Lumped-parameter

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

Mechanical systems are usually considered to comprise of the linear lumped parameter elements of stiffness, damping and mass. 

The PC version runs comparatively slow on large problems. FIRAC can perform lumped parameter/control volume-type analysis but is limited in detailed multidimensional modeling of a room or gaa dome space. Diffusion and turbulence within a control volume is not modeled. Multi-gas species are not included in the equations of state. 

EXP AC analyzes an interconnected network of building rooms and ventilation systems. A lumped-parameter formulation is used that includes the effects of inertial and choking flow in rapid gas transienl.s. The latest version is specifically suited to calculation of the detailed effects of explosions in the far field using a parametric representation of the explosive event. A material transport capability models the effects of convection, depletion, entrainment, and filtration of... 

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

TABLE V. EXPERIMENTAL VALUES FOR LUMPED PARAMETER PROPAGATION ... 

Lumped parameter propagation function Denoting difference Average residence time... 

The lumped parameter model of Example 13.9 takes no account of hydrodynamics and predicts stable operation in regions where the velocity profile is elongated to the point of instability. It also overestimates conversion in the stable regions. The next example illustrates the computations that are needed... 

In what follows we share our practical experience of the design of difference schemes for problems with lumped parameters by means of the IIM. Suppose, for instance, that a single heat source of capacity Q is located at a point x — so that a solution of problem (l)-(2) satisfies the conditions... 

One significant result from the studies of stretched premixed flames is that the flame temperature and the consequent burning intensity are critically affected by the combined effects of nonequidiffusion and aerodynamic stretch of the mixture (e.g.. Refs. [1-7]). These influences can be collectively quantified by a lumped parameter S (Le i-l)x, where Le is the mixture Lewis number and K the stretch rate experienced by the flame. Specifically, the flame temperature is increased if S > 0, and decreased otherwise. Since Le can be greater or smaller than unity, while K can be positive or negative, the flame response can reverse its trend when either Le or v crosses its respective critical value. For instance, in the case of the positively stretched, counterflow flame, with k>0, the burning intensity is increased over the corresponding unstretched, planar, one-dimensional flame for Le < 1 mixtures, but is decreased for Le > 1 mixtures. 

The basic concepts of the above lumped parameter and distributed parameter systems are shown in Fig. 1.7. 

In some cases, where the wall of the reactor has an appreciable thermal capacity, the dynamics of the wall can be of importance (Luyben, 1973). The simplest approach is to assume the whole wall material has a uniform temperature and therefore can be treated as a single lumped parameter system or, in effect, as a single, well-stirred tank. 

Figure 3.9. Model representation of temperature distribution in the wall and jacket, showing wall and Jacket with four lumped parameters.

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

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). 

Wulff, W, 1978, Lump-Parameter Modeling of One-Dimensional Two-Phase Flow, in Transient Two-Phase Flow, Proc. 2nd Specialists Meeting, vol. 1, pp. 191-219, OECD Committee on Safety of Nuclear Installations, Paris. (3)... 

In series with a desolvation energy barrier required to disrupt aqueous solute hydrogen bonds [14], the lipid bilayer offers a practically impermeable barrier to hydrophilic solutes. It follows that significant transepithelial transport of water-soluble molecules must be conducted paracellularly or mediated by solute translocation via specific integral membrane proteins (Fig. 6). Transcellular permeability of lipophilic solutes depends on their solubility in GI membrane lipids relative to their aqueous solubility. This lumped parameter, membrane permeability,... 

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. 

Qualitatively, all proposals indicate a linear dependence on ml (linewidth over a hyperfine pattern increases from low to high field or vice versa cf. Figure 9.4) plus a quadratic dependence on m, (outermost lines more broadened than inner lines). Multiple potential complications are associated with the lump parameters A, B, C, notably, their frequency dependence (Froncisz and Hyde 1980), partial correlation with g-strain (Hagen 1981), and low-symmetry effects (Hagen 1982a). The bottom line quantitative description of these types of spectra has been for quite some time, and still is, awaiting maturation. 

Comparison of the Experimental and Simulation Results. The preceding discussion has shown that both the experimental anthracene fluorescence profiles and the simulated anthracene concentration profiles decrease in a manner which closely follows an exponential decay. Therefore, the most convenient way to compare the simulation results to the experimental data is to define an effective overall photosensitization rate constant, kx or k2, as described above. Adoption of this lumped-parameter effective kinetic constant allows us to conveniently and efficiently compare the experimental data to the simulation results by contrasting the rate constant obtained from the steady-state fluorescence decay with the value obtained from the simulated decrease in the anthracene concentration. 

The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

Only for the intermediate cases - those with velocities in the range of about 100 m yr-1 to 1000 m yr-1 - does silica concentration and reaction rate vary greatly across the main part of the domain. Significantly, only these cases benefit from the extra effort of calculating a reactive transport model. For more rapid flows, the same result is given by a lumped parameter simulation, or box model, as we could construct in REACT. And for slower flow, a local equilibrium model suffices. 

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