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Thermal blending time associated with a charge of hot particles added to a fluidized bed of uniform temperature.

The process of heat transfer between particles in a fluidized bed is important for many industrial fluidized bed processes. The problem associated with studying this phenomenon is the confounding effect of particle mixing on heat transfer. The work described here was undertaken to describe the process in which heat is added to a fluid bed process by adding a hot charge of particles to a colder fluidized bed. The rate of heat transfer in this instance can have a significant impact on performance of the fluid bed process, depending upon its application. Both the method of analysis and the results of the work are applicable to other fluidized bed processes, particularly those associated with the thermal upgrading of heavy oil.

The method of data analysis, based on binomial statistics, allowed useful data to be extracted from a complex system without the need for a large number of experiments. The analysis also allowed for some assessment of the relative importance of mixing and heat transfer, which has not been possible with other approaches. The results of the experiments were further explored using a bubbling bed model that incorporated both heat transfer and solids mixing. This allowed for the formation of a conceptual model, validated by the experimentation, that explains the relative functions of the two transfer processes in the dispersion of heat from a hot charge of particles to the bulk of a fluidized bed.

Le procede de transfert de chaleur entre les particules dans un lit fluidise joue un role important dans de nombreux procedes industriels en lit fluidise. Le probleme associe a l'etude de ce phenomene est l'effet de confusion du melange des particules sur le transfert de chaleur. Le travail decrit ici a ete entrepris pour decrire le procede dans lequel un lit fluidise est chauffe en ajoutant une charge chaude de particules a lit fluidise plus froid. Le taux de transfert de chaleur dans cet exemple peut avoir un impact significatif sur la performance du procede en lit fluidise, selon son application. Autant la methode d'analyse que les resultats du travail sont applicables a d'autres procedes de lits fluidises, en particulier ceux associes a la valorisation thermique de l'huile lourde.

La methode d'analyse des donnees, basee sur des statistiques binomiales, permet d'extraire des resultats utiles d'un systeme complexe sans avoir besoin de beaucoup d'experiences. L'analyse permet egalement de jauger l'importance relative du melange et du transfert de chaleur, ce qui n'a pas ete possible avec d'autres approches. Les resultats des experiences ont ete analyses de maniere plus approfondie au moyen d'un modele a lit bouillonnant qui incorpore a la fois le transfert de chaleur et le melange de solides. Ceci permet l'etablissement d'un modele conceptuel valide, qui explique les fonctions relatives des deux procedes de transfert dans la dispersion de la chaleur a partir d'une charge chaude de particules dans le coeur d'un lit fluidise.

Keywords: heat transfer, fluidization, petroleum coking, solids mixing

INTRODUCTION

The process of heat transfer within a fluidized bed is important for many aspects of the fluidized bed thermal upgrading of petroleum. In most fluidized upgrading processes the heat required by the process is provided by the fluidized solids. The feed to the reactor is sprayed into the bed of hot solids where it coats the particles and reacts. Recent work has suggested that the coating mechanism is not necessarily one where a droplet of feed engulfs a single particle but rather larger agglomerates are initially formed, which then breakup under the influence of bed dynamics (Gray, 2002). It is often assumed that the heat transfer from solids to the feed is very fast; however, if the length scale of the feed/solid agglomerate is increased then heat transfer may be limiting until the point where the agglomerate is broken up by mixing in the bed. This effect may impact both the reaction time and the selectivity of the process. In addition to the heat up of feed, industrial upgrading process such as fluid-coking and flexi-coking rely on large solids temperature drops in the coking reactors and supporting equipment. These temperature drops involve the mixing of hot solid streams into a relatively cold fluidized bed. The mechanism of particle-particle heat transfer plays an important role in the design of these vessels and the solids injection systems.

Building on recent advances in the understanding of coking chemistry and technology, Envision Technologies Corp. has developed a novel thermal upgrading process based on a cross-flow fluidized bed. In the cross-flow design, the fluidized solids are advected through the reactor in a direction perpendicular to the fluidizing medium. This results in both a narrow solids and a short gas phase residence time distribution. These attributes lead to significant gains in liquid yield through the virtual elimination of product losses from short circuiting and a reduction in both the gas and liquid phase severities. In addition, the increased control over the solids residence time distribution allows for efficient operation at lower temperatures, which in itself results in improved liquid yields. Advantages of the design were characterized based on gas and solid tracer studies performed in a scaled cold flow model of the commercial unit, and are reported elsewhere (Envision Technologies, 2003).

The process of particle-bed heat transfer is important to the Envision Technologies Upgrader (ETX Upgrader) for the same reasons as in fluid and flexi-coking. In addition, one aspect of the ETX design calls for the relative flux of solids to feed to be much larger than in other fluidized bed coking processes. This increased solids flux allows the reactor to operate at lower temperatures without fear of defluidization (bogging) of the bed due to agglomeration, which is a common concern with this type of process (Briens et al., 2003). To reduce the absolute solids flux requirements of the process, multiple coking units can be operated in series. In this incarnation of the design, the solids are heated back up between reactors units. One efficient method for doing this is to add the heat to the solids stream using a second hot stream of solids. Minimizing the mass of hot solids added by maximizing their temperature is advantageous. Doing this requires knowledge of the time required for the solids to come to thermal equilibrium. The time required for this to occur will also dictate where the hot solids can be added to the reactor system. Understanding the mechanism of heat transfer between the hot and cool particles is critical to designing this system.

Many aspects of heat transfer in fluidized beds have been well documented in literature. Primarily, the heat transfer between the fluidizing medium and the bulk bed (Kunii and Levenspiel, 1991), and the heat transfer between the bulk bed and fixed surfaces (Molerus et al., 1994; Sunderesan and Clark, 1995) have received considerable attention. The heat transfer between individual particles and the bulk bed is not as well documented (Parmar and Hayhurst, 2002). In one extreme, the heat transfer between individual particles may be comparable to the heat transfer between particles and fixed surfaces within the fluidized bed. However, particles are not subjected to the same environment as fixed surfaces: unlike the particles, fixed surfaces are washed by the different pseudo-phases within the bed (bubble, emulsion), which all have different heat transfer properties, and fixed surfaces do not circulate with the particles, which could have a large effect on the convective element of heat transfer. Turton and Levenspiel (1988) attempted to study the heat transfer between a charge of particles and the bulk bed using a change in the magnetic properties of the particles. Their work showed the confounding effects of particle mixing and local heat transfer on the temperature of the tracer particles. While this made it difficult for them to deduce the true particle-bed heat transfer coefficient, it did show the importance of micro-mixing on the process of heat transfer to a particle charge added to a fluidized bed, which is the type of system under investigation here. In Turton's experiments, the time for the particle charge to mix into the bed was approximately 4 s, which was slightly longer than the time it took the particle charge to come to thermal equilibrium.

Many researchers have also attempted to study particle-bed heat transfer using the very direct approach of embedding small probes with fine thermocouples (Linjewile et al., 1993; Parmar and Hayhurst, 2002). This technique is limited to particles of a minimum size of 2-3 mm because the thermocouple must be inserted into the probe. In addition, the probe may behave differently than the bulk particles in the bed either due to its differing properties necessitated by the technique, or due to the thermocouple and wires. In general, heat transfer coefficients measured using thermocouple imbedded particles vary between 200 and 600 W/[m.sup.2]-K. Heat transfer coefficients have been shown to increase with increasing fluidization velocity and decreasing particle size, which seems to agree with both boundary layer and surface renewal theories of heat transfer.

The degree of dispersion of a charge of hot particles introduced into the fluidized bed determines the characteristic length of the packet. This dimension is critical in determining the time required to reach thermal equilibrium, and is dictated by the micro-mixing behaviour of the fluidized bed. As was observed by Turton and Levenspiel (1988), the two phenomena of heat transfer and micro-mixing within the fluidized bed are both coupled, and act on similar time scales, making their effects difficult to determine independently.

The work described here investigated the process of heat transfer from a hot charge particles added to a colder fluidized bed. The primary goal of this work was to establish a thermal blending time for use in the design of the ETX Upgrader; however, insight into this process has additional application in other fluidized bed processes, specifically those related to the upgrading of heavy oil.

MATERIALS AND METHODS

The fluidized bed used in this study had a square footprint, with linear base dimension of 0.75 m. Including the freeboard region, the bed was 2 m in height. The fluidizing medium was air, distributed by means of bubble caps. Glass particles were used (P008, Potters Beads) having a size range of 150-210 [micro]m, a sphericity of 0.9, and a specific gravity of 2.5. The system was capable of accommodating fluidized bed heights up to 1 m, with superficial velocities up to 0.5 m/s. For this study, the fluidized bed height was kept constant at 0.4 m. Under these conditions, the minimum fluidization velocity was measured at 0.03 m/s.

A volume of particles of 4 L, representing 2% of the total bed volume, was heated in an oven to a temperature of approximately 80[degrees]C. The device used to add the hot solids to the bed was made of a 5 cm tube (Figure 1). The hot particles were held in the tube by plugs connected by a threaded rod. The entire assembly was inserted into the fluidized bed through the roof. The particles were released by holding the threaded rod and plugs stationary and quickly pulling the tube upward; this released a column of hot particles into the bed. The plugs were spaced so that the column of hot particles spanned the height of the fluidized bed. The temperature of the charge of solids was measured with a thermocouple integrated into the sample delivery tube. This value was used in the subsequent analysis, for scaling purposes. Following release, the hot particles were allowed to mix in the fluidized bed for a predetermined amount of time, at which point the fluidization gas was turned off. Each experiment was captured on video so that the exact duration of the mixing process could be determined. A grid of eight probes was then introduced into the bed from the upper surface. A thermocouple was placed at the tip of each of the eight probes. The probes were spaced as shown in Figure 1, and reached the grid plate on full travel. As can be seen from the figure, the probe grid covered only one quadrant of the bed; this was done to decrease the distance between probes so that some information on the size of hot spots in the bed could be inferred. The location of the charge injection point (Figure 1) along the wall of the bed meant that temperature readings taken with the thermocouple grid were not representative of the entire bed. Locating the charge injection point in the centre would have alleviated this, but was not possible due to the physical design of the bed.

[FIGURE 1 OMITTED]

The grid of probes was inserted and retracted from the bed at a constant rate of approximately 1 cm/s using a linear traverse. Data from the thermocouples were captured at an interval of 0.01 s, using a data acquisition system (National Instruments PCI-6036E coupled with an SCXI-1000 multiplexer and 1303 thermocouple terminal block module), and compatible software (VI Logger, National Instruments). For some experiments the thermocouple grid was inserted ("pushed") and retracted ("pulled") vertically through the bed multiple times. Data were logged during these cycles. This sequence was performed in order to assess the degree to which the movement of the probes disturbed the bed solids.

The experimental data were first treated to remove spikes introduced as artifacts of the measuring equipment. The data associated with each trace were then processed using a recursive exponential smoothing algorithm, with a smoothing parameter of 0.9. Statistics associated with each set of eight experiments, one associated with each thermocouple, were extracted using an algorithm written in Visual Basic for Applications (VBA). Statistics associated with all experiments at a given mixing time were then combined using a different algorithm, again coded in VBA.

Experiments with gas tracers were used to validate the fluidized bed mixing model. Propylene was used as the gas tracer for all such experiments. Tracer gas was introduced directly into eight of the 16 bubble caps in the distributor of the experimental bed. A photo ionization detector (PID) was used to quantify the concentration of propylene both above the surface of the bed and within the bed (MiniPID, Aurora Scientific, Aurora, ON, Canada). Withdrawal of gas from the bed into the PID was accomplished by means of a tube 0.99 m in length and 0.794 mm in diameter (i.d.). Gas was drawn into the PID at a rate of approximately 1 L/min, which was less than 0.02% of the gas flow through the bed. Negative step change experiments were performed in which the flow of tracer gas was turned on until the concentration of tracer in the bed reached steady state. The flow of tracer gas was then stopped and PID measurements were recorded with time until no change in the output signal was detected. The raw data from the PID was first scaled and normalized linearly between the maximum and minimum recorded output values. To reduce the noise in the data, the normalized data from three experimental runs were combined, then filtered using a centred 100 point moving average.

RESULTS

Conceptual Model

The thermocouple readings associated with each rod represent a discrete sampling of the bed. To interpret the data generated by these experiments, the following assumptions are applied:

* Isotropic conditions within the bed;

* Constant fluid and particle properties;

* Heat losses to the gas phase are assumed to be negligible over the period of experimentation. The validity of this assumption will be addressed later on.

With these assumptions, the amount of heat contained in the bed is constant, and over any time interval:

Q(t) - Q(0) = [H.sub.b] (t0 - [H.sub.b,0] = [H.sub.c], (1)

Taking the initial bed temperature as the reference:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If the temperature of the charge is uniform, then the equation reduces to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

To compare results obtained under different conditions, it is convenient to work with non-dimensional variables. The natural upper scaling temperature is the hottest value measured in the charge of heated particles introduced into the fluidized bed. This temperature is associated with the initial temperature of the charge. The lowest temperature that is encountered in any experiment is the starting temperature of the fluidized bed. Therefore the dimensionless temperature is defined as:

[eta] = [T.sub.b] (x,y,z,t) - [T.sub.b,0]/[t.sub.c -[T.sub.b,0] (5)

The initial bed temperature is uniform, and is therefore not a function of position. With the assumption of adiabatic conditions, the temperature of the particles in the bed will converge to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, [eta] will not converge to zero, but rather to:

[[eta].sub.[infinity]] = [T.sub.b,[infinity] - [T.sub.b,0]/[T.sub.c] - [T.sub.b,0] (7)

In all experiments the volume of the charge of the hot particles was small relative to the volume of particles in the bed, on the order of 2% in all cases, so the simplifying assumption [T.sub.b,[infinity] [congruent to] [T.sub.b,0] can be made. As a result the dimensionless temperature [eta] shows values between zero and unity.

With this definition, assuming a uniform temperature in the charge of particles, Equation (4) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [V.sup.*]= V/[V.sub.C].

A means of extracting the value of the integral in Equation (8) from the experimental data collected is required. Depending upon the degree of mixing, the hot particles will remain somewhat associated, moving around in packets. The characteristic dimension of each packet is dependent upon the degree to which the original charge of particles is broken up due to the internal dynamics of the bed. As heat is transferred, surfaces of constant temperature can be identified within the bed. These isotherms can be viewed as continuous "shells." Each shell has its own characteristic length, depending upon its temperature. This description follows naturally once isotropic conditions are assumed within the bed.

Each time the bed is probed with a rod, the probability of encountering a shell of particular temperature is related to the relative volume encompassed by the shell. The definition of "hot" is somewhat arbitrary, being constrained only by the requirement that [eta] > 0. The probability of encountering particles hotter than a particular temperature for each traverse of a single thermocouple is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [V.sub.[eta],i] is the volume engulfed by a particular shell, and [N.sub.[eta]] is the number of shells. From binomial statistics, the probability of encountering a shell of temperature [eta] within the bed exactly r times in q experiments is given by the equation:

p (q,r,[eta] = q!/r! 9q - r)! [[pi].sup.r] [[1 - [pi].sup.q-r] (10)

where r [less than or equal to]q and [r=q.summation over (1)] p(q,r,[eta]) =1. If many groups of q experiments were performed, then the probability density function associated with the results would be described by Equation (10). Knowledge of this function alone does not help with the analysis. However, the expected (most probable) value associated with distribution does, since it provides an estimate of [pi] ([eta]), which is related to the experimental parameters through Equation (9). The binomial mean and standard deviation are given, respectively, by [mu] = q[pi] and [sigma] = [square root of (q[pi](1 - [pi])]. Thus, the expected value of r is E(r)=[mu], and E([pi])=r/q.

To take advantage of this statistic, each trace of the bed can be viewed as a discrete event, with a probability of encountering a particular temperature given by Equation (9). Since a temperature trace of the bed is continuous, each represents the compilation of an infinite number of experiments in which a particular temperature is sought. An estimate of the fraction of the bed encompassed by shells of a given threshold temperature can be estimated from the experimental data as [pi]([eta])[approximately equal to] r/q.

Experimental

To assess the impact of the probe movements and the transient heat transfer effects in the slumped bed of solids, data gathered from the insertion ("push") and retraction ("pull") events described in the MATERIALS AND METHODS were compared. Two series of this type of experiment were performed. In the first, five push/pull cycles were carried out, with a waiting period of 1 min between each. These results were compared to data gathered from two push/pull events carried out 5 min apart. Based on the data (not shown), it was concluded that the temperature profiles are most affected by the movement of the probes, rather than heat transfer in the slumped bed. Data gathered from the initial insertion (push) of the probes was used for all analyses. From these control experiments, it is estimated that the temperatures measured by the probes on the first push are below the actual bed temperatures by about 10% (in [degrees]C).

Based on the conceptual model developed above, the [pi] statistic was estimated from the experimental data (Figure 2). An initial temperature of [[eta].sub.0]=0 is associated with all experiments. As discussed above, since the charge of hot particles is relatively small, [[eta].sub.[infinity]] [congruent to] 0. As a result, particles at this temperature will be encountered in every experiment, and [pi](0)=1. In keeping with the conceptual model developed above, the shell at temperature [eta] = 0 engulfs a volume equal to the total volume of the bed, and all hotter shells are contained within it. From this analysis, all data in Figure 2 must converge to unity, and were forced through this value.

[FIGURE 2 OMITTED]

With the isotropic assumption, the characteristic temperature distribution should be independent of direction. Assuming that the characteristic distribution is known, it follows that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Therefore, if the characteristic temperature distribution is known in one dimension, it can be used to estimate the heat content in the whole bed.

For a complete set of q experiments associated with a particular mixing time, the best estimate of the average temperature in Equation (11) is given by the summation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

In the current analysis it is useful to consider the amount of energy contained in the bed that is associated with temperatures greater than a specific threshold [[eta].sub.k], where 0[less than or equal to] [[eta].sub.k] [less than or equal to] 1. For this purpose it is convenient to define:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

With this definition, the energy in the bed associated with a threshold temperature of [[eta].sub.k] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The meaning of this integral is shown in Figure 3. The best estimate of this function can be extracted from the experimental data as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[FIGURE 3 OMITTED]

This equation shows that we need only consider the subset of [r.sub.k] experiments for which [[lambda].sub.0] [not equal to] 0. Events in which hot shells of temperature [[eta].sub.k] were not encountered are taken into account through inclusion of the probability function [[pi].sub.k]. This derivation gives meaning to [pi] within the context of the average properties of the bed: it is the mathematical weighting factor, linking experimental data at a particular threshold temperature to the average bed properties. With this meaning assigned to [pi], the characteristic temperature distribution can be determined by considering only the traces in which the hot shells were encountered.

To study the impact of mixing time it is convenient to consider the incremental contribution of increasing temperatures on the energy contained in the bed. Below dimensionless temperatures of about 0.05, the threshold values approach the same magnitude as the noise. Therefore, a reference threshold of [eta] = 0.05 was used rather than [eta] = 0. The fraction of energy above [eta] = 0.05 attributed to temperatures below [[eta].sub.k] is given by:

[F.sub.k] = [DELTA][H.sub.1] - [k.summation over (i=1)] [DELTA][H.sub.1]/ [DELTA][H.sub.1] (16)

where [eta] =0.05. This function is plotted against the dimensionless temperature in Figure 4. The desired mixing time is the one that yields values of [F.sub.k] close to unity at low [[eta].sub.k] values. In all cases, at constant [[eta].sub.k], the values of [F.sub.k] are smaller for shorter mixing times.

[FIGURE 4 OMITTED]

Other features of the data are useful for elucidating the impact of mixing time. Figure 5 shows a plot of [[eta].sub.max] as a function of mixing time, where [[eta].sub.max] is the maximum temperature recorded at a particular mixing time. The data followed the expected trend, with greater temperatures associated with shorter mixing times.

[FIGURE 5 OMITTED]

The shape of the peaks associated with the various mixing times was captured by measuring the total width of all peaks at a certain threshold temperature, and then dividing by the number of peaks found at that threshold (Figure 6). A similar statistic was formulated with respect to peak height. Here, the maximum height above the threshold was determined at each threshold level. These values were totalled and then divided by the number of peaks (Figure 7).

[FIGURES 6-7 OMITTED]

To assess the impact of fluidization velocity on heat transfer, experiments were performed at a superficial velocity of 0.34 m/s (Figure 8). The data from replicate experiments were not pooled, to demonstrate the repeatability of the technique.

[FIGURE 8 OMITTED]

DISCUSSION

Many aspects of the heat transfer characteristics associated with fluidized beds have been studied. Researchers have attempted to look at the particle/fluid heat transfer characteristics associated with individual particles when immersed into a fluidized bed (Linjewile et al., 1993; Turton et al., 1989). In addition, heat transfer from fluidized particles to fixed surfaces has also been addressed (Molerus et al., 1994; Sunderesan and Clark, 1995). The difficulty with any of these studies is interpreting the raw data within the context of the process of interest. The transfer of heat is coupled intimately with the transfer of mass and momentum. Due to complexity of the fluid bed operation, it is extremely difficult or impossible to decouple the various effects experimentally. As a result, one is forced to introduce models describing the various field variables in order to extract the necessary heat transfer data (Turton and Levenspiel, 1988). For example, many experiments have been carried out in which the relevant heat transfer parameters are followed at the inlet and outlet boundaries of the fluid bed (Kunii and Levenspiel, 1991). A heat transfer coefficient characteristic of the entire bed is then calculated. The value of this coefficient is dependent upon the flow patterns assumed for the gas and solid phases. Very different results are obtained depending upon the degree of mixing assumed as the gas traverses the bed. Both plug flow and well mixed models have been assumed, as well as more complex models.

The whole bed approach to heat transfer has further limitations within the current context. Local heat transfer characteristics are required when considering the transfer of heat to a charge of particles introduced into the bed. Although heat transfer studies have been carried out in which a single particle in a fluid bed was considered, this situation does not adequately represent the case in which a charge of particles is introduced into the bed. In the former, the local thermal characteristics of the bed are dominated by the bulk of the solids, which are at a uniform temperature. When a significant fraction of hot particles is added, depending upon the forces acting on the charge, the local environment may change appreciably.

Studies have been carried out characterizing the heat transfer from a fluidized bed of solids to an immersed surface (Molerus et al., 1994; Sunderesan and Clark, 1995). In these situations the solid object of interest remains stationary within the fluidized bed. In contrast, when a charge of solid particles is added to the bed, the relative motion between the particles depends upon the local dynamics that exist within the bed. Depending upon the degree of mixing within the bed, the hot particles may move in large "packets," while in other cases, the shearing action of the bed my cause the solids to quickly disperse.

In the current study, a relatively small volume of hot particles was added to the fluidized bed. Therefore, many of the linear traces of the bed did not encounter regions of hot particles. It would be reasonable to generate "average" one-dimensional temperature distributions by combining the data generated in all experiments using an expression such as Equation (12). However, since the fraction of hot particles that could be reasonably added was small, the information contained in the traces in which hot regions were encountered would be lost in the background traces. The advantage of analyzing the data in the manner presented here is that the data associated with traces in which hot regions of the bed were encountered was not lost by averaging with events in which no hot regions were encountered. This allows statistics considering only the subset of data in which peaks were found to be formulated, as in Figures 6 and 7. Information pertaining to the experiments in which only background particles were encountered was captured in the probability function [pi]. The approach adopted in this study is obviously not without limitations. The temperature profiles measured will be somewhat distorted compared to those that exist when the bed is fully fluidized, since some of the three-dimensional structure is lost when the bed is slumped and the void fractions are expelled.

A qualitative examination of the temperature traces associated with hot regions of the bed yields some interesting information (Figure 3). As expected the heights of the peaks in temperature are larger at shorter mixing times. In addition the baseline temperatures seem to be increasing with mixing time, as more heat is transferred to the bulk of the bed particles.

It would be expected that the average height of the peaks would decrease with mixing time. Indeed, this was found to be the case (Figure 7). At all temperatures, the average peak height decreased with increasing mixing times. However, comparing the mixing times, the shapes of the curves are quite different. At the shortest mixing time, the curve is relatively flat for temperatures below [[eta].sub.k] [congruent to] 0.7. At higher mixing times, the curves exhibit a maximum at intermediate temperatures, trending towards zero at lower temperatures. This behaviour can be explained by considering the source of the statistic. A single peak is defined at all threshold temperatures below its maximum. Thus, at the shortest mixing time there are only a few hot peaks, and no cold peaks, resulting in a flat profile. As the mixing time is increased, more numerous, colder peaks are introduced as a result of heat transfer and breakup of the particle packets. As a result, the average peak height decreases with decreasing temperature.

From Figure 2, it is apparent that the volume of the bed occupied by shells of hotter particles decreases with temperature. This is to be expected, and is consistent with the analysis above. However, the [pi] statistic puts no constraints on the volume engulfed by an individual shell of a particular temperature, as only the total volume of all shells is considered. The average width of all peaks encountered at a particular threshold should be indicative of the average shell size at that temperature. These data are plotted in Figure 6. At a fixed temperature, the average peak width decreases, indicating that the shell size also decreases. This finding provides some insight as to the relative impact of packet breakup. If no breakup were induced due to the mixing of the fluidized bed, then shells of lower temperature would be expected to grow in size as heat diffused from the hot packets. Since the opposite trend is observed, it suggests that packet breakup plays a significant role in distributing the heat.

Based on its definition, the [pi] statistic should be related to the average peak widths extracted from the data. Attempts made to reconcile these two quantities were met with a moderate degree of success (data not shown). A number of factors introduce enough uncertainty into the calculation to make it of limited value. First, the number of experiments performed is quite low (6-8 experiments at each mixing time), introducing significant error. This is particularly true for the short mixing time experiments, in which most of the experiments encountered only background particles. It was estimated that approximately 100 experiments would need to be performed in order to decrease the error to reasonable level (data not shown). Second, geometry needs to be chosen through which to relate the peak widths to a shell volume. The spherical shape selected in the conceptual development was introduced simply out of convenience. Furthermore, the peak width as measured is a stochastic variable, and is related to the true shape through an unknown probability density function. Even with this drawback, the conceptual model introduced seems to provide a useful means by which to analyze the problem.

The primary motivation for this study was to determine the amount of time required for the local temperature in a fluid bed to dissipate below a critical value following the introduction of a charge of hot particles. The answer to this question is most readily seen in Figure 4. After approximately 7 s, the maximum temperature in the bed is less than 10% above the dimensionless background temperature. This number is conservative, as the fluidization velocities associated with this figure are considerably less than those targeted in the commercial process.

The manner in which the experiments were carried out made it difficult to assess the amount of heat lost to the gas phase. This is an important concern, since the intent of adding the hot particles is to heat up the bulk bed solids. Practical constraints limited the charge of particles added to an amount representing 1-2% of the total amount of solids in the bed. As a result, the background temperature of the bed did not change appreciably as heat was transferred. Although no change in the exit gas temperature was measured, this result was inconclusive, again since the amount of heat added with the hot particles was small relative to the flux of gas.

To arrive at a conclusion regarding the loss of heat to the gas phase, it is necessary to develop a mechanism by which heat is transferred between hot and cold particles. The heat transfer characteristics are influenced by the local fluid dynamics that exist within the bed. Previous work has demonstrated the ability of published models to capture the gas and solids phase mixing dynamics in bubbling fluidized beds (Kunii and Levenspiel, 1991; Sane et al., 1996). The approach of Sane was used to model the gas mixing in the bed. The same approach was also applied to the solids using the conceptual model of solids mixing from Kunii and Levenspiel. Gas tracer studies were used to validate the mixing model. It was found that the gas phase mixing model described both the internal concentration decay (data not shown) and the exit concentration decay in the experimental bed very well (Figure 9). It should be noted that the model was not fit to the data, but that all of the model inputs were calculated based on common correlations from the literature. The gas phase mixing model was then adapted to include heat transfer between the gas and solid phases in the bed, also based on the work of Kunii and Levenspiel (1991).

[FIGURE 9 OMITTED]

To examine the impact of bed mixing on the distribution of hot particles, the heat transfer coefficients in the model were set to zero, preventing heat transfer between gas and solids from taking place. As a result, solids were permitted to disperse, but could not transfer heat. The temperature of the gas was also not permitted to change due to these constraints. The solids were introduced to the top of the bed, to emulate the experiments. From the results of this exercise it is apparent that some time is required to mix the hot particles in with the bulk of the bed particles, and mixing is not instantaneous (Figure 10).

[FIGURE 10 OMITTED]

To examine the impact of heat transfer, correlations for particle/fluid heat transfer from individual particles were included in the model. These correlations have been developed for fluid bed applications. All boundary and initial conditions were kept the same. The results were almost identical to the transient temperature profiles generated without convective heat transfer (Figures 10 and 11). This finding may be misinterpreted as meaning that the heat transfer process is very slow. In fact, the opposite is true, since local gas temperatures quickly achieve equilibrium with the solids (Figure 11).

[FIGURE 11 OMITTED]

These findings can be rationalized by considering the relative heat storage capacities of the two phases. While the heat capacities of the solid and gas are on the same order, the density of the gas is three orders of magnitude smaller. As a result, an insignificant fraction of heat has to be exchanged locally to achieve equilibrium conditions between the two phases. The gas and solid phases move relative to one another due to the dynamics of the bed. As a result, there is a mechanism to relocate heated gas to regions of cool solids. Again, since the energy content contained within the gas is small, this process has little impact on the local temperature of the solids and the equilibrium temperature achieved is close to the conditions before the heat transfer process occurred.

From the above discussion, the gas is not effective in transporting energy over long ranges. However, since the heat transfer process is fast, the gas does serve as a very effective heat transfer medium through which the solids can transfer heat locally. As a result, the optimum situation for heat transfer is one in which a hot solid particle is surrounded by cool particles. The gas serves to efficiently transfer heat from the hot to cold particles, permitting redistribution of the energy to occur. Unfortunately, the relative slip between solids is not large, and the hot charge of particles tends to move around the emulsion phase in parcels. As discussed above, there is no effective mechanism to get the heat out from the hot particles within the parcel, and effective heat transfer is confined to the outer boundary of the parcel. Heat transfer is therefore limited by the ability of the fluid bed to erode these parcels into smaller entities; increasing the overall area of the hot charge of particles exposed to the bulk solids within the bed. To improve the performance of the heat transfer process, efforts should be directed to ensuring adequate distribution of the hot solids upon introduction into the fluidized bed.

The model predicts similar trends to the actual data, but tends to under predict the mixing times required to achieve complete heat transfer. This is expected, since the model assumes a uniform distribution of bubbles within the bed whereas, in reality, dead zones exist in which there is poor mixing.

The mechanism for heat transfer elucidated here is not all negative, and has some positive implications on the process. Since the heat pickup by the gas is small, the amount of energy lost to the gas phase is not significant. For instance, from the simulation associated with Figure 11, only 5% of the energy added to the fluid bed in the form of hot particles exits the bed with the gas phase in the 10 s following particle addition. By this time, the heat transfer process is complete. This finding suggests that there is no need to add the hot particles into piping upstream of the fluid bed; they can be added directly to the fluid bed with an insignificant penalty in the form of heat losses.

Extrapolating from the conceptual model developed above, increases in the superficial gas velocity should assist heat transfer through increased mixing within the bed. The experimental results support this postulate, as the fraction of the bed hotter than [eta] = 0.09 decreases from approximately 30% to 10% after 3 s of mixing, when the fluidization velocity is doubled (Figure 8). The data at the higher gas velocity also serve to demonstrate the repeatability of the technique.

A major focus of the current work was on developing a methodology through which the required time scales of mixing could be extracted from the physical system. The procedure developed generates a non-dimensional representation of the thermal uniformity of the bed at any instant in time. This methodology itself is general and is independent of the temperature or volume of the hot charge of particles, and of the fluidizing parameters.

The time scale of the thermal blending process does, however, depend upon all of these parameters. To extract the time required to achieve any degree of thermal uniformity it is noted that a major conclusion of the analysis developed in the current work is that the heat transfer problem is dominated by the mixing characteristics of the bed. Using a mixing model such as the one applied to this work (Figures 10 and 11) the blending time required to attain any given energy dispersion can be estimated, given that energy losses to the gas phase have been shown to be negligible. Of course, any such model would first have to be tuned to a set of experimental data such as those generated in the current study. Once such a model has been established it can then be used to estimate the impact of such variables as fluidization velocities, bed heights, and the relative volume of the charge of hot particles.

CONCLUSIONS

Temperature profiles were measured in a fluidized bed of particles following the release of a charge of hot particles. The data was gathered for the purpose of measuring the time required for the system to reach thermal equilibrium, which is an important parameter in the design of a novel process for the thermal upgrading of heavy oil. The method of data analysis, based on binomial statistics, allowed for the extraction of useful data from a system dominated by background noise without the need for excessive experimentation. The data were compared to a bubbling bed model that described the bed mixing and heat transfer with good results. The model allowed for further insight into the roles of heat transfer and mixing on the thermal dispersion of the energy introduced through the hot particle charge. For the system studied here, the thermal equilibrium time was approximately 7 s. This time was reduced in half when the fluidization velocity was doubled. With the mechanism of the blending phenomenon elucidated in this paper, a simple approach is suggested for determining the degree of thermal uniformity within the bed at an instant in time, for a given set of fluidization properties.

ACKNOWLEDGMENTS

The authors would like to thank the National Research Council Canada's Industrial Research Assistance Program (NRC-IRAP) for providing funding for this project.
NOMENCLATURE

A horizontal area of bed ([m.sup.2])
C heat capacity of bed (J/kg-K)
[F.sub.k] fraction of total energy in bed associated with
 temperatures smaller than [[eta].sub.k] (Equation (16))
H enthalpy (J)
[DELTA]
[H.sub.k] energy in bed associated with temperatures above
 [[eta].sub.k]
H specific enthalpy (J/m3)
N number
p probability density function associated with
 Equation (10)
Q heat (J)
q total number of experiments
r number of experiments in which a given threshold
 was encountered
t time (s)
T temperature (K)
u superficial gas velocity (m/s)
V volume ([m.sup.3])
z distance from surface of bed (m)
Z total bed height (m)

Greek Symbols

[eta] dimensionless temperature defined in Equation (5)
[??] spatially averaged dimensionless temperature
 (Equation (5))
[[??]
.sub.3] average temperature in vertical direction
[[lambda]
.sub.k] dimensionless temperature with shifted origin
 (Equation (13))
[mu] binomial mean
[pi] probability of encountering a temperature [eta]
 in a given experiment (Equation (9))
[pi] density (kg/m3)
[sigma] binomial standard deviation

Subscripts

0 initial
b bed
c charge of hot particles
p particle
[infinity] value after a very long time


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Manuscript received December 22, 2005; revised manuscript received March 14, 2006; accepted for publication March 15, 2006.

W.A. Brown, (1) R. Pinchuk (1 *), M. Pinchuk (1), J. Diep (2), M. E. Weber (3) and D. Kiel (2)

(1.) Envision Technologies Corp., 7550-114 Ave. SE, Calgary, AB, Canada T2C 4T3

(2.) Coanda Research and Development Corp., Burnaby, BC, Canada V5A 3H4

(3.) Department of Chemical Engineering, McGill University, Montreal, QC, Canada H3A 2B2

* Author to whom correspondence may be addressed.

E-mail address: pinchukr@envisiontech.ab.ca
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Author:Brown, W.A.; Pinchuk, R.; Pinchuk, M.; Diep, J.; Weber, M.E.; Kiel, D.
Publication:Canadian Journal of Chemical Engineering
Geographic Code:1CANA
Date:Jun 1, 2006
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