HigginsWhitaker_AICHEJ_2012

One application of the atomic species balance given

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Unformatted text preview: centrations, thus surface area averaging16 of the nonuniform catalytic surface is the first step in the sequence suggested in Figure A2. Given the spatially smoothed surface conditions associated with the reference reaction rates, one can upscale those quantities through the hierarchy of length scales indicated in Figure A2 using the method of volume averaging.17 Appendix B: Atomic Species Balances The atomic species balance has some advantages when carrying out calculations by hand because the number of atomic species balance equations is almost always less than the number of molecular species balance equations. We begin our development of atomic species balance equations with Axioms I and II given by Z Z d Axiom I : cA dV þ cA vA Á n dA dt V A Z ¼ RA dV ; A ¼ 1; 2; …; N ðB1Þ V Axiom II : AIChE Journal Z Given that the mechanistic matrix is independent of any upscaling process, we use Eq. A16 to obtain (A16) This result is used in the design of chemical reactors, thus, it is the local volume average form that is needed.15 In terms of the volume V indicated in Figure A2, the quantity of interest is hRMi defined by 1 V A¼N X NJA RA ¼ 0; J ¼ 1; 2; … T (B2) A¼1 To develop an atomic species balance, we multiply Eq. B1 by NJA and sum over all molecular species to obtain Published on behalf of the AIChE DOI 10.1002/aic 549 A¼N X NJA A¼1 d dt Z cA dV þ A¼N X Z Z A¼N X NJA cA vA Á ndA ¼ NJA RA dV A¼1 V A¼1 V Ae (B3) Here, we have made use of the fact that vA Á n is zero everywhere except at the entrances and exits that we have denoted by Ae. On the basis of Axiom II, we see that the last term in this result is zero, and our atomic species balance takes the form A¼N X d NJA dt A¼1 Z cA dV þ A¼N X Z V cA v Á ndA ¼ 0; J ¼ 1; 2; …; T NJA A¼1 Figure B1. Batch and steady-state reactors. Ae (B4) in which we have imposed the very reasonable approximation that vA Á n ¼ v Á n at the entrances and exits. In Eq. B4, we have indicated explicitly that there are T atomic species balance equations instead of the N molecular species balance equations given by Eq. B1. When T N it may be convenient to solve material balance problems using atomic species balances. One application of the atomic species balance given by Eq. B4 is based on the total molar concentration of the J-type atoms that is given by ( cJ ¼ A¼N X ) NJA cA A¼1 8 < 9 total molar = ¼ concentration ; : ; of J -type atoms J ¼ 1; 2; … ; T (B5) Use of this result in Eq. B4 leads to the atomic species macroscopic balance given by d Atomic Species Balance : dt Z Z cJ dV þ V cJ v Á ndA ¼ 0; ðB6Þ This form of the atomic species balance is sometimes useful for solving macroscopic mole balance problems with chemical reactions. The form represented by Eq. B4 is also useful for the analysis of batch reactors and steady-state reactors such as those illustrated in Figure B1. For the batch reactor, we can express Eq. B4 as A¼1 NJA dnA ¼ 0; dt J ¼ 1; 2; ::; T (B7) in which the total number of moles of species A in the batch reactor is given explicitly by Z nA ¼ cA dV V Integration of Eq. B7 from t ¼ 0 to t ¼ tf yields 550 DOI 10.1002/aic t Z¼tf NJA DnA ¼ 0; D nA ¼ A¼1 ðdnA =dtÞdt J ¼ 1; 2; …; T t¼ 0 (B9) and in matrix form the first of these becomes AðDnÞ¼ 0 (B10) This result is analogous to Eq. 13 and one can follow the analysis from Eq. 13 to Eq. 23 to obtain the pivot theorem for a batch reaction Pivot Theorem ðbatch reactorÞ : ðDnÞNP ¼ PðDnÞP (B8) (B11) Turning our attention to the continuous flow reactor (CSTR) illustrated in Figure B1b, we express the flux term in Eq. B4 as Z Z Z  cA v Á ndA ¼ À cA jv Á njdA þ cA jv Á njdA ¼ DnA Ae Aentrance Aexit (B12) A J ¼ 1; 2; …; T A¼N X A¼N X Use of this expression in the steady-state form of Eq. B4 leads to A¼N X  NJA DnA ¼ 0 J ¼ 1; 2; …; T (B13) A¼1 and in terms of the nomenclature used in Eq. 13, we have  AðD n Þ ¼ 0 (B14) This provides another application of the pivot theorem that we express as Pivot Theorem ðCSTRÞ:   ðD nÞNP ¼ PðD n ÞP (B15) Both the Eqs. B11 and B15 provide useful tools for the treatment of batch reactors of the type illustrated in Figure B1. The derivation of these results is based on Axiom I and Axiom II along with the uniqueness proof of the row reduced echelon form of the atomic matrix. Published on behalf of the AIChE February 2012 Vol. 58, No. 2 AIChE Journal Appendix C: Conservation of Charge Ne1 ¼ 0; The axiom associated with conservation of atomic species was originally stated as Ne2 ¼ À2; ionic species such as SO2À 4 Ne3 ¼ þ1; ionic species such as Naþ A¼N X Axiom II : NJA RA ¼ 0; J ¼ 1; 2; :::; T (C1) A¼1 in which N represents the total number of identifiable stable species such as ethane (C2H6), butadiene (C4H6), water (H2O), sulfuric acid (H2SO4), ans so forth. In addition, if we have a solution of sulfuric acid, the species would include ions such as Hþ and SO2À . The matrix form of Eq. C1 is 4 given by 2 2 N11 6 N21 6 6 6 N31 Axiom II : 6 6: 6 6 4: NT 1 N12 N22 N13 :::::: :::::: N1;NÀ1; N2;N...
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