Equations numbering

Those exponents which we have discussed expUcitly are identified by equation number in Table 4.3. Other tabulated results are readily rationalized from these. For example, according to Eq. (4.24) for disk (two-dimensional) growth on contact from simultaneous nucleations, the Avrami exponent is 2. If the dimensionality of the growth is increased to spherical (three dimensional), the exponent becomes 3. If, on top of this, the mechanism is controlled by diffusion, the... 

Equation numbers refer to those relationships discussed explicitly in text. 

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

Equation derived from equation number Differential form da dt Integral form kt = Exponents in 7= kam (1 -a)n dr x (—ln(l — a))p Rising temperature expression... 

A star next to an equation number signals that it appears in the list of Key Equations on the Web site for this book www.whfrccman.com/ chemicalprinciplcs4e. 

The ambiguity over the sign in the denominator is due to the fact that both versions, (n,- and (n, -i- 1),47A9 are found in the literature ( + resp. behind the equation number (1.22) indicate the sign used in the denominator). Equation (1.22-) appears to be correct Eq. (1.22-I-) might have arisen from transcription errors. 

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

Fluid Reference Range of NRe and/or eB where applicable Correlation Equation number... 

Fortunately, none of the equations developed in this chapter, nor the figures, used any suffix, as was occasionally done in previous chapters (we do refer to equation 42-20b above, but that equation is in a previous chapter and we will not repeat the use of equation 42-20 here. We will also copy equation 43-52b from the previous column, but the b suffix does not signify a new equation, since it is the equation used previously also, a b suffix is not indicative of a copy of an equation number in this section). Therefore, we can distinguish the numbering of any equations or other numbered entities in this... 

Our equation numbering system now causes us to jump from equation 51-24 to 51-135 for our next equation number ... 

Equation for calculating result, y Combined uncertainty Equation number... 

Equations are numbered anew in each (Roman numeraled) section. Within each section, equations from within that section are referred to simply by their numbers, while equations from other sections are designated by the equation number preceeded by a Roman numeral for the section. Thus, Eq. (3) of Section II would be referred to as (3) within Section II, but in other Sections as (II—3). 

Figure 4.18 The interrelationship between experiments and the spectra. The numbers in brackets refer to the equation numbers...

Steward s algorithm, thus, can help locate errors in formulating a model of a process by identifying the set of equations that contains improperly specified design variables or parameters. The fortran program listed in Appendix A prints out the equation numbers in the subset that contains fewer variables than equations when an output set cannot be found. 

Table XI lists the equation numbers and the variables that appear in each equation. The numbers in parentheses are the assigned numbers of each variable. Only five of the material balances are independent, and those selected are designated as Eqs. 1-5 in Table XI. Altogether there are six independent material balances, but because the overall balance for NaOH can be solved directly for F, F is removed from the role of a variable and the number of independent material balances can be reduced to five. The five selected were...

PAIRS ALTERNATING BETWEEN EQUATION NUMBER AND VARIABLE NUMBER. THE ORIGINAL ASSIGNMENT OF EQUATION AND VARIABLE NUMBERS ARE ARBITRARY. ENTER IN COLUMNS 1-3 AN EQUATION NUMBER. ENTER IN COLUMNS 5-7 ONE OF THE VARIABLE NUMBERS IN THE SELECTED EQUATION. IN COLUMNS 9-11 ENTER AN EQUATION NUMBER (THE SAME EQUATION OR ANOTHER) AND IN COLUMNS 13-15 ENTER THE NUMBER OF ONE OF THE VARIABLES IN THE SPECIFIED EQUATION. 

CONTINUE INTRODUCING EQUATION NUMBERS AND VARIABLE NUMBERS UNTIL ALL THE NONZERO ELEMENTS HAVE BEEN ACCOUNTED FOR. EACH CARD MUST BE FILLED COMPLETELY, EXCEPT THE LAST CARD WHICH MUST HAVE TWO BLANK FIELDS, OR THE READING WILL STOP. 

IN PAIRS AS ON DATA CARDS IN GROUP NO. 2. ENTER IN THE FIRST FIELD OF THE PAIR AN EQUATION NUMBER AND IN THE SECOND FIELD THE NUMBER OF A VARIABLE IN THE EQUATION WHICH CANNOT BE ASSIGNED AS THE OUTPUT VARIABLE OF THAT EQUATION. REPEAT FOR... 

Type of Measure Measure Symbol Equation Number Remark... 

Organic pollutant type Sorbent type Predictive models of partition coefficient, koc and kow values Equation number... 

Where an equation number is not given, the appropriate equation including C is given in Reference (9). 

Figure 1 is the summary of the two block PLS algorithm using the equation numbers. 

X R Preparation equation number Spectroscopic evidence Reference... 

EQUATION NUMBER (DEPENDENT VARIABLE) Step Variable Entered F to Enter Variable P Step Variable Entered F to Enter Variable P... 

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