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Moment Methods for Modeling Crystallization Flowsheets by 67uWH7

VIEWS: 36 PAGES: 64

									Population Balance Modeling:
   Solution Techniques &
        Applications
     Dr. R. Bertrum Diemer, Jr.
        Principal Division Consultant
 DuPont Engineering Research & Technology
                Lecture Outline

 Introduction
 Applications to Particles
     General Balance Equation
     Aerosol Powder Manufacturing
     Design Problem

   Solution Techniques


                     R. B. Diemer, Jr., 2003
Introduction
          Definitions & Dimensions
   The population balance extends the idea of mass and
    energy balances to countable objects distributed in
    some property.
   It still holds!
                In - Out + Net Generation = Accumulation

   External & internal dimensions
        external dimensions = dimensions of the environment:
          3-D space (x,y,z or r,z,q or r,q,f) and time
        internal dimensions = dimensions of the population:
          diameter, volume, surface area, concentration, age, MW, number
            of branches, etc.

                                 R. B. Diemer, Jr., 2003
                 A Unifying Principle!
   Objects of distributed size found everywhere...
       particles:
         – granulation, flocculation, crystallization, mechanical alloying, aerosol
           reactors, combustion (soot), crushing, grinding, fluid beds
       droplets:
         – liquid-liquid extraction, emulsification
       bubbles:
         – fluid beds, bubble columns, reactors
       polymers:
         – polymerizers, extruders
       cells:
         – fermentation, biotreatment

   Population balances describe how distributions evolve
                                   R. B. Diemer, Jr., 2003
                   Multivariateness
   Multivariate refers to # of internal dimensions
   Univariate examples:
       particle size… polymer MW...cell age
   Bivariate examples:
       particle volume and surface area (agglomerated particles)
       polymer MW and # of branch points (branched polymers)
       polymer MW and monomer concentration (copolymers)
       cell age and metabolite concentration (biomanufacturing)
   Trivariate example:
       drop size and solute concentration and drop age for internal
        concentration gradients (liquid-liquid extraction)


                              R. B. Diemer, Jr., 2003
 General Differential Form, 1-D Population

   u p (V , x, t ) n(V , x, t ) 
                                         convection              “In - Out” in
                                                                   external
   D p (V , x, t )n(V , x, t ) 
                                         diffusion               coordinates

    
     G (V , x, t ) n(V , x, t )          growth (“In - Out” in internal coordinate)
   V
        + S (V ,x,t )                       sources & sinks (Net Generation)
                             
            n(V , x, t )
                                              accumulation
               t
          Note: object’s velocity may differ from fluid’s velocity owing to
                            either slip or action of external forces
                                     R. B. Diemer, Jr., 2003
              Steady-state, Axisymmetric,
                 Incompressible Flow
   Eliminates 2 physical dimensions, time dimension
   Axial Dispersion Model:
        With slip...                                                          n(V , z ) 
                                  u pz (V , z )  n(V , z )       D p (V , z )              
                               z                            z 
                                                                                     z    
                                               
                                                  G (V , z ) n(V , z ) +S (V , z )
                                             V
        Without slip...
             n(V , z )                  n(V , z )      
        uz                  D p (V , z )                 G(V , z ) n(V , z ) +S (V , z )
                z       z                  z          V

   Plug Flow Model, No Slip:
                             n(V , z )     
                        uz                  G(V , z ) n(V , z ) +S (V , z )
                                z         V

                                         R. B. Diemer, Jr., 2003
         Ideally Mixed Stirred Tank
   Eliminates 3 physical dimensions
    Batch:    n(V , t )     

                             G(V , t ) n(V , t ) +S (V , t )
                 t         V
   Continuous:
        Unsteady state…
                 
                       n(V , t )  Fo (t )no (V , t )  F (t )n(V , t ) 
                 t
                                             
                                      (t )       G (V , t ) n(V , t )  +S (V , t ) 
                                                                                        
                                             V                                        
        Steady state (eliminates time dimension as well)...
            no (V )  n(V )        d                                         
                                    G(V ) n(V ) +S (V )  0;         
                                 dV                                         F


                                    R. B. Diemer, Jr., 2003
    Example: MSMPR Crystallizer
   MSMPR = mixed suspension, mixed product removal
   Same as continuous stirred tank
   Steady state model… no particles in feed, size
    independent growth rate, no sources or sinks (no
    primary nucleation, coagulation, breakage)...
          n o (V )  n(V )        d
                                   G (V ) n(V )  +S (V )  0
                                dV

              n o (V )  0;        G (V )  G;               S (V )  0


              dn    n(V )
                         n(V )  n(0)e V / G
              dV     G

                                  R. B. Diemer, Jr., 2003
Applications to Particles

 General Balance Equation
          Particle Formation, Growth &
                  Transformation
Precursor
                              Nuclei
Molecules        Nucleation
  . .. . ..
... . . . ..
  .
                 Growth
                                                            Agglomerates
                                  Breakage


                                Coagulation
      Singlets


                              Coalescence
                                                          Partially Coalesced
                               R. B. Diemer, Jr., 2003
                                                          Agglomerates
            Sources and Sinks

 Also known as Birth and Death terms
 Types of terms:
     Nucleation(birth only)
     Breakage (birth and death terms)
     Coagulation (birth and death terms)




                     R. B. Diemer, Jr., 2003
         Full 1-D Population Balance
     (a partial integrodifferential equation)

n
       u p n      D p n  
                                                  nucleation term
t
                                                  growth term
N  (V  vo )        G n 
                   V                   coagulation terms
  1 V                                           

  2 0   b (v, V  v) n(v) n(V  v) dv  n(V )
                                                0         
                                                     b (v,V )n(v )dv
     
   V   G( ) b(V ;  ) n( ) d   G(V ) n(V )
                                                          N = nucleation rate
                       breakage terms
                                                          G = accretion rate
                                                          b  coagulation rate
                                                          G  breakage rate
                                                          b = daughter distribution
                               R. B. Diemer, Jr., 2003   vo = nuclei size
Applications to Particles

Aerosol Powder Manufacture
 Gas-to-Particle Conversion
    Aerosol Synthesis Chemistry Examples

                                                     AL y  A  yL/  y 2  L2
   Pyrolysis:
        A=Si, C, Fe…   L=H, CO…
   Halide Oxidation:
                                           MX y   y 4  O2  MO y / 2   y 2  X2
        M=Si, Ti, Al, Sn… X=Cl, Br...

   Halide Hydrolysis:                   MX y   y 2  H2O  MO y / 2  y HX
        M= Si, Ti, Al, Sn… X=Cl, Br...

   Alkoxide Hydrolysis:                 M  OR  y  y H 2O  MO y / 2  y ROH
        M= Si, Ti, Al, Sn… R=CH3, C2H5...

   Alkoxide Pyrolysis:                         M  OR  y  MO y / 2   y 2  ROR
        M= Si, Ti, Al, Sn… R=CH3, C2H5...
    Halide Ammonation:
                                         MX y   y 3  NH3  MN y / 3  y HX

        M=B, Al … X=Cl, Br...
                                    R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

 Feed #1
Preparation      Vaporization
 Feed #2         Pumping/Compression
Preparation
                 Addition of additives
    .
    .            Preheating
    .
 Feed #N
Preparation




                          R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

 Feed #1
Preparation

 Feed #2
Preparation
              Aerosol
    .         Reactor        Mixing
    .
    .                        Reaction Residence Time
 Feed #N
                             Particle Formation/Growth Control
Preparation                  Agglomeration Control
                             Cooling/Heating
                             Wall Scale Removal
                         R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

 Feed #1
Preparation

 Feed #2
Preparation
              Aerosol            Base Powder
    .         Reactor             Recovery
    .
    .                           Gas-Solid Separation
 Feed #N
Preparation




                         R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

                                               Vent or Recycle Gas
 Feed #1
Preparation   Treatment               Offgas
              Reagents                                     Waste
                                     Treatment
 Feed #2                                                          Absorption
Preparation
    .
              Aerosol              Base Powder                    Adsorption
              Reactor               Recovery
    .
    .
 Feed #N
Preparation




                           R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

                                                   Vent or Recycle Gas
 Feed #1
Preparation       Treatment               Offgas
                  Reagents                                      Waste
                                         Treatment
 Feed #2
Preparation
                  Aerosol              Base Powder
    .             Reactor               Recovery
    .
    .                                                        Coarse
                                          Powder             and/or Fine
 Feed #N                                  Refining           Recycle
Preparation      Degassing
                 Desorption                             Size Modification
                 Conveying                              Solid Separations
                               R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

                                               Vent or Recycle Gas
 Feed #1
Preparation   Treatment               Offgas
              Reagents                                     Waste
                                     Treatment
 Feed #2
                                                                        Coating
Preparation
              Aerosol              Base Powder                          Additives
    .         Reactor               Recovery
    .                                                                   Tabletting
    .                                                   Coarse          Briquetting
                                      Powder            and/or Fine
 Feed #N                              Refining          Recycle         Granulation
Preparation
                                                                        Slurrying
              Formulating            Product                            Filtration
              Reagents             Formulation
                                                                        Drying
                           R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

                                               Vent or Recycle Gas      Bags
 Feed #1
Preparation   Treatment               Offgas                            Super
              Reagents                                     Waste
                                     Treatment                           Sacks
 Feed #2                                                                Jugs
Preparation
              Aerosol              Base Powder                          Bulk
    .         Reactor               Recovery
    .                                                                    containers
    .                                                  Coarse              trucks
                                      Powder           and/or Fine
 Feed #N                                                                   tank
                                      Refining         Recycle
Preparation                                                                 cars
                                                                              Product
              Formulating            Product
              Reagents                                     Packaging          Product
                                   Formulation
                           R. B. Diemer, Jr., 2003
  General Aerosol Process Schematic

                                               Vent or Recycle Gas
 Feed #1
Preparation   Treatment               Offgas
              Reagents                                     Waste
                                     Treatment
 Feed #2
Preparation
              Aerosol              Base Powder
    .         Reactor               Recovery
    .
    .                                                   Coarse
                                      Powder            and/or Fine
 Feed #N                              Refining          Recycle
Preparation

              Formulating            Product
              Reagents                                      Packaging   Product
                                   Formulation
                           R. B. Diemer, Jr., 2003
                              TiO2
                            Processes




 R. B. Diemer, Jr., 2003
   Thermal Carbon Black Process




                   Carbon Generated by Pyrolysis of CH4
Johnson, P. H., and Eberline, C. R., “Carbon Black, Furnace Black”, Encyclopedia of Chemical Processing and Design,
J. J. McKetta, ed., Vol. 6, Marcel Dekker, 1978, pp. 187-257.


                                               R. B. Diemer, Jr., 2003
                                 Furnace
                                 Carbon
                                  Black
                                 Process
                            Carbon generated by
                                 Fuel-rich
                              Oil Combustion

                            Johnson, P. H., and Eberline, C. R.,
                            “Carbon Black, Furnace Black”,
                            Encyclopedia of Chemical Processing
                            and Design, J. J. McKetta, ed., Vol. 6,
                            Marcel Dekker, 1978, pp. 187-257.




 R. B. Diemer, Jr., 2003
Applications to Particles

     Design Problem
              Design Problem Focus

                                                 Vent or Recycle Gas
 Feed #1
Preparation     Treatment               Offgas
                Reagents                                     Waste
                                       Treatment
 Feed #2
Preparation
                Aerosol              Base Powder
    .           Reactor               Recovery
    .
    .                                                     Coarse
                                        Powder            and/or Fine
 Feed #N                                Refining          Recycle
Preparation

                Formulating            Product
                Reagents                                      Packaging   Product
                                     Formulation
                             R. B. Diemer, Jr., 2003
                                                             Gas to Recovery
                The Design Problem                                Steps

                                                       Baghouse
                                       8 psig
                                        min
     Flame Reactor
Feeds
       .25 mm particles
          10 psig

   Pipeline Agglomerator                                         25% of
    What pipe diameter     Cyclone                             particle mass
      and length?           What                                    max
                                                75% of particle
                           cut size?              mass min
                            R. B. Diemer, Jr., 2003
           Design Problem Physics
   Simultaneous Coagulation & Breakage
       initial size = .25 micron
   Coagulation via sum of:
                                                                      2kT      1/ 3  V 1/ 3 
       continuum Brownian kernel:                     b c (V ,  )        2       
                                                                      3m     V 
                                                                                             

       Saffman-Turner turbulent kernel:
                                                                          
                                                      bt (V ,  )  .31
                                                                          
                                                                            V  3V 2/ 3 1/ 3  3V 1/ 3 2/ 3   
   Power-law breakage, binary equisized daughters:
                                                                                                
                                                                                                          3/ 2
   Fractal particles:                                                                    8 2
                                                                          G( )  110 s /cm   1/ 3
                        1/ D f                                                                  
                  6V                                                                          
        d p  d0  3            ; D f  1.8                                  b(V ; )  2  V  
                   d0                                                                         2
                                       R. B. Diemer, Jr., 2003
               Design Problem Aims
   Capture particles with a cyclone followed by baghouse
   Need 75% mass collection in cyclone to minimize bag wear from
    back pulsing
   Agglomerate in pipeline… Initial pressure = 10 psig,
   Maximum allowable DP = 2 psia
   Need to design:
        cyclone - “cut size” related to design
        agglomerator - pipe diameter and length needed to get desired
         collection efficiency
   Optimize?… minimize the area of metal in pipe and cyclone to
    minimize cost?


                                  R. B. Diemer, Jr., 2003
                    Problem Setup
 Steady-state, incompressible,
  axisymmetric flow
 Plug flow, no slip
 Neglect diffusion
 Population Balance Model:
           n   
        uz
           z
               
                V
                  G( ) b(V ;  ) n() d   G(V ) n(V )

  1 V                                       
   
  2 0
      b (v, V  v) n(v) n(V  v) dv  n(V )
                                            0        
                                              b (v, V )n(v)dv


                          R. B. Diemer, Jr., 2003
                           Moments
Moments of n(V):
               

          0
                  V j n(V )dV continuous form
     Mj  
          niVi j discrete form with Vi  iV0
          i 1
Key Moments:
         
 M0    n
        i 1
               i    particle number concentration
        
 M1     nV
        i 1
               i   i    particle volume fraction

        (proportional to particle mass concentration)
                              R. B. Diemer, Jr., 2003
Solution Techniques
     Partial List of Techniques
 Discrete Methods
                         Will discuss
 Sectional Methods
 Similarity Solutions
 LaPlace Transforms
 Orthogonal Polynomial Methods              Will not
 Spectral Methods                           discuss

 Moment Methods
 Monte Carlo Methods
                  R. B. Diemer, Jr., 2003
                  Discrete Methods
  Size is integer multiple of fundamental size
 Write balance equations for every size
 Gives distribution directly
 Huge number of equations to solve
 Have to decide what the largest size is
 Example for coagulation and breakage:
                                                  d0 3
                       Vi  iV0 ; V0 
                                                    6
   dni 1 i 1                                            
uz       b j ,i  j n j ni  j  ni  b i , j n j   G j b(i; j )n j  Gi ni
    dz    2 j 1                      j 1              j i 1



                                R. B. Diemer, Jr., 2003
Discrete Example Problem Setup
                  i 1/ 3  j 1/ 3               
                 2        ; b t ,i , j  .31 V0  i  3i 2 / 3 j1/ 3  3i1/ 3 j 2 / 3  j 
        2kT
b c ,i , j
      
        3m        j
                             i                  
               8 2     
                             3/ 2
                                                             2,                         j  2i
           110 s /cm   V0 i , i  1                     
                                  1/ 3 1/ 3
 Gi                                       ; b(i; j )  1,           j  2i  1, 2i  1
                       0, i  1                            0, j  2i  1, j  2i  1
                                                           

   dni 1 i 1                                                  
uz       b c , j ,i  j  b t , j ,i  j  n j ni  j  ni   b c ,i , j  b t ,i , j  n j
   dz  2 j 1                                                 j 1

                   G 2i 1n2i 1  2G 2i n2i  G 2i 1n2i 1  Gi ni

 Need slightly more than 2106 cells to cover
        entire mass distribution range!
                                        R. B. Diemer, Jr., 2003
                    Sectional Method
   Best rendering due to Litster, Smit and Hounslow
   Collect particles in bins or size classes, with
    upper/lower size=21/q, “q” optimized for physics
             i-3 i-2 i-1           i     i+1 i+2
         2-3/q vi   2-2/q vi  vi   21/q vi 22/q vi
                               2-1/q vi            23/q vi
   Balances are written for each size class reducing the
    number of equations, but too few bins loses resolution
   And… now the equations get more complicated to get
    the balances right
   Still have problem of growing too large for top class
   Directly computes distribution
                                    R. B. Diemer, Jr., 2003
        Sectional Interaction Types

   Type 1:
       some particles land in the ith interval and some in a smaller interval
   Type 2:
       all particles land in the ith interval
   Type 3:
       some particles land in the ith interval and some in a larger interval
   Type 4:
       some particles are removed from the ith interval and some from
        other intervals
   Type 5:
       particles are removed only from ith interval

                                    R. B. Diemer, Jr., 2003
           Sectionalization Example: q=1
                    Collision of Particle j with Particle k
Particle j,16
  V/vo
         14
         12                               In which section goes the
   4                                    daughter of a collision between
         10                                Particle j in Section i and
                                            Particle k in Section n?
           8
           6
   3       4
   2       2
   1       0
                0    2    4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
     Section
   number i:
                                                                          Particle k,
 2i-1vo<V<2ivo      1 2       3         4                             5      V/vo
                                        R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1

Particle j, 16
  V/vo
          14
    4     12
                                      j+k = constant
          10
            8
                                                      Any collision between these lines
    3       6                                          produces a particle in Section 5
    2       4

    1       2
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                                      Particle k,
  2i-1vo<V<2ivo      1 2       3          4                             5                V/vo
                                         R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1
                   i,i collisions: map completely into i+1
Particle j, 16
  V/vo
          14

    4     12
          10
            8

    3       6
            4
    2
            2
    1
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                        Particle k,
   i-1v <V<2iv
  2 o          o
                     1 2       3         4                          5      V/vo
                                         R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1
            i,i+1 collisions: 3/4 map into i+2, 1/4 stay in i+1
Particle j, 16
  V/vo
          14

    4     12
          10
            8

    3       6
            4
    2
            2
    1
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                        Particle k,
   i-1v <V<2iv
  2 o          o
                     1 2       3         4                          5      V/vo
                                         R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1
            i,i+2 collisions: 3/8 map into i+3, 5/8 stay in i+2
Particle j, 16
  V/vo
          14

    4     12
          10
            8

    3       6
            4
    2
            2
    1
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                        Particle k,
   i-1v <V<2iv
  2 o          o
                     1 2       3         4                          5      V/vo
                                         R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1
         i,i+3 collisions: 3/16 map into i+4, 13/16 stay in i+3
Particle j, 16
  V/vo
          14

    4     12
          10
            8

    3       6
            4
    2
            2
    1
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                        Particle k,
   i-1v <V<2iv
  2 o          o
                     1 2       3         4                          5      V/vo
                                         R. B. Diemer, Jr., 2003
            Sectionalization Example: q=1
         i,i+4 collisions: 3/32 map into i+5, 29/32 stay in i+4
Particle j, 16
  V/vo
          14

    4     12
          10
            8

    3       6
            4
    2
            2
    1
            0
                 0   2     4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
      Section
    number i:
                                                                        Particle k,
   i-1v <V<2iv
  2 o          o
                     1 2       3         4                          5      V/vo
                                         R. B. Diemer, Jr., 2003
                  Sectionalization Example: q=1
                       i,n collisions: 3/2n-i+1 map into n+1, n>i>0
                              i,i collisions: all map into i+1
Particle j,            i,icollisions: all map into i+1                    i,i+1 collisions: 3/4 map into i+2
  V/vo 16
        14
                                                                                                  i,i+2 collisions:
        12                                                                                        3/8 map into i+3
        10
          8                                                                                         i,i+3 collisions:
                                                                                                   3/16 map into i+4
          6
          4
                                                                              i,i+4 collisions:
          2                                                                  3/32 map into i+5
          0
              0    2    4     6    8 10 12 14 16 18 20 22 24 26 28 30 32
    interval
    number i:
                                                                                                       Particle k,
    voi-1<V<voi    1 2        3                 4                                     5                   V/vo
                                               R. B. Diemer, Jr., 2003
    Sectional Coagulation Model, q=1
   Model Equation:
       dNi         i 2
                                            1                       i 1                             
    uz      Ni 1  bi 1, jq i 1, j N j  bi 1,i 1 Ni 1  Ni  bi , jq i , j N j   bi , j N j 
                                                          2

        dz          j 1                    2                       j 1                 j i         
   Tentatively:
                 3
      q i , j  C 2 j i
                 2
   Can show (via 0th and 1st moments) that:
        number balance gives correct general form for arbitrary q i,j
        mass balance only closes for C=2/3 when Vi/Vj=2i-j
        final expression:      q i , j  2 j i
        kernels evaluated via:
                                         V  2V0 3V0
               Vi  2i 1V1 with V1  0              (recovers 3/2 factor)
                                                2   2
                                        R. B. Diemer, Jr., 2003
   General 21/q Sectional Coagulation Model

dN i         i  S ( q ) 1
                                        1                           i  S ( q )                             
      N i 1  b i 1, jq i 1, j N j  b i  q ,i  q Ni  q  Ni   b i , jq i , j N j   b i , j N j 
                                                          2

 dt               j 1                  2                            j 1                  j i  S ( q ) 1 
                                    i  S ( q  k 1)  k
                                                                                                               
                                               2)k 1 bik , j q i 1, j  k  Ni k N j 
                              q
                          
                             k  2 j i  S ( q  k                                                            
                              i  S ( q  k 1)  k 1                                                          2 new terms
                                                          b i  k 1, j q i , j  21/ q k 1  Ni  k 1 N j 
                        q
                                      
                      k  2 j i  S ( q  k  2)  k  2
                                                                                                               
                                                                                                               

                                     q
                                                             2( j i ) / q 2(1 k ) / q  1
                        S (q)   m ; q i , j                1/ q ;  k  1/ q
                                   m 1                      2 1           2 1


                                                     R. B. Diemer, Jr., 2003
     Sectional Example Problem Setup (for q=1)
                                                           3V 
               2  2(i  j ) / 3  2( j i ) / 3   .31  0   2i 1  3  2(2i  j ) / 31  2( i 2 j ) / 31   2 j 1 
          2kT
bi, j 
          3m                                             2                                                              
                          
                                           3/ 2
                                            3V0 
                                                         1/ 3

           1108 s 2 /cm                                  2(i 1) / 3 , i  1                2,      j  i 1
      Gi                                2                                      ; b(i; j )  
                                                                                                  0,      j  i 1
                                         0, i  1

                         i 2 b                                  i 1 bi , j              
             dNi                         1
                  Ni 1  i  j 1 N j  bi 1,i 1 Ni 1  Ni  i  j N j   bi , j N j 
                                 i 1, j               2
          uz
              dz          j 1 2         2                       j 1 2       j i         
                                                   2Gi 1 Ni 1  Gi Ni


       Need about 22 sections to cover entire
     mass distribution range, suggest using 25-30
                                                    R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)
 Calculation of Mass Collection Efficiency
                                                                                             1/ D f
        i 1  3V0          d0
                               3
                                                        V 
                                                                                                       d 0  3  2i  2 
                                                                                                                              1/ D f
  Vi  2            ; V0        di  d 0 n p f  d 0  i 
                                             1/ D
                   
              2            6                           V0 
                                                                                                                I

             di d pc 
                          2
                                         3  2i  2 
                                                         2/ Df
                                                                  d0 d pc 
                                                                               2
                                                                                                                V N   i i       i
   i                                                                                 ;  w  100%          i 1

          1   di d pc               1  3 2                 d       d pc 
                               2                   i 2 2 / D f                     2                             I


              example
                                                                       0                                        V N
                                                                                                                 i 1
                                                                                                                         i    i
            grade efficiency
                curve
                                                             I
                                               M 1   Vi N i  M 0 V1
                                                                  o

                                                          i 1
                                                     I
                        Ni   Vi
                    ni  o ;     2   w  100%  i 2i 1 ni
                                   i 1

                        M0   V1                    i 1

                                                             1, i  1
                Suggests to do calculation using ni with n                                o

                                                             0, i  1
                                                                                            i


                                           I
Check mass closure via:                    2 R.nB.  1. IfJr., 2003 model is coded incorrectly!
                                          i 1
                                                 i 1
                                                         i
                                                     Diemer,
                                                             not true,
          Sectional Example Problem Setup (for q=1)
                    Nondimensionalization
                                                                                                    1/ 3
                                                         3V                        3V 
                     b i , j  b co  c ,i , j    b to  0   t ,i , j ; Gi  G o  0                   2(i 1) / 3
                                                         2                         2 
                                                                                                   
                                                                                                                         3/ 2
                     2kT
              b co      ;  c ,i , j  2  2(i  j ) / 3  2( j i ) / 3 ; G o  110 8 s 2 /cm  
                     3m                                                                            
                                      
                    b to  .31          ;  t ,i , j  2i 1  3  2(2i  j ) / 31  2( i  2 j ) / 31   2 j 1
                                      
                                                                                                                         1/ 3
                 b co M 0o z       t   2 b co        b b co M 0o  2 
                           ; t            ; b                 
                c    uz             c 3b t V0
                                            o
                                                      c     G o  3V0 
                                                   t ,i , j            Ni        2M 1 4M 1
                         i , j   c ,i , j                  ; ni       ; M0 
                                                                              o
                                                                                      
                                                    t                   o
                                                                        M0        3V0  d 0
                                                                                          3


            i 2                                   i 1  i , j               2(i 1) / 3 4/ 3
dni
d
                            1
     ni 1  i  j 1 n j   i 1,i 1ni21  ni  i  j n j    i , j n j  
                    i 1, j

                                                                                    b
                                                                                              2 ni1  ni 
             j 1 2         2                       j 1 2       j i          
                                                            R. B. Diemer, Jr., 2003
         Solution Technique Choices
   If analytical method works use it! (rare)
        similarity solution
        Laplace transform

   If it is crucial to get distribution detail right, and it is a 1-D
    problem, and it is a stand-alone model (typical of research)
        discrete
        sectional
        Monte Carlo
        Galerkin (orthogonal polynomial… commercial code: PREDICI)

   If an approximate distribution will do, or if the moments are
    sufficient, or if the distribution is multivariate, or if the model will
    be embedded in a larger model (typical of process simulation)
        moments
                                  R. B. Diemer, Jr., 2003
            Concluding Remarks

   Population balance applications are everywhere
   The mathematics is difficult (unlike mass & energy
    balances)
   There are many solution techniques… choice
    depends on object of model




                         R. B. Diemer, Jr., 2003
Backup Slides
                           Moments
     Moments of n(V):
                   

                 0
                      V j n(V )dV             continuous form
                Mj  
                     niVi j discrete form with Vi  iV0
                     i 1
                 M 0  particle number concentration
M 1  particle volume fraction (proportional to particle mass concentration)
     Moments of b(V;):
                  V j b(V ;  )dV
                 0
                                                 continuous form
                
                  b j    1
                         b(i;  )Vi j ; V   discrete form
                         i 1
                        
          b0  p daughters/breaking event  2 in binary breakage
                           b1   , the parent size
                               R. B. Diemer, Jr., 2003
        Particle Number Balance
    n              d 
                               
                                         dM 0
    u z z  dV  u z dz 0 n(V )dV  u z dz 
  0        
                                
0 V   G( ) b(V ;  ) n( )d dV               0   G(V ) n(V )dV

        1  V
         
        2 0 0
              b (v, V  v) n(v) n(V  v) dvdV
                    
            n(V )  b (v, V )n(v)dvdV
             0       0


                       R. B. Diemer, Jr., 2003
    Interchange of Limits
V
                                                 V=
    0 V    f ( , V )d dV 
        
                                                 goes from V to 
    0 0    f ( , V )dVd
                                                then V from 0 to 



                                                V goes from 0 to 
                                                then  from 0 to 


                                                      
                     R. B. Diemer, Jr., 2003
   Particle Number Balance (cont.)
             Interchange limits of integration in
            both coagulation and breakage terms
                                           
  uz
     dM 0
      dz
          
            
            0
              G ( ) n (  ) 
                              0
                             
                                 b(V ; ) dV 
                                             
                                               d 
                                                    
                                                    0    
                                                      G(V ) n(V )dV
                               b0  p 
                                            
  1                                     
   
  2 0 v
        b (v,V  v) n(v) n(V  v) dVdv 
                                         0 0        
                                             b (v,V )n(V )n(v)dVdv


                                    
                       ( p  1)   0   G(V ) n(V )dV
  1                                     

  2 
    0 v
        b (v,V  v) n(v) n(V  v) dVdv 
                                         0 0        
                                             b (v,V )n(V )n(v)dVdv

                              R. B. Diemer, Jr., 2003
Particle Number Balance (cont.)
          Change of variable in coagulation integral:
           = V v      dV = d at constant v
                                   
                                    
                   dM 0
                uz       ( p  1)   G(V ) n(V )dV
                    dz             0
   1                                 
    
   2 0 0
         b (v,  ) n(v) n( ) d dv 
                                      0 0  
                                          b (v,V )n(V )n(v)dVdv

          General Number Balance for p Daughters
                                   1  
                                              
        ( p  1) 0 G(V ) n(V )dV  2 0 0 b (v, V )n(V )n(v)dvdV
   dM 0                             continuous
uz                            
                                           1  
    dz                ( p  1) Gi ni   b i , j ni n j
                              i 1        2 i 1 j 1
                                     discrete
                           R. B. Diemer, Jr., 2003
     Particle Volume (Mass) Balance
         n           d 
                                        
                                            dM 1
     0
       V u z  dV  u z dz 0 Vn(V )dV  u z dz 
          z 
                                                         
0 V V       G( ) b(V ;  ) n( ) d dV                0   V G(V ) n(V ) dV

            1    V
             
            2 0
                V
                  0     
                    b (v, V  v) n(v) n(V  v) dvdV
                                  
                  0   V n(V )   0   b (v, V )n(v)dvdV


                               R. B. Diemer, Jr., 2003
     Particle Volume (Mass) Balance
                                    (cont.)
                Interchange limits of integration in
               both coagulation and breakage terms

                                                  
                                   0
                                      Vb(V ; ) dV       
                                                              
          dM 1
       uz         G ( ) n (  )                     d    V G(V ) n(V )dV
           dz    0                                       0
                                  
                                        b1      
                                                    
     1                                        
       
     2 0 v
           V b (v, V  v) n(v) n(V  v) dVdv 
                                               0 0      
                                                   V b (v,V )n(V )n(v)dVdv




   dM 1 1                                         
uz
    dz
           
         2 0 v
               V b (v, V  v) n(v) n(V  v) dVdv 
                                                   0 0        
                                                       V b (v,V )n(V )n(v)dVdv


                                  R. B. Diemer, Jr., 2003
  Particle Volume (Mass) Balance
                                   (cont.)
             Change of variable in coagulation integral:
              = V v      dV = d at constant v


   dM 1 1                                           
uz
    dz
            
         2 0 0                                    
               (v   ) b (v,  ) n(v) n( ) d dv 
                                                     0 0
                                                         V b (v, V )n(V ) n(v) dVdv
                                            
           b (v,  ) n(v) n( ) d dv    V b (v, V ) n(V ) n(v) dVdv
           0 0                               0 0
                b ( v , ) symmetric




               General Mass Balance for p Daughters
                           dM 1
                      uz         0 mass is conserved
                            dz
                                 R. B. Diemer, Jr., 2003

								
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