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10.34, Numerical Methods Applied to Chemical Engineering

Professor William H. Green

Lecture #36: TA-Led Final Review.

BVP: Finite Differences or Method of Lines

xC

= Forward/Upwind/Central difference formulas

2

2

x

C

= Central difference-like

Understand when to use the different formulas.

Boundary Condition (Flux) D

boundaryx

C

=Reaction per surface area [moles/m2s]

[m2/s] Internal Flux [(mol/m3)/m]

A

BThe flux is the same for these two arrows

can solve even if A and B are not known

Partition

functioncoefficient Figure 1. The flux is the same for arrows at A and B.

Method of Lines

Solve a differential equation

along line i = 2, , N-1

x

CC

x

C

2

13

2

=

Sparse

Discretize

y

1 2 3

xBoundary Condition

may need to useshooting method

Initial Condition

gradient stiff in y-directon

Figure 2. Example problem good for method of lines.

If this is the B.C.:x

CC

x

C

=

12

1

Use this additional equation with rest to solve for C1 D.A.E.

Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DDMonth YYYY].

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Models vs. Datay = f(x,) y1 = f(x1,)

y2 = f(x2,)

|

yn = f(xn,)

Assumption: 1) y distributed normally around y

2) x are known exactly

P(y)

y y y

10.34, Numerical Methods Applied to Chemical Engineering Lecture 36Prof. William Green Page 2 of 6


WANT:

1) Find the best 2) Is the model consistent?3) Error bars on parameters

Assume model is exact

xi yi f(=iy xi,) data will be distributed around model

x1 y1 x2 y2 xn yn

P(yi) ( )

2

2

2

),(exp

ii xfy

P(y) ( )

( )

==

N

i

ii

N

i

ii xfyxfy

1

2

21

2

2

),(2

1exp

2

),(exp

FIT: Max P(y) Min

( )=

N

iii

xfy1

2),(

k = Aexp(-Ea/RT)

ln k = ln A - Ea/R (1/T) Linear in parameters ln k, ln A,

Ea/R

y = xn = [xTx]-1xTy

xn: n rows (measurements), m parameters

Figure 3. A normal distribution.

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S.V.D.: x = UmxmmxnVnxn

= =

N

i

i

i

iv

yv

1 Sample variance guess for : s2 =

)dim(

)(1

2

=

N

yyN

i

i

y is mean y, f(x,)

If non-linear, use optimization methods.

For correctness, compare s to . Quantitatively, use 2 (chi squared)

2 =( )

=

N

i

ii xfy

12

2),(

Transform to z

P(y)

y y y

1

0

mean of 0, = 1

z[N-dim()] 2min

2

Goodness of fit: areaunder curve 2min to

P(y)

y y y

Error Bars Difficult

If linear in parameters and is known, covariance() = 2[xTx]-1 (diagonal mxm matrix)

i=

min,iz

2,5[xTx]

i,i

-1/2 m = # parameters

min,i point that boundsfrom 2 error on min,i

0.0252.5%

Figure 4. Usually we will accept a model with the integral greater than 5%, but we wouldlike it higher. If 99% chance it is wrong, reject.

Figure 5. Chi-squared distribution.

Non-linear: [xT

x]i,i xi,j =j

ixf

),(Find xi,j



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I

F

I

y

x

F

n MATLAB, use nlinfit, nlparei



igure 6. Location of chi-squared and 95% confidence interval in 1-2 space.2 [1

2 min2] = 2 additional degrees of freedom: let 1, 2 vary

f unknown, use student t distribution based on s.

normal

student tbroader

Report T(,), being N-dim. as N increases, student t approaches normal distribution

i = ( you want to calculate )

is known, yi is to be measured.

Average value of parameter: m = (yi)/N

= xN

Tx = [ = N [x]

Tx]-1/2/N

Global OptimizationConvex function H 0 (Hessian Matrix is positive definite)

Figure 7. Comparison of normal and

Student-t distributions.

igure 8. Example of a convex function.Only 1 minimum

Non-convex:

2,m2

95% confidence

2min

1,m 1 1

2

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Branch and bound

Professor Barton Non convex function guarantees global minimum

Split

Divide domain

Bound from aboveUnderestimate below

Find mimina.

Bound again

Figure 9. An example of a

non-convex function.

Figure 10. An illustration of the branch and bound algorithm.If new upper bound is lower than the lower bound, use other region; can stop considering

that section.

Multistart:

Take a bunch of initial guesses and then run local minimization.

No guarantee.

100 points, 6 variables 1006 calculations.

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Simulated annealing

E

0 2

Dihedral angle

Figure 11. The energy varies with dihedral angle.

Start at high temperature, decrease T eventually can sample wells once the point is caught

in a minimum.

Genetic Algorithms

Hybrid system: integer variables and continuous variables

Sample space by allowing function values to live, die, replicate, switch values, etc.

Monte Carlo: Metropolis Monte Carlo

Gillespie Kinetics Monte Carlo

StochasticsLook at homework solutions to 10 and 11.

Additional TopicsFourier Transforms and operator splitting may make a showing.



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