Concentration, profile

The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form  

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, x2 . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f x2 . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J x2 (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f x2 is equal to the total area under the curve minus the area under the curve up to 2f x 2 . This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

7The Gaussian or normal distribution is generally written as 

Solution From the initial distribution and the number of crystals (nS = 67) seeded initially, we can calculate the initial values for moments as given in Table 10.2 using Equations 10.15-10.19. 

The final concentration is found to be 0.134 gms of potassium sulfate/per gram of solvent since the solution is 100 kg with 0.1743 grams/per gram of solvent. Therefore, total solvent S in the mixture is given by  

---

In the preceding section, the choice of reactor type was made on the basis of which gave the most appropriate concentration profile as the reaction progressed in order to minimize volume for single reactions or maximize selectivity for multiple reactions for a given conversion. However, after making the decision to choose one type of reactor or another, there are still important concentration effects to be considered. 

As evident from Fig. XI-6, the mean field produces concentration profiles that decay exponentially with distance from the surface [66]. A useful approximate solution to Eq. XI-18 captures the exponential character of the loop concentration profile [67], Here a chain of length iV at a bulk concentration of (j>b has a loop profile that can be estimated by... 

The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

Figure Bl.28.3. Concentration profiles of an electroactive species with distance from the electrode surface during a linear sweep voltaimnogram.

SECM is a scaiming-probe teclmiqiie introduced by Bard et aJ in 1989 [49, and M ] based on previous studies by the same group on in situ STM [ ] and simultaneous work by Engstrom et aJ [53 and M], who were the first to show that an amperometric microelectrode could be used as a local probe to map the concentration profile of a larger active electrode. SECM may be envisaged as a chemical microscope based on faradic current changes as a microelectrode is moved across a surface of a sample. It has proved iisefiil for... 

Figure C2.1.18. Schematic representation of tire time dependence of tire concentration profile of a low-molecular-weight compound sorbed into a polymer for case I and case II diffusion. In botli diagrams, tire concentration profiles are calculated using a constant time increment starting from zero. The solvent concentration at tire surface of tire polymer, x = 0, is constant.

In a solution of molecules of uniform molecular weight, all particles settle with the same value of v. If diffusion is ignored, a sharp boundary forms between the top portion of the cell, which has been swept free of solute, and the bottom, which still contains solute. Figure 9.13a shows schematically how the concentration profile varies with time under these conditions. It is apparent that the Schlieren optical system described in the last section is ideally suited for measuring the displacement of this boundary with time. Since the velocity of the boundary and that of the particles are the same, the sedimentation coefficient is readily measured. 

Fig. 7. (a) Impurity elements are rejected into the Hquid between the dendritic solidification fronts, (b) Corresponding impurity concentration profiles. Cq, weld metal composition k, impurity partitioning coefficient in the Hquid maximum impurity soHd solubiHty eutectic composition at grain... 

The Aromax process was developed in the early 1970s by Toray Industries, Inc. in Japan (95—98). The adsorption column consists of a horizontal series of independent chambers containing fixed beds of adsorbent. Instead of a rotary valve, a sequence of specially designed on—off valves under computer control is used to move inlet and withdrawal ports around the bed. Adsorption is carried out in the Hquid phase at 140°C, 785—980 kPA, and 5—13 L/h. PX yields per pass is reported to exceed 90% with a typical purity of 99.5%. The first Aromax unit was installed at Toray s Kawasaki plant in March 1973. In 1994, IFP introduced the Eluxyl adsorption process (59,99). The proprietary adsorbent used is designated SPX 3000. Individual on-off valves controlled by a microprocessor are used. Raman spectroscopy to used to measure concentration profiles in the column. A 10,000 t/yr demonstration plant was started and successfully operated at Chevron s Pascagoula plant from 1995—96. IFP has Hcensed two hybrid units. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Fig. 12. (a) Development of the physically unreasonable overbanging concentration profile and the corresponding shock profile for adsorption with a favorable isotherm and (b) development of the dispersive (proportionate pattern) concentration profile for adsorption with an unfavorable isotherm (or for... 

One potential problem with this approach is that heat loss from a small scale column is much greater than from a larger diameter column. As a result, small columns tend to operate almost isotherm ally whereas in a large column the system is almost adiabatic. Since the temperature profile in general affects the concentration profile, the LUB may be underestimated unless great care is taken to ensure adiabatic operation of the experimental column. 

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

Although molecular diffusion itself is very slow, its effect is nearly always enhanced by turbulent eddies and convection currents. These provide almost perfect mixing in the bulk of each Hquid phase, but the effect is damped out in the vicinity of the interface. Thus the concentration profiles at each... 

Fig. 3. Concentration profiles near an iaterface where the arrow represents the direction of mass transfer, = concentration of C in A-rich phase,...

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

Fig. 9. Schematic view of the development of the concentration profile of ions implanted from low (L), medium (M), and high (H) doses. The projected...

Fig. 9. Bubble-wake interactions in a gas—Hquid-soHd reactor (a) soHds concentration profile within bubble-wake domain, where A—A and B—B represent planes through the bubble, vortex, and wake (b) projected impact of interactions on reaction rate as function of particle si2e and Hquid velocity, where (—)...

Fig. 2. (a) Particle concentration profile of liquid flowing in a pipe, where YjD = the ratio of the distance along the diameter to the diameter ( ) (b)... 

To derive the concentration profile for progressive freezing, a material balance is employed for solidification of a small fraction dg of melt, as shown in Figure 1. Integration from the beginning of solidification gives (1,4,8) ... 

Fig. 3. Impurity concentration profiles resulting from progressive freezing with different values of distribution coefficient k (from eq. 2).

Fig. 5. Concentration profiles for different numbers N of 2one passes for k = 0.5 and a rod containing 10 2one masses. Obtained by numerical methods of...

---