6. Well-Stirred Reactors III

Transcription

1 6. Well-Stirred Reactors III Reactors reaction rate or reaction velocity defined for a closed system of uniform pressure, temperature, and composition situation in a real reactor is usually quite different from this ideal situation types of reactors: mass exchange: batch reactor flow reactor heat exchange: adiabatic isothermal mechanical variables: constant pressure constant volume residence time: unique distribution (ideal: exponential) operation: transient stationary residence time = time spent in the reactor by each volume element of the reacting mass ideal reactor for the direct measurement of reaction rates is a flow, isothermal, constant-volume reactor Adv PChem 6.1

2 operating in the stationary state with such thorough mixing that the composition is the same everywhere in the reactor: continuous stirred tank reactor (CSTR) [continuous = continuous-flow] c i V T c i,f volumetric flow rate q; units L/s feed concentration of species i : c i,f ; units M reactor volume V ; units L reaction velocity v; units M/s reaction in the reactor: ν i J i = 0 mass-balance equation for each species (i = 1,2,..., N) d(v c i ) dt = qc i,f qc i + ν i V v(c 1,c 2,...,c N,T ) Adv PChem 6.2

3 divide by V : dc i dt = q V ( ci,f c i ) + νi v(c 1,c 2,...,c N,T ) define α q V flow rate units s 1 dc i dt = α( c i,f c i ) + νi v(c 1,c 2,...,c N,T ) stationary state 0 = dc i dt = α( c i,f c i ) + νi v(c 1,c 2,...,c N,T ) ν i v(c 1,c 2,...,c N,T ) = α ( c i c i,f ) v(c 1,c 2,...,c N,T ) = α ν i ( ci c i,f ) tracer experiment: CSTR initially filled with dye, then Adv PChem 6.3

4 flow distilled water dc dt = αc c(t) = c(0)exp( αt) distribution of residence times of particles in the reactor: exit age distribution E(t) E(t)dt is the probability that a particle exits the reactor between t and t + dt the number of tracer particles leaving the reactor between t and t + dt is dn (t) = N A qc(t)dt the number tracer particles initially in the reactor is N 0 = N A V c(0) which equals N 0 = 0 dn (t) = 0 N A qc(t)dt = N A q 0 c(t)dt Adv PChem 6.4

5 E(t)dt is the fraction of particles exiting between t and t + dt E(t)dt = or E(t)dt = So dn (t) N 0 = N Aqc(t)dt N A V c(0) = αc(t)dt c(0) dn (t) N 0 = N Aqc(t)dt N A q 0 c(t)dt = c(t)dt 0 c(t)dt E(t) = αc(t) c(0) = c(t) 0 c(t)dt Adv PChem 6.5

6 For a CSTR we find E(t) = αc(t) c(0) = αc(0)exp( αt) c(0) = αexp( αt) or c(t) E(t) = 0 c(t)dt c(0) exp( αt) = 0 c(0)exp( αt)dt exp( αt) = [ α 1 ] exp( αt) 0 = exp( αt) 1 α = αexp( αt) residence times in a CSTR are exponentially distribut- Adv PChem 6.6

7 ed τ = t = = 0 = α te(t)dt tαexp( αt)dt t exp( αt)dt x n exp( ax)dx = n! a n+1, a > 0 τ = α 1 α 2 τ = 1 α τ = residence time of CSTR [somewhat of an abuse of language] = inverse of flow rate Adv PChem 6.7

8 Gray-Scott model Gray, P. & Scott, S. K.: Sustained oscillations and other exotic patterns of behavior in isothermal reactions, J. Phys. Chem., 89, (1985), doi.org/ /j100247a009; Gray, P. & Scott, S. K.: Chemical Oscillations and Instabilities, Clarendon Press, (1990) U + 2V V k 1 k 1 3V (R1) k 2 k 2 W (R2) rate equations du dt = k 1u v 2 + k 1 v 3 + α ( u 0 u ) dv dt = k 1u v 2 k 1 v 3 k 2 v + k 2 w + α ( v 0 v ) dw dt = k 2 v k 2 w + α ( w 0 w ) Adv PChem 6.8

9 non-dimensionalize the rate equations u = u u, v = v 0 u, w = w 0 u 0 t = t k 1 u 2 0 k = k 2 k 1 u 2 0 α = α k 1 u 2 0 η 1 = k 1 k 1 η 2 = k 2 k 2 non-dimensionalized rate equations du dt = uv 2 + η 1 v 3 + α(1 u) dv dt = uv 2 η 1 v 3 kv + kη 2 w + α(v 0 v) dw dt = kv kη 2w + α(w 0 w) (1a) (1b) (1c) Adv PChem 6.9

10 adding the three rate equations we find d(u + v + w) dt = α ( 1 + v 0 + w 0 u v w ) define z(t) u(t) + v(t) + w(t), z v 0 + w 0 then dz dt = α( z 0 z ) thermodynamic equilibrium can only be attained if the net flow of matter through the reactor vanishes: α = 0 thermodynamic equilibrium state: α = 0, u = 0, v = 0, ẇ = 0 0 = u eq v 2 eq + η 1 v 3 eq 0 = u eq v 2 eq η 1 v 3 eq kv eq + kη 2 w eq Adv PChem 6.10

11 0 = kv eq kη 2 w eq either v eq = η 2 w eq u eq v 2 eq = η 1 v 3 eq v eq = 1 η 1 u eq [if v eq 0] w eq = 1 η 1 η 2 u eq u eq arbitrary or v eq = 0 w eq = 0 u eq arbitrary the second equilibrium state is a somewhat pathological feature of the reaction scheme: if initially no Adv PChem 6.11

12 V and W is present in the reactor, then no reaction will occur for α = 0, there is a further condition dz dt = 0 = z(t) = const = z(0) u eq + v eq + w eq = u(0) + v(0) + w(0) = z(0) therefore u eq + 1 η 1 u eq + 1 η 1 η 2 u eq = z(0) u eq = otherwise z(0) η η 1 η 2 if v(0) 0 or w(0) 0 u eq = u(0) the interesting case is the non-trivial first equilibrium state u eq = z(0) η η 1 η 2 Adv PChem 6.12

13 v eq = 1 η 1 u eq w eq = 1 η 1 η 2 u eq as α is increased from zero, the system is driven away from thermodynamic equilibrium determine nonequilibrium steady states and other asymptotic states of the reactor exploit the first integral of the CSTR dz dt = α( z 0 z ) z(t) = z(0)exp( αt) + z 0 [ 1 exp( αt) ] z(t) t z 0 in other words, the total concentration relaxes to the total concentration in the feed streams in experiments, one usually takes t to mean t = 4/α = 4τ; after that time, the initial condition has essentially been washed out of the reactor Adv PChem 6.13

14 therefore we will assume from now on that z(0) = z 0, which implies z(t) = z 0 eliminate w: w = z 0 u v du dt = uv 2 + η 1 v 3 + α(1 u) dv dt = uv 2 η 1 v 3 kv + kη 2 ( z0 u v ) + α(v 0 v) study the dynamical behavior of Gray-Scott model under the following simplifying assumptions: (i) the back reactions are negligible, irreversible Gray- Scott scheme η 1 1, η 2 1 Set η 1 = η 2 = 0 in the rate equations du dt = uv 2 + α(1 u) Adv PChem 6.14

15 dv dt = uv 2 kv + α(v 0 v) (ii) the autocatalytic intermediate V is not fed, v 0 = 0 du dt = uv 2 + α(1 u) dv dt = uv 2 kv αv find the steady states: 0 = u v 2 + α(1 u) 0 = u v 2 (k + α)v = v [ u v (k + α) ] consequently either u = 1, v = 0 [trivial steady state, no reaction] or u = k + α v Adv PChem 6.15

16 0 = k + α ( v v 2 + α 1 k + α ) v 0 = (k + α)v 2 + αv α(k + α) v 2 α k + α v + α = 0 v 2,3 = 1 α 4(k + α)2 1 ± 1 2 k + α α these two roots exist if 4(k + α)2 1 α 0 they coalesce into one double root at 4(k + α) 2 α = 1 4k 2 + 8kα + 4α 2 = α ( ) α 2 + α ± = 2k 1 4 ( k 1 8 α + k 2 = 0 ) ± k k k2 Adv PChem 6.16

17 α ± = ( k 1 8 ) ± k = the two non-trivial steady states (u, v) 2,3 appear and disappear at α and α + of course α ± must be a non-negative real number; = k 1 8 < 0 and k > 0 k < 1 8 and k < 1 16 consequently, if k < 1 16 then over the interval (α,α + ) of the flow rate the two non-trivial steady states exist note: α v 2,3 (α ) = k + α Adv PChem 6.17

18 α + v 2,3 (α + ) = k + α + = these two new branches of steady states do not appear via a bifurcation from the branch of trivial steady states, v = 0 the branches of non-trivial steady states form an isola for the non-trivial steady states u v (k + α) = 0 u v = k + α u = k + α v δ u 2,3 = u 2,3 = 4(k + α)2 α 1 2 k + α α 2(k + α)2 α [0 δ 1 see above] k + α [ 1 1 δ 1 ] 1 1 δ Adv PChem 6.18

19 u 2,3 = 1 4(k + α) α 1 1 δ u 2,3 = 1 2 δ δ u 2,3 = 1 ( 1 )( 1 δ 1 + ) 1 1 δ δ u 2,3 = 1 [ 1 ± ] 1 δ 2 steady states of the irreversible Gray-Scott model: (u 1, v 1 ) = (0,1) ( 1 (u 2, v 2 ) = 2 ( 1 (u 3, v 3 ) = 2 [ δ [ 1 ] 1 δ ], 1 α 2 k + α, 1 α 2 k + α [ 1 ] ) 1 δ [ 1 + ] ) 1 δ stability of the steady states Adv PChem 6.19

20 Jacobian ( ) v 2 α 2u v J = v 2 2u v (k + α) for the trivial steady state, (u 1, v 1 ) = (0,1): J = ( ) α 0 0 (k + α) the Jacobian is diagonal and the eigenvalues are λ 1 = α λ 2 = (k + α) the trivial steady state is stable for all flow rates this implies that a small initial amount of the autocatalyst V will decay to zero non-trivial steady states J = ( v 2 α 2(k + α) v 2 k + α ) Adv PChem 6.20

21 sign structure J = ( ) + + the Gray-Scott model is a cross activator-inhibitor scheme; V is the activator T = trj = v 2 + k = detj = ( v 2 + α ) (k + α) + 2(k + α) v 2 = (k + α) ( v 2 α + 2v 2) = (k + α) ( v 2 α ) ( for[ the lower branch of the isola, (u 2, v 2 ) = ] 1 δ, 1 2 k + α [ 1 ]) 1 δ : α = (k + α) ( 1 α 2 [ 4 (k + α) 2 1 α 2 [ 4 (k + α) 2 1 ] 2! 1 δ α < 0 1 ) ] 2 1 δ α Adv PChem 6.21

22 1 α 2 [ 4 (k + α) 2 1 ] 2! 1 δ < α [ 1 ] 2! 4(k + α)2 1 δ < = δ α δ + 1 δ! < δ 2 2δ! < 2 1 δ 1 δ! < 1 δ true for 0 < δ < 1 the determinant is negative for all steady states on the lower branch of the isola and becomes zero at α ± = these states are saddle points and are unstable for ( [ the upper branch of the isola, (u 3, v 3 ) = 12 1 ] 1 δ, 1 2 k + α [ 1 + ]) 1 δ : α ( 1 α 2 [ = (k + α) 4 (k + α) ) ] 2 1 δ α 1 α 2 [ 4 (k + α) ] 2! 1 δ α > 0 Adv PChem 6.22

23 1 α 2 [ 4 (k + α) ] 2! 1 δ > α [ 1 + ] 2! 4(k + α)2 1 δ > = δ α δ + 1 δ! > δ 2 1 δ! > 2δ 2 1 δ! > δ 1 true for 0 < δ < 1 the determinant is positive for all steady states on the upper branch of the isola and becomes zero at α ± = stability is determined by T for α near α +, T is negative; the steady states are stable for α near α, T is positive; the steady states are unstable The upper branch of the isola undergoes a Hopf bifur- Adv PChem 6.23

24 cation, T = 0: v 2 + k = 0 v 2 = k v = k 1 α [ 1 + ] 1 δ = k 2 k + α 1 α 4(k + α) = k 2 k + α α 4(k + α) = 2(k + α) k α α 4(k + α)2 1 = 2(k + α) k 1 α α [ 4(k + α)2 2(k + α) k 1 = 1 α α ] 2 4(k + α)2 1 α = 4(k + α)2 k α 2 4(k + α) k α + 1 Adv PChem 6.24

25 α(k + α) = (k + α)k α k ( α 2 + 2k ) k + k 2 = 0 α 1,2 = 1 ( 2k ) 1 ( k ± 2k ) 2 k k [ k 2k ± 4k 2 4k ] k + k 4k 2 α 1,2 = 1 2 [ k 2k ± α 1,2 = 1 2 k 4k k ] α 2 is the acceptable root; α 1 is a parasite root [the original equation is not zero at that solution]; so α H = 1 2 [ k 2k k ( 1 4 ) ] k as the flow rate α is decreased from α +, the upper branch of the isola undergoes a Hopf bifurcation at α H examples for two values of k: k = = 0.9 (1/16), α = , α + = , α H = Adv PChem 6.25

26 v α k = = 1/32, α = , α + = , α H = v α Adv PChem 6.26

27 the dynamical behavior of two-variables systems is restricted since trajectories cannot cross the possible asymptotic states are stationary states and simple oscillations (simple limit cycles) additional dynamical behavior and asymptotic states occur in systems with three or more variables consider the following reaction scheme which is related to the Gray-Scott model and known as the autocalator [Merkin, J. H.; Needham, D. J. & Scott, S. K.: Oscillatory Chemical Reactions in Closed Vessels, Proc. R. Soc. Lond. A, 406, (1986), royalsocietypublishing.org/content/406/1831/299. abstract] P U + 2V V U k 0 U (R3) k 1 3V (R4) k 2 W (R5) k u V (R6) (R7) Adv PChem 6.27

28 Taking into account that the precursor P decays with a first-order rate law, the nondimensionalized rate equations are given by du dt = µexp( ɛt) uv 2 u dv dt = uv 2 + u v this describes a closed system and provides a simple model to study transient phenomena with the pool chemical assumption, supply of the precursor, inexhaustible du dt = µ P uv 2 u dv dt = uv 2 + u v two-variable system, no new dynamical behavior Showalter and coworkers introduced a modification of the pool chemical autocatalator model which leads to a three-variable scheme [Peng, B.; Scott, S. K. Adv PChem 6.28

29 & Showalter, K.: Period doubling and chaos in a three-variable autocatalator, J. Phys. Chem., 94, (1990), /j100376a014] P U + 2V V U P + W k 0 U (R8) k 1 3V (R9) k 2 W (R10) k u V (R11) k c U + W (R12) W k 3 Z (R13) nondimensionalized rate equations du dt = κ + µw uv 2 u σ dv dt = uv 2 + u v Adv PChem 6.29

30 δ dw dt = v w this scheme dispalys simple and complex oscillations and a period-doubling sequence to chaos: aperiodic deterministic motion; strange attractor; sensitive dependence on initial conditions Lorenz model [Lorenz, E. N.: Simplified dynamic equations applied to the rotating-basin experiments, J. Atmos. Sci., 19, (1962), %3C0039%3ASDEATT%3E2.0.CO%3B2] simplified model of Rayleigh-Bénard convection; shallow fluid layer between two plates, upper plate has uniform temperature of T and lower plate a uniform temperature of T + T, T > 0; if T is sufficiently small, then no fluid motion and the temperature profile between the lower and upper plate is linear; if T > T c, convective motion in the form of rolls nondimensionalized evolution equations dx dt = σ(y x) Adv PChem 6.30

31 dy = x(r z) y dt dz dt = x y bz x velocity amplitude of convective motion, y difference in temperature between rising and descending currents, z amplitude of distortion of the temperature profile from linearity; σ, r and b are positive parameters steady states 0 = σ(y x) 0 = x(r z) y 0 = x y bz The first equation implies that y = x Substitute into second and third equation 0 = x(r z) x Adv PChem 6.31

32 0 = x 2 bz 0 = x(r 1 z) = x = 0 or z = r 1 Case 1: x = 0 = y = 0 = z = 0, no convection (trivial steady state) (x, y, z) 0 = (0,0,0) Case 2: z = r 1 = 0 = x 2 b(r 1) x = ± b(r 1) y = ± b(r 1) ( (x, y, z) 1,2 = ± b(r 1),± ) b(r 1),r 1 steady states 1 and 2 do not exist for r < 1 linear stability analysis Adv PChem 6.32

33 Jacobian σ σ 0 J = z + r 1 x y x b stability of the trivial steady state σ σ 0 J = r b eigenvalues det ( ) σ λ σ 0 J λi 3 = r 1 λ b λ = σ λ σ r 1 λ ( b λ) = [ λ 2 + (σ + 1)λ + σ(1 r ) ] ( b λ) = 0 Adv PChem 6.33

34 λ 0;1 = b [ λ 0;2,3 = 1 (σ + 1) ± 2 [ = 1 2 (σ + 1) ± ] (σ + 1) 2 4σ(1 r ) ] (σ 1) 2 + 4σr λ 0,1 and λ 0,3 are always real and negative λ 0,2 is always real and crosses 0 at r = r c : σ + 1 = (σ 1) 2 + 4σr c (σ + 1) 2 = (σ 1) 2 + 4σr c σ 2 + 2σ + 1 = σ 2 2σ σr c 4σ = 4σr c r c = 1 as expected at r = r c = 1 the quiescent state state becomes unstable and convective rolls appear Adv PChem 6.34

35 stability analysis of the convective branches (x, y, z) 1,2 : J = σ σ 0 r r 1 b(r 1) ± b(r 1) ± b(r 1) b σ σ 0 = 1 1 b(r 1) ± b(r 1) ± b(r 1) b eigenvalues det ( σ λ σ 0 ) J λi 3 = 1 1 λ b(r 1) ± b(r 1) ± b(r 1) b λ = 0 characteristic polynomial λ 3 + c 1 λ 2 + c 2 λ + c 3 = 0 c 1 = 1 + σ + b > 0 Adv PChem 6.35

36 c 2 = b(r + σ) > 0 c 3 = 2σb(r 1) > 0 linear stability analysis for systems with n 3 variables: characteristic polynomial ( 1) n det ( J λi n ) = 0 λ n + c 1 λ n 1 + c 2 λ n c n 1 λ + c n = 0 Routh-Hurwitz theorem: All roots of the characteristic polynomial have a negative real part, if and only if c 1 c 3 1 c 2 c 4 0 c l = 1 c c 2 > 0, l = 1,...,n 0 0 c l together with the condition c n > 0 Adv PChem 6.36

37 Set c l = 0 in l for l > n generic bifurcations of steady states in n-variable systems, n > 2: stationary bifurcation, where a real eigenvalues passes through zero, and the Hopf bifurcation, where a pair of complex conjugate eigenvalues crosses the imaginary axis stationary bifurcation: c n = 0 Hopf bifurcation: n 1 = 0 c n > 0 l > 0, l = 1,...,n 2 Lorenz model, the convective branches (x, y, z) 1,2 : 1 = c 1 = 1 + σ + b > 0 2 = c 1 c 3 1 c = c 1c 2 c 3 = b [r (1 + b σ) + σ(3 + b + σ)] 2 Adv PChem 6.37

38 c 1 c 3 c 5 c 1 c 3 0 c 3 = 1 c 2 c 4 = 1 c 2 0 = c 1 c c = c c 1 c 3 0 c 1 c 3 2 since c 3 = 2σb(r 1) > 0, the convective states (x, y, z) 1,2 cannot undergo a stationary bifurcation the convective states (x, y, z) 1,2 undergo a Hopf bifurcation, if σ > b + 1, at r H = σ(σ + b + 3) σ b 1 where 2 = 0 and 1 > 0 and c 3 > 0 it turns out that the Hopf bifurcation is subcritical all trajectories remain bounded above r H ; the following figures illustrate the behavior of the Lorenz model for σ = 10.0, b = 8/3 and r = 28.0 [the Hopf bifurcation occurs at r H = 24.74] initial condition: (x(0), y(0), z(0)) = (x, y, z) 1 + (0.001, 0.001,0.001) Adv PChem 6.38

39 time series x y t t z t Adv PChem 6.39

40 the asymptotic state is a strange attractor z x sensitive dependence on initial conditions: at t = 500 we have (x, y, z) = ( , , ); start a second trajectory nearby (x, y, z )(500) = ( , , ); figure shows the Euclidean distance [x(t) (t) = x (t) ] 2 [ + y(t) y (t) ] 2 [ + z(t) z (t) ] 2 Adv PChem 6.40

41 t main routes to chaos in chemical systems: (i) perioddoubling, (ii) mixed-mode oscillations, (iii) intermittency (regular oscillations are interrupted by chaotic bursts of random duration), (iv) quasiperiodicity (two incommensurate frequencies) Adv PChem 6.41