Lumped-parameter method

In order to have detailed understanding of heat emission of each part of PM motor, the thermal model of motor is built in the AMEsim softwcue which its thermal library is bcised on lumped parameter method. The thermal model of wheel-rim PM motor is shown in Figure 9. 

In the lumped-parameter method, the reactor is divided into equal stages and each segment is considered to be a perfectly mixed vessel. This is shown schematically in Figure 21.11. The equation is now written as... 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

London and Seban (L8) introduced the method of lumped parameters in melting-freezing problems, whereby the partial differential equation is converted into a difference-differential equation by differencing with respect to the space variable. The resulting system of ordinary differential... 

We may take another viewpoint by taking into consideration the variation of the direction factor in a process. A simple method to analyze the effect of the distribution of D to decompose the process into multistaged lumped-parameter processes in such a manner that each subsystem may satisfy Eqs. (16) and (17), as shown in Fig. Al. Therefore we have the following equation for the i-th subsystem. 

In addition to the interpretation of lumped parameters, Eqs. 6.133-6.139 may be used to calculate the model parameters from experimentally determined moments c and ot c. This method is described here for the TD model but can also be applied to the ED model. 

Dynamic simulations represent the temporal and the spatial behavior of a chemical process unit in the presence of perturbations or at process startup. There is a natural division in the types of numerical methods used to solve the equations describing the dynamic behavior of the process. In lumped parameter descriptions of the process units, the resulting equations are ordinary time evolution differential equations, whereas for distributed parameter descriptions of process units the resulting equations are parabolic partial differential equations. The numerical methods used to solve these equations are very different and necessitate a separate discussion. Numerical methods used to solve ordinary differential equations describing the dynamics are considered first followed by a discussion of the methods employed to solve evolution equations of the parabolic type. 

Optimization requires that at/R have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 33 jR be large. Thus the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivi-ties. The calculations needed to optimize distributed-parameter systems (i.e., sets of PDEs) are much longer than needed to optimize the lumped-parameter systems (i.e., sets of ODEs) studied in Chapter 6, but the numerical approach is the same and is still feasible using small computers. 

Despite the undisputable power of this approach, it should be noted that in cases with few tracer data and little knowledge about the ground-water system, the simple lumped-parameter models still have their justification (Richter et al. 1993). In any case, it is clear that the full potential of the tracer methods can only be exploited by simultaneous application of several tracer methods and by interpretation of the data in the framework of a model. 

Jones (1974) used the moment transformation of the population balance model to obtain a lumped parameter system representation of a batch crystallizer. This transformation facilitates the application of the continuous maximum principle to determine the cooling profile that maximizes the terminal size of the seed crystals. It was experimentally demonstrated that this strategy results in terminal seed size larger than that obtained using natural cooling or controlled cooling at constant nucleation rate. This method is limited in the sense that the objective function is restricted to some combination of the CSD moments. In addition, the moment equations do not close for cases in which the growth rate is more than linearly dependent on the crystal size or when fines destruction is... 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

Many methods are a vailable to designers. The most common [3-7] are based on the finite element, finite difference or dynamic relaxation, lumped parameter and limit state methods. This book gives earlier the step-by-step approach of the finite element method. In some service and fault conditions it is necessary to consider the influence of external hazards and environmental conditions. Major... 

It has been proposed to represent the TF with 10 CSTR staged units 0.3 ft in length. Develop solutions to this problem using a finite difference method of solving an ordinary differential equation and a lumped parameter model employing a method of solution of simultaneous linear algebraic equations. 

Different models deseribing DEs are derived and verified in this chapter to address the required mathematieal deseription of the transducers behavior. Two inherently different methods based on the finite-strain electromechanics on the one hand and on lumped parameters on the other hand are presented in the following sections. 

Fig. 12 Combined electrical and mechanical lumped parameter model showing the full signal transduction from electrical stimuli to mechanical output. Calculation method for Pe w) may be chosen from Eqs. 38 or 40, according to Sect. 3.2.1 in this chapter...

In some eases, the independent and dependent process variables can vary along a spatial coordinate. A well-known example is a tnbnlar reactor, where temperatures, concentrations and other process variables vary with the axial coordinate of the reactor. In the absence of micro-mixing, the system variables eonld even vary with the radial coordinate of the reactor. Spatial variations of process variables lead to eomplex models and consequently the solution of the model is difficult. Models that aeeonnt for spatial variations are called distributed parameter models. A widely used method that approximates the behavior of distributed parameter systems is the division of the spatial coordinate into small sections, within each section the system properties are assumed to be constant Each section can then be considered as an ideally mixed section and the entire process is approximated by a series of ideally mixed sub-systems. Such a system approximation is cdled a lumped parameter system. In the case of a chain process, such as a distillation tray colmnn, usually every tray is considered to be lumped, i.e. process conditions on each tray are averaged and assumed constant. 

There exist powerful simulation tools such as the EMTP [35]. These tools, however, involve a number of complex assumptions and application limits that are not easily understood by the user, and often lead to incorrect results. Quite often, a simulation result is not correct due to the user s misunderstanding of the application limits related to the assumptions of the tools. The best way to avoid this type of incorrect simulation is to develop a custom simulation tool. For this purpose, the FD method of transient simulations is recommended, because the method is entirely based on the theory explained in Section 2.5, and requires only numerical transformation of a frequency response into a time response using the inverse Fourier/Laplace transform [2,6,36, 37, 38, 39, 40, 41-42]. The theory of a distributed parameter circuit, transient analysis in a lumped parameter circuit, and the Fourier/Laplace transform are included in undergraduate course curricula in the electrical engineering department of most universities throughout the world. This section explains how to develop a computer code of the FD transient simulations. 

The methods listed in Sections 5.1.1-5.1.7 are illustrated by a tubular reactor with a first-order reaction and laminar flow. Models for species, heat, and momentum have been formulated and simplified. In addition to showing the methods, we discuss the assumptions in a traditional 1D lumped-parameter model for a tubular reactor with axial dispersion,... 

The previous simple analysis example follows a pre-computing classical approach where a simple linearized lumped parameter model of the system was developed. In the pre-computing or classical approach this simple model was solved by the application of analytical methods such as Laplace transforms and frequency-response analysis. For completeness, these methods will be briefly introduced here. The interested reader should refer to the texts that take this pre-computing classical approach, such as... 

EIS is an electrochemical measurement tool that can be used to predict performance of electrochemical power electronic systems, such as supercapacitors and batteries [89-92]. EIS method is used to find the equivalent circuit models, where each lump parameter can be explained by physical phenomena occiuring during... 

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