Mathematical Modeling of Mass and Heat Transport Processes Occurring within the Heat Exchanger System of Alumina Rotary Kilns

V. Zs. Baranyai∗, I. Szűcs

Department of Combustion Technology and Thermal Energy, University of Miskolc, Hungary

Abstract The aim of our research is the mathematical modeling of the material and heat transport processes in the pre-heaters of a rotary kiln for calcination of alumina. The gas flow and the heat and material transfer has been predicted within the pre-heater system via solving numerically the set of mathematical equations by using a CFD solver based on finite volume method. The theoretical knowledge of the physical and chemical processes is indispensable for creating mathematical models.

∗ Corresponding author: tuzbvzs@gmail.com Associated Web site: http://combustion.uni-miskolc.hu European Combustion Meeting 2009

Introduction Rotary kilns are used in many fields of material

technologies, such as iron ore pelletizing, lime calcining, alumina calcining and cement clinker production. The counter flow rotary kiln is a rotating inclined cylinder where the solid material and the combustion gases flow in the opposite direction. Rotary kiln is the most widely used plant for calcination of alumina, and almost the only one for calcining of high quality aluminas.

For utilizing the heat of the flue-gas modern rotary kilns have pre-heaters on the feed end. The key component of the gas-suspension pre-heater is the cyclone. A cyclone is a conical vessel with a dust-bearing gas-stream passed across it tangentially. The gas leaves the vessel through a co-axial "vortex-finder". The solids are thrown to the outside edge of the vessel by centrifugal action, and exit through a valve in the vertex of the cone. To improve the efficiency of the pre-heater systems a number of cyclones are connected in series.

In this study a part of the pre-heater system of the rotary kiln no. 4 of Hungarian Aluminum Ltd. is analyzed. The model of the pre-heater section has been prepared via Fluent CFD solver modeling the case of alumina recalcination (the raw material is low calcined and washed alumina to resulting low soda end-product).

The possibilities of modeling of industrial furnaces

The modeling of energetic equipment has two fundamental ways: physical and mathematical. In case of physical modeling some kinds of phenomena (flow, heat transmission) is analyzed. The analyses are made on some kinds of equipment, e.g., on a pilot or industrial equipment.

Mathematical models can be analytical and/or statistical. In case of statistical modeling one can originate mathematical relations from the survey data of the existing furnace.

Analytical modeling has two ways: discrete and universal. With the universal model the furnace is viewed as a homogeneous system. One can examine the

effect of the parameters with the descriptive equations of the processes.

In practice the processes are too complicated; therefore they can be solved with excessive simpli-fications only. They increase the negligence of the calculation [1,2].

The other analytical method is the discrete or finite element method. The finite element method is used for finding approximate solution of partial differential equations as well as of integral equations such as the heat and mass transport equations. The solution approach is based either on eliminating the differential equation completely, or rendering the PDE into an equivalent ordinary differential equation. In practice the apparatus is decomposed into finite volume elements, the parameters of state of the bordering elements are the initial and boundary conditions. The accuracy of the method depends on the quality of the decomposition. The capacity of the used computer keeps the decom-position within limits [3].

Description of the examined pre-heater system

The pre-heater system of rotary kiln No. 4 is a two-stage pre-heater. The exhaust gases leave the kiln at 400-500 °C, the temperature setting after the first cyclone is 100-150 °C, the gases leave the second cyclone at around 150-200 °C. Pre-heater exhaust gas pressure drops from 400 Pa to 500 Pa, with gas duct velocities typically 20 - 50 m/s in the pre-heater and cyclones. The raw material (generally aluminum hydroxide, in case low calcined alumina is recalcined is) is charged into the flue gas line between the first and second cyclones. The dusts delivered by gases are separated by the second cyclone. The relatively low temperature flue gases dry the wet material (the moisture content is around 10 %). The dry material is blown by air into the flue gas line of the system followed the kiln. The gas and dust phases are separated by the second cyclone. The material after the second cyclone is preheated, precalcined. The preheated and precalcined dust is blown into the end of the kiln. The finest fraction of dust is separated by an electrostatic

filter. The filter dust is fed back to the kiln or collected by big-bags. The flow scheme of the materials is shown in Figure 1.

The part of the pre-heater examined is between the end of the kiln and the first cyclone.

Figure 1 Mass flows in a rotary kiln and its pre-heater

The equations of physical processes within the pre-heater

The dust fraction is delivered by the flue gases within the pre-heater. The flue gases leave the kiln at 400-500 °C, the gas temperature is setting via cooling effect of dust and air.

The most effective way of transferring heat is by convection from the gases to the material, because the gas temperature is relatively low, and the particles of the material can mix on the whole surface with the gases.

The heat transfer from gases to the particles:

dqc = αc (Tg – Ts ) dA dt (1) where:

dq – transferred heat via convection, J; αc – convection heat transfer coefficient,

W/m2K; Tg – temperature of flue gas, K; Ts – temperature of solid particles, K; dA – elemental unit of area , m2; dt – elemental unit if time, s [4].

The heat adsorbed by the particle warms up the solid

material: qs = cs (4/3) rs

3 π ρs dTs (2) where: qs – adsorbed heat, J; cs – heat capacity of the particle, J/(kg K); rs – radius of the particle, m; ρs – density of the particle, kg/m3; dTs – temperature rise of particles, K. The two equations can be made equalransfer equal: qs = qc (3)

cs (4/3) rs

3 π ρs dTs = αc (Tg –Ts) 4rs2 πd (4)

The elemental unit of area is the ratio of the

elemental path and the velocity of particle: dt = dl / ws (5) where:

dl – elemental path, m; ws – velocity of solid particle, m/s.

After simplification of (4) and substitute (5) for area: cs (1/3) rs ρs dTs = αc (Tg – Ts ) dl / ws (6) The rising of temperature: dTs = 3 αc (Tg – Ts ) dl / (cs rs ρs ws) (7) The velocity of particle is the difference of the

velocity of gas and the settling velocity (figure 2): ws = wg – wse (8)

where: wg – velocity of gas, m/s; wse – settling velocity, m/s.

Figure 2 Particle velocity

The ideal resulting settling velocity is given by

Stokes’s law: wse = (2/9) r2 g (ρs – ρg) / η (9) where:

wse – settling velocity; m/s r – radius of particle; m ρs , ρg – density of solid and gas phase; kg/m3

2

g – acceleration of gravity; m/s2 η – dynamic viscosity of fluidum; Ns/m2

The equation for viscous resistance or linear drag is

appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, Re < 1) [3]. In this case, the force of drag is approximately proportional to velocity, but opposite in direction.

The determination of Reynolds number: Re = ρg wül d / η (10)

where: d – diameter of particle, m [5]. Within an arbitrary pipe the velocity of gas is the

ratio of the volume flows of the gas components and the area of the segment:

wg = (dVfl + dVair) / Ase (11) where: dVfsg – volume flow of flue gas , m3/h;

dVlev – volume flow of air, m3/h; Acs – area of the segment, m2.

Conditions of modeling

A part of the pre-heater system of the rotary kiln No. 4 of Hungarian Aluminum Ltd. was examined. The simulated segment is between the kiln and the first cyclone. The dry dust from the second cyclone enters into the conical part of the pipe. The solid and gas phases are separated in the first cyclone.

The case of alumina recalcination was analyzed, where the raw material is low calcined and washed alumina.

Figure 3 shows the geometry of the pipe line. Figure 4 shows the entry and discharge points of the

pipe line. The 740 K flue gas enters into the pipe through the segment marked (1). The velocity of flue gas is 10 m/s that correspond to 16 000 m3/h volume flow. The flue gas drags dust from the kiln, its concentration is 5 V/V %.

The dust from the second cyclone is blown by air (300 K) through the two pipes marked (2) into the system. During the computation five different dust and air concentration and velocity were adjusted as shown in Table 1.

Table 1 Dust concentration and blow velocity values of the

different program runs adjusted on the small pipes

Velocity (m/s) Dust

concentration (V/V %)

1 20 45 2 25 36 3 30 30 4 35 25,7 5 50 18

The flue gas – dust mixture leaves the pipe through the segment (3) and goes to the first cyclone.

Figure 3 The geometry of the examined pipe line

Figure 4 The examined pipe line; 1: flue gas inlet, 2: dust

inlet, 3: flue gas and dust outlet

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The geometry and meshing are produced by Solid Edge V20 and Gambit 6.2, the results of calculations by Fluent 6.2. Results and discussion

One can see from the dust phase concentration distribution that one part of the charged dust comes along drug by the flue gas, at the same time the other part is settling, and concentrating on the external arch of the high diameter pipe.

The less concentrated dust drug by the flue gas from the kiln is heading to the external arch as well (figure 5-8).

First of all the fed dust falls to the lower pipe segment opposing to the flue gas, due to the higher blowing velocity, and from that place the hot flue gas removes it (figure 9).

The critical blowing velocity, under witch the flue gas drags the particles at once, or above the dust particles falls down to the lower segment, is ranging from 35-50m/s.

Figure 5 Dust phase concentration in case of 20 m/s blowing

velocity

Figure 6 Dust phase concentration in case of 25 m/s blowing

velocity

It can bee seen, that the dust concentration within the small diameter segment of the pipe after the conical segment is higher in the external side than in the internal, due to the velocity difference between the two side of the pipe (figure 10-14).

The residence time within the pre-heater depends on the velocity and the displacement, so the residence time and the preheating grade can be different.

Figure 7 Dust phase concentration in case of 30 m/s blowing

velocity

Figure 8 Dust phase concentration in case of 35 m/s blowing

velocity

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Figure 9 Dust phase concentration in case of 50 m/s blowing

velocity

Figure 10 Dust phase velocity in case of 20 m/s blowing

velocity

Figure 11 Dust phase velocity in case of 25 m/s blowing

velocity

Figure 12 Dust phase velocity in case of 30 m/s blowing

velocity

Figure 13 Dust phase velocity in case of 35 m/s blowing

velocity

Figure 14 Dust phase velocity in case of 50 m/s blowing

velocity

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Conclusions Figure 15 shows the correspondence between the air and dust blowing velocity and the maximal dust flow velocity in the pipe. Excess air causes higher volume of gas mixture flow with higher velocity trough the pipe. The residence time of the part of the particles is lower, due to the higher blowing velocity. It causes lower preheating grade.

This paper presents a numerical simulation of a part of an existing pre-heater system. The solid particles are blown into the flue gas line by air.

Five cases were analyzed for five adjusted different dust and air concentration and velocity. The dust mass flows are invariable.

The path and residence time of the particles depend on the blowing velocity. It is a critical blowing velocity, under witch the flue gas drags the particles at once, or above the dust particles falls down to the lower segment, and from that place the hot flue gas removes it.

Figure 16 shows the temperature distribution of the dust in case of 50 m/s blowing velocity. The equalization of temperature difference between the flue gas and dust is relatively fast, due to the intensive convectional heat transfer.

Cold air brings down the preheating grade, higher volume flow of air causes lower temperature of preheated particles.

One can see that the dust temperature within the small diameter segment of the pipe after the conical segment is higher in the internal side than in the external.

References The flue gas is cooled by the cold air and dust particles. The excess air brings down the preheating grade, so its volume needs be reduced.

[1] Mikó, J.,; Kapros, T.: Kemencék hőtana III.; Budapest: Tankönyvkiadó, 1985

[2] Takei, M.; Weber, R.; Niioka, T.: Mathematical Modeling of Industrial Furnaces Considering Detailed Oil Spray Characteristics; Combustion Science and Technology, Volume 175, Number 7, July 2003 , pp. 1237-1262(26)

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20 30 40 50

Air and dust blowing velocity (m/s)

Max

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vel

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of p

artic

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/s)

[3] Scharler, R.; Obernberger, I.: Numerical Modelling of Biomass Grate Furnaces; In: proceedings of the 5th European Conference on Industrial Furnaces and Boilers, April 2000, Porto

[4] Cebeci, T.: Convective Heat Transfer; Long Beach: Horizons; Berlin: Springer, 2002

[5] Saleh, J. M.: Fluid flow Handbook; New York: McGraw-Hill, 2002

Figure 15 Maximum velocity of the particles as a function of

blowing velocity

Figure 16 Dust phase temperature in case of 50 m/s blowing

velocity

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