Berichte zur Umweltphysik

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1 (Print) ISSN (Internet) ISSN Nummer/Number 4 Auflage/Edition 2 Berichte zur Umweltphysik Reports on Environmental Physics A note on the distribution function of density fluctuations in the model COFIN Eine Notiz zur Verteilungsfunktion der Dichtefluktuationen im Modell COFIN Lutz Janicke März/March 26 Ingenieurbüro Janicke, Alter Postweg 21, Dunum, Germany

2 Autoren/Authors : Dr. Lutz Janicke Ing.-Büro Janicke Alter Postweg 21 D Dunum ( lj@janicke.de) 1st Edition: August 24 2nd Edition: March 26 (figure corrected) Berichte zur Umweltphysik (Print), ISSN Berichte zur Umweltphysik (Internet), ISSN Herausgeber: Ingenieurbüro Janicke Alter Postweg 21, Dunum Deutschland Internet: Die Berichte dieser Schriftenreihe erscheinen in unregelmäßigen Zeitabständen. Sie werden im Selbstverlag gedruckt und außerdem auf der Internet Seite in Form von Pdf Dateien kostenlos zur Verfügung gestellt. Alle Rechte sind dem Herausgeber vorbehalten. Publisher: Janicke Consulting Alter Postweg 21, Dunum Germany Internet: The reports of this series are published at irregular time intervals and are printed by the publisher. In addition, the reports are made available free of charge in the Internet on page in form of pdf files. All rights are reserved to the publisher.

3 Lutz Janicke: Density fluctuations in the model COFIN 1 A note on the distribution function of density fluctuations in the model COFIN Lutz Janicke März 26 Abstract In the model COFIN the density fluctuations in a plume are derived from LIDAR measurements. However, the distribution function given is neither normalized nor non-negative. It is shown how these deficiencies can be removed. Preface to second edition The only change with respect to the first edition is a correction of the graphical representation of the Weibull distribution. Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)

4 Lutz Janicke: Density fluctuations in the model COFIN 2 1 Analysis of the COFIN model The report Risø-R-1329(EN) 1 on the model COFIN describes in chapter 3.11 a model for the probability density function (pdf) P(c, y) of concentration fluctuations in the moving system of a plume. c is the concentration and y the (horizontal) distance from the plume axis. We cite from this report. A pdf is normalized, i.e. it has to fullfill the equation P(c, y) dc = 1. The following definitions are made: (1) c n (y) = P(c, y) c n dc for n = 1, 2,..., (2) c n, = c n (), (3) c = c 1,, (4) σ 2 n = + c n (y) y 2 dy/ + c n (y) dy. (5) The following relations have been observed analysing LIDAR measurements: c n (y) = A n f (y/σ n ), σ 1 = n + β with β 1.9, (7) σ n 1 + β P(c, ) = g 2 (c), g p (c) = p c Γ(p) ( ) p 1 pc exp( pc/c ). (9) c From these relations the following pdf is deduced: P(c, y) = [1 γ(y)] δ(c) + γ(y)p c (c, y), (1) P c (c, y) = 4c 1 + (β 2) + 4se c/c exp [ e 2c/c c 2 + (3 β) ], (11) 1 + β ( γ(y) = y ) exp( 2 y /σ ), (12) σ σ = 1 + β β σ 1, s = y σ 1 (1 + β). 1 M.Nielsen, P.C.Chatwin, H.E. Jørgensen, N.Mole, R.J.Munro, S.Ott: Concentration Fluctuations in Gas Releases by Industrial Accidents Final Report.Risø-R-1329(EN), Risø National Laboratory, Roskilde, Denmark (May 22) (6) (8) (13) (14) Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)

5 Lutz Janicke: Density fluctuations in the model COFIN 3 The general structure of P c (c, y) is P c (c, y) = a 1 (y)c[1 + a 2 (y)c] exp[ a 3 (y)c], (15) a 1 (y) = 4 c 2 a 2 (y) = 1 + (β 2) 1 + β exp[(3 β)], (16) 4se c [1 + (β 2)], (17) a 3 (y) = 2e c. (18) Equation (1) applied to equation (1) yields 1 γ(y) + γ(y) P c (c, y) dc = 1. (19) That means P c itself must be normalized. We try to verify this condition in the following. Using the relations we get: c exp( a 3 c) dc = a 2 3, (2) c 2 exp( a 3 c) dc = 2a 3 3 (21) P c (c, y) dc = a 1 + 2a 1a 2, (22) a 2 3 a 3 3 = a 1 (a a a 2 ), (23) 3 = a 1 a 3 3 { 2e c + 8se, (24) c [1 + (β 2)] = a 1 a 3 3 2e c 1 + (β 2) + 4s 1 + (β 2), (25) = 4 c (β 2) c β 8 2 c 1 + β exp[(3 β) 6s + ], (26) 1 + (β 2) = exp[(1 β)]. (27) Therefore, the pdf P c (c, y) is correctly normalized only if s =, i.e. on the plume axis. Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)

6 Lutz Janicke: Density fluctuations in the model COFIN 4 2 Correction of the deficiencies The deficiency seems to have its origin in a typing error in equation (COFIN.163) which correctly should read P(c, y) =[1 γ(y)] δ(c) + exp[(1 β)]h(ζ)[1 (1 β)] 2cs exp[(2 β)]h (ζ), ζ =ce. (28) (29) This can be written as (1 + β)h ζh P(c, y) = [1 γ(y)] δ(c) + γ(y)e (3) using γ(y) = () exp( 2βs). (31) Therefore, we have P c (c, y) = e {H(ζ) [H(ζ) + ζh (ζ)]. (32) The function H(ζ) is normalized, therefore { P c (c, y) dc = H(ζ) [H(ζ) + ζh (ζ)] = 1 dζ, (33) [1 1], (34) = 1 (35) using partial integration for the evaluation of ζh (ζ) dζ. Starting from equation (32), also the higher moments of the pdf can be checked: c n = = γ(y) P(c, y)c n dc, = γ(y)e 2ns (36) P c (c, y)c n dc for n > 1, (37) { ζ n H(ζ) [ζn H(ζ) + ζ n+1 H (ζ)] dζ, (38) = exp[ (β + n)] {()H n [H n (n + 1)H n ], (39) with H n = H(ζ)ζ n dζ, (4) = H n [1 + (β + n)] exp[ (β + n)], (41) = H n [1 + 2 y /σ n ] exp[ 2 y /σ n ]. (42) Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)

7 Lutz Janicke: Density fluctuations in the model COFIN 5 This is the required expression of equation (6). Replacing H(ζ) in equation (32) by the Gamma distribution g 2 (ζ) of equation (9) g 2 (ζ) = 4ζ c 2 exp( 2ζ/c ), (43) [ 4 ζg 2 (ζ) = ζ exp( 2ζ/c c 2 ) 8ζ c 3 [ = g 2 (ζ) 1 2ζ ] c yields the pdf P c (c, y) = e g 2 (ζ) P c (c, y) = 2 c e ϕe ϕ {1 ] exp( 2ζ/c ), (44) (45) { ] 1 [2 2 cc e, (46) (2 ϕ), (47) ϕ = 2 c c e. (48) It is easily verified that this form of P c (c, y) is normalized: P c (c, y) dc = ϕe {1 ϕ (2 ϕ) dϕ, (49) = 1. However, P c is not always non-negative. From equation (47) it can be seen, that P c is negative if 1 + [β 2 + ϕ] is negative. This occurs for small values of c (small ϕ) and large values of s because β < 2. The same holds for the P c of equation (COFIN.166). This deficiency can be avoided if the general form of the Gamma distribution with p < 2 is used. If we replace H(ζ) in equation (32) by g p (ζ) = (5) p c Γ(p) ϕp 1 exp( ϕ), (51) with ϕ = pζ/c, (52) ( ) p 1 ζg p pζ p(ζ) = [p 1 pζ/c ] exp( pζ/c ), c Γ(p) (53) = c we get the following relation p c Γ(p) ϕp 1 [p 1 ϕ] exp( ϕ) (54) pϕp [β p + ϕ] P c (c, y) = e exp( ϕ). (55) c Γ(p) Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)

8 Lutz Janicke: Density fluctuations in the model COFIN 6 It can be seen that P c is non-negative only if p β 1.9. It might be possible on the base of the available data to adjust the value of p or β appropriately. A better solution probably is to replace the Gamma distribution by a Weibull distribution w(c): w p (c) = p ĉ ( c p 1 [ ( c p ] exp, (56) ĉ) ĉ) ĉ = c /Γ(1 + 1/p), (57) cw p(c) = p ( c p 1 [ ( c p ] [ ( c p ] exp p 1 p. ĉ ĉ) ĉ) ĉ) (58) In this case, equation (32) takes the form P c (c, y) = e p ĉ ψ p 1 exp ( ψp ) [ (1 + p 1 pψ p )], (59) with ψ = ζ ĉ = c c e Γ(1 + 1/p), (6) = e p c ψ p 1 Γ(1 + 1/p) exp ( ψ p ) [1 + (β p + pψ p )]. (61) The distribution of the concentration fluctuations on the plume axis in the moving frame is well approximated by a Weibull distribution with p 1.6, as can be seen from the following picture: 1. Distribution functions.8 probability density C/Cm Using this Weibull distribution the pdf P c is non-negative. Comparison of distribution functions: Gamma distribution with p = 2 (solid, red), Weibull distribution with p = 1.6 (dashed, green) and experimental data (dots). Berichte zur Umweltphysik, Nummer 4, Auflage 2, März 26. ISSN (Print), ISSN (Internet)