An Analytical Model for Atmospheric Distribution. and Transport of Pollutants from Area Source

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1 nt. J. Contemp. Math. Scences, Vol. 6, 0, no. 6, An Analytcal Model for Atmospherc Dstrbuton and Transport of Pollutants from Area Source D.V.S. Kushwah, V. S. Dhakad and 3 Vandana Kushwah Department of Mathematcs dvs.kushwah@yahoo.com 3 Department of Chemstry nsttute of nformaton Technology and Management, Gwalor (M.P.), nda Department of Mathematcs ntegral nsttute of nformaton Technology and Management Gwalor (M.P.), nda Abstract: Fnte element method has been used for study of dsperson of ar pollutants emtted from area source. Ths method generally deals wth the problem wth two dmensonal (area) doman, where n one drecton the nature of the process s regular and t s rregular n other drecton. The area under consderaton s non homogeneous. The mathematcal study has been made through varatonal fnte element method. Model ncludes the chemcal reacton of the pollutants whch s consdered to vary wth vertcal heght through troposphere. As troposphere has been dvded nto three layers stuated at dfferent heght from the datum each of these layers vares n the concentraton of pollutants and hence chemcal reacton wll take place wth dfferent rate n these layers. The result supports that the concentraton decreases rapdly up to a certan heght from the surface, ths decreases become smooth and takes more tme to become gradual. The model s supported by approprate boundary and ntal condtons. Keywords: Concentraton, Pollutants, Troposphere NTRODUCTON The dffuson process n the atmosphere from a pont area source has been studed n varous contexts such as ar polluton [] [6] and pheromone dffuson of bologcal organsm n the atmosphere [8]. n partcular, the dsperson of an ar pollutant from a pont source has been nvestgated by many workers. Durng

2 98 D.V.S. Kushwah, V. S. Dhakad and V. Kushwah dsperson n the atmosphere the prmary speces s sometmes converted to secondary speces and both prmary and secondary speces may be removed by ranout/washout and the lke. Alam and Senfeld [] studed the dsperson of sulfur doxde and sulfate n the atmosphere by solvng steady state three dmensonal dffuson equatons and dscussed the effects of the converson and removal processes by consderng these as frst order process. On the other hand at the ndustral agrculture ste, where the sources of pollutant emsson are two dmensonal, t may be possble that under certan crcumstances the emttence of pollutants from these sources may be non steady and thus t affects the reacton rate whch vares at dfferent heghts n the atmosphere [9]. For example, n atmosphere the concentraton of speces such as NO and SO vares extensvely wth respect to tme. Thus, study of transent problems would be more approprate n dealng wth such cases. n the followng work an attempt has been made to nvestgate transent state dsperson of pollutants. The purpose of ths work s to develop an analytcal model for atmospherc dstrbuton and transport of pollutants emtted from area source. However, the concentraton dstrbuton of pollutants s assumed to be unsteady. One dmensonal unsteady state model ncludes the chemcal reacton of the pollutant whch s consdered to vary wth vertcal heght throughout the troposphere. Hence, the fnte element method would be more approprate n the present case as the troposphere has been dvded nto three layers (Heubner [7]). Obvously each of these layers vares n the concentraton of the pollutants and hence chemcal reacton wll take place wth dfferent rate n these layers. The model s supported by approprate boundary and ntal condtons. MATHEMATCAL FORMULATON A mathematcal equaton descrbng the tme averaged concentraton of chemcally reactve speces wth wnd velocty and pollutant generaton s gven as: C C C + U = D + Q + R... () t x Where s vertcal dstance from the ground ( = 0). For the steady atmospherc condton.e. when there s no wnd, equaton () reduces to C C = D + Q + R... () t Where C s the pollutant concentraton n the ar, D, Q and R are eddy dffusvty, pollutant generaton rate and rate of chemcal reacton respectvely. The mathematcal descrpton of the model () s completed by the followng ntal and boundary condtons. ntally, t s assumed that concentraton of pollutants n the atmosphere s neglgble and therefore, the ntal concentraton s taken as C = 0 at t = 0... (3)

3 Model for atmospherc dstrbuton and transport of pollutants 99 At the heght = 0, polluton concentraton s prescrbed as C = C 0 at = 0... (4) where, 0 s the vertcal dstance from datum. The pollutant concentraton approaches ero far from the source. C = 0 at large... (5) SOLUTON Equaton () on comparng wth Euler Lagrange's equaton gves the followng varatonal form for layers. Z () () () C () () () () C = D Q C R C + d t Z... (6) The lower atmosphere (troposphere) s dvded nto three layers. The generaton s assumed to be takng place at constant rate n dfferent layers. n lght of dscrmnaton of lower atmosphere, the layer wse descrpton of parameters s gven as follows: Frst Layer: 0 D () = D, Q () = Q, R () = R... (7) Second Layer: D () = D, Q () = Q, R () = R... (8) Thrd Layer: 3 D () = D 3, Q () = Q 3, R () = R 3... (9) Here we are assumng that the pollutant concentraton C (say) form e th layer, vares lnearly wthn the layer, however t has defnte unknown value C. ( e) ( e) ( e) a + b, < < + C = c, = for = 0,,... () Equaton () can be put n matrx form as C = P T N C... () where, P T = [ ] Expressng the functonal n (6) symbolcally as follows: N = and C + C = C+ Expressng the functonal n (6) symbolcally as follows: () () () = d k + C... () where, () () () C d = D d... (3) () k () () () = (Z + R )C d... (4)

4 300 D.V.S. Kushwah, V. S. Dhakad and V. Kushwah () C () () C = D d... (5) Now use the parameter values from equaton (7), (8) and (9) and nterpolatng functon defned n () nto (3), (4) and (5) n order to evaluate these ntegrals and hence n equaton (). The ntegrals are then assembled to obtan = 3 = Followng Rt method to exteme wth respect to C by puttng d = 0... (6) dc dc U KC = K... (7) dt where, C C = C K and U are the matrces of order and K s a vector of order, whose values for e th element s gven below : K K D = (Z j Z ) Q + R = (Z j Z ) and U = Z j Z 6 Solvng the system (7) analytcally, employng the method of Laplace transform, to obtan the concentraton C, =, UC + K C = K t... (8) NUMERCAL RESULTS AND DSCUSSON The equaton (8) s computed numercally for concentraton dstrbuton usng the parameter values gven n the table below:

5 Model for atmospherc dstrbuton and transport of pollutants 30 Table - Layer D (Km /sec.) Q (Kg./sec. ) R (Km 3 /gm/ sec.) (Km) n ths paper the concentraton of the pollutants has been computed at dfferent heghts n the atmosphere (troposphere), n partcular for dfferent tme. The results are drawn graphcally n (fg. ). The results are compatble wth the boundary condtons as s shown n fg.. The concentraton at the hghest level.e. h=5 km. becomes ero and as we move downward t ncreases gradually and becomes maxmum at the surface (known concentraton, 90 μgms.). At t = 0 C's ( =,) are ero due to ntal condton (fg. ). t s observed from the graph that there s gradual ncrease n the pollutant concentraton upto the heght of 5 km for tme 30, 60, 90 and 0 mn. After ths heght the concentraton becomes stead Heght Z (km.) 5 Concentraton C (μgms.) C C Concentraton C (μgms.) Fg. : Graph between Concentraton and Heght for Dfferent Values of Tme Tme T (Mnutes) Fg. : Graph between Concentraton and Tme

6 30 D.V.S. Kushwah, V. S. Dhakad and V. Kushwah REFERENCES [] Alam, M.K. and Senfeld, J.H., 98 Soluton of steady state three dmensonal atmospherc dffuson equaton for sulphurdoxde and sulphate dsperson from pont sources. Atmospherc envronment, 5, -5. [] Carmchael, G.R. and Peters, L.K., 980 The transport chemcal transformaton and removal of SO and sulphate n the Eastern Unted State, n atmospherc polluton, studes n envronment scence, MM Benarc Ed., Elsever, Amstredan, [3] Cruten, P.J. and Fshman, J., 977 Average concentraton of OH n the northern hemsphere troposphere and the budgets of CH4, CO and H Geophyscs Res. Lett., 4, [4] Crank, J., 975 The Mathematcs of dffuson Clarendon press, Oxford, England, -57 [5] Ermak, D..,977 An analytcal model for ar pollutants transport and deposton from a pont source. Atmospherc envronment,, 3-37 [6] Falls, A.H., and Senfeld, J.H., 978 contnued development of a knetc mechansm for photochemcal smong. Envronmental Scence Technology,, [7] Heubner, K.H., 975 Text book of fnte element method for engneers, pp: 5-73 [8] McMohan, T.A., Denson, P.J. and Flemng, R.A., 976 A long dstance ar pollutons transport mode ncorporatng wash out and dry deposton component J. Appl. Met. 7, [9] Tokgolu, A., Saxena, V.P., Ocak, S., Erthuak, F., 00 Transent concentraton dstrbuton of pollutants over sparta n the proc. Of sec. nt. sym. On ar qualty management urban regonal and global scales stambul (Turkey). Receved: December, 0