Lumped-parameter systems

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. 

Flow patterns in a stirred tank (lumped parameter system) and a tubular reactor (distributed parameter system). 

The PSR is a lumped parameter system, where the temperature is uniform over the entire reactor. As a result, the fuel and NOj, emissions are strongly temperature dependent. As extinction is approached in the PSR, the radical mole fractions decrease sharply, and so do the NO, species. Thus, turbulent mixing in a PSR is responsible for the high sensitivity of NOj, to temperature. 

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., 1982b, Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter-systems. Chem. Engng ScL 37,433-445. 

We consider three very simple lumped parameter systems, described by similar algebraic equations, namely... 

Models are either dynamic or steady-state. Steady-state models are most often used to study continuous processes, particularly at the design stage. Dynamic models, which capture time-dependent behavior, are used for batch process design and for control system design. Another classification of models is in terms of lumped parameter or distributed parameter systems. A lumped parameter system... 

The steady-state behavior of lumped parameter systems is characterized by a set of algebraic equations that have the form ... 

We consider a generic class of reaction-separation process systems, such as the one in Figure 3.1, consisting of N units (modeled as lumped parameter systems) in series, with one material recycle stream. 

Assuming that the individual process units can be modeled as lumped-parameter systems, the mathematical model that describes the overall and component material balances of the process takes the form... 

Assuming that individual process units are modeled as lumped-parameter systems and that kinetic and potential energy contributions are negligible, the... 

The qualitative multiplicity features of a lumped-parameter system in which several reactions occur simultaneously can be determined in a systematic fashion by finding the organizing singularities of the steady-state equation and its universal unfolding. To illustrate the technique we determine the maximal number of solutions of a CSTR in which N parallel, first-order reactions with equal and high activation energies occur as well as the influence of changes in the residence time on the number and type of solutions. 

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]. 

The steady-state equations describing lumped parameter systems in which several reactions occur simultaneously contain a very large number of parameters. Thus, it is impractical to conduct an exhaustive parametric study to determine their features. The new technique presented here predicts qualitative features of these systems such as the maximum number of solutions, parameter values for which these solutions exist and all the local bifurcation diagrams. Construction of the three varieties enables the division of the global parameter space into regions with different bifurcation diagrams. 

With the dimension of multivariable MFC systems ever increasing, the probability of dealing with a MIMO process that contains an integrator or an unstable unit also increases. For such units FIR models, as used by certain traditional commercial algorithms such as dynamic matrix control (DMC), is not feasible. Integrators or unstable units raise no problems if state-space or DARMAX model MFC formulations are used. As we will discuss later, theory developed for MFC with state-space or DARMAX models encompasses all linear, time-invariant, lumped-parameter systems and consequently has broader applicability. 

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 implications of the non-monotonic behaviour phenomenon were studied on a lumped parameter system (Elnashaie and Yates, 1973 Bykov and Yablonskii, 1981 Luss, 1986), and found to give a multiplicity of the steady states. However, for the steam reforming reaction, this non-monotonicity does not give rise to multiplicity of the steady states. 

Stability, on the other hand, is strongly connected with the multiplicity phenomenon as discussed in chapter 4 for simple lumped parameter systems. In fixed bed catalytic reactors, the situation is much more complicated and may give rise to extremely complex behaviour. A relatively simple and practically oriented discussion of the problem is given by McGreavy (1984). 

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. 

Example 9.1 used a distributed-parameter-system of simultaneous PDEs for the phthalic anhydride reaction in a packed bed. Axial dispersion is a lumped-parameter system of simultaneous DDEs that can also be used for a packed bed. Apply the axial dispersion model to the phthalic reaction using D as determined from Figure 9.7 and = 1.33 D. Compare your results to those obtained in Example 9.1. 

While a rigorous treatment of the evaporator would make due allowance for its distributed nature, very useful and surprisingly accurate results are achieved by treating it as a lumped-parameter system, using the boiling model presented in Section 12.2. This may be used to determine the temperature and pressure inside the evaporator, and the split between water and steam. 

The relatively simple, lumped-parameter system model described above has been tested against and used in earnest to analyse the behaviour of the boiler recirculation loops of a number of power stations. It has been found to give excellent quantitative predictions of all the variables whose trends are important for control engineering purposes, namely steam drum pressure and temperature, feedwater flow, steam production, downcomer flow and, very important, drum water level. 

Regarding region B as a lumped parameter system of volume (with Vb=Xe A) fed by the outlet from region A, the average enzyme concentration... 

Figure 7.24 Actual and lumped parameters system. VA is the dominating volume portion where enzyme depletion takes place VB is the constant volume in which enzyme accumulation occurs.30...

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... 

Jones (1974) used control vector iteration on the lumped parameter system resulting from the moment transformation of the population balance to determine the cooling policy that maximizes the terminal size of the seed crystals subject to 7 (0e[7o. Tf] for all te[0, tf. This problem, along with the additional constraint... 

---