Modeling of the Solids Transportation within an Industrial Rotary Dryer

May 15, 1996 - occupied by the dead zones, modeled very well the industrial RTD curve. .... dead zones, is given by d(ma,i) dt. ) k(ma,i-1. - ma,i)[1 ...
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Ind. Eng. Chem. Res. 1996, 35, 2334-2341

Modeling of the Solids Transportation within an Industrial Rotary Dryer: A Simple Model Carl Duchesne,† Jules Thibault,*,† and Claude Bazin‡ Departments of Chemical Engineering and Mining and Metallurgy, Laval University, Sainte-Foy, Quebec, Canada G1K 7P4

To better understand the underlying phenomena taking place in an industrial rotary dryer and to determine the optimum operating conditions, a simulator in which the solids transportation, the gas flow, and the heat and mass transfer are modeled is currently being developed. This paper describes the use of interactive perfect mixers in series to model the solids transportation within an industrial rotary dryer, on the basis of an experimental residence time distribution curve (RTD). Two simple models are proposed : a series of perfect well-mixed interacting tanks and a modified Cholette-Cloutier model. The first model is not able to account for the nonideal behavior of the solids transportation in the rotary dryer. To account for the characteristic extended tail of the RTD curves observed in industrial dryers, in a second model, the solid phase is divided between an active and a dead zone. This model, with 36 cells and 25% of the volume occupied by the dead zones, modeled very well the industrial RTD curve. In addition, the model produces bed depth and axial velocity profiles that are consistent with those reported in the literature. Introduction In most ore dressing plants, ore is mixed with water to facilitate its transportation and treatment in the grinding and flotation processes. The final product, a mineral concentrate, sold for its metal content, must be dewatered and dried prior to be shipped to smelters. Although pressure filtration is gaining operator’s confidence, the usual vacuum filter followed by a rotary dryer continue to be used in many operations. An experimental program has been initiated to model the transportation of solids and the heat and mass transfer in an industrial rotary dryer to optimize the operating conditions and to assess various control strategies. In rotary dryers, evaporation of water takes place as hot gas is contacted with the wet concentrate. The understanding and description of solids transportation within the rotary dryer is paramount to modeling heat and mass transfer because it determines the time of contact between the hot gas and the solid phase (Mujumdar, 1987). In addition, the amount of material retained in the dryer influences the heat and mass transfer occurring within the rotary cylinder as well as the power required to drive it (Friedman and Marshall, 1949). This paper will primarily report on the simulation of the solids transportation within an industrial rotary dryer. Rotary kilns are also widely used for heating or cooling, reducing, calcining, and sintering in a large number of industries, and it is therefore not surprising to find that the modeling of solids transportation in rotary kilns has spurred the interest of researchers for many decades. The first significant contribution was presented by Friedman and Marshall (1949) where correlations for the residence time distribution were obtained over a wide range of operating conditions. These correlations were derived as a function of kiln geometry (dryer slope) and operating variables (direction and flow rate of gas, nature of the bulk solids, speed * To whom correspondence should be addressed. Tel: (418) 656-2443; Fax: (418) 656-5993; e-mail: [email protected]. † Department of Chemical Engineering. ‡ Department of Mining and Metallurgy.

S0888-5885(95)00625-7 CCC: $12.00

of rotation, solids feed rate). A similar approach was also used by Sai et al. (1990). Later Kelly and O’Donnell (1977) proposed a model where the number of phases in the cascade cycle (falling, bouncing, kiln action, and overload kiln operation) were considered individually for their contribution to the axial movement of solids through the rotating kiln. Mu and Perlmutter (1980) divided the rotary dryer into several subregions consisting of reactor models. The number of stages or subregions, the volume fractions of the mixed flow and plug flow in each stage, the recycle ratio, and the bypass ratio were adjusted to represent experimental RTD curves. Henein et al. (1983) developed a mathematical model to predict the conditions giving rise to different forms of bed motion to simulate the movement of solids through a rotating cylinder. The concept of a force balance equation with the introduction of a gas drag coefficient on the falling solid phase was proposed by Kamke and Wilson (1986). Another model has been proposed where the flow of solid materials through the dryer is treated as consisting of two different streams: the airborne phase produced by internal flights which lift the solids and control its cascade through the gas stream and the dense phase at the bottom of the dryer (Matchett and Sheikh, 1990; Sherritt et al., 1993). Representing the solid phase as a fluid flowing through the rotary dryer has also been considered. In one case, the fluid conservation and movement equations were solved (Ferron and Singh, 1991), and in another case, it has been modeled with dimensional and rheological analysis where an apparent viscosity was defined (Perron and Bui, 1990, 1994). Bui et al. (1995), in a thorough three-dimensional steady-state modeling of a rotary kiln, used three apparent viscosities to represent the granular bed motion. Two viscosities were used to represent the motion of the active top layer and the underlayer in the transverse direction, and one viscosity was used to represent the bed motion in the axial direction. More recently, the stochastic modeling of RTD was reformulated (Fan et al., 1995; Gossen et al., 1995). The complexity of the movement of particles through a rotary dryer makes difficult the derivation of a model © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2335

Figure 1. Schematic representation of the industrial rotary dryer owned by Brunswick Mining and Smelting Inc.

that is valid over a large number of situations. The quantity of solids in a rotary dryer is influenced by several variables (Friedman and Marshall, 1949): the geometry of the dryer itself (diameter, length, number and shape of lifters, slope of the dryer), the operating variables (gas and solids flow rates, speed of rotation), and the characteristics of the solids (nature of bulk solids, moisture content, composition). In addition, most of the studies were performed on small scale rotating cylinders with dry material. Therefore, models mentioned above are usually limited to specific type of rotary dryers and specific range of operating conditions, and as a consequence, it is preferable to have a RTD model, that is easy to understand and visualize, with only a few parameters to determine. In this investigation two simple models, both based on well-known reactors, are used to model the flow of wet solids through an industrial rotary dryer. This paper is divided as follows: A brief description of the industrial dryer and the experimental procedure are firstly presented, followed by the derivation of two models for ore transportation. In the last section, the two models are analyzed and a sensitivity analysis of the second model is performed. Equipment and Experimental Procedure Brunswick Mining and Smelting is mainly owned by Noranda Inc. The company exploits and processes ore from a massive sulfide orebody in New-Brunswick (Canada) as well as operates a lead sulfide smelting plant also in New-Brunswick. Ore processing leads to the production of four marketable concentrates respectively of zinc, lead, copper, and bulk lead-zinc sulfides. Except for copper, the concentrates are dried in rotary dryers prior to shipment to smelting plants around the world. Tracer experiments were conducted during normal operation of a zinc concentrate rotary dryer. Zinc Concentrate Dryer Brunswick Mining operates three zinc concentrate dryers. The tracer test was conducted on zinc dryer No. 3, which consists of a horizontal Enertec furnace connected to a Ruggles-Coles rotary cylinder (2 m in diameter and 15 m long). A schematic representation of the industrial dryer is presented in Figure 1. The slope and the speed of rotation of the rotary cylinder are respectively 5° and 5.8 RPM. Straight 15 cm lifters are bolted to the internal surface of the cylinder to facilitate solids transportation and to enhance airsolids contact during operation. Two fans blow air in the furnace where it is heated by burning Bunker C oil.

Figure 2. Schematic representation of the proposed models: (a) interactive perfect mixers in series and (b) modified CholetteCloutier model.

The wet concentrate is fed into the rotary cylinder by a screw feeder at a throughput varying from 25 to 35 metric tons/h. The moisture content at the inlet varies from 15 to 17% weight and is reduced to approximately 7% weight at the discharge. The gas flow rate passes cocurrently with the solid flow at a value in the range of 3-5 kg/s. The gas enters the rotary cylinder at a temperature of approximately 600 °C and leaves with a temperature in the vicinity of 80 °C. The damped air is scrubbed with water jets and the cleaned gas exhausted to the atmosphere. Recovered dust is recycled to the process where it is thickened, filtered, and returned to the dryer. Tracer Experiments On the basis of literature information (Perry, 1963), the dryer load was estimated at approximately 10 tons, which yields a 20 min mean residence time for a 30 tons/h throughput. The first tracer test was conducted using a color dye to detect the presence of dead zones within the dryer volume. For the test, 1.5 kg of fluoresceine was mixed with 2 L of water and injected into the dryer feed with the wet concentrate. Samples of the dryer discharge were collected for a 1 h period, repulped, and decanted, and the water was visually examined to detect the presence of fluoresceine which produces a greenish coloration of the water. The test confirmed the presence of dead zones in the dryer, requiring a test period longer than 1 h. The second test was conducted with lithium chloride. Lithium chloride was chosen because it could be determined more accurately and the other metals present in the concentrate did not interfere with its determination. A solution of 500 g/L was prepared and mixed with the inlet concentrate. Samples of the dryer discharge were collected for a period of 2 h. The dryer feed rate during that period was constant at 30 tons/h. Each sample was repulped to 50% solids, mixed thoroughly to ensure a complete dissolution of the lithium chloride, and filtered to separate water from solids. The lithium content of the water samples was assayed using an atomic absorption spectrometer. Derivation of the Models Two models were used to simulate the solids transportation through the rotary dryer : a model based on a series of interactive perfect mixers, referred to as model A, and the modified Cholette-Cloutier model which accounts for the presence of dead zones, referred to as model B.

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Interactive Perfect Mixers in Series

where

The solids transportation within the rotary dryer is represented as a fluid passing through a series of perfect mixers, which can be visualized as cylindrical slices of the industrial dryer. A schematic representation of this model is presented in Figure 2a. This model has only two parameters: the number of perfect mixers N and the conductance k that modulates the flow of solids from one mixer to the next and characterizes the ease of the solids to move along the dryer. The conductance is assumed constant for all cells. The outlet flow of a cell is given by the product of the mass gradient between consecutive cells multiplied by the conductance. Solids and water material balances, for cell i, are given by

d(mi) )m ˘ i-1 - m ˘ i ) k(mi-1 - mi) - k(mi - mi+1) (1) dt d(miwi) )m ˘ i-1wi-1 - m ˘ iwi ) dt k(mi-1 - mi)wi-1 - k(mi - mi+1)wi (2) where mi and wi are respectively the mass and water content of the concentrate in cell i. This model does not account explicitly for the nonideality of solids flow caused by the presence of water and the different motion modes such as cascading, kiln action, and gas drag (Foust et al., 1960). This discrepancy has led to the development of a model with more degrees of freedom. Modified Cholette-Cloutier Model Model B (shown in Figure 2b) is used to account for the presence of dead zones (Cholette and Cloutier, 1959). The volume occupied by the solids in a cell is divided into two zones: an active zone which contributes to the axial transportation of solids along the dryer and a dead zone where solids are not exchanged with neighboring cells but only with the corresponding active portion of the cell. In passing through the rotary dryer, some portion of the solids is delayed a various number of times by flowing into the dead zones which leads to the extended tail of the RTD curve (Levenspiel, 1972). This second model requires two additional parameters: the fraction R of the total volume of solids that is occupied by the active zone and a parameter β which corresponds to the steady state ratio of the exchange rate between the active and dead zones to the cell mass feed rate. These two parameters are assumed constant for each cell. The mathematical formulation of model B, based on total mass and water balances for both active and dead zones, is given by

d(ma,i) ) k(ma,i-1 - ma,i)[1 + βd,i - βa,i] dt k(ma,i - ma,i+1) (3) d(ma,iwa,i) ) k(ma,i-1 - ma,i)[wa,i-1 + βd,iwd,i dt βa,iwa,i] - k(ma,i - ma,i+1)wa,i (4) d(md,i) ) k(ma,i-1 - ma,i)[βa,i - βd,i] dt

(5)

d(md,iwd,i) ) k(ma,i-1 - ma,i)[βa,iwa,i - βd,iwd,i] (6) dt

[ [

ma,i ma,i + md,i -1 βa,i ) β + R

] ]

ma,i ma,i + md,i βd,i ) β -1 R and

R)

ma,i ma,i + md,i

(under steady state)

(7)

(8)

(9)

Model B assumes that the active mass fraction R is constant and independent of the solids mass flow rate. If a change in the inlet mass flow rate occurs, a new steady state will be achieved with a different rotary dryer loading. Since the ratio of the mass in the active zone to the total mass in the cell is constant, it is necessary to have different exchange flow rates between active and dead zones during a transient period to reduce or increase the volume of the dead zones, until the correct value for R is obtained. This correction is performed using eqs 7 and 8. Results and Discussion Model Calibration. The calibration of the two models requires one to estimate respectively parameters N and k for model A and parameters N, k, R, and β for model B, in order to minimize the sum of squares of the errors between the experimental and predicted RTD curves:

J)

∫0∞

(∫

0

C ˆ

C

∞

C dt

∫0 Cˆ dt ∞

)

2

dt )

∫0∞(E - Eˆ )2 dt

(10)

where C is the tracer concentration and E is the normalized RTD curve, or the exit age distribution, that is defined in such a way that the area under the curve is unity (Levenspiel, 1972). Since parameter N can only assume integer values, a nonlinear regression routine was used to estimate the model parameters (k, R, β) that minimize the objective function for different values of N. The complete set of parameters was chosen for the value of N that minimizes the objective criterion. The predicted RTD curve was obtained by imposing an impulse in the inlet water content of the mineral concentrate, in the absence of heat and mass transfer between the solids and the gas phase. Figure 3a presents the variation of the objective function J as a function of the number of cells. This graph has been instrumental in the choice of the best sets of parameters for the models A and B. The values of the optimum parameters are presented in Table 1. Model A has a clear minimum for a number of cells of 22 and the corresponding value of k is 0.285 s-1. Figure 4 presents the experimental RTD curve and the calculated one, obtained with the optimum parameters of model A. This two parameter model is not able to account satisfactorily for deviations from ideal perfect mixers. To represent the symmetrical gaussian-like curve with an extended tail, a four-parameter model (model B), comprised of cells with active and dead zones, was used. Since the objective function of this model decreases monotonously with the number of cells (Figure

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Figure 5. Evolution of the value of k′ with respect to the number of cells N. Table 2. Values of the Pseudo Diffusion Coefficient of Solids

Figure 3. Results of the calibration of the models: (a) dependency of the objective function J and (b) dependency of the conductance k on the number of cells N.

model

k h ′ (m2/s)

σ (m2/s)

A B

0.137 0.139

0.0018 0.0029

industrial RTD curve when 36 cells, each having 25% of its volume occupied by the dead zones, are used. Figure 3b gives the variation of the conductance k as a function of the number of cells for both models. The conductance values, for a given number of cells, are almost identical for both models whereas the objective function of model B is definitively better for a number of cells greater than 20. This clearly shows the importance of the two fine-tuning parameters R and β to model properly the RTD curve. In addition, there appears to be a strong relationship between the number of cells N and the conductance k. For each number of cells, there exists an optimal conductance necessary to fit the experimental RTD curve. The conductance can be expressed as a function of N:

k)

Figure 4. Comparison of the experimental and predicted normalized residence time distribution curves. Table 1. Optimum Model Parameters and Objective Function model

N (-)

k (s-1)

R (-)

β (-)

J (×106)

A B

22 36

0.285 0.794

0.751

0.013

2.013 0.597

3a), the value of N was chosen as a compromise between model accuracy and computation time. As shown in Figure 4, model B is able to represent very well the

k′ k′ ) 2 (LD/N) (∆z)2

(11)

where k′ is a constant depending on the model used and the RTD curve to be fitted. Figure 5 shows the dependence of k′ on the number of cells, and Table 2 presents its average value and standard deviation. The symbols used in Figure 5 correspond to those used in Figure 3 respectively for each model. It was found that the value of k′ is nearly constant and equal for the two models. Taking into account the relationship between N and k, eq 1 can be reformulated to obtain the following differential equation:

d(mi) ) k(mi-1 - 2mi + mi+1) ) dt k′ (mi-1 - 2mi + mi+1) (12) (∆z)2 It is interesting to observe that eq 12 can be viewed as the discrete representation of the following mass diffu-

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Figure 6. Sensitivity analysis of the model B. Table 3. Sensitivity Analysis of Model B relative change (%/%) parameters

J

τ

N k R β

190 65 3 3

1.84 -0.97 0.61 0.11

sion differential equation:

∂mi ∂2mi ) k′ 2 ∂t ∂z

(13)

A parallel can be drawn with the model presented by Ferron and Singh (1991) who used a similar diffusion equation to represent the movement of particles in a rotary kiln. In their model, the surface particle density has been used instead of the mass. The same development can be performed for model B, where the mass flow exchange term, between the active and the dead zones, can be added to eq 13. Model Sensitivity. The prime objectives of modeling are to provide a better understanding of the underlying phenomena taking place in a process and to assist on the determination of the optimal operating conditions. To achieve these objectives, one should analyze which parameters have the most impact on the accuracy of the model. A sensitivity analysis, which gives the relative importance of each parameter, can guide one to design experiments that would allow the accurate determination of the most sensitive model parameters. Such a sensitivity analysis was performed on model B, and the results are presented in Figure 6a-d and Table 3. For parameters N and k, a change of plus and minus 20% of the optimal values was done, while the changes for R and β were of higher magnitude because the model is

less sensitive to those parameters. Each parameter step change was performed separately to compare their respective influence on the RTD curve, while the other parameters were kept constant. When the number of perfect mixers is increased (Figure 6a), the global resistance to solids transportation is increased due to the additional flow resistance between consecutive cells. As a result, the mean residence time of the solids within the dryer is also increased. Table 3 presents the sensitivity of each parameter on both the objective function J and the average residence time τ using a relative percentage change, defined as the ratio of percent change on J or τ to the percent change of a given parameter. The number of cells N is the most sensitive parameter and has to be chosen properly in order to lead to the best representation of the experimental data. The conductance has the opposite effect on the RTD curve, as shown in Figure 6b. A larger value of the conductance leads to a higher solids flow rate between active zones, thereby decreasing the mean residence time of the solids in the dryer. This explains the results of Figure 3b where the conductance value had to be increased with the number of cells to fit the same RTD curve. The conductance is the second parameter that most affects the model accuracy, and as discussed in the previous section, it is almost perfectly correlated with the number of cells (eq 11). The additional parameters (R and β) of model B merely serve for the fine tuning of the model. Figure 6c shows the influence of R on the shape of the RTD curve. When the proportion of the active zone decreases, a greater accumulation of solids in the dead zones results, leading to an increase of the mean residence time. It is clear from Figure 6c that for a smaller value of R, a portion of the area under the RTD

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(b)

(a)

Figure 7. Further validation of the modified Cholette-Cloutier model: (a) calculated solids bed depth profile and (b) calculated solids axial velocity profile.

curve is displaced from the peak section to the tail. Because parameter R has a direct impact on the tail of the RTD curve, the mean residence time is more sensitive to this parameter than the objective function is. Figure 6d shows the effect of parameter β on the RTD curve. This parameter is the least significant on the model accuracy. When β is small, there is almost no mixing between the two zones and model B becomes identical to model A. Since there is no exchange between the two zones, the mean residence time is small. Indeed, the solids have to flow through a smaller volume thereby spending less time in each cell. An excessively large value of β causes a higher exchange mass flow rate between the active and dead zones, so that the entire cell can now be considered as an active zone. This leads to an increase of the mean residence time. Model Consistency. The literature abounds of rotary dryer models, based on experiments conducted at laboratory scale and on pilot plant dryers. Several correlations have been developed in order to predict the mean residence time, the solids bed depth and axial velocity profiles. However, these correlations cannot be used for the case of large dryers akin to the one described in this investigation. Only few articles deal with the modeling of industrial size rotary dryers. Even though experiments that can be performed on an industrial dryer are limited, it is interesting to compare our results to those obtained on smaller rotary dryers to get a physical consistency comparison of the models. The bed depth profile in the rotary dryer depends upon the inlet solids mass flow rate, the solids physical properties, the slope, and the speed of rotation of the cylinder. The use of interactive mixers allows the estimation of the bed depth profile shown in Figure 7a. This bed depth profile was calculated with model B using the parameters of Table 1 and assuming a

constant solids bed void volume of 40%, that is the density of the bulk solids is 40% lower than the solids per se. Figure 7b presents the solids axial velocity profile that provides useful information on the residence time at several positions along the rotary dryer, and it represents an additional way to validate the solids transportation models. Both profiles are in good agreement with the experimental investigation performed by Lebas et al. (1995) on a 6 meter long by 0.6 meter diameter rotary cylinder. It is important to note, however, that their experiments were conducted with dry solids, in a rotary cylinder without lifters. The dependency of the solids bed depth profile on the inlet mass flow rate is illustrated with the three simulations shown in Figure 8. As the concentrate feed rate increases, the rotary dryer loading increases. The same trend was also observed by Lebas et al. (1995). However, for the three feed flow rates of Figure 8, an identical mean residence time prevails. This is the main limitation of the proposed models, even though Friedman and Marshall (1949) and Sai et al. (1990) report a very weak dependency of the mean residence time on the feed flow rate. Indeed, in this investigation, the mean residence time remains constant for a given set of parameters. The solids transportation model could have used equal volumes for all cells, to allow the RTD curve to vary with the feed flow rate. However, this type of model would not have led to the calculation of the nonuniform solids bed depth profile that is normally observed in rotary dryers. To overcome this limitation and still maintain the simplicity of the models, some further experiments have to be conducted to formulate a relationship between the feed flow rate and the parameters of the model to generate RTD curves adapted to the changing flow behavior. The dynamic mode of the solids transportation model was tested using a concentrate feed flow rate that varied

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Figure 8. Dependency of the dryer loading on the mass feed rate.

rates. If this model is to be used under those extreme conditions, it would be desirable to adapt the model parameters with the prevailing concentrate rate within the dryer. Model B developed in this investigation was able to model appropriately the experimental RTD curve despite the continuously changing moisture content of the ore concentrate, that prevailed when the tracer test was performed. However, because of the influence of the dynamic nature of the solids phase on the solids flow behavior along the dryer, other attempts have been made to improve the solids transportation model. For instance, linear profiles for the conductance and/or active zone volume fraction were evaluated. However, all these attempts did not reduce significantly the value of the objective function, thereby not justifying additional parameters. Conclusion

Figure 9. Dynamic performances of model B: (a) measured concentrate feed flow rate and (b) measured and predicted discharge concentrate flow rates.

significantly over a period of more than 7 h of operation. The concentrate feed flow rate is presented in Figure 9a, and the experimental and predicted discharge flow rates are given in Figure 9b. This window of solids flow rate is atypical of a normal operation. It was chosen purposely to test the dynamic performance of the solids transportation model, over a wide range of flow rates. In general, model B is able to follow fairly accurately the variations of the experimental concentrate discharge flow rate. The greatest deviations between the experimental and predicted values occur under extreme flow

This paper was concerned with the modeling of solids transportation within an industrial rotary dryer. Two simple models were proposed: a series of perfect interactive mixers and the modified Cholette-Cloutier model. The model that best represented the observed RTD curve, on an industrial rotary dryer, was the modified Cholette-Cloutier with 36 cells and with 25% of each cell volume occupied by the dead zones. This model takes care appropriately of the nonideality of the flow of solids observed in the industrial rotary dryer and produces bed depth and axial velocity profiles akin to those found in the literature. The sensitivity analysis performed on the model was very useful to understand the contribution of each parameter on the accuracy of the models. The proposed models can be readily used to possibly assess the effect of the operating conditions on the rotary dryer performance but cannot in their present state suggest improvement in the design which would lead to a better dryer operation. Further studies are necessary to correlate the parameters of the models mainly with the solids feed flow rate and gas flow rate in order to take into account the variation of the RTD curves with these two variables. To achieve this objec-

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2341

tive, it is required to perform additional tracer experiments at different operating conditions. Acknowledgment The authors would like to thank a consortium of eight mining companies under the umbrella of MITEC, as well as NSERC, the Centre de recherches mine´rales and CANMET of respectively Natural Resources Quebec and Canada, for their support of the KBAC research program. A special thanks goes to Brunswick Mining and Smelting for their cooperation during the sampling campaign. Nomenclature C ) observed tracer concentration, ppm C ˆ ) predicted tracer concentration, ppm ∆z ) incremental length of dryer, m E ) observed exit age distribution, s-1 E ˆ ) predicted exit age distribution, s-1 J ) objective function, s-1 k ) the conductance of wet solids, s-1 k′ ) parameter in eq 11, m2 s-1 LD ) dryer length, m m ) mass of wet solids, kg m ˘ ) mass flow rate of wet solids, kg/s N ) number of dryer slices, t ) time, s w ) humidity of the solids, kg of water/kg of wet solids Greek Letters R ) fraction of the total solids volume occupied by the active zone, β ) mass flow rate exchange ratio between active and dead zones, βa ) corrected mass flow rate fraction entering a dead zone, βd ) corrected mass flow rate fraction going out of a dead zone, σ ) standard deviation of k′, m2 s-1 τ ) observed mean residence time, s Subscripts a ) active zone d ) dead zone i ) indicates properties associated with the ith cell

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Fan, L. T.; Shen, B. C.; Chou, S. T. Stochastic Modeling of Transient Residence-Time Distribution During Start-Up. Chem. Eng. Sci. 1995, 50, 211-221. Ferron, J. R.; Singh, D. K. Rotary Kiln Transport Processes. AIChE J. 1991, 37, 747-758. Foust, A. S.; Wenzel, L. A.; Clump, C. W.; Maus, L.; Andersen, L. B. Principles of Unit Operations; Wiley: New York, 1960. Friedman, S. J.; Marshall, W. R. Studies in Rotary Drying, Part I - Holdup and Dusting. Chem. Eng. Prog. 1949, 45, 482-493. Gossen, P. D.; Ravi Sriniwas, G.; Schork, F. J. Determining Residence Time Distributions in Complex Process Systems, A Simple Method. Chem. Eng. Educ. 1995, 106-111. Henein, H.; Brimacombe, J. K.; Watkinson, A. P. The Modeling of Transverse Solids Motion in Rotary Kilns. Metall. Trans. B 1983, 14B, 207-220. Kamke, F. A.; Wilson, J. B. Computer Simulation of a Rotary Dryer, Part I : Retention Time. AIChE J. 1986, 32, 263-268. Kelly, J. J.; O’Donnell, P. Residence Time Model for Rotary Drums. Trans. Inst. Chem. Eng. 1977, 55, 243-252. Lebas, E.; Hanrot, F.; Ablitzer, D.; Houzelot, J.-L. Experimental Study of Residence Time, Particle Movement and Bed Depth Profile in Rotary Kilns. Can. J. Chem. Eng. 1995, 73, 173-180. Levenspiel, O. Chemical Reaction Engineering; Wiley: New York, 1972. Matchett, A. J.; Sheikh, M. S. An Improved Model of Particle Motion in Cascading Rotary Dryers. Trans. Inst. Chem. Eng. 1990, 68, 139-148. Mu, J.; Perlmutter, D. D. The Mixing of Granular Solids in a Rotary Cylinder. AIChE J. 1980, 26, 928-934. Mujumdar, A. S. Handbook of industrial drying; M. Dekker: New York, 1987. Perron, J.; Bui, R. T. Rotary Cylinder: Solid Transport Prediction by Dimensional and Rheological Analysis. Can. J. Chem. Eng. 1990, 68, 61-68. Perron, J.; Bui, R. T. Fours rotatifs: Mode`le dynamique du mouvement du lit. Can. J. Chem. Eng. 1994, 72, 16-25. Perry, R. H.; Chilton, C. H.; Kirkpatrick, S. D. Chemical Engineer’s Handbook; McGraw-Hill: New York, 1963. Sai, P. S. T.; Surender, G. D.; Damodaran, A. D.; Suresh, V.; Philip, Z. G.; Sankaran, K. Residence Time Distribution and Material Flow Studies in a Rotary Kiln. Metall. Trans. B 1990, 21B, 1005-1011. Sherritt, R. G.; Caple, R.; Behie, L. A.; Mehrotra, A. K. The Movement of Solids Through Flighted Rotating Drums. Part I: Model Formulation. Can. J. Chem. Eng. 1993, 71, 337-346.

Received for review October 13, 1995 Revised manuscript received April 4, 1996 Accepted April 4, 1996X IE950625J

X Abstract published in Advance ACS Abstracts, May 15, 1996.