1. CHEM-E7130 Process Modeling Lecture 6

2. Outline Numerical integration
Integral equations and integral functions
Distributions and their properties
Population balances

3. Numerical integration
In numerical integration, objective is to find a value for a definite integral (integral over a known interval)
One-dimensional integration is considered here only.
How to evaluate y if a, b, and f(x) is known but cannot be integrated analytically?

4. 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
y=ex

5. Midpoint and trapezoid
Exact value
Rectangle
4% relative error
Trapezoid
8.2% relative error

6. Midpoint and trapezoid
Simpson’s 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)

7. Newton-Cotes formulas
Estimate function by polynomials through points with equal distances from each other
Trapezoid and midpoint: first order Newton-Cotes
Simpson’s rule: second order N-C
3/8 Simpson: third order N-C
Boole’s rule: fourth order N-C
etc...

8. Quadratures Newton-Cotes: quadrature points xi from constant intervals
Gauss-Legendre quadratures: quadrature points from zeros of Legendre polynomials. This is typically optimal way of distributing the points where the function is evaluated

9. Gauss-Legendre quadrature

10. Gauss-Legendre quadrature

11. Quadratures
In practice, quadratures are applied piece-wise in the interval
Division into sub-intervals could be constant or adaptive
For a very high number of quadrature points, floating point calculation error starts to dominate
X= What is Y = X – ?

12. 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
c(0,t)=c0 c(,t)=0
c(x,0)=0
”diffusion equation”

13. Integral functions c(0,t)=c0 c(,t)=0 c(x,0)=0 Analytical solution:
Also: probability theory (cumulative normal distribution)...

14. Distributions
Often variable values are somehow distributed. In these cases, (numerical) integration is often needed to get some distribution properties
Distributed variable number density
1/Length (1/m)
Distribution. Often scaled so that the area under the curve = 1
Distributed variable
Length (m)

15. Distributions Standard deviation? Median? Average? Distribution (1/m)
Size(m) number
Distribution (1/m)
Standard deviation?
Particle size (m)
Median?
Average?

16. Integral equations
Any equation where an 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

17. Integral equations, exampley
Overall sample property f(s) is measured, 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

18. Integral equations, example
Example continues. L indicates athlete’s 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).
g(L) for a basketball team
y(L)
contribution function
g(L) for a chess team
team success
athlete length distribution L

19. 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

20. What is the population balance concept?
Population balance is about counting: 6 in

21. Compare to molar concentration:
Number of
per unit
is the number density, similar to concentration. mol / m3 = NA molecules in unit volume
However, not all particles are similar, they can e.g. vary in their size, forming a size distribution that can be modelled based on breakage, agglomeration, growth etc.

22. Two kinds of coordinates: external and internaly p1 x p2

23. Every dispersed phase property that is not assumed constant, adds one dimension
Adding one more dimension to your model typically increases the computational load by an order of magnitude.
Choose such coordinates (internal and external) which are the most important to overall process performance
Here only size is considered as the internal coordinate

24. Distribution properties
Density function f(L)
Number density, total number of particles (per unit volume)
Moments of the distribution

25. Mean diameters from moments
mi/mi-1 is a characteristic length (if moments are for length coordinate)
m1/m0 number average diameter
m2/m1 length weighted average diameter
m3/m2 area weighted average diameter

26. Use of size distribution moments: Mass transfer area
Size(m) number
Mass transfer area is assumed to be the total surface area of the particles

27. Mass transfer area where s is a shape factor
L32 is the Sauter mean diameter (m3/m2)
 is the particle phase volume fraction
V is the total volume of the dispersion

28. Other characteristic numbers
Standard deviation
describes the width of the distribution
Also other characteristic numbers can be calculated from the moments: skewness, kurtosis

29. Example
Calculate moments and average diameters for bubble size distribution that follows normal distribution with average size 0.02 m and standard deviation m. Use Gauss quadrature.

30. average size
0.02 m
standard deviation
0.002 m

31. How to scale the quadrature points?
Distribution
Points and weights (1000) for 6-point quadrature

32. Sum = 8.84E-10. Not correct. It should be one.xi wi
f(xi)
f(xi)wi
1.03E-08
8.84E-10
Sum = 8.84E-10. Not correct. It should be one.

33. Scale the points from 0.013 m to 0.027 m
What would be a linear variable transformation L=f(x)?

34. Scaling
This is similar what you have done when non-dimensionalizing models. Here you are actually doing the other way around, as the points were originally non-dimensional (0 to 1), but you transform them into physical sizes

35. Sum = 0.9935. Almost correct (should be 1). xi wi Li
f(Li)
f(Li) wi
Sum = Almost correct (should be 1).
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.
Matlab: q = integral(fun,xmin,xmax)

36. Example: Other moments and average diameters

37. m0 m1 m m2 m2 m3
E-06 m3
d10=m1/m0
0.02 m
d21=m2/m1 m
d32=m3/m2 m

38. Population balances
In population balances, the number balance equation is formulated for a fraction of the distribution
Material balance in a reactor:
Population balance: y
y(L) L

39. Population balance equation
”reaction” i.e. birth and death due to breakage and coalescence
Time rate of change
Convection
where v is velocity (internal + external coordinates) B is birth rate D is death rate
What is convection along an internal coordinate (size)?

40. 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

41. 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

42. Continuous distribution is discretized for computational purposes

43. ”Easy” problem: initial value ODE
Linear terms
Nonlinear terms
Flow in
Birth by breakage
Birth by agglomeration
Flow out
Death by breakage
Death by agglomeration
Growth
Primary nucleation

44. Discretization tables
Construction of these tables is characteristic to the discretization method
Breakage
Agglomeration
Growth
These depend only on discretization and chosen daughter size distribution in breakage and should be calculated prior to the simulation

45. All phenomena occurring in the system are modeled based on particles at discrete classes. For example bubble coalescence: L La Lb

46. Generally, the size of the bubble resulting from the coalescence does not coincide with 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

47. Distribute the new bubble in two closest categories
Distribute the new bubble in two closest categories. A reasonably good method, 2 properties can be conserved, typically number and volume. Perhaps the most popular method nowadays. L
How to calculate fractions y1 and y2 that goes to the two neighbouring categories?

48. 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

49. Distribute the new bubble in several categories
Distribute the new bubble in several categories. Method order increases as more properties are conserved (e.g. moments) L

50. 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

51. 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 can be solved e.g. by discretizing the internal coordinate into several size classes, and following number of particles in them