Fixed beds model

Analytical determination of the hydraulic resistance of the medium is difficult. However, for the simplest filter medium structures, certain empirical relationships are available to estimate hydraulic resistance. The relationship of hydraulic resistance of a cloth of monofilament fiber versus fiber diameter and cloth porosity can be based on a fixed-bed model. 

Fig. 8.7 Results of the fixed bed model. Spatial profiles of the temperature, T(x), for mgu = 4.9-14.6 x 10-3 g cm-2 s-1 in steps of 2.4 x 10-3 g cm-2 s-1 (left), and the solid mass flux, msu = ps(l - e)us, into the reaction zone as a function of gas mass flux (right).

Table 2.2 gives examples of mass transfer coefficients determined from both the single particle and fixed bed models for the evaporation of water from particles of the same diameter and density as in Table 2.1, assuming the diffusivity of water in air to be 3 x 10 m s h Once... 

As can be proved (see Section 5.3.4), the pressure drop becomes independent of the conversion and thus from the material balance if the expansion factor is near zero, and then the two differential equations are decoupled. Furthermore, for nearly isothermal operation, the pressure drop is not a function of temperature. Under these conditions, the fixed-bed model is greatly simplified. 

The following equations constitute the approximate solutions of the fixed-bed model under the constant pattern and plug-flow assumption for the favorable Langmuir isotherm and linear driving forces (Perry and Green, 1999) ... 

This model covers the case where we have combined resistances to diffusion (fluid-film and solid diffusion). In this case, the concentration in the main phase of the fluid (bulk concentration) is different from the one at the interface due to the effect of the fluid film resistance. The following equations can be used for Langmuir and Freundlich equilibrium equations (Miura and Hashimoto, 1977). The solutions of the fixed-bed model are the following ... 

In conclusion, the maximum adsorption capacity should be measured in fixed-bed experiments under dynamic conditions, and if models are applicable, diffusion coefficients should be also determined in fixed-bed apparatus. Due to the fact that the equilibrium isotherms require extended data series and thus are time-consuming experiments, the latter are quite difficult to be conducted in fixed-bed reactors and from this point of view, it is more practical to evaluated equilibrium isotherms in batch reactor systems. Then, it is known that when applying fixed-bed models using an equilibrium isotherm obtained in batch-type experiments, the equilibrium discrepancy (if it exists) can be compensated by a different estimate for the solid diffusion coefficient (Inglezakis and Grigoropoulu, 2003 Weber and Wang, 1987). 

Measurement provides only limited information about the combustion and pollutant emission process. To explore further details about the flow turbulence, the interaction of different processes and influencing factors, numerical calculations are performed. Different aspects need to be considered in the modeling of fixed-bed furnace combustion, including fixed-bed modeling, gas-phase and particle-phase simulation, as well as interaction between the different phases. 

Table 6.1 Phenomena evaluated and quantified in mass balances of chromatographic fixed-bed models.

Sharma A, Philippe R, Luck F, Schweich D. A simple and realistic fixed bed model for investigating Fischer-Tropsch catalyst activity at lab-scale and extrapolating to industrial conditions. Chemical Engineering Science 2011 66 6358-6366. 

Figure 12. Recovery and purity versus the iso fraction purge to feed ratio for a PSA separation of n/iso-paraffins based on patent data of Minkkinen et al. [3]. Simulations are performed with equilibria, kinetic and a fixed bed model shown in this work.

Basic equilibrium and kinetic data were experimentally measured in view of adsorber design for the separation of n/iso-paraffins. Based on those data a fixed-bed model was set-up and experimentally validated for the separation of n/iso-paraffins mixtures. A cyclic PSA process was successfully simulated based on data from patent literature [3]. 

Reactor modelling is an essential step in a scale-up of new steam reforming reactors. Fixed-bed modelling has always been a challenge to chemical engineers and it is one of the first applications where computers have been used extensively for the tedious property calculation in the numerical solution. An early example by Kjaer is shown in [274], and later in [275]. Modelling of steam reformers is widely used and among the early ones are Hyman [250] from 1968 and an early Topsoe publication [493]. 

Most physical models in tsunami studies consist of scaled offshore bathymetry and a shoreline to simulate a beach, island, or coastal structure such as a breakwater or seawall. The models are usually fixed bed since sediment transport issues are not considered. The fixed bed model is typically constructed using templates and sand with a thin mortar cap or veneer. If a movable bed is used, then scahng issues become a concern since it is difficult to properly scale both the sediment and the model with the same scale factor. Sediment grains can be minimized only so much before changing from noncohesive (i.e., sand) to cohesive behavior (i.e., mud). 

For simple cases of kinetics, an analytical solution of the model is possible but, in general, a numerical solution of Equation A9.7 or A9.9 is preferred during the estimation of the kinetic constants. It should be noticed that the analytical solutions obtained for various kinetics in a homogeneous BR (Equation A9.1) can be used for the special case of fixed beds, Equation A9.9, but the reaction time (f) appearing in the solution of the BR model (Equation A9.1) is replaced by the product pbt (x = V/Vo) in the fixed bed model. A special case of fixed bed is considered in the next section. 

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