# Question: Fuel Cell Thermodynamic Equations

• Scorael
In summary, the equations for the partial pressure of hydrogen and oxygen in the fuel cell system and for water diffusivity are derived from Dalton's Law and the Arrhenius equation, respectively. The numbers in the equations are constants and coefficients that are obtained through experimental data and mathematical calculations.
Scorael
I'm an undergrad working on fuel cells for a project and I have been using a book to aid me. There are several equations in the book that were not explained properly so I was hoping someone on this forum could help identify how they were derived.

The first 2 is related to the pressure within the system of the fuel cell. Hydrogen is pumped in 1 end and air in the other and their partial pressure is found using these formulas:

$$p_{H2}=0.5*\frac{P_{H2}}{exp(1.653*i/T^{1.334})}-P_{H2O}$$

$$p_{O2}=\frac{P_{air}}{exp(4.192*i/T^{1.334})}-P_{H2O}$$

From what I gather, this appears to be Dalton's Law but I am not sure how did the numbers 1.653, 4.192 and 1.334 pop out from.

The next one is water diffusivity which the author defined as:
$$D_{\lambda}=10^{-6}exp(2416(\frac{1}{303}-\frac{1}{353}))*2.563-0.33*10+0.0264*10^{2}-0.000671*10^{3}$$

For this one I am not sure what is the significance of 10 in this equation such that is appears 3 times in the equation without being simplified into a constant.

I'm studying electrical engineering so when it comes to thermodynamics this is way pass me. I hope someone on this forum will be able to help out.

Dear fellow scientist,

I can understand your frustration with the lack of explanation in the book. Let me try to shed some light on the equations for you.

The first two equations are indeed based on Dalton's Law, which states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of each individual gas. In this case, we are looking at the partial pressure of hydrogen and oxygen in the fuel cell system.

The numbers 1.653 and 4.192 are actually constants that are derived from the ideal gas law and the van der Waals equation, which take into account the non-ideal behavior of gases at high pressures. The number 1.334 is a coefficient that is used to account for the effect of temperature on the gases. These numbers are obtained through experimental data and mathematical calculations.

As for the equation for water diffusivity, the number 10 is simply a unit conversion factor. The equation is using the Arrhenius equation, which relates the rate of a chemical reaction to temperature. The number 2416 is the activation energy for water diffusion, and the remaining terms are constants and coefficients that are derived from experimental data.

I hope this helps clarify the origins of these equations for you. Keep up the good work on your project!

Hello,

I can provide some insight into the equations you have mentioned. The first two equations you have provided are indeed related to Dalton's Law, which states that the total pressure of a mixture of gases is equal to the sum of the partial pressures of each individual gas. In this case, the partial pressures of hydrogen and oxygen are being calculated based on their respective partial pressures in the fuel cell system.

The numbers 1.653, 4.192, and 1.334 represent constants that are specific to the system being studied. These values are derived from experimental data and are used to accurately calculate the partial pressures of hydrogen and oxygen. They may vary depending on factors such as temperature, pressure, and composition of the gas mixture. It is important to note that these values are not arbitrary and have been determined through scientific research.

Moving on to the equation for water diffusivity, the number 10 is not a constant in this equation, but rather it is being used as a multiplier for the exponential term. The exponential term is a common way to represent the effect of temperature on a physical property. In this case, the temperature difference between 303 K and 353 K is being used to calculate the diffusivity of water. The other terms in the equation, such as 2.563 and 0.33, are also derived from experimental data and represent the contribution of other factors to the overall diffusivity.

I understand that these equations may seem complex and difficult to understand, but they are based on scientific principles and have been extensively tested and validated through experiments. I would recommend consulting with your professor or a thermodynamics expert for further clarification and understanding. Additionally, there are many resources available online that can help explain these equations in more detail. I wish you the best of luck with your project.

## 1. What is a fuel cell?

A fuel cell is a device that converts the energy from a chemical reaction into electricity. It consists of two electrodes, an anode and a cathode, with an electrolyte in between. Fuel cells are commonly used as an alternative source of energy to traditional combustion engines.

## 2. What are thermodynamic equations?

Thermodynamic equations are mathematical expressions that describe the relationship between temperature, energy, and other thermodynamic properties. These equations are used to calculate the efficiency and performance of a fuel cell.

## 3. How do thermodynamic equations apply to fuel cells?

Thermodynamic equations are used to analyze the energy conversion process that occurs in a fuel cell. These equations help determine the amount of energy that can be produced, as well as the efficiency and limitations of the fuel cell.

## 4. What are the most important thermodynamic equations for fuel cells?

The most important thermodynamic equations for fuel cells include the Nernst equation, which calculates the cell voltage, and the Gibbs free energy equation, which determines the energy available for the reaction. Other important equations include the enthalpy and entropy equations, which describe the heat and disorder of the system.

## 5. How are thermodynamic equations used in fuel cell research?

Thermodynamic equations are used in fuel cell research to understand the fundamental principles behind fuel cell operation and to optimize their performance. These equations can also be used to predict the behavior of different types of fuel cells and to design more efficient and cost-effective systems.

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