5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z = 0 dc dz R c = 0 () = c dc () Pe dz z = dc dz = 0 (3) where c = C A C Ao = is the dimesioless reactat cocetratio z = Z L = is the dimesioless distace alog the reactor measured from the etrace Pe = UL D = is the axial dispersio Peclet umber L = total reactor legth U = Q A = is the liear mea velocity i the reactor (superficial velocity i packed beds) Q = volumetric flow rate through the reactor A = cross-sectioal area of the reactor D = dispersio coefficiet i the reactor (defied per total cross-sectioal area i packed beds). Note: If D is defied per cross-sectioal area uoccupied by solids i packed beds the Dε will appear i the Pe istead of D. ε is bed porosity. R = k τ C Ao - dimesioless reactio rate group (Damkohler umber). Note: I packed beds this implies that the rate was defied per uit reactor volume.
τ = L U - reactor space time (space time based o superficial velocity). Equatios (), () ad (3) describe the behavior of tubular reactors ad catalytic packed bed reactors uder isothermal coditios, at steady state for a -th order irreversible, sigle reactio. Applicatio of these equatios to packed bed reactors assumes that exteral ad iteral mass trasfer limitatios (i.e mass trasfer from fluid to pellets ad iside the pellets) are oexistet or have bee properly accouted for i the overall rate expressio. Solutios to eqs (-3) for a -th order reactio (( 0, ) ca be obtaied oly by umerical meas. However, the problem is a difficult oliear two-poit boudary value problem. [See: P.H. McGiis, Chem. Egr. Progr. Symp. Ser. No 55 Vol 6, p (968), Lee, E.S., AIChE J., 4(3), 490 (968), Lee, E.S., Quasiliearizatio (969)]. Sice i practical applicatios Pe umbers are quite large ( Pe 5), ad ofte Pe = O (0 ), it is of iterest to develop approximate solutios to the dispersio model for large Pe umbers. Remember, large Pe σ Pe meas small variace ad, hece, small variatio from plug flow, ad is exactly the coditio uder which the dispersio model is applicable. Such approximate solutios ca allow us to estimate well the departure from plug flow performace ad will save a lot of effort which is ecessary for umerical evaluatios of the model. We are iterested i large Pe, the Equatios (-3) ca be writte as: = ε ad ε is very small ε << Pe ( ). ε d c dz dc dz R c = 0 z = 0; = c ε dc dz () () z = ; dc dz = 0 (3) A. Outer Solutio Let us assume that a solutio of the followig form exists:
c = Fz ()= ε F ()= z F 0 ()+ z ε F ()+ z ε F ()+ z... (4) =0 If we ca fid such a solutio the, due to the fact that ε <<, we ca hope that the first few terms will be adequate to describe our solutio. [Strictly speakig this would be true oly if the series give by Eq (4) coverges fast. Remarkably, a good approximatio to the solutio is obtaied eve for diverget series! (For this ad further details, see Nayfeh, A.H., Perturbatio Methods, Wiley, 973) We do eed i Eq () a expressio for c. However, as we are goig to keep oly the first few terms of the series (4) raisig it to a -th power is ot that difficult. c = [ F o + ε F + ε F + ε 3 F 3 +..] = F 0 + ε F 0 F + ε ( )F 0 F + F o F +ε 3 6 ( ) ( )F 0 3 F 3 + ( )F 0 F F +F 0 F 3 + O ε 4 ( ) (5) Differetiate eq (4) oce ad twice ad substitute both derivatives ad eq (5) ito eq (): ε d F 0 dz + ε d F dz + ε d F dz + ε 3 d F 3 dz +... df 0 dz + ε df dz + df ε dz +... - R F 0 + ε F 0 F + ε ( )F 0 F +F 0 F +... = 0 (6) Substitute also eq (4) ad its derivative ito eq (): At z = 0 = F 0 + ε F + ε F +... ε df 0 dz + ε df dz + ε df dz +... (7) Group ow the terms with equal powers of ε i eq (6) ad eq (7) together, ad require that they?? to zero: ε 0 : df 0 dz + R F 0 = 0 (8) 3
z = 0, F 0 = (8a) ε : df dz + R F 0 F = d F 0 (9) dz z = 0, F = df 0 dz (9a) ε : df dz + R F 0 F = d F R dz ( )F 0 F (0) ε 3 : etc. z = 0, F = df dz (0a) Notice at this poit that the differetial equatios (D.E.s) (eq 8-, etc.) are first order D.E.s, while the origial equatio () was secod order. Eq (8) is first order for F 0. Oce eq (8) is solved ad F 0 determied, the right had side of eq (9) is kow ad the left had side is a first order D.E. for F, etc. Thus, we ca successively determie all the F i s. However, because all of these are first order equatios, we ca oly satisfy oe of the origial boudary coditios. If we tried to satisfy the boudary coditio at the reactor exit, give by eq (3), we would have that df i dz = 0 at z = for all i, implyig that all F i s are costat due to the form of D.E.s ( eq 8- ). Thus, we would ot be able to get ay iformatio. This idicates that we have to satisfy the coditio at the reactor etrace give by eq () as idicated by eq (7). This results i a set of coditios give by eqs(8a - 0a). The solutio of eq (8) with I.C. (iitial coditio) (8a) is readily obtaied: [ ] F 0 ()= z + R ( )z () Verify that this ideed is a cocetratio profile i a plug flow reactor for a -th order reactio! Differetiate eq () twice, substitute ito eq (9) ad solve eq (9) with I.C. eq (9a): F z [ ( )z] ()= R + R l + R ( )z [ ] () Differetiate eq () twice ad substitute together with eq () ad eq () o the right had side of eq (0). Solve eq (0) with I.C. eq (0a): 4
F ()= z R where u = + R ( - ) z 3 u + u ( ) l u l u + 7 5 (3) (4) Now we have foud F 0, F, F ad we could write our solutio as F 0 + ε F + ε F. However, we realize that such a solutio oly satisfies the B.C. eq () at the reactor ilet ad does ot satisfy coditio eq (3) at the reactor outlet. Ultimately we are iterested ot i the cocetratio profile alog the reactor but i its value (cocetratio value) at the reactor exit. Sice the B.C. at the exit is ot satisfied we have a lot of reasos to doubt the validity of our solutio at the exit ad must fid ways of improvig it. I perturbatio theory the solutio give by eq (4) is called a "outer" solutio B. Ier Solutio c 0 ()= z F 0 ()+ z ε F ( z) + ε F ( z) + ε 3 F 3 ( z)+...c O outer solutio (5) We have see before that the reaso why the "outer" solutio caot satisfy the B.C. at the exit (z= ) give by eq (3) is that all the terms F i s of the "outer" solutio were obtaied from first order D.E.s. Now we must expad ("stretch") our coordiate system i such a way that we ca take a "closer" look at what happes i the very viciity (i the "boudary layer") ext to z =. Let η = z ε α α > 0 (6) Sice ε << eq (6) gives reasoably large values of η for z very close to. Thus, we are stretchig the coordiate system ear z =. By chai rule we have: dc dz = dc dη dη dz = ε α dc dη (7) d c dz = d c ε α dη (8) Substitutig eq (7) ad eq (8) ito eq () we get: 5
ε α d c dc + ε α dη dη R c = 0 (9) We require ow that the derivatives should stad by equal powers of ε i.e α = α ad plus that we will get a d order D.E. This gives α =. Hece, ad becomes: d c ε dη d c dη η = z ε + dc ε dη R c = 0 (0) + dc dη ε R c = 0 () We assume ow that the solutio to eq () is the "ier" solutio c i ad that it ca be represeted by a power series i ε : c i = = 0 ε c i = c 0 i + ε c i + ε c i + ε 3 c 3 i +... () Remember agai that we are developig the ier solutio i the very viciity of z =, so the "outer" solutio i that regio is so close to that it ca be represeted by a Taylor series expasio aroud z = i.e η = 0 c o ()= z F 0 ()+ F 0 ()z + ε F + ε F ( )+ F 0 ( )+ F ()+ F ()z ()+ F ()z ( )+ F ( ) ( z ) +... ( z ) +... ( z ) +... + O ε3 ( ) (3) where F i = df i dz Now from eq (0) z = εη ; ( z ) = ε η (4) Substitute eq (4) ito eq (3) ad group together the terms with the same powers of ε : 6
c oi c oi c oi c o ()= z c oi η ()= () + ε F () F 0 ()η F 0 [ ] + ε F () F ()η + F 0 ()η (5) This is ofte called a "ier-outer expasio", c oi, i.e the expasio of the outer solutio i the ier (boudary) layer. Now the ier solutio give by the series represeted by eq () ca be represeted by the sum of two series: oe from the ier-outer expasio ad the other from the true ier solutio udetectable by the outer oe. c i η c i ()= ε =O = ε c oi η = O [ ()+ G () η ] = F 0 ()+ G 0 ()+ η ε F () F 0 + ε F [ ()η +G () η ] () F ()η + F 0 ()η +G () η +O( ε3) (6) Differetiate eq (6) oce ad twice, raise it to the -th power ad substitute everythig ito eq () ad require that the sum of the terms with equal powers of ε be zero: d G 0 dη + dg 0 dη = 0 (7) d G dη + dg dη = F 0 ()+ R F 0 [ ] (8) ()+G 0 d G dη + dg dη = F 0 ()+ F () F 0 ()η + R F 0 [ ] F ()+G 0 [ () F 0 ()η + G ] (9) This is a set of d order D.E.s for the G i s. Whe the G i s are determied successively the right had side of the above equatios is always kow. For d order equatios we eed B.C.s. Oe B.C. is obtaied from eq (3) i.e. which leads to dc i = 0 at η = 0 (30) dη 7
dg 0 dη = 0 at η = 0 (7a) dg dη = F 0 () at η = 0 (8a) dg dη = F () at η = 0 (9a) Remember that away from the boudary at z = we do ot sese or detect the G i s as F i s icely satisfy the B.C. at z = 0 ad are probably a good represetatio of the actual solutio for >z 0 but ot at z =. This implies that as we move away from z =, i.e as η icreases, all G i s go to zero. This is the d boudary coditio: η G i 0 for i = 0,,,3,... (7b,8b,9b) The solutio of eq (7) with eq (7a) is G 0 ( η)= 0 (3) The solutio of eq (8) with eq (8a) ad eq (8b) is: [ ] G ( η)= R + R ( ) e η (3) The solutio of eq (9) with eq (9a) ad eq (9b) is: [ ( )] G ()= R η + R 3+ η +l + R [ ( )] If we substitute eq (0) ito eqs (3-33) we will get G i s i terms of z. e η (33) The cocetratio profile close to the reactor exit is give by equatio (6) (where η ca be substituted i terms of z). We are especially iterested i the outflow cocetratio at the exit, i.e at z =, η= 0. Sice ε = /Pe we get: [ ( )] c exit = x A = F 0 ( z =)+ Pe F z = ( )+G η = 0 + [ F Pe ( z =)+G ( η = 0) ]+O Pe 3 (34) 8
Let ρ = [ + ( )R ] c exit = x A = ρ + Pe R ρ l ρ + (35) 3 Pe + ρ lρ + ρ + + O Pe 3 (36) After some additioal algebra: 9
c exit = ρ + Pe ρ R lρ ρ + R Pe lρ + ( ) ρ + R +O Pe 3 (37) where R = kt C 0 ad ρ = + ( )R Notice that the leadig term is a solutio for the plug flow reactor, thus as Pe, plug flow occurs. Straightforward algebra (although admittedly tedious), would produce higher order terms 3, etc. However, there is o eed for that. Experiece from fluid mechaics, heat Pe trasfer, solid mechaics ad reactio egieerig shows that perturbatio series are ofte ot uiformly coverget ad that d order perturbatio solutios, as the oe give by Eq (37), or eve first order perturbatio solutios (droppig the term with ) provide satisfactory Pe aswers which are ofte better at lower Pe tha higher order solutios. The perturbatio solutio for the dispersio model has bee worked out i detail by Burghardt ad Zasleski, Chem. Eg. Sci., 3, 575-59 (968), but their secod order term cotais a error due to improper matchig of perturbatio solutios. See their paper for compariso of approximate perturbatio solutios with umerical solutios. Let us ow examie the coditios uder which the deviatio of the tubular reactor legth from the legth required by plug flow is less tha 5%. I other words we are lookig for the coditios uder which the tubular reactor, the legth of which is withi 5% of the PFR, would give the same coversio as the PFR. L L p L = L p 0.05 (38) L For a fixed give flow rate ( ) p L p L = τ R p τ = = R ( ) p ρ ρ 0
L p L = ρ ρ p ρ p ρ = ρ 0.05 (39) ρ ρ p 0.05 (40) ρ ρ I the plug flow reactor we have x Ap = ρ p (4) For the axial dispersio model we will use the first order perturbatio solutio: x A = ρ + Pe R lρ ρ = x Ap (4) Therefore: ρ p ρ =+ Pe R lρ ρ ρ p = + R lρ ρ Pe ρ (43) The usig eq. 40) we get ρ p = + R lρ ρ Pe ρ 0.05 (44) ρ + R lρ Pe ρ 0.05 (45) ρ For > + R lρ Pe ρ 0.05 0.05 ρ ρ (46a)
For 0 < < + R lρ Pe ρ 0.05 0.05 ρ ρ (46b) For > (From eq. (46a)) R lρ 0.05 Pe ρ ρ (47a) For 0 < < (From eq. (46b)) R lρ 0.05 Pe ρ ρ (47b) Multiply (47a) ad (47b) o both sides by (-). I (47a) - > 0 ad the iequality is preserved, i (47b) - < 0 ad whe multiplyig with the egative umber the iequality sig is reversed. Thus, for all,. Pe R lρ ρ 0.05 (48) ρ which gives Pe 0 lρ (49) Thus, whe Pe umber satisfies iequality (49) the differece i the legth of a tubular reactor which ca be described by the axial dispersio model ad plug flow reactor, both givig the same coversio, will be withi 5%. Eq. (49) may be writte as: I the last approximatio (usig x A ρ ) we get: Pe 0 lρ (50) Pe 0 l (5) x A
Formula (5) clearly shows that the magitude of the Pe umber which will guaratee small discrepacies betwee tubular reactors ad plug flow depeds ot oly o the reactio order but also o coversio! For packed beds Bo = Ud p D (5) Equatios (49) ad (5) become: L d p 0 Bo( ) lρ (53) L 0 d p Bo l (54) x A How log the reactor should be with respect to the size of packig i order to elimiate dispersio effects is depedet o Bo (Bodestei) umber, reactio order,, ad dimesioless reactio umber R (Damkohler umber) or coversio. Similarly oe may develop a criterio that will show how large Pe should be i order to guaratee that the exit cocetratio from the tubular reactor will be withi 5% of the oe predicted by plug flow desig. c exit c p c p 0.05 (55) ρ + Pe R ρ lρ ρ ρ R lρ 0.05 Pe ρ 0.05 (56) Pe 0 R lρ ( )ρ (57) The reactor legth to particle diametger ratio i packed beds that will guaratee maximum 5% deviatio i exit cocetratio based o plug flow desig is: 3
L d p B o 0 R lρ ( (58) )ρ For referece ad idustrial usage of the above see:. D.E. Mears, Chem. Eg. Sci., 6, 36 (97). D.E. Mears, Id. Eg. Chem. Fudametals, 5, 0 (976) 4