HigginsWhitaker_AICHEJ_2012

A6 in the form 2 axiom ii a r 0 3 ra 6 rb 7 67 r6 7

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: sic continuum point of view and assume that this result is valid everywhere. That is to say that Axiom II is valid in homogeneous regions where RA changes slowly and it is valid in interfacial regions where RA changes rapidly. We follow the work of Wood et al.13 and consider the cÀj interface illustrated in Figure A1. The volume V encloses the cÀj interface and extends into the homogeneous regions of both the c-phase and the jphase. The total net rate of production of species A in the volume V is represented by Z RA dV ¼ V Z RAc dV þ Vc Z RAj dV þ Vj V NJA RA dV ¼ A¼1 Z þ Acj Z A¼N X Vc A¼N X A¼1 Vc Z þ Vj A¼N X RAs dA NJA RAc dV þ A¼1 Z A¼N X Vj NJA RAs dA; NJA RAj dV A¼1 NJA RAs dA; J ¼ 1; 2; :::; T A¼N X NJA RAc ¼ 0; A¼1 A¼N X NJA RAj ¼ 0; J ¼ 1; 2; :::; T J ¼ 1; 2; :::; T A¼1 (A5) This leads to the obvious constraint on the heterogeneous rates of production, RAs, and we summarize our results associated with Axiom II as A¼N X NJA RA ¼ 0; J ¼ 1; 2; :::; T (A6) A¼1 Axiom II ðhomogeneous; c-phaseÞ A¼N X NJA RAc ¼ 0; A¼1 J ¼ 1; 2; :::; T ðA7Þ Axiom II ðhomogeneous; j-phaseÞ A¼N X NJA RAj ¼ 0: A¼1 J ¼ 1; 2; :::; T ðA8Þ NJA RAj dV A¼1 ðA4Þ At this point we require that the homogeneous net rates of production satisfy the two constraints given by (A2) Acj A¼N X A¼1 Axiom II (general) Here, the dividing surface that separates the c-phase from the j-phase is represented by Acj, and the heterogeneous rate of production of species A is identified by RAs. This quantity is also referred to as the surface excess reaction rate.14 Multiplying Eq. A2 by the atomic species indicator and summing over all molecular species leads to Z A¼N X Z NJA RAc dV þ Acj NJA RA ¼ 0 A¼1 Z A¼N X 0¼ Axiom II ðheterogeneous; c-j interface A¼N X NJA RAs ¼ 0: A¼1 J ¼ 1; 2; :::; T ðA9Þ (A3) A¼1 From Eq. A1, we see that the left-hand side of this result is zero, and we have These results can be obtained for a more general system (see Ref. 14) in which the cÀj interface represents a moving and deforming surface. For a reactor in which only homogeneous reactions occur (see Figure 1), we make use of Eq. A6 in the form 2 Axiom II: A R ¼ 0; 3 RA 6 RB 7 67 R¼6 : 7 67 4:5 RN (A10) For a catalytic reactor in which only heterogeneous reactions occur at the cÀj interface (see Figure 2), we make use of Eq. A9 in the form 2 Axiom II: Figure A1. Catalytic surface. 548 DOI 10.1002/aic Published on behalf of the AIChE A Rs ¼ 0; 3 RAs 6 RBs 7 6 7 Rs ¼ 6 : 7 6 7 4:5 RNs February 2012 Vol. 58, No. 2 (A11) AIChE Journal The pivot theorem associated with homogeneous reactions is obtained from Eq. A10, and the analysis leads to Eq. 23, which is repeated here as Pivot Theorem (homogeneous reactions): RNP ¼ P RP (A12) The pivot theorem associated with heterogeneous reactions is obtained from Eq. A11 and is given here as Pivot Theorem (heterogeneous reactions): ðRs ÞNP ¼ P ðRs ÞP (A13) Here we note that the axiom for the conservation of atomic species takes exactly the same form for homogeneous reactions (Eq. A10) and for heterogeneous reactions (Eq. A11). In addition, the application of the principle of conservation of atomic species takes exactly the same form for homogeneous reactions (Eq. A12) and for heterogeneous reactions (Eq. A13). The fact that both the axiom and the application take exactly the same form for homogeneous and heterogeneous reactions has led many to ignore the difference between these two distinct forms of chemical reaction. In general, measurement of the net rate of production is carried out at the macroscopic level, thus, we normally obtain experimental information for the global net rate of production. For a homogeneous reaction, this takes the form Z RA ¼ RA dA; A ¼ 1 ; 2 ; ::: ; N (A14) V whereas the global net rate of production for a heterogeneous reaction is given by Z RA ¼ RAs dA; A ¼ 1 ; 2 ; ::: ; N (A15) A cj Here, we note that the global net rates of production for both homogeneous reactions and heterogeneous reactions have exactly the same physical significance, thus, it is not unreasonable to use the same symbol for both quantities. Given this simplification, the global version of the pivot theorem represented by Eq. 40 applies to both homogeneous and heterogeneous reactions. Elementary Stoichiometry and Upscaling Moving on to the applications of elementary stoichiometry indicated by Eq. 41, we remark that upscaling of the results from elementary stoichiometry is routine and follows the procedure outlined by Eqs. A14 and A15. However, the key quantity of interest is the representation given by Eq. 46 that we repeat here as RM ¼ M r Figure A2. Upscaling the reference reaction rates. hRM i ¼ February 2012 Vol. 58, No. 2 RM dV (A17) V h R M i ¼ M h ri (A18) Here, the simple upscaling results given for Axiom II are lost, and the upscaling process becomes quite complex. For heterogeneous reactions, the reference reaction rates, rI, rII, rIII, and so forth, will depend on surface con...
View Full Document

Ask a homework question - tutors are online