Important Expressions in Air Pollution Engineering

The following equations are frequently used in Air Pollution Engineering for the design of settling chambers, prediction of pollutant dispersion, determination of plume rise, effective stack height, and chimney height calculations.

1. Largest Particle Size Removed in a Settling Chamber

The largest particle diameter that can be removed with 100% efficiency in a settling chamber is:

d = C √[ (18 V_a μ H) / {g L (ρ_p − ρ)} ]

Where:

Symbol	Description	Unit
d	Largest particle diameter removed	m
C	Correction factor	—
Va	Gas velocity through chamber	m/s
μ	Dynamic viscosity of gas	kg/m·s
H	Height of settling chamber	m
L	Length of settling chamber	m
g	Acceleration due to gravity	9.81 m/s²
ρp	Particle density	kg/m³
ρ	Gas density	kg/m³

This equation is used in the design of gravity settling chambers to determine the maximum particle size that can be completely removed.

2. Gaussian Dispersion Model

The Gaussian Dispersion Model is used to predict pollutant concentration at any point downwind from a source.

C = [Q / (π σ_y σ_z u)] e^{-(H² / 2σ_z²)} e^{-(y² / 2σ_y²)}

Where:

Symbol	Description
C	Pollutant concentration (g/m³)
Q	Pollutant emission rate (g/s)
u	Mean wind velocity (m/s)
σy	Horizontal dispersion coefficient (m)
σz	Vertical dispersion coefficient (m)
y	Crosswind distance (m)
H	Effective stack height (m)

Used for air quality studies, environmental impact assessment, and pollutant concentration prediction.

3. Effective Height of Stack

The effective stack height is the sum of the actual chimney height and the plume rise.

H = h + Δh

Symbol	Description
H	Effective stack height (m)
h	Actual stack height (m)
Δh	Plume rise above stack (m)

Holland's Equation for Plume Rise

Δh = (V_sD/u) [1.5 + 2.68 × 10^-3P D ((T_s − T_a)/T_s)]

Where:

Symbol	Description
Δh	Plume rise (m)
Vs	Stack gas velocity (m/s)
D	Inside stack diameter (m)
u	Wind speed (m/s)
P	Atmospheric pressure (millibar)
Ts	Stack gas temperature (K)
Ta	Ambient air temperature (K)

For unstable atmospheric conditions, increase plume rise by 10–20%. For stable conditions, decrease plume rise by approximately 20%.

4. BIS Recommended Plume Rise Equations

(a) For Hot Effluent Releases

Applicable when heat release is of the order of 10⁴ cal/s or more.

Δh = 0.84 (1.24 + 0.09u) Q_H^1/4 / u

Symbol	Description
QH	Heat release rate (cal/s)
u	Wind speed (m/s)

(b) For Very Hot Releases Acting as Momentum Sources

Δh = 3V_sD/u

5. Minimum Chimney Height

(a) For Particulate Matter Emission

h = 74(Q_p)^0.27

Symbol	Description
h	Chimney height (m)
Qp	Particulate matter emission rate (tonne/hr)

(b) For Sulphur Dioxide (SO₂) Emission

h = 14(Q_s)^0.3

Symbol	Description
h	Chimney height (m)
Qs	SO₂ emission rate (kg/hr)

Quick Revision Sheet

Expression	Formula
Settling Chamber	d = C √[(18VaμH)/(gL(ρp-ρ))]
Gaussian Dispersion Model	C = [Q/(πσyσzu)] e-(H²/2σz²) e-(y²/2σy²)
Effective Stack Height	H = h + Δh
Holland Equation	Δh = (VsD/u)[1.5 + 2.68×10⁻³PD((Ts-Ta)/Ts)]
BIS Hot Effluent	Δh = 0.84(1.24+0.09u)QH1/4/u
Momentum Source	Δh = 3VsD/u
Chimney Height (PM)	h = 74(Qp)0.27
Chimney Height (SO₂)	h = 14(Qs)0.3

Author: Mohan Dangi ( Gold Medalist)

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