HigginsWhitaker_AICHEJ_2012

8 of noble5 this means that we can use a series of

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Unformatted text preview: begin this analysis with the idea that the atomic matrix can always be expressed in row reduced echelon form, and a uniqueness proof is given in Section 3.8 of Noble.5 This means that we can use a series of ‘‘elementary row operations’’ and column/row operations to express Eq. 12 in the form (10) Axiom II : 2 3 3 R1 N1N 6 7 6 R2 7 7 N 2 N 76 76 76 R3 7 7 76 76 : 7 7 76 7 76 76 : 7 7 56 7 4 RN À 1 5 NTN RN 23 0 67 607 67 ¼ 607 ð12Þ 67 67 4:5 0 (11) A¼1 2 3 R  36 17 N1N R  76 27 7 N2N 7 6 6 R3 7 7 7 : 76 76 : 7 6 7 7 : 76 76 :7 7  76 NWN 7 6 7 6 RT 7 7 7 0 76 76 7 6 RT þ 1 7 7 56 : 7 4:5 0 RN 23 0 607 67 67 607 ¼6 7 ð14Þ 6:7 67 67 4:5 0 We refer to NJA as the ‘‘atomic species indicator,’’ and we identify the array of coefficients associated with NJA as the ‘‘atomic matrix.’’2 Here, we note that Axiom II has been identified by Eqs. 2, 5, 8 and 11, and this is acceptable because each one of these representations can be derived from the others.3 The earliest reference that the authors have found for Eq. 11 is given by Eq. 159A.3 of Truesdell and Toupin.4 AIChE Journal February 2012 Vol. 58, No. 2 Here, we see that we have W rows of nonzero values and T À W rows of zeros. This indicates that the rank of the atomic matrix is r ¼ rank ¼ W and that we have T À W linearly dependent equations in the set of T equations. As the row rank and the column rank must be the same, we have N À W linearly dependent columns. The rank of the atomic matrix may be less than T when two or more rows of the atomic Published on behalf of the AIChE DOI 10.1002/aic 539 matrix are a linear combination of each other. This occurs if two or more atomic elements appear in each molecular species in the same ratio. For example, in a system containing benzene (C6H6) and acetylene (C2H2) the atomic matrix has r ¼ rank ¼ 1 with T ¼ 2. In compact notation we express Eq. 14 as Aà R ¼ 0 (15) in which A* is the row reduced echelon form of the atomic matrix. The form of the atomic matrix given in Eq. 14 is crucial to the efficient application of Axiom II, and this form is ‘‘not universally identified’’ as the ‘‘row reduced echelon form.’’ Sometimes one encounters the following definition of a row reduced echelon matrix: 1 The first nonzero element in any nonzero row is 1 (called a ‘‘leading 1’’). 2 The leading 1 in each nonzero row appears in a column in which every other element is 0. 3 In any two successive rows with nonzero elements, the leading 1 of the lower row occurs farther to the right than the leading 1 of the higher row. 4 Rows containing only zero elements are grouped together at the bottom. An example of an atomic matrix having these particular characteristics is the following 2 3 12030405 60 0 1 7 0 8 0 97 7 A¼6 (16) 40 0 0 0 0 0 1 15 00000000 and we refer to this as simply a ‘‘row reduced form.’’ In the atomic matrix, one can interchange columns without affecting the result as long as the attendant rows of R are also interchanged. In terms of Eq. 11, this column/ row interchange is expressed as NJB RB ! NJD RD ; B; D ¼ 1; 2; :::; N and J ¼ 1; 2; :::; T ð17Þ Returning to Eq. 16, we follow this rule column #2 with column #3 to obtain 2 102304 60 1 0 7 0 8 0 C2 ! C3 : A ¼ 6 40 0 0 0 0 0 000000 We can now interchange column provide 2 1003 60 1 0 7 C3 ! C7 : Aà ¼ 6 40 0 1 0 0000 and interchange 0 0 1 0 3 5 97 7 15 0 (18) #3 with column #7 to 0 0 0 0 4 8 0 0 2 0 0 0 3 5 97 7 15 0 DOI 10.1002/aic (20) In compact notation, this can be expressed as ! RNP ¼0 ½ I WŠ RP (21) Here, I represents the (T À W)  (T À W) identity matrix, W represents the W  [N À (T þ 1)] matrix of coefficients, RNP represents the ‘‘nonpivot species’’ column matrix of net rates of production, and RP represents the ‘‘pivot species’’ column matrix of net rates of production. Carrying out the matrix multiplication in Eq. 21 leads to IRNP þ WRP ¼ 0 (22) We now define the pivot matrix as P ¼ ÀW to obtain the pivot theorem given by RNP ¼ P RP Pivot Theorem: (23) Although Eqs. 2, 5, 11, and 12 often provide useful information, it is the pivot theorem that represents the most important result one can obtain from Axiom II. In the development of the pivot theorem, the choice of pivot and nonpivot species is not arbitrary. To obtain the row reduced echelon form indicated by Eq. 14, it is a necessary condition that all the atomic species be present in at least one nonpivot species. This condition can be achieved by the initial arrangement of the atomic matrix or by the operations indicated in Eq. 17. When dealing with single independent reactions, such as the complete combustion of ethane by ‘‘homogeneous reaction’’ with oxygen illustrated in Figure 1, one can ‘‘balance’’ the chemical reaction in terms of a ‘‘picture’’ by counting atoms to obtain 1 7 3 C2 H6 þ O2 ! H2 O þ CO2 2 4 2 (19) which is the author’s version of a ‘‘row reduced echelon form.’’ This form follows directly if we replace Statement #3 in the above definition of t...
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