Dimensionless Mathematical Model

Consider the exothermic first-order reaction A — B taking place batchwise at reactor temperature Tr and coolant temperature 7j. The mathematical model describing the system is given by the mass balance on reactant A and the energy balance in the reactor  

the usually small variations of the parameters (pT, cr, U, A Hr) and of the concentration CA with respect to temperature is neglected. 

For a more systematic approach, the model equations can be rewritten in terms of dimensionless time, concentration, and temperature defined, respectively, as 

The dimensionless numbers introduced above can provide, in the light of their physical meaning, some preliminary information about the system behavior  

According to the definition (4.8), operation with very low P results in higher reactor temperatures and higher risk of runaway, because of ineffective reactor cooling. On the contrary, in the limit P - oo, the reactor temperature approaches 7] during the entire reaction cycle. 

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Table 3.12 Dimensionless mathematical models for the case of the heating of a block of four bricks.

Table 3.14 Dimensionless mathematical model forthe heat and mass transfer in a plug flow reactor for a fast exothermic reaction.

For the case when the dimensionless state of a global function is preferred for the computation, then we can operate with dimensionless process variables and with a dimensionless mathematical model of the process. However, we can also operate with dimensional variables but with partly dimensionless functions. 

MATHEMATICAL MODELING DIMENSIONLESS NUMBERS GOVERNING ELECTROCHEMICAL PROMOTION AND METAL-SUPPORT INTERACTIONS... 

Zhou et al. [55], The most effective method to assess the capacity is the flow simulation which includes volumetric formulas and more reservoir parameters rather than other methods [56], Mass balance and constitutive relations are accounted in mathematical models to capacity assessment and dimensional analysis consists of fractional flow formulation with dimensionless assessment and analytical approaches [33], From the formulations demonstrated by Okwen and Stewart for analytical investigation, it can be deduced that the C02 buoyancy and injection rate have affected the storage capacity [57], Zheng et al. have indicated the equations employed in Japanese and Chinese methodology and have noted that some parameters in Japanese relation can be compared to the CSLF and DOE techniques [58]. 

Mathematical modeling of the three-stage bed collapsing process led to a more quantitative characterization of powders, in the form of a dimensionless number called th e dimens ionless subsidence time ... 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 53, in terms of the mass fraction x2 of fines, and two empirical parameters n, and n2 ... 

The solution of mathematical model (Eqs. 17,18) is controlled exclusively by two dimensionless parameters... 

In order to simplify the mathematical model of the reactor, the following dimensionless variables are introduced... 

In order to investigate this behavior, we consider the mathematical model of the reactor given by Eq.(l) or (4), and assuming that the reactor is at the steady state corresponding at point P3 of Figure 2 in [1]. The disturbances of the dimensionless inlet stream temperature and the coolant flow rate are the following ... 

Eq.(34) are a set of differential equations, which lead a flow in a four dimensional phase space R. This flow can be simplified to three dimensional phase space R when the dynamics of the jacket can be considered negligible respect to the reactor s dynamics. Putting dxA/dr = 0 the dimensionless jacket s temperature X4 can be eliminated from Eq.(34), and the simplified mathematical model of the reactor can be written as... 

The viability of one particular use of a membrane reactor for partial oxidation reactions has been studied through mathematical modeling. The partial oxidation of methane has been used as a model selective oxidation reaction, where the intermediate product is much more reactive than the reactant. Kinetic data for V205/Si02 catalysts for methane partial oxidation are available in the literature and have been used in the modeling. Values have been selected for the other key parameters which appear in the dimensionless form of the reactor design equations based upon the physical properties of commercially available membrane materials. This parametric study has identified which parameters are most important, and what the values of these parameters must be to realize a performance enhancement over a plug-flow reactor. 

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

Two mathematical models have been described in the literature for characterizing normal flow.4" Each of these defines the RTD curve, as a function of a single parameter. The dispersion model uses the dimensionless Pec let number for this parameter since this is a ratio between convective and dispersive forces. 

Keywords Mathematical modelling dimensionless parameters aesthetics. 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

The following reviews scale-up of chemical reactors, considers the dimensionless parameters, mathematical modeling in scale-up, and scale-up of a batch system. 

A differential equation describing the material balance around a section of the system was first derived, and the equation was made dimensionless by appropriate substitutions. Scale-up criteria were then established by evaluating the dimensionless groups. A mathematical model was further developed based on the kinetics of the reaction, describing the effect of the process variables on the conversion, yield, and catalyst activity. Kinetic parameters were determined by means of both analogue and digital computers. 

We denote by 07 = Hi/HijS the dimensionless variables corresponding to the energy flow rates Hiy i = 1,..., N (the subscript s denotes steady-state values). Appending a generic representation of the overall and component material-balance equations, with xtfc IRm being the material-balance variables, the overall mathematical model of the process in Figure 6.1 becomes... 

In this case study we will model, simulate and design an industrial-scale BioDeNOx process. Rigorous rate-based models of the absorption and reaction units will be presented, taking into account the kinetics of chemical and biochemical reactions, as well as the rate of gas-liquid mass transfer. After transformation in dimensionless form, the mathematical model will be solved numerically. Because of the steep profiles around the gas/liquid interface and of the relatively large number of chemical species involved, the numerical solution is computationally expensive. For this reason we will derive a simplified model, which will be used to size the units. Critical design and operating parameters will be identified... 

The reactive absorption was modeled based on the film theory. After transforming in dimensionless form, the mathematical model was solved numerically. This required a considerable computing effort. The results obtained in this way allowed the formulation of reasonable assumptions, which were the starting point of a... 

Dimensionless group ratio Simplification of mathematical model to be allowed... 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 9b. Thus, the dimensionless time 0X for the coarse particles could be assumed to exert a mass-fraction-based influence on the fine particles, proportional to 0 1 — x2), which is affected by certain interaction by the fines, inclusive of their ability to adhere to the surface of the coarse and form clusters among themselves, lumped in certain appropriate form, for instance, [1 + /(x2)], where the function /(x2) may again be assumed to possess certain appropriate form, for instance, exponential, x2, where n1 may be called the interactive exponent. This results in an overall contribution by the coarse particles, suitably corrected for the interaction of the fine particles, 0 1 — x2Xl + x2l). This function has the property of accommodating the following boundary conditions ... 

Thin solid films of polymeric materials used in various microelectronic applications are usually commercially produced the spin coating deposition (SCD) process. This paper reports on a comprehensive theoretical study of the fundamental physical mechanisms of polymer thin film formation onto substrates by the SCD process. A mathematical model was used to predict the film thickness and film thickness uniformity as well as the effects of rheological properties, solvent evaporation, substrate surface topography and planarization phenomena. A theoretical expression is shown to provide a universal dimensionless correlation of dry film thickness data in terms of initial viscosity, angular speed, initial volume dispensed, time and two solvent evaporation parameters. 

For a correct perception of relation (3.128), we must notice that this is a heat sink that keeps its constant temperature due to a rapid heat exchange between the surface with a cooling medium maintained at constant (t j,) temperature. The assembly of relations (3.124)-(3.126) represents in fact an abstract mathematical model for the above described heating case because the numerical value is given neither for the system geometry nor for the material properties. Apart from the temperature, all the other variables of the model can be transformed into a dimensionless form introducing the following dimensionless coordinates ... 

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