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4.22: Untitled Page 77

  • Page ID
    18210
  • Chapter 4

    and the last of these quickly leads us to the result for

    .

    m

     780 lb / h

    (4‐111)

    5

    m

    The algebraic complexity associated with the simple process represented in Figure 4‐10 encourages the use of matrix methods. We begin a study of those methods in the following section and we continue to study and to apply matrix methods throughout the remainder of the text. Our studies of stoichiometry in Chapter 6 and reaction kinetics in Chapter 9 rely heavily on matrix methods that are presented as needed. In addition, a general discussion of matrix methods is available in Appendix C1.

    4.9 Matrix Algebra

    In Sec. 4.7 we examined a distillation process with the objective of determining molar flow rates, and our analysis led to a set of two equations and two unknowns given by

    Species A:

    (4‐112a)

    Species B:

    (4‐112b)

    Some thought was necessary in order to set up the macroscopic mole balances for the process illustrated in Figure 4‐7; however, the algebraic effort required to solve the governing macroscopic balances was trivial. In Sec. 4.8, we considered the system illustrated in Figure 4‐10 and the analysis led to the following set of three equations and three unknowns:

    species A:

    (4‐113a)

    species B:

    (4‐113b)

    Total:

    (4‐113c)

    The algebraic effort required to solve these three equations for

    ,

    and

    was considerable as one can see from the solution given by Eq. 4‐108. It should not be difficult to imagine that solving sets of four or five equations can become exceedingly difficult to do by hand; however, computer routines are available that can be used to solve virtually any set of equations have the form given by Eqs. 4‐113.

    Multicomponent systems

    136

    In dealing with sets of many equations, it is convenient to use the language of matrix algebra. For example, in matrix notation we would express Eqs. 4‐113

    according to

    (4‐114)

    In this representation of Eqs. 4‐113, the 3  3 matrix of mass fractions multiplies the 3  1 column matrix of mass flow rates to produce a 3  1 column matrix that is equal to the right hand side of Eq. 4‐114. In working with matrices, it is generally convenient to make use of a nomenclature in which subscripts are used to identify the row and column in which an element is located. We used this type of nomenclature in Chapter 2 where the m n matrix A was represented by (4‐115)

    Here the first subscript identifies the row in which an element is located while the second subscript identifies the column. In Chapter 2 we discussed matrix addition and subtraction, and here we wish to discuss matrix multiplication and the matrix operation that is analogous to division. Matrix multiplication between A and B is defined only if the number of columns of A (in this case n) is equal to the number of rows of B . Given an m n matrix A and an n p matrix B , the product between A and B is illustrated by the following equation: (4‐116)

    Here we see that the elements of the ith row in matrix A multiply the elements of the jth column in matrix B to produce the element in the ith row and the jth column of the matrix C . For example, the specific element

    is given by

    11

    c

    137