An application of semi-infinite programming to air pollution control A. Ismael F. Vaz 1 Eugénio C. Ferreira 2 1 Departamento de Produção e Sistemas Escola de Engenharia Universidade do Minho aivaz@dps.uminho.pt 2 IBB-Institute for Biotechnology and Bioengineering, Centre of Biological Engineering, University of Minho, Campus of Gualtar, 4710 - 057 Braga, Portugal ecferreira@deb.uminho.pt EURO XXII - July 8-11, 2007 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 1 / 36
Contents Introduction and notation 1 Dispersion model 2 Problem formulations 3 Numerical results 4 Conclusions 5 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 2 / 36
Introduction and notation Contents Introduction and notation 1 Dispersion model 2 Problem formulations 3 Numerical results 4 Conclusions 5 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 3 / 36
Introduction and notation Semi-infinite programming (SIP) Consider the following semi-infinite programming problem u ∈ R n f ( u ) min s.t. g i ( u, v ) ≤ 0 , i = 1 , . . . , m u lb ≤ u ≤ u ub ∀ v ∈ R ⊂ R p , where f ( u ) is the objective function, g i ( u, v ) , i = 1 , . . . , m are the infinite constraint functions and u lb , u ub are the lower and upper bounds on u . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 4 / 36
Introduction and notation Coordinate system Z X ∆ H θ H h b d ( a, b ) stack position a d stack internal diameter h stack height ∆ H plume rise H = h + ∆ H effective stack height Y θ mean wind direction A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 5 / 36
Dispersion model Contents Introduction and notation 1 Dispersion model 2 Problem formulations 3 Numerical results 4 Conclusions 5 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 6 / 36
Dispersion model Gaussian model Assuming that the plume has a Gaussian distribution, the concentration, of gas or aerosol (particles with diameter less than 20 microns) at position x , y and z of a continuous source with effective stack height H , is given by ” 2 � ” 2 ” 2 � Q “ “ “ − 1 e − 1 + e − 1 z + H Y z −H C ( x, y, z, H ) = 2 πσ y σ z U e 2 σy 2 σz 2 σz where Q ( gs − 1 ) is the pollution uniform emission rate, U ( ms − 1 ) is the mean wind speed affecting the plume, σ y ( m ) and σ z ( m ) are the standard deviations in the horizontal and vertical planes, respectively. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 7 / 36
Dispersion model Change of coordinates The source change of coordinates to position ( a, b ) , in the wind direction. Y is given by Y = ( x − a ) sin( θ ) + ( y − b ) cos( θ ) , where θ ( rad ) is the wind direction ( 0 ≤ θ ≤ 2 π ). σ y and σ z depend on X given by X = ( x − a ) cos( θ ) − ( y − b ) sin( θ ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 8 / 36
Dispersion model Plume rise The effective emission height is the sum of the stack height, h ( m ), with the plume rise, ∆ H ( m ). The considered elevation is given by the Holland equation � � ∆ H = V o d 1 . 5 + 2 . 68 T o − T e , d T o U where d ( m ) is the internal stack diameter, V o ( ms − 1 ) is the gas out velocity, T o ( K ) is the gas temperature and T e ( K ) is the environment temperature. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 9 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A . Moderate unstable B . Lightly unstable C . Neutral D . Lightly stable E . Moderate stable F . For example the Pasquill-Gifford equations for stability class A is σ y = 1000 × tg ( T ) / 2 . 15 with T = 24 . 167 − 2 . 53340 ln( x ) . A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 10 / 36
Problem formulations Contents Introduction and notation 1 Dispersion model 2 Problem formulations 3 Numerical results 4 Conclusions 5 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 11 / 36
Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 12 / 36
Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 12 / 36
Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 12 / 36
Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 12 / 36
Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July 8-11 12 / 36
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