Download Optimization of multistage electrochemical systems of

Optimization of Multistage Electrochemical Systems of Fuel Cell Type
by Dynamic Programming
STANISLAW SIENIUTYCZ
Faculty of Chemical and Process Engineering
Warsaw University of Technology
PL 00-645, 1 Waryńskiego Street, Warsaw
POLAND
Abstract: - In this paper we transfer to the realm of multistage electrochemical systems of fuel cells type a method of
thermodynamic optimization that was developed earlier for thermal machines (engines and heat pumps), aimed at
maximum production of power. With the thermodynamic knowledge and dynamic optimization (dynamic programming
and maximum principles) kinetic limits are estimated for the optimal work function Wmax that generalizes the familiar
maximum reversible work WE for the realm of finite rates.
Key-Words: - maximum work, fuel cells, dynamic programming, entropy, optimization.
1 Introduction
The purpose of this paper is to outline how the field of
irreversible thermodynamics (including the so-called
finite-time thermodynamics) can contribute to the
thermodynamic theory of efficiency and work generation
in irreversibly working fuel cells. A method based on
optimal control theory and finite-time thermodynamics
extends to electrical systems an optimization approach
that was recently worked out for heat engines and heat
pumps. Constraints take into account dynamics of heat
and mass transport and rate of real work production.
Finite-rate models include irreducible losses of classical
work potential, caused by resistances and overvoltage.
The performance criterion suitable for the
thermodynamic analysis of an irreversible fuel cell is its
entropy production S in a functional form that describes
a real cell from which the power delivery takes place with
a finite rate. The integral functional S or its discrete
analogue (a sum) quantify respectively continuous and
discrete models of dissipation that occurs due to chemical
reactions and mass transfer coupled with transfer of heat.
By minimizing S inevitable losses of electrical power
and limits on the work generation and reduction of the
cell voltage are determined.
thermodynamics, finite time thermodynamics and exergy
analysis had proven their potential when evaluating limits
on power production, refs. [1]-[23]. In particular, it was
shown that a vast set of energy systems which work with
finite resources can be analyzed as multistage devices that
convert the energy of heat and chemical reaction into
mechanical and/or electrical energy, thus producing
efficiently power with a finite rate and in irreversible way.
The analysis of fuel cells performed here follows this
methodology. One general achievement of these
investigations is the establishment of common
thermodynamic schemes that distinguish energy
consumption devices (separators, chillers and
electrolyzers; Fig. 1) from energy production units (cells
and engines; Fig. 2).
 chemical potentials and temperature
μ1' , T1'
j1

r1
 ---––---–––––––––---- , T
–---------------------------------1 1
-- e, T ----------------------–––
---j2 ––––––––--------------2 , T2

r2

e

G0
2 Problem Formulation
Recent research in the field of energy systems (in
particular thermal engines and heat pumps) has shown a
considerable analogy between formal descriptions of
chemical, thermal and electrical systems, in both
stationary and non-stationary cases. Network

matter and entropy
P

'
'2 , T 2

Fig. 1: Relations between basic thermodynamic
parameters on the T-S chart for an electrolyzer type
device or a heat pump (C ≤  ≤ 1).
Fig. 2: Relations between basic thermodynamic
parameters on the T-S chart for a cell type device or a
thermal engine (0≤  ≤ C).
Other successful result was the development of
analytical expressions quantifying power produced or
consumed, P, in terms of current j and predicting a
maximum of P in the regime of moderate j before the
dissipations begins to prevail (Fig. 3). This has recently
been confirmed in experiments, refs. [9, 10] as shown in
Fig. 3.
Here we apply these tools to problem of extremum of
electrochemical work at flow. Fig. 4 illustrates two basic
cases of the problem formulation.
For finite resources, a dynamical method can be
developed that leads to evaluation of extremum of power
generated in a sequential process in which a finite
resource at flow interacts with a reservoir in a finite time,
Figs. 4 - 7. From this theoretical scheme we may derive
the traditional exergy function as the reversible maximum
work and its irreversible generalizations.
Fig. 5. A scheme of a multistage control described by the
traditional backward algorithm of the dynamic
programming method. Elipse-shaped balance areas
pertain to sequential subprocesses which grow by
inclusion of proceeding units.
Fig. 3: Maximum of power in a fuel cell.
Tools of optimal control theory are central to formulating
and solving problems of optimal trajectories and optimal
decisions required by irreversible or finite-time
thermodynamics,
thermo-economics,
availability
analysis, entropy source minimization, and variational
formulations for irreversible equations of motion.
Fig. 6. Power yield in a sequence of engines or
battery-type units
Fig. 4. Two works: work associated with the energy
generation and that of energy consumption are different
in an irreversible process.
Fig. 7. Work limits for reversible and real electrolyzers
and batteries.
3 Optimization Techniques
When the affinity component of the reaction substrates,
s, is greater than that of the products, p
in the
“engine mode” the chemical affinity decreases along the
process path, and the system delivers work. In classical
problems rates and flows vanish due to the reversibility;
here, however, finite rates and inherent irreversibilities
are admitted. Methods of dynamic programming and
maximum principle are used to accomplish the multistage
optimization, Fig. 5. Both methods are discussed in detail
in refs. [21]-[23]. The dynamic programming represents
the description in terms of wave-fronts which are in our
case surfaces of constant specific work or power per unit
flow of reagents. On the other hand, the method of
maximum principle (or a similar method of variational
calculus) constitute the description in terms of (optimal)
process trajectories, which characterize the state changes
of the process reagents at flow. For a sequence of a
number of fuel-cell processes (a stack) a nonlinear model
describing the reactants evolution leads to optimal work
at flow as a finite-time exergy of the system. This exergy
has to be determined in terms of number of transfer units 
or a Hamiltonian h, the latter being a common measure of
the optimal process intensity (the same for each point of
the path). Whereas the number  is a measure of the
residence time of flowing reagents, the quantity h
quantifies the minimal irreversibilities in the system. In
the block scheme of Fig. 5, X represents the state vector of
an energy resource at flow (i.e. hydrogen) and u refers to
a set of control variables (e.g. currents). The
computational block scheme in Fig. 5 constitute abstract
(multistage) representation of the power production
process depicted in Fig. 6. A costlike criterion defined as
the sum ( l 0n + h)n is minimized, where l 0n is the
Lagrangian describing the original costs. A computer
generates tables of optimal controls and optimal costs by
solving a recurrence equation
R*n (T n , X n )  min {( lon (T n , Y n , u n , v n )  h) n
n n
n
u , v ,
 R (T  u  n , Y n  v n n )}.
n
*
n
n
n
n
for the optimal cost function. In this equation Xn= T , Y
are the state variables (temperature and concentrations),
whereas the controls un and vn are the rates corresponding
with change of these variables along the multistage
electrochemical reactor. Any equation of this sort does
n
0
not contain the time  . Some of the end coordinates (T ,
0
N
N
Y ) and (T , Y ), composed of temperature and
N
concentrations, may be fixed, but the total duration,  ,
must be free, consistent with the dimensionality
reduction. In an optimal process this duration follows for
an assumed h as a function of fixed end values and total
number of stages, N. Accuracy of results is much better
n
when the state variable  is excluded, i.e. when the
n
problem is described by only two state variables, T and
n
Y . The recurrence equation serves to generate numerical
generalizations of function R when both the transfer
coefficients and the heat capacity vary along the process
path, and an analytical solution cannot be obtained.
Enhanced limits on power production in cells (power
consumption in electrolyzers) are illustrated in Fig. 7 in
terms of internal irreversibility factor I.
4 Conclusion
In this work the fuel cell system has been analyzed as a
multistage thermodynamic device that converts the
energy of chemical reaction directly into electricity and
heat, thus producing efficiently power with a finite rate
and in irreversible way. The dynamic programming (Fig.
5) was the main tool applied in computations. The
developed analysis is similar to a primary battery, except
that the energy source is not stored internally but it is
continuously provided in the form of fuel such as
hydrogen and an oxidant such as oxygen. For practical
use individual cells are grouped into stacks (modules)
thus creating the multistage system in which stages are
connected electrically to ensure a practical voltage and
power output. A method based on irreversible
thermodynamics extends to electrical systems an
optimization approach which was recently worked out for
multistage heat engines and heat consumers. Constraints
take into account dynamics of heat and mass transport and
rates of real flows. Finite-rate models include irreducible
losses of classical work potential, caused by resistances
and overvoltage. The performance criterion in the
thermodynamic analysis of an multistage fuel cell system
is its entropy production S in a form that describes a real
stack from which the power delivery takes place with a
finite rate. The functional S in a discrete form (a sum).
constitutes a discrete model of the operation which occurs
due to chemical reactions and mass transfer coupled with
transfer of heat. A similar expression for S also appears
in the realm of other energy converters, as a
representation of their lost work divided by the
temperature. The optimization of S eliminates all
controls from S, thus generating a potential function R
(X, XB, B - A) = min S which depends only on initial
and final states and the extensive transport parameter
called the number of the transfer units. Yet, in the fuel cell
case, a more practical criterion is usually applied instead
of S; the criterion of the real work produced, W. The
optimizations of S and W are related by the familiar
Gouy-Stodola law which links the lost work with the
entropy production. Due to a finite S the actual cell
voltage U and any finite-rate work W are less than those
for the ideal cell because the losses associated with cell
polarization and Ohmic losses. In general no simple rule
exists for the optimal control of the system subject to
external adjustable decisions. Yet, the optimal solution
for work W implies often a nearly constant intensity of
the entropy production along an optimal path. Such a
simple strategy is, however, valid only when no
constraints are imposed on the control parameters.
Post-quadratic terms and nonlinearities in kinetic
equations cause the violation of this strategy. Thus, with
thermodynamic knowledge, enhanced limits are
estimated for the optimal work function, Wmax, that
generalizes the familiar maximum reversible work WE
for the realm of finite rates. As the final result of
thermodynamic analysis and optimization determined are
minimal inevitable losses of power and reduction of the
cell voltage.
X –state variable of controlled phase
x- transfer area coordinate
' - overall heat transfer coefficient
 = p/q1 -effective efficiency
n- free interval of an independent variable or time
interval at stage n
k- chemical potential of k-th component
 - nondimensional time, number of the heat transfer units
(x/HTU)
Subscripts
g-gas
i-th state variable
1,2- first and second fluid
Superscrits
e- environment, equilibrium
f - final state
i-initial state;
k or n - number of k-th or n-th stage
Acknowledgements
This work was supported in 2003 from the Warsaw TU
grant Dynamics of Complex Systems (stage II of the
project) as the introductory theoretical part to the KBN
grant, project T09.
Nomenclature
A - available energy (exergy)
c- specific heat at the constant pressure
G- mass flux, total flow rate
g1, g - partial and overall conductance
HTU-height of transfer unit
hn - solid enthalpy at stage n
k-mass transfer coefficient
N-total number of stages in the process
n - current stage number of the process
Pn, pn - cumulative power output and power output at
n-th stage
q1-driving heat in the engine mode of stage
Rn(x, t) - optimal work function of cost type
S - entropy of controlled phase
S - specific entropy production
T - temperature of controlled phase
Te -constant temperature of reservoir
T' - temperature of controlling phase
t - physical time, contact time
un - rate of change as the control variable
V  maxW -optimal work function
W  P/G - total specific work or total power per unit mass
flux
Wn - total specific work from n-stage system
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