Fundamentals of Chemical Reactor Theory

Transcription

1 UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil & Environmental Engineering Department Fundamentals of Chemial Reator Theory Mihael K. Stenstrom Professor Diego Rosso Teahing Assistant Los Angeles, 3 Introdution In our everyday life we operate hemial proesses, but we generally do not think of them in suh a sientifi fashion. Examples are running the washing mahine or fertilizing our lawn. In order to quantify the effiieny of dirt removal in the washer, or the soil distribution pattern of our fertilizer, we need to know whih transformation the hemials will experiene inside a defined volume, and how fast the transformation will be. Chemial kinetis and reator engineering are the sientifi foundation for the analysis of most environmental engineering proesses, both ourring in nature and invented by men. The need to quantify and ompare proesses led sientists and engineers throughout last entury to develop what is now referred as Chemial Reation Engineering (CRE). Here are presented the basis of the theory and some examples will help understand why this is fundamental in environmental engineering. All keywords are presented in bold font. Reation Kinetis Reation Kinetis is the branh of hemistry that quantifies rates of reation. We postulate that an elementary hemial reation I is a hemial reation whose rate orresponds to a stoihiometri equation. In symbols: A + B C + D [1] I for our purpose we will limit our disussion to elementary reations 1

2 Stenstrom, M.K. & Rosso, D. (3) and the reation rate will be defined as: -r = k ( A ) ( B ) where k is referred as the speifi reation rate (onstant). The overall order of reation III is defined as: n = [3] The temperature dependeny of k is desribed by the Arrhenius equation: a kt ( ) Ae E / R T [4] where, A E a R T = preexponential or frequeny fator = ativation energy [J/mol, al/mol] = gas onstant = J/mol K = al/mol K = absolute temperature [K] The Mass Balane Mass is a onservative entity IV, hene given a ontrol volume V the sum of mass flows entering the system will equal the sum exiting minus (plus) the onsumed (generated) or aumulated frations: rate of rate of rate of rate of rate of mass mass mass mass mass in out generated onsumed aumulated [5] shortly: IN OUT + PROD CONS = ACC [6] Equation [6] represents the key point in mass transfer: analogously to the fore balane in statis, the mass balane allows us to quantify and verify mass flows in our system. Let us now apply this fundamental balane to some ideal examples. Ideal Chemial Reators A bath reator, as its name states, is a non-ontinuous and perfetly mixed losed vessel where a reation takes plae. Figure 1 shows a shemati drawing of it. II this is the reation rate with respet to the reatants, the one with respet to the produts being -r = +k (D) (E) III rigorously, is the order of reation with respet to reatant A, and with respet to B IV physiists would orret this statement, sine energy is the only onservative entity; in all engineering appliations, mass will be by far slower than light speed

3 Stenstrom, M.K. & Rosso, D. (3), V Fig. 1. Bath reator Given its volume V, and the initial internal onentration, the total mass will be M = V. In the unit time, the onentration will be able to hange only in virtue of a hemial reation. The mass balane [6] quantifies this hange: IN OUT + PROD CONS = ACC. In this ase: dm Qin Qout rdv [7] dt V where r is the rate of generation (+) or depletion (-). Sine the assumption of no flow in or out of the reator volume (Q = ), and onstant reator volume V, dm d( V ) d V V r [8] dt dt dt where = (t) is the onentration at any time inside the reator V. Then, d r dt. [9] The differential equation [9] is the harateristi equation of a bath reator. Considering a first-order reation (r = -k ): d k [1] dt solving, or ln k t [11] e kt. [1] Equation [1] offers a relationship between onentration and time. At any point in time, then, we an know the inner onentration, known the reation onstant and the initial onentration. For a seond-order reation (r = -k ),. [13] 1 k t V Nota Bene:! in 3

4 Stenstrom, M.K. & Rosso, D. (3) This proedure may be repeated for any order of reation, just substituting the expression for r in the harateristi equation [9]. N.B.: The algebrai passages will heretofore be omitted. A Continuous-Stirred Tank Reator (CSTR) is a well-mixed vessel that operates at steady-state (Q in =Q out =Q). The main assumption in this ase is that the onentration of the inoming fluid will beome instantaneously equal to the outgoing upon entering the vessel. Fig. explains visually this onept. in, Q out, V out, Q Fig.. Continuous-Stirred Tank Reator A CSTR differs from a bath only in the fat that it is not losed. Thus, the mass flows in and out of the reator in eq. [6] will not anel: dm Q ( in out ) r dv dt. [14] V The mass balane in [14] equals to thanks to the steady state hypothesis (=no aumulation). Solving, r [15] in out H where H = V/Q = average hydrauli residene time. Eq. [15] represents the harateristi equation for a CSTR. Assuming a first-order reation, out in 1. [16] 1 k H A Plug Flow Reator (PFR) onsists in a long, straight pipe in whih the reative fluid transits at steadystate (no aumulation). The main assumptions of this model are that the fluid is ompletely mixed in any rosssetion at any point, but it experienes no axial mixing, i.e. ontiguous ross-setions annot exhange mass with eah other. Fig. 3 illustrates it. Q, Q, t Q, t+ t A Q, u t l Fig. 3. Plug Flow Reator Operating a mass balane on the seleted volume V = A l, and assuming steady-state onditions, we obtain dm Q t Q tt r dv dt [17] V hene, 4

5 Stenstrom, M.K. & Rosso, D. (3) Q Q rv Q Q rqt [18] t tt t tt whih redues to rt [19] t tt or r t [] Considering an infinitesimally thin ross-setional volume, its thikness will redue to dl, therefore: d r dt [1] whih, again, is the harateristi equation of the plug flow reator. Considering a first-order reation, the onentration equation will be e kt. [] Segregated Flow Analysis The non-ideality of industrial and natural proesses lead engineers to develop orretions to the ideal models, in order to use them with less restritions. For this reason, it is defined a residene time distribution, whih is a funtion that desribes the evolution of the average instantaneous onentration versus the elapsed time. It is very onvenient to express the residene time distribution as the normalized funtion E, E( t) t () t () dt [3] whih has its total area under the urve equal to unity: E( t) dt 1 [4] Fig. 4 shows the evolution of E vs. time. The E urve is the distribution needed to aount for non-ideal flow. Fig. 4. The residene time distribution 5

6 Stenstrom, M.K. & Rosso, D. (3) Considering the definition of E, the average residene time will beome: H te( t) dt [5] An useful tool used in this field is the umulative residene time fration (or umulative frequeny) urve F, defined as: F( t) E( t) dt t i t () dt t () dt. [6] Equation 6 shows that the F urve at t=t i is defined as the umulative area under the E urve from zero to t i. This means that F represents the fration of flow with a residene time t i. Combining [5] and [6]: 1 H tdf [7] whih is the highlited area in fig. 5. Note that the boundaries of the integral in [7] must be to 1, sine the area under the E urve equals unity. Fig. 5. The umulative frequeny urve The reason why we introdue the use of these funtions is to quantify the non-ideality of reators. A lassi example is the evaluation of the average residene time. Aording to the ideal reator theory, = V/Q, where V is the total volume of the reator. In ase dead zones are present in the vessel, the residene time distribution will not aount for them, showing a dereased reator volume. Hene, alulated in both ways will give an estimate of the dead zone volume. Fig. 6 illustrates the harateristi urves for various flows. 6

7 Stenstrom, M.K. & Rosso, D. (3) Fig. 6. Charateristi urves for various flows Priniples of Chemial Reation Engineering The effort to quantify non-ideal departures in hemial reators leads to treat two main non-ideal models, the dispersion model, and the CSTR in series model. It belongs to ommon sense that if we injet a olored traer into a flowing reator, some of the dye will exit before the expeted time (shot-iruiting), while some other will reside longer in the reator (bakmixing). In the ase of a plug flow, the dispersion model aounts for axial dispersion D, whih is the physial parameter that quantifies axial bakmixing and short-iruiting of fluid. In a differential volume of flowing fluid: D t x [8] analogous to Fik s first law on diffusion. Considering the whole flowing fluid, [8] will be orreted for the fluid motion through the introdution of an advetion term: D u t x x [9] whih, in its dimentionless form beomes z D t u L z [3] where, L u D = reator length = fluid veloity in the reator = dispersion oeffiient [m /s] 7

8 Stenstrom, M.K. & Rosso, D. (3) u L and the dimentionless group D dispersion, with limits: is alled Pelét number, the parameter that quantifies the extent of axial u L D negligible dispersion (plug flow) u L D large dispersion (mixed flow). Fig. 7 shows onentration urves in losed vessels for various extents of bakmixing quantified through the dispersion model. Fig. 7. Conentration urves at different Pelét numbers The CSTR-in-series model onsiders a sequene of ompletely mixed tanks as a fit to sequential zones of the non-ideal reator. In a sense, this is similar to onsidering the PFR as a sequene of non-interating differential volumes. Thus, both the E and F urves will be the sum of a series: E N N 1 Nt / N N t e ( N 1)! [31] and N N Nt / e i1 ( i 1)! F 1 i1 1 N t. [3] Fig. 8 visualizes this onept for the F urve. 8

9 Stenstrom, M.K. & Rosso, D. (3) F urve s measured N = 1 N = N = 3 N = 4 PFR time (min) Fig. 8. Cumulative onentration urves for different numbers of CSTR in series. N= number of reator. A PFR diagram is dashed. Bibliography The books here quoted have to be used for referene only, as they are too advaned for the purpose of this lass. We strongly reommend keeping the list as a referene for possible future need, as the books quoted represent the most important works in hemial engineering todate. 1. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. () Transport Phenomena nd ed., John Wiley & Sons, New York, NY ISBN SEL/EMS QA99.B5. Levenspiel, O. (1999) Chemial Reation Engineering 3 rd ed., John Wiley & Sons, New York, NY ISBN X - SEL/EMS TP157L57, Fogler, H.S. (199) Elements of Chemial Reation Engineering nd ed., Prentie Hall PTR, Upper Saddle River, NJ ISBN SEL/EMS TP157.F