lecture 5a: model analysis - lumped parameter & dps models · 2018-04-05 · lecture 5a: model...

Lecture 5a: Model Analysis - Lumped

Parameter & DPS Models

Rafiqul Gani

(Plus material from Ian Cameron)

PSE for SPEED

Skyttemosen 6, DK-3450 Allerod, Denmark

[email protected]

www.pseforspeed.com

2

Lecture 5a: Computer Aided Modelling

Key issues to consider

❖ Must analyze before solving

▪ Degrees of freedom analysis

▪ Incidence matrix; singularity of equation

system; index-number

▪ Stability of the model

3


Degrees of freedom analysis - 1

Classify equations

Algebraic equations (AEs)

•explicit (a = b*x + c with b, c, x as known)

•implicit (0 = b*x(a) + c – a with b, c as

known)

Ordinary differential equations (ODEs)

•first order ODEs (dy/dt = f(y, t)

Partial differential equations (PDEs)

•time plus one-three spatial directions for

different coordinate systems

Note: Mathematical model can have any combination of above

4



Classify variables

Known

variables fixed by system

variables fixed by problem

variables fixed by model

Unknown

dependent (differential)

dependent (partial differential)

implicit (algebraic)

explicit (algebraic)

Independent

time, length, ..

5



Definition (NDF or NS; degrees of freedom or number of variables

to specify):

-NV = total number of variables (not counting

independent variables)

-NU = number of unknown variables

-NE = number of independent equations

Assign dependent (partial) to PDEs; dependent (differential)

to ODEs; implicit & explicit algebraic to AEs

DF V EN N N S V UN N N OR

6


Example DoF analysis

0

0

0

)(

02

322

211

21

gzPP

PPCF

PPCF

A

FF

dt

dz

V

V

1 2 0 1 2 3

V

State (dependent-differential) variable

Algebraic variables , , , , ,

Parameters C , ,

Constants

Equations 1 + 3

11 4 7DF V E

z

F F P P P P

A

g

N N N

z

P1 P

2P

3

P0

F1

F2

Model DOF analysis

Assign z to ODE; Cv, A,

, g as fixed by system;

select 3 variables from

”algebraic” as fixed &

3 as unkown

Introduction to MoT

•Model construction

•Model analysis

7Lecture 5a: Computer Aided Modelling

Different specifications (DoF selection)

z

P1 P

2P

3

P0

F1

F2

Select 3 variables from

“algebraic” as fixed,

making the remaining 3

as unknown and assigned

to the AEs

Specification 1

3101 ,, PPPS

Specification 2

2102 ,, PPPS

x

xx

xx

3

2

1

221

g

g

g

PFF

3

2

1

321

xx

x

g

g

g

PFF

Unsolvable

system?


High index DAEs

❖Only for Differential-algebraic equations

(for PDEs, first discretize the PDEs)

❖ Pure ODE systems are index 0

❖Many index 1 DAE solvers

❖ Few higher index solvers

❖Ensure index 1 models are used


DAE index definition

❖The “index” of the DAE system is the

number of times the algebraic sub-

system must be differentiated to give a

set of ODEs.


General DAE system

General system

),,(0

),,(

tzyg

tzyfdt

dy

Differentiate the algebraic equations

rankfullgifffggdt

dz

dt

dzgfg

dt

dzg

dt

dyg

zyz

zy

zy

0

0

1

What is full

rank?


High index example

! model 1index

once ateDifferenti

1211

211

121

1212

1211

3

2

20

zyyz

yyz

zyy

zyyy

zyyy

! model 2index

...again and

once ateDifferenti

1211

121

121

21

21

1212

1211

442

30

30

20

20

zyyz

zyy

zyy

yy

yy

zyyy

zyyy

Case 1 Case 2

12


What is the index of this model?

0

0

0

)(

02

322

211

21

gzPP

PPCF

PPCF

A

FF

dt

dz

V

V

z

P1 P

2P

3

P0

F1

F2

Model

Specification 1

3101 ,, PPPS

x

xx

xx

3

2

1

221

g

g

g

PFF

• Insert F1 and F2 into Eq. 1

• Differentiate Eq. 4 with

respect to t

• What happens if P0, P1, P2

are specified?


Exercise - 4 : What is the index of this model?

2211ˆˆ hFhFQ

dt

dH

Conservation

Constitutive

DzA

TMcH

PTfh

PTfh

TTUAQ

P

H

),(ˆ

),(ˆ

)(

22

11

Tank

QhFhFdt

dHHoHHiH

J ˆˆ

Conservation

Constitutive

2)(

),(ˆ

),(ˆ

)(

HoHiH

HPHHJJ

HoHo

HiHi

HoHiPHH

TTT

TcMH

PTfh

PTfh

TTcFQ

Jacket

Tank

contents

jacket

fluid

Q

T

THo

THi

F1, T

1

F2, T

System

14


Stability of the model

❖Indicated by the dynamic modes- time constants of the system

- related to physico-chemical phenomena

•fluid flow (generally fast)

•mass transfer rates (can be slow)

•heat transfer (usually slow)

•reaction kinetics (very fast to very slow)

i

i

1

15


Linear model analysis

Linear equation system

Solution given by

0

25.1000

11

75.9992000

2

1

2

1

y

y

y

y

Ayy

10025.0999.2)(

1499.0449.1)(

5.20005.0

2

5.20005.0

1

tt

tt

eety

eety

Slow mode (component) Fast mode (component)

16


Computing eigen-values of the J-matrix

❖ In MoT, write the

ODE system; and then

ask for calculation of

the eigen-values

❖ For problem of slide

13, the solution is:

Solution with MATLAB

» A=[-2000 999.75;1 -1];

» b=eig(A)

b =

1.0e+003 *

-2.0005

-0.0005

Two eigen-values

(-2000.5, -0.5)


General linear system analysis

01

1

1

0

:

)(

)exp()(

)0( ,

kjk

n

k

ijij

tn

j

iji

xVVZ

where

eZtx

tZtx

xxAxdt

dx

j

Linear set of equations

Solution to the equations

Individual component

solution to the equations

– note the eigenvalues !

– note the eigenvectors !

The eigen-values

contain the stability

information

18


Eigen-value analysis

❖ Dynamic modes determined by eigen-values of

the linear system

❖ Nonlinear problems must be linearised first to

get Jacobian J(y,t) on the trajectory

❖ Eigen-values of Jacobian computed using MoT

or MATLAB eig(A) or similar function.


Behaviour of linear systems

❖ Phase plane analysis illustrates motion of system

❖ Second order systems plot

❖ Typical second order system (cf. valve actuator) :

yyyy vsor vs 21

2/4

2/4

0

2

2

2

1

baa

baa

with

byyay

Set y1 = dy/dt = ý

y2 = dy1/dt = d2y/dt = ÿ

So,

y2 + a y1 + by = 0

Therefore,

dy/dt = y1

dy1/dt = - (a y1 + by )


Phase plane analysis

❖ Several cases dependent on eigen-values

parts real zero with conjugatescomplex and

parts real zero-non with conjugatescomplex and

sign oppositebut realboth and

sign same and realboth and

21

21

21

21


Phase plane diagrams

22


General nonlinear model

❖Linearize at some point

❖Obtain linear model

❖Solution is

),( tyfy

)()( 1 tgvectyi

tn

i i

i

))(,())(( tgtftgyAy

Derive the linear model and solution for:

dy1/dt = y1*y2 + (y1)2

dy2/dt = y1 + (y2)2

23


Stability cases

❖ Stable model

❖ Unstable model

❖ Ultra-stable (“stiff”)

i allfor 0i

large"" Some

small"" Some

0

i

i

i

0 be will Somei

24


Stability implications for numerical solution

❖ The problem determines the choice of numerical

method

❖ “Stiff” problems need special attention

❖ All simple explicit methods have limitations on

allowable step lengths

maxi

max

boundstability method

h


Multiscale, lumping, simplification, stability: Example

• Fluid zone

• Particle zone

• Heat transfer

zone

• Catalyst

recovery zone


Multiscale balance relations



Constitutive relations



Stability analysis

• Multiple steady states

• Unstable state


lecture 5a: model analysis - lumped parameter & dps models · 2018-04-05 · lecture 5a: model...

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