Rigorous mathematical modeling of the process dynamics associated with the construction and filling of density gradient columns is presented in this article. These models incorporate the hydrostatic driving forces for fluid flow, friction losses associated with this flow, and the unsteady-state behavior of the liquid levels in the filling vessels and in the column itself. Four different filling arrangements are considered, corresponding to the density order of the two fluids in the filling vessels and two methods for introducing the fluid of varying density into the column. Time requirements for filling of the column and the resulting calibration curve for liquid density versus height in the gradient column are both obtained as a result of this modeling procedure. Further, extremely important operating guidelines for the final achievement of a linear density gradient in the column, which is normally the desired objective in most laboratory applications, are derived and presented. Conversely, the causes leading to nonlinear gradients are elucidated and quantified.

S olid-polymer-electrolyte fuel cells have received much attention recently because of dramatic improvements in the performance of single laboratory cells and the subsequent potential for application in both power generation and transportation. Mathematical models of the fuel cells that include several simultaneous transport processes would be useful in future design and scale up.Detailed one-dimensional models of solid-polymer-electrolyte fuel cells that include the transport of both water and dissolved hydrogen and oxygen across the cell are those of Verbrugge (1991, 1992), Springer et al. (1991), andEikerling et al. (1998). In the former, the membrane was assumed to be uniformly hydrated with the constant transport properties, while the latter two references focused on the effects of non-uniform membrane water content on mass transfer. Two-dimensional models are those of Newman (1993a, 1993b) and Yi and Nguyen (1998) who used concentrated solution theory and the work of Nguyen and White (1991) who adapted the results of Springer et al. (1991) to the membrane transport where these studies include heat and mass transfer in the fl ow channels. The latter work was extended to provide the computational framework for the design of fuel cell stacks (1997 Finite element analyses for multi-dimensional effects have been developed by Lee and Lalk (1998) to include thermal properties in fuel cell stacks and by Futerko and Hsing (2000) to include the effects of temperature and pressure on membrane water content and current density. All of the work above including limitations due to membrane and catalyst degradation by impurity ions is summarized by Okada (2001).The memory and computational requirements for the design of fuel cell systems can be great, particularly if the onedimensional model of mass transport across the cell require simultaneous solution of coupled ordinary differential equations. The purpose of the present paper is to illustrate an analytical procedure that would eliminate the latter requirement. A rigorous mathematical model for an ion-exchange membrane attached to a gas-fed porous diffusion electrode was published some time ago by Bernardi and Verbrugge (1991). In particular, the model was applied to simulate the oxygen electrode of a solid-polymer-electrolyte fuel cell, with emphasis on representing cell polarization characteristics, water transport and platinum A simplifi ed mathematical model for an ion-exchange membrane attached to a gas-fed porous electrode is developed to simulate the oxygen electrode of a solid-polymer-electrolyte fuel cell. In particular, the present model is derived from an earlier rigorous one of Bernardi and Verbrugge(1991) by neglecting the Peclet number for the transport of dissolved oxygen within the catalyst. The advantage of this simpler model is that it can be solved analytically, eliminating the need for numerical simulation. Longitudinal profi les for the dissolved oxygen concentration and catalyst current density calculated from the present model are in good agr...

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