ChE 471: LECTURE 4 Fall 2003


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1 ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins. This requires a mathematical mdel f the system, which in turn rests n applicatin f cnservatin laws t a welldefined cntrl vlume f the reactin system and n use f apprpriate cnstitutive expressins fr the reactin rates. The cncepts f ideal reactrs allw us t quantify reactr perfrmance as a functin f its size and selected perating cnditins. T illustrate this useful cncept we deal here with a single, hmgeneus phase, single reactin at cnstant temperature. We intrduce then the ideal batch reactr, and tw ideal cntinuus flw reactrs. In each case we apply the cnservatin f species mass principle which states (Rate f ccumulatin (Rate f Input (Rate f Output + (Rate f Generatin ( Equatin ( is applied t an apprpriately selected cntrl vlume, the largest arbitrarily selected vlume f the system in which there are n gradients in cmpsitin.. Batch Reactr The ideal batch reactr is assumed t be perfectly mixed. This implies that at a given mment in time the cncentratin is unifrm thrughut the vessel. The vlume, in the develpment belw is assumed equal t the vlume f the reactin mixture. This is then equal t the reactr vlume R in case f gas phase reactin but nt in case f liquids (< R, then. The batch reactr can be an autclave f cnst (Figure.a and a cnstant pressure, P cnst (Figure.b vessel. The frmer is almst always encuntered in practice. Our gal is: a T find a relatinship between species cncentratin (reactant cnversin and time n stream. b T relate reactr size and prductin rate.
2 Let us cnsider a single irreversible reactin P with an nth rder irreversible rate f reactin n kc ( t t 0 a batch f vlume is filled with fluid f cncentratin C. Reactin is started (n C. Find hw reactant cnversin depends n reactin time? ls determine the prductin rate as a functin f reactin time. We apply (eq  t reactant : (R dn dt d(c dt (3 a cnst b P cnst FIGURE : Schematic f Batch Reactrs In ur case due t the fact that υ j 0, cnst irrespective f the batch reactr type, s that eq (3 becmes j dc dt R (4 dc dt ( kc n ; t 0 C C (5 Separatin f variables and integratin yields: t dt dc n C dc n C 0 kc k (6 C C C
3 t t k C n C n C t 0 k( n C t n n [ C ] (7 C n [ k( n ( ] n (8a r t k(n C n [( n ] (8b Once rder f reactin, n, is specified (as shwn belw fr n0,,,.5, the relatin between t and is readily fund n 0 t C k n t k ln n.5; t 0.5k C ( 0.5 n t kc (9 Prductin Rate f Prduct P can be related by stichimetry t he cnsumptin rate f as ml F P S F The prductin rate f P is given by: (mles f P prcessed per batch (reactin time + shut dwn time per batch (0 C t + t s n k(n C C ( [( n ]+t s NOT BENE: Equatin ( is valid nly fr systems f cnstant density. Thus, it is valid fr all systems, gas r liquid, cnducted in an autclave at cnst (see Figure a. It is als valid fr gaseus systems with n change in the 3
4 number f mles ( ν j 0 cnducted in P cnst. system at T cnst (Figure b. The first equality in equatin ( gives the general result, the secnd equality presents the result fr an nth rder irreversible reactin with resect t reactant. T use this equatin the shut dwn time, i.e. the time needed between batches, t s, must be knwn. Cnsider nw the fllwing secnd rder reactin with stichimetry P. ml 0.C Lmin a Find the batch reactr vlume needed t prduce 38 kml/min if reactr shut dwn time is 60 minutes and the desired cnversin is Initial reactant cncentratin is C (ml L. Using the right frm f equatin (9 fr n we get the reactin time. t (min Then, slving equatin ( fr the vlume we get (t + t s.38( ,000L 0m 3 C x0.95 b What is the maximum prductin rate,, achievable in the abve batch reactr f vlume 0m 3 if t s, T, C all are fixed at previus values. Cnsider eq ( fr prductin rate as a functin f cnversin kc C +t s x 0 3 ( x (
5 This expressin has a maximum which we can lcate by differentiatin d 0 ( ( ( x 0 d x 0, 6 ± Clearly, the psitive sign is nt permissible as cnversin cannt exceed unity. We need t check whether the answer is a maximum r a minimum. d > 0 fr < 0.70 d d < 0 fr x > 0.70 Maximum at max x ml min n increase in prductivity f unreacted t be recycled x00 % can be achieved at the expense f mre 38 One must include the cst f separatin int the real ecnmic ptimizatin.. Cntinuus Flw Reactrs (Steady State.. Cntinuus Flw Stirred Tank Reactr (CFSTR r CSTR r STR The CSTR is assumed perfectly mixed, which implies that there are n spatial gradients f cmpsitin thrughut the reactr. Since the reactr perates at steady state, this implies that a single value f species cncentratin is fund in each pint f the reactr at all times and this is 5
6 equal t the value in the utflw. The utflw stream is a true representative f the reactin mixture in the reactr. F O F F ( C C O FIGURE : Schematic f a Cntinuus Flw Stirred Tank Reactr (CSTR What des the abve idealizatin f the mixing pattern in a CSTR imply? It pstulates that the rate f mixing is instantaneus s that the feed lses its identify instantly and all the reactin mixture is at the cmpsitin f the utlet. Practically this implies that the rate f mixing frm macrscpic level dwn t a mlecular scale is rders f magnitude faster than the reactin rate and is s fast in every pint f the vessel. Then the mass balance f eq ( can be applied t the whle vlume f the reactr recgnizing that at steady state the accumulatin term is identically zer. gain, taking a simple example f an irreversible reactin P applicatin f eq ( t reactant yields: F F + ((R 0 ( Mlar flw rate f unreacted in the utflw by definitin is given by F F ( Q C (. The prductin rate f P is given by ( (R P (3 Reactr vlume is given by eq ( F ( Q C x ( (4 6
7 Reactr space time is defined by τ Q C ( (5 Using stichimetry we readily develp the relatin between prductin rate,, and reactr vlume,. Let us cnsider again the example f ur nd rder reactin, P, with the rate belw: kc n ml 0.C L min Find CSTR vlume needed t prcess 38 ml/min. Suppse we chse again 0.95 fr ur exit cnversin. Frm eq (3 we get 0. C ( nd slving fr vlume F P 38 5,000L 5m 3 0.C ( 0.x( 0.95 If we cnsider eq (3 it is clear that nw the maximum prductin rate is btained when the reactin rate is the highest. That fr nth rder reactins is at zer cnversin. S the maximum frm CSTR,000 L is btainable at 0. max 0. x x 5,000 5,00 ml/min. The penalty r this enrmus prductin rate is that the prduct is at zer purity. Hence, the separatin csts wuld be enrmus. The average rate in a CSTR is equal t the rate at exit cnditins. ( ( exit 0.C ( exit 0.x( x0 4 (ml / Lmin 7
8 .. Plug Flw Reactr (PFR The main assumptins f the plug flw reactr are: i perfect instantaneus mixing perpendicular t flw, ii n mixing in directin f flw This implies pistn like flw with the reactin rate and cncentratin that vary alng reactr C O C O F O F d ( d F F ( FIGURE 3: Schematic f a Plug Flw Reactr (PFR Since there are nw cmpsitin gradients in the directin f flw, the cntrl vlume is a differential vlume t which eq ( is applied. Let us again use the mass balance n reactant F F + +R 0 (6 F + R 0 F lim 0 ( lim( R 0 df d ( (7 Since F F ( then df F d s that F d d ( (8 With initial cnditins: 0 0 (9 8
9 Upn separatin f variables in (eq 8 and integratin: d F (0 ( Fr an nth rder reactin (with ε 0 we get F n kc d Q ( n n kc [( n ] (n ( The expressin fr the PFR space time τ Q kc n (n ( n [ ] ( is nw identical t the expressin fr reactin time t in the batch reactr. Fr the example f the secnd rder reactin used earlier we get F F kc d kc ( F kc F kc F Q C τ Q kc (Same as the expressin fr reactin time t in the batch reactr Let us cnsider ur example f the secnd rder reactin and find the PFR vlume needed t prduce 38 ml/min ml ( 0.C Lmin when C ml L and desired cnversin Frm stichimetry it fllws that F F 9
10 Substitutin in the expressin fr reactr vlume (eq ( we get: F P kc kc ( 38 7,600L 7.6m3 0.x( 0.95 The maximum prductin rate frm that vlume can be btained at zer cnversin kc ( 0.xx ml / min max verage rate in PFR F ( entrance 0.C 0. 0 ml Lmin ( exit 0.C ( x0 4 ml L min 38 7, x0 3 ml min Clearly there is a big variatin in the reactin rate between the entrance and exit f the plug flw reactr (PFR..3 STY Space Time Yield lumetric Reactr Prductivity  RP Reactr vlumetric prductivity (RP is defined by: R P (3 Fr ur nd rder reactin example f stichimetry P, RP fr the tw cntinuus flw reactrs is: CSTR R P (R P exit ( exit kc ( (4a PFR R P kc ( (4b 0
11 Fr the same exit cnversin (R P PFR kc ( (R P CSTR kc ( t 0.95 (R P PFR (R P CSTR 0 Indeed 0 x 7,600 L 5,000 L This is why higher CSTR vlume is needed. t 0 (R P PFR R P ( CSTR There is n difference! Let us cnsider anther example t illustrate sme imprtant pints. Ex: + 3B P + S stichimetry ml r 0.C C B  rate f reactin L min C ml L and 0 ml/min is the desired prductin rate are the feed reactant cncentratin and desired cnversin ssume first that we will perate at stichimetric rati s that C B 3 (ml/l. The reactin ccurs in the liquid phase s that ε 0. Find the needed reactr vlume. a Batch (t s 60 min 0.C ( (C B b a C 0.C 3 ( 3 ( 3 0.C 3 ( 3
12 Reactin time is: t 3 0.C t g 0.x a ( 3 ( x 3.8 ( x t 3.6 ( 36 ( t 0.83 min C t + t s 0.95x x0,798 L.8m b CSTR F  frm stichimetry  basic design equatin (4 F F ( 0.C C B 0.C 3 ( C B 3 C x C ( ( ( ( exit 0.xC 3 ( exit 0.xx9( x04 (ml/l min 3 3 4,444(L 44.4m 0.05
13 c PFR F  frm stichimetry Basic design equatin ( F d F P C F 0.C 3 ( x 3 ( x ( x x0.95 ( 00 8x 0.95 ( x ( 0.95,67(L.7 m Nw the rate, at stichimetric feed rati, alng the PFR as a functin f cnversin is C ( x 3 3.6( 3 PFR reactr vlume as functin f cnversin at stichimetric feed rati is F P 3.6 F P ( x 3.8x ( x 3 Hence, the prductin rate frm a given PFR vlume as a functin f cnversin (at stichimetric feed rate is stich.8 ( x ( Hw much can we increase the prductin rate by dubling C B t C B 6 (ml/l, i.e. by using B in excess? 3
14 Nw the rate as a functin f cnversin is: 3 0.C ( C B C 3.6( ( 3 0.x 3 x 3 ( x ( x a Batch t C ( 3.6 ( x( x T integrate use partial fractins: x + B +Cx ( x ( x + (B +Cx( x ( x( x 4 4 x + x + B Bx + Cx Cx 4+B B+C03B0 B3 B3 C0 C ( x( x + x 3 x ( x x x ( x ( x ln( x + ln( x x ln( ln( ln + ln ( + ln x ( ( t.8 ln x ( ( 0.95 t.05 ln.8 x x ln t.055 min Batch ill advised at these cnditins since t s >> t! 4
15 new C t +t s 0.95x,798 ml min % ld 0 By perating at duble the stichimetric requirement f B we increase, at same, the prductin rate f the batch reactr by 80%. b CSTR 3.6( ( 3.6( 0.95( x0.05x ml L min new ld R P ,494 4,40 ml min 00 4, x00 44,000% In a CSTR we increase the prductin rate by 44,000%! c PFR F new ( x( x ( x( x 3.8 ( x( x 3.8x new x ln x ( (.8x0.95x,67 new ln x0.95x,67 ( ln.05 ( new,05 ml / min 5
16 x00,05 0 x00 0,49% ld 0 In a PFR ver 0,000% increase in is btained. We present belw these ratis f prductin rate btainable at nnstichimetric rati f C B C ( C B C stich and at stichimetric rati f C B /C 3/ fr ur example reactin. This rati is: Fr a PFR:.8 ln x (nn stich ( ( F 3.6x P( stich ( ( ln x ( ( Specifically fr 0.95 we get (nn stich (stich 0.05 x x.05 ln0.5.0 ln ln Fr a CSTR (nn stich 3.6( ( ( (stich 3.6( 3 ( ( nn stich.05 ( stich Let us examine the situatin when the reactin just cnsidered ccurs at P cnst, T cnst in the gaseus phase. Then due t stichimetry we have B + 3B P + S υ j
17 Cnsider stichimetric feed f reactants at C B /C 3/. y ( υ ε y υ j ( υ 0.4 ( x C C 0.6 C B C C B 3 C xC 3 3 ( 3 ( ( 3 ( CSTR ( R x0( 0.6x ( ( ,444x ,534(L Tremendus reductin in required vlume cmpared t the ε 0 case ccurs! PFR F d 0.95 F p ( 0.6x 3 3.6( x 3 ( 0.6x ( x 3.8x (L x x 3 gain a significant reductin in PFR reactr vlume requirement is bserved. Why? 7
18 .4 Graphic Cmparisn f PFR and CSTR F F CSTR PFR ( exit d The graphic representatin f the abve tw design equatins is represented belw fr an nth rder reactin. Clearly, fr fixed feed cnditins and feed rate and fr chsen desired cnversin the vlume f the CSTR will always be larger than r equal t the PFR vlume. F CSTR area f bx F PFR area under the curve FIGURE 4: Graphical Cmparisn f CSTR and PFR 8
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