MIT OpenCourseWare
http://ocw.mit.edu
1.020 Ecology II: Engineering for Sustainability
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
1.020 Ecology II: Engineering for Sustainability
Lectures 08_12 & 08_13 Outline: Mass Transport, Diffusion, Air Quality
Motivation/Objective
Develop a model to examine spatial variations in the concentration of a chemical plume emitted
from a line source (e.g. vehicles traveling on a road).
Approach
1. Generalize bulk system concept to provide for spatial variability. Define particles, identify
source, emission rate, diffusion/dispersion process.
2. Construct rule for particle transport over time and space.
3. Relate chemical concentration to particle density.
4. Simulate transport in MATLAB and plot particles and concentrations.
5. Examine impact of emissions rate, mean velocity and dispersion coefficient.
Concepts and Definitions Needed:
Define point particles, each identified with a small fixed mass (m) of chemical and described by
its changing spatial coordinates x ni = [ x1i,n , x 2i ,n , x3i ,n ] (particle i, time step n)
Particles move with fluid that contains them (advection). Fluid velocity = Vn
Particles move randomly within fluid due to molecular motion (diffusion) and small variations in
fluid velocity not included in Vn (dispersion).
Random particle displacement ωn = [ω1i,n , ω 2i ,n , ω3i ,n ]
Mean( ω ij,n )=0, Variance (ω ij ,n ) = 1 , all ω ij,n uncorrelated
Particle transport eq. describes changes in particle coords. over time Δt :
x1i,n +1 = x1i,n + V1,n Δt + d1ω1i,n , x2i ,n+1 = x2i ,n + V2,n Δt + d 2ω 2i ,n , x3i ,n +1 = x3i ,t + V3,n Δt + d 3ω3i ,n
d j = 2 D j Δt = Dispersion distance coord. j, (m), D j = Dispersion coef. coord. j (m2 sec-1)
Spatial characteristics of sources: point vs distributed sources
Temporal characteristics of sources: pulse vs continuous sources
Bulk properties of the plume can be described by its first and second spatial moments.
For point source, constant Vn (see attached supplement for 1 dimensional derivation):
First moment (plume center):
x1,n +1 = V1t, x 2 ,,n +1 = V2 t, x3 ,n +1 = V3t , t = nΔt
Second moment (plume length/width/height):
S1,n = σ ω21t = 2 D1t, S 2 ,,n = σ ω2 2 t = 2 D2 t, x3 ,n = σ ω2 3 t = 2 D3 t
Concentration computed from particles in specified volumes:
mN np
gm m-3 = approximate chemical concentration in cell (pixel) p at time t
C np ≈
Vp
Modeling Example -- Air Quality
Note how particle spreading depends on dispersion coefficient and mean velocity. Plot spatial
moments vs. time.
4
Supplement: Derivation of spatial moments for point source in one dimension (x1):
Assumptions:
1) Point source at origin: x1i, 0 = 0
1 P i
∑ ω1,n = 0
P i =1
3) Uncorrelated ω1i,n : Correlation =
2) Mean( ω1i,n ) = ω1,n =
∑ (ω
i ≠ j or m ≠ n
i
1, m
− ω1,m )(ω1j, n − ω1, n ) = 0
4) Constant V1n = V1 (same for every particle at all times
1 P
1 P
5) Variance( ω1i,n ) = σ ω2 = ∑ [(ω1i,n − ω1,n ) 2 = ∑ [(ω1i, n ) 2 = 1
P i =1
P i =1
If ω is distributed uniformly betweeen − ω max and + ω max , ω max = 3 gives σ ω2 = 1
First and second moments are derived from the solution to the particle transport equation:
x1i,n +1 = x1i,n + V1 Δt + d1ω1i,n → x1i,n = V1 nΔt + d1
n
∑ω
i
1, m
1 m =1
First spatial moment, invoke Assumption 2 :
n
d P ⎡ n
1 P
⎤
⎡1 P
⎤
x1,n = ∑ x1i,n = V1 nΔt + 1 ∑ ⎢∑ ω1i,m ⎥ = V1 nΔt + d 1 ∑ ⎢ ∑ ω1i,m ⎥ = V1 nΔt = V1t
P i =1
P i =1 ⎣ m =1
m =1 ⎣ P i =1
⎦
⎦
Second spatial moment:
1 P
S1,n = ∑ ( x1i,n − x1,n ) 2 =
P i =1
n
d12
1 P
i
2
ω1,m − V1 nΔt ) =
∑ (V1nΔt + d1 ∑
P i =1
P
m =1
2
d12
⎡n i ⎤
∑
⎢∑ ω1,m ⎥ = P
i =1 ⎣ m =1
⎦
P
P
n
n
∑∑∑ ω
i =1 l =1 m =1
i
1,l
ω1j,m
Separate terms of final summation into 2 parts, invoke Assumption 3:
S1,n =
d12
P
P
n
n
∑∑∑ ω1i,l ω1j,m =
i =1 l =1 m =1
d12
P
∑∑ [ω ]
P
n
i =1 m =1
2
i
1, m
+
d12
P
⎡
⎤ nd12
i
j
ω
ω
⎢ ∑ 1,m 1,n ⎥ =
∑
P
i =1 ⎣i ≠ j or m ≠ n
⎦
P
∑ [ω ]
P
i =1
Plume characteristic “size” is square root of second moment: L1 = S1,n = 2 D1t
5
2
i
1, m
= 2nD1 Δt = 2 D1t