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L. A. Farrow and T. E.
Graedel
Steady State Approximations and Urban Atmospheric Chemistry L. A. Farrow” and T. E. Graedel Bell Laboratories, Murray Hili, New Jersey 07974 (Received May 2, 1977) Publication costs assisfed by Bell Laboratories
A computer simulation of the urban troposphere in northern New Jersey based on a chemistry of 143 reactions in 76 species has been performed without using any steady state approximations. After the fact, certain steady state relationships have been tested. The “photostationary state” relationship among NO, NOz, and O3is found to be valid, but the free-radicalHO-, which plays a vital part in the photochemistry of the troposphere, is shown to go in and out of the stationary state over the diurnal cycle. Thus the existence of the steady state can be corroborated only by reference to the complete solution. The approximate solution is not supportable by itself and should be avoided wherever possible in doing chemical kinetic calculations. 1. Introduction in this area are from west to east, and the effect of wind has been included as bulk transport from county to county The differential equations that describe the kinetics of of all species. a chemical system invariably exhibit a type of behavior As is shown in Figure 1, three additional counties are commonly known as stiffness. This phenomenon occurs present in the computational matrix. These are fictional when the time derivative of a species is the small difference counties, added so that a study could be made of the of two large numbers which represent the close balance atmosphere downwind of a dense urban area without the between creation and destruction of that particular species. actual meteorological complication of simulating the New For many well-known adaptive integrating rules which are York metropolitan area. As has been described in another widely used on digital computers, stiffness leads to an p~blication,~ these three fictitious counties are successively instability such that a very small time step is set by the more rural as is indicated by their names: Shopview, integrating scheme over a very long portion of the total Deerfield, and Farmland. In the first three counties, the integration interval. Since systems of the order of a t least anthropogenic emissions of such contaminants as oxides 100 chemical equations are often found to be necessary to of nitrogen and hydrocarbons have been simulated acdescribe a process of interest, the short time step usually cording to county by county data supplied by the N. J. renders the computing time prohibitive. Department of Environmental Protection. The three The need to obviate this difficulty has inspired two fictitious counties have had their emissions adjusted acdifferent approaches. The first of these makes the apcordingly. proximation that the aforementioned small difference of Since three counties are real, and since actual physical the two large numbers appearing in the expression for the conditions such as mixing height, emissions, sunlight, and derivative is actually zero. Thus the differential equation wind direction and velocity have been simulated as closely is converted to an algebraic equation and the stiffness is as possible, comparison with field data was capable of removed as far as the integrator is concerned, usually with being made for such major photochemical constituents as a corresponding saving of computer time. The difficulties ozone, NOz, and NO. Agreement was found to be good3 inherent in this method, known as the quasi-steady state as to diurnal pattern and peak magnitudes; so it is felt that approximation (QSSA),have been dealt with extensively the total computer study contains the essential features e1sewhere.l The newer approach consists of construction of atmospheric chemistry in urban areas. of an integration scheme, which, while dealing with the In addition to the simulation reflecting the actual system of stiff differential equations with no approxiemissions, it was decided to vary the emissions in all six mations, maintains a reasonable step size such that recounties so as to assess the effect on the total chemistry quired computer time is no longer excessive. A t the of possible changes in vehicular and industrial activity present time, the most successful such algorithm is due to which would result in increasing or decreasing oxide of Gear,2and it has been the basis for a full implementation nitrogen and/or hydrocarbon emissions. In all, seven such of a chemical kinetics code which has been applied to a variational studies in each county have been carried out. simulation of the contaminated urban troposphere in Once this work was completed using no a priori steady northern New Jersey. state assumptions, it was then possible to test a posteriori As has been described in detail el~ewhere,~ the chemical to see if any commonly used steady state assumptions hold. basis for this simulation is a set of 143 reactions in 76 The most important of these is the so-called photospecies. The computation represents the atmospheric stationary state among ozone, NO, and NO2. In addition, conditions on sunny summer days in three counties of since it is usually assumed that free radicals are in a steady northern New Jersey: Morris, Essex, and Hudson. Morris state, HO. was chosen for scrutiny, since it plays an imis the most rural of the counties, Essex more urban, and portant part in urban chemistry as the main mechanism Hudson the most urban, with population density, vehicular for attack on reactive hydrocarbons.6 traffic, and industrial concentration increasing from west (Morris) to east (Hudson). The counties are represented 2. The Photostationary S t a t e by the geographical matrix shown in Figure 1,where each The formation of ozone starting with photodissociation county has been rectangularized with an area equal to its of NOz and its destruction by NO is embodied in the actual area. The volume in which the chemistry takes well-known trio of equations: place is determined in the vertical dimension by the hv meteorological mixing height; this height has a diurnal NO,NO + 0 (1) variation which is included in the calculation according to (M) 0 + 0,0, (2) the best experimental measurements available. Complete mixing is assumed within the volume. Prevailing winds NO t 0,-NO, + 0 , (3) The Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
Urban Atmospheric Chemistry 0.4
WIND ____L_
Dl RECTlON 0.3
0.2
h
0.1
E 30
0
60 90 120 DOWNWIND DISTANCE ( K M )
150
Figure 1. Geographical matrix of counties used in the computer simulation of the troposphere of northern New Jersey.
E
a a
0.0
0
v
E- -0.1 irl
PHOTOSTATIONARY STATE
d
a: 65 -0.2
A = HUDSON COCNTY 60
-0,:
0 = FARMLAND C O U N T Y
t
2
55 P
a
.-e
;I
I
50
5 0
z E
w b
*
Lr. 45
40
35 0 4
4
6
08
klNO./NO
I .2
in min-'
Flgure 2. [O,] vs. k,[NO,]/[NO] for all counties for all emission variations. The left vertical axis pertains to the left set of points representing concentrations at approximately 7 a.m. The right vertical axis pertains to the right set of points representing conditions at approximately 3 p.m. The solid lines are the least-squares fits to the points in each case, and the dashed lines pass through the origin with a slope of l l k 3 .
The stationary state assumption can be applied to NO, using these three equations, or, alternatively, it can be applied to 0 and 03 to yield two equations between which 0 can be eliminated. In either case, the result is the simple relationship
(4) where kD is the photodissociation time dependent rate constant of (1)and k3 the rate constant corresponding to (3). This equation ignores the large number of other source and sink reactions for these species in urban atmospheres, relying on the ability of the three reactions to dominate the species concentration behavior. At first glance, it would seem that the validity of this equation can be tested by plotting ozone vs. the ratio [NO,]/[NO] for all of the counties of Figure 1 for the different emission variations; the slope should be simply ~. kD varies over the ratio of rate constants k ~ ) / kHowever,
10
8
12
14
16
18
20
22
I
HOUR OF DAY
a
0
0
I
I
-0.'
Figure 3. Rates of formation and destruction of the HO. radical over a diurnal cycle for Morris County; New Jersey. The reaction numbers correspond to those in the text.
the day as the sun rises and sets; hence it is necessary to plot ozone vs. ~ D [ N O ~ ] / [ N and O ] expect a slope of l/k3. This has been done for two selected times of day. The first is around 7 a.m., when NO has a peak in its concentration due to the morning vehicular rush hour; the second is in the late afternoon, around 3 p.m., when ozone has its peak concentration. In both cases, the exact time varies because of emission pattern and transport of ozone from county to county, and this variation has been taken into account in assigning the value of kD used to construct the plot. Figure 2 shows the result, with the afternoon data to the right corresponding to the right vertical axis. In each case, a solid line has been drawn through the points representing the least-squares fit; the dashed curve passes through zero and has a slope equal to l / k 3 . It is clear that the steady state assumption underlying eq 4 is in good agreement with the calculation done without this assumption. This is in accord with the experimental observations of Stedman and Jackson.6 Calvert' also tested the photostationary relationship against experimental data and found reasonably good agreement except in the early morning hours. This discrepancy was ascribed to poor mixing in this time interval, whereas the present calculation assumes perfect mixing a t all times.
3. The Free Radical HO. Since free radicals have a tendency to react very quickly following their production, their corresponding chemical kinetic equations tend to fall most often into the stiff category. Figure 3 shows a linear plot over a diurnal cycle of the rates of formation and destruction of the HO. radical for Morris County. The most important formation reactions are seen to be NO
+ HO,. hu
+
NO,
NO
"0,
--+
O('D)
+ H,O
-+
+ HO.
+ HO. HO. + HO.
(5)
(6) (7)
The most important destruction reactions are seen to be The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
L. A. Farrow and T. E. Graedel
STATIONARY STATE OF HO' -
(8b)], gives
d[ H O * ] / d t = k5 [NO][ HO,. ] ks[C3H6][HO.] 0 (10) where k5 is the rate constant corresponding to (5) and h8 is the sum of the rate constants for (8a) and (8b). A rearrangement of eq 10 gives
-
i -
G=
t= -= )i=
o=
0
/"
100
0
200
300
$00
500
600
700
I
HO ' / H O '
Figure 4. [C,H,]/[NO] vs. [HOp.]/[HO.] for all counties and all emission variations at approximately 7 a.m. The solid line is a least-squares fit to the points and the dashed line passes through the origin with slope kS/ks.
STATIONARY STATE OF HO' 30
20
0
z \ T V
10
NOO$ 0
0
100
ZOO
300
400
500
600
700
HO2'/HO'
Figure 5. Same as Figure 4 at noon.
C3H6 I HO.
-
CO t HO.
CO, t H
+
k5 IHO2.1 (11) [NO] - k 8 [HO.] Thus a plot of [C,H,]/[NO] vs. [HO,.]/[HO.] should be a straight line passing through the origin and having a slope of k 5 / k 8 . Since neither rate constant is time dependent, identical results should be derived at any time of day. Such a plot is shown in Figure 4 for the time of the NO peak, around 7 a.m. As in Figure 2, the solid line represents a least-squares fit to the points for the various counties and emission variations and the dashed line is the theoretical one with slope k5/k8passing through the origin. Agreement is seen to be moderately good. Figure 5 shows a similar plot at noon. It is immediately obvious that agreement is poor, and that the steady state approximation represented by eq 10 and 11does not hold. This is confirmed by noting on Figure 3 that a t noon the sum of the rates for reactions 8a and 8b are greater than the rate for (51, in contradiction to eq 10, whereas at 7 a.m. the sum of the first two rates does indeed equal the rate of ( 5 ) , in agreement with eq 10 and Figure 4. [C,H,I --
products
(8% b) (9)
This ordering of the rate magnitudes holds for all cases studied. A typical QSSA formulation, written by including only the major source and sink equations for HO. [ ( 5 ) ,(8a), and The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
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4. Discussion
The mixed results of our examination of atmospheric steady state conditions support previous investigations of the inappropriateness of QSSA for large chemical kinetic calculations. The photostationary state relationship among NO, NOz, and O3 is found to hold at both times examined, but the free radical HO. is shown to go in and out of the stationary state at different points of the diurnal cycle. It is obvious that a necessary condition for QSSA to approach validity is that all significant source and sink terms be included in the formulation. The increasing use of very large reaction sets, however, often makes such a selection difficult. The alternate approach, that of including euery source and sink term in the QSSA equations soon leads to the formation of complex algebraic constructions whose solutions impose computational difficulties equal to or surpassing those of the stiff differential equations themselves. The result for the HO. radical illustrates the danger in using the steady state approximation a priori without having the complete solution available for corroboration. At one time of day, the approximation is shown to be reasonably valid, but a t another time in the same calculation it ceases to be so. It might be possible to restore the validity by including more of the reactions of Figure 3 in eq 10, thus replacing the differential equation for HO. by a system of algebraic equations. A complete formulation of the HO. QSSA relationship for our computation would involve 14 source and 18 sink terms. Again, however, it would only be possible to test the applicability of any such scheme by having the results of the complete solution, thus obviating the need for the steady state approximation. We conclude, therefore, that QSSA introduces errors that cannot be predicted a priori, and should be avoided wherever possible in doing chemical kinetic calculations. References a n d Notes (1) L. A. Farrow and D. Edelson, Int. J . Chern. Kinet , 6, 787 (1974).
Computer Modeling of Photochemical Smog (2) C. W. Gear, ”Numerical Initial Value Problems in Ordinary Differential Equations”,Prentice Hall, Englewood Cliffs, N.J., 1971, Chapter 11. (3) T. E. Graedel, L. A. Farrow, and T. A. Weber, Atmos. Environ., 10, 1095 (1976). (4) T. E. Graedel, L. A. Farrow, and T. A. Weber, Atmos. Environ., in
press. (5) B. J. Finiayson and J. N. Pitts, Jr., Science, 192, 111 (1976). (6) D. H. Stedman and J. 0. Jackson, Int. J . Chem. Kinet. Symp., 1, 493 (1975). (7) J. G. Calvert, Environ. Sci. Technol., 10, 248 (1976).
Discussion W. H. DUEWER (Lawrence Livermore Laboratory). The stationary state assumption for various free radicals has been tested by other modelers and found to be satisfactory for several. The large discrepanciesfound by the authors suggest errors either in the formulation of the model or of the steady state expression. Because only a few terms were included in the steady state approximations used, I suspect the latter. If there is no error (e.g., omitted significant terms) then the significant differences between your model and other models that cause your model not to behave
2483
as a steady state species should be described. In the case of NO/NOz/O, local NO emissions might do this in the evening, but 03 transport is unlikely. The time scales for reaching steady state in that system is =10-100 s. Interbox transport should be very much slower under most circumstances. L. A. FARROW: The more complete discussion in the published paper shows that for HO- the poor agreement at noon can indeed be attributed to the need for inclusion of more terms in the SSA formulation. However, the ability to show this deficiency in terms can only exist because the complete solution, done without any approximations, is available for comparison. In addition once the complete solution is obtained,approximate ones are obviously superfluous. Since the complete solution can be obtained with no sacrifice in computer time, particularly for large systems of equations, and since it gives results without mathematical errors, it should be used for chemical kinetic problems. Upon recalculating the photostationary relationship, I found that taking a mean value for the photodissociativerate for NOzled to an error. When the exact rate corresponding to the time of day is used, the photostationary state is found to hold remarkably well, as is described in the paper submitted for publication.
Photochemical Smog. Rate Parameter Estimates and Computer Simulations Alan C. Baldwin,+ John R. Barker,* David M. Golden, and Dale G. Hendry” Chemistv Laboratory, SRI International, Menlo Park, California 94025 (Received May 12, 1977) Publication costs assisted by the U.S. Environmental Protection Agency
Evaluation of experimental data and estimation techniques have been used to obtain rate coefficients for classes of alkoxyl radical reactions: (1) unimolecular decomposition, (2) reactions with 02,and (3) unimolecular isomerization. The uncertainties in log h (300 K,1 atm air) are estimated to be fO.9, f1.2, and f1.8,respectively, reflecting the current state of knowledge about these reactions. The reaction HOZ + NO2 + M HOzNOz + M is also discussed, and the results of an RRKM calculation are presented; this reaction is shown to be of great potential importance in the urban atmosphere, and experimental data are needed. Using the rate constants obtained in the first portion of this paper, explicit smog mechanisms for n-butane and propene were developed. Several simulations of smog chamber data are presented and the effects of rate constant uncertainties are illustrated. The mechanisms are shown to give good agreement with the experimental data, but several areas of uncertainty remain.
I. Introduction Extensive efforts are underway throughout the world to accurately model atmospheric chemistry. Whether the efforts are directed toward modeling tropospheric or stratospheric phenomena, there is a need t o evaluate existing rate constant data and to estimate unknown rate parameters in a way t h a t is consistent with current knowledge for similar reactions. We are developing an explicit reaction mechanism t o explain data obtained in smog chambers and we are constantly faced with the fact t h a t too few accurate rate data for elementary reactions are available. Since the kinetic data on particular families of reactions ranges from “excellent and extensive” t o “nonexistent”, a variety of methods must be used for evaluation and/or estimation. As a general guiding principle, it has been found t h a t best results are obtained when reactions are treated as “families” rather than as individuals, because only rarely are enough data available on an individual reaction t o make a meaningful evaluation. A second guiding principle often overlooked is that an estimate or evaluation is not useful unless accompanied by a fair assessment of its probable accuracy. Since A factor and activation energy combine t o give a single rate ‘Postdoctoral Research Associate.
constant, we have adopted the method of propagation of errors to estimate the uncertainty in rate constants, based on the probable uncertainties associated with A factor and activation energy. This estimate of uncertainty is very useful since it is a quantitative measure indicating whether an estimated rate constant can be legitimately varied and over what range. In the next three sections, some reactions of alkoxyl radicals will be discussed. Alkoxy1 radicals are important intermediates in photochemical smog as illustrated by the schematic reaction sequence: OH + RH- H,O t R* R*t 0,-aRO, RO, + N O - RO t NO, RO t 0, HO, + carbonyl compound HO, + NO OH + NO, Three of the possible reactions of alkoxyl radicals in photochemical smog will be discussed in sections 11-IV: decomposition, reaction with oxygen, and isomerization. In section V, estimates for the reactions of HOz with NOz will be presented, and in section VI, several simulations of smog chamber data will be presented for comparison t o the experimental data. Our general conclusions are presented in the last section.
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+
The Journal of Physjcal Chemistry, Vol. 8 1, No. 25, 1977
