Estimating atmospheric dilution of brief, infrequent, random releases

@30-6981/79p101-oo29 soz.aop

Atmospheric Erwironnrnt Vol. 13,pp.B-33. 0 Pergamoo PressLtd.1979. Printedin GreatBritain.

ESTIMATING ATMOSPHERIC DILUTION OF BRIEF, INFREQUENT, RANDOM RELEASES TIMOTHY

L. CRAWFORD

and

WILLIAM

B. NORRIS

Division of Environmental Planning, Tennessee Valley Authority, Muscle Shoals, AL 35660, U.S.A. (First received 23 March 1978 and infinalform

I8 July 1978)

Abstract - Brief, infrequent, random releases of pollutants into the atmosphere occur in a number of industrial operations. Their irregular occurrence requires that their impact be assessed stochastically. Probability distributions of the possible normalized concentrations resulting from these releases are approximated by a method that scales easily calculated distributions in a simple manner. The technique is explained through a series of progressively more comprehensive examples - from a single, l-hour random release to multiple, random releases lasting several hours. Potential impacts defined by these distributions span a broad range of values and depend nonlinearly on the number of releases and the level of probability. Applications, limitations, and possible extensions and improvements of the method are discussed.

1.

Even though NCR has recently developed a method (Sagendorf and Gall, 1976) for estimating the impact of purges, it has not formally proposed the method as a rigorous technique to be used for this purpose. The following method is presented as a practical alternative. It has the advantage of starting with distributions easily calculated by already available continuous emission techniques, then applying simple scaling factors derived from results from elementary statistics. The development of the method is presented in considerable detail so that the connection between the distributions arising from continuous releases and those arising from purge releases can be seen, while the fundamental difference in the nature and treatment of these two types of emissions is brought out. Ideally, to assess the impact of purge releases, one needs, for each point at ground level in the vicinity of a source, the cumulative probability distribution, a, of time-integrated dilution factors (formed by accumulating from the larger to the smaller values) resulting from every possible arrangement in time of a given set of purges. Each probability in such a distribution represents a certain level of risk. For any such level, the corresponding dilution factor, when multiplied by the strength of the purge, yields the maximum expected concentration associated with that risk. In the method presented in this paper, these distributions are approximated by combining characteristic information about the purges with short-term, average dilution factors associated with a discrete number of receptors near the source. In practice, these factors will have been computed by use of an appropriate dispersion model (e.g. as proposed by NCR, 1977) on the basis of local, historical meteorological data collected over a period of one or more years. For certain important applications the method can easily handle a small number of purges. Because the distributions are made up of time-integrated, normalized concentrations,

INTRODUCTION

Relatively brief, infrequent, random releases (hereafter referred to as “purge releases”, “purges”, or “releases”) are necessary in the routine operation of nuclear reactors and many industrial facilities. For example, in the nuclear utility industry, purging the interior of turbines and condensers, holdup tanks, and containment areas occurs only a few times a year for only a few hours at a time. The total strength of such releases may approach or exceed that of continuous, routine emissions. Although the United States Nuclear Regulatory Commission (NRC) is obliged to ensure that the radiological consequences of all types of emissions are adequately assessed before licensing a nuclear plant, only the methodology for evaluating the impact of routine and accident emissions has been promulgated. The realization of the need to assess the impact of purge releases has arisen only recently. The procedure used to evaluate the consequences of routine emissions does not apply directly to the case of purges. Any approach to analyzing the impact of purges will be more complex than that for analyzing continuous discharges because acceptable levels of risk, and factors such as number, randomness, duration, and subsidence of the release strength must be considered. The complexities arise chiefly because the times at which purges will occur are unknown, forcing their occurrence to be treated stochastically. Furthermore, because the results obtained are nonlinear, they cannot be easily extrapolated to larger numbers of sources or to more frequent releases. Similarly, the procedure used to evaluate the consequences of accident emissions is inappropriate for analyzing purge effects. For unlike accident emissions, purges will definitely occur and will be released from specific source locations with known strengths at frequencies that can be readily estimated. 29

either doses or time averages of concentration can easily be obtained by applying appropriate factors.

The method is computationally efficient since it takes advantage of the unusual form of the distribution of dilution factors (only a few nonzeros). 2. DESCRIPTION OF TtiE METHOD In general terms the problem to be solved is one of determining the frequency distribution of a random variable X, defined by h x= &l/‘(f) dt. (1) j 0 wherefis a known function of time defined on the interval [a, b], and p is a random function of time defined on the same interval. Thus, p may be viewed not as a function in the usual sense, but as a collection of functions. A particular function from thiscollection (i.e. a realization ofp), when used in Equation (l), leads to a specific value for X. In the special cases to be considered, f will represent the relative concentration (i.e. the concentration that would result from an emission rate of unity) at time t at a designated receptor in the vicinity of the source, and p will represent the time-varying relative emission rate (i.e. the ratio of the emission rate to the total emission over the period of release) of the source at the same time. Hence, values of X are normalized, timeintegrated concentrations. The approach will be to make approximations to Equation (1) after suitable assumptions about p andJ have been made. The general approach

Suppose that, during the interval of time {u,b], n identical purges of a facility occur randomly. In addition to the fact that the periods of release cannot overlap, it will be assumed that any moment is as likely as any other to be the time at which a purge begins. A typical realization of p for the case n = 3 is shown in Fig. 1, where each purge lasts for the brief period t. For a single realization ofp, because p(t) = 0 except during purges, Equation (1) becomes li+’ X=i (2) pW,fV)d~ i=) Yr 'i

which local values ofstability, wind speed,and wind direction are available in the form of averages over N consecutive. nonoverlapping time intervals of length At (usually 1 hour). With an appropriate dispersion model. one can USCthis information to compute the N corresponding relative concentrations of averaging time A\t near the purge source. This procedure approximates the continuous function f, with a step function~having step-width Ar, as shown in Fig. 2. The function p may likewise be approximated by a step function fi af step-width Ar. For convenience the time variable I will now be considered to be integer-valued and will be counted as the tth period of length At in [a. b]. It is further assumed that purges can be initiated and terminated only at the beginning and end of these periods; hence, T, if expressed in units of At, must be integer-valued. With these assumptions, Equation (2)may be approximated by

where tl, t2,. , t, are n integers having properties (a),(b), and (c)in Equation (2). By computing Equation (3) for all possible choices of ft, t,, , t,, one can approximate the continuous distribution of X by the discrete dist~bution of 2. Single 1-hour releuse As a simple example, suppose that [u, b] is a year-long Deriod. that At = 1hour (thus, N = 8760). that T = I. and ;hat n = 1. If the purge ratk is constant, th& c(t) = 1 when I; is nonzero, Furthermore, because n = 1, the list tl, tl,. , r, reduces simply to tl, for which there are 8760 possibilities. With these assumptions, Equation (3) collapses to B =At). Thus, the values of 2 are just the 8760 hourly values off. This fact leads immediately to (o(8) = cD(fl.

(4)

Actuafly, this result does not depend on ,V = 8760; hence, any period of meteorology could be used to develop @(_%?I. However, the more meteorology on which this dist~bution is based, the more confidence that can be placed in the ability of the predicted distribution to reproduce accurately the actual distribution. Determination of U@) by directly computing each individual value from which it is formed can quickly become

where (a) rI is the beginning time of a purge, (i = I.. . , ~1, (b) tt + r is the ending time of a purge. and (c) c, i- z 5 t,, 1’ Thus, a particular value of X may be viewed as the sum of areas of n regions under the graph off, weighted by p, that are nonoverlapping and have the same base-width T. The totality of values of X consists of all such possible sums. In practice, the distribution of X may be approximated as follows. Suppose [a, b] is a period of time (usually one or more years) during which n random purges occur and for

Fig. 2. Approximation of relative concentration functionfby the step function f having stepwidth of AL

Fig. 1. Typical realization of three random purges of duration r over the time period [u,b].

31

Estimating atmospheric dilution of brief, infrequent, random releases Table 1. Sample 2-purge distribution

t=1

Relative emission : b(t) t=2 t=3 t=4 t=s

1 1 1 1 0 0 0 0 0 0

t=l

1 0 0 0 1 1 1 0 0 0

0 1 0 0 1 0 0 1 1 0

0 0 1 0 0 1 0 1 0 1

f(2)

N! _

bw30)

j(l) +f(2) + 0 + 0 + 0 f(l)+ 0 +o+o+o i(l)+ 0 +o+o+o j(l)+ 0 +o+o+o 0 +f(2)+0+0+0 0 +f(2)+0+0+0 0 +f(2)+0+0+0 0 + 0 +o+o+o 0 + 0 +o+o+o 0 + 0 +o+o+o

0 0 0 1 0 0 1 0 1 1

0

0

0

unmanageable for large N if n > 1, since the number of possibilities for t,, cl,. ., t,, when r = 1, is (N

,=I

Relative concentration : f(t) t=2 t=3 t=4 t=s

f(l)

(.“I=

9 = i

“)ln!

’

(5)

For example, if N = 8760 and a = 1, the number of possibilities is only 8760; but, if n = 2, the number of possibilities is 38,364,420; and if n = 3, the number is 111,998,530,120.For n > 2, the computer resource requirement becomes prohibitive for forming distributions for this number of calculations, even for a few tens of receptors. Therefore, when multiple purges are considered, techniques must be sought to minimixe the amount of computation required. Multiple I-hour releases In most cases the wind only occasionally blows into a given sector. If the wind direction is uniformly distributed within the 16 principal compass sectors, the probability that the wind direction and receptor sector will be coincident is l/16. Thus, for a given receptor,j@) is usually zero. This fact is used to reduce the amount of computation required to compute all possible values of Equation (3). Consider an example in whichfand 6 are defined only over a 5-hour period (N = 5); during this time, two l-hour purges (n = 2, T = 1) of constant stren th occur randomly. Suppose that only two hourly values of _!are nonzero. In determining CD(X),no generality is lost in supposing that these values are the first two in the 5-hour period (i.e.fil) > 0 andfi2) > 0, but fi3) = a4) = 85) = 0). Every possible combination of starting times, t, and tr, for these two purges is obtained by choosing every possible combination of pairs of integers from the list 1, 2,3,. . . , 10, such that the integers of the pair differ (in absolute value) by at least T. Because T = 1 in this example, the ten possibilities are 1 and 2,1 and 3,1 and 4,1 and 5,. . . , and 4 and 5. Corresponding to each pair there is a realization of p^,which can be. represented by a onedimensional array of five entries consisting of two l’s and three O’s- the value of 1 indicating the relative emission rate and the position of 1 indicating an hour during which a purge occurs. Table 1 shows all possible ways for computing Equation (3); each value of d in the table is the sum of five p^.3 products. A practical method for approximating a(R) becomes apparent when the composition of an n-purge distribution is considered. For example, the distribution in Table 1 contains ten elements. Seven of these are nonzero, yet there are only threedistinct values among the seven. These. three values are a single 2-term sum composed off(l) +3(2) and two l-term sums involving either 3(l) orj(2) repeated three times. The distribution of 2 can be easily constructed if one knows the nonxero values of3 occurring in the period and that n = 2 and N = 5.

To generalize, an n-purge distribution is composed of(J) distinct sums of k terms (k = 0,1,2.. . . , n) each repeated (f:T) times, where m is the number of nonxero values offduring N periods of duration At. By convention, a O-term sum is zero. The total number of sums of k terms (Feller, 1957) is (:) = i (X:k?. L=O

(6)

Given N, n, and the m nonzero values of3 one can use this information to determine the relative composition of any npurge distribution. The composition of a typical n-purge distribution (N = 8760 and m = N/16) is illustrated by Table 2. Examination of Table 2 shows that, for a given number of purges, the frequency of occurrence of any value of 2 resulting from the sum of k terms decreases rapidly as k increases. For small n, depending on the specific application, sums of two or more terms may be dropped because their contribution to the total distribution is insignificant. Hence, Equation (6) can be written as c) z (“;“) + m(iI;).

(7)

When these higher-order sums are neglected, the nonxero portion of @(g) may be expressed as a multiple of @(tf): @(R) = s(n)

@ttz,,

(8)

where

s(n) =

N(.“--Fl (“3 + m(i:;I)

Nn

I

= N + (m - l)(n - 1)

The remainder of the probability is assigned to the zeros. The factor g(n) is shown as a function of a alone, because for a chosen receptor, m and N are constant. Typically, g(n) = 1.00, 2.67, and 4.00 for n = 1,3, and 5, respectively. Therefore., the probability that a given value of 2 is exceeded increases with increasing numbers of purges. The distribution @(tf) may be Table 2. Composition of a typical n-purge distribution n

k-term sums in an n-purge distribution (%) k=O k=l k=2 k=3 k=4 k=5

1 2 3 4 5

93.74 87.88 82.38 77.23 72.39

6.26 11.73 16.50 20.62 24.17

0.39 1.10 2.06 3.22

0.02 0.09 0.21

nil 0.01

nil

1:1

49:11

36:lO

12!04

2.40

0.32

0.63

md defined with any period of meteorology, but for annual assessment with Ar = 1 hour, m must be normalized to IV = 8760. Figure 3 illustrates typical graphs of Q(d) for n = 1, 3, and 5. Further examination of Table 2, approximation (8). and Fig. 3 shows that, when n = 1, it is unlikely (CT’,,) that a receptor will even experience the effect of a purge; but if n > 11, it becomes a likely event ( r 50”;). Also, it is apparent that the consequences of n-purge releases are very extreme: that is, values ofX may range from zero (the most likely event for n < 10) to the sum of the n largest values of 7 (a very unlikely event). Also from Fig. 3, the nonlinear nature of@(f) is apparent. Because of nonlinearities, concentrations at a given probability level resulting from n purges cannot be linearly scaled to obtain concentrations at the same level resulting from a different number of purger Multiple r-hour releusrs Because most intermittent releases from nuclear facilities result from purging the interiors of turbines. holdup tanks. and containment areas, the function p is usually characterized by an exponential decay. If r is a few tens of hours, the effect of the time-varying nature ofp becomes important. In this case, approximation (8) is not directly applicable because it was developed on the assumption that purges last 1 hour (or, more generally, At). However, by using Equation (?). a function k can be defined such that a(6) can replace Q,( ) in approximation (8). Define

WILLIAM

B. NORRIS

that correspond to these periods are set to zero before computing @(f) or Co(L). The method presented is based on the assumption that a single facility is purged and, hence, that simultaneous purges cannot occur. However, it 1s not uncommon for nearly identical facilities to be built at the same location - for example, identical units at a nuclear generating plant. In this case, simultaneous purges are possible. If purges from each facility have the same characteristics (exit height, exit velocity, strength, manner ofdecay, etc. 1and if@(f) is about the same for each facility. then the results presented above can be used to analyze the impact of single or multiple purges from a small number of sources. When simultaneous releases can occur. Equation (6) is replaced by (Feller. 1957)

If n cc N and n CCN - m, then N + n - 1 z N, m + k -l~m,andN-m+n-k-lsN-m;thatis, Equation (11) is approximately Equation (6). Thus, single source results are sufficient. As examples, the single source results for n = 2 can be used to assess the I;(r) = ‘+c I &$((r) (9) impact of single purges from two facilities; the singlesource results for n = 4 can be used to assess the for I = 1, 2,. , N. Then, ’ ’ impact of two purges from each of two facilities. @(ri) = y(n)w?. (10) The applicability of approximation (8) depends on Here his a r-hour dilution factor that has been weighted by p. three factors : II, m/N, and the probability level at which For example, if; has a constant value of l/r for r hours, h(t) is d is to be chosen. These factors determine the accuracy simply a r-hour average. In any case, h(t) is a time-integrated of m(f) or @,(A)relative to the desired probability level. concentration. Applications of approximation (8) requiring the use of As illustrated by Table 2, these factors determine the Equation (9) become more limited as T increases because m importance of higher-term sums. Usually, a proalso increases, causing the higher-term sums to attain much bability level and n are specified. The ratio, m/N, is more importance in the overall composition of the about 1.‘16and increases with T and wind persistence distribution. as a result of synoptic or channeling effects. Neglecting 3.

DISCUSSION OF THE METHOD

Partially nonrandom releases may be considered by altering the O(f) or Q(h) distributions. For example, if unfavorable meteorology is to be avoided by excluding releases during these periods, then those values of,for h

sums of two or more terms is nonconservative, but is so, for small n, usually by an insignificant amount. This

nonconservative effect is probably more than offset by the conservative assumption that a receptor (person, animal, etc.) is always present. There appear to be two ways to extend or improve approximation (8). The first is to add 2-term sums to Q(8). This is accomplished by forming the cumulative probability distribution @(f?^2)of all possible 2-term sum combinations of the ,f values. With this new distribution, the nonzero portion of an approximation to (D(g), more accurate than approximation (X), is given by

Proboblllty

that

X exceeds

o concentrokr

Fig. 3. Typical distributions of 2 for n = I. 3, and 5.

The remainder of the probability is assigned to the zeros. As noted with approximation (S), the values off may be replaced with values of 6 for r-hour releases. Also, because only k-term sums greater than or equal to three are neglected, approximation (12) is exact for n = 2 and, as illustrated by Table 2, applies to much higher values of n. Unfortunately, this approach is

Estimating atmospheric dilution of brief, infrequent, random releases iimitcd because addition of higher-term sums in this manner becomes computationally difficult. Tho problem presented is essentially that of finding the probability distribution of the sum of independent events (purge releases). Although this problem is fully treat&i in standard textbooks on statistics, the applicationto discrete data is nontrivial in terms of actual computation. This difficulty is not characteristic of the method presented above as expressed primarily by Equations (8), (10) and (12). A less limited but computationally more complicated approach is to convolute the probability density function associated with @_?) or @((i;)using Fast-Fourier-Transform techniques. With this procedure the discretely defined density functions are transformed, then manipulated in the transformed space. The results are obtained by

33

applying the inverse transform. This approach, although more difficult than the approach presented here, would remove most of the limitations mentioned above. REFERENCES

Feller W. (1957)An Inrroduction to Probability Theory and Its Applications, pp. 26-64. John Wiley, New York. Sagendod J. and Goll J. (1976)XOQDOQ- Programfor the Meteorological Evaluation of Routine Efluent Releases at Nuclear Power Stations (draft). U.S. Nuclear Regulatory

Commission, Washington, DC. U.S. Nuclear Resulatorv Commission 11977)Methods for Estimating Atiospherk Transport aA Dispersion of kaseous Ejluents in Routine Releases from Light-WaterCooled Reactors, Regulatory Guide 1.111. Washington,

l\r(