Gaussian Pollutant Dispersion Model

C B. R. Manandhar

Models are used to predict ambient concentrations based on a planned set of emissions. Models are, thus, very useful tools in air quality management programme. Among available models, Gaussian Plume Dispersion Model is the most widely used one. The equation, which describes this model, is as follows:

C _{(x, y, z)} = Q/(2× π ×s_y×s_z×u_s) × e ^{(-0.5 ×} y²/s²_y⁾ * e ^{(-0.5 ×} (z-H)²/s²_z⁾

It is a normal distribution curve, which is bell-shaped.

When a plume exits a stack, it gets dispersed in the atmosphere. Dispersion means diffusion and transportation. Dispersion of atmospheric pollutants is accomplished by two major mechanisms: the wind and atmospheric turbulence. The wind simply carries the pollutants downwind while turbulence causes pollutants to fluctuate from the mainstream concentration in the vertical and crosswind directions. Turbulence is the result of two specific effects: atmospheric heating that causes convection currents and mechanical turbulence resulting from wind shear. To achieve maximum dispersion, either the stack should be tall enough or the effluents from the stack (the plume) should exit the stack with sufficient momentum and buoyancy so that they continue to rise from the stack tip.

Major assumption in the development of Gaussian dispersion model:

• Constant point source
• Diffusion in vertical and cross-wind (horizontal) directions and diffusion along x direction neglected
• Transportation takes place in x direction (in wind direction) only
• No change in concentration with respect to time (dc/dt = 0)
• No chemical reactions between pollutant species and no dry/wet removal of pollutants

Thus no model, including Gaussian model, is perfect one. They are merely prediction tools and need verification with ambient air monitoring.

In the Gaussian dispersion model, the key input parameters are:

• Emission rate of pollutants, Q
• Spatial location of the receptor (x, y, z), with the origin at the base of the stack on the ground level
• The dispersion coefficients or plume standard deviations, s_y and s_z, which depend on the atmospheric stability and the x value
• Wind velocity, u_s
• Effective stack height, H, which is the sum of stack height (h_s) and plume rise (Dh)

Plume Rise (Dh)

When a plume comes out of a stack, it has certain exit velocity and is also hotter than the surrounding air. So the kinetic energy due to the exit velocity and the thermal buoyancy due to the temperature difference make the plume rise vertically up to a certain height. It then starts bending slowly and eventually flattens horizontally in the wind direction away from the stack.

Dh = f (V_s, d_s, u_s, P, T_s, T_a)

where,

V_s = Stack exit velocity, m/s

d_s = Inner diameter of the stack at the top, m

u_s = Wind velocity, m/s

P = Atmospheric pressure, millibars

T_s = Temperature of the stack gas, K

T_a = Temperature of the surrounding air, K

Stability class as given by Pasquill

Stability class	Actual atmospheric condition
A	Very unstable
B	Unstable
C	Slightly unstable
D	Neutral
E	Slightly stable
F	Stable

Turner’s stability categories

Surface wind speed (at 10 m), m/s	Day			Night
	Incoming solar radiation
	Strong	Moderate	Slight	Thinly overcast	Clear
0 – 2	A	A – B	B	-	-
2 – 3	A – B	B	C	E	F
3 – 5	B	B – C	C	D	E
5 – 6	C	C – D	D	D	D
>= 6	C	D	D	D	D

Dispersion coefficients, s_y and s_z, can be found out by using graphs. These graphs are log-log plots with x (in meters) on abscissa and s_y or s_z (in meters) on ordinate. The atmospheric stability class (for urban or rural condition) and x position away from the stack should be known beforehand to find out the values of s_y and s_z by using the graphs.

In the vertical and cross wind profiles of concentration, which is a normal distribution curves, the maximum concentration occurs at the plume center line (C_CL). At the distance of s_y or s_z measured from the plume center, concentration is 0.61×C_CL. Similarly, at the plume outline (boundary line), concentration is 0.1×C_CL.

So, the farther the receptor away from the stack, the flatter the concentration profile curve becomes both in z and y directions at the receptor location.

More turbulent the atmosphere, even much flatter the profile curves become at a given x position, implying that with more atmospheric turbulence, GL concentration rises.

Concentration at the ground level, GL (x>0, y = 0, z = 0)

The concentration profile along x–direction (in the wind direction) is not a normal distribution curve.

Along x-direction on GL (y = 0 and z = 0), concentration is zero up to a certain distance away from the stack base. It then slowly rises and attains the maximum value at x_Cmax (meters). After the peak, the concentration again decreases to zero.

The maximum GL concentration and the position of its occurrence can be determined by using a graph prepared by Turner. The graph is a log-log plot with x_Cmax on abscissa and (C×u_s/Q)_max on ordinate. To use such graph, the stability class and effective stack height, H should be known beforehand.

With the stability class and H given, a point can be found on the Turner’s graph, where the line corresponding to the given H and the inclined line corresponding to stability class cross each other. This point in turn is related to the x_Cmax on the x-axis and to the (C×u_s/Q)_max on y-axis. So with the value of (C×u_s/Q)_max known from the graph, C_max can be calculated for the given u_s and Q.

Effect of Ground (Reflection from ground)

When the plume hits the ground, the pollutants in the plume are reflected back into the air by the ground. In other words, the ground doesn’t work as the sink for the pollutants.

The reflection from the ground can be dealt with by considering a mirror image of the real stack. That means a virtual stack just below the real one is emitting another identical plume. The effective stack height of the virtual stack would be “-H” if H is the effective stack height of the real stack above the ground.

Thus, there will be an overlap of two identical vertical concentration profiles through the ground level and the net concentration value exactly at the ground level will be just double the concentration that would be there with the real stack alone.

So, with the plume from the virtual stack considered, the Gaussian formula for the concentration at any receptor point in space (x, y, z) will be modified as follows:

C _{(x, y, z)} = Q/(2×π ×u_s×s_y× s_z) × exp -(0.5×y²/s_y²) × [exp {-0.5×(z – H)²/s_z² +

 exp {-0.5×(z +H)²/s_z²}]

So, for the ground level (i.e. z = 0, y = 0) at a horizontal distance of x from the stack base,

	C (x, 0, 0), reflection = Q/(2× π ×us×sy× sz) × 2×exp {-0.5× H2/sz2} or C (x, 0, 0), reflection = Q/( π ×us×sy× sz) × exp {-0.5× H2/sz2}