2 A fuel cell based total energy system can be broken down into, for example, a

Borja Oyarza´bal

Michael W. Ellis

1

Michael R. von Spakovsky

Center for Energy Systems Research, Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 e-mail: [email protected]

Development of Thermodynamic,

Geometric, and Economic Models

for Use in the Optimal

Synthesis/Design of a PEM Fuel

Cell Cogeneration System for

Multi-Unit Residential

Applications

Thermodynamic, geometric, and economic models are developed for a proton exchange membrane (PEM) fuel cell system for use in cogeneration applications in multi-unit residential buildings. The models describe the operation and cost of the fuel processing sub-system and the fuel cell stack sub-system. The thermodynamic model reflects the operation of the chemical reactors, heat exchangers, mixers, compressors, expanders, and stack that comprise the PEMFC system. Geometric models describe the performance of a system component based on its size (e.g., heat exchanger surface area), and, thus, relate the performance at off-design conditions to the component sizes chosen at the design condition. Economic models are based on data from the literature and address the cost of system components including the fuel processor, the fuel cell materials, the stack assembly cost, the fuel cost, etc. As demonstrated in a forthcoming paper, these models can be used in conjunction with optimization techniques based on decomposition to determine the optimal synthesis and design of a fuel cell system. Results obtained using the models show that a PEMFC cogeneration system is most economical for a relatively large cluster of residences (i.e. 50) and for manufacturing volumes in excess of 1500 units per year. The analysis also determines the various system performance parameters including an elec-trical efficiency of 39% and a cogeneration efficiency of 72% at the synthesis/design point. 关DOI: 10.1115/1.1647130兴

Introduction

The development of proton exchange membrane fuel cell

共PEMFC兲 systems promises to bring the advantages of

cogenera-tion to relatively small-scale applicacogenera-tions such as residential or commercial buildings. Successful development of fuel cell cogen-eration systems will depend on optimizing the synthesis/design of a complex system comprised of chemical reactors, heat exchang-ers, fuel cell stacks, compressors, pumps, and expanders. This paper presents a comprehensive model of a PEMFC system uti-lized in a cogeneration application for a multi-unit residential complex. As demonstrated in a forthcoming paper关1兴, the model presented here can be used in conjunction with optimization tech-niques based on decomposition to determine the optimal synthesis and design of a fuel cell system.

PEMFC System Model

There are a wide variety of fuel cell system configurations for stationary applications in the literature. The present work focuses on PEMFC technology coupled to the steam reforming of natural gas since this combination seems to be the most prevalent and promising based on the literature关2,3兴.

The fuel cell system is one of several sub-systems that comprise a cogeneration or total energy system共TES兲2and is in turn made up of four major sub-systems: a stack sub-system 共SS兲, a fuel processing sub-system共FPS兲, a thermal management sub-system

共TMS兲, and a load management sub-system 共LMS兲. The focus

here will be on the modeling, simulation, and eventual optimiza-tion共described in detail in a companion paper 关1兴兲 of the first two sub-systems listed, namely, the SS and the FPS.

To develop the SS and FPS configuration used in this study, four key factors were taken into account, i.e. thermal manage-ment, the ability to satisfy a wide variety of loads, the ability to keep capital costs relatively low, and the ability for adequate con-trol at off-design. Based on these four factors, the fuel cell sub-system configuration shown in Fig. 1 was developed and is the configuration used in the modeling presented in this paper. The components of this configuration are summarized in Table 1 and are indexed k⫽1,2, . . . ,27 in the discussion that follows. These system components can be classified as belonging to one of the following basic types:

Chemical reactor Heat exchanger

Mixer Water separators

Compressor/Expander/Pump PEMFC stack

1_{Corresponding author.}

Contributed by the Advanced Energy Systems Division for publication in the JOURNAL OFENERGYRESOURCESTECHNOLOGY. Manuscript received by the AES Division February 2003; revised manuscript received October 2003. Associate Edi-tor: H. Metghalchi.

2_{A fuel cell based total energy system can be broken down into, for example, a} fuel cell sub-system, a vapor compression and/or absorption heat pump sub-system, a thermal storage sub-system, and an electric storage sub-system. Obviously, each of these sub-systems can undergo a further breakdown as is done for the fuel cell system above. A depiction of several TES options can be found in Gunes关4兴.

Models for these basic types of equipment are summarized in the following sections. More detailed descriptions are provided in Oyarzabal关5兴. Key assumptions employed in the models include the following:

1. Gases are assumed to behave as ideal gases with specific heat values that are constant but evaluated at the midpoint temperature of the process of interest.

2. Gas mixtures are assumed to be ideal. 3. Liquids are assumed to be incompressible.

4. System operation is assumed to be steady at each operating point of interest.

Chemical Reactors. The fuel processing sub-system 共FPS兲 includes five types of reactors–combustion (k⫽26), steam re-forming (k⫽4), high temperature shift (k⫽5), low temperature shift (k⫽7), and preferential oxidation (k⫽9) reactors. The chemical reactions and the assumptions associated with each re-actor are summarized in Table 2. The steam reforming, low tem-perature shift, and high temtem-perature shift reactors use catalysts to promote the desired reaction; and it is assumed that the catalyst surface area is sufficient to achieve equilibrium at the exit of the reactor at design conditions3. The outlet composition from each of the three equilibrium controlled reactors (k⫽4, 5, and 7兲 is 3_{This assumption, of course, may lead to oversizing of the reactor components} and subsequent work found in Georgopoulos关6兴 and Georgopoulos et al. 关7兴 has overcome the limitations associated with this assumption by moving to kinetically based models.

Fig. 1 PEMFC system configuration„components are identified in Table 1…

Table 1 Components of the configuration given in Fig. 1 Component

numbers Component descriptions k⫽1 Natural gas compressor

2 Mixer for natural gas and steam

3 Heat exchanger prior to the steam reformer reactor 4 Steam reformer共SR兲 reactor

5 High temperature shift reactor共HTS兲

6 Heat exchanger prior to the low temperature shift reactor 7 Low temperature shift reactor共LTS兲

8 Heat exchanger prior to the preferential oxidation reactor 9 Preferential oxidation reactor共PROX兲

10 Heat exchanger prior to the PEMFC stack 11 PEMFC stack

12 Water separator for the reformate stream leaving the PEMFC stack

13 Mixer for the reformate and methane entering the combustor

14 High temperature cogeneration heat exchanger 15 Low temperature cogeneration heat exchanger 16 Heat rejection unit

17 Pump

18 Compressor for the air intake 19 Air-to-exhaust heat exchanger 20 Water injection共humidifier兲

21 Water separator for the air exhaust from the stack 22 Expander for power recovery from the air and combustion

gas streams

23 Water separator for the system exhaust 24 Water tank

25 Steam generator 26 Combustor

given by solving balances for each atomic species appearing in the overall reaction and simultaneously solving the equilibrium equation k共T兲⫽ y_C␯C_y D ␯D y_A␯A_y B ␯B

冉

p p0

冊

␯D⫹␯C⫺␯A⫺␯B (1) where yiis the mole fraction of a product species in the overall reaction,␯Aand ␯B are the stoichiometric coefficients of the re-actants in the equilibrium reaction and␯Cand␯Dare the stoichio-metric coefficients of the products in the equilibrium reaction. The overall reaction and the equilibrium reaction for each of the equi-librium controlled reactors are given in Table 2.4

Equation共1兲 indicates that, for a given pressure, the tempera-ture of the exit gas from the reactor controls the exit composition. As an illustration, Fig. 2 shows the results of a set of chemical equilibrium equations for the steam reforming reactor with an inlet steam to methane ratio of 3 and a mixture pressure of 3 atm for a range of outlet temperatures from 530°C to 800°C.

Combustion is assumed to go to completion so that the exit gas from the combustor (k⫽26) contains no H2, CO, or CH4. The preferential oxidation共PROX兲 reaction, which occurs in reactor (k⫽9), is assumed to oxidize all of the CO using the

stoichio-metric amount of O2so that neither of these species appears in the product stream. In this simplified PROX model, we neglect the hydrogen oxidation reaction and the reverse water gas shift reac-tion. With these assumptions, the product streams from the com-bustion and PROX reactors can be determined from atomic bal-ances on the carbon, hydrogen, nitrogen, and oxygen species.

Application of conservation of energy to each of the reactors yields

兺

i

N˙ka,i¯hka,i⫺

兺

i

N˙kb,ih¯kb,i⫹Q˙k⫽0 (2) where N˙ka,iis the molar flow rate of species i through port a of component k, h¯ka,iis the total enthalpy of this species on a molar basis, Q˙kis the heat transfer to the reactants from the reactor heat exchanger. The combustor (k⫽26) and the shift reactors are mod-eled as adiabatic reactors for which Q˙k⫽0. The steam reforming and preferential oxidation共PROX兲 reactors are modeled as packed bed reactors with embedded heat exchangers. With this simplifi-cation, the heat transfer from these two reactors is given by

Q˙k⫽N˙kc共h¯kc⫺h¯kd兲 (3) For the steam reforming reactor, the heat transfer, Q˙4, is con-trolled to maintain the desired reformer exit temperature. For the PROX reactor, the heat transfer, Q˙9, is controlled to maintain isothermal conditions in the reactor.

Heat Exchangers. The PEMFC system includes nine heat ex-changers. Six of these heat exchangers transfer energy from one stream within the system to another. With the heat exchangers assumed to be well insulated, the associated energy balance for each of these six heat exchangers (k⫽3, 6, 8, 10, 19, 25兲 is given by

兺

i N˙ka,i¯hka,i⫹

兺

i N˙kc,i¯hkc,i⫺

兺

i N˙kb,i¯hkb,i⫺

兺

i N˙kd,i¯hkd,i⫽0 (4) Heat exchangers k⫽3, 6, 8, 10, and 19 exchange energy between mixtures of ideal gases. The steam generator (k⫽25) is a heat exchanger for which liquid water enters at the temperature of the storage tank (k⫽24) and leaves as a superheated vapor at the discharge pressure of the natural gas compressor (k⫽1).

The high temperature cogeneration heat exchanger (k⫽14), the low temperature cogeneration heat exchanger (k⫽15), and the heat rejection unit (k⫽16) exchange energy with the load or the surroundings. For these heat exchangers, conservation of energy is given by

Q˙k⫽

兺

i

N˙ka,i共h¯ka,i⫺h¯kb,i兲 (5) 4_{Values of⌬G used to calculate the equilibrium constant were determined based}

on property data from关8兴.

Table 2 Chemical reactions occurring in FPS reactors

Reactor Overall reaction

Equilibrium reaction and Equil. constant共k兲4_or Product assumptions Steam reforming N ˙_{4a,CH4⫹N˙4a,H2O} →N˙4b,CH4⫹N˙4b,H2O⫹N˙4b,CO⫹N˙4b,H2 CH4⫹H2O⇔CO⫹3H2 k⫽exp关⫺⌬GSR(T,p0)/RT兴 High temperature shift

N˙5a,CH4⫹N˙5a,H2O⫹N˙5a,CO⫹N˙5a,H2

→N˙5b,CH4⫹N˙5b,H2O⫹N˙5b,CO⫹N˙5b,H2⫹N˙5b,CO2

CO⫹H2O⇔CO2⫹H2

k⫽exp关⫺⌬GSH(T,p0)/RT兴

Low temperature shift

N˙7a,CH4⫹N˙7a,H2O⫹N˙7a,CO⫹N˙7a,H2⫹N˙7a,CO2

→N˙7b,CH4⫹N˙7b,H2O⫹N˙7b,CO⫹N˙7b,H2⫹N˙7b,CO2

CO⫹H2O⇔CO2⫹H2

k⫽exp关⫺⌬GSH(T,p0)/RT兴

PROX N˙_{9a,CH4⫹N˙9a,H2O⫹N˙9a,CO⫹N˙9a,H2⫹N˙9a,O2⫹N˙9a,N2}

→N˙9b,CH4⫹N˙9b,H2O⫹N˙9b,H2⫹N˙9b,CO2⫹N˙9b,N2

N˙_{9b,CO⫽0; N˙9b,O2⫽0} N˙9a,CH4⫽N˙9b,CH4

Combustion N˙26a,CH4⫹N˙26a,O2⫹N˙26a,N2⫹N˙26a,H2⫹N˙26a,H2O

→N˙26b,H2O⫹N˙26b,CO2⫹N˙26b,O2⫹N˙26b,N2

Complete combustion

Fig. 2 Equilibrium compositions at the outlet of the SR reactor calculated using the SR model„outlet pressure of 3 atm; steam to methane ratio of 3…

where Q˙_kis the heat transfer from the fluid flowing through the heat exchanger. Heat exchangers (k⫽14 and 16兲 transfer energy from a mixture of ideal gases. Heat exchanger k⫽15 transfers energy from a circulating heat transfer fluid.

For all nine heat exchangers, there is no reaction so that the inlet and outlet flows are equal共e.g., N˙ka,i⫽N˙kb,i). In addition, it is assumed that the pressure drop across each heat exchanger is two percent of the inlet pressure at both design and off-design conditions.

Mixers. Mixers are used for injecting liquid water for hu-midification (k⫽20) and mixing gas streams prior to chemical reaction (k⫽2, 13兲. The pressure at the outlet of a mixer is sumed to be the lowest of the inlet pressures. The mixer is as-sumed to be adiabatic and non-reacting. The resulting mass and energy balances are given by Eqs.共6兲 and 共7兲, respectively,

兺

inlets

N˙k,i⫺N˙ka,i⫽0 for each species i (6)

兺

i N˙ka,i¯hka,i⫺

兺

i N˙kc,i¯hkc,i⫺

兺

i N˙kb,i¯hkb,i⫺

兺

i N˙kd,i¯hkd,i⫽0 (7) where the subscript ‘‘a’’ indicates the outlet and all other sub-scripts refer to inlets. The humidifier (k⫽20) model assumes adiabatic saturation of the exit gas stream.

Water Separators. Water separators (k⫽12, 21, 23兲 are used to remove liquid water from the gas stream at the exits from the stack and the expander. The molar flow rate of water in excess of the flow rate assuming saturated conditions is considered to be liquid at the temperature of the gas stream. The liquid water drains to the storage tank (k⫽24) and is pumped from there to the steam generator and humidifier.

Compressors, Expanders, and Pumps. The PEMFC system includes compressors and pumps that use power from the stack and an expander that supplies a portion of the compressor power. The air supplied to the fuel cell is typically compressed to 2–5 atmospheres because fuel cell stack performance generally in-creases with reactant pressure. The fuel stream must be operated at approximately the same pressure to minimize the pressure dif-ference across the cell membrane. Since natural gas is usually available at very low pressures, a second compressor is required to compress the fuel to the cell operating pressure. While higher stack pressures improve stack performance, the power required to operate the compressors共particularly the air compressor兲 can rep-resent a substantial parasitic power requirement for the system. This can be partially offset by using an expander to recover en-ergy from the reformer and fuel cell exhaust streams to power the air compressor. In addition to the air and natural gas compressors, power is required to pump the heat transfer fluid through the fuel cell stack. However, the power associated with the coolant pump (k⫽17) was determined to be less than 1% of the compressor power and is, thus, neglected in the present model. Likewise, the power for the pump distributing water to the steam generator and the humidifier was neglected.

Compressors and expanders are modeled as adiabatic devices with a constant isentropic efficiency. With this assumption and the assumption of constant specific heat, the model for the compres-sors (k⫽1 and 18兲 is given by

␩k⫽

兺

i N˙k,i¯hkb,i s ⫺

兺

i N˙k,i¯hka,i

兺

i N˙k,i¯hkb,i⫺

兺

i N˙k,i¯hka,i ⫽

兺

i N˙k,i¯hkb,i s ⫺

兺

i N˙k,i¯hka,i W˙k (8) and for the expander (k⫽22) is expressed as

␩k⫽

兺

i N˙k,i¯hka,i⫺

兺

i N˙k,ih¯kb,i

兺

i N˙k,i¯hka,i⫺

兺

i N˙k,ih¯kb,i s ⫽ W˙k

兺

i N˙k,i¯hka,i⫺

兺

i N˙k,ih¯kb,i s (9) where W˙kis the magnitude of the power for each device and the superscript ‘‘s’’ indicates the outlet condition if the process were isentropic. The changes in enthalpy are determined by assuming a constant specific heat evaluated at the average of the inlet and exit temperatures. The exit temperature for the isentropic process is found from Tb s ⫽Ta

冉

pb pa

冊

兺iNk,iR/兺iNk,ic¯p,i

(10) Psychrometric calculations were used to evaluate whether con-densation occurred at the outlet of the expander. Operating condi-tions that led to condensation were avoided due to the potential damaging effects on the expander.

PEMFC Stack. The model of the PEMFC stack is a lumped-parameter model incorporating elements of the model by Barbir and Gomez关9兴. The family of polarization curves which comprise the stack model is based on GCtools 关10兴 and is given by the following expressions:

For J⭓0.001 A/cm2_{and T⬎303.15 K:}

Vcell* ⫽1.05⫺0.055 log10共1000J兲⫺共1.064⫺0.002493T兲J

⫹0.055 log10共p11a,O2/p0兲 (11) For J⭓0.001 A/cm2and T⭐303.15 K:

V_cell* _{⫽1.05⫺0.055 log}10共1000J兲⫺共8.966⫺0.02857T兲J

⫹0.055 log10共p11a,O2/p0兲 (12) For J⬍0.001 A/cm2:

Vcell* ⫽1⫹0.055 log10共p11a,O2/p0兲 (13) where J is the current density, Vcell* is the uncorrected cell voltage, and pO2is the partial pressure of oxygen in the inlet air. The cell voltage predicted by Eqs.共11兲–共13兲 must be corrected to reflect the influence of mass transfer limitations at high current density. The expression for the corrected cell voltage for J⭓0.001 A/cm2 and for all temperatures is

Vcell⫽Vcell* ⫹0.1 log10

冉

1⫺ J Jmax

冊

(14) where the limiting current density Jmaxis

Jmax⫽1.4⫹0.2

冉

p11a

p0 ⫺3

冊

(15) and p11ais the total pressure of the air entering the stack.

Once the current density is known, the molar flow rates of hydrogen, oxygen, and nitrogen supplied to the stack can be found from

N˙11c,H2⫽

ncellJAcell␾H2

2F (16)

N˙11a,O2⫽

ncellJAcell␾air

4F (17)

N˙11a,N2⫽

3.76ncellJAcell␾air

4F (18)

where ncellis the number of cells, Acellis the area of each cell, F is Faraday’s constant, 3.76 is the number of moles of N2per mole of O2in standard air, and␾H2and␾airare the stoichiometric ratios

for fuel and air, respectively. Equations共16–18兲 assume that there is no reactant crossover and that any fuel that reacts produces current when it does so共i.e. the Farradic efficiency is 100%兲.

Net Electrical and Thermal Output. The gross electrical power from the fuel cell is given by

W˙gross⫽ncellVcellJAcell (19)

A portion of the gross power is used within the PEMFC system to operate the air and natural gas compressors. Some of the power required to drive the air compressor is provided by an expander in the exhaust stream from the fuel cell and the reformer. The net parasitic power, used within the system, is

W˙par⫽W˙SS⫹W˙FPS (20a)

where the power is divided between the FPS and the SS共in pro-portion to the respective airflow tates兲 to facilitate the decompo-sition approach used for optimization.

W˙SS⫽共W˙18⫺W˙22兲共N˙11a/N˙18b兲 (20b) W˙FPS⫽W˙1⫹共W˙18⫺W˙22兲共1⫺N˙11a/N˙18b兲 (20c) The net PEMFC system power is

W˙net⫽W˙gross⫺W˙par (21)

The PEMFC system also provides thermal energy from the fuel cell stack and from the FPS. Application of an energy balance to the stack (k⫽11), which is assumed to be insulated, yields

兺

i N˙11a,i¯h11a,i⫹

兺

i N˙11c,i¯h11c,i⫺

兺

i N˙11b,i¯h11b,i⫺

兺

i N˙11d,i¯h11d,i

⫹N˙11e¯h11e⫺N˙11f¯h11f⫺W˙gross⫽0 (22)

where streams a and c are the reactant streams, b and d are the exhaust streams and the stack coolant flows from inlet e to outlet f. Additional thermal energy is transferred to the stack coolant from heat exchanger k⫽10 and from the PROX reactor (k⫽9). Energy is transferred from the coolant loop to the thermal load through the low temperature cogeneration heat exchanger (k

⫽15). If the thermal load is met, energy is rejected to the

sur-roundings through the heat rejection device (k⫽16).

Thermal energy from the FPS can be supplied to the thermal load through the high temperature cogeneration heat exchanger (k⫽14). If the thermal energy from the high temperature cogen-eration heat exchanger is not required by the load, the exhaust gases from the FPS bypass the heat exchanger through valve V2 and are exhausted through the expander to provide power for the air compressor.

The total rate at which thermal energy can be supplied from the PEMFC system to the thermal load is

Q˙cogen⫽Q˙14⫹Q˙15 (23)

The heat associated with the fuel processor exhaust stream (Q˙14) is available at a much higher temperature than that available from the PEMFC coolant loop. However, since the temperature require-ments for domestic water and space heating are relatively low (⬍60 C), both of these streams are useful.

Cost Models. The capital cost model for the FPS is based on the volumetric rate of hydrogen, G˙H₂, produced by the FPS. This type of model instead of a more desirable one based on the ge-ometry of the principal components involved was assumed due to the limited information available in the open literature at the time of this work. This has since been updated to a geometric model in the work of Georgopoulos关6兴 and Georgopoulos et al. 关7兴.

A number of relations between G˙_H

2and the capital cost, CFPS, appear in the literature. The relation used here is based on the work of Ogden et al.关11兴 for large scale H2production and is

CFPS⫽50,900共G˙H₂兲0.7 (24) where the volumetric flow rate of hydrogen is expressed in thou-sands of standard m3per day and the capital cost in dollars. This cost model assumes that the PROX reactor adds 20% to the cost of the primary H2 production line. This assumption is valid for relatively small values of H2production, i.e. 3 to 9 kmols/hr. The capital cost has been updated to 2000 year dollars.

The function used for calculating the capital cost of the stack was based on the active area per cell, Acell(m2), the number of cells, ncells, and the number of stacks manufactured per year, nunits. This cost function is an extrapolation of data found in Oei et al.关12兴. Thus,

Cstack⫽␥

兺

i

Ci (25)

where the Civalues are as follow:

CMEA⫽共0.00777Acell⫹0.0616兲ncells (26) Cbipolar⫽共0.001384Acell⫹0.154兲ncells (27) Ccooling⫽共0.000692Acell⫹0.154兲ncells (28)

Cmix⫽124.48 (29)

Cassembly⫽共15.4⫹0.1733ncells兲 (30)

␥⫽500,000

nunits

(31) CMEAis the capital cost of the membrane electrode assemblies; Cbipolarthe capital cost associated with the bipolar plates; Ccooling, the capital cost associated with the cooling system in the stack; Cmixthe capital cost of the endplates, plastic insulators, current collectors, plastic housing, and tie bolts; Cassemblythe part of the capital cost associated with assembly; and␥ a scale factor which accounts for the number of stack units manufactured per year.

Costs given by Eq. 共25兲 to 共31兲 have been escalated to year 2000 dollars. For the base case of 1482 units manufactured per year, the price of the stack predicted for various combinations of active area and number of cells and a gross power output of 126 kWe5compares well with the cost predicted by other cost func-tions found in the literature共e.g., Ekdunge and Raberg, 关13兴兲 for an active area per cell of about 500 cm2_.

The annual operating cost associated with the use of natural gas by the fuel processing system is based on the amount of gas used and a unit gas cost of $7.39/GJ共$7.52/MCF兲6, which corresponds to $6.58/kmol of natural gas. The annual cost in dollars is thus

CA⫽6.58N˙1ahA (32)

where N˙1a is the molar flow rate of natural gas and hA is the number of annual operating hours.

The life cycle cost of the PEMFC system is given by the sum of the first cost and the present value of the annual costs, i.e.

LCC⫽CFPS⫹Cstack⫹CA

冋

共1⫹i兲n_⫺1

i共1⫹i兲n

册

(33) where i is the discount rate共taken to be 5.5%兲, n is the life of the PEMFC system共taken to be 15 years兲, and costs are expressed in constant year 2000 dollars.

5_{The average gross power produced over the entire environmental profile for the} optimum synthesis/design of the PEMFC system for 50 residences is close to 100 kWe, with a peak electric gross power of 149 kWe.

6_{Rates assumed for gas and electricity correspond to typical rates for the} mid-Atlantic region of the US. Other assumptions could lead to a significantly different synthesis design.

Evaluation of Optimal SynthesisÕDesign. The thermoeco-nomic model described by Eqs 共1兲 to 共33兲 and the associated assumptions is a non-linear system of algebraic equations. This system of equations is supplemented by a number of parameters that are assumed to be known and constant. Table 3 lists these parameters as well as the electrical and thermal load conditions at the synthesis/design point. With the data of Table 3 known, only the variables listed in Table 4 remain to be determined. At the synthesis/design stage of the problem, the variables of Table 4 can be determined by minimizing the life cycle cost of the PEMFC system. Specifically, the optimization problem to be solved is ex-pressed as

Minimize . . . LCC⫽CFPS⫹Cstack⫹CA

冋

共1⫹i兲n_⫺1 i共1⫹i兲n

册

(34) with respect to the set of synthesis/design variables in Table 4 subject to the set of equations 共1兲 through 共32兲 that define the system model and the set of parameters in Table 3.

The optimization problem described by Eq. 共34兲 is not only a design problem in which design and operating variables are deter-mined but also a synthesis problem because, while the system level configuration of Fig. 1 remains fixed, the configurations of the components such as the number of cells in the fuel cell stack, the number of heat exchanger tubes, the number of tube passes, the number of shell passes, and the number of compact heat ex-changer plates are determined by the optimization共as dependent variables兲.

The optimization problem expressed by Eq.共34兲 is amenable to optimization by decomposition techniques. Three types of decom-position are used: conceptual, time, and physical. With respect to the latter, a technique called Local-Global Optimization共LGO兲, described in detail in Mun˜oz and von Spakovsky关14–16兴,

Geor-gopoulos et al.关7兴, and El-Sayed 关17兴 is used. Its application, as

well as that of conceptual and time decomposition, to the model developed here is presented in Oyarzabal关5兴 and Oyarzabal et al.

关1兴.

Modifications to the Model for Off-Design Operation Once the synthesis/design problem is solved, the geometric pa-rameters of the various system components can be determined. For example, solution of the optimization problem leads to the specification of an optimum value for the heat transfer from the reformate stream to the reformer feed stream. With the optimal value of the heat transfer known, a heat exchanger geometry can be determined. Once all of the geometric parameters are known, it is possible to evaluate the optimum performance of a particular solution at off-design conditions. This information can be used to determine which of the candidate solutions 共the optimal plus a number of near-optimal solutions兲 found at the synthesis/design point is chosen as the overall system optimum. In this way, both synthesis/design and operation are optimally taken into account. In order to proceed to the off-design optimization, the thermody-namic model used at the synthesis/design point is supplemented with additional relationships, which reflect the influence of the geometric parameters.

For the steam reformer, steam generator, and heat exchangers, the performance at off-design conditions is modeled using an effectiveness-NTU approach关18兴 where

Q⫽␧Qmax (35)

␧⫽␧

冉

Cmin Cmax

,NTU

冊

(36)

In Eq.共35兲, Q is the heat transfer at off-design, Qmaxis the maxi-mum possible heat transfer, and␧ is the heat exchanger effective-ness at off-design conditions. The effectiveeffective-ness is determined as a function of the ratio of the minimum and maximum specific heat rates at the off-design condition and the number of transfer units, NTU. The number of transfer units is based on the heat exchanger UA value which is sized to meet the design load and the specific heat rate at off-design. The functional relationship between the effectiveness and NTU depends on the geometric flow configura-tion of the heat exchanger 共i.e. counterflow, crossflow, parallel flow兲.

For heat exchangers, the effectiveness-NTU method is applied in the conventional manner. For the steam reformer at off-design, it is assumed that the heat transfer process共not the chemical ki-netics兲 is the limiting process. Thus, the heat transfer is deter-mined from the effectiveness-NTU method and used to establish the rate of the reaction. Also, in the steam reformer, the specific heat rate must be modified to reflect the effect of the endothermic chemical reaction. The steam generator is modeled by separating

Table 3 Fixed parameters for the PEMFC system.

Parameters Value

Isentropic efficiency of the compressors 0.7 Isentropic efficiency of the expanders 0.85 Temperature of fluid entering the stack 70°C Temperature rise for the stack cooling fluid 10°C

Ambient pressure 101.325 kPa

Pressure drop in the HXs 2%

Efficiency of the water separators 90% Stoichiometric ratio of hydrogen 1.17 Stoichiometric ratio of oxygen 2

Design共summer兲 electrical load 2.7 kW per residence Design共summer兲 thermal load 0.7 kW per residence

Table 4 Decision variables for the PEMFC system at the synthesisÕdesign point.

Fuel Processing Sub-System共FPS兲

Decision Variables Coupling Functions

Stack Sub-System 共SS兲 Decision

Variables Steam to methane ratio at the inlet to the

steam reformer (rSM⫽N˙4a,H20/N˙4a,CH4)

Rate of hydrogen produced by the FPS (N˙9a,H2)

Active area (Acell)

Steam temperature at the exit of the steam generator (T25b)

Power required to operate the FPS (W˙FPS)

Reformate temperature at the inlet to the PROX reactor (T9a)

Stack operating pressure (pstack)

Reformate temperature at the inlet to the LTS reactor (T7a)

Reformate temperature at the exit of the steam reformer (T4b)

Fraction of methane combusted to heat the steam reformer (N˙26a,CH4/N˙1b,CH4)

Change in temperature of the reformate in the steam reformer (T4b⫺T4a)

it into three zones–an economizer, an evaporator, and a super-heater. Heat exchanger calculations are performed for each zone to size the generator共i.e. determine its geometry兲 and, thus, de-termine its performance at off-design.

For the shift reactors, combustor, and PROX, the performance at off-design is determined by the product composition which is found based on the assumptions of Table 2. For the fuel cell stack, the current density at off-design is determined from Eq. 共19兲. Based on the current density, the fuel cell operating conditions are determined by Eqs.共11兲–共18兲. Finally, the compressors are mod-eled using compressor maps. The compressors are assumed to have variable speed control so that, within limits, both flow rate and pressure ratio are independent variables.

The set of additional relationships applied to evaluate the off-design performance changes the independent variables to those that can act as control variables during optimization at off-design. For example, the stack active area, optimally determined at the synthesis/design point, remains fixed during the off-design opti-mization. The independent variables used for optimizing system operation at off-design include:

• The steam to methane ratio at the reformer inlet

• The positions of the three system control valves–V1, V2, and V3

• The compressor airflow expressed as a fraction of design air-flow

• The compressor pressure ratio

The solution of the off-design problem including a discussion of the geometric considerations is described in detail in Oyarzabal

关5兴 and Oyarzabal et al. 关1兴. Results

The thermoeconomic model in the previous section was used in Oyarzabal关5兴 and Oyarzabal et al. 关1兴 to demonstrate the power of combining decomposition techniques with both heuristic and gradient based optimization algorithms to extend the power of such algorithms to the synthesis/design optimization of highly complex, highly dynamic energy systems with a large number of degrees of freedom. In fact, in many cases, it is decomposition that even makes this possible. Such an optimization allows one to evaluate the effect of system size共expressed as a function of the number of residences兲 and manufacturing volume 共expressed as units manufactured per year兲 on the economic characteristics of the system. In addition, the companion paper 关1兴 optimizes the

system at different load conditions to evaluate how optimal off-design system performance affects the choice of system at the synthesis/design point.

Effect of System Size on the Optimal Life-Cycle Cost. The effect of system size on life cycle cost was investigated by apply-ing the decomposition techniques described in Oyarzabal关5兴 and Oyarzabal et al. 关1兴 to the synthesis/design optimization of PEMFC systems serving 1, 50, and 100 residences. For this study, the manufacturing volume was fixed at 1482 units per year. The resulting optimum synthesis/design life cycle cost expressed in $/residence and the influence of the number of residences is pre-sented in Fig. 3. This data quantifies how much more expensive the synthesis/design for a single residence is compared with 50 and 100 residences. Note that the synthesis/design total cost has an asymptotic behavior with respect to the number of residences. The results also suggest that the economies of scale are most significant for the fuel cell stack, for which the cost per residence declines dramatically as the size increases from 1 to 50 resi-dences. Another important result is that the fuel used共and, thus, the fuel cost兲 per residence does not change as the number of residences served increases共see Fig. 3兲. Thus, the efficiency of the system is not significantly affected by increasing system size. Fur-thermore, if one looks at the values of the independent variables for the optimum synthesis/design for the different number of resi-dences, one can observe that the values of independent variables such as temperature, active cell area, or stack pressure tend to have relatively similar values at the optimum conditions, indepen-dent of system size.

Fig. 3 Effect of the number of residences on the optimal PEMFC system costs

Fig. 4 Effect of manufacturing volume on the optimal PEMFC system costs

Table 5 Optimal values for decision variables and coupling functions at the synthesisÕdesign point

Decision variable/coupling function Optimal value Steam to methane ratio, rSM共molar basis兲 4.38

Steam temp. leaving the steam generator, T25b 724 K

Reformate temp. at the PROX inlet, T9a 468 K

Reformate temp. at the LTS inlet, T7a 474 K

Reformate temp. at the reformer exit, T4b 1021 K

Fraction of CH4combusted for the reformer 0.248

Reformate temperature change, T4b-T4a 115 K

Rate of hydrogen production, N˙9e,H2 4.61 kmol/h

Power required by the FPS, W˙FPS 5.50 kW

Stack operating pressure, pstack 2.17 bar

Effect of Manufacturing Volume on the Optimal Life Cycle Cost. The system-level optimization problem for the PEMFC system was solved for 50 residences and three different annual manufacturing volumes 共100, 1482, and 10,000兲. As expected, costs reduce considerably when the number of units manufactured per year increases. For a small number of units produced, the capital costs of the SS and FPS are the main contributors to the total optimum cost. However, when the number of units manufac-tured is increased, the fuel consumed by the PEMFC system be-comes by far the biggest contributor to the total optimal synthesis/ design cost. These tendencies are shown in Fig. 4.

Characteristics of the Optimal PEMFC System for 50 Resi-dences and 1482 Units. Every simulation of the PEMFC sys-tem generates detailed thermodynamic data that corresponds to a

particular system synthesis/design. The values of the decision variables and the coupling functions for the optimal synthesis/ design as determined in Oyarzabal关5兴 are presented in Table 5. Results from the thermoeconomic model obtained using these val-ues of decision variables and coupling functions are presented in Figures 5 and 6 and Table 6共design SD1兲. Figure 5 shows results from the model for the chemical composition of the methane/ reformate at different stages of the FPS.

The model also determines the magnitudes of the various en-ergy streams both internal and external to the system. Internal to the system, the optimal power needed to run the methane and air compressors at the design point is 28.8 kWe, which represents 21.3% of the total net power output 共135 kWe兲. However, the expander (k⫽22) recovers an important part of this energy 共14.2 kWe兲 by expanding the combustion gases and the air before hausting them to the atmosphere. Energy recovered from the ex-pander reduces the net parasitic loss to 14.6 kWe 共11% of net power兲, which represents an important improvement in perfor-mance for the configuration. Figure 6 illustrates the optimal mag-nitudes of the principle energy flows entering and leaving the PEMFC system at the synthesis/design point.

Finally, Table 6 summarizes the major energy and economic characteristics of the optimal synthesis/design for the PEMFC sys-tem serving 50 residences when 1482 units are manufactured per year. Table 6 also presents results from the second and fifth best system designs for comparison. These results show that the per-formance of the two best and the fifth best syntheses/designs are quite close so that their off-design behavior may help in determin-ing which is the optimal synthesis/design when the life cycle cost of the configuration is evaluated over the entire environmental/ load profile. The results for the optimal off-design performance can be found in Oyarzabal et al.关1兴.

Table 7 summarizes the performance of synthesis/design SD1 at the synthesis/design conditions and at a net power output

corre-Fig. 5 Variation of fuel stream composition within the optimal FPS

Fig. 6 Energy end use for the optimal PEMFC cogeneration system

Table 6 Main characteristics of the most promising synthesesÕdesigns at the synthesisÕ design point (summer peak electrical load)

Parameter Optimal syn-thesis/design 共SD1兲 2nd best syn-thesis/design 共SD2兲 5th best syn-thesis/design 共SD5兲

Total methane used共kmol/hr兲 1.56 1.58 1.67

Hydrogen consumed共kmol/hr兲 4.6 4.8 4.84

Power to run the FPS共kW兲 5.5 6.7 6.1

Electrical system efficiency共%兲 38.7 38.24 36.2

SS capital cost共$兲 227,906 222,363 217,425

FPS capital cost共$兲 145,019 151,780 153,044

Fuel cost共$, over 15 years兲 899,909 912,168 962,965

Synthesis/design total cost共$兲 1,272,834 1,286,311 1,333,434 Note: All syntheses/designs are based on serving 50 residences共total net power requirement of 135 kWe兲 with an annual manufacturing volume of 1482 units.

sponding to 53 percent of the synthesis/design net power.7The results show that the electrical efficiency of the system decreases significantly at very low load, i.e. in this case, at a partial maxi-mum load capacity of about 36%. This is attributable, of course, to the increasing significance of parasitic losses such as compres-sor power at part load.

Conclusions

Thermodynamic and economic models have been developed for a fuel cell cogeneration system, which consists of a fuel process-ing sub-system and a PEMFC stack sub-system. The models de-scribe the behavior of the fuel compressor, air compressor, ex-haust expander, steam reformer, shift reactors, PROX reactor, fuel cell stack, and various mixers and heat exchangers. The models have been developed in sufficient detail to permit the optimization of synthesis, design, and operating parameters. A companion pa-per, Oyarzabal et al.关1兴, describes the application of decomposi-tion techniques to the optimizadecomposi-tion of the models described here and discusses the optimization of off-design performance.

The major energy and economic characteristics of the most promising syntheses/designs are presented for fuel cell cogenera-tion systems serving 50 residences when the manufacturing vol-ume is 1482 units per year. The results indicate a capital cost of $2800/kW, and an electrical efficiency of 39% at the synthesis/ design point of 135 kWe net共149.6 kWe gross兲. Studies of the effects of the number of residences served and manufacturing vol-ume suggest that when the number of connected residences is small (Ⰶ50) or the annual manufacturing volume is small (Ⰶ1500), the system cost is prohibitive. When the annual manu-facturing volume is large (⬎10,000), the first cost becomes a relatively small component of the life cycle cost of the system.

These results suggest that fuel cell cogeneration systems are likely to be economical in clusters of homes or apartment complexes first and then applicable in single family homes as the manufac-turing volume increases.

References

关1兴 Oyarzabal, B., von Spakovsky, M. R., and Ellis, M. W., 2002, ‘‘The Optimal Synthesis/Design of a PEM Fuel Cell Cogeneration System for Multi-Unit Residential Applications—Application of a Decomposition Strategy,’’ ASME J. Energy Resour. Technol., ASME, N.Y., N.Y., accepted for publication. 关2兴 Jianguo, X., and Gilbert, F. F., 1989, ‘‘Methane Steam Reforming,

Methana-tion and Water-Gas Shift: I. Intrinsic Kinetics,’’ AIChE J., 35_共1兲.

关3兴 Jianguo, X., and Gilbert, F. F., 1989, ‘‘Methane Steam Reforming: II. Diffu-sional Limitations and Reactor Simulation,’’ AIChE J., 35共1兲.

关4兴 Gunes, M. B., 2001, ‘‘Investigation of a Fuel Cell Based Total Energy System for Residential Applications,’’ Masters Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.

关5兴 Oyarza´bal, B., 2001, ‘‘Application of a Decomposition Strategy to the Optimal Synthesis/Design of a Fuel Cell Sub-system,’’ M.S. Thesis, Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia.

关6兴 Georgopoulos, N. G., 2002, ‘‘Application of a Decomposition Strategy to the Optimal Synthesis/Design and Operation of a Fuel Cell Based Total Energy System,’’ M.S. Thesis, Department Of Mechanical Engineering, Virginia Poly-technic Institute And State University, Blacksburg, Virginia.

关7兴 Georgopoulos, N., von Spakovsky, M. R., and Munoz, J. R., 2002, ‘‘A Decom-position Strategy Based on Thermoeconomic Isolation Applied to the Optimal Synthesis/Design and Operation of a Fuel Cell Based Total Energy System,’’ International Mechanical Engineering Congress And Exposition— IMECE’2002, ASME Paper No. 33320, N.Y., N.Y., November.

关8兴 Moran, M. J., and Shapiro, H. N., 1996, Fundamentals of Engineering Ther-modynamics, 3rd edition, New York: John Wiley & Sons.

关9兴 Barbir, F., and Gomez, T., 1997, ‘‘Efficiency and Economics of Proton Ex-change Membrane共PEM兲 Fuel Cells,’’ Int. J. Hydrogen Energy, 22共10/11兲, pp. 1027–1037.

关10兴 Geyer, H. K., and Ahluwalia, R. K., 1998, ‘‘GCtool for Fuel Cell Systems Design and Analysis—User Documentation,’’ Argonne, IL: Argonne National Laboratory.

关11兴 Odgen, J. M., 1996, ‘‘Hydrogen Energy Systems Studies,’’ Princeton Univer-sity for U.S. Department of Energy, August.

关12兴 Oei, D., 1997, ‘‘Direct Hydrogen Fueled Proton Exchange Membrane Fuel Cell System For Transportation Applications,’’ Ford Motor Company For U.S. Department Of Energy, July.

关13兴 Ekdunge, P., and Raberg, M., 1998, ‘‘The Fuel Cell Vehicle Analysis of Energy Use, Emissions and Cost,’’ Int. J. Hydrogen Energy, 23共5兲, pp. 381–385. 关14兴 Mun˜oz, J. R., and von Spakovsky, M. R., 2002, ‘‘Decomposition in Energy

System Synthesis/Design Optimization for Stationary and Aerospace Applica-tions,’’ AIAA J., 39_{共6兲, Nov–Dec.}

关15兴 Munoz, J. R., and von Spakovsky, M. R., 2001, ‘‘The Use of a Decomposition Approach for the Large-Scale Synthesis/Design Optimization of Highly Coupled, Highly Dynamic Energy Systems,’’ International Journal of Applied Thermodynamics, 4共1兲.

关16兴 Munoz, J. R., and von Spakovsky, M. R., 2001, ‘‘The Application of Decom-position to the Large-Scale Synthesis/Design Optimization of Aircraft Energy Systems,’’ International Journal of Applied Thermodynamics, 5共1兲. 关17兴 El-Sayed, Y., 1989, ‘‘A Decomposition Strategy for Thermoeconomic

Optimi-zation, ASME Application,’’ ASME J. Energy Resour. Technol., 111, pp. 1–15. 关18兴 Incropera, F. P., and DeWitt, D. P., 1990, Fundamentals of Heat and Mass

Transfer, 3rd_{edition, New York: John Wiley & Sons.} 7_{Note that the synthesis/design point is a maximum efficiency point and not a}

maximum power point. In the case of SDI, for example, a maximum efficiency of 39% is achieved at a gross electrical power output of 149.6 kWe which is about 65% of the total gross load共230 kWe兲 which the fuel cell can handle. Unlike in most conventional energy systems, fuel cell system efficiencies between this maximum efficiency point and the maximum power point tend to remain fairly flat. Table 7 Summary of performance for synthesisÕdesign SD1 at synthesisÕdesign and off-design conditions

Characteristic Design Conditions

Off-design Conditions Percent of design net power, % 100 53

Net electrical power, kW 135 72

Gross electrical power, kW 149 90.5

Heat available, kW 118 104

Net electrical efficiency, % 39 28