Ind. Eng. Chem. Res. 2004, 43, 5663-5669
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Injection of a Liquid Spray into a Fluidized Bed: Particle-Liquid Mixing and Impact on Fluid Coker Yields Peter K. House,*,† Mohammad Saberian,† Cedric L. Briens,† Franco Berruti,† and Edward Chan‡ Department of Chemical and Biochemical Engineering, University of Western Ontario, London ON N6A 5B9, Canada, and Syncrude Canada Ltd., Edmonton Research Centre, 9421 17th Ave, Edmonton AB T6N 1H4, Canada
In industrial fluid cokers, the feedstock, consisting of heavy bituminous hydrocarbons, is atomized with steam and injected into the hot fluidized bed of coke. Good and uniform contact of the liquid droplets with the solid particles is required to provide heat for the cracking reactions while mass transfer effects are minimized. Experiments in a pilot plant coker have suggested that the initial particle/liquid mixing, in the spray jet, is rather poor. Experiments in a X-ray scanner showed that liquid droplets and entrained particles accumulate just below the tip of the jet plume. To illustrate the importance of the initial liquid-solid mixing, a simple model has been proposed to predict its effect on coker yields. An experimental technique was developed to quantitatively determine the quality of mixing with current nozzle technology and a new enhanced solids entrainment (ESE) device. The model has been used in combination with the experimental results to show the effect of ESE on product yields. The ESE nozzle has been shown to greatly improve liquid/solid mixing by increasing the amount of wetted solids and achieving a more uniform initial liquid/solid mix. The model suggests that ESE will improve liquid yield by up to 0.6 wt % and reduce the coke yield by up to 2 wt %. Introduction Fluid coking is a process that utilizes a fluidized bed of hot coke particles to thermally crack bituminous feeds. Coking proceeds on the surface of coke particles at temperatures ranging from 510 to 550 °C. Heat for cracking the bitumen is supplied by partially combusting coke in a separate burner and recirculating it to the reactor at a temperature of approximately 650 °C. The hot coke particles heat the bitumen feed, injected through several gas-liquid spray nozzles, from its injection temperature of 350 °C to the reaction temperature of 520-540 °C (see Figure 1). During this rise in temperature, a small fraction of the bitumen feed is flashed off, and cracking of the bitumen feed commences upon contacting the coke particles. Bitumen cracking produces a mixture of gas oil, naphtha, lighter products, and coke. Gas oil, naphtha, and lighter liquids are the desirable products, which can then be mixed to form synthetic crude oil. The coke produced from the cracking of bitumen deposits on the surface of the existing coke particles and the product vapors leave the reactor. The coke is then recycled to the burner, where part of it is burned off. Cracking must occur rapidly to maximize the yield of valuable products and avoid poor (or loss of) bed fluidity. Both mass and heat transfer limitations can slow the rate of cracking. Ideally, the desired products, gas-oil and naphtha, would form and enter the vapor phase immediately, thus preventing secondary reactions leading to coke formation. If mass transfer is limiting, the desired products that form on the surface of coke particles do not enter the vapor phase quickly enough and undergo further reactions, resulting in undesirable products, especially coke.1 Mass transfer † ‡
University of Western Ontario. Syncrude Canada Ltd.
becomes limiting when the liquid thickness on the surface of a particle exceeds a threshold value of just a few micrometers.2 Similarly, heat transfer becomes limiting when coke particles do not contact the bitumen quickly enough to provide the heat for cracking to proceed at an optimal rate. If cracking does not proceed quickly, liquid accumulates in the bed, causing poor bed fluidity and particle agglomeration. Under these conditions, operator experience has shown that the bed temperature must be raised significantly to reduce the risk of bogging and defluidization. As a result, more coke is burned and the bed is operated at temperatures that do not provide optimum liquid yields. Moreover, increasing the rate of coke combustion increases undesirable SOx emissions. To prevent slow cracking and prevent heat or mass transfer limitations, the feedstock must contact a large number of particles quickly and uniformly. Ideally, within the gas-liquid jet, the contact between liquids and solids would be such that the concentration of liquid in contact with solid coke particles would be negligible. In other words, the ideal liquid-to-solid ratio (L/S) should be close to zero. Under such conditions, the heat capacity of the particles trapped in the temporary liquid-solid agglomerates, formed when the liquid spray contacts the bed, is large enough to allow the coking reaction to proceed rapidly. A mechanism for dispersion within a gas-liquid jet of coke, bitumen, and steam has been proposed by Gray et al.2 according to which large liquid droplets contact a large number of coke particles, temporarily forming an agglomerate that, due to forces within the bed, breaks up rapidly into individual coke particles containing thin films of liquid. If this is the case, heat transfer limitations will not occur, since the heat provided by the bed and particle-liquid contact will be sufficient to
10.1021/ie034237q CCC: $27.50 © 2004 American Chemical Society Published on Web 06/25/2004
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Figure 1. Fluid coker.
provide a roughly isothermal reaction environment. However, it has been shown that mass transfer limitations will occur if the film thickness exceeds 20 µm or an L/S of 1 g/g. The traditional approach to improve liquid/coke uniformity has been to design spray nozzles that produce small droplets.3 This paper presents a new approach: the development of spray nozzles that directly enhance the initial contact and mixing of the liquid with the hot coke particles. At present, there is no method to directly measure the initial contact between a nonvaporizing liquid and a fluidized bed. Gray4 used the layered growth of coke particles and reactor coke yields as evidence of a L/S ratio. This work was performed under the assumption that the liquid feedstock is spread on the surface of coke particles, without agglomeration. The liquid would then react and form a layer of coke. This coke particle would then go to the burner and return to the reactor, where a new layer of coke would form. An onion-like structure is observed in coke particles and would appear to confirm the formation of liquid layers. However, a similar structure has been observed in coke from delayed cokers; therefore, this cannot be viewed as a proof of the formation of liquid layers.5 Leclere et al.6 have developed a method to characterize the contact between a vaporizing liquid and a fluidized bed by measuring the vaporization rate of the injected liquid. Unfortunately, this method cannot be readily adapted to applications with a nonvaporizing liquid. This paper proceeds in three sections. First, a new simple model shows the effect of the initial liquid-solid mixing and heat transfer on product yields. Second, a new measurement technique is developed to characterize the initial liquid-solid mixing. Third, the model and the measurements are combined to quantify the benefits of using a device that enhances the initial liquid-solid mixing.
Theory The traditional approach to modeling primary dispersion effects on product yield is to consider mass transfer limitations. These models have been developed under the assumption that large liquid droplets entrain a number of particles and form loose liquid-solid agglomerates that break up rapidly, due to forces provided by the intense fluidization. This breakup is assumed to produce individual particles with thin, uniform films of liquid.4 However, recent work has demonstrated the likelihood that these agglomerates survive in a free jet.7 In addition, Gray4 has already demonstrated the large forces required to breakup these agglomerates if the surface tension at reactor temperature resembles that at 60 °C. To demonstrate the effect of agglomeration on heat transfer limitations, a few simplifying assumptions were made. The agglomerate that forms at the tip of the jet is considered to be large enough for heat transfer from the bed to be negligible over the time required for reaction. In addition, the L/S ratio is assumed to be uniform over the volume of each agglomerate, and the coke and the bitumen temperatures are assumed to equilibrate before reaction starts. Thus, the following heat balance is obtained:
mcCp,c(Tc-T) ) mbCp,b(T-Tb)
(1)
Then, if a control volume is drawn around the coke, bitumen, and vapor produced in the agglomerate, the heat balance, upon reaction becomes
-
dmb Tref dmv Tref [H ˆ v + Cp,v(T - Tref)] ) [H ˆb + dt dt dmc Tref Cp,b(T - Tref)] + [H ˆ c + Cp,c(T - Tref)] + dt dT (2) (mbCp,b + mcCp,c) dt
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where the left-hand side of the equation represents the energy leaving the control volume with the vapors, and the right-hand side of the equation represents the change in energy within the control volume. If we incorporate the kinetics provided by Gray et al.8 for vacuum residue cracking, a system of nonlinear ordinary differential equations is developed. These equations can be solved in MATLAB using the ode23t function for state dependent mass matrixes. The purpose of the above model is merely to demonstrate the effect of temperature in an agglomerate and the inherent disadvantage of high L/S values. A more realistic model would include mass transfer limitations and consider internal heat transfer. It should be mentioned that mass transfer in an agglomerate, unlike the thin film case, would be exceedingly poor, even at low L/S ratios. The addition of mass transfer limitations to the heat transfer limitations considered by the present model would further increase the coke yield. Heat transfer limitations occur as a result of high L/S values in an agglomerate, as there is insufficient heat contained in the agglomerate to maintain a constant temperature. The resulting drop in temperature that results causes an increase in coke yield from the primary reactants that do not readily enter the vapor phase, as their boiling points are, in some cases, well in excess of reactor temperatures. This drop in temperature also slows the reaction and increases the risk of bogging due to an increased liquid holdup. To verify the validity of neglecting the heat transfer from the bed, an unsteady-state calculation was performed to determine the heat up time for an agglomerate in a fluid coker. The agglomerate was chosen to be a sphere, 0.02 m in diameter (agglomerates of this size are normally produced in the industrial coker). The conductivity of coke was determined on the basis of literature values for coke produced in fluid cokers.9 This value was then multiplied by a fixed bed voidage, as the conductivity in the agglomerate is analogous to a fixed bed with no flow. The external heat transfer coefficient was calculated by the method of Gabor and Botterill.10 A Fourier series approximation presented by Martin and Saberian11 was used to calculate the temperature increase. The results of these calculations are presented in Figure 1. The heat up time obtained is a multiple of the solids residence time. Therefore, these results clearly illustrate the validity of neglecting heat transfer from the bed into the agglomerate, as a first approximation. In fact, after one solid’s residence time (average value), the agglomerate remains 35 °C from reactor temperature. To demonstrate the importance of the L/S ratio, the heat transfer model is employed. Figure 2 shows the effect of L/S on the temperature drop of the coke particles, first as they are wetted, at time zero, and then as reaction and vaporization proceed. The rapid drop in temperature is a result of the kinetic model used, as infinite mass transfer is assumed. Also, the initial period where the coke and bitumen reach an equilibrium temperature (before reaction) causes a significant drop in temperature. At high L/S values there is a significantly higher drop in temperature, and more importantly, there is a higher probability that this agglomerate would survive under reactor conditions. In Figure 3, the effect of temperature on reaction rate is clearly demonstrated by examining the evolution of the converted fraction of the higher boiling components
Figure 2. Results of unsteady-state heat conduction calculations: Dimensionless temperature vs time.
Figure 3. Temperature drop in an agglomerate for L/S ) 0.062, 0.073, 0.1, 0.15, 0.2.
Figure 4. Conversion of asphaltenes in an agglomerate for L/S ) 0, 0.062, 0.073, 0.1, 0.15, 0.2.
(asphaltenes, 650 °C+) in the feed. Clearly, the L/S has an enormous effect on reaction time. This evidence illustrates quite clearly the potential problems with bogging and defluidization due to an increase in liquid hold up. In addition to these detrimental effects, the coke yield from the primary reactants is increased with increasing L/S (Figure 4). Figure 5 illustrates the yield of coke relative to the asphaltenes (the primary coke precursor) reacted. This figure demonstrates that even if the agglomerate eventually breaks up, or if enough
5666 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 5. Coke yield in an agglomerate for L/S ) 0, 0.062, 0.073, 0.1, 0.15, 0.2.
Figure 6. Coke yield relative to the extent of reaction in an agglomerate for L/S ) 0, 0.062, 0.073, 0.1, 0.15, 0.2.
heat is transferred from the bed to raise its temperature, its initial low-temperature period will have an irreversible impact. Consequently, high initial L/S ratios cause an irreversible increase in coke formation. In each case, relative to the extent of reaction, the coke yield is higher at every point for higher L/S values. Therefore, the effect of heat transfer limitations should be considered in addition to mass transfer limitations. Apparatus and Procedure Experiments were performed using a fluidized bed with a height of 50 cm and a rectangular cross section of 10 cm × 50 cm. The bed was operated with 18.75 kg of coke having a Sauter mean diameter of 140 µm and fluidized at a superficial velocity of 5 cm/s and a temperature of 33 °C. The nozzle used for injection was a gas-liquid spray nozzle having an internal diameter of 1.6 mm, which was operated with an air-to-liquid ratio (ALR) of 3% and a pressure of 105 psig (725 kPa) to ensure uniform atomization. The nozzle supplied liquid to the bed at a rate of 1.5 kg/min. Experiments were conducted with both a regular spray nozzle and a new nozzle configuration designed to enhance the primary contact between the injected liquid droplets and the bed particles. The new nozzle design, called the enhanced solids entrainment (ESE) nozzle, consists of the regular spray nozzle fitted with a coaxial draft tube located at same distance from the spray nozzle tip (Figure 6). The draft tube used was 1/2 in. in diameter (13 mm) and 11/2 in. long (38 mm). The distance of the draft tube from the nozzle exit was easily varied and was chosen such that the jet would make contact with the inner draft tube wall based on a 10° angle of jet expansion that was measured by analysis of X-ray images performed by another researcher using the same apparatus.7 The distances of the draft tube from the nozzle chosen for experimentation were 1/2 in. (13 mm) and 3/4 in. (19 mm). The distances could theoretically be increased up to 11/4 in. (32 mm) while achieving contact with the draft tube wall. However, under these conditions, any fluctuation in the expansion angle would result in a jet that, at its boundary, would not completely enter the draft tube.12 A unique technique was developed to characterize the initial liquid-solid contact between the sprayed liquid droplets and the fluidized bed particles. It employed a binding solution to trap the initial liquid-solid mixture.
Figure 7. ESE nozzle setup.
To accomplish this, 13 mL of nonbinding n-propanol was injected to stabilize the spray of the nozzle (pressure transducers indicated that the spray stabilized in 0.5 s). It was followed immediately by 12 mL of binding liquid: a 21 wt % aqueous sucrose solution. The sucrose concentration of the binding solution was selected such that the viscosity of the solution was equal to that of the n-propanol, so as to prevent instabilities within the jet during injection. The viscosity of these two liquids is believed to simulate the viscosity of bitumen reasonably at injection temperature; however, no accurate value for the viscosity of bitumen under injection conditions is available.13 The injection assembly used to accomplish the consecutive injection of the n-propanol and sucrose solution from the liquid line is shown in Figure 7. During experimentation, the gas was first introduced through the nozzle by opening ball valve 1. Then a series of ball valves on the liquid line were successively opened. As depicted in Figure 7, the ball valve that allowed for the pressurization of the liquid line (valve 2) was first opened. Then the valve isolating the n-propanol from the sucrose binding agent (valve 3) was opened, followed immediately by the opening of the valve connecting to the nozzle (valve 4), thus initiating liquid flow through the nozzle. After all the liquid had been injected, the valves on the liquid and gas lines entering the nozzle
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5667 Table 1. Mass Balance on Binding Agent
trial free jet ESE
distance of draft tube entrance from nozzle
% sucrose recovery
n/a 1/ in. (12.7 mm) 2 3/ in. (19.1 mm) 4
77.5 98.9 95.1
(valves 1 and 4) were shut off, preventing any flow into the nozzle. The fluidization gas was stopped 10 s afterward: immediate defluidization of the bed ensued, thus preventing any secondary mixing due to fluidization. The bed was then allowed to dry under fixed bed conditions, with gentle aeration for 3 h. During this time strong sucrose-coke bonds formed that solidified the agglomerates created by the primary liquid-solid interaction within the jet. Agglomerates were then recovered from the bed with a 250-µm sieve. These recovered agglomerates were subsequently broken up and 20 4-g samples were taken at random from the fragments. The sucrose concentration of these samples was determined by gravimetric analysis. This was accomplished by washing the samples, twice, with 10 mL of distilled water. The water was then evaporated and the mass of sugar was measured using a balance with an accuracy of 0.1 mg. Following the determination of the amount of sucrose in a sample, the amount of liquid initially contacting these coke particles could be determined by mass balance.
Figure 8. Injection assembly.
Results As previously indicated, a low L/S value indicates that the amount of solids (S) contacted by the liquid (L) in the jet is relatively high; therefore, heat transfer would not limit cracking. Conversely, a high L/S value indicates that the number of solid particles contacted by the liquid in the jet is relatively low; therefore, cracking would be limited by heat transfer and, possibly, by mass transfer. The L/S value will vary throughout the jet, depending on the effectiveness of mixing throughout its cross section. The initial testing focused on confirming the viability of the method. As shown in Table 1, over 95% of the injected sucrose was recovered in both experiments with the draft tube by sieving the bed contents for agglomerates. In the case of a free jet, without an ESE tube, the balance of the sugar was likely lost in the bed, as relatively dry regions of coke in the jet (with a low L/S) would not be in contact with enough liquid to form solid bridges upon drying. To further validate the method, several tests were conducted to confirm the consistency and reproducibility of this technique for given sets of operating conditions, as displayed in Figures 8-10. These figures show that a clear trend can be established in all cases. It should be noted that much of the discrepancy between trials is attributed to the inherent variability of the jet-bed interactions in a fluidized bed. Numerous trials are therefore required to establish an accurate average behavior of the jet. Figures 8-10 already clearly demonstrate that the ESE configuration consistently reduced the spread of the L/S. In addition, a slight decrease is seen in the average L/S value with the ESE configuration. This is illustrated in Table 2, where the average values are listed for each configuration. In this table, the free jet
Figure 9. Cumulative distribution of the liquid-to-solid ratio in a free jet.
Figure 10. Cumulative distribution of the liquid-to-solid ratio in ESE with a distance of 1/2 in. (12.7 mm) between the nozzle exit and draft tube entrance.
is shown to wet fewer particles (corresponding to a higher L/S), while the draft tube (ESE) wets more particles (corresponding to a lower L/S) with less variation in the degree of wetness (as indicated in Figure 11). If the averages presented in Table 2 are indicative of
5668 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
Figure 11. Cumulative distribution of the liquid-to-solid ratio in ESE with a distance of 3/4 in. (19.1 mm) from the nozzle exit to the draft tube entrance.
Figure 12. Cumulative distribution of liquid and solids within the jet.
Table 2. Average Liquid to Solid Ratio (L/S) and the Coefficient of Variation
trial free jet ESE
distance of draft tube entrance from nozzle
L/S within the jet (g/g)
n/a 1/ in. (12.7 mm) 2 3/ in. (19.1 mm) 4
0.073 0.070 0.065
the effect of draft tube distance from the nozzle, it would appear that the effect of the added distance from the nozzle is to allow for more entrainment of coke from the bed into the jet, thus resulting in further reduction in L/S.12 However, more work is required to fully understand the effect of this and other geometric parameters. As is evident from Figure 11, in the free jet case some solids remain relatively dry, while the remaining coke contacted by liquid in the jet is much wetter. The very low L/S values observed in the case of the free jet demonstrate that some very dry regions exist within the jet. This is supported by the relatively low sucrose recovery in the free jet experiments, as shown in Table 1. This low recovery can be attributed to dry regions of the jet that, upon drying of the bed, do not result in sustainable agglomerate formation, and as a result, the relatively dry solids (low L/S) pass through the sieve as individual particles. In addition, dry boundaries may exist in the agglomerates formed in a free jet. Large agglomerates likely split at these boundaries and in the process lose many relatively dry particles that cannot be detected upon sieving. The uniformity of contact within the jet is apparent in both ESE cases, as the L/S ratio has a very narrow range of values. This corresponds to the elimination of both very dry regions and very wet regions in the jet and supports the near complete recovery of sucrose shown in Table 1. More importantly, in both draft tube/ free jet and draft tube cases the L/S ratio is never shown to approach the 1 g/g critical level that Gray et al.2 identified as necessary for mass transfer limitations in a film of liquid. However, observed coke yields in fluid cokers are well above the yields that would occur in the absence of mass or heat transfer limitations. This suggests that the agglomerates formed in a free jet survive for a sufficiently long time to be a major source of heat and mass transfer limitations. What appears to be important, then, is not a critical L/S ratio based on a film thickness, but rather a critical L/S ratio for
Figure 13. Effect of a critical L/S value on the coke yield of the average 3/4-in. ESE and free jet runs.
sustainable agglomerate formation. In addition to surface tension, the stability is related to the L/S ratio. At low L/S values, it is likely that the agglomerate will break apart. Figures 2-5 were all generated under the unrealistic assumption that all liquid contact results in the formation of stable agglomerates and thus the heat transfer model applies to the entire distribution of liquid injected. However, the agglomerates that form at low L/S ratios break up, as they are below a critical L/S for stable agglomerate formation in the fluid coker. Therefore, the model does not apply to liquid that reacts with an L/S below this critical value. A more accurate comparison of ESE and the free jet can be made by considering a critical L/S value for stable agglomerate formation. Figures 13 and 14 show the coke yield and liquid yield predicted by the model for a series of potential critical L/S values. The average distribution of the 3/4-in. ESE and the free jet L/S values were used to generate these figures. On the basis of this figure and preliminary results in a hot pilot plant, it can be postulated that the critical L/S for the formation of stable agglomerates lies somewhere between the point where the ESE curve and the free jet curve diverge (L/S ≈ 0.06) and the maximum L/S observed (L/S ≈ 0.155). From this figure it can be seen that there is potentially a difference of more than 2 wt % in coke yield and 0.6 wt % liquid between the free jet and ESE runs, purely
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Nomenclature Cp ) heat capacity, J/(kg‚K) m ) mass, kg T ) temperature of agglomerate, K Tbed ) coker temperature, K T* ) dimensionless agglomerate temperature, (Tbed - T)/ (Tbed - Ti) Tref ) reference temperature for reaction, K H ˆ Tref ) enthalpy of formation at reaction temperature, J/kg Subscripts c ) coke b ) bitumen i ) agglomerate state at time zero v ) vapor
Literature Cited Figure 14. Effect of a critical L/S value on the liquid yield of the average 3/4-in. ESE and free jet runs.
as a result of heat transfer limitations. When this effect is considered, in addition to mass transfer limitations, the superiority of ESE is clear. Conclusion The ESE nozzle was shown to greatly improve liquidsolid mixing by increasing the amount of wetted solids and achieving a more uniform primary liquid-solid mix. With the free jet, without ESE, the average liquid to solid mass ratio of the primary mixture was 0.073 and its coefficient of variation was 38%. With ESE, the average liquid to solid mass ratio dropped to 0.068 and the coefficient of variation was only 13%. The use of the ESE nozzle will therefore help prevent agglomeration and, therefore, heat and mass transfer limitations in fluid cokers. A model that accounts for heat transfer limitations within agglomerates formed upon contact of the spray with the bed particles shows that ESE can substantially reduce the coke yield (2 wt % decrease) and increase liquid yields (0.6 wt % increase) by reducing agglomerate formation. Acknowledgment Syncrude Canada Ltd. and the Natural Sciences and Research Council of Canada funded this work. Special thanks must be given to Souheil Afara and Mike Gaylard for the use of laboratory facilities. Also, Siva Ariyapadi, Vittorio Felli, Lenin Mejia, Jeff Wood, and David Zhou provided valuable assistance in the lab and in the reviewing process.
(1) Gray, M. Upgrading of Petroleum Residues and Heavy Oils; M. Dekker: New York, 1994. (2) Gray, M.; Le, T.; McCaffrey, W. C.; Berruti, F.; Soundararjan, S.; Chan, E.; Huq, I.; Thorne, C. Coupling of Mass Transfer and Reaction in Thin Films of an Athabasca Vacuum Residue. Ind. Eng. Chem. Res. 2001, 40, 3317-3324. (3) Chan, E. W.; Base, T. E.; Kennett, R. D.; Emberley, D. A.; Jonasson, K.; McCracken, T. W.; Bennett, A. J. CA Patent 2224615, 1997. (4) Gray, M. Fundamentals of Bitumen Coking Processes Analogous to Granulations: A Critical Review. Can. J. Chem. Eng. 2002, 80, 393-401. (5) Chan, E. Syncrude Canada Ltd., Edmonton, Alberta, Canada. Personal communication, 2003. (6) Leclere, K.; Briens, C.; Gauthier, J.; Guigon, P.; Bergougnou, M. Chem. Eng. Prog., in press. (7) Ariyapadi, S.; Holdsworth, D.; Norley, C.; Berruti, F.; Briens, C. Int. J. Chem. Reactor Eng.Chem. React. Eng., manuscript submitted. (8) Gray, M.; McCaffrey, M. C.; Huq, I.; Le, T. Kinetics of Cracking and Devolatilization During Coking of Athabasca Residues. Ind. Eng. Chem. Res., in press. (9) Michaelian, K. H.; Hall, R. H.; Bulmer, J. T. Photoacoustic Infrared Spectroscopy and Thermophysical Properties of Syncrude Cokes. J. Thermal Anal. Calorim. 2002, 69, 135-147. (10) Gabor, J. D.; Botterill, J. S. M. In Handbook of Heat Transfer Applications, 2nd ed.; Rohsenow, W. M., et al., Eds.; McGraw-Hill: New York, 1985; Chapter 6. (11) Martin, H.; Saberian, M. Improved asymptotic approximations for transient conduction and diffusion processes. Chem. Eng. Process. 1994, 33, 205-210. (12) Hulet, C.; Briens, C.; Berruti, F.; Ariyapadi, S. Int. J. Chem. React. Eng., manuscript submitted. (13) Aminu, O. M.; Elliott, J. A. W.; McCaffrey, W. C.; Gray, M. R. Ind. Eng. Chem. Res., in press.
Received for review November 6, 2003 Revised manuscript received April 4, 2004 Accepted April 7, 2004 IE034237Q
