Variables tabulated

Although enthalpies of substances are relatively independent of pressure (for gases) and of concentration (for dissolved species), their entropies, and thus the free energies as well, depend markedly on these variables. Tabulated values for 5 and G usually refer to the idealized state of 1 bar or 1 atm pressure for gases, 1 M concentration for solutes, and to the.pure substances for liquids and solids. Table... 

Tb date the aerosol chamber studies are underway and the final results will be published at a later date. However, preliminary data are shown for several variables tabulated above, including chamber volume, air flow, and inserts. 

A surface whose equation is a quadratic in the variables x, y, and z is called a quadric surface. Some of the more common such surfaces are tabulated and pictured in Figs. 3-29 to 3-37. 

Basically, the tabled values represent area (proportions or probability) associated with a scaling variable designated by Z in Fig. 3-56. The normal cui ve is centered at 0, and for particular values of Z, designated as z, the tabulated numbers represent the corresponding area under the cui ve between 0 and z. For example, between 0 and 1 the area is. 3413. (Get this number from Table 3-4. The value of A includes the area on both sides of zero. Thus we want A/2. For z = 1, A = 0.6827, A/2 = 0.3413. For z = 2, A/2 = 0.4772.) The area between... 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

Effective wavelengths have been included in Table 3-1 to show the changes that occur in this important variable when one gas is substituted for another. These wavelengths correspond to mass absorption coefficients calculated from Equation 3-14 and were obtained by interpolation from tabulated values of absorption coefficients for different wavelengths.15... 

Maximum values of specific conductance are often not achievable without exceeding maximum T alkalinity values, especially in boilers below 900 psig (6.21 MPa) with greater than 20.0% MU water whose alkalinity is >20% of TDS naturally or after pretreatment by lime-soda or sodium cycle ion exchange softening. Actual permissible conductance values to achieve any desired steam purity must be established for each case by careful steam purity measurements. The relationship between conductance and steam purity is affected by too many variables to allow its reduction to a simple list of tabulated values. 

Tabulate T and all state variables at each communication interval. 

PROC MEANS, PROC SUMMARY, and PROC TABULATE are other SAS procedures that you can use to get descriptive statistics and place them into output data sets. However, those procedures do not offer any descriptive statistical variables that you cannot get from PROC FREQ or PROC UNIVARIATE. 

If we let and t2 represent the times corresponding to reaction progress variables and <5J, respectively, the time ratio t2/tl for fixed values of <5 and <5 will depend only on the ratio of rate constants k. One may readily prepare a table or graph of <5 versus k t for fixed k and then cross-plot or cross-tabulate the data to obtain the relation between k and ktt at a fixed value of <5. Table 5.1 is of this type. At specified values of <5 and S one may compute the difference log(fe1t)2 — log f) which is identical with log t2 — log tj. One then enters the table using experimental values of t2 and tx and reads off the value of k = k2/kv One application of this time-ratio method is given in Illustration 5.5. 

In the following program, my table is called table(nent, nvar), in which the first column is the independent variable, time, which must increase monotonically down the table, and the other columns are one or more tabulated variables. In this example there is only one dependent variable,... 

Equation 2-4 provides a relationship between the probability P and the probit variable Y. This relationship is plotted in Figure 2-9 and tabulated in Table 2-4. 

From equation 14.3-19, with MA defined by equation 4.3-4. From equation 13.52, with 3.4-10 or 14.3-20 and 13.4-2. dE is an exponential integral defined by E x) = y le y dy, where y is a dummy variable the integral must be evaluated numerically (e.g., using E-Z Solve) tabulated values also exist. 

POLYMATH ODE is used for the integration in preference to Constantinides ODE because it allows printout of algebraic as well as differential variables. In the tabulation, x = C, f = 0 and e = tj. 

The variables in Eq (4) are separable. Some of the integrated results of P and na are tabulated. Then Vr is found with Eq (2), using trapezoidal integration. Some of those results also are shown. 

The version of ISAT described here requires that all tabulated points have the same At and pressure p. However, by adding extra variables in the definition of , this restriction can easily be overcome. For example, by defining 0JVi+l(t) = t — to and. V, = 1, the last component of the reaction mapping will be Un i 16/>o) = At. The... 

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