Mathematical and experimental

Georgakopoulos and Broucek (1987) investigated the effect of recycle ratio on non-ideality, both mathematically and experimentally. They investigated two cases from which the bypass case b was completely uninteresting, because total bypass of the catalyst bed could be avoided by feeding the makeup directly to the location of highest sheerfield, at the tip of the impeller blade. For their case a they showed on their Fig. 3. that from a recycle ratio of about 10 = 32 there was no observable falsification effect. This matched well the conclusion of Pirjamali et al. 

On the other hand, in a sufficient number of cases a definite equilibrium is undoubtedly reached in a short time, and if we confine ourselves to these, it becomes possible to approach the second question we have put, that referring to the connection between concentration and amount adsorbed. Among the investigators who have treated this problem both mathematically and experimentally Freundlich deserves to be mentioned particularly. 

This paper presents mathematical and experimental studies of solids deposition characteristics in the horizontal pipeline of a dense phase slug flow pneumatic conveying... 

Piringer O, Franz R, Huber M, Begley T H and McNeal T P, 1998, Migration from food packaging containing a functional barrier mathematical and experimental evaluation. J. Agric. Food Chem. 46, 1532-1538. 

Formal development of HMRDE, leading to the general method of EHD spectroscopy notably pioneered by Delouis and Tribollet [1], was given by Tokuda et al. [29], Albery et al. [15], Deslouis et al. [18], Tribollet and Newman [30] and Deslouis and Tribollet [31], who have also summarised and reviewed the mathematical and experimental details of the method and its applications [1, 19]. 

Common principles occur in different areas of science, often under different names, and are introduced in conceptually radically different guises. In many cases the driving force is the expectations of the audience, who may be potential users of techniques, customers on courses or even funding bodies. Sometimes even the marketplace forces different approaches students attend courses with varying levels of background knowledge and will not necessarily opt (or pay) for courses that are based on certain requirements. This is especially important in the interface between mathematical and experimental science. 

Mathematically and experimentally there is no reason to connect the vessels linearly, to restrict the source to the left-hand vessel, or to keep the washout rates D equal so long as the volume of the fluid in each vessel is kept constant (see [S7]). We next describe a class of gradostat models which is sufficiently general to include all cases of biological interest and yet remain mathematically tractable. 

Djakovic, L.M. Dokic, P.P. Sefer, I.B. Mathematical and experimental essentials of the emulsification process optimal parameters determination. J. Disp. Sd. Tech. 1989,70(1), 59-76. Continuous Ultrasonic Processing Cell Misonix Corporation Farmingdale, NY, 1998. 

Although the MM equation is a powerful kinetic form to which the vast majority of enzyme kinetics has been fitted, one should not forget the assumptions and limitations of the model. As a basic example, feedback inhibition, whereby the product of the reaction inhibits the enzyme-substrate cooperativity, multiple-substrate reactions, allosteric modifications, and other deviations from the reaction scheme in equation (1) are treated only adequately by the MM formalism under certain experimental conditions. In other words, enzyme kinetics are often bent to conform to the MM formalism for the sake of obtaining a set of parameters easily recognizable by most biochemists. The expUcit mathematical and experimental treatment of reaction mechanisms more complex than that shown in equation (1) is highly involved, although a mathematical automated kinetic equation derivation framework for an arbitrary mechanism has been described in the past (e.g., ref. 6). 

The program of catastrophe theory has been formulated in the Thom s book Structural Stability and Morphogenesis. The philosophical, mathematical and experimental incentives to the program of catastrophe theory may be found there. On the position of catastrophe theory among methods of investigation of stability of solutions of non-linear equations (models) one may learn from a book by Thompson. Stewart s and Zeeman s papers on the perspectives of the development of generalized catastrophe theory are also worth study. 

Kim CS, Brown LK, Lewars GC, Sackner MA. Deposition of aerosol particles and flow resistance in mathematical and experimental airway models. J Appl Physiol 1983 55 154-163. 

Isothermal frontal polymerization (IFP) is a self-sustaining, directional polymerization that can be used to produce gradient refractive index materials. Accurate detection of frontal properties has been difficult due to the concentration gradient that forms from the diffusion and subsequent polymerization of the monomer solution into the polymer seed. A laser technique that detects tiny differences in refractive indices has been modified to detect the various regions in propagating fronts. Propagation distances and gradient profiles have been determined both mathematically and experimentally at various initiator concentrations and cure temperatures for IFP systems of methyl methacrylate with poly(methyl methacrylate) seeds and wilh the thermal initiator 2,2 -azobisisobutryonitrile. 

Figure 9. Mathematical and Experimental Front Propagation at 5(fC (Mathematically) and 47-52°C (Experimentally). The Mathematical Propagation extends off the graph to 3.02 cm in 12.24 hours for 0.15% AIBN and to 4.64 cm in 28.6 hrs for 0.03% AIBN. The standard deviation on the experimental data, if larger than the markers used, is shown by vertical bars. (The velocity is the slope of the propagation.)...

Figure 10. Mathematical and Experimental Gradient Profiles (dC/dy) at 50 °C (Mathematically) and 47-52 °C (Experimentally) for 0.03 % AIBN in MMA. Curves with markers of 0, 0, and represent experimental runs. The mathematical results are for Dm-2.5X10 cm /s,f = 0.5, h = 0.03% (2 mmol/L), Mo = 9.0 mol/L, and the Arrehenius rate parameters forAIBN/MMA...

Mathematical and experimental propagation distances and times reveal qualitative trends where an increase in initiator concentration and/or cure temperature causes an increase in the propagation and velocity. Experimental propagation is on the order of 0.55 0.11 cm, except for 0.03 % AIBN in MMA at 42 to 47 (1.03 0.08 cm), and ranges from less than one hour to nineteen... 

AIBN in MMA at 42 to 47 °C. Mathematically, the propagation ranges from 1.4 cm to 7.4 cm for systems of 0.15% AIBN in MMA at 67 C to systems of 0.03% AIBN in MMA at 40 °C. Both the mathematical and experimental velocities (the slopes of the propagation curves) increase during the course of the reaction regardless of reaction conditions. This fact is in agreement with predicted results that the monomer solution polymerizes and increases in viscosity with time shortening the necessary time for the diffused monomer in the reaction zone to reach the desired viscosity for the gel effect, and thus the front, to occur. 

The differences between the mathematical and experimental results can result from the fact that various parameters of the mathematical model (e.g. the monomer diffusion coefficient) do not change with time as they do over the course of the experiment. Also, various parameters such as the monomer diffusion coefficient and the initiator efficiency, f, are obtained from literature where the experimental conditions, e.g. the solvent and/or cure temperature, are not the same as our experimental conditions. 

In this article, the mathematical and experimental methods for generation and evaluation of derivative spectra are discussed. 

Two other types of undesirable time behavior can occur in homogeneous reactors, but have not yet been seriously studied because of mathematical and experimental difficulties. Both arise because of the turbulent nature of the flow through the reactor core, in which the fluid is necessarily unconstrained by piping. The first is probably relatively trivial both in its effects and in its computational difficulty. The core fluid is non-uniform in density because different portions of it have suffered varying thermal histories since entrance into the core. The eddying which necessarily accompanies turbulent flow will consequently cause forced oscillations of reactivity, which could in principle be violent enough to damage the reactor. In practice, however,... 

A goal of IFF research is to obtain a model that accurately describes the experimental results to be used as a predictive tool [6,11, 30-32]. In the next sections, we review the most recent mathematical and experimental IFF research by discussing the mathematical models developed and their characteristics (e.g., properties calculated, trends within the data, and theories involved) and by discussing the experiments performed (e.g., data obtained, trends within the data, and techniques used to obtain the information). 

Here the mathematical and experimental aspects of linear flooding will be considered. In this case, the only polymer flooding mechanism that may operate is the improved microscopic displacement that is possible when a lower viscosity drive water displaces a more viscous oil. First a simple mathematical fractional flow approach to analysing the improved oil recovery mechanism will be examined. Results from corresponding 1-D oil displacement experiments using water and polymer flooding are then presented in order to illustrate some of the points that arise. 

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