Mass transfer flux, steady state

The ferrous Fe+ ions are not included in the analysis since the crevice cathodic protection is for preventing the formation of this t5 e ion. For a coupled diffusion and migration molar flux under steady-state conditions, dCjIddt = 0, the molar flux becomes the continuity equation for mass transfer under steady state conditions... 

SimpleBox is a multimedia mass balance model of the so-called Mackay type. It represents the environment as a series of well-mixed boxes of air, water, sediment, soil, and vegetation (compartments). Calculations start with user-specified emission fluxes into the compartments. Intermedia mass transfer fluxes and degradation fluxes are calculated by the model on the basis of user-specified mass transfer coefficients and degradation rate constants. The model performs a simultaneous mass balance calculation for all the compartments, and produces steady-state concentrations in... 

At steady state, both the mass transfer flux and the heat transfer flux are balanced according to... 

The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. 

In this case of transient difiiision as compared to the steady-state film model, the mass transfer flux is piopoftional to the square root of the difftisivity rather than the first power. 

The fllm theory is the simplest model for interfacial mass transfer. In this case it is assumed that a stagnant fllm exists near the interface and that all resistance to the mass transfer resides in this fllm. The concentration differences occur in this film region only, whereas the rest of the bulk phase is perfectly mixed. The concentration at the depth I from the interface is equal to the bulk concentration. The mass transfer flux is thus assumed to be caused by molecular diffusion through a stagnant fllm essentially in the direction normal to the interface. It is further assumed that the interface has reached a state of thermodynamic equilibrium. The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. Consider a gas-liquid interface, as sketched in Fig. 5.16. The mathematical problem is to formulate and solve the diffusion flux equations determining the fluxes on both sides of the interface within the two films. The resulting concentration profiles and flux equations can be expressed as ... 

A proper resolution of Che status of Che stoichiometric relations in the theory of steady states of catalyst pellets would be very desirable. Stewart s argument and the other fragmentary results presently available suggest they may always be satisfied for a single reaction when the boundary conditions correspond Co a uniform environment with no mass transfer resistance at the surface, regardless of the number of substances in Che mixture, the shape of the pellet, or the particular flux model used. However, this is no more than informed and perhaps wishful speculation. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

We have seen that purely diffusion-controlled biouptake fluxes may require time spans of O(103) s to decay to their eventual steady-state values (see Section 2.3.6). In reality the situation of pure diffusion as the mode of mass transfer in... 

Primary outputs are produced essentially by sedimentation and (to a much lower extent) by emissions in the atmosphere. The steady state models proposed for seawater are essentially of two types box models and tube models. In box models, oceans are visualized as neighboring interconnected boxes. Mass transfer between these boxes depends on the mean residence time in each box. The difference between mean residence times in two neighboring boxes determines the rate of flux of matter from one to the other. The box model is particularly efficient when the time of residence is derived through the chronological properties of first-order decay reactions in radiogenic isotopes. For instance, figure 8.39 shows the box model of Broecker et al. (1961), based on The ratio, normal-... 

The results in sections 2 and 3 describe the adsorption isotherms and diffusivities of Xe in A1P04-31 based on atomistic descriptions of the adsorbates and pores. The final step in our modeling effort is to combine these results with the macroscopic formulation of the steady state flux through an A1P04-31 crystal, Eq. (1). We make the standard assumption that the pore concentrations at the crystal s boundaries are in equilibrium with the bulk gas phase [2-4]. This assumption cannot be exactly correct when there is a net flux through the membrane [18], but no accurate models exist for the barriers to mass transfer at the crystal boundaries. We are currently developing techniques to account for these so-called surface barriers using atomistic simulations. 

If the DMS inventory in Salt Pond is at steady state in summer (5), production should approximately balance removal. Tidal removal of DMS to Vineyard Sound is minimal. Outflow from Salt Pond is thought to be primarily surface water, and using a maximum tidal range of 0-0.2 m/d and a mean surface water concentration of 10 nmol/L, we calculate an export rate of less than 2 /imol/m2/d. The water-air flux of DMS may be calculated using the two film model of liss and Slater (22 flux = -ki C, ). With the same surface water DMS concentration (C ) and an estimated mass transfer coefficient (ki) for DMS of 1.5 cm/h, the projected flux of DMS from the pond into the atmosphere would be 4 /unol/m2/d. This compares with the range of estimated emissions from the ocean of 5-12 /imol/m2/d (1). 

Various fluxes and processes considered in the following discussion are represented schematically in Figure 9.20, adapted from Wollast and Mackenzie (1983). Steady-state conditions require that the mass balance for each element is fulfilled for the entire system and also separately for the water and sedimentary columns. To write these mass balances, we must consider the global rate of the reactions occurring in the two subsystems. Therefore, we define R and D as the annual amount of a given element transferred from the solid phase to the aqueous phase or vice versa. The fluxes are taken as positive if there is a net input to seawater, and negative if there is a net output. To maintain the concentration of an element constant in seawater, the net flux resulting from Rj + + Lj + Pj in... 

If heat and mass transfer processes inside the treated solid are not sufficiently rapid, in comparison with the rates of heating and pore formation, and the temperature changes significantly with time, one obtains a nonsteady-state situation, whereas enough rapid heat/mass transfer and pore formation assure steady-state in systems with regular fluxes... 

Mass transfer in catalysis proceeds under non-equilibrium conditions with at least two molecular species (the reactant and product molecules) involved [4, 5], Under steady state conditions, the flux of the product molecules out of the catalyst particle is stoi-chiometrically equivalent (but in the opposite direction) to the flux of the entering reactant species. The process of diffusion of two different molecular species with concentration gradients opposed to each other is called counter diffusion, and if the stoichiometry is 1 1 we have equimolar counter diffusion. The situation is then similar to that considered in the case of self-or tracer diffusion, the only difference being that now two different molecular species are involved. Tracer diffusion may be considered, therefore, as equimolar counter diffusion of two identical species. 

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