EPA-R4-73-016c
March 1973
Environmental Monitoring Series
A Coupled Two-Dimensional Diffusion
and Chemistry Model for Turbulent
and Inhomogeneously Mixed
Reaction Systems
Office of Research and Monitoring
U.S. Environmental Protection Agency
Washington, D.C. 20460
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EPA-R4-73-016c
A Coupled Two-Dimensional
Diffusion and Chemistry Model
for Turbulent and Inhomogeneously
Mixed Reaction Systems
by
Glenn R. Hilst, Coleman duP. Donaldson,
Milton Teske, Ross Contiliano, and Johnny Freiberg
Aeronautical Research Associates of Princeton, Inc.
50 Washington Road
Princeton, New Jersey 08540
Contract No. 68-02-0014
Program Element No. A-11009
EPA Project Officer: Kenneth L. Calder
Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND MONITORING
U. S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, B.C. 20460
March 1973
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This report has been reviewed by the Environmental Protection Agency
and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
11
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TABLE OP CONTENTS
Nomenclature
1. Introduction
2. The Problem
3. An Evaluation of the Effects of
Inhomogeneous Mixing
4. Closure of the Chemical Sub-Model
5. The Construction of a Two-Dimensional Coupled
Diffusion/Chemistry Model for a Binary
Reaction System
6. Some Calculations of the Interactions of
Turbulent Diffusion and Chemistry
7. Conclusions and Recommendations
Appendices
A. Effect of Inhomogeneous Mixing on Atmospheric
Photochemical Reactions
B. Chemical Reactions in Inhomogeneous Mixtures: The Effect
of the Scale of Turbulent Mixing
iii
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NOMENCLATURE
A. moment term defined by Equation 4.5
B, ratio of mean concentrations
ratio of reaction rates
C, concentration of ith chemical species
f moment term defined by Equation 4.7
F horizontal flux defined by Equation 6.3
g acceleration due to gravity
k reaction rate constant
M value of -=- when C ' CA = 0
77 TT a B
n. frequency distribution of ith species
N ^n±
p pressure
q square root of twice the turbulent kinetic energy
r correlation coefficient defined by Equation 3.15
S wind shear
t time
t. intermittency factor
T absolute temperature
T adiabatic temperature
u velocity.
x axial coordinate
IV
-------
z vertical distance
X micro-scale length
A-, length scale
AO length scale
A,, length scale
v kinematic viscosity
a standard deviation
Z
Superscripts
— mean component
1 fluctuating component
Subscripts
chem reaction rate due to chemistry
I reaction rate neglecting third-order correlations
(Table 1) • . •
s steady state value (see e.g. Equation 3.12)
a, &,-y,f) chemical species
o initial value
1,2 species.designation
v
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1-1
1. INTRODUCTION
This section of the A.R.A.P. final report on Contract'
EPA 68-02-0014 covers the work on modeling of chemical
reactions in turbulent and inhomogeneously mixed binary
reaction systems performed during the period. September 1972
through January 1973. The primary intent of the EPA-supported
portion of this work has been the assessment of the combined
effects of turbulent diffusion and inhornogeneous chemistry on
the dispersion and chemical alteration of reactive pollutants
and natural constituents of the lower atmosphere. Parallel
programs aimed at similar assessments in the lower strato-
sphere,, particularly as they pertain to • the impact of proposed
SST exhaust emissions on the natural environment, have been
supported (under Contracts NAS1-11433 and NAS1-11873) by
NASA/Langley and, by transfer of funds, by the DOT/CIAP
program. Credit for the support of the basic technological .
developments common to these problems is therefore shared by
EPA, NASA, and DOT. The applications of this technology
reported here are restricted to the EPA orientation, however,
and have been supported solely "by that Agency.
At the time this work was undertaken in mid-1972, a
general assessment of the potential importance of inhomo-
geneous mixing in chemical reaction rates, and the basic
approach to modeling these effects via second-order closure
of the chemical kinetic equations had been developed by an
in-house program at A.R.A.P. These earlier developments
were reported in two papers by Donaldson and Hilst, one
entitled "Effect of Inhomogeneous Mixing on Atmospheric
Photochemical Reactions, " which was published in ENVIRONMENTAL
SCIENCE AND TECHNOLOGY, Volume 6, September 1972, and a second
entitled "Chemical Reactions in Inhornogeneous Mixtures: The '
Effect of the Scale of Turbulent Mixing, " which appeared in
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1-2
the Proceedings of the 1972 Heat Transfer and Fluid Mechanics
Institute. Stanford University Press, June 1972. Since the
work reported here stems directly from these earlier considera-
tions, reprints of these papers are appended to this report.
The reader is urged to read these appendices (A and B) first,
if he is not already familiar with their contents.
In addition to this preliminary work on chemical kinetics
modeling, several years of effort at A.R.A.P. have been devoted
to invariant modeling (second-order closure) of the structure
of turbulence and turbulent diffusion in boundary layer shear
flows under various conditions of hydrostatic stability and
surface roughness. This facet of model development has also
been sponsored by several agencies, including work under the
EPA Contract EPA 68-02-0014, and is reported extensively in
Volumes I and II of this final report. In the present report
on coupled diffusion and chemistry models, familiarity with
the material in Volumes I and II will be assumed.
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2-1
2. THE PROBLEM
The objective of this program, and therefore the primary
problem which we have addressed, has been to fabricate a use-
ful coupled model which, can simulate the combined effects of
turbulent diffusion and chemical depletion on the concentra-
tion patterns of reactive chemical species emanating from
common or separate sources. However., against the background
of assessment and modular model development described in the
previous section, it was evident at the initiation of the
present program that two major problems had to be solved first,
1. More rigorous analyses were required in order to
determine the magnitude of the effects of inhomogeneous mixing
on chemical reaction rates, the conditions under which these
effects could be.realized, arid an evaluation of the likelihood
that these conditions actually occur in atmospheric pollutant
situations. For example, if it could be shown that these
effects were either always insignificant, or constituted only
a transient perturbation of the chemical kinetic rates, a
coupled diffusion/chemistry model could be readily constructed
using the conventional mean-value chemical kinetic equations.
2. Given that the results of the above analyses were
not totally negative, i.e., negative in the sense that no
important real-world situations could be found in which
concentration fluctuations played a significant role, it
was recognized that the second immediate problem was the
development of a useful closure scheme for the third-order
correlations inherent in the complete chemical kinetic
equations. It was also recognized that under some circum-
stances the third-order correlations could be neglected.
However, this assumption restricts strongly the range of
joint frequency distributions of reacta.nt concentrations
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2-2
which can be considered, and denies any semblance of generality
in the chemical sub-model. A more appropriate, although
approximate, closure scheme was required.
With these considerations in mind, first efforts were
devoted to these two problems. By mid-November 1972 both, had
been resolved, and attention was focussed on the assembly of
the first coupled diffusion/chemistry model. The results of
the earlier work on the analyses of the magnitude and signifi-
cance of inhomogeneous mixing on chemical reactions and the
development of a closure scheme at the level of third-order
correlations of concentration fluctuations have been assembled
as a technical paper which was presented at the llth Aerospace
Sciences Conference of the AIAA in'Washington, D. C.,
January 10, 1973. A slightly modified version of this paper
is included as Section 3 and 4 of the present report. The
major results discussed there are:
1. There are indeed real-world atmospheric pollution
problems in which neglect of the fluctuations of concentrations
of reacting species introduces significant errors. These
effects associate primarily with multiple source situations,
many of which are very common.in the urban pollution arena.
2. An approximate closure scheme, for the chemical
sub-model which conforms to the principles of invariant
modeling and which accounts for the effects of inhomogeneous
mixing over a wide range of conditions (concentration
variance-to-mean-squared ratios up to 100) has been developed.
This sub-model predicts reaction rates to within a factor of
two of the exact chemical kinetic solutions for situations
where the mean-value chemical kinetic approximation Incurs
errors of a factor of 100. On the other hand, the chemical
kinetic sub-model recovers the mean-value approximation when
the concentration fluctuations are indeed insignificant, in
G. R. Hllst, "Solutions of the Chemical Kinetic Equations
for Initially Inhomogeneous Mixtures, " AIAA Paper No. 73-101,
January 1973, Washington, D. C.
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2-3
chemical reaction rates. This second-order closure model may
therefore be considered as a generalized (but still approxi-
mate) solution of the chemical kinetics equations.
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3-1
3. AN EVALUATION OF THE EFFECTS
OF INHOMOGENEOUS MIXING
The Basic Chemical Kinetic Equations for
Inhomogeneous Mixtures
Following Donaldson and Hilst (Appendix A) vie. assume an
isothermal, irreversible, two-body reaction between chemical
species a and p to form -y and 6 .
a + p -•• 7 + 6 (3-1)
Further, we assume that the reaction rate for any joint values
of the concentrations of the reacting chemical species are
correctly specified by
and
where C. denotes the concentration of the ith chemical
"species (expressed as a mass fraction), and k, and kp
are the reaction rate constants.
Equations (3.2) and (3.3) specify the local instantaneous
rate of change of the concentration of the reactants. In order
to determine the average rate of change, we assume the. local
history of the joint values of Ca and Cg at a fixed' location
comprises a stationary time series and each may be dissected into
.its mean and fluctuating components
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3-2
and
(3.6)
and, by definition, C ' = CA = 0 . Under these assumptions the
chemical kinetic equations for the average rates of change of
the concentrations of a and p at that location are readily
shown to be
and
(3.7)
(3.8)
In order to solve Equations (3. 7) and (3.8) we need a predic-
tion equation for C 'C£ . This is readily derived (Appendix A)
as
Q P
Ca
(3.9)
which introduces four new terms. C1 . CA . C'CA , and C' CA
a p a p ap
The prediction equations for C' and CA are
TE—
a = - 2k, (CAC|
Ivf3a
and
C|2C')
x
(3.10)
v -^ • '
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3-3
and they do not introduce any more new terms. In order to
close Equations (3.7) through (3.11), and thereby achieve a
chemical sub-model for reactions in inhomogeneous mixtures,
we require prediction equations for the third-order correla-
tions C^2CU and C^CA2 .
Before proceeding to the closure problem, however, it
is instructive to examine more closely the limits of the
effects of concentration fluctuations on chemical reaction
rates and the conditions under which these effects become
significant. This examination may be made in two steps;
1. when are the fluctuations of concentration negligible
(i.e., when are the reaction rates predicted satisfactorily
by the mean values of concentration alone?) and 2. when may
the third-order correlations be neglected? For the latter
cases, Equations (3.7) through (3.11) comprise the closed
set discussed by Donaldson and Hilst (Appendix A).
The Limits of Errors in Reaction Rate Predictions
if Concentration Fluctuations are Neglected.
Since the neglect of concentration fluctuations in
determining reaction rates is equivalent to the assumption
that the local values of C and CQ are constant in time,
a p
C'1 = CA = 0 and the reaction rates predicted under this
assumption are simply
and
/ V7T \
_ ^ 7T 7T (3.13)
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3-4
where the subscript s denotes the steady state assumption.
Then the ratio of reaction rates predicted from the inclus-
ion of concentration fluctuations to those predicted neglect
ing these terms are
(3.14)
and an identical equation for the relative rates of. depletion
of the (3 species.
The limits on Equation (3.14) are readily determined
from Equations (3.7) or (3.8) and elementary statistics.
First, we note from Equation (3.7) that for irreversible
reactions, ScT/dt < 0 and therefore
-°L > -i (ci)
Ca°P
Further, from elementary statistics we note that
^Cl
-1 < a P .- .. < + i (C2)
fC'2C'2)^
• a p '
and therefore
(C3)
Substituting conditions (Cl) and (C3) into Equation (3.14),
we establish the limits
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3-5
(04)
Conditions (C4) set the maximum errors in the prediction of
reaction rates which the neglect of concentration fluctuations
can produce. These limits are set by the individual variance-
to-mean-squared ratios of C and Cg and are therefore
f\
C,
functions of the marginal frequency distributions of C and
The limits established by conditions (C4) are shown
graphically in Figure 1 and we note immediately that the
limiting errors in reaction predictions occasioned by neglect
p |2 p ,2
A
— § -- 73— I < 0.5
~ 7
but increase to highly significant values as this ratio exceeds
1.0. The potential for order-of -magnitude errors in the
prediction of the reaction rate exists whenever the product
of the variance-to-mean-squared ratios greatly exceeds 1.0.
The actual error depends, of course , on C 'CA/C CA .
Co p ~ Ct p
This actual error may be examined by forming the ratio
,.
(3
where r is the ordinary correlation coefficient and in this
usage expresses the ratio of the actual error in reaction rate
predictions to the maximum possible error for any given Joint
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3-6
r-1.0, /
Figure 1. The limits of errors of prediction of chemical
reaction rates incurred by the neglect of
concentration fluctuations. (See text for
explanation of terms.)
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3-7
distribution of C and CM . Selected values of r are
a p
also graphed in Figure 1. In the limit of r = 0 no error
in reaction rate predictions is occasioned by the neglect of .
concentration fluctuations. This is, of course, the situa-
tion when C and CB are randomly distributed and c'c« = 0
However, it is clear from Figure 1 that even modest values of
r produce significant errors.in the reaction rate prediction
In 8 \
when -~ —|— \ > 1 , particularly when r < 0 .
V r r I
\Ca CP /
We shall return to this analysis, and identify joint
distributions of C and CM for which the fluctuations of
a p
concentration must be included later. For the purpose of
model development, we now turn attention to the importance of
the third-order correlations in the chemical kinetics equations
The Role of the Third-Order Correlation Terms
Returning1 to Equations (3.9) to (3.11), it is evident
that the primary role of the third-order correlations is to
be found in their control of the rate of change of CT'C^ ,
both directly and through the rates of change of the variances,
C ' and CM . The effects of the third-order correlations
a p
on the reaction rates will therefore appear primarily as a
time-integrated effect on Cf^CM and any cumulative error in
2 2
the estimates of C ' CM and C'CM will produce a cumulative
a p a p
error in C 'CM .
a P
We may deduce immediately from Equations (3.9) through
(3.11) that if
C '2CM « CRC |2 + C C 'CM
a p p a a a p
and
(C5)
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3-6
their effect on C 'C£ is negligible and vie may close the
model. equations by setting C^CA = C&CQ2 = ° • To illustrate
that conditions (C5) are met under any given circumstance, we
must evaluate the joint distributions of Ca and Cg from
which these moments are derived since there are now no limit-
ing conditions • on their marginal distributions. In other
words, we must turn attention to the distribution functions
from which the means and moments have been derived if we are
to determine the importance of third- or higher-order correla-
tions in chemical reaction rates. Ideally, we would examine
simultaneous experimental measurements of C . and Cgi to
make this assessment; unfortunately, very few such data exist.
However, so long as we assume that the basic chemical kinetic
equations are correct (and our whole theory is based on this
assumption) we can proceed by solving these basic equations
for various initial distributions of C . and Cg. , determin-
ing in the process the time histories of all of the relevant
moments of these distributions.
The Moment Generating Model , ...
Under the assumption that only chemical reactions are
operative in changing the concentrations of a and £ , the
chemical kinetic equations can be written as total derivatives
and integrated directly as a function of reaction time.
dC ,
dC
- -k2caicf3i
and
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3-9
K2
pi kn di
i
k2 Ca.l(Q)
ir f f rn
K! cB1ioj
expf- (CB,(0) - T-£ C ,(0))k,tl
^ pi K^ ai i j
while
r d- ^ -
'"fti \ ^ ) ~
(3.19)
Equations (3.18) and (3.19) specify the j.oint values of C .
and Cg. at -time t , given their initial values and the
reaction rate constants. They may be used to specify the
frequency distribution of (C ., Cg. ) at any time t ., given
their initial frequency distribution, n.(0) since, in the
absence of mixing, n. is conserved as C , and Cg. change
value due to chemical reaction. Equations (3.18) and (3.19)
provide the information necessary to calculate all of the
relevant moments of n.(C . , Cg.) and their rates of change.
We may introduce any arbitrary initial distribution ^(C^, Cgj
subject only to the constraints
0 < C . <
— ai —
(06)
and
0 < C .
— ai
In order to illustrate the general behavior of the
distribution of C and Cg and the associated first-, second-,
and third-order moments, we have chosen a simple distribution
of points along the line C = 1 - Cg and weighted each point
equally (HI = 1/N) . The time history of n1(C ±, Cg.) is
shown in Figure 2 and the moments of these distributions are
plotted in Figure 3. Note particularly the distortion of the
originally linear distribution of (C , Cg) and the associated
decrease (from zero) of the third-order moments. The relative
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3-10
RUN NO. 1
A<
.2<
0<
h N
> \
x
<<
\ •
N
^
V
\
> W2
^
tco
J 1^
) .2
v^^
yv
.4
x°
\
XX
^^>
^
"*"*""'-C
s±
.6
>>
>>v
^
_^\_
.8
1.0
Figure 2. Example of the time history of the joint frequency
distribution of (cm> CPi^ given the initial
distribution shown for t0 and k, = kp = 1.
Each point was weighted equally (n^ = i) for
calculation of the moments of these distribu-
tions (shown in Figure 3).
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10'
10°
10
-2
IO'3
3-11
RUN NO. 1
— &/-\ == -L
\
\
V
— o
/2 -p/2
C/2 -
~
s
I
8
10
Kt
Figure 3. .Time history of the first-, second-, and third-
order moments of ^(CtfL, C8i) for t'ne distribu
tions shown in Figure 2.
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3-12
Q p __ __
magnitudes of C ' CJ, and CRC ' + C C 'CU are also plotted in
a p pa a a p
2 2
Figure 3 and, as can be seen there, C1 CA and C 'CA
completely dominate the initial change of C ' C A . However,
this latter effect is too short-lived to be significant in
the prediction of the time history of mean concentrations.
This fact is shown in Figure 4, where the predictions of C"
OX
and C"6 as a function of time, first, neglecting the fluctua-
tions completely, and then neglecting only the third-order
moments, are compared with the exact solution. The latter
assumption produces an error of approximately 10 per cent at
kt = 10 while the total neglect of the fluctuations produces
an error of 300 per cent at that time.
As a further example, and one which illustrates the
importance of the.third-order correlations, we have constructed
the distribution functions which simulate the case of intermit-
tent sources. For physical perspective, imagine a free-way,
oriented across the wind and on which the automobile traffic
ranges from a steady, bumper-to-bumper stream to only an
occasional vehicle. We assume that each vehicle emits approx-
imately the same amount of pollutants per unit time, but.that
the ratio of the a and p1 species emitted is slightly
variable from one vehicle to another. Now we ask, "What is
the average reaction rate for these exhaust materials
immediately downwind from the roadway as a function of the
intermittency of the traffic?"
We simulate this situation by the frequency distribution
for (C , Cft) shown in Figure 5. The variability of C and
ct p - ct
CQ due to variable exhaust -emissions is portrayed as a
circularly symmetric distribution and we take C = CR = 0
when there is no traffic upwind of our observation line.
(Small background concentrations have been assumed in another
calculation but produce no significant effect.) We assume
further that the pairs of nonzero values of (C , .Cft) occur
ct p
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3-13
IOV
QQ.
10
•»
e
10
10"
10"
O = Exact Solution
Q = Solution neglecting 3rd-order moments
Solution neglecting concentration
fluctuations
8
Kt
10
Figure 4. Comparison of the predictions of C and UQ
under assumptions listed, with exact values
from the moment generating model using the
.distributions shown in Figure 3.
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3-14
•7r
.6
.5
t
03. ,
O -3
.2
.1
OO
7O
03
.1
.5
.6 .7
Figure 5. Joint distributions of (Ca, Co) chosen to
simulate chemical reaction rates immediately
downwind of a roadway on which traffic is
variably intermittent.
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3-15
with equal frequency and we measure the intermittency of the
traffic as the fraction of time there is a vehicle upwind of
the observation line, t. . The moment-generating model has
been used to determine the chemical reaction rates as a
function of kt for t± = 1.0, 0.5, 0.33, 0.2, 0.1 (t± = 1.0
corresponds to a steady, bumper-to-bumper stream of traffic).
The values of An =[-2 i- \ r = £_E and
C ' C '
a P - at kt = I
C~eC ' 2 + C" C 'C '
pa a a p
and for .each value of t.. are shown along with the observed
ratios of cKT /dt to the reaction rates predicted assuming
steady values of C and Cg , (BcT/dt) , and to the rates
predicted neglecting the third-order correlations, (d(T /dt)..
in Table 1. These results, which are now firmly grounded in
reality, show clearly that the potential errors in the predic
tion of reaction rates neglecting the concentration fluctua-
tions can be realized. Although the effects of concentration
fluctuations are negligible for a steady stream of traffic,
t. = 1 , chemical kinetics based on this assumption under-
estimate the initial depletion rate of C~ and (To by a
factor of 9 when t. = 0.1.'
The effect of neglecting the third-order moments is not
evident at kt = 1 , however, since we started with the
correct value of (T'C^ and the integral effect of neglecting
ip Ct P
C ' CA and C 'CA is not yet large at kt = 1 . Their
integral effect on the predictions of C" and d(T/dt can
be seen, however, in the time histories of these quantities
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3-16
when the third-order correlations are neglected. Values of
B! = C'Q/C^J and B2 = (dU^/dt )/(dC"a/^t )I for each value of
t. and kt = 1, 5, 10, 16, and 20 are listed in Table 2.
[( ) indicates "C ,- or (dcT/dt)... has wronS sign.]
An inspection of this table shows that the first effect
of neglecting the third-order correlations, while retaining
the first- and second-order moments, is to produce significant
errors in the reaction rate. This is, of course, the first
integral effect on C ' C ' .. The second effect is on the mean
a p
values predicted for the concentrations, an error which
depends upon the time integral of C'CA . We also note that
not only can the.errors be large (for example, the reaction
rate is over-predicted by a factor of 100 when t. = 0.1
and kt = 20) but we get the totally erroneous results of
positive values of <3C" /dt and negative values of C" .'
Prom these results we conclude that a generally useful
chemical kinetic model must include the representation of
the moments of .the concentration fluctuations through the
third-order.
Summary
The various considerations and examples of the effects
of concentration fluctuations on chemical reaction rates
discussed above may be summarized as follows:
1. The effects of concentration fluctuations can be
significant, to the point of dominating chemical reaction
rates. The situations under which they are significant are
characterized by joint distributions of the reactant concentra-
tions which are skewed toward large values of these concentra-
tions, since this is the condition under which the variance-to-
mean-squared ratios can be large with respect to one.
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3-17
2. The chemical reaction rate is significantly
accelerated when the concentration fluctuations are
positively correlated and is depressed when these fluctua-
tions are negatively correlated. Combining .these two
requirements for significant.effects of concentration
fluctuations, we may sketch the general character of
the joint distributions of C and Cg which require
the inclusion of concentration fluctuations in chemical
kinetic models. These are shown in Figure 6. The first,
when C 'CA > 0 is essentially the distribution of concen-
trations used for the intermittent traffic situation in
the last section. The second, when C "C'j < 0 , may be
CM P
quickly identified with the situation when the chemical
reaction is retarded or stopped by local depletion of one
of the reactants. This latter case depends strongly upon
the relative rates of chemical reaction and of local
diffusive mixing. Further study of this situation depends
upon an appropriate coupling of the chemical and diffusion
equations.
3. For strongly skewed distribution of C and CA
• ct p
the case when concentration fluctuations become dominant in
the determination of the chemical reaction rate, the third-
order correlations of these distributions mus_t be included
in the generalized chemical kinetic model. The neglect of
these terms leads to the highly erroneous result that
either the reactants are produced rather than depleted, or
the mean concentrations go to negative values.
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TABLE 1
•f-
ri
1.0
0.50
0.33
0.20
O.10
A
Al
0.19
1.40
2.7
5.0
11.0
-0.41
0.6l
0.67
0.74
0.75
A
Ag.
-0.04
0.02
0.50
1.48
4.00
(d^/dt)'
(^cc/at)s
0.92
1.86
2.80 .
4.70
9.25
( So ydc )
(5c(/at)I
1.0
1.0
1.0
1.0
1.0
TABLE 2
•1-
ti
1.0
0.50
0.33
0.20
0.10
•kt = l kt = 5 kt = 10 kt = 16 kt = 20
Bl
1.0
1.0
1.0
1.0.
1.0
B2
1.01
1.00
1.00
1.00
1.00
Bl
1.0
1.05
1.21
1.67
2.31
B2
1.33
l.6o
1.17
0.54
0.40
Bl
1.0
0.92
1.24
7.20
(-1.42)
B2
1.25
2.60
1.50
0.30
0.41
Bl
1.0
0.83
1.08
(-1.33)
(-0.30)
B2
1.40
(-6.80)
(-1.50)
0.09
0.03
Bi
1.0
0.77
0.96
( -o . 50 )
(-0.14)
B2
1.60
(-2.35)
(-0.50)
0.02
0.01
(JO
M
OO
-------
I
QQ.
O
3-19
t
QQ.
O
m
Figure 6. General types of joint frequency distributions of
Ca and Co for which concentration fluctuations
are significant in determining chemical reaction
rates.
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4-1
4. CLOSURE OF THE CHEMICAL SUB-MODEL
Having demonstrated the need for a more general
chemical kinetic model for reactions in inhomogeneous
mixtures, we may proceed along either of two fronts.
The more general of these is to write down prediction
equations for the thirdTorder correlations and seek
closure by suitable assumptions regarding the fourth-
order moments. If no such assumptions exist at the
fourth order, we may proceed to higher-order moments
until they are found. This line of inquiry has been
pursued to the level of the fifth-order moments in the
present study but has been set aside, primarily because
the proliferation of simultaneous partial differential
equations for the higher-order moments poses staggering
computer requirements. This latter feature of higher-
order closure becomes prohibitive when one considers
the requirement for an equivalent level of closure of
the diffusion equations with which this chemical sub-
model is to be coupled.
With these facts in mind, we have turned attention
to the development of approximate closure schemes in
which maximum possible information regarding the third-
order correlations is sought from the first- and second-
order moments. As is noted later, such a closure scheme
cannot be exact^. We begin by reviewing the definitions
Of *H ' CcfCp and CaCe2'
First, by definition,
:i°£ = If ni
-------
.4-2
where n. is the frequency of occurrence of the joint values
of (C ,, Cft, ) and N = Z n. . Expanding Equation (4.1). .and
cjt j- p -i_ T JL
making use of the definitions of C"^ and CL , i.e.,
C . = Z n C,( j = a,p) , one. obtains
.
1 ,±
a P
n. C .Cft.
i ai pi
(4.2)
We note in particular from Equation (4.2) that, since
n
i
(C7)
C'CA/C Cg = -1 only when the lower limit of conditions (C?)'
is satisfied. This can occur only when any nonzero values of
g^ , and vice versa.
C 'C/c c = -1 , which is the condition for the
C . are coupled with zero values of
"
Then, when
c =
termination of the chemical reaction, all joint moments of
nt(C
., Cg.) about the origin must be zero.
Now consider the definition of C' CA
~ Ca)2(°pi - Cp)
n. C2.C,
i ai
C '2
-i a
+ ^a + '
*""
C 'C '
-i Ct P
C" € R
a P
(4.4)
-------
4-3
using Equation (4.1) and the definitions of C^ and Cg .
It is immediately evident from Equation (4.4) that our
2
problem of approximating C ' C^ by first- and second-order
moments reduces to finding a suitable approximation for the
first term on the right-hand side of Equation (4.4) in terms
of these moments. For convenience we denote this term by
A. . i.e . .
ior '
A, = — — Z n. C.CR. (4.5)
ia X ai Pl
In seeking this approximation we may. note the following
conditions which A. must satisfy
1. A= 0 when C/C( -1
2. A1Q= 1 when C/C = 0
C' C'C,
3. A = 1 + -- + 2-- when C'2CA = 0
ia cd c ^ a p
a a P
4. Ala 2. 0 at all times. (C8)
Condition 4 in (C8) operates primarily to constrain the joint
values which C ' /C and C 'CA/C Cft may assume. For example,
ct ct ct p ct p
when conditions 3 and 4 in (C8) are applied jointly, we derive
directly the further condition that
-------
4-4
c •
a
C2
a
C 'C '
2 -SL&
C
> - 1
or
a P
> - i
" 2
a
1 x
(C9)
2 2
Finally, we must note that C' C' and C'C ' cannot be
a p a p
specified exactly from first- and second-order moments alone.
Consider, for example, the two distributions of C . and Cg.
shown in Figure 7. All of the first- and second-order moments
are identical for these two distributions, but the third-order
correlations are of opposite sign. Clearly, we are looking for
useful approximations, not exact relationships, and any such
approximations must be delimited as to their regions of applica-
bility. The criteria for usefulness of these approximations
must be based on the accuracy with which the chemical model
within which they are incorporated predicts the chemical
reaction rates. However, in this development stage we shall
be primarily interested in the accuracy of specification of
the third-order correlations.
Approximations of A. From First-
and Second-Order Moments"
Utilizing conditions (C8) and (C9), it is not difficult
to formulate approximate expressions for A. and A.ft from
a
CV
and C '2 and C'2
a p
la
For example, an early
approximation which has been tested is
-------
4-5
1.0
.8
.4
.6
.8
1.0
Figure 7. An example of two joint distributions of Ca and
CR for which all first- and second-order moments
are equal but the third-order moments are not
equal.
-------
4-6
A
ia ~
CaCp
a
(4.6)
Since C 'C' = 0 when C ' = 0 and C 'C '/C CA > - 1 ,
cxp ct a. p a p —
it is evident from Inspection of Equation (4.6) that this
approximation satisfies conditions 1, 2, and 4 of (C8), but
satisfies condition 3 only if C^CA - 0 when C^2CA = 0 .
According to condition (C9), the latter result is admitted
but is not required, i.e., C 'CA may be nonzero when
p
ex p
The development of an improved approximation of A. ,
improved in the sense of more appropriate realization of
condition 3 in (C8), proceeds as follows: Let
A
ia
2
1 +
a
+ 2
f (C ,C0;C 'C ',C |C') (4.7
ax a' P' a $' a P ; v
where the function f satisfies the conditions
a
- 0 when
f = 1 when C' CA = 0
a a p
'.CIO)
A simple function which satisfies conditions (CIO) is
+ 1
f =
a
M + 1
(4.8)
where M is defined as the value of C 'CA/C; CQ when C^ CA
We may note immediately that M as defined must satisfy
condition (C9), i.e., •
0
-------
M > -
1 +
a
As a further measure of M , however, we may also
note that jsince the ordinary correlation coefficient,
must lie between i 1 (condition (C3))
72^
(Oil)
C ' C'
_o P_
o o
7T^ 7T^
Ca °P
a
C
oTP
p i PI
_«_ _JL
772 ^2
Ca Cp
and we expect M to be related to both
p~ —p
C' /C and
a ' a
(C12)
This expectation is reinforced by the fact that, as defined,
.2
M must have the same value for both
C'2CA and C'C •'
a P a P
p —p p —p
although C ' /C is not necessarily equal to CA /Co
Some values of M versus
—o-
p i pi'
a °P
Ca CP
as determined
from the moment generating model, are shown in Figure 8.
(Each of these solutions was derived from a log-normal or
Cpi
CR .) As
"^0 P
can be seen from Figure 8, within the range tested, M is
a composite of log-normal distributions of C . and
which were skewed toward large values of C and
a relatively well-behaved function of
r' PI
Gq ,°p
—2 —2
a P
For our
present stage of approximation, however, we have chosen a
dichotomous relationship for M , namely,
M = 0 when
C ' CA
a P
P
< 1
(C13)
M = 1 when
>
-------
M
2.5
2.0
1.5
1.0
.5
10
-i
Figure 8. Values of M vs
C'2 C'2
a P
C"2
-a
C/
10'
as determined by solution of the basic
•*=•
CD
chemical kinetics equations for various initial log-normal frequency
distributions of C and C
a
-------
4-9
Using Equations (4.4), (4.7), and (4.8), our approximate
predictor equations for
and C 'C/4 are
a p
T^ v ft
Ca CP = M + 1
and
+ 2
o
a P
2 o B
°icfT - M¥T
1 +
and'-we specify M by conditions (C13) .
(4.9)
(4.10)
As a first, but severe test of this approximation,
2
comparison of the predicted and observed values of CM CM for
the most extreme value of
CA
a
shown in Figure 8 (Run
L24) is shown in Figure 9, and the predictions of
from this model are compared with those predicted by 1) neglect-
ing the fluctuations completely and 2) neglecting only the third-
order correlations, in Figure 10. .
The Approximate Chemical"Sub-Model
For Inhomogeneous Mixtures,
Employing .-Equations (4.9) and (4.10) and conditions (CIS),
the chemical .sub-model for two-body reactions in inhomogeneous
mixtures is
(4.12)
- It,
> c«c '2
-Pa
Ca
(4.13)
-------
4-10
RUN L24
^^ ^^ ^^
(
700
600
QQ. 500
10
M O >
'^ 4°°
b
csj H
o 30°
200
100
0
-100
C|2C'
_o^ a p
" r} — Exact Solution
v "r ~r
J W O r\
a p-
t r — P
\ C C ' G •'
\i Q y i > B
jk T O ^
c"2 ~ ~,
I U a fi
\ C C,1
\ U I' ,
— - - - - -L
\ r r
\ °a°P
- \ L J
\
\ ^
\\
\V
- \v
\ ^^^
°^^_^ ^""'^--^^ ^
"°--— o 0
1 1 1 1 1 1 1
0 10 20 30 40 50 60 7(
Kt
Figure 9. Approximation of the third -order correlation for an
<-> pi
C
-------
16
00
O
X
a
10
I
Figure 10.
14
12
10
8
4
2
i
\ \
P \ RUN L24
\ ^
]\ \ ,r-Mark I
\\ \ /
v ^<
\\ \
V /-Mark'M. \
\\ / \
y x
• \
\ v
V
\X ^-Mom. Gen. Model
Uniformly \ ^-o^ \
Mixed — v ^^ ^V
\ \ ^^^
l""°""a °^ ^^~ ~":=s^-
Q! i i i i i i
0 20 40 60 80 100 120
Kt
Comparison of predicted vs actual chemical reaction
for Run L24 under the following assumptions:
1. Neglect of concentration fluctuations.
(uniformly mixed)
2. Neglect of third-order correlation (Mark I)
3. Inclusion of approximate estimates of third-
order correlations (Mark VII)
»
^^
•••=
140
rates
-------
4-12
= - 2k,
C^C'2 +
P a
(4.14)
C ''
a
a P
M =
- M
(4.15)
(4.16)
(4.17)
M = 1 when
and the sub-model is closed at the level of the first- and
second order moments.
The only rigorous test of this approximate model
available to us now is a comparison with the exact solu-
tions of the chemical kinetic equations, as a function
of reaction time kt , and for various initial distribu-
tions of (C ., CQ.) . The much more realistic case of
v ai pi
coupled chemical depletion and dilution by turbulent
mixing must await the coupling of the chemical and
diffusion sub-models. However, if the local diffusive
mixing is very, very slow compared with chemical deple-
tion, this sub-model must "track" the. chemical depletion
correctly. The following tests of the chemical sub-model
(C14)
-------
4-13
are therefore restricted to this circumstance so far as any
degree of reality.is concerned.
Since our primary concern in the coupled chemistry-
diffusion models will be accuracy in the prediction of the
local mean reaction rates, we are particularly concerned
with this facet of the chemistry sub-model. The comparison
of interest is between the local depletion rates of the
reacting chemical species as measured by dC~ /dt . Prom
a variety of initial distributions for n.(C ., Cg.) , we
have chosen four which exhibit varying degrees of the effects
of inhomogeneous mixing. Their initial distributions are
shown in Figure 11 and the comparisons of reaction rates as
a function of reaction time are shown in Figures 12 to 15>
(including the initial reaction rates predicted .when the
concentration fluctuations are neglected).
These comparisons, although by no means exhaustive,
show that the approximate chemical sub-model developed here
captures a very large fraction of the effects of inhomogeneous
mixing on chemical reaction rates. Over a very wide range of
reaction rates, this model predicts the exact rate to within
a factor of two, while the neglect of the fluctuation terms
in the chemical kinetic equations incurs errors of up to a
factor of 100. On this basis it seems safe to proceed to
the coupling of this chemical sub-model and the invariant
(second-order closure) diffusion sub-model.
-------
4-14
O
Run No. 2
n. =. 1
01
oa.
o
Run No. 308
Oi
CO.
o
Run No. 380
1
X> Oi
o
O
Run "A "
O
O i
0 10
v 1000
O 100
Figure 11. Schematic of four distributions used to test
the approximate chemical model.
-------
io-
4-15
IO-2
<_
'O
a
10
io-3
in-4
1 RUN NO. 2
?
• O Mom. Gen. Model
\j A Mark VII
\ (dCg/dt) at kt = 1 is 2.0 x
\
1 V
IN
- \\
V\
"A
v^
\
\
V
\
\
\
\
\
\
\
\
\
\
\
1 1 1 1 I X 1 1
10
-1
10
20
30
40
50
60 70
Kt
Figure 12.
Comparison of reaction rates predicted by the
chemical kinetic model (Mark VII) with exact
solution for initial distribution of (Ca,
shown in Figure 11.
-------
4-16
iu •
10-2
I0~3
Ti
a
10
t>
io-4
I0'5
(
& RUN NO. 308
6
\\ O Mom<) Gen. Model
\Q
- JA A Mark VII
\
^j (^c/^t)B at kt . 1 is 6.2 x io"^
\
- X
\X
X
^^
\^xx^
Vv
A
\
\
1 1 1 1 1 1
D 10 20 30 40 50 60
Figure 13,
Kt
Comparison of reaction rates Predicted by the
chemical kinetic model (Mark VII) ^thr®xacj
solution for initial distribution of ^Ca, op
shown in Figure 11.
-------
10
r2
RUN NO. 380
4-17
10
I0~3
io-4
1C
' 5
O Mom. Gen. Model
A Mark VII
\\
\
\
\
\
\
10
6
10
20
I
30
Kt
40
50
•^-1
60
Figure 14.
Comparison of reaction rates predicted by the
chemical kinetic model (Mark. VII) with exact
solution for initial distribution shown in
Figure 11.
-------
io-9
4-18
RUN NO. A
O Mom. Gen. Model
A Mark VII.
at kt =. 1 is 1.0 x 10
I0~6-
a
10
-7
100
200 300
400
500 600
700
Kt
Figure 15.
Comparison of reaction rates predicted by the
chemical kinetic model (Mark VII) with exact
solution for initial distribution shown in
Figure 11.
-------
5-1
5. THE CONSTRUCTION OF A TWO-DIMENSIONAL
COUPLED DIFFUSION/CHEMISTRY MODEL
FOR A BINARY REACTION SYSTEM
The development of the closure scheme for the chemical
sub-model described in the previous section, coupled with
the models for prediction of turbulence structure, fluxes,
and turbulent diffusion of matter described in Volume I of
this report, provides the necessary modules for a coupled
diffusion/chemistry model, the objective of this program..
In particular, these developments make possible coupled
models which permit examination in detail of the processes
of generation and decay of the fluctuating components, as
well as the mean values, of turbulent diffusion, and inhomo-
geneous reactions of a binary or two-body reactive system.
As a starting point for these coupled models we have
chosen the relatively simple, but realistic, situation of
two reactive but otherwise passive pollutants emanating
from either common or separate cross-wind line sources.
This choice reduces the diffusion calculations to two
dimensions and permits the decoupling of the diffusion/
chemistry model from the turbulence model, since there is
no feedback into the turbulence field due to either pollutant
density or exo- or endothermic reactions. The decoupled
turbulence model is used to generate the field of turbulent
motions and fluxes, which are then used, along with source
specifications and reaction rate constants, as input to the
coupled diffusion/chemistry model. As can be seen from
the full derivation of the two-dimensional model presented
in the next section, this system, involving as it does
nine simultaneous partial differential equations, is already
.rather complicated. However, one of the primary reasons for
starting at this basic level of complexity is to permit close
-------
5-2
examination of the Interactions between turbulent diffusion and
chemical reactions in the simplest realistic mode in which they
could occur. When these interactions are understood,.extension
of the models to three-dimensional configurations, nonpassive
pollutants, and three-body reactions can be undertaken with a
much better appreciation of the complex nonlinear system within
which they will operate.
Due to the limited time and funds for this project, only
a few test calculations of the combined effects of turbulent
diffusion and chemistry have been possible. Some basic calcula-
tions, which begin to define the effects of turbulence vis a vis
chemical reaction rates, and two sets of calculations for the
NO - 0- patterns to be expected in a multiple freeway situa-
^ ->
tion are presented and discussed in Section 6. Despite their
limited number, these examples already point up sharply the
effects of inhomogeneous mixing (produced by the turbulence field)
and the effects of diffusion-limited conditions on fast reactions
characteristic of photochemical chains.
Derivation of the Modeled Equations for the Mixing^of
Two Chemically^ Reacting Materials Emanating
from Cross-Wind Line Sources
For an atmospheric shear layer in which the Schmidt number
is equal to one and the adiabatic density is constant, we may
follow Donaldson (Vol. I) and write the equation governing the
diffusion and chemistry of a reacting species, a, , as
o
Af ^f r\ C*
'"' = - u_. -^ + v. —T& - k,C Co (5-1)
'dF" ' ~ j dx . o^2 a a
J OXj
where k is the reaction rate of the a species with a
second species p . A similar equation may also be written
for CB , but this will not be done until the modeling is
completed.
-------
5-3
We may express our variables in terms of their mean
and fluctuating parts as
C = C" + C'
a a a
u. = u. + u! . (5.2)
Substituting these expressions into Equation (5.1) and averag
ing, we obtain the mean local rate of change of concentration
in terms of convection, molecular and turbulent diffusion and
chemical reaction
= _.u. _ (IC~0 + v — -;
dt j d5cj °^j J a ° ^x2
J
(5'3)
In deriving Equation (5.3) we have used the continuity equation
chl
- 0
x
J
du !
Prom Equation (5-3) we see that we now require expressions
for u'C ' and C 'C ' . We first subtract Equation (5.3)
J Ot CX p
from Equation (5.1) to obtain an equation for the fluctua-
tion C ' ,
-------
5-4
dC ' dC ' dC dC ' x d2C '
* = - u. ° - u. :r-2 - u! :r-2 + !— (u^T) + v 2
dt j dxT j d3cT j dxT dx7 v j a'
Following Donaldson (Vol. I), we may write the expression for
u.1 as
du.1 [ du.1 du. du^ du^
, uJ^ruJ^J
d2u,'
ST- +v/,l^-|£i (5.6)
° dxj
Multiplying Equation (5.6) by C^ and (5-5) by u^ , adding,
and averaging, we obtain an expression for u ')
' wu. v-/ ______ uu rv
_a = -1J. ^-i-2 - 'uTTTT . -r^ - ujc.V .
J dx^. i j
, , v32^7c7 du ' dCT
S / ... ,n . \ K TTTsrr ,. J i a 0 i __a
BT:
j
.
Equation (5.7) introduces various third -order correlations,
pressure correlations and dissipation terms that have been
modeled previously. We must, however, obtain an expression
for C "T ' . Returning to Volume I , we may write the equa-
tion governing the temperature fluctuation T1 as
-------
5-5
dT . dT ' . dTr1 d2T'
dT' f- dT' , dT , dT ' . 6T H
-vr— = - < U . -r + U -s + U J -T - U -T \ +
dt \ j dx, j oxT j dx. j dx. J
(5.8)
We then multiply this equation by C' , Equation (5-5) by T'
\JL
add them together and average to obtain the equation governing
the rate of change of C 'T
- fc CpC'T1 + C C'T' + T'C'C' (5.9)
v->^'/
The triple correlation u !C 'T ' and the dissipation term have
already been modeled. The correlation T 'C 'CA will be
a p.
discussed below.
Returning to Equation (5.3)> ™e see that we must determine
the governing equation for C 'C~I . If we multiply Equation (5.5)
by CA and add to it the equation obtained by multiplying C '
by the fluctuation equation for CA - a replaced by p and (3
by a in (5.5) - then the average of that expression gives the
equation for C 'CM
-------
5-6
dC ' dCA
2 ^- -a P
5x, dx,
(5.10)
We see from Equation (5.10) that we must model the third-order
• 2 2
correlations-.- C..'CA and C ' CA , but also that we require
ct p ct p
2 2"
expressions for € ' and CA . These two equations are
identical except for the transposition of ct and P . The
2
C1 equation is found by multiplying (5.5) by C' and
Ct " CX
averaging to obtain
- 2k fCpC |2 + C C 'C ' + C'2C' > (5.11)
a|Pa aap a PJ v
- Now, since we are dealing with an atmospheric shear layer,
and our initial attention will be directed to cross-wind line
sources of a and P , we expect that the only derivative of
importance is the one normal to the mean flow ~ in the x
cartesian direction. Thus, only x . = x^ = z will be important
J j —
in the equations. Also, we may set t = x/u without loss of
generality. The modeling for the third-order correlations,
pressure correlations, and dissipation terms is prescribed by
Donaldson in Volume I as
-------
5-7
oj3
u'C'T' = - A
dC 'T'
a .
n '
^ dx, A, K." a
du.1 dC ' u'C'
i a i a
where q = (u 'u ' + v >v ' + w 'WT)? and X, A, , A2 and A_ are
length scales. Also, A0 = (A0 A0 )5 where a and P
^B da ^P
correspond to the species in question. When Equations (5.12)
are substituted into Equations (5-3), (5.7), (5-9), (5.10),
and (5.11), we obtain our final equation set governing the
simultaneous diffusion and chemical reaction of a and p
dC d C aC 'w '
— a a a
(5.13)
(5.14)
-------
5-8
u
g
= - w'w '
x- UU '
{(2A2 + A ) q^
L a Ja
d2C 'w '"
-
A
c 'w ' + v
- 2v
l
a
- k,
C 'w '
a
0 X2
a
a p1 1
— )
(5.15)
A^ C6W ' + vo
'P
C^w '
~2~
i ip i
U
~5x
-r - O 'W ' -3—
dz a oz
{. a
^2c-nf7
+ v.
- 2v
C 'T1
a
a
(5.16)
, < C CAT1 + CftC'T' + T'C 'C'
llap pa ap
(5.17)
-------
d2C~^
v °L£ - 2v
C 'C
dC
A
•f V
d2C'
a
o , 2
dz
C1
X
a
ac-*
'•3^= - 2^
^-/.A. i
/ ^
-5-c- + -3- < A0 q -T-S
dz dz 2^ dz
P
d";
o
5-9
u
dCAT
~5x
=-• - w 'T
+ v
-P
- 2 v
cpT'
o
r_
•V2
C0C'.
(5.18)
u
a
dCg
3F~
dCjC
"d~z
'CA >»
g P I
1 a P
- k.J'C C*1
c'i a a
p
''a + Ca CP
(5.19)
(5.20)
-------
5-10
We have written
k = k,
a 1
and
kQ =
p
Since we have
decoupled the diffusion/chemistry model from the back-
ground turbulence model, a solution of Equations (5.13) -
(5.21) requires knowledge of the initial distributions of
C" and
and
Cfg and of the flow parameters
u
T, T ,
WW
q,
w'T * . The macro scale lengths A and micro lengths
X must also be known as functions of the background turbu-
lence or the plume characteristics.
Finally, the coupled diffusion/chemistry model is
closed by modeling the third-order chemistry correlations
as described in Section 4.
CaC
^L& - Ml (5.22)
- M
(5.23)
where
M = 1 when
C '
a
"
>
M = 0 when
p
< 1
(5.24)
A generalization of (5.22) and (5.23) gives the appropriate
modeling for T'C'CL
T'C
T C
- M
(5.25)
With the properly modeled equations, we can now proceed to a
discussion of some of the results of computer solutions of
these equations.
-------
6-1
6. SOME CALCULATIONS OF THE INTERACTIONS
OP TURBULENT DIFFUSION AND CHEMISTRY
As must be evident from the derivation and recapitula-
tion of the closed diffusion/chemistry equations (Section 5),
it is virtually impossible to trace the effects of variations
of any one variable through this simulation system. This
being the case, the validation of the model must.involve
multiple iterations with a systematic, step-by-step variation
of each of the input variables, and comparison of these model
predictions with observed values of concentration patterns,
chemical depletion rates, turbulent flux divergences, and the
like. Neither time nor resources have permitted this valida-
tion of the model, of course, and the calculations presented
here must be regarded as suggestive rather than authoritative
as to how real chemically reactive and turbulent systems may
operate. The verification of these predictions must be
deferred, but the results which are presented here argue
strongly that at least in some circumstances the interactions
of turbulent diffusion and chemical reactions are highly
significant and, if verified from observations, models of
this type may also improve predictions of air quality in
the lower atmosphere significantly.
An Illustrative Calculation
In view of novelty of simultaneous consideration of the
turbulent diffusion of reactive chemical species and their
reactivity in inhomogeneous mixtures, it appears highly
desirable to examine in detail the individual processes by
which turbulent diffusion and chemical reactions produce
observed patterns of reactant concentrations and reactant
depletion in a simple but realistic system. To this end we
have chosen to calculate the combined processes of diffusion
and chemistry for the case of a plane jet of reactant a
-------
6-2
released continuously and isokinetLcally into a uniform environ-
ment of reactant p . The environment of £ is characterized
by a uniform transport speed TT = io m/sec and an isotropic
and homogeneous field of turbulence characterized by the verti-
2 22
cal intensity of turbulence w ' = 1 m /sec . The plane Jet
of the a species is oriented across the mean field of flow
and the initial vertical distribution of the concentration of
a., C" , is taken as gaussian with a central value of one and
ct
standard deviation a = 0.4 m . In view of the requirements
z
that the mass fractions of a and £ equal one, this "Jet"
of the a species displaces the ambient (3 species at the
source in such a way that the initial distribution of the
concentration of the (3 species is the complementary gaussian,
Cg0 = 1 - C" 0 . This geometry of the initial distributions
of Cf and Cg is shown in Figure 16. Note that no initial
fluctuations of C and C0 are introduced at the source.
a P
Finally, we take k, = kp = 1.0 .
In keeping with the constraints imposed in the construc-
tion of the model, we assume the reaction of a with P
proceeds isothermally. For our present purposes we shall also
assume that this reaction is irreversible, even though this
assumption is not mandatory? With these input conditions the
model calculates the redistribution and the chemical depletion
of a and (3 as a function of travel distance or time after
emission.
As a first partial view of the coupled effects of diffusion
and chemistry .in this flow reactor, we may compare the predicted
axial concentrations of the a species as a function of distance
from the source and under the following conditions:
*
The effects of reversibility of reactions and catalytic
cycles may be accommodated to a certain extent by approp-
riate choices of the reaction rate constants, k]_ and k2-
Similarly, three body reactions of the type ~^-±^oPeP^,
may also be simulated, if dCM/dt « 0 , by taking fc = kj_CM
where M is the third body.
-------
t
0
1.0
Plume axis
Ci/C,
Figure 16.
Source configuration for a plane jet of pure
a species injected isokinetically into an
environment of pure |3 species.
-------
6-4
1. a does not react with £ (diffusion only)
2. Diffusion and chemistry occur, but the
chemical reaction rates are calculated on
the basis of the local mean values of
concentration only. (We have termed this
"homogeneous chemistry" since C "Cl is
U» }~J
neglected.)
3. Diffusion and inhomogeneous chemistry are
operative. (The full model described in
Section 5.)
This comparison is shown in Figure 17 and it is immediately
evident that the neglect of C 'CA leads to a significant
C£ p
over prediction of the rate of decrease of the axial concentra-
tion of the a species. 'For example, the ratio of the
predicted concentrations at x = 40 m is two, and, as is clear
from Figure 17, this ratio is increasing with x . This .
result reflects primarily the effect of C'CA on the. chemical
reaction rate. However, such a simple portrayal of the results
of coupled chemistry and diffusion does not portray the balance
of the diffusive and chemical processes at work. In -order to
gain this insight we must examine in detail the'balance of
turbulent diffusion and chemical reactions going on across the
plume.
In order to examine this balance we have neglected the
molecular diffusion terms and plotted each of the rates which
determine dcT/dt and dCL/dt as a function, of distance from
the plume centerline at x. = 37 m or t = 3,7 sec. The profiles
of mean concentrations of a 'and p are shown.in Figure 18
and the diffusion and chemical reaction rates"predicted by the
model are shown in Figure 19. In order to discuss and interpret
these results,, we recall the balance equations for dCf/dt and
-------
Diffusion-
no chem.
X
o
E
o
loe
X
o
E
10°
6-5
Inhomogeneous
chem. -+-
diffusion
Homogeneous
chem. +
diffusion
x (m)
Figure 17. Comparison of the axial concentrations of the a
species as a function of travel distance from the
plane jet as estimated for 1) diffusion only,
2^ inhomogeneous chemistry plus diffusion, and
3) homogeneous chemistry and diffusion. (See
text for details.)
-------
fi, t=3.7 sec
oi
E
-------
Height above plume Cj_ (m)
2
Above this height the chemical reaction is
essentially "diffusion" limited, i.e., the total
reaction rate is determined by the rate at
which the a species is supplied by turbulent
diffusion.
Below this height the chemical reaction rate
is increasingly "mixedness" limited, i.e., C'aCjj
becomes significant.(Compare curves 182)
-6
-4
Figure 19.
-2 0 2 4
Rates for terms indicated (ppp/sec) (xlO2)
8
Calculated values of. local rates of change of the concentrations of
a and P due to diffusive flux divergence and to inhomogeneous
chemical reactions.
-------
6-8
and
c>C
-,
Iz
where we have neglected v —3— and v —**- as small in
comparison with ^— C 'w ' and -r— C^w ' . Each of the retained
terms is plotted as a function of height above (or below) the
plume centerline in Figure 19. (Recall that d(T/dt and
are the local rates of change of the mean concentra-
tion of a and £ due to both turbulent diffusion and
^ _ ^ _
chemical reaction; -r— C 'w ' and -r— C^w ' are the local diver-
gences of the turbulent flux of the a and £ species^.
-k(C~ Cfg) is the average local chemical reaction rate due to
the local mean concentrations of a and P ; -k CfMjJ is
the average local chemical reaction rate due to correlated
fluctuations of the concentrations of a and p .)
A patient inspection of Figure 19 reveals the following
facts regarding the diffusion and chemistry processes through
the plume :
1. The diffusion of the a species is removing a
from the plume core from the center line to z.= 1.25 m and
is causing an accumulation of a from 1.25 to 5 m (Curve 4).
2. The diffusion of the p species into the plume is
operating to remove p from the height zone 2 to 8 m and
accumulate p in the height zone 0 to 2 m (Curve 6).
-------
6-9
3. The chemical reaction is depleting both the a and
P species between the plume center line and z = 6 m (Curve l),
the upper height being the limit of a penetration into the
P environment at this time.
4. The chemical reaction diminishes the rate of increase
of the concentration of P below z = 1.5 m and accelerates
the decrease of concentration of P from 1.5 to 5 m (Curve 5).
5. Below 2.75 ni the chemical reaction accelerates the
depletion of a in the plume core, but above 2.75 m the
reaction rate very nearly balances the diffusive transfer rate
for a , i.e., above z = 2.75 m the chemical reaction is
diffusion limited (Curve 3).
6. Below z = 5 m the diffusive mixing of P with
a becomes increasingly inhomogeneous so that at the plume
center line the chemical reaction rate is proceeding at only
60 per cent of the rate computed on the basis of mean values
of P and a concentration at that height. (Comparison of
Curves 1 and 2)
Prom these detailed comparisons we can immediately deduce
that the a plume is not only being rapidly depleted by reaction
with P but that it is also growing in vertical width only
slowly because of the balance between diffusion and reaction
rates in the outer limits of the plume. On the other hand, the
P deficit in the initial plume is being filled in with only
minor interference from the local chemical reaction with a .
Further, the depletion of the a species in the core of the
plume is significantly slower than would have been expected
from mean-value chemical kinetics. In this region the chemistry
is "mixedness " limited while above this region it is clearly
"diffusion" limited.
-------
6-10
This simple example is intended only to illustrate the
balance of diffusive and chemical processes (and, incidentally,
points to a laboratory experiment which may verify these
predictions). However, the example does illustrate the model's
power to simulate and portray complex chemistry/diffusion
processes.
The Sensitivity of Chemical Reactions to Turbulent
Diffusion Rates
The example discussed above is, of course, only one
particular case and cannot provide any real insight into the
sensitivity of the combined diffusion and chemical depletion
of reactive species to variations in the input variables.
Extensive sensitivity analyses have not been possible, but we
have done a partial analysis of the effect of turbulence.
intensity on the chemical depletion rate of the a species
for the flow reactor described above.
Returning to Equation (6.1) we note that since U is
constant
n oo dc -\ r> °° dF
where F is the horizontal flux of the a species at
ct
distance x . Then
/•> 0° SC 'W ' .-> oo
- dz -
and since the first term on the right of Equation (6.4) is zero,
the rate of change of the flux of the a species is determined
by the total reaction rate over the height of the plume at any
distance x . In order to compare the model 's .prediction of
this rate against a limiting condition, we may note that as
-------
6-11
C 'CA ' goes to zero and Cfi tends to be uniformly distributed
through the a plume (a condition which can only be approached
asymptotically), the basic chemical reaction goes over to a
first-order reaction and for U - constant
where C"D~ is the environmental concentration of the (3 species.
pO
We can rewrite Equation (6.. 4) as
< 6 •
and compare Equations (6.5J and (6.6) from the model calculations
2
using various values of w '
This comparison is shown for the flow reactor problem and
at x = 40 m in Figure 20. As can be seen from Figure 20, the
relative chemical depletion rate of the a species is quite
sensitive to the intensity of turbulence for small values of
w ' but becomes quite insensitive to this input parameter when
p
w ' becomes large ^ We may also note that even with the vigorous
2 22
turbulence of w ' = 3 m /sec , the chemical depletion rate
has only achieved about 60 per cent of the limiting, first -order
reaction rate at x = 40 m or kt = 4 sec,
A Simulation of NO - 0~, Reactions and Diffusion Downwind
of a Multiple Freeway System
The preceding calculations illustrate the basic capabilities
of the A.R.A.P. coupled invariant diffusion/chemistry model and,
of course, provide only a minimal excursion into the possible
interactions of diffusion and chemistry processes in a turbulent
flow reactor. The stage is set for in-depth analyses of this
kind, but neither time nor resources have permitted the multiple
-------
6-12
1.0
.8
'P -6
-La -4
-Limit of first-order reaction
x=40m
— — -O
I
I
I
2 3
w/2 (m2/sec2)
Figure 20. The partial dependence of the chemical depletion *
of the a species on the intensity of turbulence..
-------
6-13
calculations which are required. However, as an exercise of
the coupled dimensional model we have constructed an initial
simulation of the diffusion and chemical reaction of NO
(emitted from vehicles travelling on surface roads or free-
ways) with the ambient 0_ . In order to examine the effec'ts
of multiple sources of NO , we imagine four parallel freeways,
oriented across the mean wind and separated by a distance of
300 m. Steady traffic is assumed on each freeway so that
each represents a continuous line source of NO . The geometry
of the freeways and the assumed initial source distribution of
NO at each freeway are shown in Figure 21. Shown there also
are the mean wind profiles and the potential temperature lapse
rate chosen for this calculation (a neutral lapse rate and a
realistic boundary layer wind profile). Just upwind of the
first freeway (Si)we assume that 0~ is uniformly distributed
in the vertical at a concentration of 10 ppm. At the first
freeway we displace 0., with NO so that the initial street
level value of 0- immediately downwind of (Si) is zero.
For the purposes of this calculation we assume that NO
reacts with 0- irreversibly so that the only sources of NO
are the freeways (NO) and the reservoir of 0_ above the
NO plumes. With this assumption (which, of course, denies
the multiple reactions and catalytic cycles which enter into
the photochemical problems associated with NO and 0,) we
•3 c;
choose the value of the reaction rate constant as k = 5 x 10^
(l/ppm-sec). The calculation proceeds from the first upwind
freeway to a distance of 1200 m (300 m downwind from the
fourth and final freeway. From these calculations we can
recover the predicted vertical profiles of the mean concentra-
tions of NO and 0~ and of the diffusion and chemical rates
at any distance downwind.
Before looking at some of the details of these calcula-
tions, we may see the general result by examining the mean
-------
20
3
O
0)
O
.0
O
*
10
I10'
1 8
o 6
UJ
§ 4
m
°<
^f
%
SOURCE CONFIGURATION
? OF NO AT S,,S2,S3,S4
X
^\
. . . . \
) 2 4 6 8 R)
?^CONC. OF NO (PPM)
' \\
-9
U
j
;
/
) 600 900 1200 0 _2 4 6 8
83 84 u (m/sec)
I
downwind
first freeway (m)
Figure 21
The source geometries and the boundary layer meteorological conditions
chosen for simulation of a multiple freeway problem.
i
M
-pr
-------
6-15
values of the concentration of NO and 0^ at 1.625 m above
street level and as a function of distance downwind from the
first freeway. These values are plotted in Figure 22 and we
pee immediately the gradual accumulation of NO and the
depletion of 0 in the surface layer as the air moves
across multiple freeways.
A closer examination of Figure 22 also reveals the
balance between diffusive mixing and chemical reactions as
reflected in the surface layer concentrations of the 0-, .
Note that the residual 0,, entering each freeway is drastic-
ally depleted by reaction with the fresh NO during the
first 50 m of travel, and then the 0~ concentration
increases due to diffusive mixing from above at a rate
which exceeds the chemical destruction rate. Recalling the
illustrative calculation discussed in the previous sections,
we see here the suggestion of a fine balance between diffusion
of the reactants into each other and the chemical depletion
rate. In this case the chemistry is restricted by the supply
rate of 0,, .to the NO plume.
With this general result in hand, we may turn attention
to the details by examining the predicted vertical profiles
of NO and 0_. concentration and of their reaction rates
at selected positions in the array of freeways. For this
purpose we have elected to plot these profiles at x = 400
and 1200 m ; these are shown in Figures 23 and 24 (Note the
difference in height scales when comparing these two figures).
Examination of these figures shows immediately the
surface layer depletion of 0,, and the inability of diffusive
mixing to maintain a uniform distribution of 0_ with height.
Even at 1200 m the surface layer concentration of 0- is
less than 1/10th its value outside the NO plume. And we
may note in passing that this diffusion limitation can be
-------
to
CVJ
(0
•
II
Kl
0
10
in
CVJ
c
M
10
300
600
900
1200
x, meters downwind
Figure 22. Surface layer concentrations of NO and 0-z predicted for
multiple freeway simulation.
-------
6-17
readily checked by vertical profile measurements of the
concentrations of suitably chosen reactants.
Of considerably greater interest, however, is the
portrayal of the chemical reaction rates with height at
400 and 1200 m. These are plotted on the right-hand side
of Figures 23 and 24, and for comparison we have also
plotted the rates determined by the local mean values of
concentration alone. As is more than evident, thes-e .•
reactions are strongly "mixedness" limited; the total
reaction rate over the plume height is only 19 per cent
of the mean value rates at 400 m and 13 per cent at 1200 m.
This result points to the "folding" nature of turbulent
mixing', a process in which discreet volumes of each reactant
are folded into one another, but are not intimately mixed.
and therefore react chemically at a much slower rate than
their local average values of concentration suggest. The
calculation also points to the maintenance and even enhance-
ment of this "mixedness " limitation when we are dealing with
multiple sources. Its true role in a complex photochemical
system needs much further study, of course. But the portrayal
of a diffusion/chemistry interaction which may alter the
chemical reaction rates by a factor of five or more can
hardly be ignored.1
This demonstration of the potential significance of
the correlations of concentration fluctuations in the
determination of chemical reaction rates also points up the
necessity for second-order closure models if these effects
are to be simulated. The possibility that important multiple-
source urban air pollution problems exist in which this effect
will be significant commends second-order closure models,
either for the direct simulation of these situations or for
the development of correction factors which can be applied
to first-order closure models.
-------
10
8
- 6
N
01 yi
« 4
-\
0
0
\
X
/
\
/
8
It ,
J /(CaC)8-f-C/aC/)9)dz
l\ p— — =^'
19
CaC/9 , A
Units = ppm/sec
o\
i
t->
CO
Figure 23. Vertical profiles of the concentration of NO and 0
and of their chemical reaction rate at
from the first freeway.
-,
x = 400 m downwind
-------
8 0
= 0.13
flC^ 4- C^C^ ,
CaC^f A
Units = ppm /sec
VO
Figure 24. Same as Figure 23 except at x = 1200 m.
-------
7-1
7. CONCLUSIONS AND RECOMMENDATIONS
Given the objectives of this program to construct a
coupled invariant diffusion and chemistry model and to
exercise such a model sufficiently to show its applicability
and advantages in real-world situations, we can only conclude
that this effort has been more than successful. This work
has demonstrated the feasibility of incorporating the
stochastic nature of turbulent diffusion and chemistry in
dynamic models, and has provided the first working version
of such a model.. Most importantly, perhaps, even this first
and relatively simple version has revealed interactive effects
between turbulent diffusion and chemical reactions which
could not possibly be revealed by mean-value or first-order
closure models. It seems clear to us that, on the one hand,
the atmospheric chemists must reexamine their traditional
assumptions of a well-mixed system in quasi-equilibrium-,
and on the other the atmospheric dynamicists must extend
their considerations beyond the classical calculations of
average values of pollutant concentrations. The further
exercise and development of these second-order closure models
can provide the tools necessary for the joint evaluation of
turbulent, multi-source flow reactors, such as the atmospheric
boundary layer in urban areas.
We make this latter recommendation with a full awareness
of the complexities of the chemistry of urban air pollution
and the proliferation of the invariant model equations." as
multiple or chain reactions are introduced. However, 'this
Increasing complexity need not deter the development and use
of these concepts, since they may first be used to analyze
critical turbulent reactions, then to define areas where
simpler models are quite adequate, and finally to provide
simulation capabilities for those situations where first-
order closure models are demonstrably inadequate.
-------
7-2
As a desirable prelude to further development of these
models, such as their extension to coupled three-dimensional
systems, the present model should be subjected to a rigorous
sensitivity analysis wherein the input variables of the
turbulence field, the initial plume geometries, and the
chemical reaction coefficients are systematically varied., and
the outputs of reaction and diffusion rates and the concentra-
tion distributions are tested for sensitivity to these, input
variations. Second, critical experiments, first in .the
•
laboratory and then in the atmosphere, should be designed
and conducted to verify and validate not only the basix; model
output such as mean values of concentration, but also the
processes internal to the model's workings. In an atmospheric
experiment, a minimum measurement program would require an
array of towers oriented downwind from cross-wind line sources
and equipped to measure the simultaneous means and fluctuations
of at least two reacting chemical species (coming from well
defined sources) and the turbulent flux ©f these materials,
all as a function of height. The conduct of such an experi-
ment within the broader measurements program of the EPA/RAPS
appears particularly desirable.
-------
APPENDIX A
EFFECT OF INHOMOGENEOUS MIXING ON ATMOSPHERIC
PHOTOCHEMICAL REACTIONS
-------
Reprinted from
ENVIRONMENTAL
Science & Technology
Vol. 6, September 1972, Pages 812-816
Copyright 1972 by the American Chemical
Society and reprinted by permission of
the copyright owner
Effect of Inhomogeneous Mixing on Atmospheric Photochemical Reactions
Coleman duP. Donaldson and Glenn R. Hilst1
Aeronautical Research Associates of Princeton, Inc., Princeton, N.J. 08540
• The conventional assumption of local uniform mixing of
reactive chemical species is reexamined by derivation of the
chemical reaction equations to include the effect of locally
inhomogeneous mixtures on the reaction rates. Preliminary
solutions of a simplified version of these equations show that
inhomogeneities in reactant concentration generally tend to
slow the reaction rate. Estimates of the relative roles of local
diffusive mixing and chemical reactions in inhomogeneous
mixtures show that there are several relatively fast photo-
chemical reactions which may be limited by local diffusive
mixing. In these cases, the reaction proceeds much more
slowly than would be predicted if the reactants were uniformly
mixed.
The importance of chemical reactions in the atmosphere
has been increasingly recognized in the problems of air pollu-
tion. These are probably most acute in dealing with photo-
chemical smog formation (Worley, 1971). We have, therefore,
drawn our examples from photochemistry, but we have not
attempted to go beyond an examination of the possible im-
portance of inhomogeneous mixing in these processes.
Basic Chemistry Model
We assume a bimolecular reaction
+ 0
+ S
(1)
where a, /3, y, and 5 denote chemical species and that the re-
action of a with /3 to form y and 5 is stoichiometric and is
governed by equations of the form
In developing either mathematical simulation models or
laboratory chambers for the study of chemical reactions
in the atmosphere, it has been generally assumed that the
reacting materials are uniformly mixed. However, observations
of the time history of concentrations of trace materials show
quite clearly that uniformly mixed materials are the exception
rather than the rule in both air and water (Nickola et al., 1970
Singer etal., 1963; CsanadyandMurthy, 1971). Local fluctua-
tions of concentration are particularly significant during the
early stages of atmospheric mixing, immediately following dis-
charge of trace materials into the atmosphere, and when there
are multiple point sources of pollutants. Our purpose here is to
make a preliminary estimate of the importance of these fluctua-
tions on atmospheric chemical reaction rates and determine,
at least approximately, the relative roles of reaction rates and
diffusive mixing in the control of atmospheric chemical re-
actions.
1 To whom correspondence should be addressed.
812 Environmental Science & Technology
(2)
ri vm
— = -K[a]{p]
[/] denotes the molar concentration of the rth chemical
species and K is the reaction rate constant in units of (sec
mol/cm3)"1. It is convenient to transform the concentration
terms in Equations 2 to dimensionless mass fractions, C,, by
= Mt[i]
(3)
where p0 is the density of the mixture (g/cm3) and Mt is the
molecular weight of the ith chemical species. Then the de-
pletion rates for the a and /3 species may be written
and
Ma
(4)
(5)
-------
where Ka = KpJM^ and has dimensions (sec-ppm)"1 when C0
and C0 are expressed in parts per million (ppm) by wt.
We may now examine the relative contributions of the means
and fluctuations of Ca and Q to the chemical reaction rate by
assuming the time history of these quantities at a fixed point
constitutes a stationary time series and that
= C
cfl'
(6)
where the overbar indicates a time average and the prime
indicates the instantaneous fluctuation about the average.
Noting that Ca' = Ce' = Oand that
ac, &c,
— = — H
dr dr
we obtain directly from Equations 4-6
and
(7)
(8)
(9)
where we have suppressed the dependence of Ka on the tem-
perature and pressure. (This analysis can be extended to in-
clude the fluctuations of Ka owing to significant fluctuations of
temperature and pressure. For our present purposes, we shall
assume an isothermal reaction at ambient pressure.)
The role of concentration fluctuations in chemical reactions
is immediately evident from Equations 8 or 9. The second-
order correlation in the joint fluctuations of Ca and Cp either
enhances the reaction rate (when the correlation is positive) or
suppresses the reaction when Ca 'C$' is negative. Only when
these fluctuations either do not exist or are uncorrelated is the
average reaction rate governed by the average concentrations.
As a simple example of the importance of this correlation
term, imagine that the materials a and /3 pass the point of
observation at different times—i.e., they are never in contact
with each other. Values of Ca and C0 would be observed, but
it is readily seen that CaCp = — Ca'Cp' in this case, a result
which correctly predicts no chemical reaction.
If we assume no diffusive mixing of the reacting materials,
we may model the chemical reactions by noting that
(10)
and
"\y^ / "\ f*
oCt oC
- - = —
d/ dr
V - r ' —"-'
— t-a
(12)
Performing the necessary operations and time-averaging, we
get, repeating Equations 8 and 9,
(13)
a ,
cac»
CCa'C
'Q")
(14)
(15)
(16)
(17)
Equations 13-17 provide a closed set, except for the third-order
correlation terms €„'€/* and C3'Ca'J.
The appearance of the third-order correlations complicates
the modeling problem very considerably since the statistical
description now requires consideration of the distribution
functions for Ca and Cft. In an independent study, O'Brien
(1971) has proceeded from Equations 13-17 by assuming the
form of these distribution functions. Another approach, which
we are pursuing, is to model the third-order correlations in
terms of the second-order correlations. However, for our
present purpose of determining whether or not the effects of
inhomogeneous mixtures on chemical reactions may be
significant, we may neglect the third-order correlations by
assuming C0' and C^' are symmetrically distributed about Ca
and Cf,, respectively. This assumption is, of course, untenable
for more general cases but it does permit a solution of Equa-
tions 13-17 by numerical techniques.
For an initial test of the significance of inhomogeneities in
chemical reactions, we assume a reaction box in which the
initial concentration distributions of a and ft are arbitrarily
specified by ^, Q,, C^, C/*, and €„'€/,'. As a further con-
straint which isolates the chemical reaction process, we assume
there is no mixing in the reaction vessel and no wall effects.
As a reference case, let us assume a completely uniform
10
^.
10
UNIFORMLY
MIXED CASE
10°
Kl
Figure 1. Chemical depletion of randomly
mixed reactants (Ca'Cp' = 0) for various
initial degrees of inhomogeneity, as measured
by C-VC.'
Volume 6, Number 9, September 1972 813
-------
Figure 2. Chemical depletion of initially
inhomogeneously mixed reactants (C'2/C02
= 0.40) for various degrees of initial correla-
tion between Ca' and C0', as measured by
Cg'Cf,'
0.0
-1.0
10°
Kl
mixture of a and /?—i.e^ no Jluctuations in concentration,
Ma ^ Mp, and initially Ca = Cp = C0. The predicted values
of Ca are shown in Figure 1 as a function of time normalized
by the reaction rate constant. As can be seen, the reaction
proceeds to exhaustion of the reacting materials.
Now let us assume that a and |3 are initially inhomogene-
ously mixed but that there is no initial correlation between
Ca' and cy—i.e., initially Ca'Cy ss 0. As a measure of these
fluctuations, we take GyVO* = 0.2, 0.4, 0.6, 0.8, and 1.0.
The results of these calculations are also shown in Figure 1,
and it is immediately evident that any inhomogeneities
operate to suppress the chemical reaction rate and to stop it
completely before the reacting materials are exhausted.
Mathematically, the model predicts that, in the absence of mix-
ing, initial inhomogeneities operate to produce values of
Ca'Cp' which eventually become equal to —CaCp and the re-
action ceases. Physically, the local reactions have everywhere
proceeded to exhaustion of one of the reactants, leaving a
residue of the other reactant and products at that site.
It is of special interest to note, from Equation 15, that the
suppression of the reaction rate by C~'Cy depends only on
one of the reactants being nonuniformly distributed initially.
A negative rate of change of Ca'Cy can be generated by non-
zero values of either Ca'2 or Cy*, since the terms C>Ca'2 and
CQCy2 are positive definite. The presence of concentration
inhomopeneities in one of the reactants generates inhomogene-
ities in the other.
The effect of an initial correlation between Ca' and Cy may
now be examined by assigning initial nonzero values to Ca'Cy,
CV^jind Cy2. To illustrate this effect, we have chosen
c7~2/c02 = cyvc,2 = o.4 and c7cy/(c72c7')1/2 =
+ 1.0, +0.5, 0.0, -0.5, and -1.0 where Ca'Cy/(Ca'2Cy2)"2
= Ra#, the ordinary correlation coefficient. The resulting
predictions of CjC0 are shown in Figure 2 and are again com-
pared with the uniformly mixed case. As might have been ex-
pected, initial positive correlation accelerated the reaction
rate, but only when this initial positive correlation was per-
fect did the reaction go to exhaustion of the reacting materials.
In this case, although there were concentration fluctuations,
stoichiometrically equal amounts of a and /3 were initially
placed in each local volume. In all other cases, the reaction
was again halted when one of the reactants was exhausted
locally, leaving a residue of the other reactants and products
of the reaction.
The combined effects of initial inhomogeneities and correla-
tions between the fluctuations are summarized in Figure 3 by
plotting the depletion of C during the first normalized time
step as a function of Raiff and C7"2/^2. The effect of the mag-
Figure 3. Joint effect of initial correlation and inhomogeneity on de-
pletion of reacting materials at Kt = 1.0
nitude of the fluctuations, as measured by C'2, reverses as one
goes from large positive toward small positive and negative
values of Ca'Cf'.
These results point toward an important role for fluctua-
tions of concentration in controlling chemical reaction rates.
For example, if two reacting materials are discharged si-
multaneously from a point source, during their initial mixing
with the atmosphere their concentration fluctuations should
be large and positively correlated. We would then expect, on
the basis of this effect, that their reaction rate would be con-
siderably faster than if they were uniformly mixed from the
start. The emission of hydrocarbon and NO* from auto ex-
hausts is a case in point. Discharge of SO2 and particulate
matter from power plant stacks is another.
On the other hand, if reacting materials are randomly
mixed or if positive fluctuations in one are associated with
negative fluctuations in the other, the reaction should be
suppressed, compared to the uniformly mixed case. Both of
these cases could be important, but their true importance de-
pends critically on the rate at which atmospheric diffusion
tends to mix chemical species and, hence, to diminish these
fluctuations, as compared with the rate of chemical reaction
produced by the concentration fluctuations.
Estimates of Local Mixing Rates in the Atmosphere
The only way in which the correlation Ca'Cy can be elim-
inated, if it exists, in given flow situations is by the process
of molecular diffusion. To estimate the rate at which this can
occur, we may write the expressions for the contribution of
814 Environmental Science & Technology
-------
molecular diffusion to the time rate of change of Cn' and
They are
= Da
n ;X.r_«
\ d/ JAM dyjdyi
(18)
(19)
Multiplying Equations 18 and 19 by Cp' and Ca', respectively,
adding, and time-averaging gives
/
V
-2 D
(20)
(We have assumed Da =^ £>0 = Z), consistent with the assump-
tion Ma ~ M0. See O'Brien (1971) for a discussion of this
assumption.) The first term on the right-hand side of Equation
20 is nondissipative—i.e., it measures the transfer of the Ca'Cp'
correlation by gradients in the value of this correlation within
the field. The second term is dissipative—i.e., it measures the
local diminution of Ca'Cp' by the action of molecular diffu-
sion. The appropriate expression for this term is
2 D Cg'CV
X2
(21)
In this expression, the dissipative scale length X must be chosen
as it is chosen for the calculation of other turbulent correla-
tions when performing calculations of the structure of turbu-
lence (Donaldson, 1969).
For such calculations, 1/X2 is given approximately by
°-05 p°q
X2
where p0 is the atmospheric density, q1 = V2 + e'2 + tv'2, n0
is the molecular viscosity of air, and A is a length scale related
to the integral scale of the atmosphere, and is of the order of
1000 cm in the earth's boundary layer.
If we choose typical values of the parameters involved in
evaluating the magnitude of the expression for (dCa'Cy/d/)dift
given in Equation 20, we have
A = 1000cm
p0 = 10-' g/cm8
Ho = 1.7 X 10-
14. C3H6 + HO2 =
CH3O + CH3CHO 3.4 X lO"2 10-'
15. C2H3O+ M =
CH3 + CO + M 1.7X10-' 2 X 10~2
and the reaction rate will be suppressed. In this case, the re-
action is controlled by the rate of species mixing and will de-
pend on parameters other than Ka and C0Q.
We may estimate which two-body reactions will proceed as
though C'acy ~ 0—i.e., in the usual manner, and which will
be modified by having values of |C<,'Cy| of the same order as
\CaCe\ by forming the ratio
/dCa'Cy\
N = \ df Jdift = 2D = 3.4X1
\ d? /Ch
(22)
When N » 1.0, Ca'C/,' will tend to zero and the reaction will
be controlled by the reaction rate constant and the mean con-
centrations; when N and the reaction will proceed at a rate deter-
mined largely by the rate at which one reactant can be mixed
with another and will depend on the scale of the patches of
unmixed reactants.
Typical values of Ka for various reactions which enter into
the photochemical chains are listed in Table I along with esti-
mates of N. For these two-body reactions, only the first and
second are sufficiently slow for conventional kinetic models
to apply. The propylene reactions with O3,02, and HO2 (num-
bers 12, 13, and 14 in Table I) tend to represent a transition
stage between diffusive mixing control and chemical reaction
control of the reaction rate. The remaining reactions are all
clearly diffusion-limited in inhomogeneous mixtures and
should proceed at a rate which is much slower than conven-
tional chemical kinetics would predict.
Conclusion
These results indicate there are clearly reactions in the ph>; to-
chemical chain which will be suppressed by the inability of the
atmosphere to mix the reacting materials rapidly enough to
prevent serious local depletion of one of the reacting materials.
In these cases, conventional models of the reaction will tend to
Volume 6, Number 9, September 1972 815
-------
seriously overestimate the reaetion rate, and, therefore, the
production rate of the chemical species which enters into the
next reaction in the photochemical chain. On the other hand,
the enhancement of the reaction rate for two materials
emanating from a common source and, therefore, occupying
the same volume of the atmosphere, during the initial period
of incomplete mixing, may also represent a significant de-
parture from conventional simulation models.
It is hoped that this brief and necessarily incomplete dis-
cussion will serve to demonstrate the importance of turbulent
fluctuations of concentrations in atmospheric chemical re-
actions. Consideration of these effects in refining simulation
models of these reactions appears to be important.
Nomenclature
Ci = mass fraction of /'th chemical species, ppm
Di = molecular diffusion coefficient for /th chemical
species, cm2/sec
K = chemical reaction rate constant, cm3/sec-mol
K«, K/t — chemical reaction rate constants, 1/ppm-sec
MI = molecular weight of /th chemical species, g/mol
N = nondimensional ratio of characteristic times
q2 = ur* + vi* + W7*, cm2/sec2
^a,/s = ordinary second-order correlation coefficient
/ = time, sec
u',v',w' = orthogonal components of turbulent fluid
motion, cm/sec
yt = length along /th direction of a cartesian coordinate
system, cm
[ ], - averaged quantity
' - departure from the average of the primed quantity
GRI-UK LIUTKRS
a, #, 7, 8 = chemical species
A = dissipation scale length, cm
A = macroscale of atmospheric turbulence, cm
Hi, = dynamic viscosity for air, g/cm-sec
p,, = fluid density, g/cm3
Literature Cited
Csanady, G. T., Murthy, C. R., J. Pliys. Oceanogr., 1, I,
17-24 (1971).
Donaldson, C. duP., J. AIAA, 7, 2, 271-8 (1969).
Nickola, P. W., Ramsdell, J. V., Jr., Ludwick, J. D., "De-
tailed Time Histories of Concentrations Resulting from
Puff and Short-Period Releases of an Inert Radioactive
Gas: A Volume of Atmospheric Diffusion Data," BNWL-
1272 uc-53 (available from Clearinghouse, NUS, U.S.
Dept. of Commerce), 1970.
O'Brien, E. E., Phys. Fluids, 14, 7, 1326-9. (1971).
Singer, I. A., Kazuhiko, I., del Campo, R. G., J. Air Pollut.
Contr. Ass., 13, 1, 40-2 (1963).
Worley, F. L., Jr., "Report on Mathematical Modeling of
Photochemical Smog," Proceedings of the Second Meeting
of the Expert Panel on Modeling, No. 5, NATO/CCMS
Pilot Project on Air Pollution, Paris, July 26-7, 1971.
Received for review November 26, 1971. Accepted May II,
1972.
-------
APPENDIX B '
CHEMICAL REACTIONS IN INHOMOGENEOUS MJXTUKFiS:
THE EFFECT OF THE SCALE OF TURBULENT MIXING
-------
15
CHEMICAL REACTIONS IN 1NHOMOGENEOUS MIXTURES: THE EFFECT OF THE
SCALE OF TURBULENT MIXING
Coleman duP. Donaldson* and Glenn R. Hilst**
ABSTRACT
Recent studies by O'Brien [1] and the authors of this paper [2]
have provided a theoretical framework for the assessment of
chemical reaction races when the readmit!.; are embedded in a
turbulent fluid and are inhomof.eneour..! y mixed. The results of
ther.e studies, which are reviewed here, point towards a profound
effect on chemical production and depletion rate:-, when the charac-
teristic reaction time, as measured hy the product of the chemical
kinetic reaction rate constants and the average and fluctuating
concentrations of the reactions, is short compared with the
characteristic molecular diffusion time. The latter time is
measured by the ratio of the molecular diffusion coefficient and
the square of the dissipation scale length, and is, therefore,
dependent upon the scale of the turbulent motions. Both the fact
of inhomogeneous mixtures and this dependence upon turbulent
scales of motion pose significant problems when extending labora-
tory results to other scales of motion, such as the free atmosphere.
These theoretical results, which are partially substantiated
by observations, point towards the need for simultaneous measure-
ments of turbulence and chemical reaction rates over a range of
turbulence scales and reaction rate constants. If substantiated
by such new experimental measurements, tiie theoretical results
point towards a clear requirement for- joint consideration of the
chemical reactions and the scale of turbulence in such diverse but
critical problems as the design of large combustion apparatus and
the calculation of photochemical reactions in the atmosphere.
INTRODUCTION
Although the effects of inhomogeneous mixing of reacting
chemical species on the reaction rate, as measured by either the
depletion of reacting species or the production of new species, have
been recognized for at least ten years [3], methods for accounting
for this effect have only recently emerged [1,2]. Neither of
these methods are as yet fully developed, but they are sufficiently
advanced that we may make some preliminary estimates of the
situations under which the effects of inhomogeneous mixing will be
pronounced or perhaps even completely dominate the reaction.
In particular, we find for .the case of an irreversible two-
body reaction at constant temperature that the following limita-
tions are imposed by inhomogeneous mixing of either or both of the
reacting species:
* President, Aeronautical Research Associates of Princeton, Inc.
50 Washington Road, Princeton, New Jersey 085*10 (A.R.A.P.)
**Vice President for Environmental Research, A.R.A.P.
Reprinted from PROCEEDINGS OF THE 1972 HEAT
TRANSFER AND FLUID MECHANICS INSTITUTE,
Raymond B. Landis and Gary J Hoffmann. fHitorl.
Stanford University Press, 1972. © 1972 by th»
Board of Trustees of the Leland Stanford Junior
University.
-------
254 Donaldson and Hilst: Chemical Reactions and Turbulent Mixing
• If the chemical reaction rate Is slow compared with the
molecular diffusion rate, no effect is noticed, and the
reaction proceeds according to conventional chemical kinetics.
• If the chemical reaction rate is fast compared with the
molecular diffusion rate, the reaction rate is limited by
the diffusive mixing rate, tending, in the limit of very
slow diffusive mixing, to zero before the reactants are
exhausted.
A large number of reactions in combustion processes and photo-
chemical smog formation fall within this latter category. It is,
therefore, of considerable interest to investigate further just
how much the reaction rate is curtailed by inhomogeneous mixing
under such circumstances. In the following pages, we derive the
basic equations for prediction of the joint effects of chemical
reactions and molecular diffusion, examine the effects of the
dissipation scale length of the turbulent motions, and identify,
on a preliminary basis, the two-body reactions inherent in photo-
chemical smog formation for which inhomogeneous mixing is a
limiting condition.
MODELING OF CHEMICALLY REACTING FLOWS
For most computations of chemically reacting turbulent flows,
it has been customary for engineers to proceed with the calculation
according to the following scheme. First, the engineer develops by
some method (mixing length, eddy diffusivity, or other method)
equations for the tirne-averaged or mean values of the concentrations
of the reacting species of interest (say, species a and P ) at
each point in the turbulent flow under consideration. He also
obtains an equation for the mean value of the temperature that is
expected at each point in this flow. It is then customary, if the
equations that generally govern the reaction between a and P , are
BF* = -klCaCp , p p . ( ?)
Dt K2 a P .
to assume that valid equations for the time rates of change of the
mean values of the mass fractions of a and P are
DC\
~ 1 n A \ 3 J
Dt
n+- ^o^nr-'a
IJt £ Ct p
In these equations, C and CR are the time-averaged mass
fractions of the two species and k- and k0 are the reaction
rates k, and k? evaluated at the mean temperature T , i.e.,
CO
-------
Donaldson and Hilst: Chemical Reactions and Turbulent Mixing 255
and k2 = k2^ '
Although equations such as (3) and (^) are used extensively at
the present time, it is not difficult to show that they are
incorrect when reaction rates are fast and the scale of the turbu-
lence is large. This may be done by considering the proper forms
of Eqs. (1) and (2) when they are averaged. The well-known results
are*
and
DC
DT
DC
To demonstrate the character of these equations, let us discuss
them under the assumption that k'= k' = 0 . Equations (5) and
(6) then reduce to .
DC
a _ 57 (r P"
and
Dt
DC
DF
o:Cp)
(8)
It is clear from these equations that, if one wishes to calculate
the reaction of a with P , it will bep necessary to have an
equation for the second-order correlation CCi unless one can
C'Ci «
show that
conditions required for
for the particular flow in question. The
can be derived in the follow-
C 'C '
er p
« (J CQ
ct p
ing way. First, by following the method used by Reynolds for the
derivation of the equation for the turbulent stress tensor, one
finds the following equation for the substantive derivative of the
correlation C'Cl :**
a p
- u°
-chem
(9)
* For a discussion of these equations that is related to the
present treatment, reference should be made to O'Brien [1] which
was published after this work on the modeling of chemically
reacting turbulent flows was started.
** For the purposes of this illustrative discussion, the flow is
treated as incompressible.
T The notation is that of general tensor analyses.
contravariant form of the metric tensor gmn .
mn
is the
-------
256 Donaldson and Hilst: Chemical Reactions and Turbulent Mixing
where the term (DC 'CA/Dt )cnem
is the contribution of chemical
kinetics alone to the substantive derivative of
C'C
a
This
expression can be found from Eqs . (1) and (2), and is
DCaCP
K
Dt
2
'
2
'
It is instructive to discuss the behavior of the correlation
for the case of turbulent reactions in the absence of any
appreciably -large gradients. In this case, Eq. (9) becomes
chem
(11)
The second term on the right of Eq . (11) is the destruction of the
correlation C^CX by the action of molecular diffusion. In line
with our previous work [5]5 we will model this term by means of a
diffusion scale length X so that Eq. (11) becomes
Dt
chem
C'C'
a P
(12)
The diffusion term in this equation is such that C'C'
approach zero with a characteristic time that is
tends to
diff
What is the overall effect of the first term on the right-hand side
of Eq. (12)? The effect is difficult to see from an inspection of
Eq . (11) , but we may derive an expression for what this term _
accomplishes from Eqs. (7) and (8). First, multiply (7) by Co and
(8) by C/a and then add the resulting equations. The result is
chem
This equation can be interpreted by saying that the effect of chem
istry alone is to drive CaCft to the negative of C^CL (or C/CA
to the negative of cCo) with a characteristic time
Equation (14) s_tates that the reaction between a and p will
always stop, i.e., C.,C + C'C' will become zero, short of the
\Jf p Up
exhaustion of a or P unless a and P are perfectly mixed wherever
they occur in the turbulent flow under consideration. The physical
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Donaldson and Hilst: Chemical Reactions and Turbulent Mixing 257
reason for this is that, in the absence of diffusion, if a and p
are not perfectly mixed to start with, the final state of the gas
in any volume element will be a and products, P and products,
a alone, or P alone, but never any region containing both a and
P . It is easy to see that, no matter what the values taken on by
C and Cg are as a function of time, if Ca is never nonzero
when Co is nonzero and vice versa_ so that no reaction is possible,
it is mathematically true that C CQ + C'C' = 0 . . Thus, Eqs. (7)
dp Cl p
and (8) state that no reactions are possible as required by the
physics of the problem.
An actual example may make the meaning of C^CJ more clear.
Consider that the flow of material by a given point is such that
alternate blobs of a and p pass the point. Let us suppose that
half the time the flow is all a and half the time it is all p .
The resulting concentrations are sketched in Figure 1. If this
pattern keeps repeating, the average values of CQ and C_ are
obviously C~a= 1/2 and C~6= 1/2 . Whenever the flow is all a ,
C =+1/2 and C =-1/2. Whenever the flow is all P , C =-1/2
and C' = +1/2 . We find then that the average value of C'C'
p _ _ __ a p
must be C'C' =-1/4 . Since C'C' = C C0 , no reaction is possible
a p a p a p ' ^
according to Eqs. (7) and (8). and obviously no reaction should
occur.
THE EFFECT OF SCALE LENGTH
We may now return to Eq. (12). If, in this equation, the
scale X is small enough and the reaction rates are slow enough,
the second term on the right-hand side of the equation will be
dominant and the flow will be such that C'C^ is always almost
zero. This means that molecular diffusion is always fast enough
to keep the two species well mixed. On the other hand, if the'
reaction rates are very fast and X is very large, the first term
on the right-hand side of Eq. (12) will be dominant and C'C' will
tend to be approximately equal to -C CR and the two species will be
poorly mixed. The rate of removal from the flow of a and p by
reaction will then not be governed by reaction rates but will be
limited by molecular diffusion. To put these, notions into quanti-
tative form, let us consider the ratio of the two characteristic
times
N = ;
and a contact index
C C_ + C'C'
I = a P _ a P
C CQ
a P
We note that if N is much smaller than one, diffusion will be
very rapid and the two species a and P will be in intimate
-------
258 Donaldson and Hilst: Chemical Reactions and Turbulent Mixing
contact with each other. In this case C'C'/C CL. will be small
u. p up
and the contact index will approach one. If, on the other hand,
N is much larger than one, mixing will be poor and C'C' will
ct p
approach -C Cg . The contact index will then approach zero.
In this case The reaction will be diffusion-limited.
In many laboratory flows, the dissipative or diffusive scale
of turbulence is very small and N is, indeed, small so that the
neglect of C'CA in the kinetic equations is permissible. On the
other hand, if the laboratory experiment is just increased in size,
holding all other parameters such as velocity, temperature, etc.,
constant, one soon finds that the character of the flow changes.
This may be seen by examining the expression for the dimensionless
quantity N in more detail.
Let us assume the diffusive scale of a turbulent flow is of
the order of the dissipative scale so that we may relate X to the
integral scale length of the turbulence A, by (cf. Ref. [5])
bpqA1/u) (18)
where a and b are constants and pqA,/|i is the turbulent
Reynolds number. Substituting this expression into Eq. (16) gives
AiN% * kA) U9)
For relatively high Reynolds numbers, this expression becomes
If an experiment is performed in the laboratory and a value of N
for this experiment is determined or estimated and is found to be
small compared to one, then we know that the diffusive mixing of
the flow is such that the species a and (3 are in contact. The
reaction rate of these species is then chemically controlled. Now
if the apparatus is just scaled up in size, all other things being
equal, N will increase linearly with size since the scale A
increases linearly with the size of the apparatus. When the scale
has been increased sufficiently, so that N is no longer very
small compared to one, the nature of the flow in the device must
change, for the species a and p will no longer be in intimate
contact, at equivalent positions in the apparatus.
The turbulent atmospheric boundary layer is- a good example of
a flow in which it is essential to keep track of the correlation
Cg-Ci if one is to be able to make sense of the reaction of species
which are introduced into the flow. To demonstrate this, we list
in the table some of the second-order reactions responsible for the
production of photochemical smog. We have also listed in this
table the reaction rate recommended for each reaction [4] and an
estimate of the number N for each reaction if it occurs in the .
atmospheric boundary layer where a typical value for X is 10
centimeters. It is interesting to note that it is, in general,
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Donaldson and Hilst: Chemical Reactions and Turbulent Mixing 259
those reactions listed in the table for which N is greater than
one that investigators have found to proceed more slowly than pre-
dicted by formulas such as Eqs. (3) and (4) when the reaction rate
determined from laboratory experiments is used. This difficulty
has led some investigators to search for other chemical reactions
that might be considered which would explain this discrepancy.
•CONCLUSION
It certainly appears unwise to follow this course until such
time as one has at least developed a viable scheme for properly
computing turbulent reacting flows. It is the authors' opinion
that an acceptable method of computing such flows can be developed
through the use of second-order correlation equations such as
Eqs. (9) and (10). Methods of modeling the third-order correlations
that appear in these equations can be found that are similar to
those used to study the generation of turbulence and turbulent
transport [5]. The development of a viable method for computing
chemically reacting turbulent flows according to such a scheme is
under active development by the authors. It is important to note
in this connection that it is essential in developing this general
method to consider fluctuations in density and in the reaction rate
constants when the chemical rate equations are considered.
NOMENCLATURE
a, b = constants
C., CJ = concentration of subscript species, expressed as a
mass fraction
& = molecular diffusion coefficient
I = contact index (Eq. (17))
k, , k« = chemical reaction rate constants
N = dimensionless ratio of characteristic times for mole-
cular diffusion and chemical reaction
q = rms value of turbulent kinetic energy
A, = integral scale length of turbulence
X = dissipative scale length
M. = viscosity
p = density of the fluid
T ; = characteristic time
': REFERENCES
1. O'Brien, Edward E. Turbulent Mixing of Two Rapidly Reacting
Chemical Species, Physics of Fluids, 1971, 11(7), 1326-1331
2. Donaldson, Coleman duP. and Hilst, Glenn R. The Effect of
Inhomogeneous Mixing on Atmospheric Photochemical Reactions.
Submitted to Environmental Science and Technology, 1972.
3. Toor, H.L. Mass Transfer in Dilute Turbulent and Nonturbulent
Systems with Rapid Irreversible Reactions and Equal D.lf fusivlty.
J.Amer.Inst. Chem. Eng., 1962, 8, 70-78.
-------
260 Donaldson and Hilst: Chemical Reactions and Turbulent Mixing
*). Worley, Frank W. "Report on Mathematical Modeling of Photo-
chemical Smog," paper presented at Panel on Modeling, NATA/CCMS
Pilot Project on Air Pollution, Paris, July 1971.
5. Donaldson, Coleman duP. and Rosenbaum, Harold. "Calculation of
Turbulent Shear Plows Through Closure of the Reynolds Equations
by Invariant Modeling," presented at NASA Symposium on
Compressible Turbulent Boundary Layers, Hampton, Virginia, •
December 1968 and published in NASA SP-216, pp. 231-253.
Some Second-Order Reactions Responsible for Photochemical Smog
Reaction k (ppm-sec)~ N
0 + NO = N02 + 02 * 8.3 x 10"^ 0.25 *
N02 + 03 = NO, + 02 1.7 x 10~5 5.0 x 10~3
N03 + NO = 2N02 4.8 1.1 x 103
NO + H02 = N02 + OH 1.7 x 10"1 50.0
OH + 03 = H02 + 02 1.7 . 5.0 x 102
OH + CO = H + C02 5.0 X 10~2 1.5 y 102
CH 02 + NO = CH30 + N02 1.7 5-0 x 102
C2H302 + NO = C2H30 + N02 1.7 5-0 x 102
C2H402 + NO = CH3CHO + N02 1.7 5.0 x 102 • .
CH30 + 02 = HCHO + H02 1.7 5.0 x 102
C0H,- + 0 = CH, + C.H..O 6.0 x lO'1 1.8 x 102
jo J ^ J
C3Hg + 03 = HCHO + C2H1|02 8.3 x 10~3 2.5
C..H, + 0_ = CH00 + C0H00 1.7 x 10~2 5.0
3o 2 3 ^3
C3Hg + H02 = CH 0 + CH3CHO 3.4 x 10~2 10.0
C2H30 + M = CH3 -I- CO + M 1.7 X 10'1 . 50.0
*Those reactions for which N is small compared to one are those
which can be treated using mean quantities in the basic equations
of chemical change, i.e., correlations in fluctuating quantities
may be neglected.
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DonaJdson and Hilst: Chemical Reactions and Turbulent Mixing 261
ca\ ii - •• g/3u '
r'-4. '
Ca"1" 2
c' — i.
P 2
/ s+l] / =+l
«i-|j «j9-i
^'^
«i-i
- - I
Figure 1. Simple problem illustrating
reactions are possible
a P
= -C CD when no
a P
------- |