--
I
- 3 ----
activity of component gas in main fluid phssc bcd cmtrntainEq~tion3 dimensionleas nut for mha8 traoafer gmsa outaide mea of catalyst pellets, q.ft./cu. fb. = compentA gmsa outside mea of catalyst pellets, aq. ft./lb. -com entB D, av. %et, of catalyst particle D. -diEwivity msss velocity of fluid flowing, Ib./(sq. ft.)(hr.) G forward reaction velocity constant k forward reaction velocity constant k’ K adsorption uilibriumconatant L tow molal%*on sitea per unit mam of catslyst M- = mesn molecular waght of in muin fluid nheam PA. p ~etc. , partiel prmnwe o r m p o n e n t gsa in main Enid
a4 old, etc.
2m%
R R B I
---
a
6 D
h p
---
number of esuidiaant sitea adjacent to each active center consrant in Equation 4 pariable in Equation4 densitv of flud hulk d d t y of cablyst, lb./cu. ft.
--
-viwGdty
subamipta A componentA y d r o p B component B $ooctene/ B component R (Isooctane i -interiace 0 -gsa
Sheam
gaelawcon~tant = parameter re d&ed by Equation 3 componentB reaction rate, lb. moles isooctene convertdflb. catalystfir.
Conduction, Convection and Heat Release in
CATALYTIC CONVERTERS R. H. Wilhelm, W.C. Johnson, F. S. Acton PIIUiCEIUN U N N R B m , PRINCEIDN. N. 1.
FAT transfer considerations form an important part of a catalytic converter design. The productivity of a continuous converter greatiy upon proviaions for m v h g the exothermic heat of reaction or, in the CBBB of an endothmnic reaction, for supplying it. An increased e5ciency in heat transfer generslly permits a larger output of product per unit volume of catalyst. Heat transfer also is essential in maintaining a temperatwe uniformity within the oatalyst bed. The rates of heat generation in the catalyst granules must be balanced by corresponding rates of heat removal such that the point-bpoint temperatum in the bed 88 a whole will lie within speci6ed limit& In extreme cases these limits may be 88 narrow 88 10-20° C. When the thermal balance is not properly arranged by de&@ or is diaturbed by faulty operation, an excessive and local rise or fall in temperature will result. In the former cam an excessive “hot spot” m y lead to the destruction of the catalyst or to by-product formstion; in the latter, cooling the bed may lead to a stoppage of the reaction. Thus, from the viewpoint of e5ciency and control, some measure of heat exchange is e8-
H
sential in a catalytic reactor. This paper deals with the kdutions of differenti thermsl equations for cylindrical and elsb-shaped catalyst beds. The specific solutions d d with conditions in the c a w m e a section, and relate point and average catslyst temperature with (a) the temperature of the gea, (a) the average temperature of the catalyst at the periphery of the bed, (c) the rata of heat release by the chemical reaction, and the rate of heat transfer hoth (4by transvem conduction tbmugh the d i d 562
c a t & t and (e) by convection through the gea 6lm between the cakdyst and the Eowing gas stresm. Fice the rata of reaction (and of heat release) is a function not only of temperature, but SLSO of concentration of reactants and time of contact, this rate will vsry down the length of the catalyst bed. The temperature of the gse stresm, which is mumed uniform a t any craw section, will also vary along the catalyst bed. Thus, a complete reador design involves a stepbystep analysis in the direction of gse Eow from one croea section to the next with the aid of heat and material bslances. By this means the following itemu m y be computed for beds af dSerent dimensions: 1. Catalyst and gsa tamperaturea at b y point in the bed. 2. ”numerue heat Eow raka at an c m d~o n . 3. Pmduct formation in a volnmc of catalyst per unit of time.
The following fundamental information is required for the computations: 1. Variation of rata of reaction with tam atura rsactsnt ooncentretions, and time of contact, u s u p ~ yoGLned pilot plant data. 2. Molar h a t of reaction. 3. Thermsl conductidy of the cstslystgss Byatem. 4. Gaa-iihn heat tnuurfer ooe5cient m broken solids of c a b lyst siee 88 a fnuction of Eow conditions and gas propextien. 5. Thermal pmpertiea of the gae stream.
tram
The following design variables must be set: 1. Cat$ walltemperatwefrom involve tnd%d-enor methods such
INDUSTRIAL AND ENGINEERING CHEMISTRY
(Thiswill Etinttopoint. the rata of tnrnsverae
Vd. 35, No. 5
t bed at a dven ma8 d i o n equds heat trannferfromthe ca the rate of its m d by% k tM e r d u m . ) bed dismetcoor thckness. of cak3lpt.bed. 4. Inlet pwmpoetron, 00w rate, and tempersture.
:E%
It is also suggested that the thermal equations presented afTod opportunitiea for obtaining betta insight to the complexity of converter vsriables through enslysie of the dect of vsriations of the major vsriablea, taken singly. Moreover, the equations are not limited to heat release by ohemid re. action, but may he applied to electrid heating problem or any other ween for which the aemnnptions listad below am valid. Furthermore, they may be applied to a combine.tion involving any two of the three basic rak-conduction, convection, and heat release. Thus, conduction and conveation alone, for example, would apply to c r u w w t'~o d conditions
in a poked heat exchanger, and also to certain disk and strip fin heat exchanger problems (4). The present d y s i q and computations based upon it are on an a priori basii and are subject to experhental check. The problem of heat tranmiasionin a catalyst bed ha been extensively diaaussed with qualitative concluaiona by Damkoahler (1). Patason (6)presented a mathematical treab ment for the conduction of heat in an in6nite or mni-idnite medium generating heat. F'mblema relating to the measure. ment of gan and solid in a catalyst bed were pointed out by Jdcob (8,and temperature ditrerenoes hetwesn catalyst and gas were measured under various conditions for the hydrcgenation of ethylene. Conveation heat transfer caetkients in beda of subdivided solids in the catalyst rsnge have reoently been measured over a wide range of conditiqns and corm lated by Hougen, Gamson, an&Thodos (8). ASSUMPTlONS
1. The rate of beat releaze at any point in a catalyst bed is a function of catalyst activity ko, catalyst tempraturee-E/"T,
heat of d o n Q,time of contsct 8,and reactant concentrstion C. Thus the rate of heat release per unit volume of catalyst is:
Ruation i m a y therefore be rewrittan:
whm c.
-
-
- c*+
CI1
(a)
dn (Cn i CaOdVQ,e, CI
(4)
-
VC&
8,e, C1 Co and CIvary with longitudinal pasition in a c a t d p t bed, but within a thin mwa-emh'd slice they may be amlmed CI
pb,
Equation 8 thmefore 7 the linear variation of beat release with temperature m a catalyst cmss mation.
cor&.&.
2. Temperatures at all pinta in the catalyst are constant with respect to time. Heat released or absorbed by the re. action at any point is removed ot introduced by heat conduction and convection. Radiation is neglected. 3. Heat trsnsfer by conduction through solid catalyst and gan is entirely in a direction prpendicnlar to gan flow. (This transfer is easentialy independent of longitudinal gan d o c ity.) Temperature gradienta and heat flow rbts in a longitudinal direction are considered negligibly amall and are not included. 4. Because of mixing, the temperature of the gas within any thin croaksectional slice of catalyst bed is uniform and constant. CYLINDRICAL CATALYST BED SECTION
Tmpemturc Distrihth~~.Coneider an elementary ring in a catalyst section having an outer rad+ R and th~cknem AI 1). The ring with radius r, Wrdth dr, and thicknem Ai have a volume:
F
dVd = ZrAlrdr
(5)
Combiie with Equation 4 to obtain rate of heat release: dg = (Ce
+ Ctt)(2wAlr)dr
(6)
rate pf heat ralease by an exothermic reaction in a disk hasingadusrandthickness Alia: Over the limited rsnge of temperaturea of intereat in a catalyst bed the specifio reaction rate may be expressed as a
linear function of temperature as in Equation 2. 1943
h
-
~
(
C
+OCd(2wAl)dr
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
(7) 563
ta; at r tm r: qa*-%Alk-
dt
-
0, dt/dr
-
0. Ditlersntists Equation 14 with rsapact
(8)
dr
In the above. d t i v e heat is e x ~ eadBow from h k h to
disk volume, th&
fM8titutii N in Equation 14 yields a particular mlutian:
’
Substitute in Equation 11:
Equation l.!2 ia a st.ndard form of Basael’e equation: ita is positive, is:
general mlutmn. ahen p/k
N and M are condents of integration and J I and Yo W Bsaasl funations of tbe aexo order. Exp- &ustion 13 in tmmn of originalvariabh, 1:
- -
BOUNDARY CONDITION& Let radius to Outer limit Of aR,t &ion be R. The temperature at R is la. Then at r
564
Thim funotion ww evaluated p d is plotted w a Series of solid lima in Figure 2. The dimen~~ordess variablp. are called z, u,
and a, and 81‘8 defined ea follows:
Y = r/R
I N D U S T R I A L A N D B N G l N B B R l N G CHEMISTRY
Vol. 3s. No. 5
The Bolution of E d o n 12 when p / k is negative follown hmediately from the &ddation: J.
(id
-
Io
(18)
(2)
Substit.u+g Equatiqn 16 in 16,when p f k is negative, (he lollowing part~cularsolutron IE ohtsmed:
In dimensionlens form, Equation I9 b m e s :
Fi re 3 is a plot of function 20 in terms of the dimensionlean VariXIFa 2,y, z. As the values of functiona 16 and 19 ap msoh sem the evalus-
-
tion of ternperatwea hy the function &mea inheterminate. Therefore an alternate solution.as $ @ / k ) 0 is mquired. Start sdth Equation 10 and mmphfy mta the followug:
The solution of Equation 21 yields: 1 = tp
n +(R' - r') 4k
which is a parabolic Bolution. Integral Mean Temperature. A true average cmsbaection temperature is of value for catslpt design computations. The following is a derivation of an mte& mean temperatum obtained by dividing the volume of the temperature dome (formed by plottmg catalyst temperature at any m e a d o n ) by the area of the seation:
L=
*R'
(21)
Substitute Equation 16 fort in 21 and integrate. The solution in dimeneionleasform is:
Catnlytic Connerter for Oxidation of Naphtholem and Lierueme
-
The integral mean temperature when *(p/k) 0 may be found by substituting 1 from Equation 22 in Equation 23 and integrating:
Total Heat Generated The total h a t of reaction released p x unit of time in the total volume of a thin catalyst &ion is of mtemt in design computations. It may be expressed:
SLAB-SHAPED CATALYST BED SBCnON
Equation 24 has been evaluated and is pmsented as a dotted line in Figure 2. The solution for Equation 24 when the value of p l k is negative is as follows:
Tamperatme Distribution. The followin derivation dede with a thin section diced from a dab-shapd catalyst bed in a direction perpendicnlar to the gaa stream (Figure 4). The thiaknens of the &h is ZB,the length 1, and the height 8. Them are no temperature gradients along the height of the slab. Thew& of a heat tranefer system are in contact with the sides of the dab. h a w of symmetry the analpsis of heat transfer relations in the thin emtion is based upon the center line as the line of refer-
"*"...."*, T+ e p t i o n has been evaluated and p ~ t e asd a dottad line m Figure 3. The range of this solution IS limited,however, by the available Z, tables. May, 1943
The thin section has a width B, thicheea Al, and height 8. Within this thin section consider an elements strip at a dk" p c ~b from the center, with a width db, a t%ekness A1, and height 8. The volume of this strip will he:
INDUSTRIAL AND ENGINEERING CHEMISTRY
965
In dimtmsionlca form thia equation is: The ratap'heatmhse by an exotbsrrm' 0 wction in a d o n h a w a d t h b m: uation ais plotted adotted line in Tb temperature 6. heL p l k is negative, the integral mean WmeS: 88
The rata of h t l q flpm the d o n by outward oonduction through the area @Ai m a m by:
The rata of heat Laa from the d o n by w n d o n transfer betwarn d i d and gaa follows:
K e e p i n g t h e u u n e d o f n ~ d p u k d i n ~lindrid is:
catalyst bed cam, the particular mluhon of Equabon
3
-
Generated The uation for the total heat mpw unit of time in the t o ~ v d u m eof the thin d o n is similar to that for a cylindrical catalyst be8 (Equation 27):
T0t.L Xi&
CYUNDBICAL CATALYST BED WlTR A LIMIWNG ASSUMPTION
The solution in dimensionless form is:
M o w derivations and all wmputed examplea which follow this d o n are based on sssumptiop 4; i. e., the t e r t u m of tbSgMwithinany(hi0 ondeliceofca s t b e d m d o r m and wnstant. h G t i U E sssumotion &t can be made, however is tbat the &as temperaik at in? CTWenction follows the mgd temperatum, point by point, wrth a constant M e m c e between them:
Equation a6 has +an evduated for pitin values of p l k and is
pRsenteda?aeaneaofmlidcurresmFlpm5., , The solution for Equabon 83 when p / k m negabve m:
l-l.-u
(
The M e r a c e , 0, would be wnstant in any om88 d o n , but would vary and even e k @ e s i at Mere?! SeDtiom dong the M. me ~ e r e equation% n ~ this c m m:
The dution is the eantw 88 .that for the derivation of tam mtun, W b u t i o u in a cyhdncal -tal bsd d o n (pape &I, when C, and C, are tnJ constsnt. %%rea 2 and 3 may then be used for the values y, and a:
012,
Equation 37 appears 88 a series of d i d linea of varisble b1B in
%&tion
for Equation 84 or ab 88
*( p l k )
+
0 is:
Integral Mdean Temperature. The derivations for integrd mean temperature for a slab prdlel thw -ted for the c r ~ i c caw. a consequently, o& the d a o l u t i o m sill be
m-
Intagrd mean slab tamparatum with positin p l k ia:
(m)
However. to watulate a constancy of COand Ct when the gaa tem a t m h amumed to vary bver the crow section is Undour@ not d i d . At wnstant precam,,opncentrationwould varv temmratum and also wrndete mwns wuld no l o w be k e d (& is in y p t i o n 4)- Since Co-and C, are fuiictiOM O f the CO~C8lltrS~lO~, they Would DO b D @ I be W&t throughout a ~ 0 8 8d o n .
wd
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vd. 35, No. 5
I DISCUSSION
For m n s of p d o n , Epuations 17 and 20 for the cylindrical c a w c888 have been plotted to Merent d e s in Figurea 2 and 3. The m e is true for the comsponding equations and plots for the dsbnhaued bed. Actuallv t 11 the t e m m t u r e gmup, isa smooth function for C. 4'
-
f,
+
+
real and d u e s of the rate group, R.\/(Ci- h)/k, correapondmg to positive and negative d u e s of (C,- h)/k.
beat release, conduction, and canveotion. Since the fun* tim am general, they are applicable to any combinstion of two raten,or todl three. Thus we may consider the fobfour cambinationB. Systemhlving On4 h md k (C, = 0). In this cane R m is always imgimry and the solution lies entirely in the lower half of Figure 8. The oross-sadional tempera, turea of a packed heat excbanger would be an example. Car&t thin-& heat transfer problem may slao be solved in this manner.
-
Thie is shown in Figure 7, a rectilinesr plot of the variables. System InvoMng Only CI and k ( h = 0). If the rate The curves for real values of the rate group and positive of heat relesse is independent of h (which is not probable d u e s of the temperature group lie in the 6rst quadrant. in a ohemical resotion), the solution for an exothermic They run smmthIy through the origin and branch in the third reaction lies in the upper half of Figure 8, and that for an quadrant for imsgiOarg and negativa vslues. The cornputsendothermic reaotion, in the lower half, becsuse CI changeu tion of temperature becomes indehninate as the f u n ~ t i ~sign ~ witb type of resation. This solution may be of intereat in paan through the origin, and in thisregion the nee of parahlic electrical heating problems. Equations 22 and 38 is essential. System Involving M y G and h (k = 0). In The functions reach limits For eaample' as R4-k 88 k -t 0, the imaginary . d u e of the group R.\/(CI - h)/k approaches 2.4, the temperature group a p aPPm*m infinity and f -ca t l+ = I . Thismaybe proaches infinity; and as the temperature gmup approaches I=+ -1, R.\/(Ci - h)/k approaches imgimry infinity. The meaning of this becam- clear in Figure 8, where the tempera, R%mWed to give twe group is plotted againet the dimmnionka group (r/R) at Co Cd h(t - tJ psnuneters of the rate group. As the rate group approaches which is the heatbalance. 2.4, the temperature rim becomes extremely large and the thermal situation becomes unstsble. For imaginary d u e s System Involving C,, h, and k. For an endothermic mof the rate group (i. e., (CI- h)/k is negative) the tempera, action Ci is nurneridy negative and the solution involvea ture p u p temds toward a stable limit. o d y the imsginsrJr values of R.\/(Cl - h)/k. For an exThe equations developed relate the rks of temperature in a thermic reaction either the real or im,,g+q solution may bed messection, mong other variables, with the three rates: occur, depending upon the relative msepltudes of Cland A.
+
May,
leu
p
I N D U S T B I A I A N D E N Gi IN E E R IXG C H E Y I S T B I
s6l
WNVEPTEE DESIGN
Two types of information am wnired in the deaign of a catalytic&nvertm: heat tramfer eqdionn supportu-by the necesmy thermal data, and kinetic equations (or graphical correlations) and cormpondhg data. The former are m p plied by Equations 17,20,21,24,25,28,27,and Figuresland
3 for a cylindrid catalyat bed, and by Equstiona 35,87,aS, 40,41,43, and Figures 5 and 6 for a slabshaped bed); witbim the stated limitations, they are general in nature. Tbe latter must be determined experbntdly. It is apparent that numerous oombinationa of design procedure will result from the nee of the t h e d quatiom with the varioun function8 for sera-, firsts and seeon&mIar d o n s . plnposesof illub tration, however, may be served well by a aecond-order BOtion of the form: A+B---tC+D
The following equation8 are developed for this reaction in a continuous catalytic flow agatem. For A = B,
s88
e-%,-
but
AlP and&--
Vt
n&T,
WRIT,
dl
(49)
SubatitUte for de in Equation 48:
&! ‘
dn. && ‘
-
(,
+
(-) (e
7kP.p’
+
- %)* & -P)’
&)(a
(W
or since n, d, €!!Adl .
- (.+&)T(& (51)
Now,to introduce heat quantities, P
-
4anddn.
-
&/Q
Substitute for dn. in huation 50:
INDUSTRIAL A N D ENQINEBRING CHEMISTRY
Vol 38. No. 5
+
3. Using an average f = f ( A j / 2 ) compute af. over the total interval Aj, since all other terms m equation of step VI1 areksown. Ab0 calcnlnte t, at end of intarvaI(- b
+ % ' 4. eck conjectured value of T, in step 2 and COW if n-.
-
5. Compute Coand C, with the aid of uat~ona55 and 56 at f fe A j and at T. fmm step 4. %owing CO CI h k &.(from 3). R,and 68 compute the value of C.wifh Ad'oi
+
ofLatthebegmnmgofthe i n M equals the 86 value of an average 6, in step 2, the operation in completed, and computatio&l stsps e n g at I am rapeated for the next 4f. If the average 18not that -4, a newgnm of 1,must be made in step 2. -2and3. mterval and
Computational Steps. The foUowing are the computational step based upon the above equations, upon Figurea 2,3,5, or 6 and upon heat balancss. I. At the bed entrance Co and C, may be computed through Equation8 55 and 56; &, tE, and R are design vanablea which must be set; and h (#) is calculable through the Reynolds number. Knowing thew, the value of t, may be determined from Figure 2 or 3 for the catalyst in the cylindrical bed. II. The rate of heat relea& per volume per unit of time a t the bed entrance is:
c.
+C A
111. The rate of heat transfer per volume per unit of time from solid to gas is:
-
h (1, &)
.
IX. F'revioua &pe lead to estimatad valuea of 1, and b, rate of heat d e w , and radialheat transfer rate 88 a function of the fraction converaionf. It is of greater intareatto relate these to the length of catalyst bed, 1. This may be done by graphically integrating Equation 51. SAMPLE CAJ.CULATION
Problem. On the bask of the assumptions which follow, compute the mean and maximum s o l i b temperatures, the gas temperature, the fraction converaion, and the radial rata of heat flow through the cataly& periphery 88 a function of bed length. As~umptiom. The quantities which are design variables
are marked with an asterisk *.
IV. The rate of heat loss by conduction is: (Co
Jfdmav
L at the e n d T t h e
+ C A ) - [h& - &)I
V. The radial rate of heat transfer through the catalyst wall in heat units per unit a m per unit time is:
This value must balanca with the exchenger . cdicienta when the wall tamperatme is t ~ . VI. Write a heat balanca for the heat received by the &ss dong the bed:
CATALYST Bm: *Le& -3ft. 'DGeter 3 in. 'Diameter of packing Void 40%
-
-
0.0167 ft. (Tylermesh 7-8)
-- &-
GASh m e ~ n uAT l Ban Coworno~s: Swci6c heat C . 0.33 C. h. u./flb.)f' . . .. C.) V d t y P 0.018 centipoise k d t l g r o u p * C f k 0.8 Molecular +ht 30
I
Subtitute dJ from Equation 51 in the above:
VII. Carry out computations in terms of A& and Af:
VIII. Assuming a w y mrdl value of
4 (eqeciaUy at the entrance of the bed),
compute Ab as fdom:
-
1. (1, &) iB saaumsd'conatsnt over the interval and in aet qual to ita d w a t the beginning of the mtsrssl 2. Ouess at an amrage d u e of Th. and
T,,.. for the interval.
. b y , 1943
Figure 4.
Thin Section Sliced from D SrOb-Shqmd Catalyst Bed
INDUSTRIAL AND ENGINEERING CHEMISTRY
E69
hACllON:
--
Heat of d o n !2&WC. h. &/pound moL (exothermic) of activation 21,W C. h. u./pound mol. m n d order, and of the general form
Cdculntiom. To evaluate the constants in the a p proximata equation k' P 6, plot the function k' againat temperatm, from 'k = ~ - W E , T ,over the range desired. (A maximum temperatm rise of 80' C.means an avtemperature rise of app&tely 50" C.) For greater precision in later dculations, d e h e a new temperature &, using the centigrade degree, but with a displaced e m point such that 0' 8.on the new d e 350" C. Draw the best straight line thmugh the curve from IO" to BO" 8. The linear function becomes: 'k 737 106.3 1.
+
-
-
+
CAICC~JATION OF Co IWD C,. From Equation8 55 and 56 calmdata COand Claa functions off and T,. Since n, = %,
-
c0 !no
T,'
(-&
2,a~a,000 1 -
*(T.mwt be expres~edin K.) 0
-
At f = 0.00, t, CALCULATION OF B ~ TE~IAPEXWCIEES. D 10"S. (380" C.) and tn = IO'S. Substituting T,= 633' K.: C,
-
c.
-
426,200
(g)'
2,982,000
=
1.046
(-)1 - 0 ' = 7.45
INDUSTRIAL A N D ENGINEERING CHEMISTRY
VoL 35. No.
5
INUUSTRIAL REACTION RATES
Fiyre 6. solutionfor T e m p m t w e s in clw &tion
-
-
+ hl, 7.45 + (Z.arO)(lO) p = Ct - h = -1.376 n -13.16 IR + p = 10 + n
C.
-
sf S h b - S h a p d Cotllbrt Bed for Nesatioe V a l w of (Ct - h)/k
31.85
1,
-
18
-
(-13.14)(-6.17) 1,
-
-
18.12'9.
-
8.13
-
Continue the osloulations by taking an increment 4 0.01. Fmm computational step MI,
For precision it is advisable to use following alternate mmpubtion:
c o + Crt. -h
Co+lu.=b-t
' + C,
-
lo
(~.064)(10) - lo + 7.45 +-1.376
mo1
Sice n, = 2a and G = n,M,/A, if P is constant a t 1a t m w Phm, Ab
--13,14
C. h. u.
If+>I,~".,~ --.,\ ".,> > I O P
4.63 i(dimensionless) Since p is negative (hence R f l is imaginsry), use Figure 3 (dotted line) for mean temperature group:
Max 1943
+ pn
-0.617
(1
C)(T.)'.. .
f.."'...)
The latter bracket in constant throughout the bed length and may be precomputed. ..'k . variw rapidly with temperature, it is necemq Since to estimate the value of T, (absolute value of tJ for the next c m section (stf, fi An. Estimate t. z-27.5' S. and 1, = 19.0'8.; than L-. 22.8' S. and C . = 14.7" S. --.. or 637.7"K. From the plotof Quation 2,.,'k = 3050. Thus,
- +-
- 10.00)(637.7)' (1.w- +)9(3ow)
(O.Ol)(lS.lZ Ab-
la
- (40(1, --
(2)(2.440)(359) [(0.33)(W73)~
At. = 9.40' 9.
INDUSTRIAL AND ENGINEERING CHEMISTRY
571
+
Therefore t, atf = 0.01 = 10.00 9.40 = 10.40" S., and the 88sumption o f f c h h . In computationsof the above tspe specislcare must be tsken to we small valuea of the computational interval 4when the curvature of f 08. f is sharp Osge second derivative). Now,atf = 0.01, Co = 7.09, CI = 1.012,andp -1.428, 1.
+ np
-
10
(1.012)(19.40) - 19.40 + 7.09 + -1.428
= as,09
-
The aaaumption of t. checks. If it had not, a recalculation of 4 f r o m f = 0.00tof 0.01 wouldhave had to be made an well an the recalculation of the oonditions in the cross wettion atf = 0.01. Repest the above pmcedurea, letting 4 = 0.02. Assume . f and :,for f = 0.03 and proceed. A ~umm&lyof the reedta of the calculation of bhh and subsequent step is even in Table I.
CAICULATIONOF~ nsaFnrrcmomorf. FromEquation51: From Figure 3, 1.
+ n-P
-
-0.022;
1,
-
27.46
Since n, = Za, R, = 359/273, and P
.,
l atmosphere,
ALL data
SIB now h o r n except 1. Integrate g r a p h i d y by plotting 4aT.t
-
(359)'/k'(l f)3(273)' agSinat f, take cumulative areaa under the ourve from0 to e8veral valuea off, and thus obtain AZ. Plot these values of I SgSiaat f, pass a m m t h m,and thw obtain I for any value off. Now dl computed reeulta may be referred to the length position in the bed instead of the f position.
Haat Releasad through Periphery of Cat.1Jlt Bed. The method of computation is described in oomputatiod Btepe II,IV,V,and VI. At f = 0.00, the rate of hest release per volume per second is: C.
+ Ct1,
-
7.45
+ (1.084)(18.12)
-
C. h. u.
26.7 (sec.)(cu. ft.)
TABWI. Smoranr or A C o w m CALCUUTIOU *I-ta
oTg.
*k.
6
Q
!:E Z:!;:E ::E :Lo 0.918 0.w
84.80 u.30
s.is
0.807 0.m
ua.80 m.aa
0,701 O.M 0.871
6X.W
eos.~~
4.oa 4.81 4.70 4.60 4.41 4.14 8.79
047.0
8.01
0.01 0.05
087.8
6.49
w.9
0.07
0.10
m .8 w.8
8.0~ 6.66
0.11 0.18 0.14 0.16 0.17 0.10
a6.i
o.as 0.88
0.110
0.40
6m.g
W.1 W.7
w.a 678.9
w.8
w.0 w.4
. -
0.4s
O' 8.
5la
860. C
6a.u 6a.n
61.66 0.W 0 . ~ 0 60.19 SS.89 0.W1
8.a
0 . ~ 1 4asi 0 . a ~ 80.78 m.91 0.m
am am
o
.
0.816
~a i m 16.89
b 0. h.u./Cx.Ik.
&zh
la+;
1.m 1.m i.6ia 1.m 1 . w i.mi
18.14
4.m
49.69
4.88 4.w
m.0~
w.70 Tam m.a
1.787 80.01 1 . 7 ~ m.81 78.29 1.716.98 1.788 1.810 1.849 1.m 47.u 1 . ~ 1 ao.a6 a.oii ~ 1 . 1 9 a m i ir.05 ¶.la6 9.w
8%
0%
4.78
0.617 0.w
O.W
om9
0.m 6 . 1 ~ 0.661 6.08
6.oa 6.w
6.26
3.28 6.89
6.88 6.u 6.u 6.61 6.-
6.77
0.0118 o.w 0.087 0.UO 0.wo 0.W 0.w 0.67~ 0.611
o.es1
0.w
8.1) 17.u 8i.w 40.70 4e.m 61.w
41.87
ma6 6a.m 6124 W.70
ma6 m.00 61.W Bo.70
48.70 81.47
M.70 41.41
48.M
p0.m
14.71 11.61 6.a
18.11 n.4e
so.m w 6 1 .m w
ss.66
80.m m.71 ai.61 16.64
a5
AI,
m.47 88.70 67.46 71.88 m.66 87.w
9.40 16.00 10.90 7.w 7.40 1.88
87.70 87.w 88.10 84.86 81.60 M.80
ss.00
89.60
a1.as a6.W 19.18
- 0.09 --- 6.87 8.01 - 6.61 ... 0.1s -0.41 1.u) 4.80 -12.98 -1a.18
%e Bdi.l mor)
Eat
0
o.mo
0 . 1 ~ 0.m 0.880
0.870 0.1116
0.470
0.600 0.680
0.6W 0.780 1.00 1.405 1 . w 1.m
8.675
1.MO
84sn 6f90
8.170 9450
1o:m 10.870
::a 9.m 8,170
a m
...
K).
I N D U S T R I A L A N D E N Q I N E E R I N Q CHEMISTRY
Val. 35, No. 5
The rate of heat transfer from solid to the &as is: h(t.
- &)
= 2.44 (18.12
C. h. u - 10) = 19.8 (eec.)(ss. ft.)
The rate of heat loas by radial conduction is: 26.7
C. h. u. - 19.8 = 6.9 (see.)(cu. ft.)
The rate of heat transfer through the catalyst wall is: [Co
+ C& 2rRA1 - h ( t . - t o ) ] AV = Co + Cdm - h(t. - &)E 2 (6.9)(1.5) = o,431 C. h. u. (2)(12) (sec.)(Sq. ft.)
1550
C.h.u. (W(Sq.ft.)
The major variables computed in the sample calcnlationare presented in Figure 9. The maxima curvea obtained conform to the expected typ+ of tempersturedistribution in a catalyst bed. The m p l e calculation provides a rough check upon the validity of the thiid a e m p t i o n in the thermal derivations: i. e., heat conduction is in 8 radial rather than lengthwise direction. In the region of the maximum tempewturr, the radial temperature gradient is approximately ten times that along the bad, and the wumption is therefore justised an a first approximation. The results of the sample computations in this paper have been compared with those obtained by Damkwhler’s a p proximate equation (citation 1, paragraph 667, Equation C19). For a maximum temperature rise of 78’ C., Damkoehler’s equation leads to an estimated ca+t tube diameter of 1.5 inches. The deign method in this paper would predict a diameter of 3.0 inches for the same temperature rise and m e auxilisry operating and denign wumptions. NOMENCLATURE
-
D i m e n s i d Unit8 Length L Mass-M Usntity Of heat = H
Le=e a
C
Cn C., C.
e
E
(We)
any pomt, (+I = ‘/a mdth of tlnnsbb catalyst wncentration of pa wmponents intercept of 8tmght-line &&on
B
$A.
moles of reactant A entering bed per uqit of t i e ,
CMllkBeCtion area of catalyst bed, (L’) = lateral d?.ance from center of tlnn-slsh catalyst
A b
C,
--
Temperature = d
(mw. concentrataon of
of q/Vu
bed to with
1,
componsnts A and C at point (MV) . slope of 8trpightLneVariation of f V,,. H/0LW) m y
in the bed,
q with t, = heat.capaclty of p e a at bed opnditioni?, (H/ Afd) u = centwade heat u t , heat requued to mse temp. of 1lh Water 1 = basic of natural logarithms 2.718
6h h‘
--
c.
-
enerw of activation. (H/M fraction conversion of &t A at any amss mation = m e a docit of gas, (M/eZ*) = volnmetric heat transfer aoeliioient. (EIeZW d a c e film heat transfer coffioient (E)eZ@ ’
&
t
Figure 8. Solutionfor Tern mtums in Cylindricnl ComIyst-Bed Cmss G t i o n for P m i t i a , and Negatia, Valuu, of (G h)/k, with R4 an Parameter
-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
573
a5
-
/a
1.0
u
to
dC.ulmt B d h h p . M sFrt Figure 9. 6 vted VariOtioM of Mean and Marlmum Catutyst Temperature#, Meon Gas Tampamture,nj&dhl Best Flow Rate, and Fnction ConDsrrion at w e r e n t Length PoddOM in a 5-Inch Diameter Catalyst Bed
-
i
6 1 = ~esselfunction, first kind, morder, imasinary mgn-
I&)
-
ment I,@) Beasel function, first kind, first order, imsSinary ruwment I&) ru~.. = Beasel function. first kind, seoond order, inwinrm -
k
mat Beasel function, first kind, zam order Beasel funKion, first kind, 6rat order Beasel function first kind, second order thermal condukity of catalyst under bed conditions IHICITA ,---, reaction rate coeEicient, depend& on t8mperatnre. Dimensions dewnd uDon order of renation; for semnd order (L*/Me) reaction rate conntant inde Dimensionsdenend uwn o E % s o ? r & % order (~:/Mej k q t h along catalyst bed in dimction of gas tlow, (L) total -1 of catalyst bed, (L) = alone of stmkht-line variation of 'k with t. (L*/MBdfor kond-O& reaction) integration constant molecular weight \-
I IT m
MI MI n
,.n
rn
----
rh =
N p
P
574
C~+ht, moles A and C passing any point per unit *e, MB) total m o b gas passing any pomt per u t tune, M / e )
integrationconstgnt -Cr-h absolute pressure of gas at any point, (M/L')
I
rate of heat release (H e) heat ofresction, , , , radialdistance to any pomt In cyllndnoal catalpt cram section (L) outer r d w of c y ~ c a ~ . c a t abed, l e L) law constant..BDDMDNL~~ ute,. (L/ . . space velocity, (i/e) am.' C., an arbitrary tempsratnre scale for oompnta-
(&/M(,
__
m
t l O d DUmOBes
wall tem+&ture of tbin&h catalyst bed, ( d ) temperature of adid catalyst at canter of bed, (d) temperature of &lid ca,tal t. (4 temoerature of cvlindnc$mtal~st bed at wall. (4 at& of solid catal* at any cmea section
temparatwa of gas stream at any crm section in catalyst bed, (4 absolute tem ture of @a at any cmea section in catalyst volume of catalyst, (La) total volume of flowmg par unit time at any point in catalyst bed, E / e )
rf R
I
For a oylinder (dimensionless, w e consistant units)
I N D U S T R I A L A N D E N G I N E E R I N G CHBMISTRT
Vol. 35, No. 5
-
e
for &cand-n&
time,
(e)
d o n )
Vapor-Phase Esterification Rates H . F. Hoerig', Don Ramon, 0. L. Kowalka ~
'
E
IOF WI%WNIIIN. l ~ MADISON, WIS.
IIE esterification reaction between organic acidn and alcoholn has been atudied quantitatively in the liquid p h from the standpoint of equilibria and of reaction rates. The reaction in the vapor phsse has not been inveatigated 80 thoroughly, however.
The equilibrium constante for the v a p o r - p h reaetion have bean accurately determined (#), and various catalynta have been employed; silica gel apparently has been the moat s u c c d u l (I, 8, 6,6). No data are available to correlate the eEect of maas velocity of the gases and the eEect of temperature on the reaction rate. In thia investigation the esterification reaction between ethyl alcohol and acetic acid in the vapor phsse was studied in a Bow system, operated at stmonpheric pressure, and employing a silica gel catalyst. ESTERIFICATION UNIT
The a paratus consisted, in general, of a calibrated delivery aptem L r the maetantn. a propofiioning pump, vaporiaen,
k n t sdd-..
E. I. du Poot d. Nsmoun & Comp.ny. Io&. BuUdo.
Y.
reaction chamber, Dowthmn heating 8ptem. condenser, and contml panel (Figure 1 The calibrsted delivery w&.m for acetic acid and for e t b k almhol wan constructed of glass. AU the other unite in the reaction system ware mmtructed of
KAZSM08tainlesssteal. The reactmtm were pumped separately from the calibrated glaae burets by the proportioning pump m order to ad'& the D ~ toDthe desired Bow rate. After the rate wan aet, th, ram&ti wem pumped fmm storaga mb0 to avoid repeated in of the tla& which nu lied the rmts. Calcium chloride tu& were connected to slf%r inleb of the feed swlem so that
A se&wate vaporizer wan built for each m.iant from a l ' t ~ inch stainless steel nip le, 18 inchas Ion These nip lea were canned on the ends. an8were cast in n,arr%el into two &minum the Ilqtdds at an initial, mntrolled, constant temperature. In addition, a second similarly controlled region wan rovided at the disc6ane end of each vanorher tu control the of sumrheat. %e aluminum b1ocb acted BB constan~tempraiure wmim of heat, EreFntii +ow, vapo&ation of the resctsnta and consequent co ection o hqud m pools in the vaporiaenr. The superheating section WBB neceseary to obtsin close control of the temperature of the reactants entering the reaction chamber.
8-
I
I The esterification rate of acetic acid with ethyl alcohol in the vapor phose UXM investigated in a flow system using a dica gel catalyst. The mass velocity of the vapor did not affect the constants in the rate equations representing conditions a t spe&fie looations in the bed. I t appears then that maas t r a d e r in the vapor phase UXM not significant. Furthermore, with temperature increase, these constants were augmented linearly, which indicated that the sunface reoction
Ha):
1943
rate UMII not a controlling factor. The experimental evidence shows that the rate of vapor-phose esterifieotion is controlled by the rate of maas transfer or diffusion through a condensed phose present in the capillaries of the silica gel. The temperature effect is in agreement with this obserwtion, as ia the magnitude of the activation energy. The second-order reaction rate ordinnrily encountered in liquidphase esterification UXM negligible in the catalyzed reaction.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
575
