Atmospheric dispersion of a chemically reacting plume

AmMs*C &nui?onment Vol. 12, pp. 1895-1900. Q Rrgamon Press Ltd. 1978. Printed in Great Britain.

ATMOSPHERIC CHEMICALLY

DISPERSION OF A REACTING PLUME

D. J. ~~~rnent

&WLEY

of Mechanical Engineering, The University of Sydney, N.S.W. 2006, Australia (First received 28 November 1977 and infinalform 10 February 1978)

Abstract - The atmospheric dispersion of a plume of pollutants, undergoing fast enough chemical reactions so that equilibrium is reached, is modelled. The effects of turbulence on the mean concentrations, due to turbulence-chemistry interactions and dispersion, are found by combining tha probability density function method and the Gaussian plume model. The results, for a plume of NO and NO, in ambient ozone on a sunny day, show that the NO concentration is higher than the value expected for a nonturbulent chemistry model. The effects of temperature fluctuations are shown to be small. The method depends upon the assumption of instantaneous equilibrium and the conditions for this breaking down are pointed out.

Atmospheric dispersion models for pollutants range from simple Gaussian plume models for inert species (Pas&II, 1968) to complex models for photochemical smog (Eschenroeder and Martinez, 1972; Reynolds et al., 1973) which consider many simultaneous chemical reactions. Recently the Gaussian plume model has been extended by Peters and Richards (1977) to include species undergoing fast reversible reactions. They take advantage of the widely used plume model for inerts to describe the transport of conserved scalars and use the ~~ib~~ relationships between the species. For convenience, they assume that the timeaveraged ~~iib~urn expression is the same as the expression utiiising time-averaged species concentrations. This paper uses recently developed methods (Bray and Moss, 1974, 1977; Bilger, 1976) of analysing turbulent reacting flows to assess the validity of the above assumption. The probability density function (p.d.E) method is applied to a NO, plume reacting with ambient ozone on a sunny day. This paper can be regarded as an application of the work by Biiger (1978).

MODEL

The pollutant plume is emitted by a source at an effective height H into an atmosphere with an average wind velocity ti in the x-direction. There are no crosswinds and so ii, KJare zero. A representation of the plume is given in Fig. 1. The instantaneous balance equation for a conserved scalar, which has no source terms, is

-Y

Fig. 1. Repr~entation

of the plume and its coordinates.

To construct the time-averaged 4 balance equation it is convenient to split each variable up into its mean and fluctuating component, e.g. 4 = 6 + #. The balance equation for the time-averaged conserved scalar is therefore

where E,,and sz are the eddy diffusivity in the y and z directions respectively. The balance equation for the variance of the eonserved scalar is found by multiplying (1) by #‘, adding the resulting equation to itself and then time averaging. Thus

- 2D,V&*Vtj where D+ is the diffusion coefficient.

(3)

where v’ and w’ are the veiocity fluct~tions in the y and z directions respectively and the last term is the 1895

1896

D. J.

dissipation tensor of #‘. When the usual order of magnitude assessments (see e.g. Bray, 1973) are made to (3) and the gradient transport assumption

already applied to the diffusion terms in (2) and (3), is used then

KEWLEY

using balance equations for #, and #:. Based upon the review (Bilger, 1976) of experimental data in a variety of flows and its application to the atmosphere, Bilger (1978) shows that typical shapes for the p.d.f. are expected to be Gaussian-like in the centre of the plume and have an inte~ittency spike at the plume edges. The clipped-Gaussian p.d.f., in which the tails ( - .f , min(&,)) and (max(4,), 7 ). of a Gaussian distribution -_with mean 4i and variance (p’,’ are redistributed as delta functions at min(+,) and max(4,), has been utilised in turbulent flame models by Bray and Moss (1974, 1977). This p.d.f. is adopted here, with rn~n(#~~ = #t and max(#,) = x:, because it covers the range of expected p.d.f. shapes. Thus P(41) = (1 -r)&PY)+r&#-Q)V(&)

(4) where the dissipation term is simplified to the form given in Donaldson and Hilst (1972) (a = 3.4 x lo-3s-1). For the example considered here of NO, reacting with ambient ozone on a sunny day the three main reactions are NOZ+kv+NO+O

(6)

0,+NO+N0,+02.

(7)

These reactions are usually fast enough that the assumption of steady state is valid and so (Leighton, 1961)

co31WI = 5 [NO,1 k, ’

&)QzIQI

(9)

where superscript 0 refers to ambient values of 4. Using the photostationary state relationship and Equation (9) it can be shown that [NO], PO,] and [O,] are functions of &I (Bilger, 1978). Once the p.d.f. of #l is known the mean and variances of each species can be found (Bilger, 1976) as follows

mO3’* = f ([NOlddl

(13) r&i)

1 -exp(-(4, =&2n&2)

--. - Cp1)2/2#$).

(14)

P(#~)d~~

- [=])2W,b-W,.

METHOD

OF SOLUTION

The balance Equations (2) and (4), can be solved using finite differences but a simplism approach using the Gaussian plume solution is applied here. The solution of Equation (2) is given by

(8)

The quantities [NO] + [NO,] (z (B1) and [O,] + L-NO,] + [O] are not affected by chemistry and are therefore conserved scalars. Since [O] has a very small concentration 4s = [0,] + [NO,] is also a conserved scalar. These Cpsare linearly related and have essentially the same balance equation and so the values of each throughout the plume only depend upon their boundary conditions. If the source strength of the plume, Q, is constant (a good approximation for a stack) then

[NO] = j [NO(&)]

where 6 and 4 are the Dirac delta and Heaviside functions respectively; the intermittency ;’ and the Gaussian function g being defined by

(5)

O+O,+M+O,+M

rPz - cb! = (& -

(12)

(10) 01)

P(b,) is the p.d.f. of 4r and its variation can be calculated, to second order, throughout the plume by

where Q is the source strength of (b at x = 0 when d, =&“ + (@ti)&( y)&z - H) and cri =25x/U. The values of c are determined by empirical data for Gaussian models (Pasquill, 1968; Turner, 1970). The solution of Equation (4) is more difficult because of the gradient source terms. If these terms were negligible then its solution shows that in 300 s the value of p will be reduced by molecular diffusion to 63% of its original value. Peters and Richards (1977) use this result, first given by Donaldson and Hilst (1972), to point out that the effects of initially nonuniformly mixed reactants would be dissipated within the first km or so from the source. However, Equation (4) shows that the initial variance cars be increased by the gradient terms which must therefore also be considered in the solution. Ifwe assume that the NO, variance from the stack is negligible compared to that produced in- the plume mixing processes then the magnitude of #” is calcu-

1897

Atmospheric dispersion of a chemically reacting plume

lated by equating the production and dissipation terms in Equation (4). This approach has been adopted by Bray and Moss (1974,1977) and Borghi and Moreau (1977) for turbulent flames and is analogous to the method leading to the classical eddy viscosity in turbulent flows (see Borghi and Moreau, 1977). Therefore

E 5 Q

16-

where the values of cy are determined by the Gaussian plume model used for (15). DISTANCE

RESULTS

=4E

or H2U

H2=2a~.

(17)

&

(18)

Y We also find that

Wxlmax=CNOxl” +

Pasquill (1968) gives cf = x2-” C$2 and so H xlnax

STACK

cc,

km

Fig. 2. Downwind variation of maximum variance and the corresponding mean.

Calculations are presented for a plume where [NOlO = 1.33pphm, [O,]” = 10 pphm, [NO,]” = 6.67 pphm and PO,] = 28 pphm. These conditions correspond to maximum ground level concentrations appropriate for a power plant in an urban photochemical smog episode. The stack emission is 227 ppm of NO,, 200ppm of NO and its height is 16.3m. The position of the maximum NO, on the ground is given by putting y=z =0 in (15), differentiating with respect to x and setting the result to zero. This gives, assuming uy = Q,, x,,,

FROM

21(2-n)

(19)

=

(-> CY where C, is the generalised eddy diffusion coefficient and n is the wind profile parameter. For this example n = 0.3, C, = 0.1 and U= 2 m s- l, corresponding to conditions typical for a smoggy day, and so x,, is 4OOm, or is 0.332m2s-’ and eY is 11.5m.

which for this example -is 344 m downwind with a value of 15.3pphm for ,/(#“). Therefore the maximum variance does not occur at the maximum mean position but it is close by. Figure 2 shows the downwind dependence of the maximum variance. The variance is predicted by Equation (20) to be zero at the plume centreline. It is clear that this value is not realistic because the diffusion and advection terms in (4) must be important here. Therefore the assumption is made that thecentreline variance is the same as the value at y= f c,,. This assumption is consistent with experimental profiles ofconserved scalar variance for a jet in a coflowing stream (Antonia and Bilger, 1976) which show a small dip in variance at the centreline. Figure 3 shows the profile of the ground level NO, mean and variance for this example. The integrals used to determine the other mean concentrations are found by using Simpson’s Rule once the p.d.f. is determined. The results are presented in Fig. 4. For comparison the concentrations predicted by ignoring the species fluctuations are also given. The

30,

I

I

I

I

I

I

I 2

I

n

I

At ground level, the variance of NO, can be found, using (15) and (16)

u,’ xa

(20)

Note that the z derivative is zero. The maximum value of J(p) occurs at y= + cry and is therefore 14.7 pphm. This very large value of NO, fluctuation at 0.4 km downwind- of the stack shows that the gradient source terms of q2 are certainly not negligible in (4). The x position of maximum variance is given by

.

2 += ISd Li

E

lo-

z ” 5

I 0

x”

=

(@;_-$yy-2

(21)

NO2 x h-7 h--,:

_

1 1 DISTANCE

FROM

PLUME

I 00

3 CENTRE

LINE

, y/my

Fig. 3. Profiles of NO, mean and r.m.s. at x,,,

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D.

J. KEWLEY

20-

I

I

I

0

DISTANCE

I

I

FROM

I

3

2

1

PLUME

CENTRE

LINE,

y/cry

Fig. 4. Profiles of mean concentrations and photostationary state parameter. ~ Nonturbulent chemistry method. non-turbulent chemistry method uses (8) and (9) with means replacing instantaneous concentrations. The results show that the values of [NO,]

and [o,]

are

reasonably close but the [NO] values are underestimated by up to 2Spphm. The photostationary state parameter, & (Bilger, 1978) is defined by -__ I= k,[NGl E0~lh~~CNOzl) (22) and is also plotted on Fig. 4. f is seen to vary between 1 and 1.8 across the plume. This result is similar to the values obtained by Bilger (1978) for a Gaussian p.d.f. for NO, fluctuations due to fresh emissions in photochemical smog. Experimental measurements in Sydney smog have shown (Kewley and Post. 1978) that values of I up to 2.0 can often occur. The p.d.f. at y = by is plotted in Fig. 5. The clipped Gaussian shape varies from y=O.9138 at the centreline to y=O.601 at y=3Sa,. The maximum value of NO, is 235ppm and so it is unnecessary for this upper bound to be included in the p.d.f. shape. 1.0

I

I

I

I

I

I I

zi z

[Gx]=

0.8 -

203pphm

_

If the ambient NO, value is decreased to 3 pphm, the calculations show that maximum difference between the predicted [NO] values is 2.6 pphm. The maximum A’becomes 2.14 showing that this parameter is much more sensitive to NO, fluctuations. For the above calculations it is assumed that k,/k, is constant throughout the plume. k, is a function of solar U.V. radiation which can be assumed to be uniform throughout the plume. However k, is temperature dependent and therefore the temperature of the stack emissions could affect the concentrations. Temperature is a conserved scalar and so it will be determined by Equation (15) if the Lewis number is assumed to be one. Thus the maximum temperature on the ground will be i&p

7-O+ -.

2QT

ne iiH2

(23)

For a stack temperature of 473 K Tm,, is only 0.4 K higher than ambient and so temperatureeffects are not important for this example. However to illustrate the effect of temperature fluctuations, calculations were made with T.,= 298 + 10 K. The rate coefficient k, is given by Schofield (1967) as k, = (7.5 x iO’/T)exp( - 1320/T)ppm-‘min-‘.

(24)

2 0.6-

2 g Oki

$

p.d.f. method. - - - - -

0.2-

L-

O NOx CONCENTRATION,

pphm

Fig. 5. Probability density function for NO, at y = uY.

For an ambient temperature of 298 K the ratio k5/k, is fixed at 2.0 pphm as before. The results show that, at y = uy, JT’” is 7.3 K and that while the NOz and NO speciesconcentrations are virtually unchwthe O3 concentration is reduced by 5%. n^(utilising .& in Equation (22)) is increaskd to a maximum of_1.85. Therefore any experimental measurements of ;t will only be affected by temperature at distances less than 1OOm downwind of the stack. These measurements would also have to be well above the ground.

Atmospheric dispersion of a chemically reacting plume DiSCUSSlON

The results show that inclusion of turbulence induced fluctuations of the plume pollutants can lead to diierent values for [NO] than would be predicted with nonturbulent chemistry. This effect is the result of the negative correlation of NO and 0s. Using (8) it can be shown that

[Ej]

= (/c&&-j - k,[NOl~CO,lW~~o,1~ (25)

Turbulent mixing of NO in the plune with ambient 0s will cause NO and 0, to be, on average,not together at the same point and time and so [NO]’ [Os]’ is negative. Neglect of turbulence effects gives a lower value of NO concentration. The effect of temperature fluctuations on 2 is due to other covariances. If k, is not constant then

(26) At y = uY the values of these normalised covariances are -0468,0.0597, -0.0374 and -0.0128. Therefore the main effect of temperature fluctuations is to increase the 0s and NO covariance which then leads to a higher 2. The temperature e&cts in this example are sufhciently small to be neglected in experimental measurements of X. These results for three species in chemical equilibrium show that turbulent chemistry in the atmosphere is important in assessing downwind con~ntrations. Generalising to a four species example A+BeC+D

where k

_ --

cc1 PI

EAIPI

not be significant. The method presented here can be extended to include many species which are in equilibrium instantaneously. This method has been used by Bilger (1976) for hydrogen-air diffusion flames. However, it is critical to the above models that the species must instantaneously be in equilibrium at that point in the plume. Donaldson and Hilst (1972) and Bilger (1978) have shown that the important parameters are the reaction time and the time for molecular (not turbulent) mixing. Both show that the time for reaction (7) can be of the same magnitude as molecular mixing time and therefore Bilger deduces that the instantaneous photostationary state relation (8) can be in error. Its validity awaits an experimental study of the type now being undertaken at Sydney University. For the example presented here the reaction and dilB.rsion times are (k&NO] + [OJ))- ’ and c1- r, respectively. The reaction time is about 15s and so Equation (8) should be valid for the plume. If the equilibrium relationships are no longer valid the p.d.f. method can still be used but the concentrations are no longer dependent solely upon &. Complex flows have been modelled using this technique but become more sensitive to the p.d.f. which is no longer one dimensional (Kewley, 1976; Varma et al., 1977). The model presented here is only applicable far downstream of the stack. A model for the chemistry and mixing during the plume rise will require the integration of equations like (2) and (4) along with the velocity and temperature equations. CONCLUSIONS The application of a turbulent reacting flow model to atmospheric dispersion of pollutants und~going fast chemical reactions has shown the effects of turbulence-chemistry interactions. Experimental determination of NO, p.d.f. shapes and the validity of the equilibrium assumption in plumes is needed before the theory can be q~ntitatively used.

(27)

then the error in using Equation (27) with no fluctuating species to calculate the mean concentration of C is Am

1899

Acknowfec!gements - This work and the author are supported

by a grant from the Australian Research Grants Committee to Professor R. W. Bilgcr and Dr. J. H. Kent. The assistance of Professor R. W. Bilgcr is appreciated.

= (k[Aj’ [B‘J’ - CC3 [D])/i%

(28) If k is constant and there are no ambient concentrations then the error is zero since the covariances will be positive and cancel. However, if there are ambient concentrations then clearly the covariances cannot cancel and the error can be important. These examples show that the turbulence effects upon the chemistry, neglected by Peters and Richards (1977), can be important in theoretical calculations of pollutant concentrations. When actual experimental comparisons are to be made then unless the plume dispersion can be measured accurately, for example with an inert tracer, the differences in the theory may

REFERENCES

Antouia R. A. and B&r R. W. (1976) The heated round jet in a coflowing stream. A.I.A.AJ. 14, 1541-1547. Bilgcr R. W. (1976) Turbulent jet diffusion flames. Frog. Energy Combustion Sci. 1,87-109. Biker R. W. (1978) l%e effect of admixing fresh emissions ou the phot~~tiona~ state ~lations~p in ~ot~h~i~ smog. ArmospkericEnotronment12,1109-l 118. Borghi R. ad MOIWU P. (1977) Turbulent combustion in a premixed flow. Actu astro. 4,321-341. Bray K. N. C. (1973) Equations of turbulent combustion - 11. Boundary layer approximation. A.A.S.U. Report NO. 331, University of Southampton, U.K.

1900

D. J.

Bray K. N. C. and Moss J. B. (1974) A unified statistical model of the premixed turbulent flame. A.A.S.U. Report No. 335, University of Southampton, U.K.; (1977) Acta asrro. 4, 291-319. Donaldson C. dup. and Hilst G. R. (1972) Effect of inhomogeneous mixing on atmospheric photochemical reactions. Enuir. Sci. Tech&. 6, 812-816. Eschenroeder A. Q. and Martinez J. R. (1972) Concepts and applications of photochemical smog models. In Photochemical Smog and Ozone Reacrions, pp. 101-168. Advances in Chemistry Series 113, American Chemical Society, Washington. Kewley D. J. (1976) A model of the supersonic HF chemical mixing laser including turbulence effects on the chemistry. Proc. Int. Symp. on Gas dynamic and Chemical Lasers, pp. 212-223. DFVLR-Press, Koln. Kewley D. J. and Post K. (1978) Photochemical ozone formation in the Sydney airshed. Atmospheric Environment (to be published).

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Leighton P. A. (1961) Photochemistry of‘ Air Pollution. Academic Press, New York. Pasquill F. (1968) Atmospheric D&ion. Van NostrandReinhold, Princeton, N.J. Peters L. K. and Richards L. W. (1977) Extension of atmospheric dispersion models to incorporate fast reversible reactions. Atmospheric Enoironmenr 11, 101-108. Reynolds S. D., Roth P. M. and Seinfeld J. H. (1973) Mathematical modelling of photochemical air pollution I. Atmospheric Environment 7, 1033-1061. Schofield K. (1967) An evaluation of kinetic rate data for reaction of neutrals of atmospheric interest. Planet. Space Sri. 15, 643-670. Turner D. B. (1970) Workbook of Atmospheric Dispersion Estimates. Environmental Protection Agency, Publication No. AP-26. Varma A., Fishboume S. and Donaldson C. dup. (1977) Aspects of turbulent combustion. A.I.A.A. Paper No. 77100.