Water Transport Equation

The water transport equation in gas flow chaimels and in electrode gas diffusion layer is similar to the gas species transport equations presented in Chapter 6, and it is given as follows  

A typical water concentration distribution across a PEM fuel cell. 

It is evident that the pickup of water is higher on the cathode side channel compared to the anode side channel. 

Bemardi, D. M. and M. W. Verbmgge. Mathematical model of a gas diffusion electrode bonded to polymer electrolyte. AIChE Journal 37 1151-1163,1991. 

Fergus, J. W., R. Hui, X. Li, W. R Wilkinson and J. Zhang. Solid Oxide Fuel Cells— Materials Properties and Performance. CRC Press, 2009. 

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S. Loeb, G.D. Mehta, A two-coefficient water transport equation for pressure retarded osmosis, Journal of Membrane Science 1978,4, 351-362. 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

Salt flux across a membrane is due to effects coupled to water transport, usually negligible, and diffusion across the membrane. Eq. (22-60) describes the basic diffusion equation for solute passage. It is independent of pressure, so as AP — AH 0, rejection 0. This important factor is due to the kinetic nature of the separation. Salt passage through the membrane is concentration dependent. Water passage is dependent on P — H. Therefore, when the membrane is operating near the osmotic pressure of the feed, the salt passage is not diluted by much permeate water. 

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

In addition to material balance, two transport equations can be used to predict the flux of water and solute. For instance, the following simplified model can be used (Dandavati etai, 1975 Evangelista, 1986). 

An important assumption was that the solution was dilute (in this case natural water of approximately lOOp.p.m. total dissolved solids) since there are difficulties in applying mass transport equations for certain situations in concentrated electrolyte solution, where a knowledge of activities is uncertain and this can lead to large errors. 

Introducing the values into the equation, using a minimum Kd-value of >300, gives a retention factor of >750. If this value is combined with a representative water transport time from repository to recipient (>1000 years for a distance >100 m), the transport equation indicates that it will take the plutonium >750,000 years to reach the recipient which is the water man may use. This estimate is supported by findings at the ancient natural reactor site at Oklo in Gabon (67). 

A general conclusion from the review of the distribution of plutonium between different compartments of the ecosystem was that the enrichment of plutonium from water to food was fairly well compensated for by man s metabolic discrimination against plutonium. Therefore, under the conditions described above, it may be concluded that plutonium from a nuclear waste repository in deep granite bedrock is not likely to reach man in concentrations exceeding permissible levels. However, considering the uncertainties in the input equilibrium constants, the site-specific Kd-values and the very approximate transport equation, the effects of the decay products, etc. — as well as the crude assumptions in the above example — extensive research efforts are needed before the safety of a nuclear waste repository can be scientifically proven. 

It is probable that capillary flow of water contributes to transport in the soil. For example, a rate of 7 cm/year would yield an equivalent water velocity of 8 x 10-6 m/h, which exceeds the water diffusion rate by a factor of four. For illustrative purposes we thus select a water transport velocity or coefficient U6 in the soil of 10 x 10 6 m/h, recognizing that this will vary with rainfall characteristics and soil type. These soil processes are in parallel with boundary layer diffusion in series, so the final equations are... 

A major problem in the quantification of air-water transport phenomena in terms of the rate expression [Equation (4.18)] is to find appropriate values for Kl. As far as sewer systems are concerned, the most well-established knowledge concerning air-water mass transfer is on reaeration (Section 4.4). 

Neglecting the movement of water relative to the surrounding sediment, we write the steady-state transport equation in one dimension with burial, e.g., in a medium... 

In this equation, C is the concentration of element i in pore water at depth z below the seafloor and A is a reaction (sink and source) term. For reactions involving the oxidation of organic matter, A can be evaluated independently. For constant porosity , the sulfate transport equation becomes... 

Quantitative estimation of ventilation by indirect methods in mussels requires four assumptions (16) a) reduction of concentration results from uptake, b) constant ventilation (pumping) rate, c) uptake of a constant percentage of concentration (first order process), d) homogeneity of the test solution at all times. Our transport studies have utilized antipy-rine (22, 23) a water soluble, stable chemical of low acute toxicity to mussels. It is readily dissolved in ocean water or Instant Ocean and is neither adsorbed nor volatilized from the 300 ml test system. Mussels pump throughout the 4 hour test period and this action is apparently sufficient to insure homogeneity of the solution. Inspection of early uptake and elimination curves (antipyrine concentration as a function of time) prompted use of Coughlan s equation (16) for water transport. 

After the tests, Djilali s group used mathematical assumptions and equations to correlate the intensity of the dye in the image with the depth in the gas diffusion layer. With this method they were able to study the effect of compression on diffusion layers and how fhaf affects water transport. Water removal in a flow charmel has also been probed with this technique and it was observed that, with a dry DL slug, formation and flooding in the FF channels followed the appearance and detachment of water droplets from the DL. Even though this is an ex situ technique, it provides important insight into water transport mechanisms with different DLs and locations. 

It is indicated that these transport parameters are functions of A. The hydraulic permeability (D Arcy coefficient), /Cp (A), exhibits strong dependence on A because larger water contents result in an increased number of pores used for water transport and better connectivity in the porous network, as well as in larger mean radii of these pores. A modification of the Hagen-Poiseuille-Kozeny equation was considered by Eikerling et aU- to account for these structural effects ... 

With the increased computational power of today s computers, more detailed simulations are possible. Thus, complex equations such as the Navier—Stokes equation can be solved in multiple dimensions, yielding accurate descriptions of such phenomena as heat and mass transfer and fluid and two-phase flow throughout the fuel cell. The type of models that do this analysis are based on a finite-element framework and are termed CFD models. CFD models are widely available through commercial packages, some of which include an electrochemistry module. As mentioned above, almost all of the CFD models are based on the Bernardi and Verbrugge model. That is to say that the incorporated electrochemical effects stem from their equations, such as their kinetic source terms in the catalyst layers and the use of Schlogl s equation for water transport in the membrane. 

The addition of the gas and water mass balances (eq 39 with eq 23) along with the above transport equation (eq 46) and constitutive relationships com-... 

Um et al. also examined a transient using their complex model. They saw that in a matter of tens of seconds the current density response reached steady state after a change in potential. However, their model did not include liquid water. The most complex model to examine transients is that of Natarajan and Nguyen. It should be noted that although the model of Bevers et al. has transient equations, they do not report any transient results. Natarajan and Nguyen included liquid saturation effects and water transport in their model. They clearly showed the flooding of the diffusion media and that it takes on the order of a couple of minutes for the profiles to develop. 

Six coupled governing equations listed in Table 1 are valid in all regions of a PEFC, and fluxes at an internal boundary between two adjacent regions are automatically continuous. Such a single-domain model is well suited for CFD implementation. In contrast, multidomain models, such as the one developed by Dutta et al., compute separate solutions for the anode and cathode subdomains, respectively, and then patch the two solutions through the water transport flux across the MEA interface. Numerically, this model is characterized as a solver-in-solver situation. 

How do we handle sorption in our transport equation For particles that are not transported with the flow field, like sediments and groundwater flow, we are interested in the water concentrations. The sorbed portion of the compound is not in the solute phase and should not be considered in the transport equation, except when transfer of the compound between the water and particles occur. Adsorption would then be a sink of the compound, and desorption would be a source. 

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