A simplified diffusion-deposition model

C0W6981/80/0801-0953

Atmospheric Enuironmunr. Vol. 14, pp. 953 956. 0 Pergamon Press Ltd. 1980. Printed in Great Britain.

A SIMPLIFIED

DIFFUSION-DEPOSITION

W!.CQ/O

MODEL

NIELS OTTO JENSEN Ris$ National Laboratory, DK-4000 Roskilde, Denmark (First received 2 April 1979 and in.finalform

21 February 1980)

Abstract - The use of a simple top hat plume model facilitates an analytical treatment of the deposition

problem. A necessary constraint, however, is that the diffusion velocity (e.g., in terms of the plume growthrate) is large compared to the deposition velocity. With these limitations, explicit formulae for downwind deposition amounts and ground level atmospheric concentrations are given.

INTRODUCTION Various models for the combined processes of diffusion and deposition have previously been published in the literature and different assumptions and methods have been employed (Chamberlain, 1953; Overcamp, 1976) ; the approach of Horst (1977) gives the exact solution to deposition from a Gaussian plume. All of these models, however, lack the appeal of simplicity. This paper seeks to present the problem in as basic a manner as possible while still retaining the essential physical representation. Our model is a so-called source-depletion model, which, especially under conditions where the deposition is fast and the atmospheric diffusion is slow, becomes inferior to Horst’s (1977) surface-depletion model. He demonstrates that the source-depletion model under such conditions may be in error by a factor of 3-4. In our opinion this discrepancy is insufficient to dismiss the source-depletion model taking into account the current uncertainties in the deposition velocity ud and the a-values for turbulent diffusion. Under less extreme conditions (i.e., good mixing and moderate deposition) the two models are in better agreement. DEPOSITION Sedimentation, or gravitational settling, is significant if the diffusing plume consists of particles of the size of 10 pm or more in aerodynamically equivalent diameter. What is dealt with here is primarily the effect of surface removal: dry deposition. The processes responsible for this include: inertial impaction of particles onto the surface roughness elements ; Brownian motion of fine particles with resulting collection at the surface; Coulomb attraction to the surface of particles with an electric charge different in polarity to the surface; and diffusion of vapors with subsequent adsorption or condensation on the surface. If the deposited aerosol reacts chemically with the surface, as if it, for example, is highly soluble in it, the surface may be considered a perfect sink, with the consequence that the surface concentration is

effectively zero. In this case the present approach is a poor one because the deposition is as large as atmospheric diffusion permits. Under such conditions it can be shown that the deposition velocity, vd,is given by l],j u:/ti (Chamberlain, 1953), where II* is the turbulent f:ction velocity. Defined on the basis of the concentration at a height of 2 m say, typical values of l)din such turbulent controlled situations is in the range of ems-‘. A condition in addition to relatively slow dry deposition which is ideally suited to the present approach is direct precipitation scavenging known as washout. By removing material throughout the plume at a rate proportional to the concentration (and the precipitation intensity), this kind of deposition acts as a perfect example of downwind source depletion. This is, furthermore, a way in which very large deposition rates occur (a value of ud = 1 m s-l would be typical for a rainfall rate of 10mm h-l, as can be deduced from Fig. 10 in Slinn, 1977). A kind of precipitation scavenging which is not dealt with here is the so-called rainout. In this case the plume expands freely into the cloud layer, where material is “removed”, either if it is a gas soluble in water or if it consists of particles which will collect in water droplets. This “removal” causes a lowering of the surface air concentration. In this respect it has the same effect as dry deposition, but the deposition itself may or may not take place according to whether the cloud is going to form rain or not. This kind of deposition can be viewed as the opposite of sedimentation: one influences the immediate deposition but not the air concentration, and vice versa. In the remainder of this paper we deal with deposition caused by washout and by dry deposition limited by the rate of uptake rather than by the ability of the atmosphere to diffuse the matter.

953

A SIMPLEPLUMEMODELWITH DEPOSITION

For simplicity we should like to advance the concept of the well mixed plume. The concentration profiles in such a plume are square shaped in the lateral as well as in the vertical direction. This may seem crude, but it

954

NM s OTTO JENSEN

has the great advantage of simple algebra while still retaining features such as the location and the magnitude of the ground level concentration maximum. Thus, the effect of, for example, a change in the effective source height or a change in the distance to a virtual release point, as well as the combined influence of stability, can easily be demonstrated. The effect of deposition also is most easily demonstrated in this manner. In this and the following section we shall make a further simplification by assuming that the plume diffuses linearly with downwind distance given as x. In such a plume the concentration x is given by

(1) where Q is the source strength, and ccxand /Lx are the half-widths 6, and e,, of the plume in the vertical and lateral directions, respectively. Equation (1) assumes that the plume expands freely. In the event that ax becomes larger than the effective source height h, (1) must be replaced by

Q/G ___-I_-.

x = Zfix(h + ax) The deposition, D (amount (unit time)-’ area)-‘), is usually assumed to be given as D = v,x.

dimensional distance X =- a.w;h,Equation (6) assumes the form 7 =: x0x d.

(71

which clearly shows the simplicity of the present approach. The problem has been reduced to a dependence of two nondimensional parameter-s: one depending on the deposition velocity compared to the turbulent diffusion velocity, the other depending on downwind distancecompared to the distance of plume intercept with ground level. Both parameters depend on stability. For neutral (Pasquill type D) spread conditions, r may be taken to be approximately l/33 in the interval 0.1.-1Okm. With ud = 10 ‘rn s-’ and ti = 1 m s- ‘, this results in a deposition parameter, d = r,;rti, of 0.33. With this value ofd, Equation (5) has been plotted for two values of h (Fig. 1). It is seen that the present model compares favorably with the far more complicated Gaussian model. ESTIMATES OF DE~SITIO~

By combining Equations (2), (3) and (5) we obtain

(2) for large distances the deposition (unit

QO

D = dazX-d,

which is the rate at which material is deposited on a unit area. The amount of deposited material on area A during time t is found by multiplying Equation (8) by A and t respectively. Again it is noted that the entire effect of the deposition is given through the two parameters d and X. For thedeposition to be large, d must be large and X small. Thus the most noticeable effect of deposition dQ vd dx -._ = - - ..--------- I ax 2 /r, (4) obtains from a ground level release, during stable u (h + c(x) Q conditions, and with light winds. In calculations of population doses from large which by integration and appropriate use of boundary hypothetical accidents with nuclear reactors, the conditions yields amount of deposited radioactivity gives the most Q = Q&/(1 + ax/hV, (5) important contribution. The question will often be what the worst conditions are at a given distance from where the exponent d, the deposition parameter, is the reactor. Regarding deposition, the maximum defined as vJaG. For 0 c x < h/a, Q is equal to Qo. amount will result when d = (In X)-i, which is From Equation (5) it is seen that a moderate obtained from differentiation of Equation (8). deposition rate has the apparent effect of depleting the The essential effect of an elevated release is to allow original source, giving rise to the name sourcefor some dilution before the plume makes contact with depletion model for this type of approach. Once again, this deposition model is perfectly valid in the limit of olU the surface, thereby decreasing the amount of deposition which can take place per unit area. Compared >> od or d << 1, and probably a fairly good approxito a lower level source, this results in larger downwind mation where this condition is violated to some degree concentration levels. It follows from Equation (7) that (say, where d is as large as of order I). Note that the the concentration resulting from a higher level (11~) lateral diffusion does not enter into Equation (5). release will be (h1/h2)d times larger than the conThe downwind ground level concentration is then centration resulting from a lower level (h,) release, found using Equation (5) as everything else being equal. (6) K = x&/(1 + axl@)P, A real source will never be a true point source and so the release point is effectively at some virtual upstream where x0 is the concentration at the same location, but position. The deposition only begins to take place at without deposition. For 0 < x < h/a the ground level the physical release point. concentration is zero. At large values of the non(3)

Assuming that the diffusion velocity, given by Gin this linear model, is much larger than o,, the differential equation for the balance of mass reads : - dQ = DdA where dA is the di~erential area (2gx)dx. By insertion from (2) one obtains

A simplified diff~ion~e~sition

955

modei

change. Q is a non-dimensional constant of order one. The growth rate d&dx is found to be proportional to w,$i where w is a characteristic turbulent velocity, for example the standard deviation of the vertical velocity ;- 0.1 component, a,. For diabatic conditions, estimates can be obtained from Monin-Obukhov scaling arguments: in very unstable stratification (free convection*) u, is procm portional to 2ii3 and ii is constant with height, z, hence 105 104 103 102 10’ Z&d 6 - x3/2; in very stable stratification (z-less stratification*) Q, is constant with height and U is proFig. 1. The apparent source strength (or the ground level portional to height, hence 6 - xl”. The slopes con~ntration) relative to the case without deposition for two different release heights as a function of downwind distance predicted for these two diabatic extremes, as well as the from the release point. The full fines represent results from a result (Equation (10)) for the neutral case, may be Gaussian model (figure taken from Van der Hoven, 1968). compared to the slopes of the Pasquill-Gifford curves The dotted lines represent the simple linear model (Equation (Turner, 1970). (5)) for the same choice of parameter values. We may add that the exponent 1.5 is probably not of For a virtual source the equation corresponding to very great practical significance, since as soon as the Equation (5) is internal boundary layer under such conditions reaches the mixed layer, which it does pretty quickly with an Q = QoWU + x/Oh x 2 1, (9) exponent as large as 1.5, it starts to grow like 6 - x where I is the distance from the real to the virtual becauxe both ET,and U are constant with height. release point and h -c cd. The small roughness effect ((z0)if5) present in Equation (10) does not enter in the diabatic extremes POWER LAW PLUMES mentioned above. For a moderate variation in stability, however, one may in general assume that 6 - xp, Although the model described above behaves where the constant of proportionality is somewhat qualitatively right for moderate downwind distances dependent on zo, and where the exponent p is bounded (x < 20 km) it cannot produce the large curvature in between the limits 0.5 and 1.0 with a value of 0.8 in the Gaussian concentration curves in Fig. 1. This is neutral conditions. because the linear plume model assumes a larger The same type of power law relationship may be dilution at large distances than actually takes place, assumed for the vertical spread of plumes, especially whereby a smaller deposition results. The linear asfor fairly low sources. Hence we assume sumption is especially poor during stable conditions. a, = az; -pxp. Compared to Gaussian models better quantitative results are obtained if the spread parameters are Maint~ning the assumption about a well mixed approximated by power law relations: instead of plume, the differential equation for the balance of mass assuming the half-width of the plume to be pronow reads portional to x, we will now assume that it is prodQ vd dx portional to x9 The proportionality constant, how--“.-..=az~-pxp 2 h. (11) ever, must now have units of length to the power of (1 U (h + az; -px”) ’ Q - p). Some comments will be offered on the choice of a Assuming, for simplicity, that h/z, is relatively small proper scale. compared to ii/v,,, integration gives (a relation which is Pasquill(l972) drew an analogy between the growth of internal boundary layers resulting from a change in easily derivable only for p = 4) the surface roughness (diffusion of momentum) and Q = Qoev{ the vertical growth of plumes (diffusion of matter). The result of internal boundary layer investigations (numerous sources, e.g. Jensen, 1978) is that in neutral conditions the height of the internal boundary layer, 6, is to a good approximation given by where the deposition parameter in this case has been taken as d = v,faii. In Equation (12) we have used the x 415 boundary condition that Q = Q. at the distance 6 --rra--, az:-PxP = h, where the plume first touches the ground. zo 20 If xp < h/azh-P, then Q = Qe. For stable (Pasquill type F) spread conditions, u, where z. is the surface roughness of the rougher area increases roughly as the square root of x @= 4). Hence uz can be expressed as UZ~~~X”~~From the and x is the downwind distance from the roughness Pasquill-Gifford curves (Turner, 1970) we find (by assuming z0 to be 10cm) a = 1.33. For this case and * For a definition of these terms see Wyngaard (1973). I.0

-&(~~-p)

0

NIELS OTTO JENSEN

956

FkSOUILL TYPE F 0001~ 10’

102

lO3

x(m)

104

105

Fig. 2. Same as Fig. 1 except that stability here is Pasquill, F. and the dotted lines represent results from the power-law type of plume model (Equation (12)).

with II,,= lo-’ m s-l and ri = 1 m s-‘, Equation (12) has been plotted (Fig. 2) for two different values of h. The present model is seen to produce a very strong progressive decrease in Q as the downwind distance increases in agreement with a Gaussian modet. It even gives roughly the same numericai values. Quantities such as ground level concentration and deposition rate can readily be calculated by the use of Equation (12) together with Equations (2) and (3).

sull fairly clear although not quite as obvious as in the linear case. The plume models used above assume an unlimited vertical growth of the plume height. In the presence of a capping inversion at height W. however, the vertical extent is limited to H. To be able to compare the present models with results from Gaussian plume models with source depletion, as for example that given by Van der Hoven (196X). this aspect has not been considered. In the presence of a capping inversion the linear model (Equation (1)) would have a l/x dependence instead of a l/x’ dependence for distances x 1~ (H h)/a. If then h > H/2 the source depletion is simply given by (instead of Equation (5))

Q = Q. exp(-r,(x

- h/a)/lZH),

.X 2 bin.

(13)

If h I H/2 then three possibilities arise: if x 2 hia then Q = Qo; if hju I x $ (H - h)/a then Equation (5) applies; if, finally, x 2 (H - h)/a then Q = Q~(2h~H)dexp~

--I:~ x [.Y - (H - h)~a]~~~H)~.

(14)

REFERENCES DISCUSSION

Chamberlain A. C. (1953) Aspects of travel and deposition of aerosol and vapor clouds. Atomic Energy Research Estab-

Using a linear well mixed plume model, a simple lishment HP/R 1261. Harwell. Berkshire. England. 35 on illustration is given of the relevant parameter com- Horst T. W. (1977) A surface depletion model for deposit& from a Gaussian plume. Atmospheric Enoironment 11, binations in cases where the combined effect of 41-46. diffusion and deposition has to be taken into account. Jensen N. 0. (1978) Change of surface roughness and the The problem is reduced to a dependence of two planetary boundary layer. Q. JI R. Met. SK 104,351-356. dimensionless parameters : a deposition parameter Overcamp T. J. (19%) A general Gaussian diffusiondeposition model for elevated point sources. f. appl. Met. and a distance parameter. The simpIe analytical form 15, 116771171. of this model allows for easy estimates of the conPasquill F. (1972) Some aspects of boundary layer desequences when the magnitude of the physical variscription. Q. J1 R. Met. Sot. 98, 469-494. ables : deposition velocity, wind speed, stability, source Shnn W. G. N. (1977) Some approximations for the wet and height, and downwind distance, are changed. dry removal of particles and gases from the atmosphere. War. Air Soil Pollut. 7, 5133543. At large downwind distances (- 20 km or more) a Turner D. B. (1970) Workbook of Atmospheric Dispersion realistic model must take into account the gradual Estimates. Public Health Service Publication No. 999-Apdecrease of the rate of spread of a plume, however. In 26, Cincinnati. U.S. Department ofHealth,Education, and doing so by assuming power-law relations for the Welfare, 56 pp. Van der Hoven I. (1968) Deposition of particles and gases. plume width, a relatively simple analytical solution ~ereorQ~~y and Atomic Energy 1968 (Edited by D. H. may stili be obtained. This solution, however, now has Slade), available as TID24190. U.S. Atomic Energy Comthree parameters because the distance, x, and the mission, Oak Ridge, Term., 00. 202-208. source height, h. have become inde~ndent arguments. Wyngaard J. C. (1973) On surface-layer turbulence. The three nondimensional parameters are : d, x,&, and Workshop on ~jcromete~r~~u~y (Edited by D. A. Haugen), Am. Met. Sec., Boston, Mass., _. pp. 101-149. h/z,. The physics of the deposition process, however, IS