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Bergische Universit¨
at Wuppertal
Fakult¨
at f ¨
ur Mathematik und Naturwissenschaften
Institute of Mathematical Modelling, Analysis and Computational
Mathematics (IMACM)
Preprint BUW-IMACM 21/08
Jens J¨
aschke, Matthias Ehrhardt, Michael G¨
unther and Birgit Jacob
A Port-Hamiltonian Formulation of Coupled Heat Transfer
March 6, 2021
http://www.imacm.uni-wuppertal.de
A Port-Hamiltonian Formulation of Coupled Heat Transfer
Jens J¨aschkea, Matthias Ehrhardta, Michael G¨untheraand Birgit Jacobb
aIMACM, Lehrstuhl Angewandte Mathematik und Numerische Analysis, Bergische
Universit¨at Wuppertal, Wuppertal, Germany
bIMACM, Lehrstuhl Funktionalanalysis, Bergische Universit¨at Wuppertal, Wuppertal,
Germany
ARTICLE HISTORY
Compiled March 6, 2021
ABSTRACT
Heat transfer and cooling solutions play an important role in the design of gas turbine
blades. However, the underlying mathematical coupling structures have not been
thoroughly investigated. In this work, the port-Hamiltonian formalism is applied to
the conjugate heat transfer problem in gas turbine blades. A mathematical model
based on common engineering simpliﬁcations is constructed and further simpliﬁed to
reduce complexity and focus on the coupling structures of interest. The model is then
cast as a port-Hamiltonian system and examined for stability and well-posedness.
KEYWORDS
port-Hamiltonian system; heat transfer; cooling channel; coupled systems;
well-posedness
1. Introduction
With the German government’s plans to increase the share of renewable energies in
the power grid to over 60% by 2050, gas turbines will most likely gain in importance
and take on new roles and tasks in the power grid. Due to their short start-up times
and high eﬃciency, they are particularly well suited as reserve power plants and can
cushion power drops and demand peaks. They are also relevant in the 100% renewable
energy sources scenario, when hydrogen and methane are used as energy storage.
These changing application roles go hand in hand with changing design requirements,
especially in terms of eﬃciency, reliability and ﬂexibility of operation.
In order to accurately incorporate these requirements into the design process, the use
of high-level multiphysics simulations is required, combining ﬂuid dynamics, structural
mechanics, heat conduction, convective heat transport and 1D ﬂow networks, among
others. The GivEn project [cf. 1] aims to integrate multiphysics simulation with a
multicriteria shape optimisation process.
To achieve this goal and obtain useful results, other requirements must ﬁrst be
met. One of them is to ensure a suitable coupling structure between the diﬀerent
parts of the multiphysics simulation. In this work we want to model and investigate
a special coupling structure - the heat transfer at the walls of the cooling channels of
CONTACT Jens J¨aschke. Email: jaeschke@uni-wuppertal.de
the turbine blade. It couples the heat conduction in the metal of the turbine blade
with the transport of the cooling ﬂuid ﬂowing through the cooling channels.
To improve the thermal eﬃciency, modern gas turbines are operated at very high
temperatures (1200 °C to 1500 °C). Since these temperatures greatly exceed the range
in which the turbine blade’s metal can be used safely, active cooling of the turbine
blades is necessary for them to withstand these temperatures. One of the techniques
used for this purpose is convection cooling, i.e. cooling channels are installed within
the blade. These internal cooling channels are small ducts within the turbine blade
that are ﬁlled with a stream of cooling ﬂuid, usually (comparatively cool) air extracted
from the compressor.
This approach leads to a so-called conjugate heat transfer (CHT) problem (strong
thermal interactions between solids and ﬂuids) , which is the heat transfer between
the turbine blade, the internal ﬂow in the cooling channel and the hot external ﬂow.
In this work we will focus on the role of the cooling channels, the heating of the blade
by the external ﬂow is discussed in . In order to maximize heat transfer between the
blade and the ﬂuid, the ﬂow within the cooling channels is kept deliberately turbulent,
for example by use of so-called rib turbulators, periodic protrusions and recessions in
the channel walls. For a more in-depth review of turbine blade cooling, see e.g. .
We will formulate a so-called port-Hamiltonian system (pHs) modeling this conju-
gate heat transfer and study in detail the resulting coupling structure. Since it is closely
related to the Hamiltonian formalism originally developed in theoretical physics, this
port-Hamiltonian framework is a natural ﬁt for modeling physical systems and, in
particular, their interconnections, since two port-Hamiltonian systems connected by
a suitable coupling structure in turn form a port-Hamiltonian system. The formalism
also makes conservation laws, a fundamental property of virtually any physical sys-
tem, explicit. Moreover, a suitable port-Hamiltonian formulation makes the process
of discretizing a continuous system for numerical simulation relatively simple, while
ensuring that conservation laws still hold in the discretized system [cf. 4]. Moreover,
several desirable properties such as stability or controllability are either inherent to
port-Hamiltonian systems or can be guaranteed by some easily checked additional
conditions.
The paper is structured as follows. First, in Section 2 we motivate and introduce
the mathematical model of the coupled system to be investigated including a rescaling
of the variables. Next, in Section 3 we rewrite each subsystem of our model in the
port-Hamiltonian framework, after which we combine these port-Hamiltonian systems
and study the properties of the resulting coupled system. Finally, we summarize and
interpret the results in Section 5 and give some concluding remarks.
2. The Model System
Our model system is based on a highly simpliﬁed model of heat transfer within the
blade of a gas turbine, since we are mainly interested in the mathematical coupling
structure.
In the design process of gas turbine blades, much work is done with two-dimensional
slices, which are then stacked and interpolated to form the ﬁnal three-dimensional
blade. This is done because simulation and optimization is much easier and faster that
way than with a full three-dimensional model and still gives ”good enough” results.
The cooling channel, which is more or less perpendicular to the slices, then has exactly
one point of contact with each slice (neglecting 180hairpin curves) where heat transfer
2
can occur.
Since we are primarily interested in the thermal coupling structure between the
turbine blade and the cooling channel, and a 2D-slice introduces additional complexity
into the model, we decided to further simplify the system and use a one-dimensional
rod instead. Thus, our model consists of a thermally conductive rod (axmb) and
a cooling channel (ixco) through which a ﬂuid is pumped. The left end of the
rod at xm=ais in contact with a thermal reservoir with a given temperature Text(t).
The right end of the rod at xm=bis in contact with the wall of the cooling channel.
metal rod Tm
thermal
reservoir
Text Tin
Tout
cooling
channel
Tinﬂow
Toutﬂow
xm
a b
xc
c
i
o
The temperature of the metal rod Tm=Tm(xm, t) is modeled by a heat equation
supplied with Robin boundary conditions at xm=aand xm=b. The temperatures
Tin =Tin(xc, t) and Tout =Tout (xc, t) of the inﬂowing and outﬂowing parts of the
cooling channel are described by simple transport equations. This is, again, a simpliﬁ-
cation, as it assumes that the convective heat transport dominates and we can neglect
the diﬀusion in the cooling channel medium. For the usual ﬂow rates and cooling ﬂuids
used in gas turbines, such as air or water vapor, this is a valid assumption, as they
have a very low thermal conductivity compared to their heat capacity. The coupling
at point xc=cis such that the outﬂowing temperature at point cis determined by
the inﬂowing temperature at this point plus the heat ﬂowing out of the metal rod due
to the boundary condition at xm=b.
3
2.1. The System of Equations
The above setting leads to the following equations for the temperatures in the metal
rod and in the cooling channel:
∂Tm
∂t =k
cm
2Tm
∂x2
m
, a < xm< b, t > 0,(1a)
∂Tin
∂t =vTin
∂xc
, i < xc< c, t > 0,(1b)
∂Tout
∂t =vTout
∂xc
, c < xc< o, t > 0,(1c)
k∂Tm
∂xm
(a, t) = haText(t)Tm(a, t), t > 0,(1d)
k∂Tm
∂xm
(b, t) = hbTm(b, t)Tin(c, t), t > 0,(1e)
Tin(i, t) = Tinﬂow(t), t > 0,(1f)
ccvTout(c, t)Tin(c, t)=hbTm(b, t)Tin(c, t), t > 0.(1g)
Hence, to summarize, the temperature ﬁeld in the metal rod is described by the heat
equation (1a) supplied with the two Robin boundary conditions (1d), (1e) modelling
a Fourier-type heat transfer, the Newton’s law of cooling. Next, in the inﬂow part
of the cooling channel (i < xc< c) a given temperature proﬁle Tinﬂow(t) at the left
boundary xc=iis convected with the speed v, see (1b), (1f). Finally, in equation
(1g) the coupling of the two systems at the point xm=bequals xc=cis described:
the heat ﬂux from the metal rod to the cooling channel, depending on the inﬂowing
temperature Tin at xc=c.
We remark that, because of equation (1e), the last boundary condition (1g) can
alternatively be written as
ccvTout(c)Tin(c)=kTm
∂xm
(b).(1h)
2.2. The Rescaled System
A rescaling of system (1) such that the temperatures are deﬁned on a unit interval [0,1]
allows us to write the system (1) in a more compact way and we see how geometric
dimensions enter the system. For that purpose, we introduce the new space variables
ξm=xma
ba=xma
lm
, a xmb,
ξin =xci
ci=xci
lin
, i xcc,
ξout =xcc
oc=xcc
lout
, c xco,
and the rescaled temperature functions ϑj(ξj(xm/c), t) = Tj(xm/c, t) for each of the
indices j∈ {m, in, out}respectively. Restating the system in the three scaled spatial
4
variables ξjthen results in
∂ϑm
∂t =k
cml2
m
2ϑm
∂ξ2
m
,0< ξm<1, t > 0,(2a)
∂ϑin
∂t =v
lin
∂ϑin
∂ξin
,0< ξin <1, t > 0,(2b)
∂ϑout
∂t =v
lout
∂ϑout
∂ξout
,0< ξout <1, t > 0,(2c)
k
lm
∂ϑm
∂ξm
(0, t) = haText(t)ϑm(0, t), t > 0,(2d)
k
lm
∂ϑm
∂ξm
(1, t) = hbϑm(1, t)ϑin(1, t), t > 0,(2e)
ϑin(0, t) = Tinﬂow(t), t > 0,(2f)
ccvϑout(0, t)ϑin(1, t)=hbϑm(1, t)ϑin(1, t), t > 0.(2g)
Now we are prepared to rewrite this system (2) as a port-Hamiltonian system in
the next section.
3. The Port-Hamiltonian Formulation of inﬁnite systems
As mentioned earlier, we now want to formulate our model system in the port-
Hamiltonian framework, as this makes it easier to check for certain properties, es-
pecially stability.
Since the state variables ϑm,ϑin,ϑout in the model system (2) are continuous in
space, we cannot apply the usual ﬁnite-dimensional port-Hamiltonian framework as
described in, for example [5, 6]. Instead, we use a generalisation for distributed para-
meter systems, as presented in, for example, [7–9]. Since the system (2) only contains
ﬁrst-order derivatives w.r.t. time, we also restrict ourselves to linear ﬁrst-order systems
for simplicity.
Port-Hamiltonian systems (pHs) can be viewed as a combination of a Dirac struc-
ture (or Stokes-Dirac structure in the case of inﬁnite dimensional systems) and a
Hamiltonian. The Dirac structure deﬁnes the relation between the so-called ﬂow vari-
ables fand eﬀort variables e, while the Hamiltonian connects eand fto the state
variables Θ and ”contains the physics”. We start with a deﬁnition of the underlying
Dirac structure.
Deﬁnition 3.1 ((Stokes-)Dirac structure [7, 9, 10]).
Let Fbe a linear space, Eits dual and he, f itheir dual product. Further let
hhe1
f1,e2
f2ii =he1, f2i+he2, f1i,e1
f1,e2
f2∈ E × F.
Then D ⊂ (E × F ) is a Stokes-Dirac structure if D=Dwith
D={b∈ E × F|hhb, b1ii = 0 b1∈ D}.
Remark 1.
D={(e, f )∈ E × F|f=J e}
5
is a (Stokes-)Dirac structure if Jis a skew-adjoint operator.
This results in the following deﬁnition of a pHs, as given in [8, Deﬁnition 7.1.2]:
Deﬁnition 3.2 (port-Hamiltonian system ).Let P1Rn×ninvertible and self-
0=P0and H ∈ Rn×nsymmetric such that
mI ≤ H ≤ M I with constants m, M > 0. Further, let X=L2([a, b],Rn×n) be a
Hilbert space with the inner product
hf, gi=1
2Zb
a
g(ξ)H(ξ)f(ξ) dξ.
Then the diﬀerential equation
Θ
∂t (ξ , t) = P1
∂ξ H(ξ)Θ(ξ , t)+P0H(ξ)Θ(ξ, t), a < ξ < b, t > 0,(3)
is a linear ﬁrst-order port-Hamiltonian system (pHs) with the associated Hamiltonian
H(t) = 1
2Zb
a
Θ(ξ, t)H(ξ)Θ(ξ, t) dξ. (4)
Remark 2. With the usual choice of f=Θ
∂t and e=HΘ, we can see the connection
between the port-Hamiltonian system of Deﬁnition 3.2 and the Stokes-Dirac structure
of Deﬁnition 3.1, since the operator P1
The most important diﬀerence between a Dirac structure and a Stokes-Dirac struc-
ture is the presence of a boundary port that governs the power ﬂow across the boundary
and takes the place of boundary conditions in ‘regular’ PDEs. For a port-Hamiltonian
system as in Deﬁnition 3.2, the boundary port takes the following form [cf. 8, eqs.
(7.26) and (7.27)]:
f
e=1
2P1P1
I I
| {z }
R0
HΘ(b)
HΘ(a).(5)
While this is essentially a simple variable substitution, it is making the power ﬂow
across the boundary obvious, since now dH
dt=e>
fholds (in the absence of other
external ports).
We equip the port-Hamiltonian system with boundary conditions of the form
u(t) = WBf
e,(6)
where u(t) is a time-dependent input function, WBRn×2nhas full rank, WBΣW>
B
0 and Σ = 0I
I0. We note that the above property guarantees that the port-
Hamiltonian system always has a unique (classical and mild) solution which is non-
decreasing in the energy norm [cf. 8, Theorem 11.3.5]. Requiring WBΣW>
B>0 would
6
be suﬃcient for uniform exponential stability [8, Lemma 9.1.4], however, there are also
other, less restrictive criteria [cf. 8, Theorem 9.1.3] for uniform exponential stability.
Sometimes it is more convenient to write the boundary conditions in a form directly
dependent on the eﬀort variables at the boundary, in the form of u(t) = f
WBHΘ(b)
HΘ(a).
By comparison with equations (5) and (6), we see that
WB=f
WBR1
0.(7)
Not all port-Hamiltonian systems directly ﬁt into the formalism above, especially
if the order of space and time derivatives doesn’t match. However, the formalism can
be extended to dissipative systems [cf. 9, Chapter 6]:
∂ϑm
∂t (ξm, t) = J − GRSG
RHϑm(ξm, t),0< ξm<1, t > 0 (8)
G
Ris the formal adjoint operator of GR(i.e. the adjoint of GRneglecting boundary
conditions) and Sis a coercive operator on L2([0,1],R). Since we are only considering
linear ﬁrst-order systems, the operators take the following form:
Je=P1
∂e
∂ξ +P0e, GRf=G1
∂f
∂ξ +G0f, G
Re=G>
1
∂e
∂ξ +G>
0e,
As shown by Villegas [9, Chapter 6.3], the operator J − GRSG
Rin equation (8)
is equivalent to the expanded skew-symmetric operator Jetogether with the closure
relation er=Sfrin the expanded system
f
fr=Jee
er=J GR
−G
R0e
er,(9a)
er=Sfr.(9b)
In this formulation, the Dirac structure induced by Jehas an additional port, called
the resistive port. This port is then terminated with the resistive closure relation (9b).
Obviously, this also means that the boundary conditions can also depend on er.
Since WBonly gives us nconditions, this leaves the other nopen for use as outputs,
which we can deﬁne as
y=WCf
e.(10)
To ensure that we do not use quantities as output that are already set by the input,
we require that WB
WCis of full rank.
4. The PHS-Formulation of the Model System
In the following, we will ﬁrst model each part of our model system as a port-
Hamiltonian system, and then look at the combined system in Section 4.3.
7
4.1. The Heat Equation
The heat equation (2a) cannot be written as a port-Hamiltonian system of the form
given in Deﬁnition 3.2. However, it can be written as a system with dissipation of the
form given in equation (8).
With the usual choice of ﬂow variable f=∂ϑm
∂t and eﬀort variable e=Hϑm, as well
as the choices of
P1=P0=G0= 0, G1=1
lm
,S=kand H=1
cm
we have J= 0, GR=1
lmξ,G
R=1
lmξand thus we recover the heat equation (2a)
from equation (8) and obtain the associated quadratic Hamiltonian
Hm=1
2Z1
0
ϑ
m(ξ, t)Hϑm(ξ, t) dξ=1
2cmZ1
0ϑm(ξ, t)2dξ. (11)
Note that in this case the Hamiltonian (11) is not the physical energy, so the dis-
sipation present in this system does not automatically violate the law of conservation
of energy. If we wanted to explicitly use the physical energy as our Hamiltonian, we
would also need to ensure that the second law of thermodynamics is satisﬁed, i.e.,
the system becomes irreversible. It is possible to extend port-Hamiltonian systems for
irreversible cases, cf. , but this would add additional complexity and is not needed
here.
Putting the heat equation into the form of (9), we obtain the following equations
f
fr=01
lm
1
lm0
∂ξ e
er,(12a)
er=kfr.(12b)
This formulation makes it clear how spatial derivatives of ϑcan occur in the boundary
conditions. Remembering that f=∂ϑm
∂t and e=Hϑm, it follows that er=k
cmlm
∂ϑm
∂ξ .
The corresponding boundary conditions (2d), (2e) can now be rewritten in a for-
mulation with inputs:
u(t) = ha
cmText(t)
hb
cmϑin(1, t)!=0 0 ha1
hb1 0 0
| {z }
=
f
WB
e(1, t)
er(1, t)
e(0, t)
er(0, t)
.(13)
With R0=1
2P1P1
I I we ﬁnd
WB=f
WBR1
0=1
2lmhalmha 1
lmhblmhb1.(14)
WBΣW>
B=2halm0
0 2hblm(15)
8
WBis obviously of full rank. As ha,hband lmare all physical constants of the system
and thus positive, we ﬁnd that WBΣW>
B>0 holds. Therefore, the heat equation with
the chosen boundary conditions is has unique (classical and mild) solutions which are
non-increasing in norm [cf. 9, Theorem 6.9].
As outputs, we choose the temperatures at the boundaries (scaled by some constants
for convenience), so we get
y= ha
cmϑm(0)
hb
cmϑm(1)!=0 0 ha0
hb0 0 0
e(1)
em(1)
e(0)
em(0)
(16)
Then the combined matrix f
WB
f
WC!has full rank, which means we are not measuring
quantities we already set as input.
4.2. The Transport Equations
For the cooling channel, which we divide into an incoming and an outgoing channel
(indices in and out), we do not have any dissipative terms, so we can write it directly
as a linear ﬁrst-order port-Hamiltonian system as deﬁned in Deﬁnition 3.2
∂ϑin
∂t
∂ϑout
∂t != v
lin 0
0v
lout !
∂ξ ϑin
ϑout!,(17)
i.e. with the choices
P1= v
lin 0
0v
lout !, P0= 0,H=1 0
0 1.
Next, rewriting the boundary conditions (2f) and (2g) in a formulation with inputs,
we obtain
u(t) = Tinﬂow(t)
hbϑm(1, t)=0 0 1 0
(hbccv)00ccv
| {z }
=
f
WB
ϑin(1, t)
ϑout(1, t)
ϑin(0, t)
ϑout(0, t)
.(18)
As outputs, we choose the temperature at the ends of each cooling channel part, i.e.
y(t) = ϑin(1, t)
ϑout(1, t)=1 0 0 0
0 1 0 0
| {z }
f
WC
ϑin(1, t)
ϑout(1, t)
ϑin(0, t)
ϑout(0, t)
,(19)
so we have again f
WBas well as f
WB
f
WC!of full rank.
9
Calculating WBas in (7), we obtain
WB=f
WBR1
0=lin
v0 1 0
lincclin hb
vcclout (hbccv)ccv,(20)
WBΣW>
B=lin
v0
0c2
cloutv(ccvhb)2lin
v.(21)
From the last line we can see that this system is not stable for all variable choices, since
the stability condition WBΣW>
B0 is not always satisﬁed. Positive-semideﬁnitness
is only given for
lout
lin (ccvhb)2
c2
cv2
In the special case of lin =lout, this condition can be simpliﬁed to 2ccvhb.
4.3. The Combined system
When we combine the two Port-Hamiltonian systems discussed before, we get the
following system of equations. As you can see in equation (24), not all boundary
conditions are connected to an input anymore, but are instead set to zero. These are
the coupling conditions between the two subsystems. Accordingly, we also have only
two outputs. In detail, the system has the form given by the equations (9) resp. (8),
with the choices of
P1=
0 0 0
0v
lin 0
0 0 v
lout
, P0=G0= 0, G1=
1
lm
0
0
,S=k,
H=
1
cm0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Inserting these choices into equation (9), we get the following form for the combined
10
system:
∂ϑm
∂t
∂ϑin
∂t
∂ϑout
∂t
fr
=
0 0 0 1
lm
0v
lin 0 0
0 0 v
lout 0
1
lm0 0 0
∂ξ
1
cmϑm
ϑin
ϑout
er
,(22)
er=k
fr
,(23)
u
0=
ha
cmText(t)
Tinﬂow(t)
0
0
=
0 0 0 0 ha0 0 1
0 0 0 0 0 1 0 0
hbhb
cm0 1 0 0 0 0
hbcm(hbccv) 0 0 0 0 ccv0
1
cmϑm(1)
ϑin(1)
ϑout(1)
er(1)
1
cmϑm(0)
ϑin(0)
ϑout(0)
er(0)
,
(24)
y=1
cmϑm(0)
ϑout(1) =00001000
00100000
1
cmϑm(1)
ϑin(1)
ϑout(1)
er(1)
1
cmϑm(0)
ϑin(0)
ϑout(0)
er(0)
.(25)
f
WB,f
WCand f
WB
f
WC!obviously have full rank. Note that we could also choose the
heat ﬂux at the left boundary em(0) as an output, without changing the rank of the
matrices.
It remains to check whether the stability condition WBΣW>
B0 holds:
WB=1
2
lm0 0 lmhaha0 0 1
0lin
v0 0 0 1 0 0
lmlinhb
cmv0lmhbhbhb
cm0 1
0(ccvhb)lin
vloutcclmhbcmhbcmccv+hbccv0
(26)
WBΣW>
B=
2lmha0 0 0
0lin
v0 0
0 0 2 lmhblinh2
b
c2
mvlmcmhb(ccvhb)linhb
cmv
0 0 lmcmhb(ccvhb)linhb
cmvloutc2
cv(ccvhb)2lin
v
(27)
We can see from equation (27) that both eigenvalues are non-negative, if the following
11
two conditions hold:
a+c0,
ac b2
with
a= 2 lmhblinh2
b
c2
mv,
b=lmcmhb(ccvhb)linhb
cmv,
c=loutc2
cv(ccvhb)2lin
v.
Since the boundary condition (1h) is equivalent to (1g) due to (1e), we can use that
one to develop our coupling structure instead. If we do that, we ﬁnd:
f
WB=
0 0 0 0 ha0 0 1
0 0 0 0 0 1 0 0
hbhb
cm0 1 0 0 0 0
0ccv0 1 0 0 ccv0
(28)
WB=1
2
lm0 0 lmhaha0 0 1
0lin
v0 0 0 1 0 0
lmlinhb
cmv0lmhbhbhb
cm0 1
lmcclin loutcc0 0 ccv ccv1
(29)
WBΣW>
B=
2halm0 0 0
0lin
v0 0
0 0 2hblmh2
blin
c2
mvcchblin
cm+lmhb
0 0 cchblin
cm+lmhbloutc2
cvlinc2
cv
(30)
Here it becomes immediately clear that the matrix for lin =lout can never be positive
deﬁnite, but only semi-deﬁnite, if 2hblmh2
blin
c2
mv0 and cchblin
cm+lmhb= 0 holds,
which yields cchb
2cmv. Thus it is more restrictive in this respect than the previous
coupling.
Summarising, the combined system possesses unique (classical and mild) solutions
on [0,) and these solution are non-increasing in the energy norm. We remark, that
it is an open question whether exponential stability of the extended port-Hamiltonian
system implies exponential stability of the combined system, that is, whether the
condition WBΣW>
B>0 guarantees exponential stability of the combined system.
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Table 1. Variable naming conventions
Variable Quantity Unit
Ttemperature K
cvolumetric heat capacity J m3K1
hheat transfer coeﬃcient W m2K1
kthermal conductivity W m1K1
llength m
vﬂow speed m s1
5. Conclusion
While the heat equation as described in Section 4.1 is exponentially stable for all
(physically meaningful, i.e. positive) values of the constants involved, this is not the
case for the transport equations of the cooling channel described in Section 4.2, nor
for the combined system of Section 4.3. For both systems, it is possible to ﬁnd values
of the constants that are physically reasonable but do not satisfy the stability criteria,
regardless of which formulation is chosen for the coupling conditions.
It is also noteworthy that the two formulations of the coupling conditions studied,
while technically equivalent, imply diﬀerent regions of stability. Although some of the
stability conditions have a clear, physical motivation, others – particularly those re-
lated to the coupling condition – are seemingly nonsensical. The most obvious example
of the latter would be that the ratio between the lengths of the cooling channel parts
can determine the stability of the system.
A likely explanation or interpretation for this is that our model system is oversim-
pliﬁed and does not properly capture the properties of the real system. The fact that
all heat transfer occurs in a single point, as opposed to an extended region with a
non-zero physical dimension, causes a discontinuity in the temperature distribution
within the cooling channel, making it a likely candidate for the source of the above
problems. In future work we will investigate whether the observed stability problems
persist if we consider instead the coupling of a two-dimensional heat equation with
heat transfer along the entire length of the cooling channel.
In any case, the port-Hamiltonian formalism has proved to be a very useful tool
to study the properties of a model system describing several interconnected physical
processes and to ﬁnd problems and limitations of the model. With its help, it has been
shown that simpliﬁcations – even those widely used in industry – are not always useful
from a mathematical point of view and must be carefully evaluated before use.
Our future work will focus on investigating new discretisation techniques based on
this coupling strategies and studying a more realistic two dimensional setting.
Funding
The authors were partially supported by the BMBF collaborative research project
GivEn under the grant no. 05M18PXA, see https://www.given-project.de/.
Nomenclature/Notation
Th naming of variables denoting physical quantities follows the conventions set in the
Table 1. Variables that are not directly referencing physical quantities are generally
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deﬁned when introduced.
The index mis used for variables that refer to the metal subsystem, i.e. the heat
conducting rod. The indices in and out are used for the inﬂowing and outﬂowing parts
of the cooling channel, respectively. Finally, the index cis used for variables that refer
to the entirety of the cooling channel.
References
 J. Backhaus, M. Bolten, O.T. Doganay, M. Ehrhardt, et al.,GivEn – Shape optimization
for gas turbines in volatile energy networks, in Mathematical MSO for Power Engineering
and Management, S. G¨ottlich, M. Herty, and A. Milde, eds., Mathematics in Industry
Vol. 34, Springer, 2021, pp. 71–106.
 O.R. Imam-Lawal, Adjoint based optimisation for coupled conjugate heat transfer, Ph.D.
diss., Queen Mary, University of London, 2020.
 J.C. Han, Recent studies in turbine blade cooling, Int. J. Rotat. Machin. 10 (2004), pp.
443–457.
 V. Mehrmann and R. Morandin, Structure-preserving discretization for port-Hamiltonian
descriptor systems, in 2019 IEEE 58th Conference on Decision and Control (CDC). 2019,
pp. 6863–6868.
 D. Jeltsema and A.J. van der Schaft, Memristive port-Hamiltonian systems, Math. Com-
put. Modell. Dynam. Syst. 16 (2010), pp. 75–93.
 A. van der Schaft, Port-Hamiltonian systems: an introductory survey, in Proceedings of the
International Congress of Mathematicians: Madrid, August 22-30, 2006: invited lectures.
2006, pp. 1339–1366.
 A. Macchelli, A.J. van der Schaft, and C. Melchiorri, Port Hamiltonian formulation of
inﬁnite dimensional systems I. Modeling, in 2004 43rd IEEE Conference on Decision and
Control (CDC) (IEEE Cat. No.04CH37601), Vol. 4. 2004, pp. 3762–3767 Vol.4.
 B. Jacob and H.J. Zwart, Linear port-Hamiltonian systems on inﬁnite-dimensional spaces,
Vol. 223, Springer Science & Business Media, 2012.
 J.A. Villegas, A port-Hamiltonian approach to distributed parameter systems, Ph.D. diss.,
University of Twente, 2007.
 W. Zhou, B. Hamroun, F. Couenne, and Y.L. Gorrec, Distributed port-Hamiltonian mo-
delling for irreversible processes, Math. Comput. Modell. Dynam. Syst. 23 (2017), pp.
3–22.
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