Overview of course FM 1015 Modeling of Dynamic Systems€¦ · MATLAB to solve group “project ... Bernt Lie: Overview of course FM 1015 Modeling of Dynamic Systems. Title: Sensor

University of South-Eastern Norway Page 1August 10, 2018 |

FM1015: Modeling of Dynamic Systems

© Bernt Lie

Overview of course

FM 1015 Modeling of Dynamic SystemsBernt Lie

Professor

Faculty of Technology, Natural Sciences and Maritime Sciences



© Bernt Lie

Content

• Learning material• Model forms: ODE vs. DAE• Notation and units• Balance laws• Constitutive models• Models in Modelica• Reduction of DAE to ODE• Analysis• Exam tips• Modelica Example• Summary

Bernt Lie: Overview of course FM 1015 Modeling of Dynamic Systems



© Bernt Lie

Learning material …includes

• Lecture notes (free PDF file, syllabus is ca. 350 pp. out of 650+ pp.)

• A number of video tutorials

• 3 sets of quizzes (to get started)

• A compulsory group task/”project”, with oral presentation (counts 40% in grade)

• Written exam (based on group task), counts 60% in grade)

• Notes on the Modelica language (optional). Use Modelica, Python, Julia, and/or MATLAB to solve group “project”




© Bernt Lie

Model forms: ODE vs. DAE

ODE:

• State – system history

• Input – influences system

• Parameter – constants

• Output – system response

DAE (simplified):

• As for ODE +

– Algebraic variable

• State in ODE and DAE may be different

• Functions in ODE and DAE are different


dxdt

fx,u;

y gx,u;

dxdt

fx, z,u;

0 gx, z,u;

y hx, z,u;



© Bernt Lie

Notation and units

• Notation

– Single letter – avoid assumed multiplication

– Decoration or indices to distinguish

• Units

– Terms that are equal or terms that are added/subtracted same unit!!

– Argument of transcendental ++ functions should be dimensionless


Re R e ?

Re ?

NRe !

dm

dt m i m e

m e K h

h

sin xx

, ln xx



© Bernt Lie

Balance laws

Material balance• Total mass balance:

• Mass species balance:

• Molar species balance:

With 𝑛_𝑠 species, we can only have 𝑛_𝑠independent material balances!!• Level + concentration for liquid• (Partial) pressure for gas

Momentum balance

Energy balance

Thermal energy balance

Use:

• Velocity, flow rate (MB)

• Temperature (EB)

Accumulated variables 𝑥 in DAE


dm

dt m i m e

dm j

dt m j,i m j,e m j,g

dnj

dt n j,i n j,e n j,g

dMdt

M i M e F

dE

dt Ė i Ėe W Q

dU

dt H i H e W v W f Q



© Bernt Lie

Constitutive models I

Types of constitutive models:

• EOS – Equation of State, e.g. ideal gas law

• Introduction of intensive quantities

• Enthalpy:


pjV n jRT

p j

pj

mV

m

V

ĉ j mj

V

mj

V

c j n j

V

n j

V

j mj

m

x j n j

n

m V

m j ĉ jV jm

n j c jV x jn

H U pV

H mĤ nH

H m Ĥ nH

dĤ ĉpdT V 1 pTdp

dH c pdT V 1 pTdp

ĉp ĤT p

; c p HT p

p 1

VVT p

1

VVT p

H j1

ns

Hj; H j1

ns

H j



© Bernt Lie

Constitutive models II

• Bernoulli’s law

• Reaction

• Diffusion

• Kinetic and potential energy

• Friction

• Heat flow


m Cv fup

p C v fu h

h

v j v 1ĉ jĴ j v 1

c jJ j

Ĵ j D jĉ j

x, J j D j

c j

x

K 12

mv2, K 12

J2

P mgy, P ksy0 y

F f KAf, K 12|v |v, NRe

vD

f

16NRe

, NRe 2.1 103

1

f 4 log

10NRe f 0.4, NRe 2.3 103, 4 104

Sr

0, r Rnr , Qnrns

r j,g n j,g

V

n j,g

V, rg Tr

Q c k T

x, Q w2b

hTw Tb

1U

j1

nb

1hj

j1

ns j

k j

Q r T4



© Bernt Lie

Reduction of DAE to ODE

• Reduction of DAE by elimination of algebraic qtys.

• Example:

• Classification:

• Resulting ODE


dxdt

fx, z,u;

0 gx, z,u;

dxdt

fx,u;

dmdt

m i m e

m V

V Ah

m e K hh

x m

u m i

,A,K,h

z m e,V,h

dhdt

m iK h

h

A, dm

dt m i K m

Ah

x h, x m

K,h,,A, K,h,,A

u m i



© Bernt Lie

Models in Modelica

Modelica: language for model description

– Equation based, i.e., order of equations immaterial

– Supports general DAEs

including special case:

Modeling vs. Modelica

– Accumulation quantities from balance laws are differential variables:

– States need initial values, e.g.:

– Inputs and parameters are easy to find

– Remaining variables are algebraic:


F dx

dt,x, z,u; 0

dxdt

fx, z,u;

0 gx, z,u;

x m,mj,nj,M,U

xt 0

z m i,m e,m j,i,n j,i, ,n j,g,

M i, ,H i, ,c j ,V ,v,Ĥ,



© Bernt Lie

Analysis

• Solution of linear ODE

• Linear model asymptotically stable if:

• Time constant

• Linearization:


dxdt

Ax Bu

xt expAt x0 0

t

expABud

expAt I At 12At2 1

n!Atn

j : jA 0 dxdt

fx,u; dx

dt Ax Bu

A f

x , B

f

u



© Bernt Lie

Exam tips

• Before exam:

– Re-work project

– [Exam will be based on/extend project]

– Look over course summary

– Try to solve past exams

– Work with Examples in lecture notes

– Only read those parts of lecture notes needed to understand problems

• During exam:

– Come in good time

– Bring blue or black pen. Do not use pencil or red/green colored pens.

– Answer briefly and straight to the point – there is no time for wordy prose

– Answer clearly – you don’t score by using «fancy»/ unclear explanations

– You don’t pay for the paper – don’t write densely

– Don’t write in the margin




© Bernt Lie

Modelica Example: Outline

• Liquid tank: Model Modelica code

• Focus:

– DAE model

– input-output

– two objects from one class

– time varying input

• Conclusions




© Bernt Lie

Modelica Example: Liquid tank model Modelica code

Bernt Lie: Overview of course FM 1015 Modeling of Dynamic SystemsVideo

https://1drv.ms/v/s!AsaVncxXzg_3qIFBsKtQZjijl3y88A



© Bernt Lie

Modelica Example: Conclusions

• Encoding DAE model to Modelica code

• Model with both input and output

• Instantiating two models for comparing systems

• Time varying input with if-else structure




© Bernt Lie

Summary

• Modeling of dynamic system:– System: well defined subset of world

– Dynamic: inertial effects

– Model: mathematical model

– Relates inputs to outputs

– Based on mechanistic understanding/classical physics

– Basic analysis

– Need to know: balance laws, Need to understand: constitutive eqs.

• Particular characteristics:– Little emphasis on model manipulation

– Keep model as DAE

– Complete examples: model development + description using Modelica + simulation using OpenModelica


Documents

Overview of course FM 1015 Modeling of Dynamic Systems€¦ · MATLAB to solve group “project ... Bernt Lie: Overview of course FM 1015 Modeling of Dynamic Systems. Title: Sensor