Some characteristic parameters of Gaussian plume model

MAUSAM, 63, 1 (January 2012), 123-128 551.510.42 : 551.511.61 Some characteristic parameters of Gaussian plume model M. ABDEL-WAHAB, KHALED S. M. ESSA*, M. EMBABY* and SAWSAN E. M. ELSAID* Meteorology and Astronomy Department, Faculty of Science, Cairo, University * Mathematics and Theoretical Physics Dept., NRC, Atomic Energy Authority, Cairo- Egypt (Received 11 June 2010, Modified 5 January 2011) e mail : [email protected] lkj & m/okZ/kj lkanz.k forj.k dh izFke pkj fLFkfr;ksa ¼dsanzd] Hksn] oS"kE; vkSj dVksZfll½ dks Kkr djus gsrq jSf[kd lzksr ds fy, folj.k lehdj.k ds fy, xkSfl;u lkWY;w’ku dk mi;ksx fd;k x;k gSA bles vf/kdre lkanz.k Lrj dh fLFkfr ,oa ifjek.k dh x.kuk dh xbZ gSA blesa fiPNd vfHkogu iou xfr dk Hkh vkdyu fd;k x;k gSA Hkwry Lrj lkanz.k ds lehdj.kksa dh rqyuk iou Vuy ekiuksa ls dh xbZ gSA ABSTRACT. The Gaussian solution of the diffusion equation for line source is used to have the first four moments of the vertical concentration distribution (centroid, variance, skewness, and kurtosis). The magnitude and position of maximum concentration level were evaluated. Also the plume advection wind speed is estimated. Equations for the ground level concentration were compared with wind tunnel measurements. Key words ‒ Moments, Diffusion equation, Plume advection wind speed. 1. Introduction The modeling of dispersion has been performed by a Gaussian approach that takes account of atmospheric turbulence assuming simple formula for concentration distribution, in which the dispersion parameters depend on downwind distance and the Pasquill-Gifford scheme (Arya, 1999). The statistical description distribution for non Gaussian concentration model is studied by Brown et al. (1997). The moments and maximum ground level concentration in two dimensions are obtained by Tirabassi et al. (2009). In this paper, we derive the first four moments of the vertical concentration distribution from Gaussian plume model (Pasquill and smith 1984). Equations for the maximum ground level concentration along the centerline and its locations were calculated. The effective wind speed of a plume is derived. 2. The Gaussian model diffusion equation can be written as (Pasquill and Smith, 1984). U           K y C C    K z C  S x y  y  z  z  (1) where x, y, and z are the longitudinal, lateral, and vertical directions, respectively, U is the mean longitudinal wind speed, C is the mean concentration, K y and K z are the lateral and vertical eddy diffusivities, respectively, and S is a sink or source term. The solution of equation (1) can be written as (Pasquilll (1974) : C  x, y , z   Q  y2   Bz  s  exp     exp 2  2 2  y zU   z    y A     (2) From this solution, one can concentration of a point source where, One assumes that the turbulent mass flux can be described by a gradient-transfer closure. The steady-state estimate the Q - Actual emission rate of the point source (μg/s). (123) 124 MAUSAM, 63, 1 (January 2012) y - Crosswind dispersion plume spread (m). hence   2   1       B    s   s   x  Inz  z x   exp     0  1    B      s  U - Mean wind speed (m/s). z - Mean plume vertical height (m). s - Parameter depends on the stability.  A, B - Parameters depend on the stability, where B  2 / s  / 1 / s  [Van Ulden (1978)]. 3. Moments of the Gaussian distribution Where :  p    x p 1e  x dx 0 To measure the variance  z2 x  of the concentration distribution, put m = 2, in equation (3) and substituting from equation (2) in equation (3), we obtain that: The moments of centroid, the variance, skewness and kurtosis can be obtained from the following equation:    Bz  s    Bz  s  2   exp 2 exp z dz z z     dz        z     z   0 0   dz  dx m  z  z  C x, z dz 0 z (3)   C x, z dz 2 dz  dx 0 is  - function. To measure the centroid z of the concentration distribution, put m = 1 and substituting from equation (2) in equation (3), we obtain that:   Bz  s  exp    z  dz 0      Bz  s   exp  z  dz 0    (6) Using separation of variables and integrating equation (6), we obtain the variance as follows:    Bz  s    Bz  s  exp exp z dz z       dz         z     z   dz 0 0   dx   Bz  s  exp    z   dz 0     3 2 2  1   s   2 B s   B  s   x        dx  2   1  0   2 z0 z B     s   z (4) dz Using separation of variables and integrating equation (4), we obtain the centroid as follows:   3  2 2  1       2 B   B     1 1   s   s  x s      z z 0  2 1  B     s  Hence, z  z0  2 1     B dz s s      1 z B   s        x  dx (7) 0 1 z 0 B 2   s  z m2  z2 x    3  2     2 BT    s  s 1 B 2    z 0 x    s   2 1    B   s    2  1      B   s  s   x  Inz   In z     0   1 B      s    (5) ABDEL-WAHAB et al.: PARAMETERS OF GAUSSIAN PLUME MODEL   Bz  4  z exp  z  0  To measure the skewness sk (x) of the concentration distribution, put m=3 in equation (3) and substituting from equation (2) in equation (3), to have:    Bz  s    Bz  s  3 2   exp 3 exp dz z z z     dz         z     z   0 0     Bz  s    Bz  s  2 3  3z  z exp     dz  z  exp     dz   z     z   dz 0 0  s  dx   Bz    exp  z   dz 0   (8)   Using separation of variables and integrating equation (8), we obtain the skewness as follows: z  z0  4 3 2 2 2  1    s   3B s   3B  s   B  s   x          dx  3 1   0 3   z  B     s   dz  4  1 2 3    3B   3B 2    B 2    1  1 1    s  s  s x s    2  z 02 z 2    3 1 B     s     (9) 1 2              1   z 02 B 3     s  : sk x      z m 3  4  3    2   6 B    s s          2   3 1  2 2  z0 x   B  s    6 B  s          3 1    2B      s       To measure the kurtosis "ku" of the concentration distribution, put m=4 in equation (3) and substituting from equation (2) in equation (3), we get:  4z dz  dx 2   Bz   dz  4 z  z 3 exp     z   0 s  125 s dz  s   Bz   z 2 exp     dz    z      0 2z 3    s s  4  Bz    Bz    dz  z exp     dz   z   z      0  z exp   0    Bz  s  exp    z   dz 0    (10) Separating of variables and integrating eqn.(10), we obtain the kurtosis as follow :  5 4 2  3   s   4 B s   4 B  s            3 2 4 1 x   2 B     B   z dz s s  dx  4  1   4 z z0 B   0  s          5 4 2  3   s   4 B s   4 B  s            2 1 3   4      2 B    B   1 s s x  In z  z0     4 4 1 B     s              (11)   5   4 2  3  4   16 B   16 B     s  s    s      3 2 4 1    8 B     4 B      s s  x  z   z m  4 ku  exp 0  1   B 4       s              126 MAUSAM, 63, 1 (January 2012) 0.20 0.25 Observed Variance Observed Centroid Predicted Variance 0.16 0.15 0.12 Variance (m) Centroid (m) predicted Centroid 0.20 0.10 0.08 0.04 0.05 0.00 0.00 0 2 4 6 8 0 2 4 6 8 Downwind distance (m) Downwind distance (m) 2.00 6.00 Observed Kurtosis Observed Skewness Predicted Kurtosis Predicted Skewness 1.60 Kurtosis (m) Skewness (m) 5.00 1.20 4.00 0.80 0.40 3.00 0.00 2.00 4.00 6.00 8.00 Downwind distance (m) 0.00 2.00 4.00 6.00 8.00 Downwind distance (m) Fig. 1. Predicted centroid, variance, skewness and kurtosis of the vertical concentration distribution for near surface are compared with wind tunnel measured via downwind distance (Khurshudyan et al. 1981) 4. where σv is the standard deviation of the crosswind velocity component. Maximum ground level concentration Substituting with z = 0 and  y  vx in equation U (2), to have: CGLC   y 2U 2 exp   2 2 x 2 2  v x z v  AQ     (12) To estimate the maximum downwind distance (x), differentiating the above equation with respect to x, setting the result equal to zero, and then solving for x, to get:   y 2 U 2  C  AQ   exp   2 2 x 2  x  2  v z v     y 2U 2 1   (13)     2 x2 x2    v ABDEL-WAHAB et al.: PARAMETERS OF GAUSSIAN PLUME MODEL C  0 , then the maximum downwind x distance of (x) becomes: 1.20 Putting xmax  yU v Calculated data (14) Substituting from equation (14) in equation (12) to get the maximum ground level concentration "CGLCmax" by putting y = 0 as follows: CGLC max  Laboratory data 0.80 GLC/GLC max  y 2U 2 1      2 x4  x2   0   v  127 0.40  y 2U 2   exp  2 v2 x 2  2  v xmax z   AQ 0.00 CGLC max  0.00 AQ 2 ey U z Dividing equation (12) by equation (15) to get the ratio between CGLC and CGLC max as follows:  e xmax CGLC z  0   CGLC max z  0   x    y 2 U 2   exp      2 2 x 2  v    (16) Fig. 1 Shows the computed centroid, variance, skewness, and kurtosis for surface that are compared with measurements made through wind tunnel in neutral stability ( Khurshudyan et al. 1981). Recall that skewness equal 0.7 and kurtosis equals 3 for reflected Gaussian distribution. The downwind variations of the measured centroid, variance, skewness and kurtosis agree well with model predictions. 5. Calculating plume advection velocity The mean plume wind speed U is defined as:  U  U z C y x, z dz 0 (17)   C y x, z dz Where the wind speed is given as: U z   u* kv   z  In   z 0    (18) 0.80 1.20 1.60 2.00 Fig. 2. Non–dimensional maximum ground level concentrations as function of non-dimensional downwind distance. Measurements include wind-tunnel experiments Where u* is the friction velocity, kv is von-Kaman constant = 0.4 and z0 is the roughness height (m). We substitute from equations (5), (18) in equation (17) to get the mean plume wind speed in the boundary layer: 1 1  1 1   Inz     Inz 0     1 s s s s    U 1   s (19) Fig. 2 shows a non-dimensional plot of ground level concentration and downwind distance for surface releases from neutral USEPA wind tunnel point source dispersion experiment (Brown et al. 1993). The figure illustrates ground level concentration by "GLCmax" [equation (16)] and the downwind distance "xmax" [equation (14)] have good agreement between the predicted and measured data. 6. 0 0.40 x/xmax (15) Conclusion Equations of the centroid, variance, skewness, and kurtosis compared well with wind tunnel plume dispersion measurements. The equations derived for the magnitude and location of "GLCmax" were found to be agreement well with wind-tunnel measured data. 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