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GAUSSIAN MODELSumeet KhiradeKabani.K.SM E Environmental Engineering (semester 1)Sinhgad College of Engineering, Vadgaon, Pune

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INTRODUCTION

• The Gaussian plume model is a (relatively) simple mathematical model that is typically applied to point source emitters, such as coal-burning electricity-producing plants to determine the pollution.

• Occasionally, this model will be applied to non-point source emitters, such as exhaust from automobiles in an urban area.

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What is mathematical modeling?

When the process of problem reduction or solution involves transforming some idealized form of the real world situation into mathematical terms,it goes under generic name of mathematical modeling.

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Air Quality Modeling (AQM)

• Predict pollutant concentrations at various locations around the source.

• Identify source contribution to air quality problems.

• Access source impacts and design control strategies.

• Predict future pollutant concentrations from sources after implementation of new regulatory programs.

05/03/2023 5System approach to air quality model

What is air quality model ?

A mathematical relationship between emissions and air quality that incorporates the transport, dispersion and transformation of compounds emitted into the air.

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Air Quality Models

DETERMINISTIC STATISTICAL PHYSICAL

STEADY STATE TIME DEPENDENT

REGRESSION EMPIRICAL

WINDTUNNELSIMULATION

GAUSSIAN PLUMEBOX GRID PUFF TRAJECTORYSPECTRAL

LAGRANGIANEULERIAN

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The deterministic mathematical models calculate the pollutant concentrations from emission inventory and meteorological variables according to the solution of various equations that represent the relevant physical processes.

Deterministic modeling is the traditional approach for the prediction of air pollutant concentrations in urban areas.

What is deterministic approach?

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Gaussian Dispersion Models• Most widely used• Based on the assumption

– plume spread results primarily by molecular diffusion – horizontal and vertical pollutant concentrations in the plume are

normally distributed (double Gaussian distribution)• Plume spread and shape vary in response to meteorological

conditions

H

X

Y

Z

u

Q

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Factors Affecting Dispersion of Pollutants In The Atmosphere

Source Characteristics

Emission rate of pollutant

Stack height

Exit velocity of the gas

Exit temperature of the gas

Stack diameterMeteorological Conditions

Wind velocity

Wind direction

Ambient temperature

Atmospheric stability

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Model Parameters The model is based on our knowledge of the

following parameters:The emissions characteristics (stack exit

velocity, plume rise, temperature, stack diameter)

Terrain (surface roughness, local topography, nearby buildings)

State of the atmosphere (wind speed, stability, mixing height, wind direction)

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Model Assumptions• Gaussian dispersion modeling based on a number of

assumptions including– Steady-state conditions (constant source emission strength)– Wind speed, direction and diffusion characteristics of the

plume are constant– Mass transfer due to bulk motion in the x-direction far

outshadows the contribution due to mass diffusion– Conservation of mass, i.e. no chemical transformations

take place– Wind speeds are >1 m/sec. – Limited to predicting concentrations > 50 m downwind

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The Diffusion Equation and the Gaussian Plume Model

The mass rate of diffusion Nx of a gaseous species in the x-direction at some cross-sectional area A is given by the expression

Nx = -A(∂(DxC)/ ∂x)Nx is mass transfer per unit timeDx is mass diffusivity in X direction, area/timeC is concentration in mass per unit volumeA is cross sectional area in X direction

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Development of Gaussian Plume Model

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Where; x = along- wind coordinate measured in wind direction from the source y = cross-wind coordinate direction z = vertical coordinate measured from the ground C(x,y,z) = mean concentration of diffusing substance at a point (x,y,z) [kg/m3] Dy,Dz = mass diffusivity in the direction of the y- and z- axes [m2/s] U = mean wind velocity along the x-axis [m/s]

Time rate of change and advection of the cloud by the mean wind

Turbulent diffusion of material relative to the center of the pollutant cloud.( the cloud will expand over time due to these terms.)

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The rate of transfer of pollutant through any vertical plane downwind from the source is a constant in steady state, and this constant must equal the emission rate of the source, Q.

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Where Q is the strength of the emission source, mass emitted per unit time

After integrating,

Gaussian parameters

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Where; c( x, y, z ) = mean concentration of diffusing substance at a point ( x, y, z ) [kg/m3]

     x = downwind distance [m],       y = crosswind distance [m],       z = vertical distance above ground [m],       Q = contaminant emission rate [mass/s],      σx = lateral dispersion coefficient function [m],       σy = vertical dispersion coefficient function [m],       U = mean wind velocity in downwind direction [m/s],       H = effective stack height [m]. 

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2-D STEADY DISPERSION MODEL GROUND REFLECTION

• From the release height of H above ground, dispersion can progress upward towards the mixing height. In the downward direction the ground acts as a mirror unless the pollutant gets deposited.

• The effect of the ground can be handled mathematically by treating the reflection as another point source located below ground (at - H)

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2

2

2

2

2

2

2exp

2exp

2exp

2,,

zzyzy

HzHzyu

QzyxC

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Gaussian Dispersion Equation

If the emission source is at ground level with no effective plume rise then

2

2

2

2

21exp,,

zyzy

zyu

QzyxC

Ground level concentration( when Z = 0 )

The point of maximum concentration occur along plume centre line.

2

)0,0( 5.0exp 2

zzyyz

HuQc

22

)0( 5.0exp5.0exp 2

zyzyz

HyuQc

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CARAVAY’S METHOD

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Plume Rise

stackactualriseplume hhH

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Effective Stack Height

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Advantages of Gaussian modelProduce results that match closely with experimental data

Simple in their mathematics

Quicker than numerical models

Do not require super computers

Disadvantages of Gaussian model Not suitable if the pollutant is reactive in nature

Unable to predict concentrations beyond radius of approximately 20 Km

For greater distances, wind variations, mixing depths and temporal variations become predominant

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Conclusion

• Air pollution in cities is a serious public health problem. Therefore, there is need for reliable air quality management system for abatement of urban air pollution problem

• Gaussian plume model is a very effective method in determining pollutant concentrations in atmosphere.

• Gaussian model is the most widely used AQM to predict pollutant concentrations.

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REFERENCES• Weber, E., “Air pollution assessment modeling methodology”, NATO,

challenges of modern society, vol.2, Plenum press, 1982 • Chastain, J.P. 1999. Air Quality and Odor Control from Swine Production

Facilities. chapter 9 in Confined Animal Manure Managers Certification Program Manual, Clemson University, Clemson SC, pp 9-1 to 9-11, http//hubcap.clemson.edu/scafrs/Peedee/certifi/CAMM.html.

• www.mfe.govt• http://www.csiir.ornl.gov• Rao, M.N. and Rao, H. V. N., 1993. Air Pollution, Tata Mc-Graw Hill,

New Delhi.• Murty, B. P., 2004. Environmental Meteorology, I.K. International Pvt.

Ltd., New Delhi.

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THANK YOU