Dimensioning variables

In a sense, it is like trend analysis it looks at the relationship of sets of data from a different perspective. In the case of Fourier analysis, the approach is by resolving the time dimension variable in the data set. At the most simple level, it assumes that many events are periodic in nature, and if we can remove the variation in other variables because of this periodicity (by using Fourier transforms), we can better analyze the remaining variation from other variables. The complications to this are (1) there may be several overlying cyclic time-based periodicities, and (2) we may be interested in the time cycle events for their own sake. 

Since there are many trays and most are described by Eqs. (5.29) through (5.32), it is logical to use dimensioned variables and to evaluate derivatives and integrate using FORTRAN DO loops. It also makes sense to use a SUBROUTINE or FUNCTION to find given x , because the same equation is used over and over again. 

More ordinary difl erential equations must be added per tray. We need one per component per tray. But this is easily programmed using doubly dimensioned variables X(N, J), where JV is the tray number and J is the component number. 

Ascostromata enclosing axillarys buds at the nodes of culms, dimensions variable, black, corniform, papillate perithecia immersed, lageniform to obpyriform, with a protruding ostiole, no paraphyses, asci cylindrical with a hemispherical apical cap ascospores in each ascus, filiform, hyaline, 7-septate, ultimately becoming 3-septate conidiophores simple, occasionally branched,... 

Secondly, the design of the sampling and testing is chosen to provide information that is fit for purpose . To achieve this careful consideration is given to both the sampling strategy and the choice of measurement methods employed in the field or the laboratory (see Section 4). A critical consideration is that contaminated land can only be properly described by considering three dimensions. Variability in composition and properties with depth is as important as that with surface position. 

Simultaneous solution of the convective diffusion equations for mass and heat must be done numerically in all but trivial cases. The solutions can be based on dimensioned variables like z and T, and this has the advantage of keeping the physics of the problem close at hand. However, the solutions are then quite specific and must be repeated whenever a design or operating variable is changed. Somewhat more general solutions, while still numerical, can be obtained through the judicious use of dimensionless variables, dimensionless parameters, and dimensionless functions. Table 8.1 defines a number of such variables. 

Dimensioned Variable Dimensionless Variable Type of Variable... 

The usual way of defining a dimensionless variable is to divide the dimensioned variable by a quantity with the same dimensions that characterize the system. There are sometimes several possibilities. Reasonable choices for Tref include Tin, Twaii,... 

Having obtained a solution to a given problem in dimensionless terms, we might wish to go back to dimensioned variables, in order to compare with an experiment. This simply reverses the normalizing equations. Thus, we have... 

Although it is possible to derive the necessary mathematical equations in real , dimensioned quantities, it is often advantageous to use a more general, dimensionless formulation. This may involve both the current, and the time/potential variables. Often, however, in the literature only the current is normalized, while the potential is retained as a dimensioned variable, only referred to the formal potential of the electrode process. 

Eliminating q and q i and turning back to the dimensioned variables leads to... 

Using these dimensionless results, now back substitute to recover the dimensioned variables in which the problem was stated. 

To evaluate the capability of the given formulas in practice, one has to rewrite them in dimension variables. 

Returning to the dimension variables y, z, the solution (4.5) takes the form... 

Thus, when e is small and jo < 1 the polarization curve of the CCL is linear. Equation (2.46) in dimension variables takes the form... 

In dimension variables this equation has the form (2.29), obtained above for an ideal catalyst layer. As seen, Eq. (2.29) holds when oxygen transport is ideal, 1 and jo > 2. The notion of ideal oxygen transport will be rationalized in Section 2.5. 

In dimension variables this equation takes the form... 

Hence the critical time is given by the quantity L/A. Transforming Eqn. 156 back into dimensioned variables, we obtain... 

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