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Sensitivity analysis of the kinetics of acid rain models
Annospheric Environmenr Vol. 17, No. 3, pp. 64453. Printed in Great Britain.
@304-6981/83/0306409 103.0010 Pergamon Press Ltd.
1983
SENSITIVITY ANALYSIS OF THE KINETICS OF ACID RAIN MODELS D. GOLOMB, S. BATTERMAN,J. GRUHL and W. LABYS* Energy Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A (First
in final form
received 5 April 1982 and
27 Jury 1982)
Abstract-The great number of variables in acid rain modeling makes it difficult to pinpoint those parameters to which the model output (rain acidity) is most sensitive. The approach taken here separates the kinetic and dynamic parts, and analyzes the sensitivity of the kinetic module alone. Further simplifications and linearimtions are introduced, however, the essential steps of the t~sformation processes are believed to be preserved. The major conclusions are: (a) rain acidity is most sensitive to both the oxidation rates of SO2 to SOi- and NO, to NO; ; (b) dry deposition of the emitted gases, but not the formed anions, is important in determining the wet-deposited fraction; (c) wet deposition is much faster than oxidation, and acidic matter is removed very rapidly from the air, but it is the oxidation rate that determines the total amount of acidity in
rain and (d) for similar initial concentrations of SO2 and NO,, nitrate ions may be the predominant species in wet deposition due to the faster oxidation and slower dry deposition of NO, compared to SO,.
INTRODUCTION Air quality modeling (AQM), in general, is a mathematical formulation of the physico-chemical processes occurring in the transport, dispersion and chemical transformation of a pollutant from its source(s) to the receptor(s). An “acid rain model” is a more complicated form of AQM, as the geographic scale between the source and the receptor can reach several hundred, perhaps even several thousand, kilometers. This scale is predicated on the relatively long life time in the atmosphere of the precursors of acid rain: sulfur dioxide and nitrogen oxides. The large geographic and time scales require the modeling of several non-linear functions, including: (1) wind vector variability, (2) variable dispersion coefficients, (3) topography, (4) temperature gradients, (5) chemical transformations and (6) physical transformations, including precipitation. In the literature (Machta and Ferguson, 1981a, b) there exist about twenty models of long-range transport and transformation of air pollutants purporting to describe the emission/pr~ipitation relationship of acid rain. Such models encompass various degrees of complexities. Some have elaborate treatment of the transport-dispersion part of the phenomenon; some use more sophisticated chemical-physical transformation schemes. Because of the large number of variables in acid rain modeling and their high degree of interaction, it is difficult to pinpoint those factors to which the model output (acid concentration in precipitation) is most sensitive. Also, it is important to identify
* Permanent address, College of Mineral and Energy Resources, West Virginia University, Morgantown, WV 26506, U.S.A.
the critical factors where further research may reduce uncertainties. The thrust of this work is to perform a sensitivity analysis which quantifies to what extent the model output changes for given changes of the input parameters, and identifies those parameters where further measurement validation will lead to increased model accuracy. The purpose is not to construct a better model or to select and propose parameters with greater validity or reliability than those presently used. While a set of “nominal values” for the parameters is chosen, these values do not have particular weight or confidence levels. The acid rain literature contains very little in the way of sensitivity analysis. A limited sensitivity analysis was performed by Omstedt and Rodhe (1978) of a onedimensional, time~e~ndent diffusion model for the purpose of bringing the model coefficients in consonance with some empirical data. Freiberg (1978) investigated the sensitivity of sulfate yield to different ratios of “chemistry” and “dispersion.”
CONCEPTUAL APPROACH
This study is limited to the sensitivity analysis of the kinetic module of acid rain modeling. Generally, acid rain models contain a dynamic module which includes advection of the pollutants by winds, dispersion by turbulent diffusion and pr~ipitation climatology. The kinetic module includes chemico-physical transformations such as oxidation and hydrolysis, particle (aerosol) formation, dry deposition and wet deposition. The separation of the kinetic and dynamic parts reduces the dimensionality of the problem to onetime. We adapt from chemical kinetics the concept of 645
D.
646
GOLOMB
the “well-stirred reactor” or “isotropic box.” With this assumption, the reactants in the reactor or box are well mixed, and diffusion and advection (i.e. spatial gradients) play no role. Only the time coordinate determines reactant and product concentrations. In order to reduce the parameters to a manageable number, a highly simplified kinetic module is used for the governing physico-chemical processes; a sort of broad-brush picture. While it is recognized that the actual transformation processes are more complex and manifold, we believe that the simplified kinetic module retains the essential steps in the transformation sequence, and enables the identification of the critical parameters and areas of greatest uncertainty in acid rain modeling. In addition to breaking up the process into relatively few steps, each step is defined as a quasi-first-order (QFO) process with respect to the reactant concentration. This is not a prerequisite for solving the governing chemical kinetic differential equations, but sufficient empirical evidence does not exist to invoke higher-order chemical kinetics. It is also assumed that the concentrations of the ambient reactants, such as hydroxyl, peroxide, ozone, water, etc. are timeindependent, i.e. constant for a particular run. This is not to say that their concentration could not vary, e.g. over a diurnal cycle or different geographic regions. Their variations are reflected in the range of the QFO rate constants for transformation and removal.
et al.
Table 1. Quasi-first-order (QFO) rate constants for the four basic steps in sulfur and nitrogen chemistry Step (1) so, + SOa
(4) so1 -+ so,w
lxh-’ 0.8cm s - 1 0.08 ems-’ 10-b s-1
0.05 1W
(1) (2) (3) (4)
5%h-’ O.l6cms-’ 0.03 cm SC’ lo-“ s-’
15 1 0.02 100
(2) SO; + SO;d (3) ._ so, _ + SO&D NO, -+ NO, NO,-tNO,d NO, -+ NO,D NOB + NO,w
3t
5$
* These are commonly accepted average values (Machta and Ferguson, 198la, b) used in some acid rain models. The listed entries are used as the nominal values in the sensitivity analysis, but without any specific statistical weight to their validity. The sensitivity analysis considers a distribution covering a factor of ten on each side of these nominal values. t For conversion from percent h- ’ to s-l multiply by 1/3600x loo. $ For conversion from ems- ’ to s-‘, we use Garland’s (1978) “residence time” rd = h/u,, with h = 1.5 x lo5 cm. The QFO rate constant is s, = l/t,. 8 Here Garland’s (1978) “time constant” for washout is assumed to be the QFO rate constant.
the greatest number of processes. For example, step (1) may include gas and aqueous phase oxidation as well as catalyzed oxidation on the surface of an aerosol. For the aqueous oxidation (the so-called in-cloud process), additional steps would have to be inserted: soz---,
Sulfur chemistry
QFO s-l x lO-6
Rate*
so,w
(lb)
so,w.
(lc)
followed by
The following simplified scheme is representative of the transformation of emitted sulfur dioxide (SO,) into the final deposition of sulfate ions: Step (I), oxidation
s0 soz hSO‘$ OH,I-LO,
(1)
Step (2), dry deposition
SO2 &SOzd
(2)
Step (3), dry deposition
SO,*bD
(3)
Step (4), wet deposition
SO4 % ra,n
SO,w.
(4)
Here, d (or D) and w stand for dry- and wet-deposited matter, respectively. The QFO rate constants are listed in Table 1. The double negative sign of the sulfate ion and the single negative sign of the nitrate ion is omitted in these steps and subsequent text. A discussion of these steps follows: Oxidation. Gaseous sulfur dioxide SOZ is emitted by a source, and oxidized by atmospheric oxidants such as hydroxyl, peroxide, ozone, etc. into hexavalent sulfur of the sulfate ion SO,+.The latter may be a vapor molecule such as HzS04 or (NH&Sod, the sulfate portion of a condensation nucleus or an aerosol particle. It is perhaps in step (1) that we lump together
so,w---•
A great deal of uncertainty exists at present as to the importance of this route. The work of Schwartz (1982) shows that step (lc) may be indeed fast, provided there is a sufficient supply of oxidant (H,O,, 0,) in the cloud droplet. However, due to the low solubility of SO, (and NO,) in water, step (lb) would be rate determining. Step (lb) may also be pH dependent. In view of these uncertainties, the aqueous phase oxidation route is not considered separately in this sensitivity analysis. Implicitly it is assumed that step (1) includes both homogeneous and heterogeneous oxidation. The transformation of SO2 ---3 SO4 which usually is reported in units of percent per hour refers to the overall oxidation rate. The wide range of quoted rates probably reflects the various atmospheric conditions such as oxidant concentrations and humidity. The conversion factor from percent h-’ to units of s-l is l/3600 x loo. Dry deposition ofSOp. Step (2) accounts for the dry deposition or adsorption of SO2 onto surfaces such as soil, water, vegetation and structures. It is one of the most uncertain parameters in the sulfur cycle modeling because many variables affect vapor adsorption: (1) the meterological conditions, such as wind, turbu-
647
Sensitivity analysis of the kinetics of acid rain models
lence, temperature gradient and boundary layer thickness and (2) the surface conditions, such as character of soil, water roughness, character of vegetation, moisture content, etc. Dry deposition is usually given in units of velocity, cm s- I, defined as vd = F/C
(3
where F is flux in g cm -’ s- ’ and C is concentration gcmm3. Velocity is converted by: sd = v,/h
in (6)
where h is the thickness of the boundary layer. The SO2 concentration is assumed to decrease following an exponential time law C, = C,(l -emsdf), where C, and C, are the concentrations zero.
Nitrogen chemistry
Although nitric acid may constitute 30-40 per cent of the total acidity in rain (Hales, 1982; Perhac, 1981) the nitrogen chemistry is even less certain than that of sulfur. In the nitrogen cycle, sunlight (photochemistry) and organic compounds may play an important role. Nevertheless, for this acid rain model sensitivity analysis, the complex nitrogen chemistry is also represented by four basic steps: Step (la), oxidation
Step @a), dry deposition
(7)
NO, u
NO,d
(24
at time t and Step (3a), dry deposition N03LN0,D
Dry deposition ofSOd.
Step (3) accounts for the dry deposition of the formed sulfate vapor or particles. This process is also fraught with large uncertainties. Generally, it is believed that sulfate losses due to dry deposition are about l/lOth that of SO2 (Garland, 1978). Wet deposition. The incorporation of acidic matter into precipitation can occur basically by two pathways: rainout and washout. Rainout is the absorption of pollutants into cloud droplets or ice crystals; washout is the absorption of pollutants during precipitation, much as the scrubbing action of a shower. Since step (1) includes a lumped gas and aqueous phase oxidation rate, rainout is not considered here separately as a rate determining step. In this analysis washout is the only explicit mechanism whereby acidic matter (sulfate and nitrate) enters precipitation. Washout is also represented here as a quasi-first-order process which assumes that in the presence of rain (aqueous phase), the rate of absorption is linearly dependent on the concentration of sulfate (and nitrate) in the gas phase. In the absence of precipitation, step (4) is inoperative, s, = 0. A similar removal time law as Equation (7) is assumed
C, = C,(l -e-“-‘),
“d
surface
(34
surface
Step (4a), wet deposition NO3 -
%
NO,w.
rain
(4a)
Here NO, is a mixture of NO +N02, NO3 is pentavalent nitrogen, either HN03(N205 +H20) or a component of a particle. NO,d and NO,D are dry deposited molecules and particles, and N03w is nitrate dissolved in a cloud or rain droplet. The conversion from commonly used units of per cent h-’ and cm s-i to QFO rate constants is analogous to the sulfur steps. Since no “hard” literature values could be found for the wet removal rate, we assumed that n, is similar to s,. The appropriate QFO rate constants are also listed in Table 1. Rate equations
We define normalized concentrations in terms of fractions of the initial concentrations. This will allow a sensitivity analysis without reference to actual concentrations or emission levels. Thus,
(8)
where C, and Co are the concentrations at time t and zero, respectively, and s, is the QFO rate constant. Garland (1978) estimates that on the average, sulfate is removed with a time constant of about 10m4s-r, and SO2 with a time constant l/lOth that of S04. This is corroborated by the experiments of Davies (1976) who found that precipitation removes only a small fraction of atmospheric S02. Also, the MAP3S (Hales, 1982) and SURE (Perhac, 1981) precipitation chemistry networks find typically less than 3 per cent of the total dissolved sulfur in precipitation to be absorbed S02. These results lend further justification for not including rainout of SO2 and NO, in this analysis.
6 = 1 -(a+/?+?). The square brackets denote concentrations in arbitrary units, the subscripts t and 0 represent concentrations at time t and zero, respectively; 6 is the fraction of dry deposited sulfur and nitrogen, respectively, in the gaseous and particulate form. The steps (14) and (la+) represent first-order, concurrent, coupled differential equations. Their sol-
D. GOLOMB et al.
648
utions with respect to the fractional concentrations defined above are as, N = exp - (so + s,)t k.N
=
SO[(sO+sd)-(SD+s,)]-l x
YS,N
(9)
[exp - (sn + sw)t - exp - (se + s,)t] (10)
=SOS,[I(SO+Sd)-(SD+SW)]-l
x[l -exp-(sD+sW)t](sD+sw)-’ -[l
-exp-(s,+s,)t](s,+s,)-‘1.
(11)
For obtaining the appropriate values for nitrogen, substitute n’s for the s’s. The conditions for which solutions (9-l 1) apply are: (a) no spatial gradients, i.e. isotropic reactant and product distribution; (b) oxidizing agent concentration is constant and incorporated into the quasi-first-order rate constant; (c) continuous presence of dry deposition substrate; variable deposition rates reflected in choice of dry deposition rate constants and (d) continuous presence of wet deposition substrate, i.e. aqueous phase. Variable wet removal rates are reflected in the choice of QFO rate constants. In the absence of aqueous phase (no precipitation), s, = 0 and y = 0. SENSITIVITY
ANALYSIS
Investigations of the sensitivity analysis of model parameters have taken a number of forms. A generalized theory and method is given by Rabitz (1981). Although many complex designs are available, we are mostly interested in a sensitivity analysis which permits one to decide what aspects of model formulation are most crucial for further research. This approach involves executing the model in one or more of the following modes: (a) Nominal mode, In the nominal mode each uncertain parameter is approximated by a single central value (mean, median or nominal). This is primarily for initial verification of a model.
(d) Probabilistic mode. In the probabilistic mode, a Monte Carlo simulation is used to compute probability distributions for requested parameters. This mode may appear as the most comprehensive for determining sensitivities and uncertainties. However, this mode requires the assumption of parameter independence, and a probable distribution function of individual parameters. A survey of the literature revealed that the parameters are highly interdependent in any particular investigation. Also there are insufficient data which were obtained under similar circumstances to construct a most probable distribution function. Therefore, this mode was not pursued in this study. Nominal mode Sulfur. Figure 1 represents a time plot of the sulfur species using the nominal QFO rate constants in the continuous presence of an aqueous phase (rain). It is seen that the fraction of the emitted SO2 in air (as) falls exponentially with a half-life of approximately 24 h. The sulfate concentration in air increases and reaches a maximum value of only 3 per cent of the emitted SO2 at about 12 h; thereafter declining exponentially. The sulfate concentration in air (/Is) remains very low because it is scavenged by rain as soon as it is formed. Washout is a relatively fast process. In effect, rain is a very efficient cleansing mechanism of the atmosphere. The fraction of sulfate in rain (ys) increases to about 32 per cent at 72 h. The fraction of dry deposited SO2 and Sod(&) increases so that at 72 h about 55 per cent of the sulfur is removed by dry deposition. Figure 2 is a time plot in the absence of an aqueous phase. In this case no ys curve is obtained since s, is set zero. The SO1 decay curve is similar to Fig. 1. The converted SO4 mostly remains in the air, except for the small fraction which is dry-deposited. The sulfate concentration in air reaches a value of 32 per cent of the initial SO* concentration at 72 h. Nitrogen. Figure 3 represents a time plot of the nitrogen species using the nominal QFO rate constants
(b) Sensitivity mode. In the sensitivity mode most parameters are held at nominal values and one or a few at a time are varied over a range to test the sensitivity of a selected outcome variable. The range may be chosen to reflect confidence intervals or plausible extreme values for uncertain parameters with results tabulated or graphed in the order of the magnitude of sensitivities. (c) Multiple mode. In the multiple mode each execution is carried out in parallel for each assigned alternative value for different parameters. Compounding of changes in parameter values also is possible. This provides a convenient form of parametric or factor analysis.
0
IO
20
30
40 t (hours)
50
60
70
60
Fig. 1. Normalized concentrations of SO2 and products vs time using nominal QFO rate constants; continuous presence of rain.
649
Sensitivity analysis of the kinetics of acid rain models
._.
t ( hours
1
1
I
I
I
30
40 t (hours)
I
50
60
70
80
Fig. 4. Normalized concentrations of NO, and products vs time using nominal QFO rate constants; absence of rain, n, = 0, hence yN = 0.
sulfate (note that sulfate is bivalent). Since the dry deposition rates of NO, and NO3 are smaller than for sulfur species, only about 6 per cent of the emitted NO, is dry-deposited. Figure 4 is the time plot of nitrogen species without rain. Contrary to the sulfur case, very little NO, is drydeposited.
I ____--
9
.--
20
1
Fig. 2. Normalized concentrations of SO1 and products vs time using nominal QFO rate constants; absence of rain, s, = 0, hence ys = 0.
IO
SN
IO
___---
Sensitivity mode Sulfur; continuous precipitation. Figure 5 displays graphically the sensitivity of the sulfur chemistry in the continuous presence of rain. We shall call such a display a sensogram. The vertical axis is the logarithm of the normalized ratio of wet-deposited sulfate. The normalization is based on the amount which would be obtained with the nominal QFO rate constants. The horizontal axis is the logarithm of the normalized ratio of the QFO rate constants. The scale from zero to one corresponds to a ten-fold variation of the rate constant from the nominal. One set of plots could be generated for any particular time; the times selected in Figs. 5(a) and (b) are 24 and 48 h after “start” of the reactions, Four curves are obtained, one each for the variation of the four QFO rate-constants of the steps (14). The variation of the dry deposition rate of the formed sulfate has very little effect on the fraction of wet-
‘& 20
30
40 t L hours)
:50
60
70
80
Fig. 3. Normalized concentrations of NO, and products vs time using nominal QFO rate constants; continuous presence of rain. in the presence faster oxidation
of an aqueous phase. Because of the rate of NO, to N03, the curve cq,, is
steeper than the sulfur curve with an NO, half-life of approximately 12 h. Also, the fraction of NO, in rain reaches a higher value of 92 per cent at 72 h. Thus, with equimolar initial concentrations of SO;, and NO,, and with the nominal rate constants, at about 3 days after emission, rain acidity (the hydrogen ions) would be associated 58 per cent with nitrate, and 42 per cent with
=
-I
.8
.6
4
,2
I 0
I .2
normalized
I .4
I .6 rate
I .8+1-.6 constant,
I
I .6
1 .4 log
I .2
I 0
I .2
I .4
I .6
I .8
_-I I
St /sno
Fig. 5. Sensogram of normalized wet deposition of sulfate, ys/yno as a function of normalized QFO rate constants; continuous presence of rain; (a) after 24 h; (b) after 48 h.
650
D.
GOLOMB
deposited sulfate; therefore, the fourth curve labeled sD coincides with the horizontal axes of the sensograms. The curve labeled s,, represents the sensitivity to the variation of the oxidation rate. At 24 h after onset, a lo-fold increase of the oxidation rate over the nominal value would produce 4.6 times as much wet-deposited sulfate. On the other hand, a lo-fold decrease of this rate would deposit 9 times less sulfate. At 48 h after onset, the oxidation rate has a somewhat smaller effect on the wet-deposited fraction with a concomitant larger effect on the dry-deposited fraction. The curve labeled s,, represents the sensitivity to the variation of the gaseous SO2 dry deposition rate constant. An increase of s,, leads to smaller fractions of wet deposition, and vice versa. The sensitivity to increasing s, is not very pronounced; a lo-fold increase of the wet deposition rate over the nominal value would lead only to approximately 10 per cent more wet deposition. However, if the nominal wet deposition rate constant would decrease IO-fold, about 2.5 times less wet deposition would result. The sensitivity of the fraction of wet-deposited sulfate to the rate constants decreases in the order: (1) oxidation, (2) dry deposition of gaseous SO1, (3) wet deposition and (4) dry deposition of S04. Without precipitation. In the absence of an aqueous phase, s, = 0 and ys = 0. Figure 6 plots the sensitivity of the normalized sulfate concentration to the variation of the oxidation and dry deposition rates at 24 h after reaction onset. A lo-fold increase of the oxidation rate will produce 4.3 times as much sulfate in the air; and lo-fold decrease, 0.11 times as much. The sensitivity to the dry deposition rate is less pronounced: a lO-fold increase reduces the sulfate concentration 0.3 times; a lO-fold decrease produces 1.2 times as much sulfate as the nominal value. Delayed precipitation. This sensitivity analysis simulates the case where initially there is no rain, and the pollutants encounter a precipitation event at a later time. Since the nominal value of the wet deposition rate constant is much greater than the oxidation and dry deposition rate constants, it can be assumed that during the precipitation event the sulfate in the air does not change further by oxidation or dry deposition. Sulfate is removed by wet deposition with a rate law analogous to Equation (8), /Is = 1 -exp - (s,J) where the time scale T starts with the onset of precipitation. Figure 7 shows a family of “washout” curves. For the nominal rate s, = 1 x 10e4 s- ‘, 50 per cent of sulfate is removed in 2 h; for 5 times that rate in 0.5 h; for 0.5 times that rate in 4 h. We conclude that for the probable range of washout rates and durations of precipitation events, the total amount of sulfate wet deposition is largely insensitive to the washout rate; however, if s, were much smaller than the nominal value and/or the duration of the precipitation event were short, not all the sulfate present in the air would be wet-deposited. It should be emphasized that the
et al.
P .0 ?
v) .6 Q P .4 a
I -I
8 .6 normollzed
4
2 rate
I I 0 .2 consfont,
I I I 4 .6 .6 log s, /sno
I
Fig. 6. Sensogram of normalized concentrations of sulfate in the air, /?s/&,; absence of rain, s, = 0; at 24 h.
I
0
I 2
I
T
I 4 hours
,
I I 6 6 of precipitation I
,
I, IO
12
Fig. 7. Washout rates of sulfate (“delayed rain”) as a function of the wet deposition rate constant.
time distribution of sulfate (and consequently, acidity) within a precipitation event is highly sensitive to the washout rate s,, and the rate of rainfall. Recall also that rainout is not considered in this analysis. The case of “delayed precipitation” implies that the oxidation occurred entirely in the gas phase. The product (sulfate and sulfuric acid) is washed out by precipitation. Nitrogen The nominal oxidation rate constant for step (la) NO, + NO, is already much larger than the nominal dry deposition rate constant of NO,. Therefore, increasing this rate constant alone will not affect greatly the amount of formed nitrate; however, decreasing this rate constant and simultaneously increasing the dry deposition rate constant would have a significant effect. Altogether, the nitrate output is more sensitive to changes of the rate constants at earlier times. These effects are illustrated in the following figures.
Figures
8(a) and (b) give the sensitivity
of
Sensitivity analysis of the kinetics of acid rain models
651
-.4 0
z
-6
1-
-.8 I
,+ it
-I
.8
I .6
I 4
I .2
I 0
I .2
I
.4
normohzed
I .6
I I 8 +I-.8
rate
constant,
I
I .6
I 4 log
I .2
I 0
I .2
I
.4
I 6
I
.8
-,
I
ni /nno
Fig. 8. Sensogram of normal&d wet deposition of nitrate, yN/yno,as a function of normalized QFO rate constants; continuous rain; (a) after 24 h; (b) after 48 h.
“d
t?-II
I
-I
I
.a .6 normolued
I
I
4
.2 rate
-
I
I
I
I
I
0
2
4
.6
.8
constant. log n,/nno
I I
Fig. 9. Sensogram of normalized concentrations of nitrate in the air; /3N/&,o; absence of rain, n, = 0; at 24 h.
nitrate wet deposition to varying single rate constants. At 24 h, a lo-fold increase of the oxidation rate would produce only about 50 per cent more wet-deposited nitrate; at 48 h, only 14 per cent more. At 12 h (not shown in the Figures), yN/ynowould increase 2.5-fold. However, at 24 h, a decrease of n, by a factor of 10, would decrease yN/y.,, by a factor of 6.3. Figure 9 shows the effect on nitrate formation without precipitation. A lO-fold increase of n, would increase the relative amount of nitrate in the air by only 40 per cent; on the other hand, a lO-fold decrease will reduce the nitrate concentration by a factor of 6. The case of “delayed rain” would be analogous to sulfate wet deposition, as we assumed nw = s,. Multiple
mode
When two rate constants are varied simultaneously, the resulting sensograms are three-dimensional. The single mode analysis showed that the two most critical parameters are the oxidation and dry deposition rate constants. The combined effects of varying these rate constants are examined below. Figures 10(a) and (b) show the effect on sulfate
Fig. 10. Threedimensional sensogram of multimode sensitivity analysis; sulfate in the air; continuous rain; (a) after 24h; (b) after 72 h. Superscript o designates nominal value. Scales explained in text.
concentration in the air when the oxidation and dry deposition rates are varied up to lO-fold from the nominal values. The horizontal axes represent the two rate parameters and the vertical axis scales the sulfate concentration. The center of the surface, marked o, corresponds to the nominal values of both rate constants and has a height of one unit on the vertical scale. Maximum concentration is obtained with the lowest dry deposition rate and an intermediate oxidation rate after 24 h, as seen in Fig. 10(a). The maximum shifts slightly after 72 h of reaction time, shown in Fig. 10(b). Increasing the dry deposition rate over
D.
652
GOLOMB~~~.
Fig. 11. Sulfate in wet deposition; conditions as in
Fig. 13. Nitrate in wet deposition; conditions as in Fig. 10.
Fig. 10.
a.
the nominal rate makes very little difference on sulfate concentrations. Figures 11(a) and (b) show the effect of varying the rates on sulfate concentrations in precipitation. Here no local maxima are observed; the concentration increases continuously as the oxidation rate increases and the dry deposition rate decreases. Figures 12(a)and (b) show theeffect on nitrate in the air of varying the oxidation and dry deposition rates of NO,. In contrast to sulfate, these surfaces show little sensitivity to increasing the oxidation rate over the nominal value. This results since the ratio of the nominal oxidation/dry de~sition rate is very large for NO,. A significant effect is observed when the oxidation rate is decreased simultaneously with increasing dry deposition. Figure 13(a) and (b) show the effect on nitrate in wet deposition. Again, increasing the oxidation rate has very little effect. Also, the effect of dry deposition is much less pronounced than for sulfate wet deposition. An overview of Figs 11(a) and 13(a) is given in Table 2. The table corroborates that wet deposition is most sensitive to lower than nominal oxidation rates and higher than nominal dry deposition rates. With 0.1 times the oxidation rate and 10 times the dry deposition rate, only 3 per cent of the nominal value of sulfate is wet-deposited, and 11 per cent of the nitrate.
CONCLUSIONS
Fig. 12. Nitrate in air; conditions as in Fig. 10.
The sensitivity analysis concerned only the kinetic module of acid precipitation modeling with great
Sensitivity analysis of the kinetics of acid rain models simplification of the physico-chemical processes. However, several important conclusions can be reached in regard to the sensitivity of the model output to the input parameters. These conclusions are summarized as follows. (1) Among the four basic steps of the transformation process considered in this analysis (oxidation, dry deposition of the precursor, dry deposition of the product and wet deposition of the product) the rate of oxidation is the most important factor in determining rain acidity. In order to reduce uncertainties of acid rain modeling, it is imperative to accurately determine the rate of SO,---+ SOi- and NO,---> NO; conversion. There may be large natural variations in the oxidation rate constant depending on oxidant concentrations and ambient conditions (humidity, temperature, etc.). Accurate modeling must include time- and space-variability of the oxidation rate constant. (2) The second most important factor is the dry deposition of the precursor gases. Obviously, what is removed by dry deposition does not appear as wet deposition. Accurate modeling must include the timeand space-variability of the SO* dry deposition rate constant; the NO, dry deposition rate appears to have a lesser effect on nitrate wet deposition because of the fast oxidation rate. (3) The wet deposition rate constant (i.e. the washout coefficient) is important in determining the acidity distribution within a rain episode. The total amount of wet deposition, however, is only slightly affected by this rate constant, provided it rains for a sufficient period to wash out most of the pollutants. Rainout of the precursor gases with subsequent oxidation were not considered as separate steps in this analysis. If the rainout process were rate determining, the distribution of acidity within a rain episode would Table 2. Effect on wet deposition of simultaneously varying two rate constants: oxidation and dry deposition (Time: 24 h) s0ls.o or n&“,
Sd/%0 or n,/n,,
YslY.O
YNh.o
(1)
0.1
(2) (3)
0.1 0.1
1 10
0.13 0.11 0.03
0.16 0.16 0.11
(4)
1
(5) (6)
1 1
0.1 1 10
1.17 1.00 0.32
1.03 1.00 0.77
(7)
10
0.1 1 10
5.2 4.6 2.2
1.5 1.5 1.4
0.1
653
be dependent on the cloud lifetime and cloud characteristics rather than on the washout rate. (4) Nitric acid is an important ingredient of total precipitation acidity, provided the precursor NO, is present in similar quantities as the precursor of sulfuric acid, SOz. Because of the experimentally observed larger oxidation rate constant of NO, ---3 NO; than that of SOz---9 SO:-, the contribution of nitric acid at early times (i.e. closer to the source) may be predominant. Due to the large ratio of the nominal oxidation and dry deposition rate constant of NO,, further increases of the oxidation rate constants will not greatly enhance the nitrate concentrations in precipitation; however, large decreases of this rate will reduce nitrate concentrations significantly.
Acknowledgement-We thank Dr. Neil Goldman of the MIT Energy Laboratory for providing the analytic solutions of the rate equations. The contribution of Stuart Batterman, James Gruhl and Walter Labys was, in part, sponsored by EPRI contract RP1484-1, a project to which William Balson of DFI, Inc. provided some valuable initial assistance.
REFERENCES
Davies T. D. (1976) Precipitation scavenging of sulphur dioxide in an industrial area, Atmospheric Enoironment 10, 879-890. Freiberg J. (1978) Conversion limit and characteristic time of SO, oxidation in plumes. Atmospheric Enoironment 12, 339-347. Garland J. A. (1978) Dry and wet removal of sulphur from the atmosphere. Atmospheric Environment 12, 349-362. Hales J. M. (Ed.) (1982) The MAP3S/RAINE precipitation network: statistical overview. Atmospheric Environment 16, 1603-1631. Machta L. and Ferguson H. (Eds) (1981a) Atmospheric modeling. Interim Report under the U.S./Canada Memorandum of Intent on Transboundary Air Pollution, U.S. National Oceanic and Atmospheric Administration, Silver Spring MD 20910 and Atmospheric Environment Service, Downsview Ontario M3H5T4. Machta L. and Ferguson H. (Eds) (1981b) Atmospheric Sciences Review, Interim Report. (See Machta and Ferguson 1981a). Omstedt G. and Rodhe H. (1978) Transformation and removal processes for sulfur compouhds in the atmosphere as described by a onedimensional timedependent diffusion model. Atmospheric Environment 13, 503-509. Perhac R. M. (Ed.) (1981) EPRI Sulfate Regional Experiment: results and implications, Electric Power Research Institute, Palo Alto CA 94304, U.S.A. Report EA-216%SY-LD. Rabitz H. (1981) Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics. Comput. Chem. 5, 167-180. Schwartz S. E. (1982) Gas-aqueous reactions of sulfur and nitrogen oxides in liquid-water clouds. Paper ENVI-51 presented at the American Chemical Society Acid Rain Symposium, Las Vegas, NV.
@304-6981/83/0306409 103.0010 Pergamon Press Ltd.
1983
SENSITIVITY ANALYSIS OF THE KINETICS OF ACID RAIN MODELS D. GOLOMB, S. BATTERMAN,J. GRUHL and W. LABYS* Energy Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A (First
in final form
received 5 April 1982 and
27 Jury 1982)
Abstract-The great number of variables in acid rain modeling makes it difficult to pinpoint those parameters to which the model output (rain acidity) is most sensitive. The approach taken here separates the kinetic and dynamic parts, and analyzes the sensitivity of the kinetic module alone. Further simplifications and linearimtions are introduced, however, the essential steps of the t~sformation processes are believed to be preserved. The major conclusions are: (a) rain acidity is most sensitive to both the oxidation rates of SO2 to SOi- and NO, to NO; ; (b) dry deposition of the emitted gases, but not the formed anions, is important in determining the wet-deposited fraction; (c) wet deposition is much faster than oxidation, and acidic matter is removed very rapidly from the air, but it is the oxidation rate that determines the total amount of acidity in
rain and (d) for similar initial concentrations of SO2 and NO,, nitrate ions may be the predominant species in wet deposition due to the faster oxidation and slower dry deposition of NO, compared to SO,.
INTRODUCTION Air quality modeling (AQM), in general, is a mathematical formulation of the physico-chemical processes occurring in the transport, dispersion and chemical transformation of a pollutant from its source(s) to the receptor(s). An “acid rain model” is a more complicated form of AQM, as the geographic scale between the source and the receptor can reach several hundred, perhaps even several thousand, kilometers. This scale is predicated on the relatively long life time in the atmosphere of the precursors of acid rain: sulfur dioxide and nitrogen oxides. The large geographic and time scales require the modeling of several non-linear functions, including: (1) wind vector variability, (2) variable dispersion coefficients, (3) topography, (4) temperature gradients, (5) chemical transformations and (6) physical transformations, including precipitation. In the literature (Machta and Ferguson, 1981a, b) there exist about twenty models of long-range transport and transformation of air pollutants purporting to describe the emission/pr~ipitation relationship of acid rain. Such models encompass various degrees of complexities. Some have elaborate treatment of the transport-dispersion part of the phenomenon; some use more sophisticated chemical-physical transformation schemes. Because of the large number of variables in acid rain modeling and their high degree of interaction, it is difficult to pinpoint those factors to which the model output (acid concentration in precipitation) is most sensitive. Also, it is important to identify
* Permanent address, College of Mineral and Energy Resources, West Virginia University, Morgantown, WV 26506, U.S.A.
the critical factors where further research may reduce uncertainties. The thrust of this work is to perform a sensitivity analysis which quantifies to what extent the model output changes for given changes of the input parameters, and identifies those parameters where further measurement validation will lead to increased model accuracy. The purpose is not to construct a better model or to select and propose parameters with greater validity or reliability than those presently used. While a set of “nominal values” for the parameters is chosen, these values do not have particular weight or confidence levels. The acid rain literature contains very little in the way of sensitivity analysis. A limited sensitivity analysis was performed by Omstedt and Rodhe (1978) of a onedimensional, time~e~ndent diffusion model for the purpose of bringing the model coefficients in consonance with some empirical data. Freiberg (1978) investigated the sensitivity of sulfate yield to different ratios of “chemistry” and “dispersion.”
CONCEPTUAL APPROACH
This study is limited to the sensitivity analysis of the kinetic module of acid rain modeling. Generally, acid rain models contain a dynamic module which includes advection of the pollutants by winds, dispersion by turbulent diffusion and pr~ipitation climatology. The kinetic module includes chemico-physical transformations such as oxidation and hydrolysis, particle (aerosol) formation, dry deposition and wet deposition. The separation of the kinetic and dynamic parts reduces the dimensionality of the problem to onetime. We adapt from chemical kinetics the concept of 645
D.
646
GOLOMB
the “well-stirred reactor” or “isotropic box.” With this assumption, the reactants in the reactor or box are well mixed, and diffusion and advection (i.e. spatial gradients) play no role. Only the time coordinate determines reactant and product concentrations. In order to reduce the parameters to a manageable number, a highly simplified kinetic module is used for the governing physico-chemical processes; a sort of broad-brush picture. While it is recognized that the actual transformation processes are more complex and manifold, we believe that the simplified kinetic module retains the essential steps in the transformation sequence, and enables the identification of the critical parameters and areas of greatest uncertainty in acid rain modeling. In addition to breaking up the process into relatively few steps, each step is defined as a quasi-first-order (QFO) process with respect to the reactant concentration. This is not a prerequisite for solving the governing chemical kinetic differential equations, but sufficient empirical evidence does not exist to invoke higher-order chemical kinetics. It is also assumed that the concentrations of the ambient reactants, such as hydroxyl, peroxide, ozone, water, etc. are timeindependent, i.e. constant for a particular run. This is not to say that their concentration could not vary, e.g. over a diurnal cycle or different geographic regions. Their variations are reflected in the range of the QFO rate constants for transformation and removal.
et al.
Table 1. Quasi-first-order (QFO) rate constants for the four basic steps in sulfur and nitrogen chemistry Step (1) so, + SOa
(4) so1 -+ so,w
lxh-’ 0.8cm s - 1 0.08 ems-’ 10-b s-1
0.05 1W
(1) (2) (3) (4)
5%h-’ O.l6cms-’ 0.03 cm SC’ lo-“ s-’
15 1 0.02 100
(2) SO; + SO;d (3) ._ so, _ + SO&D NO, -+ NO, NO,-tNO,d NO, -+ NO,D NOB + NO,w
3t
5$
* These are commonly accepted average values (Machta and Ferguson, 198la, b) used in some acid rain models. The listed entries are used as the nominal values in the sensitivity analysis, but without any specific statistical weight to their validity. The sensitivity analysis considers a distribution covering a factor of ten on each side of these nominal values. t For conversion from percent h- ’ to s-l multiply by 1/3600x loo. $ For conversion from ems- ’ to s-‘, we use Garland’s (1978) “residence time” rd = h/u,, with h = 1.5 x lo5 cm. The QFO rate constant is s, = l/t,. 8 Here Garland’s (1978) “time constant” for washout is assumed to be the QFO rate constant.
the greatest number of processes. For example, step (1) may include gas and aqueous phase oxidation as well as catalyzed oxidation on the surface of an aerosol. For the aqueous oxidation (the so-called in-cloud process), additional steps would have to be inserted: soz---,
Sulfur chemistry
QFO s-l x lO-6
Rate*
so,w
(lb)
so,w.
(lc)
followed by
The following simplified scheme is representative of the transformation of emitted sulfur dioxide (SO,) into the final deposition of sulfate ions: Step (I), oxidation
s0 soz hSO‘$ OH,I-LO,
(1)
Step (2), dry deposition
SO2 &SOzd
(2)
Step (3), dry deposition
SO,*bD
(3)
Step (4), wet deposition
SO4 % ra,n
SO,w.
(4)
Here, d (or D) and w stand for dry- and wet-deposited matter, respectively. The QFO rate constants are listed in Table 1. The double negative sign of the sulfate ion and the single negative sign of the nitrate ion is omitted in these steps and subsequent text. A discussion of these steps follows: Oxidation. Gaseous sulfur dioxide SOZ is emitted by a source, and oxidized by atmospheric oxidants such as hydroxyl, peroxide, ozone, etc. into hexavalent sulfur of the sulfate ion SO,+.The latter may be a vapor molecule such as HzS04 or (NH&Sod, the sulfate portion of a condensation nucleus or an aerosol particle. It is perhaps in step (1) that we lump together
so,w---•
A great deal of uncertainty exists at present as to the importance of this route. The work of Schwartz (1982) shows that step (lc) may be indeed fast, provided there is a sufficient supply of oxidant (H,O,, 0,) in the cloud droplet. However, due to the low solubility of SO, (and NO,) in water, step (lb) would be rate determining. Step (lb) may also be pH dependent. In view of these uncertainties, the aqueous phase oxidation route is not considered separately in this sensitivity analysis. Implicitly it is assumed that step (1) includes both homogeneous and heterogeneous oxidation. The transformation of SO2 ---3 SO4 which usually is reported in units of percent per hour refers to the overall oxidation rate. The wide range of quoted rates probably reflects the various atmospheric conditions such as oxidant concentrations and humidity. The conversion factor from percent h-’ to units of s-l is l/3600 x loo. Dry deposition ofSOp. Step (2) accounts for the dry deposition or adsorption of SO2 onto surfaces such as soil, water, vegetation and structures. It is one of the most uncertain parameters in the sulfur cycle modeling because many variables affect vapor adsorption: (1) the meterological conditions, such as wind, turbu-
647
Sensitivity analysis of the kinetics of acid rain models
lence, temperature gradient and boundary layer thickness and (2) the surface conditions, such as character of soil, water roughness, character of vegetation, moisture content, etc. Dry deposition is usually given in units of velocity, cm s- I, defined as vd = F/C
(3
where F is flux in g cm -’ s- ’ and C is concentration gcmm3. Velocity is converted by: sd = v,/h
in (6)
where h is the thickness of the boundary layer. The SO2 concentration is assumed to decrease following an exponential time law C, = C,(l -emsdf), where C, and C, are the concentrations zero.
Nitrogen chemistry
Although nitric acid may constitute 30-40 per cent of the total acidity in rain (Hales, 1982; Perhac, 1981) the nitrogen chemistry is even less certain than that of sulfur. In the nitrogen cycle, sunlight (photochemistry) and organic compounds may play an important role. Nevertheless, for this acid rain model sensitivity analysis, the complex nitrogen chemistry is also represented by four basic steps: Step (la), oxidation
Step @a), dry deposition
(7)
NO, u
NO,d
(24
at time t and Step (3a), dry deposition N03LN0,D
Dry deposition ofSOd.
Step (3) accounts for the dry deposition of the formed sulfate vapor or particles. This process is also fraught with large uncertainties. Generally, it is believed that sulfate losses due to dry deposition are about l/lOth that of SO2 (Garland, 1978). Wet deposition. The incorporation of acidic matter into precipitation can occur basically by two pathways: rainout and washout. Rainout is the absorption of pollutants into cloud droplets or ice crystals; washout is the absorption of pollutants during precipitation, much as the scrubbing action of a shower. Since step (1) includes a lumped gas and aqueous phase oxidation rate, rainout is not considered here separately as a rate determining step. In this analysis washout is the only explicit mechanism whereby acidic matter (sulfate and nitrate) enters precipitation. Washout is also represented here as a quasi-first-order process which assumes that in the presence of rain (aqueous phase), the rate of absorption is linearly dependent on the concentration of sulfate (and nitrate) in the gas phase. In the absence of precipitation, step (4) is inoperative, s, = 0. A similar removal time law as Equation (7) is assumed
C, = C,(l -e-“-‘),
“d
surface
(34
surface
Step (4a), wet deposition NO3 -
%
NO,w.
rain
(4a)
Here NO, is a mixture of NO +N02, NO3 is pentavalent nitrogen, either HN03(N205 +H20) or a component of a particle. NO,d and NO,D are dry deposited molecules and particles, and N03w is nitrate dissolved in a cloud or rain droplet. The conversion from commonly used units of per cent h-’ and cm s-i to QFO rate constants is analogous to the sulfur steps. Since no “hard” literature values could be found for the wet removal rate, we assumed that n, is similar to s,. The appropriate QFO rate constants are also listed in Table 1. Rate equations
We define normalized concentrations in terms of fractions of the initial concentrations. This will allow a sensitivity analysis without reference to actual concentrations or emission levels. Thus,
(8)
where C, and Co are the concentrations at time t and zero, respectively, and s, is the QFO rate constant. Garland (1978) estimates that on the average, sulfate is removed with a time constant of about 10m4s-r, and SO2 with a time constant l/lOth that of S04. This is corroborated by the experiments of Davies (1976) who found that precipitation removes only a small fraction of atmospheric S02. Also, the MAP3S (Hales, 1982) and SURE (Perhac, 1981) precipitation chemistry networks find typically less than 3 per cent of the total dissolved sulfur in precipitation to be absorbed S02. These results lend further justification for not including rainout of SO2 and NO, in this analysis.
6 = 1 -(a+/?+?). The square brackets denote concentrations in arbitrary units, the subscripts t and 0 represent concentrations at time t and zero, respectively; 6 is the fraction of dry deposited sulfur and nitrogen, respectively, in the gaseous and particulate form. The steps (14) and (la+) represent first-order, concurrent, coupled differential equations. Their sol-
D. GOLOMB et al.
648
utions with respect to the fractional concentrations defined above are as, N = exp - (so + s,)t k.N
=
SO[(sO+sd)-(SD+s,)]-l x
YS,N
(9)
[exp - (sn + sw)t - exp - (se + s,)t] (10)
=SOS,[I(SO+Sd)-(SD+SW)]-l
x[l -exp-(sD+sW)t](sD+sw)-’ -[l
-exp-(s,+s,)t](s,+s,)-‘1.
(11)
For obtaining the appropriate values for nitrogen, substitute n’s for the s’s. The conditions for which solutions (9-l 1) apply are: (a) no spatial gradients, i.e. isotropic reactant and product distribution; (b) oxidizing agent concentration is constant and incorporated into the quasi-first-order rate constant; (c) continuous presence of dry deposition substrate; variable deposition rates reflected in choice of dry deposition rate constants and (d) continuous presence of wet deposition substrate, i.e. aqueous phase. Variable wet removal rates are reflected in the choice of QFO rate constants. In the absence of aqueous phase (no precipitation), s, = 0 and y = 0. SENSITIVITY
ANALYSIS
Investigations of the sensitivity analysis of model parameters have taken a number of forms. A generalized theory and method is given by Rabitz (1981). Although many complex designs are available, we are mostly interested in a sensitivity analysis which permits one to decide what aspects of model formulation are most crucial for further research. This approach involves executing the model in one or more of the following modes: (a) Nominal mode, In the nominal mode each uncertain parameter is approximated by a single central value (mean, median or nominal). This is primarily for initial verification of a model.
(d) Probabilistic mode. In the probabilistic mode, a Monte Carlo simulation is used to compute probability distributions for requested parameters. This mode may appear as the most comprehensive for determining sensitivities and uncertainties. However, this mode requires the assumption of parameter independence, and a probable distribution function of individual parameters. A survey of the literature revealed that the parameters are highly interdependent in any particular investigation. Also there are insufficient data which were obtained under similar circumstances to construct a most probable distribution function. Therefore, this mode was not pursued in this study. Nominal mode Sulfur. Figure 1 represents a time plot of the sulfur species using the nominal QFO rate constants in the continuous presence of an aqueous phase (rain). It is seen that the fraction of the emitted SO2 in air (as) falls exponentially with a half-life of approximately 24 h. The sulfate concentration in air increases and reaches a maximum value of only 3 per cent of the emitted SO2 at about 12 h; thereafter declining exponentially. The sulfate concentration in air (/Is) remains very low because it is scavenged by rain as soon as it is formed. Washout is a relatively fast process. In effect, rain is a very efficient cleansing mechanism of the atmosphere. The fraction of sulfate in rain (ys) increases to about 32 per cent at 72 h. The fraction of dry deposited SO2 and Sod(&) increases so that at 72 h about 55 per cent of the sulfur is removed by dry deposition. Figure 2 is a time plot in the absence of an aqueous phase. In this case no ys curve is obtained since s, is set zero. The SO1 decay curve is similar to Fig. 1. The converted SO4 mostly remains in the air, except for the small fraction which is dry-deposited. The sulfate concentration in air reaches a value of 32 per cent of the initial SO* concentration at 72 h. Nitrogen. Figure 3 represents a time plot of the nitrogen species using the nominal QFO rate constants
(b) Sensitivity mode. In the sensitivity mode most parameters are held at nominal values and one or a few at a time are varied over a range to test the sensitivity of a selected outcome variable. The range may be chosen to reflect confidence intervals or plausible extreme values for uncertain parameters with results tabulated or graphed in the order of the magnitude of sensitivities. (c) Multiple mode. In the multiple mode each execution is carried out in parallel for each assigned alternative value for different parameters. Compounding of changes in parameter values also is possible. This provides a convenient form of parametric or factor analysis.
0
IO
20
30
40 t (hours)
50
60
70
60
Fig. 1. Normalized concentrations of SO2 and products vs time using nominal QFO rate constants; continuous presence of rain.
649
Sensitivity analysis of the kinetics of acid rain models
._.
t ( hours
1
1
I
I
I
30
40 t (hours)
I
50
60
70
80
Fig. 4. Normalized concentrations of NO, and products vs time using nominal QFO rate constants; absence of rain, n, = 0, hence yN = 0.
sulfate (note that sulfate is bivalent). Since the dry deposition rates of NO, and NO3 are smaller than for sulfur species, only about 6 per cent of the emitted NO, is dry-deposited. Figure 4 is the time plot of nitrogen species without rain. Contrary to the sulfur case, very little NO, is drydeposited.
I ____--
9
.--
20
1
Fig. 2. Normalized concentrations of SO1 and products vs time using nominal QFO rate constants; absence of rain, s, = 0, hence ys = 0.
IO
SN
IO
___---
Sensitivity mode Sulfur; continuous precipitation. Figure 5 displays graphically the sensitivity of the sulfur chemistry in the continuous presence of rain. We shall call such a display a sensogram. The vertical axis is the logarithm of the normalized ratio of wet-deposited sulfate. The normalization is based on the amount which would be obtained with the nominal QFO rate constants. The horizontal axis is the logarithm of the normalized ratio of the QFO rate constants. The scale from zero to one corresponds to a ten-fold variation of the rate constant from the nominal. One set of plots could be generated for any particular time; the times selected in Figs. 5(a) and (b) are 24 and 48 h after “start” of the reactions, Four curves are obtained, one each for the variation of the four QFO rate-constants of the steps (14). The variation of the dry deposition rate of the formed sulfate has very little effect on the fraction of wet-
‘& 20
30
40 t L hours)
:50
60
70
80
Fig. 3. Normalized concentrations of NO, and products vs time using nominal QFO rate constants; continuous presence of rain. in the presence faster oxidation
of an aqueous phase. Because of the rate of NO, to N03, the curve cq,, is
steeper than the sulfur curve with an NO, half-life of approximately 12 h. Also, the fraction of NO, in rain reaches a higher value of 92 per cent at 72 h. Thus, with equimolar initial concentrations of SO;, and NO,, and with the nominal rate constants, at about 3 days after emission, rain acidity (the hydrogen ions) would be associated 58 per cent with nitrate, and 42 per cent with
=
-I
.8
.6
4
,2
I 0
I .2
normalized
I .4
I .6 rate
I .8+1-.6 constant,
I
I .6
1 .4 log
I .2
I 0
I .2
I .4
I .6
I .8
_-I I
St /sno
Fig. 5. Sensogram of normalized wet deposition of sulfate, ys/yno as a function of normalized QFO rate constants; continuous presence of rain; (a) after 24 h; (b) after 48 h.
650
D.
GOLOMB
deposited sulfate; therefore, the fourth curve labeled sD coincides with the horizontal axes of the sensograms. The curve labeled s,, represents the sensitivity to the variation of the oxidation rate. At 24 h after onset, a lo-fold increase of the oxidation rate over the nominal value would produce 4.6 times as much wet-deposited sulfate. On the other hand, a lo-fold decrease of this rate would deposit 9 times less sulfate. At 48 h after onset, the oxidation rate has a somewhat smaller effect on the wet-deposited fraction with a concomitant larger effect on the dry-deposited fraction. The curve labeled s,, represents the sensitivity to the variation of the gaseous SO2 dry deposition rate constant. An increase of s,, leads to smaller fractions of wet deposition, and vice versa. The sensitivity to increasing s, is not very pronounced; a lo-fold increase of the wet deposition rate over the nominal value would lead only to approximately 10 per cent more wet deposition. However, if the nominal wet deposition rate constant would decrease IO-fold, about 2.5 times less wet deposition would result. The sensitivity of the fraction of wet-deposited sulfate to the rate constants decreases in the order: (1) oxidation, (2) dry deposition of gaseous SO1, (3) wet deposition and (4) dry deposition of S04. Without precipitation. In the absence of an aqueous phase, s, = 0 and ys = 0. Figure 6 plots the sensitivity of the normalized sulfate concentration to the variation of the oxidation and dry deposition rates at 24 h after reaction onset. A lo-fold increase of the oxidation rate will produce 4.3 times as much sulfate in the air; and lo-fold decrease, 0.11 times as much. The sensitivity to the dry deposition rate is less pronounced: a lO-fold increase reduces the sulfate concentration 0.3 times; a lO-fold decrease produces 1.2 times as much sulfate as the nominal value. Delayed precipitation. This sensitivity analysis simulates the case where initially there is no rain, and the pollutants encounter a precipitation event at a later time. Since the nominal value of the wet deposition rate constant is much greater than the oxidation and dry deposition rate constants, it can be assumed that during the precipitation event the sulfate in the air does not change further by oxidation or dry deposition. Sulfate is removed by wet deposition with a rate law analogous to Equation (8), /Is = 1 -exp - (s,J) where the time scale T starts with the onset of precipitation. Figure 7 shows a family of “washout” curves. For the nominal rate s, = 1 x 10e4 s- ‘, 50 per cent of sulfate is removed in 2 h; for 5 times that rate in 0.5 h; for 0.5 times that rate in 4 h. We conclude that for the probable range of washout rates and durations of precipitation events, the total amount of sulfate wet deposition is largely insensitive to the washout rate; however, if s, were much smaller than the nominal value and/or the duration of the precipitation event were short, not all the sulfate present in the air would be wet-deposited. It should be emphasized that the
et al.
P .0 ?
v) .6 Q P .4 a
I -I
8 .6 normollzed
4
2 rate
I I 0 .2 consfont,
I I I 4 .6 .6 log s, /sno
I
Fig. 6. Sensogram of normalized concentrations of sulfate in the air, /?s/&,; absence of rain, s, = 0; at 24 h.
I
0
I 2
I
T
I 4 hours
,
I I 6 6 of precipitation I
,
I, IO
12
Fig. 7. Washout rates of sulfate (“delayed rain”) as a function of the wet deposition rate constant.
time distribution of sulfate (and consequently, acidity) within a precipitation event is highly sensitive to the washout rate s,, and the rate of rainfall. Recall also that rainout is not considered in this analysis. The case of “delayed precipitation” implies that the oxidation occurred entirely in the gas phase. The product (sulfate and sulfuric acid) is washed out by precipitation. Nitrogen The nominal oxidation rate constant for step (la) NO, + NO, is already much larger than the nominal dry deposition rate constant of NO,. Therefore, increasing this rate constant alone will not affect greatly the amount of formed nitrate; however, decreasing this rate constant and simultaneously increasing the dry deposition rate constant would have a significant effect. Altogether, the nitrate output is more sensitive to changes of the rate constants at earlier times. These effects are illustrated in the following figures.
Figures
8(a) and (b) give the sensitivity
of
Sensitivity analysis of the kinetics of acid rain models
651
-.4 0
z
-6
1-
-.8 I
,+ it
-I
.8
I .6
I 4
I .2
I 0
I .2
I
.4
normohzed
I .6
I I 8 +I-.8
rate
constant,
I
I .6
I 4 log
I .2
I 0
I .2
I
.4
I 6
I
.8
-,
I
ni /nno
Fig. 8. Sensogram of normal&d wet deposition of nitrate, yN/yno,as a function of normalized QFO rate constants; continuous rain; (a) after 24 h; (b) after 48 h.
“d
t?-II
I
-I
I
.a .6 normolued
I
I
4
.2 rate
-
I
I
I
I
I
0
2
4
.6
.8
constant. log n,/nno
I I
Fig. 9. Sensogram of normalized concentrations of nitrate in the air; /3N/&,o; absence of rain, n, = 0; at 24 h.
nitrate wet deposition to varying single rate constants. At 24 h, a lo-fold increase of the oxidation rate would produce only about 50 per cent more wet-deposited nitrate; at 48 h, only 14 per cent more. At 12 h (not shown in the Figures), yN/ynowould increase 2.5-fold. However, at 24 h, a decrease of n, by a factor of 10, would decrease yN/y.,, by a factor of 6.3. Figure 9 shows the effect on nitrate formation without precipitation. A lO-fold increase of n, would increase the relative amount of nitrate in the air by only 40 per cent; on the other hand, a lO-fold decrease will reduce the nitrate concentration by a factor of 6. The case of “delayed rain” would be analogous to sulfate wet deposition, as we assumed nw = s,. Multiple
mode
When two rate constants are varied simultaneously, the resulting sensograms are three-dimensional. The single mode analysis showed that the two most critical parameters are the oxidation and dry deposition rate constants. The combined effects of varying these rate constants are examined below. Figures 10(a) and (b) show the effect on sulfate
Fig. 10. Threedimensional sensogram of multimode sensitivity analysis; sulfate in the air; continuous rain; (a) after 24h; (b) after 72 h. Superscript o designates nominal value. Scales explained in text.
concentration in the air when the oxidation and dry deposition rates are varied up to lO-fold from the nominal values. The horizontal axes represent the two rate parameters and the vertical axis scales the sulfate concentration. The center of the surface, marked o, corresponds to the nominal values of both rate constants and has a height of one unit on the vertical scale. Maximum concentration is obtained with the lowest dry deposition rate and an intermediate oxidation rate after 24 h, as seen in Fig. 10(a). The maximum shifts slightly after 72 h of reaction time, shown in Fig. 10(b). Increasing the dry deposition rate over
D.
652
GOLOMB~~~.
Fig. 11. Sulfate in wet deposition; conditions as in
Fig. 13. Nitrate in wet deposition; conditions as in Fig. 10.
Fig. 10.
a.
the nominal rate makes very little difference on sulfate concentrations. Figures 11(a) and (b) show the effect of varying the rates on sulfate concentrations in precipitation. Here no local maxima are observed; the concentration increases continuously as the oxidation rate increases and the dry deposition rate decreases. Figures 12(a)and (b) show theeffect on nitrate in the air of varying the oxidation and dry deposition rates of NO,. In contrast to sulfate, these surfaces show little sensitivity to increasing the oxidation rate over the nominal value. This results since the ratio of the nominal oxidation/dry de~sition rate is very large for NO,. A significant effect is observed when the oxidation rate is decreased simultaneously with increasing dry deposition. Figure 13(a) and (b) show the effect on nitrate in wet deposition. Again, increasing the oxidation rate has very little effect. Also, the effect of dry deposition is much less pronounced than for sulfate wet deposition. An overview of Figs 11(a) and 13(a) is given in Table 2. The table corroborates that wet deposition is most sensitive to lower than nominal oxidation rates and higher than nominal dry deposition rates. With 0.1 times the oxidation rate and 10 times the dry deposition rate, only 3 per cent of the nominal value of sulfate is wet-deposited, and 11 per cent of the nitrate.
CONCLUSIONS
Fig. 12. Nitrate in air; conditions as in Fig. 10.
The sensitivity analysis concerned only the kinetic module of acid precipitation modeling with great
Sensitivity analysis of the kinetics of acid rain models simplification of the physico-chemical processes. However, several important conclusions can be reached in regard to the sensitivity of the model output to the input parameters. These conclusions are summarized as follows. (1) Among the four basic steps of the transformation process considered in this analysis (oxidation, dry deposition of the precursor, dry deposition of the product and wet deposition of the product) the rate of oxidation is the most important factor in determining rain acidity. In order to reduce uncertainties of acid rain modeling, it is imperative to accurately determine the rate of SO,---+ SOi- and NO,---> NO; conversion. There may be large natural variations in the oxidation rate constant depending on oxidant concentrations and ambient conditions (humidity, temperature, etc.). Accurate modeling must include time- and space-variability of the oxidation rate constant. (2) The second most important factor is the dry deposition of the precursor gases. Obviously, what is removed by dry deposition does not appear as wet deposition. Accurate modeling must include the timeand space-variability of the SO* dry deposition rate constant; the NO, dry deposition rate appears to have a lesser effect on nitrate wet deposition because of the fast oxidation rate. (3) The wet deposition rate constant (i.e. the washout coefficient) is important in determining the acidity distribution within a rain episode. The total amount of wet deposition, however, is only slightly affected by this rate constant, provided it rains for a sufficient period to wash out most of the pollutants. Rainout of the precursor gases with subsequent oxidation were not considered as separate steps in this analysis. If the rainout process were rate determining, the distribution of acidity within a rain episode would Table 2. Effect on wet deposition of simultaneously varying two rate constants: oxidation and dry deposition (Time: 24 h) s0ls.o or n&“,
Sd/%0 or n,/n,,
YslY.O
YNh.o
(1)
0.1
(2) (3)
0.1 0.1
1 10
0.13 0.11 0.03
0.16 0.16 0.11
(4)
1
(5) (6)
1 1
0.1 1 10
1.17 1.00 0.32
1.03 1.00 0.77
(7)
10
0.1 1 10
5.2 4.6 2.2
1.5 1.5 1.4
0.1
653
be dependent on the cloud lifetime and cloud characteristics rather than on the washout rate. (4) Nitric acid is an important ingredient of total precipitation acidity, provided the precursor NO, is present in similar quantities as the precursor of sulfuric acid, SOz. Because of the experimentally observed larger oxidation rate constant of NO, ---3 NO; than that of SOz---9 SO:-, the contribution of nitric acid at early times (i.e. closer to the source) may be predominant. Due to the large ratio of the nominal oxidation and dry deposition rate constant of NO,, further increases of the oxidation rate constants will not greatly enhance the nitrate concentrations in precipitation; however, large decreases of this rate will reduce nitrate concentrations significantly.
Acknowledgement-We thank Dr. Neil Goldman of the MIT Energy Laboratory for providing the analytic solutions of the rate equations. The contribution of Stuart Batterman, James Gruhl and Walter Labys was, in part, sponsored by EPRI contract RP1484-1, a project to which William Balson of DFI, Inc. provided some valuable initial assistance.
REFERENCES
Davies T. D. (1976) Precipitation scavenging of sulphur dioxide in an industrial area, Atmospheric Enoironment 10, 879-890. Freiberg J. (1978) Conversion limit and characteristic time of SO, oxidation in plumes. Atmospheric Enoironment 12, 339-347. Garland J. A. (1978) Dry and wet removal of sulphur from the atmosphere. Atmospheric Environment 12, 349-362. Hales J. M. (Ed.) (1982) The MAP3S/RAINE precipitation network: statistical overview. Atmospheric Environment 16, 1603-1631. Machta L. and Ferguson H. (Eds) (1981a) Atmospheric modeling. Interim Report under the U.S./Canada Memorandum of Intent on Transboundary Air Pollution, U.S. National Oceanic and Atmospheric Administration, Silver Spring MD 20910 and Atmospheric Environment Service, Downsview Ontario M3H5T4. Machta L. and Ferguson H. (Eds) (1981b) Atmospheric Sciences Review, Interim Report. (See Machta and Ferguson 1981a). Omstedt G. and Rodhe H. (1978) Transformation and removal processes for sulfur compouhds in the atmosphere as described by a onedimensional timedependent diffusion model. Atmospheric Environment 13, 503-509. Perhac R. M. (Ed.) (1981) EPRI Sulfate Regional Experiment: results and implications, Electric Power Research Institute, Palo Alto CA 94304, U.S.A. Report EA-216%SY-LD. Rabitz H. (1981) Chemical sensitivity analysis theory with applications to molecular dynamics and kinetics. Comput. Chem. 5, 167-180. Schwartz S. E. (1982) Gas-aqueous reactions of sulfur and nitrogen oxides in liquid-water clouds. Paper ENVI-51 presented at the American Chemical Society Acid Rain Symposium, Las Vegas, NV.