7/28/2019 1-s2.0-0255270196800122-main

1/9

E L S E V I E R Chemical Engineering and Processing 35 (1996) 323-331

C h . m i ~Ingin..nlllandProcessing

An improved equation for the overall heat transfer coefficient inpacked bedsAnthony G. Dixon

Department of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USAReceived 8 December 1993: accepted 15 January 1996

AbstractA one-point collocation analysis is performed on the two-dimensional pseudohomogeneous model of a fixed bed heat

exchanger. The analysis is based on a single radial collocat ion point whose position depends on the wall Biot number. Theresulting improved equation for the overall heat transfer coefficient U is1 I R, Bi+ 3-=-+---U Itw 3kr Bi+ 4which gives an error less than 3.8% in the exact asymptotic value of U over the entire range of Bi. A formula is also given forU', the overall heat transfer coefficient based on the difference between tube wall and bed center temperatures. The synthesis ofvinyl acetate in a fixed bed reactor was simulated. Use of the conventional one-dimensional model with the improved equationfor U, combined with the equation for U' to obtain bed center temperatures, gave predictions of acetylene conversion and bedcenter temperature in good agreement with those of the two-dimensional model. A criterion for the neglect of radial temperaturegradients in a fixed bed reactor is derived which is more stringent than criteria currently available in the literature.Keywords: Packed beds; Heat transfer coefficient; Fixed bed reactors

1. IntroductionThere must surely by now be almost as many fixedbed reactor models in the vast l ite rature of this subjectas there ar e fixed bed reactors actually in operation

around the world. Nevertheless, the continuing use o fthis relatively simple, safe an d easy to operate catalyticreactory by industry ha s assured a parallel interest inthe development o f mathematical models to predict thereactor behavior. This interest has no t abated in the last40 years, an d is perhaps fueled by th e position of thefixed bed as the prototype tubular catalytic reactor, an dpoint of departure for more exotic relatives, such as thetrickle bed reactor, the moving bed reactor and morerecently the fixed bed catalytic membrane reactor , toname bu t a few.

Th e state of the ar t in fixed bed reactor modeling hasbeen periodically reviewed [I -4]. Recent trends, in thelight of higher computational capabilities, have beentowards more complex, two-dimensional , heterogeneous models with spatially-variable velocities and0255-2701/95/$15.00 1996- Elsevier Science SA All rights reservedPII S0255-2701(96)04146-3

transport properties [5,6]. Th e goals of such simulationshave usually been to improve understanding o f fixedbed reactor behavior by detailed modeling of the fundamental phenomena occurring within the reactor. Anequally valid goal, however, is to represent a fewselected features of reactor behavior using a simplemodel suitable for fast, repetitive calculations in studiesof design, control, parameter sensitivity, etc.

For these reasons, there is continued interest in theone-dimensional pseudohomogeneous model, using anoverall tube-side heat transfer coefficient U to accountfor radial heat t ransfer. Th e coefficient U depends onmany process variables, an d the difficulty o f findinggeneral empirical correlations for it has led to thegenerally accepted approach of relating U to theparameters of the two-dimensional model. Th e usualrelation ha s taken the form of a resistances-in-seriesequation

(I)

7/28/2019 1-s2.0-0255270196800122-main

2/9

324 A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331in which k, and h.; are independently correlated, or,indeed, may be further related to even more fundamental parameters [3,7,8].The factor p in Eq. (I ) has been determined invarious ways. Beek and Singer [9] did not state Eq. (I )explicitly, however a value of p= 4 is easily deducedfrom their results. A rigorous derivation leading top= 4 was given by Froment [10], while Beek's review[II] re-stated the earlier result more clearly, emphasizing its limitations. Crider and Foss [12] claimed thaterrors of up to 27.7% in I/V were possible with p= 4,in the limit of bed resistance control which occurs forhigh values of the Biot number Bi (=hwRtlkr ) . Theyrecommended P= 3.067,which gave a maximum errorin IIV of 5.7% over the whole range of conditions, thebest possible with a constant value of p. Westerink etal. [13] also preferred p = 3.06 for their predictions ofregions of parametric sensitivity of the homogeneousone-dimensional model. In a recent experimental study,however, Borkink and Westerterp [14] found that avalue of p= 3.7 gave the best empirical fit to their data,for the range of Bi covered in their work.Finlayson [15] applied one-point orthogonal collocation to the two-dimensional pseudohomogeneous modelat the root of p ~ O . O ) ( , 2 ) , i.e, at ,2 = 1/2, to obtain theresult p= 4, and at the root of p ~ 1 . 0 ) ( , 2 ) , i.e. at ,2 = 1/3,to obtain p = 3. Villadsen and Michelsen [16] pointedout that this latter choice of collocation point wasoptimal only in the limit Bi -400 , which reduces thewall boundary condition to a first kind, or Dirichlet,condition, T = Tw For a finite Bi, and a third kind, ormixed, boundary condition, the optimal choice of collocation point was the zero of p ~ O . O ) ( , 2 ) , which gave theresult p= 4. Villadsen and Michelsen [16] then developed a one-point collocation method where the position of the collocation point depended on Bi, similar tothe earlier work of Stewart and Sorenson [17]; this willbe the point of departure for the new formula for V tobe developed below.An alternative route to the overall heat transfercoefficient V lies through the heterogeneous one-dimensional model and its parameters Vr and U, [18]. Therelations between U, and Vs' and the more fundamentaland independently correlated parameters of the twodimensional heterogeneous model, have been providedby the same group '[19]. Although in theory this approach is as equally valid as that of Eq. (I), it has notfound wide acceptance, probably because of the shortage of correlations for U, and U, and the use of therelatively unfamiliar one-dimensional heterogeneousmodel with a continuous solid phase.A slightly different approach has been adopted byWellauer et al. [20] and Cresswell [3], who used anoverall heat transfer coefficient V', defined with respectto the bed center-line temperature. This coefficient wasalso introduced by Froment [10]. Although the one-

dimensional model is strictly applicable only if there arenegligible radial gradients, in which case T; = Tav andV = V', in reality the model will undoubtedly be usedas a simplification or approximation in the presence ofradial temperature gradients, when only limited information is of interest. Mears [21] has presented a criterion for replacing a two-dimensional model by aone-dimensional model based on making less than 5%error in the calculated linearized average reaction rateat the bed hot spot. It is thus anticipitated that in manycases there will be significant differences between T;and Tav ' and between V and V'. Wellauer et al.'smotivation to consider V' sprang from their experimental set-up, in which the center-line' temperature wasmeasured by a thermocouple in a thin centered thermowell, a fairly common situation in pilot plant studies[22]. It is also the center-line temperature that will givethe highest 'hot-spot' temperature in the bed, for reactor safety studies. It seems more reasonable for reactorsimulations, however, to evaluate the reaction rate atthe radial average temperature Tav , or at a suitableaverage temperature to give the same radially-integrated reaction rate.In all the studies cited above, the relations for V havebeen derived in the absence of chemical reaction(Pereira Duarte et al. [19] did include a thermal sourceterm in their analysis, but it was assumed constant).Several workers have argued that the one-dimensionalmodel cannot adequately reproduce the essential features of the two-dimensional model unless V varies withaxial position. McGreavy and Turner [23] proposed asemi-empirical relation for V which accounted for thedistortion in the temperature profile caused by thechemical reaction. Sundaram and Froment [24] analyzed the empty tubular reactor, which was complicatedby the presence of a non-uniform radial velocity profile(turbulent or laminar), and showed that the overall heattransfer coefficient significantly depended on the reaction parameters and the axial position. An even morerecent article by Westerink et al. [25] has examined thelength dependence of V. They concluded that the approach of V(z) to the asymptotic value of V dependson the Peclet number and the Biot number, and anempirical relation was presented for this dependence.The empirical relation predicted the constant value ofV to within 2% for all Bi. The effect of using V(z) tocalculate axial temperature profiles was studied andcompared to calculations using constant V. It wasconcluded that the axial variation of V could beneglected for low to moderate values of Pe and Bi,typical of either low dtldp cooled tubular reactors withhigh flow rates or short, thick laboratory tubes at lowflow rates. There was a significant effect at high Bi andPe values typical of adiabatic bed reactors with highReynolds number. A criterion was presented for theneglect of axial dependence.

7/28/2019 1-s2.0-0255270196800122-main

3/9

A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331 325

where

2. Classical development

Note that in general both overall coefficients must befunctions of x in order to match the radial energy fluxat every x. Numerical evaluation of the infinite seriesgives Vex) and V '(x) . At x = 0, it is easy to show that

(2)

(5)

(6)

(7)

(8)

(3)(4)

(10)

( I I)

(13)

(12)

The two-dimensional pseudohomogeneous model fora fixed bed heat exchanger with cold incoming fluid atx = 0, being heated from a constant temperature wall(x> 0), is1 020 1 (0 20 1 (0) 00PeAox 2+ PeR oy2+yoy = oxwitho-+ 0 as x -+ - 00e ~ 1 as x -+ 00

oe = 0 aty = 0oyeo B'(oy = 1 1- e) at y = 1where the axial conduction term and the axialboundary conditions have been included for completeness, and have no effect on the results developed here.The analytical solution to Eqs. (2)-(6) has been givenby Gunn and Khalid [33] and for x> 0 it isO(X'y) = 1 - f ( B B i . ~ l + ~ ; ) p J o ( ) . . ; Y ),= I 1 + A, ) tlo(A,)

[ _ P . . . : . . : ~ P . . . : . . .---...:1):..-xJx exp - 2

4 ' 2P, = 1+ A,PeAPeRand the eigenvalues A satisfy the characteristic equationBiJo(A;) = Atll(A,) (9)

V'(x)(O(x, 0) - I) = hw(O(x, 1) - I)

From this it is straight forward to evaluate eat y= 0 orat y = I, or to obtain the radial average U.Define overall heat transfer coefficients V (with respect to bed average temperature) and V' (with respectto bed center temperature) by

V(x)(U(x) -I) = hw(O(x, I)-I)and

V(O) = V'(O) = h;

V(x) -+ V and V'(x) -+ V'and as x -+ 00 ,

Ahmed and Fahien [26,27] demonstrated thedifficulty of accurately simulating the behavior of anS02 oxidation reactor using the conventional onedimensional model with a constant overall heat transfercoefficient, and evaluating the reaction rate at theradial-average temperature. They developed a newmodified one-dimensional model by using a problemspecific collocation method to write the radial profilesof conversion and temperature in terms of center-line,wall and bed average values. A cross-sectional averagereaction rate was used. This model required five ODEsto be integrated instead of the usual two, but gavemuch improved agreement with both experimental dataand the predictions of a two-dimensional model.Extending the earlier work of Pereira Duarte et al.

[18,19], Mascazzini and Barreto [28] provided formulasfor the fluid and solid phase overall heat transfercoefficients of the one-dimensional heterogeneousmodel, in which the thermal source term varied axiallyas a function of solid temperature. They also claimed'improved agreement with two-dimensional model calculations, away from conditions of parametric sensitivity.In a recent series of publications, Hagan and coworkers [22,29-32] have presented a new one-dimensional fixed bed reactor model, termed the a-model.The conventional radial average temperature is replaced by a 'reaction-averaged' temperature Tr , and theoverall heat transfer coefficient V is a decreasing function of the temperature rise The authors presentedimpressive agreement between their one-dimensionalmodel and the full two-dimensional model, while theconventional one-dimensional model did not do so well[31]. The conventional model, however, used V fromEq. (1) with P= 4, which may not have been the bestchoice for the cases studied since Bi was in the range7-10. Also, it may be noted that the new a-model wassomewhat more computationally expensive than theconventional one-dimensional model, since an algebraicequation had to be solved iteratively for a at each axialpoint z.In the present work, it will be shown that an im

proved formula for V in terms of k, and h.; is available,if the constant P in Eq. (1) is allowed to vary with Bi.This results in accurate prediction of V for the wholerange of Bi, in contrast to equations of the form of Eq.(1), which are accurate only for a limited range of Bi. Itis then interesting to compare the predictions of theconventional one-dimensional model with the improvedformula for V, to those of the more detailed twodimensional model. This comparison allows the reaction engineer to judge whether improved predictivecapabilities of more complicated models are worth theextra computational effort.

7/28/2019 1-s2.0-0255270196800122-main

4/9

326 A.G. Dixon I Chemical Engineering and Processing 35 (/996) 323-331The variation of V(x) and V'(x) and their approach toasymptotic values is shown for a typical case in Fig. I.Ulh.; is within 1% of its asymptotic value at x = 9, i.e.after 4.5 tube diameters; U'lh.; appears to take a littlelonger to reach its limiting value. The approach to theasymptotic values will depend on the other modelparameters, however the general trend is similar.The approach of V(x), V'(x) to constant asymptotescoincides with large enough values of x that only thefirst term of the infinite series is needed:

Note that with only one term the exponentials cancel,removing the dependence on x; also the terms in Picancel, removing the remaining dependence on PeA-The smallest eigenvalue Al depends only on Bi, nevertheless it is inconvenient to have to solve Eq. (9) foreach Bi. The evaluation of V' also requires the calculation of Jo(A 1) , a relation given previously [3,20]. Weseek simpler expressions that do not involve Besselfunctions.Following Crider and Foss [12] we note that for

small values of Bi, Al is also small and we may use thesmall argument approximations

, I ' i i i ' i I I I I8 12 16 20X = (zlR)

(17)i AtJo(A I ) ~ 1-"4+ 64and

AI AiJI(A\) ~ 2 - 1 6 (18)Substituting these into Eq. (9) givesA 2 _ ~ (19)1 - Bi +4and then using Eq. (15)V 4 1 1 s;hw = Bi+ 4 or V = hw + 4kr (20)as obtained earlier [9,10]. Along similar lines,V' ( 4)2 1 1 Bi+ 8 s,hw = Bi+4 or V,= hw + - 8 - 2 k r (21)for small Bi.

Note tha t if we use only two terms in Eq. (17) andone term in Eq. (18), then Ai = 4Bi/(2 + Bi) and weobtainI I 1 s;V = V' =h + 2k (22)w rwhich is inaccurate even for fairly small Bi, as has beenremarked by Froment [34]. Hence V = V' only in a(fairly poor) first approximation.As Bi increases, Al approaches 2.4048 which is notlarge enough to allow a large argument expansion ofJo(A\). We may proceed formally by allowing hw ~ 00 so

that 11hw ~ 0 and write1 1 s;---+ (23)V - h.; (Ai",,/2)krwhere Ai"" ~ 5.783186, to obtain a formula similarto Eq. (20), that can be used as an approximationfor large Bi-c co. For V', write V'=hwlo(A I ) = (krlRt)BiJo(A1) = (krIRt) x AIJI(A I) and evaluating AIJ\(A\)yields1 1 s;---+ a ~V' - h; 1.2483krand comparison of Eqs. (23) and (24) again emphasizesthe difference between V and V', especially as Biincreases.The classical development above can yield only

expressions in the form of Eq. (1), for the limits ofBi ~ O and Bi ~ 00. A more flexible approach is neededthat can give an equation for the entire range of Bi.

(15)

(14)

(16)

I4

1.0

0.8

.c 0.6 Ulhw--::>.J 0.4 U'/h- w::> - - - - - - - - - - -0.2

1-0(x,y)_ Bi (I + PI) JO().'lY) [- PeA(PI - I)X]- BF +Ai PI Jo(A I) exp 2and thenV 1 - O(x, 1) Ain; 1 - (J(x) = 2BiV' = 1 - O(x, 1) = Jo(A 1)s; 1 - O(x, 0)

Fig. 1. Comparison of the approach to the asymptotic values of theoverall heat transfer coefficients defined with respect to bed averagetemperature (V ) and bed center temperature (V'). Parameter valuesare PeR = 40 and Hi= 3.

3. One-point collocation developmentThe symmetry of the problem is reflected by the

change of variable u = y 2 giving

7/28/2019 1-s2.0-0255270196800122-main

5/9

A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331 327

thus

( (I - U)(Bi/2) + 1) (H ' ) (34)O(x, u) = 1 - (I _ u.)(Bi/2) + 1 exp xFrom this, the terms required in the definitions of Vand V' follow easily. For V' we can immediately show

o() = Bi (I _ () at u= 1 (29)au 2Applying one-point collocation at u = U. ' we set

1- u U - UIO(U)= - - ( ) I+--02 (30)1 - U. 1 - UJand substitute into Eq. (29) and solve for ()2 in terms of0.We then collocate the differential Eq. (25) to obtaind2() . d()1 () 1) 0 (31)dx2 - PeA dx - H( ) - ='whereH = 4PeA Bi/2 (32)PeR(I - u,) Bi/2+ 1/(1 - U.)Without loss of generality, we neglect axial conduction(PeA- . 00) and solve the resulting first order ODE toget, setting H' = H /peA

which is expected to be accurate in the limit of small Bi.(39)

(38)

Exacto BBiI(B+4)tI 6BiI(Bi+3) Thiswork

2 3 5 10 2 3 5 100Bi

105

2A12

532

00 2 3 5

which has the limit Ai - .6 as Bi - . 00 . An approximateformula for V' may also be found by returning to Eqs.(16) and (17) and substituting Eq. (38), givingV' Bi4 + 24BP + 240Bi2+ 1152Bi+ 2304hw = 16(BF + 6Bi+ 12)2

Fig. 2. Comparison of approximations for the leading eigenvalue .n1 1 R t Bi+ 3V = hw + 3kr Bi+ 4 (37)This formula reduces to Eq. (I ) with P= 4 in the limitBi -.0, and to Eq. (1) with P= 3 in the limit Bi - . 00 .As a corollary to this result, a new approximation toAi is obtained2 Bi +4Al = 6BI BF + 6Bi + 12

(33)

(25)

(26)(27)(28)

O.= 1 - exp(H'x)

0-.0 asx- . - 00o-. 1 as x -. 00

1 02() 4 (02() O()) eoPeAox 2 + PeR u au2 + au = oxwith() finite at u = 0

(35)

(36)

V' O(x, I) - 1 2hw = O(x, 0) - 1= Bi+ 2independent of choice of collocation point, so that thecollocation method allows us to recover only the poorfirst-order approximation of Eq. (22) for V'.To obtain lJ(x), we choose Gauss-Jacobi quadrature

rather than Radau quadrature, as the boundary valueO2 was no t given directly but was approximated bydiscretizing the boundary derivative; thus it is no tknown to any special degree of accuracy. We obtain thesimple result lJ=, and henceV O(x, I) - 1 Ihw = lJ(x) - 1 = 1+ (I - u .)(Bi/2)Choosing U. to be 1/2 or 1/3 gives Eq. (I ) with p= 4and P= 3, respectively [15J. We prefer to follow Stewart and Serensen [17] and Villadsen and Michelsen [16]and choose U. = (Bi+ 6)/(3Bi + 12); substitution intoEq. (36) gives our main result

4. Comparison of prediction formulasThe first test of the new formulas presented here willbe the calculat ion of Ai as a function of Bi. Fig. 2shows the results from the corol lary to Eq. (I ) and Eq.

(15), Ai = 2P Bi/(Bi +P), for the two limiting collocation cases p= 4 and p= 3, and the result of the presentwork, Eq. (38). The comparison in Fig. 2 indicates thatall methods are quite accura te for low Bi, in the regionBi < 1. For Bi > 2 the choice of the p= 4 approximation becomes less satisfactory, as noted before [12,16],and the formula is highly inaccurate for Bi > 4-6. Boththe formula p= 3 and the present formula give goodapproximations over the enti re Bi range. The p= 3formula makes er rors of up to 6.3% for low andintermediate Hi, and is very accurate for high Bi. Asexpected, these results are comparable to those ofCrider and Foss [12], who obtained a maximum errorof 5.7% with P= 3.067 rather than p= 3. Our new

7/28/2019 1-s2.0-0255270196800122-main

6/9

328 A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331Table I given by Eq. (39) indicates that it is caused somewhatComparison of prediction formulas for U/hw more by the low-argument expansion of JO(A), Eq. (17),Bi Exact This work Low Bi limit High Bi limit than by the approximation of AI' Eq. (38). Attempts to

O?!2Bi) [Eq. (37)] 4/(Bi+4) 3/(Bi+3) include more terms in the expansion of JO(A), however,gave results that became limited by the AI approxima-0.2 0.9516 0.9517 0.9524 0.9375 tion, and only a marginal improvement was obtained.1.0 0.7885 0.7895 0.8000 0.7500 The bed-center temperature may be obtained' from3.0 0.5332 0.5385 0.5714 0.50006.0 0.3499 0.3571 0.4000 0.3333 the bed-average temperature by the relation10.0 0.2375 0.2442 0.2857 0.2307 V20.0 0.1309 0.1353 0.1667 0.1304 (Tc - Tw) = V' (Tav - Tw ) (40)

formula, Eq. (38), has an overall error less than 3.8%,and is especially good for low and intermediate Bi < 10,where the maximum error is less than 2.8%. The newformula makes its worst errors at high Bi, where theone-dimensional model is a poorer representation of thetemperatures in the bed. Eq. (38) is to be preferred forthe prediction of Af since it is very accurate in the lowto intermediate Bi region which is where most fixed bedprocesses with strong heat effects will operate, i.e. atrelatively high Re.Table I presents a comparison of the three formulasfor V/hw , corresponding to the three choices of collocation point in the previous section. Results are presentedfor a selected set of Bi values, and the conclusions fromthese results echo those from Fig. 2. Eq. (1) with P= 4predicts Vjhw accurately for low Bi but is poor forBi > 2. Eq. (1) with P= 3 predicts Ulk; accurately forBi > 10 but is less good below that value. Our newresult, Eq. (37), provides a simple approximation that isaccurate to within 3.8% over the entire range of Bi.The collocation method did not yield a good approximation of U'lh.; and therefore Eq. (39) was based ona return to the small argument expansion of Eq. (17). Itis no surprise, therefore, that the prediction results forU'lb; that appear in Table 2 show accurate values forlow Bi and poorer performance as Bi increases. Eq. (22)is included to confirm that it gives inaccurate valuesover the entire range of BLEq. (39) of the present workgives the best agreement with the exact values, whichbecomes progressively worse as Bi increases. The intermediate Eq. (21) performs reasonably, but is still not asgood as Eq. (39). An analysis of the error in the valuesTable 2Comparison of prediction formulas for U'/hwBi Exact This work Eq. (21) Eq. (22)[loO,)] [Eq. (39)] [4/(Bi +4)]2 2/(Bi+2)0.2 0.9071 0.9071 0.9071 0.90911.0 0.6429 0.6442 0.6217 0.66673.0 0.3466 0.3554 0.3265 0.40006.0 0.1957 0.2156 0.1600 0.2500

10.0 0.1218 0.1517 0.0816 0.166720.0 0.0620 0.1045 0.0278 0.0909

with V and V' given preferably by Eqs. (37) and (39).The one-point collocation method does give a parabolicradial profile, however direct use of this is equivalent tousing the inaccurate Eq. (35) for V' and is not recommended. It may be noted from Tables I and 2 thatthere is a definite difference in the values of Ulb; andU'lii; at the same Bi, thus the distinction between thesetwo coefficients is one worth making, even for fairlylow values of Bi.A considerable amount of earlier work has beenbased on the use of u. = 1/2 as the collocation point,including model matching studies [7,8,35]. The successof this work, despite the inadequacies of Eq. (20) andthe use of the root of P\o.O)(u) as the collocation point,lies in the fact that the model matching formulas for k,and hw were derived for the limiting case of high Re [7]and the predictions of k, and hw have generally beencombined to obtain V for processes with reasonablyhigh Re. Since this corresponds to low values of Bi, theearlier work gave quite acceptable results and the use ofUI = (Bi+ 6)J(3Bi + 12) as the collocation point gaveonly marginal improvement, as noted by Dixon [36].

5. Reactor simulation using improved formulasA rigorous analysis of the one- and two-dimensionalpseudohomogeneous models with a reaction termincluded will show that the lumped parameter V shoulddepend on the kinetics and heat of reaction. To investigate this, the collocation analysis was repeated with areaction term present in the energy balance equation.

As is frequently done (see Mears' article [21] for example) the reaction rate expression was linearized, whichled to the result that the reaction kinetics constantscancelled out in the final expression for Ulh.; So to afirst approximation the same formula holds for V inboth the reacting and non-reacting cases. The approximation involved in linearizing the reaction rate expression may be questioned, but it is likely to beappropriate given that the radial profile is representedby a one-point collocation approximation (i.e. aquadratic in the radial coordinate).Herskowitz and Hagan [31] presented the results ofsimulations of vinyl acetate synthesis in a tubular fixed

7/28/2019 1-s2.0-0255270196800122-main

7/9

A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331 329

(41)

(42)

1.0.8

Tw=To=455.1K

0.4 0.6X (=zIL)0.2

... ------ 2-D- - - 1-0 (OLDU)1-0 (NEWU)

,,,, - - - -"., ",, vr,,0-f-r-T'""T""T-r-r-T'"T"T-r-r-rT"T-r-T""'T'T"T-+0.0

40

20

30

10

6. Criteria for the neglect of radial temperaturegradientsAs mentioned above in the Introduction. Mears [21]developed a criterion for the neglect of the radialtemperature gradients. A one-dimensional energy balance was written at the hot-spot, which allowed an

analytical solution for the temperature profile when thereaction rate was linearized around the wall temperature. Under some reasonable assumptions. the temperature profile became parabolic. and expressions weredeveloped for the ratio of the cross-sectional bed-average reaction rate to the reaction rate at the wall temperature. By requiring these two to be within 5% of oneanother, Mears obtained criteria for the neglect of theradial gradients, for the two cases of negligible andsignificant heat transfer resistance at the wall.The one-point collocation method was applied to thesimplified model considered by Mears and a similardevelopment to that explained above was carried out toderive a general criterion in terms of the location of thecollocation point. In terms of the notation already usedin this article, the criterion is

E 1- L\HI R ~ R ; [ 4 ]R T ~ 8kr 1+Bi < 0.05

For the collocation point at u. = 1/2 which correspondsto the parabolic radial profile found by Mears' approximate method, Mears' result is recovered exactly .Again, in the present notation the result is

with R; being the reaction rate at the collocation point.which is identified with the radial-average reaction rateR' used in Ref. [21].

Fig. 4. Model predictions of axial profiles of temperature rise at bedcenter, for the vinyl acetate synthesis example, for two differentinlet/wall temperatures.

1.0.8.4 0.6X (=zIL)

2-D1-0 (OLDU)1-0 (NEWU)

0.2

0.20Z 0.180 0.16Ciic:: 0.14~ 0.120U 0.10WZ 0.08W-I 0.06~ 0.04~ 0.02

0.000.0

Fig. 3. Model predictions of axial acetylene conversion profiles forthe vinyl acetate synthesis example, for two different inlet/wall temperatures.

bed reactor. They used the conditions reported byValstar et al. [37], and compared a two-dimensionalmodel, the conventional one-dimensional model andtheir new a-model. The new one-dimensional a-modelwas found to agree very well with the acetylene conversion and temperature rise given by the twodimensional model, but the conventional one-dimensional model under-predicted both. The overall heattransfer coefficient used in the conventional modelwas given by Eq. (20), although the Biot number wasquite high, 7.22. It is interesting to investigate howthe conventional model would perform with a different estimate of U.

The calculations of Herskowitz and Hagan [31]have been repeated using the conventional model, forthe kinetics of Valstar et al. [37]. The kinetic expressions and values of the other parameters are given inTable I of Herskowitz and Hagan [31]. The results ofthe present calculations are compared to the two-dimensional model in Fig. 3 for acetylene conversion.and in Fig. 4 for the predicted centerline temperaturerise. There is good improvement in the predictedacetylene conversion using the U value from Eq. (37),so that the conventional one-dimensional model isclose to the two-dimensional results. The predictedcenter-line temperature rise using Eqs. (39) and (40)also improves significantly. underpredicting the hotspot temperature by approximately four degrees. Thea-model overpredicted the temperature rise by abouttwo degrees [31], so that for this case in terms ofaccuracy there is not a lot to choose between themodels. The overprediction of the a-model makes itthe more conservative for design purposes, and thedetermination of runaway conditions. at the cost of alittle extra computation.

7/28/2019 1-s2.0-0255270196800122-main

8/9

330 A.G. Dixon / Chemical Engineering and Processing 35 (1996) 323-331

Greek lettersP parameter, Eq. (1)Pi parameter, Eq. (8)AH heat of reaction, kJ mol- 1A, ith eigenvalue, defined by Eq. (9)A,oo value of first eigenvalue, as Bi -+ 00o dimensionless temperature,

(T - To)/(Tw - To)01 value of 0 at collocation point UIO2 value of 0 at U= I{f radially-averaged dimensionlesstemperature

For the general Bi-dependent collocation pointlocation used in this paper, UI = (Bi+ 6)/(3Bi+ 12) andthe result isE 1- AHI R ~ R ; [4Bi + 12 4 ]R T ~ 8kr 3Bi+ 12+Bi < 0.05 (43)

As Bi becomes small, Eq. (43) reduces to Eq. (42) since(4Bi+ 12)/(3Bi+ 12) -+ I. As Bi becomes larger, thisfraction tends to 4/3. The criterion of Eq. (43) is morestringent than Mears' criterion for large values of Bi, i.e,when the resistance to heat transfer is within the bedrather than at the wall, and a large temperature gradientthrough the catalyst bed is to be expected. I t is of interestthat the vinyl acetate synthesis example of the previoussection fails by both the old and new criteria, confirmingthat the discrepancies seen in Figs. 3 and 4 are reasonable.7. ConclusionsThe new formula

I I n; Bi+ 3-=-+--V hw 3krB i +4should be used to combine estimates of k, and b; intoan overall heat transfer coefficient V, in preference to theprevious lumped form given by Eq. (1).The new formulais accurate for all values of Bi, and involves little or noextra computation.Accurate formulas were harder to obtain forthe overall heat transfer coefficient defined withrespect to bed center temperature, V'. It is recommendedthat V should be used in the 10 model, calculated fromthe formula directly above, and that Eqs. (39) and (40)should be used to obtain T; for comparison with experimental thermowell temperature data, from the values ofTav given by the one-dimensional model.A conventional one-dimensional model with anaccurate prediction of V was used to simulate a fixed bedreactor with moderate radial temperature gradients.With V well-predicted, the model was nearly as accuratecompared with a two-dimensional model as the ~ - m o d e lof Herskowitz and Hagan [31], and was a little easier touse.An extended version of Mears' criterion for theneglect of radial temperature gradients in a fixed bedreactor was derived. The extended criterion reduces tothe earlier result for small values of Bi, and is morestringent for the large values of Bi that usually give riseto stronger radial temperature gradients.

8. NotationBi tube Biot number, hwRt/krcp fluid specific heat, kJ kg K- I

P\O,O)P\I,O)rRRtR'

U

V'o,o.xyz

activation energy, kJ mol- 1superficial mass flow rate, kg m - 2 S - Iwall heat transfer coefficient, kJ m - 2 sK- 'dimensionless parameter, Eq. (32)effective axial thermal conductivity, kJm " ' S - I K- 'effective radial thermal conductivity, kJm " ' S - I K- 'tube length, maxial Peelet number for heat transfer,GCpRtka-Iradial Peelet number for heat transfer,GCpRtkr - 1Legendre polynomial of first orderJacobi polynomial of first orderradial coordinate, mgas constant, kJ mol - I K - Itube radius, mreaction rate per unit bed volume, molm -3 S -Iradial-average temperature, Kinlet temperature, Kcenter-line temperature, Kwall temperature, Kreaction-averaged temperature of Hagan etal. [29], Ktransformed dimensionless radialcoordinate (y2)first radial collocation pointoverall heat transfer coefficient [Eq. (10)],kJ m- 2 S - I K- 'overall heat transfer coefficient [Eq. (11)],kJ m- 2 S - I K- 'fluid-phase overall heat transfer coefficient,kJ m- 2 S - I K - 1solid-phase overall. heat transfer coefficient,kJ m- 2 S - I K- 1dimensionless axial coordinate, zlLdimensionless axial coordinate, r/Raxial coordinate, m

7/28/2019 1-s2.0-0255270196800122-main

9/9

A.G. Dixon / Chemical Engineering and Processing 35 (/996) 323-331 331References[I) W.R. Paterson and J.J. Carberry , Fixed-bed cata lytic reactormodeling. The heat transfer problem, Chern. Eng. Sci., 38 (1983)

175.[2) O.M. Mart inez, S.1. Pereira Duarte and N.O. Lemcoff, Modelling of fixed bed catalytic reactors, Comput. Chern. Eng., 9(1985) 535.

[3J D.L. Cresswell, Heat transfer in packed bed reactors, in H.1. deLasa (ed.) Proc. NATO ASI Chern. React. Design Technol.,Ontario, Canada, 1986, Vol. 110, p. 687.[4) G.F. Froment and H.P.K. Hofmann, in J.J. Carbe rry and A.Varma (eds.) Design of Fixed-Bed Gas-Solid Catalytic Reactors,

Chemical Reaction and Reactor Engineering, Marcel Dekker,New York , 1986, p. 373.

[5) D. Vortmeyer and E. Haidegger, Discrimination of threeapproaches to evaluate heat fluxes for wall-cooled fixed bedchemical reactors, Chern. Eng. Sci., 46 (1991) 2651.

[6) T. Daszkowski and G. Eigenberger, A reevaluation of fluid flow,heat transfer and chemical reaction in catalyst filled tubes, Chern.Eng. Sci., 47 (1992) 2245.[7) A.G. Dixon and D.L. Cresswell, Theoretical prediction of effective heat transfe r parameters in packed beds, AIChE J., 25(1979) 663.[8) A.G. Dixon, Wall and particle-shape effects on heat t r a n s ~ e r inpacked beds, Chern. Eng. Commun., 71 (1988) 217; A.G. Dixon,Chem. Eng. Commun., 87 (1990) 241.[9J J. Beek and E. Singer, A procedure for scaling-up a cata lyticreactor, Chern. Eng. Prog., 47 (1951) 543.

[10) G.F. Froment, Design of fixed-bed catalytic reactors based oneffective transport models, Chern. Eng. Sci., 17 (1962) 849.

[II) J. Beek, Design of packed catalytic reactors, in T.B. Drew, J.W.Hoopes, Jr. and T. Vermeulen (eds.) Advances in ChemicalEngineering, Academic Press, New York, 1962, Vol. 3, p. 223.

[12) J.E. Crider and A.S. Foss, Effective wall heat t ransfer coefficients and thermal resistances in mathematical models of packedbeds, AIChE J., 11 (1965) 1012.[13) E.J. Westerink, N. Koster and K.R. Weste:terp, The choicebetween cooled tubular reactor models: analysis of the hot spot,Chem. Eng. Sci., 45 (1990) 3443.[14J J.G.H. Borkink and K.R. Westerterp, Influence of tube andparticle diameter on heat transport in packed beds, AIChE J., 38(1992) 703.[15J B.A. Finlayson, Packed bed reactor analysis by orthogonalcollocation, Chem. Eng. Sci., 26 (1971) 1081.

[16J J. Villadsen and M.L. Michelsen, in S o l u t i ~ n of Dif!erentialEquation Models by Polynomial Approximation, Prentice-Hail,Englewood Cliffs, 1978, p. 248.[17) W.E. Stewart and J.P. Serensen, Transient reactor analysis byorthogonal collocation, Fifth European Symposium on ChemicalReaction Engineering,Elsevier, Amsterdam, 1972, pp. B8-75-B888, C2-8-C2-9.

(18) S.l. Pereira Duarte, O.A. Ferret ti and N.O. Lemcoff, A heterogeneous one-dimensional model for non-adiabatic fixed-bed catalytic reactors, Chern. Eng. Sci., 39 (1984) 1025.[19) S.1. Pereira Duarte, O.M. Martinez, O.A. Ferretti and N.O.Lemcoff, Theoretical prediction of heterogeneous one-dimensional heat transfer coefficients for fixed-bed reactors, AIChE J.,31 (1985) 868.

[20) T. Wellauer, D.L. Cresswell and E.J. Newson, Heat transfer inpacked reactor tubes suitable for selective oxidation, ACS Symp.Ser. No. 196, Am. Chern. Soc., Washington, DC, 1982, p. 527.[21) D.E. Mears, Diagnostic criteria for heat transport limitations infixed bed reactors, J. Catal., 20 (1971) 127.

[22) J.C. Pirkle, Jr., H.S. Kheshgi and P.S. Hagan, An accurateone-dimensional model for nonadiabatic annular reactors,AIChE J., 37 (1991) 1265.

[23) e. McGreavy and K. Turner , Model reduction of fixed bedcatalytic reactors, Can. J. Chern. Eng., 48 (1970) 200.

[24) K.M. Sundaram and G.F. Froment, A comparison of simulationmodels for empty tubular reactors, Chem. Eng. Sci., 34 (1979)117.

[25) EJ . Westerink, J.W. Gerner, S. van der Wal and K.R. Westerterp, The lumping of heat transfer parameters in cooled packedbeds: effect of the bed entry, Chern. Eng. Process., 32 (1993) 83.

(26) M. Ahmed and R.W. Fahien, Tubular react ion design-I. Twodimensional model, Chern. Eng. Sci., 35 (1980) 889.[27] M. Ahmed and R.W. Fahien, Tubular reaction design-II. A

modified one dimensional model, Chern. Eng. Sci., 35 (1980) 897.[28) N.O. Mascazzini and G.F. Barreto, Development of overall heat

transfer coefficients from the heterogeneous two-dimensional(2D) model for catalyt ic fixed bed reactors . A comparison between ID and 2D models, Chern. Eng. Commun., 80 (1989) II .[29) P.S. Hagan, M. Herskowitz and e . Pirkle, A simple approach tohighly sensitive tubular reactors, SIAM J. App. Math., 48 (1988)1083.[30) P.S. Hagan, M. Herskowitz and e. Pirkle, Runaway in highlysensitive tubular reactors, SIAM J. Appl. Math., 48 (1988) 1437.

(31) M. Herskowitz and P.S. Hagan, Accurate one-dimensional fixedbed reactor model based on asymptotic analysis, AIChE J., 34(1988) 1367.[32) H.S. Kheshgi, P.S. Hagan, S.e. Reyes and J.e. Pirkle, Jr.,Transients in tubular reactors: comparison of one- and twodimensional models, AIChE J., 34 (1988) 1373.

[33) D.J. Gunn and M. Khalid, Thermal dispersion and wall heattransfer in packed beds, Chern. Eng. Sci., 30 (1975) 261.

[34) G.F. Froment, Fixed-bed catalytic reactors-current designstatus, Ind. Eng. Chem., 59 (1967) 18.[35) A.G. Dixon, Thermal resistance models of packed-bed effectiveheat transfer parameters, AIChE J., 31 (1985) 826.

[36) A.G. Dixon, Heat t ransfer in packed beds of low tube/parti clediameter ratio, Ph.D. Thesis, University of Edinburgh, 1978.

[37) J.M. Valstar, P.J. Van Den Berg and J. Oyserman, Comparisonbetween two-dimensional fixed bed reactor calculations and measurements, Chern. Eng. Sci., 30 (1975) 723.