Pseudophysical Compartment Modeling of an Industrial Rotary Dryer

Article pubs.acs.org/IECR

Pseudophysical Compartment Modeling of an Industrial Rotary Dryer with Flighted and Unflighted Sections: Solids Transport O. O. Ajayi and M. E. Sheehan* School of Engineering and Physical Sciences, James Cook University, Townsville, Queensland 4811, Australia S Supporting Information *

ABSTRACT: A novel pseudophysical compartment model describing the solids transport within a case-study cocurrent industrial rotary dryer is presented. The model is novel and distinctive because it combines both multiscale partial differential equation and ordinary differential equation modeling techniques to represent the five separate sections occurring within the dryer, including both flighted and unflighted sections. The solid phase in the unflighted sections was modeled as an axial dispersed plug flow system. The solid transport in the flighted sections was partitioned into the usual series−parallel formulation of well-mixed tanks representing airborne and flight-borne solids. Parameter values and compartment numbers were estimated using mechanistic geometric modeling, dryer design loading constraints, and solids flow properties. Geometric modeling enabled the effects of internal scaling arising from solid materials sticking to the internal walls of the dryer to be accounted for in a realistic manner. Industrial residence time distribution data were collected for a range of operational scenarios and were used to estimate the remaining model parameters (axial dispersion coefficient and kilning velocity). The model results were well-matched to the collected set of industrial residence time distributions (RTDs), and estimated parameter values were within expected limits. The axial dispersion coefficient and kilning velocity were modeled as functions of operating variables in order to fully embed physical realism into the compartment model, and new correlations are presented. The effect of operating conditions on RTDs and solids distribution within the dryer was investigated. The results matched past observations and provide insights into dryer efficiency. within the flights and drum base (referred as the passive phase). Clearly, when modeling flighted rotary dryers it is important to be able to distinguish between these two phases of solids and to accurately characterize solids transport. In order to characterize the solid transport and dispersion within a dryer, residence time distributions (RTDs) are determined.2,3 Modeling approaches are different for flighted dryers and unflighted dryers. In the literature describing unflighted dryer modeling, studies have assumed the solids movement was via plug flow either with4−7 or without axial mixing.8,9 Experimental studies have shown that axial dispersion occurs in both unflighted and flighted dryers.7,10 Fan and Ahn5 used the classic diffusion model to simulate the dispersion and residence time distribution of material in a rotating cylinder. Sai et al.7 proposed the use of an axial dispersion model with an appropriate Pe number estimated via experiments. In their study, the effect of operating conditions such as solid feed rate, rotational speed, and dam height on the mean residence time was investigated. In another study, Kohav et al.11 used stochastic algorithms to determine the effect of segregated rolling or slumping distance on the axial dispersion in rolling and slumping beds. There has been a much more substantial body of work describing the modeling of the flighted rotary dryers, which has included empirical and black-box correlations, geometric and mechanistic modeling, as well as empirical and mechanistic

1. INTRODUCTION Flighted rotary dryers are commonly used in the food and mineral processing industries for drying granular or particulate solids. The flighted rotary dryer consists of a cylindrical shell slightly inclined at the inlet and fitted internally with an array of flights. The arrangement and type of flights vary with the nature of granular solids and are intended to facilitate gas−solid interaction. As the dryer rotates, solids are picked up by flights, lifted for a certain distance around the drum, and fall through the gas stream in a cascading curtain. Gas used as a drying medium is introduced as either cocurrent or countercurrent to the solid flow. Dryer internals vary widely and may consist of a single flight geometry and arrangement along the entire drum length, or may consist of individual drum sections with different flight geometries, or may even incorporate unflighted sections. These units are both capital intensive and also energy intensive during operation, which leads to motivation to model these units for both design and optimization purposes. The movement of solids through the dryer is influenced by the following mechanisms: lifting by the flights, cascading from the flights through the air stream, bouncing, rolling, and sliding of the particles on impact with the bottom of the dryer.1 This latter mechanism is often referred as kilning flow. The performance of rotary dryers is dictated by three important transport and thermodynamic mechanisms: solids transportation, heat transfer, and mass transfer. The heat and mass transfer that occurs is dependent on the solids transportation. For example, heat transfer and mass transfer are commonly thought to occur only between the solids in direct contact with gas phase (referred as the active phase) rather than between those solids contained © 2014 American Chemical Society

Received: Revised: Accepted: Published: 15980

July 2, 2014 September 15, 2014 September 15, 2014 September 15, 2014 dx.doi.org/10.1021/ie502607m | Ind. Eng. Chem. Res. 2014, 53, 15980−15989

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic representation of industrial rotary dryer operated by MMG, Karumba.

In this paper, the modeling of unflighted and flighted sections within the case-study industrial dryer is described. Industry RTD and solids property experiments were carried out under different dryer operating conditions and were used to define solids properties, to undertake parameter estimation, and to validate model predictions. Two model parameters were estimated in order to best fit the measured RTDs, and these included the axial dispersion coefficient and the kilning velocity. The model is used to examine the effects of operating conditions such as the extent of solid adhesion to the internal surfaces of the dryer and rotational speed on solids transport and loading characteristics. Multiscale simulations were undertaken using gProms process modeling software17 and Excel regression and table tools.

compartment modeling. The empirical models typically do not account for dryer loading capacity limitations or the effect of flight configuration and solid properties. They are also unable to reproduce the dryer residence time distributions that are observed experimentally. A comprehensive review of empirical correlations, semiempirical correlations, and mechanistic models for predicting residence times and RTDs can be found in Sheehan et al.3 Key early papers that have informed the development of the compartment models described in this paper have been those by Duchesne et al.12 and Matchett and Baker.13 Matchett and Baker13 proposed a semiempirical model which first made the clear distinction between the airborne and flightborne phases in a flighted rotary dryer. Duchesne et al.12 proposed a twin tanks in series compartment model as an effective structure for reproducing the classic shape of the flighted rotary dryer RTD. Sheehan et al.3 combined these ideas and proposed a pseudophysical compartment (PPC) model for flighted rotary dryers. The PPC model developed by Sheehan et al.3 and described in Britton et al.14 is a compartment model involving twin tanks in series where the tanks represent the airborne (active phase) and flight/drum-borne (passive phase) solids. In their model, a physically realistic structure is imposed which includes observable exchanges of solids between the active and passive phases and between passive and passive phases, both between adjacent and neighboring tanks. Model parameters dictating solids flows, capacity constraints, and compartment numbers/ dimensions were estimated based on solids physical properties and drum operational properties, as well as drum and flight geometric properties. The parameter values were determined using geometric modeling of the flight unloading process, an approach that was validated experimentally by Lee and Sheehan.15 The PPC modeling approach has also been used in modeling drum granulation processes such as that presented in Rojas et al.16 Since the PPC model structure and model parameters are governed by geometric properties such as flight dimensions, it is particularly suitable for modeling dryers with varying flight dimensions. The case-study cocurrent industrial rotary dryer examined and modeled in this paper is used in drying zinc and lead concentrates (MMG, Karumba, Australia, 2008−2010 seasons). To facilitate granulation, the dryer has both unflighted and flighted sections. Each flighted section has a different flight configuration and geometry, although all flights are standard twostage designs. Further geometric complications include adhesion of solids to the internal surfaces of the drum and flights. To the best of the authors’ knowledge, there are no examples of dryer models that account for varying flight geometry or that combine both unflighted and flighted dryer sections.

2. INDUSTRIAL TESTING 2.1. Industrial Dryer. Figure 1 shows the schematic diagram of the cocurrent drying process. The drying process consists of a combustion chamber and a rotary dryer. In this study, the cocurrent dryer is 22 m long with diameter of 3.9 m. The slope and the typical rotational speed of the dryer are 4° and 3 rpm, respectively. The dryer is divided into five sections. Sections A and E are unflighted and sections B, C, and D are flighted but have different flight dimensions. Flight geometry measurements are presented in Table 1, and a schematic diagram defining the flight geometry is provided in Figure 2. A granulation process takes place over the last unflighted section. Table 1. Geometric Configuration of the Drum section

section length (m)

flight base (m)

flight tip (m)

anglea (deg)

no. of flights

A B C D E

2.1 2.4 3.3 6.6 7.5

− 0.120 0.130 0.120 −

− 0.210 0.220 0.210 −

− 135 150 130 −

30 30 30

Refers to the angle between the flight base and flight tip. All flights are at 90° to the drum wall. a

The concentrate is fed into the dryer via a screw feeder. The typical wet solid feed rate ranges from 36 to 50 kg/s. The solid inlet moisture content on wet basis varies between 16 and 17%, and the target outlet moisture content on a wet basis is 12%. Hot gas enters the dryer at an approximate temperature of 500 °C and leaves at a roughly 140 °C. A distributed control system (DCS) based on feed forward control is used to control the dryer. The control algorithm of the DCS calculates the amount of water to be removed using the inlet and outlet target moisture contents, solids flow rate, and air flow into the combustion chamber, as 15981

dx.doi.org/10.1021/ie502607m | Ind. Eng. Chem. Res. 2014, 53, 15980−15989


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Figure 3. Moisture content profile along the length of the dryer. Figure 2. Schematic representation of typical two-staged flight defined by the flight base length (s1), flight tip length (s2), angle between the flight base and the drum wall (α1), and angle between the flight segments (α2).

judged by fan opening. These measurements are used to adjust the quantity of fuel oil required in the combustion chamber. High rates of water removal lead to increased fuel oil consumption, which leads to increased gas inlet temperatures. In this study, the air flow rate into the combustion chamber was determined from industry correlations for percentage of fan opening (PI data) versus measured experimental volumetric flow rates of both dilution and combustion air.18 Air flow and drum gas velocity were inferred through reconciliation using mass and energy balances around the combustion chamber and were based on drum diameter and assumed plug flow. The sticky nature of the concentrate leads to the adhering of the solids to the internal walls of the dryer, including the inside wall of the dryer and on the flight surfaces. We refer to this accumulation of solids as scaling. The buildup of hard deposit affects dryer dyer performance and leads to poor controllability. Frequent shutdowns are required so as to remove the hard deposits from the internals of the dryer, resulting in further costs and production loss. 2.2. Physical Properties of Zinc Concentrates. A number of important physical properties are dependent on the solids moisture content, which varies along the length of the dryer. In order to simplify the solids transport modeling process, an internal moisture content profile was assumed. The assumed moisture content profile (Figure 3) was based on moisture contents determined after one RTD test (test 3), by opening up the dryer and sampling solids along the entire dryer length. This opening was also used as an opportunity to determine the degree of solid adhesion to the internal surfaces of the flight and drum. The industry moisture content data in Figure 3 was fitted using a rational polynomial function to derive eq 1, relating the moisture content to the dryer length. Moisture content is expressed on a wet basis (kilograms of water per kilogram of wet solid).The flowability of solids within the flights is typically characterized by determining the material’s dynamic angle of repose. Experiments were carried out in a pilot scale drum by placing a subset of sampled zinc concentrate solid in a container. The filled container was affixed to the front-end Perspex of the rotating drum (as shown in Figure 4) and rotated at 3 rpm. Photographs of the front end of the rotating drum were taken, and the dynamic angle of repose was measured using ImageJ software. Equation 2 defines the dynamic angle of repose as a function of moisture

Figure 4. Experimental apparatus to measure dynamic angle of repose.

content. The consolidated bulk densities of the inlet and outlet solids were determined in the laboratory and were assumed to change linearly with respect to moisture content (eq 3). It should be noted that the range of applicability of eqs 1−3 is limited to moisture contents between 12 and 19%. Equations 1−3 were used in the compartment model to define the solid’s properties as a function of length. The measured mean particle size varied from 18 to 6 mm. xw =

0.1006L + 2.218 L + 13.51

(1)

ϕ = 419.6x w − 7.801

(2)

ρs = 2044 − 3095x w

(3)

Here xw, L, ϕ, and ρs are the solids moisture content (kg/ kgwet solid), axial position along the dryer (m), dynamic angle of repose (deg), and consolidated bulk density (kg/m 3 ), respectively. 2.3. RTD Trials. The residence time distribution (RTD) experiments were carried out by injecting tracer at the inlet of the dryer while sampling outlet solids over a period of time. Lithium chloride was used as the tracer compound. The lithium concentrations in the samples taken at the outlet were determined using inductively coupled plasma mass spectrometry 15982



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Figure 5. Model structure.

(ICP-MS). The tracer injection approach involved mixing 2 kg of dry powdered lithium chloride (LiCl) with approximately 8 kg of inlet feed. It should be noted that LiCl is highly soluble and, because of the inlet feed’s high moisture content and sticky nature, the tracer is expected to dissolve completely and mix well with the concentrate. Better results were obtained using this method when compared to the addition of a predissolved liquid solution into the feed. The prepared material was discharged into the dryer via the inlet conveyor over a period of 45 s. Samples from the outlet were taken at intervals of 30 s for the first 30 min and every 60 s for the next 1.5 h. A series of RTD experiments were carried out under different operating conditions, and the details of each test are presented in the Supporting Information, Table S1. To determine the effect of solid adhesion to the wall and internal surfaces of flights, two tests were conducted: prior to a scheduled shutdown (test 2) and just after a scheduled shutdown where internal cleaning had occurred (test 3). Tests 4 and 5 were used to assess the effect of rotational speed. Comparison of the moisture removal and outlet gas temperatures for test 1 and test 2 (Supporting Information, Table S1) illustrate the reduction in dryer performance when there is solid adhered to the internal surfaces of the dryer. The variation in drying efficiency for test 1 and test 4 can be attributed to higher gas inlet temperature in test 1 and a difference in rotational speed. These operating conditions determine rates of convection and evaporation within the studied dryer.

Figure 6. Model structure for the flighted section (one tank set) indicating the corresponding transport coefficients that govern stream flows between the tanks.

cell. The splitting of the flow is governed by the geometrically defined forward step coefficient (CF). The mass balances on solids for each of the two phases are described in eqs 4−6. mass balance on solids in passive cell i: dmpi dt

= k4i − 1mpi − 1 + k 3i − 1C Fmai − 1 + k 3i(1 − C F)mai − k 2impi − k4impi

(4)

mass balance on solids in active cell i: dmai = k 2impi − k 3iC Fmai − k 3i(1 − C F)mai dt

3. MODEL STRUCTURE The overall model structure including both the unflighted and flighted sections is presented in Figure 5. Following typical terminology,3,13,14 airborne solids are regarded as active solids (i.e., the curtains) and flight- and drum-borne solids are regarded as passive solids. Distinguishing between these two solids phases is critical to both solids transport and defining the proportions of solids undergoing heat and mass transfer. In this study, modeling of each of the flighted sections (sections B, C, and D) was structurally similar to the countercurrent sugar dryer PPC model described by Sheehan et al.3 and Britton et al.14 Differences here include no back-mixing flow paths to account for the fact that the dryer examined in this study has cocurrent gas flow. 3.1. Flighted Section. The compartment model structure for the flighted sections is shown in Figure 6. There are some key differences in this work compared to previous studies.3,14 Since gas flow is cocurrent to solids flow in this study, the flow rate of solids from the active phase is divided into two different paths only: axial movement of falling solid forward into the next passive cell and the return of falling solid into the corresponding passive

(5)

Mass balances on tracer or moisture content are easily derived from these balances. For brevity the tracer balance on the passive tank is shown as an example: dxt,pimpi dt

= k4i − 1xt,pi − 1mpi − 1 + k 3i − 1C Fxt,ai − 1mai − 1 + k 3ixt,ai(1 − C F)mai − k 2ixt,pimpi − k4ixt,pimpi (6)

where mpi, mai, xt,ai, and xt,pi are passive mass (kg), active mass (kg), mass fraction of tracer in active phase, and mass fraction in passive phase, respectively. 3.2. Unflighted Section. The unflighted sections were modeled as an axial dispersed plug flow system, and a complete derivation is provided below. Considering a differentiable element (Δz) within an unflighted section of the dryer (Figure 7), the dynamic mass balance is given as 15983



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7

z = L,

(7)

The mass flow rate (Fs) for an axially dispersed plug flow system19 is defined as Fs = usA sρs −

7A sρs (8)

Δz

where us, 7 , As, and ρs are the solid velocity (m/s), axial dispersion (m2/s), cross-sectional area occupied by the solid (m2), and solid consolidated density (kg/m3), respectively. By substitution ⎛ 7A sρs ⎞ ∂ (ρs A sΔz) = ⎜usA sρs − ⎟ ∂t Δz ⎠ ⎝

z

⎛ 7A sρs ⎞ − ⎜usA sρs − ⎟ Δz ⎠ ⎝

z +Δz

(9)

Dividing by Δz and taking the limit Δz → 0, and assuming 7 is constant with respect to length ∂2 ∂ ∂ (A sρs ) = 7 2 (A sρs ) − (usA sρs ) ∂t ∂z ∂z

(10)

The term Asρs is defined as the mass of solids per unit length (ms) and is an important property of the model as it represents drum holdup per unit length. Asand ρs vary both in time and with length, but us was considered constant within the model. By substitution, the axially dispersed plug flow model for the unflighted drum is given as ∂ ∂2 ∂ (ms) = 7 2 (ms) − (usms) ∂t ∂z ∂z

(11)

In a similar manner, the equivalent mass balance on the tracer is ∂ ∂2 ∂ (xtms) = 7 2 (xtms) − (xtusms) ∂t ∂z ∂z

ta̅ =

Fs us

ms(0) =

z = 0,

xt(0) = x in

k2 =

(14)

At the outlet, we take a typical approach and use boundary conditions that ensure there is only convective flow and no solids dispersion. The outlet boundary conditions are stated as follows: 7

∂2 (ms) = 0 ∂z 2

∑j ṁ j

(17)

1 , t p̅

k3 =

1 ta̅

(18)

3.3.2. Loading State. It is important to recognize that there is a limit to the capacity of a given set of dryer flights and that the proportion of airborne solids to flight-borne solids dictates the extent of gas−solids interaction. The distribution of solids also affects the residence time because of the difference in the transport mechanisms and flow rates of airborne solids and flightborne solids. When dryers are run at high feed rates, the flights reach their capacity and the excess solids accumulate in the drum base. This is accounted for in the model by using a design loading constraint which limits the capacity of the airborne phase (ma). In

(13)

20

z = L,

∑j ṁ jtaj

The transport coefficients k2 and k3 in eqs 4 and 5 were defined as the inverse of the passive and active cycle times, respectively (eq 18).

(12)

where xt is the tracer concentration (kg/kgTotal). The Danckwerts boundary conditions4 for eqs 11 and 12 were used and are shown in eqs 13 −16. z = 0,

(16)

3.3. Geometric Modeling: Parameter Values. Many of the model parameters including transport coefficients and compartment numbers were derived using the geometric model of flight unloading.15 The model was developed for two-staged flights, assumes free-flowing solids, and assumes the mean solid surface angle (in this case termed the dynamic angle of repose) remains constant with respect to rotational speed. Input properties include the number of flights, flight configuration, dryer geometry and rotational speed, and the solids’ mean surface angle and consolidated density. The central calculations in the geometric model are the calculation of the cross-sectional area of solids contained in each flight as a function of the rotation angle. Areas are converted to mass using the compartment lengths and consolidation density, and flight areas are differentiated with respect to time to obtain mass flow rates from each flight. Full details of the geometric model and its validation can be found in Lee21 and Lee and Sheehan.15 3.3.1. Cycle Times. Taking geometry model predictions for the mass flow rates from each flight as a starting point, Newton’s equations of motion for a single falling particle under the influence of gravity are integrated. In these calculations, the particle curtains that cascade off each flight are assumed to behave like a single isolated sphere. In this model, the particles are assumed to exhibit negligible drag. To include the effect of air drag on solids transport, this assumption could be relaxed. Drum geometry and falling distances are used to determine both the time-of-flight and path a particle takes between leaving a flight tip and landing on the drum base. This time-of-flight is referred to as the active or airborne cycle time (ta). The time taken by a particle between landing on the drum base and returning to the original flight tip location is referred to as the passive cycle time (tp). The distance the particle moves forward, in the axial direction, is referred to as the forward step (D). Mass averaged cycle times representing the behavior of the bulk phase are determined by averaging with respect to the mass flow rate off each flight (ṁ j). The example of the mass averaged active cycle time is shown in eq 17, where j refers to the number of unloading flights. This equation would take a similar form for other mass averaged properties such as the passive cycle time (tp̅ ) and average forward step (D̅ ).

Figure 7. Model structure of the unflighted section characterized by its feed rate (Fs) and length.

∂ (ρ A sΔz) = Fs|z − Fs|z +Δz ∂t s

∂2 (xtms) = 0 ∂z 2

(15) 15984



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solid adhering to the internal surfaces of the drum and flight. Its purpose was to accurately represent the impacts of solid adhesion to the internal surfaces which were observed to reduce the flights’ capacity and internal drum diameter. Flight and drum dimensions were modified in the geometric modeling to approximate the observed solid adhesion. The assumed new drum radius (RD) is shown in Figure 8. The new flight

an underloaded dryer, the cycle times described above will vary because unloading will occur later in the rotation cycle. The point at which the flights become completely full is referred to as the design load and defines the limiting constraint. The geometric model includes the calculation of the passive phase design load via an equation described and validated by Ajayi and Sheehan,22 and shown in eq 19. In this equation Mi is the mass in each unloading flight and MFUF is the mass in the first (f) unloading flight. n

M pdesign = 1.24[(2∑ Mi) − MFUF] f

(19)

If the mass of solids in the flights and that in the drum (i.e., passive solids) are at or exceed the passive design loading (i.e., Mp ≥ Mp design), then the flows of solids from the flights into the airborne phase (moderated by the transport coefficient k2) are constrained to their maximum value (via eq 20). M p ≥ M pdesign ,

|k 2M p|max = k 2M pdesign

(20)

Figure 8. Geometric description of the reduction in drum radius as a result of solid adhering to internal walls of the drum.

3.3.3. Kilning and Tank Numbers. In both the unflighted and flighted sections of the model, the kilning flow transport coefficient (k4) is described as a function of the solids kilning velocity and the length of the compartment cell. In the modeling that follows, the value for us(i) was assumed to be constant across the entire length of the dryer, thus independent of both flight geometry and solids moisture content. us(i) k4(i) = L(i) (21) where k4(i), us(i), and L(i) are the kilning flow transport coefficient (1/s), solid velocity (m/s), and length of the compartment cell (m), respectively. In the future, researchers might wish to consider alternative forms for kilning velocity (us) and to make it a function of rotation speed, drum diameter, and solid particle diameter. However, kilning is very difficult to describe in simple mechanistic terms, and for this work, us was parameter estimated. The final model parameters were derived using the geometric model. The average forward step (D̅ ) provided bounds on the maximum number of cells since solids are constrained in the model (by enforced physical realism) not to fall (i.e., flow) further than one cell ahead. The physical length of each compartment cell was calculated by first determining the total number of compartment cells (Nc), using eq 22. The nearest integer value for Nc was selected, which constrained the numbers of compartment cells within the flighted sections. The effect of compartment numbers is well described by Lee.21 Table S2 in the Supporting Information gives the number of cells in the each flighted section.

Nc =

Ls D̅

Figure 9. Schematic diagram showing solids adhering to the flight base (S1), flight tip (S2), and flight angle (α2).

dimensions S1D and S2D are shown in Figure 9 and represent the new geometric approximations for the dryer with solids adhering on its internal surface. The lengths S1D and S2D are also reduced in the geometric model by the scale accumulation thickness (SF). An internal examination of the dryer prior to the RTD experiment (test 2) revealed significant solid adhesion to the flights and the internal walls of the drum. The length of the flight base was observed to be reduced by 80% (i.e., SF = 0.8S1). 3.4. Multiscale Model Architecture. The interaction between the geometric model and the process model is shown in Figure 10. The process model data (passive mass (Mp) and moisture content (xw)) were exported to the geometric model so as to calculate the model parameters (tα̅ , tp̅ ) and the geometric constraints (D̅ , Mp design, CF). In order to simplify the initial solids transport fitting process, energetic mechanisms such as convection, evaporation, and radiation were neglected. Instead, a moisture content profile along the length of the drum was used.

(22)

Here Ls and D̅ are the total length of the section (m) and average forward step (m), respectively. To account for the geometric implications of noninteger values of the ratio in eq 22, the flow of solids from the active phase into both the adjacent and next passive phase tanks was governed by a split fraction (CF). CF was calculated as the true ratio of the average forward step (D̅ ) to the representative physical length of the cell (Ls/Nc). 3.3.4. Scaling Effects. A scale accumulation thickness (SF) was introduced in the geometric modeling in order to account for 15985



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Γ=

z 1 ln(2π ) + 2 2 αi βj ⎡ ⎧ ⎫ (Yij̅ − Yij)2 ⎤⎪ ⎪ ⎥⎬ min⎨∑ ∑ ⎢ln(σijk 2) + ϑ ⎪ ⎥⎦⎪ σij 2 ⎩ i = 1 j = 1 ⎢⎣ ⎭

(23)

Here Z is the total number of measurements taken during the experiments, ϑ is the set of model parameters to be estimated (us, 7 ), αi and βj are the number of experiments performed and the number of variables in each experiment, respectively. Y̅ij and Yij are RTD measured and predicted values in experiment i. σij2 is the variance in the measured variables. σij2 is described by the constant relative variance model available in gPROMS. The discretization scheme and number of discretized cells within the plug flow models had significant effects on the accurate estimation of the parameters and the fitting of the RTD profile. Large grid sizes (0.3 and 0.15 m) resulted in numerical instability. The grid independency study suggested a 0.075 m grid size would be appropriate, and the number of discretized cells for sections A and E were 28 and 100, respectively. It should be noted that the two unflighted sections (sections A and E) were discretized using a second-order backward finite difference scheme. Table S3 in the Supporting Information presents the estimated model parameters for different RTD trials which are similar to those reported in the literature. In a study of countercurrent industrial sugar drying, Lee21 reported an estimated kilning velocity of 0.04 m/s, which is the same order of magnitude as that determined in this study (0.01−0.03 m/s). In another study, a noninvasive technique was used to measure the axial mixing in a rotating drum and calculated axial dispersion coefficients.20 The estimated axial dispersion coefficient values in this study (Supporting Information, Table S3) were close to their suggested range (10−7−10−4 m2/s). Considering the geometric complexity of the various flighted and unflighted sections being modeled, it is very encouraging to see that our model results are consistent with the results from models of dryers with less varied geometries (such as unflighted drums) that have appeared in the literature.20,21 We believe this consistency provides an increased degree of confidence in the physical realism of our chosen model structures. 3.6. Model Results. The fitted data and experimental data for different conditions are plotted in Figure 11. The model RTD profiles are well-matched to the experimental RTD profiles for the different operating conditions of the dryer. This further indicates that the proposed model structure was appropriate for the studied rotary dryer. The key features of initial steep rise and extended tail in the RTD profiles were also reproduced by the model. Table S4 in the Supporting Information outlines the simulated and experimental residence times and drum holdups for different conditions of the dryer and provide a good match between the model and the experimental data. It can be seen that there is a difference of 3 min between the residence times of test 1 and test 5, despite the similar feed rates and similar drum incline angles in these tests. Furthermore, the shorter residence time in test 1 results in a reduced drum holdup in comparison to test 5, and also an increase in the active to passive holdup proportions. Although there are many compounding factors in industrial drying operations, a key factor leading to these observed differences is that in test 1 there is substantially higher inlet moisture content compared to test 5. The higher moisture content will result in increased cocurrent gas flow because the automatic control algorithm of the dryer, based on the amount of

Figure 10. Interaction between the process model and geometric model.

The authors recognize that it would be more comprehensive or preferable to include energy transfers within the model (i.e., predict moisture content simultaneously) when fitting the RTD. However, there is substantial uncertainty in defining these energetic transfers (interfacial areas, for example) that results in introducing more parameters that complicate the fitting process further. This can result in substantial difficulties finding global minima during the parameter estimation optimization. We believe that by using known inlet and outlet moisture contents, and assuming what is a very reasonable approximation to the internal moisture profile, the influences of energy transfers on material properties such as the angle of repose are accurate approximations to the real system. The parameter estimation process can reasonably be considered insensitive to what would be only minor variations in the internal moisture profile, particularly considering the angle of repose variation is only 8° from inlet to outlet. In order to reflect different inlet and outlet moisture contents for each RTD test, eq 1 was modified for each test so that moisture content at the inlet (L = 0 m) and outlet (L = 22 m) correspond to the experimental values. The fitted linear equations of the dynamic angle of repose and consolidated density versus moisture content (eqs 2 and 3) were used to obtain values for these parameters for each model compartment. Rather than simultaneously solve the geometric and process models, regression models for the necessary parameters were derived within the geometric model as a function of the moisture content and loading state (i.e., Mp/Mp design and these regression relations were coded directly into the process model. 3.5. Numerical Solution and Parameter Estimation. Model equations were solved using gPROMS software on a high performance PC. Typical solution times for the parameter estimations were roughly 30 min. The two remaining model parameters, solid velocity (us) and the axial dispersion coefficient (7 ), were parameter estimated by comparing simulated tracer tests with the experimental RTD tracer tests. The parameter estimation process was executed in gPROMS software and was based on minimizing the sum of square errors between the modeled RTD and experimental RTD data points. The following objective function (eq 23) was used.17 There are different variance models in gPROMS software and an appropriate variance model is chosen based on the magnitude of standard deviation within the predicted or experimental data. In this study, the constant relative variance model was used. An initial guess value and lower and upper limits of the standard deviation are required. In this study, the initial guess value was 0.5 ppm and lower and upper limits were 0.01 and 10 ppm, respectively. 15986



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Figure 12. Solids distribution within the flighted sections for test 1. Holdup in the unflighted sections: 2237 kg/m.

Figure 13. Solids distribution within the flighted sections for test 2. Holdup in the unflighted sections: 2044 kg/m.

loading state for the dryer. Under scaled conditions the flight capacity is greatly reduced, leading to a massively overloaded (i.e., above design load) dryer compared to unscaled conditions. Minimal solids would be presented to the gas phase in these circumstances. These differences are the primary reason for the reduced drying that is observed during scaled conditions (Supporting Information, Table S1). The capacity of the developed model to provide information on how the accumulation of solids on internal surfaces of the drum affects drying efficiency is an important step in developing appropriate control strategies to manage the dryer at this condition. The model could also be utilized as a diagnostic tool to assess the degree of scaling that has occurred. These observations regarding the dryer holdup further assist in confirming the physical realism of the developed model. 3.7. Model Predictions. The effects of operating conditions on the RTD and the solids distribution were examined using the model. However, in order to utilize the model for predictive purposes, it was important to formulate the estimated parameters (us and 7 ) so that they are also responsive to changes in operating and geometric conditions. Using the RTD fitting results across all industry trials, a Pearson correlation technique24 was used to determine the significance of interaction between the estimated parameters and operational/geometric variables. Although a t test at 95% confidence interval indicated nonsignificance (more trials across a wider range of experimental conditions would help in this regard, but this can be challenging to obtain in an industry setting), a correlation between 7 and the inlet dynamic angle of repose was observed. In addition, the Pearson correlation coefficients showed that both rotational speed and the internal diameter of the drum were correlated to us. Microsoft Excel’s multiple regression analysis was used to develop eqs 24 and 25. The regression models were

Figure 11. RTD profiles of (a) test 1, (b) test 2, (c) test 3, (d) test 4, and (e) test 5.

water to be removed, would deliver higher fuel and air flow rates to the combustion chamber and dryer. Higher gas temperatures would also be a result of these control measures. While not directly accounted for within the proposed compartment model parameters, in practice, the higher gas flow is likely to increase the axial drag force experienced by the falling solid particles, shortening the residence time. The effect of gas flow on dryer residence time is commonly described in the literature.23 Higher gas temperatures in test 1 have also led to an increase in the rates of both convection and evaporation between the gas and solids. The resulting variation in the moisture content profiles between the two tests will affect the dynamic angle of repose profile for solids along the dryer, and is expected to influence the observed residence times. The capacity of the compartment model to accurately depict the solids distribution within the dryer is very valuable, particularly as a base for building in energy considerations. See, for example, Figures 12 and 13, which illustrate the solids distribution under unscaled and scaled operating conditions. It should be noted that scaled operating conditions refers to when there are solids adhered to the dryer walls and flight surfaces. In Figures 12 and 13, the design load represents the optimum 15987



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implemented in the solids transport model to study the effect of operating variables. us = 0.161318 + 0.006533ω − 0.04198D (R2 = 0.82) 7 = 0.000197ϕ − 0.01154

(24)

(R2 = 0.97)

(25)

The effects of solids adhesion (scale accumulation) onto the internal surfaces on both the RTD and solid phase distribution are examined in Figures 14 and 15, respectively. The operating Figure 16. Effect of rotational speed on RTD.

experimental RTD study for the flow of lignite particles through a rotary dryer.25

4. CONCLUSION Pseudophysical compartment modelling of solid transport within a zinc concentrate industrial rotary dryer containing both flighted and unflighted sections was undertaken. The solid phase in the unflighted sections was modeled as an axial dispersed plug flow system. The solid transport in the flighted sections was modeled using a series−parallel formulation of well-mixed tanks representing flight-borne and drum-borne solids. Compartment model parameters were estimated through geometric modeling of flight unloading and included important design loading constraints. The modeling approach accounted for variations in flight configuration and solids flow properties, flight and drum geometry, as well as operational parameters such as revolutions per minute. Model parameters characterizing kilning were estimated via parameter estimation. To validate the dynamic multiscale model of the dryer and estimate model parameters as functions of operational variables, a series of industrial residence time distribution (RTD) pulse tracer tests were performed. The estimated values for the solid kilning velocity and axial dispersion coefficient were similar to values obtained in previous studies of FRDs. The developed model was used to examine the effect of both rotational speed and the extent to which solids have adhered to internal surfaces, on both the RTD and internal solids distribution, which provided valuable insights into dryer efficiency. The solids transport model developed in this paper provides a firm basis for the realistic integration of energy considerations and the development of a full predictive dynamic mass and energy model for complex flighted rotary dryers.

Figure 14. Effect on RTD of hard scale accumulation on dryer internals.

Figure 15. Effect on active mass (airborne solids) of hard scale accumulation on dryer internals.

conditions and moisture profile for test 3 were used as the basis for these comparisons. It is worth noting that changing operating conditions would lead to changes in the moisture profile which are not included in these simulations. A truer picture of these effects would require inclusion of the energy transfers that occur in the dryer, which will be described in a later paper by the authors. However, these results still offer a reasonable approximation of the effects of scale accumulation and provide insights into operational limitations. In this study, increasing scale accumulation and the resultant reduction in drum internal diameter lead to both longer residence times and more dispersed solids. Increasing scale accumulation leads to significant reductions in the airborne phase and would result in poor drying performance, as observed during industry dryer performance monitoring. The effect of rotational speed on the RTD is examined in Figure 16. To avoid the complications of scaling, the operating conditions and moisture profile for test 4 were used as the basis for these comparisons. The mean residence time decreases and the residence time distribution narrows with an increase in the rotational speed. This substantial trend was also observed in an

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ASSOCIATED CONTENT

S Supporting Information *

Table S1, operating conditions for different RTD trials; Table S2, number of cells in each flighted section; Table S3, estimated parameters for different conditions in the dryer; Table S4, simulated dryer performance. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE As = area (m2) dx.doi.org/10.1021/ie502607m | Ind. Eng. Chem. Res. 2014, 53, 15980−15989


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(4) Danckwerts, P. V. Continuous Flow Systems Distribution of Residence times. Chem. Eng. Sci. 1953, 2, 1. (5) Fan, L. T.; Ahn, Y. K. Axial Dispersion of Solids in Rotary Flow Systems. Appl. Sci. Res. 1961, 10 (1), 465. (6) Mu, J.; Perlmutter, D. D. The Mixing of Granular Solids in a Rotary Cylinder. AIChE J. 1980, 26 (6), 928. (7) Sai, P. S. T.; Surender, G. D.; Damodaram, A. D.; Sureh, Z. G. P.; Sankaran, K. Residence Time Distribution and Material Flow Studies in a Rotary kiln. Metall. Trans. B 1990, 21B, 1005. (8) Ortiz, O. A.; Martinez, N. D.; Mengual, C. A.; Noriega, S. E. Steady State Simulation of a Rotary Kiln for Charcoal Activation. Lat. Am. Appl. Res. 2003, 33, 51. (9) Ortiz, O. A.; Suárez, G. I.; Nelson, B. Dynamic Simulation of a Pilot Rotary Kiln for Charcoal Activation. Comput. Chem. Eng. 2005, 29, 1839. (10) Sheehan, M. F. A Study of the Material Transport in a Flighted Rotary Dryer. B.Eng. (Honours) Thesis, University of Queensland, Brisbane, Australia, 1993. (11) Kohav, T.; Riachardson, J. T.; Luss, D. Axial Dispersion of Solid Particles in Continuous Rotary Kiln. AIChE J. 1995, 41 (11), 2465. (12) Duchesne, C.; Thibault, J.; Bazin, C. Modelling of the Solids Transportation within an Industrial Rotary Dryer: A simple model. Ind. Eng. Chem. Res. 1996, 35, 2334. (13) Matchett, A. J.; Baker, C. G. J. Particle Residence Times in Cascading Rotary Dryers, Part 1Derivation of the Two-Stream Model. J. Sep. Process Technol. 1987, 8, 11. (14) Britton, P. F.; Sheehan, M. E.; Schneider, P. A. A Physical Description of Solids transport in Flighted Rotary Dryers. Powder Technol. 2006, 165, 153. (15) Lee, A.; Sheehan, M. E. Development of a Geometric Flight Unloading Model for Flighted Rotary Dryers. Powder Technol. 2010, 198 (3), 395. (16) Rojas, R.; Pina, J.; Bucala, V. Solids Transport Modeling in a Fluidized Drum Granulator. Ind. Eng. Chem. Res. 2010, 49, 6986. (17) gPROMS; Process Systems Enterprise Ltd.: London, 2010. (18) Arbuckle, D.; Brash, I. Technical Report on the Monitoring Conducted at the Pasminco Century Zinc Mine Portside in Karumba; Draft report May 01078; Unilabs Environmental: Loganholme, Australia, 2001. (19) Davis, M. E.; Davis, R. J. Fundamentals of Chemical Reaction Engineering; McGraw-Hill Higher Education: New York, 2003. (20) Sherritt, R. G.; Chaouki, J.; Mehrotra, A. K.; Behie, L. A. Axial Dispersion in the Three-Dimensional Mixing of Particles in a Rotating Drum Reactor. Chem. Eng. Sci. 2003, 58, 401. (21) Lee, A. Modelling Transport Phenomena within Flighted Rotary Dryers. Ph.D. Thesis, James Cook University, Townsville, Australia, 2008. (22) Ajayi, O. O.; Sheehan, M. E. Design Loading of Free and Cohesive Solids in Flighted Rotary Dryer. Chem. Eng. Sci. 2012, 73, 400. (23) Iguaz, A.; Esnoz, A.; Martinez, G.; Lopez, A.; Virseda, P. Mathematical Modelling and Simulation for the Drying Process of Vegetable Wholesale By-products in a Rotary Dryer. J. Food Eng. 2003, 59, 15. (24) Mendenhall, W. Introduction to Probability and Statistics; Brooks/ Cole: Boston, 1979. (25) Hatzilyberis, K. S.; Androutsopoulos, G. P. An RTD Study for the Flow of Lignite Particles through a Pilot Rotary Rryer. Part II: flighted drum case. Drying Technol. 1999, 17 (4 & 5), 759.

CF = forward step coefficient D̅ = forward step (m) F = mass flow rate (kgwet solid/s) g = acceleration due to gravity (m/s2) H = holdup (kgwet solid) k2, k3, k4 = transport coefficients (s−1) L = length of the dryer (m) L(i) = length of compartment cell (m) Ls = length of flighted section (m) m = mass (kg) Mp design = design load mass (kg) ms = mass per length (kg/m) ṁ j = mass flow rate of flight j (kg/s) Nc = number of cells Nf = number of flights 7 = axial dispersion coefficient (m2/s) R = radius of clean dryer (m) RD = radius of dirty dryer (m) RF = flight tip radius (m) SF = scale accumulation factor S1 = flight base of clean dryer (m) S1D = flight base of scaled dryer (m) S2 = flight tip of clean dryer (m) S2D = flight tip of scaled dryer (m) t = time (s) ta = active cycle time tα̅ = mass averaged active cycle time (s) tp̅ = mass averaged passive cycle time (s) us = solid velocity (m/s) xt = tracer concentration (mg/kg) xw = moisture content (kg/kgwet solid) Y̅ij = measured RTD value in experiment i (eq 23) Yij = predicted RTD value in experiment i (eq 23) Z = total number of experiments (eq 23) Δz = length (m) Greek Symbols

Γ = objective function (eq 23) ϕ = dynamic angle of repose (deg) σij2 = variance of RTD data in experiment i (eq 23) ω = rotational speed (rad/s) ρs = density of solid (kg/m3) θ = slope of the drum (rad) θk = kilning angle (rad) τ = residence time (s) ϑ = set of model parameters to be estimated (us, 7 ) in eq 23 α, βi = number of experiments performed (eq 23) Subscripts

a = active phase FUF = first loading flight p = passive uf = unflighted section f = flighted section s = solid

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REFERENCES

(1) Yliniemi, L. Advanced Control of a Rotary Dryer. Ph.D. Thesis, University of Oulu, Oulu, Finland, 1999. (2) Renaud, M.; Thibault, J.; Trusiak, A. Solids Transportation Model of an Industrial Rotary Dryer. Drying Technol. 2000, 18 (4 & 5), 843. (3) Sheehan, M. E.; Britton, P. F.; Schneider, P. A. A Model for Solids Transport in Flighted Rotary Dryers based on Physical Considerations. Chem. Eng. Sci. 2005, 60, 4171. 15989


Pseudophysical Compartment Modeling of an Industrial Rotary Dryer

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