Download Modelling Thermodynamic Systems with Changing Gas Mi

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Eilif Pedersen
The Norwegian University of Science and Technology
Faculty of Marine Technology
N-7034 TRONDHEIM, Norway
E-mail: [email protected]
.(<:25'6
Modeling, Bond Graph, Thermodynamics, Gas Mixtures
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Modelling complex thermofluid processes often
require models capable of handling working media that
are changing its composition due to introduction of new
constituents into the volume of interest. Mixing different
constituents also changes the thermodynamic properties
of the mixture which must be included in the models.
This paper presents an extension to the classical
pseudo bond graph concept. The extensions introduced
extend the ability of the pseudo bond graphs concept to
also include mixtures. The proposed extension is pictured
after the classical pseudo bond graph representation and
preserves this representation only by adding the required
features.
The proposed extension is illustrated by modelling a
gas mixing experimental rig using the basic component
models developed.
,1752'8&7,21
Thermofluid processes are found in most engineering
systems and the number of applications has been increasing where a thorough analysis of both the steady-state
and dynamic behavior of a system is of vital interest . The
rapid development of computer power has also made it
possible to analyze real problems with a reasonable use
of resources. The result is that there nowadays exists
computer programs for almost any analysis of interest to
the engineer. However, often the required flexibility necessary to analyze a specific problem is not available using
a specific tool. It is therefore, mandatory to be able to
develop new or modified models and to integrate them
easily into a total model.
A modelling technique which answers most of the
requirements put forward by modelers is bond graphs.
The technique have since its invention by H. M. Paynter
proved itself within different applications such as in
mechanical, electric and hydraulic systems. Within thermofluid systems the research and applications has been
more scarce.
Despite the fact that the research on bond graph representation and applications to thermofluid systems is
somewhat limited, some important contributions deserve
to be mentioned.
One of the first researchers to model thermodynamic
systems by the way of bond graphs was Thoma [1]. He
proposed to use temperature as the effort variable and
entropy flow as the flow variable, a natural choice since
their product gives power just as true bond graphs. Later
Thoma extended this to systems with variable mass [2].
Although this approach has many advantages especially
when systems of different energy domains are present,
thermodynamic relations are complex and entropy flow
and accumulation is not very well understood by engineers.
Other suggestions for modelling thermofluid systems
is given by Brown [3]. He proposes to include a convection bond for the flow of a pure substance with two effort
variables, stagnation enthalpy and stagnation pressure,
and one flow variable, mass flow. His motivation for
introducing this additional convection bond is that two
variables, as in true bond graphs, is not sufficient to
define the thermodynamic state of the fluid. The result of
this new representation is a notation that differs significantly from conventional bond graphs and are not easily
incorporated into most bond graph modelling environments.
Yet another contribution is given by Shoureshi [4]. He
proposes a bond graph notation for modelling two-phase
variable density systems using the same effort and flow
variables as Thoma. The result of the choice of variables
is a very complicated bond graph with several signal
bonds, modulated transformers and gyrators. In addition
to a complex bond graph the choice of variables led to
additional complications computing the state of the fluid
from known thermodynamic tables. He solved his problem by developing a computer program which transformed the known thermodynamic tables to comply to his
selection of state variables in the bond graph.
The work involved in developing such transformations
are in general not straight forward. It would be nice if the
selection of state variables were such that thermodynamic
property tables or software packages could be used
directly as they are published. However, such a choice
would lead to selections of effort and flow variables that
do not multiply to yield power as required for true bond
graphs.
This selection of variables have although been suggested by Karnopp [5] in his concept called: pseudo bond
graphs. The effort and flow variables he proposes are:
at the end using the basic models developed to simulated
a gas mixing experimental setup.
02'(//,1*
Modelling a gas tank as shown in Fig. 2a, which in
general can be looked upon as a compressible fluid accumulator, using the pseudo bond concept is shown in Fig.
V
m· in
h in
• Efforts:
temperature
pressure
T
p
h out
(a)
·
Q
• Flows:
m·
·
E
·
Q
·
V
mass flow rate
&
total energy flow rate
p m·
·
T E
heat energy flow rate
p
m· in
rate of change of volume
• State variables:
mass
m
total energy
E
p
m·
T
·
E
%
Figure 1. Energy exchange between system A and B.
Pseudo bond graph representation.
Despite the fact that pseudo bond graphs do not comply with true bond graphs, they have been applied in a
number of engineering systems. Examples of use are in
modelling of complete diesel engine systems by Engja
[6], Granda [7] and Pedersen [8], one-dimensional compressible fluid flow by Strand [9], building simulation by
Thoma [10] et.al. and two-phase systems by Moksnes
[11].
Looking at the list of applications mentioned, a natural
next step would be to extend the technique to include systems where the working medium is a mixture of different
constituents, and where the composition is changing.
The next sections of this paper introduces the classical
pseudo bond graph concept and describes the extensions
developed to handle mixtures. An example are included
p
m· out
T
·
E in
(b)
Following his proposition, the exchange of energy
between two systems can be represented schematically as
shown in Fig. 1, using a double bond representation.
$
·
m out
p, T
T
·
E out
·
Q
6I
Figure 2. Accumulator, (a) sketch, (b) pseudo bond
graph
2b. The pseudo bond graph consists of a 0-junction structure representing the continuity of mass and energy, the
two state variables selected, and a C-field giving the relations between the displacements and the efforts. The fixed
causality caused by the selection of variables on the
bonds in the pseudo bond graph concept, is shown on the
bond graph. This fixed causality is one of the prices we
have to pay for the selection of variables.
The constitutive laws of the C-field is given by
p = Φ p ( m, E, V )
T = Φ T ( m, E, V )
(1)
where m and E are the state variables mass and energy
and V is the volume of the accumulator. For a perfect gas
the relations can be deduced from the thermodynamic
equation of state and the relation between the temperature
and internal energy, E = mc v T .
1E
T = ---- ---cv m
RE
mRT
p = ------------ = ---- --cv V
V
(2)
For situations where the gas can be said to follow an
ideal gas law, i.e. to follow the ideal equation of state and
where the specific internal energy is a function of temperature only, the solution of Eq. 2 involves an iterative solution. For the ideal gas assumption the equations are easily
solvable by iteration due to its monotone properties. Also
for real gas assumptions the states p and T can be found
from Eq. 2 although the iterative solution in such cases
requires more computer power.
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The number of independent intensive variables F necessary to calculate the thermodynamic state of a fluid in
equilibrium is given by the Gibbs phase rule [12] as
F = n+2–r
(6)
i=1
In Fig. 3a a gas mixture accumulator is shown with an
inflow and outflow ports. The mixture entering the accumulator at the inflow port is given as a function of time,
producing changes in the composition of the mixture in
the accumulator. The energy entering or leaving the accumulator through the ports is given by
N m
i
·
E = m· h = m· ∑ ----- h i
m
(7)
i=1
where the mass fractions and enthalpies used are calculated based on upstream parameter values.
The number of independent thermodynamic variables
necessary to specify the state of a gas consisting of N constituents is N+1. The extended pseudo bond graph of the
(3)
V
m· 1, in …m· N, in
m· 1, out …m· N, out
p, T
c 1 …c N
h in
(a)
h out
·
Q
&
p m·
· 2 p N m·
p2 m
N
...
p N, out
m· N, ou t
T in
·
Ein
....
p
m· ou t p
2, out
m· 2, o ut
.
p N, in
m· N, in
·
TE
...
p
m· in p
2, in
m· 2, in
....
where n is the smallest number of variables necessary
to describe the composition of the mixture at equilibrium
conditions and r is the number of phases present in the
volume of interest. In addition one extensive variable
must be available to calculate the amount of matter in the
control volume. For a gas mixture consisting of two constituents the number of intensive variables necessary to
calculate the state of the mixture is : F = 2 + 2 – 1 = 3 .
Specifying the volume adds the fourth extensive variable.
In modelling the gas accumulator using the pseudo
bond graph concept as described above, we have available two state variables for calculation of the thermodynamic state of the fluid. This is not enough for a mixture,
and a third state variable has to be introduced.
The equation of state for a mixture of ideal gases is
given by
pV = mRT
N m
i
E = mu = m ∑ ----- u i
m
T out
·
Eou t
(4)
where the gas constant R for the mixture is calculated
(b)
by
·
Q
6I
N
R =
∑
m
-----i R i
m
(5)
i=1
m
where Ri and -----i are the gas constant and mass fracm
tion of the individual constituents of the mixture. N is the
number of constituents of the mixture.
The relation for the total internal energy, E , of the
mixture can be expressed in i similar way by neglecting
some minor important energy contributions like kinetic
and potential energy together with chemical potential.
Figure 3. Gas mixture accumulator, (a) sketch and
(b) pseudo bond graph.
gas mixture accumulator is shown in Fig 3b using N+1
pseudo bonds. The new pseudo bond graph for mixtures
are closely related to the version proposed by Karnopp
[5].
The effort and flow variables proposed for the
extended pseudo bond graph are:
• Efforts ( e ):
10 for calculation of the temperature,
pressure
partial pressure of constituent 2
p
p2
…
…
E
T = ---------------------------------N m
i
m ∑ ----- u i ( T )
m
pN
partial pressure of constituent N
T
temperature
i=1
but with an explicit evaluation of the total and partial
pressures as given by the last two equations in Eq. 9.
Assuming real gases not obeying the ideal gas law
assumption the constitutive laws for the C-field given in
Eq. 8 are still valid. With the use of available thermodynamic property libraries the solution of Eq. 8 is possible
in a general way, but still is costly and in most cases an
overkill.
• Flows ( f ):
m·
total mass flow rate
m· 2
mass flow rate of constituent 2
…
…
m· N
·
E
mass flow rate of constituent N
total energy flow rate
• State variables (displacements) ( q ):
m
m2
total mass
total mass of constituent 2
…
…
mN
total mass of constituent N
E
total energy
and the constitutive laws for the C-field are now
expressed as:
p = Φ p ( m, m 2 …m N, E, V )
p i = Φ ( m, m 2 …m N, E, V )
(10)
i = 1…N
(8)
T = Φ T ( m, m 2 …m N, E, V )
$GGLWLRQDOHOHPHQWV
An important and fundamental element in fluid flow is
the pipe junction or restrictor. In bond graph notation this
can be represented by a R-field with continuity conditions
on the flow variables mass and energy. For gas flow the
restrictor can be modeled as an ideal nozzle where the
mass flow depends not only on the pressure difference,
but also on the pressure ratio. Extending the model to
handle mixtures is straight forward following the same
ideas as for the accumulator described above. In Fig. 4 the
extended pseudo bond graph for the ideal nozzle is shown
for the case where the number of constituents present is 2.
The effort and flow variables on the left and right hand
side of the R-field are named with subscripts a and
b .The bond graph indicates appropriate causality
required by the law-set for the R-field.
which for a perfect gas assumption gives:
pa
m·
E
T = ----------------------------N m
i
m ∑ ----- c v, i
m
2
i=1
 N m 
T
i
p = m  ∑ ----- R i --m
V
i = 1

p 2a
m·
(9)
ni
mi  N mi 1 
p i = ---- p = -----  ∑ ----- ------ p
n
m
m M i
i = 1

The selection of the partial pressure as an additional
effort variable is only one of several possible choices.
Any variable fixing the mixture composition is possible,
and another option is to use the mass fraction of the i -th
constituent directly.
For an ideal gas law assumption, i.e. when c v , c p are
functions of temperature only, the constitutive laws for
the C-field in general requires an iterative solution of Eq.
Ta
·
E
R
pb
m·
p2 b
m·
2
Tb
·
E
Figure 4. Pseudo bond graph model of gas flow restructure or nozzle for mixtures.
The constitutive laws for the R-field assuming ideal
gas conditions is given by
pu
m· a = m· b = Cd A ---------------- ψ ( π u )sign ( p a – p b )
RuTu
m· 2a = m· 2b = m· c 2u
·
E·a = E b = m· h ( p u, p 2u, T u )
p 2u M 2u
, c 2u = -------- --------p u M u (11)
2 p M
i i
, h= ∑ ---- ------ h i ( T )
pM
i=1
where
2
κ  ------------
 κ + 1
=
κ+1
-----------κ–1
1
--2
1
--2
2
, 1 ≥ π ≥  ------------
κ+1
κ
-----------κ–1
the thermodynamic properties necessary are [12].:
Properties
[KC/kg]
R
[KC/Kg]
cv
κ
------------
2 κ–1
, π ≤  ------------
 κ + 1
and where π = p d ⁄ p u , and all the thermodynamic
variables including κ , c p , h , R and M are calculated
based on the conditions on the upstream side of the nozzle.
For real gases this set of equations can be replaced
with real gas thermodynamic calculations.
The initial conditions in the mixing chamber is 1 bar
C :B2
C :B1
2
3
C :MC
A1
4
5
6
13
7
8
9
R
16
17
12
18
Mixing
valves
25
26
27
22
23 24
A2
15
11
19
R
20
21
28
Methane
bottle
B1
14
10
29
Air
bottle
Methane
518.35
1735.4
The total pressure and temperature in the bottles are
initially between 100 and 300 bar and 300 K. The partial
pressure p 2 in the air and methane bottles is 0 and 300
bars respectively.
1
02'(//,1*$6,03/(*$60,;(5
A simplified gas mixer found in a combustion experimental rig for testing combustion properties of various
natural gases is shown in Fig 5. The two bottles contain
methane and air, delivered through non-return valves to a
mixing chamber. The resulting gas composition are leaving the mixing chamber through an exit valve. The mix-
Air
287.0
716.5
30
31
R
B2
32
S
f
κ+1
2
--------------
2κ  κ
- ⋅ π – π κ 
ψ ( π ) = ----------κ–1 

A3
33 34
Se :atm
Mixing chamber:MC
Exit valve
Figure 6. Complete pseudo bond graph model for the
simple gas mixer.
Figure 5. Gas mixing system
ture which is produced is observed to be dependent on the
pressure in the bottles. For combustion purposes a fixed
composition is required. To better understand the process
and to develop a control system which will ensure a fixed
mixture, a model was build based on the basic elements
developed earlier.
The pseudo bond graph for the system in Fig. 5 is
shown in Fig. 6. The bottles and the mixing chamber are
modeled using the accumulator developed and the valves
at the mixing chamber inlet and outlet are modeled using
the ideal nozzle model developed. The opening area of
the mixing valves are 0.785 10-4 m2. The area of the exit
valve of the mixing chamber is 1.0 10-4 m2. In this example no heat loss from the bottles or from the mixing
chamber is assumed. The nozzle flow coefficient for all
the valves are set to unity.
Using a perfect gas assumption for the working media
and 300 K, completely filled with air, i.e. the partial pressure p 2 is 0. The volume of the air bottle is 2 m3 and the
volume of the methane bottle is 0.5 m3.
0RGHOHTXDWLRQV
Deriving the model equations from the bond graph in
Fig 6 is shown next, although this is a process which efficiently can be given to computer programs like MS1 [13]
or 20Sim [15]. Here we only give an outline of the equation generation process utilizing standard bond graph
techniques.
The state variables of the model are the displacements
q 1 , q 2 , q 3 , q 13 , q 14 , q 15 , q 25 , q 26 and q 27 , or m B1 ,
m 2, B1 , EB1 , m MC , m 2, MC , E MC , m B2 , m 2, B1 and EB2 as
named in the text above. Using the constitutive laws for
the mixing chamber or C-field MC, the efforts of each
bond can be evaluated using
(12)
where the methane bottle initially is at 100 bar. The molar
fraction of methane in the gas leaving the mixing chamber is also shown in the same figure. The results show that
Pressure B1
Pressure B2
Pressure MC
Pressure [Pa] 107
where PCT is a function returning the efforts on each
power bond of the C-field, i.e. the pressure, the partial
pressure of methane and the temperature, and Y is a vector identifying the constituents of the mixture. Repeating
the use of the PCT-function on C-field B1 and B2 gives
the efforts e 1 , e 2 , e 3 , e 25 , e 26 and e 27 . The effort source
Se sets the total pressure, the partial pressure of methane
and temperature using atmospheric conditions. Using the
information present in the bond graph all efforts can now
be assigned.
The state equations for the mixing chamber are:
Mass fraction CH4.
in mixing chamber
Molar fraction CH4 [-]

e 13 

e 14  = PCT ( q 13, q 14, q 15, V, Y )

e 15 

Time [s]
·
q 13 = f13 = f 10 + f 16 – f 29
·
q 14 = f14 = f 11 + f 17 – f 30
·
q 15 = f15 = f 12 + f 18 – f 31 – f 28
Figure 7. Pressure in the bottles and mixing chamber,
and mass fraction of methane leaving the mixing
chamber for an initial pressure of bottle B2 of 100
bar.
(13)
and the flows of each bond is calculated using
f10
f11
f12



 = Nozzle A1 ( e 7, e 10, e 8, e 11, e 9, e 12, A, C d, Y )



the lowered total pressure of the methane bottle results in
a cut-off period where pure air is delivered and no ignition of the mixture delivered will be possible.
In Fig. 8 the simulated molar fraction of methane leav0.6
f17
f18

f29 

f30  = Nozzle A3 ( e 29, e 32, e 30, e 33, e 31, e 34, A, C d, Y )

f31 

where Nozzle A1 , A2 and A3 is a function implementing Eq. 12 returning the total mass flow, the 2nd
constituent mass flow and the total energy flow. Writing
the state equations for the bottles B1 and B2 similar to
Eq. 14 closes the equation writing process.
5HVXOWV
The model is implemented and simulated using the
modeling and simulation tools MS1 [13] and ACSL [14].
Fig. 7 shows the simulated pressures in the bottles and
in the mixing chamber as a function of time for the case
Pressure : 100 bar
InitialPressure
pressure
: 200300
bar bar
: 300200
bar bar
InitialPressure
pressure
Initial pressure 100 bar
0.5
Molarfraction
fraction [-]CH4 [-]
Molar
f16



 = Nozzle A2 ( e 19, e 16, e 20, e 17, e 21, e 18, A, C d, Y )



0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
Time [s]
Figure 8. Mixture composition in mixing chamber
for methane bottle initial pressure of 100, 200 and
300 bar.
ing the mixing chamber is shown as a function of time for
initial pressure of the methane bottle of 100, 200 and 300
bars respectively.
The mixture compositions dependence on the pressure
in the bottles are clearly documented with the use of the
model developed, and the methane cut-off period is captured for the low initial pressure case.
&21&/86,216
The proposed extension of the pseudo bond graph concept for mixtures has several advantages.:
•
•
•
•
The extension fits nicely into the pseudo bond
graph concept and simply adds the required feature
without changing the structure of bond graph.
The extension can be generalized to any number of
mixture components and both ideal and real gas
thermodynamics.
The selection of state variables, efforts and flows
are such that thermodynamic property tables or
computer libraries can be used directly.
Although the proposed extension also inherits the
disadvantages of the pseudo bond graph concept,
the resemblance to traditionally used modelling
approaches for thermofluid systems together with
the advantages of the bond graph technique makes
it a powerful tool for modelling such systems.
The extended pseudo bond graph proposed for modeling thermofluid systems with changing fluid composition
opens new applications for modeling using the powerful
bond graph technique.
c 1 …c N
cv
cp
Cd
E
h
m
M
n
N
p
R
T
u|
V
·
Q
κ
-
-
constitutive law of C-field returning pressure
Φp ( )
-
constitutive law of C-field returning partial pressure i
ΦT ( )
-
constitutive law of C-field returning temperature
i
6XEDQGVXSHUVFULSWV
a a, a b
-
effort, flow, variables or properties to the left or right
side of the bond graph element.
a1
-
constituent number 1 of the gas mixture
-
effort, flow, variable or properties at the upstream/
downstream end of the flow, i.e. changes with flow
direction
a 2 …a N
au , ad
constituent number 2 ... N of the gas mixture
a in, a out
ai
a·
effort, flow, variable or properties at the inflow port of
an element
-
effort, flow, variable or property i
-
rate of change of any variable
a B1, a B2 -
variables related to the bottles and mixing chamber
5()(5(1&(6
[1] Jean U. Thoma, (QWURS\DQGPDVVIORZIRUHQHUJ\FRQYHUVLRQJournal of the Franklin Institute, 299(2), p 89-96, February 1975.
[2] Jean U. Thoma, 1HWZRUN WKHUPRG\QDPLFV ZLWK HQWURS\ VWULSSLQJ
Journal of the Franklin Institute, 303(4), p 319-328, 1977.
[3] F. T. Brown,
&RQYHFWLRQ ERQGV DQG ERQG JUDSKV -RXUQDO RI WKH
)UDQNOLQ,QVWLWXWHS
[4] R. Shoureshi and K. McLaughlin,
$SSOLFDWLRQ RI ERQG JUDSK WR
Trans. ASMA., J. Dynamic
Syst. Measure. Control, 107, p 241-245, December 1985.
WKHUPRIOXLG SURFHVVHV DQG V\VWHPV
[5] Dean C. Karnopp, 6WDWHYDULDEOHVDQGSVHXGRERQGJUDSKVIRUFRP
SUHVVLEOH WKHUPRIOXLG V\VWHPV Transactions of ASME, Journal of
Dynamic Systems, Measurement and Control, 101(3), Sept. 1979.
120(1&/$785(
A
Φp ( )
nozzle area [m2]
[6] H. Engja and K. Strand, 0RGHOLQJIRUWUDQVLHQWSHUIRUPDQFHRIGLH
VHOHQJLQHVXVLQJERQGJUDSKV In ISME, Tokyo, 1983.
mass fraction of constituent 1…N
-
specific heat capacity at constant volume [ J ⁄ ( kgK ) ]
-
specific
-
nozzle flow coefficient
-
total energy of fluid ( = mu ) [ J ]
-
enthalpy [ J ⁄ ( kg ) ]
-
mass [ kg ]
-
molecular weight [ kg ⁄ kmol ]
heat
[ J ⁄ ( kgK ) ]
capacity
-
number of kmol
-
number of constituents
-
pressure [ Pa ]
-
gas constant [ J ⁄ ( kgK ) ]
-
termperature [ K ]
-
internal energy [ J ⁄ ( kg ) ]
-
volume [ m ]
at
constant
pressure
3
-
heat flow [ J ⁄ s ]
-
ratio of specific heats c p ⁄ c v
-
thermodynamic function returning enthalpy given the
pressure, temperature and composition of the mixture.
[7] Jose J. Granda and G. R. Channel, 9,QWHUQDO&RPEXVWLRQ(QJLQH
ERQGJUDSKPRGHO$'HWDLOHG0RGHOLQJ3URFHGXUHIn proc. of the
ICBGM’97 conference, Editors J. J. Granda and G. Dauphin-Tanguy, Simulation Series, Volume 29, no. 1, 1997
[8] E. Pedersen and Ø. Bunes,0RGHOLQJDQG6LPXODWLRQRIWKH8OVWHLQ
%HUJHQ
%5
(QJLQH
&RPSDULVRQ
ZLWK
PHDVXUHG
GDWD
MARINTEK Report MT222502, June 1998.
[9] K. Strand and H. Engja, %RQG JUDSK LQWHUSUHWDWLRQ RI RQHGLPHQ
VLRQDOIOXLGIORZ Journal of the Franklin Institute, 328(5/6), p 781793, 1991.
[10] Jean Thoma et.al, %XLOGLQJ 6LPXODWLRQ ZLWK &RQYHFWLRQ DQG &RQ
GXFWLRQ In ICBGM’97, Simulation Series Vol. 29, no. 1, SCS 97.
[11] Paul Ove Moksnes,0RGHOLQJWZRSKDVHWKHUPRIOXLGV\VWHPVXVLQJ
ERQGJUDSK Dr.ing thesis, Department of Marine Engineering, Norwegian University of Science and Technology.
[12] R. E. Sonntag and G. J. Van Wylen, ,QWURGXFWLRQRW7KHUPRG\QDP
LFV&ODVVLFDO6WDWLVWLFDOJoh Wiley & Sons, 3rd. edition, 1991.
[13] Lorenz Simulation, SOCRAN Centre, Scientific Park, Avenue PreAily, B-4031, Liebe, Belgium
[14] MGA Software, 200 Baker Avenue, Concord, MA. 01742, USA
h( )
[15] 20Sim Reference Manual, Controllabs Products Inc., P.O.Box 217,
7500 AE Enshede, The Netherlands