Instantaneous concentration profile

Figure 9.1 Constant-size particle (B) in reaction A(g) + bB(s) - products instantaneous concentration profiles for isothermal spherical particle illustrating general case (b) and two extreme cases (a) and (c) solid product porous arrows indicate direction of movement of profile with respect to time...

The Gaussian plume illustrated in Figure 6 represents the cross-section of a time-averaged, tracer concentration. That is, if time-series concentration measurements taken at a number of points across the plume were separately averaged over their duration, then one would expect to obtain a Gaussian profile. However, at any one time the instantaneous concentration profile would look very different. Figure 12, a typical instantaneous concentration cross-section, shows the small-scale concentration fluctuations resulting from the interaction of coherent structures... 

Figure 3.2.3 shows the instantaneous concentration profiles for solutes 1 and 2 at any time t (only the z -coordinate is shown for simplicity). The concentration profile of species i i —1,2) is located around a zf = - i/>f) tf and has its mcaitna there. The profiles are Gaussian with a standard deviation af (nondimensional O ,), where... 

Mass transfer in either the stationary or mobile phase is not instantaneous and, consequently, complete equilibrltui is not established tinder normal separation conditions. The result is that the solute concentration profile in the stationary phase is always displaced slightly behind the equilibrluM position and the mobile I se profile is similarly slightly in advance of the equilibrium position. The combined peak observed at the column outlet is broadened about its band center, which is located where it would have been for instantaneous equilibrium, provided the degree of nonequllibrluM is small. The stationary phase contribution to Mass transfer is given by equation (1.25)... 

In Figure 9.1(c), the opposite extreme case of a very porous solid B is shown. In this case, there is no internal diffusional resistance, all parts of the interior of B are equally accessible to A, and reaction occurs uniformly (but not instantaneously) throughout the particle. The concentration profiles are flat with respect to radial position, but cB decreases with respect to time, as indicated by the arrow. This model may be called a uniform-reaction model (URM). Its use is equivalent to that of a homogeneous model, in which the rate is a function of the intrinsic reactivity of B (Section 9.3), and we do not pursue it fiirther here. 

The two-film model is a steady-state model that is, the concentration profiles indicated in Figure 9.4 are established instantaneously and remain unchanged. 

Equation (35) predicts that the mass transfer coefficient increases with increases in the screw speed and the number of parallel channels on the screw. The explanation for this is rather simple and is related to the fact that each time the film on the barrel wall is regenerated and the surface of the nip is renewed, a uniform concentration profile is reestablished, which means that the driving force for mass transfer is maximized. Since the instantaneous mass transfer rate decreases with time, mass transfer rates can be maximized by keeping the exposure time as short as possible, and... 

For mass transfer with instantaneous chemical reaction the concentration profiles as schematically represented in Fig. B1 are assumed where the chemical reaction only takes place at plane f. Mass transfer is described by eq. (4) with i = A, B, C and s. At the left side of plane/ Ng = Xg = 0 (no B can pass plane /) and Nc = N, = Q (product and solvent are non-volatile). This results in the following difierential equations and associated boundary conditions ... 

Figure 19.15 (a) Concentration profile at a diffusive boundary between two different phases. At the interface the instantaneous equilibrium between CAB and CB/A is expressed by the partition coefficient KB/A. The hatched areas show the integrated mass exchange after time t MA (t) = MB (<). (b) As before, but the size of KB/A causes a net mass flux in the opposite direction, that is from system B into system A. 

Equation 5.18 offers a convenient technique for measuring self-diffusion coefficients. A thin layer of radioactive isotope deposited on the surface of a flat specimen serves as an instantaneous planar source. After the specimen is diffusion annealed, the isotope concentration profile is determined. With these data, Eq. 5.18 can be written... 

Although reactions in reactive absorption processes are fast, the assumption of instantaneous reactions is usually not justified. Figure 9.5 demonstrates substantial differences in the gas phase concentration profiles of CO2 in a sour gas absorption... 

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

Figure 2-4 Typical concentration profiles of instantaneous reaction between the gas A and the reactant C, based on film theory, ids Diffusion controlled - slow reaction, (fcl kinetically controlled-slow reaction, (c) gas-film-controlled desorption - fast reaction, 0 liquid-film-controlled desorption-fast reaction, (e) liquid-film-controlled absorption -instantaneous reaction between A and C, (/) gas-film-controlled absorption-instantaneous reaction between A and C, (g) concentration profiles for A, B, and C for instantaneous reaction between A and C-both gas- and liquid-phase resistances are comparable.1 2...

Figure 2-12 shows the concentration profiles of A and C for this case and compares it with the previous case and the case of an instantaneous reaction between A and C, where C is a liquid reactant. As shown in the figure, the reaction plane moves towards the gas liquid interface as the absorption rate increases. Equation (2-75) will predict higher absorption rates than Eq. (2-71) under similar reaction conditions. 

Figure 2-13 Typical concentration profiles for A and C in the presence and absence of solids (instantaneous reaction between A and C).

Fig. 2 shows the concentration profiles for the component A at different angular positions for the cases of prolate (a) and oblate (b) spheroids-in-cell. For high Peclet numbers (Pc=1000), higher concentration gradients are found for prolate spheroids compared to those for oblate ones, as it has also been observed previously for the case of instantaneous adsorption [7]. Principally, the concentration decreases, as the angular position is closer to the stagnation point and approaches its bulk value at distances less than 25% of the envelope thickness, i.e. close enough to the solid surface in all cases. Dashed lines in Fig. 1 denote the concentration profiles for a low Pe value (Pe=15). Note that this value has been selected to be... 

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