Variables lumped

The complex exponential separates into factors involving only a single variable. Lumping the constant S with the mask, we now have for the correction term ... 

Fig. 8. The characteristic function of i vs.

Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

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. 

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. 

The Newton method can be extended to several variables in order to find the zeroes of n functions/ in n variables x which we lump as the vector x = [x,..., x ]T, i.e., to solve the system of equations... 

Given a geochemical variable y, m measurements at times tu t2,..., tm produce the unevenly spaced time series yl5 y2,..., ym, which we lump together as the vector y. In order to find out eventual periodicities, Lomb (1976) suggests fitting the data by a sine wave using a least-square criterion. For any arbitrary frequency /, the fitting function is written... 

We now proceed to m observations. The ith observation provides the estimates xi of the independent variables Xj and the estimate y, of the dependent variable Y. The n estimates xtj of the variables Xj provided by this ith observation are lumped together into the vector xt. We assume that the set of the (n+1) data (i/,y,) associated with the ith observation represent unbiased estimates of the mean ( yf) of a random (n + 1)-vector distributed as a multivariate normal distribution. The unbiased character of the estimates is equivalent to... 

This procedure of lumping all non-idealities into a few adjustable parameters is unsatisfactory for many reasons. Thermodynamic rigor is lost if experimentally determined dissociation constants or vapor pressures are disregarded. Also the parameters determined in this way are accurate only over the range of variables fitted and usually the model cannot be used for extrapolation to other conditions. The attractive feature of these models in the past was their need for little input information and the simple equations could often be solved algebraically. 

We will discuss in this book only deterministic systems that can be described by ordinary or partial differential equations. Most of the emphasis will be on lumped systems (with one independent variable, time, described by ordinary differential equations). Both English and SI units will be used. You need to be familiar with both. 

We have used an ordinary derivative since t is the only independent variable in this lumped system. The units of this component continuity equation are moles of A per unit time. The left-hand side of the equation is the dynamic term. The first two terms on the right-hand side are the convective terms. The last term is the generation term. 

Rhodes CG, Camici PG, Taegtmeyer H, Doenst T. Variability of the lumped constant for [18F]2-deoxy-2-fluoroglucose and the experimental isolated rat heart model clinical perspectives for the measurement of myocardial tissue viability in humans. Circulation 1999 99 1275-1276... 

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 first choose variables sufficient to describe the situation. This choice is tentative, for we may need to omit some or recruit others at a later stage (e.g., if V is constant, it can be dismissed as a variable). In general, variables fall into two groups independent (in our example, time) and dependent (volume and concentration) variables. The term lumped is applied to variables that are uniform throughout the system, as all are in our simple example because we have assumed perfect mixing. If we had wished to model imperfect mixing, we would have had either to introduce a number of different zones (each of which would then be described by lumped variables) or to introduce spatial coordinates, in which case the variables are said to be distributed.2 Lumped variables lead to ordinary equations distributed variables lead to partial differential equations. 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

The forcing variable is the coolant temperature jc2c as in Sincic and Bailey (1977) and more recently in Mankin and Hudson (1984). In eq. (10) jti is a dimensionless reactant concentration while x2 is a dimensionless reactor temperature. These equations hold at the limit of infinite reaction activation energy. All models were thus chosen so that extensive simulation results existed in the literature, and they cover a wide range of lumped reactor types. 

Very occasionally bricks are still made by means of hand moulding. A lump of clay is forcefully thrown into a mould and then the surplus clay is removed. The result is a lively and variable product. When large quantities of bricks are produced the process does not differ much from that of hand moulding. However, now the clay is pressed into a mould and a rather uniform product is the result, e.g. the common brick. 

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