Article pubs.acs.org/IECR
Reduced Order Inferential Model-Based Optimization of a Phosphoric Acid Fuel Cell (PAFC) Stack Saibal Ganguly,*,† Sonali Das,‡ Kajari Kargupta,‡ and Dipali Banerjee§ †
Chemical Engineering Department, Universiti Teknologi Petronas, Malaysia Department of Chemical Engineering, Jadavpur University, Kolkata, India § Department of Physics, Bengal Engineering and Science University, Shibpur, India ‡
ABSTRACT: The steady state optimization structure for the PAFC stack in the overall hierarchical optimization and control scheme is proposed in this paper. An easy to implement, low CPU time-consuming reduced order steady state PAFC stack model, that maps the PAFC performance satisfactorily, is used as the equality constraint equation block and variable bounds are used as inequality constraints. Functional dependence of the operating variables namely hydrogen and oxygen/air flow rates, humidifier, and cell temperature on the stack power generation are simulated. Electrolyte concentration, inferentially predicted by the model aids to identify acid drying and dilution during operation. For optimization two variables namely load current and electrolyte (phosphoric acid) concentration are considered as the optimization variables; the optimized values of which are communicated as set points for gas flow rates and humidifier temperature at the advanced control level. In the present paper steady state optimization is carried out using the sequential quadratic programming (SQP) algorithm with quasi-Newton line searching to enhance convergence. Two case studies have been performed (i) economic optimization for a PAFC stack resulting in maximization of profit and (ii) optimization to achieve a time variant electrical load based on market demand. For very high demand power, the optimizer converges to the maximum possible power and restricts the system from entering into a state of operational breakdown. The inferential reduced order model based optimization scheme showed promising potential for real-life utilization of fuel cells in remote rural areas, marine, submarine, desert, mountain terrains, oceanographic applications, and transport systems, where large computational facilities are neither possible nor feasible.
1. INTRODUCTION In the wake of increasing energy demands and environmental concerns, fuel cells offer an innovative alternative to current power sources since they are environmentally friendly and highly efficient with renewable fuels. A fuel cell is an electrochemical device that converts chemical energy of a fuel directly into electricity. Of the several classes of hydrogen fuel cells, phosphoric acid fuel cells (PAFCs) represent one of the most mature technologies. A phosphoric acid fuel cell (PAFC) is composed of two porous gas diffusion electrodes, namely a cathode and anode juxtaposed against a porous electrolyte matrix. The gas diffusion electrodes are composed of a porous substrate (carbon or cloth) facing the gas feed and a reactive catalyst layer consisting of platinized fine carbon powder, facing the electrolyte. At the anode, hydrogen ionizes to H+ and migrates toward the cathode to combine with oxygen and forms water. The free energy of reaction is converted to electrical energy, which is available in the form of a potential difference developed between the two electrodes. The reactions at anode and cathode are as follows:
Figure 1. Hierarchical optimization structure for a PAFC stack.
evaporation of water. In place of oxygen, air can also be used for oxidation. Fuel cell involves multitude of physical phenomena namely electrochemical reaction, mass transport, energy transport, and operational problems namely electrolyte drying and flooding, cross over through the electrolyte, starvation due to blockage of
Anode: H 2 = 2H+ + 2e− Cathode: 1/2 O2 + 2H+ + 2e− = H 2O
Special Issue: PSE-2012
A stack consists of a number of fuel cells connected in parallel. The electrical load is applied across the stack. Figure 1 presents a schematic of a composite PAFC stack with the hierarchical optimization and control layers. Hydrogen supplied is humidified in order to prevent drying of electrolyte by © 2013 American Chemical Society
Received: Revised: Accepted: Published: 7104
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reported by NASA,2 Center for Aerospace Information (CASI). In this report, an optimized cost and performance model for a phosphoric acid fuel cell power plant system has been derived and developed into a modular computer code. The mathematical model developed in this study combines appropriate cost, energy, and electrochemical analysis as applied to the major components of a phosphoric acid fuel cell system. Steam to methane ratio in the reformer, hydrogen utilization in the fuel cell stack, and the number of fuel cell plates per stack are considered as optimization variables. Grujicic et al.3 reported the design and optimization of polymer electrolyte membrane (PEM) fuel cells. A single phase two-dimensional model coupled with a nonlinear constrained optimization algorithm was used to determine an optimum design of the fuel cell with respect to the operating and the geometrical parameters of cathode such as the air inlet pressure, the cathode thickness, and length and the width of shoulders in the air distributor. Different strategies and methods of fuel cell system optimization are reported and reviewed4 in the literature. Kumar et al.5 used genetic algorithm for optimization of proton exchange membrane fuel cell at different operating and design conditions. Optimization of channel geometry in a proton exchange membrane (PEM) fuel cell has also been reported in the literature.6 Zhang et al.7 used three-dimensional steady-state electrochemical mathematical model and Powell algorithm to perform optimization of PEM fuel cells. In this paper, a novel steady state hierarchical optimization scheme is proposed for PAFC stacks. Though hierarchical optimization has been used for industrial applications,8,9 a hierarchical scheme for PAFC optimization is hardly available in international literature. A novel hierarchical scheme with successive quadratic programming based economic optimization is proposed for PAFC stack operation. Flags are introduced to prevent the solution to enter the undesirable zone of operation namely electrolyte drying, flooding, concentration polarization.
catalyst pores, etc. Computational transport phenomena modeling and simulation are essential tools for the performance mapping of fuel cell stacks over varying ranges of operating conditions like temperature, partial pressure, humidity, flow rate, etc.1 However, optimization of the fuel cell consists of challenging multiobjective, multivariable problems requiring inferencing of internal operating conditions within the optimization cycle for successful implementation. The importance of model based inferencing of electrolyte concentration or drying and dilution conditions during the optimization convergence needs to be understood with an insight into the physics of the process. The ionic conductivity decreases if the acid concentration increases during PAFC operation due to loss of moisture. Subsequently, either operating temperature should be increased to enhance ionization or the humidity level in the inlet gas should be increased to reduce the acid concentration. However, if the acid gets diluted, the volume increases and acid is lost from the system as carryover mist. Excess water increases the fractional liquids in the catalyst pores causing starvation of oxygen for the reactive catalyst site. This condition may lead to acid flooding. Figure 2 shows the schematic of a single PAFC cell to articulate the physics of the system.
2. FORMULATION OF INFERENTIAL MODEL-BASED OPTIMIZER 2.1. Hierarchical Optimizer Structure for a PAFC Stack. The power generated by the fuel cell stack is a function of load current (I), hydrogen and oxygen flow rates (QH2, QO2), cell temperature (Tcell), humidity of inlet hydrogen stream and electrolyte concentration (Cphos). Humidity of inlet hydrogen is a function of humidifier temperature (Th) and hydrogen flow rate. Increase of the cell temperature, humidifier temperature, and gas flow rate lead to a higher output power. However, this change increases the operating cost of the stack. In the phosphoric acid fuel cell, the electrolyte (phosphoric acid) concentration determines the proton conductivity and thus the power generation. For normal operation of the fuel cell, the electrolyte concentration must be controlled at the optimal value. Drying of electrolyte due to moisture evaporation and excess dilution of electrolyte drastically decreases the power generation. The load current has the most important effect on power generation. Increase in the load current results in a fall in the cell voltage. Power increases with the load current up to a certain value and then decreases due to drastic reduction in the cell voltage. It is undesirable to operate the fuel cell beyond this critical value of load current.
Figure 2. Schematic of a single PAFC.
In order to achieve the optimal performance of PAFC, it is crucial to have an adequate water balance to ensure good proton conductivity of the electrolyte while cathode flooding and electrolyte dehydration must be avoided. The performance of the PAFC stack is crucially dependent on the effective management of humidity, electrolyte concentration, and moisture balance across the stack. In the present work, a robust, comprehensive, and easy to implement reduced order inferential model is presented to optimize the performance of the PAFC stack. The “cathode−anode−electrolyte” integrated model is capable of predicting the phosphoric acid concentration based on the overall steady state balance of moisture over the PAFC assembly. Online optimization using a computational model is usually employed for the maximum utilization of the available resources and for maximization of profitability. Online optimization is also essential for maximization of performance to meet the time variant market demands for power. In a recent article, Secanell et al.1 presented a review of research work aimed at using numerical optimization to design fuel cells and fuel cell systems. Cost optimization of a PAFC power plant is 7105
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In the present work, a steady state optimization scheme has been proposed for the PAFC stack for exploitation in the hierarchical structure (Figure 1). At the topmost level, the process optimization variables [X] considered are the load current (I) and electrolyte concentration (Cphos). The set points for the operating variables [Y] in the advanced control level namely QH2, QO2, Th, and Tcell are determined by I and Cphos. The present paper deals with the steady state optimization layer only. The hierarchical advanced control applications will be available in future publications. In the present work, two case studies have been considered: case study 1, economic optimization for a PAFC stack resulting in maximization of profit; and case study 2, optimization to meet a time variant electrical load based on market demand. 2.2. Structural Blocks for Optimization of a PAFC Stack. 2.2.1. Objective Function for Profit Optimization (Case Study 1). The objective is to maximize the profit function, φ, which can be mathematically expressed as maximize φ = P1 − P2 − P3 − P4
flag1 = 0 for Cphos > 95.5%
(3)
This flag allows a gradual initial decrease followed by a rapid fall ending with zero value at Cphos = 95.5%. For flooding,
flag1 = 1 − exp( −2 × (Cphos − 85.5));
flag1 = 0 for Cphos < 85.5%
(4)
Power is calculated as power = current × voltage × flag1 × flag 2
with respect to the operating variables where, φ = profit based objective function, P1 = selling price of power generated, P2 = cost of raw materials (hydrogen and oxygen), P3 = loss factor due to unutilized hydrogen and oxygen, and P4 = operating costs including heater input to fuel cell, humidifier, and compressor cost. The different terms are described below: 1. Selling Price of Power Generated, P1.
flag 2 = a + b × voltage, where ⎧1 flag 2 = ⎨ ⎩0 ⎪ ⎪
at voltage ≥ 0.7 at voltage ≤ 0.5
Thus, a = − 2.5, b = 5, and
P1 = (power generated × unit price × w1 × flag)
flag 2 = ⎧1 for voltage > 0.7 ⎪ ⎨− 2.5 + 5 × voltage for 0.5 < voltage < 0.7 ⎪ for voltage < 0.5 ⎩0
(2)
where, power generated is in terms of kilowatt minutes, unit price is the selling price of power per kilowatt minute, w1 is the weight factor, and flag is a parameter whose value depends on the zone of operation of fuel cell. The purpose of penalty functions in the objective function is to prevent the optimum solution from being in the undesirable operation zones. The term flag is composed of two flag variables, one for keeping the operation in the desirable zone with respect to electrolyte concentration (flag1) and another with respect to load current (flag2). The different operation zones based on electrolyte concentration have been considered for using penalty functions from experimental observations, and they are listed in Table 1:
in out out P2 = (Q hyd − Q hyd )C hydw2 + (Q oxin − Q ox )Coxw2
= function of (I )
zone
performance
characteristic
good poor poor
85% < Cphos < 95% Cphos > 95% Cphos < 85%
(6)
It is not necessary to include flag for extreme hydrogen flow rates or low cell temperatures since flag is already included to account for flooding and drying, in this term. 2. Cost of Raw Materials, P2. The cost of raw materials includes cost of hydrogen and oxygen. According to the stoichiometric ratio, the hydrogen flow rate is twice the value of oxygen flow rate. Assuming that excess hydrogen and oxygen are recycled, the raw material cost is
Table 1. Operating Zones Based on Electrolyte Concentration normal operation phosphoric acid drying electrolyte dilution
(5)
where, flag2 is included to prevent the optimum power from being in an undesirable zone of very high overpotential. Usually it is seen that the power reaches a maximum with respect to the load current with high current value and high overpotential. In order to constrain the operation to the ohmic zone, a continuous linear flag has been introduced as follows:
(1)
= function of (I , Cphos)
flag1 = 1 − exp( −2 × (95.5 − Cphos));
For drying,
(7)
where, Chyd and Cox are costs of hydrogen and oxygen respectively in dollars per liter and w2 is the weight factor of raw material cost in the objective function. The amounts of hydrogen and oxygen consumed are mainly a function of the current generated. in out (Q hyd − Q hyd ) = I /2F ;
out (Q oxin − Q ox ) = I /4F
(8)
3. Loss Function for Unutilized Raw Materials, P3. A loss function is considered for the unutilized hydrogen and oxygen streams. It accounts for the recycling cost.
Acid drying may be caused by the low humidity of inlet hydrogen stream, very low hydrogen flow rate whereas flooding is caused by high hydrogen flow rates, low fuel cell temperature, etc. Both flooding and drying of electrolyte leads to a drastic fall in its proton conductivity leading to poor cell performance. Instead of using a discrete value as a flag, a continuous exponential decay function with the operating variables has been used in this work.
out out P3 = Q hyd C lH2w3 + Q ox C lO2w3
= function of (Q H2 , Q O2 , I ) = function of (I , Cphos) 7106
(9)
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nstack = number of fuel cell stacks, Nd = number of demand time cycles Power output denotes the net power output available for end use from a single PAFC stack. It is dependent on the number of stack (nstack) used. The gross power produced from the cell may not be available for the market demand since part of it is used to run the heaters and other auxiliary equipments. The remaining power constitutes the net power output. power output = gross power − power operation (14)
where, ClH2 and ClO2 are the loss coefficients corresponding to hydrogen and oxygen and w3 is the weight factor for the loss function. 4. Operating Costs of Fuel Cell Heater, Humidifier, and Compressor, P4. Operating costs include: (i) cost of heating fuel cells to required temperature, (ii) cost of heat supplied to humidifier, and (iii) compressor cost for pressurizing gas streams to the required pressure. Cost of hydrogen and oxygen separation and recycle are not included here as it has been already accounted for in P3. Cell temperature profile in the stack is predicted by a steady state heat transfer model. The cell temperature should be maintained within the range of 120−200 °C for proper functioning of fuel cell and to avoid CO poisoning. However, it has been observed experimentally and through model prediction that power generated by the fuel cell does not vary considerably with the cell temperature. This may be because of conflicting effects of reduction of ΔG of the process and enhanced kinetics at higher temperatures. In this paper, a controlled constant temperature profile for fuel cell operation has been assumed. Heat input to the humidifier is calculated as follows:
Gross power is the power obtained from the stack as a result of cell reaction as defined in section 2.2.1 in eq 5 is a function of current and electrolyte concentration. Power operation is the power required to operate heater connected to fuel cell and humidifier. The different terms are described below: power operation = ΔH + Hhumidifier = f (Cphos , I )
where, Hhumidifier is the heat input to humidifier during the operation of PAFC. It is interesting to note that the heat requirement for stable operation of the cell is mostly generated from the exothermic heat of reaction in the cell. Only the remaining heat required to maintain the cell temperature, ΔH, is used from the gross power output of the stack. During the initial startup of a stack, the temperatures of each cell is raised to the set points using heaters. During operation, the temperatures may deviate from the set values due to the transfer of sensible heat to the gaseous streams (hydrogen and oxygen), due to loss of heat to the environment and due to the continuous production of heat of electrochemical reaction. The temperature changes may be modeled using a heat balance around the stack, with heat generation from the heat of reaction and heat outflows in the form of losses into the gases:
Hhumidifier = latent heat of vaporization of water in humidifier + heat lost due to convection = λ(Th)[He − Hi]Q Hyd + hAh (Th − Tα) (10)
where, He and Hi are the exit and inlet humidity of hydrogen stream in the humidifier, respectively. Assuming that the extent of humidification is 100%, He is a function of Th, and Hi is a constant. Thus, Hhumidifier = f(Th). Since, Th can be expressed in terms of Cphos and load current I, Hhumidifier = f(I, Cphos). The compressor work required may be approximated as that of an isothermal compressor: ⎛P ⎞ W = (2.78e−4)ξQ 1Pc1 ln⎜ c 2 ⎟ ⎝ Pc1 ⎠
where, ΔHr|T = heat of reaction (kJ/mol water produced) at the stack temperature, T (K). The heat of reaction at the stack temperature (ΔHr|T) can be calculated from the standard heat of reaction at 298 K (ΔHr|T=298) by using the following formula:
(11)
where Q1 = volumetric input gas flow rate = Qinhyd + Qinox,
T
ΔHr|T = ΔHr|T = 298 +
P4 = unit price[(Hhumidifierw4) + (ΔH w5) + (Pcompressorw6)] = f (C phos , I )
∫298 (CpH O − CpH 2
2
− 1/2CpO ) dT 2
(17)
(12)
nH2O = rate of formation of water (mol/s) = I/2F. β denotes the fraction of the heat generated through electrochemical reaction which is utilized to heat the stack. (1 − β) denotes the fraction of heat generated that has been lost due to dissipation into the atmosphere. The heat accumulation term is subsequently evaluated to determine the periodic use of the heater. A positive value indicates that heat is available and the heater is switched out of action. A negative value indicates that heat must be replenished by using the heater to keep the temperature at their steady state set points values. 2.2.3. Constraint Equations for Optimizer. The variable bounds are used as the inequality constraints. The steady state PAFC stack model equations are used as equality constraints ([Ceq] = 0) that predicts the relation between the operating variables namely humidifier temperature, hydrogen and oxygen flow rates, absolute humidity of inlet hydrogen, and the optimization variables (I, Cphos). Thus, for case study 1, the optimization problem is
where, ΔH is the differential input to heater plate, Hhumidifier is the heat input to humidifier, and Pcompressor is the power input to gas compressor in terms of kilowatt mintues and unit price of power is in dollars per kilowatt mintue. Terms w4, w5, and w6 are the weight factors for humidifier cost, cost of differential heat input to heater plate, and cost of compressor cost, respectively. 2.2.2. Objective Function for Optimization to Achieve Time Variant Power Demand (Case Study 2). The objective function is the square of the difference between the time variant market demand for power and the net power output from the stack. It is mathematically expressed as Nd
minimize φ =
(15)
∑ [power demand − nstack × power output]i 2 i=1
(13)
φ = residual-based objective function, power output = net predicted power output from each stack in kilowatt minutes, 7107
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subject to:
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and the proton conductivity of the electrolyte (Ke) used in above set of constraint equations (eqs A5−A7) are the functions of electrolyte concentration (Cphos) and cell temperature (Tcell). The functional correlations are established by fitting the experimental data reported in literature10 and used in the model. The inferential prediction of electrolyte concentration allows one to identify the acid drying and dilution (Table 1). Acid drying lowers the proton conductivity and power generation. Stack model provides cell to cell variation of electrolyte concentration in a PAFC stack. An overall energy balance equation is used for stack to obtain cell to cell temperature variation. 2.4. Unit Cell Experimentation for Data Generation and Steady State Validation. PAFC unit cell comprises two electrodes of carbon with thin layers of platinum (20% Pt/C) deposited on it (anode and cathode) and glass mat soaked in phosphoric acid (electrolyte) sandwiched between the two electrodes. This assembly is placed between two grooved graphite plates for feeding oxygen to the cathode and hydrogen to the anode. Two stainless steel plates act as current collector at the two ends, and all these are pressed by two end pusher plates which are electrically isolated. Graphite gaskets are used for good fitting. A heater with a PID temperature controller is introduced to set and maintain a constant cell temperature. The cell temperature was controlled at a given set point within ±0.2 °C. Teflon insulator plates are used to reduce the heat loss. Pure hydrogen from a cylinder is passed through a humidifier to moisturize the gas that is passed to the anode through graphite grooved plate. Pure oxygen from a cylinder is directly passed into the inlet of the graphite plate which is in contact with the cathode. Throughout all experiments, the inflow and outflow hydrogen and oxygen are measured using a rotameter. All physical conditions were kept constant except cell and humidifier temperatures to find the effect regarding flooding and drying inside the cell. The active surface area (approximately 50 cm2) was kept constant throughout all experiments. After a period of 15 min from starting the experiment, DC polarization data was noted. During experimentation, first the open circuit voltage was measured without connecting the load across cathode and anode. A variable load was connected afterward between two current collectors. For different load currents, the resultant voltages were measured (polarization data). To generate a polarization curve, the cell was tested under constant voltage mode for 5 min time interval and the cell currents were averaged over these 5 min intervals. 2.5. Optimization Methodology. Nonlinear programming based steady state optimization was carried out using sequential quadratic programming (SQP) algorithm with quasiNewton line search to enhance convergence. The dependent variables namely power, Qhyd‑consumed, Qoxygen‑reacted, Qhyd‑excess, Qoxygen‑reacted, and Hhumidifier required in the objective function are found as function of optimization variables X, namely current and Cphos. The computed variable equations obtained using the PAFC stack model act as the equality constraints. Variable bounds are used as inequality constraints. The converged solution depicts the optimum variables values.
(18)
Ceq[X , Y ] = 0
[lb] ≤ [x] ≤ [ub] ⎤ ⎡ Y 1 ⎤ ⎡Th ⎢ ⎥ ⎢ ⎥ Q ⎢ Y 2 ⎥ = ⎢ H2 ⎥ ⎢ Y 3 ⎥ ⎢ power ⎥ ⎥ ⎢⎣ ⎥⎦ ⎢ ⎢⎣ Hi ⎥⎦ Y4
and
⎤ ⎡ X1 ⎤ ⎡ I ⎢⎣ ⎥⎦ = ⎢C ⎥ X2 ⎣ phos ⎦
Where Th is the humidifier temperature, QH2 is the hydrogen flow rate, and Hi is the absolute humidity of the inlet hydrogen. For the case study 2, the objective function defined in eq 13 is subjected to the same set of equality and inequality constraints discussed above for case study 1. The PAFC stack model used as equality constraint equation block comprises of computed variable blocks, single cell simulation block, and inferential prediction block. The computed variable block computes and maps the input variables to feed the simulation block from the experimental raw data. For the anode side, the computed variables like mass flow rates, molar flow rates, and partial pressure of each component entering each cell are computed using the data on inlet humidity, inlet total flow rate of pure hydrogen gas, inlet hydrogen gas temperature, and inlet total pressure. The property data of viscosity, density, and diffusion coefficients of hydrogen and moisture are also computed. Similar computations are also performed for the cathode side. In addition to these computations, the flow channel data and grid data are computed in this block. The single cell simulation block computes the current versus voltage, moisture transport rates from cathode and anode sides, and more importantly the electrolyte concentration for each individual cell of the stack. Finally, the inferential block computes the output variables namely stack voltage and power, outlet flow rates, and humidity of gas streams from PAFC stack using the simulation output data from all the individual cells. For a given current, the model computes the total overpotential for individual cell. The total voltage across the stack, Vstack, is inferred from these data. The total molar flow rates and the total individual component molar flow rates from the stack, the exit air humidity and exit hydrogen humidity, and the total exit volumetric flow rates are inferred from individual cell data obtained from predictor block model output. nc
Vstack =
∑ V (i ) i=1
(19)
The reduced order model equations for a PAFC single cell is presented in Appendix. In the present work, the PAFC stack model equations are developed considering several single cell lumped model in series. For a particular overpotential, the cathode model calculates the current density and water flux. For this calculated current density, the anode model calculates the anode overpotential and anode side water flux. From a moisture balance of anode and cathode water flux, the concentration of phosphoric acid electrolyte are calculated and updated. Using this new value of electrolyte concentration, the computation is repeated until the concentration of two successive steps are equal. 2.3. Inferential Prediction of Electrolyte Concentration. The vapor pressure of moisture over the electrolyte (psat)
3. RESULTS AND DISCUSSION 3.1. Effect of Operating Variables on PAFC Performance. Figure 3 depicts stack voltage versus load current plots with different air flow rates for a PAFC stack consisting of 32 7108
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Figure 5. Performance decay due to acid drying: effect of humidifier temperature on the performance of the phosphoric acid fuel cell. Figure 3. Stack voltage versus load current plots with different air flow rates for a 32 cell PAFC stack.
the drastic decay in performance at low humidifier temperature due to acid drying. The simulated data follows the similar trend obtained from experimental data. To achieve the optimal performance and maximum power generation, electrolyte concentration in a PAFC must be maintained at some intermediate optimum value by manipulating the operating variables namely gas flow rates and humidifier temperature. Figure 6 illustrates the functional dependence of electrolyte
cells. Increase in fuel and air/oxygen flow rate increases the power generation at the cost of operating cost. 3.2. Inferential Prediction of Phosphoric Acid Concentration. To prevent moisture loss from the electrolyte, hydrogen is humidified in the humidifier before passing it to the fuel cell. Figure 4 shows the steady state electrolyte
Figure 4. Effect of humidifier temperature and fuel cell temperature on deviation of phosphoric acid concentration from its base value of 88%. Figure 6. Electrolyte concentration versus cell voltage plots with different hydrogen flow rates.
concentration as a function of cell temperature and humidifier temperature obtained from simulation. For the purpose of demonstration of acid drying and dilution, the value of deviation of steady state electrolyte concentration from its base value of 88% is plotted as a function of cell temperature and humidifier temperature. In the phosphoric acid fuel cell, an initial concentration of 88% phosphoric acid solution was used. Depending on the operating conditions, the electrolyte may get dried or diluted. It may be inferred from Figure 4 that at higher cell temperature and lower humidifier temperature electrolyte becomes dried (concentration increases up to 95%) due to the evaporation of moisture to gas phase. Acid gets diluted at lower cell temperature and higher humidifier temperature due to condensation from the gas phase to the electrolyte acid dilution. As mentioned earlier the performance of PAFC drastically reduces due to acid drying/flooding. Figure 5 shows
concentration on hydrogen flow rates at different cell overpotential (voltage). Low hydrogen flow rate causes drying of electrolyte. 3.3. Case Study 1: Optimization of Profit from a PAFC Stack. A fuel cell stack consisting of 6 PAFCs is considered for the present study. Three heater plates each having a maximum heater capacity of 6000 W are placed between the fuel cells as shown in Figure 7. Cost optimization is carried out to calculate the optimum load current, phosphoric acid concentration, humidifier temperature, and gas flow rates for the maximum profit. 3.3.1. Contour Plot and Optimum Operating Conditions. The contour plot for the objective function and the optimum 7109
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Figure 7. Schematic of stack layout including the temperature profile.
point of operation is presented in Figure 8. Figure 8 shows the contour plot of the normalized objective function (φ) or the
Figure 9. Change of directional derivative of objective function with number of iterations.
Figure 8. Contour plot for normalized cost with respect to load current and electrolyte concentration.
normalized cost with respect to the optimization variables, I and Cphos. The minimum of the objective function corresponds to the optimum point and is shown by the white dot. The optimum load current and electrolyte concentration for this case are I * = 223.26 amps;
Cphos* = 90.83%;
Th* = 72.33 °C;
* = 104.5 L/min; Q H2
Figure 10. Maximum constraint variation vs iteration number.
* = 52.25 L/min Q O2
The optimum load current corresponds to a value of current density close to 0.2 A/cm2 for each cell cross sectional area of 1200 cm2. These results are in excellent agreement with practical observations. Figure 9 shows a plot of the directional derivative of the objective function with the iterations. It shows that the gradient approaches zero (
