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Open Systems & Information Dyn. 10: 31-49, 2003.

A synthesis of thermodynamic models unifying traditional and work-driven operations with heat and mass exchange

Stanislaw SIENIUTYCZ Warsaw University of Technology

Faculty of Chemical and Process Engineering Department of Process Separation

1 Waryńskiego Street (room 232), 00-645 Warsaw, Poland E-mail: sieniutycz@ichip.pw.edu.pl

Abstract A novel trend in the theory of thermodynamic limits for energy

convertors and traditional heat & mass exchangers is analysed where certain

special controls called the Carnot control variables play a common role. In terms of

these controls an expression for the lost work has the same form in irreversible

energy convertors and in traditional processes of purely dissipative transport. For

sequential-type equipment of a finite size, enchanced limits are obtained for the

energy production or consumption. Formal models of simplest energy convertors

and characteristics of endoreversible operations are particularly lucid in terms of

the Carnot controls. Progress in the theory of energy generation problems is

achieved; examples of applications are outlined. Efficiency decrease caused by

dissipation and finite rate bounds are estimated for work released from an engine

or work added to a heat pump. It is shown that a simplification in analysis of

energy limits in complex thermal operations is achieved when Carnot controls are

applied. Extensions involve mass transfer problems and a finite-rate counterpart of

classical available energy (exergy).

Key words: energy optimization, efficiency, irreversible engines, heat exchangers

1. Introduction

1.1 Optimization with Thermodynamics

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The main ideas of this work are derived from thermodynamic optimization [1] which is a

branch of process optimization, [2, 3], a general strategy of seeking best solutions for real

processes in presence of constraints. Thermodynamic optimization uses the following

performance criteria:

a) mechanical or electric energy (work), generated or consumed

b) losses of mechanical energy or exergy, production of entropy

c) thermodynamic state parameters at the final stage of the process (they constitute the so-called

physiochemical criteria of optimization), e.g. final concentration of a valuable product in a

chemical reactor.

Thermodynamic criteria are generally insufficient for economic purposes. Yet, they can

sometimes be useful to assess the quality of the process economics as well. This is, for example,

the case when the work flux from a thermal machine or the final concentration of the key reagent

are main indicators of the profit from the process.

1.2. Optimization with Thermoeconomics

Thermoeconomic optimization uses economic performance criteria formulated with the help

of costs engineering and engineering economics [3]. These criteria may be regarded as

generalization of performance criteria encountered in thermodynamic optimization. Yet,

thermoeconomic optimization frequently rests on the same model of physical constraints

(thermodynamic balances) as thermodynamic optimization. While formulations considered here

refer to thermodynamic optimization only, interesting and useful extensions of models presented

in this work in the spirit of thermoeconomic optimization are not excluded. They are, however,

beyond the scope of this work. Our approach here is concentrated on energy limits rather then

any engineering economics. It thus avoids economics due to the conviction that any energy limit

must be a physical quantity which is independent of the system’s economy. In fact, the

optimization related to limits and that applied in economic analysis are two different issues. In

other words, the range of optimizing applied in this paper is restricted to thermodynamic limits

exclusively

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1.3. Purpose of the paper

This paper analyses those of several new directions in thermodynamic optimization of

processes in thermal machines in which the role of nonequilibrium transports and optimal control

theory are essential. Considering energy limits as purely physical properties, i.e. staying purely

on physical grounds, we formulate and analyse an optimization approach to limits on production

or consumption of mechanical energy in operations with heat and mass exchange. We also

compare the developed optimization criteria for work-assisted operations with those for

conventional operations (without work). As our method rests on general thermodynamics, it can

deal with arbitrary participating fluids, including, in particular the radiation fluid. For that fluid

the endoreversible limits incorporate, as a sole effect, a minimum entropy production caused by

simultaneous emission and absorption of radiation. To evaluate mechanical energy limits control

processes of engine type and heat pump type are considered, both with pure heat exchange and

with simultaneous heat and mass exchange. Links with exergy analysis of nonequilibrium systems

and classical problem of maximum work are pointed out. Illustrative examples are selected

mainly from the didactic viewpoint, without an attempt to give comprehensive presentation of

applications. Amongst new results the most important seems the notion of the so-called Carnot

control variables in terms of which the expression for the lost work in a thermal machine has the

same form as in a related traditional processes of purely dissipative transport. It is shown that

considerable simplification in analysis of complicated thermal machines is achieved when these

special controls are applied.

2. Basic definitions and properties

2.1 Driving a single-stage process by pure heat transfer

Fig. 1: A single stage CAN as an operation with active heat exchange between two fluids (in the

engine mode ∆T1 ≤ 0).

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Consider a standard operation of Curzon-Ahlborn-Novikov (CAN; [4]-[7]), Fig. 1. 1Its classical

characteristics based on efficiency as possible control are well known, Fig. 2.

Fig. 2: Conventional characteristics of CAN operation: the efficiency η is the control variable.

From the energy and entropy balance of the endoreversible CAN operation a structural

property of the system follows which is called here the Carnot temperature T’. In our earlier

work, [8] and [9], T’ was also called the driving temperature due to its specific properties. In

terms of upper and lower temperatures of the circulating medium, T1' and T2’, Carnot temperature

is defined as

'2

'12 T

TTT =′ (1)

When the second reservoir is the environment, T2= Te and equation (1) is used in an equivalent

form:

'2

'1

TTTT e=′ (2)

The primary property of T’ is that in terms of T' and T2 the thermal efficiency of the

endoreversible engine is given by the Carnot formula in which T' replaces T1.

TT

TT

′−=−= 2

'1

'2 11η (3)

1 A. Bejan gives a comprehensive history of this problem in ref. [11]. He credits also Chambadal who have considered it in 1957, i.e. as early as Novikov.

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A similar expression holds when the second reservoir is the environment i.e. T2= Te. The perfect

efficiency of the Carnot (work-producing) discontinuities across a finite-resource stream and an

infinite environment is essential in the “endoreversible” case considered. Work limits then follow

in terms of the time of state change and properties of boundary layers in each reservoir. The

boundary layers and the dissipation restricted to these may be related to „endoreversibility”, a

notion which is usually associated with models of the so-called finite-time thermodynamics

(FTT), but which is of a very restricted use in case in predicting actual work characteristics of

real thermal machines. Yet, restriction to external irreversibilities is unnecessary; in fact, FTT

models can go beyond “endoreversible limits” i.e. to treat internal irreversibilities as well, see,

e.g., ref. [15]. It is most essential, however, that in either of two methodological versions of FTT,

of which the first gives up internal irreversibilities whereas the second one estimates these from a

model, the FTT limits on energy consumption or production are stronger then those predicted by

the classical exergy. In short, this results from the “process rate penalty” that is taken into

account in every version of FTT. In the hierarchy of limits resulting from more and more detailed

models, the limits of the second and higher versions of FTT are, of course, stronger than the

limits of the first (endoreversible) version. The weakest or the worst are limits of classical

thermodynamics, resulting from the classical exergy [1]. We stress again that, when calculating

energy limits, we search for (hierarchy of) diverse, purely physical extrema with no direct regard

to economic optima.

Fig. 3: Characteristics of CAN operation when Carnot temperature ′ T is the control variable.

Whenever T1' = T2' ("short circuit") the process is purely dissipative and

2TT =′ or eTT =′ (4)

Along with the efficiency formula (1), this describes absence of mechanical energy production at

the short circuit point. Otherwise at the Carnot point (open circuit point, where the process is

quasistatic)

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1TT =′ (5)

This means that the engine efficiency (3) now equals η = ηc = 1-T2/T1 in accordance with the

classical result of Carnot. The crucial issue is, however, that the both two forms of Eq. (3) hold

for diverse irreversible process, with various contribution of dissipation, stretching through

Newton-Fourier and Carnot regions, as shown in Figs. 3 and 4.

Fig. 4: Entropy production in CAN operation when Carnot temperature ′ T is the control

variable. In terms of Carnot variables classical thermodynamic formulae are extended to

irreversible operations with work production or consumption.

2.2 Controlling of a one-stage process by coupled heat and mass transfer

Consider first a power generation expression and related quantities in terms of properties of

circulating fluid. After combining energy and entropy balances the power generated or consumed

in an endoreversible machine can be obtained in the form [10]

nssThhqTTw 212211

1

2 )]([)(1 '''''''

' −−−+−=& (6)

corresponding with a bilinear structure [5]

nnqqnqwp ),(),( '1'1'1 βη ′+== & (7)

Here the second component of power emerges which is associated with the work production

(consumption) due to the mass transfer. Its interpretation is the product of mass flow n and an

exergylike function β' whose structure follows from combination of the energy and (conservative)

entropy balances. For more details regarding this issue, see an earlier work [10].

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The quantity β' is not really suitable to use, thus we shall eliminate it by passing to the

process description in terms of different fluxes. We shall leave mass flux n unchanged, but,

introduce a new energy flux, Q1’, defined by an equation

1''1''1 nsTqQ 11 +≡ (8)

It is easy to see that Q1’ is the product of temperature T1’ and the total entropy flux, the latter

being the sum of entropy transferred with heat q1’/T1’ and substance s1n1. The virtue of the flux

Q1’ is that the power production w& assumes in terms of Q1’ the most suitable form that contains

the difference of chemical potentials µ of the active component

nQTTw 1

1

2 )()(1 2'1'''

' µµ −+−=& (9)

Thus Carnot chemical potential of the active component can easily be introduced as

µµµµ 2'2 1'−+≡′ (10)

In the reversible case where µ1’ = µ1 and µ2’ = µ2 this equality reduces µ’ to µ1 as it should be, so

the chemical component of power production efficiency becomes µ1 –µ2, which is also a correct

result. The entropy balance of an endoreversible machine takes in terms of Q a simple form

TQTQ '2'2'1'1 // = (11)

which is formally the same as in the operation with pure heat transfer. The endoreversible thermal

efficiency in terms of properties of the circulating fluid is the same as in the process of pure heat

transfer, Eq. (3). Also the Carnot expression holds for this efficiency in an unchanged form, η=1-

T2/T’.

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The thermal efficiency (3) is the first component of 2D efficiency vector. The second component

is the difference of chemical potentials

2' µµω −= (12)

The power expression in "the flux representation" conforms with the general structure

nnQQnQwp ),(),( '1'1'1 ωη +== & (13)

In a traditional analysis the above quantity depends on two unknown (primed) temperatures and

two unknown (primed) chemical potentials of circulating fluid, linked by a reversible entropy

balance and a mass balance across the reversible part of the system. Thus the equations are very

difficult to use, and a method to overcome the difficulty should be designed. We will show that

the Carnot quantities play in this matter an essential role.

We shall pass now to corresponding relationships in terms of the Carnot quantities. The

definition of T' given in Sec. 2.1 applies without any changes when mass transfer accompanies

the heat transfer. However, it may be shown that Carnot chemical potential, µ', should be

redefined when a more usual energy flux ε= q + Σhini is applied instead of Q. In this case µ' is

expressed by the parameters of circulating fluid (T1', T2', µ1' and µ2') as follows:

))//)(/(/( '2'2'1'12'222 TTTTTT µµµµ −+′=′ (10’)

This quantity, which was occasionally used in some of our previous papers (see, e.g. [13]) is

equivalent with µ’ of Eq. (10) only in an isothermal case. Both quantities (10) and (10’) are

correct, still they slightly differ because each refers to a different energy flux. When the second

reservoir is the environment, the quantities subscripted with the index 2 are replaced by those

superscripted by the index e (the environment). In any process with mass transfer the work

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production is characterized by the vector of thermal efficiencies, (η, ω), such that its first

component has the Carnot structure of efficiency (3).

We shall pass now to corresponding relationships in terms of Carnot quantities associated

with (3|) and (10). The thermodynamic functions of state referred to Carnot temperatures and

chemical potentials are called Carnot functions. In this way we may deal with Carnot energy,

enthalpy, entropy, etc. Importantly, in terms of Carnot thermodynamic properties the reversible

structure of basic equations is preserved in irreversible cases, and prediction is possible of

irreversible process equations on the basis of well-known or easily derived equations of

reversible processes.

At the short circuit point the equalities T' = T2 and µ' = µ2 (or T' = Te and µ' = µe) hold and

all components of the efficiency vector (η, ω) vanish. On the other hand, at the Carnot point the

efficiencies refer to the quasistatic process. In this (reversible) case, the efficiency vector has the

components

1

21TT

−=η (14)

and

21 µµω −= . (15)

Leaving aside the above special cases and returning to the general problem in terms of Q1’ we can

state that a general optimization task is to seek for optimal T' and µ' which maximize power p in

"Carnot variables representation"

]/)()/1/1([)()/1()()/1(

),,,(),,,(),,,(),(

211'2

21'122'2

1'

1'

2'

2'

1'

1'

'2'

TnTTQTnQTTnQTTTTnTTTTQTTwp

1

11

1

µµ

µµµµ

µµµµωµµη

′−+−′−

−+−=−′+′−=

+=≡ &

(16)

The first expression of the second line is the irreversible power expression that was split into the

reversible part (the second expression in the second line, without Carnot controls) and the

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irreversible part (the expression in third line which is the negative product of T2 and entropy

production). Note that Carnot temperature and Carnot chemical potential are independent

variables and the power is extremized with respect to these variables as process controls. The

reversible balances of entropy and mass across the (perfect) thermal machine are already

included; thus the extremizing procedure works without constraints. In an alternative case, all

working quantities may be expressed in terms of the fluxes in the system, of heat q1 and of mass

n1 (see examples in the next section). In this case the extremizing takes place with respect to the

fluxes q1 and n1.

In the simplest case of the pure heat transfer

)(),,,( 1'11' TTgqTTQ 11 ′−==′′ µµ (17)

where g is the overall conductance of heat transfer satisfying 1/g = 1/g1 + 1/g2. This was, in fact,

shown by elementary calculations in many works, see, e.g., ref. [8]. It follows from Eq. (17) that

at the Carnot point, where T' = T1, the reversible heat flux equals zero which means the

quasistaticity property. On the other hand, at the short circuit point, where T' = Te, the usual

Newton-Fourier formula describes the heat flow which drives the thermal engine.

When the mass transfer is coupled with the heat transfer an exact approach to dynamics

requires using the Onsager's theory to evaluate the fluxes Q1' and n1 driven by differences of

temperature reciprocals 1/T2- 1/T’ and the ratio of chemical potential difference µ’ – µ1 and

temperature T2. The latter may be approximated by the difference in the corresponding partial

entropies when the (Ackerman's type) effect of the partial enthalpy changes of the active

component is ignored. The entropy production and its approximation are as follows

).()11()()11(

/)()/1/1(

11

11'111'1

211'

XXnRXTTQssnTTQ

TnTTQS 1

′−+−′≅′−+−′≅

′−+−′=−

µµσ& (18)

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But under such an approximation the Onsager's symmetry breaks, and the pursued structure of

exchange equations cannot be evaluated by using this symmetry as the basis. In this case, as in

Eq. (17), the pure heat flux q1’ is proportional to the difference in temperatures (or in temperature

reciprocals for a suitably modified heat conductance), and the mass flux is proportional to the

difference in concentrations of the active component [10]. In agreement with the Lewis analogy

)(),,,( 11

11 XXcgTTn ′−=′′ −µµ (19)

Here c is the specific heat of the fluid, and gc-1 is the overall mass transfer conductance.

2.3 Sequential processes and work optimization in dynamical problems

This case, which is derived from the multistage systems theory, refers to finite resources

(finite size of at least one reservoir) and involves differentials and work functionals. It is assumed

that the second fluid constitutes an infinite constant reservoir whereas the first fluid changes its

properties when proceeds through stages in time. We have to attribute changes in state variables

to each considered flux.

For mass flux a suitable state coordinate is the amount of substance, N, or concentration, X.

However, there is no such an exact state variable for the flux Q. Still, a suitable variable of state

can immediately be defined for the ratio Q1’/T1’. Clearly, this is the entropy S. In the formulae

below we use the symbols S and T for the variable entropy S1(t) and temperature T1(t) in the bulk

of fluid. For an endoreversible flow process the specific work obtained in terms of the Carnot

controls is

})()({/ dXdSTTGPW eeT

T

f

i

µµ −′+−′−=≡ ∫

})()({ dXdSTT eeT

T

f

i

µµ −+−−= ∫ })()({ dXdSTTT

T

f

i

µµ −′+−′− ∫ (20)

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This leads to a generalized or finite time exergy A= max W satisfying the formula

})()(-{max

/maxmaxA

dXdSTT

GPW

eeT

T

f

i

µµ −−−=

≡=

∫

)()( XXSSTHH fiefiefi −−−−−= µ

})()({min dXdSTTT

T

f

i

µµ −′+−′− ∫ (21)

Consequently the (maximum) work produced by the engine equals to the change of the classical

exergy reduced by the lost work or the product of the reservoir's temperature and the (minimum)

entropy production. An analogous equation is obtained for the heat pump, but then the effect of

entropy production is added to the classical exergy change, i.e. an increase of the work input is

necessary to assure the required state change in a finite time. Equation (21) is the

“endoreversible” limit for the work production between two given states and for a given number

of transfer units. Even this simple limit is stronger than that the one predicted by the classical

exergy. What can be said about a yet stronger limit which involves an internal dissipation in the

participating thermal machine? We need to recall the hierarchy of limits. For limits of higher

rank, an internal entropy generation is included in the dissipation model and then Eq. (21) is

replaced by its simple generalization which contains the sum of the “endoreversible” and

“internal” productions of the entropy, intσσ SS endo + , in agreement with the Gouy-Stodola law

which links the lost work with the entropy generation. For a still stronger limit, other components

of total entropy source are included at the expense of a more detailed information input, but with

the advantage that the limit is closer to reality. For a sufficiently high rank of the limit, it

approaches the real work quite closely, but also the cost of the related information becomes very

large. What is important then, is a proper compromise associated with the accepted limit of a

finite rank. For limits of various ranks, inequalities are related to A and real work Wreal that are

valid in the form Wreal >∆Ak > ∆A1> ∆A0, where ∆A1 refers to the change of “endoreversible

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exergy”, and ∆A0 pertains to the change of the classical exergy. The classical exergy change

consitutes then the weakest or the worst standardized limit on the real work. In the described

scheme any considerations of relations between the irreversibility and costs are unnecessary.

Equation (21) expresses - in terms of Carnot controls - the Gouy-Stodola law for the

endoreversible system [11]. The thermodynamic form of the lost work expression without

kinetics incorporated is classical in terms of the Carnot controls, and it is the sum of products of

thermodynamic fluxes and forces.

τµµσ dXSTTSTT

T

ef

i

})()({ && −′+−′= ∫ (22)

Kinetic form of this source, which incorporates the Onsager's relations, is classical as well, i.e. it

is the same as in processes without production or consumption of mechanical energy

τσ dXrXSrSrST XXSXSST

T

ef

i

}2{ 22 &&&& ++= ∫

)(2)({ 2 TTgTTg SXSST

T

f

i

−′+−′=∫ τµµµµ dgXX })()'( 2−′+− (22’)

This classical structure implies the constancy of rates and forces along an optimal path, and also

the constancy of the entropy production (the so-called "equipartition of the entropy production",

first systematically treated by Tondeur and Kvaalen [12]). Yet this property holds only for linear

processes in which conductance coefficients are state-independent constants and no constraints

are imposed on control variables. Examples are Fourier heat exchange, linear mass diffusion,

first-order chemical reactions, etc.

3. Carnot temperatures in spaces of rates and flows

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In spaces of flows or time derivatives of state variables (rates) Carnot variables are some

operator expressions which depend on process state and flow or time derivative characterizing

the process. Expressions given below show respective formulae for processes recently treated in

the literature. We restrict to the Carnot temperature operator T’ and related thermal efficiencies.

3.1 Classical endoreversible process with Newtonian heat exchange

When Fourier-Newton heat exchange takes place on both parts of the engine and the partial

conductances are g1 and g2 then the Carnot temperature

gqTT /11 −≡′ (23)

where q1 is driving heat flux (on the high-T part of the engine) and g is the overall conductance

which results as the reciprocity of the sum of the partial heat resistances, g= (1/g1 + 1/g2)-1 or

g1g2/(g1 + g2). The resulting thermal efficiency as a function of heat flux satisfies the well-known

formula (derived earlier in a number of works, see e.g. [7])

gqT

T1TT1

/1

22

−−=

′−≡η . (24)

3.2 Non-Newtonian heat exchange with power exponent α

Here we deal with the case when the heat exchange on the side of the high-T fluid is non-

Newtonian and that on the side of low-T fluid is Newtonian. The respective formula for T’ is

22/1

111 /)/( gqgqTT a −−=′ (25)

[13] whence the efficiency

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gqgqT

T1TT1 a /)/( 2

/1111

22

−−−=

′−≡η . (26)

3.3 Coupled heat and mass transfer

The Carnot temperature evaluated from the results of ref.[10] is

)( )/(

)/1(11

)(1

)/(11

)1(11

11 1

1 11

221

111

111

)/1()()/()1(

)(),(

gcRg

gnXm

X

gnXm

X

m

m

mm

m

m

gnXXgnXX

gq

ncgcgcnqT

−+

−+

−+

−+

−=′

)( )/()/1(

22)(

2

)/(22

)1(2 222

222

222

)/1()()/()1( gcRg

gnXm

X

gnXm

Xmm

m

m

gnXXgnXX −

++

++

++++

(27)

where the partial conductances indexed by m refer to mass transfer. Again, the related thermal

efficiency is calculated as η =1-T2/T'. Both results can contribute to the present theory of active

drying processes [14]. Note, however, that (27) is in terms of the pure heat flux, q1 = ε1 - Σh1in1i,

rather than in terms of the energy flux ε or Q. In this equation R is the gas constant, c1 and c2 are

the specific heats of the first and second fluid, gm1 and gm2 are related conductances of mass

transfer. X1 and X2 are concentrations of the active component in subsystems 1 and 2.

4. Carnot variables in chemical systems

Power and simplicity of models with Carnot controls is observed also in chemical systems.

Below we consider an approach in which maximum power obtained from a chemical engine is

estimated by using Carnot chemical potentials built in the definition of chemical affinity.

4.1. A simple model of power production in an isothermal chemical reactor

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Here we consider an engine driven by a set of isothermal chemical reactions, where the substance

exchange fluxes satisfy the kinetic laws

)( '1111 μμgn −= , )( 2'222 μμgn −= (28)

where g1 and g2 are the matrices of system conductances. The change in the vector of molar

fluxes n1-n2 is caused by chemical reactions in the system. Assuming a stoichiometric matrix ν

the chemical change is n1-n2 = νr, where r is a vector describing the rates of chemical reactions

referred to total volume of the system. As previously we assume that an active process of power

production takes place between the primed states (1’, 2’). These states determine, therefore, an

active chemical affinity A1’, 2’ which should be contrasted with a total available affinity A1, 2 that

is referred to points 1 and 2. Both these quantities are “inhomogeneous” or space-extended

affinities which generalise the classical chemical affinity. We determine a formula that links these

two affinities. Using (28) we shall first write down the classical chemical affinities at the states

1’ and 2’

11

1111

11'1'1 )( ngνAngμνμνA −− +=−−=−= TTT (29)

and

21

2221

22'2'2 )( ngνAngμνμνA −− −=+−=−= TTT (30)

As n1-n2 = νr and the matrix of conductivities is diagonal, these two equations can be combined

into the form

)()( 21T1

212

T11

T12

T121

T11

2T1

1T1

'2T1

2'1T1

1 nnνgggνgν

AνgAνggνgν

AνgAνg−++

++

=++ −

−−

−−

−−

−−

(31)

This equation contains two weighted affinities that are effective or „space-extended” quantities.

The first of these affinities, A1’, 2’, the left hand side of (31)) is the is the active affinity, whereas

the second, A1, 2, (the right hand side of (31)) is the total affinity available in the process. These

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affinities are connected with the reaction rate r and a resistance type matrix R= (g1 + g2)-1 the by

the relation

νrνAA TR−= 2,1'2,'1 (32)

With this formula an estimate is obtained for the power production in terms of the active affinity

)()( '2,'12,11T

'2,'1T

'2,'1 AAννArA −== −TRw& (33)

4.2. Optimum of active affinity in a chemical reactor

To get rid effect of system constraints we must now introduce Carnot variables. From Eq. (10)

the primed vector of second chemical potential satisfies μμμμ ′−+= 2 1'2' . In terms of Carnot

chemical potentials µ’ and the corresponding Carnot affinity A’ (µ’, µ2) based on these potentials

we obtain form (10) and (33) the power expression

)()( 2,11TT AAννArA ′−′=′= −TRw& (34)

which is quadratic with respect to the active Carnot affinity. As in previous examples, power is

maximized with respect to free Carnot control, thus no constraints are imposed on the variable

A’. With this observation, equation (34) proves that an optimal (power-maximizing) A’ must lie

somewhere in between A1,2 and zero.

Equation (34) is a simplest estimate of power production which enables one to find optimal

affinity and the maximum power for an arbitrary set of chemical reactions. As an example

consider a linear system that is characterised by a constant R. In this system conductances and all

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other coefficients are constant and the value of effective affinity A’ that maximizes power follows

from the formula

0)2()(/ 2,11 =′−=′∂∂ − AAννA TRw& . (35)

This means that the following estimate is valid: in linear systems at extremum power the active

affinity A’ equals one half of the total available affinity

2,121 AA =′ (36)

Accordingly, the corresponding estimate for the maximum power is

2,11T

2,1max )(41 AννA −= TRw& (37)

At this point of reasoning we should note that an estimate for the power dissipated at the short-

circuit point of the system, where both A’ and active power are zero, equals

2,11T

2,1 AA −= cRdissw& (38)

where Rc is the matrix of chemical resistances [16]. This power could, in principle, be exploited

in a non-dissipative system, and this is why (38) can measure the power available in the system

characterized by the total affinity A1, 2. An estimate for the fraction of the maximum power

utilised in the chemical system in the considered sense is

1max )tr(tr41 −= ννTRcR

availww&

& (39)

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When resources are finite extra constraints must be added that take into account the role of finite

reservoirs or flows of driving fluids. In this case results (36) and (37) are no longer valid and a

sequential (multistage) optimisation of the affinity profile is necessary. This generalization of the

optimisation problem, which involves variational calculus, will be a subject of our future effort.

5. Final remarks

It follows from the analysis of radiation engines that the overall kinetics expressed in terms

of the temperatures T’ or T" is identical in operations with and without work only when involved

transport processes are linear. In nonlinear cases, the overall kinetics differ in both kinds of

operations. Yet, the benefits that result from describing thermal machines in terms of Carnot

variables are preserved in all thermodynamic relations. They follow from a common expression

for the maximum work potential (or the entropy source), the same for operations with and

without work. For large number of processes in which linear approximations are acceptable,

models and optimization results in both kind of operations are identical. This offers the good

opportunity to predict optimal controls and optimal trajectories for difficult processes in thermal

machines with coupled heat and mass transfer by applying well-elaborated optimization results

obtained for classical heat and mass exchangers.

It should always by kept in mind that an economic problem of the system optimization and the

physical problem of work limits for a resource (considered here) are two different problems. The

real work supplied to a compressor at economically optimal conditions may sometimes be many

times larger than the mechanical energy (exergy) limit associated with the production of a key

substance; this is a well known fact from the theory of Linde operation, for example [17]. In the

context of mechanical energy limits, the trade-off between the exploitation and investment costs

and the problem of investment reduction by admission of exergy losses are irrelevant issues [18].

Moreover, the entropy source minimization, as applied here, has no relevance to an economic

optimum of a product yield. Thanks to the kinetic and time factors incorporated in dynamical

models, our method (which does not yield the standard exergy but rather its endoreversible and

20

higher-order generalizations) can be applied to yield enhanced energy limits in highly non-

equilibrium, kinetically driven processes of mass and energy transfer. We especially stress the

hierarchical nature of the FTT limits, where endoreversible limits are one step better then those

derived from the classical exergy. Application of this theory to fuel cells is predicted in future

works along this line.

Acknowledgments

The author acknowledges the financial support from a Warsaw TU grant Dynamics of Complex

Systems in the academic year 2002/2003. A referee has made several helpful comments on an

earlier version of this work in which the analysis was based on the energy flux ε rather than Q.

Nomenclature

A- generalized exergy

A - exchange area

c- specific heat

G - fluid flow

g - overall conductance

g1, g2 - partial conductances

H - enthalpy of controlled phase

n - molar mass flux

P, p - total and local power

Q- energy flux based on total entropy flow, Eq. (8)

q1' - heat flux at state 1' (Fig.1)

r - overall resistance

S – entropy

Sσ - entropy production

T’-Carnot temperature

21

T1-temperature of driving fluid

T2-temperature of reservoir fluid

T1’, T2’- tmperatures of circulating fluid

t - physical time

W - specific work

X - concentration of active component

α-heat exchange coefficient

β'-material efficiency in the frame with heat flux q

γ-cumulative conductance

ω-material efficiency in the frame with energy flux Q

ε-standard energy flux, q + Σhini

η - thermal efficiency

µ - chemical potential

µ’-Carnot chemical potential

τ - nondimensional time, number of transfer units

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22

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Figure Captions

Fig. 1: A single stage CAN as an operation with active heat exchange between

two fluids (in the engine mode ∆T1 ≤ 0).

Fig. 2: Conventional characteristics of CAN operation: the efficiency η as the

control variable.

Fig. 3: Characteristics of CAN operation when Carnot temperature ′ T is the

control variable.

Fig. 4: Entropy production in CAN operation when Carnot temperature ′ T is the

control variable. In terms of Carnot variables classical thermodynamic formulae

are extended to irreversible operations with work production or consumption.

25

Figures

A synthesis of thermodynamic models unifying traditional and

work-driven operations with heat and mass exchange S. Sieniutycz1

1 Faculty of Chemical and Process Engineering, Warsaw University of Technology

00-645 Warsaw, 1 Waryńskiego Street, Poland

e-mail: sieniutycz@ichip.pw.edu.pl

Fig. 1: A single-stage CAN as an operation with active heat exchange between two fluids (in the

engine mode ∆T1 ≤ 0).

26

Fig. 2: Conventional characteristics of CAN operation: the efficiency η is the control variable.

27

Fig. 3: Characteristics of CAN operation when Carnot temperature ′ T is the control variable.

Fig. 4: Entropy production in CAN operation when Carnot temperature ′ T is the control

variable. In terms of Carnot variables classical thermodynamic formulae are extended to

irreversible operations with work production or consumption.