# 4.31: Untitled Page 86

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## Chapter 4

*Section* 4.8

Note: Problems marked with the symbol will be difficult to solve without the use of computer software

4‐27. It is possible that the process illustrated in Figure 4‐12 could be analyzed beginning with Control Volume I rather than beginning with Control Volume II.

Begin the problem with Control Volume I and carry out a degree‐of‐freedom analysis to see what difficulties might be encountered.

4‐28. In a glycerol plant, a 10% (mass basis) aqueous glycerin solution containing 3% NaCl is treated with butyl alcohol as illustrated in Figure 4.28.

*Figure 4.28*. Solvent extraction process

The alcohol fed to the extraction tower contains 2% water on a mass basis. The raffinate leaving the extraction tower contains all the original salt, 1.0% glycerin and 1.0% alcohol. The extract from the tower is sent to the distillation column.

The distillate from this column is the alcohol containing 5% water. The bottoms from the distillation column are 25% glycerin and 75% water. The two feed

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streams to the extraction tower have equal mass flow rates of 1000 lbm per hour.

Determine the output of glycerin in pounds per hour from the distillation column.

*Section* 4.9

4‐29. In Sec. 4.8 the solution to the distillation problem was shown to reduce to solving the matrix equation, Au b , in which

Here it is understood that the mass flow rates have been made dimensionless by dividing by lb */ * h . In addition to the matrix

m

A , one can form what is known as

an *augmented matrix*. This is designated by Ab and it is constructed by adding the column of numbers in b to the matrix A in order to obtain Define the following *lists* in *Mathematica* corresponding to the rows of the augmented matrix, Ab.

1

*R*

1 *, * 1 *, * 1 *, *

1000

*R* 2 0 *. * 045 *, * 0 069

*. *

*, * 0 955

*. *

*, *

500

*R* 3 0 *. * 091 *, * 0 *. * 901 *, * 0 *. * 041 *, *

300

Write a sequence of *Mathematica* expressions that correspond to the elementary row operations for solving this system. The first elementary row operation that given Eq. 4‐135 is

*R* 2 ( 0 *. * 045) *R* 2

1

*R *

Show that you obtain an augmented matrix that defines Eq. 4‐139.

4‐30. In this problem you are asked to continue exploring the use of *Mathematica* in the analysis of the set of linear equations studied in Sec. 4.9.2, i.e., Au b where the matrices are defined by

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