5.4 Using Equation 5.10, calculate the limiting current density for a fuel cell cathode running on air at STP. Assume only O2 and N2 and ignore the presence of water vapor. Assume that the diffusion layer is 500 µm thick and has a porosity of 40%.
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![### Equation 5.10
The equation shown is:
\[ j_L = nF D_{\text{eff}} \frac{c_R^0}{\delta} \]
#### Explanation of Terms:
- \( j_L \) is the limiting current density.
- \( n \) is the number of electrons involved in the reaction.
- \( F \) is Faraday's constant, representing the charge of one mole of electrons.
- \( D_{\text{eff}} \) is the effective diffusion coefficient.
- \( c_R^0 \) is the initial concentration of the reactant.
- \( \delta \) is the diffusion layer thickness.
This equation is often used in electrochemistry and relates the limiting current density to the diffusion of reactants in an electrochemical cell. The limiting current is determined by the rate at which the reactant can be supplied to the electrode surface, which is governed by factors like the diffusion coefficient and the concentration gradient.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe801909c-59c9-4b6d-8f92-02e1e3a7b70b%2Fc2f32d3d-ba91-4be0-a44d-60cad91c17ac%2Fv4gg3sk_processed.png&w=3840&q=75)
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