Article pubs.acs.org/IECR
Pseudophysical Compartment Modeling of an Industrial Flighted Rotary Dryer with Flighted and Unflighted Sections: Energy 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 multiscale dynamic mass and energy process model was developed for an industrial rotary dryer with both unflighted and flighted sections. This paper focuses on the development and integration of energy balances into a validated solid transport model published by the authors in an earlier paper. In order to facilitate the drying process, a gas phase model is introduced. The gas phase in both the unflighted and flighted sections was modeled as a plug flow system. Simulations and parameter estimation were undertaken using gPROMS (process modeling software). Parameter estimation and model validation were carried out using the experimental moisture content profile, residence time distribution data, and gas and solid internal temperature profiles. The developed model was utilized to gain better understanding of the interactions between solids transport, operational parameters, and internal solids distribution on heat and mass transfer that occur in flighted rotary dryers. The model provides a means to quantify and solve dryer operational challenges in a real industry example.
1. INTRODUCTION Drying is an operation of great importance in industrial applications throughout the food, agricultural, mining, and manufacturing sectors. It is usually the final step in a series of operations, and the product from a dryer is often ready for final packaging. Thus, there are tight controls necessary on dryer performance, which often can only be addressed using reliable models. In process industries where rotary dryers are used, the outlet moisture content and temperature are used as performance criteria. The cocurrent industrial rotary dryer examined in this study is used in drying zinc and lead concentrate (MMG, Karumba, 2008−2010 seasons). The dryer has both unflighted and flighted sections. Each flighted section has a different flight configuration, although all flights are standard two-stage designs. Operational challenges that lead to the requirement for a dynamic model include high fuel consumption, poor gas−solids contact, and performance deteriorating that is time dependent associated with solids adhering to and accumulating on the internal surfaces of the dryer. There have been rotary dryer models described in the literature which have been used to predict the moisture content and temperature profiles inside dryers.1−4 These models differ in the way the drying rate, heat transfer, and the residence time are described, as well as model structure. Douglas et al.2 developed a model based on heat and mass balances to illustrate the effect of changes in inlet conditions on the outlet conditions of sugar dryers. Both the residence time and volumetric heat coefficient were calculated using Friedman and Marshall empirical correlations,5,6 which are invariants to important properties such as flight geometry. The effects of flight geometry and solid distribution were not considered. Wang et al.7 developed a generalized distributed parameter model for a sugar dryer. The heat transfer coefficients were calculated using three different correlations for comparison. These correlations include Freidman−Marshall,6 Hironsue,8 and Ranz−Marshall.9 The gas phase was modeled as a plug flow © 2015 American Chemical Society
system, and the residence time was again calculated via the Friedman and Marshall model. The authors concluded that a dynamic rotary dryer model should be developed to account for flight geometry and varying solid’s distribution within the dryer. Duchesne et al.10 developed a dynamic simulator of a mineral concentrate rotary dryer, which consisted of a furnace model, a solid transport model, and a gas model. Their modeling approach was different from previous studies2,7,11 in that solid transport was modeled using a compartment model, empirically fitted to measured residence time distribution data. The predicted and measured values of the outlet variables, such as moisture and gas temperature, were comparable, and the modeling approach provided a good foundation for future model development. Cao and Langrish1 also developed an overall system model for a counter-current, cascading dryer. The model used heat and mass balances around the dryer, together with the Matchett and Baker12 mechanistic residence time model. The heattransfer correlation of Ranz and Marshall9 was used, but the solids transport was again semiempirical, requiring dryer specific experiments. Shahhosseni et al.4 used an adaptive modeling strategy that combined online model identification with well-known conservation laws, but the solids transport was calculated using the modified Friedman and Marshall5 correlation. Despite numerous models in literature, there remain deficiencies and uncertainty in modeling of dryer solid transport and particularly the coupling of solids loading to energy transfer. Dynamics models of flighted rotary dryers are rarely presented. Neglecting the importance of dryer loading and internal solids distribution, which are dependent on flight and Received: Revised: Accepted: Published: 12331
August 2, 2015 November 18, 2015 November 19, 2015 November 19, 2015 DOI: 10.1021/acs.iecr.5b02833 Ind. Eng. Chem. Res. 2015, 54, 12331−12341
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Industrial & Engineering Chemistry Research
quantity of fuel oil fed to the combustion chamber in order raise or lower inlet gas temperature. A series of validation trials were undertaken where a selection of process and dryer performance data were collected. The Process Information was collected using sensors and this information is referred to as PI data. The dryer’s shell temperature was measured using infrared (IR) thermography. Other data obtained during the industry trials were: spatial sampling of the solid along the length of dryer; moisture content; particle size; and measurement of hard scale accumulation to the dryer internals. Process Information data included fuel consumption; solid feed rate; inlet and outlet solid and gas temperatures; and fan opening. Internal temperature profiles across the dryer were not determined because of the potential safety hazards. The industrial trials were carried out under conditions as close to steady state as possible.
drum geometric configuration, is common. This commonly involves the use of empirical and semiempirical correlations, which are invariant to solids moisture content and flight geometry. Thus, we consider it important to describe a dynamic model that addresses these limitations through the use of physically realistic compartment modeling. As an alternative, we have chosen to use a dynamic solids transport model, which is dependent on flight and drum geometry, as well as material properties.13,14 Of particular importance is the inclusion of physically realistic model compartments to represent airborne and drum borne solids because they provide realistic framework for embedding energy relations. This framework has been used in a variety of examples,13,14 but inclusion of energy transfers has not been discussed. In this paper, we describe the incorporation of energetic processes onto a dynamic pseudophysical compartment model (PPCM) capable of reproducing a realistic, geometrically dependent, dynamic solid distribution, as well as operationally dependent RTD’s. This paper extends the solid transport modeling presented in Part 1 of this paper (Ajayi and Sheehan)15 by including energy relations into the validated solid transport model. In the previously developed dynamic solid transport model15 a moisture content profile was assumed and no energetic processes were included. Including energy transfers into that validated model provides a more thorough understanding of the complexities involved in flight rotary drying. To model the energetics of drying process, a gas phase transport model is introduced. Simulations and parameter estimations are described, and results are assessed against industry data. The model results are used to examine critical assumptions in flighted rotary dryer modeling and to study the effect of key control and design variables that are of critical importance to industry practitioners.
3. MODEL DEVELOPMENT The model equations described in this paper predominantly describe the energy transfers that occur between solids and gas in the industrial case study flighted rotary dryer. The framework for this model is a PPCM structure developed to describe solids transport and validated in Ajayi and Sheehan.15 The model was capable of reproducing industrial RTDs and was fitted over a range of operating conditions. The model combined both partial differential and ordinary differential modeling techniques to represent the five separate sections within the dryer, including both flighted and unflighted sections. A brief selection of these equations is provided to give the reader a better understanding of the base model structure and the integration of energetic transfer within it. 3.1. Model structureEnergy transfers. The kilning phase occurring in the unflighted sections was modeled as an axial dispersed plug flow system (eq 1). According to the PPCM methodology, the solid transport in the flighted sections was partitioned into a series-parallel formulation of well-mixed tanks representing both air-borne, and drum and flight-borne solids. The air-borne solids are referred to as active solids and the drum-borne solids as passive. The mass balance on solids in both the active and passive phases of a single tank set (i) are presented in eqs 2 and 3, respectively (see also Figure 3). The compartment numbers and a limited number of model transport coefficients were derived through geometric flight discharge modeling, based on dryer geometry and solids physical properties. A comprehensive description of the multiscale solids transport model structure can be found in Ajayi and Sheehan.15
2. PROCESS DESCRIPTION Figure 1 shows the schematic diagram of the drying process. The drying process consists of the combustion chamber and
Figure 1. Schematic representation of an MMG combustion chamber and industrial rotary dryer.
∂ ∂2 ∂ (ms) = ρ 2 (ms) − (usms) ∂t ∂z ∂z
the rotary dryer. The geometrical configuration of the cocurrent dryer, such as drum and flight geometry, are full described in Ajayi and Sheehan.15 The concentrate is fed into the dryer via a screw feeder. The typical solid inlet moisture content varies between 16% and 18%, and the target outlet moisture content ranges between 12% and 12.5%. Hot gas enters the dryer at 500 °C via the combustion chamber. 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 content, solid flow rate and air flow into the combustion chamber (via the fan opening), and adjusts the
dmpi dt
(1)
= k4i − 1mpi − 1 + k 3i − 1CFmai − 1 + k 3i(1 − CF )mai − k 2impi − k4impi
(2)
dmai = k 2impi − k 3iCFmai − k 3i(1 − CF )mai dt
(3)
The major extension to the solids transport model structure is the inclusion of a gas phase, which acts as a sink or source for heat and evaporated water. Figure 2 illustrates the overall compartment model structure including the modeled energy 12332
DOI: 10.1021/acs.iecr.5b02833 Ind. Eng. Chem. Res. 2015, 54, 12331−12341
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Figure 2. Overall process model structure showing material and energy flows. Shaded tanks represent solids phases, and bold tanks represent gas phases.
As such, moisture balances for the passive and active phases are presented here, in eqs 4−5.
transfers. Transfer mechanisms included in the model are heat transferred to the solid from the gas by both convection (Qconv) and radiation (Qrad), evaporation mass transfer from the solids to the gas; and radiative heat loss from the dryer shell. Given the mean residence time of the gas phase was around 5 s (compared to the solid phase residence time of 15 min), the gas phase in both the flighted and unflighted sections was modeled as a plug flow system without dispersion. 3.2. Flighted section mass and energy balances. Figure 3 illustrates the model structure for a single compartment
Passive cell i mass balance: d(ximpi)
= xwi − 1mpi − 1k4i − 1 + xwi − 1mai − 1k 3i − 1CF
dt
+ xwimaik 3i(1 − CF ) − xwimpik 2i − xwimik4i (4)
Active cell i mass balance: dxwimai = xwimpik 2i − xwimaik 3iCF dt − xwimaik 3i(1 − CF ) − R w(i)
(5)
Energy balance across the passive cell i:
(
d mpixwi ∫
Tspi
To
CpwdT + mpi(1 − xwi) ∫
Tspi
To
Cps dT
)
dt Tspi − 1 Tspi ⎛ ⎞ ⎜ ⎟ = ⎜xwi − 1 CpwdT + (1 − xwi − 1) Cps dT ⎟·k4impi − 1 ⎜ ⎟ To To ⎝ ⎠
∫
⎛ ⎜ + ⎜xwi − 1 ⎜ ⎝
∫
Tsai − 1
∫
Tsai − 1
Cpw dT + (1 − xwi − 1)
To
∫ To
⎞ ⎟ Cps dT ⎟·k 3i − 1CFmai − 1 ⎟ ⎠
Tsai ⎞ ⎛ Tsai ⎟ ⎜ + ⎜xwi Cpw dT + (1 − xwi) Cps dT ⎟·k 3i(1 − CF )mai ⎟ ⎜ ⎠ ⎝ To To
Figure 3. Model structure for tank set i in a flighted section of the dryer.
∫
∫
Tspi ⎛ Tspi ⎞ ⎜ ⎟ − ⎜xwi Cpw dT + (1 − xwi) Cps dT ⎟·k 2impi ⎜ ⎟ To ⎝ To ⎠
or tank set within a flighted section of the dryer. A key assumption in this model is that all heat and mass transfers occur only from the active phase. The dry solid dynamic mass balances in the passive and active tanks are presented in eqs 2 and 3. Relations for the solid transport coefficients (k2,k3,k4, CF) are available elsewhere.15 In Ajayi and Sheehan,15 moisture balances were not provided because drying was not modeled.
∫
⎛ Tspi ⎜ − ⎜xwi Cpw dT ⎜ ⎝ To
∫
∫
Tspi
+ (1 − xwi)
∫ To
⎞ ⎟ Cps dT ⎟·k4impi ⎟ ⎠
(6) 12333
DOI: 10.1021/acs.iecr.5b02833 Ind. Eng. Chem. Res. 2015, 54, 12331−12341
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Industrial & Engineering Chemistry Research Energy balance across each active cell i. It was assumed gas−solid interaction occurred only in the active phase. d(maixwi ∫
Tsai
To
Tsai
CpwdT + mpi(1 − xwi) ∫
Cps dT )
To
dt = (xwi
∫T
Tspi
CpwdT + (1 − xwi)
o
− (xwi
Tspi
Cps dT )·k 2impi
o
∫T
Tsai
o
− (xwi
∫T
∫T
Tsai
o
Cpw dT + (1 − xwi)
∫T
Cpw dT + (1 − xwi)
∫T
− R w(i)(Hv +
∫T
Tsai
o
Tsai
o
Tsai
o
Cps dT )·k 3iCFmai Cps dT )·k 3i(1 − CF )mai
Cpw dT ) + Q conv(i) + Q rad(i)
(7)
Figure 4. Model structure of the unflighted section, including mass and energy flows.
In eqs 4−7, ma, mp, xw ,Cps, Cpw, Hv, To, Ts, and Rw are active mass (kg), passive mass (kg), moisture content (kg/kgwet solid), specific heat capacity of zinc concentrate (J/(kg·K)), specific heat capacity of liquid water (J/(kg·K)), latent heat of vaporization of water (J/kg), solid reference temperature (°C), solid temperature (°C) in cell i, and evaporation rate (kg/s), respectively. The specific heat capacities for liquid water and solid were assumed to be constant across the anticipated solid temperature range (25 to 55 °C). The total water and mass balance equations in the gas phase: Fg
i+1
= Fg + R w(i)
i+1
i+1
= Fg yw + R w(i) i
∂ (xwms ∂t
i+1
i+1
Tgi + 1
∫To
+ Fg yw i+1
i+1
i+1
∫To
= (1 − yw ) i
∫To
i
i+1
Cpw dT
Cpg dT
+ Fg yw Hv + Fg yw i
i+1
+ R w(i)(Hv +
∫T
i
Tsai
o
Tgi + 1
∫To
o
Cps dT )
∫
∫
∫
∂ ∂ (mg ) = − (vg mg ) ∂t ∂z
Cpw dT g
(13)
The mass balance on moisture in the gas is
Cpw dT ) − Q conv(i) − Q rad(i) − Q loss(i)
R ∂ ∂ (yw mg ) = − (vg yw mg ) + w ∂t ∂z Δz
(10)
where Fg, yw, Cpg, Tg, and Qloss are gas flow rate (kg/s), mole fraction of water, specific heat capacity of water vapor (J/(kg·K)), operating gas temperature (°C), and heat lost through the shell (J/s), respectively. The reference state for water in these equations is taken to be liquid at 0 °C. 3.3. Unflighted section. Figure 4 shows the model structure in the unflighted sections. The energy balances take a different format to the flighted sections because the solids transport is characterized by axially dispersed plug flow. To facilitate the derivation of the energy balance equations, the dry solids mass balance presented in eq 1 is augmented with the corresponding moisture balance. R ∂ ∂2 ∂ (xwms) = ρ 2 (xwms) − (usxwms) − w ∂t ∂z Δz ∂z
Ts
Assuming ideal gas behavior, the mass and energy balances on the gas phase are presented in eqs 13−15. Included in the gas phase energy balance are heat losses through the dryer shell (Qloss) as well as the usual convection and radiation terms. The mass balance equation on the gas across a differential element (Δz) is
g
Tgi
o
∫T
∫
Cpg dT + Fg yw Hv
Tgi + 1
Cpw dT + (1 − xw)ms
∫
Energy balance in the gas phase: Fg (1 − yw )
Ts
Ts Ts ∂2 (x m Cpw dT + (1 − xw)ms Cpw dT ) 2 w s To To ∂z Ts Ts ∂ − us (xwms Cpw dT + (1 − xw)ms Cpw dT ) To To ∂z Ts Q Q R − w (Hv + Cpw dT ) + conv + rad To Δz Δz Δz (12)
(9)
i
∫T
=ρ
(8)
i
Fg yw
The energy balance over the solid phase, including convection (Qconv), radiation (Qrad), and evaporation (Rw), is expressed in eq 12:
(14)
Energy balance on the gas across a differential element (Δz) is ∂((1 − yw )mg
∫0
Tg
Cpg dT + yw mg Hv + yw mg
∫0
Tg
Cpw dT g
− (nRT )g − (nRT )w )/∂t = −vg ∂((1 − yw )mg + yw mg −
(11)
∫0
Q conv Δz
Tg
−
∫0
Tg
Cpg dT + yw mg Hv
Cpw dT )/∂z + g
Q rad Δz
−
Rw (Hv + Cpw Ts) Δz
Q loss Δz
(15)
Following the established convention for boundary conditions,15,16 the Danckwerts boundary conditions17 were used to solve eqs 11−15:
where ms and Rw are the mass per length (kg/m) and the rate of moisture removal (kg/s) within a slice (Δz), respectively. 12334
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Industrial & Engineering Chemistry Research ⎛ ⎞ h P jm = ⎜⎜ m v ⎟⎟Sc 2/3 ⎝ vg ρg ⎠
At the inlet where z = 0: xw(0) = xinlet
(20)
mg (0) = Fg /vg
Thus, the mass transfer coefficient (hm) is
Ts(0) = Ts(inlet )
⎛ Stvg ρg ⎞⎛ Pr ⎞2/3 ⎟⎟⎜ ⎟ hm = ⎜⎜ ⎝ Pv ⎠⎝ Sc ⎠
yw (0) = yinlet Tg(0) = Tg(inlet )
The Stanton (St) and Schmidt (Sc) numbers were defined as follows:
(16)
At the outlet where z = L, we ensure there is only convective flow and no dispersion of either heat or mass: ρ ρ
∂ 2xwms ∂z 2
Sc =
∂z 2 ∂mg ∂z ∂yw mg ∂z
St =
=0
∂ 2(Cpw Tx s wms + Cps Ts(1 − xw)ms )
=0
=0
∂(vg yw mg Hv + vg yw mg Cpv (Tg − 100) (17)
μg Dρg
(23)
Q conv = hcAs (Tg − Ts)
(24)
Q rad = εσAs (T g4 − Ts4)
(25)
Nu = 0.33Re 0.6
(26)
The Nusselt (Nu) and Reynolds numbers (Re) were defined as follows (eqs 27 and 28). Nu =
Re =
(18)
hcdpt λg
(27)
vg ρg dpt μg
(28)
Heat loss from the dryer shell was calculated by determining individual thermal resistances according to eq 29.21
where hm, Pw, Pv, and As are the mass transfer coefficient, water vapor pressure at the temperature of the solids being dried, partial pressure of water vapor in the gas phase, and surface area of solid particles in contact with the incoming gas, respectively. This approach has been used in a wide variety of industry drying models including mineral concentrate dryers and raw sugar dryers.10,18 The advantage of this approach is that the drying rate is responsive to changes in gas and solids properties such as moisture content and temperature. Furthermore, loading and geometric configuration are intrinsically linked via the inclusion of a term describing interfacial contact area (As) between the solids and gas phases. The estimation of the interfacial contact area differs for both the unflighted and flighted sections, and its determination is described in Section 3.6. The impact of assumptions regarding interfacial contact area are particularly important and are discussed in Section 4.1. The mass transfer coefficient (hm) was calculated using the Chilton−Colburn analogy (jH = jm)
jH = StPr 2/3
(22)
In line with previous studies,18−20 the Nusselt number (Nu) correlation for spherical particles in air expressed was used.
3.4. Drying rate. One of the most important parameters in the model is the drying rate (Rw). In this study, we take a classical approach and assume that surface drying is ratedetermining step. The thin film model is used, which assumed that the drying rate is proportional to the difference between the partial pressure of water in the gas phase, and vapor pressure of water in the thin film surrounding the solids. The film is assumed to be at the same temperature as the solids (eq 18).
R w = hmAs (Pw − Pv)
hc Cpair vg ρg
3.5. Heat transfer. The study assumed heat transfer from the gas to solid by both convection and radiation as described in eqs 24 and 25.
=0
+ vg (1 − yw )Cpg (Tg − 100))/∂z = 0
(21)
⎛ Tg − Tamb ⎞ ⎟⎟ Q loss = θloss⎜⎜ ⎝ ∑ Rj ⎠
(29)
where Tg, Tamb, and Rj, are gas temperature, ambient temperature, and resistance in the radial direction, respectively. θloss was included as a tuning factor to account for uncertainties in the shell heat transfer model, and was adjusted during model fitting to match the predicted gas outlet temperature to the measured outlet temperature. In this study, the intermittent contact between the solids and the walls (and the insulating effect) was ignored in the resistance analysis. Hence, the heat loss was assumed uniform over the entire circumference of the shell. A range of heat transfer mechanisms were considered in the resistance analysis:21 forced convection from the hot gas to the dryer inside surface; radiation from the hot gas to the dryer inside surface; conduction through the drum wall; free convection from the outside dryer surface; radiation from the outside of
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Given the use of an isolated sphere approach, there are still uncertainties in this model and a correction factor (θa) was also included.
the dryer surface to ambient (eq 30).
∑ Rj =
ln(Ro/rin) 1 + (hin2πrinL + hradin 2πRL) kc 2πrinL 1 + (hout 2πR oL + hradout 2πRoL)
⎧ ⎛ 4π (0.5d )2 ⎞⎫ ⎪ ⎪ Ma p ⎜ ⎟⎬ Aa = θa⎨ ⎜ ⎟ 3 ⎪ ⎪ 2 ⎠⎭ ⎩ (4/3π(0.5dp) )ρpt ⎝
(30)
The Sieder and Tate correlation22 was used to calculate the internal forced convective heat transfer coefficient (eq 31) and the internal radiation heat transfer coefficient was estimated using eq 32.23 Emissivity was taken as 0.9. ⎛ μ ⎞0.14 ⎞ ⎛ Kair ⎞ ⎛ hin = ⎜ in ⎟ × ⎜0.027Re 4/5Pr1/3⎜⎜ ⎟⎟ ⎟ ⎝ Din ⎠ ⎜⎝ ⎝ μs ⎠ ⎟⎠
(31)
hradin = εσ(Tg + Tw)(Tg2 + Tw2)
(32)
In the unflighted sections, convection, radiation and evaporation are assumed to occur, and a different methodology is adopted. In unflighted dryers, the surface area in contact with the gas largely depends on the solid bed’s motion within the drum, which is a function of the rotational speed and dryer geometry. The most common solid movement in an industrial rotary drum is the rolling mode, because of its low rotational speed.27 The rolling mode is characterized by two distinct regions: a top active layer where there is vigorous mixing; and an underlying passive layer where there is little or no gas−solid interaction occurs. We assume that drying only occurs in the top active layer. The estimation of the mass of particles in the active layer has been a subject of interest for various researchers.28−31 Henein et al.32 measured the thickness of the active layer to be less than eight particle diameters, but others have modeled the active layer as a flat-bed.33,34 Using the known quantity of solids rolling in the base of the drum, geometric modeling was used to calculate the cross sectional area of solids (Akiln) and the length of the exposed surface (Lc), using eq 36.14 The area of the active layer was calculated using the length of the exposed surface (Lc) and assuming that the thickness of the active layer is the diameter of a single particle (dp), via eq 37.
The external heat transfer coefficient (convection (hout)) and radiation heat transfer (hradout) were calculated using the Churchill and Chu correlation24 (eq 33 and eq 34, respectively).23 ⎞2 ⎛ Kair ⎞ ⎛ 0.387Ra1/6 ⎟ hout = ⎜ out ⎟ × ⎜0.60 + ⎝ Dout ⎠ ⎝ (1 + (0.559/Pr )9/16 )8/27 ⎠ (33)
hradout = εσ(Tw +
Tamb)(Tw2
+
2 Tamb )
(35)
(34)
3.6. Interphase contact area. The interfacial surface area available for heat and mass transfer in flighted rotary drying is one of the most challenging properties to accurately estimate, and most examples of FRD modeling in the literature have avoided explicit definition of this property. Typical assumptions taken in the past have included assuming that all particles in the dryer undergo equal rates of drying (and heat transfer). In these examples interfacial contact area is derived from the total dyer holdup, via calculations to determine the total number of particles in the dryer. It is obvious that the rates of heat and mass transfer (to and from the particles) in the airborne phase, would be substantially greater those for those particles in the flights and base of the drum (referred to as the passive phase). With reliable estimates of the distribution of solids between these two phases, it becomes possible to investigate the relative rates of heat and mass transfer in these phases. The PPCM framework is particularly well-suited to this task, because it is constructed on the basis of achieving a physically realistic partitioning of these phases and being responsive to changes in dryer loading, drum and flight geometry, and operational parameters. In the modeling of the flighted sections of the dryer, we assume all drying occurs only in the air-borne phase. The contact surface area in each compartment is equal to half the sum of the total surface area of the suspended/airborne particles in that compartment, assuming only one side of each particle is directly exposed to the gas stream. We refer to this as an “isolated sphere approach”, which assumes that each airborne particle experiences the same conditions. This is a common assumption, but obviously has its shortcomings because it is well-known that both drag and heat transfer within curtains of particles is different to that experienced by isolated particles.25,26 In this paper, the surface area (Aa) of airborne (or active) particles in contact with the gas stream was calculated via eq 35.
mkiln R2 = Akiln = (θk − sin θk) ρ 2
(36)
AAL = Lc × dp
(37)
where mkiln, θk, Lc, and dp are kilning mass (kg/m), kilning angle (radian), chordal length (m), and particle diameter (m) respectively. The mass of active layer (Figure 5) was determined from the ratio of the total and active layer cross sectional areas (eq 38). ⎛A ⎞ MAL = ⎜ AL ⎟ × (ms × Δz) ⎝ Akiln ⎠
(38)
Figure 5. Active layer and passive layer in the kilning section (Akiln is the chordal area (kilning area)).
where ms and Δz are the mass of solid in each cell (kg/m) and length of the discretized cell (m), respectively. The interfacial surface area of the active layer was calculated by determining the number of solid particles within the active layer and assuming that half of the total surface area of each solid particle in the active layer is exposed to the gas stream (eq 39). 12336
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The model equations in Section 3 and the equations defining dryer solids transport (Ajayi and Sheehan15) were solved simultaneously using gProms modeling software on a high performance PC. Because of the complexity and nonlinearity of the equations, typical solution times for the parameter estimations were roughly 10 min. A single set of comprehensive performance data (RTD, moisture content, shell temperature profile, gas and solid inlet and outlet temperatures, combustion system and dryer feed rates) were collected immediately after internal cleaning of the dryer, and during a time in which the operating conditions were as close to the original design criteria as possible. Following the RTD test, internal testing was undertaken to obtain particle size as a function of length and also to quantify the extent of hard scale adherence to the dryer walls and flight surfaces. The operational conditions of this test are outlined in Table 1. Mean particle diameters used in the
A correction factor (θAL) is also introduced to account for uncertainties ⎧ ⎛ 4π (0.5d )2 ⎞⎫ ⎪ ⎪ MAL pt ⎜ ⎟⎬ AAL = θAL⎨ ⎜ ⎟ 3 ⎪ ⎪ 2 ⎠⎭ ⎩ (4/3π(0.5dpt ) )ρpt ⎝
(39)
θAL and ρPt are the correction factor for the surface area of solid particles in the active layer, and density of a particle, respectively. In this work we use the interphase contact area per unit mass as a measure with which to examine fundamental differences in the rates of heat and mass transfer that occurs for the airborne and drum borne solids.
4. MODEL FITTING All model fitting was undertaken by using data collected during industrial trials on the full scale dryer. The solids transport and its governing parameters (k4, us and ρ) were determined by fitting to RTD data and validating model predictions against industrial trial data undertaken for different operating conditions.15 A key outcome of this modeling process was to obtain confidence in the model’s ability to accurately represent internal loading states (i.e., the distribution between the airborne and drum borne solids), and to embed predictability, with respect to changing geometric and operational characteristics, such as drum rpm. Energetic processes were neglected during this fitting, although the impact of drying on important material properties, such as repose angle, was accounted for by including a measured moisture content profile. This fitting and validation process is described in Ajayi and Sheehan.15 In this section, we focus on the use of model fitting techniques (i.e., parameter estimation) to match model predictions to experimental solids moisture, and gas temperature profiles. There are a number of areas that contribute to model uncertainty, including choosing appropriate heat transfer mechanisms, including the insulating effects of solids on the drum wall, embedding suitable physical realism into the rolling solids relations, and accounting for gas flow and channelling. However, we feel that the most significant areas of uncertainty lie in the modeling of heat and mass transfer, specifically, the estimation of heat and mass transfer coefficients, and the estimation of interfacial contact areas. Although, it is possible to blend CFD modeling of simplified particle phases into process models, and potentially reduce uncertainty, methods for doing so are not well-established, and model complexity would become a significant impediment to model utility, particularly when modeling a full-scale device. Since it is not feasible to decouple uncertainty in both interfacial area terms and transfer coefficients, in the parameter estimations described in this section, we lump uncertainty in both these properties into a single interfacial area correction factor for each phase (θAL, θa and θloss). Furthermore, given the model’s accurate representation of loading states and collected particle size data, these optimized values give interesting insights into relative rates of heat and mass transfer in the different phases. It is worth noting in the results that follow, that if there were no model uncertainty then the correction terms would be determined to be unity. In our analysis, we attribute a degree of physical meaning to the fitted values for the correction factors by ignoring uncertainty in the estimated heat and mass transfer coefficients and assuming that the isolated sphere approach is valid, which is appropriate given the universal adoption of these assumptions across all dryer modeling literature.
Table 1. Operating Conditions15 Description
Test 3
Solid feed rate (kgwet solid/s) Gas inlet temperature (oC) Solid inlet moisture content (%) Gas outlet temperature (oC) Product outlet temperature (oC) Product outlet moisture content (%) Rotational speed of the drum (rpm) Internal condition of the dryer
41 500 16.3 131 46 12.4 3 Unscaled
calculation of solid surface area in contact with the gas stream were based on experimental mean particle diameter data for dryer sections A,B,C,D and E (0.02 m, 0.015 m, 0.012 m, 0.008 m, 0.007 m, respectively). Gas properties such as density, thermal conductivity and dynamic viscosity were modeled as functions of gas temperature.23 The specific heat capacity of the water vapor was also modeled as a function of operating temperature.23 Three model parameters (thermal correction factors) were adjusted: the extent of heat and mass transfer in the active layer of the unflighted sections (θAL); the interfacial area of contact in the airborne solid phase across the flighted sections (θA); and also the thermal loss coefficient moderating heat loss from the dryer shell (θloss). In initial model fitting tests, the thermal correction factors were taken to be constant across the entire dryer. The parameter estimation process was executed in gPROMS software and was based on minimizing the sum of square errors between the modeled moisture content profile and measured moisture content profile. There are different variance models in gPROMS software and the constant relative variance model was used.35 Broadly speaking, this approach resulted in poor fitting of the experimental moisture content profile and led to the adoption of different thermal correction factor values for each section of the dryer. The correction factors for the flighted and unflighted sections were ultimately obtained by manually tuning to fit the moisture content profile, using the initial estimates as starting values. The resulting match between the modeled and industrial data is shown in Figure 6 and demonstrates an excellent correlation. The manually tuned surface area correction factors for the unflighted sections (A and E) were 2.4 and 1.4, respectively. The correction factors for the flighted sections B, C and D were 1.2, 0.6, and 0.42, respectively. All of the resulting correction factors were of a similar order of magnitude, are 12337
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Figure 6. Solid moisture content profile. Figure 7. Predicted internal solids temperature profile.
close to unity, and display consistent trends with respect to particle size. As such, their values provide a degree of confidence in the model’s underlying representation of the heat and mass transfer mechanisms. The predicted corrections factors reduce as drying progresses and particle size reduces. The isolated sphere assumptions is likely to be more reliable for the larger diameter particles at the feed end, compared to the smaller free-flowing particles at the product outlet end. Recent CFD simulations26 of particle curtains undergoing convective heat transfer has demonstrated that rates of heat transfer in small diameter particle curtains are reduced compared to those for isolated spheres. Clearly there is potential to utilize CFD simulations of particle curtains to reduce uncertainty in flighted rotary dryer modeling. When taken on a mass basis, the estimated interfacial surface areas for flighted sections B, C and D were 0.086 m2/kg, 0.054 m2/kg and 0.056 m2/kg, respectively. For the unflighted sections A and E, the interfacial surface areas were 0.0047 m2/kg and 0.0029 m2/kg, respectively. As expected, the interfacial areas per unit mass in the unflighted sections of the dryer were an order of magnitude smaller than those in the flighted sections. This reinforces the commonly held assumption that the rate of drying in the airborne phase is much more significant than in the bulk kilning rolling phase. However, it is also worth noting that the actual mass of solids in the airborne phase is typically 10 to 15% of the total mass. When taken in combination, the rates of water removal per unit length are comparable in both phases. In future energetic models of flighted rotary drying, we would recommend mass and heat transfer be modeled in both the active (airborne) and passive (flight and drum borne) phases, but that reduced interfacial area in the passive phase be included. The magnitude of the thermal loss coefficient moderating heat loss from the dryer shell (θloss) was taken to be constant across the entire dryer and manually tuned to match the measured gas outlet temperature. The manually tuned value was 15 which is reasonable considering the substantial uncertainty in describing these mechanisms. In future modeling of heat loss from dryers it is recommended that the effect of solid contact on the internal surfaces be accounted for by integrating the geometric model with a thermal resistance model 4.1. Model verification. Verification and validation of the underlying solids transport model is described in Ajayi and Sheehan.15 Furthermore, under steady state conditions, both mass and energy were conserved. The internal gas and solid temperature profiles across the dryer were not available for model validation. However, the profiles produced by the model showed expected behavior and appear very reasonable (Figures 7 and 8). The flattening of the shell temperature roughly corresponds to the
Figure 8. Predicted internal gas and measured external drum shell temperature profiles.
beginning of the flighted section and illustrates the insulating effect of solids contact on the wall surface. In this work, we assess the validity of the model by comparing the predicted values for the outlet properties with measured values. . A series of industry trials were used for this purpose. The operating conditions for these trials presented in Table S1. The comparison between the predicted and measured values of product moisture content, product temperature and gas outlet temperature are presented in Tables S2−S4. The predicted values of product moisture content and product temperature at different operating conditions are close to the measured data, illustrating the strong predictability and utility of the model, particularly given that moisture content and solid temperatures are key target control variables. There is some discrepancy in the gas outlet temperature but the trends are consistent. The model appears to overpredict the convective and radiation heat loss from the gas to the solid and drum walls. A more detailed investigation of heat resistance and drum temperature could improve this in the future. 4.2. Model investigations. Pragmatic multiscale dynamic dryer models such as that developed in this paper, are well suited to developing model-based control schemes for industrial dryers, as well as assisting in dryer design and retrofit studies. Model complexity is sufficient to capture the dynamic influence on outlet moisture content of important process variables such as rpm, solids feed rate, gas inlet temperature and internal scale build-up, while being numerically simple enough to run in real time. Furthermore, having a model that is responsive to variations in drum and flight geometry enables greater confidence when assessing new flight designs, or for example, when assessing the thermal impacts on dryer wall temperature 12338
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Industrial & Engineering Chemistry Research and dryer performance of insulating the drum outside surface. These later two scenarios are very common design challenges in industries that use flighted rotary drying units. To illustrate the utility of the model, the effect of operating parameters on solid moisture content was investigated. The operational parameters investigated were gas inlet temperature, drum rotational speed and buildup of internal scale. These parameters are important to consider because they have implications for control, design and maintenance of the industrial dryer. Figure 9 shows the effect of gas inlet temperature on the
Figure 10. Effect of rotational speed on solid moisture content profile.
Figure 9. Effect of gas inlet temperature on solid moisture content profile.
solid moisture profile. In the real industrial dryer control scheme, gas inlet temperature is used as the manipulated variable and increased temperature is achieved by increased fuel rate. The gas inlet temperature has a significant effect on the loss of moisture within the dryer reinforcing the suitability of this variable for moisture control. It was observed that there was an increase in the solid outlet temperature with an increase in the gas inlet temperature, which emphasizes the important role that convection plays in enhancing drying. The coupled effects of convection and evaporation are often under reported in the dryer modeling literature, where convection may even be neglected. Related examples that illustrate the importance of this coupling include the counterintuitive use of hot water additions to raw sugar feed to increase drying rates in FRD’s in that industry. Alternative approaches to dryer control, such as hot water addition, may offer new opportunities to enhance the efficiency of drying in the minerals industry, but predictive models such as the model outlined in this paper, are essential to quantifying the potential benefits (fuel reduction) and potential problems (high solid temperature). Figure 10 shows the effect of rotational speed on the solid moisture content profile within the dryer. Rotational speed affects both the residence time and the proportion of solids in the active and passive phases. Decreasing rpm slows the solids down, but also decreases the quantity of airborne solids. Our model predicted low sensitivity to this parameter with a slight increase in product moisture content and gas outlet temperature with an increase in rotational speed. A similar observation was reported by the Iguaz et al.3 in his study of the effects of dryer speed. The increase in the product moisture content can be attributed to reduced residence time but this effect is counteracted by the increase in the airborne phase solids. In the periods between regular maintenance cleans, the internal diameter and the flight capacity are progressively decreased as solids adhere to the walls and flights of the dryer. Control of moisture as performance deteriorates and predicting the degree of scale accumulation are important to the industry. In our
Figure 11. Effect of internal diameter on solid moisture content.
previous paper on solids transport in the case study dryer, the rate of solid accumulation on the walls was modeled using geometric arguments. Figure 11 shows the effect of the hard scale accumulation on the outlet solid moisture profile. In this figure, 0% scaling represents a flighted rotary dryer that has no solids adhering the flights or dryer’s walls (i.e., immediately following a scheduled clean). The model predicts limited effect in the unflighted sections of the dryer, which makes reasonable sense given heat and mass transfer conditions would remain consistent. However, as the rate of hard scale accumulation in the flighted sections increases, the flights loading capacity is reduced and there is a decrease in the amount of solid in the airborne phase. Since most of the drying occurs within the flighted sections, there is a significant deterioration in dryer performance. There is also a decrease in solid temperature as the hard scale accumulates.
5. CONCLUSION This paper builds upon a pseudophysical solids transport compartment model of an industrial flighted rotary dryer processing mineral concentrate, by extending the model to account for energy transfers. The model equations describing the dynamic energy transfers are presented for a dryer that includes both flighted and unflighted sections. The multiscale model structure described here is well-suited to modeling complex geometrically constrained unit operations across a range of scales. The underlying solids transport model structure, which differentiates airborne and flight/drum borne solids, facilitates detailed definition of the interfacial areas available for heat and mass transfer. Correction factors linked to available interfacial area terms are used to deal with uncertainty and were parameter 12339
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estimated by matching a moisture content profile obtained through industry trials. The estimated areas appear meaningful and provide good insights into fundamental assumptions used in flighted rotary dryer modeling. Rates of heat and mass transfer per unit mass for the flighted solids were found to be an order of magnitude greater than transfer rates per unit mass experienced in the drum phase. However, heat and mass transfer for flight and drum borne solids are still important to include, but the relative magnitude compared to flight borne solids should be properly considered. The model shows good predictability, and is validated by comparison to industry trial data across a variety of input−output measurements, including temperature and moisture. Model simulations were used to demonstrate the potential for using the model in control, design, and maintenance scheduling applications. In these simulations, maximizing convection was found to be an important mechanism for enhancing solids drying, but decreasing rpm had minimal effect on drying because of the reduced solids loading in the airborne phase.
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kgout Kh L Ls m ms Nc Nf Pr
P̅ Q rin R Ra Rj Ro Re RF Rw Sc Sh St T t us U̅ V̅ vg xw xt
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b02833. Tables S1−S4, showing operating conditions for different RTD trials, product moisture content, product temperature, and gas outlet temperature (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
yw
yw′ z Δz
ACKNOWLEDGMENTS This research work was supported by MMG Limited, Australia.
thermal conductivity of gas (W/(m °C)) thermal conductivity of gas based on shell temperature (W(/m °C)) correction factor for convective heat transfer coefficient length of the dryer (m) length of flighted section (m) mass (kg) mass per length (kg/m) number of cells (−) number of flight (−) Prandtl number pressure in ideal gas equation (Pa) heat (J/s) inside radius (m) ideal gas constant (J/(mol·K)) Rayleigh number Resistance ((m2·s·K)/J) outside radius (m) Reynolds number flight tip radius (m) drying rate (kg/s) Schmidt number Sherwood number Stanton number temperature (°C) time (s) solid velocity (m/s) internal energy (J/kg) volume in ideal gas equation (m3) gas velocity (m/s) moisture content (kg/kgwet solid) tracer concentration (kg/kg) gas humidity (kg/kg) gas humidity (mol/mol) direction coordinate length (m)
Greek letters
NOMENCLATURE A area (m2) CF forward step coefficient (m) Cp specific heat capacity (J/(kg·K)) CV controlled variable D diameter (m) dP particle diameter (m) Dout outside diameter (m) mass flow rate (kgwet solid/s) F Froude number Fr g̅ acceleration of gravity (m/s2) g0 radial distribution function Gu Gukhman number H height of curtain (m) hc convective heat transfer (J/(m2·s·K)) hm mass transfer coefficient (kg/(m2·s·Pa)) H̅ enthalpy energy (J/kg) Hv latent heat of vaporization (kJ/kg) hin internal convective heat transfer coefficient (J/(m2·s·K)) external convective heat transfer coefficient (J/(m2·s·K)) hout hradin internal radiation heat transfer coefficient (J/(m2·s·K)) hradout external radiation heat transfer coefficient (J/(m2·s·K)) jH heat transfer factor jm mass transfer factor k2, k3, k4 transport coefficients (s−1)
ρ ρAB εr θ θAL θa θk θloss θs λs μ μs ρ σr ϕ
axial dispersion coefficient (m2/s) mass diffusivity (m2/s) emissivity slope of the drum (radians) area correction factor for the unflighted sections area correction factor for the flighted sections kilning angle (radians) heat loss tuning factor granular temperature (m2/s2) bulk viscosity (kg/m s) dynamic viscosity (kg/m s) shear viscosity (kg/m s) density (kg/m3) Stefan−Boltzmann constant (W/(m2 K4)) dynamic angle of repose (°)
Subscripts
a AL amb b ct conv g i, j 12340
active phase active layer ambient bulk curtain convection gas phase directional coordinates DOI: 10.1021/acs.iecr.5b02833 Ind. Eng. Chem. Res. 2015, 54, 12331−12341
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heat loss passive particle radiation solid vapor water
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