Numerical simulation of heat transfer during

International Journal of Heat and Mass Transfer 106 (2017) 263–279

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Numerical simulation of heat transfer during production of rutile titanium dioxide in a rotary kiln Ashish Agrawal, P.S. Ghoshdastidar ⇑ Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, U.P. 208016, India

a r t i c l e

i n f o

Article history: Received 24 February 2016 Received in revised form 4 October 2016 Accepted 7 October 2016

Keywords: Heat transfer Rotary kiln Rutile titanium dioxide Simulation

a b s t r a c t This paper presents a computational heat transfer model of a rotary kiln used for the production of rutile titanium dioxide by the calcination of paste-like hydrous titanium dioxide. The work details the modelling of several chemical reactions occurring in the solid bed region along with turbulent convection of gas, radiation heat exchange among hot gas, refractory wall and the solid surface, and conduction in the refractory wall. Finite-difference techniques are used and the steady state thermal conditions are assumed. The kiln is divided into axial segments of equal length. The solution is of marching type and proceeds from the solid inlet to the solid outlet. The direction of gas flow is opposite to that of the solids. Mass balance of each species in the solid charge, and mass and energy balances of the solid and gas in an axial segment are used to obtain solids and gas temperatures, and species concentration at the exit of that segment. The kiln length predicted by the present model is 45.75 m as compared to 45 m of an actual kiln reported by Ginsberg and Modigell (2011). The steady-state axial gas and solid temperature profiles have been also satisfactorily validated with the numerical results of the aforementioned paper. The output data consist of refractory wall temperature distribution, the axial solids and gas temperature profiles, axial solids composition profile, the length required for drying of the solid charge and the total kiln length required to achieve 98% conversion of anatase TiO2 to rutile TiO2. A detailed parametric study with respect to the controlling parameters such as percent water content (with respect to dry solids), solids flow rate, gas flow rate, kiln inclination angle and kiln rotational speed lent a good physical insight into the rutile-TiO2 production process in a rotary kiln. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction This paper presents a computer model of heat transfer during production of titanium dioxide white pigment in rutile form in a rotary kiln. 1.1. Production of rutile titanium dioxide (TiO2) in a rotary kiln Titanium dioxide is a white solid inorganic substance which is used as a pigment or whitener in paints, paper, plastics, textiles, and other products. It occurs in several polymorphs, among them, anatase and rutile are manufactured in the chemical industry as white pigments. The pigment properties of rutile titanium dioxide are better than that of anatase titanium dioxide and are of more economical importance. Titanium dioxide white pigments are produced from a variety of ores by two different processes, namely, the sulphate process using concentrated sulphuric acid and the ⇑ Corresponding author. E-mail address: [email protected] (P.S. Ghoshdastidar). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.024 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

chloride process using chlorine gas. The last process step of the sulphate method, named calcination is performed in rotary kiln and has been considered in the present work. 1.2. Description of rotary kiln A rotary kiln consists of a refractory lined cylindrical shell mounted at a slight inclination from the horizontal plane (Fig. 1). The kiln is rotated at a very low speed about its longitudinal axis and the raw charge comprising hydrous titanium dioxide in a moist cake form is fed into the upper end of the cylinder and a hot combustion gas mixture at 1 bar flows from the other end. The gas is a mixture of products on burning of natural gas in a separate combustion chamber. In the present study, the kiln is considered to comprise three sections. In the first section, the wet solids are heated to the saturation temperature of water. In the second section, the liquid evaporates at constant temperature until the charge is completely devoid of moisture. In the third section, the solids are heated till the required degree of conversion of anatase to rutile titanium

264

A. Agrawal, P.S. Ghoshdastidar / International Journal of Heat and Mass Transfer 106 (2017) 263–279

Nomenclature Ae;k Acg Ag ag Ai Aj,inner Cp D Dh _ k;z dm _ v;z dm Ea;k Eb f Fgj Fij Gr hcgw hcj hfg ho hws

Dhi k L L1 L2 L3 _ g;z m _ k;z m _ s;z m Mwk nk N Nk Nr Ns NuDh Pg Pr q1;z q2;z qcr;z qgp;z

frequency factor for kth reaction (s1) cross-section area of gas flow in an axial segment (m2) total surface area exposed to gas in an axial segment (m2) gravitational acceleration (9.81 m/s2) elemental area for ith element at inner wall in an axial segment (m2) contact area between wall and solids per unit element (m2) specific heat at constant pressure (J/kg K) diameter of the kiln (m) hydraulic diameter of gas flow (m) depletion rate of the reactant of kth reaction in an axial segment (kg/s) evaporation rate of moisture content in wet solid in concerned axial segment (kg/s) activation energy for kth reaction (J/mol) blackbody emission per unit area (W/m2) percentage of moisture content on dry basis in wet solids shape factor between gas and surface element j shape factor between surface elements i and j (including j = i) Grashof number average convective heat transfer coefficient from gas to refractory wall in an axial segment (W/m2 K) local convective heat transfer coefficient from gas to jth element of solids (W/m2 K) latent heat of vaporization of water (J/kg) convective heat transfer coefficient from outer wall to surroundings (W/m2 K) contact heat transfer coefficient between wall and solids (W/m2 K) heat of reaction for ith reaction (J/kg) thermal Conductivity (W/m K) length of the kiln (m) length of the first section of the kiln (m) length of the second section of the kiln (m) length of the third section of the kiln (m) gas mass flow rate at axial position z (kg/s) mass flow rate of the kth component of solid charge at axial position z (kg/s) solids mass flow rate at axial position z (kg/s) molecular weight of kth component of solids (g/mol) Order of kth reaction total number of surface elements in an axial segment rotational speed of the kiln (rev/min) number of surface elements exposed to gas in an axial segment number of surface elements covered by solid in an axial segment nusselt number based on hydraulic diameter Dh wetted perimeter of gas flow in an axial segment (m) Prandtl number net heat transfer from gas and exposed wall to solids (W) net heat transfer from covered wall to solids (W) net heat energy absorbed or released by chemical reactions at axial position z (W) thermal energy associated with the released gaseous products of chemical reactions (W)

qj qr;z R Rex ReDh Ru T g;z T j;inner T s;z Dt To,av U

vg

Vz x Xk y z

net heat transfer for jth surface element in an axial segment (wall or solids), (W) net heat transfer from gas to solids and wall (W) radius of the kiln, Fig. 2 Reynolds number based on relative velocity between wall and air outside the kiln Reynolds number based on hydraulic diameter Dh universal gas constant (8.314 J/mol K) gas temperature at axial position z (K) inner wall temperature at jth surface element (K) solids temperature at axial position z (K) residence time (s) average temperature at outer wall (K) circumferential speed of the kiln (m/s) mean velocity of gas (m/s) axial velocity of solids (m/s) radial coordinate (m), Fig. 2 mass fraction of the solid component k circumferential coordinate (m), Fig. 2 axial coordinate (m)

Greek letters a fill angle (deg), Fig. 2 arf thermal diffusivity, Eq. (4) b volumetric thermal expansion coefficient (K1) C fill angle (radian) e emissivity fk;z degree of conversion of kth reaction at axial position z r Stefan–Boltzmann constant (5.67 108 W/m2 K4) h as defined in Fig. 2 (deg) l dynamic viscosity (kg/m-s) m kinematic viscosity (m2/s) n Darcy friction factor, used in Gnielinski [35] correlation q density (kg/m3) sg transmissivity of the gas Ds time step (s) / kiln inclination angle (deg) Subscripts a air cr chemical reaction g gas gs gas to solids i element number of the wall or the solid j element number of the wall or the solid k number of reaction, also number of component in solid charge l liquid oxygen O2 s solids sh shell or outer wall SO2 sulphur dioxide rf refractory wall v vapour w water ws wall to solids z at an axial distance z from the solids inlet z + Dz at a distance z + Dz from the solids inlet Abbreviations CFD Computational Fluid Dynamics


265

order chemical reactions occur. They are modelled by a modified form of Arrhenius rate law. 1.4. Literature review

Fig. 1. Simplified schematic diagram of a rutile-TiO2 rotary kiln.

dioxide is achieved and then are released from the kiln. The calcination process produces TiO2 pigment with a rutile content of around 98%. The lengths of the first, second and third sections of the kiln are denoted by L1, L2 and L3, respectively. The total length of the kiln is L which is the sum of three individual lengths. Heat transfer processes in the kiln, the nomenclature of the kiln and the coordinate system are shown in Fig. 2.

1.3. Basic solution methodology A model of heat transfer among the gas, refractory wall and solid has been developed. The radiation heat transfer is modelled by dividing the solid surface and the wall into surface elements. The turbulent convection heat transfer from the gas to the wall and solid is estimated by a stand-alone CFD model assuming a non-rotating kiln with stationary solid bed. It is in this part that FLUENT, a commercial software which applies the finite-volume method is made use of. The local convective heat transfer coefficients obtained from the FLUENT simulation are then supplied to the main computer program developed in-house. Since the refractory wall is alternately heated and cooled during each revolution, the quasi-steady heat conduction in the wall is present. The energy equation for the wall is discretized using the finite-difference technique. The mass and energy balance equations are solved for the solid and the gas assuming only axial temperature variation, since the solid and the gas are well-mixed in the cross-sectional plane of the kiln, giving rise to uniform temperature in the transversal section. The ‘‘well-mixed” assumption is valid since the solid bed motion is in rolling mode (as the kiln speed is small) and the gas flow is turbulent. A number of higher order as well as fractional

The literature on rotary kilns can be divided into two parts, namely, (i) studies related to modelling without chemical reactions; and (ii) studies related to modelling with chemical reactions. The literature review that follows covers these aspects. 1.4.1. Studies related to modelling without chemical reactions Sass [1] developed a computer model for heat transfer in a rotary kiln dryer using empirical relationships for radiation heat transfer calculations. Kamke and Wilson [2] developed a computer model of a single-pass, rotary drum dryer for drying of wood particles with or without a center-fill flighting section. Cook and Cundy [3] presented an analytical model for the heat transfer process taking place between the heated wall of a rotating cylinder and the solid contained in it with an initial moisture content equal to 2–7% of the mass of the dry solids. Ghoshdastidar and Unni [4] developed a steady-state heat transfer model of drying and preheating of wet solids in the non-reacting zone of a cement rotary kiln. Ghoshdastidar et al. [5] have reported simulation results based on an improved heat transfer model for a rotary kiln used for drying and preheating of wet iron ore. Ghoshdastidar and Agarwal [6] performed a numerical simulation and optimization study of heat transfer in a rotary kiln used for drying and preheating of wood chips with superheated steam. Liu et al. [7] obtained an analytical solution to predict axial solid transport in rotary kilns. Schmidt and Nikrityuk [8] carried out a 2D numerical simulation of transient temperature distribution by granular mixing in a horizontal rotary kiln. Sonavane and Specht [9] presented a finiteelement method based numerical analysis of heat transfer in the wall of a rotary kiln. 1.4.2. Studies related to modelling with chemical reactions This can further be sub-categorized into two parts, namely, (i) production of titanium dioxide; and (ii) production of other chemicals. 1.4.2.1. Production of titanium dioxide. Ginsberg and Modigell [10] developed a comprehensive one-dimensional dynamic model of a rotary kiln for the production of titanium dioxide white pigment.

Fig. 2. Schematic cross-section of a rotary kiln showing heat transfer processes, the fill-angle and the coordinate system.

266


Their numerical predictions matched very well with the measurement data. Based on heat and mass transfer equations, Dumont and Bélanger [11] described a simplified process model for making of titanium dioxide in a rotary kiln. Roubal et al. [12] developed the Kalman Filter design based on the reduced model of the process in an attempt to control the titanium dioxide production in a rotary kiln. In both studies, dehydration and desulphurisation steps were not considered. Also, Koukkari et al. [13] introduced the RATEMIX model by combining multicomponent thermodynamics with chemical kinetics and applied in calcination of metatitanic acid to produce rutile titanium dioxide in a rotary drum calciner. 1.4.2.2. Production of other chemicals. Spang [14] developed a dynamic partial-differential equation model of cement kiln incorporating a flame model. Manitius et al. [15] described a mathematical model for production of aluminium oxide by calcination of basic ammonium aluminium sulphate. A steady state mathematical model of cement kiln was presented by Guruz and Bac [16] using zone method. Watkinson and Brimacombe [17] performed experimental work on pilot-scale rotary kiln for calcination of limestone with parametric study. Watkinson and Brimacombe [18] observed the effects of enriching the combustion air with oxygen in a rotary lime kiln. Hard and Mu [19] calculated sensitivities of certain process variables on fuel rate and solids throughput in modelling a phosphate nodulizing kiln. A detailed steady-state heat transfer model of burning of Plexiglas in a rotary kiln was developed by Ghoshdastidar et al. [20]. A 3D steady state model of a rotary calcining kiln was presented by Bui et al. [21] for the petroleum coke. Davis [22] reported a model for magnetite oxidation during iron ore pellet induration in a rotary kiln. A model was developed for iron oxide pelletizing to simulate effects of under bed injection on kiln fuel requirements and magnetite oxidation by Davis [23]. Georgallis et al. [24] presented a 3D model for rotary lime kiln which included evolution and combustion of species and granular bed motion with calcination reaction. Marias et al. [25] modelled pyrolysis of aluminium waste in a rotary kiln. Mintus et al. [26] predicted solid composition and temperature profiles and fuel requirement based on their one-dimensional cell model for wet process rotary cement kilns. Mujumdar and Ranade [27] presented a 1D model to analyse key processes occurring in solid bed of cement kilns. Mujumdar et al. [28] developed Rotary Cement Kiln Simulator by integrating the separate models for pre-heater, calciner, rotary kiln and cooler. Flow and transport processes were modelled in a calciner for cement production by Fidaros et al. [29]. Shahriari and Tarasiewicz [30] modelled a clinker rotary kiln using operating functions concept. 1.5. Objectives of the present work The above literature review reveals that only a few models [10– 13] of titanium dioxide rotary kiln are available. The main difference between the present and aforementioned ones is that the latter do not have the capability of predicting the kiln length required to produce rutileTiO2 for a given set of input parameters. The present work is an attempt to fill this gap in the existing literature. The objectives of the present study are as follows. 1. To develop a detailed computer model of heat transfer with chemical reactions during the production of rutile titanium dioxide in a rotary kiln; 2. To compare the total predicted kiln length required to achieve the desired solids composition as predicted by the present model with that of the actual kiln reported in Ginsberg and Modigell [10]; 3. To predict axial distributions of solid temperature, gas temperature, composition of the solid charge within the kiln;

4. To conduct a detailed parametric study with respect to moisture percent (dry basis), mass flow rate of the solid, mass flow rate of the gas, kiln inclination angle and kiln rotational speed in order to get a good physical insight into the manufacturing process of rutile titanium dioxide in a rotary kiln. 2. Problem formulation In this section the reaction processes, modelling of radiation and conduction, gas convection, reaction kinetics, and, mass and energy balances in the solids and gas are presented in detail. 2.1. Reaction Processes Occurring in a Rutile-TiO2 Rotary Kiln In the sulphate process, ilmenite or titanium slag as raw material is digested with sulphuric acid and results in a solution containing TiOSO4 and FeSO4 . After removing iron and other impurities by vacuum crystallization from the solution, titanium sulphate is hydrolysed to hydrous titanium dioxide. It largely consists of slurry of TiOðOHÞ2 (metatitanic acid) and a relatively small amount of TiOSO4 H2 O. This solid charge is then fed into a rotary kiln to undergo calcination process and finally, rutile titanium dioxide is produced. In addition to moisture evaporation at saturation temperature, following chemical reactions take place inside the rotary kiln as wet solids move from the inlet to the outlet of the kiln. Dehydration of metatitanic acid and TiOSO4 H2 Ooccurs in the range of 100–250 °C [31] (Reactions 1 and 2). Reaction 3 results in loss of sulphur trioxide (which immediately decomposes into SO2 and O2) from titanyl sulphate at about 650 °C [32] and finally, phase transformation of anatase to rutile takes place exothermally in the range of 700–950 °C [32] (Reaction 4). The above mentioned reactions are listed in Table 1a along with their respective heat of reactions, pre-exponential factors, activation energies and orders of reaction. The composition of wet solids at the solid inlet is given in Table 1b. 2.2. Radiation exchange among hot gas, refractory wall and the solid surface Since the gas temperature is high, thermal radiation plays an important role. The radiation is modelled by dividing the wall into surface elements as shown in Fig. 3. Each axial segment of the refractory surface is divided into Nr surface elements of equal size. The solid surface is divided into Ns surface elements. This can be considered as a 2-D enclosure since the surface elements are quite long. The temperature of the solid surface element and the gas are assumed to be uniform in each axial segment. The wall surface elements are assumed diffuse and gray. It is to be kept in mind that the surface elements exchange radiation only with the surfaces of the same axial segment which is sufficiently long. Since the hot gas contains CO2 and H2O and it is treated as radiatively participating. The radiation heat transfer is computed by using the theory of Hottel [33] for a gray enclosure containing a gas which emits, absorbs and transmits radiation. If the gas volume is enclosed by N gray surface elements at different temperatures, the net energy gain, qj at a particular surface can be expressed as

qj ¼

eþ1 2

"

N X Ag F gj eg Eb;g þ Ai F ij sg Eb;i Eb;j Aj

#

ð1Þ

i¼1

where e is the emissivity of the wall. Eq. (1) is valid when e is greater than 0.8, which is true in the present case. For the calculation of shape factors, F ij expressions derived by Ghoshdastidar et al. [20] have been used. eg and sg are the emissivity and transmis-

267

A. Agrawal, P.S. Ghoshdastidar / International Journal of Heat and Mass Transfer 106 (2017) 263–279 Table 1a Reactions, kinetics and heat of reaction data (Ginsberg and Modigell [10]). S. No.

Reaction Equation

Frequency Factor Ae;k (s1)

Activation Energy Ea;k (kJ/mol)

Heat of Reaction Dhk (kJ/mol)

Order of Reaction nk

1 2 3 4

TiOðOHÞ2 ! TiO2 ðaÞ þ H2 OðgÞ TiOSO4 H2 O ! TiOSO4 þ H2 OðgÞ TiOSO4 ! TiO2 ðaÞ þ SO2 ðgÞ þ 0:5O2 TiO2 ðaÞ ! TiO2 ðrÞ

4.1 103 5.0 105 5.0 109 1.7 1023

45 75 230 491

90 90 396 -5.6

3.00 0.50 0.30 0.67

Table 1b Mass fractions of reactants (Ginsberg and Modigell [10]). S. No.

Reactant

Mass fraction at solid inlet (w.r.t dry solids)

1 2

TiOðOHÞ2 TiOSO4 H2 O

0.8894 0.1106

Fig. 3. Section of the kiln showing surface elements of the wall and the solid.

sivity of the gas. The value of gas emissivity has been taken directly from Ginsberg and Modigell [10]. The gas transmissivity has been calculated using the expression,sg ¼ 1 eg , assuming gas reflectivity to be zero and Kirchhoff’s law of radiation to be valid.

Fig. 4. A two-dimensional grid showing triangular elements in a cross-sectional plane of the TiO2 kiln.

2.3. Gas convection In direct fired kilns of small diameter operating at solids temperature up to 1100 K gas convection may be significant (Brimacombe and Watkinson [34]). In order to bring more precision in the calculation of connective heat transfer from main heat source in the rotary kiln, that is hot gas mixture, local convective heat transfer coefficients on the exposed surfaces of refractory and solid bed are estimated a priori, via a secondary model of heat transfer. In this secondary model, a non-rotating kiln containing a stationary solid bed is simulated using a finite volume-based commercial CFD package ANSYS-FLUENT 15.0. An unstructured mesh with triangular cross-section prisms as elements is generated using commercial software Gambit 2.4.6. Fig. 4 shows a two-dimensional grid constructed via triangular elements in a cross-sectional plane of the TiO2-kiln. The length of the kiln used for this simulation is predicted from the main model. Since local convective heat transfer coefficients are not known in advance, average values for gas–solids and gasrefractory contact surfaces at each axial location are estimated using a correlation (Eq. (2)) for Nusselt number developed by Gnielinski [35] for turbulent flow in a smooth pipe.

NuDh ¼

ðn=8ÞðReDh 1000ÞPr 1 þ 12:7ðn=8Þ1=2 ðPr2=3 1Þ

ð2Þ

where, n is the Darcy friction factor and for smooth tubes, which is given by

n ¼ ð0:79lnðReDh Þ 1:64Þ

2

ð2aÞ

Gnielinski correlation is valid for,

0:5 6 Pr 6 2000 and 3000 6 ReDh 6 5 106 NuDh and ReDh are the Nusselt number and Reynolds number, respectively, based on hydraulic diameter for the gas region, Dh . Dh is expressed as

Dh ¼

4Acg Pg

ð2bÞ

where Acg and Pg are cross-sectional area and wetted perimeter of gas flow in an axial segment, respectively.

Acg ¼

C sin C þ 1 4 2p 2p

pD2

P g ¼ pD

a 360 a þ D sin 360 2

ð2cÞ ð2dÞ

where C and a are the fill angle of solids in radians and degrees, respectively. ReDh is expressed as

ReDh ¼

qg v g Dh 4m_ g ¼ lg P g lg

ð2eÞ

268


_ g are the average density, mean velocity, where qg , v g , lg and m dynamic viscosity and mass flow rate of gas mixture, respectively. The average convective heat transfer coefficient from gas to refractory wall in an axial segment is given by

hcgw ¼

NuDh kg Dh

ð2fÞ

where kg is the mean thermal conductivity of the gas mixture. The average convective heat transfer coefficient from gas to solids in an axial segment is assumed to be an order of magnitude higher than that from gas to refractory wall [36,37]. The Reynolds Number based on the hydraulic diameter of gas flow cross-section is found to be 101,672. The standard k-omega turbulence model is implemented. The conjugate heat transfer is considered at the interfaces of gas–solids and gas-refractory. Thermo-physical properties of gas mixture, refractory wall and bulk solid are listed in Table 2. At the solid inlet of the kiln the mass flow rate and temperature of gas mixture are as given in Table 2. At the outer peripheral wall and end walls of the kiln, a mixed boundary condition is specified with a heat transfer coefficient as estimated using a correlation given by Suryanarayana et al. [38]. The momentum, turbulent kinetic energy, turbulent dissipation rate and energy equation are solved with the second order upwind differencing scheme. The convergence criteria selected for the governing equations are 103 for the continuity, momentum, k and omega equations, and 106 for the energy equation. A grid independence test is performed based on average wall function heat transfer coefficients at the interfaces of gas–solids and gas-refractory. The optimal value of total number of cells used in the 3D computational domain is 3154424. The computations are performed on SONY workstation with core i5 processor and 4 GB memory by using ANSYS-FLUENT 15.0. The local convective heat transfer coefficients for each circumferential element (Fig. 3) in each axial segment are obtained from the aforesaid simulation. The range of the same is found to be 12.16 W/m2K to 21.08 W/m2K. These local values are imported into the main model (at the same locations in computational domain of the main model) as input data and convection heat transfer from gas to solids and gas to refractory is estimated. 2.4. Conduction in the refractory wall The conduction in the refractory wall is modelled assuming quasi-steady heat conduction since wall elements of the kiln are alternately heated and cooled during each revolution by gas and solid, respectively. Only radial heat conduction in the wall is taken into account, assuming negligible conduction in the circumferential and axial direction as the kiln dimensions are much larger in those directions. A non-rotating coordinate system is used to model the wall conduction equation (Eq. (4)). Because the refractory thickness (0.25 m) is appreciably smaller than kiln diameter (2.3 m), the ratio being about 10.9%, the effect of curvature is neglected and hence, Cartesian coordinates are used instead of cylindrical coordinates for the sake of simplicity.

U

@T @2T ¼ arf 2 @y @x

ð3Þ

where U is the circumferential speed of the wall and arf is the thermal diffusivity of the wall. The boundary condition for the inner wall depends on whether the surface element is exposed to the gas or it is covered by solids. The radiation heat transfer, qj from the gas to the inner surface elements is obtained from Eq. (1). In addition, convection heat transfer is calculated from the local convective heat transfer coefficients as discussed in Section 2.3. Surface elements of the kiln that are

covered by solids exchange heat with solid through surface contact (Helmrich and Schugërl [39]). It may be noted that the sticking of some rutile TiO2 to the inner refractory wall at various axial locations will alter the contact heat transfer coefficient which can only be obtained from an experiment on a similar kiln. The authors don’t have any data regarding this aspect available in the published literature and hence, an average value of contact heat transfer coefficient (377 W/m2 K) for the TiO2 kiln as estimated by Ginsberg and Modigell [10] has been used. The heat transfer coefficient for convection from outer wall to surroundings is calculated as (Suryanarayana et al. [38]):

ho ¼

0:11ka Dsh

"

! #0:35 Re2x þ Gr Pra 2

ð4Þ

where, Dsh is the outer diameter of the kiln, ka and Pra are the thermal conductivity and Prandtl number of the surrounding air respectively, Rex is the Reynolds number based on relative velocity between the cylindrical wall and the external air surrounding the kiln. Gr is the conventional Grashof number characterizing natural convection.

Rex ¼

pD2sh x 2ma

ð4aÞ

where, x is the angular velocity of the rotary kiln and ma is the kinematic viscosity of surrounding air.and

Gr ¼

ag bðT o;av T a ÞD3sh

ð4bÞ

m2a

where, ag is the gravitational acceleration, b is the volumetric thermal expansion coefficient, T o;av is the average temperature on outer wall of the kiln and T a is the surrounding temperature outside the kiln. 2.5. Reaction rates For any reaction, the rate of change in degree of conversion of the reaction is assumed to be governed by kinetic expression (Eq. (5)), which is a modified form of Arrhenius rate law:

dfk;z Ea;k ð1 fk;z Þnk ¼ Ae;k exp dt Ru T s;z

ð5Þ

where fk;z denotes the degree of conversion of kth reaction at axial position z, Ru is the universal gas constant, T s;z is the mean solids temperature at axial position z, Ae;k , Ea;k , nk are the frequency factor, activation energy and order of kth reaction, respectively. The values of Ae;k , Ea;k and nk are provided in Table 1a. The degree of conversion, fk;z is defined as

fk;z ¼

_ k;z _ k;0 m m _ k;f _ k;0 m m

ð6Þ

Table 1c Molecular masses of various components in solid charge. S. No. k

Component

Molecular weight Mw (g/mol)

1 2 3 4 5 6 7

TiOðOHÞ2 TiOSO4 H2 O TiOSO4 TiO2 ðaÞ TiO2 ðrÞ H2 OðlÞ H2 OðgÞ

97.88 177.94 159.93 79.87 79.87 18.02 18.02


_ k;0 and m _ k;f are the initial and final mass flow rates of the where m _ k;z is its mass flow rate at reactant of kth reaction respectively, m any axial position z. With the help of degrees of conversion of the reaction at any axial segment, using Eq. (6), the amount of depletion per second _ k;z ) are estimated. Using for different species of solid charge ðdm this information, based on stoichiometry of the chemical reactions, updated mass flow rates of solid species (that is, the mass flow _ k;z ) are estimated. rates at the exit of that axial segment, m Table 1c lists the components in solid charge with corresponding index k and their respective molecular weights. _ k;z exit are the mass flow rates of kth species at _ k;z entry and ½m If ½m _ k;z is the amount of the entry and exit of axial segment ‘z’ and dm depletion per second of reactant in kth reaction in that segment, then using molecular masses Mwk , following expressions are obtained for updated mass flow rates at the exit of the segment. Eqs. (7)–(12) show the expressions for updated mass flow rates of solid components in each axial segment and are written based on the stoichiometry of the chemical reactions. Reaction 1:

269

ichiometry of reactions 1, 3 and 4, its new mass flow rate at exit of this axial segment is given as,

_ 4;z exit ¼ ½m _ 4;z entry dm _ 4;z þ ½m

Mw4 Mw4 _ 1;z þ _ 3;z dm dm Mw1 Mw3

ð10Þ

_ 4;z entry is the mass flow rate of TiO2 ðaÞ at entry of axial segwhere ½m _ 4;z is the decomposed amount during this reaction in ment and dm that axial segment. Mw4 is the molecular weight of TiO2 ðaÞ. Mass flow rate of TiO2 ðrÞ, as it is generated from reaction 4, is updated as well in a similar way,

_ 5;z exit ¼ ½m _ 5;z entry þ dm _ 4;z ½m

ð11Þ

_ 5;z entry and ½m _ 5;z exit are the mass flow rates of TiO2 ðrÞ at where, ½m entry and exit of that axial segment. Mass flow rate of water vapour that is released as gaseous product from reaction 1 and 2, is tracked at each axial segment before it is mixed into hot gas mixture. Again based on stoichiometry of reactions 1 and 2, amount decomposed per second in segment is estimated as,

Mw7 Mw7 _ 1;z þ _ 2;z dm dm Mw1 Mw2

TiOðOHÞ2 ! TiO2 ðaÞ þ H2 OðgÞ

_ 7;z ¼ dm

In reaction 1, reactant TiOðOHÞ2 or metatitanic acid is thermally decomposed into TiO2 (a) and water vapour. TiOðOHÞ2 is denoted by index 1, as shown in table. Updated mass flow rate of this compound at exit of axial segment is given by following expression.

Here, Mw7 is the molecular weight of water. From the knowledge of enthalpies of reactions, heat generation due to chemical reactions in an axial segment can be estimated and expressed as:

_ 1;z exit ¼ ½m _ 1;z entry dm _ 1;z ½m

_ 1;z Dh1 þ dm _ 2;z Dh2 þ dm _ 3;z Dh3 þ dm _ 4;z Dh4 qcr;z ¼ dm

ð7Þ

_ 1;z entry is the mass flow rate of TiOðOHÞ2 at entry to the where ½m _ 1;z is the amount of depletion of TiOðOHÞ2 per secsegment and dm ond in that segment. Reaction 2:

TiOSO4 H2 O ! TiOSO4 þ H2 OðgÞ Similarly, in reaction 2, reactant TiOSO4 H2 O is decomposed into TiOSO4 and water vapour. TiOSO4 H2 O is denoted by index 2, as shown in table. Its new mass flow rate at exit of axial segment is given by following expression.

_ 2;z exit ¼ ½m _ 2;z entry dm _ 2;z ½m

ð8Þ

_ 2;z entry is the mass flow rate of TiOSO4 H2 O at entry to segwhere ½m _ 2;z is its decomposed amount per second during that ment and dm axial segment. Reaction 3:

TiOSO4 ! TiO2 ðaÞ þ SO2 ðgÞ þ 0:5O2 TiOSO4 is a product of reaction 3 and is used as a reactant in reaction 3. TiOSO4 is denoted by index 3, as shown in table. Based on stoichiometry of both reactions, following expression can be written for updated mass flow rate of TiOSO4 at exit to that axial segment.

_ 3;z exit ¼ ½m _ 3;z entry dm _ 3;z þ ½m

Mw3 _ 2;z dm Mw2

ð9Þ

ð12Þ

ð13Þ

where, Dh1 , Dh2 , Dh3 and Dh4 are the heat of reactions for Reactions 1 to 4 and are given in Table 1a. 2.6. Mass and energy balances in solids and gas As mentioned earlier, the kiln is considered to be consisting of three sections. In the first section, wet solid charge is heated to the saturation temperature of the entrained liquid. In the second section, liquid evaporates at constant temperature and in the third section dry solid charge is heated till the required conversion of anatase TiO2 to rutile TiO2 is achieved. The mass and energy balance of the solid in each segment either in first and third segment gives the expression for exit solid temperature of that segment, T s;zþDz for that section while the same performed on an axial segment in second section of the kiln gives _ v;z . The end of the secthe expression for the rate of evaporation, dm _ v is equal to the ond section is indicated where the cumulative m predetermined amount of water to be evaporated per second. Similarly, mass and energy balance of hot gas contained in an axial segment in any of the three sections of the kiln gives the expression for T g;zþDz for that section. 2.6.1. Mass balance in Sections I and III In solids region (Fig. 5),

_ s;z dm _ g;z _ s;zþDz ¼ m m

ð14Þ

_ 3;z entry is the mass flow rate of TiOSO4 at entry to axial segwhere ½m _ 3;z denotes the amount depleted per second in that ment and dm segment. Mw2 and Mw3 are the molecular weights of TiOSO4 H2 O and TiOSO4 respectively. Reaction 4:

TiO2 ðaÞ ! TiO2 ðrÞ TiO2 ðaÞ is produced from reactions 1 and 3. It is consumed in reaction 4, where it is converted into final product that is TiO2 ðrÞ. TiO2 ðaÞ is denoted by index 4, as shown in table. According to sto-

Fig. 5. Mass balance of any axial segment in first and third sections of the kiln.

270


_ s;z and m _ s;zþDz are the mass flow rates of the solids at any where m _ g;z is the total amount axial position z and z + Dz respectively. dm of released gaseous products from chemical reactions in an axial segment.

_ g:z ¼ dm _ H2 OðgÞ;z þ dm _ SO2 ðgÞ;z þ dm _ O2 ðgÞ;z dm

ð15Þ

_ H2 OðgÞ;z , dm _ SO2 ðgÞ;z and dm _ O2 ðgÞ;z are the amounts of H2 OðgÞ, Here, dm SO2 ðgÞ and O2 ðgÞ releasing from chemical reactions in solids in that axial segment. In gas region (Fig. 5),

_ g;z dm _ g;z _ g;zþDz ¼ m m

ð16Þ

Dt ¼

Dz Vz

ð23Þ

where V z is the axial velocity of the solids. Note that V z ¼ U tan /. For derivation of this expression see Ghoshdastidar and Agarwal [6]. So,

q1;z ¼

Ns X j¼1

qj þ

Ns X hcj Aj ðT g;z T s;z Þ

where qj is the total radiation heat transfer to the jth element of solid. Ns X

_ g;z and m _ g;zþDz are the mass flow rates of the gas at any axial where m position z and z + Dz respectively. and,

q2;z ¼

_ w;g _ w;s ¼ m m

_ H2 OðgÞ;z cp;v þ dm _ SO2 ðgÞ;z cp;SO2 þ dm _ O2 ðgÞ;z cp;O2 Þ qgp ¼ T s;z ðdm

ð17Þ

ð24Þ

j¼1

hws Aj;inner ðT j;inner T s;z Þ

ð25Þ

j¼1

ð26Þ

_ w;g is the mass flow rate of evaporated moisture content in where m first section, that is accumulated in gas from the second section of the kiln.

T g;zþDz ¼

2.6.2. Mass balance in Section II In solids region (Fig. 6),

where qr;z is the total heat transfer from gas to solids and wall. cp;g is the specific heat of gas at constant pressure.

_ s;z dm _ g;z _ s;zþDz ¼ m m

In gas region (Fig. 7),

_ g;z cp;g þ m _ w;g cp;v Þ þ qr;z qgp;z T g;z ðm _ w;g cp;v _ g;zþDz cp;g þ m m

ð27Þ

ð18Þ 2.6.4. Energy balance in Section II In solids region (Fig. 8),

and,

_ w;s;z dm _ v;z _ w;s;zþDz ¼ m m

ð19Þ

_ w;s;z is the mass flow rate of moisture content in wet solids where, m in second section of the kiln at any axial position z. In gas region (Fig. 6),

_ g;z dm _ g;z _ g;zþDz ¼ m m

ð20Þ

and,

_ v;z ¼ dm

q1;z þ q2;z qgp;z qcr;z hfg

_ v;z is the evaporation rate of moisture content in dry solid where, dm in concerned axial segment and hfg is the latent heat of vaporization of water in J/kg. For this section,

ð21Þ

T s;zþDz ¼ T s;z ¼ 373 K

_ w;g;z is the mass flow rate of evaporated moisture content where m present in gas in second section of the kiln at any axial position z.


_ v;z _ w;g;z dm _ w;g;zþDz ¼ m m

2.6.3. Energy balance in Section I In solids region (Fig. 7),

T s;zþDz ¼

_ s;z cp;s þ m _ w;s cp;l Þ þ q1;z þ q2;z qgp;z qcr;z T s;z ðm _ w;s cp;l _ s;zþDz cp;s þ m m

¼ Saturation temperature of water at 1 bar

T g;zþDz ¼ ð22Þ

_ w;s is the mass flow rate of moisture content present in wet where, m solids in first section of the kiln. It remains same throughout the section. cps and cpl refer to specific heats (at constant pressure) for dry solids and moisture content respectively. Also, q1;z is the heat transfer to the solids from gas and exposed surface elements of the kiln through convection and radiation, q2;z is the heat transferred to the solid through surface elements of kiln in direct contact with solids, qgp;z is the thermal energy associated with the released gaseous products of chemical reactions and qcr;z is the heat generated by the chemical reactions in Dz distance or Dt time, where Dt is computed from

ð28Þ

_ g;z cp;g þ m _ w;g;z cp;v Þ T s;z dm _ v;z cp;v þ qr;z qgp;z T g;z ðm _ w;g;zþDz cp;v _ g;zþDz cp;g þ m m

ð29Þ

2.6.5. Energy balance in Section III In solids region (Fig. 10),

T s;zþDz ¼

_ s;z C p;s þ q1;z þ q2;z qgp;z qcr;z T s;z m _ s;zþDz cp;s m

ð30Þ


T g;zþDz ¼

_ g;z cp;g þ qr;z qgp;z T g;z m _ g;zþDz cpg m

ð31Þ

3. Overall method of solution A computer program in FORTRAN 95 was developed to obtain the numerical results for the present problem. The input data required for the program are shown in Table 2. The specific heat of the dry solids is calculated by using Eq. (34) from the specific heat of individual components listed in Table 1c.

C p;s ¼

5 X X k C p;k

ð32Þ

k¼1

Fig. 6. Mass balance of any axial segment in second section of the kiln.

where X k and C p;k are the mass fraction and specific heat of component k respectively. k = 1 to 5 correspond to TiOðOHÞ2 , TiOSO4 H2 O, TiOSO4 , TiO2 ðaÞ and TiO2 ðrÞ respectively.


271

Fig. 7. Energy balance of any axial segment in first section of the kiln.

Fig. 8. Energy balance for solids in any axial segment in second section of the kiln.

A mixture density relation is used to calculate the effective density of wet solids from the densities of the components as given in Table 1d.

qs ¼

f qs;dry solids þ 100 ql f 1 þ 100

ð33Þ

where f is the percent of moisture content with respect to dry solids, ql is the density of liquid water and qs;dry solids is the density of dry solids.

qs;dry solids ¼

1 5 X X k =qs;k

ð34Þ

k¼1

where qs;k is the density of the component k in solid charge. The solution is initiated at the solids entrance of the kiln and proceeds to the outlet. In an axial segment, the gas and solids temperature being known, the steady state refractory wall temperature distribution is calculated by solving Eq. (3) through the False Transient Approach. The solid temperature and mass flow rate are known at the inlet. Similarly, gas temperature and gas mass flow rate are known at the solid inlet (which is actually the exit for the gas since the gas is flowing countercurrent to the solid flow). Thus, the energy balance equations (see Section 2.6) are solved for the first segment, and solid composition, solids, and gas temperature are obtained at the exit of the first section. Once

the new composition of the solids at the inlet of the second segment is determined, the new cross-sectional area of the solids and the new fill-angle are estimated and hence, new shape factors are computed. The solids and gas temperature, and species composition at the exit of the second segment can be calculated in the same way as described earlier. The solution then proceeds by the analysis of each succeeding segment. The process is continued till the end of the kiln. The end of the kiln is indicated by the position where the degree of conversion for the last reaction, that is, anatase to rutile titanium dioxide conversion has reached 98%. Thus the kiln length can be computed. A time step of Ds = 0.1 s is used for calculations. Since the initial temperature distribution of the refractory wall is not known, an arbitrary temperature is assumed at all grid points in the refractory wall. The solution converges where there is no further change in the temperature at each grid point as s ? 1. This temperature distribution represents the steady state temperature distribution of the wall. Grid independence tests have been performed to obtain the optimum grid spacing values in the circumferential and radial directions. The number of grid points used in the circumferential direction is 150 and that in the radial direction is 61. In addition, sensitivity analysis for axial segment length, Dz with respect to predicted kiln length is also performed. It is found that the predicted kiln length and hence axial solid and gas temperature distributions remain more or less unchanged while Dz is changed from 0.10 to 0.25 m. The largest value of Dz in this range is taken in order to save CPU time. Therefore, Dz used in this study is 0.25 m. The simulations are performed on a high performance computing system (having Terra-flops rating). The CPU time required for a simulation with the input data (Table 2) is approximately 46.43 h. The output data consist of refractory wall temperature, solids temperature, gas temperature, solids composition, individual lengths of first, second and third sections of the kiln (L1, L2, L3), and the total kiln length (L). The capability of this model to predict the kiln length is unique and has not been found in earlier TiO2-kiln models. The overall solution algorithm is described next in a step-bystep fashion.

Fig. 9. Energy balance for gas in any axial segment in second section of the kiln.

272


Fig. 10. Energy balance of any axial segment in second section of the kiln.

Overall Solution Algorithm 1. Input the data for various kiln geometrical, flow, thermophysical and chemical kinetics parameters. 2. Input the solids composition, solids temperature and gas temperature at solids inlet. 3. Set m =1 for the first axial segment. 4. Calculation of solid fill angle in the segment. 5. Shape factor calculations for radiative heat echange among gas, wall and solid bed. 6. Calculation of heat transfer by convection from Eq. (2) and radiation from Eq. (1). 7. Estimation of refractory wall temperature distribution by solving energy equation by false transient method using explicit finite-difference scheme. 8. Calculation of total heat transfer (convection + radiation) from gas and to solids in that segment. 9. Calculation of mass flow rates of solid species at the exit of that segment using modified Arrhenius equation for each reaction. 10. Calculation of thermal energy absorbed or released in that segment due to chemical reactions in the solid bed. 11. Calculation of the temperatures of solids and gas at the exit of that segment using global mass and energy balances for solids and gas, respectively. 12. Update the gas and solids flow rates at the exit of the segment. 13. Check whether degree of conversion of final reaction (i.e., that is conversion of anatase titanium dioxide to rutile titanium dioxide) has reached 98%. If it has reached this value, print the total kiln length, wall temperatures and axial distributions of solids composition, solids temperature and gas temperature. 14. If it has not reached, then proceed to next axial segment, m=m+1, and repeat the procedure from steps 4 to 13. 15. Grid independence test in two-dimensional refractory zone and sensitivity analysis for axial segment length with respect to predicted kiln length are performed. 16. Optimum kiln length obtained in step 15 is used in the secondary model in ANSYS-FLUENT (as discussed in Section 2.3). 17. From the secondary model, local convective heat transfer coefficients from gas to wall and solids are obtained and are imported back to the main model at the same physical locations. 18. Same procedure consisting of steps 1 to 14 are repeated in this modified model and final output is received in terms of total kiln length, wall temperatures and axial distributions of solids composition, solids temperature and gas temperature.

Table 1d Material properties of the solid components (Ginsberg and Modigell [10]). S.No.

Component

Density (kg/m3)

Specific heat (J/mol K)

1 2 3 4 5 6

TiOðOHÞ2 TiOSO4 H2 O TiOSO4 TiO2 ðaÞ TiO2 ðrÞ H2 OðlÞ

2540 2430 2890 3900 4240 1000

142 208 133 70 70 76

Table 2 Input data to the program (Ginsberg and Modigell [10]). 1. Rotary Kiln (a) Diameter (inner) (b) Refractory (i) Thickness (ii) Thermal Conductivity (iii) Specific heat (iv) Density (v) Emissivity (c) Rotational speed (d) Angle of inclination 2. Solid (a) Inlet temperature (b) Mass flow rate (dry) (c) Percent water (on dry basis) (d) Emissivity 3. Gas (a) Outlet temperature (b) Specific heat (c) Mass flow rate (d) Emissivity (e) Dynamic viscosity 4. Water Vapor (a) Latent heat of vaporization$ (b) Specific heat $ 5. Specific heat of sulfur dioxide$ 6. Specific heat of oxygen$ 7. Contact heat transfer coefficient 8. Ambient temperature outside the kiln $

2.3 m 0.25 m 1.6 W/m K 950 J/kg K 2310 kg/m3 0.8 0.33 rpm 2.29° 308 K 1.309 kg/s 75.14 0.8 648 K 1356 J/kg K 6.218 kg/s 0.422 3.58 105 kg/m-s 2.258 106 J/kg 2042.71 J/kg K 627.99 J/kgK 919.83 J/kgK 377 W/m2K 287.15 K

Chase [40].

4. Results and discussion This section firstly presents comparison of the results with that of an earlier work. Following that, results of the parametric study are shown to see the effect of variation in moisture content (dry basis), dry solids mass flow rate, gas mass flow rate, rotational speed of the kiln, and angle of inclination of the kiln on the axial solid and gas temperature distributions, axial profiles of percent conversion of anatase TiO2 to rutile TiO2 as well as on the predicted length of the kiln and the rate of production of rutile TiO2.


273

4.1. Validation of Results with the Numerical Results of Ginsberg and Modigell [10]

after towards solid outlet similar to that predicted by the present computer model.

The kiln length predicted by present model turns out to be 45.75 m as compared to the actual kiln length of 45 m reported in Ginsberg and Modigell [10]. The predicted kiln length is only 1.67% larger than the actual kiln, indicating an excellent match. Fig. 11 illustrates the comparison of the present axial solid and gas temperature profiles with those computationally obtained by Ginsberg and Modigell [10]. It may be noted that no steady-state experimental data are found in Ginsberg and Modigell [10]. It can be seen from Fig. 11 that general trends of both the profiles show a good agreement with Ginsberg and Modigell [10]. In the first section of the kiln, solid temperature increases slowly to the saturation temperature of the wet solids and it takes about 37% of the kiln length. At this point, the second section begins and evaporation of water takes place. The temperature remains constant at the saturation point. The second section covers about 47% of the kiln length and is the largest among all the three sections. The length of this section largely depends on the amount of moisture present in the wet solids. Right from the beginning of the third section, temperature increases at a higher rate due to large heat transfer from hotter gas and wall in that region. From s about 92% to 98% of the kiln length, there is decay in @T . It is caused @z by higher energy requirement of third reaction, where decomposition of titanyl sulfate occurs. The temperature increases thereafter till the final reaction, that is, the phase transition of TiO2 (a) to TiO2 (r) is completed. The final reaction is exothermic in nature and hence, unlike other reactions, it releases thermal energy to gas region. Completion of this reaction also marks the end of the third section of the kiln length. The predicted temperature profiles also qualitatively agree with Dumont and Bélanger [11] which shows that pre-heating and moisture evaporation take place in nearly 70% of the kiln length and solid temperature rapidly rises there-

4.2. Axial composition profile of the solid charge Fig. 12 shows axial composition distribution moisture content, reactants and the final product. The graph clearly shows the present model has been able to simulate the chemical reactions properly. The thermal decomposition of TiOðOHÞ2 and TiOSO4 H2 O (that is, Reactions 1 and 2 as shown in Table 1a) starts close to the kiln entrance. In the first section of the kiln, Reaction 1 proceeds at much faster rate than Reaction 2. It starts slowing down at the beginning of the second section of the kiln and remains slow till any moisture is left in solid charge. This is because of the fact that solid temperature remains constant within second section. In the third section, due to increase in solid temperature, Reaction 1 speeds up for a while and then remains slow and steady. Reaction 2 also progresses rapidly in the third section. The production of titanyl sulfate from this reaction and increasing solid temperature lead to commencement of Reaction 3 where the desulphurization produces TiO2(a) and SO3. The SO3 that is released is immediately decomposed into SO2 and 0.5O2. This reaction is over at 954 K. Before completion of Reactions 3 and 1 (Table 1a), final reaction of phase transformation of anatase TiO2 to rutile TiO2 (Reaction 4) initiates very slowly and exothermally at around 910 K. Due to the facts that, more TiO2(a) is being produced by both reactions 1 and 3 and it is an exothermal reaction in nature, the production of rutile TiO2 occurs at a very fast rate in the last section of the kiln. 4.3. Parametric study In this section, the effect of various parameters on the predicted length of the kiln, rutile TiO2 production rate, axial solid and gas temperature distributions, and axial variation in solids

Fig. 11. Axial solid and gas temperature distributions: Comparison with the numerical solution of Ginsberg and Modigell [10].

274


Fig. 12. Axial composition profiles of various compounds.

Table 3a Predicted Kiln Length and Rutile TiO2 Production Rate vs. Moisture Percent (Dry Basis). Moisture Percent (Dry Basis)

L1 (m)

L2 (m)

L3 (m)

L = L1 + L2 + L3 (m)

Rutile TiO2 Production Rate (kg/hr)

60.0 75.14 90.0

17.0 17.25 17.75

18.5 21.0 23.0

9.75 7.5 6.0

45.25 45.75 46.75

3649.68 3648.24 3646.80

Table 3b Predicted Kiln Length and Rutile TiO2 Production Rate vs. Dry Solids Mass Flow Rate. Dry Solids Mass Flow Rate (kg/s)

L1 (m)

L2 (m)

L3 (m)

L = L1 + L2 + L3 (m)


1.0 1.31 1.6

14.25 17.25 20.0

19.75 21.0 21.75

9.75 7.5 6.25

43.75 45.75 48.0

2789.28 3648.24 4458.96

Table 3c Predicted Kiln Length and Rutile TiO2 Production Rate vs. Gas Mass Flow Rate. Gas Mass Flow Rate (kg/s)

L1 (m)

L2 (m)

L3 (m)

L = L1 + L2 + L3 (m)


4.0 6.218 8.0

17.0 17.25 17.0

15.75 21.0 23.75

3.25 7.5 12.25

36.0 45.75 53.0

3641.76 3648.24 3651.12

Table 3d Predicted Kiln Length and Rutile TiO2 Productiion Rate vs. Kiln Inclination Angle. Inclination Angle (deg)

L1 (m)

L2 (m)

L3 (m)

L = L1 + L2 + L3 (m)


1.4 2.29 5.0

16.25 17.25 18.75

20.0 21.0 23.25

7.75 7.5 7.5

44.0 45.75 49.5

3651.12 3648.24 3642.48


275

Table 3e Predicted Kiln Length and Rutile TiO2 Production Rate vs. Kiln Rotational Speed. Rotational Speed (r.p.m)

L1 (m)

L2 (m)

L3 (m)

L = L1 + L2 + L3 (m)


0.21 0.33 1.0

16.5 17.25 18.25

20.5 21.0 22.25

8.25 7.5 7.25

45.25 45.75 47.75

3651.12 3648.24 3639.24

composition is discussed. In Tables 3a–3e, predicted kiln length values and rutile TiO2 production rates for various parameters are shown and the base values (Table 2) are highlighted in bold letters. 4.3.1. Predicted kiln length as a function of various parameters Table 3a reveals that, as the moisture percentage is increased from 60 to 90, the lengths of the first and second sections of the kiln increase. This is because of the fact that more amount of moisture will require large amount of heat to reach the saturated state and then to get vapourized. Therefore, lengths of the both sections will be higher for this case. This rise in length of the second section also causes delay in completion of Reaction 1 and 2 as they speed up only after evaporation is finished. On the contrary, the length of the third section decreases on increasing water content. This can be explained as follows. Though it takes longer for water to get vaporized, the fill-angle of the kiln and amount of dry solid mass available in third section remains same and it has to meet the same requirement of phase transformation in chemical reactions. Now, total heat transfer to the solids in last section for the solid with higher water content will be greater than that for solid with lower water content. This is because of exposure of the dry solid to the hotter gas in the former case as heating starts at a longer distance from the solid inlet. The combined effect on total length is that it is greater for higher moisture content case. Table 3b reveals that as dry solids mass flow rate increases from 1.0 kg/s to 1.6 kg/s, predicted kiln length increases from 43.75 m to 48.0 m. This is because higher solids residence time is needed to achieve the desire objective and hence more kiln length is needed. Here, moisture percentage remains same but amount of moisture present in wet solids increases which raises the length of the second section. However, the length of the third section decreases due to higher heat transfer from gas and wall to solids. The overall effect is that the predicted kiln length increases. As gas flow rate at inlet increases from 4 kg/s to 8 kg/s, total kiln length increases from 36.0 m to 53.0 m (Table 3c). Near solids inlet, in spite of lower gas residence time at higher gas flow rate, there is enough heat transfer to the material to reach the boiling point of water in almost same distance as in lower gas flow rate cases. However, at latter stages of second section, effects of higher gas flow rate are evident on processing of solid charge, as lower gas temperature at the same axial location results in less heat transfer to solids, and lengths of second and third section increase significantly. Table 3d reveals that as the kiln inclination angle changes from 1.4° to 5°, predicted kiln length increases from 44 m to 49.5 m. This is expected because, as inclination angle increases, axial velocity of solid also increases as V z ¼ U tan / a nd hence the residence time of the solid is less. Hence, to reach the same composition of solids at the exit, greater kiln length is needed. At higher rotational speed of the kiln, the axial velocity of the solid increases and therefore, the solid particles are exposed to gas for less amount of time. This results in slower progression of reactions and therefore, to obtain same solids chemical state of the product, length of the kiln has to be increased (Table 3e). 4.3.2. Rutile TiO2 production rate as a function of various parameters The rate of production of rutile TiO2 is calculated by dividing the mass of rutile TiO2 produced, by the residence time of the solids in

the kiln. Table 3a shows that with the increase in moisture percent from 60 to 90 the rate of production of rutile Titanium dioxide decreases slightly from 3649.68 to 3646.8 kg/hr. This is because of higher kiln length required for drying in the case of larger water content in the solids. On the other hand, as dry solids mass flow rate increases from 1 kg/s to 1.6 kg/s the production rate substantially rises from 2789.28 kg/hr to 4458.96 kg/hr (Table 3b). This is because of higher amount of solid charge being processed and faster rate of final reaction per unit time. Table 3c indicates higher gas flow rate results in larger production of rutile TiO2 per unit time. This is because solid temperature is slightly higher in the third segment of the kiln which accelerates the rate of reaction. Increasing kiln inclination angle and kiln rotational speed reduces the rutile titanium dioxide production rate (Tables 3d and 3e) since the kiln has to be longer and hence the solids residence time in the kiln is more. 4.3.3. Axial solid and gas temperature distribution as a function of various parameters Fig. 13 shows that, with the increase in moisture content in wet s solids, axial gas temperature and @T increase in third section of the @z kiln while axial solid temperature with respect to percentage kiln length decreases. The explanation of this trend is as follows. For a larger amount of water in the solid charge, the wet solid will have to travel a longer distance to be totally moisture-free. This implies that the same amount of dry mass of solid will have to be heated to the same exit temperature. It may be noted that the fill angle remains the same in the third section no matter what the original moisture content is. So, the dry solid will be exposed to hotter gas in the case of high moisture content as the heating starts close to the kiln exit which is the inlet for the hot gas. Therefore, to reach the requisite exit temperature, the dry solid will have to travel a smaller percentage length of the kiln and hence the axial temperature gradient of the solid is larger. The axial solid temperature at the same percentage kiln length will be obviously lower. Fig. 14 depicts that, for higher solid mass flow rate, gas temperature is considerably higher in the third section of the kiln. This is because for larger solid mass flow rate, the fill angle increases resulting in reduction in the mean beam length as the gas volume decreases. Hence, the loss of heat by the gas by radiation is smaller and hence the gas temperature is high. Fig. 15 indicates that increasing gas flow rate causes significant drop in gas inlet temperature. The reason is that higher gas flow rate means less gas residence time and hence the kiln has to be longer (see Table 3c). Thus, solid temperature is higher at the same percent kiln length in the third segment of the kiln. Gas on the other hand loses more heat to the solid and hence the gas temperature drops in the last section of the kiln for higher gas flow rates. Finally, lower kiln inclination angle and lower rotational speed result in lower gas temperature and higher solid temperature at the same percent kiln length in the third section of the kiln (graphs not shown). This is because of more residence time of the solids in the kiln which leads to shorter kiln length. Finally, lower kiln inclination angle and lower rotational speed result in lower gas temperature and higher solid temperature at the same percent kiln length in the third section of the kiln (graphs not shown). This is because of more residence time of the solids in the kiln which leads to shorter kiln length.

276


Fig. 13. Axial solid and gas temperature distributions for different moisture percentages (dry basis).

Fig. 14. Axial solid and gas temperature distributions for different mass flow rates of the dry solids.

4.3.4. Axial composition of solid charge as a function of various parameters Fig. 16 shows the effect of moisture content on the percent conversion of Anatase TiO2. It is revealed that both Reactions 1 and 2 (Table 1a) progress faster as solids enter the third and the last

section of the kiln. As the third section comes latter for higher moisture content case, these reactions along with the decomposition of TiO.SO4 to TiO2 (a) and final reaction of conversion occur later and relatively faster due to higher gas temperature.


277

Fig. 15. Axial solid and gas temperature distributions for different mass flow rates of the gas.

Fig. 16. Axial profiles of percent conversion of anatase TiO2 to rutile TiO2 for different moisture percentages (dry basis).

Fig. 17 depicts how solids mass flow affects the percent conversion of anatase TiO2. At higher dry solids mass flow rates Reaction 3 and the final reaction are delayed due to requirement of longer first

and second sections because of larger solid mass flow rate and also larger amount of moisture to be evaporated per unit time. However, at the same percent kiln length near the exit reactions occur

278


Fig. 17. Axial profiles of percent conversion of anatase TiO2 to rutile TiO2 for different mass flow rates of the solids in the last 20% of the kiln length.

Fig. 18. Axial profiles of percent conversion of anatase TiO2 to rutile TiO2 for different mass flow rates of the gas in the last 20% kiln length.


at a faster rate as compared to the cases of lower dry solids flow rates. Fig. 18 shows the influence of gas flow rate on the percent conversion of anatase TiO2. The conversion of anatase TiO2 to rutile TiO2 starts earlier in the kiln for lower gas flow rate due to higher inlet gas temperature resulting in higher solids temperature. Figs. 16–18 also depict that the final reaction in the kiln, or, conversion of anatase TiO2 to rutile TiO2 takes place in approximately last 12% of the kiln length. For higher kiln inclination angles and kiln rotational speeds the reaction process of conversion from anatase TiO2 to rutile TiO2 starts earlier with respect to percent kiln length (graphs not shown) because kiln length is larger and hence the solids temperature is higher near the solids outlet. The reaction rate is also faster for higher kiln inclination angles and kiln rotational speeds. It may be noted that the reaction rate should not be confused with production rate of rutile TiO2 discussed in Section 4.3.2.

5. Conclusions This paper presents a computer simulation study of calcination of hydrous titanium dioxide for the production of rutile titanium dioxide in rotary kiln. The heat transfer model includes turbulent gas convection, radiation heat exchange among the hot gas, refractory wall and the solid surface, conduction in the refractory wall, and mass and energy balances in the gas and solids. The gas convection has been simulated using a stand-alone CFD model assuming the kiln to be non-rotating and solid bed to be stationary. Based on the aforesaid CFD data the local convection heat transfer coefficients have been calculated. The present computer model predicts steady-state axial distributions of solids and gas temperature that are in excellent agreement with an earlier work [10]. The axial chemical composition profile of the solid charge is consistent with the trends found in literature. The kiln length predicted by the present model is 45.75 m as compared to 45 m of the actual kiln reported in Ginsberg and Modigell [10]. The accurate prediction of the kiln length is a major achievement of this work. The capability of the present model to predict the kiln length and the modelling of gas convection are two novel aspects of this work which can greatly help in the design and optimization of such kilns. A detailed parametric study reveals that higher moisture content in the solids results in slightly larger kiln length and lower rutileTiO2 production rate. However, higher dry solids mass flow rate requires slightly larger kiln length but gives rise to substantial rutile-TiO2 production rate. For low speed gas flow kiln length is much smaller although the rate of production of rutile titanium dioxide is marginally lower. Smaller kiln inclination angles and kiln rotational speeds result in shorter kiln length and higher rutileTiO2 production rate. On the whole the parametric study lent a good physical insight into the rutile titanium dioxide production process in a rotary kiln.

References [1] A. Saas, Simulation of the heat transfer phenomenon in a rotary kiln, Ind. Eng. Chem. Proc. Des. Dev. 6 (4) (1967) 532–535. [2] F.A. Kamke, J.B. Wilson, Computer simulation of a rotary dryer-II. Heat and mass transfer, AIChE J. 32 (2) (1986) 269–275. [3] C.A. Cook, V.A. Cundy, Heat transfer between a rotating and a moist granular bed, Int. J. Heat Mass Transfer 38 (3) (1995) 419–432. [4] P.S. Ghoshdastidar, V.K. Anandan Unni, Heat transfer in the non-reacting zone of a cement rotary kiln, J. Eng. Ind., Trans. ASME 118 (1996) 169–172.

279

[5] P.S. Ghoshdastidar, G. Bhargava, R.P. Chhabra, Computer simulation of heat transfer during drying and preheating of wet iron ore in a rotary kiln, Drying Technol. 20 (1) (2002) 19–35. [6] P.S. Ghoshdastidar, A. Agarwal, Simulation and optimization of drying of wood chips with superheated steam in a rotary kiln, J. Therm. Sci. Eng. Appl., Trans. ASME 1 (2009) 024501. [7] X.Y. Liu, J. Zhang, E. Specht, Y.C. Shi, F. Herz, Analytical solution for the axial solid transport in rotary kilns, Chem. Eng. Sci. 64 (2009) 428–431. [8] R. Schmidt, P.A. Nikrityuk, Numerical simulation of the transient temperature distribution inside moving particles, Can. J. Chem. Eng. 90 (2012) 246–262. [9] Y. Sonavane, E. Specht, Numerical analysis of the heat transfer in the wall of rotary kiln using finite element method ANSYS, in: Proceedings of 7th International Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia, 2009. [10] T. Ginsberg, M. Modigell, Dynamic modelling of a rotary kiln for calcination of titanium dioxide white pigment, Comput. Chem. Eng. 35 (2011) 2437–2446. [11] G. Dumont, P.R. Bélanger, Steady state study of a titanium dioxide rotary kiln, Ind. Eng. Chem. Proc. Des. Dev. 17 (2) (1978) 107–114. [12] J. Roubal, D. Pachner, V. Havlena, T. Hanis, Analysis of the rotary kiln model, in: Proceedings of European Control Conference, Budapest, Hungary, 2009. [13] P. Koukkari, I. Laukkanen, K. Penttila, New applications with time-dependent thermochemical simulation, in: Proceedings of 3rd Colloquium on Process Simulation, Espoo, Finland, 1996. [14] H.A. Spang, A dynamic model of a cement kiln, Automatica 8 (1972) 309–323. [15] A. Manitius, E. Kurcyusz, W. Kawecki, Mathematical model of the aluminium oxide rotary kiln, Ind. Eng. Chem. Proc. Des. Dev. 13 (2) (1974) 132–142. [16] H.K. Guruz, N. Bac, Mathematical modelling of rotary cement kilns by the zone method, Can. J. Chem. Eng. 59 (1981) 540–548. [17] A.P. Watkinson, J.K. Brimacombe, Limestone calcination in a rotary kiln, Metall. Trans. B 13 (1982) 369–378. [18] A.P. Watkinson, J.K. Brimacombe, Oxygen enrichment in rotary lime kilns, Can. J. Chem. Eng. 61 (1983) 842–849. [19] J. Mu, R.A. Hard, Simulation study of a phosphate nodulizing kiln, Ind. Eng. Chem. Proc. Des. Dev. 23 (1984) 374–381. [20] P.S. Ghoshdastidar, C.A. Rhodes, D.I. Orloff, Heat transfer in a rotary kiln during incineration of solid waste, in: Proceedings of 23rd ASME National Heat Transfer Conference, Denver, Colorado, 1985, 85-HT-86. [21] R.T. Bui, G. Simard, A. Charette, Y. Kocaefe, J. Perron, Mathematical modeling of the rotary coke calcining kiln, Can. J. Chem. Eng. 73 (1995) 534–545. [22] R.A. Davis, Mathematical model of magnetite oxidation in a rotary kiln furnace, Can. J. Chem. Eng. 74 (1996) 1004–1009. [23] R.A. Davis, D.J. Englund, Model and simulation of a ported kiln for iron oxide pellet induration, Can. J. Chem. Eng. 81 (2003) 86–93. [24] M. Georgallis, P. Nowak, M. Salcudean, I.S. Gartshore, Modelling the rotary lime kiln, Can. J. Chem. Eng. 83 (2005) 212–223. [25] F. Marias, H. Roustan, A. Pichat, Modelling of a rotary kiln for the pyrolysis of aluminium waste, Chem. Eng. Sci. 60 (2005) 4609–4622. [26] F. Mintus, S. Hamel, W. Krumm, Wet process rotary cement kilns: modeling and simulation, Clean. Technol. Environ. Policy 8 (2006) 112–122. [27] K.S. Mujumdar, V.V. Ranade, Simulation of rotary cement kilns using a onedimensional model, Chem. Eng. Res. Des. 84 (A3) (2006) 165–177. [28] K.S. Mujumdar, K.V. Ganesh, S.B. Kulkarni, V.V. Ranade, Rotary Cement Kiln Simulator (RoCKS): integrated modeling of pre-heater, calciner, kiln and clinker cooler, Chem. Eng. Sci. 62 (2007) 2590–2607. [29] D.K. Fidaros, C.A. Baxevanou, C.D. Dritselis, N.S. Vlachos, Numerical modelling of flow and transport processes in a calciner for cement production, Powder Technol. 171 (2007) 81–95. [30] K. Shahriari, S. Tarasiewicz, Modelling of a clinker rotary kiln using operating functions concept, Can. J. Chem. Eng. 89 (2011) 345–359. [31] H. Ratajska, A. Przepiera, M. Wisniewski, Thermal transformations of hydrous titanium dioxide, J. Therm. Anal. 36 (1990) 2131–2134. [32] W.F. Sullivan, S.S. Cole, Thermal chemistry of colloidal titanium dioxide, J. Am. Ceram. Soc. 42 (3) (1959) 127–133. [33] H.C. Hottel, Radiation heat transfer, in: W.H. McAdams (Ed.), Heat Transmission, 3rd ed., McGraw-Hill, New York, USA, 1954, Chap. 4. [34] J.K. Brimacombe, A.P. Watkinson, Heat transfer in a direct-fire rotary kiln-I. Pilot plant and experimentation, Metall. Trans. B 9 (1978) 201–208. [35] V. Gnielinski, Neue Gleichungen für den Wärme- und Stoffübergang in turbulent durchströmten Rohren und Kanälen, Forsch. Ingenieurwes. 41 (1) (1975) 8–16. [36] S.H. Tscheng, A.P. Watkinson, Convective heat transfer in a rotary kiln, Can. J. Chem. Eng. 57 (4) (1979) 433–443. [37] J.K. Brimacombe, A.P. Watkinson, Heat transfer in a direct-fired rotary kiln-II. Heat flow results and their interpretation, Metall. Trans. B 9 (1978) 209–219. [38] N.V. Suryanarayana, J.E. Lyon, N.K. Kim, Heat shield for high temperature kiln, Ind. Eng. Chem. Proc. Des. Dev. 25 (1986) 843–849. [39] H. Helmrich, K. Schugërl, Rotary kiln reactors in the chemical industry, Ger. Chem. Eng. 3 (1980) 194–202. [40] M.W. Chase, NIST-JANAF thermochemical tables, J. Phys. Chem. Ref. Data Monogr. 9 (1998) 1–1951.