CHAPTER 5 Concentration Models Diffusion Model Diffusion model

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

• The Gaussian Plume. Physical stack height = h The plume rise = h Effective stack height, H = h + h ir a d e t f eo na i m nta co m Plu Figure A

• The Gaussian Plume. Assumptions: • Wind blows in the x direction, with air d velocity, u and emission rate, Q, and te a n mi a t it is independent n co f o e of time, location m Plu or elevation. 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, 3 D 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, 2 D spreading • Applicable to steady-state release of plume. • Assume negligible net transfer of material in the x direction zamriab@petronas. com. my

• 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 2 D Gaussian Plume equation

Example 5: • A factory emits 20 g/s of SO 2 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 SO 2 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 Clear or overcast or 3/8 cloud 4/8 cloud Strong Moderate Slight 0– 2 A A–B B – – 2– 3 A–B B C E F 3– 5 B B–C D D E 5– 6 C C–D D 6 C D D Note: The neutral class D should be assumed for overcast conditions during day or night

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

• 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 SO 2. Estimate the ground level concentration of SO 2 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

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

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. a) What is the maximum estimated ground-level concentration ? b) 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 Vs D u P Ts Ta = plume rise, m = stack exit velocity, m/s = stack diameter, m = wind speed, m/s = pressure, mbar = stack gas temperature, K = 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 100 o. C and 15 o. C, respectively. • Solution:

End of Lecture