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Accounting for the source strength in the solution of the diffusion equation: Alternative mathematical formulations
Atmospheric Environment 33 (1999) 1327—1330
Technical Note Accounting for the source strength in the solution of the diffusion equation: Alternative mathematical formulations Maithili Sharan *, Anil Kumar Yadav, M.P. Singh, Suman Gupta Centre for Atmospheric Sciences, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India Department of Applied Mathematics, Guru Jambheshwar University, Hisar 125001, Haryana, India Govt Girls Senior Secondary School C1, Madangir, New Delhi, India Received 23 March 1998; accepted 23 July 1998
Abstract The source term is an integral part of the mathematical formulation of the problem leading to the well-known solutions to the diffusion equation. Its representation through the material balance and through the boundary of the domain is discussed. Three alternative mathematical formulations of the problem have been obtained. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Diffusion equation; Source term representation; Mathematical formulation
1. Introduction The Gaussian plume models have been used extensively for air quality predictions especially by the regulatory agencies all over the world. The popularity of these models for estimating air pollutant concentrations from various sources lies in the fact that they are simple, easy to understand and use, and are conceptually and computationally efficient. Extensive information on these models can be found in the literature (Pasquill and Smith, 1983; Seinfeld, 1986; Zannetti, 1990; Turner, 1994). Further, the form of the Gaussian plume solution and the mathematical problem associated with it have been widely discussed (Csanady, 1973; Seinfeld, 1986). An important
*Corresponding author. Fax: 00 91 11 686 2037; e-mail: [email protected].
aspect of the problem is the representation of a source in the model formulation. The source term can be accounted for in two possible ways: either through the material balance or through the boundary of the domain. The former seems quite obvious physically whereas the latter needs elaborate description mathematically. Sometimes, the source term is accounted for through the material balance equation even though it lies on the boundary (Seinfeld, 1986) of the domain. Although it is mathematically inconsistent to take the source in this manner, it gives the correct solution. The other way commonly found in the literature to account for the source is through one of the boundaries (either x"0 plane or z"0 plane in case of ground level release) and is purely from physical considerations. Quite often, it is relatively easier to obtain a closed form solution of the resulting problem by accounting for the source using one of these possible ways.
1352-2310/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 2 - 2 3 1 0 ( 9 8 ) 0 0 2 6 8 - 4
1328
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
In this note, we have considered a point source at the ground and consequently obtained equivalent mathematical formulations of the same problem. The equivalent formulations are useful in acquiring a deeper insight into this simple problem and its possible extensions.
This is a law of conservation of the flux of pollutant in the differential form. Integrating Eq. (4) from !R to x, we can obtain the law of conservation in an integral form as
2. Mathematical description
where
Assuming an incompressible flow, the equation governing the atmospheric dispersion of a pollutant from a point source located at the origin (0, 0, 0), in the absence of removal mechanisms, can be written as
* * *C * *C (ºC)" K # K W *y X *z *x *y *z #Qd(x) d(y) d(z),
(1)
where d(.) is Dirac’s delta function, C is the mean pollutant concentration, º is the mean wind speed directed towards the positive x-axis and thus we have assumed º'0, Q is the source strength, and K and K are the W X eddy diffusivities in the y- and z-directions, respectively. Generally, in the direction of mean wind, transport due to advection is dominant over turbulent diffusion and hence the latter is ignored. Following are the relevant boundary conditions: (i) C"0 at x"0 (deleted neighbourhood) (2a) Here, deleted neighbourhood means the region excluding a small neighbourhood of the point where the source is located: (ii) CP0 as "y", zPR, (iii) !K
(2b)
*C "0 at z"0 X *z (deleted neighbourhood).
(2c)
In Eq. (1), the source term, has been accounted through material balance. Alternatively, it can be accounted through one of the boundaries as mentioned earlier.
h(x)"
0, x(0 1, x'0
(5)
(6)
is the step function. Eq. (5) shows that the double integral (ºC) dy dz"0 \ for all negative values of x, but the integrated (ºC) is non-negative and so ºC"0 for all x(0. This implies C"0 for x(0. Even from physical consideration, the neglect of longitudinal diffusion results in the approximate diffusion equation (1) whose solution is zero at every point upwind of the source (x(0). For x'0, Eq. (5) may be rewritten as (ºC) dy dz"Q. \
(7)
This means that the net flux in the downwind direction across any plane normal to x-axis equals the source strength Q. For x(0, we can use C"0 as the solution of Eq. (1) and for x'0, the source term will no longer contribute in Eq. (1). Eq. (1), then can be solved with an appropriate boundary condition at x"0 accounting for the source. Because of the continuity of the flux Q of pollutant for x'0, Eq. (7) must hold even at x"0. Therefore, the boundary condition at x"0 may be written as ºC(x, y, z)"Q d(y)d(z).
(8)
2.2. Source through z"0 plane
2.1. Source through x"0 plane Integrating Eq. (1) with respect to y from !R to R and using conditions (2b), we get
* *C * (ºC) dy" K dy#Qd(x) d(z). X *z *x *z \ \ (3) Integrating Eq. (3) with respect to z from 0 to R and using conditions (2b) and (2c), we obtain * (ºC) dy dz"Q d(x). *x \
(ºC) dy dz"Qh(x) \
(4)
Now, integrating Eq. (3) with respect to x from 0 to R and using boundary conditions (2a) and (2b) along with the fact that concentration approaches zero far away from the source to obtain
* *C !K dy dx"Q d(z). X *z *z \
(9)
Integrating Eq. (9) from 0 to z, we obtain
*C !K dy dx"Q for all z'0. X *z \
(10)
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
For z'0, the double integral in Eq. (10) represents the flux across any horizontal plane. In this region, the source term will no longer contribute in Eq. (1). Eq. (1), then can be solved with an appropriate boundary condition at z"0. Because of the continuity of the flux Q of pollutant for z'0, Eq. (10) must hold even at z"0. Thus, the boundary condition at z"0 may be written as *C !K "Q d(x) d(y). X *z
(11)
Conditions (8) and (11) give the alternatives for accounting for the source term and thus help in formulating equivalent mathematical problems. Physically, condition (8) [(11)] means that material flux across the vertical (horizontal) plane, containing the source, is essentially from a single point and equals the source strength Q. Further, the relation (7) [(10)] being satisfied for all positive x [z], means that material flux across any vertical (horizontal) plane perpendicular to x-axis [z-axis] equals Q, as long as it is assumed that there is no chemical reaction or removal by any other process.
3. Alternative formulations Conditions (8) and (11) allow us to have the following three alternative mathematical formulations of the same problem: First:
* *C * *C *C º " K # K X *z *x *y W *y *z #Q d(x) d(y) d(z).
(12)
CP0 as "y", zPR
1329
¹hird: º
* *C * *C *C " K # K X *z *x *y W *y *z
CP0 as "y", zPR
(14)
C"0 at x"0 (deleted neighbourhood) *C !K "Q d(x) d(y) at z"0. X *z All the above formulations with constant K’s yield the same solution of the form
º y z Q ! # C(x, y, z)" e 4x KW KX (15) 2nx (K K W X By assuming the K’s as linear functions of downwind distance (K "bºx, K "cºx where b and c are conW X stants), it is relatively easy to find the solution using the second formulation rather than the first (Yadav, 1995). Also, recently in a similar study (Sharan et al., 1996; Sharan and Yadav, 1998), for near source dispersion, the solution for the problem accounting for the downwind diffusion has been obtained using the second formulation without much difficulty. Further, by parametrizing K W and K as power-law functions of z (Pasquill and Smith, X 1983), we expect that an analytical solution would be easier to obtain using the third formulation.
4. Conclusions This note has simply brought to light various possible ways of accounting for the (point) source in mathematical description of air pollution dispersion problems. A mathematical procedure has been provided in order to obtain alternative formulations of the problem leading to the same solution of the diffusion equation. This, essentially, would help in understanding the problem better and extending it to more general situations.
C"0 at x"0 (deleted neighbourhood) *C !K "0 at z"0 (deleted neighbourhood) X *z
The authors wish to thank Prof. John Harris and Prof. E. Genikhovich for the valuable suggestions/discussions.
Second: º
Acknowledgements
*C * *C * *C " K # K X *z *x *y W *y *z
CP0 as "y", zPR ºC"Q d(y) d(z)
References (13)
at x"0
*C !K "0 at z"0 (deleted neighbourhood) X *z
Csanady, G.T., 1973. Turbulent Diffusion in the Environment. Reidel, Dordrecht, Holland. Pasquill, F.A., Smith, F.B., 1983. Atmospheric Diffusion, 3rd Edn. Ellis Horwood, Chichester, UK, 437 pp. Seinfeld, J.H., 1986. Atmospheric Chemistry and Physics of Air Pollution. Wiley, New York, 738 pp.
1330
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
Sharan, M., Singh, M.P., Yadav, A.K., 1996. Mathematical model for atmispheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmospheric Environment 30, 1137—1145. Sharan, M., Yadav, A.K., 1998. Simulation of diffusion experiments under light wind, stable conditions by a variable K-theory model. Atmospheric Environment 32, 3481—3492.
Turer, D.B., 1994. Workbook of Atmospheric Dispersion Estimates, Lewis Publishers, CRC Press, Boca Raton. Yadav, A.K., 1995. Mathematical modelling of dispersion of air pollutants in low wind conditons. Ph.D. Thesis, Indian Institute of Technology, Delhi, 151 pp. Zannetti, P., 1990. Air Pollution Modelling, Computational Mechanics Publications, Southampton, 444 pp.
Technical Note Accounting for the source strength in the solution of the diffusion equation: Alternative mathematical formulations Maithili Sharan *, Anil Kumar Yadav, M.P. Singh, Suman Gupta Centre for Atmospheric Sciences, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India Department of Applied Mathematics, Guru Jambheshwar University, Hisar 125001, Haryana, India Govt Girls Senior Secondary School C1, Madangir, New Delhi, India Received 23 March 1998; accepted 23 July 1998
Abstract The source term is an integral part of the mathematical formulation of the problem leading to the well-known solutions to the diffusion equation. Its representation through the material balance and through the boundary of the domain is discussed. Three alternative mathematical formulations of the problem have been obtained. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Diffusion equation; Source term representation; Mathematical formulation
1. Introduction The Gaussian plume models have been used extensively for air quality predictions especially by the regulatory agencies all over the world. The popularity of these models for estimating air pollutant concentrations from various sources lies in the fact that they are simple, easy to understand and use, and are conceptually and computationally efficient. Extensive information on these models can be found in the literature (Pasquill and Smith, 1983; Seinfeld, 1986; Zannetti, 1990; Turner, 1994). Further, the form of the Gaussian plume solution and the mathematical problem associated with it have been widely discussed (Csanady, 1973; Seinfeld, 1986). An important
*Corresponding author. Fax: 00 91 11 686 2037; e-mail: [email protected].
aspect of the problem is the representation of a source in the model formulation. The source term can be accounted for in two possible ways: either through the material balance or through the boundary of the domain. The former seems quite obvious physically whereas the latter needs elaborate description mathematically. Sometimes, the source term is accounted for through the material balance equation even though it lies on the boundary (Seinfeld, 1986) of the domain. Although it is mathematically inconsistent to take the source in this manner, it gives the correct solution. The other way commonly found in the literature to account for the source is through one of the boundaries (either x"0 plane or z"0 plane in case of ground level release) and is purely from physical considerations. Quite often, it is relatively easier to obtain a closed form solution of the resulting problem by accounting for the source using one of these possible ways.
1352-2310/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 2 - 2 3 1 0 ( 9 8 ) 0 0 2 6 8 - 4
1328
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
In this note, we have considered a point source at the ground and consequently obtained equivalent mathematical formulations of the same problem. The equivalent formulations are useful in acquiring a deeper insight into this simple problem and its possible extensions.
This is a law of conservation of the flux of pollutant in the differential form. Integrating Eq. (4) from !R to x, we can obtain the law of conservation in an integral form as
2. Mathematical description
where
Assuming an incompressible flow, the equation governing the atmospheric dispersion of a pollutant from a point source located at the origin (0, 0, 0), in the absence of removal mechanisms, can be written as
* * *C * *C (ºC)" K # K W *y X *z *x *y *z #Qd(x) d(y) d(z),
(1)
where d(.) is Dirac’s delta function, C is the mean pollutant concentration, º is the mean wind speed directed towards the positive x-axis and thus we have assumed º'0, Q is the source strength, and K and K are the W X eddy diffusivities in the y- and z-directions, respectively. Generally, in the direction of mean wind, transport due to advection is dominant over turbulent diffusion and hence the latter is ignored. Following are the relevant boundary conditions: (i) C"0 at x"0 (deleted neighbourhood) (2a) Here, deleted neighbourhood means the region excluding a small neighbourhood of the point where the source is located: (ii) CP0 as "y", zPR, (iii) !K
(2b)
*C "0 at z"0 X *z (deleted neighbourhood).
(2c)
In Eq. (1), the source term, has been accounted through material balance. Alternatively, it can be accounted through one of the boundaries as mentioned earlier.
h(x)"
0, x(0 1, x'0
(5)
(6)
is the step function. Eq. (5) shows that the double integral (ºC) dy dz"0 \ for all negative values of x, but the integrated (ºC) is non-negative and so ºC"0 for all x(0. This implies C"0 for x(0. Even from physical consideration, the neglect of longitudinal diffusion results in the approximate diffusion equation (1) whose solution is zero at every point upwind of the source (x(0). For x'0, Eq. (5) may be rewritten as (ºC) dy dz"Q. \
(7)
This means that the net flux in the downwind direction across any plane normal to x-axis equals the source strength Q. For x(0, we can use C"0 as the solution of Eq. (1) and for x'0, the source term will no longer contribute in Eq. (1). Eq. (1), then can be solved with an appropriate boundary condition at x"0 accounting for the source. Because of the continuity of the flux Q of pollutant for x'0, Eq. (7) must hold even at x"0. Therefore, the boundary condition at x"0 may be written as ºC(x, y, z)"Q d(y)d(z).
(8)
2.2. Source through z"0 plane
2.1. Source through x"0 plane Integrating Eq. (1) with respect to y from !R to R and using conditions (2b), we get
* *C * (ºC) dy" K dy#Qd(x) d(z). X *z *x *z \ \ (3) Integrating Eq. (3) with respect to z from 0 to R and using conditions (2b) and (2c), we obtain * (ºC) dy dz"Q d(x). *x \
(ºC) dy dz"Qh(x) \
(4)
Now, integrating Eq. (3) with respect to x from 0 to R and using boundary conditions (2a) and (2b) along with the fact that concentration approaches zero far away from the source to obtain
* *C !K dy dx"Q d(z). X *z *z \
(9)
Integrating Eq. (9) from 0 to z, we obtain
*C !K dy dx"Q for all z'0. X *z \
(10)
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
For z'0, the double integral in Eq. (10) represents the flux across any horizontal plane. In this region, the source term will no longer contribute in Eq. (1). Eq. (1), then can be solved with an appropriate boundary condition at z"0. Because of the continuity of the flux Q of pollutant for z'0, Eq. (10) must hold even at z"0. Thus, the boundary condition at z"0 may be written as *C !K "Q d(x) d(y). X *z
(11)
Conditions (8) and (11) give the alternatives for accounting for the source term and thus help in formulating equivalent mathematical problems. Physically, condition (8) [(11)] means that material flux across the vertical (horizontal) plane, containing the source, is essentially from a single point and equals the source strength Q. Further, the relation (7) [(10)] being satisfied for all positive x [z], means that material flux across any vertical (horizontal) plane perpendicular to x-axis [z-axis] equals Q, as long as it is assumed that there is no chemical reaction or removal by any other process.
3. Alternative formulations Conditions (8) and (11) allow us to have the following three alternative mathematical formulations of the same problem: First:
* *C * *C *C º " K # K X *z *x *y W *y *z #Q d(x) d(y) d(z).
(12)
CP0 as "y", zPR
1329
¹hird: º
* *C * *C *C " K # K X *z *x *y W *y *z
CP0 as "y", zPR
(14)
C"0 at x"0 (deleted neighbourhood) *C !K "Q d(x) d(y) at z"0. X *z All the above formulations with constant K’s yield the same solution of the form
º y z Q ! # C(x, y, z)" e 4x KW KX (15) 2nx (K K W X By assuming the K’s as linear functions of downwind distance (K "bºx, K "cºx where b and c are conW X stants), it is relatively easy to find the solution using the second formulation rather than the first (Yadav, 1995). Also, recently in a similar study (Sharan et al., 1996; Sharan and Yadav, 1998), for near source dispersion, the solution for the problem accounting for the downwind diffusion has been obtained using the second formulation without much difficulty. Further, by parametrizing K W and K as power-law functions of z (Pasquill and Smith, X 1983), we expect that an analytical solution would be easier to obtain using the third formulation.
4. Conclusions This note has simply brought to light various possible ways of accounting for the (point) source in mathematical description of air pollution dispersion problems. A mathematical procedure has been provided in order to obtain alternative formulations of the problem leading to the same solution of the diffusion equation. This, essentially, would help in understanding the problem better and extending it to more general situations.
C"0 at x"0 (deleted neighbourhood) *C !K "0 at z"0 (deleted neighbourhood) X *z
The authors wish to thank Prof. John Harris and Prof. E. Genikhovich for the valuable suggestions/discussions.
Second: º
Acknowledgements
*C * *C * *C " K # K X *z *x *y W *y *z
CP0 as "y", zPR ºC"Q d(y) d(z)
References (13)
at x"0
*C !K "0 at z"0 (deleted neighbourhood) X *z
Csanady, G.T., 1973. Turbulent Diffusion in the Environment. Reidel, Dordrecht, Holland. Pasquill, F.A., Smith, F.B., 1983. Atmospheric Diffusion, 3rd Edn. Ellis Horwood, Chichester, UK, 437 pp. Seinfeld, J.H., 1986. Atmospheric Chemistry and Physics of Air Pollution. Wiley, New York, 738 pp.
1330
M. Sharan et al. / Atmospheric Environment 33 (1999) 1327—1330
Sharan, M., Singh, M.P., Yadav, A.K., 1996. Mathematical model for atmispheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmospheric Environment 30, 1137—1145. Sharan, M., Yadav, A.K., 1998. Simulation of diffusion experiments under light wind, stable conditions by a variable K-theory model. Atmospheric Environment 32, 3481—3492.
Turer, D.B., 1994. Workbook of Atmospheric Dispersion Estimates, Lewis Publishers, CRC Press, Boca Raton. Yadav, A.K., 1995. Mathematical modelling of dispersion of air pollutants in low wind conditons. Ph.D. Thesis, Indian Institute of Technology, Delhi, 151 pp. Zannetti, P., 1990. Air Pollution Modelling, Computational Mechanics Publications, Southampton, 444 pp.