Annospheric Environment Vol. IO. pp 63M35.
A SYSTEMS
Pergamon
Press 1976. Printed in Great Britain.
APPROACH
TO AIR POLLUTION SELMO
CONTROL
TAUBER
Dept. of Mathematics, Portland State Univ., Portland, OR 97207, U.S.A. (First received 31 July 1975 and in jinal form 3 February 1976) Abstract-A preliminary study for the systematic investigation and the control of the pollution of the atmosphere over a wide region is considered and discussed.
gives a variable time interval depending on the preceding variations of the variable. The determination of At, can easily be done by the computer for each of the substances A(m), m = 1,2, . . . . n. Averaging times required by ambient air quality standards are fixed time intervals (usually 24, 12 or 8 h) independent of the history of the concentration changes. The At, given by equation (1) being based on the history of the changes would be much safer for air control standards since it would automatically increase the number of measurements in case of a crisis.
I. INTRODUCTION
In Peterson (1970, 1972) a statistical approach to the study of the relative influence of meteorological factors in the concentration of sulphur dioxide in the atmosphere of St. Louis is taken. In Padmanabhamurty (1974) the same is done for the atmosphere of Toronto. It is probable that a number of other (statistically inclined) meteorologists will do similar studies for other cities and other pollutants. Although such studies are highly interesting and stimulating the results obtained are geography and time limited. On the other hand the problem of pollution control becomes more acute every day and more systematic analysis seems appropriate. In Tauber (1975) a similar problem for a limited atmosphere was analysed. It is impossible to “solve” the same problem for the atmosphere of a whole area with our present means of investigation and we known that at present we do not have sufficient information about what is going on in our atmosphere. The present paper is only a very simple preliminary outline for the systematic study of the atmosphere, but we hope it is a start in the right direction for the pollution control of the future.
2. THE BASIC GAS ANALYZER
3. METEOROLOGICAL
In addition to the automatic, computer controled gas analyzer at the location (x, y,z) there would be a completely automatic, computer-connected meteorological station giving all meteorological parameters by continuous reading and cumulative reading (for precipitation). The readings would be transmitted to the same computer periodically. 4. GENERAL
UNIT
We assume that it is possible to build an automatic, computer-controlled gas analyzer for n different components. Using the notation of Tauber (1972) let C(m, t) be the concentration at time t of the pollutant A(m), m = 1,2, . . , n at a location (x, y,z). When C(m, t) varies very slowly its determination will be repeated by the analyzer every p,, seconds. If C(m, t) varies more rapidly, the time interval between measurements will be lowered. Let At,, At,, . , be the time intervals between successive measurements i.e. At,; = tj+, - tj, at to, t,,..., and ACj = I C(WI, tj+ 1) - C(m, tj)l. A convenient expression for At, is given by the formula -1 1
1
(1)
where the constant A can be chosen experimentally and p0 would be of the order of 600 s. Equation (1)
UNIT
VIEW OF THE SYSTEM
Consider a set of units as described in Sections 2 and 3 and installed at the nodes of a grid established over a certain region or, even better, over a whole country. In areas of weak population the meshes of the grid would be at distances a x a units. In more densely populated areas the grid could be refined to meshes of (a/k) x (a/k) units. The mesh would-be extremely fine in highly industrial areas with strong pollutant emission. The determination of k can be based on a simple mathematical model. Assuming that k 2 1 and a is a unit (mile or km) for both populated and industrial areas we can make the following choice: (i) For purely residential areas k = k, = ctN,, where N, is the average number of residents to the area a2 and a a coefficient chosen empirically according to the geographical and building configuration and would take in account the facility for the air to circulate or to stagnate (as for example in the downtown area of a large city).
633
634
SELLMOTAUBER
(ii) For purely industrial areas k = k, = aNz + where N2 is the maximum number of people in the area considered a2 during working hours, a is the same as in (i), n is the number of plants or parts of plants in a’, E/f, is the total mass of polluted air emitted by each of the n plants or parts of plants, and /Ii an average pollution coefficient empirically determined according to the plant installation and air cleaning facilities varying on an established scale. (iii) For a mixed area k, and kZ can be determined separately and k = k3 = (N,kl + ~~k~)~(N~ + N,). All information from the gas analyzers and meteorological stations would be continuously transmitted (i) to a central computer, (ii) to a system of local computers, and all presently existing installations would be integrated into the system. In addition, if some pollution related information may be obtained from satellite pictures it would be fitted into the general system. The data from the satellite picture fed automatically into the central computer could be used to compare the information to the information received from the stations. The computer could compare the informations and signal any discrepancies. This would be a method for checking the stations. All information transmitted into the computer system would be stored and kept available for further analysis and the establishing of theoretical models. In addition it would permit a continuous check and control of the pollution situation. The feedback to the polluting agents could in case of an emergency permit immediate action. Since at present little is known of the upper atmosphere and no systematic and continuous monitoring is possible we limit the present aspect of our investigation to the lower atmosphere.
CjL 1 /?$fi,
5. GRAPHICAL
REPRESENTATION
Through the computer, equiconcentration curves can be traced dynamically on geographical maps for each of the major pollutants. The movements of masses of pollutants could be traced and existing but unknown sinks could be determined. A systematic view of the circulation of a given pollutant in the atmosphere could be used for the establishment of a rational control and alarm system based on forecasting. The problem of air pollution would be under a close and continuous scrutiny and serious action could be taken to improve the situation wherever and whenever necessary. 6. MODELS
The analysis of the changes in the atmosphere is a problem so complicated that one can hardly expect a general usable mathematical model. Several general equations have been established (see, e.g. Tauber and Trau, 1973). The equations are usually very complicated and a closed form solution can be found only for OversimpIified ideal cases. Since
the equations depend on the wind velocity, chances of a good prediction are almost nil. A computer approximation can be programmed based on the observation of the wind velocity or even the forecast of the wind velocity. The reliability of such a forecast would be of the order of a regular weatherforecast, i.e. very poor. A more rudimentary but someways more useful model will be an input-output model for a limited time interval (sever& hours) and for a limited region. A simple approach would be the vector-matrix representation (cf. Tauber, 1972) developed in a way similar to the case of a limited atmosphere (cf. Tauber, 1975). We shall analyse such a model in the next section. Statistical models of the kind introduced by Peterson (1970, 1971) and by Padmanabhamurty (1974) could be immediately obtained from the computer by proper programming. All possible correlations between pollutants concentrations and meteorological parameters would be obtained immediately in the computer output for any considered region. Forecasting for an immediate future based on the statistical data could be perfected and used in avoiding crises of high pollutant concentrations. The system would give no information about the chemical reactions occuring and the possibility to influence these reactions. A complete computer based model would be available from the accumulated data and its formulation in closed form would be a problem of numerical analysis.
7. A GLOBAL
MODEL
Consider a closed contour (C) determined by a set of analyzing units on (C). Let 7’be a finite time interval and H an acceptable height taken so that all sources of pollutants within (C) are at most at a level H/l0 above ground. Let V be the volume of the cylinder of base determined by (C) and of height H. The choice of H is such that the interaction of substances emitted by sources of pollutants with the atmosphere in V is assured. The emission of all gases, solids and liquids into the atmosphere of V can be found from the computer data of the analyzing units within the contour (C). Since H will not be too large we may accept an approximate plane air ciruculation model within and about V. This model can be obtained from the meteorological data fed into the computer. Under these conditions the exchanges of substances between V and the upper layers of the atmosphere can be hmited to horizontal diffusion through the portion of the plane at altitude z = H limited by the contour (C). Little is known about the absorbtion and adsorbtion of gases by the soil (cf. Tauber and Trau, 1973 for references) and by running or stagnant bodies of water.
A systems
Regional
General computer
*
i
computer-)
approach
to air pollution
maps
635
control
Di Admitted through diffusion from above D2 Eliminated through diffusion into upper layers Si Production from sources (Polluting or not) S2 Absorbtion by sinks X Input at t = to Y Output at t = to + T M is clearly the Pollution Matrix (cf. Tauber 1975). By repeated observation under conditions of weak circulation where D, and D2 are small and by estimating SZ M can be found as in the case of a closed atmosphere (cf. Tauber, 1975). Under such conditions a good approximation for the chemical reactions can be found. Although the problem is not completely determined, the systematic observations will improve the estimates of the unknown quantities such as Di, D2, and S2, which will be estimated. This model can be improved in time.
Generol +
geographic
*
analysis
8.
FLOW DIAGRAM FOR THE SYSTEM
(mops)
1
Figure 1 is rather simplified but summarizes discussion of Sections 3, 4 and 5.
General storage
the
(archives)
,,
O--Interconnection
Fig.
to other
centers
REFERENCES
w7iversities.e
I Flow chart for general air pollution control.
A set of chemical reactions occuring in V can be established as well as the physical conditions under ‘which the equilibrium will be displaced. Under such conditions we can write A-B+Dr-D2+S,-SZ+MX=Y where all the vectors have components in total masses of substances chemically active within V during the time interval T. The use of masses rather than concentrations makes the problem much simpler. The vectors have the following significance: A Admitted through (C) for 0 5 z 2 H B Eliminated through (C) for 0 6 z s H
Peterson
J. T. (1970) Distribution of sulphur dioxide over metropolitan St. Louis, as described by empirical eigenvectors, and its relation to meteorological parameters. Atmospheric Environment 4, 501-518. Peterson J. T. (1972) Calculations of sulphur dioxide concentrations over metropolitan St. Louis. Atmospheric Environment 6, 433442. Padmanabhamurty B. (1975) Eigenvectors of sulphur dioxide in metropolitan Toronto and its association with meteorological parameters. Atmospheric Environment 9. 365-366. Tauber S. (1972) Linear algebra in air pollution problems. Atmospheric Environment 6, 279-281. Tauber S. (1975) On the deterination of pollution matrices. Atmospheric Environment 9. 135-137. Tauber S. and Trau J. (1973) A vector partial differential equation model for air pollution. Atmospheric Enoironment I. 973-977.
A SYSTEMS
Pergamon
Press 1976. Printed in Great Britain.
APPROACH
TO AIR POLLUTION SELMO
CONTROL
TAUBER
Dept. of Mathematics, Portland State Univ., Portland, OR 97207, U.S.A. (First received 31 July 1975 and in jinal form 3 February 1976) Abstract-A preliminary study for the systematic investigation and the control of the pollution of the atmosphere over a wide region is considered and discussed.
gives a variable time interval depending on the preceding variations of the variable. The determination of At, can easily be done by the computer for each of the substances A(m), m = 1,2, . . . . n. Averaging times required by ambient air quality standards are fixed time intervals (usually 24, 12 or 8 h) independent of the history of the concentration changes. The At, given by equation (1) being based on the history of the changes would be much safer for air control standards since it would automatically increase the number of measurements in case of a crisis.
I. INTRODUCTION
In Peterson (1970, 1972) a statistical approach to the study of the relative influence of meteorological factors in the concentration of sulphur dioxide in the atmosphere of St. Louis is taken. In Padmanabhamurty (1974) the same is done for the atmosphere of Toronto. It is probable that a number of other (statistically inclined) meteorologists will do similar studies for other cities and other pollutants. Although such studies are highly interesting and stimulating the results obtained are geography and time limited. On the other hand the problem of pollution control becomes more acute every day and more systematic analysis seems appropriate. In Tauber (1975) a similar problem for a limited atmosphere was analysed. It is impossible to “solve” the same problem for the atmosphere of a whole area with our present means of investigation and we known that at present we do not have sufficient information about what is going on in our atmosphere. The present paper is only a very simple preliminary outline for the systematic study of the atmosphere, but we hope it is a start in the right direction for the pollution control of the future.
2. THE BASIC GAS ANALYZER
3. METEOROLOGICAL
In addition to the automatic, computer controled gas analyzer at the location (x, y,z) there would be a completely automatic, computer-connected meteorological station giving all meteorological parameters by continuous reading and cumulative reading (for precipitation). The readings would be transmitted to the same computer periodically. 4. GENERAL
UNIT
We assume that it is possible to build an automatic, computer-controlled gas analyzer for n different components. Using the notation of Tauber (1972) let C(m, t) be the concentration at time t of the pollutant A(m), m = 1,2, . . , n at a location (x, y,z). When C(m, t) varies very slowly its determination will be repeated by the analyzer every p,, seconds. If C(m, t) varies more rapidly, the time interval between measurements will be lowered. Let At,, At,, . , be the time intervals between successive measurements i.e. At,; = tj+, - tj, at to, t,,..., and ACj = I C(WI, tj+ 1) - C(m, tj)l. A convenient expression for At, is given by the formula -1 1
1
(1)
where the constant A can be chosen experimentally and p0 would be of the order of 600 s. Equation (1)
UNIT
VIEW OF THE SYSTEM
Consider a set of units as described in Sections 2 and 3 and installed at the nodes of a grid established over a certain region or, even better, over a whole country. In areas of weak population the meshes of the grid would be at distances a x a units. In more densely populated areas the grid could be refined to meshes of (a/k) x (a/k) units. The mesh would-be extremely fine in highly industrial areas with strong pollutant emission. The determination of k can be based on a simple mathematical model. Assuming that k 2 1 and a is a unit (mile or km) for both populated and industrial areas we can make the following choice: (i) For purely residential areas k = k, = ctN,, where N, is the average number of residents to the area a2 and a a coefficient chosen empirically according to the geographical and building configuration and would take in account the facility for the air to circulate or to stagnate (as for example in the downtown area of a large city).
633
634
SELLMOTAUBER
(ii) For purely industrial areas k = k, = aNz + where N2 is the maximum number of people in the area considered a2 during working hours, a is the same as in (i), n is the number of plants or parts of plants in a’, E/f, is the total mass of polluted air emitted by each of the n plants or parts of plants, and /Ii an average pollution coefficient empirically determined according to the plant installation and air cleaning facilities varying on an established scale. (iii) For a mixed area k, and kZ can be determined separately and k = k3 = (N,kl + ~~k~)~(N~ + N,). All information from the gas analyzers and meteorological stations would be continuously transmitted (i) to a central computer, (ii) to a system of local computers, and all presently existing installations would be integrated into the system. In addition, if some pollution related information may be obtained from satellite pictures it would be fitted into the general system. The data from the satellite picture fed automatically into the central computer could be used to compare the information to the information received from the stations. The computer could compare the informations and signal any discrepancies. This would be a method for checking the stations. All information transmitted into the computer system would be stored and kept available for further analysis and the establishing of theoretical models. In addition it would permit a continuous check and control of the pollution situation. The feedback to the polluting agents could in case of an emergency permit immediate action. Since at present little is known of the upper atmosphere and no systematic and continuous monitoring is possible we limit the present aspect of our investigation to the lower atmosphere.
CjL 1 /?$fi,
5. GRAPHICAL
REPRESENTATION
Through the computer, equiconcentration curves can be traced dynamically on geographical maps for each of the major pollutants. The movements of masses of pollutants could be traced and existing but unknown sinks could be determined. A systematic view of the circulation of a given pollutant in the atmosphere could be used for the establishment of a rational control and alarm system based on forecasting. The problem of air pollution would be under a close and continuous scrutiny and serious action could be taken to improve the situation wherever and whenever necessary. 6. MODELS
The analysis of the changes in the atmosphere is a problem so complicated that one can hardly expect a general usable mathematical model. Several general equations have been established (see, e.g. Tauber and Trau, 1973). The equations are usually very complicated and a closed form solution can be found only for OversimpIified ideal cases. Since
the equations depend on the wind velocity, chances of a good prediction are almost nil. A computer approximation can be programmed based on the observation of the wind velocity or even the forecast of the wind velocity. The reliability of such a forecast would be of the order of a regular weatherforecast, i.e. very poor. A more rudimentary but someways more useful model will be an input-output model for a limited time interval (sever& hours) and for a limited region. A simple approach would be the vector-matrix representation (cf. Tauber, 1972) developed in a way similar to the case of a limited atmosphere (cf. Tauber, 1975). We shall analyse such a model in the next section. Statistical models of the kind introduced by Peterson (1970, 1971) and by Padmanabhamurty (1974) could be immediately obtained from the computer by proper programming. All possible correlations between pollutants concentrations and meteorological parameters would be obtained immediately in the computer output for any considered region. Forecasting for an immediate future based on the statistical data could be perfected and used in avoiding crises of high pollutant concentrations. The system would give no information about the chemical reactions occuring and the possibility to influence these reactions. A complete computer based model would be available from the accumulated data and its formulation in closed form would be a problem of numerical analysis.
7. A GLOBAL
MODEL
Consider a closed contour (C) determined by a set of analyzing units on (C). Let 7’be a finite time interval and H an acceptable height taken so that all sources of pollutants within (C) are at most at a level H/l0 above ground. Let V be the volume of the cylinder of base determined by (C) and of height H. The choice of H is such that the interaction of substances emitted by sources of pollutants with the atmosphere in V is assured. The emission of all gases, solids and liquids into the atmosphere of V can be found from the computer data of the analyzing units within the contour (C). Since H will not be too large we may accept an approximate plane air ciruculation model within and about V. This model can be obtained from the meteorological data fed into the computer. Under these conditions the exchanges of substances between V and the upper layers of the atmosphere can be hmited to horizontal diffusion through the portion of the plane at altitude z = H limited by the contour (C). Little is known about the absorbtion and adsorbtion of gases by the soil (cf. Tauber and Trau, 1973 for references) and by running or stagnant bodies of water.
A systems
Regional
General computer
*
i
computer-)
approach
to air pollution
maps
635
control
Di Admitted through diffusion from above D2 Eliminated through diffusion into upper layers Si Production from sources (Polluting or not) S2 Absorbtion by sinks X Input at t = to Y Output at t = to + T M is clearly the Pollution Matrix (cf. Tauber 1975). By repeated observation under conditions of weak circulation where D, and D2 are small and by estimating SZ M can be found as in the case of a closed atmosphere (cf. Tauber, 1975). Under such conditions a good approximation for the chemical reactions can be found. Although the problem is not completely determined, the systematic observations will improve the estimates of the unknown quantities such as Di, D2, and S2, which will be estimated. This model can be improved in time.
Generol +
geographic
*
analysis
8.
FLOW DIAGRAM FOR THE SYSTEM
(mops)
1
Figure 1 is rather simplified but summarizes discussion of Sections 3, 4 and 5.
General storage
the
(archives)
,,
O--Interconnection
Fig.
to other
centers
REFERENCES
w7iversities.e
I Flow chart for general air pollution control.
A set of chemical reactions occuring in V can be established as well as the physical conditions under ‘which the equilibrium will be displaced. Under such conditions we can write A-B+Dr-D2+S,-SZ+MX=Y where all the vectors have components in total masses of substances chemically active within V during the time interval T. The use of masses rather than concentrations makes the problem much simpler. The vectors have the following significance: A Admitted through (C) for 0 5 z 2 H B Eliminated through (C) for 0 6 z s H
Peterson
J. T. (1970) Distribution of sulphur dioxide over metropolitan St. Louis, as described by empirical eigenvectors, and its relation to meteorological parameters. Atmospheric Environment 4, 501-518. Peterson J. T. (1972) Calculations of sulphur dioxide concentrations over metropolitan St. Louis. Atmospheric Environment 6, 433442. Padmanabhamurty B. (1975) Eigenvectors of sulphur dioxide in metropolitan Toronto and its association with meteorological parameters. Atmospheric Environment 9. 365-366. Tauber S. (1972) Linear algebra in air pollution problems. Atmospheric Environment 6, 279-281. Tauber S. (1975) On the deterination of pollution matrices. Atmospheric Environment 9. 135-137. Tauber S. and Trau J. (1973) A vector partial differential equation model for air pollution. Atmospheric Enoironment I. 973-977.
