Analytical solution models

P5.08.07. DISPERSION MODEL. ANALYTICAL SOLUTION FOR FIRST ORDER... 

Catalyst deactivation in large-pore slab catalysts, where intrapaiticle convection, diffusion and first order reaction are the competing processes, is analyzed by uniform and shell-progressive models. Analytical solutions arc provid as well as plots of effectiveness factors as a function of model parameters as a basis for steady-state reactor design. 

Only for a small number of relatively simple single-component adsorption isotherm models analytical solutions of the set of IAS theory equations can be derived. This is possible, for example, for the Langmuir model. If the saturation capacities of all components in the corresponding single-component isotherm equations (Equation 2.51) are identical, the IAS theory generates the same competitive isotherm model as given by Equation 2.57. 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

Using continuous time Markov models, analytical solutions for time-dependent state probabilities can be obtained for homogeneous Markov models. As time increments approach zero, the notation Sn(t) is used to indicate time-dependent probability for state n. 

Since the earlier treatments of this problem by Ramachandran and Sharma(4) and Uchida et.al.(7).several experimental studies and verifications of predictions of enhancement factors have been reported(7,15,16) several detailed models based on film concept have also been proposed(7-12).Recently a penetration model for an instantaneous irreversible chemical reaction has also been presented.which however differs numerically only negligibly than the film model(13).The most important modification of Ramachandran and Sharma s treatment is due to Uchida et. al.(7-9) who consider that the rate of solid dissolution may be accelerated by the absorption of gas as discussed above.They have also considered the case where the concentration of solid component in the bulk liquid phase may not be maintained at the saturation solubility(that is,"finite" slurry) which occurs of course when the rate of solid dissolution is relatively slow compared with gas absorption rate(8).The case where the solid dissolution is finite was further considered by Sada et.al.(12) both theoretically and experimentally.Uchida et.al.(8) could also explain the data of Takeda et.al.(14) by their modified model.Analytical solutions presented above are for instantaneous reactions ... 

Numerically, the solution of the model equations (PDEs subject to initial and boundary conditions) corresponds to an integration with respect to the space and time coordinates. In general, this is an approximation to the mathematical model s exact solution. In simple cases, often restricted models, analytical solutions given by some, even complex, mathematical function are available. Additional work, e.g., Laplace transformation of the original mathematical model, may be required. Generating an analytical solution is commonly not termed simulation ( modelling... without... simulation [15]). If such solutions are not practical, several techniques are applied, among these ... 

For homogeneous systems, a = 1, while for heterogeneous catalytic reactions = Wcat/ hi =/7b simple kinetic models analytical solutions are possible, while numerical solutions provide a general approach to model simulation and parameter estimation. 

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

This can be solved analytically only for a few simplified systems. The Onsager model uses one of the known analytic solutions. 

In this section we proceed to study the plate model with the crack described in Sections 2.4, 2.5. The corresponding variational inequality is analysed provided that the nonpenetration condition holds. By the principles of Section 1.3, we propose approximate equations in the two-dimensional case and analytical solutions in the one-dimensional case (see Kovtunenko, 1996b, 1997b). 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

UNIFAC andASOG Development. Pertinent equations of the UNIQUAC functional-group activity coefficient (UNIFAC) model for prediction of activity coefficients including example calculations are available (162). Much of the background of UNIFAC involves another QSAR technique, the analytical solution of groups (ASOG) method (163). 

Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control-system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonhnear processes. If dcjdi on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets ... 

In general, fiiU time-dependent analytical solutions to differential equation-based models of the above mechanisms have not been found for nonhnear isotherms. Only for reaction kinetics with the constant separation faclor isotherm has a full solution been found [Thomas, y. Amei Chem. Soc., 66, 1664 (1944)]. Referred to as the Thomas solution, it has been extensively studied [Amundson, J. Phy.s. Colloid Chem., 54, 812 (1950) Hiester and Vermeiilen, Chem. Eng. Progre.s.s, 48, 505 (1952) Gilliland and Baddonr, Jnd. Eng. Chem., 45, 330 (1953) Vermenlen, Adv. in Chem. Eng., 2, 147 (1958)]. The solution to Eqs. (16-130) and (16-130) for the same boimdaiy condifions as Eq. (16-146) is... 

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

If the puncture occurs on a pipe which is at least 0.5 m from a vessel, it is justifiable to use a homogeneou.s equilibrium model (HEM) for which an analytical solution is available. The discharge rate pre-... 

The parameters for the model were originally evaluated for oil shale, a material for which substantial fracture stress and fragment size data depending on strain rate were available (see Fig. 8.11). In the case of a less well-characterized brittle material, the parameters may be inferred from the shear-wave velocity and a dynamic fracture or spall stress at a known strain rate. In particular, is approximately one-third the shear-wave velocity, m has been shown to be about 6 for various brittle materials (Grady and Lipkin, 1980), and k can then be determined from a known dynamic fracture stress using an analytic solution of (8.65), (8.66) and (8.68) in one dimension for constant strain rate. 

The UCKRON AND VEKRON kinetics are not models for methanol synthesis. These test problems represent assumed four and six elementary step mechanisms, which are thermodynamically consistent and for which the rate expression could be expressed by rigorous analytical solution and without the assumption of rate limiting steps. The exact solution was more important for the test problems in engineering, than it was to match the presently preferred theory on mechanism. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

E. Clement, P. Leroux-Hugon, L. M. Sander. Analytical solution of an irreversible surface reaction model. J Stat Phys 65 915-939, 1991. 

Dispersion modeling equations for water systems take the same form as those presented later in this chapter for the atmosphere. Analytical solutions tire not nearly as complicated or difficult, since the bulk motion of the fluid (in this case, wtiicr) is a weak vtiriablc with respect to m.ignitude, direction, lime, and position as it is when the fluid is air. 

An analytical solution to this has already been attempted [25]. According to this model, the minimum concentration of fines would be that quantity required to coat each coarse particle with a monolayer of fines. Treating the particles as perfect spheres, the fractional change in combined particle volume due to additional film of fines is then ... 

Determination of the model parameters in Equation (7.7) usually requires numerical minimization of the sum-of-squares, but an analytical solution is possible when the model is a linear function of the independent variables. Take the logarithm of Equation (7.4) to obtain... 

The axial dispersion model discussed in Section 9.3 is a simplified version of Equation (14.19). Analytical solutions for unsteady axial dispersion are given in Chapter 15. 

Sensitivity to step size was thought to be likely due to an unnecessary simplification in the original development of the model. The simplification was to consider initiator concentration constant over a small time increment. When instead the initiator was allowed to vary according to the usual first order decomposition path an analytical solution for the variation of polymer concentration could still readily be obtained and was as follows ... 

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