Instantaneous time concentration profiles

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...

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. 

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... 

The rates of movement of foreign compound into and out of the central compartment are characterized by rate constants kab and kei (Fig. 3.23). When a compound is administered intravenously, the absorption is effectively instantaneous and is not a factor. The situation after a single, intravenous dose, with distribution into one compartment, is the most simple to analyze kinetically, as only distribution and elimination are involved. With a rapidly distributed compound then, this may be simplified further to a consideration of just elimination. When the plasma (blood) concentration is plotted against time, the profile normally encountered is an exponential decline (Fig. 3.24). This is because the rate of removal is proportional to the concentration remaining it is a first-order process, and so a constant fraction of the compound is excreted at any given time. When the plasma concentration is plotted on a logio scale, the profile will be a straight line for this simple, one compartment model (Fig. 3.25). The equation for this line is... 

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. 

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... 

We shall briefly deliberate the choice of the particle state. Since the quantity of interest is the mass transfer rate from the droplets, particle state must be chosen to yield from it the instantaneous mass flux from the droplet. The mass flux by diffusion requires the concentration profile near the surface. Indeed, drop size (say radius) is clearly important if we choose the average solute concentration in the droplet as another variable, the two together cannot yield the surface mass flux. But since the concentration at birth is uniform, specification of drop age, the time elapsed since its birth, can be used as a third particle state variable. Shah and Ramkrishna (1973) provide the details of the calculation of how drop size x, the average solute concentration c, and drop age t together help to calculate the mass flux at the drop surface. The mass flux will directly provide the rate of change of average concentration C(x, c, t) so that the rate of change of particle state required for the population balance model is also completely identified. 

We start by considering a source located at the origin of a three-dimensional Cartesian coordinate system. At time f = 0, the source releases a substance of mass Mp kg. The release is instantaneous and leaves immediately (i.e., it is of infinitesimally short duration). Thereafter, the released substance diffuses into the three-dimensional infinite space surrounding the source, giving rise to time-dependent three-dimensional concentration profiles C (x, y, z, t). The three coordinates x, y, z can be combined into a single radial distance r anchored at the origin, where + y + z. The resulting concentration... 

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... 

The finite surface kinetics were treated by considering the step function in flux produced by the potential step. Any change in the surface coverage of adsorbed hydrogen is assumed to cause an instantaneous change in the flux just inside the metal surface. The implicit rationale behind this assumption, as indicated in Section III.l, can be explained on the basis of Eq. (7), which results when the entry of hydrogen is restricted since the coverage is assumed to be constant under potentiostatic conditions, and therefore D dC/dx)x=Q, are kept constant. The subsurface concentration at the entry side increases with time until it reaches a steady state. The concentration profile and a typical anodic transient are shown in Fig. 4. 

Continuous monitoring of organ function displays in real-time the clearance profile or concentration of an organ function-specific marker in a pre-deter-mined body compartment. This approach allows the early intervention by clinicians since instantaneous deviation from the established baseline reading may indicate the onset of an anomaly. However, it may not be useful for measuring dynamic functional reserve of the organ because subsequent activities after marker elimination from plasma may not be reflected in the clearance profile. 

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