he 471 Fall 2014 LEUE 7a Nnithermal hemical eactr S far we have dealt with ithermal chemical reactr and were able, by ug nly a many pecie ma balance a there are dependent react t relate reactr ize, let and utlet cmpit. wever, even fr ithermal reactr we need the energy balance t determe what heat duty i neceary rder t keep the reactr ithermal. Fr nnithermal reactr (adiabatic and nnadiabatic) we need the energy balance tgether with the ma balance rder t arrive at reactr deign equat. he energy balance i the prciple f cnervat f energy r the firt law f thermdynamic a applied t ur react ytem. ere we cnider nly: A1 hmgeneu ytem A2 gle react A 0 1 A3 cntuu flw reactr peratg at teady tate he cnervat equat require: ate f ate f ate f ate f (1) put utput generat accumulat Apply it t the energy, the cntrl vlume beg the ttal vlume f the react mixture the reactr. Becaue abence f nuclear react, energy cannt be detryed r created (III) 0. Al, due t aumpt f teady tate (IV) 0. he energy per unit ma f the let tream may have different frm: E KE PE U O
Energy Ketic Ptential Internal Other per unit ma energy per energy per energy per energy per f a tream unit ma unit ma unit ma unit ma Amng ther, energy frm may be urface energy (nt imprtant hmgeneu flw), magnetic energy, electric energy, etc. A we already ee frm equat (1) nly change energy play a rle. Mt ften chemical reactr change ther energy frm can be neglected ce here are n magnetic r electric field ued. Nw we can write: ate f put energy ate f energy put by ate f heat addit flw f material tream frm theurrundg ate f utput energy ate f energy utput by ate f wrk dne flw f material tream by the ytem ate f energy put r utput by flw f material tream Maflw rate Energy per unit ma Nte that due t teady tate the verall ma balance dicate a cntant ma flw rate m m ut m ( kg/ ) ate f energy put by flw f material tream (2) mke PE U ate f energy utput by flw f material tream (3) mke PE U ut ate f heat addit KJ (4) q frm the urrundg
he wrk dne by the ytem cnit f three part: ate f ate f ate f wrk dne by ate f wrk dne haft wrk the urrundg wrk dne texpel dne by the tprpel by theytem the utflw t ytem theflw t the urrundg the ytem ate f wrk dne agat gravity r ther bdy frce (5) w w m m mg h h p ut p Ug eq (2), (3), (4) and (5) (1) we get 0 m ut u KE PE q m u KE PE ut p w m ut p m p p (6) m U KE PE U KE PE q w ut p m E q w t 1 law One can readily hw that chemical reactr: A4 ketic energy change are negligible cmpared t the ternal energy KE U A5 ptential energy change are negligible cmpared t ternal energy change PE U ecall the defit f enthalpy u pv u p ence equat (6) becme m q w (7) ut In almt all chemical reactr there i n cniderable haft wrk leadg t:
A6 negligible haft wrk (8) m q ut Fr bkg purpe chemical reactr we repreent (9) m ut F ; m F 1 Enthalpy per unit ma f mixture 1 Ma flw rate We will here further aume: A7 ideal mixture all pecie Partial mlal enthalpy f Mlar flw rate pecie per mle f at f pecie cndit f the mixture partial mlal mlal enthalpy enthalpy While f (, P, cmpit ), f (, P). Furthermre, fr gae we will aume ideal gae *A8 ideal ga f ( ) hi mean that mlar enthalpie are funct f temperature nly. Nte. hi aumpt mut be deleted and effect f preure cnidered when dealg with react ytem uch a: ammnia ynthei, high P plymerizat, cal cnver, etc. hi mean that reference (tandard) p (10) p pecificheat fr (10a) ( ) ( ) pd f enthalpy f frmat at tandardcndit temperature crrect With eq (9) and aumpt (7) and (8), the reactr energy balance can fally be written a: (11) F ( ) F ( ) q 1 where exit temperature i, let temperature i. he ma balance fr any pecie i
(12) F F X J Ug eq (12) t elimate F eq (11) we get (13) F X ( ) ( ) ( ) q 1 Ug equat (10a) eq (13) we get (14) F pd X f 1 1 p d q ecall that by defit the tandard heat f react i: 1 f eat f react at temperature i ( ) 1 p dt (14a) F pd X ( ) 1 ( I ) ( II ) q ( III ) Equat (14a) can be terpreted wrd a fllw: ate f heat addit t the reactr ate f heat "abrpt" by react at temperature f feed i.e.change enthalpy caued by react therate f react prgre ate f heatg the react mixture at exit cmpit frm feed t exit temperature (III) = (I) + (II) ad we elimated by eq (12) F eq (11) we wuld bta an equivalent equat t eq (14a). (14b) 1 F p d X ( ) q
eat needed t raie eat ued by let temperature t react at exit exit temperature temperature eat added In react engeerg prblem, addital aumpt are ften made. A9: - cntant mean pecific heat and denity can be ued, defed by: Q pm ( ) S 1 F p d A10: - heat f react i apprximately cntant () ( ) ( ) Under the aumpt (9) and (10), equat (14a) r (14b) are reduced t: (15) Q pm( ) X q ( I) ( II) ( III) Equat (15) i then ften terpreted a the heat balance caug great diturbance amng thermdynamicit. Senible heat eat abrbed abrbed by flwg by react tream eat added By terpretg term (II) a a generat term an energy balance, diervice i dne t the clarity f the ubect. wever, a uer can ue equat (15) a lng a he recall what the aumpt under which it wa frmulated are and make ure that they are atified. Applicat t Ideal eactr S he entire reactr i at temperature. ence eq (15) can be ued a it i Q pm With the ma balance: X V. ( ) X ( ) q
PF Equat (15) a it tand i valid between let and utlet. wever q then repreent ttal rate f heat addit. In rder t lve the reactr prblem we have the ma balance dx J r dv J where the react rate r i a funct f extent X and temperature at pit V. hu we mut apply eq (15) t a differential vlume reultg Q pm d dv dx q dv q v - rate f heat addit per unit reactr vlume at pit V. v