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=Mass Transport via Molecular Diffusion for Fuel Cell Application=

Introduction to Fuel Cell Technology


Fuel cells are a clean energy source that holds great promise for displacing conventional energy systems toward a cleaner, more efficient future. Similar to a battery, fuel cells produce DC current from an electrochemical reaction that takes place between hydrogen and oxygen via a polymer membrane electrolyte.

The basic structure of a PEM (polymer electrolyte membrane) fuel cell is given in Figure 1. It consists of a polymer electrolyte membrane sandwiched between two gas diffusion layers, known as the anode and the cathode. The electrochemical reaction takes place with the aid of two catalyst layers, located at the anode/electrolyte and cathode/electrolyte interfaces, respectively. The chemical reactions for a PEM fuel cell are given as follows :

Anode RXN: $$H_2 \Rightarrow 2H^+ + 2e^-$$

Cathode RXN: $$\frac{1}{2}O_2 + 2H^+ + 2e^- \Rightarrow 2H_20$$

Overall RXN: $$ H_2 + \frac{1}{2}O_2 \Rightarrow H_20$$

As shown in the above equations, fuel cells require both hydrogen and oxygen to enable the reactions at the catalyst layers. The product of the electrochemical reaction is simply water (and heat).

In a fuel cell system, the reactant (fuel & oxygen) flows for the fuel cell are closely monitored. Depending on the performance set point, information is needed about the required flow rates, the amount of storage needed to hold the reactants, as well as the amount of storage needed to hold the reaction product water.

Mass transport of reactants through the anode/cathode layers to the catalyst layers (reaction zones) is governed by molecular (mass) diffusion. Consideration of mass diffusion is presented here in the form of 1) a detailed derivation of the governing partial differential equation using a control volume approach, 2) a discussion about how mass diffusion is similar to the heat conduction, and, 3) an illustrative example using gas diffusion through the electrode layers of a PEM fuel cell.

It is hoped that reader will gain a deeper understanding of the transport processes within a fuel cell as well as an appreciation of the mathematics that govern this real-world physical process.

Derivation of Mass Diffusion Partial Differential Equation


The foundation of the molecular mass diffusion is that the flux, F, of a substance is related to the change in concentration, C, over the change is distance, x, via a constant of proportionality known as the diffusion coefficient, D. An expression of this is given as follows:


 * $$F = -D \frac{\partial C}{\partial x}$$

Note that the flux is given as the mass transfer of a substance per unit time per unit area.

Consider a control volume of an isotropic medium as shown in the Fig. 1. Isotropic materials have the same properties in all orientations. The control volume features a rectangular box that is parallel to the coordinate axis. The lengths for the box are given as:


 * $$AA' = BB' = DD' = CC' = 2dx$$


 * $$AD = BC = A'D' = B'C' = 2dy$$


 * $$AB = DC = A'B' = D'C' = 2dz$$

The point P = P(x,y,z) represents the center of the control volume.

If the ABCD and A'B'C'D' faces are perpendicular to the x-axis, then the flux of a substance entering the ABCD face is given as:


 * $$4dydz \left( F_x - \frac{\partial F_x}{\partial x} dx\right)$$

Respectively, the flux exiting the c.v. from the A'B'C'D' face is given as:


 * $$4dydz \left( F_x + \frac{\partial F_x}{\partial x} dx\right)$$

The rate of increase of the species between the ABCD and A'B'C'D' faces is given as:


 * $$-8dxdydz\frac{\partial F_x}{\partial x}$$

The other faces can be treated the same way, therefore:


 * $$-8dxdydz\frac{\partial F_y}{\partial y}$$, and $$-8dxdydz\frac{\partial F_z}{\partial z}$$

Lastly, the increase in the rate of the diffusing substance is given as:


 * $$8dxdydz\frac{\partial C}{\partial t}$$

The mass conservation principle that governs unsteady diffusion can be stated as :

(net rate of diffusion of species A into medium) = (rate of increase of species A in medium)

Following this principle, then the following expression results:

$$\frac{\partial C}{\partial t} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 0$$

Substituting the above expression for the flux, F, as a function of the concentration, yields:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} + \frac{\partial^2 C}{\partial z^2} \right)$$

In this equation, the diffusion coefficient is constant. For the 1D case, the equation reduces to:

$$\frac{\partial C}{\partial t} = D \left( \frac{\partial^2 C}{\partial x^2} \right)$$

This 1D formulation for mass transport via diffusion is the governing partial differential equation that will be explored deeper for the fuel cell application example that follows below.

source: J. Crank, The Mathematics of Diffusion, Oxford University Press, New York, 1956.

Diffusion Thoery: Historical Background ,

The 1D mass diffusion equation given above was first proposed by Adolf Eugene Fick in 1855. He based his derivation on the work of Jean Baptiste Fourier who created the landmark studies of heat conduction. The mass diffusion equation is analogous to the heat conduction equation which is of the form:

$$\alpha \frac{\partial^2 T}{\partial x^2} = \frac{\partial T}{\partial t}$$

where T is the temperature, the constant of proportionality $$\alpha$$ is the thermal diffusivity.

An Illustrative Example: Gas Diffusion through electrode layer of a fuel cell


This section will attempt to utilize the mathematical theory presented above to model a real-world application of hydrogen gas diffusion through the anode layer of a PEM fuel cell.

Figure 3 shows a schematic diagram of the anode layer along with the respective coordinate system that will be used in the calculations. The anode layer is treated as a 1D slad of thickness L. The boundary conditions are such that there is a known concentration, C1, of hydrogen on the bulk flow side of the anode (x = 0). The concentration, C2, represents the hydrogen concentration at the anode/catalyst interface (x = L). The anode layer has an initial concentration equal to some unknown function given by f(x).

The following is a derivation of the hydrogen gas concentration, C(x,t), as a function of space and time given the boundary conditions listed here.

Problem Statement (summary)
$$\frac{\partial^2 C}{\partial x^2} = \frac{1}{D} \frac{\partial C}{\partial t}$$

(initial condition)

IC: $$C(x,t=0) = f(x)$$, at $$t = 0$$

(boundary conditions)

BC1: $$C(x = 0,t) = C_1$$

BC2: $$C(x = L,t) = C_2$$

Splitting a Non-Homogenous Problem into a Steady-state and Homogeneous Problem
The problem as posed is non-homogeneous and must be split into two simpler problems in order to obtain an analytical solution. The first problem will be a steady-state problem that results in a simple 2nd order ODE with respect to the space variable, x. The second problem will be a homogeneous problem that results in a PDE with only the initial condition as a non-homogeneity.

Details of the two sub-problems are given below.

Steady-state Solution
$$\frac{\partial^2 C_ss}{\partial x^2} = 0$$

BC1: $$C_{ss}(x = 0) = C_1$$

BC2: $$C_{ss}(x = L) = C_2$$

Integrating once, yields:

$$\frac{\partial C_{ss}}{\partial x} = A$$ where A is an integration constant

A second integration yields yields the auxillary solution for the steady-state condition:

$$C_{ss}(x) = Ax + B$$

The constants of integration, A & B, can be determined by applying the boundary conditions as follows:

$$\blacktriangleright C_{ss}(x = 0) = C_1$$


 * $$C_{ss} = C_2 = A(0) + B$$


 * $$\therefore B = C_1$$

$$\blacktriangleright C_{ss}(x = L) = C_2$$


 * $$C_{ss} = C_2 = A(L) + C_1$$


 * $$\therefore A = \frac{C_2 - C_1}{L}$$

The steady-state solution is then

$$C_{ss}(x)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1$$

Homogeneous Solution using Separation of Variables
$$\frac{\partial^2 C_h}{\partial x^2} = \frac{1}{D} \frac{\partial C_h}{\partial t}$$

IC: $$C_h(x, t = 0) = f(x) - C_s(x) = f^*(x)$$

BC1: $$C_h(x = 0, t >0) = 0$$

BC2: $$C_h(x = L, t >0) = 0$$

Separation of Variables
Using separation of variables method, the concentration can be expressed as the product of two functions that each have a single dependent variable:

$$C_h(x,t) = X(x)\Gamma(t)$$

Substituting this expression into the homogeneous PDE, yields:

$$\frac{X''}{X} = \frac{1}{D\Gamma} \frac{d\Gamma}{dt} = -\lambda^2$$

The greek lambda in the above equation represents a constant value. The right hand side and left hand side of the the expression are both functions of single dependent variables (x and t, respectively). This can only be valid if they equal a constant. The sign of the constant lambda term has been chosen as negative in order to force the boundary value problem in the x-direction (finite).

Auxiliary solutions for each variable (x & t) are determined as follows:

$$\blacktriangleright $$Space variable, X


 * $$X''+\lambda^2 X = 0$$
 * The solution of this equation is of the form:$$X(x) = G cos(\lambda x) + H sin(\lambda x)$$
 * Applying the boundary condition at x=0, yields:
 * $$G(1) + H (0) = 0$$, therefore $$G = 0$$

$$\therefore X(x) = H sin(\lambda x)$$
 * Applying the second boundary condition at x = L
 * $$0 = H sin(\lambda x)$$

$$\therefore \lambda_n = \frac{n\pi}{L}$$ where n = 0,1,2,...

$$\blacktriangleright $$Time variable, t
 * $$\frac{d\Gamma}{\Gamma} = -\lambda^2dt$$
 * $$ln\Gamma = -\lambda^2 D t + E^*$$ where E is a constant of integration

$$\therefore \Gamma(t) = Eexp(-\lambda^2 D t)$$

The homogeneous solution is given by merging both solutions for the X and t variables and summing over all possible solutions for $$\lambda_n$$. The homogenous solution is:

$$C_h(x,t) = \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Principle of Orthogonality
The first step in finding the coefficient H is to apply the initial condition to the auxiliary homogeneous solution which states:

$$C_h(x, t=0) = f^*(x)$$

$$\therefore f^*(x) = \sum_{n=1}^\infty H sin(\lambda_n x) \cdot (1) = \sum_{n=1}^\infty H sin(\lambda_n x) $$

Observing the orthogonality of the sin function, both sides of this equation can be operated by the following expression:

$$\int_{0}^{L}sin(\lambda_n x)dx$$

Solving for the coefficient, H, yields:

$$H = \frac{\int_{0}^{L}f^*(x)sin(\lambda_n x)dx}{\int_{0}^{L}\left[sin(\lambda_n x) \right]^2 dx}$$

where$$f^*(x) = f(x) - C_{ss}(x)$$ ,or, $$f^*(x) = f(x) - \left[ \frac{C_2 - C_1}{L} \right] x - C_1$$

The values for $$H$$and $$f^*$$ complete the homogeneous solution given above.

Final Solution
The final solution is found by summing the steady-state and the homogeneous solutions

$$C(x,t) = C_{ss}(x) + C_h(x,t)$$ $$\therefore C(x,t)= \left[ \frac{C_2 - C_1}{L} \right] x + C_1 + \sum_{n=1}^\infty H sin(\lambda_n x) \left[ exp(-\lambda_n^2 D t)\right]$$

Substituting the respective values for $$H$$ is given above (see Principle of Orthogonality) as well as $$f^*$$, then grouping like terms yields :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) - C_1 - (C_2-C_1) \frac{x}{L}\right]$$

The final solution is found by integrating this expression, resulting in the following equation :

$$C(x,t) = C_1 + (C_2 - C_1) \frac{x}{L} + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)sin(\lambda_n x)\int_{0}^{L} \left[ f(x) sin(\lambda_n x) dx\right] + \frac{2}{L}\sum_{n=1}^\infty exp(-\lambda_n^2 Dt)\frac{1}{\lambda_n}sin(\lambda_n x) \left[ T_2 cos(n\pi) - T_1\right]$$

Simulation using MathCAD


The final solution to the given problem was simulated using MathCAD. The following parameters were used:

Electrode diffusion layer thickness: L = 0.005;

Diffusion Coefficient: D = 2*10^(-9);

Bulk Flow Channel Concentration: C1 = 100;

Catalyst Layer Concentration: C2 = 0;

Initial Concentration: f = 0;

Figure 4 shows a data plot of the resulting hydrogen concentration profiles throughout the thickness of the anode layer for various time elapse conditions based on the boundary (and initial) conditions listed above. It illustrates just how the hydrogen gas migrates through the anode thickness over time. As shown in the the data plot, the hydrogen concentration converges to a linear profile at large time scales.

Conclusion
Fuel cells hold great promise for a clean energy future. Fuel cells consist of multiple layers including anode/cathode, electrolyte, and catalyst layers. Mass transport within the various layers of a fuel cell is governed by Fick's equation for mass diffusion which is derived using a control volume approach for an isotropic medium.

Fick's law is analogous to the heat diffusion equation given by Fourier. Both process of mass transport and heat conduction rely on random molecular interactions. Respectively, the mathematical formulation and derivation is similar between the two physical phenomena.

A real-world application of Fickian diffusion is the gas transport of hydrogen fuel from the bulk flow stream through the anode diffusion layer to the catalyst reaction cite at the anode/electrolyte interface. Though a relatively simple case was presented here, it should be noted that mass transport for fuel cells is a very complex topic. Levels of model complexity increases with consideration of the multi-phase liquid/gas reactant streams, non-constant diffusion parameters, changing material properties from layer to layer, chemical reactions (chemical kinetics), and the interaction of reactants with the material layers.

It is a truly exciting topic!