II. PILOT ROTARY KILN

   A pilot rotary kiln is basically a cylinder which rotates around its longitudinal axis and operates essentially as a heat exchanger. The cylinder is lightly inclined (i.e., slope about 2-6%) to facilitate the axial displacement of the solid bed, which moves towards the discharge end as the hot gases circulate counter-current mode. Figure 1 shows an outline of the rotary kiln. The solid feed is a carbonized matter obtained from a variety of raw materials (e.g., eucalyptus wood). The hot gases, which arise from a central burner and are originated by combustion of natural gas, supply the necessary energy for the activation reaction. Steam is used as the activation agent and is injected in co-current mode.
   Most of the raw materials are relatively pure solids, with moisture contents around 5 to 10%. Raw material grain size was -6+20#ASTM, which is equivalent to an average particle size of 0.002 meter. Usually, the content of impurities in the carbonized solid was negligible.
   It is supposed that the solid bed moves as a pseudo fluid with axial displacement and without retro - mixing, and it rolls or slides in traverse direction as the cylinder rotates.
   The geometric parameters (i.e., L_cs, L_cu and L_li) were determined as a function of the solid flow rate, residence time, and rotary kiln dimensions, by means of geometric relationships and iterative calculation (Coulson and Richardson, 1981; Wilbrand, 1997). Equation 1, which satisfactorily predicts residence times in rotary kilns (Perry et al., 1984) was used in this work.

                                                                         (1)

III. MATHEMATICAL MODEL

   There are three phases inside a rotary kiln: gas, solid, and wall. Beside the activation chemical reaction, the phenomena occurring are: mass transfer (i.e., solid’s moisture and gaseous reaction products are transferred from the solid to the gas phase) and heat transfer (i.e., convection and radiation between gas and solid bed, gas and wall, wall and solid bed, and wall and surroundings, and conduction inside of the wall).
   In order to model the transfer phenomena for the existing phases, the differential mass and energy balances, which include chemical reaction terms, are posed in cylindrical coordinates. Based on laboratory tests, publications on the subject (Wilbrand, 1997; Kim and Srivastava, 1990), and geometric characteristic, the following assumptions were adopted:
- Both, solid and gas axial linear velocity changes are negligible.
- All variables are supposed uniform in the radial direction due to the small size of the reactor.
- There is neither solid nor gas axial mixing. Then, the two phases are considered as a plug flow model. Solid drag by gas is negligible.
- Variations in rotary kiln wall temperature in the angular direction are neglected. (i.e., the so called regenerative effect of the wall).
- Constant steam flow rate is assumed.
- Pressure in the reactor is constant.
- Chemical reaction rate obeys the Arrhenius law, and secondary reactions are rejected.
- Axial heat transfer by conduction and radiation is negligible.
- Axial heat transfer by conduction inside the wall is unimportant.
- Specific heats, reaction heat, heat of vaporization, emissivities, absorptivities, and heat transfer coefficients by conduction in the wall, are supposed constant and temperature independent.

A. Chemical reaction rate

   Equation 2 shows the solid disappearance, a consequence of the chemical reaction between carbonaceous material and steam (Yehaskel, 1978). In order to obtain the kinetic expression, the secondary reactions which occur in the reactor are not considered. The secondary reaction, formation of CO₂ from CO and steam, has not been taken account since it is a reversible reaction produced into an environment containing high CO₂ concentration because of the combustion of heating gases.

C + H₂O ==> CO + H₂ [28,5 Kcal / mol] (2)

    A kinetic equation responding to the Arrhenius law was obtained from laboratory data measured under experimental conditions similar to those of the rotary kiln operation (Martínez, 1998). Considering the steam flow rate constant, the reaction becomes pseudo first order: The obtained expression was modified in order to adapt it to the mass balance given as the mass flow rate variation along the z-coordinate.

                  (3)

B. Mass balance for moisture in the solid bed

   The drying of solids has a first stage governed by the evaporation rate, and it is considered that the solid surface remains saturated. In a second stage, the drying process is governed by diffusion of moisture from inside of the particle toward its surface (Coulson and Richardson, 1981). For critical moisture of 10% and supposing the continuity of the two stages, the moisture variation of the solid in the axial direction can be expressed by Eqn. 4 (see Table 1).

C. Mass balance for the solid bed

   The solid moves through the reactor in the positive direction of the z-coordinate. The solid flow rate variation along the reactor can be expressed by Eqn. 5 (see Table 1). The right term of this expression has two parts: the disappearance of solid by chemical reaction, according to Eqn. 3, and the loss of water by evaporation, according to Eqn. 4.

 Table 1. Equation system for the proposed model

D. Mass balance for the gas phase

   The gas flow rate increase in the negative direction of the z-coordinate is due to the appearance of water evaporated from the solid bed and the gases produced by chemical reaction. Gases and solids move along the rotary kiln in counter-current mode. This fact determines the sign of each term in Eqn. 6.

E. Energy balance for the gas phase

   The gas heat flow variation along the z-coordinate depends on the heat transferred by convection and radiation between gas and solid and gas and wall, plus the required sensible heat to overheat the vapor from the boiling temperature (i.e., 373 K) to the freeboard gas temperature (see Eqn. 7 in Table 1). The parameters C₃, C₄, C₅ and C₆, which include heat transfer coefficients and heat transfer areas, are given in Table 2.

F. Energy balance for the solid phase

   The solid heat flow variation along the z-coordinate depends on the heat transferred by radiation from solid to both gas and wall, the heat transferred by convection to gas and wall, the heat loosed as the solid’s moisture evaporates, and the heat absorbed by the solid to chemically react (i.e., endothermic reaction, see Eqn. 8 in Table 1 and parameters C₇ and C₈ in Table 2).

G. Energy balance for the wall

   It accounts for the heat transferred by conduction through the wall and the heat transferred by convection and radiation (gas-wall and wall- surroundings).
   The heat transferred by conduction and convection between the covered wall (i.e., covered by the solid bed) and the solid itself was not considered because both phases, solid and wall, reach thermal equilibrium just in the first third of the kiln length.
   The cyclic change in wall temperature was neglected (i.e., the so called regenerative effect) due to the small size of the kiln and because the high temperature zone covers the reactor length almost completely. In this zone, the main heat flow takes place fundamentally by radiation to the solid from both the gas and the not covered wall.
   The internal wall temperature was necessary to solve the model. A complex system of equations was obtained after considering the net heat flow (i.e., gas-wall, internal-external wall and external wall-surroundings). In order to solve such system, the heat transferred by radiation was neglected and only the heat transferred by convection was considered. The radiation phenomenon was included in the convection heat transfer coefficient (Gorog et al., 1982).
   The resulting equation for the internal wall temperature T_w (see Eqn. 9 in Table 1) include the external wall temperature, T_wo, which was calculated using the expression proposed by Gorog et al. (1982) (see Eqn. 10 in Table 1). The wall-surroundings heat transfer coefficient was included in Eqn. 9. The gas-wall heat transfer coefficient, h_i, is shown in Eqn. 11. This coefficient includes the heat transferred by both convection and radiation (McAdams, 1954).

 Table 2. Equation parameters

   The parameters C₃,…, C₈ in the model given in Table 1 are shown in Table 2. For the wall-solid convection heat transfer, h_w, the equation proposed by Wilbrand, (1997) was used. The gas-wall, h_gw, and gas-solid heat transfer coefficients, h_gs, were calculated by means of the expressions proposed by Gorog et al. (1982).

IV. RESULTS AND DISCUSSION

   The proposed model which is constituted by a non-linear differential and algebraic equation system, was solved by means of the finite differences method. Although this resolution method is less exact than others (e.g., finite volumes) it is suitable for a one-dimensio-nal simple geometry. Due to its user’s friendly nature, the commercial software Mathcad 2000 Professional was used to solve the model.

   The model was applied to different operation conditions of the rotary kiln, and the results compared with experimental data.
   The initial operation conditions at the rotary kiln input extreme, Q_s0, T_s0, Q_g0, T_g0, Q_h0, T_w0, Q_a and N, were known. Because the calculation starts at the solid input extreme, the output gas temperature was also supposed as an initial condition. The computation begins at z = 0 with the initial given conditions, and continues until z = L. Then, the calculated inlet gas temperature is compared with the actual data. If the convergence is not achieved, a new outlet gas temperature value is proposed. A Nelder and Mead (1965) optimization method has been used, and the rate of convergence was acceptable for the used initial conditions. Rotary kiln operation data and initial conditions assumed for the example case are reported in Table 3.
   Figure 2 shows the temperature profiles of the solid (T_s), gas (T_g) and wall (internal, T_w, and external, T_w0) versus rotary kiln axial length. Since the simulation results could not be contrasted with our experimental data, the data reported by Barr et al. (1989) were taken as a reference. A complete set of experimental results for several endothermic reactions in a pilot rotary kiln is presented in this paper. The temperature profiles obtained with the proposed model fitted acceptably the experimental data reported in the literature (see Fig. 3). The solid flow rate variation, which was due to weight losses by reaction, along the axial length is shown in Fig. 4.

   In order to analyze the production performance, two parameters were used: production yield and burn-off. The kiln’s production yield was defined as the relationship between the solid flow rate at the output extreme and the solid flow rate at the input extreme (see Eqn. 12). The burn-off, which represents a measure of the activation reaction extent, was calculated as the relationship between the solid converted and the solid fed minus the moisture lost by the solid (see Eqn. 13).

(12)

(13)

   For the particular case studied, the results for production yield and burn off were 72.5 % and 18.4 %, respectively. These results agree with those reported by Smith in 1979 for eucalyptus charcoal samples tested in a similar pilot rotary kiln. The average production yield and burn off published were 70 - 75% and 18 - 20%, respectively. Furthermore, the activated product obtained displayed very good adsorbent properties and an acceptable mechanical resistance. For a production yield of 75%, the results were: Iodine Number of 763 mg/g, Modified Phenol Value of 15.8 ppm and Abrasion Number of 54%.

V. CONCLUSIONS

   Since the physical and the chemical processes which occur inside the rotary kiln are very complex, it was necessary to adopt different assumptions in order to develop a mathematical model that could be solved easily.

Figure 4. Solid flow rate vs. reactor axial length

   Given the small size of the kiln and the operation conditions tested, the assumptions lead to a model whose resolution was of medium complexity when traditional mathematical methods were applied.
   The mass transfer due to the chemical reaction was modeled using a simple kinetic equation generated from own experimental data. This allowed to simulate the mass transfer between solid and gas, properly.
   An estimation of the heat transfer coefficients is quite difficult because of the lack of reliable experimental data. The relative importance of the heat transferred by radiation with respect to other transfer phenomena was also verified since the model was particularly influenced by changes in those coefficients.
   The results obtained from the steady state simulation allowed to observe the effects of the heat and mass transfer, when the activation reaction was carried out. They also permitted to have a model that describes the kiln performance properly. Although the model should be adjusted to our experimental data, the comparison with characteristic experimental values from the literature was reasonably good. This occurred in spite of the model middle complexity and that the regenerative effect of the wall was not considered.
    The model, once improved with our experimental data, will permit to adapt the operation conditions of the rotary kiln to reduce the energy consumption and to improve product quality. In a next paper, a performance analysis will be addressed to select the optimal operational conditions.

NOMENCLATURE

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Received: November 16, 2001.
Accepted for publication: May 14, 2002.
Recommended by Subject Editor G. Meira.