A First Course on Kinetics and Reaction Engineering Unit 17. Reactor Models and Reaction Types

Transcription

1 Unt 17. Reactor Moels an Reacton Types Overvew The focus of Part II of ths course was the moelng of reacton rates. Ths unt s the ntroucton to Part III where chemcal reacton engneerng s the focus. It ntrouces two prmary reacton engneerng actvtes: the esgn of reactors that are yet to be bult an the moelng of reactors that alreay exst. The purposes an goals for these actvtes ffer, but both utlze reacton engneerng at ther core. The reactors consere n Part III of the course are the same three eal reactor types as were use n the generaton of knetcs ata. However, when these reactor types are use n commercal processes, ther operaton s not constrane as t was urng knetcs stues. In other wors, commercal process reactors nee not operate at steay state, sothermally, wthout pressure rop, etc. For ths reason, more rgorous esgn equatons are neee n orer to accurately moel commercal process reactors, an these esgn equatons are erve n ths unt. In aton, havng spent tme examnng reacton knetcs n Part II of the course, t s useful to entfy typcal types of chemcal, to conser the typcal knetc behavor of chemcal an to classfy certan atypcal behavors here n ths ntrouctory unt. Ths s so because certan classes of are better sute to certan types of reactors. In aton, f one has a general feel for the knetc behavor of a reacton, then one can qualtatvely assess how a gven type of reactor wll perform when runnng that reacton. Ths can be partcularly useful to a practcng engneer who s calle upon when somethng unexpecte happens urng a reacton process. It allows the engneer to entfy lkely causes for the unexpecte behavor an to rule out other causes wthout havng to wrte out an solve equatons. Learnng Objectves Upon completon of ths unt, you shoul be able to efne, n wors, the followng terms: Auto-thermal Auto-catalytc Reactant-nhbte Prouct-nhbte Seres reacton networks Parallel reacton networks Seres-parallel reacton networks Upon completon of ths unt, you shoul be able to perform the followng specfc tasks an be able to recognze when they are neee an apply them correctly n the course of a more complex analyss: State the two objectves of reacton engneerng Lst examples of engneerng tasks that utlze reacton engneerng moels Explan that reactors must be optmze n the context of the overall chemcal process n whch they appear Copyrght 201, Carl R. F. Lun

2 Ientfy the mole, energy an momentum balances for eal batch reactors, eal CSTRs an eal PFRs Smplfy the mole, energy an, f approprate, momentum balances for cases where the flow reactor (CSTR or PFR) operates at steay state Ientfy nput, output, accumulaton an generaton terms n the mole, energy an momentum balances for eal batch reactors, eal CSTRs an eal PFRs, an escrbe ther physcal sgnfcance Derve an smplfy esgn equatons for reactors that are smlar to the eal reactors Descrbe the effects of temperature, reactant concentraton an prouct concentraton upon the rate of a typcal reacton Dscuss the thermal effects assocate wth exothermc an enothermc, both reversble an rreversble Informaton Reacton engneerng entals the formulaton an use of accurate mathematcal moels of chemcal reactors. The objectves of reacton engneerng are () to construct accurate mathematcal moels of real worl reactors an () to use those moels to perform some engneerng task. There are several such tasks n whch the reacton engneerng moel mght be employe. One s to smulate the effect of some change n operatng proceure n orer to etermne how t wll affect proucton, energy consumpton, etc. Another s n process smulaton an optmzaton where the reactor s one part of a much larger moel an that larger moel s beng use to fn the most effcent an proftable way to operate the overall process. Reacton engneerng moels can also be use to construct computer programs to control a chemcal process so that the reactor operates at the contons for whch t was esgne. Those tasks nvolve moelng of an exstng reactor system. Reactor esgn s an engneerng task where the purpose s to specfy a new reactor an how t shoul be operate n orer to prouce some prouct or proucts. The three ealze reactor moels, perfectly mxe batch, CSTR an PFR, can be use to moel a surprsngly large number of commercal reactors as long as we remove the restrctons we use when stuyng knetcs n Part II of ths course. Here, n Part III, we wll conser reacton engneerng usng these three ealze moel types. Then, n Part I, we wll examne moels for some selecte reactors that cannot be accurately represente usng the eal reactor moels. Before begnnng our stuy of reacton engneerng t s mportant to put t n the proper context. In ths course, we wll concentrate prmarly on the frst of the two objectves mentone above: constructon of accurate mathematcal moels of reactors that are use for commercal chemcal processng. Dong so can create a false mpresson for frst tme stuents because ths approach gnores the fact that n almost all cases a commercal chemcal reactor s one part of a larger overall process. In a course lke ths one, our narrow focus can leave the mpresson that the secon objectve mentone above, usng the moel to perform some engneerng task, nvolves only the reactor moel, an that perspectve s ncorrect. Chemcal processes are operate for the purpose of makng a fnancal proft, an therefore the goal of AFCoKaRE, Unt 17 2

3 most of the engneerng tasks that use the reacton engneerng moels s to maxmze the rate of proft of the process, not of the reactor. The key pont to recognze s that the reactor must effcently ntegrate nto the overall process; t s not a stan-alone tem. For example, t mght not be possble to have an absolutely pure fee to the reactor because that woul make the cost of reactant purfcaton prohbtve. It mght be necessary to lmt the temperature of the proucts so that ownstream separaton processes can accommoate them. The amount of materal to be processe n the reactor may be lmte by the capacty of an upstream or ownstream process, an so on. As such, reacton engneerng can be thought of as a form of constrane optmzaton. The optmzaton nvolves creatng a process that maxmzes the rate of fnancal proft. The constrants come from a varety of sources nclung safety, cost, operablty, ablty to ntegrate nto the overall process an so on. For the reactor porton of the overall process, some of the key factors to be consere when performng reacton engneerng nclue operablty an relablty of the reactor, energy usage by the reactor, space tme yel (rate of prouct generaton per volume of the reactor) an selectvty. Even wthn ths reactor-focuse group of factors there are traeoffs, e. g. gettng acceptable selectvty may requre lower space tme yel. For a general rule of thumb, n reacton engneerng the objectve s often to generate the esre prouct as fast as possble, wth the hghest selectvty possble, usng as lttle energy as possble an n as small a reactor volume as possble whle mantanng relablty, operablty, envronmental compatblty an safety. Inustral chemcal reactors can be large, complex peces of equpment. Nonetheless, a great many nustral reactors can be moele wth reasonable accuracy as ether a perfectly mxe batch reactor, a CSTR or a PFR. For the purpose of analyzng knetc ata the reactors were assume to be sothermal an n the case of the flow reactors to operate at constant pressure an at steay-state. For the most part, t also was assume that only a sngle reacton was takng place. In most cases t woul requre a lot of effort to make an nustral reactor run sothermally, an rarely s there any compellng reason for ong so. Atonally, t s common for more than one chemcal reacton to take place n an nustral reactor. As such, t s necessary to erve the esgn equatons for each of the three eal reactor types wthout makng these assumptons. The startng pont, n each case, s a balance equaton,ether on the moles of one partcular or on energy, whch takes the general form of equaton (17.1) INPUT + GENERATION = OUTPUT + ACCUMULATION (17.1) Mole balance (on an arbtrary ) for a perfectly mxe batch reactor. We begn wth the balance equaton. INPUT + GENERATION = OUTPUT + ACCUMULATION By efnton, no materal enters or leaves the system urng a batch reacton, so the nput an output terms are zero. GENERATION = ACCUMULATION AFCoKaRE, Unt 17 3

4 We exclue nuclear, so the only way moles of can be generate s by chemcal reacton. In general, there can be any number of, j, an coul be a reactant or a prouct n any of those. Lettng rj enote the generalze rate of reacton j (as efne n Unt ), the rate of generaton of, per unt flu volume, s equal to ν,jrj where ν,j s the stochometrc coeffcent of n reacton j. If we assume that there s a sngle flu n whch all take place, then the total rate of generaton of per unt flu volume s equal to all j ν, j. The reactor s perfectly mxe, an therefore the rate s the same anywhere wthn the reacton volume. Therefore, the total generaton of moles of s equal to ν, j where we have further assume that every reacton rate has been normalze per unt flu volume. ν, j = ACCUMULATION all j Fnally, we smply note that the nstantaneous accumulaton of moles of s just n. Ths leas to the complete mole balance on n a perfectly mxe batch reactor, equaton (17.2), where the equaton has been rearrange so that the ervatve s on the left han se. A mole balance of ths type can be wrtten for each present n the system. n (17.2) = ν, j Energy balance for a perfectly mxe batch reactor. Once agan, we begn wth the general balance equaton. INPUT + GENERATION = OUTPUT + ACCUMULATION Whle no mass enters or leaves the batch reactor urng the reacton, t s possble for energy to enter or leave the reactor. Recall that a batch reactor can have a jacket surrounng the reacton volume or t can have a cole tube submerge wthn the flu n the reactor. If a heatng meum (for example steam) or a coolng meum (e. g. chlle water) flows wthn the jacket or col, heat wll move to or from the reactor. Even f nether a jacket nor a col s use, there can stll be heat losses to the surrounngs, unless the reactor s perfectly nsulate. We efne!q to be the net heat nput to the reactor, nclung both elberate heat exchange an heat losses. Ths term woul also nclue energy nput from electrcal heatng elements, by raaton, etc.!q + GENERATION = OUTPUT + ACCUMULATION AFCoKaRE, Unt 17

5 Energy can also enter or leave the system n the form of mechancal work that s one on or by any shafts, movng bounares, etc. We efne! W to be the net rate at whch mechancal work s one by the reactor system on ts surrounngs through shafts, movng bounares (nclung pstons), etc. Ths means that for the very common case of a close, constant volume tank wth an agtator to mx the contents, the work term wll be negatve an equal n magntue to the rate of energy nput by the agtator.!q + GENERATION =! W + ACCUMULATION Snce we exclue nuclear, there s no way for energy to be generate n the reactor, an so the generaton term equals zero. You may be thnkng that the heat of reacton generates energy, but actually t only transforms t from chemcal potental energy to heat. As such, the heat of reacton wll appear momentarly when we conser the accumulaton term.!q =! W + ACCUMULATION Each wthn the reactor possesses energy n varous forms (knetc, potental, nternal, etc.). If we assume that the reactor s statonary, then overall there s no change n the total knetc or potental energy of the system. There can be accumulaton (or epleton) of nternal (chemcal) energy wthn the system, however. If û represents the specfc molar nternal energy of an Δũmx represents the mass-specfc energy of mxng of the entre soluton, then the total nternal energy of the system s equal to m tot Δ!u mx + n û, where mtot represents the total mass of the soluton n the reacton volume. Here we wll assume that the reactng soluton s an eal soluton (gas phase or lqu phase), n whch case the frst term s equal to zero. The nstantaneous accumulaton of nternal energy s then equal to the ervatve of the nternal energy wth respect to tme, energy balance. Q = W + n û Ths can be rearrange whle also notng that!q! W = n û n û, whch can then be substtute nto the n û = ( n û ). AFCoKaRE, Unt 17 5

6 Recall from thermoynamcs that û = ĥ P ˆ, where ĥ s the specfc molar enthalpy of, P s the pressure an ˆ s the specfc molar volume of.!q! W = ( n ( P ˆ )) The strbutve property of multplcaton can be apple to the terms wthn the outer parentheses!q! W = ( n n P ˆ ) After takng the ervatve of the term n parentheses, the resultng summaton over two terms can be splt nto two summatons, one over each of the terms.!q! W =!Q! W = n n P ˆ. ( n ) ( n P ˆ ) Accorng to the chan rule for fferentaton, ĥ ( n ) = n + n ĥ an ( n P ˆ ) = ( n ˆ ) P + P n ˆ tme are each the same n every term n the summatons, they can be factore out.. Then, because the total pressure an ts ervatve wth respect to!q! W = ĥ n + ĥ n P ( n ˆ ) P ( n ˆ ) Next note that has been assume to be eal. ( n ˆ ) = ( n ˆ ). In aton, n ( ˆ ) = snce the soluton!q! W = ĥ n + ĥ n P P Assumng that no phase changes take place urng the reacton, ĥ = Ĉ p specfc molar heat capacty of. T, where Ĉ p s the AFCoKaRE, Unt 17 6

7 !Q! W = T n Ĉ p + ĥ n P P Snce T oes has the same value n every term n the summaton, t can be factore out.!q! W = T n ( n Ĉ p ) + P P The mole balance esgn equaton, equaton (17.2) can then be substtute for n.!q! W = T ( n Ĉ p ) + ν j rectons P P Agan the volume can be factore out of the summaton. At the same tme, the strbutve property of multplcaton can be apple: ĥ out.!q! W = T ν, j r = ν, j r. ( n Ĉ p ) + ν, j The summatons can be performe n ether orer.!q! W = T ( n Ĉ p ) + ν, j P P P P Snce the generalze rate, rj, has the same value n every term n the nner sum, t can be factore!q! W = T ( n Ĉ p ) + ν, j Fnally t can be note that mxe batch reactor, equaton (17.3). P P ( ν, j ) = ΔH j. The result s the energy balance for a perfectly AFCoKaRE, Unt 17 7

8 !Q W! = T ( n Ĉ p ) + ΔH j P P (17.3) Mole balance (on an arbtrary ) for an eal CSTR. An eal CSTR s qute smlar to a perfectly mxe batch reactor. The most sgnfcant fference s that mass flows nto an out of a CSTR, an consequently atonal terms are present n the mole an energy balances. We begn the ervaton wth the general balance equaton. INPUT + GENERATION = OUTPUT + ACCUMULATION If the same assumptons are mae, the generaton term an accumulaton terms are the same as they were for the perfectly mxe batch reactor. INPUT + ν, j = OUTPUT + n As alreay note, mass oes flow nto an out of a CSTR. If ṅ 0 an ṅ are respectvely efne as the nlet an outlet molar flow rates of, they become the nput an output terms.!n 0 + ν, j =!n + n Ths equaton s not very useful because t mxes moles, n, an molar flow rates, e. g. ṅ. Frst, by efnton n = C, where C, s the concentraton of n the reacton volume.!n 0 + ν, j =!n + C The reactor s perfectly mxe, so the concentraton of s the same everywhere wthn the flu. Snce ths same flu s what leaves the reactor, t, too, must have the same concentraton. If we assume that there s a sngle flu stream leavng the reactor, then by efnton, C = where s the total volumetrc flow rate of the stream leavng the reactor.! = 0 + ν, j (Note: the equaton above wll be use below n ervng the energy balance esgn equaton for an eal CSTR). For present purposes, the chan rule for fferentaton can be apple to the ervatve: AFCoKaRE, Unt 17 8

9 ! =! +!!n!! 2. Upon substtuton, ths leas to the fnal form for the mole balance esgn equaton on for a CSTR, equaton (17.), whch has been rearrange to put the ervatves on the left han se of the equaton. One must be careful when usng ths equaton not to confuse the reacton volume,, wth the total outlet volumetrc flow rate of the flu,.! +!!n!! 2 = 0 + ν, j (17.) Energy balance for a CSTR. Once agan, we begn wth the general balance equaton. INPUT + GENERATION = OUTPUT + ACCUMULATION As wth the batch reactor, there s no energy generate wthn the reactor snce we exclue nuclear. The accumulaton term s also the same as t was for the batch reactor. Ths mples the same assumpton that the reactor s statonary so that ts knetc an potental energes o not change. INPUT = OUTPUT + n û Makng the same assumpton of a sngle flu stream leavng the reactor, the moles of can be replace: n = C =. INPUT = OUTPUT + û As wth the batch reactor, one source of heat flow nto the system s by heat transfer, an!q can be use to represent ths heat flow wth the exact same meanng as for the batch reactor. Smlarly,! W can be use to represent the rate of any mechancal work one by the system va movng shafts, etc. Agan ths term s exactly the same as t was for the batch reactor. Wth a CSTR, there are two atonal nput an two atonal output terms. Frst, every flowng nto the system has an assocate nternal energy, an every flowng from the system carres an assocate nternal energy wth t. Assumng eal solutons, these terms are smply equal to 0û 0 an û, an when they are summe for all, they gve the total nternal energy enterng an leavng the reactor. (We assume that the knetc an potental energy terms assocate wth the nlet an outlet streams are nearly equal to each other an consequently cancel each other out. If a stream entere or left the reactor n the form of a hgh spee jet, one woul nee to nclue corresponng fference between the nlet an outlet knetc energy terms.) AFCoKaRE, Unt 17 9

10 Fnally, n orer for the flu to flow nto an out of the reactor, t must o flow work (thnk of t as pushng ts way n an out of the reactor) at rates of P 0 an P, respectvely. 0!Q + 0û + P 0 = W! + û + P + û Ths can be rearrange takng note that the ervatve of a sum s the sum of the ervatves.!q! W = 0 û + P 0û P 0 + û The substtutons that P 0 =!n 0 P ˆ 0 an P = P ˆ can be mae.!q! W = û + P ˆ 0 0û!n 0 P ˆ 0 + Next note that û + P ˆ = û + P ˆ an 0 0û!n 0 P ˆ 0 =!n 0 û 0 + P ˆ 0.!Q W! = û + P ˆ!n 0 û 0 + P ˆ 0 + û û Thermoynamc relatonshps can then be substtute ( û + P ˆ = ĥ an û 0 + P ˆ 0 = ĥ0 ).!Q W! =!n P ˆ The chan rule can be apple: P ˆ = ( ĥ P ˆ ) +!n ( P ˆ ).!Q W! =!n P ˆ + P ˆ ( ) AFCoKaRE, Unt 17 10

11 The ervatve of the fference can be replace as follows: ( P ˆ ) = ĥ!q W! =!n P ˆ + ĥ P ˆ ( ) The strbutve property can then be apple nse each of the sums.!q W! =!n 0 0!n + P ˆ + The relatonshp, ĥ! P ˆ ( ) ( P ˆ ). = 0 + ν, j, from the ervaton of the mole balance esgn equaton can then be substtute nto the thr term on the rght han se of the equaton.!q W! = ( )!n 0 0 ( ) +!n 0 + ν, j P ˆ + all j ĥ The same term can be expane usng the strbutve property:!n jν j r j = + ( ν, j ) all j P ˆ ( )!Q W! = ( )!n ( ) + ( ) ( ) + ( ν, j ) P ˆ + ĥ P ˆ ( ) AFCoKaRE, Unt 17 11

12 The frst an fourth sums on the rght han se cancel each other out. The secon an thr sums on the rght han se can be combne:!n =!n 0 ĥ same value n every term n the ffth sum, t can be factore out, an 0 ( ). Snce has the ĥ can be move nse the nner sum. If we assume no phase changes take place, ĥ = Ĉ T p. The chan rule can be use n the last term, ( P ˆ ) = P ( ˆ ) + ˆ ( P).!Q W! =!n 0 ĥ 0 ( ) + ( ν, j ) P ˆ + Ĉ p T P ˆ ˆ Many of the sums nclue quanttes that have the same value n every term n the sum; (n one case after changng the orer of the sums) these can be factore out, an the terms can be re-orere.!q W! =!n 0 ĥ 0 ( ) + ( ν, j ) + P ( ˆ ) P T P!n ˆ P ˆ ( Ĉ p ) If we assume there are no phase changes wthn the reactor, the enthalpy fference n the frst term on the rght se of the equaton can be wrtten as ĥ ĥ0 = Ĉ p T. The nner sum of the secon term s, by efnton, ( ν, j ) = Δ H j. As before, the sum n the fourth term on the rght se s T = ˆ. The chan rule can be use for the ervatve n the last term, = +!n 2. T T 0 AFCoKaRE, Unt 17 12

13 T!Q W! 0 = Ĉ p T + Δ H j + T P T 0! P!n ˆ P ˆ!n P!n ˆ + P ˆ 2 T ( Ĉ p ) The volumetrc flow rate can be cancele from the term n blue above. Also, t can be factore outse the summaton, along wth the ervatve, summaton that remans s, from the term n re. After ong ths, the ˆ =. Ths wll leave a volumetrc flow rate n the numerator an n the enomnator, whch wll cancel. These changes can be mae an the equaton can be rearrange. T!Q W! 0 = Ĉ p T + Δ H j + T T 0 P P T ( Ĉ p ) P!n ˆ!n P!n ˆ + P ˆ 2 The terms whose values are the same n every term can be factore out of the fnal two terms. T!Q W! 0 = Ĉ p T + Δ H j + T T 0 P P The fnal sum can be replace,!n ˆ!n + ˆ 1 numerator an n the enomnator that wll cancel. T!Q W! 0 = Ĉ p T + Δ H j + T T 0 P P T ( Ĉ p ) P!n ( ˆ ) ˆ =, an ths wll leave a volumetrc flow rate n the T!n ˆ!n + ˆ ( Ĉ p ) P AFCoKaRE, Unt 17 13

14 Next the last ervatve can be replace by frst notng that snce = rule, =!n ˆ +!n ˆ. T!Q W! 0 = Ĉ p T + Δ H j + T P P T 0 P T ( Ĉ p )!n ˆ!n + ˆ!n ˆ!n ˆ ˆ, then by the chan Followng the substtuton, the four terms n the square brackets cancel each other out an the entre term goes to zero. Specfcally, ang the two terms n re gves zero as oes ang the two terms n blue. Ths fnally gves the energy balance equaton for an eal CSTR, equaton (17.5). T!Q W! 0 = Ĉ p T + Δ H j T + T 0 ( Ĉ p ) T P P (17.5) Mole balance (on an arbtrary ) for an eal PFR. Unlke the perfectly mxe batch reactor an the CSTR, the composton n an eal PFR s not unform. As a consequence, t becomes necessary to wrte the mole an energy balance equatons on a fferental element of reactor volume an then take the lmt as the sze of ths element goes to zero. Here, we wll assume that the reactor s cylnrcal wth a constant ameter, D, an a length, L. Fgure 1 shows such a reactor, hghlghtng a fferentally thck cross secton that wll be use n formulatng the mass an energy balances for the eal PFR. z D ṅ L AFCoKaRE, Unt 17 1

15 Fgure 1 olume element use n ervng the eal PFR esgn equatons. Recall that n a PFR, t s assume that there s perfect mxng n the raal recton an no mxng n the axal recton. Wth these assumptons, the envronmental varables wll be unform throughout the entre fferental element shown n Fgure 1 (or, put fferently, the fferental element shown n Fgure 1 s perfectly mxe). The ervaton of the mole balance on an arbtrary begns wth the balance equaton, apple to the fferental element. INPUT + GENERATION = OUTPUT + ACCUMULATION The nput an output terms are smply the molar flow rate, evaluate at the front an back of the fferental element. z + GENERATION = z+z + ACCUMULATION If we assume that there s a sngle flu wthn the reactor n whch all take place, then the total rate of generaton of per unt flu volume s equal to all j ν, j. The fferental element s perfectly mxe, an therefore the rate s the same anywhere wthn ts volume. Therefore, the total generaton of moles of wthn the element s equal to the volume of the fferental element tmes the total rate per volume, π ( D2 z) all j rate has been normalze per unt flu volume.!n + π ( D2 z) z ν, j ν, j =!n + ACCUMULATION z+z, where we have further assume that every reacton The nstantaneous accumulaton of moles of s equal to n, where n represents the number of moles n the fferental element. Snce the fferental element s perfectly mxe, the total moles of n the fferental element s equal to the concentraton of n the fferental element tmes the volume of the fferental element. In a flowng system, the concentraton of s equal to the molar flow rate of ve by the volumetrc flow rate, just as t was for the CSTR.!n + π ( D2 z) z ν, j =!n + π D 2 z z+z Ths equaton can be rearrange an the constant terms can be factore from the partal ervatve. z z+z z z = π D2 ν, j π D2 z AFCoKaRE, Unt 17 15

16 In the lmt of an nfntesmally thn volume element, lm z 0 z+z z z = z. z = π D2 ν, j π D2 Fnally, usng the chan rule to fferentate the partal ervatve on the rght leas to the mole balance esgn equaton for an eal PFR, equaton (17.6). z = π D2 ν, j π D2 + π D2 2 (17.6) Energy balance for a PFR. Once agan, we begn wth the general balance equaton. INPUT + GENERATION = OUTPUT + ACCUMULATION As wth the batch reactor an the CSTR, energy s not generate n the PFR snce we exclue nuclear. Snce we have assume a tubular reactor of fxe ameter, there are no shafts or movng bounares, an consequently the PFR oes not o any work on the surrounngs (there s no! W term). Heat can flow nto the reactor through the reactor wall. We assume that ths heat transfer can be escrbe usng an overall heat transfer coeffcent, U. If we let Te enote the temperature exteror to the reactor an let T enote the temperature wthn the reactor, then the heat nput through the ws equal to the wall area tmes the heat transfer coeffcent tmes the temperature fference. Applyng ths to the fferental volume element gves!q = π D( z)u T e. As wth the CSTR, the flu flowng nto an out of the volume element carres nternal energy wth t an t must o flow work as t enters an leaves. These substtutons can be mae for the nput, output an generaton terms. ( û ) z + P z + π D( z)u T e = ( û ) z+z + P z+z + ACCUMULATION The nstantaneous accumulaton of energy equals the nstantaneous accumulaton of moles of each (see the mole balance above) tmes ts specfc nternal energy. Once agan, we assume the reactor s statonary an that the total knetc energy an potental energy of the flowng flu remans constant. AFCoKaRE, Unt 17 16

17 + ( û ) z + P z + π D( z)u T e π D 2 z û = ( û ) z+z + P z+z Assumng an eal soluton, the volumetrc flow rate can be relate to the molar specfc volumes, = ˆ, an the nternal energy can be relate to the specfc enthalpy (û + P ˆ = ĥ ) n a manner exactly analogous to the CSTR. ( ) z + π D( z)u T e The equaton can then be rearrange. = z+z + π D2 z P ˆ ( ) ( ) z+z z z = π DU ( T e ) π D2 ĥ For a fferentally thn volume element, lm z 0 z (!n ĥ ) = π DU ( T e ) π D2 z+z z Ths equaton can be rearrange, combnng all the sums. π DU ( T e ) = z (!n ĥ ) + π D2 The ervatves nse the sum can be expane. z P ˆ ( ) ( P ˆ ) P ˆ ( ) = (!n ĥ ) z.!n = z + ĥ z + π D2 π D2 P ˆ If we assume no phase changes take place, then the term n blue s equal to Ĉ p z. AFCoKaRE, Unt 17 17

18 = Ĉ p z + ĥ z + π D2 The chan rule can next be apple to the term n re. π D2 P ˆ = Ĉ p z!n + z + π D2 + π D2 ĥ P ˆ The mole balance, equaton (17.6), can be rearrange as follows: z + π D2 π D2 2 = z + π D2 = π D2 all j ν, j, an substtute for the term n blue. = Ĉ p z + ĥ π D 2 ν, j + π D2 ĥ P ˆ Ths equaton can be rearrange, separatng the terms of the sum, changng the orer of the neste sums, an factorng constants outse the summatons. = Ĉ p π D2 As above, ĥ z = Ĉ p z. = Ĉ p π D2 z + π D2 P ˆ z + π D2 P ˆ The chan rule can be apple to the fnal term. ν, j ĥ ν, j ĥ + π D2 + π D2 ĥ ( Ĉ p ) AFCoKaRE, Unt 17 18

19 = Ĉ p π D2 P z + π D2 ˆ π D2 ν, j ĥ ˆ + π D2 P ( Ĉ p ) The terms that o not epen upon the nex can be factore outse the fnal sum, whch leaves ˆ =. The resultant volumetrc flow rate appears n the numerator an enomnator, an so t can be cancele. = Ĉ p π D2 P z + π D2 ˆ Next the chan rule s apple to the term n blue. = Ĉ p π D2 P z + π D2 1 π D2 ( ˆ ) P ν, j ĥ ν, j ĥ ˆ 2 + π D2 + π D2 π D2 P ( Ĉ p ) ( Ĉ p ) The terms that o not epen upon the nex can be factore outse the secon blue sum. = Ĉ p π D2 P z + π D2 1 ( ˆ ) 1 2 ν, j ĥ The resultng sum n re s just equal to the volumetrc flow rate, atonal smplfcaton. + π D2 ( ˆ ) π D2 ( Ĉ p ) P ˆ =, whch allows AFCoKaRE, Unt 17 19

20 = Ĉ p π D2 P z + π D2 1 Notng that snce = ˆ, then = that t cancels the one preceng t. = Ĉ p π D2 P ( ˆ ) 1 z + π D2 1 ( ˆ ) ν, j ĥ π D2 + π D2 P ( Ĉ p ) ( ˆ ), an substtuton n the term n blue shows 1 ν, j ĥ + π D2 ( ˆ ) π D2 ( Ĉ p ) Notng that the summaton n blue above s just the heat of reacton, ths, fnally, gves the energy balance equaton for an eal PFR, equaton (17.7). P = Ĉ p + π D2 z + π D2 ( Ĉ p ) π D2 P ( ΔH j ) (17.7) Mechancal energy balance for an eal PFR. A thr type of balance equaton s often neee when moelng plug flow reactors, partcularly packe be catalytc reactors. The flow of a flu through a tube, partcularly one packe wth sol partcles, requres that some of the flow energy must be expene to overcome frcton between the flowng flu an the statonary sol (reactor walls an packng, f present). The net result s a rop n pressure along the length of the tube. Chemcal engneers typcally use a mechancal energy balance to moel ths pressure rop. The complexty of the processes nvolve s such that these moels are usually emprcal n nature an nclue parameters such as the frcton factor. Ths topc s normally covere n chemcal engneerng courses on unt operatons. No attempt wll be mae here to erve the mechancal energy balance equatons. However, n the next secton two common steay-state forms of the mechancal energy balance wll be presente wthout ervaton. AFCoKaRE, Unt 17 20

21 Summary an steay-state forms of the balance equatons. In ervng the esgn equatons for a perfectly mxe batch reactor, the followng assumptons were mae: the reactor s perfectly mxe no mass enters or leaves the reactor urng the reacton there s a sngle flu volume n whch all take place the reactng flu s an eal soluton (gas or lqu) every reacton rate has been normalze per unt flu volume the reactor s statonary no phase changes occur urng the reacton Wth these assumptons for a perfectly mxe batch reactor, the resultng mole balance s equaton (17.2) an the energy balance s equaton (17.3). n (17.2) = ν, j!q! W = T ( n Ĉ p ) + ΔH j P P (17.3) It s sometmes helpful to assgn physcal sgnfcance to the terms appearng n these equatons. The mole balance s trval n ths regar, the left se of the equaton s the accumulaton of n the reactor an the rght se s the net generaton of by chemcal reacton. In the energy balance, the two terms on the left se are the exchange of energy wth the surrounngs, ether as heat or as work. The frst term on the rght s the change n sensble heat wthn the reactor, the secon term s the heat absorbe by chemcal reacton, an the last two terms are work the flu must o n expanng or contractng. The assumptons use n ervng the esgn equatons for a CSTR were as follows: perfect mxng every reacton rate has been normalze per unt flu volume the reactng flu s an eal soluton (gas or lqu) there s a sngle flu stream leavng the reactor the reactor s statonary no phase changes occur urng the reacton the knetc an potental energy of the enterng an extng flu streams are effectvely equal to each other Wth these assumptons for an eal CSTR, the resultng mole balance s equaton (17.) an the energy balance s equaton (17.5). 0 = 0 + ν, j (17.) T!Q W! 0 = Ĉ p T + Δ H j (17.5) T T 0 AFCoKaRE, Unt 17 21

22 If an eal CSTR operates at steay state, these equatons smplfy conserably. Specfcally, at steay state all the ervatves wth respect to tme vansh. Thus, equatons (17.8) an (19.9) represent the eal, steay-state CSTR mole an energy balances, respectvely. 0 = 0 + ν, j (17.8) T!Q W! 0 = Ĉ p T + Δ H j (17.9) T T 0 As wth the batch reactor, the two terms on the left se of the steay state energy balance are the exchange of energy wth the surrounngs, ether as heat or as work. The frst term on the rght s the change n sensble heat wthn the reactor, an the secon term s the heat absorbe by chemcal reacton. The assumptons use n ervng the esgn equatons for a PFR were as follows: plug flow (perfect raal mxng an no axal mxng) cylnrcal reactor wth a constant ameter every reacton rate has been normalze per unt flu volume the reactng flu s an eal soluton (gas or lqu) there s a sngle flu stream leavng the reactor the reactor s statonary no phase changes occur urng the reacton heat transfer wth the surrounng can be escrbe usng an overall heat transfer coeffcent no shafts or movng bounares the knetc an potental energy of the enterng an extng flu streams are effectvely equal to each other Wth these assumptons for an eal PFR, the resultng mole balance s equaton (17.6) an the energy balance s equaton (17.7). z = π D2 ν, j π D2 + π D2 2 (17.6) = Ĉ p + π D2 z + π D2 ( Ĉ p ) π D2 P ( ΔH j ) (17.7) If an eal PFR operates at steay state, these equatons smplfy conserably; equatons (17.10) an (17.11) represent the eal, steay-state PFR mole an energy balances, respectvely. As wth the CSTR, these equatons result from settng all of the ervatves wth respect to tme equal to zero. AFCoKaRE, Unt 17 22

23 z = π D2 ν, j (17.10) = Ĉ p z + π D2 ( ΔH j ) (17.11) Here, the term on the left se of the steay state energy balance represents exchange of energy wth the surrounngs as heat. The frst term on the rght s the change n sensble heat of the reactng flu, an the secon term s the heat absorbe by chemcal reacton. It was note above that a mechancal energy balance must be use to moel pressure rop. Equaton (17.12) s the mechancal energy balance for an unpacke tubular reactor an equaton (17.13) s the mechancal energy balance (Ergun equaton) for a packe be tubular reactor. In these equatons G represents the mass flow ve by the cross sectonal area of the tube, gc s a gravtatonal constant (not the acceleraton ue to gravty), f s the frcton factor, ρ s the flu ensty, ε s the vo fracton of the packe be, Φs s the sphercty of the be packng, Dp s the ameter of the be packng partcles, an µ s the flu vscosty. P z = G g c π D 2! z 2 fg 2 ρd P z = 1 ε G 2 ε 3 ρφ s D p g c 150( 1 ε )µ Φ s D p G (17.12) (17.13) Typcal knetc behavor an classfcaton of. Reacton thermoynamcs allows one to calculate the heat, ΔH, of a chemcal reacton at any set of envronmental contons. If the heat of reacton s negatve (heat s release by the reacton), the reacton s calle an exothermc reacton, an f the heat of reacton s postve (heat s absorbe by the reacton), the reacton s calle an enothermc reacton. To a frst approxmaton, the heat of reacton etermnes how the equlbrum constant vares wth temperature. Recall from Unt 2 that by efnton, the equlbrum constant for a reacton, j, epens upon the entropy an enthalpy changes for that reacton as gven n equaton (17.1). If the heat an entropy changes are taken to be constant, the epenence of the equlbrum constant upon temperature can be expresse as n equaton (17.15), whch can be seen to be of the same form as the Arrhenus equaton for the temperature epenence of a rate coeffcent. One sgnfcant fference s that the actvaton energy n the Arrhenus equaton s a postve number whereas ΔHj can be ether postve or negatve. K j = exp ΔG j RT = exp ΔS j R exp ΔH j RT (17.1) AFCoKaRE, Unt 17 23

24 K j = K 0, j exp ΔH j RT (17.15) Rate expressons are typcally comprse of concentraton terms, one or more rate coeffcents, an sometmes one or more equlbrum constants. Qute commonly, the rate coeffcents are taken to obey the Arrhenus equaton an the equlbrum constants can be moele usng equaton (17.15). We have seen that rate expressons can assume we varety of mathematcal forms. Nonetheless, there are some generalzatons that can be mae an that are foun to be true for a majorty of. The followng s a short lst of generalzatons about an ther rates. Typcally, the rate of a chemcal reacton wll ncrease f the temperature of the reactng system s ncrease. Typcally, the rate of a chemcal reacton wll ecrease f the concentraton of the reactants ecreases. Typcally, the rate of an rreversble chemcal reacton s not strongly affecte by the concentraton of the proucts. Typcally, the rate of a reversble chemcal reacton ecreases as the concentraton of the proucts ncreases. The equlbrum constant for an exothermc reacton wll ecrease as the temperature ncreases. The equlbrum constant for an enothermc reacton wll ncrease as the temperature ncreases. Agan, there are a multtue of rate expressons for chemcal, an the statements above are commonly, but not always, true. At ths pont, several hypothetcal chemcal wll be use, along wth corresponng hypothetcal rate expressons, to llustrate these generalzatons. Whle the an the rate expressons to be use here are mae up, one coul fn real chemcal systems that behave smlarly. Conser frst an rreversble reacton, usng equaton (17.16) as the hypothetcal reacton an equaton (17.17) as the hypothetcal rate expresson. A + B Y + Z (17.16) r = k 0,17.16 exp E RT C C A B (17.17) Supposng ths to be a lqu phase reacton that occurs at temperatures near ambent, reasonable values for the rate expresson parameters mght be k0,17.16 = 1 x 10 8 L mol 1 mn 1 an E = 50 kj mol 1. The frst generalzaton above suggeste that ncreasng the temperature woul ncrease the rate. In equaton (17.17), there s only one term that contans the temperature, namely the exponental term. It s easy to show that ncreasng T ncreases ths term an consequently ncreases the rate. For example, at 298 K, the exponental term has a value of 1.72 x Increasng the temperature to 313 K ncreases the exponental term to a value of.53 x The secon generalzaton suggests that as the AFCoKaRE, Unt 17 2

25 concentraton of a reactant ecreases, the rate wll ecrease. Ths s easly seen n equaton (17.17) because the rate s rectly proportonal to the concentraton of each reactant, an so ecreasng CA an/ or CB wll ecrease the rate proportonately. The thr generalzaton ncates that the rate of an rreversble reacton s not strongly affecte by the concentratons of the proucts. Here t can be seen that nether CY nor CZ appear n the rate expresson, so obvously changng ether of ther values woul not affect the rate at all for ths rreversble reacton. Next conser a reversble reacton, usng equaton (17.18) as the hypothetcal reacton an equaton (17.19) as the hypothetcal rate expresson. A + B Y + Z (17.18) r = k exp E RT C C 1 C Y C Z A B K 0,17.18 exp ΔH RT { } C A C B (17.19) The fourth generalzaton suggests that for reversble, the rate ecreases as the prouct concentraton ncreases. The rate expresson, equaton (17.19), nvolves the fference between two terms (nse the square brackets). The quantty beng subtracte can be seen to be proportonal to CY an CZ. Hence, ncreasng ether CY or CZ causes a larger quantty to be subtracte an therefore leas to a smaller rate. It s smple to llustrate the last two generalzatons. As an example, conser the equlbrum constant appearng n the enomnator n equaton (17.19). Supposng that the equlbrum constant has a value of 0.65 at 298 K an that the (exothermc) heat of reacton s equal to 200 kj mol 1, t woul be foun that ncreasng the temperature to 313 K woul ecrease the equlbrum constant for ths exothermc reacton to Ths s consstent wth the ffth generalzaton above. Smlarly, f the equlbrum constant ha a value of 0.65 at 298 K an the reacton was enothermc wth a heat of reacton equal to +200 kj mol 1, ncreasng the temperature of the enothermc reacton to 313 K woul ncrease the equlbrum constant to 31.1, consstent wth the sxth generalzaton. Examnng equaton (17.19) more closely shows that for an exothermc reacton there s a bt of a trae-off wth respect to temperature. Far from equlbrum, where the 1 nse the square brackets s much, much greater than the secon term, a hgher temperature means that the rate coeffcent wll be larger, an consequently the rate of reacton wll be greater. However, as the temperature s ncrease, the secon term nse the square brackets also becomes greater. Wth contnue ncrease n temperature, eventually the secon term nse the square brackets becomes sgnfcant relatve to the 1. Further ncreases n temperature beyon ths pont stll ncrease the rate coeffcent (tenng to ncrease the rate), but they smultaneously ncrease the secon term nse the square brackets (tenng to ecrease the rate). Eventually the secon effect wll come to preomnate an the rate wll ecrease to zero as the temperature ncreases. Put fferently, as the temperature ncreases wth an exothermc reacton, the equlbrum converson becomes smaller an consequently the rate goes to zero at larger AFCoKaRE, Unt 17 25

26 reactant concentratons. As a result, the reacton wll not procee as far as t woul at a lower temperature. In contrast, an enothermc reacton s favore by hgh temperature. In an enothermc reacton a hgher temperature means both a larger rate coeffcent (faster rate) an a larger equlbrum coeffcent (greater converson). An auto-thermal reacton (or system of ) s a specal case of an exothermc reacton system. Actually t s probably more accurate to speak of runnng a reacton auto-thermally than to say the reacton tself s auto-thermal, but the latter usage s commonly encountere. The stngushng feature of an auto-thermal reacton s that the heat release by the reacton ( ΔH) s suffcent to rase the temperature of the fresh reactants to a pont where the reacton wll procee spontaneously wthout the aton of heat from any external source. Many nustral take place at elevate temperatures, an consequently t s necessary to heat the reactants up to the reacton temperature before feeng them nto the reactor. For most, ths means the reactants must pass through a heat exchanger an steam or some other heatng meum must be use to rase ther temperature. In the case of an autothermal exothermc reacton, t woul not be necessary to use any steam or other heatng source. Instea, the heat release by the reacton coul (wth approprate system engneerng) be transferre to the reactants an prove all the heat necessary to heat those reactants to the nlet reactor temperature. The reacton classfcatons mentone so far have all been relate to the thermoynamc behavor of the reactng system, an the behavor that has been scusse pertans to typcal knetc behavor. In some cases reacton rates are not affecte by changes n reactant or prouct composton as s typcally expecte. That s, the rates of some o not obey the generalzatons wth respect to concentraton that were gven earler. Ths represents another bass for classfyng. Examples are auto-catalytc, prouct nhbte an reactant nhbte. In an auto-catalytc reacton, the rate of reacton ncreases as the concentraton of a prouct ncreases (contrary to pont 3 or above). Ths oesn t necessarly occur across the full composton spectrum, but for an auto-catalytc reacton there s some range of composton wthn whch an ncrease n prouct concentraton causes an ncrease n rate. A very goo example of an auto-catalytc process s that of cell growth. If a very smple cell growth process takes place wth a yel factor of 0.5, then the cell growth reacton can be wrtten as n equaton (17.20), where X s use to enote cell mass concentraton an S s use to enote substrate (. e. nutrents or foo ) mass concentraton. If the Mono equaton escrbes ths partcular cell growth reacton, then the corresponng rate expresson s gven n equaton (17.21) where the C s represent mass concentratons, µmax s a constant (for a gven reacton temperature) representng the maxmum specfc growth rate an Ks s a secon constant (the saturaton constant). Typcal values of the constants n equaton (17.21) mght be µmax = 1.0 h 1 an Ks = 0.2 g L 1. Notng that cell mass s the prouct of reacton (17.20), t s easy to see that the reacton s auto-catalytc. If you ha a system of pure substrate, no cells woul ever grow. The system nees at least one cell ntally to metabolze the substrate an eventually ve nto two cells. If more cells are present, then the growth wll be more rap because each cell can be metabolzng the substrate an vng nto two cells at the same tme. The choce of the name auto-catalytc for lke ths s unfortunate, AFCoKaRE, Unt 17 26

27 though, because the reacton nee not nvolve a catalyst. Also, whle the example gven here nvolves a bologcal process, chemcal can also be foun that are auto-catalytc. 2 S X (17.20) r = µ max C X C S K s + C S (17.21) Another behavor that s not consstent wth the generalzatons above s the prouct nhbte reacton. A hypothetcal example can be constructe agan usng rreversble reacton (17.16). Generally one woul not expect an rreversble reacton to be affecte sgnfcantly by ncreasng the prouct concentraton, as long as the reactant concentraton remane the same. (Ths woul also be true of a reversble reacton that was very far from reachng equlbrum.) However, f the rate expresson for the rreversble reacton (16.16) was lke that gven n equaton (17.22), ths generalzaton woul not be obeye, an the reacton woul be calle a prouct nhbte reacton. Supposng that the values of k17.16 an k were 0.17 L mol 1 s -1 an 8.61 L mol 1, respectvely, the rate n an equmolar mxture of A an B (1 M each) woul be 0.17 mol L 1 s -1, but f the prouct Y was also present at a 1 M concentraton, the rate woul ecrease by a factor of almost 10 (to mol L 1 s -1 ), even though the reacton s rreversble an the amounts of A an B ha not change. r = k C C A B (17.22) 1+ k C Y Some heterogeneous catalytc an other are reactant nhbte. Normally, one expects that the rate of reacton wll ncrease f the concentraton of reactant s ncrease (pont 2 above), but for a reactant nhbte reacton, the rate of reacton ecreases when the concentraton of a reactant s ncrease. A hypothetcal example of ths behavor mght agan nvolve reacton (17.16), but wth the knetcs shown n equaton (17.23). (The fference between equatons (17.22) an (17.23) s the entty of the whose concentraton appears n the enomnator, Y versus B, an the power to whch that concentraton s rase, 1 versus 2.) If k17.16 an k have the same numercal values as gven above, then the rate n an equmolar mxture of A an B at 1 M concentraton each woul be mol L 1 s 1. If the concentraton of B was ouble (CA = 1 M an CB = 2 M), the rate woul ecrease to mol L 1 s -1. r = k C C A B 2 1+ k C (17.23) B To ths pont n the present scusson, only sngle chemcal have been consere. It s very, very common for more than one reacton to be takng place n an nustral reactor. Sometmes the reactants an proucts nvolve n each reacton are fferent from the reactants an proucts nvolve n all the other. In ths case the reacton system s sa to nvolve an nepenent set of. In other cases, a reactant or prouct n one reacton s also nvolve n other. There are many AFCoKaRE, Unt 17 27

28 possble combnatons of, but there are three common combnatons that are often use n the classfcaton of reacton networks. If the prouct of one reacton n a reacton network s the reactant for a subsequent reacton, the reacton network s calle a seres reacton network. Reactons (17.2) an (17.25) schematcally represent a seres reacton network. For seres, such as those gven n (17.2) an (17.25), that are forme n one reacton an consume n a subsequent reacton are sometmes calle ntermeate proucts or ntermeate oust ntermeates (not to be confuse wth the reactve ntermeates that appear n reacton mechansms; here the ntermeates are not necessarly hghly reactve ). The names D an U appearng n (17.2) an (17.25) are not arbtrary. It frequently occurs that the ntermeate prouct s the esre (hence represente as D) that has a hgh value whereas the fnal prouct s unesre (hence U) an has lower value. The number of an ntermeates s not lmte to two, as llustrate schematcally n equaton (17.26). A D (17.2) D U (17.25) A B C D. (17.26) A secon classfcaton of reacton networks s the parallel reacton network whch s stngushe by havng one servng as the reactant n two or more of the takng place. Ths s hypothetcally llustrate, for two, n equatons (17.27) an (17.28), but ths shoul not be taken to mean the number of parallel s lmte to two. In these A s a reactant n both, an so they are parallel. Once agan, t s not uncommon for one prouct to be the esre prouct an another to be an unesre prouct. A D (17.27) A U (17.28) A seres-parallel reacton network s a combnaton of a seres network an a parallel network. Reactons (17.29) an (17.30) hypothetcally llustrate a seres-parallel reacton network. From the perspectve of one n the network (B n the use here) the appear to be parallel, whle from the perspectve of a fferent (Y n the use here) the appear to be seres. A + B Y (17.29) Y + B Z (17.30) Fnally, t s worth pontng out that n some cases, the number of n a system can become so great that t s effectvely mpossble to wrte every one of them own an treat them explctly. An example s polymerzaton where a sngle reactant mght form mllons of slghtly fferent proucts. Fortunately, n these cases t s often possble to analyze the assocate knetcs an reactors by lumpng proucts nto groups an usng an effectve stochometry to represent the whole group of. Also, AFCoKaRE, Unt 17 28

29 n such stuatons, one often lmts the analyss of the reactor to conser only the consumpton of the reactant wthout worryng about whch of the mllons of proucts t became a part of. Hypothetcal have been ntrouce n ths unt, n part, for the purpose of llustratng some generalzatons about knetc behavor as well as some cases where the generalzatons fal. It s useful for an engneer to evelop a qualtatve feel for how fferent kns of reactors perform an to te that feel to ts unerlyng physcal orgns. Ths kn of unerstanng helps when the engneer s calle upon to select whch type of reactor to use n a new process an how to operate that reactor. Ths wll be llustrate n future unts. AFCoKaRE, Unt 17 29