A Dynamic Mathematical Model

Although a dynamic mathematical model of the polymerization system has been developed (17) it is not capable of providing the necessary operating policies for the reactor in order to preselect the time-averaged MWD in the product. Hence the flow policies for the reagents were selected empirically and for experimental convenience. 

Derive a dynamic mathematical model of this system. 

Assuming each reaction is linearily dependent on the concentrations of each reacr-tant, derive a dynamic mathematical model of the system. There are two feed streams, one pure benzene and one concentrated nitric acid (98 wt %). Assume constant densities and complete miscibility. 

Derive a dynamic mathematical model of the flooded-condenser system. Calculate the transfer function relating steam flow rate to condensate flow rate. Using a PI controller with tj = 0.1 minute, calculate the closedloop time constant of the steam flow control loop when a closedloop damping coefTident of 0.3 is used. Compare this with the result found in (u). 

A dynamic mathematical model was developed to match the laboratory data. Very simple first-order kinetics and perfect mixing were assumed. The predictions of the model were found to fit experiments conducted in the 0.02-m3 (5-gal) pilot plant reactor quite well. The final yields and conversions checked well, but more importantly, the time-dependent heat transfer rates predicted by the model and measured in the pilot plant were in close agreement. 

The use of thermogravimetric analysis (TGA) apparatus to obtain kinetic data involves a series of trade-offs. Since we chose to employ a unit which is significantly larger than commercially available instruments (in order to obtain accurate chromatographic data), it was difficult to achieve time invariant O2 concentrations for runs with relatively rapid combustion rates. The reactor closely approximated ideal back-mixing conditions and consequently a dynamic mathematical model was used to describe the time-varying O2 concentration, temperature excursions on the shale surface and the simultaneous reaction rate. Kinetic information was extracted from the model by matching the computational predictions to the measured experimental data. 

A dynamic mathematical model of the three-phase reactor system with catalyst particles in static elements was derived, which consists of the following ingredients simultaneous reaction and diffusion in porous catalyst particles plug flow and axial dispersion in the bulk gas and liquid phases effective mass transport and turbulence at the boundary domain of the metal network and a mass transfer model for the gas-liquid interface. 

There is some uncertainty connected with testing techniques, errors of characteristic measurements, and influence of fectors that carmot be taken into account for building up a model. As these factors cannot be evaluated a priori and their combination can bring unpredictable influence on the testing results it is possible to represent them as additional noise action [4], Such an approach allows to describe the material and testing as a united model — dynamic mathematical model. 

A further (mathematical) model for the evolution of the genetic code was devised by Carl Woese and co-workers. This dynamic theory provides information on the evolvability and universality of the genetic code. One conceptual difficulty was due to the fact that it had been overlooked that the genetic code was highly communal... 

The dynamics of high-temperature CO adsorption and desorption over Pt-alumina was analyzed in detail using a transient mathematical model. The model combined the mechanism of CO adsorption and desorption (established from ultrahigh-vacuum studies over single-crystal or polycrystalline Pt surfaces) with extra- and intrapellet transport resistances. The numerical values of the parameters which characterize the surface processes were taken from the literature of clean surface studies ... 

A third and very important use of dynamic experiments is to confirm the predictions of a theoretical mathematical model. As we indicated in Part I, the verification of the model is a very desirable step in its development and application. 

In the current work, we present a comprehensive approach to the problem dynamic mathematical models for simultaneous reactions media, numerical methodology as well as model verification with experimental data. Design and optimization of industrially operating reactors can be based on this approach. 

Even a simple mathematical model for transport on colloids in an aquifer must include dynamic equations for the dissolved phase and for the colloids. The latter equation describes the migration, immobilization, and detachment of the colloids. More sophisticated models include dynamic equations for sorption and desorption of the chemical onto colloids and the stationary solid phase. 

We will develop detailed steady-state and dynamic mathematical models of CSTRs in Chapters 2 and 3 with several types of reactions and quantitatively explore the effect of kinetic and design parameters on controllability. For the moment, let us just make some qualitative observations. There are several features of a CSTR that impact controllability ... 

The dynamic mathematical model describing the system consists of a total mass balance, two component balances, an energy balance on the reactor liquid, and a jacket energy balance ... 

In the remainder of the chapter, wave dynamics in integrated reaction separation processes will be studied in more detail. The analysis is based on a simple mathematical model, which will be discussed in the following section. 

Mathematical models of biological processes are often used for hypothesis testing and process optimization. Using physical interpretation of results to obtain greater insight into process behavior is only possible when structured models that consider several parts of the system separately are employed. A number of dynamic mathematical models for cell growth and metabolite pro-... 

Foubert, I., Vanrolleghem, P.A., Vanhoutte, B., Dewettinck, K. 2002. Dynamic mathematical model of the crystallization kinetics of fats. Food Res. Int. 35, 945-956. 

HWE] S. B. Hsu, P. Waltman, and S. F. Ellermeyer (1994), A remark on the global asymptotic stability of a dynamical system modeling two species competition, Hiroshima Mathematical Journal 24 435-46. 

One may use an approach discussed earlier in which a comprehensive mathematical model is developed, which relates the reactor hardware to fluid dynamics and polymerization reactions in one framework. However, it is extremely difficult... 

We found a way out of this difficulty by going back and asking what it was that we wanted to find out. First, we want to know whether the dynamic response of a catalyst is "complex," that is, affected by changes in a transient chemical process, as defined above. Second, If the response is complex, we would like to be able to determine whether "accumulation-reaction" processes are present and whether "activity change" processes are present. Only after these first two questions are answered do we need more detailed information that would require a detailed mathematical model. 

Second stage - optimization. In case of having incomplete information about the system "pesticide-environment" determinated mathematical methods of analysis are of little use. That is why, in the block of optimization of the model system, a dynamic stochastic model based on Bellman s method of dynamic programming, has been used. Markov process was taken as a mathematical model of the system (Hovard, 1964). The main goal of the optimization model is to find out the optimal value of X taking into account the ecological negative influence of pesticide. In... 

C. Starbuck, D.A. Lauffenburger, Mathematical model for the effects of epidermal growth factor receptor trafficking dynamics on fibroblast proliferation responses, Biotechnol. Prog. 1992, 8, 132-143. 

For a process already in operation, there is an alternative approach based on experimental dynamic data obtained from plant tests. The experimental approach is sometimes used when the process is thought to be too complex to model from first principles. More often, however, we use it to find the values of some model parameters that are unknown. Although many of the parameters can be calculated from steady-state plant data, some must be found from dynamic tests (e.g., holdups in nonreactive systems). Additionally, we employ dynamic plant experiments to confirm the predictions of a theoretical mathematical model. Verification is a critical step in a model s development and application. 

Finegood D T, Scaglia L, Bonner-Weir S (1995). Dynamics of beta-cell mass in the growing rat pancreas. Estimation with a simple mathematical model. Diabetes. 44 249-256. 

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