Docstoc

On the Aggregation Model of Marine Particles by Quadrature

Document Sample
On the Aggregation Model of Marine Particles by Quadrature Powered By Docstoc
					On the Aggregation Model of Marine Particles by
          Quadrature Method of Moments


                Louis Y. Liu and Adrian Burd
                 Summer Graduate Student Seminar
                          July 14, 2009




    Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
The Size Distribution of Marine Particles




  Marine particles contain organic particles which can grow, like
  algae and microbes, and inorganic particles, like colloids.
  The size distribution of the diameter of particles, from
  submicron scales to macroscopic scales, can be determined by
  factors like:
       The dynamic of marine ecological system
       Ocean currents
       Chemical reactions in ocean




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
The Size Distribution of Marine Particles




  Marine particles contain organic particles which can grow, like
  algae and microbes, and inorganic particles, like colloids.
  The size distribution of the diameter of particles, from
  submicron scales to macroscopic scales, can be determined by
  factors like:
       The dynamic of marine ecological system
       Ocean currents
       Chemical reactions in ocean




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Marine Particles




  Why do we study it?
     It can aect the ecological state in a certain region of the
     ocean, like zooplankton grazing, vertical ux, light
     penetration, etc.
     It can aect the climate change and global warming,
     because it provides a way to get carbon from the surface
     ocean to the deep ocean, and so away from the atmosphere,
     and to be stored in the deep ocean for many thousands of
     years.




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Marine Particles




  Why do we study it?
     It can aect the ecological state in a certain region of the
     ocean, like zooplankton grazing, vertical ux, light
     penetration, etc.
     It can aect the climate change and global warming,
     because it provides a way to get carbon from the surface
     ocean to the deep ocean, and so away from the atmosphere,
     and to be stored in the deep ocean for many thousands of
     years.




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Aggregation Processes in the Ocean Environment




              Figure: Particles in the ocean environment

       Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Growth-aggregation Equation




  Particle growth and aggregation processes can be described by
  an integro-dierential equation which has been used previously
  to study marine particles (Hunt 1980, McCave 1984) as well as
  in industrial processes (Marchisio et al, 2003),
   dn(D,t)       α
                  ´v
     dt      =   2 0 β(¯, v − v )n(v − v , t)n(¯, t)d¯
                    ´ ∞v      ¯        ¯       v      v
                                                     ∂
                 −α 0 β(v, v )n(v, t)n(¯, t)d¯ + ∂D (G(D)n(D, t)),
                            ¯           v      v
                                                                  (1)
  where v is the volume, (directly) proportional to D3 for solid
  particles and β(¯, v − v ) is the coagulation kernel, measuring the
                  v      ¯
  frequency of particles' coagulation.



         Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Moments of Distribution




  The k-th moments of the size distribution at time t is dened as
                                   ˆ∞

                       mk (t) :=        Dk n(D, t)dD.                      (2)
                                   0

  They have particular meanings, for instance, m0 is the total
  number of particles, m2 is (proportional to) the total surface
  area of particles, m3 is (proportional to) the total solid volume,
  etc.




        Louis Y. Liu and Adrian Burd      On the Aggregation Model of Marine Particles
Growth-aggregation Equation of Moments




  Let u3 := D3 − δ 3 , by multiplying Dk and integrating, one can
  get the equation of moments

     dmk (D,t)       α
                      ´∞          ´∞                 3   3 k
        dt       =   2 0 ´ n(δ, t) 0 β(δ, u)n(u, t)(u + δ ) dudδ
                                       ´∞
                                                           3
                           ∞ k
                     −α 0 ´ n(D, t) 0 β(D, δ)n(δ, t)dδdD
                             D
                              ∞
                          + 0 kG(D)n(D, t)Dk−1 dD.
                                                                        (3)




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Quadrature Method of Moments(QMOM)



  Assuming that
                                n
                    n(δ, t) ≈         wi (t)δ(D − Li (t)).                    (4)
                                i=1

  and then using the approximation
                                       n
                        mk (t) ≈            wi (t)Lk (t).
                                                   i                          (5)
                                      i=1

  The main advantage of QMOM is that it allows us to close the
  system of equations and solve them for any kind of intial
  distributions.


        Louis Y. Liu and Adrian Burd         On the Aggregation Model of Marine Particles
Transformation of the Equation by QMOM




  By the quadrature method of moments, the integro-dierential
  equations can be transformed to ordinary dierential equations

     dmk (D,t)       α
                       ´∞          ´∞                    3   3 k
        dt       =   2  0 ´ n(δ, t) 0 β(δ, u)n(u, t)(u + δ ) dudδ
                                         ´∞
                                                                 3
                            ∞
                     −α 0 ´ k n(D, t) 0 β(D, δ)n(δ, t)dδdD
                               D
                               ∞
                           + 0 kG(D)n(D, t)Dk−1 dD
                 ≈   α    3         3                   3    3 k
                          i=1 wi    i=1 wj β(Li , Lj )(Li + Lj )
                                                                 3
                     2
                       α     3        k  3
                     − 2 i=1 wi Li i=1 wj β(Li , Lj )
                           +k 3 G(Li )Lk−1 wi .
                                  i=1        i
                                                                        (6)



        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Program to Solve the Equation




  The program of quadrature method of moments for the model
  contains:
    1 Generate weights and abscissas from a moment
      (product-dierence algorithm, Gordon, 1968);
    2 Compose the program of generating weights ans abscissas
      into the ordinary dierential equation system;
    3 Set a time step and solve the ordinary dierential equation
      system by Matlab's ode solvers;
    4 Put the solutions into the general output of moments for
      each time step.




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Numerical Experiments for the Model


  Assume the constant coagulation kernel, Brownian coagulation
  kernel, etc, in the equation.
  The Brownian kernel has the expression,
                                 1 1
                          ˜
                                   ˜
                                          ˜
                  βBr (D, D) = c( + )(D + D),                           (7)
                                 D D
  which is a coagulation kernel of the interaction between two
  particles with diameters D and D due to the Brownian motion
                                   ˜
  in particle physics.



                                  Figure:




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Initial Size Distribution




  Exponential spectrum




          Figure: Initial Distribution:   n(D) = 3D2 exp(−D3 )




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Initial Size Distribution

  Power law spectrum




                Figure: Initial Distribution:   n(D) = D−3


        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Growth

  Consider only the growth in the growth-aggregation equation.




    Figure: Size-dependent growth for the power law initial spectrum


         Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
The Aggregation

  Considering only the aggregation, we have only the rst two
  terms in the growth-aggregation equation.




                           Figure: Aggregation


        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Moments in the Growth-aggregation Process

  For the exponential spectrum




   Figure: Growth-aggregation with the initial exponential spectrum

        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Moments in the Growth-aggregation Process

  For the power law spectrum




    Figure: Growth-aggregation with the initial power law spectrum



        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
With the Brownian Kernel




   Figure: With the Brownian kernel and exponential initial spectrum



        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles
Further Work




  Consider the sinking process in the model, for which the
  equation will be
   dn(D,t)       α
                     ´v
     dt      =   2  β(¯, v − v )n(v − v , t)n(¯, t)d¯
                     0 v ´ ¯          ¯       v      v
                            ∞                       ∂
                 −αn(v, t) 0 β(v, v )n(¯, t)d¯ + ∂D (G(D)n(D, t))
                                   ¯ v        v
                                (D)
                    −n(D, t) wsh ,
                                                                        (8)
  where ws (D) is the settling velocity and h is the depth.




        Louis Y. Liu and Adrian Burd   On the Aggregation Model of Marine Particles

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:2
posted:12/29/2012
language:Unknown
pages:21