Steady-state transport equation

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... 

The flow of heat and momentum in the electric furnace was assumed to occur under steady state conditions, so that only steady-state solutions were sought. Further, the flow was mainly laminar, but where it was turbulent the conventional k-e model (9) was applied. Hence, the steady-state transport equations for momentum and heat were solved in three dimensions. The generalized steady-state three-dimensional equation for a conserved variable, q>, is ... 

Here Npe > 1 means that transport in the chemical isolation layer is dominated by advection while Wpe < 1 implies that transport is dominated by diffusion. Advection and diffusion in either the cap isolation layer or bioturbation layer are not independent because advection tends to reduce diffusion gradients and diffusion tends to reduce the advective flux. In the cap isolation layer, a reasonable approximation is to assume that the flux is well-estimated by the dominant flux (either advection or diffusion). Solutions to the steady-state transport equations considering both diffusion and advection with and without reaction are feasible, but are algebraically more complicated and deviate significantly from solutions assuming only the dominant process in the relatively narrow range of approximately 0.3 < Npe < 3. Even within this range, the dominant process correctly estimates the flux within a factor of 2. 

Nikonenko and Urtenov (54, 55] have presented an algorithm for the solution of the steady state transport equations in multicomponent systems, which is based on calculating first the electric field E and then using Eqs. (92 and 93). Their method is outlined here for the particular case when the LEN assumption is used. Unlike Schlogl s... 

Now, we can see that exclusive restrictions were used only for thermodynamic fluxes, namely they have to satisfy the steady-state balance equations. It is easy to see that D(0) is not empty, and contains the unique solutions ]wffho>->iNO> io>-> ko>-> No) of Ih steady-state transport equations. Now we can formulate the minimum principle of the steady-state system... 

Minimum principie of giobai entropy production. Deduction of the homogeneous steady-state transport equations from variationai principie... 

Models Considering Pore Diffusion. Pseudo steady state transport equation for SOj in a spherical porous solid reactant (CaO) can be written as follows ... 

A situation which is frequently encountered in tire production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

In contrast to the full equilibrium transport model, melt could be incrementally removed from the melting solid and isolated into channels for melt ascent. This model is the disequilibrium transport model of Spiegelman and Elliott (1993). Instead of substituting Equation (A7) in for Cs, the problem becomes one of separately keeping track of the concentrations of parent and daughter nuclides in the solid and the fluid. In this case, assuming steady state, two equations are used to account for the daughter nuclide ... 

The steady-state transport of A through the stagnant gas film is by molecular diffusion, characterized by the molecular diffusivity DAg. The rate of transport, normalized to refer to unit area of interface, is given by Fick s law, equation 8.5-4, in the integrated form... 

Calibration is necessary to allow correlation between collected dialysis concentrations to external sample concentrations surrounding the microdialysis probe. Extraction efficiency (EE) is used to relate the dialysis concentration to the sample concentration. The steady-state EE equation is shown in equation (6.1), where Coutiet is the analyte concentration exiting the microdialysis probe, Ci iet is the analyte concentration entering the microdialysis probe, CtiSSue> is the analyte tissue concentration far away from the probe, Qd is the perfusion fluid flow rate and Rd, Rm, Re, and Rt are a series of mass transport resistances for the dialysate, membrane, external... 

All of these simple models have in common the fact that they are accessible to mathematical analysis, while more complex models are not. Yet whether one is dealing with idealized (analyzable) models or complex three-dimensional models, it is essential that the governing equations appropriately represent the underlying physical phenomena. To serve as a resource for this purpose, examples involving time-dependent and steady state transport, simple and facilitated diffusion, and passive permeations between regions were studied. 

In principle, Eq. 8 might be solved to yield the densities fiU and /, corresponding to the various nonuniform states of the system such as those involved in steady-state transport processes. However, there is no requirement implicit in this equation which assures that the average forces acting will correspond to a drift of the system toward thermal equilibrium instead of a way from it, and no progress has been made in solving them in the absence of additional assumptions. 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process dfiN)/dt = 0 at every point in phase space, where f N) is the instantaneous phase density of the N molecules. In principle, at least, it should thus be possible to express the steady-state pressure tensor and the mass and heat currents in terms of ensemble averages constructed without preliminary time-averaging in the replica systems. Thus it is desirable to examine the possibility of obtaining solutions to the reduced Liouville equation, Eq. 8, without preliminary timeaveraging. 

Partial differential equations are involved, and a simple analytical solution is usually impossible. One has to use advanced numerical techniques and computing aids to solve such models. The two-dimensional models use the effective transport concept to formulate the flux of heat and mass in the radial direction. This flux is superimposed upon the transport by the overall continuity equation for the key reacting component A and the energy equation. For a single reaction and at steady state, the equations can be written for the pseudo-homogeneous model as follows ... 

One-dimensional models basically assume that species concentrations and fluid temperature vary only in the axial direction. The only transport mechanism operating in this direction is the overall convective flow. The conservation equations may be obtained from mass and energy balance on a reference component A, over an elementary cross section of the tubular reactor. For a single reaction, the steady state conservation equations can be written for the pseudo-homogeneous model as follows ... 

Mass transfer, an important phenomenon in science and engineering, refers to the motion of molecules driven by some form of potential. In a majority of industrial applications, an activity or concentration gradient serves to drive the mass transfer between two phases across an interface. This is of particular importance in most separation processes and phase transfer catalyzed reactions. The flux equations are analogous to Ohm s law and the ratio of the chemical potential to the flux represents a resistance. Based on the stagnant-film model. Whitman and Lewis [25,26] first proposed the two-film theory, which stated that the overall resistance was the sum of the two individual resistances on the two sides. It was assumed in this theory that there was no resistance to transport at the actual interface, i.e., within the distance corresponding to molecular mean free paths in the two phases on either side of the interface. This argument was equivalent to assuming that two phases were in equilibrium at the actual points of contact at the interface. Two individual mass transfer coefficients (Ld and L(-n) and an overall mass transfer coefficient (k. ) could be defined by the steady-state flux equations ... 

Plug flow is a simplified and idealized picture of the motion of a fluid, whereby all the fluid elements move with a uniform velocity along parallel streamlines. This perfectly ordered flow is the only transport mechanism accounted for in the plug flow reactor model. Because of the uniformity of o>nditions in a cross section the steady-state continuity equation is a very simple ordinary differential equation. Indeed, the mass balance over a differential volume element for a reactant A involved in a single reaction may be written ... 

The equations governing steady-state transport, assuming no electrically driven fluxes, are ... 

Equation (9.4-12) may be integrated for steady-state transport and combined with Eq. (9.4-10). giving, for a matrix-block fragment of radius r,-. 

R. Krishna. A simplified procedure for the Solution of dusty gas model equations for steady state transport in non-reacting systems. Chem. Eng. J 35, (1987) 75-81. 

Integration of Eq. (61.1) for the desired geometry and boundary conditions yields the total rate of permeation of the penetrant gas through the polymer membrane. Integration of Eq. (61.2) yields information on the temporal evolution of the penetrant concentration profile in the polymer. Equation (61.2) requires the specification of the initial and boundary conditions of interest. The above relations apply to homogeneous and isotropic polymers. Crank [3] has described various techniques of solving Pick s equations for different membrane geometries and botmdary conditions, for constant and variable diffusion coefficients, and for both transient and steady-state transport. 

Equation (5.84) defines the dimensionless parameter which characterizes, in forced convection systems, the transition from the non-stcady-state to the steady state transport regimes. 

We discussed above solution of the PNP equations, which couple Poisson and diffusion problems to solve for the steady-state transport." ° The PNP theory is a mean-field theory where, like in the PB equation, the ions are assumed to be pointlike and uncorrelated. In addition, the surrounding solvent is treated as a dielectric continuum. These methods are thus adequate for studying transport through pores that are much larger than the size of the ions, but unsatisfactory for some of the most important ion channels in biology where the pore size is comparable to the ion size. Nevertheless, by incorporating physically reasonable diffusion constants and dielectric profiles, decent results can be obtained in some cases, and if the pores are larger, accurate results are possible. 

Diffusion-Limited Transport of Salts. In the case of fast extraction equilibria for salts at the interfaces, the rate of transport is determined by diffusion of the complex through the membrane and the flux for steady state transport through an SLM is given by Equation 9. 

If the electrolyte has large ionic disorder and since the blocked ionic current i,o is zero. Equations (9.5) and (9.6) indicate that no gradient can exist in the electrostatic potential within the sample. Then the steady-state transport of electrons and holes occurs only due to diffusion nnder the influence of gradients in their concentrations. These gradients must be uniform if the diffusion coefficient does not depend markedly on the concentration. From Equations (9.2) and (9.3) and the ionization equilibrium, the cell voltage determines the ratio of the activities of the electronic species at both sides of the electrolyte ... 

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