CHAPTER 5 Concentration Models: Diffusion Model

Diffusion model Using the Gaussian plume idea. Consideration: The point source is the chimney or smoke stack. One need to measure concentration downwind form the point source

Plume of contaminated air The Gaussian Plume. Physical stack height = h The plume rise = h Effective stack height, H = h + h Plume of contaminated air Figure A

Plume of contaminated air The Gaussian Plume. Assumptions: Wind blows in the x direction, with velocity, u and emission rate, Q, and it is independent of time, location or elevation. Plume of contaminated air Figure A

Through material balance around a cube of space near the center of the plume, and considering the dispersion due to turbulent mixing: z x y

Diffusion Model – Gaussian Plume Gaussian puff, 3D spreading Applicable to an instantaneous shot-term release of pollutants from the chimney shown in previous figure, i.e. at x = y = 0 and z = H where K = turbulent dispersion coefficient x = the distance upwind or downwind from the center of the moving puff t = time since release  t = time duration of release

Diffusion Model – Gaussian Plume Gaussian plume, 2D spreading Applicable to steady-state release of plume. Assume negligible net transfer of material in the x direction zamriab@petronas.com.my

y = horizontal dispersion coefficient The above equation is generally used by making the following substitutions: Where: y = horizontal dispersion coefficient z = vertical dispersion coefficient zamriab@petronas.com.my

Diffusion Model – Gaussian Plume Making the substitutions, we find: Basic 2D Gaussian Plume equation

Example 5: A factory emits 20 g/s of SO2 at height H. The wind speed is 3 m/s. At a distance of 1 km downwind, the values of σy and σz are 30 and 20 m, respectively. What are the SO2 concentrations at the centerline of the plume, and at a point 60 meters to the side and 20 meters below the centerline? zamriab@petronas.com.my

Solution At centreline, y = 0 and z = H (refer Fig. A). Thus, at centreline: At the point away from the centreline, zamriab@petronas.com.my

Diffusion Model – Gaussian Plume The basic Gaussian plume equation predicts a plume that is symmetrical with respect to y and with respect to z. Different values of σy and σz mean that spreading in the vertical and horizontal directions is not equal. To find the approximated values for σ y and σ z ,

Diffusion Model – Gaussian Plume Surface Wind Speed (at 10 m), m/s Day Night Incoming Solar radiation Thinly overcast or  4/8 cloud Clear or  3/8 cloud Strong Moderate Slight 0 – 2 A A – B B – 2 – 3 C E F 3 – 5 B – C D 5 – 6 C – D  6 Note: The neutral class D should be assumed for overcast conditions during day or night

Horizontal dispersion coefficient Horizontal dispersion coefficient, y, as a function of downwind distance from the source for various stability categories

Vertical dispersion coefficient Vertical dispersion coefficient, z, as a function of downwind distance from the source for various stability categories

Diffusion Model – Gaussian Plume Some modifications The effect of the ground The ground damps out vertical dispersion and vertical spreading terminates at ground level. Commonly assumed that any pollutants that would have carried below z = 0 if the ground were not there; are ‘reflected’ upward as if the ground is a mirror

Diffusion Model – Gaussian Plume Some modifications Therefore: zamriab@petronas.com.my

Example 6: If z = 10 m, repeat the calculation in Example 5 for the cases where H = 20 m and where H = 30 m.

Solution: H = 20 m zamriab@petronas.com.my

Solution: H = 30 m zamriab@petronas.com.my

Example 7 A large, poorly controlled copper smelter has a stack 150 m high and a plume rise of 75 m. It is currently emitting 1000 g/s SO2. Estimate the ground level concentration of SO2 from this source at a distance 5 km directly downwind when the wind speed is 3 m/s and the stability class is C.

Solution Q = 1000 g/s u = 3 m/s y = 438 m – from Figure 1 z = 264 m – from Figure 2 y = h + h = 225 m zamriab@petronas.com.my

Substituting into the previous equation: Diffusion Model – Gaussian Plume Ground level concentration, simplified At ground level, z = 0. Substituting into the previous equation: zamriab@petronas.com.my

Diffusion Model – Gaussian Plume Ground level concentration, simplified At y = 0 and z = 0  correspond to the line on the ground directly under the centerline of the plume Rearrange: zamriab@petronas.com.my

We can plot a graph of cu/Q vs. distance x. Diffusion Model – Gaussian Plume Ground level concentration, simplified We can plot a graph of cu/Q vs. distance x. zamriab@petronas.com.my

Ground-level , directly under the plume centreline, as a function of downwind distance from the source an effective stack height, H, in meters, for stability Class C only

Example 8 A plant is emitting 750 g/s of particulates. The stack height is 100 m and the plume rise is 50 m. The wind speed is 7 m/s and the stability category is C. What is the maximum estimated ground-level concentration ? How far downwind it does occur?

Plume Rise Figure below shows the plume rising a distance h, called the plume rise, above the top of the stack before leveling out.

Plume Rise Plumes rise buoyantly because they are hotter than the surrounding air and also because they exit the stack with a vertical velocity that carries them upward.

Plume Rise They stop rising because: (i) they mix with surrounding air (ii) they lose velocity (iii) they cool by mixing

Plume Rise To estimate h, Holland’s formula is: where h = plume rise, m Vs = stack exit velocity, m/s D = stack diameter, m u = wind speed, m/s P = pressure, mbar Ts = stack gas temperature, K Ta = atmospheric temperature, K

Example Estimate the plume rise for a 3 m diameter stack whose exit gas has a velocity of 20 m/s when the wind velocity is 2 m/s, the pressure is 1 atm, and the stack and surrounding temperatures are 100oC and 15oC, respectively. Solution:

End of Lecture