A Low-Order Dynamic Model of Counterflow Heat Exchangers for the Purpose ofMonitoring Transient and Steady-State Operating Phases

Maik Gentsch1,∗, Rudibert King2

Technische Universitat Berlin, Chair of Measurement and Control, Straße des 17. Juni 135, 10623 Berlin, Germany

Abstract

We present a model-based real-time method to monitor a counterflow heat exchanger’s thermal performance for alloperating conditions. A first principle reference model that describes the reference counterflow process in an accuratemanner is derived first. Real gas behavior is taken into account. Without simplifications, the respective equations mustbe solved in an iterative, computationally expensive manner, which prohibits their use for real-time monitoring purposes.Therefore, we propose one-step-solvable model equations, resulting in an approximate but quick model, which is ableto track an important thermal property reliably. The monitoring, i.e., the online estimation of the thermal properties,is achieved via a nonlinear Kalman-Filter. Due to the low-order dynamic model formulation, the overall monitoringscheme is accompanied by an acceptable computational burden. Moreover, it is easy to deploy and to adapt in industrialpractice. Monitoring results, where the reference model replaces a real process with supercritical carbon dioxide, aregiven and discussed herein.

Keywords: Heat exchanger modeling, Model-based supervision, Flexible operation, Real gas process fluid, Onlinemonitoring, Extended Kalman Filter

1. Introduction

Due to modern standards and future challenges, indus-trial plants will be bound to run very flexibly across a widerange of operating points without any trade-offs concern-ing their availability and reliability. This is true, for exam-ple, for multi-stage compressors with intermediate heat ex-changer units. Formerly, these machines were designed forsteady-state operating conditions. This, likewise, appliesto the supervision methodology of such machines in indus-trial practice. Typically, supervision is based on measure-ments only, alarming the supervisor if data exceed certainthresholds concerning the expected steady-state operatingvalues. Using this methodology during transient operatingphases could lead to frequent false alarms if the supervisedmachine comprises any dynamics within the relevant timescale.

In this paper, we investigate model-based supervision ofcounterflow heat exchangers that are known to be a slug-gish plant component when it comes to industrial scale.Therefore, a proper mathematical model description is es-

∗Corresponding authorEmail addresses: maik.gentsch@tu-berlin.de (Maik Gentsch),

rudibert.king@tu-berlin.de (Rudibert King)1Graduate Research Assistant2Head of Department

© 2020. This manuscript version is made available under the CCBY-NC-ND 4.0 license https://creativecommons.org/licenses/

by-nc-nd/4.0/ .

sential. Since transient behavior is to be covered and on-line monitoring is addressed, the model must consider rel-evant system dynamics and, further, the computationaleffort to solve the model equations should be low at thesame time.

The modeling of heat exchangers has been proposed invarious studies, from simple lumped descriptions [1, 2] tocomplex spatial models using Computational Fluid Dy-namics [3]. The most common application considers one-dimensional parameter and temperature distributions rep-resented by partial differential equations, which are solvedby implementing a discretization scheme, i.e., finite vol-ume or finite difference methods [4, 5, 6, 7]. Usually, modelbuilding and equation solving is performed in the frame-work of technically mature software, such as Dymola (e.g.,[4, 8]) or gPROMS (e.g., [7]). Because of the heavy im-pact on plant efficiency, further investigations have beenconducted to model the phenomenon of fouling in moredetail (e.g., [9, 7]). The major purposes of the more com-plex models stated above are: i) achieving deeper sys-tem knowledge; ii) accomplishing off-line analyses; and iii)improving design methods for heat exchangers. Unfortu-nately, they are accompanied by a tremendous effort con-cerning model implementation and parametrization, re-sulting in a highly customized application for a specificheat exchanger.

The aim of this work is to meet certain industrial re-quirements, namely, adaptability and ease of deployment.More specifically, these requirements include: i) little nu-

Preprint submitted to Chemical Engineering Science February 16, 2021

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merical effort; ii) an algorithm proven to be reliable interms of numerical issues; and iii) the simplicity of adap-tion to various heat exchangers. Though the simpler, low-order models found in literature may pass these criteria,they require some inappropriate assumptions concerningthe scope of the application addressed here, e.g., the ne-glect of specific temperature differences between the heatexchanger’s intake and outlet cross-section. Therefore, wedecided against a model based on partial differential equa-tions, and derived a suitable low-order dynamic modelbased on simple first principles with few global parameters.However, as the model does not consider spatial tempera-ture distributions, some heuristics are utilized to form thedynamic model equations. The model formulation allowsfor incorporating arbitrary enthalpy calculation models,which is a further feature of this work. On that account,we were able to present the effect of the common perfectgas assumption on the monitoring results when appliedto real gas applications, such as the supercritical carbondioxide heat exchanger from the simulation study below.

Beside fouling, there are conceivable faults like coolantleakage or faults of the neighboring plant components (e.g.,compressor, valves) that could lead to a significant changeof the heat exchanger’s performance. Fault detection andisolation of a specific fault, however, is not within the scopeof this paper. Instead, the proposed monitoring schemeprovides the time series of parameters, filtered measure-ments, and, consequently, residuals between real and fil-tered measurements. All of them could serve as a basefor such fault detection and isolation algorithms. The pro-vision of these time series in real-time, i.e., during plantoperation, necessitates proper estimation techniques. Forthis purpose, the Extended Kalman Filter scheme is ap-plied. This scheme considers the model equations and therespective algorithm is of moderate numerical complexity,as is it formulated in a recursive manner.

Within the paper the following issues are presented anddiscussed: In Section 2, the scope of the application isstated, and all assumptions used are summarized and dis-cussed. The model building is presented in Section 3. Be-cause the final aim is to monitor unmeasurable properties,i.e., (convective) heat transfer coefficients, we develop areference model for validation purposes in Section 3 first.The reference model is accurate if the reference counter-flow process, as defined in Section 2, is valid. We derivethe respective model equations and give a proof of stabil-ity, which is an essential property for the derivation of thelow-order state equations. Based on the reference model,an approximate model is derived, which is suitable for thereal-time application of online monitoring due to its signif-icantly lower computational complexity. In this paper, theterm “monitoring” is the equivalent of an online parame-ter estimation on the basis of a model-based measurementscheme consisting of the dynamic model as a part of an Ex-tended Kalman Filter. The whole approach is presentedin Section 4. The main results of this investigation will bediscussed in the last subsections before the conclusions are

Th1

Tc2

Tw1

Th2

Tc1

Tw2

mh

mc

hot

cold

Figure 1: Reference counterflow process:h – hot fluid, c – cold fluid, w – wall, 1 – intake, 2 – outlet

drawn in Section 5.

2. Assumptions and the Scope of the Application

2.1. Reference Counterflow Process

The presented modeling approach addresses all types oftwo-fluid heat exchangers fitting the reference counterflowprocess, as shown in Fig. 1. The model allows for lumpedtemperature information on the heat exchanger’s intakeand outlet cross-section; hence, there is neither need norchance to interact with spatial parameter or temperaturedistributions. Because no further assumption concerningthe inner geometry of the exchanger is made, one could fitshell-and-tube heat exchangers into the illustrated schemeby merging the inner tube bundle to a single wall and,accordingly, the inner tube streams to a single flow. Thisappears to be valid as long as

1. the process fluid enters the exchanger near the coolantexit and vice versa,

which is quite a relaxation compared to a strict counter-flow definition. Despite the lumped temperature approach,we do not think of the inner wall as a lumped mass witha single temperature value. In fact, a wall temperaturedistinction is drawn between the intake and outlet cross-section, which is in accordance with the expected situationdrafted in Fig. 1. This truly differs from the conventionallumped models. On the other hand, a local wall temper-ature distribution in a radial direction is neglected, whichmeans

2. the local thermal conductance of the inner wall in aradial direction is infinite.

Further assumptions defining the reference process in-clude:

3. the outer casing is adiabatic;

4. the interchange of heat is an isobaric process; and

5. the response time, due to the thermal inertia of theinner wall, is many times greater than the delay dueto mass transport.

For supervision, two different rating problems could bethought of: the determination of heat transfer (thermal

2

rating) and the pressure drop performance (hydraulic rat-ing) [10]. In this research, we focus on the carbon dioxideprocess fluid at a supercritical state, which appears, for ex-ample, in a carbon capture and storage (CCS) process withthe aim of reducing greenhouse gases in the atmosphere.Concerning the relative high pressure level, we found thatthe pressure drop over the exchanger was nearly negligible,which lead to assumption 4 and the decision to accomplisha thermal rating solely. However, in the presence of rel-evant pressure losses, we argue that a thermal rating iscapable of reflecting a change of system behavior even ifthe underlaying model assumes isobaric conditions. Notethat we do not assume ideal gas behavior.

In terms of instrumentation, it is presupposed that

6. mass flow, intake, and outlet temperatures, both ofthe process and cooling fluid, are known; and

7. if one or both fluids shall be treated as a real gas,their respective pressure is known.

Quite clearly, assumption 6 provides an ideal situation,which is hard to find in industrial practice. Moreover, evenif all temperature measurements exist, they are quite oftendelayed and considerably biased due to the thick-walledshield casings. There are two ways to deal with miss-ing or unreliable measurements when it comes to modelvalidation: replacement by data sheet specifications or re-placement by peripheral model calculations. Both of thembring some uncertainty, but we chose the latter approach.Thus, the validation of the heat exchanger model is cor-related with these peripheral models. Because they arebeyond the scope of this paper, we will present the wholeissue of validation given an imperfect data status in a fu-ture article. For the sake of deriving the model equations,we accept assumption 6, stating that the model validationyields satisfactory results. Within the monitoring scheme,that assumption could be relaxed since uncertain informa-tion is permitted. Further details are provided in Section4.4.

2.2. Thermal Rating

The overall heat transfer coefficient km serves as an ap-propriate measure to accomplish the thermal performancerating. With a lack of information about the total heattransfer surface area A, it is more convenient to look forthe overall thermal conductance (kA)m as a single term:

(kA)m =

∣∣∣∣∣Q

∆Tm

∣∣∣∣∣ . (1)

For heat exchangers, the mean temperature difference∆Tm is typically calculated using the logarithmic mean(LM) of the fluid’s temperature differences at the intakeand outlet cross-section, respectively (cf. Fig. 1):

∆T1 = Th1 − Tc2 , ∆T2 = Th2 − Tc1 , (2)

∆Tm = LM(

∆T1, ∆T2

):=

∆T1 −∆T2

ln ∆T1

∆T2

. (3)

Even though the derivation of the log-mean temperaturedifference assumes strict counterflow or parallel flow con-ditions [11], it is commonly used for alternative flow ar-rangements, considering a hypothetical counterflow unitoperating at the same resistance and effectiveness [10].

The aforementioned temperatures are accessible in alloperating phases. This does not apply to the total heattransfer rate Q, the calculation of which assumes a steady-state operation. To overcome this issue, we look at theserial connection of thermal resistances, leading to:3

(kA)m =

(1

(αA)m,h+

1

(αA)m,c

)−1

, (4)

where (αA)m represents the overall convection conduc-tance at the hot fluid side (subscript h) and the cold fluidside (subscript c), respectively. Their definitions do notdepend on steady-state conditions:

(αA)m,h =

∣∣∣∣∣Qh

LM(

∆Th1, ∆Th2

)∣∣∣∣∣ , (5)

(αA)m,c =

∣∣∣∣∣Qc

LM(

∆Tc1, ∆Tc2)∣∣∣∣∣ . (6)

The temperature differences as well as the heat transferrates are no longer noted in reference to the opposite fluidstream but to the inner wall, leading to the introductionof wall temperatures at the exchanger’s intake and outletcross-section, as depicted in Fig. 1:

∆Th1 = Th1 − Tw1 , ∆Th2 = Th2 − Tw2 , (7)

∆Tc1 = Tw1 − Tc2 , ∆Tc2 = Tw2 − Tc1 . (8)

Commonly, wall temperature measurements do not exist;hence, we have derived a model to calculate these time-variant variables. Note that additional terms for foulingresistances are not introduced. As a matter of fact, a sep-arate monitoring of (αA)m,h and (αA)m,c could providemore information than a coupled monitoring of (kA)m, ac-cording to Eq. (4). Unfortunately, the presented approachdoes not allow for decoupled monitoring under most oper-ating conditions, as will be shown in Section 4.3. However,the coupled estimate of (kA)m is capable of tracking theexchanger’s overall thermal performance quickly and reli-ably.

3. Modeling

The starting point for a model-based monitoring ap-proach is a nonlinear, dynamic system description

x(t) = f

(x, u, θ, t

), x(t0) = x0 , (9)

y(t) = g(x, u, θ, t

), (10)

3Note that the thermal resistance of the heat transmitting wall isneglected due to assumption 2.

3

where y ∈ Rny , x ∈ Rnx , u ∈ Rnu , and θ ∈ Rnθ arethe measurable outputs, the states, the inputs, and theparameters of the model, respectively. In general, all ofthese values are time-variant, but the model parametersare assumed to vary much slower than the other variables.To increase the readability of the equations, the time ar-gument t is suppressed in what follows.

Only two dynamic variables, Tw1 and Tw2, composed inthe state x, are considered for an appropriate descriptionto facilitate the setup of a real-time algorithm, namely, thewall temperatures used in Eqs. (7)–(8). In summary, theproposed assignment of variables for the system (9)–(10)is:

x =

[Tw1

Tw2

], y =

[Th2

Tc2

], u =

Th1

Tc1mh

mc

, θ =

θ1

θ2

...θ7

.

(11)Based on the state x, the given inputs u (see assumption6), and parameters θ, the outlet temperatures y are calcu-lated with the output equation g. The supervision schemewill utilize the residual between these model outputs andmeasured outlet temperatures to calculate a proper esti-mate of the model parameters used for the thermal perfor-mance rating. In the input vector, mh and mc are the hotand cold fluids’ mass flows, respectively. Like the modelequations g and f , the concrete model parameters θ differdepending on whether they belong to the reference or theapproximate model. They will be introduced later.

In Section 2, it was pointed out that the sluggish re-sponse due to the thermal inertia of the inner wall is signif-icantly more dominant than secondary dynamics, like themass transport delay (see assumption 5). To that end, weuse a quasi-steady-state approach. That means the outlettemperatures y are calculated as if they would adjust afteran infinite amount of time but under the conditions of fixedwall temperatures x and fixed inputs u. As long as thiscalculation leads to an imbalance between the wall heatingand cooling fluxes, the wall temperatures will tend towardtheir respective equilibrium state, affecting the subsequentquasi-steady-state calculation of y.

We presuppose the existence of an appropriate model tocalculate specific enthalpies of the process and the coolingfluids. For the sake of generality, the dependancy on pres-sure is considered, and the enthalpy calculation is denotedas

hh(T, p

)and hc

(T, p

). (12)

With respect to availability and accuracy, one has to de-cide whether to use a calorically perfect gas, an incom-pressible fluid, a thermally perfect gas, or a real gas model.Deviations from the real fluid behavior will affect the quan-tity of the observed thermal property, as will be shown inSection 4.3. Before the dynamic model f is specified inSection 3.3, the output equations g and the steady-statesolutions for the reference and approximate models are in-

troduced each in Sections 3.1 and 3.2, respectively.

3.1. Output Equations

3.1.1. Reference Model

As mentioned in [10], only two important relationshipsconstitute the entire thermal design procedure (or, viceversa, the thermal rating problem) of two-fluid heat ex-changers. The first of them is the heat transfer rateequation represented by Eq. (1) for both fluids (steady-state only) or by Eqs. (5)–(6) in a partitioned manner.In a highly transient situation, with fast changing intaketemperatures, we have to consider cases ∆Th1 < 0 or∆Tc2 < 0, for which the logarithmic mean according to Eq.(3) is not defined due to arguments with opposite signs.To be able to describe such phases as well, we use an un-restricted formulation:4

Qh = −Q(

∆Th1, ∆Th2, (αA)m,h), (13)

Qc = Q(

∆Tc1, ∆Tc2, (αA)m,c), (14)

Q(z1, z2, z3

):=

{z3 · LM

(z1, z2

), (z1, z2) ∈ Lz2z1

z3 · AM(z1, z2

), (z1, z2) /∈ Lz2z1

,

(15)

where

AM(z1, z2

):=

z1 + z2

2and (16)

Lz2z1 := {(z1, z2) ∈ R2 | z1 > 0 , z2 > 0 , z1 6= z2} (17)

denote the arithmetic mean and the domain of the loga-rithmic mean, respectively. Calligraphic variables, such asQ(z1, z2, z3), denote functions that are specifically definedin this contribution to ensure a compact representation ofrelevant dependencies.

The second elementary relationship is given by the iso-baric enthalpy rate equations

Hh = Hh(Th2

):= mh ·

[hh(Th2, ph

)− hh

(Th1, ph

)],

(18)

Hc = Hc(Tc2)

:= mc ·[hc(Tc2, pc

)− hc

(Tc1, pc

)].(19)

Because of the adiabatic outer casing, the heat transferrates and enthalpy rates must be of equal value. Suchequalities, which will appear in different forms in this con-tribution, can always be reformulated as a root searchingtask of an appropriate residual function by bringing allterms of an equality on one side. For the specific case con-sidered here, given the current wall temperatures x, theintake temperatures, and the mass flows, all compressedin u, in addition to the overall convection conductances

θ1 = (αA)m,h and θ2 = (αA)m,c , (20)

4We always balance from the fluid point of view. Thus, a negativeQ or H denotes a cooling of the fluid.

4

which are treated as model parameters here, the roots ofthe residual functions

Rh(T ∗h2

):= Hh

(T ∗h2

)+Q

(Th1 − Tw1, T

∗h2 − Tw2, θ1

),

(21)

Rc(T ∗c2)

:= Hc(T ∗c2)−Q

(Tw1 − T ∗c2, Tw2 − Tc1, θ2

),

(22)

have to be determined, which are the unknown outlet tem-peratures Th2 and Tc2. Note that with all remaining valuesfixed, Rh and Rc are strictly increasing in their respectiveargument. Thus, there is, at maximum, one unique rootwithin the physically possible domain T ∗h2 ∈ [Tw2 ; Th1]and T ∗c2 ∈ [Tc1 ; Tw1]. In general, the root determinationof (21)–(22) necessitates a numerical multiple-step proce-dure. Especially if the included enthalpy calculation isaccomplished by real gas models, the overall computationof the reference output model (subscript r)

gr

(x, u, θ, t

)=

[root of Rhroot of Rc

](23)

comes with a high numerical burden. For this reason, wewill not deploy this reference model g

rin the scheme of

online monitoring. Advantageously, this model formula-tion is the exact description of the reference counterflowprocess without any further assumptions compared to thelist given in Section 2.1. On that account, the referencemodel will be used to calculate the time series of the tem-peratures and thermal properties, on which the validationof the approximate model and the proof of the thermalproperty ‘observability’ will be accomplished.

3.1.2. Approximate Model

This section concerns simplifying the output equation(23) so that it becomes solvable in one step without itera-tions. Two issues imply the necessity of the multiple-stepapproach above: i) the integration of an arbitrarily com-plex enthalpy calculation model into the root determina-tion problem and ii) the log-mean temperature difference.

Externalizing the Enthalpy Calculation Model. To facili-tate the one-step-solvable outlet temperature calculation,irrespective of the applied enthalpy calculation model, weintroduce mean specific heat parameters

θ3 = Ch(T−h2

):=

hh(T−h2, ph

)− hh

(Th1, ph

)

T−h2 − Th1

, (24)

θ4 = Cc(T−c2)

:=hc(T−c2, pc

)− hc

(Tc1, pc

)

T−c2 − Tc1, (25)

where T−h2 and T−c2 are the model outputs from a previoustime step, and, as above, hh/c(T, p) denotes an enthalpymodel, possibly featuring real gas behavior. Note that θ3

and θ4 are time-variant in general, but they are fixed forthe root determination step solely, where we replace the

enthalpy calculation in (18)–(19) with the approximations

hh/c(T, p

)= θ3/4 · T . (26)

This is somewhat different from using a calorically perfectgas model in general but manifests the externalization ofan unspecified enthalpy calculation model out of the rootdetermination of (21)–(22). To point out the differencesto an integrated real-gas model, a specific case study willbe considered in Section 4.3, for which the dependenciesdescribed in Eqs. (24)–(25) are depicted in Fig. 2. Thechosen reference will be a supercritical carbon dioxide heatexchanger, for which a calorically perfect gas model (con-stant specific heat) is particularly improper.

Replacing the Logarithmic Mean. Despite the simplifica-tion affecting the calculation of Hh/c, the log-mean ap-proach in Q impedes a one-step-solution. To find a properapproximation of the logarithmic mean, we look at thearithmetic-logarithmic-geometric mean inequality [12]:

GM(z1, z2

)< LM

(z1, z2

)< AM

(z1, z2

), (27)

for (z1, z2) ∈ Lz2z1 . Here,

GM(z1, z2

):=√z1 · z2 (28)

denotes the geometric mean. This relationship motivatesthe replacement of the logarithmic mean by the followingweighted mean WM:

WM(z1, z2, β

):=β · GM

(z1, z2

)

+ (1− β) · AM(z1, z2

), (29)

where β is a novel weighting parameter that has to bebounded between 0 and 1 if WM is to serve as a propersubstitution for LM in Eq. (27). A proper determinationof β concerning the domain of the approximate outputequations is given in what follows.

For the sake of convenience, we replace the sepa-rated terms for the hot and cold sides with substitutesthat cover both sides according to Table 1. Given that, itis sufficient to solve the following universal residual andrecover the side-specific expressions afterwards:

∼R(

∆T ∗II)

:=∼H(

∆T ∗II)− ∼Q

(∆T ∗II

), (30)

where ∆T ∗II ∈ [0 ; ∆TI + ∆Tw]. Here, the recently men-tioned approximations, denoted by symbols with ∼, are ef-fective and all dependencies can be reformulated in termsof temperature differences ∆T. The approximated en-thalpy and heat transfer rates are

∼H(

∆T ∗II)

:= γ · Cp ·[∆TI −∆T ∗II + ∆Tw

], (31)

∼Q(

∆T ∗II)

:= γ · (αA)m · WM(

∆TI , ∆T ∗II , β). (32)

In contrast to Eqs. (21)–(22), the root of Eq. (30) can be

5

2 2.5 3 3.5 4

40

50

60

70

Ch(

Th2)

[kJ/(kg K)]T h

2[◦

C]

3.56 3.58 3.6

40

50

60

70

Cc(

Tc2)

[kJ/(kg K)]

T c2

[◦C

]tem

pera

ture

exchanger dimension

Th1

Tc1

Th2

Th2

Tc2

Tc2

no heat transfer

max. heat transfer

Figure 2: Mean specific heat range of a supercritical carbon dioxide exchanger;process medium: carbon dioxide with mh = 30 kg/s, ph = 100 bar; coolant: glycosol-water with mc = 41 kg/s, pc = 4 bar

substitute hot side term cold side term

∆TI Th1 − Tw1 Tw2 − Tc1∆T ∗II T ∗h2 − Tw2 Tw1 − T ∗c2∆Tw Tw1 − Tw2 Tw1 − Tw2

Cp mh · θ3 mc · θ4

γ -1 1

(αA)m θ1 θ2

Table 1: Substitutions

calculated directly:

root of∼R = G

(∆TI , ∆Tw, (αA)m , Cp, β

)

:= ∆TI + ∆Tw +2 (αA)m · β ·

[∆TI · (αA)m · β − ξ4

]

ξ21

+(αA)m ·

[2 ∆TI + ∆Tw

]· (β − 1)

ξ1, (33)

where we have used the following abbreviations

ξ1 = (αA)m · (1− β) + 2Cp , (34)

ξ2 = 2 (αA)m ·[

(αA)m ·∆TI − Cp ·∆Tw], (35)

ξ3 = 4C2p · (∆TI + ∆Tw)

+ (αA)m ·[2Cp ·∆Tw − (αA)m ·∆TI

], (36)

ξ4 =√

(ξ2 · β + ξ3) ·∆TI . (37)

To guarantee a real-valued solution inside [0 ; ∆TI + ∆Tw](cf. Eq. (30)) for a given ∆TI , β must be chosen within

the domain (∆TI , β) ∈ L1 ∪ L2 , where

L1 = R× {0} , L2 =(R+ \ {0}

)×B , (38)

B = ]0 ; 1] ∩{β∣∣ ξ2 · β + ξ3 ≥ 0

}

∩{β∣∣ ∆TI · (αA)m · β − ξ4 ≤ 0

}. (39)

Inside the partial domain L1, where β = 0, the mean tem-perature difference is calculated with the arithmetic-meanapproach (cf. Eq. (29)), which is the equivalent domainextension, as can be found in Eq. (15). For most operat-ing conditions, we found that B = ]0 ; 1], and, thus, thechoice of β is not very restrictive. However, to meet theabove specified requirements for all possible operating con-ditions, we have to think of situations where B ( ]0 ; 1],and the arithmetic-mean approach is not favorable. A gen-erally applicable suggestion on how to choose β is:

If ∆TI > 0 and B 6= ∅, then choose the element of{βLM , β

∗1 , β

∗2

}∩B that is the nearest neighbor of

βLM ; otherwise, choose β = 0.5

The included terms are calculated as follows:

βLM =AM

(∆TIs, ∆TIIs

)− LM

(∆TIs, ∆TIIs

)

AM(

∆TIs, ∆TIIs)− GM

(∆TIs, ∆TIIs

) ,

(40)

β∗1/2 =ξ2 ±

√4 ∆TI · ξ3 · (αA)

2m + ξ2

2

2 ∆TI · (αA)2m

. (41)

Subscript s denotes the steady-state, the calculation ofwhich is presented in Section 3.2. For steady-state condi-tions, the favorable βLM yields an exact approximation ofthe log-mean temperature difference. Note that one has

5If B ( ]0 ; 1] is not empty, then B =]0 ; β∗

1/2

], B =

[β∗1/2

; 1],

or B =[β∗1/2

; β∗2/1

].

6

to determine ∆T ∗II and, thus, β for the hot and cold sides,respectively, by substituting the general terms accordingto Table 1. Finally, in terms of Eq. (10), the derived one-step-solvable output equations of the approximate modelare:

g(x, u, θ, t

)(42)

=

G(Th1 − Tw1, Tw1 − Tw2, θ1, mh · θ3, βh

)+ Tw2

Tw1 − G(Tw2 − Tc1, Tw1 − Tw2, θ2, mc · θ4, βc

) .

3.2. Steady State

The steady-state is of particular importance in the pre-sented modeling scheme. It is essential for the determina-tion of Eq. (40) and the derivation of the dynamic stateequations, as presented in Section 3.3. Within this section,the a priori calculation of the steady-state is presented.For a discussion on the presupposed uniqueness of thatsteady-state, the reader is referred to the appendices. Thereference model is introduced first in Section 3.2.1 beforean approximate solution is derived in Section 3.2.2.

3.2.1. Reference Model

To obtain the steady-state outlet temperatures(Th2s, Tc2s) without specifying a simple enthalpy calcu-lation model, the root of the following residual functionshas to be determined

Rs1(T ∗h2s, T

∗c2s

):= Hc

(T ∗c2s

)+Hh

(T ∗h2s

), (43)

Rs2(T ∗h2s, T

∗c2s

):= Hc

(T ∗c2s

)(44)

−Q(Th1 − T ∗c2s, T ∗h2s − Tc1, (kA)m

),

applying a numerical multiple-step procedure. The rootdetermination of (43)–(44) is a mathematical formulationstating the equality of steady heat transfer and enthalpyrates. In contrast to (23), this is a coupled problem. Nev-ertheless, the root of (43)–(44), which is (Th2s, Tc2s), isunique. This is expanded upon in Appendix A.

Further, we have to determine the steady-state values ofthe wall temperatures (Tw1s, Tw2s), obtained as the rootof

Rs3(T ∗w1s, T

∗w2s

)(45)

:= Q(Th1 − Tc2s, Th2s − Tc1, (kA)m

)

−Q(Th1 − T ∗w1s, Th2s − T ∗w2s, (αA)m,h

),

Rs4(T ∗w1s, T

∗w2s

)(46)

:= Q(Th1 − Tc2s, Th2s − Tc1, (kA)m

)

−Q(T ∗w1s − Tc2s, T ∗w2s − Tc1, (αA)m,c

).

Equations (45)–(46) state the balance between wall heat-ing and cooling fluxes and the total heat transfer, calcu-lated with the above determined steady outlet tempera-tures. Again, a unique solution is obtained. For a sketch

of the proof, see Appendix B. The solution is given by

Tw1s = Th1 +(αA)m,c

(αA)m,h + (αA)m,c· (Tc2s − Th1) , (47)

Tw2s = Th2s +(αA)m,c

(αA)m,h + (αA)m,c· (Tc1 − Th2s) . (48)

Thus, if (Th2s, Tc2s) are determined, the steady wall tem-peratures can be calculated directly without a furthermultiple-step root determination.

3.2.2. Approximate Model

Applying the same strategy here as presented in Sec-tion 3.1.2, i.e., fixing fluid properties concerning a pre-vious time step (superscript −), which led to the simpleenthalpy calculation (26), the steady-state calculation be-comes solvable in one step:

Th2s =

Tc1 +[Tc1 − Th1] [mcθ6 − mhθ5]

mhθ5 − mcθ6ξs, mhθ5mcθ6

6= 1

Tc1 (kA)m + Th1mhθ5

(kA)m + mhθ5, mhθ5mcθ6

= 1,

(49)

Tc2s = Tc1 +mhθ5

mcθ6[Th1 − Th2s] , (50)

where

ξs = exp

((kA)mmhθ5

− (kA)mmcθ6

), (51)

θ5 = Ch(T−h2s

), θ6 = Cc

(T−c2s

). (52)

Clearly, the steady outlet temperatures of the approximatemodel given the fixed inputs and parameters are uniqueaccording to (49)–(50). Likewise, this holds true for thesteady-state calculated with Eqs. (47)–(48). The afore-mentioned proof of uniqueness is valid for the approximatemodel as well because the enthalpy calculation model wasnot specified further, and, thus, the simplified enthalpyrate equation of the approximate model (cf. Eq. (31)) isalready included.

Note that the log-mean approximation does not affectthe steady-state calculation if β = βLM is set according toEq. (40). If so, the steady log-mean temperature differ-ence equals the steady weighted-mean temperature differ-ence.

3.3. Dynamic State Equations

To completely specify the model (9)–(10), the right handside of Eq. (9) has to be known, i.e., a dynamic modelis needed to describe the temporal evolution of the twowall temperatures, Tw1(t) and Tw2(t). The steady-statevalues Tw1s and Tw2s were already determined in Section3.2. Tw1(t) and Tw2(t) change as functions of time t dueto driving temperature differences concerning the processand cooling media. However, in an attempt to keep themodel order as low as possible, dynamic states for the

7

Tw2

Tw1

Tw2s

Tw1s

Tw = 0

Tw=

0

V

III

III IV

x(t)x(t)

Figure 3: The state space

temperatures of the hot and cold fluids were discarded.Therefore, a black-box-like approach was chosen.

As the system is stable, Tw1(t) and Tw2(t) tend towardsTw1s and Tw2s (see Fig. 3), which are unique values givenfixed input and parameter vectors (see Section 3.2 andappendices). As an oscillatory-like approach of the steady-state is highly unlikely, it is proposed that the gradientpoints directly to the steady state. This can be describedby

[Tw1(t)

Tw2(t)

]= a ·

[ew1(x,θ,u,t)︷ ︸︸ ︷

Tw1s(θ, u, t)− Tw1(t)Tw2s(θ, u, t)− Tw2(t)︸ ︷︷ ︸

ew2(x,θ,u,t)

], a ∈ R+ . (53)

Despite its black-box character, a physical constraint hasto be met for Eq. (53). The temperature rates, which aredetermined by a common parameter a, will depend on thetotal heat capacity of the wall θ7 and on the heat fluxes,Qh and Qc. An overall balance of the wall results in

θ7 · Tw = −Qh − Qc . (54)

Qh and Qc can be determined according to Eqs. (13)–(14)with the respective outlet temperatures of either the refer-ence or the approximate model. Further, as the parameterθ7 is either known from first principles or from an identi-fication using historical data, Eq. (54) can be utilized todetermine a mean temperature change Tw that has to bedistributed to Tw1 and Tw2 by the proper choice of theparameter a. To this end, we first look at a combinationof Eq. (53) and Eq. (54).

An intuitive way would be to assume that the averagetemperature change of the refined model equals the tem-perature change of the lumped one, i.e.,

Tw = AM(Tw1, Tw2

). (55)

This yields reasonable norm values of x within sectors Iand III, where both wall temperatures either increase or

decrease (cf. Fig. 3), bounded according to ‖x‖ < 2∣∣∣Tw∣∣∣

inside the respective area and with ‖x‖ = 2∣∣∣Tw∣∣∣ on the

borderlines of sectors II and IV. On the contrary, due tothe opposite signs of Tw1 and Tw2, the norm values areunlimited within II and IV if Eq. (55) is forced. To over-come this issue with a physically reasonable behavior, we

postulate ‖x‖ = 2∣∣∣Tw∣∣∣ within sectors II and IV. Finally,

the overall norm design in this black-box model is achievedby setting

a =

2 Twew1+ew2

within sectors I and III ,

2 |Tw|√e2w1+e2w2

within sectors II and IV, and

0 within sector V .

(56)

Note that sector V has been introduced for numerical rea-sons. It is finite but arbitrarily small concerning numericalaccuracy. Further, we suggest setting a lower bound for∣∣∣Tw∣∣∣ within sectors II and IV to guarantee asymptotic sta-

bility. Otherwise, the model could, metaphorically speak-ing, become stuck at the “Tw = 0 graph” in Fig. 3, whichis a curved line in the state space, depicting the set ofsteady-states of the lumped capacity model.6

4. Monitoring

The proposed model-based monitoring scheme will con-sist of a real-time estimation of parameters of the approx-imate model introduced above. A proper parametrizationwill be presented in Section 4.1. The real-time estimationis done in the framework of the Extended Kalman-Filter,summarized in Section 4.2. Then, Section 4.3 shows theapplication of the monitoring scheme in a steady-state aswell as in highly transient operating conditions. In Sec-tion 4.4, we address a specific situation in which informa-tion, provided for the monitoring algorithm, is uncertainor missing.

4.1. Parametrization

The literature is full of empirically derived correlationsbetween the overall convective heat transfer coefficientand the operating conditions expressed with dimensionlessnumbers (e.g., [11]). We want to offer the option to partlyintegrate such dependencies into the monitoring scheme,as we address flexible plants that will often run in non-steady-state operations during which the heat transfer co-efficients might change dynamically. All of the approachescan be reformulated as

Num = c1 · ReE1m · PrE2

m · fα(

Rem, Prm)

+ c2 , (57)

where ci, Ej , and fα denote coefficients, exponents, and afunction depending on the specific approach, respectively,

6Referring to the remarks in Section 3.2, this line must containthe unique steady-state (Tw1s, Tw2s).

8

and Num, Rem, and Prm are the well-known dimensionlessnumbers; more precisely, they are

the overall Nusselt number Num =αm · Lλm

, (58)

the overall Reynolds number Rem =wm · ρm · L

ηm, (59)

the overall Prandtl number Prm =ηm · cpmλm

. (60)

Here, wm and L denote the mean fluid velocity and a ref-erence length, respectively. The included fluid properties,such as density ρm, viscosity ηm, specific heat cpm, andthermal conductivity λm, are typically evaluated based onthe arithmetic mean of the intake and outlet temperatures.

In contrast to the reference model, which accounts forarbitrarily complex approaches in the manner of Eq. (57),we do not intend to presuppose the existence of such ageneral fluid property model within the monitoring schemethat is based on the approximative model. A simpler cor-relation is supposed instead. Discarding fα and the tem-perature dependencies of the fluid properties motivates thefollowing approach:7

(αA)m,h = υh

(mh

1 kg/s

)θh1 ( cph1 J/(kg K)

)θh2+ θh3 , (61)

(αA)m,c = υc

(mc

1 kg/s

)θc1 ( cpc1 J/(kg K)

)θc2+ θc3 , (62)

where we set (cph, cpc) = (θ5, θ6) in the scope of the apriori steady-state calculation and (cph, cpc) = (θ3, θ4)

otherwise. Here, υ =[υh υc

]Tare time-variant param-

eters that will be estimated in the context of monitor-ing, and θhc =

[θh1...3 θc1...3

]Tare time-invariant model

parameters, which should be identified using the histor-ical data of the individual heat exchanger. If there areno proper data, θhc = 0 yields the primary approach

υ =[(αA)m,h (αA)m,c

]T. Remember that slow varying

model parameters are assumed in general. The better thisassumption holds, the better the model will perform. Thatmeans, e.g., if a relevant correlation between (αA)m,h andmh is known, it would be unreasonable to set θh1 = 0.The model-based estimator introduced below would thenperform worse in tracking (αA)m,h during phases of fastvarying mh.

4.2. Joint Estimation

For the joint estimation of model states and theυ-parameters, the well-known Extended Kalman Filter(EKF) scheme is applied. It is referred to as the Joint-EKF approach by the state estimation community. Here,for online monitoring, a real-time estimation of (kA)m is

7Physical dimensions:[υh/c

]=[θh3/c3

]= W/K, [m] = kg/s,

and [cp] = J/(kg K)

accomplished. The EKF is a recursively formulated model-based estimation method, and, thus, it is an eligible onlineestimator. It assumes a stochastic system formulation inthe sense of

xυ(t) ∼ N(fυ

(xυ, u, θ, t

), Rxυ

),xυ(t0) = xυ0 , (63)

y(t) ∼ N(gυ

(xυ, u, θ, t

), Ry

), (64)

where the expression z(t) ∼ N(zm(t) ,Rz

)denotes rep-

resentatively that a vector z is normally distributed withthe mean zm, and z(t)− zm(t) is a continuous white noiseprocess with a spectral density matrix Rz. To satisfy thepostulated demand, the joint estimation requires

xυ =

[xυ

], f

υ

(xυ, u, θ, t

)=

[f(x, u, θυ, t

)

0

],

(65)

Rxυ =

[Rx 00 Rυ

], g

υ

(xυ, u, θ, t

)= g(x, u, θυ, t

),

(66)

θυ =[θ1(υ, θhc) θ2(υ, θhc) θ3 · · · θ7

]T, (67)

where θ1(υ, θhc) and θ2(υ, θhc) are the overall convectionconductances calculated with Eqs. (61) and (62), respec-tively. As seen in Eq. (65), the approximate model

(f, g

)

is part of the model used in the Joint-EKF. Although thespectral density matrices Rx, Rυ, and Ry are well-definedby Eqs. (63)–(64), they are typically unknown by valuefor a specific application. Therefore, they are interpretedas tunable design parameters of the Joint-EKF approach.In a common application and for the presented scenariosbelow, measurements y are not accessible in a continuousmanner but at discrete points in time tk. On that account,the Joint-EKF is implemented with a time-continuous pre-diction step

x−υ (t) = xυ(tk−1) +

t∫

tk−1

fυ

(x−υ , u, θ, τ

)dτ , (68)

P−(t) = P(tk−1) + Rxυ · (t− tk−1)

+

t∫

tk−1

(F(τ)P−(τ) + P−(τ)F(τ)T

)dτ

(69)

within the time interval t ∈ [tk−1; tk] and a time-discretemeasurement update

K = P−(tk)HT(HP−(tk)HT + Ry · (tk − tk−1)

−1)−1

,

(70)

xυ(tk) = x−υ (tk) + K(y(tk)− g

υ

(x−υ , u, θ, tk

)), (71)

P(tk) = P−(tk)−KHP−(tk) . (72)

9

The introduced jacobian matrices

F(t) :=∂

∂xυfυ

(xυ, u, θ, t

)∣∣∣xυ=x−

υ (t)(73)

H :=∂

∂xυgυ

(xυ, u, θ, tk

)∣∣∣xυ=x−

υ (tk)(74)

can be analytically derived for the approximate model,which is a further advantage of this model. The “ˆ” sym-bol is used to distinguish the estimate xυ from the truestate xυ, which is unknown in a real experiment. MatrixP serves as an estimate for the covariance matrix of theerror between estimated and true state. For further detailson the EKF, the reader is referred to [13]. For the experi-ments below, we set biased start conditions xυ(t0 = 0 min)and, further, P(t0 = 0 min) = 1 s ·Rxυ to initialize the al-gorithm.

4.3. Monitoring in Steady and Dynamic Operating Condi-tions

In this section, on the basis of a simulation experiment,the advantage of the suggested monitoring scheme over aconventional (model-free) thermal rating is noted, wherethe latter is more precisely the steady calculation of (kA)maccording to Eq. (1). Furthermore, the influence of the ap-plied enthalpy calculation model is discussed herein. Therelevant time series, which belong to the considered exper-iment, are depicted in Fig. 4.

The experimental setup is as follows: Within the ref-erence model, a real gas model [14, 15] is used to cal-culate specific enthalpies of the carbon dioxide processfluid at a supercritical state. The coolant is a glycosol-water-mixture that is described by a thermally perfectfluid model (dhc(T ) = cpc(T ) dT ; data from [16]). Thepreset temporal variations of (αA)m,h(t) and (αA)m,c(t)are denoted as reference trends within Fig. 4 e. Their com-bination, according to Eq. (4), yields the reference trend(kA)m(t), which is depicted in Fig. 4 f. The thermal rat-ing is stated as successful if an algorithm is capable oftracking those reference trends on the basis of the systeminputs u and measured outlet temperatures y. Here, themeasurements (Fig. 4 c) are the superposition of the ref-erence model’s output g

r(x, u, θ, t) and an artificial, nor-

mally distributed noise with a standard deviation of 0.1 K.The inputs (Fig. 4 a–b) are chosen such that the simu-lated exchanger continuously runs from a steady operatingpoint at t = 0 min to the measurable highest-frequent tran-sient phase at t = 40 min (test frequency: ∆t−1 = 1 Hz).Quite clearly, this is an unrealistic but eligible excitationto demonstrate the functionality of the monitoring scheme.The settings for the monitoring algorithm are:

Rx = 0.1 s ·(Qdesign100 · θ7

)2

· I2 , Ry = 1 s · (0.1 K)2 · I2 ,

Rυ = 0.1 s · (100 W/(K s))2 · I2 , θhc = 0 ,

(75)

40

45

50

55

60

y[◦

C]

c)

−0.5

0

0.5

ε(x,

x υ)

[K]

d)

90

100

110

120

(αA) m

[kW

/K]

e)

0 5 10 15 20 25 30 35 40

50

55

t [min]

(kA) m

[kW

/K]

f)

20

40

60

80

T h1,

T c1

[◦C

]

b)

30

35

40

mh,

mc

[kg/

s]

a)

reference / input / meas. hot side cold sideestimate (real gas model) hot side cold sideestimate (constant cp) hot side cold sidesteady calculation (real gas model)steady calculation (constant cp)

Figure 4: Model-based and model-free monitoring from steady(t = 0 min) to high-frequency (t→ 40 min) operating conditions;a) preset mass flows; b) preset intake temperatures;c) noisy measurements; d) noise canceled deviationε(x, xυ) = g

r(x, u, θ, t)− g

υ(xυ , u, θ, t); e) preset and estimated

overall convection conductances; and f) preset, estimated, andmodel-free calculated overall thermal conductance

10

where Qdesign = 1.6 MW and θ7 = 566.5 kJ/K are firstprinciple parameters of the specific exchanger, and I2 rep-resents the 2× 2 identity matrix.

Without model-based estimation, one could calculatethe overall thermal conductance from the inputs and mea-surements directly if the exchanger’s dynamic is ignored.Typically, the hot-side enthalpy rate would be preferred(cf. Eq. (18)) to replace the total heat transfer rate in Eq.(1) because of the smaller impact of potential measure-ment errors. Such rating points are depicted within Fig.4 f for two cases: i) if the (correct) real gas enthalpy calcu-lation is applied and ii) if calorically perfect gas behavioris assumed (constant cp). Despite an adequate choice forthe constant caloric heat value8, this fluid model yields bi-ased rating points, where the bias depends on the operat-ing point, which is undesirable for the flexible monitoringtask.

Even if the (bias free) real gas model is applied in theapproach without the Joint-EKF, two effects are super-posed: i) the (temporally constant) measurement noiseimpact and ii) the (temporally growing) impact of the ig-nored dynamic, causing the cumulative outliers (see Fig.4 f). In this respect, the Joint-EKF serves as a noise filterthat is insensitive to the exchanger’s dynamic behavior.That is why the filtering performance is quite stable forthe whole experiment, cf. Fig. 4 f.

In the case where the improper constant cp model isapplied within the model-based monitoring scheme, theestimates become biased as well. This is true both forthe individual estimates of (αA)m and the overall ther-mal conductance (kA)m. But in contrast to the model-free approach, the estimation error for (kA)m is sig-nificantly smaller, and the Joint-EKF offers additional,useful information in terms of the so-called innovationy(t)− g

υ(xυ, u, θ, t), which is the deviation between the

measured and estimated outputs. For the sake of clar-ity, we subtract the artificial measurement noise from theinnovation, which leads to the “noise canceled deviation”ε(x, xυ) = g

r(x, u, θ, t)− g

υ(xυ, u, θ, t), i.e., the deviation

between the true and estimated outputs, as depicted inFig. 4 d. Every time the model-based estimation of (kA)mis biased, the monitoring scheme indicates that deficit witha biased (not zero-mean) innovation (or deviation, here).This is a result of the applied fluid model (constant cp),which is improper for describing the reference counterflowprocess at that respective operating point.

Obviously, the monitoring of (kA)m is superior to a con-ventional (unfiltered) thermal rating, cf. Fig. 4 f. This es-timation is the result of the combined, estimated overallconvection conductances θ1(υ, θhc) and θ2(υ, θhc), accord-ing to Eqs. (4), (61), and (62). Although their combinationusing Eq. (4) shows satisfactory results, unfortunately, itdoes not apply to the convection conductances themselves,

8With respect to the caloric dependance, according to Fig.2 and the range of measured outlet temperatures, we sethh(T, p) = 2.3 kJ/(kg K) · T for the constant cp approach.

as can be seen in Fig. 4 e. In numerous simulation studies,we noticed that the observability of these model parame-ters was highly sensitive to the exchanger’s excitation. Forsteady-state operation, they are not separately observableat all, but their combination (kA)m is. Hence, we statethat the suggested monitoring scheme is incapable of of-fering reliable estimates of individual convection conduc-tances. Thus, these estimates should be ignored for themonitoring task.

4.4. Monitoring with a Reduced-Information Setup

As a matter of fact, the assumed knowledge above con-cerning mass flows and intake temperatures as well as theexistence of outlet temperature measurements, both onthe hot and cold sides, provides the best possible condi-tions yielding the most reliable monitoring results. Unfor-tunately, that premise often does not map the situationfound commonly in industrial practice. In this section, weaim to show the impact of unknown coolant flows mc andcanceled measurements of the coolant’s outlet temperatureTc2 on the monitoring results. Therefore, three variants ofthe Joint-EKF, adapted to a specific setup, are considered:

A) assuming certain coolant flow knowledge and coolantoutlet temperature measurements do exist,

B) assuming uncertain coolant flow knowledge andcoolant outlet temperature measurements do exist,and

C) assuming uncertain coolant flow knowledge andcoolant outlet temperature measurements do not ex-ist.

If mc is uncertain, it is estimated by the Joint-EKF aswell. In contrast, if mc is assumed to be known, the Joint-EKF uses a specified value or trajectory (here: mc(t) =41 kg/s ∀t) although the real trajectory might differ duringa faulty situation.

Following the aforementioned Joint-EKF principle—sofar, this is the joint estimation of x and υ, leading to de-sign parameters Rx and Rυ—approaches B and C realizea joint estimation of xυ and mc, leading to design param-eters Rxυ and Rmc , where dmc/dt ∼ N (0, Rmc). Here,Rx and Rυ are chosen as stated in (75); further, we set

Rmc = 0.1 s ·(1 kg/s2

)2.

Again, the reference model is used to offer the “true” ref-erence series, as depicted in Fig. 5. Because we do not in-tend to discuss the impact of the chosen fluid model again,we use the same enthalpy calculation model within the ref-erence and the monitoring model.9 The simulated scenariois an abrupt coolant flow reduction at tf = 2 min (magni-tude: 50 %), which is a critical failure in plant operations.This time, some reasonable heat transfer correlations are

9Carbon dioxide (process fluid): real gas model according to [15];glycosol-water mixture (coolant): thermally perfect model accordingto [16].

11

40

50

60

y[◦

C]

a)

10

20

30

40

mc

[kg/

s]

b)

0 1 2 3 4 5 6 7 8 9 10

40

50

t [min]

(kA) m

[kW

/K]

c)

reference / meas. hot side cold sideestimate A hot side cold sideestimate B hot side cold sideestimate C hot side cold side

t f (fault injection time)

Figure 5: Monitoring with reduced information;a) noisy measurements and respective model outputs; b) true andestimated coolant flows; and c) true and estimated overall thermalconductances;estimate A) without coolant flow adaption, estimate B) with coolantflow adaption, and estimate C) without coolant outlet temperaturefeedback

used within the reference model (cf. Eq. (57)):10

(αA)m,h = 37 W/K

(mh

1 kg/s

)4/5(cpm,h

1 J/(kg K)

)1/3

(76)

·(

ηm,h1 kg/(m s)

)−7/15(λm,h

1 W/(m K)

)2/3

,

(αA)m,c = 2 W/K

(mc

1 kg/s

)4/5cpm,c

1 J/(kg K)(77)

·(

ηm,c1 kg/(m s)

)1/15

.

Note that the mass flows and fluid properties are time-variant due to the time-variant inputs (not shown, withthe exception of mc) and outputs of the reference model.Due to the correlations stated above, the abrupt coolantflow reduction causes a spontaneous breakdown of the ref-erence trend (kA)m in Fig. 5 c at t = tf . In contrast, weuse a simpler approach, exclusive of fluid properties, andwith a different mass flow correlation within the monitor-ing setup, according to Eqs. (61)–(62), with

θhc =[0.6 0 0 W/K 0.6 0 0 W/K

]T. (78)

Despite this biased approach, all estimates (A, B, C) per-formed well for t < tf . None of the monitoring modelsreceives information about the “true” coolant flow trend,as this experiment should show the impact of uncertaininputs and reduced measurement information. As a mat-ter of missing adaptability, estimate A assumes the presetcoolant flow mc = 41 kg/s as a certain measure even af-ter the event. Thus, it is not able to adapt the correctphysical cause (step of mc) in regard to the measured ef-fect (behavior of y). This applies especially to the cold sidemeasurements, which are obviously unaccountable for withmc = 41 kg/s, leading to a high deviation from estimate A.Again, such model-based measures, i.e., the innovation incombination with the detected breakdown of (kA)m, couldserve as meaningful indicators for fault detection. Notethat this work is related to the basic monitoring schemethat provides the parameter time series. Thresholds orsophisticated fault detection approaches are not discussedherein.

If an estimator is free to adapt the uncertain input mc,it is capable of staying on the true reference tracks, as isthe case for estimate B. As a matter of course, it becomesworse if less measurements are considered. Estimate Cprovides proof of that. Note that a steady calculation onthe basis of the heat transfer balance, like the mentionedrating in the previous section, would not be capable ofcompensating for two unknown or incorrect measures atonce (here, mc and Tc2) without a superior estimation

10Mass flows and fluid properties are denoted with their respec-tive SI unit. Fluid properties are calculated by a property databaseprogram [14].

12

technique. To this point, one may interpret that setup C isproper for the monitoring purpose and that the Joint-EKFC only needs a little more time to converge to the correctvalues. Unfortunately, we cannot guarantee this desiredbehavior for setup C. Instead, we observed a higher sensi-tivity to the tuning matrices Rx, Rυ, and Ry and a slowdivergence rate (with a reasonable set of tuning matrices)concerning the estimates of (kA)m and mc for some sim-ulation scenarios, especially if the system excitation wasweak. However, a drastic event, like the one shown, wouldhave been detected by all approaches (A, B, C), at leastin a qualitative manner.

5. Conclusions

For the purpose of monitoring a counterflow heat ex-changer’s thermal performance, we derived an appropri-ate model concerning the initially stated specifications.Primarily, its ability to describe processes from steady tohighly transient operating phases as well as the ease ofdeployment within industrial practice were the main de-mands. The model considers the exchanger’s dynamic be-havior by only two ordinary differential equations. Fur-thermore, the parametrization effort is very low. Moreprecisely, there are three to nine model parameters (de-pending on a priori knowledge) and two arbitrary enthalpycalculation models, (hc(T, p) and hh(T, p)), which are notrestricted in their setup. For the suggested monitoringscheme, one merely has to choose starting values for themodel state and three additional tuning matrices, the ef-fects of which on the monitoring behavior are easy to in-terpret.

To validate the model applied in the monitoring scheme,a reference model was derived. It calculates model outputson the basis of the current model states, parameters, andinputs in an accurate manner if the reference counterflowprocess is valid, as defined in Section 2.1. The respectiveoutput equations are accompanied by high computationalburden due to an integrated root determination. This iswhy we derived approximate output equations that aresolvable in one step. Depending on the preset enthalpycalculation model and the specific root determination al-gorithm, the approximate model is substantially faster(about 50× for our setup), making it an eligible candidatefor real-time applications.

Furthermore, the provision for an arbitrary enthalpy cal-culation model within the derived equations truly differsfrom conventional modeling approaches. In this manner,the impact of an improper fluid model could be shown toresult in biased observations. As a result, the best-suitedfluid model available should be used, which is mostly basedon real gas equations.

The model-based estimation of unmeasurable systemquantities is realized by a Joint-EKF approach, which isthe joint estimation of the model states and parametersof the heat transfer correlation on the basis of the Ex-tended Kalman Filter equations. We were able to show

that the chosen online estimation technique achieves ad-missible ratings of the overall thermal conductance for thefastest transients considered as well as for steady-state op-erating points. Furthermore, a realistic situation of uncer-tain information (model inputs) and a reduced measure-ment setup were addressed. We suggested an adapted es-timation strategy for this setup. As anticipated, the modi-fied version was capable of compensating for one unknowninput information smoothly, as this fact applies for a con-ventional rating based on the steady heat transfer balanceas well. An additional canceling of one outlet’s tempera-ture measurement reduces the reliability of the monitoringresults since the estimator becomes sensitive to the tuningmatrices and prone to (slow) divergence. This motivatesa full instrumentation of heat exchangers.

In the present paper, we treated the heat exchanger asan isolated plant component with mostly known intaketemperatures and flows. The next step is to focus on aplant with integrated heat exchangers, where the inputinformation of the presented monitoring algorithm arisesfrom uncertain measurements and peripheral plant compo-nents. The basic research presented here points out, again,the advantages of a model-based monitoring scheme. Theautomated interpretation of the generated auxiliary mea-sures (innovations and estimates), i.e., a fault-detectionalgorithm, will be a topic of our future research.

Acknowledgment

This work was supported by MAN Energy Solutions SEand the Federal Ministry for Economic Affairs and Energybased on a decision by the German Bundestag as part ofthe ECOFLEX-Turbo project [grant number 03ET7091T].

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Appendix A. The uniqueness of the steady-stateoutlet temperatures

Here, we argue why the root of

Rs1(T ∗h2s, T

∗c2s

):= Hc

(T ∗c2s

)+Hh

(T ∗h2s

),

Rs2(T ∗h2s, T

∗c2s

):= Hc

(T ∗c2s

)

−Q(Th1 − T ∗c2s, T ∗h2s − Tc1, (kA)m

)

is unique. Let (T ∗h2s, T∗c2s) ∈ (A1 ×B1) be a root of the

set of roots (A1 ×B1) of Rs1. One can easily find

∂ T ∗c2s∂ T ∗h2s

< 0 , (T ∗h2s, T∗c2s) ∈ (A1 ×B1)

due to physical reasonable positive heat capacities

∂ hh/c(T, p

)

∂ T> 0 .

Thus, for a fixed T ∗c2s ∈ B1, there is a unique pair ele-ment T ∗h2s ∈ A1. To determine the roots of Rs2 which arecoexisting roots of Rs1 it is sufficient to vary one elementof the given element pairs, e.g., T ∗h2s within A1. One canfind the strict monotonicity

T ∗w2s

T ∗w1s

Tw2s

Tw1s

f4 : A4→ B4

f3 : A3→ B3

Figure 6: The uniqueness of the steady-state

∂Rs2(T ∗h2s, T

∗c2s

)

∂ Th2s∗< 0 , (T ∗h2s, T

∗c2s) ∈ (A1 ×B1) ,

which is sufficient proof of the uniqueness of the steadyoutlet temperatures (Th2s, Tc2s).

Appendix B. The uniqueness of the steady-statewall temperatures

For the derivation of the low-order dynamic model equa-tions, we postulate the uniqueness of a steady state, whichis not a self-evident fact for nonlinear systems. To atleast emphasize the shown approach, we desired to sketchan elaborated proof of that prerequisite, which is verylengthly in its entirety.

In Section 3.2.1, the steady-state wall temperatures(Tw1s, Tw2s) were introduced as the roots of

Rs3(T ∗w1s, T

∗w2s

)

:= Q(Th1 − Tc2s, Th2s − Tc1, (kA)m

)

−Q(Th1 − T ∗w1s, Th2s − T ∗w2s, (αA)m,h

),

Rs4(T ∗w1s, T

∗w2s

)

:= Q(Th1 − Tc2s, Th2s − Tc1, (kA)m

)

−Q(T ∗w1s − Tc2s, T ∗w2s − Tc1, (αA)m,c

).

Note that Th2s and Tc2s are predetermined and unique (seeAppendix A). In Fig. 6, f3 and f4 are the graphs combiningall of the roots of Rs3 and Rs4 in a separate manner:

Rs3(T ∗w1s, T

∗w2s

)= 0 , (T ∗w1s, T

∗w2s) ∈ (A3 ×B3) ,

Rs4(T ∗w1s, T

∗w2s

)= 0 , (T ∗w1s, T

∗w2s) ∈ (A4 ×B4) .

According to the qualitative situation in Fig. 6, there isa unique shared root (Tw1s, Tw2s). Obviously, the drawn

14

graphs fulfill:

i) (Tw1s, Tw2s) ∈ (A3 ×B3) ,

ii) (Tw1s, Tw2s) ∈ (A4 ×B4) ,

iii)d f3

d T ∗w1s

∣∣∣∣Tw1s

=d f4

d T ∗w1s

∣∣∣∣Tw1s

,

iv)d2 f3

d T ∗ 2w1s

∣∣∣∣T∗w1s∈A3

< 0 , and

v)d2 f4

d T ∗ 2w1s

∣∣∣∣T∗w1s∈A4

> 0 .

We can show that expressions iv–v are fulfilled in generaland, further, with

Tw1s = Th1 +(αA)m,c

(αA)m,h + (αA)m,c· (Tc2s − Th1) ,

Tw2s = Th2s +(αA)m,c

(αA)m,h + (αA)m,c· (Tc1 − Th2s) ,

the remaining expressions i–iii hold true. Consequently,(Tw1s, Tw2s) is the unique shared root of Rs3 and Rs4.

15