Methodology for the Design of Man-Portable ... - ACS Publications

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Methodology for the Design of Man-Portable Power Generation Devices Alexander Mitsos,† Benoıˆt Chachuat,‡ and Paul I. Barton*,† Department of Chemical Engineering, Massachusetts Institute of Technology, 66-464, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139, and Laboratoire d’Automatique, EÄ cole Polytechnique Fe´ de´ rale de Lausanne, Switzerland

An overview of a comprehensive methodology for the design of microfabricated fuel cell systems for portable power generation is presented. The methodology is based on a decomposition into three levels of modeling detail: (i) system-level models for process synthesis, (ii) intermediate-fidelity models for optimization of sizes and operation, and (iii) detailed computational fluid dynamics (CFD) models for geometry improvement. Process synthesis, heat integration, and layout considerations are addressed simultaneously through the use of lumped algebraic models, general enough to be independent of detailed design choices, such as reactor configuration and catalyst choice. At the intermediate-fidelity level, space-distributed models are used, which allow optimization of unit sizes and operation for a given process structure without the need to specify a detailed geometry. Finally, the use of detailed CFD facilitates geometrical improvements as well as the derivation and validation of modeling assumptions that are used in the system-level and intermediate-fidelity models. 1. Introduction The widespread use of portable electric and electronic devices increases the need for efficient autonomous man-portable power supplies (up to ∼50 W). Currently, batteries are the predominant technology in most applications. However, batteries have a large environmental impact, high cost, and relatively low gravimetric (Wh/kg) and volumetric (Wh/L) energy density (on the order of a few hundred Wh/L and Wh/kg for rechargeable batteries). Moreover, the upper limit on performance is now being reached.1 A promising alternative is to use common fuels/ chemicals, such as hydrocarbons or alcohols, as an energy source. There is great military2 and civilian interest in developing battery alternatives that are based on these fuels and portable fuel-cell systems. Recently, microchemical systems have received special attention3 and significant advances have been made. Chemical units such as reactors, separators, and fuel cells with feature sizes in the submillimeter range have been considered for a variety of applications. Microchemical systems have several advantages, compared to macroscale processes: (1) The increased heat- and mass-transfer rates at the microscale allow higher yields.4 (2) The small holdup, combined with the controlled conditions, allow reaction pathways deemed too dangerous for conventional processes. (3) The small quantities required and the possibility of parallelization have sparked interest in micro-total-analysissystems (lab-on-a-chip).5 Currently, most of the microreactors are not stand-alone devices, but rather are used within a conventional laboratory. The replacement of batteries for electronic devices requires truly man-portable systems, and, therefore, the use of microfabrication technologies is plausible, because a minimal device size is desired. * To whom correspondence should be addressed. Tel.: 617-2536526. Fax: 617-258-5042. E-mail address: [email protected]. † Department of Chemical Engineering, Massachusetts Institute of Technology. ‡ Laboratoire d’Automatique, E Ä cole Polytechnique Fe´de´rale de Lausanne.

The area of man-portable power generation is extremely active. Mitsos and Barton6 have provided a collection of ∼150 contributions, mainly in journal articles. Holladay et al. recently performed a literature review on hydrogen production;7 Maynard and Meyers co-authored another review article.8 There are several academic programs that have explored microfabricated fuel-cell systems, including Massachusetts Institute of Technology (MIT),9-11 University of Illinois at Urbana-Champaign (UIUC),12-14 Institut fu¨r Mikrosystemtechnik (IMTEK), 15 Battelle,16,17 Bell Laboratories,18 Lawrence Livermore Laboratories,19 Eidgeno¨ssiche Technische Hochschule Zu¨rich,20 and California Institute of Technology (Caltech).21 Also, several companies (such as Motorola, Toshiba, Casio, Fujitzu, NEC, Sanyo, and Mesoscopic Devices) have research projects with the objective of developing miniature fuel cells,22-27 focusing mostly on the direct methanol fuel cell (DMFC). The vast majority of publications involve fabrication issues. There are a few contributions on basic scaling considerations28-31 and some contributions on detailed modeling.32-34 Previous work35,36 has presented a framework for the comparison of alternative fuels, fuel cells, and reaction pathways, which includes heat integration and layout considerations. We previously proposed a methodology for the optimal sizing and steady-state operation of micropower devices,37 and we considered its extension to transient operation in other research38,39 and to design for variable power demand in Yunt et al.40,41 This paper combines these approaches, together with detailed modeling, into a methodology for the overall design of man-portable power generation systems. With current computational capabilities and available algorithms, it is impossible to solve for the optimal design and operation in one step, because the devices that are considered involve complex geometries, multiple scales, time dependence, and parametric uncertainty. The methodology proposed here is based on a decomposition into three levels of modeling detail, namely (i) system-level models for process synthesis, (ii) intermediatefidelity models for optimization of component sizes and operation, and (iii) detailed computational fluid dynamics (CFD) models for geometric design and justification of modeling assumptions. In the following, we first give an overview of the methodology, followed by a description of the three subtasks,

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Figure 1. Overall methodology and its subtasks.

along with case studies for each subtask. We conclude the paper by exploring the potential for future work. 2. Methodology Overview Our methodology is formulated with the goal of harnessing and adapting the knowledge basis from macroscale process synthesis, design, and operation. A one-to-one transfer is not possible, because of different objectives, relevant physical phenomena, and limitations in fabrication. For instance, process synthesis at the macroscale is usually performed in stages,42,43 by first specifying the input-output structure of the process, then the recycle structure, then the separation system, and finally, the heat recovery network. The physical layout is performed in the late stages of process design and is primarily driven by safety considerations. This hierarchical decomposition is possible because different units can operate essentially independently from each other; this is a fact that has led to the unit-operations paradigm. At the microscale, a different design paradigm is necessarysthat of closely interconnected components of an integrated process. Therefore, it is necessary to consider heat integration and layout in the early stages of the process design simultaneously with the input-output structure. Although there are recent advances in multiscale methods that are intended to couple modeling at different scales automatically (see, e.g., Braatz et al.44), with the current computational capabilities and available algorithms, it is impossible to solve for the optimal design and operation in one step, because the devices considered involve complex geometries, multiple scales, time dependence, and parametric uncertainty. Therefore, our methodology is based on a decomposition into three levels of modeling detail, namely (i) system-level models for process synthesis, (ii) intermediate-fidelity models for optimization of sizes and operation, and (iii) detailed CFD models for geometric design. This decomposition is illustrated in Figure 1, along with examples of the connection between the three subtasks. Process synthesis, heat integration, and layout considerations are addressed simultaneously with the use of algebraic models that are general sufficient to be independent of technological details, such as the catalysts used or the reactor configuration. Because the models are general and relatively simple, devices and reaction pathways at an early stage of development can be modeled. Through the use of simulation and parametric mixedinteger optimization, the most promising process structures, along with idealized layouts, are selected among thousands of alternatives.35,36 The system-level analysis provides estimates on the size of the device and limits of performance and can be used to determine at an early stage if the development of a proposed device is worth pursuing. As an example, the use of methane, which has been proposed in the literature, is shown to be marginally competitive with existing battery technologies, because of the storage requirements, even assuming high conversion efficiencies.45

At the intermediate-fidelity level, we use distributed models, which allow optimization of unit sizes and operation (steadystate and transient) for a given process structure without the need to specify a detailed geometry. The resulting models involve partial differential-algebraic equations (PDAEs) and the mathematical programming formulations applied include local and global dynamic optimization in single- and two-stage programs. The models used are rigorous and based on validated kinetic models. This level of modeling detail is particularly useful for technologies with a demonstrated proof-of-principle. Finally, the use of detailed two- and three-dimensional CFD allows geometrical improvements as well as the derivation and validation of modeling assumptions that are used in the systemlevel and intermediate-fidelity models. The development of these models requires specification of the geometry and, therefore, benefits from collaboration with fabrication efforts. The convergence of such models is time-consuming and not robust; therefore, it is only possible to consider small variations in the geometry, and this is done based on simulations, as opposed to embedded in mathematical programming formulations. One of the major findings from CFD models is that, for a class of devices, the temperature in the active regions (reactor, etc.) is essentially spatially uniform; this observation is also supported by scaling analysis and preliminary experimental results. The decomposition into three levels of modeling detail is made, with respect to the different considerations at each scale, and the coupling between the three levels is made using engineering judgment. The communication channels indicated in Figure 1 can be extended if needed. The chosen decomposition not only makes the overall problem computationally tractable, it also allows interactions with experimental efforts from collaborations or from literature results, see Figure 2 for examples. At the system level, the set of alternatives considered is based on fabrication limitations, and system-level considerations can be used to determine on which processes fabrication efforts should focus, as described in Section 3.1. Catalysis and reaction engineering efforts can provide lumped reactions to be used in the system-level models, whereas a resource allocation formulation at the system level can suggest the reactions on which catalysis effort should focus. For detailed modeling, an initial geometry can be provided by reactor engineering efforts, and CFD analysis can suggest improvements on this geometry. At the intermediate-fidelity level, kinetic models and limits of operating conditions are required and provided by reaction engineering and material characterization efforts. Conversely, intermediate-fidelity models provide optimal sizing of components and operating conditions. 3. System-Level Analysis The choice between alternatives at the system level is based on the notion of a superstructure from macroscale process design. The superstructure is a construct that contains all the alternatives to be considered in the selection of an optimal

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Figure 2. Examples of interactions with experimental efforts.

process structure.42 An actual process structure is a subset of the units and connections in the superstructure. While on the macroscale level, there are few limitations for process synthesis, at the microscale level, only relatively simple processes are possible.46,47 The set of alternatives considered here is formulated with the constraint that the realization of the processes is either currently possible, has been proposed in the literature, or is foreseeable in the short-term future (next few years). As a consequence of the inherent requirement for process simplicity and the limitations in fabrication, we chose to synthesize the set of alternatives considered manually, as opposed to using an automatic method such as that reported by Lakshmanan and Biegler.48 Unlike macroscale process synthesis, the complexity of man-portable power generation results from the large choice of fuels, fuel reforming reactions, and fuel cells, and the early stage of component development, rather than from an elaborate combination of mixing, reaction, and separation steps. In the past, alternative and/or complementary approaches to the process superstructure have been proposed for macroscale process synthesis based on attainable regions,49-51 phenomena-based process synthesis,52 and the state-space approach.53 Although these ideas could be used, we chose the superstructure approach as the most natural choice. Unlike macroscale process synthesis, layout and heat integration must be addressed simultaneously with the flowsheet creation. We consider a variety of fuels, including hydrogen, ammonia, various hydrocarbons and alcohols, and fuel cells (including solid oxide fuel cell (SOFC), polymer electrolyte membrane (PEM), single chamber solid oxide fuel cell, direct methanol fuel cell (DMFC), and a proton-conducting fuel cell based on ceramic technology (PCFC). The alternatives considered are illustrated in Figure 3 and have been described in our previous work,35,36,54 where the physical properties and the equations used to model the “unit operations” also are detailed. Technological constraints do not allow for all potential process structures; e.g., carbon monoxide is a poison for PEM and, therefore, butane partial oxidation can only be combined with a PEM after gas purification. The reader is referred to the previous publications for details. These publications demonstrate that the optimal process structure is dependent on technological advances and

product specifications. Similar to macroscale process synthesis, the notion of a superstructure can be used for comparison of alternatives through the specification of degrees of freedom and simulation of the resulting options, or for the automatic identification of optimal structures through mixed-integer optimization. The effect of modeling parameters, such as the achievable conversion, can be studied by local sensitivity analysis, parametric studies, or, as described in the following subsection, ideally parametric optimization. 3.1. Resource Allocation: Process Choice. In research and development, resources are limited and it is desirable to allocate resources in an optimal fashion. In particular, the choice between which alternative products/processes to develop is of interest and can often be addressed with relatively simple models. For man-portable power generation based on fuel-cell systems, questions of interest include “should microfabricated solid-oxide fuel cells be developed?”, “is it better to focus on separation membranes or on catalysis development?”, and “should heat insulation or heat recovery be improved first?”. We propose to address such questions through the use of parametric optimization. Mathematical programs often involve unknown parameters, and the task of parametric optimization is, in principle, to solve the mathematical program for each possible value of these unknown parameters.55 Discretization of the parameter range generally is not rigorous, because there is no guarantee for optimality between the mesh points. Moreover, discretization on a fine mesh is a very expensive procedure, especially for high-dimension parameter spaces. Suppose that, generally, a model of a system under development with many potential components is given and the uncertain parameters describe the performance of the various components. Parametric optimization quantifies the influence of these parameters on the system performance and optimal design including the arrangement of the components. Identifying the most important parameters enables determination of whether it is worthwhile to pursue improvement of a given component. Post-optimality sensitivity analysis provides the correct parameter dependence for infinitesimally small changes in the parameter values (i.e., only local information). In contrast, parametric optimization provides a correct estimate of the


Figure 3. Set of alternatives considered.

influence of parameters over an entire range (i.e., global sensitivity information). The resource allocation problem described here is presented as qualitative rather than quantitative formulations in the literature.56-58 In these references, chemical products such as pharmaceuticals with quantifiable resource needs for development and testing are considered. In micropower, on the other hand, the goal is to identify components of a microfabricated electrochemical process that should be developed further. Attempts to estimate the resources needed to advance a given technology are conceivable; however, proper quantification is not always possible. Instead, we propose that the answer provided by parametric optimization must be evaluated by the decision maker, e.g., the program manager, who can balance the tradeoff between the expenditure of resources and the potential for improving the system performance. If the effect of resource allocation can be quantified, the unknown parameters can be replaced with their function of the resources, and the resources can be added to the variable list. In that case, the resulting program would directly furnish the optimal resource allocation. 3.1.1. Solution Method. The case studies in our previous reports35,36 were performed for fixed process alternatives, i.e., for fixed choices of fuel and fuel reforming and fuel-cell type, and, only in some cases, the optimal heat integration option was obtained by considering the alternatives manually. The effect of parametric variation on process performance was obtained approximately by simulating for a finite number of parameter values; the corresponding parameter grid was chosen either a priori or manually by inspection of the sensitivity of the key results to the unknown parameter. Automatic methods to cover the entire parameter space rigorously are generally desirable. More importantly, in many cases, it is more interesting to also observe how the parameter value affects the optimal process configuration. As described previously, the most appropriate tool for such a study is parametric optimization, which is used in the following. Because of the presence of both discrete design choices (such as the choice of fuel) and continuous

decision variables (such as the fuel flow rates), a mixed-integer formulation is necessary. Moreover, there are several sources of nonlinearity resulting in a mixed-integer nonlinear parametric program. With a few simplifications, a mixed-integer linear formulation is possible and the algorithms developed recently54,59 can be used. Because the operating pressures and temperatures are considered as parameters, a common source of nonlinearity is eliminated.42 The full set of process alternatives contains both mixers and stream splitters of unknown composition, and no method is known to represent the mass and species balances with only linear constraints. On the other hand, if no splitters of unknown composition are considered, the corresponding equations are linear if the molar flow rates are chosen as the variables.60 Note that the hydrogen separation membrane has outlets of known composition and can be written using linear equations. The surface area for heat losses is nonlinear in the volume, which is assumed proportional to the molar flow rates. The calculation of the heat losses can be approximated by linearizing around a reference volume. Finally, elaborate calculations relevant to the system energy density, such as the calculation of cartridge volume, introduce nonlinearity; therefore, here, only the fuel energy density will be considered, assuming an infinite mission duration. 3.1.2. Case Study. The following case study considers the question whether the development of a microfabricated SOFC is warranted. Figure 4 shows the set of alternatives considered. To use the parametric optimization algorithms, the models are implemented in C, which is tedious and error prone; therefore, a relatively small set of alternatives is considered. Ammonia, methanol, and propane/butane are taken as the fuel choices, with the reforming options described in Mitsos et al.35 and the options of PEM and SOFC. In the PEM, the only reaction is hydrogen oxidation, whereas in the SOFC, the oxidation of both hydrogen and carbon monoxide are considered as electrochemical reactions; for the level of detail considered, the effect on system performance is approximately the same as accounting for equivalent hydrogen production.36,61 The option of hydrogen

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Figure 4. Set of alternatives considered for the parametric optimization case study.

purification is allowed for the SOFC and is mandated for the PEM. An exception is the combination of ammonia cracking with a PEM. Although traces of ammonia can result in a significant decrease in the performance of PEM,45,62,63 our models do not account for this degradation. A possibility for actual deployment in that case is to use an ammonia sorbent.64 Autothermal operation is ensured by burning the fuel cell effluents and/or a fuel, allowing for fuel combinations. For simplicity, the cooling requirement for the PEM is not considered and the simple energy balances presented in Mitsos et al.35 are used. It is assumed that all reactors operate at the same temperature and require the same residence time for a given conversion; this assumption is oversimplifying. Because of these simplifications, the results should be considered qualitative rather than quantitative. For the reformulation of the bilinear terms between binary and continuous variables, the big-M method is used.65 From the plethora of possible case studies, here, the effect of SOFC efficiency (ηSOFC) on optimal design and performance is shown. As a design objective, we use the gravimetric fuel energy density. The SOFC efficiency is defined as the maximal ratio of power (PW) produced to power output of an ideal fuel cell at the SOFC temperature:

|PW| eηSOFC |

∑r ξr ∑i νriG0i (T)|

where ξr is the extent of electrochemical reaction r, νri the stoichiometric coefficient of species i in reaction r, and G0i (T) the molar gas-phase Gibbs free energy of pure species i at the reference pressure (including the free energy of formation); the magnitudes are used to avoid confusion with sign definitions. This simplified definition of efficiency eliminates the dependence on composition. The inequality sign allows the optimizer to choose whether to produce as much power as possible or, instead, generate more heat. Similarly, the optimizer has the choice of adjusting the extents of reaction up to the maximal conversions specified. The complete set of constraints corresponding to the case study are found in Mitsos and Barton.66 Figure 5 shows the gravimetric fuel energy density, as a function of the achievable SOFC efficiency, for the parameter values indicated in Table 1. For low ηSOFC values (less than ∼40%, or approximately half the PEM efficiency), the optimal process configuration is to use ammonia decomposition, fol-

lowed by a hydrogen separation and electrochemical conversion of the hydrogen in a PEM. The energy balance is closed by burning the fuel-cell effluents and the membrane waste. This configuration is better than directly feeding the reactor products into the PEM, because the gas separation is possible at a high temperature and the heat recovery is better. Above the threshold of 40% thermodynamic efficiency for the SOFC, the optimal process is partial oxidation of propane/butane and electrochemical conversion of the syngas generated in a SOFC, with oxidation of the fuel-cell effluents in a burner. In this configuration, it is optimal to use the air effluent from the cathode for this oxidation, because it is preheated. Note that moredetailed considerations, such as results provided by the intermediate-fidelity models,37 show that providing all the air to the fuel-cell cathode, as opposed to splitting into a stream to the cathode and a stream to the burner, improves the fuel-cell performance. For ηSOFC e 67%, the heat generation in the three units suffices to overcome the heat losses, and, therefore, the maximal power possible is produced in the SOFC, using the maximal efficiency and maximal conversion. Above this threshold, there is no heat excess, and, therefore, the conversion in the fuel cell is gradually reduced to produce more heat. This explains the kink in the graph. In terms of the mathematical program, the set of active constraints changes, and, therefore, also the post-optimal sensitivity changes. More-detailed considerations are necessary to decide whether it is advantageous to lower the conversion in the fuel cell or let part of the fuel stream bypass the fuel cell. For the low-efficiency window, the change in ηSOFC is not reflected in the achievable energy density. On the other hand, for the mid-efficiency window, the process performance is highly sensitive to improvements in the SOFC performance, whereas, for the high-efficiency window, the sensitivity is lower. Generally, technological improvements of components below a threshold component performance do not affect system performance, because other technologies are better. Perfection of component performance is not warranted either, because the effect on process performance is minimal, showing an effect of diminishing returns. Parametric optimization is a valuable tool in identifying these regions and, therefore, in allocating (research) resources.


Figure 5. Optimal gravimetric fuel energy density, as a function of achievable fuel cell efficiency. Table 1. Process Parameters for Parametric Optimization Case Study in Figure 5 parameter

value

ambient temperature, Tamb power output, PW reactor temperature, Top reactor outlet temperature, Tout maximal conversion in reactor, ζ SOFC temperature, Top PEM temperature, Top residence time in reactor, τ discard temperature from SOFC, Tout maximal conversion in burners, ζ residence time in burners, τ air excess, Φ maximal conversion in fuel cells, ζ overall heat loss coefficient, U residence time in fuel cell, τ emissivity (incl. view factor), efficiency of PEM, ηPEM air excess in fuel cell, Φ compression parameter for air feed, KC burner temperature, Top discard temperature from burner, Top water factor in fuel cell, Ψ water excess in reactor, Ψ pump parameter, KP membrane efficiency, ηPd propane mole fraction in feed

298 K 1W 900 K 700 K 0.9 900 K 370 K 5 ms 700 K 0.95 5 ms 1.2 0.8 3 W/m2/K 30 ms 0.2 0.7 1.2 10 J/mol/K 900 K 700 K 1 1 100 J/L 0.8 0.5

4. Detailed Modeling Microchemical systems often have complicated geometries and coupled physical phenomena, including heat and mass transfer, as well as chemical reactions. Therefore, in principle, only three-dimensional space-distributed models can accurately characterize the temperature and concentration profiles. Stateof-the-art CFD tools are based on the discretization of the underlying PDAEs, using the finite-element or finite-volume

methods. CFD tools have been used for the characterization of designs and flow patterns in microchemical systems (see, e.g., refs 67-72). Major drawbacks of these methods are that they are only valid for a given geometry, require a high modeling effort, and are very computationally expensive; in addition, the convergence is not robust. Because some reactors have very different scales, it is advisable to couple three-dimensional models with two-dimensional models;67 however, this is not straightforward in typical commercial CFD tools. Within our proposed methodology, detailed models are used for two main tasks. The first task is to analyze proposed geometries and to also improve on these; an example of such a characterization is found in refs 54 and 73. Another equally important application of detailed models, in combination with scaling analysis,74 is the derivation and validation of simplifying assumptions. In the remainder of this section, a case study involving the validation of simplifying assumptions is presented. In particular, the effect of lumping silicon structures and gas phase to an effective single phase is studied. Mitsos54 provides the justification for simplifying assumptions further. To eliminate the dependence of the results on the geometry, a worstcase analysis is required. For instance, to estimate the maximal temperature difference in an exothermic reactor, the worst-case scenario is that the heat generation is localized at one point. 4.1. Case Study. To improve the performance of heterogeneous reactions, microfabricated reactors are constructed with catalyst support structures, such as posts or slabs.68,75 These structures also enhance heat transfer; however, on the other hand, modeling them explicitly greatly increases the computational requirements. A simple approximation is to model the reactor as if no catalyst support structures were present and use the values of the gas for all physical properties except the thermal conductivity, for which a volume-weighted average of the structure and gas conductivity values is taken. Note that the

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Figure 6. Geometry (top) and temperature profiles for explicit modeling of catalyst support (bottom left) and lumped model (bottom right).

small distance between catalyst supports in the radial direction causes approximate thermal equilibrium between the gas and solid (locally). Here, we examine the effect of averaging the thermal conductivity and the assumption of uniform temperature using a duct reactor with a width and height of 500 µm and a length of 2.5 mm, depicted in Figure 6. Exploiting the symmetry, we only model a quarter of the geometry. Our model also includes a 1-mm-long inlet and outlet to the reactor; we assume that the reactor contains four silicon slabs as a catalyst support that cover 2/ of the width. We assume an inlet velocity of 1 m/s. We 5 account for radiative heat losses to the ambient with an overall emissivity of 0.2. For the simulations, we used the finite-element package FEMLAB version 3,76 and the Navier-Stokes equations with variable density and conduction-convection equations for the energy balance. The chemistry is not explicitly modeled, but rather, a heat generation term is used; its magnitude is set to a value that leads to a temperature in the reactor of ∼1200 K. When the slabs are modeled explicitly, the heatgeneration term is introduced as a surface term; for the volumeaveraged model, a volume heat generation is used. Heat losses to the ambient are considered as boundary conditions. Figure 6 shows that using volume-averaged thermal conductivity qualitatively and quantitatively captures the effect of

increased heat transfer of the catalyst support structure and an explicit model of the slabs is not required. Modeling the slabs explicitly increases the modeling and computational requirements significantly and makes convergence much more demanding. Also, the temperature within the reactor portion is essentially uniform. Note that, to plot the temperature profiles, a stretching of the axis is performed. For the volume-averaged model, we explore three cases for the heat generation (namely, constant, linear, and exponential) dependence on the axial coordinate, always with the same overall heat generation. In Figure 7, the temperature is plotted along the axial coordinate for these three cases, as well as the case with explicit modeling of the slabs. The temperature in the reactor portion is essentially uniform, whereas in the inlet and outlet (where no silicon structure is present), there is a significant temperature gradient; also, the differences between the various heat generation terms are relatively small. 5. Intermediate-Fidelity Modeling To study the optimal sizing of units and optimal operation, models that rely on first principles, including detailed kinetic mechanisms, are required. Only such models can adequately handle the coupled design and operation problem, and, therefore,


Figure 7. Temperature profiles for explicit and average modeling of slabs.

Figure 8. Conceptual process flowsheet.

system-level models are not suitable. Because the underlying physicochemical phenomena are complicated and coupled, one cannot always rely on engineering intuition, and optimization based on mathematical programming is warranted. State-of-theart optimization algorithms require hundreds or thousands of iterations; therefore, fast, reliable, and robust solution of the models is needed. As a consequence, the CFD-based models are not suited for optimization either. This motivates the development of models of an intermediate detail level. These models consider spatial dependence whenever necessary and do not require a fully defined geometry, but rather the specification of a minimal number of design parameters, such as the volume or the surface-area-to-volume ratio of the units. The nature of the models obviously is dependent on the class of devices considered. Valid approximations can be established by detailed modeling, scaling analysis, and experimental evidence. The simplifications described below and used in previous publications are valid for a specific class of devices and should be viewed as examples within the overall methodology. For high-temperature systems with maximal characteristic dimensions on the order of millimeters and high thermal conductivity, a good approximation is to assume a spatially uniform temperature. For instance, high conductivity is achieved through the use of silicon and the presence of a catalyst support, as was described in the previous section. Note that the approximation of uniform temperature is not the case for every

microstructured reactor, particularly in the case of combustion.69,70,77 For reactors that are based on thin tubes, with diameters on the order of 100 µm, a one-dimensional distribution of the species balance is adequate.54 Finally, for components with a small residence time, whose operating conditions are changed slowly, there is a separation of time scales, and pseudosteady-state species balances can be used.38,54 The specific process flowsheets that should be considered at the intermediate-fidelity modeling level should meet two criteria. The first criterion is to have the potential for high performance, which is ascertained from system-level considerations; for instance, it would be a waste of resources to develop a detailed model for a methane-based process. The second criterion is a demonstrated proof of concept for the technology considered and availability of validated chemical kinetics. These two criteria mandated the case studies that we consider in the intermediate fidelity framework, namely, a simple process comprised of ammonia cracking for hydrogen generation, a SOFC for power generation, and butane oxidation for heat generation, as illustrated in Figure 8. The SOFC is chosen because of the longterm promise for fuel flexibility.27 Butane is chosen because of its high-energy density, which makes it a very suitable heat source; note that the oxidation of butane does not pose a significant challenge. Although butane partial oxidation, combined with a SOFC, is expected to have higher energy density (Figure 5), the corresponding chemistry has not been demon-

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strated sufficiently yet in microreactors. Therefore, ammonia is chosen instead, despite its toxicity, which limits the potential applications. 5.1. Steady-State Operation. For the class of devices described previously, a model that describe the steady-state operation requires the species balance to be distributed in space, whereas a lumped energy balance with a spatially uniform temperature can be used. The optimal sizing and steady-state operation is formulated as a mathematical program with a system of differential-algebraic equations embedded.37 The independent variables correspond to the axial coordinates of the reactors and fuel cell. One source of algebraic equations is the activation polarization equations. The case study considered by Chachuat et al.37 confirms the conjecture that design and operation are coupled and must be considered simultaneously. Moreover, it reveals several counterintuitive results. For instance, the temperature that is optimal from a perspective of process performance lies between 1400 K and 1500 K, depending on design parameters. This is too high, from a material stability perspective, and therefore bounds on the temperature must be set. The reason that the “optimal” temperature is so high is that radiative heat losses decrease as the temperature increases. This counterintuitive result is due to the increase in chemical kinetic rates and the resulting decrease in size required to achieve a given conversion. At the “optimal” temperature, the chemical kinetics are so fast and the device is so small that the dominant heat losses are no longer radiative, but rather the convective losses in the outlet streams. In contrast, equilibrium considerations would suggest that a low temperature is optimal. Although ammonia cracking is an endothermic reaction and therefore high temperatures favor the equilibrium conversion, the effect from a mass-balance perspective is quite small: in the range of 900-1300 K, the equilibrium ammonia residual mole fraction varies from ∼4 × 10-4 to ∼5 × 10-5. Similarly, the equilibrium conversion for butane oxidation is almost complete. On the other hand, the open-circuit voltage for the electrochemical oxidation of hydrogen decreases significantly as the temperature increases: in the range of 9001300 K, the voltage decreases from ∼1.04 V to ∼0.92 V. Note that voltage is a suitable metric for fuel-cell efficiency.45 5.2. Transient Operation. Because most power-consuming devices are not operated constantly and have rapidly changing power demands, the dynamics and automated operation of portable power production are very important and must be considered thoroughly. Transient considerations are indeed so important that they are likely to influence the optimal design. For example, it might be necessary to oversize certain units, relative to optimal steady-state design, or exclude processes that exhibit poor transient behavior. For transient operation, in addition to the spatial distribution, integration through time must be considered. This results in models that are comprised of PDAEs.38 Associated with the transient operation are complicated control issues, which are outside the scope of this paper; the reader is referred to reported research by Kothare and coworkers,78-80 where control issues are discussed. In particular, the optimal startup of the processes is of great interest. It is most likely that the devices will be coupled with a relatively small rechargeable battery for start-up and shutdown operations, as well as for changes in power output. Objectives for the optimal start-up procedure include minimization of the time required to reach steady state and minimization of the overall system mass and volume. Constraints include requirements regarding the emission of toxic gases, as well as

structural stability considerations, which translate mainly into constraints on temperature, temperature gradients, and their time derivatives. Detailed models of the complete operating cycle exhibit hybrid discrete/continuous behavior, because different operating modes (e.g., discharging and recharging of the battery) are described by different models. The models involve PDAEs, and no algorithms exist that guarantee the global optimal solution of such problems; therefore, we apply local optimization techniques. The PDAEs have multiple time scales, and we have developed numerical techniques based on this separation of scales to allow efficient and robust integration of the state and sensitivity equations.81 The optimal startup consists of three phases. In the first phase, the battery is used to meet the power demand, as well as to heat the fuel-cell stack; an alternative for the heatup is a spark ignition of a combustion reaction. In the second phase, the battery is used only to meet the power demand while butane is burned to further heat the stack. In the last phase, the fuel-cell stack is operated at a higher load than steady state to meet the power demand, as well as to recharge the battery; for this, an increased flow of fuel is required, resulting also in increased heat production than at steady state. The implication of the startup for design is that the sizing of the reactor/fuel cells must take into account the start-up phase, where power is generated to recharge the battery. 5.3. Design for Variable Power Demand. Most portable electric and electronic devices are not operated with a constant load over time, but rather follow a power profile. In many cases, this profile can be approximated through a small number of discrete power demands and the frequency at which they occur. For instance, a cell phone is either in standby mode, consuming on the order of 0.2 W, or in talk mode, consuming ∼2-3 W. A conceptually simple (but computationally overly expensive) way to account for this power variation is to consider the time profile and formulate a dynamic optimization problem. A more elegant and computationally more tractable method is based on two-stage stochastic programming.40,41 The formulation as a twostage optimization problem also has the advantage that only the average time spent at each power demand is needed, as opposed to a detailed power profile. The first-stage variables are the design variables, e.g., the volume of reactor and fuel cell, which are constant for each power demand. The second-stage variables are the operational variables, e.g., the operating temperature and fuel flow rates, which change depending on the current power demand. The fraction of time spent in one power demand corresponds to the probability of an event. An inherent assumption of this formulation is that changes between power demands does not significantly affect the average performance. The case studies in Yunt et al.40 clearly demonstrate that variability in the power demand should be considered at the design stage. 5.4. Resource Allocation: Component Optimization. Similar to the system-level models, intermediate-fidelity models can also be used for resource allocation studies. The intermediatefidelity models consider fixed structures and, therefore, are not as suitable for the comparison of different technologies as the system-level models. On the other hand, intermediate-fidelity models can be used to identify which components of a given system should be improved: for instance, “should we improve the electrochemical kinetics of the anode or the cathode?” or “is the development of ultrathin membranes justified?”. Although in a previous report,37 we already touched on these issues, here, we present a more-focused resource allocation case


Figure 9. Optimal fuel energy density as a function of kinetic rate acceleration.

study. More specifically, we will illustrate resource allocation by identifying which catalysts are the most important to improve. As a reference point, we use the reduced one-step ammonia decomposition kinetics from Deshmukh et al.,82 the SOFC kinetic data from Achenbach,83 and the ammonia and hydrogen oxidation kinetics proposed by Pignet and Schmidt.84,85 We then vary the kinetic rates by adjusting the pre-exponential factor, relative to these reference values, and study the effect on optimal system performance, as measured by the fuel energy density. We use the model developed in Chachuat et al.37 with a power output of PW ) 10 W, an operating temperature of T ) 1000 K, an outlet temperature of Tout ) 650 K, and the same values for the various design parameters as in the base case found therein. We consider six potential catalyst improvements, namely (i) the ammonia decomposition kinetics, (ii) the ammonia oxidation kinetics, (iii) the hydrogen oxidation kinetics, (iv) the anode half-reaction, (v) the cathode half-reaction, and (vi) both cathode and anode half-reactions simultaneously. Because the formulation for optimal design and operation is based on mathematical programs with differential-algebraic equations embedded, no suitable parametric optimization algorithms are available and we therefore discretize the parameter space. The results of the case study are shown in Figure 9. The abscissa captures the relative improvement of the various kinetic rates on a logarithmic scale, whereas the ordinate shows the optimal fuel energy density. Variation of ammonia oxidation kinetics in the range considered has no effect whatsoever on the optimal fuel energy density. This is because optimal operation mandates a very high conversion in the reactor, and a residence time in the fuel-cell effluent burner that is more than sufficient for complete oxidation of the residual ammonia. Improving the ammonia decomposition or the hydrogen oxidation kinetics has a relatively small effect on the system performance: a 100-fold increase of the former leads to a 1.4% improvement in the fuel energy density, whereas a 100-fold increase of the latter gives a 0.2% improvement in the fuel energy density. Faster fuel-cell kinetics, on the other hand, significantly improve system performance; a 2-fold improvement of either electrode gives a 10% higher fuel energy density. In all cases, the main reason for improved system performance is that faster kinetics result in a smaller residence time requirement and, therefore, lower heat losses, because of the decreased device size. Because the fuel-cell volume is dominant, the most effective way of reducing the device size is to improve the electrochemical kinetics. This result applies to other hightemperature fuel-cell systems in which the electrochemical kinetics are significantly slower than the reforming kinetics. Note also that, if we were to account for the device mass and

calculate the system energy density, we would get similar results, because the fuel cell dominates the device size. Comparing the relative merits of improving the cathode or anode kinetics is somewhat more complicated. A small improvement of either electrode has a similar influence on the system performance. However, the effect of anode kinetics on fuel energy density levels off relatively quickly, because the cathode reactions become the limiting step. Recall also that, for this particular system, the cathode overpotential is significantly higher than the anode overpotential (compare Figure 3 in the Chachuat et al. work37 ). Finally, the best possible strategy is to improve the kinetics of the anode and cathode simultaneously; this is particularly true for large improvements. Accelerating the anode and cathode kinetics by a factor of 2 achieves the same effect as accelerating only the cathode kinetics by slightly more than a factor of 4, whereas accelerating the anode and cathode kinetics by a factor of 16 achieves the same effect as accelerating only the cathode kinetics by a factor of more than 256. Generally, whenever the anode and cathode overpotentials are of similar magnitude, it is advisable to improve both electrodes simultaneously, as opposed of only one of the two. Note that resource allocation could not have been adequately studied with (local) sensitivity analysis, because local information does not predict the effect of diminishing returns. 6. Future Directions Our modeling framework at the system level is flexible and could be expanded to include more fuels (e.g., formic acid86) or different fuel processing mechanisms (e.g., autothermal reforming87). Another interesting example is the combination of an exothermic reaction for syngas generation, such as the partial oxidation of hydrocarbons, with a thermophotovoltaic cell that would transform part of the heat excess to power. Also, a more-detailed consideration of hybrid systems with a fuel cell and a battery or capacitor is of interest. At the system level, we consider separately simulation of the full set of alternatives using nonlinear models and parametric optimization of a smaller set of alternatives and simplified models via newly developed algorithms.59 Currently, models implemented in the process simulator ABACUSS88 are manually rewritten to be used in the parametric optimization algorithms. This procedure is tedious and error prone, and the full potential of the two tools can only be used effectively if a single modeling framework is used. One possibility is to automatically transform the model from one modeling language to the other, and the other is to use advanced interfaces. Moreover, the application of (parametric) mixed-integer nonlinear programming, consider-

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ing the full set of alternatives, could lead to useful insights. At the intermediate-fidelity level, the parametric studies are run by a series of dynamic optimization problems for different parameter values; a parametric optimization algorithm may be significantly more efficient and also provide some guarantees for the entire parameter range. A very interesting question is the flexibility and robustness of the designs that are obtained. We believe that the most appropriate tool to study these issues is the intermediate-fidelity models, because these include kinetic mechanisms that accurately capture the effect of temperature. Flexibility problems are formulated as max-min problems,89,90 and, because of the embedded DAEs, cannot be addressed rigorously with stateof-the-art algorithms. It would be of interest to either develop algorithms that can address these types of problems or consider these problems heuristically. An important aspect that is not considered extensively involves pressure considerations, including pressure drops and dependence of the process performance on the operating pressure (e.g., at high elevations). Similarly, the influence of peripheral components, such as valves and pumps, is neglected; in contrast to the macroscale, the energy consumption and influence of these components may be substantial and should be considered. Moreover, the mass and volume must be taken into consideration in the design objective of maximal energy density. Structural stability considerations are extremely important, and it would be interesting to include these considerations. Degradation and durability issues also could be considered. Because microfabricated fuel-cell systems are at an early stage of development, very limited experimental results are available to validate the models presented. As further experimental results become available, the model predictions must be compared with the experimental results for validation of the models. This procedure would possibly also suggest refinements to the models. The methodology proposed is flexible, in regard to the design objective; nevertheless, primarily, energy densities are considered in the case studies presented. It would be interesting to consider further design objectives, such as undesired heat signals for the dismounted soldier. Furthermore, economic and environmental calculations are not included in our case studies but are worth investigating. For instance, it would be interesting to minimize the cost of the device subject to an acceptable decrease in performance over the optimal device. Also, the influence of multiple objectives is interesting; recall that, in a previous report,45 we considered a simple case study of two objectives and the corresponding Pareto curve. Finally, note that the methodology proposed is not the only conceivable one and it would be interesting to consider alternatives. For instance, automatic connection of the different levels of modeling detail following the paradigm of multiscale modeling (see, e.g., Braatz et al.44) is conceivable in the future. At the system level, it would be interesting to study whether and how alternative ideas to the process superstructure can be applied to man-portable power generation; possibilities include attainable regions,49-51 phenomena-based process synthesis,52 and the state-space approach.53 Reduced models based on linearization methods or input-output models are also conceivable. At the computational fluid dynamics (CFD) level, it would be interesting to consider the application of structural optimization techniques. Acknowledgment This work was supported by the Department of Defense (DoD) Multidisciplinary University Research Initiative (MURI) program, administered by the Army Research Office (under

Grant No. DAAD19-01-1-0566). We would like to acknowledge our colleagues in the MURI program and coauthors in previous publications for fruitful discussions. Literature Cited (1) Linden, D. Handbook of Batteries; McGraw-Hill: New York, 2001. (2) Meeting the Energy Needs of Future Warriors; National Academy Press: Washington, DC, 2004. (3) Hessel, V.; Lo¨we, H. Mikroverfahrenstechnik: KomponentenAnlagenkonzeption-Anwenderakzeptanz-Teil 1. Chem. Ing. Tech. 2002, 74, 17-37. (4) Jensen, K. F. Microreaction engineeringsIs small better? Chem. Eng. Sci. 2001, 56, 293-303. (5) Jensen, K. F. Microchemical systems: Status, challenges and opportunities. AIChE J. 1999, 45, 2051-2054. (6) Mitsos, A.; Barton, P. I. Publications on portable power generation, http://yoric.mit.edu/download/Reports/micropowerpub.pdf. Technical report, Massachusetts Institute of Technology, Cambridge, MA, 2006. (7) Holladay, J. D.; Wang, Y.; Jones, E. Review of developments in portable hydrogen production using microreactor technology. Chem. ReV. 2004, 104, 4767-4789. (8) Maynard, H. L.; Meyers, J. P. Miniature fuel cells for portable power: Design considerations and challenges. J. Vac. Sci. Technol. 2002, 20, 1287-1297. (9) La, G. L.; Hertz, J.; Tuller, H.; Shao-Horn, Y. Microstructural features of RF-sputtered SOFC anode and electrolyte materials. J. Electroceram. 2004, 13, 691-695. (10) Baertsch, C. D.; Jensen, K. F.; Hertz, J. L.; Tuller, H. L.; Vengallatore, S. T.; Spearing, S. M.; Schmidt, M. A. Fabrication and structural characterization of self-supporting electrolyte membranes for a micro solid-oxide fuel cell. J. Mater. Res. 2004, 19, 2604-2615. (11) Hertz, J. L.; Tuller, H. L. Electrochemical characterization of thin films for a micro-solid oxide fuel cell. J. Electroceram. 2004, 13, 663668. (12) Rice, C.; Ha, S.; Masel, R. I.; Waszczuk, P.; Wieckowski, A.; Barnard, T. Direct formic acid fuel cells. J. Power Sources 2002, 111, 8389. (13) Rhee, Y. W.; Ha, S. Y.; Masel, R. I. Crossover of formic acid through nafion membranes. J. Power Sources 2003, 117, 35-38. (14) Zhu, Y.; Ha, S. Y.; Masel, R. I. High power density direct formic acid fuel cells. J. Power Sources 2004, 130, 8-14. (15) Heinzel, A.; Hebling, C.; Mu¨ller, M.; Zedda, M.; Mu¨ller, C. Fuel cells for low power applications. J. Power Sources 2002, 105, 250-255. (16) Palo, D. R.; Holladay, J. D.; Rozmiarek, R. T.; Guzman-Leong, C. E.; Wang, Y.; Hu, J.; Chin, Y.-H.; Dagle, R. A.; Baker, E. G. Development of a soldier-portable fuel cell power system. Part I: A bread-board methanol fuel processor. J. Power Sources 2002, 108, 28-34. (17) Holladay, J. D.; Wainright, J. S.; Jones, E. O.; Gano, S. R. Power generation using a mesoscale fuel cell integrated with a microscale fuel processor. J. Power Sources 2004, 130, 111-118. (18) Meyers, J. P.; Maynard, H. L. Design considerations for miniaturized PEM fuel cells. J. Power Sources 2002, 109, 76-88. (19) Morse, J. D.; Jankowski, A. F.; Graff, R. T.; Hayes, P. Novel proton exchange membrane thin-film fuel cell for microscale energy conversion. J. Vac. Sci. Technol., A 2000, 18, 2003-2005. (20) Reuse, P.; Renken, A.; Haas-Santo, K.; Go¨rke, O.; Schubert, K. Hydrogen production for fuel cell application in an autothermal microchannel reactor. Chem. Eng. J. 2004, 101, 133-141. (21) Shao, Z.; Haile, S. M.; Ahn, J.; Ronney, P. D.; Zhan, Z.; Barnett, S. A. A thermally self-sustained micro solid oxide fuel-cell stack with high power density. Nature 2005, 435, 795-798. (22) Mann, C. Near-term nanotech. Technol. ReV. 2004, (July/August), 22. (23) Takei, F.; Cooray, N. F.; Yoshida, K.; Yoshida, H.; Ebisu, K.; Suzuki, S.; Sawatari, N. Development of prototype micro fuel cells for mobile electronics. Fujitsu Sci. Tech. J. 2005, 41, 191-200. (24) Sato, Y.; Matsuoka, K.; Sakaue, E.; Hayashi, K. Material and heat management in a DMFC for portable usage. Presented at the AIChE Annual Meeting, Cincinnati, OH, October 30-November 4, 2005. (25) Sato, Y.; Ono, A.; Tezuka, F.; Isozaki, Y. Hydrogen generation by reforming dimethylether using micro-channel reactor. Presented at the AIChE Annual Meeting, San Francisco, CA, November 12-17, 2006. (26) Kawamura, Y.; Ogura, N.; Yamamoto, T.; Igarashi, A. A miniaturized methanol reformer with Si-based microreactor for a small PEMFC. Chem. Eng. Sci. 2006, 61, 1092-1101.

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ReceiVed for reView April 25, 2007 ReVised manuscript receiVed July 5, 2007 Accepted August 1, 2007 IE070586Z

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