system modeling - lecture 6€¦ · diesel cycle the combustion process in early diesel engines was...
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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 ;-)
Mauro Salazar �2
http://www.racecar-engineering.com/articles/f1/2014-f1-the-power-unit-explained/
Motivation: The F1 Hybrid Electric Power Unit
Mauro Salazar �3
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%
Mauro Salazar �4
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)
Mauro Salazar �7
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)
Mauro Salazar �8
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
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