Michael M. Whiston1 Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261 e-mail: [email protected]

Melissa M. Bilec Associate Professor Department of Civil and Environmental Engineering, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261 e-mail: [email protected]

Laura A. Schaefer Professor Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261 e-mail: [email protected]

SOFC Stack Model for Integration Into a Hybrid System: Stack Response to Control Variables Due to the tight coupling of physical processes inside solid oxide fuel cells (SOFCs), efficient control of these devices depends largely on the proper pairing of controlled and manipulated variables. The present study investigates the uncontrolled, dynamic behavior of an SOFC stack that is intended for use in a hybrid SOFC-gas turbine (GT) system. A numerical fuel cell model is developed based on charge, species mass, energy, and momentum balances, and an equivalent circuit is used to combine the fuel cell’s irreversibilities. The model is then verified on electrochemical, mass, and thermal timescales. The open-loop response of the average positive electrode-electrolyte-negative electrode (PEN) temperature, fuel utilization, and SOFC power to step changes in the inlet fuel flow rate, current density, and inlet air flow rate is simulated on different timescales. Results indicate that manipulating the current density is the quickest and most efficient way to change the SOFC power. Meanwhile, manipulating the fuel flow is found to be the most efficient way to change the fuel utilization. In future work, it is recommended that such control strategies be further analyzed and compared in the context of a complete SOFC-GT system model. [DOI: 10.1115/1.4029877] Keywords: solid oxide fuel cell, transient model, dynamic response, hybrid gas turbine system

1

Introduction

Integrating a SOFC stack into a GT system presents a number of benefits, including high efficiency [1–5], improved fuel cell performance [6], and cogeneration [1,4]. Adopting these systems in distributed generation applications, however, requires not only excellent thermodynamic performance but also the ability to follow a dynamic load. Residential, university, and office buildings experience significant load changes over the course of a day [7–9]. In order for hybrid systems to meet these needs, quick and efficient dynamic response is required. Physical processes inside SOFCs are tightly coupled [10–13], and choosing the proper combination of controlled and manipulated variables inside the SOFC stack is essential to achieving safe and efficient dynamic response. This is especially true in a multilevel controller, such as that proposed by Martinez et al. for an SOFC-GT system [5]. These authors developed a cascade control strategy that involved controlling a number of variables at varying levels of priority. At the highest priority, control of the average SOFC temperature was achieved by manipulating the air flow (or shaft speed). At a lower priority, control of fuel utilization proceeded by manipulating the SOFC voltage. At the lowest priority, control of system power was achieved by manipulating the inlet anode flow rate. A major benefit of such a multilevel controller is the minimization of interference between control loops, as lower levels are not pursued until the higher (safety-oriented) levels have been satisfied (such a control scheme is also amenable to development in a conventional programing language, such as FORTRAN or C, to coincide with a model written in one of these or similar languages). A major challenge, however, is avoiding oscillations between the various levels, as changes in one level could 1 Corresponding author. Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF FUEL CELL SCIENCE AND TECHNOLOGY. Manuscript received November 11, 2014; final manuscript received December 18, 2014; published online March 10, 2015. Editor: Nigel M. Sammes.

provoke changes in another level due to the coupled nature of physical processes inside SOFCs [5]. The present study investigates the uncontrolled (open-loop) response of the average PEN temperature, fuel utilization, and SOFC power to step changes in the inlet fuel flow rate, current density (or voltage), and inlet air flow rate on different timescales. The former set of variables typically requires control in an SOFCGT system for safety and efficiency reasons. The latter variables are often manipulated to achieve control, as manipulation of these variables is feasible and can also induce significant changes in the controlled variables [5,12,14–16]. This work identifies pairs of control variables (i.e., pairs of controlled and manipulated variables) that minimize interdependence, where interdependence may be defined as the inability of a manipulated variable to effectively control a targeted variable, unless control of another variable(s) is implemented. The reason for minimizing interdependence is to reduce the risk of oscillations between control levels in a multilevel controller [5]. Consideration is also given in this study to the time required for the SOFC to meet a power demand. Faster settling times are desired, as they enable a system to more quickly meet demand. Because emphasis is placed on SOFC (rather than system) behavior in this study, shaft speed dynamics are not considered (air flow is assumed to be instantaneous—this assumption may be likened to using an air bypass valve to adjust the air flow rate [16]) nor are balance of plant components considered. By focusing on SOFC stack behavior, the present study complements previous studies that have investigated controllers at the system level. Mueller et al. [14] developed a control strategy that involved manipulating the SOFC current to rapidly alter the fuel flow rate exiting the SOFC, thus controlling the combustor temperature. The fuel cell power, however, depended on the slower inlet fuel flow rate (because current had already been taken by the combustor control loop), leading to an unavoidable loadfollowing delay. Martinez et al. [5] similarly controlled fuel utilization using the SOFC voltage, and system power using anode fuel flow. Stiller et al. [15] instead chose to manipulate current to

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control SOFC power. In this system, power responded to disturbances in less than one second, whereas fuel utilization (controlled by system fuel flow) responded somewhat more slowly, on the order of seconds. Leucht et al. [16] also controlled fuel utilization by manipulating the fuel flow. The present study differs from these studies by focusing specifically on the SOFC stack. Results herein are intended to inform control decisions at the system level. In Sec. 2 of this work, the fuel cell model is developed based on charge, species mass, energy, and momentum balances, and an equivalent circuit is used to combine the fuel cell’s irreversibilities. Section 3 discusses features of the fuel cell stack. The stack’s operating conditions and solution procedure are presented in Secs. 4 and 5, respectively. The dynamic response of key SOFC variables to changes in the inlet fuel flow rate, current density, and inlet air flow rate is investigated in Sec. 6, and the model is also verified against results from a previous study. A discussion of possible control strategies is presented in Sec. 7. Finally, Sec. 8 summarizes the conclusions of this study, and future work is recommended.

2

2.1 Charge Balance. The reversible fuel cell voltage is described by the Nernst potential 0 1 1  PEN pH2 p2O2 D geoh ðTPEN Þ RT A þ ln@ (1) EN ¼  2F 2F pH2 O p 12 where D geoh ðTPEN Þ is the change in molar Gibbs free energy associated with the oxidation reaction at standard pressure and PEN temperature, F is Faraday’s constant, R is the universal gas constant, pi is the partial pressure of species i, and p is standard pressure [19,20]. 2.1.1 Activation Polarization. The activation polarization is determined separately for each electrode, and the total activation polarization is the sum of the electrode polarizations [19]. The Butler–Volmer equation models the activation polarization 0 1 agact ne F ð1  aÞgact ne F B  PEN C  PEN RT (2) e j ¼ j0 @e RT A

Fuel Cell Model

The SOFC model is a planar, electrolyte-supported fuel cell operating in co-flow, as shown in Fig. 1(a). A single SOFC channel is discretized into computational segments of finite length along the axial (x) direction. Each computational segment consists of several control volumes (CVs), corresponding to the anode channel, cathode channel, PEN structure, and interconnect components, as shown in Fig. 1(b). The composition of the pre-reformed mixture entering the anode channel is determined using a prereformer model based on that presented by Braun [17], assuming 30% conversion with a steam-to-carbon ratio of 2.5. Electrochemical oxidation of H2 is assumed in the SOFC, whereas CO is assumed to participate only in the water–gas shift reaction [18]. Methane steam reforming is also included in the present model.

where j is the current density, j0 is the exchange current density, gact is the activation polarization, and ne is the number of moles of electrons transferred [20]. Substituting a ¼ 0.5 for the charge transfer coefficient [20], Eq. (2) is solved for the activation polarization as follows:    PEN 2RT j (3) sinh1 gact ¼ 2j0 ne F The exchange current densities at the anode and cathode are given by the following expressions: ! tpb !   p H2 O ptpb Eact;an H2 (4) j0;an ¼ can exp   PEN p p RT !0:25   ptpb Eact;ca O2 j0;ca ¼ cca (5) exp   PEN p RT where c is a pre-exponential factor, ptpb is the partial pressure of i species i at the triple phase boundary, and Eact is the activation energy [20–23]. The cathode exchange current density is based on a formula provided in Ref. [24]. Costamagna and Honegger [21] cite Refs. [25,26] in reference to the anode exchange current density formula. In calculating the exchange densities, it is assumed that can ¼ 5:5108 A=m2 , cca ¼7108 A=m2 , Eact;an ¼100103 J=mol, and Eact;ca ¼120103 J=mol [22]. 2.1.2 Concentration Polarization. The concentration polarization is determined by calculating the difference between the Nernst potential evaluated using the bulk (channel) partial pressures and the triple phase boundary partial pressures 0 1 1 tpb ch2  PEN p pch p RT O O H H 2 A ln@ 2ch 2 (6) gconc ¼ ne F pH2 O tpb tpb12 pH2 pO2 where pch i is the partial pressure of species i in the anode or cathode channel [20,23]. Expressions for the triple phase boundary partial pressures are obtained using Fick’s law of diffusion, which is applied along the thickness of the electrode as follows: Ji ¼ CDi;eff

Fig. 1 SOFC model: (a) SOFC with channel highlighted, (b) channel with computational segment highlighted (CVs are indicated by dashed lines)

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dyi dz

(7)

where Ji is the diffusive flux of species i, C is the molar concentration of the gas mixture, Di;eff is the effective diffusion Transactions of the ASME

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coefficient through the electrode, and yi is the mole fraction of species i [27]. The oxidation reaction rate is given by the following formula: r_eoh

(8)

e Di;eff ¼ Di s

(9)

where e is porosity and s is tortuosity [28]. Representative values for both electrodes are e ¼ 0.30, and s ¼ 6 [10,29]. In Eq. (9), Di is the combined diffusion coefficient, which is calculated as follows: 

1 1 þ Dij Di;Knudsen

1 (10)

where Dij is the binary diffusion coefficient and Di,Knudsen is the Knudsen diffusion coefficient [10,23,30]. The Chapman–Enskog formula for the binary diffusion coefficient of species i and j in an ideal gas mixture is given by the following formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0:0018583 1 1 3 (11) þ T Dij ¼ PEN Mi Mj pr2ij XD;ij where rij is the collision diameter, XD,ij is the diffusion collision integral, and Mi is the molecular weight of species i [31]. The Knudsen diffusion coefficient is given by the following formula: 

gact i gconc concentration : Rconc ¼ i gohm ohmic : Rohm ¼ i activation :

i ¼ ne F

where i denotes the electric current [20]. The effective diffusion coefficient appearing in Eq. (7) is calculated as follows:

Di ¼

by an equivalent resistance [13,19,20,33,34]. Equivalent resistance is the ratio of each polarization to the current [13]

 PEN 2 8RT Di;Knudsen ¼ re 3 pMi

12 (12)

Ract ¼

(14) (15) (16)

Furthermore, during operation, layers of charge accumulate along the electrode–electrolyte interfaces (Fig. 2(b)) [19,20]. Such layers of charge are called “charge double layers” [19]. Charge double layers are represented as capacitors in the equivalent circuit (Fig. 2(a)), and the Nernst potential is represented as a voltage source [19,33]. In practice, a simplified equivalent circuit is used to calculate the fuel cell’s operating voltage, as shown in Fig. 2(c). In this circuit, the quantities Ract and Rconc represent the total activation and concentration equivalent resistances, respectively, which are the sums of the individual electrode equivalent resistances. The operating voltage is determined by applying Kirchoff’s voltage law. Application of Kirchoff’s voltage law to the entire circuit shown in Fig. 2(c) results in the following expression for Vop [13,34,35]:   dVdbl Vop ¼ EN  i  Cdbl (17) ðRact þ Rconc Þ  iRohm dt The equation above accounts for the charging and discharging of the charge double layer, as well as changes in the activation, concentration, and ohmic polarizations. 2.2 Species Mass Balance. The species mass balance applied to each CV in the gas channels is presented below on a molar basis

where re is the effective pore radius of the electrode material [30]. A representative value of the effective pore radius is re ¼ 0.5 lm for both electrodes [10,29]. 2.1.3 Ohmic Polarization. Ohmic polarization includes voltage loss caused by ohmic, contact, and spreading resistances. Since the SOFC components occur in series, total ohmic resistance is calculated by summing the resistances of the individual components. The ohmic resistance of each component is calculated using electrical conductivity formulas provided in the literature (Table 2). The estimate for the contact resistance is based on results presented in Koide et al. [32]. These authors experimentally determined the total ohmic area-specific resistance (ASR) of an SOFC (with materials similar to those used in the present model) to be approximately 0.25 ohm cm2 (higher or lower depending on the volume percent of Ni in the anode), and they further determined that contact resistance between the anode and electrolyte dominated their ASR measurement. The estimate of 0.25 ohm cm2 for the contact resistance is used in the present study, and this value is assumed to remain constant, regardless of the stack’s operating point [21]. The ohmic polarization is thus calculated as follows: gohm ¼ iðRanode þ Rcathode þ Relectrolyte þ Rinterconnect þ Rcontact Þ (13) where Ri is the resistance of component i. 2.1.4 Operating Voltage. The irreversible processes in an SOFC can be represented by the equivalent circuit presented in Fig. 2(a). In this circuit, each of the polarizations is represented Journal of Fuel Cell Science and Technology

Fig. 2 Representation of irreversible processes inside the SOFC: (a) equivalent circuit (adapted from Refs. [13,19,20,34]), (b) possible charge double layer in the SOFC (adapted from Ref. [20]), and (c) simplified equivalent circuit used to calculate the SOFC operating voltage (adapted from Refs. [19] and [33])

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@n000 @ n_ 00 @Ji 00 X i i;j r_j000 ¼ i  þ @t @x @x j

(18)

where n000 i represents moles of species i contained in the CV per 00 unit volume, n_ 00i is the molar flux of species i due to advection, Ji is the diffusive flux of species i,  i,j is the stoichiometric coefficient of species i associated with reaction j, and r_j000 is the rate of reaction j per unit volume [17,20,23,27]. The water–gas shift reaction is assumed to be in equilibrium. Changes in composition associated with the water–gas shift reaction are related as follows by stoichiometry [17,36]: Dn_ CO;wgs ¼ Dn_ H2 O;wgs

(19)

Dn_ H2 O;wgs ¼ Dn_ H2 ;wgs

(20)

 Dn_ H2 ;wgs ¼ Dn_ CO2 ;wgs

(21)

Furthermore, the methane steam reforming reaction rate is given by the following formula: r_msr ¼ ko fe pCH4 Ar e

Eact RT 

(22)

PEN

where the pre-exponential factor ko ¼ 4274 mol/s m2 bar, the equilibrium function fe  1, Ar is the electrode reaction surface area, and the activation energy Eact ¼ 8.2  104 J/mol [17,37]. The oxidation reaction rate is as previously given in Eq. (8). Diffusive flux is given by Fick’s law of diffusion applied along the fuel cell’s length (Eq. (7)), where the diffusion coefficient is calculated using the following formula for multicomponent mixtures: 0

11

n BX

Di;mixture ¼ ð1  yi ÞB @

j¼1 j6¼i

yj C C Dij A

(23)

where Dij is the binary molecular diffusion coefficient for species i and j [38], which is calculated using the Chapman–Enskog formula (Eq. (11)). Different definitions of the fuel utilization are used in this study depending on the type of simulation being performed: model verification versus dynamic response. During model verification, the fuel utilization is defined as follows: Uf;1 ¼

n_ H2;consumed 4n_ CH4 ;inlet þ n_ H2 ;inlet þ n_ CO;inlet

(24)

where n_ H2;consumed is the amount of H2 consumed by the SOFC and n_ i;inlet is the inlet molar flow rate of species i [17,39]. This definition agrees with that used by Wang and Nehrir [13], against whose results the present model is verified in Sec. 6.1. During dynamic response simulations, on the other hand, the fuel utilization is defined as follows [14,17]: Uf;2 ¼ 1 

4n_ CH4 ;exit þ n_ H2 ;exit þ n_ CO;exit 4n_ CH4 ;inlet þ n_ H2 ;inlet þ n_ CO;inlet

(25)

The latter definition accounts for fuel storage inside the SOFC (i.e., mass flow dynamic behavior) [14]. Finally, the excess air ratio is n_ O2 ;inlet k¼ 1 2n_ CH4 ;inlet þ ðn_ H2 ;inlet þ n_ CO;inlet Þ 2

(26)

which accounts for oxygen provided above the stoichiometric amount [17,39]. 031006-4 / Vol. 12, JUNE 2015

2.3 Energy Balance. The energy balance is applied to each of the gas channels, interconnect, and PEN structure. The energy balance applied to the gas channels includes contributions from advection, reactions, and convection   X X @ hi ðn_ 00i þ J_i00 Þ @TfðaÞ þ ¼ hi i;j r_j000 qfðaÞ cp;fðaÞ @x @t (27) i reactions þ q_ 000 conv where qf(a) is the fuel (or air) density, cp,f(a) is the fuel’s (or air’s) specific heat capacity, Tf(a) is the fuel’s (or air’s) temperature, hi is the molar specific enthalpy of species i, and q_ 000 conv is heat transfer due to convection [10,17,20,23,27,40,41]. Convection between the solid structures and gases is modeled using Newton’s law of cooling, where the convection coefficient is determined using a formula for a rectangular channel with laminar, fully developed flow and uniform temperature [27,42]. The bulk thermal conductivity is calculated using Wassiljewa’s formula for the thermal conductivity of a gas mixture at low pressure (with Mason and Saxena’s modification) [43]. The energy balance applied to the PEN structure includes contributions from reactions, as well as conduction, convection, radiation, and power generated by the SOFC qPEN cp;PEN

X @TPEN @ 2 TPEN þ q_ 000 ¼ hi i;j r_j000 þ kPEN conv @t @x2 reactions 000 þ q_ 000 rad  Pseg

(28)

where kPEN is the thermal conductivity of the PEN structure, and P000 seg is the power generated in each computational segment [17,20,23]. Heat transfer due to radiation is calculated according to the formula for a two-surface enclosure, assuming that both surfaces are opaque, diffuse, gray, and isothermal with uniform radiosity and irradiation q_ rad ¼

rðT14  T24 Þ 1  e1 1 1  e2 þ þ A1 F12 e1 A1 e2 A2

(29)

where r is the Stefan–Boltzmann constant, ei is the emissivity of surface i, and Ai is the heat transfer surface area associated with surface i [27]. Finally, the energy balance applied to the fuel and air-side interconnects includes contributions from conduction, convection, and radiation. This energy balance takes the following form [17,20,23]: qinterfðaÞ cp;interfðaÞ

@TinterfðaÞ @ 2 TinterfðaÞ 000 þ q_ 000 ¼ kinterfðaÞ conv þ q_ rad @t @x2

(30)

2.4 Momentum Balance. The momentum balance applied to the gas channels is presented below:     @ qfðaÞ ufðaÞ @ qfðaÞ ufðaÞ ufðaÞ @pfðaÞ ¼  @t @x @x p^fðaÞ sw (31)  Ac;fðaÞ where uf(a) is the axial velocity in the anode (or cathode) channel, p^fðaÞ is the perimeter of the anode (or cathode) channel, Ac;fðaÞ is the cross-sectional area of the anode (or cathode) channel, and sw is the wall shear stress [44–46]. The wall shear stress is a function of the Fanning friction factor, and the Fanning friction factor for fully developed and laminar flow in a rectangular channel is a function of the Reynolds number and channel aspect ratio [27,42]. Transactions of the ASME

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The mixture viscosity is calculated using Wilke’s formula for the viscosity of a gas mixture at low pressure [43]. The SOFC model also accounts for minor losses at the inlet and outlet of the gas channels [17,19,46–48]. 2.5 Geometric and Material Properties. Geometric and material properties of the SOFC are provided in Tables 1 and 2, respectively ([49], [39] (citing [50])). The material properties provided in Table 2 pertain to ceramic components, which comprise the interconnect and PEN structure.

3

Fuel Cell Stack Considerations

In addition to the governing equations presented previously, the SOFC stack model accounts for stack-specific phenomenon. In particular, the SOFC stack model accounts for thermal radiation exchange between the stack and stack’s container. The stack’s surface is assumed to be opaque, gray, and diffuse with an emissivity of e ¼ 0.8 due to the highly emissive interconnect, electrolyte, and seal materials [19,20,39,51]. Furthermore, the container is typically made of a metal alloy that is heated to approximately 650  C during operation [52], and the container’s geometry bears resemblance to a cavity approximating a blackbody. The surface energy balance applied to the stack results in the following formula for the net rate at which thermal energy leaves the stack due to radiation exchange:  4  4  Tcontainer (32) q_ stack ¼ Astack estack r Tcell where estack is the emissivity of the stack, r is the Stefan–Boltzmann constant, and Tcontainer and Tcell are the container and SOFC (interconnect, electrolyte, and seal) temperatures, respectively [27,39]. In the stack model, Eq. (32) is treated as a boundary condition, applied at the inlet and exit of the solid material (PEN and interconnect) energy balance equations. The stack surface area in Eq. (32) is calculated using the dimensions of the inlet and exit solid materials. The SOFC stack model also accounts for heat transfer between adjacent SOFCs [17,53,54]. In particular, the interconnect conducts and convects thermal energy between fuel and air channels belonging to adjacent fuel cells in the stack. It should be noted that this type of heat transfer, which we shall term “cross-channel heat transfer,” is different from intrachannel convection, which has already been included in the gas channel energy balance equation (Eq. (27)). That is, cross-channel heat transfer occurs between adjacent fuel cells, whereas intrachannel convection occurs within a single fuel cell between the solid material and bulk flow. Crosschannel heat transfer is calculated using the following equation, which is based on a thermal resistance network:

q_ crosschannel ¼

Tf  Ta 1==hf Af þ 1=kPEN S þ 1=ha Aa

where Af and Aa are the surface areas of the anode and cathode channel walls, and S is a shape factor that accounts for the 2D heat flux through the interconnect [17,27]. Using numerical analysis, Braun [17] calculated S ¼ 0.019 m, which is adopted in the present work as well.

4

Operating Conditions

The SOFC’s operating conditions are presented in Table 3. The operating conditions differ depending on the type of simulation being performed: model verification versus dynamic response. The operating conditions for model verification are intended to closely match those of the study performed by Wang and Nehrir [13], as this study has been chosen to verify the present model. The operating conditions for dynamic response, on the other hand, reflect typical SOFC-GT operating conditions found in the literature. In particular, the SOFC operates at a pressure ratio of 4:1 [1,3], and the stack is sized to meet a power demand of approximately 100 kW (assuming a typical power output of approximately 20 W per fuel cell [49]). The power rating of 100 kW is similar to that of a small microturbine [55]. In both model verification and dynamic response simulations, consideration has been given to the SOFC operating conditions presented in the 1996 IEA benchmark study [49], particularly the inlet gas temperature, fuel composition, and mean current density. No consideration, however, has been given to the balance of plant component behavior in this study. Thus, the operating parameters in Table 3 are assumed to remain constant, unless otherwise specified.

5

Solution Procedure

To solve the SOFC’s system of equations, the balance equations are applied along the length of a single channel in implicit, finite difference form [17,27,56]. The resulting linear system is solved using ENGINEERING EQUATION SOLVER (EES) [57]. The model performs calculations along the length of a single channel, and the results are applied identically to all channels (due to convergence issues, the temperature is assumed to remain constant for simulations performed on the millisecond timescale). Likewise, results from a single fuel cell are applied identically to all fuel cells in the fuel cell stack. Dynamic simulations are performed in EES using parametric tables to step through time, and lookup tables are used in EES to store previous iterations.

6

Results

6.1 Model Verification. The SOFC model is first verified against the results of a previous study from the literature. In particular, Wang and Nehrir [13] developed a dynamic, tubular,

(33) Table 2

Material properties of the SOFC

Property

Table 1

Thermal conductivity Heat capacity Density

Geometric properties of the SOFC

Property Electroactive area (PEN and interconnect) Anode thickness Cathode thickness Electrolyte thickness Interconnect thickness Channel width Channel height (fuel and air) Number of channels Rib width

Journal of Fuel Cell Science and Technology

Value 100  100 mm2 0.05 mm 0.05 mm 0.15 mm 2.5 mm 3 mm 1 mm 18 2.42 mm

Anode conductivity Cathode conductivity Electrolyte conductivity Interconnect conductivity

Value 2 W/m K 400 J/kg K 6600 kg/m3

  95  106 K S m1 1150 K exp  TPEN TPEN   6 1 42  10 K S m 1200 K exp  TPEN TPEN   10300 K 3:34  104 S m1 exp  TPEN   9:3  106 K S m1 1100 K exp  Tinter Tinter

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Table 3 Operating conditions of the SOFC Parameter Flow configuration Inlet gas pressure Inlet gas temperature Inlet fuel compositiona

Inlet air composition Inlet fuel flow rate Excess air ratio Inlet air flow rate Mean current density Number of fuel cells

Verification

Response

Co-flow 3 atm 1173 K CH4 ¼ 17.10% H2 ¼ 26.26% H2O ¼ 49.34% CO ¼ 2.94% CO2 ¼ 4.36% O2 ¼ 21% N2 ¼ 79% 2.833  106 kg/s 6 — 3000 A/m2 (initially) 1

Co-flow 4 bar 1173 K CH4 ¼ 17.07% H2 ¼ 21.95% H2O ¼ 53.66% CO ¼ 2.51% CO2 ¼ 4.81% O2 ¼ 21% N2 ¼ 79% 3.323  106 kg/sb — 9.012  105 kg/sb 3000 A/m2b 5000

a The composition of the pre-reformed fuel mixture used during verification simulations is from Ref. [49]. The composition of the pre-reformed fuel mixture used during dynamic response is determined using a pre-reformer model based on that presented by Braun [17] (assuming 30% conversion with a steam-to-carbon ratio of 2.5). b Unless otherwise specified.

pressurized SOFC model in MATLAB/SIMULINK. These authors investigated the voltage response of the SOFC to changes in current density load on small, medium, and large timescales, corresponding to electrochemical, mass flow, and thermal processes, respectively. Figures 3–5 present the voltage responses of the present model on millisecond, second, and minute timescales, and estimated settling times from Wang and Nehrir’s study are indicated by dashed lines for comparison [35]. The model’s electrochemical voltage response is shown in Fig. 3. The current density load decreases from 3000 A/m2 to 2500 A/ m2 at 50 ms, and the double layer capacitance is varied between 0.1 mF and 10 mF. The electrochemical voltage settling time for Cdbl ¼ 10 mF is found to be slightly over 50 ms. This result agrees with the settling time estimated from Wang and Nehrir’s results, who also found a settling time of approximately 50 ms [13]. Notice, also, that the voltage settling time is very close to the double layer polarization settling time, indicating that the charge double layer is associated with the SOFC’s dynamic behavior on this timescale. The model’s mass flow voltage response is shown in Fig. 4. The current density load decreases from 3000 A/m2 to 2500 A/m2 at 5 s. The mass flow voltage settling time is found to be approximately 2 s. The settling time estimated from Wang and Nehrir’s

Fig. 3 Electrochemical voltage response. The dashed line indicates the estimated electrochemical voltage settling time based on Wang and Nehrir’s results. The double layer polarization (axially averaged) is shown for Cdbl 5 10 mF.

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Fig. 4 Mass flow voltage response. The dashed line indicates the estimated mass flow voltage settling time based on Wang and Nehrir’s results. The hydrogen mole fraction (axially averaged) is also shown.

results is also approximately 2 s [13]. Notice, also, that the voltage settling time of the present model is very close to the H2 mole fraction settling time, indicating that change in the gas composition is associated with the SOFC’s dynamic behavior on this timescale. Finally, the thermal voltage response is shown in Fig. 5. The current density load decreases from 3000 A/m2 to 2500 A/m2 at 3000 s (50 min). The thermal voltage settling time is approximately 600 s (10 min). The settling time estimated from Wang and Nehrir’s results, on the other hand, is approximately 1500 s (25 min). One possible explanation for this discrepancy is the values used for the materials’ heat capacities. In general, higher heat capacities correspond to slower changes in temperature. Wang and Nehrir do not provide the heat capacity values used for the solid materials in their paper, but they do assume an inlet fuel composition of 90% H2. Hydrogen has a relatively high heat capacity, roughly an order of magnitude higher than that of other typical SOFC gases [39]. In the present model, on the other hand, H2 accounts for only approximately 26% of the inlet fuel composition (Table 3). This difference may help to explain the present discrepancy. 6.2 Dynamic Response. The verified fuel cell model is used to perform simulations of the stack’s dynamic response to changes in specified variables on different timescales. In particular, Figs. 6, 8, and 9 present the uncontrolled (open-loop) response of the average PEN temperature, fuel utilization, and SOFC power to

Fig. 5 Thermal voltage response. The dashed line indicates the estimated thermal voltage settling time based on Wang and Nehrir’s results. The PEN temperature (axially averaged) is also shown.

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step changes in the inlet fuel flow rate, current density, and inlet air flow rate. In each figure, only one variable is altered to observe the particular influence that each manipulated variable has on SOFC stack behavior. Due to the fuel cell’s varying response on different timescales, each simulation has been performed on millisecond, second, and minute timescales. The SOFC stack’s dynamic response to a step change in the inlet fuel flow rate is shown in Fig. 6. During this simulation, the fuel flow is increased by 50% (from an initial value of 3.323  106 kg/s, as indicated in Table 3) at times 50 ms, 5 s, and 50 min (Figs. 6(a)–6(c), respectively). On the millisecond timescale (Fig. 6(a)), a small increase in the fuel utilization occurs before the exit fuel flow rate has time to respond (Eq. (25)), followed by a gradual decline in the fuel utilization. This gradual decline likely demarcates the beginning of the mass flow response. Meanwhile, power and PEN temperature remain relatively constant. On the second timescale (Fig. 6(b)), the fuel utilization changes significantly, decreasing from 85% to slightly over 55% in a few seconds time. This settling time is indicative of mass flow transient behavior. The power undergoes a slight increase as well, which is due to the increasing operating voltage on the second timescale. Finally, on the minute timescale (Fig. 6(c)), the PEN temperature decreases slightly, which is likely due to increased convection between the PEN structure and fuel. This slight change in PEN temperature corresponds to a similarly small decrease in the power. Interestingly, the influence of the inlet fuel flow rate on the SOFC power under constant fuel utilization differs drastically from that when the fuel utilization is allowed to vary freely. In particular, under constant fuel utilization (i.e., ideal control of fuel utilization), changing the fuel flow changes the power significantly. Figure 7 displays the SOFC stack’s response to the same change in the fuel flow as considered previously (Fig. 6), except the fuel utilization is held fixed at Uf,2 ¼ 85%. As can be seen in Fig. 7, the power exhibits an increase of over 7 W on the second timescale. This result differs drastically from that obtained when the fuel utilization is allowed to vary freely (Fig. 6(c)), during which the power increases by only 1 W. The reason for this difference in behavior is rooted in the definition of the fuel utilization (Eq. (25)), which may be qualitatively defined as the ratio of mass consumed over mass acquired. If this ratio is held constant and fuel flow increases, then mass consumed (or current) also increases, leading to an increase in the power. The current density is also manipulated to observe its influence on the SOFC stack’s behavior. Figure 8 illustrates the stack’s dynamic response to a step change in the mean current density. The current is decreased by 50% (from an initial value of 3000 A/m2) at times 50 ms, 5 s, and 50 min (Figs. 8(a)–8(c), respectively). On the millisecond timescale (Fig. 8(a)), the power responds instantaneously to the change in the current density, decreasing sharply from 21 W to 11 W. The fuel utilization, on the other hand, decreases gradually, which is likely the beginning of the mass flow response. Meanwhile, the PEN temperature remains relatively constant. On the second timescale (Fig. 8(b)), the fuel utilization changes at a rate similar to that previously seen when manipulating the fuel flow (Fig. 6(b)), decreasing from 85% to 45% within seconds. Again, mass flow dynamics appear to be at work here. Observe, also, that the power increases slightly on the second timescale, which is due to the increasing operating voltage on the second timescale. On the minute timescale (Fig. 8(c)), the PEN temperature decreases slightly. Lower power generation (and hence lower thermal energy generation) is likely the reason for this behavior (Eq. (28)). Finally, it can be seen from Fig. 9 that a step change in the inlet air flow rate (50% increase) negligibly influences all of the SOFC variables shown, under the assumed operating conditions. This result is not surprising, however, as the inlet air temperature is specified to be 1173 K (Table 3), and the PEN temperature is already near this value at the outset of the simulation (1143 K, initially). Changing the air flow thus has a negligible effect on the Journal of Fuel Cell Science and Technology

Fig. 6 SOFC stack’s response to a step change in the fuel flow rate: (a) millisecond timescale, (b) second timescale, and (c) minute timescale

PEN temperature under the present operating conditions. During a transient event that induces more severe PEN temperature changes, however, changing the air flow could be useful for returning the PEN temperature back to its reference value [5].

7

Discussion

Based on the results presented herein, two different control strategies are conceivable. While both of these strategies have been separately considered before at the system level [5,14–16], the present work compares these strategies at the stack level. The first strategy involves manipulating the fuel flow to control the fuel utilization, and manipulating the current to control the power. The second strategy is the reverse of the first—it involves JUNE 2015, Vol. 12 / 031006-7

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manipulating the current to control the fuel utilization, and manipulating the fuel flow to control the power. Control of the fuel utilization may be achieved by manipulating either the fuel flow or current because changes in both of these variables induce significant changes in the fuel utilization (Figs. 6(b) and 8(b), respectively). Likewise, control of the power may be achieved by manipulating either the fuel flow or current because changes in both of these variables also induce significant changes in power (Figs. 7 and 8(a), respectively). In either strategy, the air flow would serve to control SOFC temperature, as the air flow was found to negligibly influence the fuel utilization and power (Fig. 9) while still having the potential to control the PEN temperature during a transient event [5]. A major difference between these control strategies is their interdependent quality. As mentioned previously, interdependence may be defined as the inadequacy of a manipulated variable to effectively control a targeted variable, unless tight control of another variable(s) is assumed. Interdependence between pairs of control variables is not desired, as it could lead to oscillations between control levels in a multilevel controller [5]. Results from this study suggest that the first strategy mentioned above (manipulating the fuel flow to control the power, and manipulating the current to control the fuel utilization) requires tight control of the fuel utilization. When the fuel utilization is maintained at 85%, in particular, Fig. 7 shows that manipulating the fuel flow influences the power significantly. However, when the fuel utilization is allowed to vary freely, Fig. 6(b) shows that manipulating the fuel flow hardly influences power at all. Hence, controlling the power using fuel flow is feasible only if tight control of the fuel utilization is implemented. In a multilevel controller, such as that proposed by Martinez et al. [5], such a control strategy may result in oscillations between the fuel utilization and power control levels, as these control loops are highly interdependent. If the current density is used to control power, on the other hand, the two main control variables—power and fuel utilization—can be controlled fairly independently. In particular, it can be seen from Fig. 8(a) that manipulating current gives rise to a significant change in SOFC power, without requiring any restrictions on fuel utilization. Likewise, Fig. 6(b) shows that manipulating fuel flow gives rise to a significant change in fuel utilization, without requiring any restrictions on the current. Because these control pairs operate fairly independently, control need not jump back and forth between the power and fuel utilization levels to satisfy control criteria in a multilevel controller. Another difference between these control strategies is the time required for the SOFC to meet a power demand. If the current is used to change the power, in particular, then the power will

Fig. 7 SOFC stack’s response to a step change in fuel flow rate assuming constant fuel utilization (Uf,2 5 85%)

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respond instantaneously to a load increase or decrease (Fig. 8(a)). If the fuel flow rate controls power, on the other hand, then the power will respond more slowly to load changes (Fig. 7). Buildings experience significant load change over the course of a day [7–9], and quick response is essential to meeting distributed generation needs. Of course, meeting power demand must be balanced with maintaining safe operation. Fortunately, neither operating strategy presents an especially high or low PEN temperature (under the present operating conditions). Based on the considerations discussed above, manipulating current density appears to be the quickest and most efficient way to control the SOFC power, while manipulating the inlet fuel flow rate appears to be the most efficient way to control the fuel utilization. It should be mentioned, however, that such a conclusion is based on results obtained from an idealized study. That is, the stack has been simulated in isolation from the rest of the system. In an actual system, the SOFC stack is subject to changes in operating conditions that heretofore have been considered static, such

Fig. 8 SOFC stack’s response to a step change in the current density: (a) millisecond timescale, (b) second timescale, and (c) minute timescale

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8

Conclusions and Future Work

This study has investigated the response of key SOFC variables to step changes in the inlet fuel flow rate, current density, and inlet air flow rate. Each simulation has been performed on millisecond, second, and minute timescales to fully capture the SOFC’s dynamic response, and each of the key SOFC variables has been analyzed and compared in terms of magnitude and rate of change. The SOFC model has been verified against results in the literature, and two main control strategies have been discussed. Results suggest that changes in the inlet fuel flow rate and current density each give rise to significant changes in the SOFC power, but they do so to varying degrees. Manipulating the current density gives rise to significant changes in the SOFC power without any restrictions placed on the fuel utilization. Manipulating the inlet fuel flow rate to control the SOFC power, on the other hand, requires tight control of the fuel utilization; otherwise, the inlet fuel flow rate has little or no influence on the SOFC power. Because the former strategy provides greater independence between control loops, it is recommended that this strategy be considered for use in a multilevel controller. Consideration has also been given in this study to the time required for the SOFC to meet a power demand. It has been found that the SOFC power responds quicker to changes in the current density (instantaneous) than it does to changes in the inlet fuel flow rate (seconds). Thus, manipulating the current density to control the SOFC power enables the system to more aptly follow a dynamic load. It is recommended that future work address the ability of these control strategies to satisfy safety and power requirements in the context of a larger system. From an SOFC stack perspective, use of the current density to control the SOFC power is well-suited for implementation in a multilevel controller. It should be remembered, however, that the control strategies discussed in this study have been simplified, and further information about the system is needed before arriving at an optimal control strategy. Such information includes shaft dynamics, part-load efficiencies of the compressor and turbine, and changes in balance of plant component operation [6]. Due to this added complexity, many variants of the strategies considered in this study are possible [5,14,15]. The present study has been carried out with the intention of elucidating the dynamic behavior of the system’s main component—the SOFC stack. In so doing, this study is intended to serve as a starting point for further development at the system level.

Acknowledgment This material is based upon work supported by the National Science Foundation under Grant No. EFRI-1038139. Fig. 9 SOFC stack’s response to a step change in the air flow rate: (a) millisecond timescale, (b) second timescale, and (c) minute timescale

as changes in pressure and shaft speed [14]. Such changes may give rise to unforeseen interactions between the PEN temperature, fuel utilization, power, and/or other key variables. Because of these complexities, findings from this study are intended to guide control development, but the stack’s behavior in the context of a larger system may differ from that presented herein. It is also worth noting that both strategies discussed in this section involve some degree of coupling between the fuel utilization and SOFC power, which appears to be unavoidable. That is, changing the current density significantly changes both the fuel utilization (Fig. 8(b)) and SOFC power (Fig. 8(a)). Such coupling arises from the relationship between the current density and fuel utilization (Eq. (25)), as well as the relationship between the current density and SOFC power (Pseg ¼ iVop). Fortunately, these processes occur on different timescales (the fuel utilization changes on the second timescale, whereas the SOFC power changes on the millisecond timescale), thus reducing the risk of instability [15]. Journal of Fuel Cell Science and Technology

Nomenclature A¼ Cdbl ¼ cp ¼ D¼ Eact ¼ EN ¼ F¼ fe ¼  geoh ¼ h ¼ i¼ j¼ j0 ¼ 00 Ji ¼ k¼ Mi ¼ n_ ¼ n_ 00 ¼

area (m2) charge double layer capacitance (F) specific heat capacity (J/kg K) diffusion coefficient (m2/s) activation energy (J/mol) Nernst potential (V) Faraday’s constant (96485.34 C/mol) equilibrium factor specific Gibbs function at standard pressure (J/mol) specific enthalpy (J/mol) electric current (A) current density (A/m2) exchange current density (A/m2) diffusive flux of species i (mol/s m2) thermal conductivity (W/m K) molecular weight of species i (kg/mol) molar flow rate (mol/s) molar flux (mol/s m2) JUNE 2015, Vol. 12 / 031006-9

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ne ¼ p¼ p^ ¼ p ¼ P000 seg ¼ 000 q_ conv ¼ q_ 000 rad ¼ R¼ R ¼ re ¼ r_i000 ¼ t¼ T¼ u¼ Uf ¼ Vdbl ¼ Vop ¼ x¼ yi ¼

number of moles of electrons pressure (Pa) perimeter (m) standard pressure (101,325 Pa) power (W/m3) convection (W/m3) radiation (W/m3) ohmic or equivalent resistance (X) universal gas constant (8.314 J/mol K) effective pore radius of electrode material (m) rate of reaction i (s1m3) time (s) temperature (K) velocity (m/s) fuel utilization charge double layer voltage (V) operational fuel cell voltage (V) coordinate axis along flow mole fraction of species i

Greek Symbols a¼ c¼ e¼ gact ¼ gconc ¼ gohm ¼ k¼ i ¼ q¼ ri ¼ rij ¼ s¼ sw ¼ XD,ij ¼

transfer coefficient pre-exponential factor (A/m2) emissivity or porosity activation polarization (V) concentration polarization (V) ohmic polarization (V) excess air ratio stoichiometric coefficient of species i (mol) density (kg/m3) electrical conductivity of component i (S/m) ˚) collision diameter for species i and j (A tortuosity wall shear stress (N/m2) collision integral for species i and j

Subscripts a¼ electro ¼ eoh ¼ f¼ inter ¼ msr ¼ PEN ¼ wgs ¼

air electrolyte electrochemical oxidation of hydrogen fuel interconnect methane steam reforming reaction pos. electrode–electrolyte–neg. electrode water–gas shift reaction

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