Las Celdas De Combustible

DERIVATION OF A GENERAL MASS

TRANSFER EQUATION

In this section, we discuss the general partial differential equations governing mass transfer;

these will be used frequently in subsequent chapters for the derivation of equations

appropriate to different electrochemical techniques. As discussed in Section 1.4, mass

transfer in solution occurs by diffusion, migration, and convection. Diffusion and migration

result from a gradient in electrochemical potential, JL. Convection results from an imbalance

of forces on the solution.

Consider an infinitesimal element of solution (Figure 4.1.1) connecting two points in

the solution, r and s, where, for a certain species j , Jip) Ф ^(s). This difference of ^

over a distance (a gradient of electrochemical potential) can arise because there is a difference

of concentration (or activity) of species у (a concentration gradient), or because there

is a difference of ф (an electric field or potential gradient). In general, a flux of species j

will occur to alleviate any difference of /Zj. The flux, Jj (mol s^cm"2), is proportional to

the gradient of /xj:

Jj oc grad^uj or Jj oc V)itj (4.1.1)

where grad or V is a vector operator. For linear (one-dimensional) mass transfer, V =

i(d/dx)9 where i is the unit vector along the axis and x is distance. For mass transfer in a

three-dimensional Cartesian space,

V = i | - + j | - + k | -

The minus sign arises in these equations because the direction of the flux opposes the direction

of increasing ~jl-

If, in addition to this ~jx gradient, the solution is moving, so that an element of solution

[with a concentration C}(s)\ shifts from s with a velocity v, then an additional term is

added to the flux equation:

In this chapter, we are concerned with systems in which convection is absent. Convective

mass transfer will be treated in Chapter 9. Under quiescent conditions, that is, in

an unstirred or stagnant solution with no density gradients, the solution velocity, v, is

zero, and the general flux equation for species j , (4.1.9), becomes

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