Pyrex Journals

Transcription

1 Prex Journal Prex Journal of Ecolog and the Natural Environment Vol 1 (1) pp June, Copright 015 Prex Journal Review Paper Solving the advection diffuion equation in three dimenion in neutral cae haled S.M Ea * and S. E. M. Elaid Mathematic and Theoretical Phic, NRC, Atomic Energ Authorit, Cairo- Egpt Accepted 9 th June, 015 In thi paper, the advection diffuion equation i olved in three dimenion pace (x,, z) uing eparation of variable technique to evaluate pollutant concentration per emiion rate, taking edd diffuivitie of pollutant and mean wind peed in neutral cae. The ued data et wa oberved from the atmopheric diffuion experiment were made at Indian Intitute of Technolog (ITT) Delhi. The trace of ulfur hexafluoride (SF 6 ) wa releaed at a rate q=50 ml min -1. The reult how that the predicated Gauian Preent, previou with u*, and lender with u* are in agreement with the obervation at 100m downwind and mot cae lie within a factor of two. e word: Diffuion, Neutral Cae; Edd diffuivit, Gauian Model. INTRODUCTION Atmopheric diperion ing refer to the mathematical decription of contaminant tranport in the atmophere. The term diperion in thi context i ued to decribe the combination of diffuion (due to turbulent edd motion) and advection (due to the wind) that occur within the air near the Earth urface (John- 011). The analtical olution of the atmopheric diffuion equation in different hape depend on Gauian and non-gauian olution. An analtical olution with power law for the wind peed and edd diffuivit with the realitic aumption wa tudied b (Demuth-1978). The olution ha been implemented in the APPA-G (Tirabai et al. 1986). (Lin and Hildemann, 1997) extended the olution of (Demuth-1978) under boundar condition uitable for dr depoition on the ground. Etimating of crowind integrated Gauian and non- Gauian concentration through different diperion cheme wa tudied b (Ea et al. 011). In thi paper the advection diffuion equation (ADE) i olved in three direction wa to obtain the crowind integrated ground level concentration uing eparation variable of technique, taking edd diffuivitie of pollutant and mean wind peed in neutral cae. The ued data et wa oberved from the atmopheric diffuion experiment were made in the Indian intitute of Technolog (ITT) Delhi. The trace SF 6 wa releaed at a rate q=50 ml min -1. Data and Methodolog The tead tate the advection diffuion equation for diperion of a non reactive contaminate releaed from continuou ource (Seinfeld, 1986). C d C d C U z z x d z d z z z C d C d C d C U z x d d z d z (1) Where u * z U z ln kv z 0 U x k w z () z v Where β= 3.61*((u*/U) ^), z 0 i the roughne height (0.78), z i the vertical height at 15 m and Q=50ml min -1 (Ea et. al. 009). Where x, and z are the coordinate in the along-wind, cro wind and vertical direction repectivel. C i the mean concentration of pollutant and U (z) i the mean wind peed in the downwind direction, and z are edd diffuivitie of pollutant in the crowind and vertical direction repectivel. Correponding Author: mohamedkm56@ ahoo.com

2 E a a n d E l a i d P Y R X. J. E c o. N a t. E n v 00 Where k v i the von arman contant (k v ~0.4), z i the tack height (115m) and w* i the convection velocit cale, and β i a parameter depend on the tabilit condition. u* i frication velocit. Let the olution of the equation (1) in the form C(x,, z) =X(x) Y() Z (z) (3) Subtituting from equation (3) after differentiation (1) repect to x,, z and divided on (X(x) Y() Z (z)), we get that:- X Y Z Z U z z z X ( x ) Y ( ) Z ( z ) Z ( z ) X X ( x ) C e X ( x ) U ( z ) x U ( z ) 1 Y Y ( ) 0 Y ( ) C co C 3in z o 0 z z z z Z ( z ) Z z Z ( z ) 0 Z ( z ) AJ BY C(x,,z) i finite at z=0 that B=0 Z ( z ) AJ o z Subtituting from equation (4) in equation (3), we get that:- x U ( z ) C x,, z e J o co in z Let α= C 1 C A, β= C 1 C 3 A 4 x x J0 x where J 0 () i zero Beel function in firt order. (Shamu, 1980). (5) Equation (5) i ubjected to the following boundar condition:- 1-The flux at the top of the mixing laer can be given b:- (4) z C 0 at z=0, z z -The flux at cro wind of the mixing laer can be given b: - C x, z 0 at =0, 3-the pollutant i releaed from an elevated point ource of trength Q located at the point (0,, z ) i written in the form:- UC(x,, z) =Q δ(- ) δ (z-z ) at x=0 (iii) where δ i Dirac delta function. Subtituting from equation (ii) in equation (5), we get that:- 0 n Subtituting from and equation (iii) in equation (3), we get that:- J 0 U z Q n z Subtituting from equation (6) and equation (7) in equation (5), we get that:- n n co x C x,, z U ( z ) n e Jo q z n U z J 0 con z Where n 4 n n J z 4 z z (i) (ii) (6) (7) (8)

3 E a a n d E l a i d P Y R X. J. E c o. N a t. E n v 003 Reult and Dicuion Table 1: wind peed and frication velocit in different run are 0.78m (roughne length ) at 15m U(m/) run no. u * (m/) (at 15m) Table : peak value of concentration in part per trillion (ppt) oberved and Predicated at 50 downwind of the ource Run No, Oberved Previou Preent Model Slender plume Gauian

4 E a a n d E l a i d P Y R X. J. E c o. N a t. E n v Table 3: peak value of concentration in part per trillion (ppt) oberved and predicated at 100 downwind of the ource Run No oberved Previou Preent Slender plume with u * Gauian With u * Table how peak value of trace concentration reult from the Previou with u*, preent, the Slender plume and Gauian in part per trillion (ppt) oberved and predicated at 50 m downwind of the ource. Table (3) how peak value of trace concentration reult from the with u*, preent, the Slender plume and Gauian in part per trillion (ppt) (Ea et al. (009) oberved and predicated at 100m downwind of the ource. Table -4- and Table -5- how the ratio predicated and oberved concentration in part per trillion (ppt) at 50 m and 100m downwind of the ource. Figure 1: Logarithmic plot of oberved at 50 downwind of the ource veru predicated concentration (ppt) Middle line i the one to one line where a other repreent the factor of two

5 E a a n d E l a i d P Y R X. J. E c o. N a t. E n v 005 Figure : Logarithmic plot of oberved at 100 downwind of the ource veru predicated concentration (ppt) Middle line i the one to one line where a other repreent the factor of two Figure 3: Scatter diagram of the ratio of predicated to oberved concentration veru the oberved concentration Fig-1- give the logarithmic plot for peak value of trace concentration reult from the with u*, the Slender plume and the Gauian.Thi figure how that the reult from Gauian with u*, Preent and with u* lie within a factor of two with the obervation at 50m downwind ditance. Fig-- give the logarithmic plot for peak value of trace concentration reult from the with u*, the Slender plume and the Gauian.Thi figure how that the reult from Gauian with u*, Preent and with u* compare well with the obervation at 100m downwind and mot cae lie within a factor of two. The ued data et wa oberved from the atmopheric diffuion experiment were made at Indian Intitute of Technolog (ITT) Delhi. The trace SF6 wa releaed at a rate q=50 ml min -1. Concluion

6 E a a n d E l a i d P Y R X. J. E c o. N a t. E n v 006 In thi paper, the advection diffuion equation i olved in three dimenion pace (x,,z) uing eparation of variable technique to evaluate crowind integrated of pollutant concentration per emiion rate, taking edd diffuivitie of pollutant and mean wind peed in neutral cae. The ued data et wa oberved from the atmopheric diffuion experiment were made at Indian Intitute of Technolog (ITT) Delhi. The trace SF 6 wa releaed at a rate q=50 ml min -1. Show that the reult from the Gauian with u*, Preent and with u* in agreement with the obervation at 100m downwind and mot cae lie within a factor of two. Comparion between different at 50,100 m downwind of the ource according to tandard tatitical performance meaure i calculated. We find mot cae lie within a factor of two. Reference Demuth, C., (1978). "A contribution to the analtical tead olution of the diffuion equation". Atom. Environ. 1, 155. Ea,. S. M. and E, A. Fouad, (011). Etimated of crowind integrated Gauian and non-gauian concentration b uing different diperion cheme. Autralian Journal of Baic and Applied Science, 5(11): Ea haled S. M. and El-Otaif Maha. S. (009). Simulation of hort range hermitzed atmopheric diperion in low wind convective condition, Diater advance Journal. (1). John M. (011). The mathematical of atmopheric diperion ing Societ for Indutrial Applied Mathematic Seinfeld, J. H. (1986). "Atmopheric Chemitr and Phic of air pollution", New York: John Wile & Son Lin, J. S. and Hildemann L. M., (1997). "A generalized mathematical cheme to analtical Solve the atmopheric diffuion equation with dr depoition", Atom. Environ. 31, Shamu- (1980). "Theorie and example in Mathematic for Engineering and Scientific". Tirabai-T. Tagliazucca, m.- zannetti, p.kappa-g, (1986). "A non-gauian plume diperion : decription and evaluation againt tracer meaurement".jaca