Process Modeling methods and tools Lecture 10 Ville Alopaeus

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Process Modeling methods and tools Lecture
10Ville Alopaeus
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Outline

• Numerical integration
• Integral equations and integral functions
• Distributed properties
• Integro-partial differential equations
  (Population balances)

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Numerical integration

• In numerical integration, objective is to find a
  value for definite integral (integral over a
  known interval)

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Midpoint and trapezoid methods

• Estimate function value over the interval either
  by a constant (rectangle rule) or by a trapezoid
  (linear approximation through the endpoints)

Trapezoid
Rectangle
yex
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Midpoint and trapezoid

• Exact value

Rectangle
3.7 relative error
Trapezoid
8.2 relative error
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Midpoint and trapezoid

• Simpsons rule

0.03 relative error
One more function evaluation per interval Each
interval is approximated with a quadratic
polynomial (higher order method than trapezoid or
rectangle)
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Newton-Cotes formulas

• Estimate function by polynomials through points
  with equal distances from each other
• Trapezoid first order Newton-Cotes
• Simpsons rule second order N-C
• 3/8 Simpson third order N-C
• Booles rule fourth order N-C
• etc

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Quadratures

• Newton-Cotes quadrature points xi from constant
  intervals
• Gauss-Legendre quadratures quadrature points
  from zeros of Legendre polynomials (Special case
  of Jacobi polynomials with weight 1)
• Other weight functions can be used as well
  (different quadrature formulas)

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Gauss-Legendre quadrature
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Gauss-Legendre quadrature
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Quadratures

• In practice, quadratures are applied piece-wise
  in the interval
• Division into sub-intervals could be constant or
  adaptive. In case of adaptive sub-interval
  division, an error estimate is required (e.g.
  Gauss-Kronrod method)
• Again multipoint integration ? high order
  method, less points ? lower accuracy
• For high number of points, floating point
  calculation error starts to dominate

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Quadratures

• Gauss-Legendre quadrature calculates numerical
  integrals exactly for polynomials of degree 2n-1
  with n quadrature points
• Numerical integration based on either equidistant
  quadrature points or points from zeros of
  orthogonal polynomials is closely related to
  solution of differential equations by equidistant
  or orthogonal collocation (see lecture 8)

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Integral equations

• Any equation where integral appears can be called
  an integral equation.
• In mathematics, an equation where unknown
  function is under integral sign is called an
  integral equation
• Integral functions are such where unknown is only
  outside the integral sign

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Classification of integral equations

• Fredholm equation of the first type

Fredholm equation of the second type
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Classification of integral equations

• Volterra equation of the first type

Volterra equation of the second type
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Classification of integral equations

• Limits of the integration fixed Fredholm
• One limit not fixed Volterra
• Unkonwn function only inside integral first kind
• Also outside integral second kind
• Known function f(x)0 for all x homogeneous
• f(x) ? 0 for some x inhomogeneous
• Not all integral equations can be classified with
  the above.

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Integral functions

• When differential equations are solved,
  integration is the final step. Sometimes there is
  no analytical solution to the integral. In those
  cases the solution is given in terms of integral
  functions

Error function

• Solution of diffusion equation
• Probability theory (cumulative normal
  distribution)

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Some integral functions

• Gamma function
• Generalization of factorial n!
• Incomplete gamma function
• Some bubble breakage models
• Complete elliptic integral
• Pendulum movement

There are solution methods (series solution etc.)
for these integrals. They are also widely
tabulated
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Distributions

• Distributed variable density
• 1/Length (1/m)

Distribution. Often scaled so that the area under
the curve 1

• Distributed variable
• Length (m)

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Distributions
y
Example we have measured an overall sample
property f(s), and we know the sample size
distribution y(s,L). Our task is to find a
size-dependent function g(L) that models how the
distributed property contributes to the sample
property
L
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Distributions
y
This is a Fredholm equation of the first kind
L
Notation does not matter, important is to
identify what is the unknown or known function,
distributed property etc. s?x, y ?K, g ??, L ?t
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Distributions
Example continues. L indicates athletes length
and y is probability that there is such an
athlete in a team. Find a function g that
predicts how well the team succeeds (f).
Basketball team
y(L)
Chess team
contribution function
team success
athlete length distribution
L
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Population balances

• As an example of integral equations
• Here only particle size distributions are
  considered
• Generally any distributions (with one or more
  distributed variables) can be modeled

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What is the population balance concept?

• Population balance is about counting

in
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Number of
Compare to molar concentration
per unit
is the number density, similar to
concentration.mol / m3 NA molecules in unit
volume
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However, the counted particles may be dissimilar
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Usually the properties associated to the counted
particles form continuous distributionsThis is
different to molar balances
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Two ways of counting

• 1) Assume (multiple) distributions within each
  control volume. Formulate balances for the
  dispersed phase distributions
• ? Eulerian approach
• 2) Count each particle with the properties
  associated with it. Properties of the particles
  may change as they flow around.
• ? Lagrangian approach
• The former is often useful for dense dispersions,
  the latter for lean dispersions with fewer
  particles to be counted

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Two kinds of coordinates external and internal
internal

• external

y
p1
x
p2
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• Every dispersed phase property that is not
  assumed constant, adds one dimension (Eulerian
  approach)
• Often considering more than one internal
  coordinate is computationally quite a heavy
  burden
• Choose the one that is most important to overall
  process performance
• Here only size is considered as the internal
  coordinate

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Distribution properties
Density function n(L)

• Moments of distribution

Total number of particles in unit volume NP
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Dimensions
Population density function n(L) 1/m (for L
as internal coordinate) Total number of
particles in unit volume NP number /
m3 Number density NPn(L) number / m4
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Dimensions (2)
In the population balance equation, number
density appears always as NPn(L). Often the
symbol n(L) is used instead of NPn(L). Be sure
not to confuse these in calculations! For
example, number of particle collisions (adopted
from the kinetic gas theory) is NPn(L1)NPn(L2)
(second order process)
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Various average sizes

• Lk,k-1 mk / mk-1 has a dimension of length
• Some typical averages (whether m01 or Np)
• L10 mean particle size
• L21 length weighted average size
• L32 area weighted average size (Sauter mean)
• Also L30 (m3/m0)1/3 volumetric average size

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L32 is called Sauter mean diameter
Lagrange
Euler
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Example

• Caluclate moments and average diameters for
  bubble size distribution that follows normed
  normal distribution with average size 0.02 m and
  standard deviation 0.002 m.

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• Gauss quadrature is used.

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• How to scale the quadrature points?

Distribution
Points and weights (?1000) for 6-point quadrature
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• Sum 8.84E-10. Not correct

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• Scale the points from 0.013 m to 0.027 m

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Scaling
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• Sum 0.9935. Almost correct.
• In practice, use several (adaptive) sub-intervals
  for numerical integration. Suitable algorithms
  are available, but a clue about the correct
  integration limits is usually needed.

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Example Other moments and average diameters
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(No Transcript)
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Population balance equation
wherev is velocity (internal external
coordinates)B is birth rateD is death rate
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Population balance equation terms for particle
volume as an internal coordinate
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...for diameter
nonlinear (n2)
Integro ...partial differential equation
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Example growth only
Similar to the advection equation Hyperbolic
partial differential equation
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Solution strategies for population balance
equations

• 1) Lagrangian or Monte Carlo methods
• - dispersed particles are tracked. Suitable for
• relatively lean dispersions
• 2) Method of moments
• - when only approximate information is needed
• 3) Analytical solutions
• - only seldom possible
• 4) Discretization of the internal coordinate
• - general but sometimes laborious

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Solution strategies for population balance
equations

• 1) Lagrangian or Monte Carlo methods
• - dispersed particles are tracked. Suitable for
• relatively lean dispersions
• 2) Method of moments
• - when only approximate information is needed
• 3) Analytical solutions
• - only seldom possible
• 4) Discretization of the internal coordinate
• - general but sometimes laborious

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Method of moments

• Most suitable if only some overall properties
  that can be expressed in terms of the moments are
  needed. For example mass transfer area or average
  size
• Moment transformation of the PB equation is
  obtained by multiplying it by Li

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Method of moments (2)
Solution of this equation gives transport
equations for moments k0...n, where n is the
largest moment to be tracked
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QMOM (quadrature method of moments)

• Quadrature points zi and weights wi fulfil the
  moment equation

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QMOM (quadrature method of moments)

• Quadrature points and weights

wi and zi can be calculated from the moments with
the PD (product difference) algorithm or directly
if relative locations of zi can be specified
(e.g. zeros of Legendre polynomials, scaled to
the correct distribution location) PD-QMOM
F-QMOM (fixed qmom) DQMOM Direct QMOM.
Transport equations written directly for
abscissas (quadrature points) and weights
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QMOM (quadrature method of moments)

• Quadrature points and weights

Gauss-Legendre quadrature weights actually fulfil
the same equation with moments calculated for a
function y(x)1, x0,1 i.e. mk1/(1k)
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Discretization of the internal coordinate

• The most common Eulerian type dispersed phase
  model is to discretize the internal coordinate
  domain into a number of classes
• This corresponds to discretization of the
  external coordinates (grid generation), but the
  numerical solution strategies differ.

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Continuous distribution is discretized for
computational purposes

• Actually the continuous distribution is not
  transformed into rectangles, but into a set of
  Dirac delta distributions

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Discretization of the distribution

• Continuous distribution is approximated by a
  number of discrete categories

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• All phenomena occurring in the system are modeled
  based on particles at discrete classes. For
  example bubble coalescence

L
La
Lb
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• Generally, the size of the bubble resulting from
  the coalescence does not coincide to any
  category. In case of equal diameter
  discretization it never does.
• There are different methods how to distribute the
  bubble resulting from the coalescence

Distribute the new bubble only in the closest
category. Very poor method, even number and
volume cannot be conserved simultaneously.
L
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Distribute the new bubble in two closest
categories. A reasonably good method, 2
properties can be conserved. Perhaps the most
popular method nowadays.
L
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Number and volume conserved
fraction of the coalesced particle to be
distributed into two closest categories
number
and volume are conserved
two closest category sizes
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Distribute the new bubble in several categories.
Method order increases as more properties are
conserved (e.g. moments)
L
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Moments are distributed
categories into which coalesced bubble is to be
distributed
category sizes
number of particles to be distributed into each
target category
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Distributed moments
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Example breakage with power-law kernel

• Breakage rate

Daughter size distribution
Initial distribution
Analytical solution
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Example breakage with power-law kernel

• Daughter size distribution moments associated to
  category i

These moments are distributed to neighbouring
categories as
Compare to
Daughter size table is obtained with a linear
transformation
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Example breakage with power-law kernel

• Daughter size table is constructed based on
  conservation of arbitrary number of daughter size
  distribution moments (high order method if high
  number of moments are set to be conserved). For
  six moments and 15 categories

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Example breakage with power-law kernel
Initial distribution
Distribution after 10000 s
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Example breakage with power-law kernelRelative
error in the simulated distribution moments
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Example breakage with power-law kernel
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Example Breakage and agglomeration

• Breakage rate

L lt 51/3
L 51/3
Daughter size distribution
Agglomeration rate
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Example Breakage and agglomeration

• Geometric discretization on the interval 1,11,
  but at least one particle intervals for smallest
  particles

Steady state distribution is sought. No
analytical solution exists. Discretization
refinement characteristics can be used to
evaluate various methods.
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Example Breakage and agglomeration

• Results Six moment conserved in both breakage
  and agglomeration processes (left), vs. two
  moments (right)

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Example Breakage and agglomeration

• Results Evolution of the size distribution as a
  function of number of categories

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Example Breakage and agglomeration

• Results Evolution of the size distribution as a
  function of number of categories

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Conclusions

• There are several methods for evaluating definite
  integrals numerically. The simplest ones are not
  usually very effective.
• Gaussian quadratures are usually quite good.
• Integral equations are such where unknown
  function is under integral sign
• Integral functions are such where an integral
  needs to be evaluated in order to calculate the
  function value

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Conclusions

• In population balances, distributions are
  considered. For example particle size
  distributions, when all particles are not of the
  same size
• Moments are important measures of distribution
  properties
• Population balances are non-linear
  integro-partial differential equations

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Conclusions

• There are several methods to solve population
  balances
• In moment methods, distribution moments are
  followed
• In category methods, continuous distribution is
  approximated by a number of discrete categories