Since the specified values of (i/,)co are to have a sum equal to the specified value of D, the desired value of 0 is that 0 > 0 that makes g(0) = 0, where m=i(d,U-D (2-27)

A graph of this function for 0 > 0 is shown in Fig. 2-4.

In the determination of 0 by Newton's method, the following formula for the first derivative, g'(0), is needed m=-t n(2"28)

After the desired value of 0 has been obtained, (b,)co may be computed by use of Eq. (2-25). (Note, Newton's method converges to the positive root of g(0\ provided 0 = 0 is taken to be the first trial value; see Prob. 2-11). For the case where the dew-point temperature of the distillate is specified instead of the distillate rate D, the g function has the form shown in Prob. 2-13.

Figure 2-4 Geometrical Representation of the function g(6) in the neighborhood of the positive root 0.

Figure 2-4 Geometrical Representation of the function g(6) in the neighborhood of the positive root 0.

The corrected mole fractions for the liquid and vapor phases are computed as follows v (Iji/ddJdiU

The development of these formulas as well as the proof of the fact that they are consistent with the definition of 0 is left as an exercise for the student (see Prob. 2-15).

Determination of a Set of Improved Temperatures by Use of the Kb Method

On the basis of the mole fractions given by Eq. (2-29) and the last temperature profile (the one assumed to make the nth trial), the new temperature profile is found by use of the method13 in the following manner. For any plate j9

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