Mauro Salazar �1

System Modeling - Lecture 6Class Content 1. Thermodynamic Systems 2. Distributed Parameter Systems

Learning Objectives 1. You can model thermodynamic systems 2. You understand the properties of ideal gases 3. You can model heat transfer phenomena 4. You can perform lumped parameter assumptions and understand their limitations

Script: Chapters 2.4.5 and 2.5

Program 10’ - Recap: Electromagnetic Systems 30’ - Theory of Thermodynamic Systems 5’ - Example: Stirred Reactor 40’ - Examples: Gas Receiver and Heat Exchanger 5’ - Distributed Parameter Systems

The entire class will be taught at the blackboard. You are supposed to take notes ;-)

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http://www.racecar-engineering.com/articles/f1/2014-f1-the-power-unit-explained/

Motivation: The F1 Hybrid Electric Power Unit

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Electrical Motor-Generator Units

Measured

Model

Ph/Ph,0

Ph,dc/P

h,dc,0

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Measured

Model

Pk/Pk,0

Pk,dc/P

k,dc,0

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Ebbesen*, Salazar*, Elbert, Bussi and Onder: “Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 2017

MGU-K - Fitting Error = 2.3% MGU-H - Fitting Error = 2.4%

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p-V Diagrams

Guzzella and Onder: “Introduction to Modeling and Control of Internal Combustion Engines”, Springer, 2004

Gasoline Engine - Otto Cycle

334 C Combustion and Thermodynamic Cycle Calculation of ICEs

Diesel Cycle

The combustion process in early Diesel engines was slow and almost com-pletely diffusion-controlled, i.e. there was almost no premixed combustion atall4 This can be best approximated by an isobaric state-change.

Seiliger (or Dual) Cycle

The Seiliger cycle combines Otto and Diesel cycles by attributing an isochoricas well as an isobaric part to every combustion process. Depending on theratio of these two parts, it allows a rather good approximation of real gasolineand Diesel combustion.

Fig. C.1. Otto, Diesel, and Seiliger cycle in double-logarithmic representation.

C.2.1 Real Engine-Cycle

If real engine cycles have to be simulated or analyzed, a more complex treat-ment is necessary. According to (2.3) and (2.4), the balances of inner energyand mass yield the following two equations

dmc

dφ=

dment cyl

dφ−

dmexit cyl

dφ(C.1)

dUc

dφ= mϕHl

dxB

dφ− pc

dVc

dφ−

dQw

dφ+

dHent cyl

dφ−

dHexit cyl

dφ(C.2)

where the subscript ent cyl describes all quantities which enter the cylinder,whereas exit cyl indicates the exiting quantities. The inner energy Uc is calcu-lated according to (2.6) and xB(φ), as defined in (3.67), is the already burntmass-fraction of fuel and thus its derivative represents the normalized burn

4 Modern Diesel engines use pilot injections before the main injection in order toincrease the premixed part of the combustion.

1: Intake

2: Compression

3: Power

4: Exhaust

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Intake Manifold of an ICE

Isothermal VS Adiabatic - Step Response

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Heat Exchanger

Causality Diagram

2.4 Basic Modeling Elements 43

Remark: Sometimes the formulation

∗Qj(t) = k ·A · (

ϑ1i,j + ϑ1o,j2

−ϑ2i,j + ϑ2o,j

2) (2.97)

is used. This approach better captures the changing temperature profile in-side a cell. However, it introduces algebraic loops that must be dealt withseparately.

The causality diagram of one cell element is shown in Figure 2.24. Sev-eral cells can be stacked together to produce finite-element models of heatexchangers of any desired order.

ϑ1o,j(t)

ϑ1o,j(t)

ϑ2o,j(t) ϑ1i,j(t)

ϑ2i,j(t)

k · A · (ϑ1o,j − ϑ2o,j)

eq. (2.94)

eq. (2.95)

-

-

-

+

+

+

+

+

∗

m1 · cp1

∗

m1 · cp1

∗

m2 · cp2

∗

m2 · cp2

∗

Qj

Fig. 2.24. Heat exchanger element causality, any desired number of elements canbe stacked.

The static behavior of one cell element can be found by setting equations(2.94) and (2.95) both to zero

ϑ1o,j − ϑ1i,jϑ1i,j − ϑ2i,j

= −

!

1 +

∗m1 · c1k ·A

+

∗m1 · c1∗m2 · c2

"−1(2.98)

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Heat Exchanger

Causality Diagram - Connected Cells

2.4

Basic

ModelingElemen

ts43

Remark

:Som

etim

estheform

ulation

∗ Qj(t)=

k·A

· (ϑ1i,j+ϑ1o,j

2−ϑ2i,j+ϑ2o,j

2)

(2.97)

isused.This

approachbettercapturesthechan

gingtemperature

profile

in-

sideacell.How

ever,it

introd

ucesalgebraic

loop

sthat

must

bedealt

with

separately.

Thecausality

diagram

ofon

ecellelem

entis

show

nin

Figure

2.24.Sev-

eral

cellscanbestackedtogether

toproduce

finite-elem

entmod

elsof

heat

exchan

gers

ofan

ydesired

order.

ϑ1o,j(t)

ϑ1o,j(t)

ϑ2o,j(t)

ϑ1i,j(t)

ϑ2i,j(t)

k·A

· (ϑ1o,j−ϑ2o,j)

eq.(2.94)

eq.(2.95)

-

-

-

+

+

+

+

+

∗ m1· c

p1

∗ m1· c

p1

∗ m2· c

p2

∗ m2· c

p2

∗ Qj

Fig.2.24.Heatex

chan

gerelem

entcausality,an

ydesired

number

ofelem

ents

can

bestacked.

Thestatic

behav

iorof

onecellelem

entcanbefoundby

settingequations

(2.94)

and(2.95)

bothto

zero

ϑ1o,j−ϑ1i,j

ϑ1i,j−ϑ2i,j=−

! 1+

∗ m1· c

1

k·A

+

∗ m1· c

1∗ m2· c

2

" −1

(2.98)

2.4

Basic

ModelingElemen

ts43

Remark

:Som

etim

estheform

ulation

∗ Qj(t)=

k·A

· (ϑ1i,j+ϑ1o,j

2−ϑ2i,j+ϑ2o,j

2)

(2.97)

isused.This

approachbettercapturesthechan

gingtemperature

profile

in-

sideacell.How

ever,it

introd

ucesalgebraic

loop

sthat

must

bedealt

with

separately.

Thecausality

diagram

ofon

ecellelem

entis

show

nin

Figure

2.24.Sev-

eral

cellscanbestackedtogether

toproduce

finite-elem

entmod

elsof

heat

exchan

gers

ofan

ydesired

order.

ϑ1o,j(t)

ϑ1o,j(t)

ϑ2o,j(t)

ϑ1i,j(t)

ϑ2i,j(t)

k· A

· (ϑ1o,j−ϑ2o,j)

eq.(2.94)

eq.(2.95)

-

-

-

+

+

+

+

+

∗ m1· c

p1

∗ m1· c

p1

∗ m2· c

p2

∗ m2· c

p2

∗ Qj

Fig.2.24.Heatex

chan

gerelem

entcausality,an

ydesired

number

ofelem

ents

can

bestacked.

Thestatic

behav

iorof

onecellelem

entcanbefoundby

settingequations

(2.94)

and(2.95)

bothto

zero

ϑ1o,j−ϑ1i,j

ϑ1i,j−ϑ2i,j=−

! 1+

∗ m1· c

1

k·A

+

∗ m1· c

1∗ m2· c

2

" −1

(2.98)

2.4

Basic

ModelingElemen

ts43

Remark

:Som

etim

estheform

ulation

∗ Qj(t)=

k·A

· (ϑ1i,j+ϑ1o,j

2−ϑ2i,j+ϑ2o,j

2)

(2.97)

isused.This

approachbettercapturesthechan

gingtemperature

profile

in-

sideacell.How

ever,it

introd

ucesalgebraic

loop

sthat

must

bedealt

with

separately.

Thecausality

diagram

ofon

ecellelem

entis

show

nin

Figure

2.24.Sev-

eral

cellscanbestackedtogether

toproduce

finite-elem

entmod

elsof

heat

exchan

gers

ofan

ydesired

order.

ϑ1o,j(t)

ϑ1o,j(t)

ϑ2o,j(t)

ϑ1i,j(t)

ϑ2i,j(t)

k·A

· (ϑ1o,j−ϑ2o,j)

eq.(2.94)

eq.(2.95)

-

-

-

+

+

+

+

+

∗ m1· c

p1

∗ m1· c

p1

∗ m2· c

p2

∗ m2· c

p2

∗ Qj

Fig.2.24.Heatex

chan

gerelem

entcausality,an

ydesired

number

ofelem

ents

can

bestacked.

Thestatic

behav

iorof

onecellelem

entcanbefoundby

settingequations

(2.94)

and(2.95)

bothto

zero

ϑ1o,j−ϑ1i,j

ϑ1i,j−ϑ2i,j=−

! 1+

∗ m1· c

1

k·A

+

∗ m1· c

1∗ m2· c

2

" −1

(2.98)

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PDE VS Chain of ODE’s

Step Response

2.5 Distributed Parameter Systems 61

−w ·−1w

· g(t−x

w) = g(t−

x

w)

Therefore, taking x at the input (x = 0) and at the output (x = L), theinput/output dynamics u(t) = ϑ(t, 0)→ y(t) = ϑ(t, L) of the simplified heat-exchanger system is described by

y(t) = u(t−L

w) = u(t− T ) (2.145)

which is, of course, a simple delay element with delay T = Lw .

Figure 2.38 shows the comparison of the solutions obtained with equation(2.145) and with 100 instances of equation (2.142) connected in series. Theinput signal used is u(t) = h(t). Despite the rather high order, the ODEapproximation is not able to capture the essential dynamics of the system,i.e., its “shock-wave” behavior and this is true for any n <∞.

time t (s)

temperatu

res(K

)

input u(t)

output ODE

output PDE

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Fig. 2.38. Comparison of the numerical solution obtained with a series connectionof n = 100 elements (2.142) and the exact solution of the simplified heat-exchangersystem with ϑ(0, t) = u(t) = h(t). Parameter values: L = 1 m, v = 1 m/s