The droplet diffusion model—An empirical model for micromixing in reacting gas phase systems

Pergamon

Chemical Enqineermy Science. Vol. 50, No. 13, pp 2061 2067. 1995 Copyright .( 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(95)00063-1

THE DROPLET DIFFUSION M O D E L - - A N EMPIRICAL MODEL FOR MICROMIXING IN REACTING GAS PHASE SYSTEMS MARTIN OSTBERG and KIM DAM-JOHANSEN Department of Chemical Engineering,The Technical University of Denmark, Building 229, DK-2800 Lyngby, Denmark

(Received 28 September 1994; accepted in revised form 18 January 1995) Abstract--An empirical mixing model for mixing of reacting species in a fluid is proposed where the mixing process is divided into macromixing and micromixing.Macromixing is assumed to be instantaneous and it is modelled by dividing the injected fluid into a number Nd of identical droplets, each surrounded by the same amount of bulk fluid. Micromixing is modelled as a molecular diffusion process in each droplet. The model can be used for any fluid residence time distribution E(t). The model is solved by means of collocation polynomials and numerical integration of the set of coupled differential equations using a third order semi implicit Runge-Kutta method. The proposed model is used to simulate pilot plant experiments of the selectivenon-catalytic reduction of NO by NH 3. The experiments have previously been simulated using an ideal plug flow reactor model without success. Combining the empirical kinetic model for the reactions with this mixing model gives satisfying results when choosing the mixing parameter corresponding to a mixing time of 20-30 ms.

INTRODUCTION Gas phase reactions are often induced by injecting the reactants separately into a reactor in which reactions proceed gradually as the reactants mix. A well known example of this kind of reaction is combustion of a gaseous fuel in a diffusion flame. In this case, the mixing process of reactants is delayed for several reasons e.g. to prevent explosions or minimize NOx formation. In high temperature gas phase reactions, many parallel reactions may occur and the mixing process may be crucial for process selectivities, as in the case of NOx formation in a flame. Thus it is important to control the mixing process. The NOx formation in a flame may be kept low if the fuel and air are gradually mixed and kept at a low combustion temperature (Beer, 1988). However, in some reaction systems an almost instantaneous mixing is required to obtain a good selectivity for the desired reactions (Jodal, 1991). In these cases, it is important that the injected gas is distributed over the entire cross section of the reactor and that mixing on a molecular level (micromixing) is rapid compared to reaction time. The mixing of gases depends on the nozzles or jets used. In the case of axisymmetric jets, where the gas is injected in the main flow direction, the fluid mechanic behaviour of the flow is well known (Hawthorne et al., 1949). The injection of gases with different densities was investigated by Era and Saima (1977), who showed that the penetration of the injected gas was dependent on the difference in the densities of the injected gas and the bulk gas. In their experiments they found that gases with low densities (hydrogen

and helium) penetrated further into air than the ones with higher densities (argon and carbon dioxide). The penetration of the jet momentum was found to depend on the ratio of the density of the injected gas to the density of the bulk gas. Only gas consisting of pure species was investigated, thus it is not possible to predict.the behaviour of a gas consisting of several species on the basis of these experiments. Regarding injection of a jet perpendicular to the main flow, Patrick (1967) studied both jet trajectories based on velocity (momentum) measurements and concentration trajectories based on measurements in which nitrous oxide (N20) was used as a tracer gas for jet momentum ratios (J = p~u]/poU~) of 6.6 and 45.5. A comparison of the N20 concentration trajectories and the jet velocity trajectories shows that the penetration of the jet momentum is deeper than the penetration of N20. This is in agreement with the results of Era and Saima (1977). Experiments by Andresen et al. (1993) have determined jet gas distribution in a crossflow by measuring glycerine tracer particle concentrations using a laser sheet visualization technique. It is assumed that the glycerine tracer particles will follow the jet flow and thus no direct comparison can be made with the behaviour of individual chemical species as described above. Thus this technique does not show the mixing of the species from the jet, only the distribution of the jet (momentum). The influence of several jets has been investigated as well, with seeding in one or three jets out of a row with five. The radioactive tracer technique has been used to investigate the jet gas distribution in a 135 MW, boiler as described by J~dal (1991), krypton isotope

2061

2062

MARTINOSTBERGand KIM DAM-JOHANSEN

85 (SSKr) was used as a tracer gas in the jet. Radioactivity was measured at different heights by four detectors placed at different horizontal positions from the nozzles. The measurements were used in selecting the optimal nozzle which was able to dose jet gas across the entire cross-section of the boiler. Jodal (1991) assumed that SSKr was distributed like the other species in the injected gas. The above mentioned papers are all about the mixing process of jets based on the fluid mechanic description. However, to be able to estimate the conversion of the species in a reacting flow, it is necessary to establish a model which can describe the mixing process and the chemical reactions. Magnussen (1981) uses the eddy dissipation concept and the k-e model (Jones and Launder, 1972) in combination with kinetic models. Chemical reactions are assumed to take place on the smallest turbulent scale, referred to as the fine structures, and all species are assumed to be ideally mixed on this scale. The model thus combines a perfectly stirred reactor (PSR) with the turbulent flow model. The model has been used to simulate flames for example. However, the computation of this flow model, in which detailed kinetic models (Glarborg et al., 1991) are used, is too comprehensive for this work; a more simple empirical model was intended. Mixing processes may generally be divided into two types: Macromixing or segregation as described by Danckwerts (1958), and micromixing. Macromixing is obtained, when physical properties like density, temperature and viscosity can be considered the same all over the fluid. However, even if perfect macromixing has been achieved, it does not necessarily imply that the concentrations of the individual molecules are homogeneous on a molecular level (perfect micromixing). Models have been developed, especially for mixing of aqueous solutions, to describe micromixing when perfect macromixing is obtained. Some of these models can be applied to gases as well. Several kinds of empirical micromixing models have been developed, e.g. environmental models (Weinstein and Adler, 1967), which describe continuous flow systems based on the concepts of age and life expectation. Assuming either total segregation or maximum mixedness in the ideal reactors, a flow model is obtained to describe the mixing process. Another kind is the diffusion-interaction models (Spielman and Levenspiel, 1965; Kattan and Adler, 1967; Rao and Dunn, 1970), according to which the fluid is divided into droplets in a flow reactor, where they coalesce two at a time and redisperse instantly to form two identical droplets. Goto and Matsubara (1975) have proposed a combination of environmental and diffusion-interaction models. However, the mixing process for all these models is described by assuming ideality (complete mixing or no mixing at all), therefore these models do not give a good description of the mixing in a gaseous system where the mixing occurs by turbulent and molecular diffusion. A diffusion model would be much more suitable for this

purpose. Nauman (19.75) has proposed a droplet diffusion model for backmix reactors in which the injected fluid is immediately dispersed into spherical droplets with a uniform size and with a concentration of each species (Ci.). These droplets remain physically intact during their stay in the reactor. The mass transfer between droplets occurs by diffusion, which takes place because the droplets are plunged into an environment with the average concentration Cou, (the exit concentration), A concentration gradient governed by reaction and diffusion will occur in the droplets. Cou, must be chosen to fulfil the reactor design equation for the backmix reactor. This model is not suitable for reactor designs other than a backmix reactor. However, the principles of the droplet diffusion model are applicable to a simple micromixing model, also in other types of reactors. Though empirical, it is not unlikely to give a good description of the actual mixing process.

THE DROPLET DIFFUSION MODEL The model proposed is based on diffusion in a droplet, as in the model proposed by Nauman (1975), but for an arbitrary residence time distribution. Where Nauman uses a concentration gradient for the diffusion process based on the difference between the inlet concentration and the average outlet concentration, this model uses the concentration of the injected gas and the concentration in the bulk gas. The mixing process is divided into two separate processes: macromixing and micromixing. Macromixing is assumed to occur instantaneously and is modelled by separating the injected fluid into Na identical spherical droplets per time unit, where Nd is determined from the mixing parameter R~.j, which is the radius of each droplet: Vi,j N,~ = (4/3)nR~.j"

(1)

V~.j is the flow rate of the injected fluid at the same pressure and temperature as the bulk fluid at the injection position since the temperatures of the two fluids are assumed to equilibrate immediately. Each injected droplet is assumed to be surrounded by the same amount of bulk fluid forming Na droplets per time unit with a total radius R0 given by: + Vbulk

Ro = Rinj ( VinJ ~i.~

)

I/3

(2)

where Vbutk is the flow rate of the bulk fluid. The mixing process for a plug flow system is illustrated in Fig. I. On the left-hand side of Fig. l is shown a cross section of the droplets at t = 0, where the inner droplet (light grey) is the injected gas and the surrounding droplet (black) is the bulk gas. All the Na droplets are identical and therefore it is only necessary to take one droplet into consideration. Micromixing is assumed to occur as molecular diffusion with no interaction between the individual

The droplet diffusion model Injection

2063

internal droplet (0 ~< r ~< Rt.j) with the following nondimensional variables where the boundary condition for r = 0 is contained in the non-dimensional radius u (Ostberg, 1991):

Plug flow reactor

1

CiR T

2

t 0 =-,

Yi = - - , p

N,

Fig. 1. Illustration of the droplet mixing model in a plug flow reactor. droplets, as illustrated in Fig. 1. This results in the following partial differential equation for each species, in which the species included is in deficit: OCi o--/ = Di

(O2Ci

+

20Ci~

/

+ vi

(3)

where C~ is the concentration of species i, t is time, D~ is the molecular diffusion coefficient for the species, r is radius, and v~is the formation rate of species i. For each droplet, the boundary conditions are based on the assumptions of no flux at the interface (r = Ro), since there is no mass transfer between droplets and symmetry about the centre of each droplet (r = 0): -~/(0, t) = 0 (4)

0C_~/(Ro, t) = 0. Or The average concentrations of the species can be found at any time by integrating the local concentration over the droplet volume:

(~i(t) =

f

ro 4rtr 2 Ci(r, t)dr =o (4/3)

nR~

I;

Cl(t) E l t ) dr.

(7)

inj

where p is the pressure, R is the universal gas constant, T is the temperature and z is the mean residence time of the gas in the system. The collocation polynomial for the surrounding droplet ( R i n j ~
CiRT P

0=-

- - ,

! z'

r -- Rinj x = - - . Ro - - Rinj

(8)

The non-dimensional variables are the same for both polynomials, except for the non-dimensional radius (u and x, respectively). The following non-dimensional equations for each species are then obtained: Oy,

D,z(

~2y,

=RL

30y,'~

+2 0"/+v;

Oyi D,r 00 (Ro -- Rinj) 2

(9)

x ( OZyi 2 \ ?~fx2 + x + (Ri.j/tRo

-

Oy,'] Ri.j) ~x J + v~

for (0 ~< r ~< Ri.j) and (Ri.j ~< r ~< Ro), respectively, where v~ is the non-dimensional reaction rate ( = v~zRT/p). The point at the interface (x = 1, r = R0) is eliminated using the boundary condition in eq. 4. The points at r = R,,j(u = 1 and x = 0) are eliminated for both collocation polynomials using the following conditions: ~u (u = l, O) = OY-~(x = O, O)

(6)

=0

If there is a temperature gradient in the system, this can be introduced into the model shown below.

OX

(10) yi(u = 1, O) = yi(x = O, 0). Thereby the two collocation polynomials are coupled together and one matrix c [ j , k ] , j, k e [1, N + M] consisting only of internal points are formed. Then the two equations in eq. 9 can be written as N + M equations for each species (Ostberg, 1991): Oy~[j] O0

SOLUTION OF THE MODEL

The set of partial differential equations which arc generally coupled through the reaction rates vi is solved using collocation polynomials for terms which depend on the radius of the droplet r (Villadsen and Michelsen, 1978) and then integrating over time t. One collocation polynomia! with N + 1 points where the (N + l)th point is for r = Ri,j is used for the CES SO-I}-O

r2 R2

T

(5)

The outlet concentrations of the species from a rector can be calculated on the basis of the residence time distribution E(t) by means of integration: Cc o., =

U

=Oi

~'+M ~. ( c [ j , k ] y / [ k ] ) + v l k: l

(11)

f o r j = [I; N + M] where c [j, k] is the same matrix for all species. The total number of N,p.,i,, x (N + M) coupled differential equations is solved by integrating over time (0) using a semi-implicit third order Runge--Kutta method (Villadsen and Michelsen, 1978). Appropriate choice for N and M is about 10.

MARTIN OSTBERGand K1M DAM-JOHANSEN

2064

A temperature profile can easily be introduced by introducing a correcting function dependent on the temperature and multiplying it with the first term on the right side ofeq. (11) and then recalculating the rate constants used in the last term (v3. The correcting function e~(T) will be dependent on the i species and is given by

1000 - -

100

NO

--:

-- - - H20

/,~'"

Z

j/

//../

v

Di(T) T 213 ei(T) = DI(To) T 2/3

.x

(12)

E //~j/J

where the ratio ( T o / T ) 2/3 is introduced to correct for the change in the radius (R+,j and R0). The developments of the concentation profile over time for species in the injected gas and in the bulk gas are shown in Fig. 2 for ammonia (NH3) and nitric oxide (NO) respectively for 1200 K as a function of the non-dimensional radius r/Ro for a non-reacting system. The average concentrations can easily be found from eq. (5) using Gaussian quadrature (Villadsen and Michelsen, 1978; Ostberg, 1991). THE INFLUENCE OF THE MIXING TIME A characteristic mixing time (t-=~,) can be calculated based on the mixing parameter Ri.j and the diffusion coefficient of the reacting species: t-mix --

R~.j

D

(13)

The mixing times for different species at 1200 K are shown in Fig. 3 as a function of Ri,j using eq. (13) and

loo 80

60

"+

,Z /

Y

0.1

0.1

1

10

Ri.j ( r a m )

Fig. 3. Mixing time as a function of the radius Ri. j for relevant species in flue gas at 1200 K and p = 1 atm.

using the binary diffusion coefficients in nitrogen as an estimation for the diffusion coefficient D calculated using the Lenard Jones potential (Bird et al., 1960). This approximation for D is acceptable for air, flue gas etc. when nitrogen is in excess. The mixing time can be used to predict the time scale for the mixing process, not the exact time. However, although it is not exact, the mixing time scale can be useful for predicting when reactions between the injected species and the species in the bulk gas will occur. In these cases, one or more characteristic reaction times (z.... ) depending on the number of species, can be determined from the kinetics of the reactions by assuming a pseudo first order reaction for the relevant species. 1

r ..... i

40

k~

(14)

where k2 is the pseudo first order rate constant for species i. The reactions will not be influenced by mixing when t-,m~ is smaller by more than one tenth of r .... . On the other hand, if t-.... is smaller by more than one tenth of t'mix, the reactions will occur before mixing has taken place.

20 6 C 0 O (jlO0 80

6O 4O 20 0 0.0

I 0.,

,o

r/Ro

Fig. 2. Concentration profiles for ammonia in the injected gas (a) and nitric oxide in the bulk gas (b) as a function of time. The solid lines are at time t = 0. The dashed lines are the profile plotted at 20 ms intervals. Calculations are performed at T = 1200 K for p = I atm and inlet concentrations of 100 ppm~.

THE MODEL USED TO DESCRIBE SNR PILOT SCALE EXPERIMENTS One process for which the proposed model can be used to predict the influence of mixing is the selective non-catalytic reduction of nitric oxide (NO), generally known as the thermal deNO~ process, the SNCR process or the SNR process. The N O concentration in a flue gas is reduced at high temperatures by injecting a reductive species, e.g. a m m o n i a (NH3) (Lyon, 1976). The mixing and the reaction of the injected fluid must take place within a narrow temperature interval depending on the choice of reductive species: about 1150 K - 1 4 0 0 K for NH3. This temperature interval

The droplet diffusion model favours the selectivity for NO reduction over the oxidation of the reductive species to form NO, which will occur as well (Ostberg and Dam-Johansen, 1994). To succeed with the SNR process it is crucial to obtain a good and rapid mixing of the injected fluid and the flue gas at the right temperature (Jodal, 1991). To use the proposed mixing model to describe the SNR process it is necessary to have a kinetic model describing the reactions. Duo et al. (1992) have proposed an empirical kinetic model for the SNR process where the very complex reactions involved are reduced to only two empirical reactions. Ostberg and Dam-Johansen (1994) have shown that this model gives a good description of the process in systems with no influence of mixing. The empirical kinetic model is based on the following reaction rates v for NO and NH 3 respectively (Ostberg and Dam-Johansen, 1994): VNO = --krCNoCNti3 + koxCNH,

(15)

vsm = -- k, CNOCNU~ -- ko, CN.,

(16)

where k, and ko, are empirical rate constants for the following empirical reactions: 4NH 3 + 4 N O + O 2

t.,4N2+6H20

4NH3 + 502 ~ - , 4 N O + 6H20

(I) (II)

and they are based on the following Arrhenius expressions k, = 2.45 x 101" exp ( - 2 9 ~ 1 0

K ) (m3/mole s) (17)

ko. = 2.21

x

-38,160K\ ~) [s- t]. (18)

101'* exp

The model does not include the influence of O2 on the reaction rates even though Kasuya et al. (1994) have shown a minor influence of 02 between volume fractions of 1 and 10%. As described above, it is possible to predict at what point the mixing will influence the reactions by comparing the mixing time with the reaction time. The time scales can be calculated on the basis of eqs (13) and (14). In the case of the SNR process, it is most relevant to compare the mixing time with the reaction time of NH3 because it is NH3 that initiates the process. Estimation of the reaction time scales for NH3 in the two reactions can, according to eq. (14), be given as to.

2065

The reaction times for Yno = 500ppmv and p = 1 atm are shown as a function of temperature in Fig. 4. By comparison with the mixing time given as a function of the mixing parameter R,,j for NH3 in Fig. 3, it can be seen that the reactions will be influenced by the mixing process for RI, J = 1 mm above 1250 K. At temperatures above 1500 K, an almost immediate micromixing, i.e. R,,j ~ 0, is necessary in order not to influence the reactions. However, these temperatures lie outside the temperature interval in which the SNR process is effective. In Ostberg and Dam-Johansen (1994), the empirical kinetic model for the SNR process given above was used together with an ideal flow reactor assumption. The results showed good agreement with SNR in the cyclone of a circulating fluidized bed eombustor (Leckner et al., 1991); a mixed flow reactor assumption was used to describe the flow in the cyclone and a plug flow reactor assumption to describe the flow in the flue gas duct. However, the model was not able to describe pilot scale experiments performed in a flow reactor (Jodal et al., 1992) when a plug flow assumption was used. Ostberg and Dam-Johansen (1994) suppose that this is explained by mixing effects, and therefore the same experiments were modelled using the proposed droplet diffusion model with binary diffusion coefficients in nitrogen as shown above. The results obtained with the mixing parameters R,,j = 2 to 4 mm, corresponding to mixing times of about 15 to 50 ms, and the results for ideal micromixing (Ostberg and Dam-Johansen, 1994) are shown in Fig. 5 together with the experimental data of Jedal et al. (1992). Using the proposed micromixing model provides an improved description of the experimental data. As expected from Figs 2 and 3, the delayed mixing has little effect on the low-temperature data (temperatures below the optimum temperature for NO reduction) where z. . . . is higher than Zm,,. At temperatures higher than the optimum temperature

10' I/(k,,,,+k.,.C,,o) ywe =

.500 ppm I/k,, . . . . . . . I / ( I ~ , C ~ ) yNo = 500 ppm

'. "

-

10 ~ ~

-

\ 10

~'"

"

t10 "'

1

"-....

(19)

= --

kox

I0

-'

\

"""

1

t" = k, COo

(20) I 0 "~

where C°o is the initial concentration of NO. For simultaneous reactions the reaction time will be 1 t .... = ko, +

k,C°o"

(21)

,1~

i

I

I

13oo

,500

17oo

Temperoture

19oo

(K)

Fig. 4. Reaction time for NH 3 calculated by eqs (19), (20) and (21) at 500 ppm, NO as a function of temperature for p = I atm.

2066

MARTIN ~)STBERG and KIM DAM-JOHANSEN 600

Ot~(~ - -

D~

Pilot I~ont e x p ~ d r v ~ m t s k:kml mixillx~] R ~ - - 0

. . . . . .Mn@ mo¢~_~ - 2 m m -- -- u i x l n g m o d ~ _R~ = 3 mm

-

500

--

Mixlng mo¢l~

Rq -

4 mm

.. ~$5- - ~ E

4oo

/ //,,"

x ~

/ / "

d 3oo

0 Z

k, ko, d M

/ /

i//,,,,,,

c 0 0

k k~

diffusion coefficient of species i in N2, m2/s residence time distribution turbulent kinetic energy, m2/s 2 pseudo first order rate constant for species i, S

/

/o/,,

E(t)

• 1

rate constant for reaction (I), m3/(mole s) rate constant for reaction (II), smomentum ratio [p~u~/poU~] number of collocation points from Rin j < r < R o

200

N

number

of

collocation

points

from

0 < r < Rin j 100

0 lOOO

I

I

I

I

11 oo

~200

~3oo

14oo

~soo

Injection temperoture (K) Fig. 5. Simulation of experimental results with an inlet concentration of 500 ppm~NO, the molar ratio fl = 1.3,temperature gradient of - 200 K/s and residencetime of 0.25 s. The calculations are performed with pressure p = I atm and the number of collocation points N = M = 20.

for NO reduction, delayed mixing has a significant effect on the reduction process due to the higher reaction rates. The values of 2-4 mm chosen for Rinj, corresponding to a mixing time of about 15-50 ms, indicate that there will be some influence on the mixing process at 1200 K and above when the initial NO concentration is 500 ppm. The mixing time found is in good agreement with the experiments made by Jodal (1991). At lower temperatures, where no reaction of NH3 occurs, the mixing time was found to be about 40-50 ms.

N~ N,p,¢~,, p r R R~.j R0 t T u u~ Uo V x

number of droplets in eq. (1) number of species included in the model pressure, Pa radial coordinate, m molar gas constant: 8.3144 J/(mole K) droplet radius of injected gas (mixing parameter), m droplet radius of injected gas and flue gas, m time, s temperature, K non-dimensional radius [r2/R~.j] velocity of the jet flow, m/s velocity of the bulk flow, m/s flow rate, m3/s non-dimensional radius [ ( r - Ri.jJ/(RoRin))]

y~

Yi[j] )?~

molar fraction of species i molar fraction of species i in collocation point j average mole fraction in droplet

Greek letters fl e,

molar ratio [NH3].,/[NO-].v o , o dissipation of turbulent kinetic energy,

el(T)

correction function for the temperature profile non-dimensional time (t/z) reaction rate of species i, mole/(m 3 s) non-dimensional reaction rate of species i density, kg/m 3 space time or residence time of plug flow reactor, s mixing time defined as R2,jD, s reaction time for reaction (II), s reaction time for reaction (I), s reaction time defined in eq. (19) s

m4/s 3 CONCLUSIONS

A micromixing model has been proposed which separates the mixing process into two parts: Macromixing, which is modelled by dividing the injected fluid into Na identical droplets, each surrounded by identical amounts of bulk fluid, and micromixing, which is modelled by diffusion in the droplets with no interactions between the individual droplets. The model was developed to describe the reactions between species from the injected fluid and the bulk fluid for any given residence time distribution. The model may include a temperature profile as well. This model has been used to simulate the mixing in the SNR process in pilot plant experiments where an ideal mixing assumption could not describe the results. Assuming a mixing time of about 20-50 ms, which is in good agreement with experimental results, a successful result was obtained.

NOTATION

c[j, i] C D

collocation matrix molar concentration, mol/m 3 diffusion coefficient, m2/s

0 vi v~ p

rmi~ rol ~, ....

Subscripts bulk

value for bulk gas (flue gas) without injected gas

in inj j 0 out

inlet value value for injected gas value for the jet flow value for the bulk flow outlet value

Superscript 0

initial value

The droplet diffusion model REFERENCES

Andresen, E., Larsen, P. S. and Dam-Johansen, K., 1993, Jets in a cross flow--mixing studies by light sheet visualization. Third Symposium on Experimental and Numerical Flow Visualization. ASME. FED-Vol. 172, pp. 293-300. Beer, J. M., 1988, Stationary combustion: The environmental Leitmotiv, 22nd Symposium (International) on Combustion, The Combustion Institute, pp. 1-16. Bird, R. B., Stewart, W. E. and Lightfoot, E. N., 1960, Transport Phenomena. Wiley, New York. Danckwerts, P. V., 1958, The effect of incomplete mixing on homogeneous reactions. Chem. Engnq Sci. 8, 93-102. Duo, W., Dam-Johansen, K. and ~stergaard, K., 1992, Kinetics of the gas-phase reaction between nitric oxide, ammonia and oxygen. Can. J. chem. En#na 709, 1014-1020. Era, Y. and Saima, A., 1977, Turbulent mixing of gases with different densities. Bull. JSME 20(139), 63-70. Glarborg, P., Lilleheie, N. I., Byggstoyl, S., Magnussen, B. F., Kilpinen, P. and Hupa, M., 1991, Modelling and Chemical Reactions. Development and Test of Reduced Chemical Kinetic Mechanism for Combustion of Methane, NGCreport, Nordic Gas Technology Center, Horsholm, Denmark. Goto, S. and Matsubara, M., 1975, A generalized two-environmental model for micromixing in a continuous flow reactor--l. Construction of the model. Chem. En.qna Sci. 30, 61-70. Hawthorne, W. R, Weddell, D. S. and Hottel, H. C., 1949, Mixing and combustion in turbulent gas jets. Third Symposium on Combustion Flame and Explosion Phenomena, pp. 266-288. Jodal, M.. 1991, Optimerization of selective non-catalytic reduction of nitric oxide in coal combustion, Ph.D. thesis, The Technical University of Denmark/Aalborg Ciserv International (in Danish). Jodal, M., Lauridsen, T. L. and Dam-Johansen, K., 1992, NO~ removal on a coal-fired utility boiler by selective non-catalytic reduction. Environ. Pro#. 11(4), 296-301. Jones, W. P. and Launder, B. E., 1972, The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transfer 15, 301-314.

2067

Kasuya, F., Glarborg, P. and Dam-Johansen K.. 1994, The thermal deNOx process: Influence of partial pressure and temperature. Chem. Engng Sci. (in press). Kattan, A. and Adler, R. J., 1967, A stochastic mixing model for homogeneous, turbulent, turbular reactors. A.I.Ch.E. J. 13(3), 580-585. Leckner, B., Karlsson, M., Dam-Johansen, K., Weinell, C. E., Kilpinen, P. and Hupa, M., 1991, Influence ofadditives on selective noncatalytic reduction of NO with NH a in circulating fluidized bed boilers. IEC Res. 30, 2396-2404. Lyon, R. K., 1976, The N H 3 - N O - O 2 reaction. Int. J. Chem. Kinet. 8, 315-318. Magnussen, B. F., 1981, On the structure of turbulence and a generalized eddy dissipation concept for chemical reactions in turbulent flow, 19th AIAA Sc. Meeting, St. Louis, USA. Nauman, E. B., 1975, The droplet diffusion model for micromixing. Chem. Engng Sci. 30, 1135-1140. Ostberg, M., 1991, The dependence of non ideal flow on gas phase reactions, Thesis, Department of Chemical Engineering, The Technical University of Denmark (in Danish). Qstberg, M. and Dam-Johansen, K., 1994, Empirical modelling of the selective non-catalytic reduction of NO---Comparison with large scale experiments and detailed kinetic modelling. Chem. Entlntl Sci. 49(12), 1897-1904. Patrick, M. A., 1967, Experimental investigation of the mixing and penetration of a round turbulent jet injected perpendicularly into a transverse stream, Trans. Instn. chem. Engrs 45, T16-T31. Rao, D. P. and Dunn, I. J., 1970, A Monte Carlo coalescence model for reaction with dispersion in a tubular reactor. Chem. Enyny Sci. 25, 1275-1282. Spielman, L. A. and Levenspiel, O., 1965, A Monte Carlo treatment for reacting and coalescing dispersed phase systems. Chem. Engn# Sci. 20, 247-254. Villadsen, J. and Michelsen, M. L., 1978, Solution of Differential Equation Models by Polynomial Approximation, Prentice-Hall, New Jersey. Weinstein, H. and Adler, R. J., 1967, Micromixing effects in continuous chemical reactors. Chem. Engng Sci. 22, 65-75.