Mathematical steady state

Mathematically, steady-state is never reached within a finite time. For practical purposes, however, one can compute the time necessary to reach steady-state by imposing the condition that a given transient magnitude (concentration or flux) differs from the steady-state value in a reasonably low relative proportion [42], For calculating the proximity to steady-state, the diffusive flux Jm is more convenient than the internalisation flux /u, because of the continuously decreasing behaviour with time of the former. 

Although it takes >711/2 to reach mathematical steady state, by convention clinical steady state is accepted to be reached at 4r-5 t1/r... 

A non-mathematical, steady-state approach to enzyme kinetics has been... 

The mathematical description of the echo intensity as a fiinction of T2 and for a repeated spin-echo measurement has been calculated on the basis that the signal before one measurement cycle is exactly that at the end of the previous cycle. Under steady state conditions of repeated cycles, this must therefore equal the signal at the end of the measurement cycle itself For a spin-echo pulse sequence such as that depicted in Figure B 1.14.1 the echo magnetization is given by [17]... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this sec tion. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. 

The effect of the disturbance on the controlled variable These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exac t, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the abihty to predict the appropriate control ac tion). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standara prac tice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. 

When a process is continuous, nucleation frequently occurs in the presence of a seeded solution by the combined effec ts of mechanical stimulus and nucleation caused by supersaturation (heterogeneous nucleation). If such a system is completely and uniformly mixed (i.e., the product stream represents the typical magma circulated within the system) and if the system is operating at steady state, the particle-size distribution has definite hmits which can be predic ted mathematically with a high degree of accuracy, as will be shown later in this section. 

Only a very few experimental studies have been made for detection of mnlti-plicities of steady states to check on theoretical predictions. The studies of multiplicities and of oscillations of concentrations have similar mathematical bases. Comprehensive reviews of these topics are by Schmitz (Adv. Chem. Sen, 148, 156, ACS [1975]), Razon and Schmitz (Chem. Eng. Sci., 42, 1,005-1,047 [1987]), Morbidelli, Vamia, and Aris (in Carberry and Varma, eds.. Chemical Reaction and Reacton Engineering, Dekker, 1987, pp. 975-1,054). 

Two types of interac tion, competition, and predation are so important that worthwhile insight comes from considering mathematical formulations. Assuming that specific growth-rate coefficients are different, no steady state can be reached in a well-mixed continuous culture with both types present because, if one were at steady state with [L = D, the other would have [L unequal to D and a rate of change unequal to zero. The net effect is that the faster-growing type takes over while the other dechnes to zero. In real systems—even those that approximate well-mixed continuous cultures—there may be profound... 

The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

The purpose of our study was to model the steady-state (capillary) flow behavior of TP-TLCP blends by a generalized mathematical function based on some of the shear-induced morphological features. Our attention was primarily confined to incompatible systems. 

Tye [38] explained that separator tortuosity is a key property determining the transient response of a separator (and batteries are used in a non steady-state mode) steady-state electrical measurements do not reflect the influence of tortuosity. He recommended that the distribution of tortuosity in separators be considered some pores may have less tortuous paths than others. He showed mathematically that separators with identical average tortuosities and porosities can be distinguished by their unsteady-state behavior if they have different distributions of tortuosity. 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

This chapter takes up three aspects of kinetics relating to reaction schemes with intermediates. In the first, several schemes for reactions that proceed by two or more steps are presented, with the initial emphasis being on those whose differential rate equations can be solved exactly. This solution gives mathematically rigorous expressions for the concentration-time dependences. The second situation consists of the group referred to before, in which an approximate solution—the steady-state or some other—is valid within acceptable limits. The third and most general situation is the one in which the family of simultaneous differential rate equations for a complex, multistep reaction... 

We shall consider the same scheme, Eq. (4-28), with a simplifying assumption. Imagine that intermediate I is so reactive that it does not accumulate at an appreciable level compared to A or P. This situation is very common, and the limit in which it is valid is referred to as the steady-state region (subscripted ss ). We shall first examine its consequences and then explore the mathematical requirements. 

One way to carry out the steady-state derivations is to set d[l]/dt = 0. As we shall see, however, this mathematical prerequisite is unduly strict (and even impossible) if taken literally. If we proceed in this manner nonetheless, Eq. (4-30) then becomes... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

A mechanical system, typified by a pendulum, can oscillate around a position of final equilibrium. Chemical systems cannot do so, because of the fundamental law of thermodynamics that at all times AG > 0 when the system is not at equilibrium. There is nonetheless the occasional chemical system in which intermediates oscillate in concentration during the course of the reaction. Products, too, are formed at oscillating rates. This striking phenomenon of oscillatory behavior can be shown to occur when there are dual sets of solutions to the steady-state equations. The full mathematical treatment of this phenomenon and of instability will not be given, but a simplified version will be presented. With two sets of steady-state concentrations for the intermediates, no sooner is one set established than the consequent other changes cause the system to pass quickly to the other set, and vice versa. In effect, this establishes a chemical feedback loop. 

Toward these ends, the kinetics of a wider set of reaction schemes is presented in the text, to make the solutions available for convenient reference. The steady-state approach is covered more extensively, and the mathematics of other approximations ( improved steady-state and prior-equilibrium) is given and compared. Coverage of data analysis and curve fitting has been greatly expanded, with an emphasis on nonlinear least-squares regression. 

The polymerization system for which experiments were performed is represented by the mathematical model consisting of Equations 1 and 7. Their steady state solutions are utilized for kinetic evaluation of rate constants. Dynamic simulations incorporate viscosity dependency. 

It gives a false impression of certainty. T5q i-cally at least one of the fluxes in a cycle is calculated by the imposed mathematical necessity of balancing a steady-state budget. Such estimates may erroneously be taken to represent solid knowledge. 

Even if satisfactory equations of state and constitutive equations can be developed for complex fluids, large-scale computation will still be required to predict flow fields and stress distributions in complex fluids in vessels with complicated geometries. A major obstacle is that even simple equations of state that have been proposed for fluids do not always converge to a solution. It is not known whether this difficulty stems from the oversimplified nature of the equatiorrs, from problems with ntrmerical mathematics, or from the absence of a lamirrar steady-state solution to the eqrratiorrs. 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Shimao, K, Mathematical Simulation of Steady State Isoelectric Eocusing of Proteins using Carrier Ampholytes, Electrophoresis 8, 14, 1987. 

R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vol. I The Theory of the Steady State, Clarendon, Oxford, 1975. 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

---