the gravitational acceleration is assumed to be -980 cm/s2.

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4 Numerical Solutions

A series of six cases were run out to steady state using a variety of numerical parameters

and physical submodels. The base case has a resolution of 1 mm and Ficks law is used for

the species diffusion. The same diffusivity is used for all species and is calculated from themixture viscosity and a Schmidt number of 0.7. The radiation model was included. The

same problem was run with 0.5 mm zones. There was no significant change in the solution,

suggesting grid independence of the solutions. It is somewhat surprising that the solution

with 1 mm resolution is so well converged. Another variation of the base case was to use a

fuel oxidation rate that is half that of equation (3). This had no effect on the solution. This

is not surprising since it takes approximately 0.01 s for the fluid to cross a computational

zone, but the chemical time scale for oxidizing the fuel is approximately three orders of

magnitude smaller. A similar insensitivity to the reaction rates, however, is not expected for

the much slower thermal NO reactions.

Another case was the same as the base case except Ficks law was replaced by the

detailed mass transport model. A comparison between these two cases is given in the next

five figures. Figure 2 shows the isotherms for both cases. The multicomponent diffusion

model produces a higher peak temperature (2422 K, as compared to 2317 K with Ficks

law). The height of the flame was not measured in the experiment, but it was observed to

be well beyond the height of this computational grid (6.0 cm), as predicted here. The axial

temperature gradient in the core of the flame is significantly smaller in the multicomponent

case.

Figure 3 shows the calculated and experimental horizontal temperature profiles. The

left edge of each plot is at the center line of the flame, and the peak temperature occurs

approximately above the edge of the slot. The Ficks law calculation does very well except

in the center of the flame. The multicomponent calculation does better in the center, but

is systematically a little hotter than the experimental values. The experimental flame was

intended to produce sooting conditions, and the calculations do not yet have a soot produc-

tion or soot radiation model. The fact that the multicomponent calculation is systematically

slightly too hot is consistent with the radiative losses expected from the soot.

Figure 4 shows the horizontal profile of axial velocity for the experiment and bothcalculations. Both calculations agree with the experiment out to 1.0 cm. It is not known

with certainty why the outer flow of air is systematically higher than the 22 cm/s flow speed

reported by Flower and Bowman, but it seems likely that the experimental air inflow speed

was closer to 30 cm/s in the cases at 1 atm pressure.

Figure 5 shows mass fractions of H2 for both calculations. Since differential diffusion

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effects are largest for the very light species, we expect some differences between these two

plots. Not only are the details of the internal distributions in the flame different, but the

peak values are different by a factor of nearly two.

Figure 6 shows mass fractions of NO. As in the case of H2, there are some significant

differences in the spatial distribution. With Ficks law, the peak NO mass fraction occursin a sheet in the outer part of the flame, coincident with the high temperatures. In the

multicomponent model, the peak NO mass fraction occurs in an island just above the lip of

the burner, and there is nearly a factor of four difference in the peak NO mass fraction. The

NO flow rates at the top of the grid are 2 .5 106 g/s-cm and 6.4 106 g/s-cm for the

Ficks law and multicomponent solutions, respectively. These results strongly suggest that

burner models that are expected to produce reliable solutions that include any significant

level of chemical detail must include a detailed mass transport model.

The base case was rerun without the radiative heat loss model. The peak temperature

is 2355 K, close to the adiabatic stoichiometric flame temperature of 2380 K. The isotherms

are qualitatively the same as in the base case. Temperatures are typically 50 to 100 K higher

at points in the flame zone as compared to the base case. The H2 mass fraction peaks at

8.6 104, a little over twice the base case value. The peak NO mass fraction is 3 .5 104

and has a different spatial distribution than in the first two cases. Far more NO production

is occurring as the hot gases rise through the grid than in the base case. The NO flow rate

is 6.8 106 g/s-cm and clearly would be higher if the grid were taller.

The importance of buoyancy forces is demonstrated by rerunning the base case with

zero gravitational acceleration. Figure 7 shows the isotherms and NO mass fractions for thezero-gravity case. The flame is now much wider than before, and the lower temperatures

suppress NO production.

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5 Conclusions

The COYOTE hydrodynamics program has been used to simulate reactive flows in a sim-

plified experimental burner. One objective of this study is validation of the program and

demonstration of its capability to simulate laminar diffusion flames. The second objective isto study the importance of several physical submodels in burner simulations.

While detailed comparison of the computational and experimental results is still in

progress, we can make the following general observations:

1) The model successfully predicts the velocity and temperature fields with surpris-

ingly coarse zoning (1 mm resolution).

2) It is critical to include gravity in the calculations as buoyancy effects are quite

pronounced. Without buoyancy forces to help lift hot combustion products away from the

burner, the flame becomes much broader and takes on a different velocity profile.

3) A comparison of the results with Ficks law mass diffusion and the Stefan-Maxwell

equations shows that both the temperature and velocity fields are predicted more accurately

with the latter model. The difference is as much as several percent. Moreover, the spatial

distributions of several radical species change significantly between the two models. This

behavior suggests the importance of using the detailed transport model in applications where

accurate predictions of some of the minor species are important, such as in the case of

emissions of NO, formaldehyde, and PAHs. In general, we recommend the use of the more

complex model whenever possible for laminar flame simulations.

4) The radiative cooling model lowers local combustion temperatures on the order of

50-100 K for non-sooting flames. While this change has little effect on the overall dynamics

of the flow, it makes approximately a factor of two difference in the thermal NO production

rate due to the high temperature sensitivity of the thermal NO mechanism.

5) Future work will concentrate on more detailed chemical kinetics, an improved

radiative transfer model, and development of a soot model (both the chemistry and the

radiative properties).

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References

[1] W. L. Flower and C. T. Bowman, Measurements of the Structure of Sooting Laminar

Diffusion Flames at Elevated Pressures, 20th Symposium (International) on Combus-

tion, The Combustion Institute, Pittsburgh, 1984, 1035-1044.

[2] W. L. Flower and C. T. Bowman, Measurements of the effect of elevated pressure on

soot formation in laminar diffusion flames, Comb. Sci. Tech. 37, 1984, 93.

[3] W. L. Flower, The Effect of Elevated Pressure on the Rate of Soot Production in

Laminar Diffusion Flames, presented at the Spring Meeting of the Western States

Section of the Combustion Institute, 1985.

[4] L. D. Cloutman, Numerical Simulation of the Laminar Diffusion Flame in a Simplified

Burner, Lawrence Livermore National Laboratory report UCRL-JC-122619, 1995.

[5] L. D. Cloutman, COYOTE: A Computer Program for 2-D Reactive Flow Simulations,

Lawrence Livermore National Laboratory report UCRL-ID-103611, 1990.

[6] J. D. Ramshaw, Self-consistent effective binary diffusion in multicomponent gas mix-

tures, J. Non-Equilib. Thermodyn. 15, 295 (1990).

[7] J. D. Ramshaw, Hydrodynamic theory of multicomponent diffusion and thermal dif-

fusion in multitemperature gas mixtures, J. Non-Equilib. Thermodyn. 18, 121 (1993).

[8] L. D. Cloutman, Numerical Simulation of Radiative Heat Loss in an Experimental

Burner, Lawrence Livermore National Laboratory report UCRL-JC-115048, presented

at the 1993 Fall Meeting of the Western States Section Meeting of the Combustion

Institute, 1993.

[9] B. H. Chao, C. K. Law, and J. S. Tien, Twenty-Third Symposium (International) on

Combustion, The Combustion Institute, 1990, 523.

[10] M. M. Abu-Romia and C. L. Tien, Appropriate mean absorption coefficients for in-

frared radiation of gases, J. Heat Transfer 89, 321 (1967).

[11] L. D. Cloutman, A Database of Selected Transport Coefficients for Combustion Stud-

ies, Lawrence Livermore National Laboratory report UCRL-ID-115050, 1993.

[12] D. R. Stull and H. Prophet,JANAF Thermochemical Tables, 2nd ed. (U. S. Department

of Commerce/National Bureau of Standards, NSRDS-NBS 37, June 1971).

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[13] M. W. Chase, J. L. Curnutt, A. T. Hu, H. Prophet, A. N. Syverud, and L. C. Walker,

JANAF Thermochemical Table, 1974 Supplement, J. Phys. Chem. Ref. Data 3:311

(1974).

[14] M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, M. A. McDonald,

A. N. and Syverud, JANAF Thermochemical Tables, Third Edition, Parts I and II.

Supplement No. 1, J. Phys. Chem. Ref. Data14 (1985).

[15] D. H. Rudy J. C. and Strikwerda,Computers & Fluids9:327 (1981).

[16] L. D. Cloutman, Numerical Simulation of Turbulent Mixing and Combustion Near the

Inlet of a Burner, Lawrence Livermore National Laboratory report UCRL-JC-112943,

1993.

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1

a ir

12 cm/s

grid

ethylene

7 cm/s

T

air

a ir

F ig. 1. Top a nd side view s of t he slot burner . The loca tion of the tw o-dimeneiona l computa tiona l

gr id is a lso sh ow n.

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TEMPERATURE

Ficks Law Multicomponent

MIN = 2,7103990 02

I.AAX= 2.316704D -03

MIN = 2.550626D -02

MAX = 2.4215020 03

F]g. 2. I sot herms for t he bsse cuse F icks la w m aa s t ra nspor t a nd mult icompon en t m sss t ra nspor t

solutions.

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Ficks Law

\

1 02 0.4 0.6 0.8 Lo - t~ -

x an

:

0.0 0.2 0.+ 0.6 O.B LO 1.2 L4 I-6

x an

3. Hor iz on ta l pr ofiles of t emper a tu re 2.0 cm a bove t he bu rn er .

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Fides Law

1

Calc

0

:

Multicomponent

0

:

z:

0

:

;

:

0

Calc =

s

0

s

0

0

0.0 02 0.4 0.6 0.8 ,0 L2 ,.+ *.6

* cm

4.

Hor izon ta l pr ofiles of a xia l v elocit y 2.0 cm a bove t he bu rn er .


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H2 MASS FRACTION

Multicomponent

L

MAX = 3.841191D-04 MAX = 6.438925D-04

F ig. 5. H z ma ss ikct ion con tours for t he ba se ca se F icks la w ma ss t ra nspor t a nd mult icomponen t

ma ss t ra nspor t solut ions.

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NO MASS FRACTION

Ficks Law

1

~

I

3

1 : 3

I

I

I

1

MAX= 9 680672005

Multicomponent

F ig. 6. NO m a ss fr act ion con tou rs for t he bsse ca se F icks la w m a ss t ra nsport a nd mult icompon en t

ma ss t r a ns por t s olu t ion s.

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8

MAX = 2.313397D 03

8

MAX = 9.079066D-05

F ig. 7. I sot herms a nd NO mass fra ct ion con tour s for t he zero-gr avit y solu tion .

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Numerical Simulation of the Laminar Diffustion Flame in a Simplified Burner

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