Your Federal Quarterly Tax Payments are due April 15th

# Moment Methods for Modeling Crystallization Flowsheets by 67uWH7

VIEWS: 36 PAGES: 64

• pg 1
```									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

 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
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:

   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
.
.            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
.
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
.         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
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
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

```
To top