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Ind. Eng. Chem. Res. 2005, 44, 3522-3528
Optimization of Reforming Catalyst Distribution in a Cross-Flow Molten Carbonate Fuel Cell with Direct Internal Reforming Peter Heidebrecht† and Kai Sundmacher*,†,‡ Process Systems Engineering, Otto-von-Guericke-University Magdeburg, Universita¨ tsplatz 2, D-39106 Magdeburg, Germany, and Max-Planck-Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany
Two optimization procedures of a molten carbonate fuel cell (MCFC) with direct internal reforming are presented. First, optimal operating conditions such as the amounts of feed gas, water, and air are calculated for a given cell current in order to obtain optimal electric efficiencies. An optimal current-voltage curve for the system is obtained by repeating this optimization for various cell currents. The second optimization balances the cooling effect of the endothermic reforming process and the heat-producing electrochemical reactions inside the cell in order to achieve a more homogeneous temperature profile. This is realized by optimization of the spatially distributed reforming catalyst density. A repeated calculation of the optimal current-voltage curve shows a significant increase of the electric efficiency by this measure. Both optimization procedures are based on a cross-flow MCFC model and consider several constraints concerning temperature, cell voltage, and carbonization. 1. Introduction Molten carbonate fuel cells (MCFCs), which are currently about to enter the market of stationary distributed power supply, are highly complex systems. This is due to the fact that several processes of chemical reaction, mass and heat transport, and heat production are integrated into one single spatially distributed apparatus. Another reason is that, in contrast to all other commonly known fuel cell types, the cathode gas composition is dependent on the anode gas composition because both parts are connected by a catalytic combustion chamber. The MCFC can be fueled by a wide range of mixtures of gaseous hydrocarbons, which are converted to hydrogen by the so-called reforming process. In the direct internal reforming concept, this endothermic process is located directly inside the anode channel of the cell. On the one hand, this increases the electric efficiency of the cell, but the coupling of the endothermic reforming process with the heat-producing electrochemical processes in the cell also leads to new problems such as cold and hot areas in the cell. To avoid expensive temperature-resistant materials, on the one hand, and to reach high degrees of electric efficiency, on the other hand, the temperature inside the cell has to be controlled carefully. This can happen in two ways: first, the input conditions can be adjusted so that all temperature limitations are fulfilled. This is the objective of our first optimization task. The second possibility lies in the system design. Because the temperature field is governed by the thermal interplay of reforming and oxidation processes, these two have to be carefully adjusted to one another. This can be * To whom correspondence should be addressed at the MaxPlanck-Institute for Dynamics of Complex Technical Systems. Tel.: +49-391-6110 350. Fax: +49-391-6110 353. E-mail:
[email protected]. † Otto-von-Guericke-University Magdeburg. ‡ Max-Planck-Institute for Dynamics of Complex Technical Systems.
realized by applying an optimal spatial distribution of the reforming catalyst density inside the anode channel, which is the objective of the second optimization in this paper. Several authors describe the spatially distributed temperature profile in DIR-MCFC. Yoshiba et al.1 compare the temperature profile of large-size MCFCs with different flow configurations using steady-state models. They assume that the feed gas is already reformed and only the water-gas shift reaction occurs. Bosio et al.2 also consider only the water-gas shift reaction in the anode channel. The simulated temperature profile from their model is qualitatively similar to our findings. Park et al.3 include the methane steam reforming reaction in their model of a cross-flow MCFC and evaluate the influence of the reforming process on the temperature profile. Because the size of their cell is rather small, it is not directly comparable to the simulations in this paper. According to our knowledge, optimization calculations for MCFCs were published only by Arato et al.4 together with Bosio. Their calculation is based on a spatially twodimensional, steady-state, single-cell model and considers the variation of one of the feed parameters at a time in order to ensure that the maximum temperature within the stack stays below a given maximum value. The optimization given in the present contribution goes far beyond that, including the simultaneous optimization of several input parameters and a number of inequality constraints at different current densities. In the following section, some information on the technical and mathematical background of the MCFC, the modeling approach, and the optimization strategy are given. After a short introduction into the working principle of the MCFC, the major features of the applied model are mentioned. For more details about the modeling, we refer to our previous work.5,6 Then, the applied objective function and the used constraints are explained, and some important details of the numerical procedure are given. In the Results section, currentvoltage curves of optimized operating points are shown
10.1021/ie048759x CCC: $30.25 © 2005 American Chemical Society Published on Web 04/08/2005
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Figure 2. Relevant input and output variables of the twodimensional, cross-flow MCFC model.
Figure 1. Working principle of the DIR-MCFC.
and the potential benefit of the catalyst density optimization is demonstrated. 2. Technical and Modeling Background 2.1. MCFC Working Principle. The working principle of the considered MCFC is illustrated in Figure 1. The anode channel is fed with a preheated mixture of desulfurized natural gas (mainly methane) and steam at a certain steam-to-carbon ratio, which usually is at about S/C ) 2.5. The anode channel contains a reformer catalyst, at which methane and water are converted to hydrogen, carbon monoxide, and carbon dioxide according to the following reaction schemes of the methane steam reforming reaction and the water-gas shift reaction:
CH4 + H2O T CO + 3H2 ∆Rhθ(T)298.15K) ) 206 kJ/mol (1) CO + H2O T CO2 + H2 ∆Rhθ(T)298.15K) ) -41 kJ/mol (2) The reforming products, namely, hydrogen and carbon monoxide, are continuously removed from the anode gas phase by direct electrochemical oxidation at the electrode, which consumes carbonate ions from the electrolyte and produces carbon dioxide, water, and free electrons according to the following reaction scheme:
H2 + CO32- T H2O + CO2 + 2e-
(3)
CO + CO32- T 2CO2 + 2e-
(4)
In addition to this mass coupling (the reforming products are directly consumed by the oxidation process), the reforming process, which is overall endothermic, is energetically coupled with the electrochemical process, which generates heat; thus, the heat required for the generation of hydrogen is provided. The integration of the reforming process into the anode channel is known as direct internal reforming (DIR). The anode exhaust gas is mixed with air, and the unoxidized components are fully oxidized in a catalytic combustion chamber. Because air is fed in excess, the exhaust gas of the burner still contains significant amounts of oxygen. This gas is fed to the cathode channel where the electrochemical reduction takes place. There, new carbonate ions are produced from carbon dioxide and oxygen according to the following reaction:
CO32- T 1/2O2 + CO2 + 2e-
(5)
The carbonate ions are transported toward the anode electrode through the electrolyte, which is a eutectic carbonate melt. One part of the cathodic exhaust gas is recycled to the catalytic combustion chamber. This is done to homogenize the cell temperature profile. The rest of the cathode exhaust gas leaves the fuel cell and can be used to preheat the anode feed gas. Afterward, the exhaust gas usually has a temperature of about 400 °C, so it may be applied for subsequent steam generation, for additional production of electric energy via a microturbine, or for other purposes. Because of the electrochemical reactions, electrons are available at the anode electrode and they are missing at the cathode electrode, so an electric circuit can be closed and the electrons can be exchanged via any arbitrary load. As was already mentioned in the Introduction, temperature is the most vital state in a high-temperature fuel cell like the MCFC. To allow the use of inexpensive materials, the temperature may not be too high at any location in the cell. Also, high temperature leads to fast degradation of the nickel catalyst. Too low temperatures lead to high polarization losses due to the Arrhenius effect, which slows down the reaction rates and the ion transport through the electrolyte. In a DIR cell, a mixture of methane and water enters the anode channel, causing a very high reforming reaction rate near the inlet region. This leads to a significant heat sink and thereby a possible cold spot in that region. This cold spot leads to low reaction rates of the electrochemical process. Also, the hydrogen production is insufficient at low temperatures. On the other hand, close to the anode outlet region, the reforming process has almost come to full conversion, which means that the only way to bleed off heat is by convective transport in the gas phases. In this region, the highest temperature is likely to occur. Another important effect in high-temperature fuel cells is the deactivation of the catalyst by carbon deposition. This reversible effect can quickly deactivate the catalyst in the anode channel and thereby render the cell useless. Although this is a more significant problem for solid oxide fuel cells because of the temperature level, it can also occur in MCFCs. To avoid carbon deposition, sufficient amounts of water have to be supplied with the feed gas. Besides the desired oxidation of hydrogen and carbon monoxide and the reduction of oxygen, other electrochemical reactions may occur in an MCFC. If the cell voltage is too low, the nickel catalyst dissolves in the electrolyte in ionic form, which also leads to a quick deactivation of the catalyst. 2.2. Fuel Cell Model. The model applied for process optimization is depicted in Figure 2 as a black-box model. It describes a spatially two-dimensional, single MCFC in cross-flow configuration. Its state variables include the molar fractions and temperatures in the anode and cathode gas phases, the temperature in the solid compartments of the cell, the overall cell voltage,
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and the current density distribution. The input variables are the steam-to-carbon ratio, S/C, the molar flow of the anode feed gas, Ga,in, the air number, λair, which determines the amount of air fed into the catalytic combustion chamber, the cathode gas recycle ratio, Rback, and the average current density, icell. The output variables include the objective function, which is the electric system efficiency, ηel, and several additional variables that are subject to the constraints: cell temperature, Ts, cell voltage, Ucell, and a carbon deposition criterion, Pcarbon. The equations are based on the description of physical processes; that means mass and energy balances are set up, and kinetics for reactions and mass and heat transport are applied. The phenomena considered include convective mass and heat transport along the gas channels, mass and heat exchange between the gas and solid phases, heat conduction in the solid phase, mass transport resistance in the electrode pores, finite reaction rates of reversible reforming (power law kinetics) and electrochemical reactions (Butler-Volmer kinetics), a set of spatially discretized Poisson equations for the description of the electric potential field and the current density, and complete combustion in the catalytic combustion chamber. For additional details about the modeling approach, we refer to previous publications.5,6 The model presented there is formulated in dimensionless terms. For easier understanding, we use dimensional expressions in this contribution. The model is roughly fitted to measurement data from an industrial-scale MCFC (Hotmodule, by MTU CFC solutions7,8). Either the parameters are calculated from available data or they are estimated in such way that the simulated states such as temperatures and concentrations are approximately equal to the measurement data. A more detailed validation of the model parameters is currently being conducted. However, the Hotmodule features external and internal reforming. Because the design of a pure DIR system is one favorable aim in MCFC development, we consider such a system here, so the exact model parameters from the Hotmodule could not be used anyway. Thus, we are using estimated parameters that, as the ongoing validation shows, are mostly quite close to the realistic values. 2.3. Optimization. A fuel cell can be optimized in many ways with different objective functions. For example, the minimization of the maximum temperature difference in a stack requires a simple objective function, but economical objective functions, which are aimed at the maximization of monetary benefits, tend to become rather complex for optimization purposes. In our case, a purely technical objective function is chosen, which is the electric efficiency of the system. It is defined as the ratio of the electric cell power, Pcell, diminished by the system’s parasitic power consumption divided by the lower heating value of methane in the anode feed. The system’s power consumption is mainly determined by the power demand of the large air blowers that circulate the cathode gas flow, Pblower. For simplification, a proportional correlation between cathode gas flow and the blower power is assumed. The enthalpy of the feed gas is the product of the total molar feed flow, Ga,in, and the molar fraction of methane in the feed, quantified by the steam-to-carbon ratio, S/C, and the molar stanθ dard enthalpy of combustion of methane, ∆ChCH . 4 Thus, the objective function reads as follows:
ηel )
Psys ) Ha,in
Pcell - Pblower 1 θ Ga,in ) (-∆ChCH 4 1 + S/C
(6)
Two different types of optimizations are performed with this objective function: First, only the input parameters are used to optimize the objective function for a given average current density, icell:
ηel(Ga,in,S/C,λair,Rback) f max!
(7)
This yields optimal operating conditions, and repeating this optimization for several different values of the average current density results in a current-voltage curve under optimal conditions. The second optimization task also includes the spatial distribution of the reforming catalyst inside the anode channel. For the purpose of this optimization, the overall two-dimensional cell area is divided into four sections each with a constant catalyst density. Starting from the so-called “base case” configuration with a completely constant catalyst density distribution, one factor, FDa, in each section is used to describe the change in the catalyst density in that section. Because the amount of catalyst that can be placed in the anode channel is limited, this factor is also limited to a maximum value of 3.0. Thus, the second optimization uses four additional optimization variables:
ηel(Ga,in,S/C,λair,Rback,FDa,1,FDa,2,FDa,3,FDa,4) f max! (8) The number of four sections was chosen for two reasons: The first reason is to keep the numerical effort in an acceptable range. Using the results from this optimization as an initial guess, one could probably increase the number of sections to 9 (3 × 3) sections. Because the numerical effort increases rapidly with the number of optimization parameters, the number of sections should not be much higher. The second reason is that very fine distributions are hardly applicable from a practical point of view. Both optimization tasks consider several constraints. Equations 9 and 10 express the fact that the cell temperature is limited to a certain window. This is to avoid material damage due to too high temperature and to guarantee a minimum temperature necessary to ascertain high reaction rates and a good ion conductivity in the electrolyte. The third constraint (eq 11) limits the maximum temperature difference in order to avoid mechanical stress and leakages due to different thermal expansion. The fourth constraint (eq 12) accounts for a minimum cell voltage, below which undesired side reactions of the nickel catalyst with the carbonate melt occur. The last constraint (eq 13) takes into account the possibility of carbonization. The criterion ensures that carbonization is thermodynamically impossible at any location in the cell.
min[Ts(z1,z2)] ) Ts,min g Tlim,min ) 591 °C
(9)
max[Ts(z1,z2)] ) Ts,max e Tlim,max ) 681 °C (10) max[Ts(z1,z2)] - min[Ts(z1,z2)] ) ∆Ts,max e ∆Tlim,max ) 60 °C (11) Ucell g Ulim,min ) 694 mV
(12)
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∃j: ∆RgCj[Ta(z1,z2),xi,a(z1,z2)] g 0
(13)
In addition to this, the cathode reflux ratio, Rback, is limited to a maximum value of 0.5. The full set of steady-state equations of the twodimensional MCFC model are considered as equality constraints to this optimization. 2.4. Numerical Aspects. For the optimization, a spatially discretized model of the MCFC5,6 is used, which is implemented in ProMoT (Process Modeling Tool9) and solved within the simulation environment DIVA.10 The sequential quadratic programming (SQP) algorithm E04UCF from the NAG library,11 offered by the DIVA package, is employed. The model is a transient formulation because directly solving the steady-state equation system requires excellent starting values and is thus not preferable. Therefore, a dynamic approach is chosen in which the equation system is integrated for every optimization step until the steady state is reached. Then, the objective function is evaluated at the steady state. The highly effective SQP optimization algorithm reaches the optimum within only a few steps, which is due to the use of parameter sensitivities. These are delivered by the underlying integration algorithm. To work properly, the integrator requires a continuously derivable equation system including the variables for the constraints. This is not the case with the minimum and maximum temperatures in eqs 9-11 and with the carbonization criterion in eq 12. As long as, say, the location of the lowest temperature in the cell stays constant, Ts,min is continuously derivable. However, as soon as the location of the lowest temperature changes, the derivability is no longer given and the integrator fails. Thus, the following two differential equations are introduced to provide continuously derivable extremal temperatures:
(14)
dTs,max 1 ) {max[Ts(z1,z2)] - Ts,max} dt τϑ
(15)
where τϑ is an arbitrary time constant, which should be significantly lower than the slowest time constant in the fuel cell model. With this, also the maximum temperature difference in eq 11 is derivable. Also, the carbon deposition criterion requires attention. Here, a kind of penalty function is applied. Consider the Gibbs energy of one of the carbonization reactions j at a certain location z and a certain time t. Then, a continuous, limited, reaction-related penalty function is defined according to
{
if ∆RgCj(z,t) g 0
pj(z,t) )
[∆RgCj(z,t)]
2
1 + [∆RgCj(z,t)]2
if ∆RgCj(z,t) < 0
local penalty function of carbonization is defined as follows:
p(z,t) )
∏j pj(z,t)
(16)
This function is 0 if the carbonization reaction is not likely to happen, and it continuously increases to 1 if the reaction is likely to happen. To decide whether carbonization is thermodynamically possible at this location, the Gibbs enthalpies of all carbonization reactions under consideration have to be negative. Thus, a
(17)
Because carbonization is not allowed to occur at any location in the cell, the overall penalty function for carbonization (or overall carbonization criterion) is expressed as the integral of the local penalties:
P(t) )
∫zp(z,t) dz ) ∫z∏pj(z,t) dz
(18)
j
The reaction-specific penalty functions depend on continuously derivable states such as concentrations and temperatures. The overall carbonization criterion is comprised of these by continuous operations; thus, this formulation is usable for the integrator. The constraint (eq 13) now reads
P(t) e 0
dTs,min 1 ) {min[Ts(z1,z2)] - Ts,min} dt τϑ
0
Figure 3. Current-voltage curve under optimal operating conditions for a system with constant catalyst distribution.
(19)
With these reformulations of the inequality constraints, it is possible for the integrator to provide the required sensitivities to the optimization algorithm. A typical optimization run with an 8 × 8 spatial discretization grid includes about 1200 ordinary differential equations, which are repeatedly integrated until steady state, 5 inequality constraints, and 4 optimization parameters. It takes between 1 and 3 h of CPU time on a standard 1 GHz machine. 3. Results Optimization of Input Conditions at Constant Catalyst Density. The optimization of the first mentioned type (eq 7) is performed at various average cell current densities. The result is a set of operating conditions for each cell current that fulfills all constraints and offers an optimal electric efficiency. These points form an optimal current-voltage curve (Figure 3). The circles in Figure 4 show the corresponding electric efficiencies. They range from more than 57% at low currents to 38% at high cell currents. Table 1 lists the detailed results of the optimizations. It not only contains the calculated input conditions but also gives information on the resulting temperatures, voltage, fuel utilization (Yfuel), and system efficiency. Figure 5 shows the temperature field inside the cell at base case conditions, that is, at icell ) 124.6 mA/cm2.
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Figure 4. Effect of the optimization of the reforming catalyst distribution: electric system efficiency over the cell current for a system with constant reforming catalyst distribution and for a system with optimized catalyst distribution.
Figure 5. Cell temperature at icell ) 124.6 mA/cm2 with optimized operating conditions and constant reforming catalyst density.
The highest temperature occurs at the outlet/outlet corner while the lowest temperature is found at the inlet/inlet corner of the anode and cathode channels. This will be of interest later on. For load cases with low current density (first two load cases with icell ) 50-70 mA/cm2), the cell temperature is at the lower end of the admissible range. Because only small amounts of methane are fed into the anode channels (see FCH4,a,in), the residence time is high enough for the reforming process to reach high degrees of conversion. This means that the electrochemical oxidation can be sustained with high concentrations of hydrogen even though the fuel utilization, Yfuel, is high. The air number, λair, is mainly determined by the need to sustain a minimum cell temperature. The addition
of cold air usually tends to decrease the cell temperature, so in these load cases, its amount is kept low. Nevertheless, sufficient amounts of oxygen are provided to the cathodic reduction process. The low air flow rate also leads to small power losses at the blowers. Because the electrochemical reaction rates are quite low and the concentrations of the educts are relatively high, the cell voltage, Ucell, is above 0.8 V. Thus, the low-load regime has a good electrochemical performance and is mainly governed by the need to sustain the minimum operating temperature. Between 70 and 100 mA/cm2, there is a transition in the operating regime. At medium-load cases from 100 to 140 mA/cm2, the cell temperature is at the upper end of the allowable range. Here the heat losses inside the cell are increased because of higher electrochemical reaction rates, as can be seen by the decreasing cell voltage, Ucell. To avoid overheating, an increased cooling effect by larger amounts of air is necessary; thus, λair is increased. The fuel utilization is decreased slightly in order to provide sufficient amounts of hydrogen to the oxidation reaction, and the cell voltage decreases because of polarization losses. This and the increased blower power lead to a decrease in the system efficiency. Overall, the medium operating regime is governed by the need to cool the cell. A third operating regime is observed at very high current densities, i.e., above icell ) 140 mA/cm2. The high flow rates through the anode channels lead to a short residence time. To supply sufficient amounts of hydrogen to the oxidation process under these circumstances, a large excess of methane must be fed, meaning that the fuel utilization, Yfuel, goes down rapidly. The fact that all excessive fuel is completely oxidized in the catalytic combustion chamber means that the heat production rate inside the system increases dramatically. To cool the cell, large amounts of air are required, leading to high air numbers, λair, which again increase the power consumption by the blowers. The high electrochemical reaction rate decreases the cell voltage further, until it almost reaches its minimal allowed value. Consequently, the system efficiency breaks down to quite low values. The upper operating regime is dominated by the need to sustain high reaction rates at low degrees of fuel utilization and the need to withdraw lots of heat with large amounts of air. It is expected that there is another operating regime beyond 180 mA/cm2 because the cell voltage reaches its minimum value and has to be sustained by additional amounts of feed gas. Presumably, the operating conditions will be governed by quite similar arguments as in the upper operating regime, which we have discussed.
Table 1. Optimal Operating Conditions and Simulated System States at Various Average Current Densities for a Single Cell with Constant Reforming Catalyst Distribution icell [mA/cm2] optimized operating conditions simulated system states
Rback Ga,in [mol/s] S/C λair FCH4,a,in [scm/s] Yfuel Tmin [°C] Tmax [°C] Ucell [mV] Psys [W] ηel [%]
53.4
71.2
89.0
106.8
124.6
142.4
160.2
178.0
0.5 1.24 3.95 1.87 22.4 0.742 591 651 822 338 57.1
0.5 1.07 2.16 1.96 30.2 0.737 591 651 812 445 56.0
0.5 1.29 2.02 1.99 38.0 0.728 612 672 799 547 54.5
0.5 1.46 1.85 2.04 45.8 0.723 621 681 783 644 53.0
0.5 2.14 2.50 2.17 54.5 0.717 621 681 760 724 50.7
0.5 2.49 2.51 2.45 63.2 0.703 621 681 735 798 47.9
0.5 2.94 2.46 2.63 75.9 0.658 621 681 717 869 43.4
0.5 3.25 2.14 2.97 92.4 0.602 621 681 696 931 38.3
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Figure 6. Result of the optimization of the reforming catalyst density at the base case, i.e., icell ) 124.6 mA/cm2. Optimal configuration of the catalyst density factors.
Depending on the design of the porous electrode structures, mass transfer may also play a role here. Because the system efficiency at these load cases is quite unattractive for application, we do not further discuss this possibility. As the discussion shows, most parameters show a certain tendency over the range of current density. Yet, there are a few that do not. The cathode gas recycle ratio, Rback, is at its maximum value at all load cases. This indicates that the effect of homogenizing the cell temperature by cathode gas recycling is advantageous even though it increases the volumetric flow through the cathode channel and thereby increases the power losses at the blower. Another parameter is the steamto-carbon ratio, S/C, which varies strongly from one load case to another. This is because the electric efficiency, which serves as the objective function of this optimization, is not very sensitive to this input parameter. Extending the definition of the efficiency in such a way that the energy required for the evaporation of water is also included could amend this. Further investigations that will also include the pretreatment of the gas will consider this input condition in more detail. Optimization of the Reforming Catalyst Density Distribution. The second type of optimization is only performed once at a base case current density (icell ) 124.6 mA/cm2). The result is not only a set of optimal operating conditions but also four factors describing the optimal reforming catalyst density distribution in the anode channel, as illustrated in Figure 6. The anode channels on the right-hand side have a slightly increased reforming catalyst density near their inlet (section 1). This leads to a fast hydrogen generation in that section, providing a certain amount of hydrogen for the oxidation reaction. The result is a moderate cooling effect and a high current density. The relatively
Figure 7. Cell temperature at icell ) 124.6 mA/cm2 with optimized reforming catalyst density distribution and optimized operating conditions.
depleted gas then enters the channels’ second half (section 3), where a maximum amount of reforming catalyst converts the remaining methane to hydrogen as quickly as possible. In the anode channels on the left-hand side, the situation is different. Near the inlet, the reforming catalyst density is very low (section 2). This section is not temperature critical (see Figure 5), so the catalyst density here is a compromise between the need for hydrogen for the oxidation reaction and the need to spare the cooling effect of the reforming reaction for the hot area in the second half of these channels. In this configuration, sufficient methane is left in the gas when it enters section 4, where it is quickly reformed. This provides a significant cooling effect in the hot area of the cell. With a reduced maximum temperature, the other input parameters can be adjusted in such a way that the system efficiency is increased, until the maximum allowable temperature difference is reached again. Unlike in the right-hand side anode channels, the catalyst density in the channels’ second half is not at its maximum value (section 4). Increasing the catalyst density in this section would lead to a higher current density here. However, then because the total cell current is constant, the current density in section 1 would decrease. Thereby, an important heat source would be diminished, the cell temperature in that section would decrease, and thus the allowable temperature difference would be exceeded. The cell temperature distribution for the optimized catalyst density is plotted in Figure 7. Optimization of the Input Conditions for a System with Optimized Catalyst Density. Using this catalyst distribution, a new optimal currentvoltage curve is calculated. Figure 4 shows the results
Table 2. Optimal Operating Conditions and Simulated System States at Various Average Current Densities for a Single Cell with Optimized Reforming Catalyst Distribution icell [mA/cm2] optimized operating conditions simulated system states
Rback Ga,in [mol/s] S/C λair FCH4,a,in [scm/s] Yfuel Tmin [°C] Tmax [°C] Ucell [mV] Psys [W] ηel [%]
53.4
71.2
89.0
106.8
124.6
142.4
160.2
178.0
0.5 1.16 3.52 1.88 22.8 0.736 591 651 825 340 57.1
0.5 1.05 2.16 1.93 29.6 0.756 591 651 812 444 56.7
0.5 1.20 1.84 1.99 37.7 0.744 600 660 796 544 55.3
0.5 1.44 1.94 1.99 43.7 0.736 621 681 784 643 54.0
0.5 1.50 1.51 2.08 53.5 0.732 621 681 766 733 52.4
0.5 1.74 1.51 2.25 62.0 0.719 621 681 745 811 49.8
0.5 2.79 2.48 2.49 71.6 0.702 621 681 712 868 46.3
0.5 3.01 2.17 2.79 84.7 0.655 621 681 694 933 41.8
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of these optimizations and compares them against those of the system with constant catalyst distribution. It turns out that the catalyst distribution increases the system efficiency not only at the point for which it has been optimized but for all medium and high current density load cases. Only at low average cell currents is its performance equivalent to the system with a constant configuration. From an economic point of view, low currents are not attractive, so these operating points are of minor importance. Table 2 lists the optimal operating conditions for the system with optimized catalyst distribution. 4. Conclusions and Outlook Solutions for two MCFC optimization tasks are presented in this paper. The optimization of input parameters for an MCFC is successfully performed based on a spatially two-dimensional, cross-flow model. The operating conditions obtained from this optimization not only yield optimal efficiency at any given cell current but also guarantee safe operation of the system with regard to temperature constraints, cell voltage, and carbon deposition. Three different operating regimes are identified in which the operating conditions have to be set according to a different argumentation. In a system mainly governed by temperature effects, the second optimization balances the cooling effect of the reforming reaction with the heat source from the electrochemical reactions by optimizing the reforming catalyst density distribution. With the catalyst distribution obtained at one single operating point, the system performance is significantly increased over the whole range of cell currents, especially at high currents. This strongly indicates that this measure is advisable in future DIR-MCFC systems. In this paper, the electric efficiency is used as an objective function. Nevertheless, more complex objective functions can be applied, including the energetic costs of the evaporation and preheating of steam, aging effects of the catalyst, economic aspects such as fuel costs or prices for electric energy, and influences of the operating conditions on the expected lifetime of the stack. Furthermore, additional analysis of the parameter sensitivity and robustness of the optimization results are necessary. The validation of the applied fuel cell model to experimental data is currently in progress. While we expect the basic optimization results to be qualitatively similar for the actually used and validated model, the sensitivities may not be. Thus, these detailed analyses and the use of more complex objective functions are future objectives. Acknowledgment The authors gratefully acknowledge the financial support of this research project by the German Federal Ministry of Education and Research (BMBF) under Grant 03C0345A (“Optimierte Prozessfu¨hrung von Brennstoffzellensystemen mit Methoden der Nichtlinearen Dynamik”). Many thanks go to Prof. Pesch and his colleagues (Mathematics in Engineering Sciences, University Bayreuth, Bayreuth, Germany) for their fruitful discussions about the numerical implementation of the inequality constraints. List of Symbols FCH4,a,in ) volumetric feed flow of methane, Nm3/h FDa,k ) catalyst density factor in section k Ga,in ) anode inlet molar flow, mol/s
Ha,in ) enthalpy flow at the anode inlet, J/s icell ) average current density, mA/cm2 Pblower ) electric power consumption by the blower, W Pcell ) electric cell power, W Pcarbon ) carbonization criterion Psys ) overall electric system power, W Rback ) cathode recycle ratio S/C ) steam-to-carbon ratio at the anode inlet t ) time, s Ts ) cell temperature, °C Ta ) anode gas temperature, °C Tlim,min ) minimum allowable cell temperature, °C Tlim,max ) maximum allowable cell temperature, °C Ts,min ) minimum cell temperature, °C Ts,max ) maximum cell temperature, °C Ucell ) cell voltage, V Ucell,min ) minimum allowable cell voltage, V xi,a ) molar fraction of component i in the anode channel Yfuel ) fuel utilization z ) dimensionless spatial coordinate Greek Symbols θ ∆ChCH ) standard enthalpy of methane combustion, 4 J/mol ∆RgCj ) Gibbs energy of carbonization reaction j, J/mol ∆Tlim,max ) maximum allowable cell temperature difference, °C ∆Ts,max ) maximum cell temperature difference, °C ηel ) electric efficiency λair ) air number τϑ ) time constant for extremal temperature equations, s
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Received for review December 22, 2004 Revised manuscript received March 3, 2005 Accepted March 9, 2005 IE048759X
