Initial conditions, dispersion model

Equation 5-9 together with appropriate boundary and initial conditions forms the fundamental basis for dispersion modeling. This equation will be solved for a variety of cases. 

The model is referred to as a dispersion model, and the value of the dispersion coefficient De is determined empirically based on correlations or experimental data. In a case where Eq. (19-21) is converted to dimensionless variables, the coefficient of the second derivative is referred to as the Peclet number (Pe = uL/De), where L is the reactor length and u is the linear velocity. For plug flow, De = 0 (Pe ) while for a CSTR, De = oo (Pe = 0). To solve Eq. (19-21), one initial condition and two boundary conditions are needed. The closed-ends boundary conditions are uC0 = (uC — DedC/dL)L=o and (dC/BL)i = i = 0 (e.g., see Wen and Fan, Models for Flow Systems in Chemical Reactors, Marcel Dekker, 1975). Figure 19-2 shows the performance of a tubular reactor with dispersion compared to that of a plug flow reactor. 

Brenneman and Nair (2001) proposed a strategy that combines their modified version of Harvey s method with joint location and dispersion modeling for a log-linear dispersion model. After fitting a location model with ordinary least squares regression, they recommended an initial check to see if there are sufficient degrees of freedom even to consider looking for dispersion effects. The condition they... 

For the mathematical solution of the dispersion model flow, we add the univo-city conditions that include the signal input description for the initial conditions to Eq. (3.97) or (3.98). A more complete description of this mathematical model... 

The size distribution of particles containing Cs-137 is not known and here only mono-disperse particles with a radius of 0.5 pm and a density of 1.88 g cm were considered. The model area corresponded to the G45 domain (Fig. 5.1). The start time was at 25 April 1986 at 18 UTC and the model was mn 2 days ahead and then reinitialized and restarted until 7 May at 18 00 UTC. Surface analysis and 3DVAR upper air analysis was used as initial conditions for the meteorology at the beginning of each cycle and 6 hourly boundaries were post-processed from the IFS model. 

Peak profiles can be calculated with a proper column model, the differential mass balance equation of the compound(s), the adsorption isotherm, the mass transfer kinetics of the compound(s) and the boundary and initial conditions [13], When a suitable column model has been chosen, the proper parameters (isotherm and mass transfer parameters and experimental conditions) are entered into the calculations. The results from these calculations can have great predictive value [13, 114], The most important of the column models are the ideal model , the equilibrium-dispersive (ED) model , the... 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

For mass balance reasons, Cbfl = Cafi. The problem is to find a relationship between the isotherm parameters and the parameters and of the f-th harmonic. In the calculations made to relate the equilibrium isotherm and the response of the system (Eq. 3.102), the equilibrium-dispersive model is used (Chapter 2, Section 2.2.2) and the mass balance equation is integrated with the Danck-werts boundary conditions (Chapter 2, Section 2.1.4.3) and with the initial conditions C = Ca,o, q = qiCa,o). 

The model of Tayakout et al. [117,118] in addition accounts for the possibility of axial dispersion effects in the tubeside and shellside. The inclusion of axial dispersion effects in regions (1) and (4) necessitates a different set of initial conditions at Z = 0 and a companion set of conditions at Z = L. The effect of pressure drop through the catalytic bed could be included in this type of model using Ergun s equation. 

Initial conditions (t=0) Initially there is no flow in the reactor v = Vro = Vz = Vzo = 0), the tube is filled with stagnant gas having prescribed composition and temperature, as for the 2D dispersion model simulations. 

A computer model has been developed which can generate realistic concentration versus time profiles of the chemical species formed during photooxidation of hydrocarbon polymers using as input data a set of elementary reactions with corresponding rate constants and initial conditions. Simulation of different mechanisms for stabilization of clear, amorphous linear polyethylene as a prototype suggests that the optimum stabilizer would be a molecularly dispersed additive in very low concentration which can trap peroxy radicals and also decompose hydroperoxides. 

The initial and boundary conditions are given in Chapter 9. The present treatment does not change the results of Chapter 9 but instead provides a rational basis for using pseudohomogeneous kinetics for a solid-catalyzed reaction. The axial dispersion model in Chapter 9, again with pseudohomogeneous kinetics, is an alternative to Equation 10.1 that can be used when the radial temperature and concentration gradients are small. 

To describe the peak shapes of a separation under overload conditions a clear understanding of how the competitive phase equilibria, the finite rate of mass transfer, and dispersion phenomena combine to affect band profiles is required [ 11,66,42,75,76]. The general solution to this problem requires a set of mass conservation equations appropriate initial and boundary conditions that describe the exact process implemented the multicomponent isotherms and a suitable model for mass transfer kinetics. As an example, the most widely used mass conservation equation is the equilibrium-dispersive model... 

We have previously pointed out that the use of the dispersion model changes the reactor analysis from an initial-value (PFR) to a boundary-value problem. As a result, we should worry about the form of the boundary conditions to use for equation (5-77). This is illustrated in Figure 5.16, where several possible configurations of inlet and outlet conditions are shown. Hopefully, this is not to make a... 

In spite of this enticing come-on, we will not solve this problem for the moment, being content with its illustration of a typical two-phase reactor balance formulation using the PFR model. We hasten to add, however, that the solution to the set of equations (7-140) and (7-141) with the initial and boundary conditions given is identical to that for the much simpler set of (7-54) and (7-139). In the following sections we shall pursue in detail the developments using the by-now familiar dispersion model for tubular reactors, and in Chapter 8 will treat a number of other multiphase reactor models. 

In the following the relevant models for liquid chromatography are derived in a bottom-up procedure related to Eigure 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. In all cases the mass balances must be complemented by initial and boundary conditions (Section 6.2.7). Eor the so-called transport-dispersive model a dimensionless representation will also be presented below. 

Analytical solutions are those whose precision depends only on the accuracy of the initial data. They do not contain errors associated with the approximation due to simplification of the computation process. This approach is applicable to the simplest models, which are often represented by a restricted number of relatively simple equations. These may be solved without specialized program software. Analytical solutions, as a rule, are used in modeling of processes with minimum participation of chemical reactions, in particular in the analysis of distribution of nonpolar components, radioactive decay, adsorption, etc. Under such conditions for modeling often are sufficient equations of advective-dispersive mass-transport, which are included in the section Mixing and mass-transport . 

The proper application of the dispersion model is bedevilled by difficulties with the initial boundary condition and in obtaining accurate values for the axial disperaon coefficient D. Sherman [1964] realised that e q>res g the initial condition by the step function, although convenient for the mathematical analysis, was not easy to do in practice especially with thin cakes and he contrived to measure the solute concentration at the inflow face of Ms beds. The initial boundary condition was th represaited by a power-eTqtonential function with six e erimental constants ... 

A computer analysis was performed of the loss-of-load event with delayed reactor trip, similar to that used in safety valve capacity evaluation, except that a conservative 20% safety valve blowdown and initial conditions biased to maximize pressurizer liquid level were assumed. The purpose of this analysis was to determine the pressurizer liquid level response and the RCS subcooling under these conservative conditions. For additional conservatism, adjustments were made to the computer-calculated pressurizer level on the basis of a very conservative pressurizer model. This model assumed that the initial saturated pressurizer liquid did not mix with the cooler insurge liquid, that the initial liquid remained in equilibrium with the pressurizer steam space, and that the steam which flashed during blowdown remained dispersed in the liquid phase and caused the liquid level to swell. The adjusted pressurizer water level vs time curve showed a maximum level of 78%, Reference 2, (1874 ft" ), below the safety valve nozzle elevation which is greater than 100% level, so that dry saturated steam flow to the safety valves is assured throughout the blowdown. The computer analysis also showed that adequate subcooling was maintained in the RCS during the blowdown, so that steam bubble formation is precluded. 

Wheatley, C. J. 1987. Discharge of Liquid Ammonia to Moist Atmospheres Survey of Experimental Data and Model for Estimating Initial Conditions for Dispersion Calculations, UKAEA Report SRD/HSE/ R410, London. 

It has been calculated that a spill during tanker filling or unloading operation as outlined above could release approx. 1.0 ton of SO3 as sulphuric acid mist, which would form the initial cloud. As the cloud is blown along by the wind, it will be diluted by entrained air until the concentration is no longer hazardous. The distance to this point will depend on the wind and weather conditions. In stable weather conditions (Pasquill category D and at 5 m/s windspeed) a dispersion model predicts that the clouds could be hazardous up to about 1 kilometre from the point of release ... 

However, in various coating and granulation experiments a change in the shape of the distribution is observed, for example a dispersion of the distribution. In Fig. 7.45 the results of five different coating experiments are shown, which were conducted in Wurster equipment, a conventional fluidized bed apparatus (FB) in top and bottom spray configuration, and a spouted bed apparatus (SB) in top and bottom spray configuration. Although identical initial conditions were used and the process conditions are comparable (see Tab. 7.6), different final distributions are achieved. This effect cannot be explained by the common model of Eq. 7.34, but it underlines the influence that different types of equipment with different flow patterns may have on the process result. 

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