[ocw] optimal design of energy systems chap 15

Min Soo KIMDepartment of Mechanical and Aerospace Engineering

Seoul National University

Optimal Design of Energy SystemsChapter 15 Dynamic Behavior of Thermal Systems

Chapter 15. Dynamic Behavior of Thermal Systems

focus on thermal systems

automatic control

block diagram

with respect to time

on (start up), off (shutdown), under control, disturbance

off-design

design


15.3 Dynamic Simulation

compressor ref. capacity

2 ( , )e cP f T T

condenser ,/, ( )(1 )p aUA mc

c p a c ambq mc T T e

evaporator ( )( )e s e eq T T UA

energy balance c eq P q

heat transferto chamber

( )e tr ch amb sq q UA T T

1 ( , )e e cq f T T

steady-state

compressor power


15.3 Dynamic Simulation

pull-down ( )tr ch amb sq UA T T

eq trq

sT

str e p

dTq q mcdt

, pm cdynamic : during pull-down tr eq q

0

0

0 0

0

0 0

2

{ ( )} ( ) ( )

{ ( )} ( )

( ) ( )( )

(0) ( )

{ ( )} ( )

( ) ( )( )

(0) [ (0) ( )]

(0) (0) ( )

st

st

st st

st

st st

L F t F t e dt f s

L F t F t e dt

e F t F t s e dt

F sf s

L F t F t e dt

e F t F t s e dt

F s F sf s

F sF s f s


15.4 Laplace transform Powerful tool in predicting dynamic behavior

One way to solve ODE


15.5 Inversion

<example 15.4> Invert

<solution>

1{ ( )} ( )L f s F t

2

10( 2) ( 1)

ss s

2 2

2

10( 2) ( 1) ( 1) ( 2) 2

: 10 4 2

: 1 4

: 01, 4, 1

s A B Bs s s s s

constants A B B

s A B B

s A BA B B

1 2 22

10 4( 2) ( 1)

t t tsL e te es s


15.5 Inversion

<another solution for example 15.4>

1{ ( )} ( )L f s F t

( )( )

N s A BD s s a s b

For non-repeated roots

2

( )( ) ( )

N s A B BD s s a s b s b

( )( ) ( )( )( ) ( )s a s b

N s s a N s s bA BD s D s

For repeated roots

2 2( )( ) ( )( )( ) ( )s b s b

N s s b d N s s bB BD s ds D s

22 21

10 10 101 4 1( 2) 1 1s ss

s s d sA B Bs s ds s


15.7 Block diagram and transfer functions

{ ( )} ( ){ ( )} ( )

L O t O sTFL I t I s

Transfer function :

ratio of the output to the input

- Variable in S domain(not in time domain)


15.8 Feedback control loop

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) 1 ( ) ( )

C s G s E sE s R s H s C s

C s G sTFR s G s H s


15.9 Time constant blocks

( )

[ ( ) (0)] ( ) ( )

f

f

dTm c T T hAdt

m c sL T T L T L ThA

m cBhA

1 (0)( )( )

( ) 1f

f

BTT sT sTF

T s Bs

Standard technique for developing transfer function

1. Write differential equation

2. Transform equation

3. Solve for transfer function ( ){ } / { }L O L I

For special case :(0) 0T 1

1TF

Bs


15.9 Time constant blocks

00 0

0

0

( ) [( ) ( )]

{ } 1{ } 1

f

f

d T Tm c T T T T hAdt

L T TTFL T T Bs

( )fT ss

: unit step increasefT

: time constant

0 01 1{ } { }

( 1) ( 1) 1 1fBL T T L T T

Bs s Bs s Bs s Bs

0, 1,B B

/0 (1 )t BT T e

m cBhA

desiredtemperature


15.9 Time constant blocks - additional

• Control

• Feedback

• Block diagrams

Referencesensor

Actuator process

output sensor

2-

+controlinput output

disturbance

desired roomtemperature T → V Inverter

Refrigeration system

Thermometer

2-

+ Actual Temp.

Ex)



• Dynamic model – a set of differential equations to describe the dynamicbehavior of the system

q pm cT

pdTq m cdt

Linear case

State of the system:( , )( , )

f uy h u

u : input

y : output

FX G uy H X Ju

1

2

: [ ], : [ 1]: [1 ], :

F n n G nH n J scalar

xX

x

• Differential equation in state variable form (modern control)



Ex)

Ex) Heat exchanger

1 2

1

1 1 2

( )

( )

p

p

q h T TdTq m cdt

hT T Tm c

1 2( )q h T T

1pq m c T

1T 2T

( )

( )in s p si s

out w p w w i

q m c T T

q m c T T

,

,

, ,

( ) ( )

( ) ( )

ss p s in steam water

s p s si s s w

ww p w w p w wi w s w

dTm c q qdt

m c T T UA T T

dTm c m c T T UA T Tdt

Steam

Water

siT

sT

wiT

wT


15.10 Cascade time-constant blocks

Heat balance equation :

1 1 2 2

2 2

1 21 1 2 2

( ) ( )

( )

,

ba b b L

Lb L

dTT T h A m c T T h AdtdTT T h A m cdt

m c m cLeth A h A

1

11s 2

11s

0{ }bL T T 0{ }LL T T0{ }aL T T

Air to bulb Bulb to liquid

Cascade time-constant block

subscript 1 : air to bulb

subscript 2 : bulb to liquid

neglect


15.10 Cascade time-constant blocks

For unit step input

01 2

1 1{ }1 1LL T T

s s s

Inversion

1 2/ /0 1 2

1 2 2 1

1 t tLT T e e

1

0

0

/2 1 0

//0

2 1

0, 0( ) 0 0

(1 )

1

L

L

tL

ttL

t T Td T T at t

dtif T T e

T T teif e

①

②

③

④


15.11 Stability analysis

- Frequency response / Bode plot (diagram)

Response to sinusoidal input

Sinusoidal input : 0( ) sin(2 )fT t T ft

0 2 2 2 2

2 2 2 2 2

2

2 2 2 2 2

0 2 2

1( )1 1

, ,1 / 1 / 1

1 1 / 1( )1

a A Bs CL T Ts s a s s a

a a aA B Ca a a

a a asa a aL T T

s s a

<Example in 15.9>

<From example in 15.9>

2 2sin 2 , 2aL ft a fs a



21

0 2 2 2 2

/2 2

0 2 2

2 22 2 2 2

1

11 1

1cos( ) sin( )1

sin( ) cos( )1

1 sin( ) sin( )1 1

tan ( )

t

a sT T La s s a

a e at ata a

as t

T T at a ata

a at ata a

a

a

1

2 21 a

2

11 (2 )f

Amplification ratio :

Phase lag : 1tan (2 )f



• Stability criterion

At the frequency f where sum of phase lags = 180o Fig.(a)

product of amplitude ratio > 1 Fig.(b)

→ the loop is unstable

Phase lag =180o Amplitude ratio > 1

Fig. Bode diagram


15.11 Stability analysis - additional

*

1

0

02 2

*0 01

1

1 01 0

0

0

( ) ( )( )

( ) sin

( ) ( )

Im( )( ) 2 sin( ), tanRe( )

, ( ) sin( )

n

n

n

a ta tn

Y s G sU su t U t

UY s G ss

s a s a s j s j

y t e e t

as t y t U A t

magnitude

phase

2 2

1

( ) ( ) {Re[ ( )]} {Im[ ( )]}Im[ ( )]( ) tanRe[ ( )]

A G j G s G j G jG jG jG j

: input


15.12 Stability analysis using loop transfer function

Roots are the poles of the transfer function 1+KG(s)

→ root locus : graph of all possible roots

KG(s)2-

+Y(s)R(s) ( ) ( )

( ) 1 ( )Y s KG sR s KG s

root < 0

＂ = 0

＂ > 0

＂ = a+ib ( )

1

at

at

a ib t

e

ee


15.12 Stability analysis using loop transfer function

Root locus form : ( )1 ( ) 1 0 ( ) ( ) 0( )

b sKG s K a s Kb sa s

Ex) 2( ) 1 0( 1) ( 1)

K KKG s Denominator s s Ks s s s

Root locus is a graph of the roots of the quadratic equation

1 21 1 4,2 2

0 1 / 4 1 0

1 / 4

Kr r

K

K complex

11

11

1 2

1 2

( )( )( )( )

( ) ( )( ) ( )( ) ( )( ) ( )

m mm

n nn

m

n

K s b s bKb sKG s n ma s s a s a

b s s z s z s za s s p s p s p

-1 0-1/2

as K increases

Breakaway point

poles


15.14 Restructuring the block diagramconverting complex block diagram → feedback loop

simplifying


15.14 Restructuring the block diagram


15.15 Physical situation → block diagram

aq( )

( )(1 ) ( )

h set s

hAwc

a h i h i

q K T T

q wc T T e W T T

( ) ( ) [ ( ) (0)]

hh a

h a h h

dTq q mcdt

q s q s M sT s T

heater → airwc W


15.15 Physical situation → block diagram


15.16 Proportional control

( )h p set sq K T T

errorK↑ unstable

K↓ offset


15.17 Proportional – Integral (PI) control

- to eliminate the offset

- PI control

( )IK error dt

s

2IK

s

2//

I IK K sTFs s


15.18 PID control

IP D

KK K ss


15.18 PID control

Ziegler-Nichols Tuning of PID controller (1942)

1( ) 1 DI

ID P D

I

D s K T sT s

KKK KT s K K sT s s

Reaction curve

TF may be approximated by

Time delay of td seconds

1

dt sKeTFs

/te

dL t

KKR

: reaction rate

open loop control


15.19 Feed forward control

(a) No control

(b) Feed forward control

(c) Feedback control

[ocw] optimal design of energy systems chap 15

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