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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 References: [1]K.G.Denbigh, The Second Law Efficiency of Chemical Processes, Chem. Eng. Science 6, 1956, 1-9. [2] A. Bejan, N. Dan, Analogy Between Electrical Machines and Heat Transfer–Irreversible Heat Engines, Intern. J. Heat Mass Transfer, 39, 1996, 3659-3666. [3] Bejan A, Dan N, Maximum Work from an Electric Battery Model, Energy, 22, 1997, 93-102. [4] K.W. Bedringas, I.S. Ertesvag, S. Byggstoyl, and B.F. Magnussen, Exergy Analysis of Solid-Oxide Fuel Cell (SOFC) Systems, Energy, 22, 1997, 403-412. [5] J.H Hirschenhofer, D.B. Stauffer, R.R. Engleman and M.G. Klett, Fuel Cell Handbook, Fourth Edition, Parsons Corporation, Reading PA, 1998. [6] D. Singh D., D.M. Lu and N.A. Djilali, Two-Dimensional Analysis of Mass Transport in Proton Exchange Membrane Fuel Cells. Int. J. Engng Science 37, 1999, 431-452. [7] Ch. Fellner and J. Newman High-Power Batteries for Use in Hybrid Vehicles, J. Power Sources 85, 2000, 229-256. [8] C. Haynes, W.J. Wepfer, “Design for Power” of a Commercial Grade Tubular Solid Oxide Fuel Cell. Energy Convers. Mgmt, 41, 2000, 1123-1139. [9] M.S. El-Genk and J.M. Tournier, Design Optimization and Integration of Nickel/Haynes-25 AMTEC Cells into Radioisotope Power Systems. Energy Convers. Mgmt, 41, 2000, 1703-1728. [10] Y. Ando, T. Tanaka, T. Doi and T.A.Takashima, A Study on a Thermally Regenerative Fuel Cell Utilizing Low-Temperature Thermal Energy, Energy Convers. Mgmt, 42, 2001, 1807-1816. [11] Cownden R., Nahon M., Rosen M. Exergy Analysis of Fuel Cells for Transportation Applications, Exergy Int. J. 1, 2001, 112-121. [12] J.K. McKusker, Fuel from Photons, Science 293, 2001,1599-1600. [13] Z. Shi, J. Chen and Ch. Wu, Maximum Work Output of an Electric Battery and its Load Matching. Energy Convers. Mgmt, 43. 2002, 241-247. [14] M.R von Spakovsky and B. Olsommer, Fuel Cell Systems and System Modeling and Analysis Perspectives for Fuel Cell Development, Energy Convers. Mgmt, 43, 2002, 1249-1257. [15] N. Lior, Thoughts about Future Power Generation Systems and the Role of Exergy Analysis in their Development, Energy Convers. Mgmt, 43, 2002, 1187-1198. [16] K.-H. Hoffmann J. M. Burzler and M. Schubert, Endoreversible Thermodynamics, J. of Non-Equilibrium Thermodynamics 22, 1997, 311-355. [17] S.Sieniutycz, Hamilton-Jacobi-Bellman theory of dissipative thermal availability, Physical Review 56 1997, 5051-5064. [18]S. Sieniutycz and R.S. Berry, Discrete Hamiltonian Analysis of Endoreversible Thermal Cascades. Chap. 6 in book: Thermodynamics of Energy Conversion and Transport, eds. S. Sieniutycz and A. de Vos, Springer N.Y., 2000, pp. 143-172. [19]F. L. Curzon and B. Ahlborn, Efficiency of Carnot engine at maximum power output. Amer. J. Phys., 43 (1975) 22-24. [20]J.Chen, Z.Yan, G. Lin, and B. Andresen, On the Curzon-Ahlborn efficiency and its connection with the efficiencies of real heat engines, Energy Convers. Mgmt, 42, 2001, 173-181. [21] R.S. Berry, V.A. Kazakov, S. Sieniutycz, Z. Szwast and M.A. Tsirlin, Thermodynamic Optimization of Finite Time Processes, Wiley, Chichester, 2000. [22] S. Sieniutycz, Optimization in Process Engineering. 1-st edn, Wydawnictwa Naukowo Techniczne, Warsaw, 1978. [23]S. Sieniutycz, Hamilton-Jacobi-Bellman Framework for Optimal Control in Multistage Energy Systems. Physics Reports 326, issue 4, March 2000, pp. 165-285, Elsevier, Amsterdam, 2000 (ISBN: 0370-1573).