We present, for a class of industrially relevant chemical reactions with two educts the dependence of stability on important chemical parameters, such as coolant, dilution and diffusion rates. The main analytical tools are generalized upscaling balance condition for the equilibria concentrations and spectral properties of corresponding operators. Although we illustrate the stability analysis for a model reactor (2 educts, E1 and E2), it should be emphasized that our approach is applicable to more complex reaction mechanisms

Efendiev, Messoud A.

Hebermehl, Georg

Lasser, Rupert

English

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

WeierstraÿInstitutfür Angewandte Analysis und Stochastikim Forschungsverbund Berlin e.V.Preprint ISSN 0946  8633Generalized necessary scaling condition andstability of chemical reactors with several eductsMessoud Efendiev,1Georg Hebermehl,2Rupert Lasser3submitted: 19th November 20011Universität StuttgartMathematisches Institut APfaenwaldring 57D - 70569 StuttgartGermanyE-Mail: efendiev@mathematik.uni-stuttgart.de2Weierstrass-Institute forApplied Analysis and StochasticsMohrenstraÿe 39D - 10117 BerlinGermanyE-Mail: hebermehl@wias-berlin.de3GSF - National Research Centre for Environment and HealthIBB - Institute for Biomathematics and BiometryIngolstadter Landstr. 1D - 85758 Neuherberg, München, GermanyPreprint No. 699Berlin 2001WIAS2000 Mathematics Subject Classication. 35Q60, 65F15, 65N22.Key words and phrases. Reaction-Diusion equation, stability, chemical radical reactions.Edited byWeierstraÿInstitut für Angewandte Analysis und Stochastik (WIAS)Mohrenstraÿe 39D  10117 BerlinGermanyFax: + 49 30 2044975E-Mail (X.400): c=de;a=d400-gw;p=WIAS-BERLIN;s=preprintE-Mail (Internet): preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractWe present, for a class of industrially relevant chemical reactions with twoeducts the dependence of stability on important chemical parameters, such ascoolant, dilution and diusion rates. The main analytical tools are general-ized upscaling balance condition for the equilibria concentrations and spectralproperties of corresponding operators. Although we illustrate the stabilityanalysis for a model reactor (2 educts, E1and E2), it should be emphasizedthat our approach is applicable to more complex reaction mechanisms.Contents1 Introduction 22 Generalized necessary scaling condition 43 Necessary scaling exponent for a model reactor with two educts:full picture 64 Stability of chemical reactor in the presence of diusion 9List of Figures1 Eigenvalues 1,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Eigenvalues 1,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 113 Eigenvalues 2,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Eigenvalues 2,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 115 Eigenvalues 3,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Eigenvalues 3,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 127 Eigenvalues 4,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Eigenvalues 4,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 129 Eigenvalues 5,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 Eigenvalues 5,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 12111 Eigenvalues 6,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Eigenvalues 6,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 1313 Eigenvalues 7,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314 Eigenvalues 7,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 1315 Eigenvalues 8,  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316 Eigenvalues 8,  = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 131 IntroductionAs an example of stability of chemical reactor we consider a class of chemical radicalreactions with starter S, educts E1, E2and chemical product of formS ! 2S;S+ E1! E1+D1;S+ E2! E2+D2;E1+ E2! P;E2+ E1! P;P+ E1! P + E1;P+ E2! P + E2;E1+ E2! P;2E1! D3;2E2! D4;E1+ S! D5;E2+ S! D6;(1)where by Di; i = 1(1)6, denote by-products. The three blocks in (1) characterizestarter, chain and reactions, respectively. Note, that the educt radicals E1and E2act catalytically even in low concentrations. Therefore these reactions are usuallyrun with low starter concentrations S, Therefore these reactions are usually run withlow starter concentrations S, already producing sucient quantities of E1, E2.We emphasize that, a reaction mechanism (2) is industrially relevant, and arise forexample, in photosensitive chlorivation of aromates, where E1= Cl2; E2Toluoene.The chemical product of these reactions, Chlorbenzene, plays an important role bothin food and pharmaceutical industries (see [5]).In this paper we study stability of chemical reactor from important parameters, suchas diusion, dilution rates, when  := S0goes to zero.We use the following designationsS;E1; E2concentrations of starter and educts;s; e1; e2corresponding radical concentrations;S0; E1;0; E2;0feed concentrations;P; p product and product radical;ki(T ) = kiexp  iT; i = 1; :::; 12; Arrhenius law;dS; dE1; dE2; ds; de1; de2; dp; dTdiusion rates;T; hj; j = 1; :::; 12; temperature and reaction enthalpy;v velocity due to Navier-Stokes equations:2The spatially heterogeneous evolution equations for (1) take the form (see [2])@S@t= dSS + (v;r)S + S0  k1S@E1@t= dE1E1+ (v;r)E1+ E1;0  k2sE1  k5e2E1  k6pE1@E2@t= dE2E2+ (v;r)E2+ E2;0  k3sE2  k4e1E2  k7pE2@s@t= dss + (v;r)s+ 2k1S   k2sE1  k3sE2  k11e1s  k12e2s@e1@t= de1e1+ (v;r)e1+ k2sE1  k4e1E2+ k6pE1  k8e1e2 2k9e12  k11e1s@e2@t= de2e2+ (v;r)e2+ k3sE2  k5e2E1+ k7pE2  k8e1e2 2k10e22  k12e2s@p@t= dpp+ (v;r)p+ k4e1E2+ k5e2E1  k6pE1  k7pE2@T@t= dTT + (v;r)T + h1k1S + h2k2sE1+ h3k3sE2+ h4k4e1E2+h5k5e2E1+ h6k6pE1+ h7k7pE2+ h8k8e1e2+ h9k9e12+ h10k10e22+h11k11e1s+ h12k12e2s:(2)To investigate the stability of chemical (model) reactor (1) we distinguish laminarand turbulent zones of our reaction. In the laminar zone we consider partial dier-ential equations of the form (2). In the turbulent zone, where fast mixing prevails,Equation (2) is averaged, spatially, to yield (see [2])@S@t= S0  k1S  D0S@E1@t= E1;0  k2sE1  k5e2E1  k6pE1 D1E1@E2@t= E2;0  k3sE2  k4e1E2  k7pE2 D2E2@s@t= 2k1S   k2sE1  k3sE2  k11e1s  k12e2s D3s@e1@t= k2sE1  k4e1E2+ k6pE1  k8e1e2  2k9e12  k11e1s D4e1@e2@t= k3sE2  k5e2E1+ k7pE2  k8e1e2  2k10e22  k11e1s D5e2@p@t= k4e1E2+ k5e2E1  k6pE1  k7pE2 D6p@T@t= h1k1S + h2k2sE1+ h3k3sE2+ h4k4e1E2+ h5k5e2E1+ h6k6pE1+h7k7pE2+ h8k8e1e2+ h9k9e12+ h10k10e22+ h11k11e1s+ h12k12e2s (T   T0):(3)Here~D = (D0;    ; D6) and  indicates dilution and external cooling rate withcoolant temperature T0, respectively.32 Generalized necessary scaling conditionFollowing Feinberg [1] we write chemical reactions of species x0;    ; xnasy(j)0x0+   + y(j)nxnkj ! y(j)0x0+   + y(j)nxn(4)with integer stoichiometric coecients y(j); y(j) 0 and positive real reaction ratecoecient kj. Then the associate dynamics is given by_x =MXj=1kjxy y(j)  y(j): (5)Here x = (x0;    ; xn) 2 Rn+1is the vector of concentrations and xy:= xy00  xynnrepresents the usual mass action kinetics.We emphasize that the external feed concentrations can be incorporated in thismodel by adding formal reactions0ki0 ! xi: (6)Following mathematical convention we denote by  := S0(0k0=  ! S) the feedconcentration of the radical starter and  = (0;    ; n) the scaling dened by0BBB@x0x1...xn1CCCA=0BBB@00 : : : 00 1: : : 0............0 0 : : : n1CCCA0BBB@01...n1CCCA: (7)Substituting (7) into (5) we obtainj_j=MXj=1kj0y(j)0++ny(j)ny(j)00   y(j)nn y(j)  y(j): (8)Then the equilibrium equation takes the formMXj=1kjy(j)y(j) y(j)  y(j)= 0 (9)for the scaled steady state  = (), where by y(j)and y(j)we denotey(j):= 0y(j)0+   + ny(j)nand y(j):= y(j)00   y(j)nn, respectively.In general, for a class of chemical reations (1) the reaction rates kjcan often bedeterminated only up to the order of magnitude. Therefore we make followingansatz:kj= kj;0j+O(j) with  := S0: (10)4Denition 1 We call a pair (; ) = (0; : : : ; n; 1; : : : ; M) 2 Rn+1 RMa gen-eralized scaling exponent for the steady state concentrations x = (x0; : : : ; xn) of (5),if there exists a sequence of scaled equilibria (l) = (0(l); : : : ; n(l)); l! 0; l!1 of (9), with kjgiven by (10), such that for all m = 0;    ; n the following nonde-generacy conditions holds:0 < liml!1inf m(l)  liml!1sup m(l) <1: (11)Theorem 1 Any generalized scaling exponent (; ) 2 Rn+1 RMsatisesminj:yjm6=0f(y(j)) + jg = mini:y(i)m6=0f(y(i)) + ig: (12)Proof. Let (; ) 2 Rn+1 RMbe any generalized scaling exponent. We rewriteEquation (9) for the scaled steady state  = () in components m = 0;    ; n asMXj=1kjy(j)y(j)y(j)=MXi=1kiy(i)y(i)y(i)(13)or equivalentlyXj:y(j)m6=0kjy(j)y(j)y(j)m=Xi:y(i)m6=0kiy(i)y(i)y(i)m: (14)Let kj= kj;0j+O(j);  := S0(see (10)), and x m. We sort the terms on eachside of last equality by increasing powers y(j)+ j; y(i)+ iof , respectively.Note, that all terms are strictly positive by the nondegeneracy assumption (11). Letminj:yjm6=0f(y(j)) + jg = y(jo)+ joand mini:y(i)m6=0f(y(i)) + ig = y(io)+ ioin (14). Then the leading (alias: lowest) power of  in the m-th component of(14) is given by y(jo)+ joon the left, as compared to y(io)+ ioon the right.Note, that these leading powers may in fact be realized by several other termsy(j)+j; y(i)+iin addition. Comparing coecients, we divide (14) by  to thepowersmin fy(jo)+ jo; y(io)+ iogand let ! 0. This immediatly yieldsy(jo)+ jo= y(io)+ ioby positivity of all terms in the sums. This proves the Theorem 1.53 Necessary scaling exponent for a model reactorwith two educts: full pictureIn this section we apply Theorem 1 to concrete reaction mechanism (1). We startour analysis in the case kj= O(1), that is j= 0; j = 1;    ;M , and D = 0. Forvery spatial case, that is chemical reactions for 1 (one) educt (see [4]). Based onmass action kinetics we add to reaction mechanism (1) following terms0k0=  ! S;0k10=E1;0    ! E1;0k20=E2;0    ! E2:9>>=>>;inputHence we consider the following radical reaction0k0=  ! S;0k10=E1;0    ! E1;0k20=E2;0    ! E2;9>>=>>;input (15)Sk1 ! 2S;S+ Eiki;k3  ! Ei+Di;)starter; i = 1; 2; (16)E1+ E2k4 ! P;E2+ E1k5 ! P;P+ Eik6;k7   ! P + Ei;9>=>;chain; i = 1; 2; (17)E1+ E2k8 ! P;2Eik9;k10   ! Di+2;Ei+ Sk11;k12    ! Di+4;9>=>;by reactions; i = 1; 2; (18)with  := S0> 0. Note, that photosensitive chlorivation of aromate is a specicexample of mechanism (15) - (18) ( see [5]).With x = (x0;    ; x6) = (S;E1; E2; s; e1; e2; p) and j= 0 and  = S0, the general-ized necessary scaling condition (12) for the scaling exponent  = (0;    ; 6) :=(S; E1; E2; s; e1; e2; p) takes the following formS= 1; (19a)min fE1+ s; E1+ e2; E1+ pg = 0; (19b)min fE2+ s; E2+ e1; E2+ pg = 0; (19c)min fE1+ s; E2+ s; s+ e1; s+ e2g = S; (19d)minfE2+ e1; e1+ e2; 2e1; s+ e1g = min fE1+ s; E1+ pg; (19e)min fE1+ e2; e1+ e2; 2e2; s+ e2g = minfE2+ s; E2+ pg; (19f)minfE1+ p; E2+ pg = min fE2+ e1; E1+ e2g: (19g)6Theorem 2 Let  = (S; E1; E2; s; e1; e2; p) be any scaling exponent of(15) -(18). Then  = (1; E1; 12; 1  E1;12; 12  2E1; E1) with E1  12.Proof. We distinguish two cases:(a) E1< E2,(b) E1= E2.Let us start with the case (a). Then we rewrite (19e) and (19f) ase1+min fE2; s; e1; e2g = E1+minfs; pg;e2+minfE1; s; e1; e2g = E2+minfs; pg:(20)From (20) follows that e1< e2.Next we prove e1< p. Assume contrary, i.e. e1 p. Then from (19b) and(19c) we obtainE1+min fs; pg = 0;E2+min fs; pg = 0:Hence E1= E2contradicting E1< E2. Note, that s+E2 1 due to Equation(19d) and (19a). Thus Equation (19c) takes the formE2+minfe1; pg = E2+ e1= 0;that is e1=  E2.As a result Equation (19g) can be written asp+ E1= minfe1+ E2; E1+ e2g = min f0; E1+ e2g:Last equality implies p+ E1= 0, since E1+ e2 0 (due to Equation (19b)),that is p=  E1. Next we show E2 0. To this end we emphasize that e1< s,because otherwise we get contradiction0 = e1+ E2 s+ E2 S= 1due to (19a) and (19d). To obtain E2 0 we rewrite Equation (19e) in the forme1+minfE2; e2; e1; sg = minfs+ E1; p+ E1g:With p=  E1, e1< e2, e1< s, s+ E1 1 last equality yieldse1+min fE2; e1g = 0or equivalently, with e1=  E2min fe1+ E2; 2e1g = min f0; 2E2g = 0;7which in turn implies E2 0. Let us prove that s= 1   E1. Indeed (19d) canbe written ass+min fE1; E2; e1; e2g = 1; (21)so that taking into account E1< E2 0 and e2> e1=  E2 0, we obtainfrom (21) that s= 1   E1. To express e2through E1and E2we rewriteEquation (19f) as followse2+minfE1; e1; e2; sg = E2+minfs; pg (22)Due to e1< e2, e1< s, E1< e1, p=  E1, and p< s, it follows from(22) thate2+ E1= E2  E1or equivalently e2= E2  2E1. Thus we obtain = (1; E1; E2; 1  E1; E2; E2  2E1; E1) with E2 0:To prove Theorem 2 it remains to show that E2=  12. To this end, we considerthe equations _e1= 0, _e2= 0, _p = 0, that is0 = _e1= k2sE1  k4e1E2+ k6pE1  k8e1e2  2k9e12  k11e1s;0 = _e2= k3sE2  k5e2E1+ k7pE2  k8e1e2  2k10e22  k12e2s;0 = _p = k4e1E2+ k5e2E1  k6pE1  k7pE2:(23)Adding these equations we obtaink2sE1+ k3sE2= 2k8e1e2+ 2k9e12+ 2k10e22+ k11e1s+ k12e2s: (24)Comparing powers  in (24)1 = minf 2E1; 2E2; 2(E2  2E1); 1  E1  E2; 1  3E1+ E2g:Therefore E2=  12and =1; E1; 12; 1  E1;12; 12  2E1; E1with E1<  12: (25)This proves Theorem 2 in the case (a) E1< E2.In the same manner (even simple) we prove that, if E1= E2(case (b)), thenscaling exponent =1; 12; 12;32;12;12;12:84 Stability of chemical reactor in the presence ofdiusionFor simplicity we restrict ourselves to the case, when the dilution vector~D :=(D1; :::; D6) = 0 and  = 0. Moreover we assume that the bulk of reactor to bewell-mixed. Let G be a reactor volume and   G be a domain where laminar owprevails. We denote byu() = (S(); E1(); E2(); s(); e1(); e2(); p(); T())homogeneous equilibrium of (3). Due to the Theorem 2 we have following (forsimplicity we restrict ourselves to the case E1= E2=  12) asymptotic expansionsfor equilibrium u(), that isS() s  +O(); E1() s  12+O( 12);E2() s  12+O( 12); s() s 32+O(32); (26)e1() s 12+O(12); e2() s 12+O(12);p() s 12+O(12):The equation_T = 0 yields an -expansion of the steady state temperature T(),that isT() sE1;0+ E2;02+ T0+O(): (27)We rewrite Equation (2) in the absence of transport terms (r; v) in the followingabstract form@~u@t= D~u+~F (u); ~uj@= u(); (28)where D = diag(dS; dE1; dE2; ds; de1; de2; dp; dT) and u() is a homogeneous equi-librum mentioned above. Both due to equilibria concentrations and the bulk ofreactor to be well-mixed. Let~u(x; t) = u() +W (x; t):Then W (x; t) satises@W@t= DW + F0(u())W; W j@= 0; (29)where F0(u()) is the (8; 8)-matrix at u(), that isF0(u()) = (30)0BBBBBBBBBB@ k1 0 0 0 0 0 0 k01S0 10  k2E10  k5E1 k6E110 0 2 k3E2 k4E20  k7E222k1 k2s  k3s 3 k11s  k12s 0 30 k2s+ k6p  k4e1k2E1  k11e14 k8e1k6E140  k5e2k3s+ k7p k3E2  k12e2 k8e25k7E250 k5e2  k6p k4e1  k7p 0 k4E2k5E166h1k112345671CCCCCCCCCCA;9where1=  k2s  k5e2  k6p; 2=  k3s  k4e1  k7p;3=  k2E1  k3E2  k11e1  k12e2; 4=  k4E2  k8e2  4k9e1  k11s;5=  k5E1  k8e1  4k10e2  k12s; 6=  k6E1  k7E2;1= h2k2s+ h5k5e2+ h6k6p;2= h3k3s+ h4k4e1+ h7k7p;3= h2k2E1+ h3k3E2+ h11k11e1+ h12k12e2;4= h4k4E2+ h8k8e2+ 2h9k9e1+ h11k11s;5= h5k5E1+ h8k8e1+ 2h10k10e2+ h12k12s;6= h6k6E1+ h7k7E2;1=  k02sE1  k05e2E1  k06pE1;2=  k03sE2  k04e1E2  k07pE2;3= 2k01S   k02sE1  k03sE2  k011e1s  k012e2s;4= k02sE1  k04e1E2+ k06pE1  k08e1e2  2k09e12  k011e1s;5= k03sE2  k05e2E1+ k07pE2  k08e1e2  2k010e22  k012e2s;6= k04e1E2+ k05e2E1  k06pE1  k07pE2;7= h1k01S + h2k02sE1+ h3k03sE2+ h4k04e1E2+ h5k05e2E1+ h6k06pE1+ h7k07pE2;+ h8k08e1e2+ h9k09e12+ h10k010e22+ h11k011e1s+ h12k012e2s  withk0j:=ddTkj(T ); j = 1; :::; 12:Let 0 < 1 2     N    ; N!1 as N !1 be eigenvalues of   (theLaplacian) with Dirichlet boundary conditions, that is ek(x) = kek(x); ekj@= 0: (31)Hence stability of chemical reactor (1) in the presence of diusionD = diag(dS; dE1; dE2; ds; de1; de2; dp; dT) will be governed bydetfF0(u())  1diag(dS; dE1; dE2; ds; de1; de2; dp; dT)  Ig = 0; (32)where 1is the rst eigenvalue of  , and I is the indentity matrix.Note that in the absence of diusion stability of (1) will be governed byF0(u()) = : (33)We begin our stability analysis by grouping the matrix F0(u()) according to itsleading order in  := S0. To this end we use asymptotic expansion (26). ThereforeF0(u()) can be decomposed asF0(u()) =  12hA0+O12i: (34)10It remains to compute j(A0); j = 1; ::: . Simple linear algebra computations showthat for   cwe have stability and 0   < cbelongs to the instability interval.Here csome positive number. Recall that by  is denoted the coolant rate.Now we are interested in stability of (1) in the presence of diusion. To this endwe consider equation (32), where F0(u()) is the same matrix as in (33). Takinginto account that 1> 0, it is not dicult to see that  > c  Æ, where cthesame number as above (in the absence of diusion) and Æ is some positive numberdepending on 1and~D = (dS; dE1; dE2; ds; de1; de2; dp; dT). We emphasize that theinterval c  Æ <  < cwhich provides stability of chemical reactor in the presenceof diusion, belongs to the instability interval in the diusionless case.Remark 1. In the same manner one can analyse inuence of dilution rate to thestability of (1).The following gures show the role of diusion for the stability of chemical reactor(1) and conrm our theoretical observations. The graphs in the Figures 1 - 16represent the eigenvalues (ordinate) of the matrices F0(u()) for the coolant rates = 0 and  = 100 with E1=  2:00; 1:99; :::; 0:49; 0:50 (abscissa, see (25)), = 10 4, ki= 1; hi= 1; k0i= 1; i = 1; :::; 12. The 8 eigenvalues of each matrixare sorted in ascending order.Acknowledgement: We thank Gerd Reinhardt (Weierstrass Institute for AppliedAnalysis and Stochastics) for his support with the graphics.Figure 1: Eigenvalues 1,  = 0 Figure 2: Eigenvalues 1,  = 100Figure 3: Eigenvalues 2,  = 0 Figure 4: Eigenvalues 2,  = 10011Figure 5: Eigenvalues 3,  = 0 Figure 6: Eigenvalues 3,  = 100Figure 7: Eigenvalues 4,  = 0 Figure 8: Eigenvalues 4,  = 100Figure 9: Eigenvalues 5,  = 0 Figure 10: Eigenvalues 5,  = 10012Figure 11: Eigenvalues 6,  = 0 Figure 12: Eigenvalues 6,  = 100Figure 13: Eigenvalues 7,  = 0 Figure 14: Eigenvalues 7,  = 100Figure 15: Eigenvalues 8,  = 0 Figure 16: Eigenvalues 8,  = 10013References[1] Feinberg, M.: Chemical reaction network structure and the stability of complexisothermal reactors. Part I, Part II, Chem. Eng. Sci., 42 (1987) 2229 - 2268, 43(1988) 1 -25.[2] Laidler, K. T.: Chemical Kinetics. Harper Collins Publishers, (1987).[3] Efendiev, M. A., Schuppert, A., Wolfrum, M.: Isothermal upscaling of exother-mic reactors: A example. Journal Nonlinear Analysis, TMA, 30 No. 6 (1997)3455 - 3461.[4] Efendiev, M. A., Schuppert, A.: Stability analysis of chemical reactors. (Eds.Keil, F., Mackens, W., Voÿ, H., Werther, J.) Scientic computing in chemicalengineering II, Springer-Verlag, (1999) 225 - 230.[5] Efendiev, M. A., Fiedler, B., Schuppert, A.: Upscaling of some exothermic re-actions from chemical industry. Zeitschrift für Angewandte Mechanik, Suppl. 2,(1996) 153 - 156.14

Generalized necessary scaling condition and stability of chemical reactors with several educts

Efendiyev, M.A.

Hebermehl, G.

Lasser, R.

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Generalized necessary scaling condition and stability of chemical reactors with several educts.

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Generalized necessary scaling condition and stability of chemical reactors with several educts

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