Concentration steady-state profile

Equations (8.1)-(8.13) can be solved to provide transient- or steady-state profiles of O2 and CH4 concentration, reaction rates and surface fluxes for any combination of the controlling variables 9q,0], v,k, a, Vm,Vq and Vr. Where, as is usual, one or more of the controlling variables may be further simplified, approximated or neglected, process-based simulation of CH4 emission becomes possible using a relatively limited set of input data. 

Figure 26. Predictions of the Adler model shown in Figure 25 assuming interfacial electrochemical kinetics are fast, (a) Predicted steady-state profile of the oxygen vacancy concentration ( ) in the mixed conductor as a function of distance from the electrode/electrolyte interface, (b) Predicted impedance, (c) Measured impedance of Lao.6Cao.4Feo.8-Coo.203-(5 electrodes on SDC at 700 °C in air, fit to the model shown in b using nonlinear complex least squares. Data are from ref 171.

Comparison of steady-state profiles shows that neglecting axial mass diffusion has very little effect on the temperature and concentration profiles even though the axial gradients are significant. However, Figure 16 shows that neglecting the axial thermal dispersion in the gas does affect the solution... 

Due to the spherical geometry of the surface, the concentration profile across the boundary layer is no longer a straight line as was the case for the flat bottleneck boundary (Fig. 19.4). We can calculate the steady-state profile by assuming that CF and CFq = Cs/Fs/F are constant. Then, the integrated flux, ZF, across all concentric shells with radius r inside the boundary layer (r0 < r < r0 + 8) must be equal ... 

If the catalyst were not to decay but for some other reason, perhaps temperature control, the particles were taken out and recycled, then each might be supposed to be in pristine condition on entering the bed. Each particle would then undergo a transition during which the steady state profile of reactant within the particle would be built up. The analysis of Amundson and Aris (1962 this part is not tainted with the error mentioned above in fn. 13) may be used. We assume spherical particles of radius R, and call the profile of concentration at time a, c r, a). If D is the diffusivity of the reactant and k the rate constant per unit volume of catalyst,... 

Experimental data on multiple steady-state profiles in tubular packed bed reactors have been reported in the literature by Wicke et al. 51 -53) and Hlavacek and Votruba (54, 55) (Table VI). The measurements have been performed in adiabatic tubular reactors. In the following text the effects of initial temperature, inlet concentration, velocity, length of the bed, and reaction rate expression on the multiple steady state profiles will be studied. 

This concentration difference can be obtained by solving Fick s second law under the appropriate boundary conditions of constant concentrations or concentration gradients. The initial values are determined by the constant concentration profile in the case of the polarization, and by the linear steady-state profile (linear in this simplest approximation) in the case of the depolarization. The relevant solutions are given in Appendix 2. The desired U(t) functions of the polarization cells 3 and 4 simplify for long and short terms to... 

The assumption of a steady-state profile in the oil laminates and small concentration drops in the water layers may be used to derive asymptotic solutions for the permeation problem. It may be shown that (2) for P P y and t[Pg.36]

A comparison of the steady-state profiles predicted by the wave model and those predicted by a rigorous tray-by-tray column model is shown in Fig. 5.16 for a coupled column system which serves for the separation of a mixture of methanol, ethanol, and 1-propanol. The approximation by the wave model in Fig. 5.16 is fairly good, although the reduction of the system order is considerable. The state variables of the rigorous model are the concentration and temperatures on each column tray. In contrast to this the state variables of the wave model are only the front positions. 

This relationship is exactly the same as the one that was found in the TMB case. Similar conditions apply to the two compoimds in the other three columns. As a consequence, Eqs. 17.10a and 17.11a apply as well. Combining the node equations and the propagation equation (Eq. 17.15), we can derive the concentration profiles of both components in each column. This calculation must be done at the end of each period, as is explained in the next few subsections. These concentration profiles are complex functions of the experimental conditions that depend on the rank of the period considered. One interesting result is that these functions have asymptotic limits that are easily derived from simple theorems on the infinite limits of suites, series, and products. This allows the calculation of the steady-state profiles. 

Shock Layer Because the efficiency of actual columns is finite, concentration shocks are not stable. They are eroded by axial dispersion and the finite rate of mass transfer. A steep concentration gradient is formed instead. The steepness of the profile depends on the axial dispersion and the mass transfer resistance. In frontal analysis, a constant pattern, or steady-state profile, forms after an infinite period of time and an infinite migration length. In this case, the shock layer profile... 

In the absence of supporting electrolyte, the electroneutrality principle demands that the local concentrations of the electroactive and electroinactive ions remain equal (56) in the tip-substrate gap when the concentration of the former is depleted by electrolysis at the tip UME. In the case of AgCl dissolution, the mass transport problem was shown to reduce to the consideration of a single species (7). Figure 33 shows steady-state profiles that illustrate the interfacial undersaturations, obtainable for a range of first-order dissolution rate constants, with no added supporting electrolyte. Although the saturation ratio at the substrate/solution interface is close to unity for AT, = 100 (Fig. 33i), i.e., the dissolution kinetics are close to the diffusion-... 

Computed as explained in the text using initial stationary concentration (Ct 0) given by profile 1 in Figure 12 and new steady-state given by profile 2. Time to steady-state shown for different values of eddy diffusion coefficient (K) and advective velocity (U). Time to steady-state defined as the time when the oxygen concentration has attained 95% of the concentration difference between the new and initial steady-state profiles C — Ct 0 = 0.95(Ct x — Ct= 0). 

Fig. 40. Plots showing successive NHi concentration profiles predicted at 22°C for the bioturbated zone at NWC by Eq. (6.19). A basal flux rather than concentration constraint at X = L = 15 cm is used in this case. The sequence shown would be similar to that which would occur in a box core collected at 10°C, warmed rapidly to 22°C and allowed to sit (except F = 0 in that case). A regular periodic oscillation of temperature instead of a single warming would produce corresponding oscillating profiles similar in shape to the steady-state profile no maxima would occur.

A linear, steady-state profile of the vacancy concentration is established between the source and the sink. 

Let us assume that we introduce a particle of radius Rp consisting of a species A in an atmosphere with a uniform gas-phase concentration of A equal to c. Initially, the concentration profile of A around the particle will be flat and eventually after time xdg it will relax to its steady state. This timescale, xdg, corresponds to the time required by gas-phase diffusion to establish a steady-state profile around a particle. It should not be confused with the timescale of equilibration of the particle with the surrounding atmosphere. We assume that cx remains constant and that the concentration of A at the particle surface (equilibrium) concentration is cs and also remains constant. 

The collision rate is initially extremely fast (actually it starts at infinity) but for t 4Rp/nD, it approaches a steady-state value of /coi = 8nRp DNq. Physically, at t — 0, other particles in the vicinity of the absorbing one collide with it, immediately resulting in a mathematically infinite collision rate. However, these particles are soon absorbed by the stationary particle and the concentration profile around our particle relaxes to its steady-state profile with a steady-state collision rate. One can easily calculate, given the Brownian diffusivities in Table 9.5, that such a system reaches steady state in 10-4 s for particles of diameter 0.1 pm and in roughly 0.1 s for 1 pm particles. Therefore neglecting the transition to this steady state is a good assumption for atmospheric applications. 

Figure 8.12 shows the transient profiles. We see the reactor initially has zero A concentration. The feed enters the reactor and the A concentration at the inlet rises rapidly. Component A is transported by convection and diffusion down the reactor, and the reaction consumes the A as it goes. After about t = 2.5, the concentration profile has reached its steady value. Given the low value of dispersion in this prob lem, the steady-state profile is close to the steady-state PFR profile for this problem. o... 

Figure 3.9 Concentration of solute diffusing with first-order elimination, (a) Steady state profiles for Dp, = lO cm /s and k varying from 0, 10 , 10 , 10 s approach to steady state for a solute with Dp, = 10 — ...

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