Available online at www.sciencedirect.com Computers and Chemical Engineering 32 (2008) 2365–2381 Review Dynamic modelling and control of planar anode-supported solid oxide fuel cell A. Chaisantikulwat, C. Diaz-Goano ∗ , E.S. Meadows Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6 Received 11 October 2006; received in revised form 23 July 2007; accepted 14 December 2007 Available online 5 January 2008 Abstract Most solid oxide fuel cell (SOFC) modelling efforts emphasize steady-state cell operation. However, understanding the dynamic behaviour is essential to predict the performance and limitations of SOFC power systems. This article presents the development of a SOFC dynamic model and a feedback control scheme that can maintain output voltage despite load changes. Dynamic responses are determined as the solutions of coupled partial differential equations derived from conservation laws of charges, mass, momentum and energy. To obtain the performance curve, the dynamic model is subjected to varying load current for different fuel specifications. From such a model, the voltage responses to step changes in the fuel concentration and load current are determined. Low-order dynamic models that are sufficient for feedback control design are derived from the step responses. The development of the partial differential equation model is outlined and the limitations of the control system are discussed. © 2008 Elsevier Ltd. All rights reserved. Keywords: Solid oxide fuel cell; Dynamic modelling; Simulation; Load change; Control Contents 1. 2. 3. 4. ∗ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOFC operating principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Thermodynamics of solid oxide fuel cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Voltage and overpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Electrochemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Ionic charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Electronic charge transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Mass transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Momentum transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Steady-state simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Ionic and electronic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Mass diffusion and convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Fluid velocity and pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Temperature distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corresponding author. Tel.: +1 403 294 4366. E-mail address: carolina@ualberta.net (C. Diaz-Goano). URL: www.ualberta.ca/∼carolina/ (C. Diaz-Goano). 0098-1354/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2007.12.003 2366 2366 2366 2367 2367 2368 2368 2368 2368 2369 2369 2369 2370 2370 2371 2371 2372 2372 2373 2373 2366 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 4.2. 4.3. 5. 6. Nominal operating condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374 Dynamic modelling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374 4.3.1. Step changes in load current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374 4.3.2. Step changes in hydrogen molar fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2376 Process control of SOFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2377 5.1. Low-order dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378 5.2. Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378 5.2.1. Proportional controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379 5.2.2. PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379 5.2.3. Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2380 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2381 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2381 1. Introduction A fuel cell is a device that converts a constant supply of fuel directly to electrical power. Solid oxide fuel cells (SOFC) have emerged as one of the leading fuel cell technologies which can be used in a wide range of commercial applications. Their solid electrolyte is made of a ceramic material which requires the operating temperature range of 800–1000 ◦ C. In recent years, the number of computational models of SOFC has been gradually increasing. Since SOFC operations are often subjected to transient condition such as changes in power demand, fuel cell dynamics have been increasingly considered in modelling activities. By developing a physically based dynamic model, the transient behaviour of SOFC can be accurately predicted and the design envelopes can be optimized. The dynamic model is especially beneficial for control testing in the development stage of SOFC. Most of the existing dynamic models were developed for prediction of SOFC performance and limitations. Additionally, the majority of dynamic models for process control have focused on large-scale operation such as an integrated-SOFC power plant system. For instance, Stiller, Thorud, Bolland, Kandepu, and Imsland (2006) and Thorud, Bolland, and Kvamsdal (2002) have presented a dynamic model for control of the integrated SOFC and turbine systems. It has been shown that the power supplied by the SOFC system can be controlled by manipulating the fuel flow using a proportional–integral–derivative (PID) type controller. In other work by Aguiar, Adjimana, and Brandon (2005), the temperature control of a stack-level SOFC model was presented. A PID controller was implemented to maintain the outlet fuel temperature and the fuel utilization during load changes by varying the air flow rates. The findings from these models emphasize the need for the process control to enhance the reliability and minimize the degradation of SOFC. A physically based three-dimensional (3D) dynamic model of a single SOFC is presented in this article. To investigate the transient performance and limitations of SOFC, this dynamic model is subjected to step changes in inlet gas concentrations and external load currents. Low-order models capable of capturing the main dynamic behaviour of the SOFC system are derived from the step responses. Feedback PI controllers are simulated with the low-order models in the voltage control-loop. An approach to control the output voltage such that it is close to the set-point voltage despite external load changes is outlined. The remaining of this article is organized as follows: Section 2 presents a review of the SOFC operating principles; Section 3 presents the numerical formulation for the dynamic model; Section 4 discusses the steady-state and dynamic modelling results; Section 5 addresses the control of the SOFC output voltage in the presence of varying load by implementing a PI controller. The concluding remarks are presented in Section 6. 2. SOFC operating principles In SOFC, the oxygen ion (O2− ) is the mobile ion transferred through a solid electrolyte in the following half-cell reactions at the cathode and anode, respectively. + 2e− ↔ O2− (1) H2 + O2− ↔ H2 O + 2e− (2) 1 2 O2 The overall reaction is then H2 + 21 O2 → H2 O (3) A schematic diagram presenting the flow of mass and charges for a SOFC is shown in Fig. 1. 2.1. Thermodynamics of solid oxide fuel cells The amount of voltage that an electrochemical fuel cell produced is determined from the change in Gibbs free energy of an overall chemical reaction. The change in Gibbs free energy is dependent on the partial pressure of the reactants and products. For a hydrogen–oxygen fuel cell, the change in Gibbs free energy is P H2 O ◦ G(T ) = G (T ) + RT ln (4) 1/2 PH2 PO2 where G◦ (T ) is the Gibbs free energy change at standard state. At equilibrium, the change in Gibbs free energy is related to the electrochemical work done by electrons according to the equation G(T ) = −ne FE(T ) (5) A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Nomenclature a Cp Dik D̃ik DK E F F̂ h hi ji J J◦ k KP KL n ne P Q r̄i R RΩ S t T u V wi xi stoichiometric coefficient isobaric molar heat capacity (J/kg K) binary diffusivity between species i and k (m2 /s) multicomponent Fick diffusivity (m2 /s) Knudsen diffusivity (m2 /s) electrical potential (V) Faraday’s constant (96,484.56 C/mol e− ) radiative view factor convective heat transfer coefficient (W/m2 K) enthalpy of species i (J/kg) mass flux of species i (kg/m2 s) current density (A/m2 ) exchange current density (A/m2 ) thermal conductivity (W/m K) process steady-state gain disturbance steady-state gain unit normal vector number of moles of electrons pressure (Pa) heat source (W/m3 ) mean pore radius (m) universal gas constant (8.314 J/mol K) resistivity ( m2 ) entropy (J/mol K) unit tangential vector temperature (K) velocity (m/s) voltage or potential (V) mass fraction of species i molar fraction of species i Greek symbols δ layer thickness (m) emissivity ε porosity ηact activation overpotential (V) ηconc concentration overpotential (V) κ permeability (m2 ) μ dynamic viscosity (kg/m s) νi mean molecular velocity (m/s) ρ density (kg/m3 ) σ conductivity (S/m2 ) τ tortuosity io load 2367 ionic property load resistance where F is the Faraday’s number. Since two electrons are transferred for every one mole of reacted hydrogen, ne = 2. Combining the above equations, the Nernst voltage is obtained. RT PH2 O ◦ (6) ln E(T ) = E (T ) − 1/2 2F P H2 P O2 E◦ (T ) Here, is the standard electrode potential. This Nernst potential corresponds to the thermodynamically reversible open circuit voltage, VOCV and E(T ) = VOCV,A . 2.2. Voltage and overpotentials SOFC output voltage is calculated from the reversible voltage and the sum of activation loss due to non-equilibrium condition, concentration loss by mass transport limitation and voltage loss due to ohmic resistance. In this study, the concentration loss is accounted for in the mass transport model and is incorporated in the calculation of the reversible, electronic and ionic potentials at the electrode–electrolyte interface. Vcell = VOCV − ηact − ηconc − JRΩ (7) The activation loss or overpotential is determined from the difference between electronic potential at the electrode and ionic potential at the electrolyte over the equilibrium potential. ηact = (Vel − Vio ) − VOCV (8) The activation loss at the cathode is known to be larger than that at the anode, Chen (2003). This activation overpotential contributes to the kinetics of the electrode reactions occurring at the electrode–electrolyte interfaces. 2.3. Electrochemical kinetics The rate of current density produced at the interface is related to the activation overpotential as described by the Butler–Volmer equation. F F (9) ηact − exp −αC ηact J = J◦ exp αA RT RT Subscripts A anode AI anode interconnect cell fuel cell property C cathode CI cathode interconnect el electronic property eff effective parameter in porous medium E electrolyte f fluid property of reactive gas Fig. 1. Schematic diagram of mass and charges flows in SOFC components. 2368 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 According to Costamagna and Honegger (1998), the value of the apparent charge transfer coefficients, αA and αC , are 1.4 and 0.6, respectively, for the cathode current density, JC . For the anode current density, JA , their values are 2.0 and 1.0, respectively. The exchange current density, J◦ , is related to the partial pressure of chemical species. The cathode and anode exchange current density are given as PO2 0.25 Eact,C exp − Pref RT P H2 P H2 O Eact,A exp − = γA Pref Pref RT J◦,C = γC (10) J◦,A (11) Here, γC and γA are the pre-exponential coefficients. Eact,C and Eact,A represent the activation energies of the cathode and anode exchange current densities, respectively. To account for the concentration loss, the partial pressure of gas species at the electrode–electrolyte interface is used in the above equations. 3. Modelling approach 3.1. Assumptions Mathematical models of SOFC were derived from coupled partial differential equations describing the transport of charges, mass, momentum and energy. Additionally, knowledge of thermodynamics and electrochemical kinetics were essential for the development of the model. The dynamic model was developed for a 3D anode-supported planar SOFC with a counter-current flow direction of air and fuel as shown in Fig. 2. The following necessary assumptions were made: (1) The SOFC components were H2 –H2 O–N2 , Ni–YSZ/8YSZ/ LSM, air. The interconnects were metal alloys. (2) The electrochemical reactions occurred at the electrode– electrolyte boundaries. (3) Bulk diffusion and Knudsen diffusion occurred in the flow channels and the porous electrodes, respectively. (4) Gas streams were preheated with uniform temperatures and velocities at the flow inlets. The outer SOFC boundaries were thermally insulated. (5) The density and heat capacity of the solid components were temperature independent. (6) Heat transfer was convection-dominated in the gas channels, while it was conduction-dominated in the solid phase. 3.2. Transport equations Formulation of the transport equations was required to model the SOFC potentials, species concentrations, flow profiles and temperature gradients. The electronic and ionic charge transports were assumed to be in steady-state since, as stated by Haynes (2002), these were instantaneous phenomena. Therefore, in the current work, the time-derivative term was applied only to the mass, momentum and heat equations. The proper boundary conditions were specified to solve the partial differential equations accordingly. 3.2.1. Ionic charge transport The following ionic charge transport equation applied to the electrolyte layer. −∇ · (σio ∇Vio ) = 0 (12) Here, σio and Vio represented the ionic conductivity and the ionic potential, respectively. At the electrode–electrolyte boundary, the normal ionic current density was equal to the Butler–Volmer current density. For the other boundaries where no ionic charge was transferred, the normal ionic current density was specified as zero. Fig. 2. SOFC modelling domain and its dimension. A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 2369 3.2.2. Electronic charge transport The electronic model coupled with the ionic model provided solution to the local potential losses in the electrolyte, electrodes and interconnects. The electronic charge balance equation at the electrodes and their interconnects was given as Combining the Knudsen diffusivity with the binary diffusivity, a modified diffusivity was obtained. Dik DK,i (19) D̂ik = Dik + DK,i −∇ · (σel ∇Vel ) = 0 For mass transport in porous electrodes, the effective binary diffusivity depended on the material properties such as the porosity, tortuosity and pore size. The effective diffusivity for the porous material was corrected by the ratio of the porosity to the tortuosity. (13) At the electrode–electrolyte boundary, the magnitude of the normal current density was equal to the Butler–Volmer current density. However, the ionic and electronic charges flowed in the opposite direction to one another. At the anode interconnect boundary, the net current demanded by the external load resistance was specified. −n · Jel = −Jload (14) At the outer boundary of the cathode interconnect, the voltage was specified as a zero reference voltage. 3.2.3. Mass transport Mass transport in the porous electrodes and the flow channels was modeled using the continuity equation. ∂ (15) (ρwi ) = −∇ · (ji + ρwi u) ∂t where wi was the mass fraction. The mass flux, ji , was calculated using the Maxwell–Stefan model, which accounted for molecular interactions between gas mixtures in a multicomponent system. N ∇P D̃iK ∇xk + (xk − wk ) ji = −ρwi (16) P k=1 Here, D̃ik was the multicomponent Fick diffusivity between species i and k. Since the density of gas mixture was relatively low in the hydrogen SOFC application, the multicomponent Fick diffusivities were calculated from the binary diffusivities for all pairs of gas species. The method of calculating the multicomponent Fick diffusivity can be found in COMSOL (2004). The binary diffusivities can be determined from the Chapman–Enskog gas kinetic theory (Bird, Stewart, & Lightfoot, 1960) 0.5 1 (1.8583 × 10−7 )T 1.5 1 Dik = (17) + 2 Ω̂ MWi MWk P σ̂ik D,ik where σ̂ik was the average collision diameter and Ω̂D,ik was the collision integral based on the Lennard–Jones potentials. Diffusion mechanism in the porous electrodes differed from that in the flow channels. In the porous electrodes, the average pore size was considerably smaller than the gas particle’s mean free path. As a result, the molecule-to-wall of gas particles collision dominated the molecule-to-molecule collision. Hence, Knudsen diffusion was incorporated in the mass transport model in the porous electrodes. The Knudsen diffusivity was calculated as follow 8RT 2 DK,i = r̄i (18) 3 πMWi eff D̂ik = ε D̂ik τ (20) Since mass transport in the flow channels was convectiondominated, the convective flux boundary condition was applied to the channel outlets. This boundary condition represented a zero diffusive component across the channel outlets. −n · ji = 0 (21) At the electrode–electrolyte interfaces, the rate of the electrochemical reaction was related to the rate of current generation according to the Butler–Volmer expression, JA or JC . ±a JA/C MWi −n · ji + ρwi u = (22) ne F Here, a was the stoichiometric coefficient, which carried a negative and positive sign when the species i was being consumed and produced, respectively. The solution to this boundary condition returned the partial pressure of chemical species as a result of the concentration loss throughout the porous electrodes. The inlet mass fraction was specified at the inlet of the air and fuel channels. 3.2.4. Momentum transport To model the flow profile in the gas channels and the porous electrodes, the general Navier–Stokes equation was employed. ∂u 2 T ρ + ρu · ∇u = −∇P + ∇ · μ(∇u + (∇u) ) − μ∇ · u ∂t 3 (23) The above equation was used in conjunction with the continuity equation. ∂ρ + ∇ · (ρu) = 0 ∂t (24) In the porous electrodes, the flow was modeled using the Darcy’s Law κ u = − ∇P (25) μ where κ was the permeability of the electrode material. The uniform inlet velocity was specified at the channel inlets. At the outlets, the pressures and the normal flow boundary condition were specified. P = Pout , t·u=0 (26) 2370 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 At the electrode–electrolyte boundaries, the change in the gas velocity dependeds on the net rate of the species produced and consumed at that interface. N a JA/C (27) MWi −n · u = − ne F ρ i=1 Pressure conditions at the electrode–gas channel interfaces were specified to be equal to the Navier–Stokes pressure at the channel walls. PDarcy = PNavier−Stokes (28) 3.2.5. Heat transport Many mechanical properties of the SOFC components were temperature dependent. Therefore, it was essential to develop a heat transport model that could account for various heat effects in both of the solid structure and the flow channels. The following equation accounted for conductive heat transfer in the solid structure ∂ (29) (ρCp )T = −∇ · (−keff ∇T ) + Q ∂t where Cp was the specific heat capacity, keff was the effective thermal conductivity and Q was the heat source. The subscript eff denoted the effective parameters associated with the porous electrodes. The effective thermal conductivity of a porous electrode was determined from keff = εkf + (1 − ε)ks (30) where ε was the electrode porosity and the subscript f and s denoted the fluid and solid properties, respectively. For SOFC with hydrogen as a fuel source, the followings heat sources were generated (VanderSteen, Kenney, Pharoah, & Karan, 2004): • Ohmic heating due to resistances in ionic and electronic conducting materials. • Heat generated by the activation overpotentials under a nonequilibrium condition. • Heat loss through entropy change in the electrochemical reactions. The Ohmic heat was generated by the material resistance due to current flow. Qohm = J2 σ (31) This ohmic heat represented the heat source Q in Eq. (29) which applied to the entire solid structure. The activation overpotential contributed to the heat generation at the electrode–electrolyte boundaries. It represented the irreversible heat loss which was not recoverable from the electrochemical reaction. Qact = Jηact (32) Some part of the energy provided by the overall reaction cannot be completely converted to electrical energy. As a results, the change in entropy of reaction contributed to heat generation at the electrode–electrolyte interfaces. The entropy heat effect was given by T (S) (33) Qrev = J − 2F where S was the entropy change of the half-cell reaction. Thus, the boundary condition along the electrode–electrolyte interface became T (S) −n · (−keff ∇Ts ) = − (34) + ηact J 2F The entropic heat effects were generated in unequal amounts at the interfaces. The amount of entropic heat was greater at the cathode–electrolyte boundary than the anode–electrolyte boundary due to the exothermic heat effect. The fluid temperature model applied to the flow channels. To account for the effect of mass transport, the enthalpy change as a result of species diffusion was included. The heat equation for the fluid was given as N ∂Tf = −∇ · −kf ∇Tf + hi ji − (ρCp )f u · ∇Tf (35) (ρCp )f ∂t i=1 where hi was the species enthalpy in J/kg and ji was the mass flux of species i. It was determined by VanderSteen et al. (2004) that surface radiation contributed to the overall heat effect in the SOFC. Therefore, the radiative effect was incorporated in the heat model as a form of surface heat transfer. The convective and radiative heat effects were implemented as a boundary condition along the flow channel walls. 4 −n · (−k∇Ts ) = hf (Tf,bulk − Ts ) + σ̃ F̂ (Tf,bulk − Ts4 ) (36) Here, hf was the convective heat transfer coefficient, Tf,bulk was the bulk fluid temperature, was the emissivity of the solid component and σ̃ was the Stefan-Boltzmann constant. F̂ was the radiative view factor which accounted for the radiative interaction between surfaces. The view factor for surface-to-surface radiation was calculated according to the modelling geometry specified in Section 3.1. Since convective heat transfer dominated in the flow channels, the outlet heat flux was entirely convective and the conductive flux was zero. 3.3. Model implementation The model was implemented in COMSOL (2005), a partial differential equation solver based on a finite element method. Structured mesh elements were composed of 3D rectangular parallelepiped as shown in Fig. 3. The mesh consisted of 1152 elements with 42,487 degrees of freedom. The direct solver (UMFPACK) was used for steady-state and dynamic calculations. A relative tolerance of 1 × 10−5 was specified for the nonlinear system solver. In the dynamic simulation, changes to the system input variables with respect to time were generated using a step function. The automatic time steps were generated using the adaptive time-stepping solver. The SOFC properties required for solving the models are shown in Table 1. The simulation were carried out for a SOFC operating condition as shown A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 3. Mapped mesh for 3D geometry. in Table 2. The steady-state model results were obtained for five different fuel compositions, each containing 10%, 24%, 49%, 73% and 97% hydrogen. The model was solved by using the specified current density as a forcing function for each fuel composition. 4. Numerical results 4.1. Steady-state simulation results The solutions corresponding to each fuel composition at various current density were determined. The model was validated according to the experimental results from Keegan et al. (2002) which had a similar operating parameters to the model. However, 2371 the experiment was conducted on a button-type SOFC which had a different geometry than the planar cell. Therefore, the discrepancy between the simulated and experimental results were expected. However, the purpose of this work was to develop a dynamic model for the process control. For this reason, the validation goal was not to precisely fit the experimental data, but rather to capture the overall trend of the fuel cell performance curve. Additionally, the exact match between the model and the experiment was not required in the development of controller, since the presence of feedback control would compensate for the model error. To implement the controller in the voltage control-loop, it was necessary for the SOFC to have a stable voltage under normal operating condition. Since the output voltage always decreased with increasing load current, the voltage stability was always guaranteed in the event of the process being uncontrolled. Therefore, the controller stability was attainable even when model errors existed. The numerical results from this model were then compared to the experimental results for qualitative purposes. A comparison of the simulated and experimental cell voltages is shown in Fig. 4. The current-voltage curve was generated using the following parameters in the exchange current densities; γC = 9.61 × 108 , γA = 7.55 × 109 , Eact,C = 110 × 103 and Eact,A = 120 × 103 . It was observed that the predicted voltage curve had a similar shape to that of the experimental data. Although the limiting currents did not agreed well with each other, the predicted voltages showed similar tailing effects at high current densities. To produce the tailing effect in the simulated voltage, an anode tortuosity of 8.5 was used. This value was within the typical value of the anode tortuosity in the range of 2–10 as reported by Williford, Chick, Maupin, Simner, and Stevenson (2003). The voltage drop at high current density was Table 1 SOFC properties for the SOFC model Parameters Electrolyte Cathode Anode σ (S/m) 3.34 × 104 e−10,300/T 4.0×107 T 9.0×107 T ε τ r̄ (m) – – – 0.3 2.5 [3] 1.0 × 10−6 [4] 0.3 8.5 1.07 × 10−6 [5] – – – Cp (J/kg K) k (W/mK) ρ (kg/m3 ) 100 2.7 6000 377 2.37 4640 377 11.0 4760 300 2.2 Same as electrode [1] e−1200/T [1] e−1150/T [1] Interconnect Source 4 × 106 [1] Ferguson, Fiard, and Herbin (1996) and [2] Costamagna and Honegger (1998) Costamagna and Honegger (1998) [3] Chan, Chen, and Khor (2004) [4] Zhu and Kee (2003) and [5] Ackmann, de Haart, Lehnert, and Stolten (2003) Chyu (2005) Chyu (2005) Chyu (2005) [2] Table 2 Simulation condition for the half-channel SOFC model Parameters Fuel Air Composition Flowrate Inlet velocity 10–97 mol% H2 , 3 mol% H2 O, balancing N2 10 standard cm3 /min 1.25 m/s 21 mol% O2 , 79 mol% N2 15 standard cm3 /min 1.87 m/s Inlet temperature Outlet pressure Load current density 1023 K (750 ◦ C) 1 atm 0–1.8 A/cm2 2372 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 4. Comparison of simulated and experimental cell voltages at 750 ◦ C (Keegan et al., 2002). largely due to the concentration loss, especially at the anode. The mass transport of the fuel in the porous anode was limited by the diffusion mechanism and the anode thickness. In the anode reaction, the fuel utilization increased with increasing load current. However, the amount of hydrogen at the reactive site was not sufficient to supply the electrode reactions due to the diffusion limitation. As a result, a sharp drop in the voltage was evident in the high current region. In the low current region, a voltage drop was present as a result of the kinetic limitation of the electrode reactions. This limitation represented the activation loss which was prominent in the high current density operation. Overall, the simulated voltages at various fuel concentrations were within the same magnitude as those from the experiment. Although the performance curve contained the same trend as the experimental results, the mismatch in the limiting current density contributed to the largest deviation. The average percent differences between the simulated and experimental results for the hydrogen fuel of 10%, 24%, 49%, 73% and 97% were 5.8%, 4.9%, 3.2%, 12.5% and 12.4%, respectively. On average, the predicted voltages for the SOFC operating with 49% hydrogen fuel agreed best with the experimental values. The main contribution to the discrepancy was likely due to model parameter errors since the parameters used in the model were collected from various sources. The difference in the component thicknesses and the cell geometry between the proposed model and the experiment was also a major contribution to the overall discrepancy. The steady-state modelling results for a 49% hydrogen fuel at the load current of 0.80 A/cm2 are shown in the following sections. The corresponding cell voltage was 0.743 V. 4.1.1. Ionic and electronic potentials The cross-sectional plot of the electronic SOFC potential is presented in Fig. 5. The overall cell voltage was calculated from the potential difference between the cathode and anode interconnects. The arrows represented the direction of the current flow. The magnitude of current density across the SOFC component is shown in Fig. 6. The magnitude of the current density was largest as the current flew through the corners of the electrode–gas channels due to the geometry effect. This was more prominent at the Fig. 5. Cross-sectional voltage (x = 9.5 mm) with arrows showing the direction of the current flow. cathode where the layer thickness was much smaller than the anode. 4.1.2. Mass diffusion and convection The hydrogen and water distributions in the anode and fuel channel are illustrated in Figs. 7 and 8, respectively. The concentration gradients of hydrogen and water in the fuel channel were not substantial, whereas the gradients were large in the porous anode. It was observed that hydrogen was consumed mostly in the anode location close to the fuel channel outlet. This was also the area where the most water was produced. The outlet molar fraction of hydrogen and water were 39% and 13%, respectively. The oxygen distribution in the porous cathode and the air channel is shown in Fig. 9. The gradient of oxygen along the air channel was significantly less than that along the porous cathode. Although the cathode thickness was smaller than that of the anode, the diffusivity of oxygen in the cathode was much smaller than the hydrogen diffusivity in the anode. For this reason, a large Fig. 6. Cross-sectional SOFC showing streamlines of current density (x = 0 mm) for 49% H2 fuel at 0.80 A/cm2 . A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 7. Molar fraction of hydrogen in the anode and the fuel channel for 49% H2 fuel at 0.80 A/cm2 . 2373 Fig. 10. Fluid velocity in the electrodes and the gas channels. 4.1.3. Fluid velocity and pressure The gas velocities in the flow channels and the porous electrodes are shown in Fig. 10. The arrow showed the direction of the fluid flow with the magnitude being proportional to the fluid velocity. The fuel and air flows reached their maximum velocity of approximately 3.3 m/s and 5.5 m/s, respectively, at their channel outlets. The velocity changes in the porous electrodes relative to the gas channels were considerably less noticeable. The pressure gradient at the air channel was found to be significantly larger than that at the fuel channel. The pressure drop in the fuel channel was 30 kPa, whereas the pressure drop in the air channel was 65 kPa. This larger pressure drop occurred as a result of greater fluid velocity. gradient of oxygen concentration was observed through out the cathode. The outlet molar fraction of oxygen was approximately 16%. The results from the mass transport model showed the hydrogen and air utilization of 19.2% and 18.3%, respectively. 4.1.4. Temperature distribution The temperature distribution in the solid phase is presented in Fig. 11. The maximum temperature difference for the overall SOFC was found to be 40 K. The highest temperature was observed in the electrode–electrolyte layer in the middle region of the cell. The temperature distribution along the anode, cathode and electrolyte (x = 9 mm, y = 1 mm) is illustrated in Fig. 12. Since the cathode reaction was exothermic, the highest solid Fig. 9. Molar fraction of oxygen in the cathode and the air channel for 49% H2 fuel at 0.80 A/cm2 . Fig. 11. Solid temperature of the electrolyte, electrodes and interconnects for 49% H2 fuel at 0.80 A/cm2 . Fig. 8. Molar fraction of water in the anode and the fuel channel for 49% H2 fuel at 0.80 A/cm2 . 2374 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 12. Solid temperature across the electrodes and electrolyte for 49% H2 fuel at 0.80 A/cm2 . temperature was seen at the interface between the cathode and electrolyte. However, the maximum temperature difference between the electrodes and electrolytes was insignificant (0.56 K). The fluid temperature in the flow channels is shown in Fig. 13. The outlet air and fuel temperatures were found to be 1314 K and 1298 K, respectively. It was evident that the greater temperature gradient of the air stream resulted from the heat of cathode half-cell reaction. 4.2. Nominal operating condition After the steady-state analysis was completed, the nominal operating parameters were selected for the time-dependent simulation. The nominal operating condition was used as the initial condition for the dynamic model. Selecting the operating condition for the SOFC was often not a trivial process. To achieve the maximum power output, many trade-offs were involved. From the current-voltage characteristics, the maximum power can be supplied by a fuel cell in a medium to high voltage range. According to Aguiar et al. (2005), SOFC was normally designed to operate within a voltage range of 0.60–0.70 V, although a higher or lower voltage was acceptable. From the steady-state model validation, it was observed that the predicted cell performance for 49% hydrogen fuel contained the least discrepancy when compared with the literature data. The model was able to predict acceptable cell voltages in the current density range of 0.2–1.0 A/cm2 . To operate the SOFC within the designed voltage range, the operating current density of 0.80 A/cm2 was chosen. This current density corresponded to the cell voltage of 0.743 V. Therefore, 49% hydrogen fuel and a load current density of 0.80 A/cm2 were chosen as a nominal operating condition in the dynamic SOFC model. In transient operations, the SOFC were often subjected to sudden changes in the load resistance, which resulted in variable power demand. This step change in the load current was simulated in the dynamic model by using a step function. The dynamic response of the cell performance was then investigated. Also, the effects of varying the molar fraction in hydrogen fuel was simulated. By doing so, it was determined if the voltage could be maintained constant in the presence of load changes by varying the fuel composition. 4.3. Dynamic modelling results 4.3.1. Step changes in load current density Positive and negative step changes with different magnitude were made to the load current and the dynamic SOFC responses are shown in Fig. 14. The load current density was varied from 0.80 A/cm2 to 0.90 A/cm2 , 0.85 A/cm2 and 0.70 A/cm2 . The inlet hydrogen concentration was maintained at the nominal value of 49%. The step changes in the load current density was introduced at 10 s. The temperatures shown were obtained from the average temperature over the cathode–electrolyte interface where maximum temperature occurred. It was observed that the voltage responded immediately to the change in load demand according to Ohm’s law. In Fig. 14(b), a small overshoot in the voltage response was observed after the load current was increased from 0.80 A/cm2 to 0.90 A/cm2 . This overshoot was likely a result of a numerical error caused by a discontinuity in time. Small dynamic effects were observed and the response time for the output voltages to reached new steady-states were approximately 330 s. The dynamic response provided the steady-state gain information which was used to develop a simplified dynamic model. The gain, KL was calculated from the change in the output voltage with respect to the change in the load current at the final steady-state. KL = Fig. 13. Fluid temperature in the gas channels for 49% H2 fuel at 0.80 A/cm2 . V Jload (37) The steady-state gains, KL , from each load step changes are summarized in Table 3. It was observed that the gains were consistent for the positive step changes in the load current. However, the gain from the negative step change was approximately 10% lower. The slight inconsistency in the steady-state gains indicated that the effect of load change on the output voltage was nonlinear in this operating region. A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 2375 Fig. 14. Model output responses to step changes in load current density. Table 3 Steady-state gains in the output voltages with respect to the load step changes Final load current (A/cm2 ) Final voltage (V) Steady-state gain (KL , cm2 ) 0.90 0.85 0.70 0.711 0.728 0.770 −0.315 −0.302 −0.271 In Fig. 14(c), it was observed that the solid temperature had the largest response time of approximately 400 s. The cell temperature had a substantial effect in the transient operation of SOFC. Although the overall voltages were not significantly influenced by the temperature distribution, the material stresses could potentially be a problem. The largest temperature increase of 36 K was obtained from the load change from 0.80 A/cm2 to 0.90 A/cm2 . At such temperature gradient, stresses would likely 2376 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 be developed. A complete thermal stress analysis was required to investigate the effect of temperature gradient on the material stresses. Fig. 14(d–f) present the responses of the outlet species flux to load step changes. By increasing the load current, the reactant utilization was increased. This was observed in the concentration profiles of the reactants in the presence of the positive load changes. As the load current was increased, more hydrogen and oxygen were consumed and vice versa. Although, the concentrations changed instantaneously, some dynamic effects were observed. 4.3.2. Step changes in hydrogen molar fraction Fig. 15 presents the dynamic responses of the outlet gas to step changes in the inlet hydrogen concentration from 49% to 60%, 55% and 38%. The load current density was kept at Fig. 15. Model output responses to step changes in the inlet hydrogen concentration. A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 2377 Table 4 Steady-state gains in the output voltages with respect to the changes in the inlet hydrogen compositions Final mol% of hydrogen (%) Final voltage (V) Steady-state gain (KP , V/%) 60 55 38 0.780 0.765 0.675 0.334 0.365 0.622 its nominal value of 0.80 A/cm2 . The step tests in the hydrogen molar fraction were introduced at 10 s. The temperature responses were obtained from the overall solid temperature at the cathode–electrolyte interface. When the hydrogen concentrations were step changed, the sudden changes in the voltage were observed in Fig. 15(b). The dynamic effects in the voltage changes were unnoticeable. The steady-state gain, KP , was calculated for the change in the cell voltage with respect to the change in hydrogen molar fraction. KP = V xH2 (38) A summary of the steady-state gains, KP , is provided in Table 4. The gains for the positive step changes in hydrogen concentration were consistently in agreement with one another. However, the negative step change in hydrogen produced a larger gain, twice the magnitude of those from the positive step changes. This was indicative that nonlinear effects were present around operating condition. For control design purpose, the output voltage could be difficult to control in this region with low hydrogen molar fraction. In Fig. 15(c), the response time of the temperature change of approximately 400 s was observed in all cases. By increasing the hydrogen content, a lower temperature was observed. This could be attributed to the change in the activation overpotentials shown in Fig. 16. It was seen that the dynamic effect of hydrogen concentration change was more pronounced on the cathode overpotential than the anode overpotential. Initially, the overshoot in the cathode activation overpotential was observed when the change in the hydrogen content was introduced. However, the cathode activation overpotential then decreased below the original value as the new steady-state was obtained. The heat sources along the electrode–electrolyte boundaries were calculated from the entropic (reversible) and the activation (irreversible) heat effects. Since the current was maintained constant, the change in the entropic heat became less significant and the heat source was more dependent on the activation overpotential term. When both of the anode and cathode activation overpotentials were reduced, less irreversible heat effect was generated and the temperature was lowered as a result. The dynamic responses of the outlet species flux to changes in hydrogen concentrations are illustrated through Fig. 15(d–f). The dynamic effects were not observed. However, overshoots in the outlet water mass flux were observed right after step changes were introduced. These overshoots could be explained by the numerical errors, which were generated by the discontinuity in time when the step changes were introduced. By simulating the dynamic model Fig. 16. Time responses of the activation overpotentials to step changes in inlet hydrogen concentration. using a smaller time increment, these numerical errors could be eliminated. The simulated voltages from the dynamic model were validated with the interpolated voltages from Keegan et al. (2002). Using linear interpolation on the performance curve shown in Fig. 4, the voltages were determined for the fuel with hydrogen content of 38%, 55% and 60%. At the load current of 0.80 A/cm2 , the percent differences between the simulated and experimental results were found to be 2.9%, 3.6% and 4.5%, respectively, for 38%, 55% and 60% hydrogen. Overall, the results were in agreement with the experimental data and the dynamic model produced physically reasonable results. These dynamic responses provided useful information on the dynamic characteristics of the SOFC which could be used for the process control. 5. Process control of SOFC In practice, SOFC was often subjected to load changes which resulted in voltage drops. Therefore, the load current was accounted for as a disturbance to the SOFC system. It was observed that the cell voltage was increased by increasing the composition of hydrogen in the fuel. Based on these findings, the control objective was proposed to maintain a constant voltage despite of load changes by manipulating hydrogen concentration in the fuel. The controller design was carried out on the 2378 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 17. Open-loop and closed-loop feedback control diagram in MATLAB Simulink. simplified dynamic models which were capable of representing the main dynamic properties of the system. 5.1. Low-order dynamic model In this section, first-order transfer functions were derived from the dynamic step responses of the SOFC to represent the relationship between the output voltage, load current and fuel composition. The following Laplace transfer functions were defined as process models for the output voltage with respect to the input variables and the disturbance, which were the hydrogen content and the load current, respectively. KP KL y(s) = u(s) + d(s) (39) τP s + 1 τL s + 1 Here, y was the output voltage, u was the hydrogen content in the fuel and d was the load disturbance. The process and disturbance gains, KP and KL , were the steady-state gains introduced in Section 4. The parameters τP and τL were the time constants of the process and the disturbance, respectively. These parameters in the transfer function model can be derived from the dynamic step responses. To determine the process and disturbance gains for the simplified model, the steady-state gains were selected from Tables 3 and 4. It was observed that the nonlinear effect of changes in the inlet hydrogen concentration on the output voltage was more prominent in the low concentration region. It was difficult to control the voltage within this region since the nonlinear effect could not be accurately accounted for in the first-order transfer function model. For the purpose of control simulation, the steady-state gain of 0.334 V/% was chosen for KP . This was the gain corresponding to the change in the hydrogen content from 49% to 60%. The steady-state gain of the voltage with respect to the load current of −0.315 cm2 was selected for KL . The process time constant represented the dynamic component of the output voltage response. It was observed that the output voltage responded to the changes in the hydrogen content and load current instantaneously. Since the dynamic components in the output responses were negligible, the process and disturbance transfer functions could be developed as gain-only models. However, the value of τP and τI were taken as 0.1 s to simulate various controller settings for the first-order transfer functions. These time constants were sufficiently small to provide an instantaneous voltage response to step changes. Additionally, they allowed for the dynamics of the controller responses to be investigated. The nominal operating condition of these transfer function models were the same as that specified in the previous section. 5.2. Feedback control Three basic modes of feedback control were proportional (P), integral (I) and derivative (D) control. According to the A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 2379 PID control algorithm, the controller output was the process input, u, which was calculated from the error, e, in the following relationship 1 (40) + τD s e(s) u(s) = Kc 1 + τI s Here, e was the difference between the actual output and its set-point value. The controller tuning parameters, Kc , τI and τD were the controller gain, integral time and derivative time, respectively. They each represented the proportional, integral and derivative actions in the controller accordingly. Controllers with P-only and PI settings were implemented in the voltage control-loop. The derivative term was omitted due to the lack of dynamic effects in the voltage responses. The voltage controller for the SOFC system was implemented in MATLAB Simulink (MathWorks, 2002). The Simulink diagram of the open-loop process and the closed-loop feedback control is shown in Fig. 17. 5.2.1. Proportional controller The closed-loop voltage control simulation was first subjected to various proportional controller settings. The step change in load current from 0.80 A/cm2 to 0.90 A/cm2 was introduced at 10 s. The set-point voltage was specified at 0.743 V, which was the initial voltage for the SOFC system operating with 49% hydrogen fuel and the load current of 0.80 A/cm2 . The closed-loop voltage responses and the controller output responses obtained from various proportional control settings are shown in Fig. 18. The controller parameters were selected to investigate the control effects on the closed-loop system. They were not comprehensively tuned to achieve the optimum controller performance. From Fig. 18(a), it was observed that the uncontrolled (open-loop) voltage dropped instantaneously after the change in the load current was introduced. With the controller, the output voltage was brought closer to the desired voltage. However, offsets between the steady-state and the set-point voltages were observed. This was the expected output response from the process with proportional control. In the P-only control, the controller output was calculated from the product of the controller gain, Kc and the offset, e. Therefore, the presence of the offset was always required to generate the output for the P-only controller. By increasing the controller gain, the offset was reduced and faster response to load change was obtained. In proportional control, a large value of Kc was required to achieve perfect control. However, this also required a high controller effort. Generally, the P-only control yielded steady-state errors that occurred after a change in the set-point or the disturbance. This offset can be eliminated by incorporating the integral action, τI , with the proportional controller. 5.2.2. PI controller The proportional plus integral (PI) controllers were implemented in the voltage control-loop. The step changes in the load current from 0.80 A/cm2 to 0.90 A/cm2 at 10 s were simulated. The voltage responses and the controller output Fig. 18. Closed-loop proportional-control responses to the load step change from 0.80 to 0.90 A/cm2 . responses for various PI controller settings are presented in Fig. 19. With the integral action, the steady-state offsets were eliminated and the desired set-point voltage was obtained. The dynamic voltage responses were reflected by the controller responses of the hydrogen molar fractions. Slow responses from the controller with lower integral time were observed. A robust control was obtained for the PI controller setting with a small controller gain (Kc ) and a large integral time (τI ). It was observed that the controller settings of Kc = 1 and τI = 10 yielded the most robust controller performance. 5.2.3. Disturbance rejection The PI controller with the controller gain of 1 and the integral time of 10 s was used to maintain the output voltage in the control-loop under various load disturbances. At 10 s, the load disturbance was changed from its nominal value of 0.80 A/cm2 to 0.85 A/cm2 , 0.70 A/cm2 and 0.65 A/cm2 . The voltage responses and the controller output responses of the inlet hydrogen content are shown in Fig. 20. After the load disturbances were introduced, the output voltages increased for the negative load changes and vice versa. The PI controller successfully restored the output voltage to the desired set-point value after approximately 2 s. Therefore, the PI controller gave a satisfactory performance in rejecting the load disturbances. 2380 A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 Fig. 19. Closed-loop PI-control responses to the load step change from 0.80 to 0.90 A/cm2 . In Fig. 20(c), it was observed that the PI controller responded to changes in the load disturbance by manipulating the controller output, i.e. the inlet hydrogen concentrations, such that the set-point value of the output voltage was restored. For a positive change, the inlet hydrogen concentration was increased to obtained the desired output voltage. As seen from the figures, a range of hydrogen concentration of 35–54% was required to maintain the voltage of 0.743 V when the SOFC was subjected to the load current range of 0.65–0.80 A/cm2 . The feedback PI controller was able to to maintain a constant SOFC voltage for small changes in the load current. For the case where significant dynamic effects were observed, the presence of the derivative control action often improved the settling time of the system. It was demonstrated that the low-order dynamic models developed from the physically based model allowed for the implementation of controller design without involving a significant amount of computational effort. A low-order model provided a useful tool to investigate the effect of the output response when a small change was applied to the input variables. However, the simplified models must be used with caution when representing highly non-linear process such as the SOFC. Since the low-order models were derived under a specific operating range, the predicted cell performance for a different operating condition would be less accurate. A new simplified model should be derived if there was a change in the nominal operating condition. Fig. 20. Closed-loop responses under various load changes with the PI controller settings: Kc = 1, τI = 10 s. 6. Conclusion In this article, a 3D dynamic model of an anode-supported planar SOFC was presented. The model was derived from the partial differential equations representing the conservation laws of charges, mass, momentum and energy. The steady-state cell performance curve was qualitatively compared with the experimental data from the literature. Overall, the simulation results agreed with the experimental data despite the difference in SOFC geometry and model parameter error. The steady-state simulation showed that the cell performance was strongly dependent on the solid temperature. The mass diffusion limitation contributed to voltage loss at high current density, whereas the kinetic limi- A. Chaisantikulwat et al. / Computers and Chemical Engineering 32 (2008) 2365–2381 tation of the electrode reactions were responsible for the voltage loss at low current density. A nominal operating condition was chosen such that it became the initial condition for the dynamic simulations. Step responses were obtained from the step changes in the system input and disturbance around the nominal operating condition. The following observations were noted: • A stable controller was implemented. The errors in the model were compensated by this feedback controller. • A settling time for the temperature of approximately 400 s was reported, whereas the dynamic effects of the output voltage and mass diffusion were substantially smaller. • For a step change of small magnitude, the main dynamic properties of the SOFC were captured in the derived first-order transfer function models. • P-only and PI controllers were implemented in a voltage feedback control-loop using the information from the dynamic step responses. • In the event of load disturbance, the controller maintained a constant output voltage by adjusting the hydrogen content in the fuel source. The derived low-order dynamic SOFC model could be used to assess the controller’s performance at other operating conditions without much computer effort. However, if the operating condition significantly deviated from the selected nominal value, the simplified model would likely produce larger error. The control system of a SOFC presented in this paper focused on maintaining a constant voltage by manipulating hydrogen concentration in the fuel. Since temperature also played an important role in the performance and failure of the fuel cell, the present work could be extended to take into considerations thermal gradients in the SOFC. By identifying a suitable temperature control strategy, material stresses could be maintained within an acceptable range. Acknowledgements This research was made possible in part by funding from the Alberta Energy and Research Institute (AERI) and the Depart- 2381 ment of Chemical and Materials Engineering at the University of Alberta. 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