71
I n d . E n g . C h e m . R e s . 1990, 29, 71-81 Muhrer, C. A. Study of the Dynamics and Control of Vapor Recompression Columns. Ph.D. Thesis, Lehigh University, Bethlehem, PA, 1989. Nielsen, C. S.; Andersen, H. W.; Brabrand, H.; Jorgensen, S. B. Adaptive Dual Composition Control of a Binary Distillation Column with a Heat Pump. Presented at IFAC Symposium on Adaptive Control of Chemical Processes, ADCHEM-88, Lyngby, Denmark, 1988. Null, H. R. Heat Pumps in Distillation. Chem. Eng. Prog. 1976, 73, 58-64. Quadri, G. P. Use of Heat Pump in P-P Splitter. Part 1: Process Design. Hydrocarbon Process. 1981,60,119-126; Part 2: Process Optimization. 1981, 60, 147-151.
Robinson, C. S.; Gilliland, E. R. Elements of Fractional Distillation; McGraw-Hill: New York, 1950. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Haggblom, K. E.; Gustafsson, S. E. An Experimental Comparison of Four Control Structures for Two-Point Control of Distillation. Znd. Eng. Chem. Res. 1988, 27, 624. Yu, C. C.; Luyben, W. L. Design of Multiloop SISO Controllers in Multivariable Processes. Znd. Eng. Chem. Process Des. Deu. 1986, 25, 498-503.
Received for review February 15, 1989 Revised manuscript received August 17, 1989 Accepted September 6, 1989
Simulation and Control of an Industrial Calciner Nam K. Kim* and Rajeev Srivastava Department of Chemistry and Chemical Engineering, Michigan Technological University, Houghton, Michigan 49931
Rotary calciners, when employed in chemical ore refining and manufacturing processes, are responsible for the conversion of the raw materials into crude products. This paper presents a mathematical model of an industrial calciner used for the endothermic decomposition of sodium sesquicarbonate (trona ore) into crude soda ash. The calciner is equipped with lifters to enhance the contact efficiency of solids with the hot gas stream. The model-simulated dynamic response of the calciner in unreacted sodium sesquicarbonate content, to the step and sinusoidal disturbances, also agrees qualitatively with the observed system behavior. A PID feedback control scheme for the calciner is presented. The quality of control accomplished is severely limited by the large dead time inherent to the process. T h e phase plane analysis for a sinusoidal disturbance (period = 2td) reveals t h a t the closed-loop system slowly approaches a limit cycle. Despite extensive use, rotary calciners have remained a difficult system to analyze. Their design and performance have been simply covered by empirical procedures with relatively superficial insight into the theory of operation. The model described in this paper is a part of the continuing research aimed at developing more effective control schemes for the calciner than is possible by using empirical techniques.
Calciner DescriDtion The calciner under study is essentially a cylindrical steel shell, rotated about a slightly inclined horizontal axis. Fourteen sections of closely spaced lifting flights are installed inside the calciner to shower the solids through the gas stream and improve the contact efficiency. Trona ore is fed at the high end of the calciner and advances to the lower discharge end due to the rotation of the shell and the lifting action of the internal flights. Heat for the reaction is provided by burning a mixture of natural gas and air, using a single centerline burner. The hot calcining gases move cocurrently in direct contact with the solids. The first section of flights start at 6.0 m from the burner end in order to minimize showering of solids through the flame and promote full flame development (combustion zone). The lifters continue for 18 m (lifter zone), followed by 3.0 m of bare section to allow for dust disengagement from the flue gas (heat soaking zone). The dimensional characteristics of the calciner and arrangement of flights in a cross section are shown in Figure 1 (Kim and Srivastava, 1988). Mathematical Model The mathematical model for the calciner is a set of heat and mass balance equations derived in this section. The 0888-5885/90/2629-0071$02.50/0
following assumptions were made in order to construct the model. 1. All parameters in a radial section of the calciner are constant. 2. Axial heat transfer due to conduction and radiation is insignificant, including the conduction in the axial direction of the wall. 3. Coefficients of convection, emissivities, specific heats, latent heat, heat of reaction, and activation energy are independent of temperature. 4. The change in the velocity of the gas and solids as they move down the length of the calciner is negligible. 5. There is no axial mixing of the solids. 6. Carrying away of solid particles by the flue gas is not explicitly included in the model. 7. The decomposition of sodium sesquicarbonateto soda ash follows Arrhenius’ law and is first order with respect to sodium sesquicarbonate. These assumptions are aimed at simplifying the otherwise cumbersome heat- and mass-transfer equations. Distribution of Material in a Cross Section. Bare Zones. In the feed and discharge ends of the calciner, solid material rolls or slides on the inside surface of the lower shell and is distributed in a crescentlike section with fixed ends, as shown in Figure 2a (Kim and Srivastava, 1988). The exposed heat- and mass-transfer area of the solid is small. The material distribution can be approximated by a circular segment. If given the solids flow rate and the dimensions of the calciner, arc lengths L1,Lp,LB,and L4 can be readily calculated (Perry, 1963). Lifter Zone. In this zone of the calciner, particles of the solid are continuously spilled from the flights. This occurs from the time the flights leave the rolling bed until they are empty. Heat and ;ass transfer takes $ace as the particles fall through the hot gas stream. The exposed area 0 1990 American Chemical Society
72 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 h
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trolled primarily by the mechanism of moisture movement within the solid, the rate of drying during the falling rate stage is directly proportional to the free moisture content of the solids. Thus,
where Kh is a constant determined at the boundary between the two drying stages, which occurs when the moisture content of the solids equals the critical moisture content. Because of continuity, eqs 1 and 2 can be equated at this point, and for a known value of critical moisture content, Kh can be calculated. Assuming a critical moisture content of 1070,on a wet basis
Figure 2. Distribution of solids in the (a, top) bare zones and (b, bottom) lifter zone.
of the solids drastically increases. Critical loading conditions are assumed, which implies that the solid flow rate is just sufficient to fill up the flights completely as they leave the rolling bed. Distribution is shown in Figure 2b (Kim and Srivastava, 1988). L,' and L,' are the counterparts of L , and L3 in the bare zones. Mass-Transfer Equations. Drying. Drying of the solids in a rotary calciner occurs in two stages, the constant rate stage and the falling rate stage. Constant rate drying occurs when the moisture content of the solids is greater than the critical moisture content. During this stage, moisture movement within the solid particles is rapid enough to maintain a saturated condition at the surface. The rate of drying is controlled by the rate of heat transfer to the evaporating surface (Coulson and Richardson, 1978). In any given cylindrical section of the calciner, the rate of change of moisture contained in the solids must be equal to the rate at which moisture enters the section with the solids minus the rate of drying. Thus, the mass balance equation for free moisture is of the form
where Qh is the flow rate of the free moisture contained in the solids, L, is the latent heat of vaporization of water, ht is the total heat-transfer coefficient, A is the exposed surface area of the solids per unit length of the calciner, V, is the velocity of the solids inside the calciner, and T g and T , are the temperatures of the calciner gas and the solid. For the bare zones, A is simply equal to L2 For the lifter zone, however, A is a complex function of flight design and material characteristics as derived in Appendix 2. Correlations used to estimate ht in the bare sections and the lifter section are also given in Appendix 2. When the moisture content of the solids is less than critical, the entire evaporating surface can no longer remain saturated. Therefore, the drying rate decreases. Con-
where Q, is the flow rate of the solids in the calciner. The mass balance equation for the free moisture in the falling rate period can, therefore, be written as
Since the free moisture content of the feed ore varies between 0.5% and 5%, it is reasonable to assume that all drying takes place in the falling rate period. Reaction Kinetics. The function of the calciner is to decompose trona ore into crude soda ash according to the reaction 2(Na2CO3.NaHCO3.2H2O)= 3Na,C03 + C 0 2 + 5 H 2 0 (5) The reaction is endothermic. For temperatures lower than 95 "C, the activation energy, E , is 33.8 kcal/mol of trona. The high value of E suggests that the chemical reaction is the rate-limiting step in this range of temperatures. A t temperatures higher than 95 "C, the reaction exhibits an activation energy of less than 12 kcal/mol of trona. This indicates that another resistance of lower activation energy, the diffusion of reaction products out of the particles, becomes increasingly important at higher temperatures. Again, it is assumed that the reaction follows Arrhenius' law and is first order with respect to sodium sesquicarbonate. The mass balance equation for the total amount of sesquicarbonate contained in the solids can be written as
i a
- -(S,Q,)
v, at
= - -(S,Q,) a
ai
- Qm-Sqe-E/(RTm) K* (6)
Vm where K, is the preexponential factor and S , is the sodium sesquicarbonate content (7') of the solids at any instant. Gas a n d Solid Flow Rates. Drying of solids and evolution of carbon dioxide and water vapors due to decomposition of trona result in an increase in the gas flow rate, Qg, and decrease in the solids flow rate, Q,, down the length of the calciner. Considering the stoichiometry of
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 7 3
H,, = honD(T, - To)+ unDe,(TW4- T t ) (12) where D is the diameter of the calciner, T, and Toare the
air
Figure 3. Heat fluxes.
the reaction, the mass balance equation for the solids can be written as
temperatures of the calciner wall and the surroundings, and ho is the coefficient of convection determined in Appendix 2. Heat fluxes H (gas to material), H (gas to wall), and H,, (inner w a f t 0 material) depenrdirectly upon the geometry of the distribution of solids. They must be written separately for the bare zones (Dumont et al., 1978; Manitus et al., 1974) and the lifter zone of the calciner. Bare Zones. H , = h h 2 ( T ,- T,) + oL2e,eg(T,4 - Tm4) (13a)
+ uL3eweg(T,4
H m = hgL3(Tg- T,) H,, = h,Ll(Tw
- T,)
,-
TW4) (14a)
+ oL2ew(l- eg)e,(TW4- Tm4) (154
Lifter Zone. Since the gaseous products of the reaction and evaporated moisture join the gas phase, the mass balance for the calciner gas becomes
In the combustion zone, however, the mixing of the flame combustion products with the calciner gas must be considered. Therefore,
i a v, -(Qg) at
a
= - -(QJ ai
+ 100 v,
where Q, is the rate of flow of combustion products in the flame. Heat-Transfer Equations. Heat fluxes considered in the development of the model are shown in Figure 3 (Kim and Srivastava, 1988). Drying solids results in the heat flux, He,, which can be expressed as dQh dl Since the latent heat of vaporization, L,, is provided by the solids, drying has a cooling effect on the solids in the calciner. H,,, the heat flux due to the reaction, is evaluated as
He, = --L,
where AH is the heat of reaction. The reaction is endothermic; therefore, H , represents the rate at which energy is consumed. Hvg, the heat flux from the vapor to the gas, can be expressed as dQm Hvg = - -C,T, dl
Hgm = UA(Tg- T,)
(13b)
+ uL3/eweg(Tg4- TW4)(14b)
H , = h,L3’(Tg- T,)
H- = h&l’(Tw - 7”) (15b) where e f , eg, e,, and e, are the emissivities of the flame, the calciner gas, the solids, and the calciner wall, respectively. U is the unit surface conductance which combines the effects of convection and radiation for heat transfer between the solids and gas in the lifter zone, and h, and h, are coefficients of convection. The correlations used to estimate U and h are given in Appendix 2. In addition to the a%ove-mentioned fluxes, heat transfer from the flame has to be incorporated in the combustion zone of the calciner. The flame is assumed to be cylindrical in shape, having a length of 6.0 m and a diameter equal to half the diameter of the calciner. It is further assumed that combustion is complete as soon as the fuel-air mixture leaves the burner. Adiabatic combustion temperature is thus achieved adjacent to the burner. The combustion products then lose heat to the surrounding calciner gas as they move down the length of the combustion zone. Linear mixing of the combustion products and the calciner gas is also assumed. Heat fluxes H f , (flame to material), Hf, (flame to inner wall), and H f g(flame to calciner gas) can be expressed as Hfm = gL,eF,(l - eg)Ff,(T+- Tm4) (16)
H f w= uL3efe,(l H f g = UTD~~&,(T;L - 7’:)
- e,)Ffw(Tf4-
Tw4)
(17)
dQC + hfnDf(Tf- Tg)- -C,Tf dl
(18) where Ff, and Ffware geometric shape factors, Df is the diameter of the flame, T f is the temperature of the flame, and hf is the coefficient of convection. Considering the flow of heat by virtue of the abovementioned fluxes, the heat balance equations for the calciner gas, the solids, the flame, and the calciner wall can be represented by the equations
a d ( Q C T ) = --(Q vgat ai g
g
g
g
C T ) + H f g+ Hvg - HBm- Hm g
g
(19) where C, is the specific heat of the vapor. This represents the rate a t which heat enters the gas stream with the vapors produced by drying and reaction. Loss of heat from the outside surface of the calciner wall to the surroundings can be expressed as
74 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
(22)
where Cc, C,, C,, and C, are the specific heats of the combustion products, the calciner gas, the solids, and the calciner wall, respectively. M , is the mass of the calciner wall per unit length. Retention Time Equations. Bare Zones. The retention time of solids in a bare, rotating, cylindrical shell is controlled by parameters such as diameter, length, slope, the rate of revolution of the shell, and the loading and physical characteristics of the solid material. The following relation has been used for the bare zones of the calciner: C(ang1e of in ft) Tb (min) = (slope in in./ft)(rpm)(D in ft) (23) where Tb is the retention time of the solids in the bare zone, L b is the length of the bare zone, and D is the diameter of the calciner. C is a constant which has been concealed for confidentiality purposes. This equation has been experimentally verified for the calciner under study and has proved to be in close agreement with the actual plant measurements. Lifter Zone. The retentiom time model for this zone is dominated by the dynamics of particle movement in a gas stream, but it is also influenced by the geometry of flights and the slope and revolution of the calciner. Relative motion of the gas imposes a longitudinal drag on the particles as they fall from the flights. Retention time in the lifter zone is calculated by solving the equations of motion of the particles. It is assumed that the particles are spherical. This results in Lf T I f(min) = (t + (24) 1 Yavgsin N + -KV,2t2 2 where Yavg is the weighted mean length of fall of the particles and 6 is the ratio of the circumference to the distance an average particle travels on the periphery before it falls from the flight (in rpm). Both Yavgand 6 are complex functions of the flight dimensions and material characteristics. Tlfis the retention time of solids in the lifter zone of length LIP K is a function of particle Reynolds number, and V gis the velocity of the calciner gas. In eq 24, t is the average time of fall of the particles, given by
i)
(25) Detailed analysis can be obtained elsewhere (Schofield and Glikin, 1962).
Steady-State Simulation Steady-state equations for the calciner can be obtained by setting the time derivatives in eqs 4,6-8, and 19-21 equal to zero. This results in a set of nonlinear differential equations and an algebraic equation, which can be solved by using the Runge-Kutta and the Newton-Raphson techniques. Comparison of the steady-state simulation against the plant operating data was discussed in detail elsewhere (Kim and Srivastava, 1988). Algorithm. 1. S,, T,., T f ,T,, Qh, and Q, are known at the hot (1 = 0) end. Using T,, T,, and T,, calculate T , from eq 22 by using the Newton-Raphson technique.
2. Using now known values of s,, Tg,T,, Qh, Q,, and Tw,integrate forward the system of eqs 4,6-8, and 19-21, employing the fourth-order Runge-Kutta method with an appropriate step size. 3. After each integration step, calculate T, from eq 22 using the newly calculated values of T,and T,. 4. Continue until the exit end is reached, and then compare the computed soda ash content to the actual value. Adjust K , in eq 6 until the simulation matches the plant’s measured values. Simulated Steady-State Profiles and Discussion. Computed temperature and sodium carbonate concentration profiles, along with the measured plant data (0, 0, A , A ) , are shown in Figure 4. Sharp variations in the profiles are evident at 6 m from the feed end of the calciner. The first row of lifters starts at this point, resulting in a dramatic increase in heat- and mass-transfer rates due to showering of solids through the hot gas stream. This explains the steep decline in the calciner gas temperature. The solid temperature, on the other hand, rises sharply until a point is reached where the rate of endothermic reaction balances the rate of heat transfer from the gas. The temperature of the solids then remains constant until the reaction is almost complete. The actual temperature of the calciner product and the flue gas agree well with the simulated values. The bands over the sodium carbonate concentration curve are due to the operating range of the feed ore quality. The simulation confirms the shell overheating problem previously encountered in industry. The calciner shell in the feed end severely warped and eventually cracked. The greatest thermal stress was imposed on the area that was 5-6 m from the feed end. The simulated shell temperature profile indicates that the mild steel must have suffered the upper limit temperature (Kim et al., 1986). The heart of the rotary calciner is the section from 6 to 16 m where most of the reaction takes place. The last row of the lifters may be safely removed. This is because the temperature driving force in this section diminishes and the role of holdup for longer heat soaking period becomes far more important than showering of the solids. The removal of the last row of lifters will further reduce the dust entrainment. This, in turn, will alleviate the load on the dust recovery system.
Dynamic Simulation Computation of the dynamic response of the calciner requires the solution of a set of partial differential equations (4), (6)-(8), and (19)-(22). These equations have been simplified by averaging out any variations in the radial direction. Thus, the system variables depend exclusively upon time and the position along the length of the calciner. This is fairly representative of actual conditions, especially for the solid and the gas since they are well mixed. A first-order finite difference scheme was used to solve this system of equations. Algorithm. 1. Run a steady-state simulation. This generates the initial temperature and concentration profiles of the system along the length of the calciner for time t = 0. The steady-state profiles are the starting point for the transient study. 2. Introduce step change in the input for which the dynamic response is desired. In other words, increase or decrease its value by a known amount, at 1 = 0, for all time t. J. Run the dynamic simulation, starting at t = AT and 1 = U. At each time and length increment, make an initial guess for the values of the various system variables, based
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 75 1
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on their previous values, and iterate the solution of the discretized dynamic equations until a specified convergence criterion is met between two successive iterations. A trade-off exists between the computation time and accuracy of the simulation. By imposing a stringent convergence criterion, the accuracy can be increased at the expense of computation time and cost. Simulated Open-Loop Response and Discussion. Open-loop response of the system to step changes in the manipulated variable along with some selected inputs is presented in this section. Also presented is the response of the system to a sinusoidal disturbance in the sesquicarbonate content of the feed ore. Other than the discharge end, the response is computed a t 5 and 15 m from the feed end, thereby covering all three zones of the calciner. Concentration profiles are presented in terms of unreacted sesquicarbonate content of the solids. The sesquicarbonate content of the product should always be close to 0% for satisfactory calciner operation. This serves as the common reference value in determining the extent
to which various disturbances affect the system. In order to show the effect clearly, the magnitude of the disturbance in all cases is relatively large. A disturbance of smaller magnitude will provide the same effect, but to a lesser degree. Step Changes in the Free Moisture Content of the Feed, Variations in the free moisture content of the feed ore are commonly encountered. These usually result from changes in the climatic and storage conditions. Response of the system to step changes of +1.5% and +3.5% in the free moisture content of the feed ore, from an initial value of 0.72%, is shown in Figure 5. An increase in the free moisture content of the feed leads to an increase in the unreacted sesquicarbonate content of the product. The magnitude of the response is proportional to the size of the step change introduced. Evaporation of additional moisture reduces the temperature of the solid below its steady-state value. Gas temperature also decreases as a larger mass of evaporated moisture now joins the hot gas stream, absorbing a greater
1
76 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 I...
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amount of heat from it. Reduced temperatures of the gas and solids result in a lower reaction rate, which explains the increase in the sesquicarbonate content of the product. In the combustion zone, the exposed surface area of the solids is small, so the increased moisture content of the solids does not affect the evaporation rate significantly. The change, therefore, has little effect on the state of the calciner in this zone. No significant drop in the temperature of the solids and the calciner gas is observed. However, a sizable drop in the sesquicarbonate content of the solids is observed. This does not signify an increase in the reaction rate. Increased moisture content results in an increased total mass of the solids. Since the mass of sesquicarbonate remains the same, its percentage decreases. In the lifter zone, the exposed area of the solids is large and the effect of an increased evaporation rate is felt more prominently. This can be seen in the dynamic profiles of the system variables at 15 m from the feed end. The sesquicarbonate content of the solids increases, and the temperatures of the solids and the gas drop from their respective steady-state values. A dead time is observed in the gas temperature response. This is of the order of the retention time of the solids in the combustion zone, where, as mentioned earlier, the change has little effect. The gas temperature drops markedly only after the first batch of solids with an increased moisture content enters the lifter zone. Since the gas moves with high velocity, the drop is registered throughout the calciner length without further delay. The dead time observed in the response of sesquicarbonate content and temperature of the solids is larger, as solids move slowly in the calciner. Response of the system a t the discharge end is similar to that at 15 m from the feed end. The dead time observed in the dynamic profiles of the solids temperature and concentration is now essentially of the order of the retention time (18-25 min) of the solids in the calciner (Kim, 1977). Step Changes in the Sesquicarbonate Content of the Feed. The content of sesquicarbonate in natural trona ore from the same site can vary between 85% and 98%. This is one of the major causes of perturbation in the
operation of the calciner. Response of the system to step changes of +5% and +8% in sesquicarbonate content of the feed ore, from an initial value of 90.83%, is shown in Figure 6. The ultimate effect of these changes is an increase in the unreacted sesquicarbonate content of the product. An increase in the sesquicarbonate content of the feed is actually an increase in the total amount of reactant entering the system. Therefore, the energy requirement of the system increases if no change in the product quality is desired. Since no manipulations are made to supplement the energy input, some of the sesquicarbonate passes through the calciner unreacted and shows up in the product. It is interesting to note, however, that all of the additional sesquicarbonate does not end up in the product. This is because the reaction is first order with respect to sesquicarbonate and its rate increases with an increase in sesquicarbonate concentration. In the following discussion, the response of the system is analyzed in reference to this increased reaction rate. Since no reaction occurs in the combustion zone, a 5% (or 8%) increase in the sesquicarbonate content of the solids is observed at 5 m from the feed end. The dead time observed in the response is the time elapsed before the first batch of solids, with increased sesquicarbonate content, reaches the 5-m position. Temperatures of the gas and solids are not affected in this zone, as the reaction does not proceed to any appreciable limit. This is evident from the straight line response of these variables. Once the affected solid enters the lifter zone and heats up to the reaction temperature, reaction proceeds at a faster rate, due to increased sesquicarbonate content. Since the reaction is endothermic, the temperature of the solids drops from its steady value. The temperature of the gas also drops as it now loses more heat to the colder solids. These effects can be seen in the temperature profiles of the system at 15 m from the feed end. The sesquicarbonate content of the solids now shows an increase of less than 5% (or 8%). The dead time observed in the gas temperature is the time elapsed before the reaction starts, a few meters inside of the lifter zone. The dead time
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 77
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Figure 7. Dynamic response of the system to step changes of -300 and -450 kg/h in the natural gas flow rate. (left) At 5 m from the feed end; (middle) a t 15 m from the feed end; (right) a t the discharge end.
observed in the response of solids temperature and concentration is of the order of the time taken by the first batch of solids with increased sesquicarbonate content to reach the point of observation. Response a t the discharge end is similar, with larger dead times in solids temperature and concentration profiles. Step Changes in the Natural Gas Flow Rate to the Burner. Changes in the natural gas flow rate do not occur naturally. They are the result of manipulations made by either a human operator or a controller in order to rectify the adverse effect of other disturbances. The computed response of the system to step changes of -200 and -450 kg/h in the natural gas flow rate, from the initial value of 5563 kg/h, is shown in Figure 7. A decrease in the fuel flow rate leads to lower gas and solid temperatures in the calciner. Since decomposition of sodium sesquicarbonate proceeds slowly at low temperatures, the unreacted sesquicarbonate in the product increases. A change in the fuel rate directly affects the gas temperature instead of the solids. Therefore, the response of all system variables is fast. The degree of the effect varies from zone to zone. In the combustion zone, the area of heat transfer between the solids and the gas is small; even a steep drop in the gas temperature does not affect the solid temperatures much. Sesquicarbonate content of the solids also remains unaffected, as the reaction does not start in the combustion zone. A quick drop in the solid and gas temperatures is observed in the lifter zone, where the heat-transfer area is large due to showering. The sesquicarbonate content of the solids increases as the reaction rate decreases. No dead time is observed in the response of any of the system variables. As opposed to the lifter zone, the sesquicarbonate content of the solids at the discharge end remains unchanged for about 3 min; then it starts increasing. This dead time is of the order of retention time of solids in the heat soaking zone. Since the original operating conditions are such that the reaction is virtually complete in the lifter
zone, solids that had already reached the heat soaking zone at the time of application of the change are not affected by the change. This explains the delay in the response of the system. No dead time is observed in the gas temperature profile at the discharge end. Sinusoidal Disturbance in the Sesquicarbonate Content of the Feed. The open-loop response of the system to a sinusoidal disturbance of amplitude 8% and period 2td is shown in Figure 8. As expected, the system variables oscillate about their steady-state values with a frequency equal to the frequency of the disturbance. Phase lag exists between the response of the variables at different positions along the length of the calciner. This lag is induced by the dead time, which increases as we move further away from the feed end. Lag in the gas temperature response is smaller than that in the solids temperature and concentration response. The straight line response of the gas and solid temperatures at 5 m is due to the fact that the reaction does not start in the combustion zone; therefore, changes in the sesquicarbonate content of the solids has no effect on the state of the calciner.
PID Control Scheme for the Calciner The control objective for the PID control scheme discussed in this section is to maintain the unreacted sesquicarbonate content of the product close to 0%. The scheme uses natural gas flow rate to the burner as the manipulated variable and sesquicarbonate content of the product as the direct measured and controlled variable. In the simulation program, this controller can be implemented as Qng = Qng,s+ K,(error + erint + erder) (26) and the feedback control algorithm can be programmed in part as error = S, - S,,s 1
erint = erint
+ error(Ar/rI)
erder = sd(error - erold)/AT where Qngis the natural gas flow rate, S , is the sesquicarbonate content of the product (subscript s denotes
78 Ind. Eng. Chem. Res.. Vol. 29, No. 1, 1990
480
Tim# ( m i d
IO
1
8
i
e
Figure 8. Dynamic response of the system to a sinusoidal disturbance of amplitude 8% and period 2td in sesquicarbonate content of the feed. (left) At 5 m from the feed end; (middle) at 15 m from the feed end; (right) at the discharge end.
..
"i t.8
//
/
i
0
I,
Figure 9. Extraction of process parameters: Cohen-Coon method
steady-state values), AT is the integration step size, and erold is the error that existed a t the preceding integration step. K,, T I , and T D are the controller parameters. Before this algorithm can be incorporated in the simulation, the controller has to be tuned to the system. The Cohen-Coon method of controller tuning was used. The extraction of process parameters (&, t d , and T ) from the process reaction curve (open-loop response of the system to a step change of -450 kg/h in the natural gas flow rate) is shown in Figure 9. Controller parameters are then evaluated as (Stephanopoulos, 1984)
Closed-Loop Profiles. Closed-loop profiles of the system to step changes in the free moisture and sesquicarbonate content of the feed are shown in Figures 10 and 11. Figure 12 shows the closed-loop profile for a sinusoidal disturbance in the sesquicarbonate content of the feed. For step changes in the inputs, the designed control scheme meets the control objective of keeping the unreacted sesquicarbonate content of the product a t its steady-state value, close to 0%. For a sinusoidal disturbance, however, the closed-loop system seems to approach a limit cycle. This is confirmed in the phase plane plot of Figure 13. The scheme uses sesquicarbonate content of the product as the measured variable. On-line composition analysis of the calciner product is very difficult to perform. A sample has to be taken to a nearby laboratory, where it is analyzed by titration methods; there is a gap of a t least 10-15 min before its composition is known. Although this gap can be reduced by using more sophisticated analytical techniques (such as X-ray diffraction, etc.), a substantial delay can still be expected. On the other hand, the sim-
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 79
0
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40
00
12
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Time ( m i d Figure 10. Closed-loop response of the system to a step change of +3.5% in free moisture content of the feed.
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12
24
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40
60
72
04
100
06
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Figure 11. Closed-loop response of the system to a step change of +5% in the sesquicarbonate content of the feed a.0-
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80 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
Figure
content of the feed.
13. Phase
ulation uses an integration step size of 10 s, which means that the controller gets an update on the product concentration every 10 s. Because of this discrepancy, the actual response of the system will be much more unstable than the one shown in Figures 10, 11,and 12, if this scheme is implemented in the real world. Another disadvantage of such a scheme originates from the fact that the system exhibits a large dead time before the effect of certain disturbances is felt on the sesquicarbonate content of the product. In the open-loop analysis of the system (Figures 5 and 6), it was observed that this dead time is of the order of the retention time of the solids in the calciner. This implies that the controller will not take any corrective action until one whole batch of solids, whose mass is equal to the holdup of solids in the calciner, is adversely affected by the disturbance. At this point, another problem arising from the large dead time of the system needs to be commented upon. Naturally occurring disturbances are sporadic; i.e., a disturbance may occur for a short while and then die out. If the time of occurrence of the disturbance is smaller than the dead time, it is possible that the controller may, in the future, take action to rectify a disturbance that does not exist any more. Occurrence of such a phenomenon can throw the system completely out of control. From the above discussion, it is evident that satisfactory control of the calciner cannot generally be accomplished if deviations in product quality alone are used to decide the control action. The open-loop analysis of the system revealed that the response of the gas temperature shows very little dead time. Thus, gas temperature measurements a t the discharge end can probably be used in conjunction with product concentration measurements in order to achieve better control of the calciner. This opens up a whole new chapter of inferential control and prospects for further research in this field.
Concluding Remarks Our objective, to simulate the steady-state and dynamic operation of the commercial scale calciner with trona ore decomposition, has been successfully completed. Simulated steady-state temperature and sodium carbonate concentration profiles agree well with the plant data (Kim and Srivastava, 1988). Also, the problem of shell overheating in the combustion zone is now better understood. Plant data to evaluate the dynamic simulation are not immediately available. The simulated response, however,
agrees qualitatively with the actual calciner behavior in the retention (dead) time. It WBS shown that the large dead time exhibited by the process makes it difficult to control the calciner if the product quality is used as the directly measured and controlled variable. Further research is being done to develop an inferential control scheme for the calciner using gas temperature as the measured variable. Although attempts have been made to make the mathematical model as rigorous as possible, further improvement appears to be in order.
Acknowledgment We gratefully acknowledge the Smead Manufacturing Company for financial support and FMC Wyoming Corporation for providing useful information about the calciner under study.
Nomenclature A = area of solids exposed to the gas, per unit length of the calciner, m2/m C,, C,, C,, C, = specific heat of the combustion products, calciner gas, solids, and vapors, kcal/(kg K) D = diameter of the calciner, m Df = diameter of the flame, m d, = average diameter of the solid particles, m E = activation energy, kcal/kmol ef, e,, e,, e, = emissivities of the flame, calciner gas, solids, and calciner wall Ffm,Ffw= form factors for radiative heat transfer g = acceleration due to gravity, m/min2 hf, h,, h,, h,, h, = heat-transfer coefficients used in the model equations, kcal/(h m2 K) He,, Hfg,Hfm,Hf,, HP, Hgwl H,,, Hvg,H-, HW= heat fluxes defined in the text, kcal/(h m) K = function of particle Reynold number, used in the retention time equation for the lifter zone, m-l K , = thermal conductivity of air, kcal/(h m2 K) K, = proportional gain of the controller, kg/(h %) K , = thermal conductivity of the calciner gas, kcal/(h m2 K) Kh = proportionality constant used in drying equation, m-l K , = process gain constant, (% h)/kg K , = frequency factor in the reaction rate expression, h-’ L = length of the calciner, m Lb = length of the bare zones of the calciner, m (or ft wherever specified) Lu = length of the lifter zone of the calciner,m (or f t wherever specified) L,, L 2 , L,, L4, Li’, L,’ = arc lengths shown in Figure 2, m
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 81
L, = latent heat of evaporation of water, kcal/kg M , = mass per unit length of the calciner wall, kg/m N,,, = Nusselt number Pr = Prandtl number Q,, Q,,Q,, Qh,Q, = flow rate of combustion products, calciner gas, solids free moisture, and natural gas, kg/h R = ideal gas constant, kcal/(kmol K) R e , ReD = Reynolds number rpm = rotational speed of the calciner, min-' S, = sodium sesquicarbonate content of the solids, % t = time, h (or min wherever specified) t d = process dead time, min Tb, TIf = retention time of the solids in the bare zones and the lifter zone of the calciner, min Tf, T8,T,, To,T, = temperature of the flame, calciner gas, solids, surroundings, and calciner wall, K U = combined heat-transfer coefficient for convective and radiative heat flow between solids and gas in the lifter zone, kcal/(h m2 K) V,, V,. = calciner gas velocity and solids velocity through the calcmer, m/h (or m/min wherever specified) YaVs= weighted mean distance of fall of a particle as it drops from a flight in the lifter zone, m Greek Symbols = kiln axis inclination, rad 6 = ratio of the circumference of the calciner to the distance an average particle travels on the periphery before it falls from the flight AH = heat of reaction, kcal/kg AL, = integration step size with respect to length, m AT = integration step size with respect to time, h pg = density of the calciner gas, kg/m3 pp = particle density, kg/m3 u = Stefan-Boltzmann constant, kcal/(h m2 K4) T = process time constant, h T D = derivative time constant, h q = integral time constant, h CY
Appendix 1: Data Used for Simulation The following values were assigned to some of the coefficients used in simulation: C, = 0.16 kcal/(kg K), C, = 0.27 kcal/(kg K), C, = 0.12 kcal/(kg K), d, = m, el = 0.19, eg = 0.1, e , = 0.50, e, = 0.85, Ff, = 0.3, Ff,= 0.3, h, = 4.0 kcal/(h m2 K), hf = 10.0 kcal/(h m2 K), ho = 20.0 kcal/(h m2 K), h, = 20.0 kcal/(h m2 K), K, = 1.9 X lo7 h-l, L1 = 4.398 m, L2 = 3.861 m, L3 = 11.434 m, L1' = 3.40 m, L3/ = 12.40 m, M , = 650 kg/m, U = 50 kcal/(h m2 K). For the steady-state simulation, the following values have been assigned to some of the input variables: A = 100 m2/m, Qh = 1600 kg/h, Q, = 222323 kg/h, S, = 90.83%. Appendix 2: Estimates of the Model Parameters A: In the Lifter Zone (Schofield and Glikin, 1962). From the retention time model, the mass of solid in air a t any instant is LfQm Yav, sin
CY
*
+ 1KV,2t2 2
t
assuming that the particles are spherical. The surface area,
A , in contact with gas, per unit length of the calciner, is given by
In the bare zones, A is simply equal to L2. h, can be estimated by the correlation (Kreith, 1976) N,,, = 0.036Pr1/3Re0.8 Reynolds number (Re) and Prandtl number (Pr) can be estimated from the known gas flow rate and the average gas composition. ho can be estimated by the correlation (Kreith, 1976) h& _ -- 0.0239Re0.805 Ka where K , is the thermal conductivity of the air and L is the calciner length under consideration. h,. In the bare zones, h, can be estimated from the following relation (Coulson and Richardson, 1978) h, = C(G')0.8
where G'is the mass flow rate of gas in kg/(s m2). The value of C is different for different geometries. In the lifter zone, h, is equivalent to U. U is estimated from the correlation (Kreith, 1976)
-UdP -
- 0.37Re~"~ Kg where d, is the average particle diameter, K, is the thermal conductivity of the calciner gas, and ReD is particle Reynolds number. Registry No. Na2C03, 497-19-8; trona, 15243-87-5. Literature Cited Coulson, J. M.; Richardson, J. F. Unit Operations, 3rd ed.; Pergamon Press, Ltd.: Oxford, England, 1978; Vol. 2. Dumont, Guy; Belanger; Pierre R. Steady State Study of A Titanium Dioxide Rotary Kiln. Ind. Eng. Chem. Process Des. Deu. 1978, 17, 107-117. Kim, Nam K. Retention Time, Memo to R. Waggener; FMC Corp.: Green River, WY, Oct 6, 1977. Kim, Nam K.; Srivastava, Rajeev. Simulation of an Industrial Calciner with Trona Ore Decomposition. Ind. Eng. Chem. Res. 1988, 27, 1194-1198. Kim, Nam K.; Lyon, Jay E.; Suryanarayana, Narasipur V. Heat Shields for High Temperature Kiln. Ind. Eng. Chem. Process Des. Deu. 1986,25, 843-849. Kreith, Frank. Principles of Heat Transfer, 3rd ed.; IEP-A DunDonnelley Publishers: New York, 1976. Manitus, Andrzej; Kurcyvsz, Ewa; Kawecki, Wieslaw. Mathematical Model of the Aluminium Oxide Rotary Kiln. Ind. Eng. Chem. Process. Des. Deu. 1974, 13, 132-142. Perry, R. H. Mathematical Tables. In Chemical Engineers' Handbook, 4th ed.; Perry, J. H., Chilton, C. H., Kirpatrick, S. D., Eds.; McGraw-Hill: New York, 1963. Schofield, F. R.; Glikin, P. G. Rotary Dryers and Coolers for Granular Fertilizers. Trans Inst. Chem. Eng. 1962, 40, 183-190. Stephanopoulos, George. Chemical Process Control; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1984.
Received for review February 10, 1989 Accepted October 9, 1989
