MIT10_626S11_lec18

# MIT10_626S11_lec18 - IV Transport Phenomena Lecture 18...

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Unformatted text preview: IV. Transport Phenomena Lecture 18: Forced Convection in Fuel Cells II MIT Student (and MZB) As discussed in the previous lecture, we are interested in forcing fluid to flow in a fuel cell in order to increase the limiting current, and thus the power output of the device. The model problem considered has a steady uniform flow of fluid carrying the reactant at a constant concentration, c , into a 2-D channel of height H . In this model we will assume that the fluid velocity profile across the channel is uniform (i.e. plug flow). We are interested in analyzing the transport of the reactant carried into the system by the convective fluid stream to the active membrane located at y = 0, which has a length of L . y Uniform Flow c = c 0 δ (x) Diffusion Layer Wall Membrane c(y=0) = 0 x = 0 x = L x y = H + x = l 1 Entrance Length x = l 2 Figure 1: Model problem of forced convection in a fuel cell From conservation of mass, there is a balance between convection and diffusion terms: [1] Since the flow is steady, the transient term can be neglected. In the region when x >> l 1 , convection dominates over axial diffusion: [2] From scaling analysis, we can define l 1 as the point where convection occurs at the same scale as axial diffusion. That is: [3] Nondimensionalizing the position variables as follows leads us to the Péclet Number, Pe . Let: [4] In the region where x >> l 1 or , we can neglect axial diffusion and define a “time” variable t = x/U along the streamlines to obtain a 1D diffusion equation in terms of y only. This approach is valid because the fluid flow rate is invariant with location in our model. It is worth restating that the Péclet Number is a dimensionless value that represents the ratio of the rate of convection to the rate of diffusion in the system. We can redefine Equation [1] as: [5] and the “initial condition” at the inlet becomes: c ( x = 0, y ) = c ( t = 0, y ) = c . We are now ready to calculate the limiting current I lim as a function of the fluid velocity, U , under the assumption that reactions are fast enough everywhere for c ( t, y = 0) → 0 along the entire membrane when I → I lim . The problem has now become: y Wall y = H c = c 0 Membrane c ( y=0 ) = 0 x = 0 x = L t = x / U Figure 2: 1D diffusion model for forced convection in a fuel cell Lecture 18: Forced convection in fuel cells (II) 10.626 (2011) Bazant The limiting current is found by integrating the transverse mass flux along the membrane: [6] Below we will obtain an exact solution as a Fourier Series , but it is more accurate and easier...
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MIT10_626S11_lec18 - IV Transport Phenomena Lecture 18...

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