The following data were obtained in a homogenous batch reactor for the esterification of butanol (B) with acetic acid (A):

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1 Departmet of Eerg Politecico di Milao ia Lambruschii MILNO Exercises of udametals of hemical Processes Prof. Giapiero Groppi EXERISE Reactor for the esterificatio of butaol The followig data were obtaied i a homogeous batch reactor for the esterificatio of butaol () with acetic acid (): H OOH + H H 2 H 2 H 2 OH H OOH 2 H 2 H + H 2 O The experimets were carried out at a, startig from the followig iitial coditios: =.22 mol =.6 mol The data collected durig the experimets are reported i the table below. The batch volume is equal to 2 cm. Determie the parameters of the power law tpe kietic equatio that describes the experimetal results. urthermore, assume that the reactio is also carried out i a flow-tpe reactor. Specificall, compare the results obtaied i the case of a STR reactor ad a PR reactor. t the ilet of the reactor, the flow rate of acetic acid is mol/h (MM = 6 g/mol, desit = 5 kg/m ) ad that of butaol is 5 mol/h (MM = 74 g/mol, desit = 8 kg/m ). ssumig that the mixture is ideal ad that the total volumetric flow rate (acetic acid ad butaol) maitais costat, estimate the volume required b each reactor i case the target coversio of acetic acid is 2%. Data overted t [h] moles of

2 Solutio The reactio rate (r) is expressed b a power law tpe expressio: r k Give that the cosumptio stoichiometr is equimolar, at the ed of the reactio.798 mol of butaol are coverted. Therefore, the amout of butaol maitais almost costat, passig from to.6 mol at the begiig, to.86 mol at the ed of the reactio. The cocetratio of butaol ca be assumed as costat durig the experimet. ased o the data, it is ot possible to determie the value of the expoet of. It is the possible to iclude i the reactio costat. r k The material balace o the batch reactor is writte as: d dt r k k k k The data at disposal are expressed as moles of acetic acid, ot i terms of cocetratio. Despite the volume of the batch reactor is kow, it is coveiet to rearrage the material balace i order to iclude the reactor volume i the costat. lso, the data are i itegral form, whereas the material balace is i differetial form. It is the possible to: ) tur the data ito a differetial form, b applicatio of the fiite differece approach; i this wa, the ucertait due to the fiite differece approximatio is itroduced i the estimates; 2) tur the material balace equatio ito a itegral form (ad, if possible, ito a aaltic form); this approach however icreases the complexit of the parameter estimatio, via o-liear regressio. I the followig, the parameters are determied via the rearragemet of the itegral data ito differetial data, followed b a liear regressio o the expressio of the material balace ito a logarithmic form: l d dt l l The rate of cosumptio of the acetic acid d /dt is determied as follows: ' ' ' x x x x x x x x x x x x 2 x I the iitial istat of the reactio (t = ), the + expressio must be used. I the fial istat (t = 8h), the expressio - must be used. It is importat to ote also that the data are expressed i term of moles of coverted acetic acid. t ever istat t of the reactio, the moles of acetic acid preset i the sstem are calculated as follows: 2

3 mol mol mol cov The results are reported i the table ad i the plot below. t [h] moles of moles r coverted of [mol/h] l(-r ) l( ) verif The liear iterpolatio gives the followig kietic parameters: exp.4.79 mol.627 m h The costat icludes also the volume of the reactor (expressed i m ). The costat k is the determied: it icludes the reactio costat ad the term raised to the reactio order β. k k k m mol h It is worth to ote that the uits of the k costat do ot express a phsicall soud relatio, but are a cosequece of the fact that the global rate has the uits of mol/m /h. It is fiall possible to verif whether or ot the estimates of the kietic parameters are correct, b trasformig the material balace equatio

4 ito a itegral form. Give the value of the reactio order, it is possible to itegrate the material balace equatio: d dt d t t t,.627 t.627 t.627, I the last colum of the table, the values of the costat estimated via the itegral form are reported. The arithmetic average of the values (excludig the iitial ull value) leads to a estimate equal to.722, which verifies the hpotheses of the liear equatio. I the secod part of the problem, the same reactio is assumed to be performed i a STR ad i a PR. The problem asks for the estimatio of the volume of each reactor. The kietic equatio is kept ualtered with respect to the batch problem. efore writig the material balace for each reactor, it is coveiet to evaluate the volumetric feed flow rate. Give that the liquid mixture is ideal (that is, there are o variatios of the volume as a cosequece of the mixig of the two currets), the volumetric flow rate is obtaied from the molar masses ad the desities of the species: PM PM m h I the case of the STR, the material balace is: R STR k STR It is importat to ote that the reactio rate i the STR balace is a fuctio of the outlet cocetratio of the stream at that exits the reactor. It is also possible to express the phsical quatities preset i the balace as a fuctio of the coversio of the acetic acid ad of the volumetric flow rate of the itel feed : rom which oe obtais: The estimatio of the cocetratio leads to: 4

5 945.78mol m Ol i case the total volumetric flow rate is costat (i.e. = ), it is possible to estimate based o the coversio as a fuctio of the cocetratio, as follows: substitutio of the latter relatio i the balace equatio, it results: STR rom which, it is obtaied: STR I the problem, the volume of the STR results equal to 6.8 liters. dditioall, it is possible to estimate the residece time iside the reactor: STR. 25 h I the case of the PR reactor, the balace equatio is: d d PR R k I the equatio, the reactio rate is a fuctio of the cocetratio of the stream, which is varig alog the reactor volume, startig from the feed sectio up to the outlet sectio. assumig that the total volumetric flow rate is costat, ad b substitutig the cocetratio as a fuctio of the coversio, it results: d d d d PR The equatio has to be itegrated betwee ad the fial coversio χ,f (equal to 9%), obtaiig: 5

6 6,, f PR PR PR d d d f The volume of the PR is equal to 4.6 liters. s i the previous case, the residece time is estimated as: h PR 7 8. It is importat to ote that the residece time i the PR is lower tha that obtaied i the STR, at equal outlet coversio of the acetic acid. This result is coheret with the fact that, i the case of the STR, the reactio rate is evaluated at the outlet coditios i the whole volume of the reactor (specificall, is the relevat quatit) ad therefore the reactio rate value is alwas lower tha that obtaied i the PR balace, which varies from the highest value at the ilet cocetratio ( ) to the lowest value at the outlet cocetratio ( ).