current research A Dynamic Model of Photochemical Smog S. K. Friedlander and J. H. Seinfeld Division of Chemistry and Chemical Engineering and W. M. Keck Laboratories of Environmental Health Engineering, California Institute of Technology, Pasadena, Calif. 91109
A simplified kinetic scheme is proposed as a model for the photochemical smog reactions. Calculations based on the model lead to concentration dependences on time, similar in form to the experimental results for laboratory reaction chambers. To take into account the effect of atmospheric mixing processes on the chemical reactions, the Lagrangian similarity hypothesis for the diffusion of nonreactive components is extended to reacting species. This leads to a set of ordinary differential equations for the reactive species, of the type describing a chemical reactor of variable volume. As a preliminary example of the application of the model, a calculation is made for a single bimolecular reaction.
T
he United States is currently being divided into about 100 air quality control regions, each one incorporating two or more communities having a common air pollution problem. These regions will have the responsibility for the development of local air quality standards. A major emphasis will be placed on diffusion modeling in the delineation of the regions (Sec’y., HEW, 1968); forecasting the effects of variations in the strength and location of pollutant sources will also depend on the existence of reliable methods for predicting the spread and behavior of pollutant clouds. The object of this paper is to formulate a general dynamic model for photochemical smog. Unlike previous diffusion models, which have been concerned almost exclusively with nonreacting pollutants. the formulation of the model takes into account both the chemical reaction and turbulent mixing aspects of the photochemical smog problem. The need for such a model has been pointed out by Hilst (1967). The considerable quantities of source inventory information and pollutant concentration data currently being gathered in many urban areas makes the validation of a model a practical possibility. In addition, the factors which most heavily influence smog formation can be determined by a sensitivity analysis of the model. Finally, a model will enable one to pose and answer questions relating to the control of smog formation. In the first part of the paper, a simplified kinetic mechanism is presented for the formation of photochemical smog from NO and unburned hydrocarbons. In the second part, diffusion models based on the general equation of conservation of species are discussed. The dynamic model is a combination of the transport and chemical kinetic equations, and predicts the behavior of a reacting pollutant cloud. The roles of oxides of sulfur and aerosols are not considered in the present study.
Photochemical Reactions in the Urban Atmosphere Photochemical smog is produced by reactions involving N O and unburned hydrocarbons, introduced into the atmosphere in automobile emissions. The smog formation process is characterized by the oxidation of N O to NOn,the oxidation of unsaturated and aromatic hydrocarbons to aldehydes and ketones, and the formation of O3 and peroxyacyl nitrates (PAN’S). Considerable progress has been made in recent years towards understanding the principal reactions in smog formation (Leighton, 1961; Wayne, 1962; Altshuller and Bufalini, 1965; Stephens, 1966, 1969; Haagen-Smit and Wayne, 1967). In developing a chemical reaction scheme which can be combined with a diffusion model to simulate atmospheric processes, the following points should be kept in mind. The details of the hydrocarbon oxidation process are not known, even for the relatively well defined laboratory systems which have been studied. The complexity of the chemical mixtures present in urban air makes a detailed understanding of the atmospheric processes very difficult. Moreover, the theory of turbulent diffusion with simultaneous chemical reaction is not well understood for homogeneous turbulence, much less for multicomponent reactions in the atmosphere. Finally, the role of the smog aerosol particles in the chemical reaction processes and as condensation centers is not known. For these reasons, it will be useful to begin with a simplified kinetic model which, while not involving a large number of chemical species whose concentrations cannot be measured, permits simulation of the behavior of an irradiated mixture of automobile exhaust. The study of reaction mechanisms for smog formation should include the following steps. First, the compilation of published mechanisms for the photochemical smog reactions and integration of the rate equations for a constant volume batch reactor to produce curves of concentration us. time for the key constitutents; second, the compilation of published experimental data on the irradiation of mixtures typical of automobile exhaust and air, and comparison of the results generated in the first step with the experimental observations; and third, the derivation of new kinetic models to incorporate effects or information which might be lacking in current proposed mechanisms. Following this procedure, a simplified kinetic mechanism for the smog formation process has been developed which possesses the features common to most mechanisms currently proposed. Simpli3ed Kinetic Model It has been established that absorptipn by NO2 of the ultraviolet portions of sunlight (300Ck4000A) results in the followVolume 3, Number 11, Nowmber 1969 1175
ing set of reactions in air (Ford and Endow, 1957; Blacet, Hall, and Leighton, 1962; Pitts, Sharp, and Chan, 1964; Stephens, 1966): NOs+hv*NO+O
+ M 2 0 3 $. M O3 + NO -k NOe + 0,
0
+
0 2
(1) (2)
RH
+ 0 2 R . + products
(4)
+ O3+.products (including R . )
(5)
ka
(2) The propagation and branching steps which follow may or may not involve the nitrogen oxides:
NO
+ R . NO^ + R .
+ R . E products (incl. PAN)
k? >> k4[RH] k s[NO]>> k JRH] Then, [OI = k-4N021 = m02l k?
(7)
(10)
A consequence of assumptions 4 and 5 is that rl = rn = r 3 ,the usual pseudo-steady state attained among NOe, NO, 03,and 0 in the absence of organic vapors. If we assume, in addition, that all the free radicals are in a pseudo-steady state, then
where E is the number of free radicals generated as a result of propagation and branching in reaction 6. This yields
Finally, if it is assumed that: k4/(ki[N02] - (E - l)ks[NO]) is approximately constant, the free radical concentration is [ R . ] = k4’[RH][O] The rate equations are then
mor1 - r6 - r; dt
(c) The collision of a radical with the surface of a particle (which may contribute to particle growth). The rate constants k 4 ,k s ,kg, and k7 depend on the particular hydrocarbon species present. The proposed kinetic model consists of reactions 1-7. Summarizing, the following assumptions have been made in this mechanism: 1. The principal initiation step is reaction 4. 2. Reaction 6 embodies all the propagation steps. (Actually considerable branching, which is not represented in reaction 6, takes place. This will be accounted for shortly.) 1176 Environmental Science & Technology
Prom the orders of magnitude of kz, k3, k4, kc9one can make the further assumptions that
(6)
where a reaction of this type occurs for many different radical species R . with different values of kg.R . indicates that another free radical species is a product of the reaction. (3) The radicals are lost by chain terminating steps. As suggested by Haagen-Smit and Wayne, these may be of three main types : (a) Reaction of two free radicals (unlikely at atmospheric concentrations) (b) The reaction of a free radical with NO or NOn. In the latter case, one product is PAN NOn
(9)
(3)
where kl = aka, with k , the rate of absorption of ultraviolet by NO2 (per unit of NOz concentration), @ the ratio of molecules dissociating to molecules absorbing (the quantum yield), and M a third body. When a reactive hydrocarbon is added to the system, many additional reactions take place producing such characteristic products as formaldehyde, acetaldehyde, acrolein, PAN, and alkyl nitrites. Olefins and aromatics are most reactive, but saturates (paraffins and naphthenes), although less reactive than either unsaturates or aromatics, may also play a significant role. It is generally agreed that hydrocarbon oxidation plays a key role in the conversion of NO to NOi. However, current mechanisms advanced to explain the conversion of NO to NOz in the presence of hydrocarbons predict rates of hydrocarbon consumption two to ten times less than observed (Schuck, Doyle, and Endow, 1960; Leighton, 1961, p. 264). A number of detailed mechanisms have been suggested for the hydrocarbon photooxidation process (Haagen-Smit and Wayne, 1967). While all are considered highly speculative, they have in common a chain-like behavior: (1) In the initiation step, hydrocarbon molecules react with atomic oxygen to give oxygen-containing free radicals and other products. In later stages, hydrocarbons also react with ozone to give a variety of products. If we represent any free radical species by R . , then these two reactions can be written: RH
3. The principal termination step is reaction 7. If it is assumed that 0 and O3 are in a pseudo-steady state (Stephens, 1966):
which, in terms of [NO,],[NO], and [RH] become
(1 3)
integrated numerically, to give the results shown in Figures 1 and 2. The parameter values CY = 0.1 p.p.m.-2 min.-1, 0 = 1.83 X p.p.m.-' min.-', F = 2.45 X lo-* min.-l and X = 0.02 p.p.m.-2 min.-l were used. These were based on rate coefficients given by Leighton (1961), but were adjusted to simulate the experimental observations. Figure 1 represents the dynamics for [NO210 = 0.2 p.p.m., [NO10 = l.Op.p.m.,and [RHIO= 2.0 p.p.m. Figure 2 represents the case of [NO&, = 0.2 p.p.m., [NO10 = 0.68 p.p.m. and [RHIO= 1.15 p.p.m. The shapes of the curves are in general agreement with those found in irradiation chamber experiments. General Equarion of Diffusion and Chemical Reaction The calculations of the previous section apply to a constant volume, batch chemical reactor. In the formation of smog over an urban basin area, the transport and reaction of the gaseous pollutants are further influenced by the rate of introduction of each species into the atmosphere, the locations of the sources of pollution in the city, the wind speed and turbulent mixing, and the temperature and intensity of the solar radiation. The spatial and time history of all chemical contaminants could, in principle, be obtained by solving the equation of conservation of mass for each species dC*
at
+ u'
.V
C.I
= V
DiVcf
+ ri
(i
=
1,2,....,n)
1
t
,
'7
1
11
TIME (MIN)
Figure 1. Pollutant concentrations us. time, based on simplified kinetic model for photochemical smog reactions-constant volume (0) = 0.2, CNO(0) = 1.0, CRR(0) = 2.0 p.p.m. batch reactor. CKO?
I
.
(24)
where cf = concentration of species i, Z = wind velocity vector, Df = molecular diffusivity for species i in the atmosphere, and rt = rate of generation of species i by homogeneous chemical reaction. (Molecular diffusion cross coefficients have been neglected.) In the case of a turbulent flow with a surface source whose strength is a function only of time and distance in the wind direction, x, these equations become
8C
IO0
I20
140
160
180
TIME ( M I N )
Figure 2. Pollutant concentrations us. time, based on simplified kinetic model for photochemical smog reactions-constant volume batch reactor. Cxo2 (0) = 0.2, CKO(0) = 0.68, CRB(0) = 1.15 p.p.m.
with the usual simplifying assumptions, including the neglect of diffusion in the wind direction compared with convective transport (Sutton, 1953). Atmospheric transport models have generally been concerned with inert species (such as CO) for which rt = 0. Solutions based on different assumptions concerning the wind speed and eddy diffusivity are reviewed by Sutton. The solution of even a moderate number of simultaneous, nonlinear, partial differential equations of the type Eq. 24 or 25 represents a formidable task for even the largest of modern computers, if, indeed, a solution can be obtained at all. It is thus desirable to simplify the problem, taking care to retain the essential aspects of the physical situation. These simplifying considerations fall into two categories : First, the identification of the principal reacting species from among the large number of reactants, intermediates, and products actually present (this has already been done in a previous section) ; second, the identification of meteorological situations in which the general conservation equation can be simplified. This can be done as follows: Consider the case of an instantaneous point source at height h above ground level, from which a particle of contaminant is released into the turbulent atmospheric surface layer. If the experiment is repeated many times, the mean position of the particle at any time t is f ( r ) , ~ ( t = ) 0, i(t) where z(t) is measured in the direction downwind from the source and i(t) is the height above ground level. According to the Lagrangian
similarity hypothesis (Batchelor, 1964; Gifford, 1962; Cermak, 1963),the average concentration at any point in the cloud is given by the relation
where Q f is the strength of the instantaneous point source. The relation is expected to hold for t of order h/u* or greater where u* is the friction velocity. Let us assume that the same form can be used in the case of a reacting pollutant but with Q a function of time. This assumption appears reasonable if the chemical reaction processes are slow compared with the mixing processes, so that Q is a slowly varying function of time. A more quantitative criterion can be obtained by an examination of the equation of conservation of species:
where the velocity ZR is measured relative to the velocity of the average cloud position, and all distances are measured relative to cloud center. Now the following similarity forms are assumed:
Volume 3, Number 11, November 1969 1177
The function ft depends not only on q2, vu, q 2 but also on the time. Substitution in Eq. 27 gives
_1 dQi _
Q@A (9- A ) 1 di __
- 3\k - ?).v79 +
i dt
P
V',\k
+ V,,
*
G
=
0 (29)
Approximate similarity exists if
1-
partially uncoupled. The set of nonlinear, coupled, second order partial differential equations (Eq. 25) has been reduced to a set of nonlinear, coupled, first order ordinary differential equations suitable for numerical computations. Eq. 34 corresponds to the kinetic expression for a variable volume batch reactor (VVBR) and is related to chemical reactor models often studied in chemical engineering (Denbigh, 1965). The term Q t / i 3is equivalent to the concentration of component i in the reactor and 23 to the reactor volume. The diffusional aspect of the problem appears in the rate at which the reactor volume expands with time; i.e., the dependence of i3 on t. This is the information necessary to find Qt as a function of time. In the general case, for both neutral and diabatic surface layers, Gifford (1962) has proposed the following relation for the average vertical velocity
I _1 d_Q t ' Qt dt ldi
I iz
i
