Ind. Eng. Chem. Res. 1988,27,963-969 z = z-transform variable
Greek Symbol AU = uk - uk-1
Literature Cited Cutler, C. R. Ph.D. Thesis, University of Houston, Houston, 1983. Cutler, C. R.; b a k e r , B. L. Proc. Am. Control Conf. San Francisco 1980,WP5-B (also presented at 83rd National AIChE Meeting, Houston, 1979). Demuster. A. P.: Schatzoff. M.: Wermuth. N. J. Am. Statist. Assoc. 1i77,72,77. ' Garcia, C. E.; Morari, M. Znd. Eng. Chem. Process Des. Dev. 1982, 2, 308-323. Hoerl, A. E.;Kennard, R. W. Technometrics 1970,12, 55. Marchetti, J. L. M.S. Thesis, University of California, Santa Barbara, 1981. I
,
963
Marchetti, J. L. Ph.D. Thesis, University of California, Santa Barbara, 1982. Marchetti, J. L.; Mellichamp, D. A,; Seborg, D. E. Znd. Eng. Chem. Process Des. Dev. 1983a,22, 488. Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. Proc. Joint Autom. Con,trol Conf., San Francisco, 1983b, 193. Maurath, P. R. Ph.D. Thesis, University of California, Santa Barbara, 1985. Maurath, P. R.; Seborg, D. E.; Mellichamp, D. A. Proc. Am. Control Conf., Boston, 1985,1059. Otto, R. E. Paper presented at the 1986 AIChE Annual Meeting, Miami Beach, 1986. Richalet, J.; Rault, A,; Testud, J. L.; Papon, J. Automatica 1978,14, 413. Smith, G.; Campbell, F. J . Am. Statist. Assoc. 1980, 75,74. Received for review April 23, 1987 Accepted October 13, 1987
Estimated Effects of Process Variables on Jet Temperature in a U-Gas Reactor J. M. Beeckmans* Faculty of Engineering Science, The University of Western Ontario, London, Ontario, Canada N6A 5B9
J. F. Large Centre d'Etudes et Recherches des Charbonnages de France, BP No. 2, Verneuil-en-Halatte, 60550 France
A steady-state model was developed of the jet in a fluidized bed containing coke and ash particles. The combustion of the particles of coke is included in the model, which was used to predict longitudinal gas and solids temperature profiles for different operating conditions and scales of operation. T h e model was developed to assist in scale-up design in going from pilot plant to demonstration or commercial plant size and also in extrapolating effects of increases in pressure. The temperature profile in the jet is important because of its effects on the kinetics of agglomeration of ash particles. It was concluded that a scale-up in size and/or pressure in going from a pilot plant to a full scale plant will result in a significantly higher solids temperature in the jet. The principle of the U-gas process (developed at the Institute of Gas Technology in Chicago) has been described in numerous publications (e.g., Patel (1980), Palat et al. (1983), and Schora et al. (1985)). A steam/oxygen or steam/air jet is injected at the base of a fluidized bed having a conical bottom. The bed is composed of a mixture of ash and char particles originating from the devolatilization and gasification of particles of coal in the body of the bed. The char ignites and burns in the jet, providing sensible heat required to balance losses and to drive the endothermic gasification reactions occurring in the bed. The jet temperature is sufficiently high to cause fusion and agglomeration of particles of ash, but the bed itself is below the ash fusion temperature (Mason and Patel, 1980). This feature permits controlled agglomeration of the ash and prevents the formation of large agglomerates and defluidized masses of clinker in the bed, which would prevent it from functioning as intended. Instead, agglomerates which reach a certain size fall into an ash pit through a venturi constriction at the jet inlet and are thus prevented from growing beyond a certain size. At steady state the rate at which inert matter enters with the coal must be in approximate balance with the rate of discharge of ash agglomerates, although some inert material is also recovered in the cyclones. This implies that the rate of agglomeration of ash must be controlled, since too rapid a rate will cause the inventory of fluidizable ash particles in the bed to diminish to the point that the system becomes inoperable. In addition, the rate of ag0888-5885/88/2627-0963$01.50/0
glomeration must be controlled in order to discharge agglomerates having a low carbon content and therefore achieve a high carbon conversion in the gasifier (Mason and Patel, 1980). It is difficult to study precisely the effects of the operating variables, especially the temperature of the jet, on the rate of agglomeration in a gasifier. However, it has been shown that the temperature range between incipient deformability and fusibility in coal ash is of the order of 125 O C (Mason and Patel, 19801, which suggests that the temperature range in the jet of the gasifier for maintaining controlled agglomeration is relatively small. Studies on model systems have shown that the rate of agglomeration of a fusible particulate material in a fluidized bed with a central jet is very sensitive to jet temperature, bed temperature, and cone angle (Arastoopur et al., 1986). For instance, it was found that the rate of agglomeration of polyolefin particles increased 5-fold for an increase in jet temperature of only 3 "C. To date, stable, extended operation of the U-gas process has been demonstrated with various coals at IGT's pilot plant, which has a capacity of approximately 25 ton of coal per day at a maximum pressure of 5 bar, and in a smaller Process Development Unit which is operable at 20 bar. All commercial reactor would, however, involve considerable scale-up in the size of the reactor and in the severity of operating conditions. In view of the demonstrated sensitivity to temperature of agglomeration kinetics in a fluidized bed, it appeared 0 1988 American Chemical Society
964
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 mensionless radical position coordinate (6 = r/b). The velocity profile for the solids is similar in form:
Table I. Values of Parameters Remaining Constant throughout the Calculations parameter value
Pa2
Tb "6
c
Em A E n 8
mean particle size of ash particles mean particle size of char particles density of ash particles density of char particles bed temperature radial gas velocity from jet to bed bed porosity emissivity constant in eq 33 activation energy in eq 33 reaction order initial half-angle of jet envelope
v ( ~ ,=~v)~ l . + 0 p+ + (1.0- p 5 ) 2 5 d + 1
0.4 mm 1.0 mm 2500 kg/m3 800 kg/m3 1250 "C 5 mmjs 0.5 1.0 80 kg/(m2 s) 16000 kcal/kmol 1.0 10 deg
Fluxes of mass and momentum across a plane perpendicular to the jet axis for each component are as follows: mass flux for component i in the gas: 1
W,i = 2ab2pJi(1 - y )
U 6 dt
(4)
(5) gas total momentum flux:
Jet Mechanics A jet in a fluidized bed is usually pulsatile (Knowton and Hirsan, 1980), but in some cases a flame-like, steady portion extends a considerable distance from the base. Although it is possible to compute the behavior of a pulsatile jet numerically (Gidaspow and Ettehadieh, 1983), the problem is difficult and very demanding of computer time. The case of a jet in which particles are undergoing combustion is necessarily even more complex, and given the limitations on time and computer resources available when the study was undertaken, an unsteady-state approach to the problem was not deemed possible. Given that a steady-state jet model has been used successfully by Massimilla and his co-workers (1981) to predict particle velocities and concentrations in the nonpulsatile lower portions of small jets, it seemed reasonable to apply a steady-state model to describe the behavior of jets with large momentum fluxes, which might be expected to be more stable due to their large inertia.
solids total momentum flux:
is the mass fraction of component i in either phase, pg is the gas density, y is the volume fraction of solids in the jet and p , is the mean density of the solids at that point in the jet. Overall mass fluxes for both phases are obtained by summing component mass fluxes. The gas phase was assumed to contain four components: carbon dioxide, carbon monoxide, oxygen, and water vapor. The solids phase was assumed to contain two components: ash and carbon. Differential mass, momentum, and enthalpy flux balances were obtained over an element of jet of thickness 6x. It was assumed that the solids enter the jet at a constant radial velocity. This assumption receives partial support from the results of Yang and Keairns (1982), who observed that the radial solids entry velocity into the jet increased slightly with distance along the jet axis. These results were valid for distances up to 0.6 m in a jet at ambient conditions and were obtained with a geometry which differed significantly from the geometry of the U-gas reactor. The differential solids mass flux equation for inert solids then has the form
fi
Mass and Momentum Flux Balances Two versions of the model were developed. In the simpler version, gas and solids temperature and composition were assumed radially uniform. In the more complex model, radial variations in these variables were taken into account, as were radial heat and mass transfer. The latter model was developed for use in conjunction with a laboratory study which was to have provided data on radial mixing of gas and solids in the jet. However, the experimental study was not pursued; hence, only the simpler model will be discussed in this paper. In previous jet models, the surface of the jet was taken as a truncated cone; in the present model, a paraboloid of revolution was used, which reduces to a cone as a special case. The equation relating jet radius to axial position has the form x = a. + a,b + a2b2 (1)
U , equals the velocity at the axis of the jet; $. is a di-
0
mass flux for component j in the solids phase:
to be the better part of wisdom to attempt to obtain an indication of the effects of scale-up in size and pressure on the temperature profile in the jet of the U-gas reactor.
where x equals the axial distance from the jet inlet, b equals the jet radius, and ao, ul, and u2 are constants. Values for ul and a2 may be determined by specifying the diameter of the jet at its point of entry (do),the angle of the jet envelope with the vertical (8) at its base. The velocity profile and the axial mass and axial momentum flux equations are similar to those used by Donsi et al. (1980) and Massimilla et al. (1981). The velocity profile for the gas is U(x,r) = U,[~.O+ + (1.0 - ~ ~ . ~ ) ~ t ; ~ 0(2)' ~
(3)
dW,, = 2 a b ( l - E ) p b ( l - xb)U,dx
(8)
where t equals bed porosity, P b equals bed density, x b equals mass fraction of char in the bed, and u, equals the solids radial entry velocity. The corresponding equation for the char is CY is the rate of combustion of char (in kg/ms) per unit volume in the jet at that location. It was assumed that some gas leaks out of the jet through the envelope at a constant velocity ug. In addition, gasphase component balances must take into account the reversible equilibrium between carbon oxides and oxygen and the combustion reaction which consumes oxygen and generates both carbon oxides. The overall gas-phase differential mass flux equation then has the form
]
equals the rate of generation or consumption of component i per unit volume due to chemical transformations. Values of K~ were calculated as follows: K~
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 965
for carbon dioxide (component 1): ~1
+
= 3 . 6 7 ~ ~ 4 1.571r
(11)
for carbon monoxide (component 2):
2.33a(l - 4){ for oxygen (component 3): KZ
K3
= -1.33a(l - 4) - 0.571r
(12) (13)
4 equals the molar ratio of carbon dioxide to carbon monoxide produced at the surface of the burning char, and f equals the rate of gas-phase oxidation of carbon monoxide per unit volume. The numerical coefficienb in eq 11-13 are ratios of molecular weights (e.g., 3.67 is the ratio of the molecular weight of carbon dioxide to that of carbon, and 1.571 is the corresponding ratio for carbon dioxide and carbon monoxide). The following equation, due to Arthur (1951), was used to evaluate 4:
4 = l.O/(l.O
+ 2500e+240Tx)
(14)
T, being the temperature at the surface of the particles of char. {was estimated by using the equation (Howard et al., 1972)
above the base. It was arbitrarily assumed that the jet terminates at the point that the volumetric concentration of solids reaches 25%.
Rate of Combustion of Char The temperature at the surface of the particles of char, T,, must be known in order to calculate the rate of combustion. T, was obtained on the basis of a heat balance on unit area of surface: Hg = H, + H, (18) where H g equals the rate of generation of heat by combustion, H,equals the rate of dissipation of heat by radiation, and H,equals the rate of loss of heat by conduction through the laminar film surrounding the particle. Hg was calculated from the heats of combustion of carbon to carbon monoxide and carbon dioxide and the proportion (4) of these two gases which is assumed to be formed at the surface in accordance with Arthur's (1951) expression. The following expression was obtained:
H~ = p.28 x 1074 + 9.49 x
ioyi - 4 ) l p
(19)
where p equals the rate of combustion of carbon per unit area. Heat of combustion values for CO and C 0 2valid at 2000 K were used in eq 19, and no allowance was made for f = 3.64 X 1012(P/RT,)2y2(y3y4)1/2e-15048/T~ (15) dissociation. H,was estimated by y l , y2, and y3 are mole fractions of carbon monoxide, oxygen, and water, respectively. H, = .E,(Tx4 - T:) (20) The differential momentum flux equations are based on those proposed by Massimilla et al. (1981) and Donsi et where u is the Stefan-Boltzmann constant, E, equals the al. (1980) but are modified due to transfer of momentum emissivity of the particles, and T, equals the local tembetween phases due to combustion. The gradient of moperature of the inert solids in the jet. It is implicitly mentum flux of the solids is given by assumed that the particles of char exchange thermal radiation predominantly with surrounding particles in the jet rather than with the walls of the jet whose temperature would be expected to approximate the temperature of the bed. Calculation of the rate of diffusion of oxygen to the G equals the axial pressure gradient in the bed. The f i t surface of the particle is complicated by two factors, term on the right represents the effect of drag on the namely, the large variation in values of the transport particles by the gas; the second term represents the effect properties of the gas from one end of the laminar film to of gravity, the pressure gradient, and the loss of solids due the other and the effects of Stefan flow. The former factor to combustion. The axial momentum of solids entering arises because the transport properties, especially diffuthe jet is considered to be negligible. The pressure gradient sivity, are very temperature dependent, and there are large was found from temperature differences across the film, and the second factor is important because of the relatively high oxygen concentration (75% by volume at the beginning of the jet) and because most of the oxygen is converted to CO, causing a net molar flow away from the particle. The system of eq 1-17 is closed, when the surface temThe distribution of temperature across the film was first perature of the particles of burning char (T,) and the rate calculated by using an assumed value for T,. The thermal of combustion of char per unit volume (a)are known. conductivity of a gas is proportional to and indeAside from the complications arising from combustion, pendent of pressure: the mechanical equations differ in two respects from the equations proposed by Massimilla and his co-workers k = ko(T/273)0.75 (21) (Donsi et al., 1980; Massimilla et al., 1981). (1) The moving Then layer of solids at the jet envelope was not included in the present analysis; Le., it was considered to be a part of the H , = ko(T/273)0~75(dT/dx) (22) bed rather than part of the jet. (2) The earlier workers If eq 22 is integrated from x = 0 to x = h and from T assumed that the pressure gradient in the jet was the same = Tgto T = T,, we obtain as that in the bed and calculated the injection rate profile for solids along the jet, whereas in this study solids were H, = kO(Tx1.75 - Tg1.75)/(1.75hToo.75) (23) assumed to enter the jet at constant radial velocity and the pressure profile was calculated. The total pressure If the same integration is performed between the limits drop along the jet was then determined by integrating the x = 0 and x = x , and T = T and T = T, we obtain an pressure gradient profile. This procedure was repeated analogous expression for H,. eliminating H,and simpliusing different values of the injection velocity (us) to find fying yields the value for usgiving the same overall pressure drop across T [Tg1.75+ ~ ( T ~ 1 . 7 5T 1.75)Ah]0.571 (24) B the jet as that expected across the bed at the same height
966 Ind. Eng. Chem. Res., Vol. 27, No. 6 , 1988
t
The molar flux of oxygen toward the surfaces is given by N = -D,(dc/dx) + iic (25)
-08
ii equals the net molar-average velocity away from the surface; it is easily shown that = -(I - 4)N/CT
CT equals the molar gas concentration. Note that if only COz were to be produced, 4 = 1.0 and ii = 0, since there would be equimolar and opposite fluxes of O2and of COz. Alternatively, if only CO is produced, ii = -N/CT, since the molar flux of CO is twice the molar flux of 02,and hence the net molar flux equals the oxygen molar flux. In a gas, diffusivity depends on temperature and pressure as follows: D , = Do(T/273)'.75/P
moo^
(26)
\
\
L
0
0-
05
A
-,
'O xim)
Figure 1. Solids temperature (T,) and oxygen mole fraction Cy3) in the jet versus distance from the origin (do = 0.05 m, Uj = 50 m/s, P = 20 bar, xb = 0.2).
io3
(27)
Also, CT is given by CT = 273CJ'/T
(28)
Substituting eq 26-28 into eq 25, and noting that c = cTy3 leads, after simplification, to the following equation: N[Trg1.75+ ~ ( T ~ ~ 1-. T 7 5)1.75/Ah]o.429= rg
-Doc0 dY3/[1 + ( 1 - 4)~31(29)
Trgand T,, are, respectively, the reduced gas and surface temperatures. Equation 29 may be integrated between the limits x = 0 to x = Ah and T = T gto T = T,, giving
y3, equals the oxygen mole fractions in the gas and at the particle surface. The carbon burning rate ( p ) is related to the molar oxygen flux by p = 12(2 - 4)N (31) The thickness of the laminar film is given by (Steinberger and Treybal, 1960) Ah = d , , / ( 2
+ 0.552Re0.53S~0.33)
(32)
Re is the particle Reynolds number, and Sc is the Schmidt number evaluated at the mean temperature across the film. Equations 30-32 permit p to be calculated ify, is known. The intrinsic burning rate depends on a number of factors, particularly temperature, but also the nature of the coal and its porosity. A general equation may be written, having the form (Smith, 1982) p
= Ae-E/RTx(Py,,)n
(33)
A is a constant, E is the activation energy, and n is the reaction order with respect to oxygen partial pressure at the surface. Equations 30-33 together permit y3, and p to be calculated using an assumed value for T,. H,,H,, and H,were then calculated, and if eq 18 waa not satisfied, the calculations were repeated using a different estimate for T,. An algorithm was used to find the correct value for T,, satisfying eq 18 by this method. The combustion rate of the char per unit volume of jet may now be calculated from the following expression: CY
= 6YPef2P/dp2P2
(34)
Figure 2. Pressure drop ratio and solids volume fraction in the jet versus distance from the origin (same conditions as for Figure 1).
where the subscript 2 refers to the second component (char) in the solids phase. It was assumed in deriving eq 34 that the particles of char are spherical. The integration of the equations of mass and momentum is straightforward when local values have been found for T,, p, and a,since conditions at any point in the jet do not depend on downstream conditions. The solutions were found to be stable provided that sufficiently small increments ( 6 x ) were chosen. It was found that the smallest values of 6x were needed at the low end of the jet, where the gas velocity was highest and solids acceleration was largest. In this region, an increment of 0.005 mm was used, which increased progressively to 0.4 mm along the jet.
Results and Discussion Table I gives values of parameters whose values were maintainedconstant throughout all the calculations. These included the mean size of inert and char particles, the kinetic parameters pertaining to the rate of combustion of char, and the leakage velocity of gas from the jet into the bed. Unless explicitly stated the contrary, it was assumed that the gas entering the jet was composed of 25% oxygen and 75% steam on a molar basis, at a temperature of 900 oc. Figures 1 and 2 show typical profiles of solids temperature, oxygen concentration, solids volume concentration, and pressure ratio along the jet. Solids temperature rises as the particles burn but eventually reaches a maximum as the oxygen becomes depleted and the solids in the jet are cooled by particles which enter the bed from the jet. The preasure drop ratio (i.e., static pressure drop along the jet divided by pressure drop across an equivalent length of bed) initially falls below zero because of pressure recovery in the virtually empty jet as it decelerates due to expansion; however the pressure drop ratio rapidly rises due to entry of solids into the jet and quickly reaches values greater than unity because of large drag forces. In the upper parts of the jet, the drag forces are much smaller, and since the concentration of solids in the jet is significantly smaller than that in the bed, the pressure gradient
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 967 15001
I I
12001
5
10
50
100
P (bar)
Figure 3. Effect of pressure on maximum solids temperature in the jet. Solid curve: inlet velocity, 50 m/s. Stippled curve: inlet velocity, 100 m/s (do = 0.05 m).
t 13001
I
I
I
I
50
1 I
1
\ I
100
500 d. (mm)
Figure 7. Maximum solids temperature in the jet as a function of jet inlet diameter, at constant carbon combustion rate (do2Pconstant). Upper curve, 16 ton/h; middle curve, 2.5 ton/h; lowest curve, 1.0 ton/h.
c
1300
1200-
0.5 xb
Figure 4. Effect of scale-up on maximum solids temperature in the jet. Solid curve: Uj = 100 m/s.
1350
-
1300
50
130-
50
loo
do(")
Figure 8. Maximum solids temperature in the jet as a function of jet inlet diameter, at constant carbon combustion rate (do2Ujconstant).
c
I30
U, (m/s)
Figure 5. Effect of carbon content in the bed on maximum solids temperature in the jet (do = 0.05 m, Uj = 100 m/s).
t
I
k = 3 0 bar
'3001 Figure 9. Maximum solids temperature in the jet as a function of the percentage of steam in the gas at the jet inlet, at constant carbon combustion rate (in ton/h). y3p constant, Uj = 50 m/s.
-031
io
50
100 d,k"
Figure 6. Effect of gas velocity at jet entry on maximum solids temperature in the jet.
in the jet is smaller than in the bed, and the pressure drop ratio eventually falls to unity at the point that the jet is deemed to terminate. Figures 3-6 show curves obtained for maximum solids temperature in the jet, each curve corresponding to a single, continuouslyvarying parameter. These results show
that pressure, jet inlet diameter, and bed carbon content are variables which have a strong positive correlation with bed temperature. The effect of jet entry velocity appears to be much smaller, and at higher pressures the slope of the curve becomes negative over part of the range of the argument. Figure 7 shows the effect of varying jet entry diameter and pressure in such a manner as to maintain constant throughput of gas (Pdo2is constant), for three different values of gas throughput. These results suggest that for a given throughput the solids temperature in the jet can be lowered by operating with a larger jet at a lower pressure.
968 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
Figure 8 shows that when jet entry velocity and jet entry diameter are varied in such a manner as to keep gas throughput constant (d,2Uj constant) there is relatively little change in maximum solids temperature in the jet. Finally, Figure 9 gives maximum solids temperature in the jet, as a function of initial mole fraction of oxygen. The principal conclusion from these results is that scaling up from a pilot plant operating at a relatively low pressure to a full-scale plant operating at a much higher pressure may result in significantly higher jet temperatures. The results do not lend encouragement to the notion that maximum solids temperature can be significantly lowered by manipulating jet inlet diameter and inlet velocity at constant throughput (Figure 8). However, Figure 5 suggests that if the feed rate of coal were to be reduced, causing the bed carbon content to fall, this should result in a lowering in the maximum jet temperature. Lowering the pressure while simultaneously increasing jet inlet diameter can result in lower maximum solids temperature (Figure 7); however, these two factors will have detrimental effects on the performance of the gasifier. In summary, the results of this study suggest that a scale-up in size and pressure in going from a pilot plant to a demonstration or commercial plant may result in significantly higher solids temperatures in the jet, and future designers of commercial plants are advised to take this possibility into account.
Acknowledgment The authors gratefully acknowledge financial support for this project from the Centre d’Etudes et Recherches des Charbonnages de France.
Nomenclature A = constant in eq 33 ao,al,a2 = constants in eq 1 b = radius of the jet, m c = local molar concentration of oxygen in a film around the char particles, k-mol/m3 CD = drag coefficient Co = gas molar concentration at standard conditions, kmol/m3 CT = molar gas concentration, kmol/m3 do = diameter of the jet at its point of entry, m D, = diffusivity of oxygen, m2/s Do= diffusivity of oxygen at standard conditions, m2/s dPi= mean particle size of component i in the solids phase, m E , = emissivity of char particles fi = mass fraction of component i g = gravitational constant, m/s2 G = pressure gradient in the jet, Pa/m Ah = thickness of the laminar film surrounding a particle of char, m H, = rate loss of heat by conduction from the surface of a burning particle of char, J/(m2 s) Hg = rate of generation of heat by combustion at the burning surface, J/(m2 s) HI= rate loss of heat by radiation from the surface of a burning particle of char, J/(m2 s) k = thermal conductivity of gas in the film around the char particle, J/(m K) ko = thermal conductivity of gas in the film at standard conditions, J/(m K) A& = momentum flux of gas in the jet, (kg m)/s2 nir, = momentum flux of solids in the jet, (kg m)/s2 MT = total momentum flux in the jet, (kg m)/s2 N = molar oxygen flux per unit area in the film around a char particle, kmol/ (m2s) P = pressure, Pa
r = radial coordinate T = temperature at a point in the gas film around the char
particles, K Tb = temperaure of the fluidized bed, K Tg= absolute temperature in the gas phase, K T, = absolute temperature at the surface of the char, K T, = temperature of inert particles in the jet, K Trg,TI, = reduced temperatures U = gas velocity in the jet, m/s U, = jet inlet velocity U,,, = center-line gas velocity in the jet, m/s us = radial velocity at which bed particles enter the jet, m/s ug = radial velocity of gas entering the bed from the jet, m/s V = velocity of particles in the jet, m/s V,, = center-line particle velocity in the jet, m/s = mass flux of component i in solids, kg/s W,= mass flux of gas in the jet, kg/s W,,= mass flux of gas of component i in gas, kg/s x = axial distance from the point of origin of the jet, m xb = mass fraction of char in the bed y, = mole fraction of component i in the gas y3, = mole fraction of oxygen at the surface of the char Greek Symbols a = rate of combustion of char per unit volume in the jet, kg/(m3 s) y = volume fraction of solids in the jet e = void fraction in the bed { = rate of gas-phase oxidation of carbon monoxide, kmol/(m3
4
B = angle of jet envelope with the axis, at its origin, rad K & = rate of generation or consumption of component i, per
unit volume, kmol/ (m3s)
= dimensionless radial coordinate ( E = r / b ) p = rate of combustion of carbon per unit surface, kg/(m2s) p, = density of component i in the solids phase, kg/m3 /sa = average density of solids, kg/m3 p b = bed density, kg/m3
gas density, kg/m3 4 = molar ratio of carbon monoxide to carbon dioxide at the surface of char particles
p, =
Literature Cited Arastoopur, H.; Gu, A. Z.; Weil, S. A. The Effect of Temperature and Gas Velocity Distribution on the Fludization of the Sticky Particles; Ostergaard, K., Sorenson, A., Eds.; Engineering Foundation: New York, 1986; pp 209-216. Arthur, J. R. “Reactions Between Carbon and Oxygen”. Trans. Faraday SOC.1951,47, 164. Donsi, G.; Massimilla, L.; Colantuoni, L. The Dispersion of AziSymmetric Gas Jets in Fluidized Beds; Grace, J. R., Matsen, J., Eds.; Engineering Foundation: New York, 1980; pp 297-304. Gidaspow, D.; Ettehadieh, B. “Fluidization in Two-Dimensional Beds with a Jet. 2. Hydraulic Modeling”. Ind. Eng. Chem. Fundam. 1983,22, 193-201. Howard, J. B.; Williams, G. C.; Fine, G. H. “Kinetics of Carbon Monoxide Oxidation in Postflame Gases”. Proceedings of the 14th Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, PA, 1972, pp 975-986. Knowlton, T. M.; Hirsan, I. The Effect of Pressure on Jet Penetration in Semi-Cylindrical Gas-Fluidized Beds; Grace, J. R., Matsen, J., Eds.; Engineering Foundation: New York, 1980; p 315. Mason, D. M.; Patel, J. G. “Chemistry of Ash Agglomeration in the U-GasProcess”. Fuel Process Technol. 1980, 3, 181-206. Massimilla, L.; Donsi, G.; Migliaccio, N. “The Disperion of Gas Jets in Two-Dimensional Fluidized Beds of Coarse Solids”. AIChE Symp. Ser. 1981,205, 17-27. Patel, J. G. “The U-Gas Process”. Energy Res. 1980, 4, 149-165. Palat, P.; Schora, F. C.; Patel, J. G. “The Versatility of the U-Gas Process: A French Perspective”. Synfuels 3rd Worldwide Symposium, Washington, D.C., 1983. Schora, F. C.; Palat, P.; Patel, J. G. “Present Status of the U-Gas Process”. Symposium on Coal Gasification and Synthetic Fuels for Power Generation, 1985. Smith, I. W. “The Combustion Rates of Coal Chars: a Review”.
Ind. Eng. Chem. Res. 1988,27, 969-973 Proceedings of the 19th International Symposium of The Combustion Institute, 1982, pp 1045-1065. Steinberger, R. L.; Treybal, R. E. “MassTransfer from a Solid Soluble Sphere to a Flowing Liquid Stream”. AZChE J. 1960,6, 227-232. Yang, W.-C.; Keairns, D. L. ‘Solid Entrainment Rate into Gas and
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Gas-Solid, Two-PhaseJets in a Fluidized Bed”. Powder Technol. 1982,23,89-94.
Received f o r review May 18, 1987 Revised manuscript received January 20, 1988 Accepted February 22, 1988
Improved Multiloop Single- Input/Single - Output (SISO) Controllers for Multivariable Processes Thomas J. Monica,?Cheng-Ching Yu,’ and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Three methods are proposed for improving the performance of multiloop single-input/single-output (SISO) controllers which have been tuned by an approach illustrated in an earlier paper. In that paper, multiloop PI controllers were designed by detuning the controllers equally from SISO Ziegler-Nichols settings until a multivariable stability criterion was satisfied. In the present paper, several methods are described for adding derivative action to the PI controllers. An empirical procedure is developed for weighting the detuning of the loops by the predicted final ITE of the process after load and set-point disturbances. The combination of weighted detuning and derivative action provides a significant performance improvement, with little added complexity. The results of both the earlier method and the enhancements described in this paper are compared to Dynamic Matrix Control for several multivariable systems, from 2 X 2 up t o 4 X 4.
In a recent paper, Luyben (1986) proposed a method for tuning PI SISO controllers for multivariable processes, which was called BLT tuning. The method consists of detuning all the controllers equally from Ziegler-Nichols settings until a safe margin of closed-loop stability is obtained, as indicated by a multivariable Nyquist plot. In the earlier work, good performance was obtained with very little expenditure of engineering or computing effort. However, comparisons of diagonal controllers with fully multivariable controllers such as DMC-which are demonstrated in this paper-indicate some incentive for developing improved versions of the diagonal controller. This paper is directed at improving the diagonal BLTtuned controller performance so it approaches the fully multivariable controllers, without sacrificing the simplicity of the controller design and function. As stated in the earlier work, we do not claim that this method provides better control than full multivariable controllers. Our objective is to demonstrate a simple technique that provides reasonable performance for little engineering cost. In this work, we assume that selections of controlled variables, manipulated variables, and variable pairings have been made, for example by the method of Yu and Luyben (1986). We start with a matrix of transfer functions of order N X N . Simulations of three linear multivariable processes are presented. The performances of four multiloop SISO diagonal controllers are compared to the performance of a fully multivariable DMC controller for both load and set-point disturbances. The original BLT method is labeled BLT-1 in this paper. It uses only PI modes in the diagonal controllers and equal detuning of all loops. Two techniques were developed and combined that improve the performance of BLT control, while retaining its *Author to whom correspondence concerning this paper should be addressed. Present address: Department of Chemical Engineering, University of Houston, Houston, TX 77004. Present address: Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, R.O.C. 0888-5885/88/2627-0969$01.50/0
straightforward nature. In the first (BLT-2), derivative action is added to all controllers. A single F factor is used to adjust the integral time constant (q)and the controller gain (K,)of each loop, as in the BLT-1 method. Then, a separate F D is used to tune the derivative action (rD)of each loop in a similar manner. The second method (BLT-3) compensates for the fact that multiloop interactions are not normally symmetric. The detuning of each loop uses a weighting factor that depends on a ratio of a prediction of its integral total error (ITE) and the ITE of the other loops, for both load and set-point disturbances. The best results are obtained when these two modifications are applied in combination (BLT-4). This approach yields a decided improvement over BLT-1 tuning alone and provides a clear method for weighting the performance of each controlled variable according to it’s importance. As was true with the original BLT method, these techniques are limited to open-loop stable systems, and the question of integrity is only partially addressed: when all loops are on automatic, the system is stable; when any one loop is on automatic, the system is stable.
Discussion of BLT Procedures 1. BLT-1. We begin with a multiloop SISO system tuned by the BLT-1 method as described in the earlier work (Luyben, 1986). This relatively straightforward procedure, described below, was tested successfully in simulations on 10 multivariable process models. 1. Compute the Ziegler-Nichols tuning parameters of t,he diagonal elements of the process transfer function matrix, gii(s),as though the diagonal elements represented SISO systems. 2. Choose a detuning factor F. F should be greater than one. 3. Compute K , and T I J for each loop by
0 1988 American Chemical Society
