Model distributed parameters

The optimal eontrol profiles identified by the solution of the non-linear programme were used to simulate the network through rigorous distributed parameter models on SPEEDUP to obtain a detailed deseription of its... 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Examples 9.1 and 9.2 used a distributed parameter model (simultaneous PDEs) for the phthalic anhydride reaction in a packed bed. Axial... 

Development of a distributed parameter model will rely on data obtained in vivo. Time and spatial dependencies of drug concentration in a target organ are used as the basis to estimate parameters by nonlinear regression analysis. Distribu-... 

RK Jain, J Wei. Dynamics of drug transport in solid tumors Distributed parameter models. J Bioeng 1 313-330, 1977. 

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model 1561 can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model 1541 should be used. 

Distributed models (usually called distributed parameter models , which is again wrong terminology since the state variables are distributed, not the parameters). These are models for systems for which the state variables are distributed along one or several spatial directions. They are described by ODEs or PDEs for the steady-state and by PDEs for the unsteady-state case. 

Another situation when the use of the statistical model can be a good choice over the RSM is when the deterministic model is excessively complex. For example, when the process is described by a distributed parameters model, the steady-state mass and energy balances are differential equations. The use of differential equations as constraints in an optimization problem makes its solution difficult and increases the incidence of convergence problems. In this case, solving the optimization problem using the statistical model is much simpler. The statistical model can also be used when the computational effort to solve the optimization problem using the deterministic model is too high, as can be the case for real-time optimization problems. 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

This model can be applied not only to a tank-type equipment but also to a tubular type equipment. Mathematically, the use of the SPMV model implies that the mixing process is expressed by a lumped parameter model and not by a distributed parameter model. The RTD function in this model is expressed as... 

In addition, there are a lumped and distributed parameter models. 

Distributed parameter model The parameter is continuously distributed in the system. 

Most commonly, distributed parameter models are applied to describe the performance of diesel particulate traps, which are a part of the diesel engine exhaust system. Those models are one- or two-dimensional, non-isothermal plug-flow reactor models with constant convection terms, but without diffusion/dispersion terms. 

The aforementioned controllers were implemented on the full-order 2006-dimensional discretization of the original distributed-parameter model, and their performance was tested through simulations. The relevant Matlab codes are presented in Appendix C. 

Freedman, B. G., Paper No 61C, "Nonlinear Distributed Parameter Model of a Gaseous Feed Ethylene Furnace", the 68th AIChE National Meeting, Houston, 1971. 

These authors [32, 33] have considered an alternative classification based on the nature of the variables involved in the model. They classify models by grouping them into opposite pairs deterministic vs. probabilistic, linear vs. non-linear, steady vs. non-steady state, lumped vs. distributed parameters models. In a lumped parameters model, variations of some variable (usually a spatial one) are ignored and its value is assumed to be uniform throughout the entire system. On the other hand, distributed parameters models take into account detailed variations of variables throughout the system. In the kinetic description of a chemical system, lumping concerns chemical constituents and has been widely used (see Sects. 2.4 and 2.5). 

However, such a complex system would not be helpful to describe organic-removal wastewater-treatment processes because of its high degree of complexity and, therefore, in an attempt to achieve a useful model, some assumptions could be made in order to simplify the model. Hence the transformation of this distributed-parameter model in a simpler lumped-parameter model is very common in the modeling of wastewater-treatment processes, because it is not very important to obtain detailed information about what happens in every point of the cell but simply to know in a very simple way how the pollution of a influent waste decreases at the outlet of the electrochemical cell. In this context, there are three types of approaches typically used ... 

Freedman, B. G., Nonlinear Distributed Parameter Model of a Gaseous... 

Automatic control of the flow rates or the switching time to meet purity specifications is difficult due to the extremely long time delays and complex dynamics described by nonlinear distributed parameter models, and mixed discrete and continuous dynamics, leading to small operating windows and a strongly nonlinear response to input variations. 

Hackenberg, J., Krobb, C., Marquardt, W. An object-oriented data model to capture lumped and distributed parameter models of physical systems. In Troch, I., Breitenecker, F. (eds.) Proceedings of the 3 MATHMOD, IMACS Symposium on Mathematical Modelling, Vienna, Austria, pp. 339-342 (2000)... 

K), nj is the effectiveness factor for reaction j (to be computed from the diffusion reaction equation at each point in the reactor), (—AHj) is the heat of reaction for reaction j (KJ/Kg mole), Rt is the catalyst tube radius, m, U is the overall heat transfer coefficient in (KJ/mP.K) and To> is the wall temperature (K) which is determined through the coupling between the above model equations for the catalyst tube and the model equations for the combustion chamber (the furnace). A distributed parameter model for the combustion chamber is being developed by (CREG) for both top fired and side fired furnaces. 

The steady states which are unstable using the static analysis discussed above are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. That is to say simply that branch 2 in Figure 4.8 is always unstable, while branches 1,3 in Figure 4.8 and branch 4 in Figure 4.8 can be stable or unstable depending upon the dynamic stability analysis of the system. As mentioned earlier, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped parameter models or a distributed parameter model made discrete by a technique such as the orthogonal collocation technique. However, in the latter case, the system dimensionality will increase considerably, with n dimensions for each state variable, where n is the number of internal collocation points. 

This example found the reactor throughput that would give the required annual production of product B. For prescribed values of the design variables T and V there may not be a solution. If there is a solution, it is unique. The program uses a binary search to And the answer, but another root finder could be used instead. For the same accuracy, Newton s method (see Appendix 4.1) requires about 3 times fewer function evaluations (i.e., calls to the Reactor subroutine). The saving in computation time is trivial in the current example but could be important if the reactor model is a complicated, distributed parameter model such as those in Chapters 8 and 9. 

We developed a distributed parameter model that describes the spatial and temporal growth of granulation tissue during wound-healing (Zawicki... 

Analogous to the experimental approaches discussed in the previous section, mathematical models have been developed to describe mass transfer at all three levels—cellular, multi-cellular (spheroid), and tissue levels. For each level two approaches have been used—the lumped parameter and distributed parameter models. In the former approach, the region of interest is considered to be a perfectly mixed reactor or compartment. As a result, the concentration of each region has no spatial dependence. In the latter approach, a more detailed analysis of the mass transfer process leads to information on the spatial and/or temporal changes in concentrations. Models for single cells and spheroids were reviewed in Section III,A and are part of the tissue-level models (Jain, 1984) hence, we will focus here only on tissue-level models. 

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