ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Literature Cited (1) Barrer, R. hI., "Diffusion In and Through Solids." Macmillan Co., New York, 1941. (2) Dryden, C. E., and Kay, 77'. R.,Am. Doc. Inst., Doc. 4314, Library of Congress, Washington, D. C. (3) Eagle, S., and Scott, J. TV., IND.ENG.CHEM.,42, 1287 (1950). (4) Edeskuty, F. J., and L\mundson, IS. E.. I b i d . , 44, 1698 (1952). ( 5 ) Freundlich, H., "Colloid and Capillary Chemistry," E. P. Dutton. Kern York, 1952. (6) Koeeny, J.,Wasserhuf't u. Wasserwirtsch., 22, 0 7 , 80 (1927). ( 7 ) Lemieux. It. U.. and hIorrlson, J. L., Can. J . Resemch, 24B, 137 (1946). ( 8 ) Paterson, S..Proc. Phys. SOC.(London), 59, 50 (1947).
(9) Piret. E. I>..and associates, Chrm. Erg. I'rogr., 47, 405 (19sl). (10) lioehl, E. J . , King, C. V.. and Kymen, S.J.. J . A m . C h e m . Snc., 63. 284 (1941). (11) TTicBe. E . , Kolloid-Z., 86, 167 (1939). RECEIVED for reviex January 16, 1534. ~ C C E P T E D June 30, 155-1. From the thesi3 of C. E . Dryden presented in partial fulfillment of the degree of Doctor of Philosophy from the Ohio State University. Material snpplenentary to this article has been deposiied as Document No. 4311 with the -4121 Auxiliary Publications Project, Photoduplication Rcrvice, Lihrrry of Conaress. JTaslijngton 2 5 , D. C. A copy may be secured by citing the document number and by remitting $2.50 for pl;otoprints o r $1.75 for 33-mm. microfilm. Advance payment is required. M a k e checks or money orders payable t o Chief, Photoduplication Service. Library of Congress.
Temperature Distribution for Low
xygen Concentrations
J. J. VAN DEEMTER Koninklijke/Shell-Labora~orium,Amsferdam, Holland
IK
AN earlier study ( 1 ) it was shown that transient teniperature profiles during fixed bed regeneration (coke oxidation) can be derived mathemat,ically when thc folloiying assumptions are introduced: 1. Constant rate of reaction 2. Infinitely high gas-particle heat transfer coefficient 3. Adiabatic reactor 4. No diffusion of heat in longitudinal direction 5 . Xegligible gas density variations due to temperature gradients, pressure drop, or chemical conversion
It was found that there exists a certain critical oxygen concentration above which the velocity of the oxidation zone is equal to the velocity of heat transport by convection in the bed; then t,he teniperature in t,he oxidation zone increases linearly as the zone proceeds through the bed. Below the critical oxygen concentration the velocit,y of the oxidation zone is smaller than the velocity of heat t,ransport because it is determined by the oxygen supply. The temperature profiles become much more intricate. Bn expression for the magnitude of the temperature peak has been given for the case Ivhen t,he oxygen concentrat'ioii is lower than half the critical value (f). Formulas of more general validity for oxygen concentrat,ion below the critical value are presented in this paper. Transient Temperature Profiles below Criticall Oxygen Concentration Are Obtained b y Construction
Temperature profiles may be obtained by construction rather than by mathematical derivation. This m-ill be demonstrated for = the special case that the oxygen concentration co = 213 co 2/3 CY cd. c6 is the initial coke concentration expressed in equivalents of 1 mole ol oyygen. P&O is equal to the ratio of the heat capacity (1. - E ) P & EpQce per unit volume of incoming gas to the hcat capacity per unit volume of the bed.
a =
+
It has been pointed out (1) that the velocity of hcat transport by convection is equal i o ov,vvheie i) is the superficial gas velocity. I n tho initial stage the oxidation zone nil1 compriae the region 0 6 z 6 ovE which has been heated by the gas I n this region the temperature increases linearly according to 2300
+
1' = To H V ~ / p ~ c , v .To is the temperature of the incoming gas. The initial temperature of the bed is chosen as the zero level. H is the heat of reaction per mole of oxygen. U is the oxygen rcaction rate. The temperature gradient, HC/p,c,v, is characteristic for the conditions prevailing and will also occur in the oxidation zone a t more advanced h g e s of the process. The oxygen mill be consumed completely in the oxidation zoiic when the latt'er has reached a depth 20 = c ~ v / U . This corresponds t o a time to = a ~ / a v= c O / a U . The time for coke removal tl = cdj7,. S17hen to < tl: as is the case in this example, the oxygen viill be consumed before the coke has been removed from the entrance (Figure la). (For the crit'ical orygen concentration cO = CYC,', the time ts, is equal to ti.) During the next stage to t tl the depth of the oxidation zone does not increase, as there en present beyond z = zn. There is, however, a removal om the oxidation zone by convection at a level 0'2 IlUxO/p,c,l; until a t t = il the coke a t the entrance has been r e moved (Figure l b ) . After that both rear and front of the oxidation zone are moving through the bed n-ith the same velocity, a ~because , of removal of coke a t the rear (Figure IC). During this stage t,he velocit,y of the oxidation zone is equal to the ve1ocit)y of convect,ive heat transport, so no heat will be removed from the oxidation zone. Therefore its temperature incrcases at a linear rate. Behind and ahead of it the temperature remains at'the same level. At t = tl i o (Figure I d ) thc discontinuity in the coke concentration a t the place a = 20 causes a etandstill of the oxidation to 6 t 2 1 , until a t t = 211 (Figure l e ) the coke zone during tl a t zo has been removed and a situat>ion similar to the one a t t = tl has been obtained. The temperature profile, however, has becoine quite different. During the standstill. gas of temperature T o ent,era the o atmionzone and causes a lowering. of the tcmperat,ure prevail t8here(Figure le!. This cooling a t the rear side of the oxidation zone proceeds also n-ith the velocity CLU as it is in fact, a heat transport, by forced convection. I n the cooled part of the oxidation zone the tcmperaturc gradient reniains equal to HL'/pseUu. The hcat produced is, at, thc same 2HIUzo/pUc,v. time, removed in a forwad direction a t n level 1'0 Between 2ti and 2ti io t,hc oxidation zone is moving again, so that its temperature increases a t a linear rate t o the value 3HUao/p,c,v for z = 320, t = 2t1 t o (Figure If).
2t1 + t o )
November 1954
the temperature remains below a maxi-
Co
=
+
Oxygen Concentration-Temperature Relationship M a y Be Used at Bed Depths Greater Than 10 to 100 Particle Diameters
The importance of the assumption of infinitely high heat transfer coefficient may be estimated with the aid of the Schumann-Furnas theory (4,6 ) on the heating of a packed bed by a fluid flowing through it. This theory shows good agreement with experimental results and does not rely upon assumption of infinitely high gas-particle heat transfer coefficient. From the theoretical curves of Schumann-Furnas it follows that the temperature difference between fluid and solid is smaller than 10% of the final temperature increase if
h z H.T.U. = z> 8 pQcgv where h is the heat transfer coefficient (per unit volume of the bed) and H.T.U. = poc,v/h is the height of a transfer unit. Heat transfer experiments (6) have s h o m that for gases H.T.U. is of the order of 10 d, (d, = particle diameter), for a Reynolds number (referring to particle diameter) equal to about 8000, and becomes of the order of one particle diameter for Re = 10. The temperature difference between gas and particles will, therefore, become relatively unimportant a t a depth in the bed greater than 10 to 100 particle diameters, depending on the flow conditions. This means, for instance, that with a bed height of 3 feet and a particle diameter of 0.1 inch the assumption of
INDUSTRIAL AND ENGINEERING CHEMISTRY
230 1
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT author. Furnas ( 2 > 3 ) carrieil out temperature measureinent,s during limestone calcination. The present theory, howcver, does not apply to this case, since the air used for heating does not talce part in t,he reaction. a,+in oxidation. Besides, the ratmeof reaction n-as so slow that the' hr,;rt iieetird could be easily supplied by convection. Conelusions
It appears from the formulas derived in this paper thiLt below the critical oxygen concentration e,, orit. = aci the temperature in the oxidation zone initially increases linearly until at a drfiiiite distance from the entry a certain maximum value is attained. This maximum is independent of the rate of reaction id the gas velocity. The distance from the entry itt, which permanent conditions are attained (the inlet lengt,h) is proportional to the gas velocit,y, u, and inversely proportional t'o the reaction rate, U. Both inlet length and height of the temperature peak may be easily controlled with the aid of the oxygen concentration. When bhe oxygen concentration approaches the critical valuc t,he inlet length becomes infinite and no maximum for thn temperature \vi11 be attained. As has been shown ( I ) , this also occurs for oxygen concentrations higher t>hnnthe critical value. Nomenclature
1
Figure 2.
.
Temperature Profiles for CO = 1 '3
COUrlt
Oxidation zone indicated by shading
infinite heat transfer bdl be v d i d for t,he greater part of the bed. With d, = 1 inch this will usually not be tmhecase. Unless the Reynolds number is very small. It has to be emphasized, however, that, although the kmperature difference between gas unci solid might be negligible, the finite rate of heat transfer between gas and particles will cause a spreading of heat in the longitudinal direction. The discontinuous temperature profiles, as derived, will therefore change into smoot,h curves of a similar shape when the finit,e rate of heat transfer is taken into account. I n this respect there is a close analogy to chroinatography-ctiromatoRrains consisting oE step functions are obtained for complete adsorption equilibrium; smooth Gaussianlike curves result, from theories that assume finite rates of mass transfer. The equilibrium theory, although mathematically much simpler, h:Ls many esseiitial features of a chromatographic procem. It may therefore be expected that in t'he p a m e way the heat equilibrium theory results in a fair description of the t,cmperaturc history during yegeneration. Experimental temperature profiles have not been deternlined so far, nor could they be found in t,he literature available t'o the
dT = inass L = lengt,h t = time r , 1 = t,emperature fl = heat, energy, J4LYt-2 co = oxygen concentration of incwming gas, mole X L--.3 ( ; = initial colce concent c o = specific heat of gas, cs = specific heat of solid niat,ciri:tl, ),':If - - I T d,, = particle diameter, L h = heat transfer coefficient per unit volume of the bed, EL-3T-lt-1 H.T.G. = height of transfer unit, I, H = heat of reaction per mole of oxygen, I:' X To = temperature of incoming gas U = rate of reaction of oxygen, mole x 1:--3t - 1 II = superficial gas velocity, Lt-1 z = longitudinal coordinate, L B = fractional void space (no dimrnsioti) p o = gas density, ML-3 pa = density of solid materialj Acknowledgment
The author is indebted to the man:igoincxiit of the Koninltlijke/ Shell-Laboratorium, Amsterdam, for prrmiwion to publish this paper. Literature Cited (1) Deernter, J. J. van, IND. ENG.C I ~ M 45, . , 1227 (195R). (2) Furnas. C. C., Ibid., 22, 721 (1930). (3) Ibid., 23, 534 (1931). (4) Furnay, C!. C., Trans. Am. Inst. Chem. Engrs., 21, 143 (1930). (6) Gamson, B. W., Thodos, G., and Hougen, 0 . A,, Ibid., 39, 1
(1943). (6) Schumanii, 'F. E. W., .J. Pranklin I n s t . , 208, RECEIVEDf o r review January 5 , 19.54.
405 (1929).
ACCEPTED June 19, 1954.
END OF ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT SECTIQN
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 11
