Radial concentration profiles analysis

Although additional radial collocation points increase the dimensionality of the resulting model, they may be necessary to accurately express the radial concentration profiles. Preliminary analysis in this section considers only one interior radial concentration collocation point, although a detailed analysis of this assumption is presented in Section VI,E. 

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

Use of Ficks law to describe the diffusion process requires the solute particle to be small compared with the diffusion boundary layer. The analysis presented above suggests that, for Peelet numbers greater than 100, the ratio 8o/0p is proportional to (Pe)ua/R. The solid curves in Figure 3 are truncated at the value of the Peelet number corresponding to Pe/R3 10 "2, where an inspection of the radial concentration profile revealed that the ratio 8d/Op is about ten. 

A full analysis of the efficiency of particle separation in CHDF gives the appropriate criteria for the development of a steady state radial concentration profile [68]. Particle transit time is a logarithmic function of particle size. Pressures of up to 30,000 Pa are required and give a separating range from 0.2 to 200 nm [69]. 

The at erage oullei conversion becomes 68.8 55 . not much differem from the one in pan (a) in agreement with the Aris-Taylor analysis. However. due to the laminar flow assumption in the reactor, the radial concentration profiles are very different throughout the reactor,... 

The results of the modelling of liquid flows, based on Navier-Stokes equations and the C- model of turbulence, demonstrate that the turbulent diffusion coefficient increases decreases) for reactors with a radial input of reactants (P2- and P3-type mixers), especially for reactors with conical widening at the input of the liquid flows (P4, P5). For P5-type reactors, the time of mixing decreases approximately tenfold compared with the Pl-type at given flow parameters (Figure 2.5). An increase of Dfuji, results in a faster equalisation of the reactant concentration profile. Analysis of the construction of the reaction devices confirms that drops of hydraulic pressure... 

B.2, Here, we demonstrate once more how Brownian dynamics relates to diffusive behavior, by simulating spherical particles of radius 1 mm in water at room temperature. At time f = 0, a particle is released at the origin and undergoes 3-D Brownian motion. Write a program that repeats this simulation many times and plots the radial concentration profile of particles as a function of time. It is easier to do the data analysis if you do the simulations concurrently. Then, solve the corresponding time-dependent diffusion equation in spherical coordinates and compare the results to that obtained fi om Brownian dynamics. 

Optical fiber measurement of local solids concentrations of FCC catalyst fluidized in a 9-cm-i.d. column gave the results shown typically in Fig. 26. Analysis of these data showed that the radial voidage profile could be described solely by the cross-section-average voidage e, calculated as shown in Sec. 5.1, and the reduced radial coordinate r/R ... 

In his analysis of the open tube distillation column Westhaver (1942) goes into a detailed consideration of radial concentration gradients which is very similar to Taylor s approach. His final formula, however, is the same as if he had assumed a constant velocity profile and an effective diffusion coefficient (Dt + llU2r2l48Dt). This is just the diffusion coefficient that we have found for viscous flow in the presence of a film on the tube wall in which the solute concentration is infinitely greater than in the fluid. This is clearly the case for... 

As described in further details in Section 5, we analyze the scans using the software DCDT+ (Philo, 2006), which converts the raw concentration profiles into time derivatives (dc/dt) and fits these values to approximate unbounded solutions of the Lamm equation (Philo, 2000 Stafford, 1994). As the rotor speed (ft)) and the concentration of the macromolecules (c) are known, and the time (t) and the radial concentration distribution [c(x, f)] are obtained from the scans of absorbance profiles, the fitting yields values of s and D. As both parameters are dependent on the solvent viscosity and temperature, they are transformed to standard values with reference to a standard temperature (20 °C) and a standard solvent (water) and reported as 52o,w and /92o,w This standardization allows analysis of the changes in the intrinsic properties of solute molecules with changes in solution condition and is a prerequisite in cation-mediated folding studies of RNA molecules. 

A dynamic experimental method for the investigation of the behaviour of a nonisothermal-nonadiabatic fixed bed reactor is presented. The method is based on the analysis of the axial and radial temperature and concentration profiles measured under the influence of forced uncorrelated sinusoidal changes of the process variables. A two-dimensional reactor model is employed for the description of the reactor behaviour. The model parameters are estimated by statistical analysis of the measured profiles. The efficiency of the dynamic method is shown for the investigation of a pilot plant fixed bed reactor using the hydrogenation of toluene with a commercial nickel catalyst as a test reaction. 

Moonen R.H.W, Radial and axial gas concentration profiles in a CFB biomass gasifier - measurements and analysis (1999), MSc. thesis University of Twente... 

Problem 3-37. Taylor Dispersion with Streamwise Variations of Mean Velocity. We consider steady, pressure-driven axisymmetric flow in the radial direction between two parallel disks that are separated by a distance 2h. We assume that the volumetric flow rate in the radial direction is fixed at a value Q and that the Reynolds number is small enough that the Navier-Stokes equations are dominated by the viscous and pressure-gradient terms. Finally, the flow is ID in the sense that u = [nr(r, z), 0, 0]. In this problem, we consider flow-induced dispersion of a dilute solute. We follow the precedent set by the classical analysis of Taylor for axial dispersion of a solute in flow through a tube by considering only the concentration profile averaged across the gap, ( ) = f h dz. 

In these equations D represents the corresponding diffusion coefficients, and Q the permeate flow rate. The first term of each equation gives the radial dispersion and the second one corresponds to the radial convection. The authors [5.103] used in their model, a biological kinetic rate expression (cp), which was obtained by independent experiments and analysis of a batch reactor, and also made an effort to account for and correlate the permeate flow decrease with the amount of produced biomass. The simulation curves obtained matched well the experimental results in terms of permeate flow rate evolution and product concentration. One of the important aspects of the model is its ability to theoretically determine the biomass concentration profiles, and the relation between the permeate flow rate and the calculated biomass concentration in the annular volume (Fig. 5.24). Such information is important since the biomass evolution cannot be determined by any experimental methodology. 

Figure 31 shows the model analysis of the effects of radial gas dispersion coefficient on radial profiles of propylene concentration. The radial mass transfer has a significant effect on the conversion and yield. When the radial Peclet number decreases from 1400 to 200, the conversion of propylene increases by over 10%, and the yield of acrylonitrile increases by about 7%. Since the reaction is first order with respect to propylene, risers are operated under dilute conditions at Pe = 200, so the radial concentration distribution of propylene is uniform and radial mass transfer is not... 

Martin et al. (1992) He Gas radial velocity profiles and dispersions in a CFB studied by gas chromatograph analysis of concentration of injected He... 

For each region a mean value of the void fraction was calculated and a hydraulic radius was defined which was used in a pressure drop correlation. Martin [20] divided the bed into two regions a wall and a bulk region. He calculated for both different flow rates and a different rate of heat transfer. Carbonell [2] also used a two zone model for his analysis of the dispersion phenomena. In more recent work Vortmeyer et al. [5>6] tried to use the complete radial void fraction profile, and so did Chang [3]. They followed the same itinerary outlined by Lerou and Froment [l] and Marivoet et al. [2l]. Starting from the void fraction profile the radial velocity profile is calculated. With both profiles the effective thermal conductivity is established and the temperature and concentration profiles can be calculated by means of a two dimensional pseudo homogeneous model for the reactor. 

As the radial concentration gradient at the tube wall determines the mass flux supplied by diffusion a good approximation of this gradient is necessary to obtain an accurate description of the concentration profiles in the tube. In the numerical analysis this is effected by reduction of the radial step width near the tube wall. During the calculations it turned out, however, that a non equidistant radial grid resulted in persistent numerical instabilities. 

Fig. 6.33 The concentration profiles (top 893 K, bottom 848 K) obtmned as a Unction of space and time. The profiles are obtained by fitting with a constant D. The introduction of a concentration dependence only alters the result slightly. It can be seen that the assumption of a pure difiPusion control (solid lines) is better fulfilled for diffusion from fresh surfaces (bottom) than is the case for the relaxed surfaces (top). In this case the fit only becomes perfect if the surface control is taken into account (broken line). The analysis of the above set of curves then yields — instead of 1.4 x 10 cm /s — more precisely O = 2.0 x 10 cm /s with a rate constant P = 2 x 10 cm/s. The fact that a radial geometry was analyzed (see abscissa) top and a cartesian geometry below does not affect the result.

The simplest way computationally of obtaining a sedimentation coefficient distribution is from time derivative analysis of the evolving concentration distribution profile across the cell [40,41]. The time derivative at each radial position r is d c r,t)/co /dt)r where cq is the initial loading concentration. Assuming that a sufficiently small time integral of scans are chosen so that Ac r t)/At= dc r t)ldt the apparent weight fraction distribution function g (s) n.b. sometimes written as (s ) can be calculated... 

Figure 4 [29] shows the (s) versus profiles for potato amylose and the amylose/amylopectin mixture from wheat starch corresponding to the concentration versus radial displacement data of Fig. 3. The s data used in the concentration dependence plot of Fig. 3 for wheat amylopectin comes from (s) vs. s analysis data of Fig. 2b and similar. The concentrations shown in the abscissa in Fig. 4 have been obtained from the total starch loading concentration normalised by the weight fraction of the amylopectin component estimated from the (s) vs. s profiles.

Although the subsequent discussion describes the stereoselection at the steady state through the example of radical reactions, the analysis and principles are general for any reaction profile that fits into the scheme of complex stereoselective reactions. In the process proposed and analyzed by Curran et al., the activation of compounds of type 1 is done, for example, by radical formation. The group selectivity in this first step has again no effect on the stereomeric nature of the product. To obtain a stereoconvergent process it is crucial, however, that the reaction is operating at the steady state. This means that the concentrations of the radial intermediates (compounds in brackets in Scheme 2) is low and stationary, while their absolute concentrations are determined by the different rates of reaction. 

An analysis of radial flow, fixed bed reactor (RFBR) is carried out to determine the effects of radial flow maldistribution and flow direction. Analytical criteria for optimum operation is established via a singular perturbation approach. It is shown that at high conversion an ideal flow profile always results in a higher yield irrespective of the reaction mechanism while dependence of conversion on flow direction is second order. The analysis then concentrates on the improvement of radial profile. Asymptotic solutions are obtained for the flow equations. They offer an optimum design method well suited for industrial application. Finally, all asymptotic results are verified by a numerical experience in a more sophisticated heterogeneous, two-dimensional cell model. 

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