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A fuel cell is an electrochemical device in which hydrogen reacts with oxygen to produce water and DC electricity. A 1-watt proton-exchange membrane fuel cell (PEMFC) could be used for portable applications such as cellular telephones, and a \(100-\mathrm{kW}\) PEMFC could be used to power an automobile. The following reactions occur inside the PEMFC:Anode: \(\quad \mathrm{H}_{2} \rightarrow 2 \mathrm{H}^{+}+2 \mathrm{e}^{-}\) Cathode: \(\quad \frac{1}{2} \mathrm{O}_{2}+2 \mathrm{H}^{+}+2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2} \mathrm{O}\) Overall: \(\quad \overline{\mathrm{H}}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}\) A flowchart of a single cell of a PEMFC is shown below. The complete cell would consist of a stack of such cells in series, such as the one shown in Problem 9.19.The cell consists of two gas channels separated by a membrane sandwiched between two flat carbonpaper electrodes- -the anode and the cathode- -that contain imbedded platinum particles. Hydrogen flows into the anode chamber and contacts the anode, where \(\mathrm{H}_{2}\) molecules are catalyzed by the platinum to dissociate and ionize to form hydrogen ions (protons) and electrons. The electrons are conducted throughthe carbon fibers of the anode to an extemal circuit, where they pass to the cathode of the next cell in the stack. The hydrogen ions permeate from the anode through the membrane to the cathode.Humid air is fed into the cathode chamber, and at the cathode \(\mathrm{O}_{2}\) molecules are catalytically split to form oxygen atoms, which combine with the hydrogen ions coming through the membrane and electrons coming from the external circuit to form water. The water desorbs into the cathode gas and is carried out of the cell. The membrane material is a hydrophilic polymer that absorbs water molecules and facilitates the transport of the hydrogen ions from the anode to the cathode. Electrons come from the anode of the cell at one end of the stack and flow through an extemal circuit to drive the device that the fuel cell is powering, while the electrons coming from the device flow back to the cathode at the opposite end of the stack to complete the circuit. is important to keep the water content of the cathode gas between upper and lower limits. If the content reaches a value for which the relative humidity would exceed \(100 \%,\) condensation occurs at the cathode (flooding), and the entering oxygen must diffuse through a liquid water film before it can react. The rate of this diffusion is much lower than the rate of diffusion through the gas film normally adjacent to the cathode, and so the performance of the fuel cell deteriorates. On the other hand, if there is not enough water in the cathode gas (less than \(85 \%\) relative humidity), the membrane dries out and cannot transport hydrogen efficiently, which also leads to reduced performance. 400-sell 300-yolt PEMFS anerates at stady state witha nonwer outnul of 36 k W, The air fod to It is important to keep the water content of the cathode gas between upper and lower limits. If the content reaches a value for which the relative humidity would exceed \(100 \%,\) condensation occurs at the cathode (flooding), and the entering oxygen must diffuse through a liquid water film before it can react. The rate of this diffusion is much lower than the rate of diffusion through the gas film normally adjacent to the cathode, and so the performance of the fuel cell deteriorates. On the other hand, if there is not enough water in the cathode gas (less than \(85 \%\) relative humidity), the membrane dries out and cannot transport hydrogen efficiently, which also leads to reduced performance.A 400-cell 300-volt PEMFC operates at steady state with a power output of 36 kW. The air fed to the cathode side is at \(20.0^{\circ} \mathrm{C}\) and roughly 1.0 atm (absolute) with a relative humidity of \(70.0 \%\) and a volumetric flow rate of \(4.00 \times 10^{3}\) SLPM (standard liters per minute). The gas exits at \(60^{\circ} \mathrm{C}\). (a) Explain in your own words what happens in a single cell of a PEMFC. (b) The stoichiometric hydrogen requirement for a PEMFC is given by \(\left(n_{\mathrm{Hz}}\right)_{\text {conanmad }}=I N / 2 F,\) where \(I\) is the current in amperes (coulomb/s), \(N\) is the number of single cells in the fuel cell stack, and \(F\) is the Faraday constant, 96,485 coulombs of charge per mol of electrons. Derive this expression. (Hint: Recall that since the cells are stacked in series the same current flows through each one, and the same quantity of hydrogen must be consumed in each single cell to produce that current at each anode.) (c) Use the expression of Part (b) to determine the molar rates of oxygen consumed and water generated in the unit with the given specifications, both in units of mol/min. (Remember that power = voltage \(\times\) current.) Then determine the relative humidity of the cathode exit stream, \(h_{\mathrm{r} \text { rout. }}\) (d) Determine the minimum cathode inlet flow rate in SLPM to prevent the fuel cell from flooding ( \(h_{\mathrm{r}, \text { out }}=100 \%\) ) and the maximum flow rate to prevent it from drying \(\left(h_{\mathrm{r}, \text { out }}=85 \%\right)\) .

Step by step solution

01

Understanding the PEMFC

In a single cell of a Proton-Exchange Membrane Fuel Cell (PEMFC), hydrogen gas (H2) enters and is split into protons and electrons. The protons pass through a membrane to reach the cathode, while the electrons are routed externally to form an electric current. At the cathode, oxygen (from the air) combines with the protons and electrons to form water.

02

Deriving the Stoichiometric Hydrogen Requirement

The Faraday’s law of electrolysis states that the amount of chemical change induced in an electrolytic cell is proportional to the amount of electricity passed through the cell. The amount of electricity required to produce a given amount of substance at an electrode during electrolysis (I) is proportional to the amount of substance (nH2) by a constant factor which is the product of the Faraday's constant (F) and the number of cells (N). The hydrogen molecule splits into two electrons at the anode, hence the stoichiometric number of 2 in the equation.

03

Determining Molar Rates and Relative Humidity

Based on the power output of the PEMFC and derived stoichiometric hydrogen requirement, we can calculate the current across the cell. From the current and relations provided in the problem, we can further calculate the molar rates of the oxygen consumed and the water generated. The relative humidity of the stream at the cathode exit can be calculated based on the water content of the stream.

04

Determine minimum and maximum cathode inlet flow rates

The minimum cathode inlet flow rate is determined by the condition where the relative humidity of the exit stream is 100%. Any lower and the cell risks being flooded. The maximum flow rate is determined by the condition where the relative humidity at the outlet is 85%. Any higher and there is a risk of drying out.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electrochemical Processes

In a proton-exchange membrane fuel cell (PEMFC), the underlying process is electrochemical in nature. These fuel cells use hydrogen as the primary fuel and oxygen from the air, combining them to produce electricity, water, and heat. The beauty of this setup is in its efficiency and the relatively clean energy output.

In a single cell of a PEMFC, hydrogen gas enters the anode. Here, it undergoes a reaction facilitated by a catalyst, typically platinum, which splits the hydrogen molecules into protons and electrons. This reaction can be represented by the equation:\[\mathrm{H}_{2} \rightarrow 2 \mathrm{H}^{+} + 2 \mathrm{e}^{-}\]

The protons, now free, travel through the proton-exchange membrane to reach the cathode. Meanwhile, the electrons take an alternative route, traveling through an external circuit. This movement of electrons generates an electric current, which can be harnessed to power devices.

At the cathode, oxygen from the air participates in the reaction with the protons and electrons to form water, the benign byproduct of this electrochemical process:\[\frac{1}{2} \mathrm{O}_{2} + 2 \mathrm{H}^{+} + 2 \mathrm{e}^{-} \rightarrow \mathrm{H}_{2} \mathrm{O}\]

The overall reaction in the fuel cell is the combination of these two processes:\[\mathrm{H}_{2} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}\]

Stoichiometry in Fuel Cells

Stoichiometry is vital in understanding and determining the exact amounts of reactants and products in chemical reactions, especially in fuel cells. In the context of a PEMFC, the stoichiometric hydrogen requirement is critical to ensure the fuel cell operates efficiently.

The stoichiometric relationship tells us about the ratio in which hydrogen must be supplied relative to the current drawn. This requirement is derived as:\[(n_{\mathrm{H}_2})_{\text{consumed}} = \frac{I N}{2 F}\]

Here, \(I\) is the current (in amperes) flowing through each cell. It's important to note that since the cells are in series, the same current flows through every single cell. \(N\) is the number of cells in the stack, and \(F\) is Faraday’s constant, roughly 96,485 coulombs per mole of electrons.

This equation shows us that for each coulomb of charge flowing through the system, specific amounts of hydrogen are consumed. It's crucial for the reaction to balance perfectly in order to maintain efficient operation of the PEMFC.

Faraday's Law of Electrolysis

Faraday's Law of Electrolysis is a principle that links the amount of electrical charge flowing through an electrolyte to the amount of substance that undergoes electrochemical transformation. This is incredibly significant in fuel cells since it underpins the concept of converting chemical energy into electrical energy effectively.

According to Faraday's Law, the mass of a substance altered at an electrode during electrolysis is directly proportional to the total electric charge passed through the substance. For PEMFCs, this means that the amount of hydrogen consumed or the oxygen used is directly proportional to the electrical output of the fuel cell.

The law is quantitatively given by:\[m = \frac{Q}{F} \cdot (\text{equivalent weight})\]
Where:

• \(m\) is the mass of the substance (in grams) produced or consumed.
• \(Q\) is the total electric charge (in coulombs) that passed through the substance.
• \(F\) is Faraday's constant (approximately 96,485 coulombs per mole).

For hydrogen in a PEMFC, this principle simplifies to relating the flow of current to hydrogen consumption, emphasizing efficiency and effectiveness in converting fuel to electricity. Without understanding Faraday’s laws, optimizing these cells would be far more challenging.