asme2e v3 - GNS Science · efﬁciency variations of an existing plant. Here it is demons trated how these can be found automatically. A natural extension of diagnostics is the ﬁnding

Proceedings of the ASME 2013 International Mechanical Engi neering Congress andExposition

IMECE2013November 15-21, 2013, San Diego, USA

IMECE2013-63369

ON THE ADDITIVITY OF ENTROPY, THE GEOMETRY OF THERMODYNAMICS, ANDTHE MAXIMIZING OF POWER-PLANT THERMAL EFFICIENCY AT NO COST.

JOINT VENTURE WITH IGNS LTD, NEW-ZEALAND

E. YasniEntropy NZ Ltd Dunedin, New-ZealandIGNS, Ltd. Wellington, New Zealand

Email: [email protected]

ABSTRACT

Predictions of power plant efficiency improvement due tomachinery repair, modification, replacements, or new design canbe made by way of mathematical models (What If). No suchmodel can solve the converse question of finding root-causesofefficiency variations of an existing plant. Here it is demonstratedhow these can be found automatically. A natural extension ofdiagnostics is the finding of values for available set-points whichmaximize thermal efficiency - a decades-old quest. The solutionis demonstrated for a steam-turbine coal fired plant.Methodof approach. The facilitating underlining physics are the Bal-ance Equations and the additivity of entropy. These yield ther-modynamic availability with no further definitions and assump-tion. The automatic maximization of thermal efficiency is tied tothe geometrical formulation of thermodynamics, which is brieflydiscussed.Results. The example of automatic diagnostics wasrecorded on line under routine running of a Combined Cycleplant. The example shows graphics of automated diagnosticsofefficiency drop, and remedy action. In passing a natural way tonormalize thermal efficiency against variable environmentis de-vised. Another example shows a 4% gain in efficiency recordedduring the maximization tests. The analysis of the gains pro-vides a quantitative proof of why a generic efficiency maximiz-ing set of fixed values for set-points (which is the entrenchedconventions) does not exist.Conclusions. Remedies followingreal-time automatic diagnostics can improve plant thermalper-formance in a measurable way mostly at no capital cost. Effi-

ciency maximization can be fully automated by integrating thetuning algorithm into plant’s controls. Even if controllable pa-rameters are set manually 1% improvement is achievable.

NOMENCLATUREA surface area

Aentex the surfaces area across which matter enters and exits thecontrol-volumeB,b exergy, specific exergyck concentration of chemical specieskEN N-dimensional thermodynamic manifoldE3+1,E3 3+time, 3-dimensional Galilean manifold

(e), 0, (e), 0 subscripts, superscripts denoting environmentH,H f ,h enthalpy, enthalpy of formation, specific enthalpy~jk diffusion vector of chemical speciesk~Js entropy flux= −

~QT +∑ µk

T~jk

L f loss factor = exergy loss divided by exergy of fuel~n unit vector normal to the surfaceP thermodynamic pressure.~Q heat fluxQ rate of radiative heat per unit massRi the ith intensive parameterR

n n-tuples of real numbersS,sentropy, specific entropyT absolute temperaturet time

1 Copyright c© 2013 by ASME

V volume~v velocity vector~vs velocity of surface vectorU,u internal energy, specific internal energy~vs velocity vector of a surface of a control-volume{

γαβ

}

Christoffel Symbols

η1, η2 “1st Law”, thermodynamic (“2nd Law”) efficiencyσ rate of entropy production =σQ +σV +σD

σQ, σV , σD rate of entropy production due to heat-transfer, fric-tion, diffusionνk, j the kth coefficient of the jth chemical reaction~~σ stress tensor~~τ shear components of the stress tensorρ densityψ specific potential energyµ chemical potential

Introduction.The current global energy problems comprise the depletion

of both source and sink with the latter brought about mostly bycoal firing power plants. The relaxation of the current energyconstraints does not lie with the cutting of demand. Advancedsocieties are recognized by greater consumption of primaryen-ergy, not lesser. Civilized advanced societies however, are ear-marked by theefficient consumption of energy.

Experience shows that significant efficiency increase due toreal time remedies and tuning of fossil power-plants is typicallyfeasible. This work outlines how the absolute practical limitsto cost-free thermal efficiency improvements can be reached. Itis demonstrated how the introduction of a consistent and uniformformulation of thermodynamics leads vertically to significant im-provement in thermal efficiency of existing power plants, eventhough the said formulation is quite theoretical, calling uponsuch abstract disciplines as Riemannian Geometry. The intendedaudience of this paper are power station managers, engineers, de-signers, and regulators concerned with global warming. Famil-iarity with power plant general layouts, ingredients and terminol-ogy is assumed; no prior knowledge of differential geometryisrequired.

The quest for maximum efficiency can be categorized intotwo sub-quests. The first is concerned with tracking down theroot-causes of a generator unit efficiency drop relative to the im-mediate past. Such drop is sometimes caused by machinery fail-ure, but by far more often by erroneous operation. It is demon-strated how such tracking, normally a formidable task, can beachieved automatically, in real time, and how the better efficiencycan be restored. The algorithm constitutes proper application ofclassical thermodynamics, in particular the non-conservation ofentropy. Examples off real plants are given.

Attaining maximum efficiency of a plant subject to existing

operational constraints is a more challenging problem. This pa-per shows how such maximizing can be done for the restrictedcase, where only set-points can be manipulated (i.e. efficiencyimprovement at no capital cost). It is shown how this maximiza-tion can be tied to the geometrical structures on the thermody-namic manifold.

The examples provide an insight into the failure of the en-trenched recipes to maximize efficiency (like setting steamtem-peratures to maximum, attemperating water flow to minimum)can easily be self defeating. Examples in this paper exhibitval-ues of parameters different from those conventionally prescribed,and yet yielded significantly higher efficiency when appliedtoreal plants - 4×250Mw belonging to Huntly power station, NZ.Even though said units are equipped with state of the art controlsystems they were overridden to manually set the maximizingvalues.

As a side product, this paper offers an alternative, based onsound physics, to the“heat-rate correction curves”. Thesehavetheir origin in the Salisbury method of plant analysis developedin 1958 - 1961 (under the sponsorship of Bailey Meters Co),known as “method of heat-rate deviation”. Even though it ismore often than not inconsistent with reality this method isstillwidely used (see [1]).

1 The fundamental manifolds.

A typical power plant constitutes turbo machines and recip-rocating machines. Both are governed by the model of continu-ous media. conventionally thermodynamic variables are inter-preted as invariants defined on the Galilean 4-space (spannedby x,y,z,t). Here it is proposed to interpret the thermodynamicvariables as coordinates spanning an enveloping thermodynamicN-spaceEN, the Galilean 4-spaceE3+1 embedded in it. TheGalilean 4-space is a Euclidean manifold, whilst the thermody-namic is a Riemannian manifold endowed with the Weinhold [7]metric. If N ≤ 4 then the embedding is replaced by mapping be-tween manifolds. These mappings are a manifestation of the factthat there is a one-to-one correspondence between the motion ofa system (an elementary volume, say) and an associated thermo-dynamic processes given by an ordered pair of its states. Be-cause of the smooth map to the real numbersR

N or R4 (by way

of coordinates) the spaces are in fact differential manifolds. Thecorrespondence between thermodynamic space and the Galilean4-space is given by a set of functions which are solutions of the(differential) Balance Equationscombined with eqns of state,and assumptions like constituent eqns. For an existing plant thesolution is partially given by measurements, and one is mostlyconcerned with the integral version of the balance equations.


2 Balance equations.A complete set of general balance differential equations

covering a control-volumes enveloping thermodynamic pro-cesses which are associated with continuous fluid are spelled byGyarmati [2] of which Sun & Carrington [3] give a good sum-mary. Slattery, [4] gives also the all important integral versions.Here is a recoup using mostly the same notation(s). Amount ofsubstance. The integral balance

∫

V

∂ρk

∂ tdV +

∫

A(~jk +ρk~v) ·~ndA

=∫

V

∂∂ t

[

Mk

r

∑j=1

νk j(ρn j)

]

dV (1)

=∫

V

(

Mk

r

∑j=1

νk jJj

)

dV (2)

where n j is number of Kmoles of chemical reactionj per kgof total mass, (Callen, [5]). Gyarmati [2] introduces the rate of

chemical reactionj J j =∂ (ρn j )

∂ t .Integral balance of internal energy.

∫

V

[

∂∂ t

(

ρ(u+12

v2 +ψ)

)]

dV

+

∫

Aentex

ρ(h+12

v2 +ψ)(~v−~vs) ·~ndA

= −

∫

A~Q·~ndA−

∫

A∑ψk~jk ·~ndA

−

∫

A−Aentex~v·~~σ ·~ndA

−

∫

Aentex

~~τ ·~v·~ndA

+∫

AentexP·~vs ·~ndA (3)

where~jk = −ρ(~v−~vs)+ρk~vk. For thermal efficiency analysis itis common practice to ignore potential energyψ since it is negli-gible compared to enthalpy, as is in most cases the kinetic energy.The term

∫

Aentex~~τ ·~v·~ndA involves shear of the fluid at the termi-

nals; it is also ignored. The way probes are installed in powerplant implies an assumption that properties are uniform andthat~v ·~n = 1 (co-linear) on allAentex. With these approximations inplace, the energy balance equation for a single component fluidwith no chemical reactions and∂∂ t

[

ρ(u+ 12v2 +ψ)

]

= 0 every-where assumes the familiar format

n

∑i=1

mihi +∫

(A−Aentex)i

~v·~~σ ·~ndA

+∫

Ai

~Q·~ndA=n

∑i=1

mihi +Wi

+∫

Ai

~Q·~ndA= 0 (4)

whereWi includes shaft power and work done against a pressurereservoir. Then there is the entropy balance:

ddt

∫

VρsdV=

∫

V

∂ (ρs)∂ t

dV

+∫

Aentexρs(~v−~vs) ·~ndA

= −

∫

A

~QT·~ndA+

∫

A∑ µk

T~jk ·~ndA

+∫

V

[

1T

(

R

∑j=1

K

∑k=1

µkνk jJj +K

∑k=1

~jk ·~Fk

)]

dV

−

∫

V

[

∇(

1T

)

· ~Q

]

dV

+∫

V

[

1T~~τ : ∇~v+∑

k

∇(µk

T

)

·~jk

]

dV (5)

= Js+σC +σQ +σV +σD = Js+σC +σ

whereσC is defined as the entropy of formation of all species ofall chemical reactions;TσC = GT,P = the rate of Gibbs functionof formation. If the reactants and the products of the chemi-cal reaction are unmixed then the reaction is reversible andTσC

takes the form of pure mechanical work (this is significant for theanalysis of the expansion stroke of a Diesel engine). The othervolume integral on the left, designated asσ , is known as the rateof entropy production.σQ ≥ 0, σV ≥ 0, σD ≥ 0.

3 The availability “function”.The literature gives several categories for thermodynamic

availability. For example Sun & Carrington [3] define “non-flow exergy function”, “steady-flow exergy function” and thenby adding mechanical energy ”total exergy function”(s). Theseare formally substituted into the Master Transport Law (a math-ematical identity) and combined with yet another definition“anon-flow exergy flux” to yield non-operational results, i.e.a tau-tology of known physics. Other authors sited in [3] notably Be-jan, follow similar procedures. Here it is proposed to derive thenon-flow and flow availability directly from the balance equa-tions. It is shown that these are in fact all and the same.

3.1 Flow availability.Consider a subprocess of a power plant which “lives” on a

single continuum. Theordered pairwhich is the Clausius notion


of a process, are the states onAent andAex respectively. Thethermodynamic availability is the maximum magnitude of workinteraction of said continuum. It is delivered by a reversible pro-cess which may include interactions with reservoirs subject toclosure of entropy. That is, the original flow is replaced by areversible flow which is compensated by a generalized Carnotheat-pump (or engine) to maintain theAentexstates. General inthe sense that it can transfer not only heat but also mass, anddis-place volume; The forerunner for this idea is Keenan [6]. Now,the kinetic energy differences at theAentexof this reversible sub-stitute cannot be ignored; furthermore some of it is drawn todrivethe Carnot heat-pump. We call this combination of kinetic en-ergy differences and Carnot heat-pump Carnot Engine. The ge-ometrical setting here is one of a product manifoldEN ×EN(e)((e) denotes environment) endowed with a mapg : EN 7→ EN(e)such thatS+S(e) = const, M +M(e) = const, V +V(e) = const;g is physically manifested by the Carnot engine. In additionEN 7→ E3+1 , EN(e) 7→ E3+1(e), see section1.

Next, one extends the physical boundaries such that thereversible system includes the primary flow plus Carnot heatpump which interact with an environment constituting reser-voirs atT(e),P(e),µk(e). They share a common boundary where~Q ·~n = −~Q(e) ·~n(e). By definition the thermodynamic propertiesof the environment are uniform and (hence) it generates no work,

and all surface integrals of eqn (3) overA(e)entexvanish. Hence en-

ergy balance eqn around the environment reduces to

ddt

∫

V(e)

(ρu)(e)dV

=∫

V(e)

∂ (ρu)(e)

∂ tdV =

∫

V(e)

−∇ · ~Q(e)dV (6)

By this setting and the balance of mass and the conservation ofentropy for a reversible process it is sufficient to prove

ddt

∫

Vρ

(

12~v2 +ψ

)

dV

+∫

V

(

ρQm+∑~Fk ·~jk)

dV

= −

∫

V

∂ (ρu)

∂ tdV +

∫

V

[

T(e)∂ (ρs)

∂ t

+ ∑µk(e)∂ (ρck)

∂ t

]

dV

+∫

Aentex[−(ρh)+T(e)(ρs)

+ ∑µk(e)(ρck)](~v−~vs) ·~ndA (7)

Note,∫

Aentexρs(~v−~vs) ·~ndA 6= 0 because∫

A

~Q(e)T(e)

·~ndA 6= 0 is en-

tering the Primary System reversibly (via a Carnot heat pump).The first integral on the lhs of (7) is the time variation of differ-ence in kinetic energy. The expression in the square brackets onthe rhs of (7) is the flow-availability−ρb which is a “definition”according Sun & Carrington [3] and Bejan). The integrals on therhs are the exergy change. Finally,ρ,h,s,ck on the one hand andT(e),P(e),µk(e) on the other, are defined on different manifoldshence the availability is not strictly a function.

Now move the reversible work due chemical reactionddt

∫

V ∑ µk(e)(ρck)dV to the lhs of (7), and subtract the actual en-ergy balance eqn (3) form (7). The thermodynamic states at theboundaries (in 4-space) are the same for both processes, andonthe common surface~Q = ~Q(e), T = T(e), µk = µk(e). Hence

[

ddt

∫

Vρ

(

12~v2 +ψ

)

dV

]

rev

−

[

ddt

∫

Vρ

(

12~v2 +ψ

)

dV

]

irr

−

∫


+

[

∫

Aentex(P−P(e))~vs ·~ndA

]

+

[

∫

V

(

ρQm+∑~Fk ·~jk)

dV

]

rev

−

[

∫

V

(

ρQm+∑~Fk ·~jk)

dV

]

irr

+ddt

∫

V∑(µk−µk(e))(ρck)dV

= T(e)(σQ +σV +σD = T(e)∆S (8)

If there is no diffusion and radiative heat interaction, then (8)reduces to

[

ddt

∫

Vρ

(

12~v2 +ψ

)

dV

]

rev

−

∫

A−Aentex~v ·~~σ ·~ndA

+

[

∫

Aentex(P−P(e))~vs ·~ndA

]

+ddt

∫

V∑(µk−µk(e))(ρck)dV

= T(e)σ = T(e)∆S= L (9)

Clearly the lhs of (9) is by definition theLost WorkWI , and the rhsis theExergy Loss. Eqns (9,8) are the celebrated Gouy-Stodolatheorem.


3.2 Non-Flow availabilityBy the so called “non-flow availability” for “isolated sys-

tems” one means that(~v−~vs) ·~n = 0 on every point on the sur-face, that is there is no Aent or Aex. If so, then the PrimarySystem and sink energy balances reduce to simply

ddt

∫

Vρ

(

12~v2 +ψ

)

dV +∫

A(P~v) ·~ndA

+

∫

V

(

ρQm+∑~Fk ·~jk)

dV

= −ddt

[

∫

VρudV

]

dV−

∫

V∇ · ~QdV

= −ddt

[

∫

VρudV

]

dV +∫

V(e)

∇ · ~Q(e)dV

= −ddt

[

∫

VρudV

]

+ddt

[

∫

V(e)

ρudV

]

−dUdt

−dU(e)

dt(10)

where∫

A(P~v) ·~ndA is recognized as the rate of reversible shaft-work Wrev plus work against a pressure reservoir.

−dWrev

dt=

dUdt

+dU(e)

dt= (T −T(e))

dSdt

+ (−P+P(e))dVdt

+∑k

(µk−µk(e))dMk

dt

= TdSdt

−T(e)dSdt

−PdVdt

+P(e)dVdt

+ ∑k

µkdMk

dt−µk(e)

dMk

dt

=dUdt

−T(e)dSdt

+P(e)dVdt

−∑k

µk(e)dMk

dt

=ddt

∫

VρudV−T(e)

ddt

∫

VρsdV+P(e)

ddt

∫

VdV

− ∑k

µk(e)ddt

∫

VρckdV (11)

Note, the non-flow availability function is the lhs of the MasterBalance Eqn of sum of the energy of the sink and Primary Sys-tem, it covers both kinetic energy and work done against a pres-sure reservoir. In contrast the flow-availability does not includework done against a pressure reservoir. Eqn (11) always holds, itdoes not matter at all if the system(s) is ”isolated” or open;how-ever for a continuous system theT,P,µk have a meaning dictatedby the mean value theorem for integrals.

4 Efficiencies.The main concern of thermal-performance engineers is with

the fuel cost. They define various efficiencies in order to mon-itor the performance of different types of machinery. Thesead-hoc efficiencies are not yielded by the physics in any uniquemanner, hence are arbitrary and in general inconsistent witheach other. The efficiency parameter of interest to managersisthe (generator-)Unit-Heat-Rate or Specific Fuel Consumption;at given load unit heat-rate equals the fuel charges up to a fac-tor. It is desirable therefore, that all other efficiencies definedthroughout the plant are consistent with the heat-rate as well asamongst themselves. The Second-Law (or thermodynamic) ef-ficiency loosely defined as “Exergy picked up” divided by “Ex-ergy given up” conforms with these requirements. To make thisdefinition precise define an ensemble to be a control volume en-compassingN interacting subprocesses each of which can be un-ambiguously identified by its boundaries. Consider an ensem-ble which obeys 1-dimensional fluid dynamics model. It can bediagrammed as multi-terminal nodes, called planar graph, withedgesAi , i = 1...K. All such models are constructed by com-bining in one way or another the balance equations with con-stituent equations resulting in second order PDEs in the vector-components of the velocity field; the best known are the Navier-Stokes eqns. The integral curves of velocity fields are stream-lines. Even though different models yield different stream-linesthey have the samemeasuredendpoints regardless of the modelused. Now, by eqn (7)b=−(ρh)+T(e)(ρs)+∑ µk(e)(ρck). The2nd Law efficiency of an ensembleη2 is defined as

η2 =

[

=I

∑i

∫

Aientex

ρb(~v−~vs) ·~ndA

+I

∑i

∫

Ai−Aientex~v·~~σ ·~ndA

]

×

[

J

∑j−

∫

A j−A j entexρb(~v−~vs) ·~ndA

−J

∑j

∫

A j−A j entex~v·~~σ ·~ndA

]−1

(12)

such that∆Bi > 0 ∀i, ∆B j < 0 ∀ j. For example, thei index fora feed-water-heater corresponds to the feed-water,I = 1. The jindex corresponds to the bled steam and the drain from the up-stream heater,J = 2. For a turbine stageI = 1, J = 1 but∆BI = 0.By the interpretation of entropy generation as lost workη2 rep-resent the ratio between actual to potential (reversible) work or2 potential works. One important property of thermodynamicefficiency is that it can easily be normalized against the sink(ocean, atmosphere) temperature. Indeed should the plant re-main unchanged, that isσD,σV ,σQ all remain unchanged as the


sink varies fromT0 to T ′0, then atT ′

0

L′ = T ′0∆S= L

T ′0

T0(13)

by eqn (12)

η ′2 =

η2T0

[T ′0 +(T0−T ′

0)η2](14)

Formulas (14) are valid for both subplants and the entire plant.In section 6.1 it is shown how these 2 simple formulas rid oneselfof of “corrections curves” or likewise rules of thumb.

5 The utility of the additivity of entropy.It is important to emphasize that thermodynamic efficiencies

or their trend do not in general correspond to conventional effi-ciencies if such are defined, with the notable exception of theentire plant. That is, if the dimensionless Thermal Efficiency ofthe entire generator-unitη1(t1) > η1(t2) then the thermodynamicefficiencyη2(t1) > η2(t2). To see why consider the plant as con-trol volume comprisingN subinteractions, an ensemble boundedby K thermodynamic boundaries. LetS=

∫

V ρsdV be the en-tropy of the ensemble at any given moment (“snapshot”) of time.By additivity of entropy and eqns (5)

S=N

∑i=1

Si ⇒ S=ddt

∫

VρsdV=

N

∑i=1

Si

=N

∑i=1

∫

Vi

∂ (ρs)∂ t

+∫

Ai

ρs(~v−~vs) ·~ndA

=N

∑i=1

∫

Ai

~Js ·~ndA+σi (15)

The Lost Work (Irreversibility) associated with the ensemble is

WI = L = T0

N

∑i=1

σi

= T0

[

∫

V

∂ (ρs)∂ t

+∫

Aρs(~v−~vs) ·~ndA

−

∫

A~Js ·~ndA

]

= T0σ (16)

Eqn (16) states that one can either sum up the individual irre-versibilities of the subinteractions or carry out an entropy bal-ance about the entire ensemble to obtain the same result. The

time honored name of eqn (16) is The 4th Lost Work Theorem.Now, T0σ = ∑N

i=1T0σi is the formal statement that the individualdissipationT0σi equals exactly the heat flux into the sink due toirreversibilities in processesi, which may be due to friction, heattransfer, combustion etc. Ifσ(t2) > σ(t1) thenη2(t2) < η2(t1),and furthermore sinceT0(σ(t2) − σ(t1)) = ∆(T0σi) = ∆Qsink,then extra fuel has to be supplied to maintain the load; henceη1(t2) < η1(t1). A quantitative comparison ofη1,η2 is given insection 6.1 below.

6 Scope of diagnosticsThe finding of root-causes of efficiency variations of an ex-

isting plant is a labour intensive task given the multitude of plantcomponents. The very many possible combinations renders amodel driven trial and error procedure prohibitive. To facilitatethe additivity of entropy in the sense of eqn (16) one constructs atree-graph of a plant; since every branch, i.e level of aggregation,contains all branches below it, and branches and finally leavesare disjoint, and since the loss of a higher level of aggregationequals the sum of losses of its branches, one can automatically“drill down” to track the root-cause of loss excursion of thehigh-est level of aggregation (entire plant) to the lowest. For trendplots the “Drill Down” is identified by way of correlations forwhich there are ample numerical methods. It remains to be seenhow plant heat-rate relates to plant loss; on this in section(6.1)below.

6.1 Long term diagnostics. Normalizing against vari-able environment

Now, linear combination of Exergy Audits is also an ExergyAudit. Applying the “Drill-Down” to trends of deviation frombenchmarks (∆) depends on the availability of realistic bench-marking data. Commissioning Acceptance-Tests yield such data;otherwise periodic performance tests can be called upon. Thesemay prove expensive for steam-turbine or combined cycle plants,but are routine in Diesel plants. Hence eqn (14) normalizing∆L f (t) is quite beneficial to marine Diesels which operate un-der variable load, speed and rapidly varyingT0.

Considerη1 the conventional “1st Law” thermal efficiencydefined as shaft-power divided by the flow fuel energy. It is theultimate fuel-cost criterion of a power plant; however it varieswith environmental conditions over which operators have nocon-trol. Therefore managers are interested only in normalizedcost,that is the trend ofη1 deviation which factors out the variableenvironment. Here it is proposed how to do this.

η1 =

∫


H f


=H f +

∫

Aentexρh(~v−~vs) ·~ndA− Q0

H f

where Aent is the throughput of fuel-air, Aex the flue-gas exhaustand leaks. DeriveQ0 from the entropy balance eqn (5) around theentire unit including the condenser (whereQ0 = −

∫

A~Q0 ·~ndA);

somewhat laborious algebraic manipulations yield

η1 =−

∫

Aentexρb(~v−~vs) ·~ndA−T0σH f

(17)

whereσ = σQ + σV + σD. Eqn (17) says that the “1st Law”thermal efficiencyη1 depends on the availability change, the ex-ergy loss and the enthalpy of formation. Compare to eqn (12)for η2 of the entire unit. It is essentially the same as eqn (17)except thatBf is used rather thanH f ; hence the trends ofη1,η2

are the same. Note, the critical variable here isQ0; thereforethe same considerations do not hold for “1st Law” efficienciesof subinteractions. Finally, due to the correlationη1, η2 one canrid oneself of the ad-hoc “correction curves” for variable weatherconditions, by normalizing the widely used “1st Law” efficiencyη1 or unit-heat-rate against the sink temperature using single for-mula like eqn (14) p. 6. Long term disgnostics are useful for theassessment of plant maintenance and modifications. See ref [8].

6.2 ExamplesThe following example demonstrates how eqn (16) applied

facilitates the automatic tracking of losses down the planttreehierarchy both in real time and for historical data. Rather thenormalized loss factorL f = T0σ

Bf uelis used.

Data modelling. The raw data input to the installed Exergy Au-dit(s) is processed by a moving-time-average which filters thedata and averages it over a frame of 30 readings, made up of10 new readings and 20 old; a reading being sampled every 30seconds. (I.e output is generated every 5 minutes; this is ad-justable). Graphic presentation of data. In addition to trendplots, a most common presentation of plant losses is the Sankeydiagram. this is a special kind of a flow diagram where the widthof the arrow is proportional to the flow quantity. It is usefulcomparing say, the flows of energy to flows of exergy as is wellknown. It is less convenient for the purpose of comparing onestate of the plant to another as there may be many variations ondifferent scales of magnitude, see ref [8]. For this purposeit isintroduced the Pseudo Sankey Diagram where rather the mag-nitude of loss flows is proportional to the length of a bar; thedirection is indicated by colour increase red, whilst decrease ofloss green. Because Exergy Audits are closed under addition,the Pseudo-Sankey diagram represents in fact the redistributionof losses from plant state to another.

Example 1 (real-time diagnostics). This example wasrecorded on line under routine running of the Kwinana CCGTcogeneration plant. The plant is located 40 kilometers south ofPerth, Western Australia. It supplies steam and electricalpowerto a nearby oil refinery as well as electricity to the state ownedutility. The plant was commissioned in December 1996, ratedat140 MW of electricity. Kwinana is primarily fuelled by naturalgas from gas fields in the northwest of Western Australia. It is a3-shaft plant, constituting two gas turbines each driving its ownalternator, and exhausting into HRSG. The sets are marked as1A and 1B. Each HRSG utilizes duct firing burning refinery fuelgas. The steam from the HRSGs is used to run a steam-turbinedriving its own alternator. Steam bled off that turbine is suppliedto the refinery.

Fig [1] is the the code output showing an automated diag-nostics of efficiency drops followed by graphic illustration, andthen followed by graphic illustration of the remedy action.Theshortfall could not be timely diagnosed by any other methods.The time of the efficiency excursion of the plant (unit 1) and itscost are displayed. The cost is proportional to the heat-rate in-crease. The shortfall is associated with the duct firing of eachHRSGs The recommendation is that controllable parameter i.erate of duct firing be scaled back; in this case it was followedup,with the pre-alarm efficiency partially restored. Fig [2] top showsthe redistribution of losses caused by the extra duct firing of0.190 Kg/sec of gas at each HRSG (shaded; tags are 01431.FW,02421.FW respectively). The values are averages over the peri-ods of 45 minutes before and after the alarm. The shaded linetagged 1900.1L shows an increase in heat-rate of 302 Kj/kgwith an equivalent increase of unit loss-factor (tagged 1900.LF)of 10.950 0/00. Clearly the biggest loss occurs in HRSG 1Aat 11.281 which is greater than the gas turbine 1A gain at 9.139.The biggest contributor to HRSG 1A loss is the duct burner com-bustion process 1A at 9.095. A similar picture takes place for set1B. Bottom, the redistribution after remedy action was taken atabout 2 hours after the alarm at 18:00. The shaded line tagged1900.1L shows a reduction in heat-rate of 83 Kj/kg with anequivalent reduction of unit loss-factor (tagged 1900.LF)of 2.880/00. Here the redistribution is in essence a mirror image with thebiggest gain occurs in HRSG 1B (say) at 9.548 which is greaterthan the gas turbine 1B loss at 8.859. The biggest contributor toHRSG 1B gain is duct burner combustion process 1B at 6.767.Here the 1A behaves differently than 1B which actually ends upin the red.

Example 2 (long term diagnostics). This example was anoff-line set-up of the Pseudo Sankey Diagrams calculated fromhistorical data recorded on-line. The purpose of the exercise wasto assess the efficiency related effect(s) of major overhauls (out-ages) which took place between April 7 2013 and May 28 2013at the Kwinana CoGen plant. The trend plots are by and large in-effective for this purpose, hence only the Pseudo-Sankey presen-


tation is shown. As indicated in fig [3], input and output dataofpost-outage were averaged over the period of June 1 to June 13tocompare to pre-outage data averaged over March 20 to March 30.Much of the measured data of the pre-outage period was filteredaway, hence the longer time interval. The net result of the re-pairs is a small increase in heat-rate of 24 Kj/Kg with equivalentloss-factor of the entire plant increase of 5.539 0/00 shownat thetop, tags 1900.1L and 1900.LF respectively). The increase is thedifference between the loss decrease of the gas turbines tagged2700.LF, and the increase in the steam plant tagged 2800.LF.Themain loss decrease is sue to gas turbine 1B which in turn boilsdown to∆L fexpander=−3.013, ∆L fcombustor=−4.188 which arepartially effaced by∆L fcompressor+ f ilter = 3.90. Station’s plantengineer confirmed overhauling GT 1B rotor. Loss increase andpower consumption increase of the compressor is associatedwithincreased air flow. The root-cause of the unit increased (heattransfer) losses is the deaerator, caused in turn, by the 16.279%increase in the “process-steam to da” control-valve strokeresult-ing in extra steam flow of 0.165 Kg/sec.


Figure 1. TOP: ALARM DISPLAY, MIDDLE: ALARM DISPLAY IN TRENDS, BOTTOM: REMEDY DISPLAY IN TRENDS


Figure 2. TOP: ALARM LOSS REDISTRIBUTION, BOTTOM: REMEDY LOSS REDISTRIBUTION


Figure 3. REDISTRIBUTION OF LOSSES DUE TO THE KWINANA PLANT 2013 OVERHAULS.


Summary A number of more examples are given in ref [10];obviously the method is generic: it is applicable to plants likecombined cycles, co-generators, clusters of generators, Rankineregenerative plants, and Diesel plants. Since it all depends on thenon-conservation and additivity of entropy, there is no substituteto the invoking of the Second Law in order to obtain such di-agnostics. By a conservative estimate the associated annual fuelcosts run well in six digit figures; these could have been, or weresaved by cheap or costless remedies. A 0.5% improvement inCO2 emission in the industrialized countries will come a longway towards the Kyoto targets, and with an obvious financialbonus at that. The methodology can fail only if field probes fail,which is not altogether impossible, and the much more likelyfailure of human intervention to remedy the root-cause of sig-naled efficiency excursion. Finally by examples (1,2) it appearsthat savings due to tuning are by scales of magnitude greaterthanthose introduced by maintenance.

7 On the fundamentals of maximizing thermal effi-ciency.The quest for optimal values of controllable parameters is

paramount. Two of the most sacred procedures dear to plant en-gineers is to drive the steam pressure to its lowest possiblevalueand temperature to its highest. Engineers attribute cost todevia-tions of as low as 0.1oC to below the standard value of 540.0OC,however, there is no shortage of occurrences where hefty varia-tion of these temperature had unexpected or no impact on the unitheat-rate. Such events tend to bring about some red faces. Theyshould not. It is true that due to lowerσQ of the boiler, a stateof the plant of higher steam temperature is more efficient, pro-vided that the only other different parameter is fuel mass-flow-rate. However, it is in general not possible to have a processwitha sole effect of raising the steam temperatures. To predict allside effects, one must employ a detailed mathematical modelofthe plant with accuracy of up to the third significant digit. Suchmodels do not exist at the present time; even if one comes byoperators are in no position to interrogate models. Hence theuse of the easy but wrong Salisbury’s like “heat-rate deviation”method. One alternative is the resetting of controllable parame-ters described in section 5; it does yield better efficiency but notnecessarily the best.

Yasni & Carrington [1], suggested that under the sliding-pressure mode of operation there exists an optimal steam-drumpressure such that the sum of evaporator and throttling lossesare minimized and consequently the efficiency of the entiregenerator-unit is maximized. In spite of persisting efforts andtries the present author never managed to identify such pressure;at best a break-even pressure was achieved at less than 50% loadof a 660Mw power plant. The reason is that this too, cannot bedone in isolation; there are side effects (pointed also in [1]). Infact the thermal efficiency of a generator-unit is maximizedby a

combinationof values of controllable parameters which dependson the plant’s state.

7.1 Elements of theory and methodology leading tomaximizing efficiency

The problem of finding maximizing values of controllableparameters cannot be resolved by an Exergy-Audit alone. Onemust devise a method which predicts the impact of loss variationof one component, on the rest of the plant without any ad-hoc as-sumptions. To that end consider the most basic definition of lossi.e. the difference between maximum power output and actualone. Since the latter is measured this evaluation is reducedtosimulating of an equivalent reversible plant, a much easiertaskthan to simulate the real thing. To that end 2 new independentfundamental eqns are identified, given here without proof. Thefirst is the Reversible Energy Conservation (REC) eqn:

∂∂ t

(

ρ12~v2 +ψ

)

+∇ ·

(

ρ12~v2 +ψ

)

+∂

(

∑r jk=1 ρgk

)

∂ t−ρQ

=

(

∑µk∂ρck

∂ t+∇ ·ρ(h−Ts)~v

− ∇ ·∑µk~jk)

(18)

Note the similarity with eqn (7) p. 4. The second is the equationsof a geodesic curve over the thermodynamic manifold referred toin section (1):

d2Ri

dt2+

{

ijk

} dRj

dtdRk

dt= 0 (19)

whereR1 = T, R2 = P, R3 = ρ(1),R4 = ρ(2)...RN+2 = ρ(N),at boundary valuesRi(t = 0) = Ri

0, Ri(t = t2) = Ri2, ρ2 =

ρ(T,T(P)) Here the Christoffel Symbols are in the Weinhold [9]metric. Eqns (18, 19) are reconciled by considering the eqnsofthe geodesic vector field which generates the geodesic curve

∂ Rk

∂Rj Rj +

{

ki j

}

RiRj = 0;dRk

dt= Rk (20)

and recalling that the fundamental manifolds map into each other.Hence one maps eqn (20) onto the Galilean manifold to obtainthe eqns of a geodesic vector field over said manifold:

∂vγ

∂qβ vβ +{

γαβ

}

vαvβ = 0 (21)


here the Christoffel Symbols are in theembeddedWeinhold met-ric; the solution to both sets of PDEs (21) and (18) are the veloc-ity vector field which generates the streamlines of the reversiblecontinuum.

Now the optimizer is a computer implemented method ofcontrolling any energy conversion plant, in particular powerplants of all kinds, having a plurality of measured parameters(which may include temperatures, pressures, partial pressures,mole numbers, as well as flows, in and out electrical and/or me-chanical powers, position of valves and other actuators), thatmaximizes the plants’ thermal efficiency with respect to a subsetof plant’s measured parameters called Variables. As the namesuggest, these can be manipulated, subject to operational struc-tural financial and environmental constraints; this constrainedmaximization is called optimizing. said computer can be eitherintegral part of the plant’s control system (DCS or other) inthesense that it both reads measured parameter from the DCS’s DataAcquisition System (DAS) and writes the Variables maximizingvalues into the DSC as set-points (closed loop optimization), orjust reads measured parameter from the DAS whilst human oper-ators apply maximizing value of the Variables as set-pointsor bymodifying the plant itself (open loop optimization), said methodconstitutes the following steps:

1. Determine set of all relevant measured thermodynamicproperties throughout the plant (e.g. temperatures, pres-sures, partial pressures, mole number, liquid and gas flowsand electrical input and output values position of valves andother actuators), and its Variables subset (that is, those mea-sured parameters which can be independently manipulated),by reading them via an interface with the DAS, or manualinput.

2. Derive from the measured parameters in accordance to step(1) and corresponding thermodynamic properties (e.g spe-cific volume enthalpy and entropy), the state of the plantwhich constitutes all relevant thermodynamic propertiesthroughout the plant, as well as energy mass and entropyflows, temperatures, pressures, partial pressures, mole num-bers, liquid and gas flows and electrical input and outputvalues, real velocity vector fields, by way of relevant bal-ance equations and particular expressions.

3. From the plant state determined in step (2) partition the plantinto a finite number of real, irreversible physical continuumsin the context of continuum-mechanics, which may or maynot deliver useful work, and which correspond to disconti-nuities of measured and derived parameters making up thestate of the plant as established in step (2), and satisfyingconservation-of-mass condition(s).

4. construct an isometric (in the thermodynamic metric) mapinthe context of differential geometry, of each real continuumas established in step (3) from the thermodynamic manifoldwhich is spanned by thermodynamic coordinates (for exam-

ple pressure temperature and chemical potentials) and time,to a region of the Galilean manifold spanned by the spatialand time coordinates (for example Cartesian x,y,z,t).

5. Construct the plant model using the partition into physicalcontinuums determined in accordance to step (3) and in ac-cordance of the physical real arrangement of the plant’s ac-tual hardware , as interfacing (i.e incedenting) physical con-tinuums, exhibited as a graph in the context of graph the-ory which can be reduced to a planar graph, wherein eachboundary plays the role of an edge and each continuum therole of a node.

6. Convert each partitioned real, irreversible physical contin-uum into a (virtual) corresponding reversible continuum orReversible Masking of the real continuum, subject to theconstraint(s) that the real continuum and reversible con-tinuum assume the same boundary values of thermody-namic properties in accordance to all previous steps andthat their derivatives are continuous and equal at the bound-aries, as well as mass-flow-rate, such that the partitioned ir-reversible real continuums which are governed by a systemof conventional balance equations of section (2) and con-stituent (phenomenological) equations, are converted intopartitioned Reversible Maskings substitute which are gov-erned by a system of (partial) differential equations exclud-ing any constituent equations, but including either the equa-tion of Thermodynamic Geodesic Field (TGF) in the ther-modynamic metric, or a direct Reversible Energy Conser-vation (REC), uniquely describing the reversible continuum(called a Mathematical Model of the reversible continuum).Specifically these are

(a) Euler’s balance of momentum equation; and(b) the conservation of mass equation ; and(c) the reversible conservation of energy (REC) equation;

and(d) the thermodynamic eqns of state; or(e) the thermodynamic geodesic field (TGF) equations,(f) and a module to carry out the numerical solution of

the simultaneous equations (a) to (e) subject to revisedboundary conditions based at least partly on CentralSchemes methods, or Method of Lines, or reduced ver-sions of these thereof.

7. construct an equivalent reversible (virtual) plant by map-ping the partitioned Reversible Masking substitute equationsfrom step (6) into the plant model constructed in step (5),such that the graph of step (5) is maintained, that is, mapthe partition of the real plant of irreversible real continuums,into a partition of Reversible Maskings substitute equations,with the same incidence matrix in the context of graph the-ory.

8. Solve the mapped equations from step (7) for the currentplant state in terms of velocity (vector) fields across the re-


versible continuums and store the values of the solutionswithin the spaces defined by the reversible continuums.

9. Construct the objective function (called Loss) to be mini-mized by a surface integral of kinetic energy obtained fromthe velocity field derived according to step (8) over theboundary of each Reversible Masking, inputing to the (nu-merical) integration the velocity field, density field(s) of(5)values at the boundary, outputing the difference in kineticenergy between boundaries where matter leaves and entersthe Reversible Masking, subtracting from the sum of all ki-netic energy increment the actual work delivered by the realplant.

Summary. The objective function is simply the difference be-tween the maximum work that can be delivered by a reversibleplant and the actual work. It is by definition the exergy loss,yetthe environment is not invoked here rather eqn (18) or (20) isused. If the electrical load is predetermined as is common thenthe objective function reduces to the sum of all kinetic energy dif-ferences of all Reversible Maskings defined in step (6), i.e.oneis minimizing the maximum work. Intuitively speaking, by mini-mizing the maximum work one is ridding oneself of unnecessarypowerful but costly potential. Finaly the most notable advantagein term of numerics over conventional simulators is the solvingof Euler’s eqns for which there off-the shelf numerical solutions,over Navier-Stokes for which there are none.

8 Examples of maximizing thermal efficiency on line.Scope.The following 2 examples generated by a prototype

optimizer demonstrate the finding of the maximum efficiencywith respect to main steam pressure and temperature, reheattem-perature, all 4 attemperating flows, combustion air temperature,flue-gas exhaust temperature, and the burners tilting angle, sub-ject to plant’s limitations. Other controllable parameters are keptfixed under the present prototype. The illustrations comprise out-puts of the Exergy-Audit-Optimizer supporting graphics, withpainted-over markers.Set-up.The host plants are Hunlty units whose layout and char-acteristics are given in [8],[1]. The scene however is not one ofa formal heat-rate test, but rather of normal operation, that is, themeasured values are given on-line by commercial probes. Thisis less than a meticulous experimentalist would require, but thenagain, optimization in real-time cannot wait for formal test preci-sion probes. Further, managers decision-making is based ontheorganic instrumentation; moreover, appropriate data-modellingeliminates shortfalls of the cheaper instruments. The conditionsfor the tests were for the load to be constant subject to control-system fluctuations only, the firing to be coal only, and no distur-bances (like soot-blowing). Tests ran typically for several hoursallowing for operator action-time, stabilization, and therun-timeof the code. No other restrictions were needed. Data modelling

is described in section (6.2) p. 6.2

Example 3. On May 3 2012 the recommendation listed in fig(4) below were issued, to minimize the heat-rate of Huntly Unit2. The significant ones were to reduce main steam temperature,to increase main steam pressure, to increase flue gas shtr inlet byincreasing the burner tilt angle, to increase reheat steam temper-ature, and increase air heater B exhaust temperature. All but the2 latter recommendations were heeded; since this was done man-ually the actual values achieved where somewhat different thanthe recommanded. Fig [5] top is the trends plot which showsthe clear correlation between the Variables on the one hand andthe decline of unit heat-rate, unit loss, and coal flow-rate on theother. The divide line is at 16:35, where the recommended val-ues take effect. Let D2 be the time duration 16:35 to 21:04 andD1 duration 11:04 to 16:35. The Pseudo Sankey Diagram showsthe D2 to D1 redistribution of average losses as well as the dif-ference in other averaged parameters. Variables conform withrecommendations at 16:35 to 21:04 the period before of 11:04to 16:35. Note the main steam pressure increase and tempera-ture reduction, contrary to conventional recommendations. Theburner tilting angle (increased) is a major factor of combustionlosses and heat-transfer losses in the various boiler banks. Thenet result is a 15.48 0/00 reduction in plant loss-factor whichtranslate to about 444.27 decrease in plant heat-rate (about 4%).Coal flow decreased by 1.33 kg/sec.Huntly’s Unit 1 was fit-ted recently with a modern Distributed Control System, yetit had to be overridden to carry out the Optimizer’s recom-mendations. Three more maximizing efficiency examples fromHuntly power station are shown in ref [10].


Maximizing efficiency for data collected at 3-MAY-2012 11:46 for Unit 2at net load of 221.05 MW and gross load of 230.29 MW

Current Recommended1 03616 MAIN-STEAM BEFORE THROTTLE"A"Tag=03616.PR EPN= PRESSURE BAR 164.80487 164.80346

2 03616 MAIN-STEAM BEFORE THROTTLE"A"Tag=03616.TP EPN=3603 TEMPERATURE oC 527.15637 521.58948

3 03618 MAIN-STEAM BEFORE THROTTLE"B"Tag=03618.PR EPN= PRESSURE BAR 164.80487 181.26773

4 03618 MAIN-STEAM BEFORE THROTTLE"B"Tag=03618.TP EPN=3602 TEMPERATURE oC 527.15637 527.46717

5 05627 HOT REHEAT:REHEATER#2 OUTLET"A"Tag=05627.TP EPN=3413 TEMPERATURE oC 536.70416 540.84355

6 05628 HOT REHEAT:REHEATER#2 OUTLET"B"Tag=05628.TP EPN=3617 TEMPERATURE oC 536.70416 538.39989

7 03105 REHEATER ATTEMPERATOR "A" WATER INLETTag=03105.FW EPN=3617 FLOW kg/s 0.16781 0.16541

8 03106 REHEATER ATTEMPERATOR "B" WATER INLETTag=03106.FW EPN= FLOW kg/s 0.29054 0.33307

9 03102 S.H. ATTEMPERATOR "A" COOLING-WATER INLETTag=03102.FW EPN=2618 FLOW kg/s 1.42583 14.97013

10 03104 S.H. ATTEMPERATOR "B" COOLING-WATER INLETTag=03104.FW EPN= FLOW kg/s 1.41207 0.68852

11 00614 FLUE-GAS: AIR-HEATER "A" MIXER OUTLETTag=00614.TP EPN=0021 TEMPERATURE oC 141.08414 140.20094

12 00616 FLUE-GAS: AIR-HEATER "B" MIXER OUTLETTag=00616.TP EPN= TEMPERATURE oC 137.66989 151.87002

13 00215 PRIMARY-AIR: PRIMARY AIR-MIXER OUTLETTag=00215.TP EPN= TEMPERATURE oC 337.23041 336.83951

14 00420 FLUE-GAS: SECONDERY SUPERHEATER INLETTag=00420.TP TEMPERATURE oC 1295.60156 1308.45126

Figure 4. OPTIMIZER’S RECOMMENDATIONS IN EXMAPLE (3)


heat−rate, loss

loss

T P

Figure 5. UPPER: TREND PLOTS OF PLANT HEAT-RATE, AND LOSSES AS FUNCTIONS OF ADJUSTED MEASUREMENTS.LOWER: PSEUDO SANKEY OF LOSSES REDISTRIBUTION DUE TO OPTIMIZATION, AND RELATED MEASUREMENTS.


9 Closing remarks and future developments.The generation cost in thermal power stations comprises

three principal aspects: fuel charges, operating and maintenanceexpenses and capital costs. The relative importance of these fac-tors depends on the maturity of the station and its role in thegen-eration network. The relevance of exergy methods to all threeaspects has been acknowledged for more than 70 years (at leastsince Keenan [6]). The decisive advantage exergy methods offerover existing, well entrenched approaches to maintenance andcosts-benefit evaluations is speed, semi-automated decision mak-ing, and above all verifiable validity of predictions.

At the final analysis maximizing of power plant efficiencyreduces to trade offs between various exergy losses. Matteritdoes not which technique is ultimately used to solve the con-strained maximization problem, the correct answer always cor-responds to the smaller sum of exergy losses. Hence any solvingtechnique which does not call directly upon the entropy balanceeqn in addition to the conservation laws, is bound to be an ap-proximation at best, or more likely, altogether wrong. Phrasingthis in higher level wording, often there exist a loss distribution,yielded by an operating mode different from the current or con-ventional, such that unit-heat-rate is significantly improved. Thatmuch was demonstrated by this work.

Yet, 2nd Law methods are not widely practiced by powerplant owners and operators to this very day. There are a num-ber of reasons for this reluctance of the electric power indus-try to apply theoretically sound thermodynamics. These have todo with the industry’s traditional conservatism, a chroniclackof accountability with respect to fuel efficiency (no enforcing ofbenchmarks), the politicizing of fuel efficiency issue bothlocallyand globally with various interest groups each pulling in theirown direction.

However it appears that the main reason for ignoring theExergy-Audit tool has nothing to do with economics or ideology;rather it lies with a saying due to Max Planck:“new scientifictruth does not triumph by convincing its opponents and makingthem see the light, but rather because its opponents eventuallydie, and a new generation grows up that is familiar with it”. In-deed, plant operators find it hard to fathom that their decadesold practices to improve efficiency are in fact wrong in a funda-mental way. As for electric power industry as a whole, this israther typical; more than one Planck-type generations has lapsed(since Keenan [6]). However, neither their customers, nor theirshareholders, nor Planet Earth are too keen on waiting that muchlonger.

Hence, one great leap towards maximizing thermal effi-ciency of plants, is the full automation of it. True, one cannot au-tomate workshop repairs, but automation can be introduced withrespect to controllable parameters. All one needs to do is feedsthe values calculated by the Optimizer into the setting(-points)mechanism - closing the loop, that is. The efficiency maximizingwill then be locked in. Another, more comprehensive approach is

for ASME to dump the heat-rate and the associated mysterious“correction curves” as a measure of plant performance; ratheradopt the thermodynamic efficiency together with eqn (14) p.6.

REFERENCES1. Yasni E., Carrington C.G. 1988Off Design Exergy Audit of

A Thermal Power Station.Transaction ASME J.Eng PowerVol 110 no 2 April pp 166-172

2. Gyarmati I. 1970 Non-equilibrium Thermodynamics.Springer-Verlag Berlin

3. Sun Z.F, Carrington C.G 1991Application of Non-Equilibrium Thermodynamics in Second Law Analysis.Journal of Energy Resources Technology, Vol 113/33

4. Slattery J.C. 1972Momentum Energy and Mass Transfer inContinua .McGraw-Hill, New York

5. Callen H. 1985Thermodynamics and Introduction to Ther-mostatistics.John Wiley & Sons, New York

6. Keenan J. H. 1932“A Steam Chart for Second Law Analy-sis” Mechanical Engineering Vol. 54 pp. 195-204

7. Gibbs J. W. 1873 “Methods of Geometrical Representationof the Thermodynamic Properties of Substances By Meansof Surfaces” Transaction of the Connecticut AcademyII :382-404

8. Yasni E., Carrington C.G. Dec 1987The Role For Ex-ergy Auditing in A Thermal Power Station.ASME Win-ter Annual Meeting,Heat Transfer Division, Boston, Mas-sachusetts. HTD. Vol 80.

9. Weinhold F. 1975“Metric Geometry of Equilibrium Ther-modynamics.”(followed by II,III,IV,V) Journal of ChemicalPhysics 63 :2479-2483

10. www.gns.cri.nz/content/download/6808/37186/file/cepsi.pdfOff proceedings of CEPSI 2008, Macau, China

ACKNOWLEDGMENTMy gratitude to Dr D.M. Warrington of the Department of

Physics, University of Otago for guidance with spectral methods,and to Mr Phil Scadden Senior Scientist with the Institute ofGe-ological and Nuclear Sciences, New-Zealand for MMI program-ming and software engineering. Also thanks to GenesispowerNew-Zealand, owners of Huntly Power Station. A great deal ofgratitude to Mr Andrew Usher Plant Engineer Kwinana Cogen-eration Plant GDF SUEZ Australian Energy


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asme2e v3 - GNS Science · efﬁciency variations of an existing plant. Here it is demons trated how these can be found automatically. A natural extension of diagnostics is the ﬁnding