1855
Ind. Eng. Chem. Res. 1991,30, 1855-1864 Knight, J. R.; Doherty, M. F.Optimal Design and Synthesis of Homogeneous Azeotropic Distillation Sequences. Znd. Eng. Chem. Res. 1989,28, 564-572. Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. The Curious Behavior of Homogeneous Azeotropic Distillation-Implicationsfor Entrainer Selection. AIChE Annual Meeting, Chicago, 1990. Levy, S. G.;Doherty, M. F. Design and Synthesis of Homogeneous
Azeotropic Distillation. 4. Minimum Reflux Calculations for ‘Multiple-feedColumns. Znd. Eng. Chem. Fundam. 1985,25, 269. Skogestad, S.;Morari, M. Understanding the Dynamic Behavior of Distillation Columns. Znd. Eng. Chem. Res. 1988,27,184&1862. Received for review January 31, 1991 Accepted April 29, 1991
Unsteady-State Retention of Sulfur Dioxide in a Fluidized Bed with Continual Feeding of Lime and Limestone Miloslav Hartman,* Karel Svoboda, and Otakar Trnka Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, 165 02 Prague 6-Suchdol, Czechoslovakia
Two unsteady-state models are proposed for sulfur dioxide retention in an isothermal, fluidized bed reactor with sorption of SO2by calcium oxide. Both models include continuous feeding of the sorbent and differ with respect t o possible accumulation (addition model) or withdrawal of the solids from the bed (overflow model). Transient characteristics of the exit gas were measured in a fluidized bed, 85-mm4.d. reactor. The continuous feeding of limestone sorbents was approximated by a semicontinuous introducing of very small doses of the particles into the bed. The measured transient curves are compared to the model predictions. The influence of the type of stone and calciumto-sulfur mole ratio on the dynamic behavior of the reactors as well as on their steady-state performance is also explored. Several models have been proposed for describing the performance of fluidized bed limestone reactors in removal of sulfur dioxide from flue gas. These include two-phase models with different fluid flow behavior (e.g., Lee and Georgakis, 1981; Zheng et al., 1982; Fee et al., 1983; Ho et al., 1986; Hartman et al., 1979, 1984, 1987, 1988). In contrast to catalytic reactors (Fan and Fan, 1980; Choi and Ray, 1985), little has been done toward understanding the transient behavior of these reactors with continuous feeding of limestone particles. Dimensions of the small-size reactors with a fluidized bed do not usually make it possible to install a downcomer for particle withdrawal from the bed. However, practically every laboratory reactor can be provided with continuous or batchwise adding of sorbent particles. If the rate of solids feeding is low, the volume and properties of the bed do not substantially change in the course of such an operation. This arrangement thus offers a possibility for the approximation of a fluidized bed reactor working under conditions of continuous operation. Sulfur dioxide removal from flue gas with the use of calcium oxide or magnesium oxide are examples in which the ratio of volume flow rates of the solids and gas is very low. This work is a sequel to aforementioned studies of ours on the modeling of sulfur dioxide removal in different types of reactors. The main aim of this work is to formulate and verify a model of the desulfurization reactor with continual feeding of calcium oxide and its accumulation in the bed. The model is also extended to the unsteady-state behavior of the reactor under conditions of continuous feeding and withdrawal of the solids.
Models A simple two-phase theory of fluidization is employed to describe the fluid-bed hydrodynamics. Conservation equations include convection, interfacial mass transfer, dispersion, chemical reaction, and accumulation. Assuming
that the solids are ideally mixed, the conditions over the cross section of the bed are uniform, the hydrodynamic parameters in the volume of the bed are constant, and reaction occurs only in the emulsion phase, models are proposed for the two following situations: 1. The first is model “addition”, in which feeding of the sorbent particles starts at a given time; no solids are withdrawn from the bed. Both the bed height and concentration of the sorbent change with time. 2. The second is model “overflow”,in which the introduction of the sorbent particles starts a t a given time. Simultaneously, solids are withdrawn from the bed at the same rate. Therefore, the height of the bed remains constant but the sorbent concentration in the bed is a function of time. Figure 1shows a schematic diagram of the fluidized bed with ideally mixed solids. The concentration profiles of a reacting gaseous component in the bubble and emulsion phase are described by eqs 1 and 2:
The independent variables t and h are defined in the regions t E (0, +-) and h E (0, H ( t ) ) . The initial conditions are expressed by CB(~,O) = f(h), W h o ) = g ( h ) (3) for h E (0, H ( t ) ) . The boundary conditions are given by eqs 4 for t > 0. The definitions of the coefficients in the
* Author to whom correspondence should be addressed. 0888-5SS5/91/2630-1855$02.50/0
0 1991 American Chemical Society
1856 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
JIR dw = MUpGCO x Hin(l- eB)(l- ed)ph + MUpGCo(t- to)
dX +(12) dt
If the bed contains only the inert particles at t = to, Le., eq 12 simplifies to %=&‘Rdw-X t - to
pin = 0, then
with the initial condition X ( 0 ) = x, Model “Overflow”. In this situation it holds that S1 = Sz and H = constant. For the unreacted solids, Le., Xo = 0, eq 5 takes the form FUPGCo
Figure 1. Schematic diagram of fluidized bed reactor with ideally mixed particles.
above equations are given in the Nomenclature section. As follows from their definitions, the coefficients Al-A3, As, A6,B1,B2,and are always constants. The coefficient B 4 ( t )and quantity H ( t ) are, in general, given functions whose forms depend on the mode of operation. The aforementioned equations hold for both the situation “addition” and the case “overflow”. The differences between the two situations result in the different expressions for the sorbent concentration in the bed and height of the bed and in the different conservation equation for the solids. These differences are described in the following paragraphs. Model ’Addition”. With respect to the situation depicted in Figure l, the unsteady-state mass balance on sulfur dioxide reacting in the solid phase is
d nJIR dw = SlpsX + -(nX) dt which can be rewritten into L’Rdw =
X dX 1 dPs + - + - -X fs dt ps dt
The symbol fs denotes the mean residence time of the solids in the bed introduced as
The unsteady-state mass balance for the solid is
where Vs = FH(1 - eB)(l- emf)= constant
(19)
and leads to dt = fs
For S, = 0 and Xo = 0 we obtain dX X dn X I R dw = - + - dt n dt As n = H(t)F(l - eg) (1 - emf)ps(t) eq 6 can be rewritten into the form
(16)
dPs Pfeed
- PS
(20)
with the initial condition ~ ~ ( =0 Pin 1
(7)
Provided that the sorbent feeding is started at t = to, then the sorbent concentration in the bed and height of the bed are given by Hin(1 - eB)(1 - emf)Pin + MUPGCO(~ - to) pS(t) = Hin(l- eB)(1 - emf)pfeed + MUpcCo(t - t o )Pfeed (9)
(21)
The concentration of the sorbent in the bed is then given by & ( t ) = Pfeed - (Pfeed - Pin) exp(-(t - t o ) / f S ) (22) On substituting
in eq 16 we get after some arrangements the final equation for the solid phase:
where
with the initial condition eq 14. The models are represented by a set of nonlinear partial differential equations of the parabolic type. The system of model equations is of the ”stiff“ character, and its numerical solution is outlined in the Appendix.
On substituting from eqs 9 and 10 in eq 8, we get
Experimental Section The experimental apparatus consisted of three fundamental parts: electrically heated reactor, facility for the
Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991 1857 container
container
reactor
Ill
Figure 2. Slide feeder of the particulate solids.
continuous withdrawal and analysis of gas samples, and feeder of solids. The reactor was constructed of a heat resistant steel tube, 0.50 m high and of 0.085-m diameter. Fluidization air with an addition of sulfur dioxide was preheated and passed upward through the bed via a perforated plate distributor. The detailed description of the experimental apparatus can be found elsewhere (Svoboda and Hartman, 1981). The concentration of sulfur dioxide in the inlet and outlet gas was measured by the infrared gas analyzer (Infralyt 4) with the time lag of approximately 1 min. The time taken for the SOz concentration to attain the 95% value of the final size of a signal was always less than 1 min. The total concentration of SO,, as the sum of SO2 and SO3, was determined from the analyzer readings according to the equation Cso = Cm,/(l - z), where z is the conversion of SOz to SO3. bnder our experimental conditions, Le., 850 O C and 21% Oz by volume, we found that z = 0.2. This result is close to the equilibrium value. No effect of the oxygen concentration in the gas phase was detected on the process of desulfurization in a concenof 0.5% by tration span of 2-21% Ozby volume for Csol volume. Both sulfur dioxide concentration and temperature were continuously recorded on two pen recorders. Sorbent Feeding. Continual feeding of the particulate solids often poses a problem even in large-scale processes. The rates of feeding required in our work are very low and range from about 5 to 50 g/h. Attempts to develop a reliable, continuous plate feeder provided with the transport screw were not successful. Thorough testing disclosed some difficulties such as short- and long-term variations in the feed rate, stability problems with the disk and screw revolutions, formation of the powder foulings on the walls, and long times required to stabilize the process. Therefore, experimentationwith this feeder was discontinued. Three types of discontinuous feeders were designed and constructed for further testing: slide feeder, rotary feeder, and disk feeder. Figures 2-4 show the principles of these facilities. All the types also make it possible to separate the space of the reactor with harmful gases from the surroundings. These feeders were carefully tested with narrow fractions of limestone and lime CI (Ciskovice quarry). The mean particle size ranged from 0.42 to 0.90 mm. Apart from the operational reliability, the reproduction of doses, range of application, and grinding of particles were observed. The diameter of the holes or pockets varied from 3 to 5 mm, and their depth ranged from 2.8 to 8 mm. The mass of a single dose changed from 39 to 110 mg with limestone and from 23 to 56 mg with lime. If the frequency of 2-20 doses min-' is considered, any of the above discontinuous facilities can feed limestone
reactor
Figure 3. Rotary feeder of the particulate solids.
container
Ill I
I
mox.dev.1
0
%
I
0
0
4
0
L
x
00 I
6
Q
X
I
-a, dh
10
Figure 5. Variations in single doaes of slide feeder and rotary feeder as a function of ratio of hole diameter to particle size: ( 0 )slide feeder, limestone; ( 0 )slide feeder, calcine; (X) rotary feeder, limestone.
and lime at the rates of 5-130 and 2 . 7 a g/h, respectively. As illustrated in Figure 5, the reproduction of the dose is strongly effected by-the ratio of the hole diameter, dh, to mean particle size, d,. No problems occur for dh/dp >
1858 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 Table I. Physical and Chemical Properties of the Solids Physical Characteristics ceramsite calcine CI mean particle size, mm 0.565 (0.5-0.63) 0.565 (0.5-0.63) density, g/cm3 1.47 1.29 porosity 0.5 mean pore size, nm 200 LTd at 850 O C , cm/s 10.2 8.0 ~
Chemical Characteristics limestone CI CaO, % by wt 42.4 MgO, % by wt 0.63 Fez03,% by wt 1.4 P f d r mol/cm3 0.015
calcine CI 65.0 0.96 2.14 0.015 I
7. A t lower values some difficulties appear with the filling and mainly with the emptying of the pockets, which results in a substantially reduced accuracy of feeding. When the feeder is not grounded, electrostatic charges develop, which leads to the repulsion of particles and, consequently, to some nonuniformities in the filling of the pockets. In contrast to the parent limestone, the particles of lime are soft and somewhat friable. This makes feeding of the calcine more difficult than that of limestone. The slide feeder, shown in Figure 2, was chosen for the desulfurization experiments. The sliding plate was 8 mm thick, constructed of PTFE, and provided with a 4-mmdiameter hole. The mass of a single dose of lime particles (d = 0.565 mm; dh/dp= 7.1) was as large as 56.5 mg. The refative deviations of the doses were less than 1.5%. The frequency of the doses ranged from 2 to 22 min-'. Preliminary testing showed that the intermittent feeding of the sorbent results in a nearly smooth curve of exit SO2 concentration if the frequency of the doses is higher than or equal to 2 min-'. The experimental materials employed in this work were ceramsite as an inert and the commercial limestone CI and its calcine with a mean particle diameter of d, = 0.565 mm. The physical and chemical properties of the solids are given in Table I. The calcine was prepared by the thermal decomposition of the particulate carbonate in a muffle furnace at 850 "C and by subsequent screening of the calcined particles. The ceramsite particles (calcined claystone) were saturated with sulfur dioxide prior to their use in the desulfurization experiments. To start up a run, the bed of about 500 cm3of ceramsite was fluidized and heated to 850 "C, and required flow rates of air and sulfur dioxide were adjusted. When steady-state conditions were attained, the feeding of sorbent particles was started. In dependence on the rate of feeding, the initial drop of the SO2 concentration is quite rapid and it is followed by a gradual decrease to a new steady-state value. The character of this transient process is mainly influenced by the inlet concentration of sulfur dioxide in the gas, gas flow rate, and amount and reactivity of the sorbent. The feed rate of limestone and lime was adjusted according to the different calcium-to-sulfur mole ratios required. In the experiments with limestone and lime CI, the exit concentration of sulfur dioxide changed very little 1. after 60-70 min of feeding at M
-
Results and Discussion The transient curves of the exit concentrations of sulfur dioxide were measured at the carbonate or calcine feeding. The calcium-to-sulfur mole ratio was constant in a given run and was varied from M = 0.876 to M = 2.92 in the course of the work. The superficial velocity of gas varied from 0.206 to 5.243 m s-l, and the height of fixed bed of
1330
2003
t s
300C
Figure 6. Dependence of exit gas SO2 concentration on elapsed time of calcine feeding: temperature 850 "C;calcium-to-sulfur mole ratio M = 1.46; superficial gas velocity U = 24.3 cm/s; inlet gas concentration of SOz Co = 0.003 by volume; mean gas residence time in the bed, fc = 0.4 s. (0) Experimental data poinb. The solid line shows the values predicted by the model 'addition".
i
XL __
, '003
2000
I
+ s
33GO
Figure 7. Dependence of exit gas SO2 concentration on elapsed time of calcine feeding: 850 "C; M = 0.876; U = 24.3 cm/s, Co = 0.005 by volume; fc = 0.4 s. (0) Experimental data points. The solid line shows the values predicted by the model 'addition".
05c ' O \
1000
2000
+
3000
Figure 8. Dependence of exit gas SOz concentration on elapsed time of limestone feeding: 850 OC; M = 2.92; U = 20.6 cm/s; Co = 0.003 Experimental data points. The solid line by volume; fG = 0.5 s. (0) shows the values predicted by the model 'addition".
the inert particles was approximately 8.5 cm. Typical transient curves of the exit gas SOz concentration in experiments with the quasi-continual feeding of sorbent are shown as examples and compared with the model predictions in Figures 6-9. The mean concentra-
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1859 l
I
1
I
i
I
0
1000
J
I
2ooo
t, s
3000
Figure 9. Dependence of exit gas SO2 concentration on elapsed time of limeatone feeding: 850 OC; M = 1.82; U = 24.3 cm/s, C, 5 0.00306 by volume; = 0.4 8. (0)Experimental data points. The solid line shows the values predicted by the model "addition".
tions were computed from the concentrations in the bubble and emulsion phase
The overall coefficient of gas exchange between the bubble and emulsion phase was estimated with the aid of
which is often employed in similar systems (Ho et al., 1986, Zheng et al., 1982). The diffusivity of sulfur dioxide in air at 850 O C predicted by the method of Wilke and Lee (Perryand Chilton, 1973) amounts to 1.4 cm2/s. The correlation of Mori and Wen (1975) was employed for estimating the mean bubble size. Systematic computations showed that the bubble size and mass-transfer coefficient usually occur in the range 1-5 cm and 80-10 s-l, respectively, under the conditions of our measurements. In a recent work of ours we developed a formula for predicting the height of expansion in bubbling beds, H (Hartman et al., 1987):
*[
Ao.6 + ( A - &mH)o.6 H = H , f + - m Ao.6 1- 1 In Ao.6 + ( A - B)05 Ao.6 - ( A - B8mH)O.S 1 + Ao.6 + 1 In Ao.6 - (A - B)O.6 1 + ( A - BPH)o*6 2 (26b) 1- A 1 ( A - B)'"
+
]
This equation, which incorporates the semiempirical correlation of bubble size by Mori and Wen (1975), was also used for estimating the fraction of bed occupied by the bubbles, eg, given as eB = 1 - Hmf/H
(264
The dependence of the bubble size on the distance above the distributor was not considered, and the value of db predicted at H/2 was used. The reaction term, R, developed in our earlier work (Hartman et al., 1979) for limestone CI is in the form
for d = 0.565 mm, 850 "C. It gas been established that the flow characteristics of gas and solids are particularly complex in larger fluidized beds (e.g., Lehmann and Schiigerl, 1978). Nevertheless, our recent experience with small-scale beds suggests that the assumptions of plug flow of gas and perfect mixing of the particles are quite realistic for such systems (Hartman et al., 1984, 1988). The values of the axial dispersion coefficients were set somewhat arbitrarily as DGB = DGE = lo4 cm2s-l, which correspond to plug flow of gas in the bed. As can be seen in Figures 6-9, there is reasonable agreement between the measured and computed curves. In case of the lime (precalcined) particles, the experimental and model curves are nearly parallel. The computed values of the exit concentrations are about 10-20% higher than the measured values. A somewhat different picture can be seen when the limestone (carbonate) particles are introduced into the bed. The experimental and computed curves are intersected, and lower concentrations are predicted for the advanced stages of the process. Comparison of the experimental curves for the lime and carbonate solids indicates that the precalcined particles are more reactive than those calcined in situ. Although features such as the surface area, pore size distribution, and porosity of the two solids are not much different, we believe that the appreciable difference in reactivity bears some relation to the conditions of formation of calcium oxide. It appears that the slow decomposition of a mass of carbonate in the muffle furnace provides the calcine which is more reactive than that produced by the nearly instantaneous calcination of small doses introduced into the fluidized bed. The rate equation 27 was proposed on the basis of the experimental data measured on a differential reactor with a thin, fixed layer of sorbent particles. The thermal decomposition of carbonate took place quite rapidly in situ in flue gas containing 10% carbon dioxide by volume. It is most likely that the rate of calcination increases in order from the muffle furnace to the fluidized bed. It appears, and the measured performance of the reactor supports the idea, that the reactivity of calcium oxide decreases with the increasing rate of the thermal decomposition of calcium carbonate. When possible causes of the differences in reactivities are considered, another point should not be neglected. The data correlated by eq 27 were measured with limestone from the same quarry, but from a different batch of the rock CI. Comparison of Different Kinetic Equations. Zheng et al. (1982) proposed simplified kinetic expressions to describe the sulfation of calcium oxide. On eliminating the reaction time, the original kinetic equations of the authors have been rewritten into the following, very simple forms: R = 0.2004C - 0.00407X (%a) for
dp = 0.62 mm, 800 O C , C E (0.004, 0.0061) and R = 0.9875C - 0.0035X
(28b)
for
dp = 0.15 mm, 800 OC, C E (0.002,0.0061) Zheng et al. (1982) assumed the analogy between particle sulfation and fist-order deactivation of catalyst particles. The numerical constants in eqs 28a and 28b were evaluated from the transient curves of exit SO2concentration mea-
1860 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
1
0.2
,
I
0.4
0.6
I X
0.8
Figure 10. Comparison of correlations for rate of adfation of different materials at C, = 0.003: curve 1, limestone CI, d, = 0.56 mm, Hartman et al. (1979), eq 27; curve 2, limestone 1359, d, = 0.15 mm, Zheng et al. (1982),eq 28b; curve 3, limestone 1359, d, = 0.62 mm, Zheng et al. (1982),eq 28a; curve 4,a, = 0.5 mm, Fee et al. (19831, eq 29.
UI
I
I
0.2
0
0.6
0.4
X
0.8
Figure 11. Comparison of reaction rates predicted by different correlations under the conditions at which high fraction of SO2 is removed from the gas: Y = 0.9;C = 0.0012. The labeling of the curves is the same aa in Figure 10. The dashed lines outline an approximated practical range of the solid reactivity for the desulfurization. The parameter R’ is defined by eq 30.
sured on introducing a small batch of limestone into a 10-cm4.d. fluidized bed of sand particles. Fee et al. (1983) developed a simple empirical relationship for predicting the reaction rate which can be rewritten as R = 0.1356C(0.34 - X) (29) for d, = 0.5 mm, 850 “C. The parameters in eq 29 were derived from TGA data amassed with the use of a commercial thermogravimetric apparatus. As can seen, there are considerable differences in the above kinetic equations. Equations 28a and 28b suggest that the maximum conversion of sorbent is a linear function of the gas SO2concentration. In contrast to these relationships, eqs 27 and 29 imply that the maximum utilization of sorbent does not depend on the gas SO2 concentration. In order to compare all the presented kinetic equations, we carried out systematic computations of the reaction rates. Some of the results are plotted in Figures 10 and 11as a function of conversion for C = 0.003 and C = 0.0012. As shown, the rate of reaction is strongly affected by both the conversion and the gas concentration. Recent AFBC (atmospheric fluidized bed combustion) experience makes it possible to outline a practical range of the reactivity of sorbent. Fee et al. (1983) suggested as a useful parameter (30)
where p c is the sorbent concentration (mol m-3) and C, is the SO2 concentration (mol m-9. Values of R’lower than 10 s-l usually result in insufficient sulfur dioxide removal. On the other hand, values higher than 50 s-l are mostly achieved by excessive sorbent feed rates which are too high from the standpoint of process economy. If plug flow of the gas is assumed and Co = 0.003 and Y = 0.9, then the average SO2concentration in the bed CS is as large as 0.0012 (C= (Co - C,J/ln (Co/Cd). Under these conditions, the range of interest is marked for calcium oxide in Figure 11. It is shown that the conversion of calcium oxide is close to ita maximum value at the practical conditions. In this stage of reaction the average sorbent reactivity is only a
0
1000
2000
t, 5
30GG
Figure 12. Dependence of exit gas SO2 concentration on elapsed time of sorbent feeding. The lines show the predictions of the model ‘addition” with different kinetic equations for M = 0.876, C, = 0.005, U = 24.3 cm/s, and tG = 0.4 s. The labeling of the curves is the same as in Figures 10 and 11.
small fraction of the reactivity of the fresh particles. In order to explore the effect of different kinetic equations on the transient c w e s , the model computations were also carried out in which the relationships 28a, 28b, and 29 were incorporated. Two runs with the limits of M = 0.876 (calcine) and M = 2.92 (carbonate) were chosen for comparison. The predicted c w e s are presented in Figures 1 2 and 13. The curves computed with the use of eqs 27, 28a, and 28b decrease rapidly at the initial stages. In case of eqs 28a and 28b the exit concentrations level off and are virtually constant after about 15-20 min. The curves for eqs 27 and 29 tend to decrease even at time in hours. Apart from variations in the sorbent reactivities, one should also consider the different experimental techniques employed by the authors. While Hartman et al. (1979) and Fee et al. (1983) measured the reaction rate as weight gain, Zheng et al. (1982) deduced the kinetic equations from the transient curves of exit gas concentration in the experiments with batch addition of limestone to the fluidized bed. Comparison of the Models “Addition” and “Overflow”. The computed c w e s X = X(t) and C = C ( t )
Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991 1861 0.91
,
I
I
I
io3
IO2 I
I
I
0
loo0
2000
I t, s
3000
Figure 13. Dependence of exit gas SO2 concentration on elapsed time of sorbent feeding. The linea show the predictions of the model 'addition" with different kinetic equations for M = 2.92, Co= 0.003, U = 20.6 cm/s, and €0 = 0.5 s. The labeling of the curves ia the same as in Figures 10-12.
io5
104 t.s
Figure 16. Sensitivity of gas exit concentration to overall mats transfer coefficient between the bubble and emuleion phase. The curves were computed for the same parameters as in Figure 6 (M= 1.46).
I
t nc, -.-
-0
I
I
f:
/
0
0
0
1
'
M
Figure 17. Fraction of sulfur dioxide removed from gas under IO2
io3
10'
105 t, S
Figure 14. Comparison of models 'addition" (curve A) and *overflow" (curve 0). The curves are computed for the same conditions as in Figure 7 (M= 0.876) and Figure 8 (M = 2.92).
0.21
0
\
M= 2.92
I
102
io3
IOL t. s
1os
Figure 15. Comparison of models 'addition" (curve A) and 'overflow" (curve 0). The curvea are computed for the same conditions as in Figure 14.
are shown in Figures 14 and 15 for the situations at M = 0.876 and M = 2.92. As can be seen, there are slight differences in the model predictions. These differences are visible only at longer times and are more appreciable for
steady-state conditions as a function of calcium-to-sulfur ratio for different limestones. The presented resulta are the predictions of the model "overflow" for to = 0.5 s. The solid curve representa the solutions with the kinetics of eq 27, Hartman et al. (1979); ( 0 )solutions with eq 28a, Zheng et al. (1982); (0)solutions with eq 28b, Zheng et al. (1982); (a) solutions with eq 29, Fee et al. (1983).
lower values of M,i.e., for higher conversions of sorbent. The computed cleaning effect of bed in the situation of "overflow" is lower than that in the situation "addition", but the difference is very small. Using the model "addition", we explored how the performance of the reactor is affected by the rate of mass transfer between the bubble and emulsion phase. The model equations were also solved by using a value of KBE by an order of magnitude smaller than the normal value predicted by eq 26. The remaining parameters were maintained unchanged. The results of the computations shown in Figure 16 indicate that the increased masstransfer resistance results in reduced overall performance at lower concentrations (C/Co C 0.6). Steady-StateEfficiency under Conditions of Continuous Operation. The proposed sets of the model equations make it possible to describe a number of situations. One of the most important needs is to predict the steady-state desulfurization efficiency of the bed in the situation "overflow". It is apparent that the solutions of the model equations at suffkiently long times describe with good accuracy the steady states of a reactor working under conditions of continuous operation. The period of time needed to attain the steady-state situation depends upon a number of circumstances, e.g., the initial state of the bed and reaction kinetics. With the use of the initial condi-
1862 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
tions, very distant from the steady-state ones ( ~ ~ ( =0 0, ) Xo= 0), simulation computations of the model "overflow" were carried out until a time of 1.3 X lo5 s was reached. The above kinetic equations 27, 28a, 28b, and 29 were employed in the computations. The results are plotted in Figure 17 as a function of the calcium-to-sulfur mole ratio. In the steady-state operation, a fraction of SOz removed from the gas in the reactor can be evaluated from the compositions of either stream at the inlet and outlet of the reactor:
Y=
co - C0"t -- X o u t - xo M CO
1-
x,
(31)
The differences in Y computed from C or X were always less than 5 % . The results shown in Figure 17 also provide a test of the employed numerical procedure. Some years ago we proposed a simple, steady-state model of a continuous, desulfurization reactor in which the same kinetic equation 27 was incorporated (Hartman et al., 1979). The predictions of the former, steady-state model, obtained by a completely different algorithm provided the values of Y between Y = 0.75 and Y = 0.80 for M = 1 and lG = 0.5 s in dependence on the intensity of mixing. The corresponding value shown in Figure 17 amounts to Y = 0.74, which is in agreement with the former results. The model predictions presented in Figure 17 can be confronted with the experimental measurements of other researchers. Zheng et al. (1982) measured the sulfur retention in a fluid bed combustor with continuous feeding of coal and limestone 1359. Working with 0.62- and 0.14-mm particles of limestone, the authors achieved the retention Y = 0.25 and 0.60, respectively, at M = 3. The corresponding values predicted from the model amount to Y = 0.3 and 0.7, respectively, which is in good agreement with the experiment. The results in Figures 11 and 17 illustrate how the steady-state performance of the reactor is markedly affected by the sorbent reactivity. Both figures document very good reactivity and desulfurization efficiency of the limestone CI. As mentioned elsewhere (Hartman et al., 1979), this cryptocrystalline rock contains considerable amounts of silica, aluminum oxide, and ferric oxide. The experience shows that the calcium-to-sulfur ratio is a major parameter governing the desulfurization efficiency of the reactor. As indicated in Figure 17, the high sulfur retention ( Y = 0.90-0.95) can be achieved at the calcium-to-sulfur ratio of 1.5-2 when the reactive limestone CI is employed. Postlude We wish to emphasize that desulfurization is carried out in much larger fluidized combustors. Such beds are operated, for example, at higher gas flow rates, work with solids of large size and density distribution, and temperature profiles are not often uniform throughout the bed volume. In general, however, both the experimental findings and the theory indicate that the reaction kinetics and calcium-to-sulfur ratio are parameters of major importance in any situation. Conclusions Continuous feeding of sorbent particles can be approximated by adding small doses, if the frequency of doses is greater than 2 min-'. Particles can be introduced into a laboratory reactor without difficulties at rates ranging from 3 to 130 g/h with the use of the slide, rotary, or disk feeder. If the ratio of the hole diameter to the particle diameter is greater than 6-7, single doses of the slide and
rotary feeder can be reproduced with accuracy better than 1%.
Two unsteady-state models have been developed for the sorption of sulfur dioxide from a gas stream. The measured transient curves of exit gas SO2 concentration depend on the calcium-to-sulfur ratio and vary according to which sorbent is introduced into the bed. The precalcined particles, prepared by the slow thermal decomposition of the limestone CI, are somewhat more reactive than the particles formed by rapid, in situ calcination within the fluidized bed. Although the reactivities of these sorbents are appreciably different, their physical properties such as the porosity, specific surface area, and pore size distribution are very similar. The duration and course of the transient process is strongly influenced by the kinetics of the sulfation reaction, which varies widely with the type of stone. The differences in predictions of the models "addition" and "overflow" are very small and can be noted only at long times ( lo4 s). The conditions of a continuous, desulfurization reactor with the solids feeding and withdrawing can be approximated by the feeding of sorbent to the fluidized bed and accumulation of spent sorbent in the bed. At the limit of long time, the model "overflow" describes the steady-state performance of the continuous reactor. The SOz retention predicted for such conditions is in good agreement with the experiments of Zheng et al. (1982). It is also supported by the predictions of our much simpler, steady-state model for the removal of sulfur dioxide from flue gas (Hartman et al., 1979). During practical use of fluidized bed with limestone, every effort should be made to attain high sulfur retention at reasonable consumption of the sorbent. Under such conditions the sulfation reaction is slow. Nevertheless, the sensitivity of the model to the interphase exchange coefficient indicates that the mass transfer between the bubbles and the emulsion phase has to be taken into consideration. The limestone C1 has been found as an efficient sorbent for the SO2 removal from flue gas. Using the 0.56-mm particles of this material, it is possible to obtain 90% sulfur retention at the calcium-to-sulfur ratio M = 1.5-2 and the mean gas residence time tG= 0.4-0.5 s. N
Nomenclature A = (22.27/(11- umf))2Db, A, = D G B P G ~ B - Umf)PG A3 = K B E P G ~ B A5 = PCeB A , = DGBeB/(U - u m f ) B = (22.27/(U - Umf))2(&m- Db) = D G E (-~ edemfPG
A, =
(u
B, = UmfPG B , = As B, = ps(t)(l B, = pce,Al
--
e d l - emf) eB)
= (DcEe,r(l - eB))/Umf C = gas concentration defined
by eq 25, mol of SO,/mol CIB = concentration of SO2 in bubbles, mol of SOz/mol CE = concentration of SO, in interstitial gas, mol of S02/mol C, = concentration of SO, in gas feed, mol of SOp/mol c,,, = concentration of SO, iri exit gas, mol of SOz/mol db = average bubble diameter in bed, cm c& = hole diameter of feeder, cm d, = average particle size, cm Dbm= maximum bubble diameter defined by Mori and Wen (1975), cm D, = initial bubble diameter defined by Mori and Wen (1975), cm
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1863 DG = molecular diffusion coefficient of gas, cm2/s DGB= effectiveaxial dispersion coefficient for bubble-phase gas, cm2/s DGE= effective axial dispersion coefficient for interstitialphase gas, cm2/s Dt = diameter of fluidized bed, cm eg = volume fraction of bed occupied by bubbles e& = void fraction of bed at minimum fluidization velocity F = cross section of reactor, cm2 f ( h )= starting concentration profile in bubbles g = acceleration due to gravity, cm/s2 g(h) = starting concentration profile in interstitial gas h = distance above distributor, cm H,H ( t ) = height of fluidized bed, cm Hi,= initial height of bed, cm H& = height of bed at the point of minimum fluidization KBE= overall coefficientof gas exchange between bubble and emulsion phase, s-l m = O.3/Dt M = calcium-to-sulfur mole ratio, eq 11 M, = molecular weight, g/mol n = mass of sorbent in bed, eq 7, mol O(At) = relative deviation of balance, eq A3 R = dX/dt = reaction rate given by eqs 27,28a, 28b, and 29,
1
model “addition” = H:eB
- dw + H t V ) (1U- eB)emf X
‘CB
CO
model “overflow”
5-1
R’ = reactivity parameter given by eq 30, s-l SI= feed rate of sorbent, cm3/s S2 = withdrawal rate of solids, cm3/s t = time, s to = time instant when feeding is started, s fG = (H/U)(eB + (1- eB)emf)= mean residence time of gas in bed, s fs = mean residence time of solids in bed, eq 17, s At = time step, s U = superficial gas velocity, cm/s Ud = minimum fluidizing velocity, cm/s Vs = volume of solids in bed, cm3 w = h / H ( t ) = relative distance above distributor X = conversion of sorbent to sulfate Xo = inlet conversion of sorbent Xout= outlet conversion of sorbent X, = initial conversion of sorbent y = content of calcium oxide, weight fraction Y = fraction of SOz retained in bed, eq 31 Greek Letters
and
The balance time, t b i , given by eqs Al-A4 is compared with the real time, td,.in the course of the computation. The purpose of controhg the time step is to maintain the deviation of both times within a chosen interval, Le., to ensure for
I
O(At) = 100
tred - tbil
t1d
1
E ( m ,N ,
(A51
where m and N are the limits given in percent. Then the control of At takes place by monitoring the relative error of the time increments Atred and Atbil,i.e.
I
AO(At) = 100
Atred
- Atbil
I
mol/cm3 p~ = 0.01219/7’ = density of gas, mol/cm3 pin = initial content of calcium oxide in bed solids, mol/cm3 pp = density of particles (mercury), g/cm3 p~(t)= n/Vs = content of calcium oxide in bed solids,mol/cm3 \k = ratio of actual visible bubble flow rate to gas flow rate given by (U- Umf)F
646) At4 If AO(At) > N, then At is shortened and it is enlarged when AO(At) < m. In case of AO(At) E (m,N), the time step, At, remains unchanged. Details on the finite-difference approximation and the iteration solution of a similar set of equations are presented in a recent work of ours (Trnka and Hartman, 1987). Registry No. SO2, 7446-09-5.
Appendix. Numerical Solution of the Model Equations
Literature Cited
pfd = ppy/Mw= content of calcium oxide in introduced solids,
With the aid of the finite-differenceapproximations, the model equations were recast into a set of simultaneous, nonlinear algebraic equations. The employed difference scheme made it possible to solve the resulting set of the algebraic equations in successive steps by the method of interval halving. With the use of this procedure, the “stiff character of the problem is entirely eliminated. Thus the time step can be chosen quite large, only with respect to the accuracy of the approximation of the solution. The good approximation necessitates, however, controlling the size of the time step. As a useful criterion of the approximation, the balance equations were employed as follows:
Choi, K.-Y.; Ray, W. H. The Dynamic Behaviour of Fluidized Bed Reactors for Solid Catalyzed Gas-Phase Olefin Polymerization. Chem. Eng. Sci. 1985,40, 2261-2279. Fan, LA.;Fan, L. T. Transient and Steady State Characteristicsof Gaseous Reactant in Catalytic Fluidized-Bed Reactors. AZChE J . 1980,26, 139-144. Fee, D. C.; Wilson, W. I.; Myles, K. M.; Johnson, I. Fluidized-Bed Combustion: In-Bed Sorbent Sulfation Model. Chem. Eng. Sci. 1983, 38, 1917-1925. Hartman, M.; Hejna, J.; Beran, Z. Application of the Reaction Kinetics and Dispersion Model to Gas-Solid Reactors for Removal of Sulfur Dioxide from Flue Gas. Chem. Eng. Sci. 1979, 34, 475-483. Hartman, M.; Svoboda, K.; Trnka, 0. Sulfur Dioxide Removal in a Batch Fluidized Bed Reactor. Znst. Chem. Eng. Symp. Ser. (special issue on ISCRE-8) 1984, No. 87, 509-516.
Ind. Eng. C h e m . Res. 1991,30, 1864-1869
1864
Hartman, M.; Veseli, V.; Trnka, 0.; Svoboda, K. The Height of Expansion in Bubbling Fluidized Beds. Collect. Czech. Chem. Commun. 1987,52, 1178-1185. Hartman, M.; Svoboda, K.; Trnka, 0.;Vesely, V. Reaction of Sulfur Dioxide with Magnesia in a Fluidized Bed. Chem. Eng. Sci. (special issue on ISCRE-10) 1988,43, 2045-2050. Ho, T.-Ch.; Lee, H.-T.; Hopper, J. R. Simulation of Desulfurization in a Fluidized-Bed Limestone Reactor. AIChE J. 1986, 32, 1754-1759. Lee, D. C.; Georgakis, Ch. A Single, Particle-Size Model for Sulfur Retention in Fluidized Bed Coal Combustors. AIChE J . 1981,27, 472-481. Lehmann, J.; Schiigerl, K. Investigation of Gas Mixing and Gas Distributor Performance in Fluidized Beds. Chem. Eng. J. 1978, 15,91-109. Mori, S.; Wen, C. Y. Estimation of Bubble Diameter in Gaseous
Fluidized Beds. AZChE J . 1975,21, 109-115. Perry, R. H.;Chilton, C. H. Chemical Engineer's Handbook, 5th ed.; McCraw-Hill: Tokyo, 1973;pp 230-233. Svoboda, K.; Hartman, M. Influence of Temperature on Incipient Fluidization of Limestone, Lime, Coal Ash and Corundum. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 319-324. Trnka, 0.; Hartman, M. Numerical Solution of the Model for Sulfur Dioxide Removal in a Fluidized Bed of Sorbent. Chem. Eng. Sei. 1987, 42, 1919-1925. Zheng, J.;Yates, J. G.; Rowe, P. N. A Model for Desulfurization with Limestone in a Fluidized Coal Combustor. Chem. Eng. Sci. 1982, 37, 167-174.
Received f o r review December 1, 1990 Revised manuscript receioed April 2 , 1991 Accepted April 15, 1991
Simple PI and PID Tuning for Open-Loop Unstable Systems Guillermo E.Rotstein a n d Daniel R. Lewin* Department of Chemical Engineering, Technion, IIT, Haifa 32000, Israel
Simple tuning methods for traditional low-order schemes are lacking for open-loop unstable systems. In this paper, the limits to the applicability of proportional-integral (PI) and proportional-integral-derivative (PID)controllers are derived for unstable processes in the presence of gain uncertainty and unmodeled dynamics (such as dead time or neglected poles), Simple IMC-based tuning rules are employed that, when combined with appropriate robustness analysis, provide a useful alternative to lengthy on-line tuning. 1. Introduction
Simple systematic rules for the tuning of PI and PID controllers for open-loop unstable systems are necessary in order to avoid extensive on-line tuning. The stability limitations of these control schemes should be delineated in order to serve as a basis for comparison with more advanced control strategies. These limitations will serve as criteria to indicate when upgrading the PID controller becomes unavoidable. A design method has recently been presented for first-order systems with time delay (De Paor and O'Malley, 1989) where gain and phase margin criteria were used. A modified version of the Smith predictor has also been proposed (De Paor, 1985). Quinn and Sanathanan (1989) have suggested a more general design method that will lead to control schemes of the same order as the process. The effect of rational polynomial approximations for dead time has been investigated by Stahl and Hippe (1987), who employed pole placement control design. The problem of delineating simple tuning methods for PI and PID controllers for open-loop unstable processes with parameter and unmodeled dynamic uncertainty was not addressed by any of these studies. Tuning rules for PI and PID controllers have been derived for open-loop stable systems (Rivera et al., 1986) employing IMC (internal model control) as the design tool. The rules have been extended for simple open-loop unstable systems (Rotstein and Lewin, 1990). In this paper, tolerance limits for robust stability tuning based on the developed rules are derived. The uncertainty resulting from dead time, pole neglection, and gain modeling errors is considered. Combining the tuning rules and tolerance limits, a tuning method that helps to reduce lengthy on-line search is obtained. The conservativeness introduced by robust tuning on the basis of the multiplicative uncertainty bound when applied to open-loop unstable systems will also be shown. 0888-5885/91/2630-1864$02.50/0
2. Tuning for Open-Loop Unstable Systems 2.1. IMC Controller Design. Control design is intrinsically a model-based activity. This is true even when tuning rules for PI or PID control are employed [Such methods include the industrial standard Ziegler-Nichols method (Ziegler and Nichols, 1942).] to determine controller settings, since such rules are implicitly based on first- or second-order lag approximations of the process. Clearly, the control quality that can be expected is related to the sophistication of the process model; this is the reason that PID control usually improves on the performance achieved using a PI controller. However, even the most detailed model is still only an approximation of reality, and an important property of the designed feedback controller is that it be insensitive to modeling error. The relationships between designed controller sophistication, model uncertainty, and achievable closed-loop performance can be elucidated by using the concept of internal model control (Morari and Zafiriou, 1989). The IMC design procedure utilizes the structure shown in Figure la, in which p ( s ) represents the "true" process, P(s) is the process model, and q(s) is the IMC controller. After defining the model, the design of q(s) consists of two steps. First, the nominal controller, Q(s), is computed in terms of the proposed linear model, thus tacitly assuming that it is accurate. The second step is to append a low-pass filter, f ( s )(q = qf), in order to attenuate the effect of model uncertainty, which usually increases with frequency. This has the effect of d e t u n i n g the controller. For open-loop unstable systems, the IMC structure is internally unstable and thus can only be utilized as a design tool. The parametrization for the feedback classical controller 4s) =
q(s)
(1 -
m) d s ) )
(1)
is used to effect the implementation in the classical control 0 1991 American Chemical Society
