Power generation in thermochemical and electrochemical systems – A thermodynamic theory

International Journal of Heat and Mass Transfer 55 (2012) 3984–3994

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Power generation in thermochemical and electrochemical systems – A thermodynamic theory Stanisław Sieniutycz a,⇑, Marcin Błesznowski b, Agata Zieleniak b, Janusz Jewulski b a b

´ skiego Street, PL 00-645 Warszawa, Poland Department of Process Separation, Faculty of Chemical and Process Engineering at Warsaw University of Technology, 1 Waryn Fuel Cell Department, Institute of Power Engineering, 36 Augustówka Street, PL 02-981 Warszawa, Poland

a r t i c l e

i n f o

Article history: Available online 21 April 2012 Keywords: Power limits Thermal machines Fuel cells Energy Entropy

a b s t r a c t In this paper power limits and other performance indicators are investigated in various power generation systems with downgrading or upgrading of resources. Energy flux (power) is created in a power generator located between a resource fluid (‘upper’ fluid 1) and the environmental fluid (‘lower’ fluid, 2). Transfer phenomena, fluid properties and conductance values of dissipative layers or conductors influence the rate of power yield. While temperatures Ti of participating media are only necessary variables to describe purely thermal systems, in the present work both temperatures and chemical potentials lk are essential. This case is associated with engines propelled by fluxes of both energy and substance (chemical and electrochemical engines). Optimization methods are applied to determine power generation limits which are important performance indicators for various energy converters, such as thermal, solar, chemical, and electrochemical engines. Methodological similarity is shown when analysing power limits in thermal machines and fuel cells. Numerical approaches are based on the methods of dynamic programing (DP) or Pontryagin’s maximum principle. In view of the limitation of DP to systems with low dimensionality of state vector, we focus here on the Pontryagin’s method, which involves discrete canonical algorithms derived from the process Hamiltonian. Some new or relatively unknown properties of these algorithms are described in the context of their application to power systems. In fuel cells and other electrochemical systems downgrading or upgrading of resources may also occur. However, we restrict here to the steady-state fuel cells. An approximate (topology-ignoring) analysis shows that, in linear systems, only at most 1/4 of power dissipated in the natural transfer process can be transformed into mechanical or electric power. This indicator may be viewed as a new form of the second law efficiency. The relevant experimental data obtained at the institute of Power Engineering are also presented in this paper. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Power limits are important indicators of engine performance and their potential. The present paper is focused on a synthesizing approach to evaluate power limits in various power generators working as steady or dynamical systems. Power limits should be evaluated by optimization methods in which total power yield is maximized with respect to system controls. Mathematical treatment of differential optimization models describing dynamical power systems (such as Eqs. (27) and (28) in the present paper) is generally a very difficult task. Functionals of total power must be maximized by variational methods, which are typical for dynamical systems.

⇑ Corresponding author. Tel.: +48 22 8256340; fax: +48 22 8251440. E-mail address: [email protected] (S. Sieniutycz). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.030

In comparison with the previous publications, in the present paper a novel point is the Hamiltonian treatment of optimal dynamical processes. This is especially important for dynamic energy systems with resources decaying in power yield mode (engine mode), such as those in right part of Fig. 1. Hamiltonian treatment is opposed to the dynamic programing treatment applied in the previous publications. Discrete difference equations are derived here as canonical equations for the system Hamiltonian, and solved by numerical methods. Therefore the ‘‘curse of dimensionality’’ observed for the DP algorithms is omitted. Summing up, in this paper, we abandon DP approaches and develop Pontryagin’s type (Hamiltonian-based) approaches [15–21], as outlined in the Section 5. The novelty and innovative aspects of the paper also lie in its synthesizing mathematical formalism whose outcome is the common thermodynamic model applicable to the thermal, radiative, chemical, and electrochemical power generators. The inclusion of the imperfect fuel cells into the common model, while quite

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Nomenclature A1 and A2 substrate and product of a simple isomerisation reaction a temperature exponent in an exchange equation a0 = 4r/c radiation constant related to the Stefan–Boltzmann constant [Jm3K4] av exchange area per unit volume [m1] Nernst and idle run voltages [V] E0, E0 G Gibbs energy flux driving chemical engine (J s1) G_ resource flux [g s1], [mol s1] g1, g partial and overall conductance [J s1 K a] profit rate and process rates [–] f0, fi H Hamiltonian function HTU height of transfer unit [m] h numerical value of Hamiltonian [J m3 K1] specific and volumetric enthalpies [J g1, J m3] h, hv i electric current density [A m2] n flux of fuel reagents [g s1, mol s1] _ p¼W power output [J s1] p0m molar constant of photons density [mol m2 K3 s1] q heat flux between a stream and power generator [J s1] Q total heat flux involving transferred entropies [J s1] S, Sr entropy and entropy produced [J K1] s, sv specific and volumetric entropy [J K1 g1, J K1 m3] T variable temperature of resource [K] T1, T2 bulk temperatures of reservoirs 1 and 2 [K] T10 , T 20 temperatures of circulating fluid (Fig. 1) K Te temperature of the environment [K] Carnot temperature control [K] T0 T_ ¼ u rate of control of T in non-dimensional time [K] s u and t ate controls, dT/ds and dT/dt [K, K s1] V voltage, maximum work function, respectively [V, J mol1] v velocity of resource stream [m s1] W work produced, positive in engine mode [J] w specific work at flow or power per unit molar flux [J mol1] x mass fraction [–] z adjoint variable [–]

simple, is original, unique and beneficial to the FC works that have appeared in the literature. The limitation of the approach manifests, however, whenever the FC topology differs significantly from that of thermal or chemical system. Techniques for variational power maximization used previously in dynamical systems (e.g. those applied in Refs. [1,3,4,8,13]) treat numerically Bellman’s recurrence equation of a suitable dynamic programing model (DP model). However, the Bellman’s approach is effective only for low dimensionality of the state vector. Since in chemical systems the number of state variables is usually large (concentrations and catalyst activities may accompany temperature), DP algorithms become inefficient and inaccurate for real chemical systems. This fact substantiates the development of suitable Hamiltonian approaches in the present paper. The size limitation of our paper does not allow for inclusion of all derivations to make the paper self-contained, thus the reader may need to turn to some previous works, [1–5]. In view of difficulties in getting analytical solutions in complex systems, approximations by difference equations and numerical approaches are treated in a separate paper [3], which, in particular, discusses convergence of numerical algorithms to solutions of HJB equations and role of Lagrange multipliers in the dimensionality reduction.

Greek symbols partial and overall heat coefficients referred to respective cross-sections [J m2 s1 K1] b coefficient of radiation transfer related to molar constant of photons density p0m and Stefan–Boltzmann con0 1 stant; b ¼ rav c1 [s1] h ðpm Þ e total energy flux, conservative along a conductor [J s1] gp/q1 first-law thermal efficiency [–] vqcv(a0 av)1 time constant assuring the identity of ratio t/v with number of transfer units [s] l chemical potential [J mol1] 0 l Carnot chemical potential [J mol1] r factor of internal irreversibility [–] r Stefan–Boltzmann constant for radiation [J m2 s1 K4] rs entropy production of the system [J K1 s1] n intensity index [–] f chemical efficiency [–] s dimensionless time or number of transfer units [–]

a1, a0

Subscripts C Carnot point m molar flow s entropy v per unit volume 1, 2 bulks of first and second fluid 10 , 20 circulating fluid 0 idle run voltage Superscripts e environment i initial state f initial state 0 ideal (equilibrium) voltage ‘ Carnot state Abbreviations CNCA Chambadal–Novikov–Curzon–Ahlborn engine HJB Hamilton–Jacobi–Bellman equation.

In power systems many various controls can be applied, which accomplish the effect of propelling fluxes of heat and mass transfer and help to satisfy the principle of energy conversion presented in Fig. 1. Here we shall recall and use definitions of some special control variables (Carnot controls; [1,2]). As in [2] it is convenient to begin with the simplest case of no mass transfer, i.e. to consider first a steady, internally reversible heat engine with a perfect internal power generator characterized by temperatures of circulating fluid T10 and T20 . 2. Entropy generation and power production Consider entropy generation in a diffusion limited power system i.e. the system in which contributions following from the role of convective (bulk) motion are negligible. The present analysis refers to one-stage operation of the overall dynamic systems shown in Fig. 2 (with stage indices neglected). Balance of fluxes involving intensive thermal parameters in the bulks of the streams yields

rs ¼

q2 q1  þ ðs2  s1 Þn1 : T2 T1

ð1Þ

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Note that, as the consequence of the definition of Q, the internal entropy balance of a perfect engine driven by both heat and mass transfer has in terms of total heat flux Q the same structure as in purely thermal systems.

Q2 Q1 ¼ T 20 T 10

ð4Þ

Eliminating heat flux q2 from Eq. (1) with the help of the engine’s energy balance

e1 ¼ e2 þ p

ð5Þ

and applying the mass balance of a reacting system with a complete conversion in terms of conserved fluxes through cross-sections 1– 10 and 20 –2

n1 ¼ n2

ð6Þ

(compare primed generator states with non-primed bulk states in Figs. 1 and 2) we obtain

rs ¼ ðq1 þ h1 n1 Þ Fig. 1. Real work associated with energy generation is different from that of energy consumption.



   1 1 l1 l2 p þ n1  ;   T2 T1 T2 T1 T2

ð7Þ

where the sum q1 + h1n1 is the total energy flux e1 in the system with a complete conversion. A transformed form of this equation is the power formula

    T2 l l þ T 2 1  2 n1  T 2 rs ; p ¼ e1 1  T1 T1 T2

ð8Þ

which links power output p with bulk parameters of reservoirs and entropy production rs in the system. Eq. (10) may be compared with the same power evaluated for the endoreversible part of the system. Combining the mass balance (6) with an equation describing the continuity of the reversible entropy flux

e1  l10 n1 T 10

¼

e2  l20 n2

ð9Þ

T 20

and eliminating from this result e2 and n2 with the help of Eqs. (5) and (6) yields

e1  l10 n1 T 10

¼

e1  p  l20 n1 T 20

;

ð10Þ

which leads to the power expression

    T 0 l10 l20 n1 : p ¼ e1 1  2 þ T 20  T 10 T 10 T 20

Fig. 2. A discrete scheme of a dynamical thermo-chemical engine used for power calculations.

ð11Þ

Power p is expressed in terms of fluxes continuous through the conductors. Eq. (11) may be regarded as the consequence of Eq. (8) when it is applied to the reversible zone of the system. After comparing Eqs. (8) and (11) we obtain an equality



   T2 l l þ T 2 1  2 n1  T 2 rs T1 T1 T2     T 20 l10 l20 þ T 20 n1 ¼ e1 1   T 10 T 10 T 20

e1 1  This expression contains classical heat fluxes q1 and q2 identified with the so-called senstive heat (the heat attributed to thermal agitations in the continuum medium). In this paper we shall also define and use other flux related to heat q, the so-called total heat flux which contains the product of temperature T and the sum of the partial entropies of the species multiplied by the involved mass fluxes

Q ¼ q þ Tðsa na þ sb nb    þ sk nk Þ

ð2Þ

We shall also use total energy flux, e, defined in accordance with the usual definition, that is as the quantity

e ¼ q þ ha na þ hb nb . . . þ hk nk

ð3Þ

ð12Þ

from which entropy production can be expressed in terms of bulk driving forces and active driving forces (measures of process efficiencies). Finally we arrive at the result

rs ¼



e1 T 20 T 2 T 10



     T2 l T 0 l10 l20 l þ n1 1  2  2 ;  T1 T 1 T 2 T 10 T 20 T2

ð13Þ

which expresses entropy production in terms of bulk properties of streams and (primed) properties of the power producing zone in the system. Introducing an effective temperature called Carnot temperature

S. Sieniutycz et al. / International Journal of Heat and Mass Transfer 55 (2012) 3984–3994

T 0  T 2 T 10 ðT 20 Þ1 :

ð14Þ

We note that the endoreversible thermal efficiency in Eq. (11) satisfies the Carnot formula in terms of T0 and T2,

g¼1

T 20 T2 ¼1 0 : T 10 T

ð15Þ

The name Carnot temperature is used for the quantity T0 simply because the efficiency of an internally reversible engine expressed in terms of T0 and T2 satisfies the Carnot formula. When an input energy flux is given, Eq. (15) is suitable to determine power production or consumption in various steady and unsteady systems. Whenever internal dissipation effects prevail in the power generation zone, an effective environment temperature T eff 2 ¼ /T 2 should appear in Eq. (15) in place of T2 (see Section 3). In dynamical thermal systems a state trajectory is a curve T1 = T(t) which describes the change of temperature of the resource fluid in time, whereas a control curve may be represented by efficiency g(t) or Carnot temperature T0 (t). The latter quantity, defined by Eq. (14), is particularly suitable in describing driving forces and resource relaxations in dynamical energy systems. Whenever T0 (t) differs from T(t) the resource relaxes to the environment with a finite rate associated with the efficiency deviation from the Carnot efficiency. Only when T0 (t) = T(t) the efficiency is Carnot, but this corresponds with an infinitely slow relaxation rate of the resource to the thermodynamic equilibrium. In real processes (those with finite relaxation rates) thermal efficiencies are always lower than Carnot, corresponding with Carnot temperatures T0 (t) lower than the resource temperatures T(t). In chemical systems the role similar to T0 is played by the Carnot chemical potential, described below. Also, in terms of Carnot T0 , the first part of the entropy production (13) for a pure heat process, takes the following simple form

rs ¼ q 1



1 1  T0 T1

 ð16Þ

Consequently, Eq. (13) generalizes the familar entropy production expression (16) for the case when a single reaction A1  A2 = 0, undergoes in the system [1,2]. Eq. (13) also leads to the definition of Carnot temperature (14) and to Carnot chemical potential of the first component

l0 T0

¼

l2 T2

þ

T 20 T2



l10 T 10



l20

 ð17Þ

T 20

In a special case of an isothermal process the above formula yields a chemical control

l0 ¼ l2 þ l10  l20

ð18Þ

which has been used earlier to study an isothermal chemical engine [4]. After introducing the Carnot potentials in accordance with Eqs. (14) and (17), total entropy production of the endoreversible power generation by the simple reaction A1  A2 = 0 (isomerisation or phase change of A1 into A2), takes the following simple form

rs ¼ e1

   1 1 l1 l0 þ n1   T1 T0 T0 T1

3987

exchange of energy and matter between two bodies with temperatures T1 and T0 and chemical potentials l1 and l0 . Examples considered in the further part of the present paper refer first to classical and radiation thermal machines and next to chemical and electrochemical power generators (fuel cells). Ideas referring to endoreversible systems are occasionally generalized to those with internal dissipation [1]. 3. Basic results for steady thermal systems Majority of research on power limits published to date deals with stationary systems, in which case both reservoirs are infinite. To this case refer steady-state analyses of the Chambadal– Novikov–Curzon–Ahlborn engine (CNCA engine [1,2,5]), in which energy exchange is described by Newtonian law of cooling, or the Stefan–Boltzmann engine, a system with the radiation fluids and the energy exchange governed by the Stefan–Boltzmann law [7]. Due to their stationarity (caused by the infiniteness of both reservoirs), controls maximizing power are lumped to a fixed point in the state space. In fact, for the CNCA engine, the maximum power point may be related to the optimum value of a free (unconstrained) control variable which can be efficiency g or Carnot temperature T0 . For a steady-state heat operation of CNCA type [1,2,5], with bulk temperatures T1 and T2 and internal irreversibility U the propelling heat in terms of Carnot temperature is [1]

q1 ¼ gðT 1 ; T 2 ; UÞðT 1  T 0 Þ;

ð21Þ

where g is an effective overall conductance which may be function of bulk state and r. Hence the power output is

  UT 2 p ¼ gðT 1 ; T 2 ; UÞ 1  0 ðT 1  T 0 Þ: T

ð22Þ

Setting to zero the partial derivative of p with respect to T0 one finds at the maximum power point

T 0opt ¼ ðT 1 UT 2 Þ1=2 :

ð23Þ

Since the effective environment temperature equals rT2 and the Carnot structure holds for thermal efficiency g in terms of Carnot T0 , the power-maximizing efficiency follows as

gmp ¼ 1  UT 2 =T 0opt ¼ 1 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UT 2 =T 1 :

ð24Þ

This equation represents a generalization of the classical CNCA formula for the case when internal imperfections (coefficient U) persist in the heat power system. For the Stefan–Boltzmann engine an exact expression at the optimal power point cannot be determined analytically, yet, the temperature can be found graphically from the chart p = f(T0 ). A pseudo-Newtonian model [6–8], which treats state dependent energy exchange with coefficient a(T3), omits to a considerable extent analytical difficulties of the Stefan–Boltzmann equation. Moreover, we can extend the present approach to dynamical systems, (Fig. 2), as outlined below.



ð19Þ

Introducing into the above formula total heat flux Q1, satisfying in the considered case an expression Q1  e1  l1n1, we finally obtain

rs ¼ Q 1



1 1  T0 T1

 þ n1

l1  l0 T0

;

ð20Þ

where the total heat flux Q1 can also be expressed as the sum Q1 = q1+T1s1n1. The resulting Eqs. (19) and (20) are formally equivalent with an expression obtained for a process of purely dissipative

4. Selected results for dynamical thermal systems In this case power maximization problem requires the use of variational metods (to handle extrema of functionals) in place of static optimization methods (which handle extrema of functions). Obtained non-exponential shape of the relaxation curve is the consequence of nonlinear properties of the radiation fluid. Non-exponential are also other curves describing the radiation relaxation, e.g. those from based on the Stefan–Boltzmann equation [6–9]. Dynamical energy yield, Figs. 2 and 3, is associated with a limited amount of the resource fluid. The operation may be viewed as

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Fig. 3. A scheme of a multistage control with variable time intervals hn, described by an algorithm of the discrete maximum principle. Elipse-shaped balance areas pertain to sequential sub-processes which grow by inclusion of following units.

the sequence of small elementary engines of CNCA type, i.e. the sequence of small units shown in Fig. 3. This way of power yield is associated with the continual decrease of the resource temperature T1 = T(t), approximated by the sequence Tn. In the scheme in Fig. 3 control un may be represented by the rate of temperature change in time, whereas the control hn refers to the size of the stage measured in terms of the holdup times of the resource fluid in stages of the system. The associated optimization searches for an extremal curve rather than an extremum point. Power maximizing T(t) is accompanied by optimal control T0 (t); they both are components of the dynamical solution which satisfies a Hamilton–Jacobi–Bellman equation for the optimal control. Expressions extremized in HJB equations of dynamical problems are Hamiltonians, H. With the feedback control, the optimal temperature T0 is obtained as the quantity maximizing H with respect to Carnot temperature at each point of the path. For example, after using the pseudo-Newtonian model, we obtain an optimal dynamics of relaxing radiation in the form

T_ ¼ nðhr ; T; UÞT:

ð25Þ

Eq. (25) describes an optimal law, i.e. the trajectory in terms of state T and Hamiltonian constant hr. The related optimal Carnot control has the structure 0

T ¼ ð1 þ nðhr ; U; TÞÞT:

ð26Þ

Comparing with linear systems, the pseudo-Newtonian relaxation curve is not exponential. When standard boundary conditions for exergy are used, optimal work functions become generalized (rate dependent) exergies [1,2]. A more exact approach to radiation engines, described below, abandons the pseudo-Newtonian approximation and uses the exact Stefan–Boltzmann equations from the beginning. 5. Radiation engines using the Stefan–Boltzmann equations We shall continue our considerations of power limits in dynamical engines. In the dynamical case power integral involves the product of differential heat and imperfect efficiency

_ ¼ W

Z

 e _ 1  UT dT: Gc 0 T

ð27Þ

[1,2,8]. When the propelling medium consists of the radiation fluid, power maximization problem is described by Eqs. (27) and (28) below. They describe a symmetric model of power yield from radiation (both reservoirs consist of radiation). In the physical space, power exponent a = 4 for radiation and a = 1 for a linear resource. The integrand of Eq. (27) represents power intensity as the intensity of a generalized profit, f0.

In the engine mode integral (27) has to be maximized subject to the dynamical constraint (‘state equation’)

dT T a  T 0a ¼ b 0 0 e a1 dt ðU ðT =T Þ þ 1ÞT a1

ð28Þ

derived in recent publications [6–8]. As it follows from the general theory of dynamic optimization, extremum conditions for the problem involving Eqs. (23) and (24) are contained in the HJB equation of the problem

( )   e @V T a  T 0a _ c ðTÞ 1  U0 T þ @V b  max ¼ 0; G @t @T T 0 ðtÞ T0 ðU0 ðT 0 =T e Þa1 þ 1ÞT a1 ð29Þ _ and U0 ¼ Ug ðg Þ1 . As it is impossible to solve where V ¼ maxW 1 2 this equation analytically, except for the case when a = 1, we outline here a way for numerical solving based on Bellman’s method of dynamic programing (DP). Considering computer needs we introduce a related discrete scheme

 e N _ N ¼ P  G_ c ðT k Þ: 1  UT ðT k  T k1 Þ W 0k T k¼1

ð30Þ

a

T k  T k1 ¼ hk b tk  t k1 ¼ hk

T 0ka  T k

a1

ðU0 ðT 0k =T e Þa1 þ 1ÞT k

ð31Þ ð32Þ

We search for maximum of the sum (30) subject to discrete constraints (31) and (32). Applying to this problem the dynamic programing method, the following recurrence equation is obtained _ for the minimum power function R ¼ minðWÞ

(   UT e T na  T 0na Rn ðT n ; tn Þ ¼ minun ;hn G_ c ðT n Þ: 1  0n b 0 0n e a1 hn T ðU ðT =T Þ þ 1ÞT na1 !) hn bðT 0na  T na Þ n n ð33Þ ; t  h þRn1 T n  0 0n e a1 ½U ðT =T Þ þ 1T na1 While the analytical treatment of Eqs. (27) and (28) is a tremendous task, it is quite easy to solve recurrence Eq. (33) numerically. Low dimensionality of state vector in Eq. (33) assures a decent accuracy of DP solution. Moreover, an original accuracy can be improved after performing the so-called dimensionality reduction associated with the elimination of time variable tn. In the transformed problem, without tn, accuracy of DP solutions is high. Yet, if the number of state variables increases (e.g. when several concentrations xni may accompany temperature Tn as in chemical engines), DP algorithms become inefficient and inaccurate. We

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must then abandon DP approaches and turn into the Pontryagin type (Hamiltonian-based) approaches. For the Hamiltonian approaches, which apply Pontryagin’s canonical equations, problems of large dimensionality of state vectors are inessential, as outlined in the section below. Taking into account computational needs, in this paper discrete difference equations are derived as canonical equations for the system Hamiltonian, and solved by numerical methods. 6. Hamiltonian algorithms The dynamic changes for the discrete state of a multistage system can be described by a set of ordinary difference equations called the state transformations which describe the discrete state from the stage n  1 in terms of the state from the stage n and some control variables Un. The set of discrete state transformations can be written in the following general form

xn1 ¼ Tn ðxn ; t n ; Un Þ

ð33Þ

and

t

n1

n

ð34Þ

where

Un ¼ ðun ; hn Þ

ð35Þ

is an enlarged vector of control variables which includes the discrete interval of time hn, and the time variable tn is identified with any state variable growing monotonically. After defining the function n

f ¼ ðxn  Tn ðxn ; t n ; Un ÞÞ=hn

ð36Þ

the above state transformations can be transformed into the form [9,10] n

xn1 ¼ xn  f ðxn ; t n ; un ; hn Þhn

ð37Þ

and

t n1 ¼ tn  hn

ð38Þ n

As they involve the discrete rates (f , 1), we call this form the ‘‘standard form’’. A performance index describing a generalized profit is in this formalism (total power in our case) is defined by following equation N P n¼1

e n1 ðxn ; zn1 ; hn ; un Þ=@hn ¼ 0 @½hn H

ð43Þ

and

e n1 @H ¼0 @unj

ð44Þ

e n1 along a Eq. (43) implies constancy of the enlarged Hamiltonian H discrete optimal path whenever discrete rates fi are independent of hn. In addition, the energy-like Hamiltonian (without zt term) is constant for the process whose rates are independent of time tn. Under convexity properties for rate functions and constraining sets the optimal control is

e n1 ðxn ; zn1 ; un ; hn Þg hn ¼ arg maxfhn H

ð45Þ

hn

and

e n1 ðxn ; zn1 ; un ; hn Þg un ¼ arg maxf H

n

¼t h ;

_N¼ PN  W

(adjoint equations), and the equations which describe the necessary optimality conditions for decision variables un. For example, if the optimal control lies within an interior of admissible control set

f0n ðxn ; tn ; un ; hn Þhn ;

ð39Þ

where f0 is the generation rate for the generalized profit (power in the case of energy yield problems). To solve the optimization problem of extremum W, an (enlarged) Hamiltonian is defined in the following form

e n1 ðxn ; tn ; zn1 ; un ; hn Þ  f n ðxn ; tn ; un ; hn Þ H 0 þ

s P

zn1 fin ðxn ; tn ; un ; hn Þ þ zn1 ; i t

ð40Þ

i¼1

where zi are adjoint (Pontryagin’s) variables. e n1 satisfies in In an optimal process the enlarged Hamiltonian H the enlarged phase space x = (x, t) and z = (z, zt) the following equations:

e n1 xni  xn1 @H i ¼ n1 n h @zi

ð41Þ

(state equations) and

e n1 zni  zn1 @H i ¼ n @xni h

ð42Þ

ð46Þ

un

(n = 1, . . . N; i = 1, . . . s + 1 and j = 1, . . . r.) Note that the algorithm can also be used in the special case of a one-stage process, and, in this case, is capable of handling a model of a single, steady-state, power-producing unit, or, at least its reasonable approximation. Optimization theory for generalized (hn-dependent) costs and rates provides the bridge between constant-H algorithms [11–13] and more conventional ones such as those by Katz, Fan and Wang [14], Halkin [15], Canon et al. [16], Boltyanski [17], and many others [10,18]. Since, as shown by Eq. (43), control hn can be included in the Hamiltonian definition, i.e. an effective Hamiltonian can be used

Hn1 ¼ hn ;

ð47Þ n1

extremum conditions (41)–(46) can be written in terms of H related canonical set is that of Halkin

xni  xn1 ¼ i

@Hn1 @zn1 i

zni  zn1 @Hn1 i ¼ n @xni h

. The

ð48Þ ð49Þ

[15,16]. Qualitative difference between the role of controls un and hn in the optimization algorithm is then lost since they both follow from the same stationarity condition for Hamiltonian Hn1 in an optimal process. For example, in the weak maximum principle

@Hn1 @Hn1 ¼ ¼0 @unj @hn

ð50Þ

in agreement with Eqs. (43) and (44) above. Moreover, Eqs. (43) and e n1 ¼ 0 if H e n1 is independent of time (50) imply the condition H interval h. Until now Hamiltonian algorithms were used in power systems for models with h-independent discrete rates [19]. Yet, Pos´wiata and Szwast have shown many their applications in exergy optimization of thermal and separation systems, in particular fluidized dryers [9,20,21]. Sieniutycz has shown some other applications for energy and separation systems and for a minimum time problem [10]. In view of diversity of discrete rates, which may contain explicit time intervals hn as the consequence of various ways of discretizing, applications of Algorithm (36)–(44) in power or separation systems may be quite appropriate and useful. In particular, the algorithm is suitable in numerical studies of the optimal solutions for the discrete equations of the radiation

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6000

Temperature [K]

ζ

5000

4000

3000

Teng Thp T = 300 K T = 5800 K

ζ

ζ

2000 Fig. 5. Fuel flux n in a chemical engine in terms of the efficiency of power production f.

1000

Non-dimensional time

0 0

20

40

60

80

100

120

140

160

Fig. 4. Decreasing temperatures of radiation relaxing in engine mode and increasing temperature of radiation utilized in heat pump mode in terms of time, for a constant value of Hamiltonian H = 1⁄108 [J K1 m3] [8].

engine, Eqs. (30)–(32), for which the enlarged Hamiltonian follows in the form a e   T na  T 0n e n1 ¼ zn1  G_ c ðT n Þ 1  UT  b H þ zn1 0n a1 T t a1 0n e 0 T ðU ðT =T Þ þ 1ÞT n

ð51Þ Optimal dynamics corresponding to this function are shown in Fig. 4. 7. Chemical systems The simplest model of power producing chemical engine is that with an isothermal isomerization reaction, A1  A2 = 0 [3,5]. Power expression and efficiency formula for the chemical system follow from the entropy conservation and energy balance in the powerproducing zone of the system (‘active part’). Thermodynamic approaches can also be applied to chemical [3,5] and electro-chemical [22,23] engines. In chemical engines mass transports drive transformation of chemical energy into mechanical power. Yet, as opposed to thermal machines, in chemical ones generalized streams or reservoirs are present, capable of providing both heat and substance. Large streams or infinite reservoirs assure constancy of chemical potentials. Problems of maximum of power produced or minimum of power consumed are then the static problems. For a finite ‘‘upper stream’’, however, the amount and chemical potential of an active reactant decrease in time, and considered problems are those of dynamic optimization and variational calculus. Because of the diversity and complexity of chemical systems the area of power producing chemistries is broad. In an ‘endoreversible chemical engine’ total entropy flux is continuous through the active zone. When a formula describing this continuity is combined with energy balance we find in isothermal case

p ¼ ðl10  l20 Þn1 ;

ð52Þ

where the feed flux n1 equals to n, an invariant molar flux of reagents. Process efficiency f is defined as power yield per molar flux, n. This efficiency is identical with the chemical affinity of reaction in the chemically active part of the system. While f is not dimensionless, it describes correctly the system. In terms of Carnot variable, l0 , which satisfies Eq. (18),

f ¼ l0  l2 :

ð53Þ

For a steady engine the following function describes chemical Carnot control l0 in terms of fuel flux n1 and its mole fraction x

l0 ¼ l2 þ f0 þ RT ln



 x1  n1 g 1 1 : 1 n1 g 2 þ x2

ð54Þ

As Eq. (53) is valid, Eq. (54) also characterizes the efficiency control in terms of n and fuel fraction x. Eq. (54) shows that an effective concentration of the reactant in upper reservoir x1eff ¼ x1  g 1 1 n is decreased, whereas an effective concentration of the product in lower reservoir x2eff ¼ x2 þ g 1 2 n is increased due to the finite mass flux. Therefore efficiency f decreases nonlinearly with n, Fig. 5. When effect of resistances is ignorable or flux n is very small, reversible Carnot-like chemical efficiency, fC, is attained. The power function, described by the product f(n)n, exhibits a maximum for a finite value of the fuel flux, n. A related dynamical problem may also be considered [4]. Application of Eq. (54) to the Lagrangian relaxation path leads to a work functional

W ¼

  Z sf  1 X=ð1 þ XÞ þ dX=ds1 dX f0 þ RT ln ds i d x  jdX=d s s1 1 2 1 s

ð55Þ

1

whose maximum describes the dynamical limit of the system. Here X = x/(1  x) and j equals the ratio of upper to lower mass conductance, g1/g2. The path optimality condition may be expressed in terms of the constancy of the following Hamiltonian

_ ¼ RT X_ 2 HðX; XÞ

  1þX j : þ X x2

ð56Þ

For low rates and large concentrations X (mole fractions x1 close to the unity) optimal relaxation rate of the fuel resource is approximately constant. Yet, in an arbitrary situation optimal rates are state dependent so as to preserve constancy of H in Eq. (56). 8. Electrochemical engines: fuel cells Experimental work has been completed at the Institute of Power Engineering. A fuel cell, Fig. 6, is an electrochemical energy converter which directly and continuously transforms a part of chemical energy into electrical energy by consuming fuel and oxidant. Fuel cells have attracted great attention by virtue of their inherently clean and reliable performance. Their main advantage as compared to heat engines is that their efficiency is not a major function of device size. Power maximization approaches can be applied to many electrochemical systems, in particular to fuel cells [22,23]. Validation of the presented thermodynamic modeling in the context of fuel cells is based on the organization of FC power

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S. Sieniutycz et al. / International Journal of Heat and Mass Transfer 55 (2012) 3984–3994 1.2

Cell voltage [V]

1

Temperature 8000C

0.8 0.6 100% H2 20% H2

0.4

60% H2 40% H2

0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Current density [A/cm 2 ] Fig. 7. Voltage performance of the SOFC cell at the temperature 800 °C. Fuel: H2+N2. Points refer to experiments in the Fuel Cell Department at the Warsaw Institute of Power Engineering, [24]. These data were applied for the purpose of the validation of the SOFC thermodynamic model in the forthcoming Blesznowski’s PhD thesis, supervised by one of the present authors (SS).

Fig. 6. Principle of a solid oxide fuel cell.

experiments and identification of power maxima in terms of control variables. These issues are described in the forthcoming Blesznowski’s Ph.D. thesis and Wierzbicki’s Ms.D work [23], both supervised by one of the present authors (SS). In these experiments a number of voltage and power measurements was made in terms of the following control variables of SOFC:  working cell temperature,  various flows of fuels,  different components of fuel. A complete content of the theory and experiments also includes application of the Aspen Plus™ software for simulation purposes, and is presented in the report [24]. Voltage lowering in fuel cells below the reversible value is a good measure of their imperfection. Reversible cell voltage E0 is usually a reference basis calculated from the Nernst equation. Yet, in more general cases, actual voltage without load must take into account losses of the idle run, which are the effect of flaws in electrode constructions and other imperfections. In [23] the operating voltage of a cell is evaluated as the departure from the idle run voltage E0

V ¼ E0  V int ¼ E0  V act  V conc  V ohm

ð57Þ

Losses, which are called polarization, include three main sources: activation polarization (Vact), ohmic polarization (Vohm), and concentration polarization (Vconc). As shown by the above equation, ratio of the actual voltage and the reversible voltage E0 is a suitable measure of cell efficiency, whereas the corresponding voltage decrease is the quatitative measure of the dissipation effects in the FC systems, Fig. 7. Power density is the product of voltage V and current density i. Large number of approaches for calculating polarization losses has been presented in literature, as reviewed in [22]. Experiments

show power maxima in fuel cells [22,23]. Activation and concentration polarization occurs at both anode and cathode locations, while the resistive polarization represents ohmic losses throughout the cell. As the voltage losses increase with current and power is the product of voltage and electric current, the initially increasing power begins finally to decrease for sufficiently large currents, so that maxima of power are observed [22,23]. The data include the losses of the idle run attributed to the flaws in electrode constructions and other imperfections. Entropy and power generations in fuel cells are compared in Fig. 8. Power curve has two points in which power production vanishes in the system. The first point, corresponding to reversible behavior of the system is also called the open circuit point; the electric current and entropy production vanish at this point. The second point at which power vanishes is called Newton–Fourier point or short circuit point. It corresponds with the situation when currents are so large that only irreversible phenomena are present in the system. In the ‘‘short circuit’’ case all currents flow only by resistances, and there is no power production for any value of efficiency f0, in spite of possible chemical reaction. Only entropy is then produced. In a special case when f0 = 0 entropy production at the ‘‘short circuit point’’ corresponds with the situation without chemical reaction and power generation. Only at the short circuit point and for the associated absence of power yield, the entropy production at this point (modulo to multiplier T1) is equal to the product of the reaction rate and its chemical affinity. This is a classical result, which, however, does not hold when the system produces power (i.e. belongs to the class of ‘‘active systems’’). Fig. 9 depicts curves of power density of a SOFC fuel cell for various hydrogen content in the fuel at the temperature 800 °C, whereas Fig. 10 characterizes the thermodynamic efficiency of the SOFC system for various hydrogen content in the fuel. Limiting parameters of a reference fuel cell at the temperature 800 °C are summarized in Table 1 below. 9. Assessing power limits in thermo-electro-chemical engines Validity of FC models in the thermodynamic framework allows for assessment of power limits in general thermo-electrochemical systems, as outlined below. Let us focus on fuel cells described by the formalism of inert components [25,26] rather than the ionic description [27]. Assume, for simplicity, that the active (power producing) driving forces

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Fig. 8. Comparison of qualitative characteristics of entropy production and power in terms of electric current in fuel cells.

300

300 100% H2

250

Temperature 8000C

Power density [mW/cm2 ]

Power density [mW/cm 2 ]

100% H2

60% H2

200 20% H2

40% H2

150 100 50 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

250 200 40% H2

150 20% H2

100 50 0

0.8

0.1

Current density [A/cm 2 ]

0.2

0.3

0.4

0.5

0.6

0.7

Thermodynamic efficiency of SOFC electrochemical reaction.

Fig. 9. Power density curves of a SOFC fuel cell for various fuels at the temperature 800 °C points refer to experiments in the Fuel Cell Department at the Institute of Power Engineering, [24]. These data were applied for the purpose of the validation of the SOFC thermodynamic model in the forthcoming Blesznowski’s Ph.D. thesis, supervised by one of the present authors (SS).

Fig. 10. Example of data describing power density of a SOFC in terms of the firstlaw cell efficiency in temperature 800 °C [24].

Table 1 Summary of limiting parameters of a reference fuel cell at the temperature 800 °C.

involve only: one temperature difference T 10  T 20 , single chemical affinity l10  l20 and the operating voltage /1  /20 . Total power production is the sum of thermal, substantial and electric components, i.e.

P ¼ ðT 10  T 20 ÞIs þ ðl10  l20 ÞIn þ ð/10  /20 ÞIe ¼ ðT 1  T 2 ÞIs þ ðl1  l2 ÞIn þ ð/1  /2 ÞIe  Rss I2s  Rnn I2n  Ree I2e  Rsn Is In  Rse Is Ie  Rne In Ie :

Temperature 8000C 60% H2

ð58Þ

Eq. (58) represents linear thermo-electro-chemical systems. Linear systems are those with constant (current independent or flux independent) resistances or conductances. They satisfy Ohm type or Onsager type laws linking thermodynamic fluxes and thermodynamic forces (dissipative driving forces which are represented by products RikIk in Eq. (58)). While many fuel cell systems are nonlin-

a

Gas flow

Maximum power density of fuel cell [mW/cm2]

Current density at MDPa [A/cm2]

Voltage of fuel cell [V]

(a) 200 ml/min H2 (b) 120 ml/min H2 + 80 ml/min N2 (c) 80 ml/min H2 + 120 ml/min N2 (d) 40 ml/min H2 + 160 ml/min N2

260 254

0.55 0.53

0.47 0.48

246

0.49

0.50

191

0.34

0.56

MDP–Maximum Density of Power.

ear, i.e. possess current dependent resistances, the dependence is often weak, so the linear model can be a good approximation. Below we develop a simple theory of power limits for these systems.

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After introducing the enlarged vector of all driving potentials

l~ ¼ ðT; l; /Þ, the flux vector eI of all currents and the overall resis-

e Eq. (58) can be written in a simple matrix–vector tance tensor R, form

e : eIeI: ~1  l ~ 2 Þ:eI  R P ¼ ðl

ð59Þ

Maximum power corresponds with the vanishing of the partial derivative vector

e eI ¼ 0: ~1  l ~ 2  2 R: @P=@eI ¼ l

ð61Þ

This result means that the power-maximizing current vector eI mp in strictly linear systems equals one half of the purely dissipative current at the Fourier–Onsager point, eI F at which no power production occurs. Moreover, we note that Eqs. (59) and (61) yield the following result for the maximum power limit of the system

Pmp ¼

1 e 1 ðl ~1  l ~2Þ  R ~1  l ~ 2 Þ: ðl 4

ð62Þ

In terms of the purely dissipative flux vector at the FourierOnsager point, eI F the above limit of maximum power is represented by an equation.

Pmp ¼

1e ee R : IF IF : 4

ð63Þ

Of course, the power dissipated at the Fourier-Onsager point equals

e : eI F eI F : PF ¼ R

different way than the power ratios Pmp/PF satisfying Eqs. (63) and (64). In fact, the considered power ratios represent some specific, second-law efficiencies of the overall thermo-electro-chemical process. Other second law efficiencies can also be defined. One of the most correct, simplest, and practical definition of efficiency for a fuel cell operating near ambient temperature is the ratio of the actual voltage to the reversible voltage. 10. Conclusions

ð60Þ

Therefore, the optimal (power-maximizing) vector of currents at the maximum point of the system can be written in the form

1 ~Imp ¼ 1 R e 1 :ðl ~1  l ~ 2 Þ  ~IF : 2 2

3993

ð64Þ

Eqs. (63) and (64) prove that, in linear systems, only at most 25% of power (64), which is dissipated in the natural transfer process, can be transformed into the noble form of the mechanical power. This is a general result which, probably, cannot be easily generalized to the nonlinear transfer systems where significant deviations from Eq. (63) are expected depending on the nature of diverse nonlinearities. Despite of the limitation of the result (63) to linear transfer systems its value is significant because it shows explicitly the order of magnitude of thermodynamic limitations in power production systems. The above analysis also proves that a link exists between the mathematics of the thermal engines and fuel cells, and also that the theory of fuel cells can be unified with the theory of thermal and chemical engines. Explanation of related physical effects is in order. While power ratios involving Eqs. (63) and (64) can be regarded as some efficiency measures, they should not be mixed with commonly used, popular efficiencies, especially with first law efficiencies. There is a number of definitions of FC efficiencies, based on first or second laws, proposed for measuring and comparing the performance of electrochemical processes. Only second-law efficiencies are correct measures which show how close the process approaches a reversible process. Efficiencies based on the first law such as reversible effi_ 1 _ DHÞ ciency g ¼ DGðDHÞ1 or tank-to-wheel efficiency g ¼ Wð often found in the literature) can generate efficiency values greater than 100% for certain systems depending on whether the change in entropy for the overall chemical reaction involved in the process is positive or negative. See, for example, paper [28] on various definitions of FC efficiencies. _ DHÞ _ 1 The popular fuel cell efficiencies gFC = DG/DH or g ¼ Wð which are commonly applied to many fuel cell systems, can easily achieve numerical values much higher than 1/4 [power ratio of Eqs. (63) and (64)]. They are first-law efficiencies defined in a

The main methodological novelty of the paper lies in its synthesizing nature of its approach which includes fuel cells to the common class of thermodynamic power yield systems. In the thermodynamic optimization, considered here, i.e. the optimization applying thermodynamic constraints and performance criteria, thermodynamic synthesis means an idea of combining various partial optimization models into a ‘‘synthesizing’’ (not necessarily ‘‘generalizing’’) model from which performances of all the component units can be predicted. We have shown that, with irreversible thermodynamics, we can predict the performance behavior and power limits for quite diverse practical systems. In comparison with the previous publications, canonical (Hamiltonian) treatment of optimal dynamical processes constitutes a novel approach which should be contrasted with the dynamic programing approaches (DP approaches) developed in the earlier publications (such as those in Refs. [1–5]). Since the analytical treatment of differential optimization models (such as those in Eqs. (28) and (29) of the present paper) is most often a very difficult task, solving techniques used previously treated numerically Bellman’s recurrence equation of a dynamic programing model (DP model). However, DP approaches are effective only for low dimensionality of the state vector. As the number of state variables in chemical systems is usually large (many concentrations may accompany temperature and catalyst activities), DP algorithms become inefficient and inaccurate in real systems. Therefore, in this paper, we abandoned DP approaches and developed the Pontryagin type (Hamiltonian-based) approaches, Section 5. The novel technical core of the present work is the development and application of Pontryagin’s type (Hamiltonian-based) approaches to optimal power generation systems. As opposed to the dynamic programing algorithms (DP algorithms, applied earlier), Hamiltonian algorithms, which involve discrete or difference equations rather than DP recurrence equations, are particularly effective in power systems with large dimensionality of the state vector. There is also a novelty in the mathematical structure of Hamiltonian algorithms applied here, which admit discrete process rates as entities explicitly dependent on time intervals, h. Until now Hamiltonian algorithms were used in power systems quite seldom, and were limited to models with h-independent discrete rates [19,29,30]. This research provides data for power production limits which are enhanced in comparison with those predicted by the classical thermodynamics. In fact, thermo-static limits are often too far from reality to be really useful. Generalized limits, obtained here, are stronger than those predicted by the thermostatic theory. As opposed to classical thermodynamics, generalized limits depend not only on state changes of resources but also on process resistances, process direction and mechanism of heat and mass transfer. Extending ideas initiated in [31] common methodology was developed for thermal, chemical and electrochemical systems. Fuel cells are included into this methodology. Acknowledgment This research was supported by a Grant NN208 019434 from The Polish Ministry of Science, entitled Thermodynamics and

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Optimization of Chemical and Electrochemical Energy Generators with Applications to Fuel Cells.

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