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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) [2], 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 [2]. 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. [3].

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 180◦hairpin 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 (a≤xm≤b) and

a cooling channel (i≤xc≤o) 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 =−v∂Tin

∂xc

, i < xc< c, t > 0,(1b)

∂Tout

∂t =−v∂Tout

∂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)=−k∂Tm

∂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=xm−a

b−a=xm−a

lm

, a ≤xm≤b,

ξin =xc−i

c−i=xc−i

lin

, i ≤xc≤c,

ξout =xc−c

o−c=xc−c

lout

, c ≤xc≤o,

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=D⊥with

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 [8]).Let P1∈Rn×ninvertible and self-

adjoint, P0∈Rn×nskew-adjoint, i.e. P>

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∂

∂ξ +P0is skew-adjoint.

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

√2P1−P1

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>

∂f∂holds (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, WB∈Rn×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

WBR−1

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∗

RHϑm(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

lm∂ξ,G∗

R=−1

lm∂ξand 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. [10], 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 ha−1

hb1 0 0

| {z }

=

f

WB

e(1, t)

er(1, t)

e(0, t)

er(0, t)

.(13)

With R0=1

√2P1−P1

I I we ﬁnd

WB=f

WBR−1

0=1

√2lm−halmha −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

0−v

lout !∂

∂ξ ϑin

ϑout!,(17)

i.e. with the choices

P1= −v

lin 0

0−v

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

(hb−ccv)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

WBR−1

0=lin

v0 1 0

lincc−lin hb

vcclout (hb−ccv)ccv,(20)

WBΣW>

B=lin

v0

0c2

cloutv−(ccv−hb)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>

B≥0 is not always satisﬁed. Positive-semideﬁnitness

is only given for

lout

lin ≥(ccv−hb)2

c2

cv2

In the special case of lin =lout, this condition can be simpliﬁed to 2ccv≥hb.

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

0−v

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

0−v

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

hb−hb

cm0 1 0 0 0 0

−hbcm(hb−ccv) 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>

B≥0 holds:

WB=1

√2

lm0 0 −lmhaha0 0 −1

0lin

v0 0 0 1 0 0

lmlinhb

cmv0lmhbhb−hb

cm0 1

0(ccv−hb)lin

vloutcc−lmhbcm−hbcm−ccv+hbccv0

(26)

WBΣW>

B=

2lmha0 0 0

0lin

v0 0

0 0 2 lmhb−linh2

b

c2

mv−lmcmhb−(ccv−hb)linhb

cmv

0 0 −lmcmhb−(ccv−hb)linhb

cmvloutc2

cv−(ccv−hb)2lin

v

(27)

We can see from equation (27) that both eigenvalues are non-negative, if the following

11

two conditions hold:

a+c≥0,

ac ≥b2

with

a= 2 lmhb−linh2

b

c2

mv,

b=−lmcmhb−(ccv−hb)linhb

cmv,

c=loutc2

cv−(ccv−hb)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

hb−hb

cm0 1 0 0 0 0

0−ccv0 1 0 0 ccv0

(28)

WB=1

√2

lm0 0 −lmhaha0 0 −1

0lin

v0 0 0 1 0 0

lmlinhb

cmv0lmhbhb−hb

cm0 1

lmcclin loutcc0 0 −ccv ccv1

(29)

WBΣW>

B=

2halm0 0 0

0lin

v0 0

0 0 2hblm−h2

blin

c2

mv−cchblin

cm+lmhb

0 0 −cchblin

cm+lmhbloutc2

cv−linc2

cv

(30)

Here it becomes immediately clear that the matrix for lin =lout can never be positive

deﬁnite, but only semi-deﬁnite, if 2hblm−h2

blin

c2

mv≥0 and −cchblin

cm+lmhb= 0 holds,

which yields cc≥hb

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.

12

Table 1. Variable naming conventions

Variable Quantity Unit

Ttemperature K

cvolumetric heat capacity J m−3K−1

hheat transfer coeﬃcient W m−2K−1

kthermal conductivity W m−1K−1

llength m

vﬂow speed m s−1

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

13

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.

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