Download Transport Equations: An Attempt of Analytical Solution and Application

THE TRANSPORT EQUATION:
ANALYTICAL SOLUTION AND APPLICATION FOR A
CONTROL PROBLEM
CE291 Term Project
Di Jin
Zeshi Zheng
Motivation and Strategy
• Predict air pollutant
concentration by solving 2D
transport equation
• Apply the solution to a control
problem
Advection-Dispersion Equation
Initial condition: c(x,y,0)=f(x,y).
C: solute concentration.
DL, DT: the longitudinal and transverse dispersion coefficients
u, v: wind speed in the x and y direction
q: a source/sink term
Overview of Solving Strategy
• Non-dimensionalization
• Transformation
• Analogy of Green’s Function
• Fourier Transform of Green’s Function
• Solution of Green’s Function
• Ref: Aral, Mustafa M., and Boshu Liao. "Analytical solutions for two-
dimensional transport equation with time-dependent dispersion
coefficients." Journal of Hydrologic Engineering 1.1 (1996): 20-32.
Non-Dimensionalization
Assumptions and Transformation
Assume DL = a2DT = D, and D scales with L2/T, we have:
Let X = x’-ut, Y = y’-v’t
𝜕𝑐(𝑥′, 𝑦′, 𝑡) 𝜕𝑐(𝑋, 𝑌, 𝑡)
𝜕𝑐
𝜕𝑐
=
+ −𝑢
+ (−𝑣′)
𝜕𝑡
𝜕𝑡
𝜕𝑋
𝜕𝑌
Introduce T:
Assumption and Transformation
Where:
𝑄 𝑋, 𝑌, 𝑇 =
𝑞{𝑋 + 𝑈𝛼
𝑌 + 𝑉𝛼 𝑇
,𝛼
𝑎
𝐷[𝛼 (𝑇)]
𝑇 ,
Initial condition:
𝑐 𝑋, 𝑌, 0 = 𝑓 𝑋,
𝑌
, −∞ < 𝑋, 𝑌 < ∞
𝑎
(𝑇)}
Green’s Function
• Our Equation
𝜕𝑐 𝜕 𝑐 𝜕 𝑐
=
+
+ 𝑄(𝑋, 𝑌, 𝑇)
𝜕𝑇 𝜕𝑋
𝜕𝑌
• Heat equation with sources on an infinite domain
𝜕𝑢
= 𝑘𝛻 𝑢 + 𝑄(𝒙, 𝑡)
𝜕𝑡
• Green’s Function for the heat equation without boundaries satisfies
𝜕𝐺
= 𝑘𝛻 𝐺 + 𝛿 𝒙 − 𝒙𝟎 𝛿 𝑡 − 𝑡
𝜕𝑡
subject to:
𝐺 𝒙, 𝑡; 𝒙𝟎 , 𝑡
=0
𝑓𝑜𝑟 𝑡 < 𝑡
• To solve this, apply Fourier Transform. The reason is it will transform the PDE into
an ODE, which is much easier to solve.
Fourier Transform
• Introduce Fourier Transform of the Green’s Function
𝐺̅ 𝝎, 𝑡; 𝐱 𝟎 , 𝑡
1
=
2𝜋
𝐺 𝒙, 𝑡; 𝐱 𝟎 , 𝑡
• Transforming
=
𝐺 𝒙, 𝑡; 𝐱 𝟎 , 𝑡 𝑒
𝐺̅ 𝝎, 𝑡; 𝐱 𝟎 , 𝑡 𝑒
𝝎𝒙𝑑
𝝎𝐱𝑑
= 𝑘𝛻 𝐺 + 𝛿 𝒙 − 𝒙𝟎 𝛿 𝑡 − 𝑡 gives
𝜕𝐺̅
𝑒 𝝎 𝐱𝟎
= −𝑘𝜔 𝐺̅ +
𝛿(𝑡 − 𝑡 )
𝜕𝑡
2𝜋
• For 𝑡 > 𝑡
𝜕𝐺̅
= −𝑘𝜔 𝐺̅
𝜕𝑡
• Solving this we will have:
𝐺̅ = 𝐶 𝝎 𝑒
𝝎
𝑥
Fourier Transform
• From 𝐺 𝐱, 𝑡; 𝐱 𝟎 , 𝑡
• Integrate
̅
= 0, 𝑓𝑜𝑟 𝑡 < 𝑡
𝐺̅ 𝝎, 𝑡; 𝐱 𝟎 , 𝑡
= −𝑘𝜔 𝐺̅ +
𝝎 𝐱𝟎
=0
𝑓𝑜𝑟 𝑡 < 𝑡
𝛿(𝑡 − 𝑡 ) from 𝑡 = 𝑡
to 𝑡 = 𝑡
𝑒 𝝎 𝐱𝟎
𝐺̅ 𝑡
=
2𝜋
𝝎 𝐱𝟎
𝑒
𝐺̅ 𝝎, 𝑡; 𝐱 𝟎 , 𝑡 =
𝑒
2𝜋
1
𝐺 𝐱, 𝑡; 𝐱 𝟎 , 𝑡 =
2𝜋
• From conversion between 𝑢 and 𝐺:
𝜋
𝑘 𝑡−𝑡
𝑒
𝐱 𝐱𝟎
(
)
𝑢 𝒙, 𝑡
=
1
4𝜋𝑘 𝑡 − 𝑡
𝒙 𝒙𝟎 𝟐
𝑒
𝑄 𝒙 , 𝑡 𝑑 𝑥 𝑑𝑡 +
where n is the number of dimensions
1
4𝜋𝑘𝑡
𝑒
𝒙 𝒙𝟎 𝟐
𝑓 𝒙 𝑑 𝒙𝟎
Solution
• Point Source Transport:
where M is the mass of the release.
Control Problem-Definition
• Factory releasing pollutants at
m(t)
• Number of people entering the
road during the day P0(t)
• Health impact
• cost function
• Want: m(t) that minimizes the
cost function
• Constraints:
Analytical Approach
1) Superposition to model continuous source
Analytical Approach
2) Rewrite the cost function (product of number of people and
concentration, integrated over time and space):
Numerical Approach- Discretization
Superposition principle: treat continuous release as i individual releases of mi
at infinitesimal time intervals δt, where δt=24hr/i
(Recall the cost function:)
Numerical Approach- Discretization
constant
Recall the constraints:
Solution: rank ki from smallest to largest, then assign mmax to mi in this
order until
Numerical Solution
Given:
u=100; uw=10; D=100; L1=1000; L2=200; mmax=15; M=150; δt=1; δy=u*δt=100;
P0 generated by random function
References
• Aral, M. M.,& Liao, B.(1996). Analytical Solutions for Two-
Dimensional Transport Equation with Time-Dependent
Dispersion Coefficient. Journal of Hydrologic
Engineering,1:20-32.
• Prentice-Hall, Inc. (1987). Elementary Applied Partial
Differential Equations (2nd ed.). New Jersey: Richard
Haberman