Lumped-parameter analysis

First, we note that with few exceptions RTD studies represent lumped parameter analysis. Only in a few instances, such as that of fully developed laminar flow of fluids in tubes, one can attempt distributed parameter analysis in that the expressions for the RTD functions C and F can be derived from known velocity profiles. 

The Sandia code HECTR (Hydrogen Event Containment Transient Response) is a lumped parameter analysis code for modeling the containment atmosphere under accident conditions involving release, transport, and also combustion of hydrogen [37]. It can handle saturated and supe eated conditions and it covers both short-term transients and long-term convection up to several hundred days. 

This chapter discussed the art of modeling, with a few examples. Regardless of the type of model developed, a mathematical model should be validated with experimental results. Validation becomes very important in the black-box type of models such as the neural network models. Moreover, the model results are valid only within certain regimes where the model assumptions are valid. Sometimes, any model can be fit to a particular set of data by adjusting the parameter values. The techniques of parameter estimation were not presented in this chapter, the presentation was limited to lumped-parameter analysis, or macroscopic modeling. 

Gomez A. E. et al. Lumped parameter analysis of cyhndrical prestressed concrete reactor vessels. University of Illinois, Chicago Vols 1,2, USA, 1968. 

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. 

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. 

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. 

Define the composition of the (waste) water. Which individual compounds organic as well as inorganic are present Can the type of compound at least be determined How can it be quantified (lumped parameter, individual analysis) ... 

V. Balakotaiah and D. Luss. Analysis of multiplicity patterns of a CSTR. Chem. Eng. Commun. 13, 111 (1981) 19, 185 (1982). See also these authors papers Structure of the steady state solutions of lumped-parameter chemically reacting systems. Chem. Eng. Sci. 37, 43,1611 (1982) Multiplicity features of reacting systems. Dependence of the steady-states of a CSTR on the residence time. Chem. Eng. Sci. 38, 1709 (1983) Global analysis of multiplicity features of multi-reaction lumped-parameter systems. Chem. Eng. Sci. 39, 865 (1984). 

Balakotaiah, V. and Luss, D., 1984, Global analysis of the multiplicity features of multi-reaction lumped-parameter systems. Chem. Engng ScL 39, 865-881. 

The results from this analysis can now be used to construct geometrically accurate models of the diffusive transport in porous polymers. Previous models of diffusion in these polymers have used an empirically determined tortuosity factor as a lumped parameter to account for the retardation of release by all mechanisms (7-8). 

We describe here a new technique based on the singularity and bifurcation theories for predicting the multiplicity features of lumped-parameter systems in which several reactions occur simultaneously. Our purpose is mainly to illustrate the power of the technique and present some novel results. A more detailed analysis is presented elsewhere [jL, 2]. 

Balakotaiah, V., Luss, D., and Keyfitz, B., Steady State Multiplicity Analysis of Lumped-Parameter Systems Described by a Set of Algebraic Equations, Chem. Eng. Comm. 36 (1982) 121-147. 

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. 

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. 

In this analysis, the dielectric elastomer transducer and energy harvesting circuits were modeled with highly simplified, experimentally validated lumped parameter models that did not include the interactions resulting from Eq. 3.1. Such models also did not include aU of the nonlinear effects detailed in Sect. 3.2 above. The circuits were also modeled separately using more specialized circuit modeling... 

Even though the cells present in an activated sludge bioreactor represent a mixture of microorganisms, the analysis of such facilities is normally based on a lumped-parameter approach that simplifies the kinetics of BOD removal. In many respects the resulting analysis resembles that developed in Section 13.2.4, the major difference being the inclusion of a term to account for the kinetics of cell death. For example, in the present analysis, we can express the net rate of cell growth per unit mass of microorganism as... 

Leaks, back pressure, and actuator dynamics all influence the performance of peristaltic pumps. Leaks and back pressure effects will alter the distribution of fluid as actuators open and close. The dynamics of the actuators determines the maximum actuatitm rate, which in turn limits the maximum flow rate. These effects can be incorporated into lumped-parameter models for analysis and simulation. We do not pursue this further in this entry, but refer the reader to the literamre. One general approach is presented in Ref. [7]. Also relevant for further smdy are Refs. [6, 8], and [13], which present dynamic models for pneumatic and electrostatic pumps, respectively. All of these works are applied to liquid pumps. For gas pumps, or robustness to bubbles, compressibility becomes a factor. Some considerations of micropumps for compressible fluids may be found in Ref. [1]. Finite-element analysis of individual chambers can also be used to obtain detailed predictions of pump dynamic performance. 

The most difficult data to obtain for the lumped parameter equation with reasonable accuracy are those forradiative and convective heat fluxes, flame emissiv-ity and the convection heat transfer coefficient. These data are also key for the transient thermal and structural collapse analysis. It is therefore important to know how the possible inaccuracies in these data influence the time-to-loss-of-strength of a structure. This is examined by the following simulations. 

WILLIAMS, W.R., ANDERSON, J.C., Estimation of time to rupture in a fire using 6FIRE, a lumped parameter UFg cylinder transient heat transfer/stress analysis model ,... 

In this analysis, making various assumptions, we have formulated a lumped-parameter deterministic model to predict the number of engineers (biomedical) present in the United States at any given time. If we want to know the geograi cal distribution, we can take two a noaches. We can divide the entire United States into a number of compartments (e.g., northeast, east, west, etc.) and study the intercompartmental diffusion. Alternatively, we can make E a continuous variable in space and time, I(x,y,t), and account for spatial diffusioiL... 

With an effective thermal model of the cells, modules and overall system, an analysis of the performance under different situations and load conditions can be evaluated. This proves to be a very useful tool in the development of the pack as these thermal models can be input into computational fluid dynamic (CFD) models to determine how the cells will heat during operation. A good CFD model can be used to determine flow rates, turbulence, and heat transfer within a pack. In addition, it is possible to use a lumped parameter model to develop a simplified model where the external parameters are basically ignored and the model is designed using fully adjustable parameters to do high-level evaluations of the thermal effectiveness of a system. 

Lumped-parameter elements. High-frequency and microwave filters may rely on distributed-parameter elements however, much of the analysis and many of the design procedures are applicable to distributed-parameter filters. 

Many methods are available to the designers. The most common are [14, 15] based on the finite element, finite difference or dynamic relaxation lumped parameter and the limit state methods. Appendices A and B give the step-by-step approach of the finite element analysis, Chapter 3 and limited state method used for these vessels is given in this chapter. In some service and fault conditions it is required to consider the influence of external hazards and environmental conditions. Major external hazards are due to seismic disturbances, wind/local generated missiles and aircraft crashes. These are fully dealt with by a number of researchers or designers [231]. 

Thus, the output of the omnidirectional condenser microphone depends on the minute displacement Asound pressure Pq. The diaphragm displacement can be obtained by an equivalent circuit analysis in a lumped-parameter system as follows. 

---