I n d . Eng. Chem. Res. 1987,26, 343-347 McHugh, M. A. Ph.D. Thesis, University of Delaware, 1980. McKetta, J. J.; Katz, D. L. Ind. Eng. Chem. 1948, 40, 853. Meskel-Lesavre, M.; Richen, D.; Renon, H. Ind. Eng. Chem. Fundam. 1981,20, 284. Orr, F. M., Jr.; Taber, J. J. Science (Washington, D.C.) 1984, 224, 563. Radosz, M. Ber. Bunsenges. Phys. Chem. 1984,88, 859.
343
Rousseaux, P.; Richen, D.; Renon, H. Fluid Phase Equil. 1983,11, 153. Tiffin, D. L.; DeVera, A. L.; Luks, K. D.; Kohn, J. P. J. Chem. Eng. Data 1978, 23, 45. Received for review January 21, 1986 Accepted June 19, 1986
Statistical Analysis of Mass Transfer in Liquid-Solids Fluidized Beds N. Yutani and N. O t o t a k e Department of Chemical Engineering, Tokyo University of Agriculture a n d Technology, Koganei, Tokyo 184, Japan
L. T. F a n * Department of Chemical Engineering, Kansas S t a t e University, Manhattan, Kansas 66506
T h e correlations of wall-to-bed mass-transfer coefficients appearing in a number of recent papers have been mainly obtained experimentally. Their accuracy is about *30-50%. I t seems likely that deviations of such magnitude have arisen from randomness of particle movement. T h e present work takes into consideration the effects of the fluctuations of particle velocity near the wall on the mass- transfer rate. These fluctuations are estimated from a so-called partition function of solid particle velocities. New experimental data, obtained electrochemically, deviate less than 10% from the resultant correlation. Mass is transported from the walls of a liquid-solids fluidized bed to the bed itself in various process operations, including those employed in coal liquefaction and electrochemical processing (Le Goff et al., 1969; Morooka et al., 1980). Thus, the mechanism of the wall-to-bed mass transfer in a liquid-solids fluidized bed has become a subject of intensive investigation. However, earlier publications appear to contain insufficient information for understanding the mechanism of the mass transfer from the walls to the bed proper (Jagannadharaju and Rao, 1965; King and Smith, 1967; Storck et al., 1975). Furthermore, the mass-transfer coefficients contained in these publications have been determined without taking into account the effect of turbulence caused by the movement of particles near the walls. This may be one reason why the published correlations for these coefficients are reliable only to *30-50%. Mass is transported from the walls of a liquid-solids fluidized bed to the bed itself in various process operations, including those employed in coal liquefaction and electrochemical processing (Le Goff et al., 01969; Morooka et al., 1980). The instantaneous movements of the solid particles are quantitatively treated by means of a statistical method. Theoretical Mechanism of t h e Wall-to-Bed Mass Transfer. It has been known (King and Smith, 1967; Itoh et al., 1973) that the rate of mass or heat transfer between the wall surfaces and bed itself in a liquid-solids fluidized bed, including the fluidizing medium and particles, is strongly influenced by fluctuations of the fluid velocity induced by fluctuations of the velocities of particles near the wall. Two relatively simple mechanisms are proposed to account for the causes of such fluctuations. First, the displacement of particles constantly stirs the fluid near the walls of the bed, and second, the fluid flowing through the void spaces in the bed is inherently unstable especially when the flow rate is high.
Numerous correlations between the mass-transfer factor,
JD,and the Reynolds number, Re, are available in publications on the subject of the wall-to-bed mass transfer in liquid-solids fluidized beds (Jagnnadharaju and Rao, 1965; Carbin and Gabe, 1974; Storck et al., 1975; Morooka et al., 1980). In general, they have the form
tJD = f(Re)
(1)
JD = StSc2I3
(2)
where
k = mass-transfer coefficient, U , = superficial velocity of the fluidizing medium, D, = particle diameter, and t = average void fraction in the fluidized bed. Equation 1 assumes implicitly that the behavior of fluidized particles in the system under consideration is governed by the average void fraction, t; the effects of the instantaneous change in the void fraction on the wall-tobed transfer coefficients have been neglected (King and Smith, 1967; Storck et al., 1975; Morooka et al., 1980). The continual local change in the void fraction generated by the displacement of particles gives rise to the fluctuations of local velocity of the fluidizing medium, as depicted in Figure 1. The movement of particles in the system under consideration can be explained in terms of the partition function of particle velocities, described in the succeeding section. Partition Function of Particle Velocities. Let U,, be the instantaneous velocity of an individual particle, e.g., particle j , in the gravitational or z direction relative to UO
0888-5885/87/2626-034~~0~.50/00 1987 American Chemical Societv
344 Ind. Eng. Chem. Res., Vol. 26, No. 2, 1987
1
/--,Current
position
0
0
fluidizing medium
Figure 1. Local particle displacement and flow of the fluidizing medium.
0 @
Under the condition of steady-state fluidization, the average velocity of particles relative to a fixed coordinate or the wall of a fluidized bed is obviously zero. The average particle velocity relative to the upward superficial velocity of the fluidizing medium, Ufo,is denoted by Uz.Then,
U, = o - Ufo or Uz = -Ufo
(5)
a @
m 6
I '
INITIAL STEbDY STATE ONSETOF HETEROGENEOUS BED-EXRWSION HETEROGENEOUS UNSTEADY BED-EXPANSION ONSET Of HOMOGENEOUS BED-EXPANSION HOMOGENEOUS UNSTEADY BED-EXPANSION FINAL S T E M Y STATE
I
TlmO
Figure 2. Interpretation of the relationship between the transfer and bed expansion in a liquid-solids fluidized bed (Yutani et al., 1982).
Furthermore, we define where Uzjis the instantaneous velocity of particle j at any instant. U,, is the minimum of UZj;the minimum exists because the amount of kinetic energy imparted to a particle to set it in motion is finite under the condition of steady fluidization. As the bed density approaches that of the packed bed, the fluctuations of particle velocities in the bed become negligibly small. Then, U,, becomes equal to the negative of the minimum fluidizing velocity, U d With these definitions in hand, the partition function is obtained based on the so-called "cell model" (Yutani and Ototake, 1980). With the number of cells, each containing a single particle, at any cross section of the bed known, the velocity is partitioned to each cell according to the principle of entropy maximization. To apply this principle, the overall entropy of the system is maximized under the constraints that the total number of particles or cells and their overall momentum in the axial direction of the bed are fixed. These constraints are removed by resorting to the Lagrange multiplier method as follows (Reif, 1965; Yutani and Ototake, 1980):
Model. A statistical model consisting of two parts is proposed here to correlate the mass-transfer coefficients in terms of the flow conditions. The first relates these to the mass-transfer coefficients; the second defines the modified Stanton number, St ', and the modified Reynolds number, Re', both of which are expressed in terms of the standard deviation of fluctuations of the particle velocities near the wall of the bed given by eq 11. The two numbers are defined, respectively, as
and
Re'=
q0,+ (UZ- UZC)lP P(1 -
The Stanton number, St', is employed in defining the modified mass-transfer factor as EJD' =
~St'Sc'i~
Substitution of eq 7 into this expression gives
=
0,'+ (UZ- Uzc)2
(9) Thus, the variance of particle velocities relative to the average velocity of all the particles, u ~ ( U , ]can ) ~ , be expressed as u2(uzj)r= 02(Uzj) -
UZ2
= (U,- U Z C ) 2 (10) Therefore, cr(UZJ),, which is the standard deviation of the instantaneous particle velocities, [J,,, is given by u(uZJ)T
=
(Tz(uZJ)r
=
(t:- U Z C )
(11)
(14)
In turn, this modified mass-transfer factor is correlated as a function of the modified Reynolds number, Re', in the form
tJD' = f ( R e ' ) The magnitude of U,, in this equation can be estimated empirically by carrying out unsteady expansion of the bed (Yutani and Ototake, 1980). The variance of the instantaneous particle velocities UzJ, u2(Uzj),is obtained as
(13)
(15)
Determination of the Partition Function of Particle Velocity. The partition function of the particle velocities can be empirically determined by carrying out unsteady expansions of liquid-solids fluidized beds. This is accomplished by suddenly changing the superficial velocity of the fluidizing medium from its initial velocity, .Yo, to Um(>Uo); the bed expands as depicted in Figure 2 (Yutani et al., 1982). Consider an arbitrary point on the bed-expansion curve as illustrated in Figure 2. A t this point the superficial velocity of the fluidizing medium is fixed to U,, and the bed has an average void fraction of t; naturally, the value of t is different from one point to another on the expansion curve, even though the superficial velocity, U,, remains unchanged. Prior to the completion of expansion, the average t of the bed will correspond to that of a bed under steady state at a superficial velocity, which is less than U,; in fact, during the expansion, the average velocity of particles relative to the superficial velocity, iJ,,increases continually until it reaches U,. The fluidized bed illustrated in Figure 2 is divided into two compartments; compartment A is located below and compartment B above the static pressure sensor. The movement of a particle irom one compartment to another can be related to the partition function of particle velocities
Ind. Eng. Chem. Res., Vol. 26, No. 2, 1987 345 I Fluidized bed proper 2 Cathode 3 Anode
7
0
1
c
L
10
A
2
uZ
20 or
L
-vi
-
-
30
40
4 5 6 7
Orifice meter Rota meter Pump Head tank S Slrage tank 9 Valve I O Mearurement section
13
Icmisoc 1
Figure 3. Relationship between U,, and iJz or -Ut for glass beads of various diameters (Yutani and Ototake, 1980).
(Yutani et al., 1982). Assume that this function is applicable t~ any "quasi-steady-state" point on the expansion curve where e is less than the void fraction of the new steady state, and the corresponding superficial velocity, U , is less than U,. Then, the number of particles which are transported from compartment A to compartment B as a function of the quasi-steady-state particle velocity can be computed as
Substituting eq 7 into this expression and integrating the result, we have AWA WA
- -1 - ex.( e
--)u,u-u,u, -
(17)
where AWA is the weight of particles which have passed into compartment B from compartment A and WAis the weight of particles in compartment A under the initial condition of the average particle velocity, U. As indicated in the previous work (Yutani and Ototake, 1980), substitution of the experimentally obtained WAand AWAicto eq 17 gives the value of U, or U,, as a function of U or U,, as illustrated in Figure 3.
Experimental Section Facilities. The experimental setup employed is shown schematically in Figure 4. It consisted of a head tank, an orifice flow meter, flow-regulation valves, and a fluidized bed. The platinum plate with a width of 1.15 cm, a length of 11 cm, and a thickness of 0.3 cm served as the cathode for measuring the so-called diffusional current. The cathode electrode was located in the center of the fluidized bed; the anode electrode covered the inside wall surface of the bed column from the distributor to a height of 2.5 cm. A porous glass filter, fabricated by sintering glass beads with a diameter of 74 pm, was used as the distributor. The fluidized bed was made of acrylic resin with an inner diameter of 8.0 cm and a height of 90.0 cm; the height of the calming section was 29 cm. N) or Materials. An aqueous solution of iodine potassium iodide N) served as the fluidizing medium. The physical properties of the fluidized particles are summarized in Table I. This table also contains the properties of the particles used by some other investigators. Procedure. A weighed quantity of particles was poured into the bed and fluidized by maintaining the superficial
Figure 4. Experimental apparatus. Table I. Experimental Conditions a n d Materials Used materials used glass beads D, = 203, 273, 459 gm pp = 2.5 g/cm3 I2 water fluidizing medium area of electrode (1 cm X L [cm]) L = 10 cm range of void fraction 0.55-0.95
+
velocity of the fluidizing solution at a fixed value. A specified voltage was applied to the electrical system, consisting of an anode electrode, a cathode, a current meter, and a constant-voltage power supply. The diffusional current was measured by means of the electric current meter after the fluidization had become steady. The mass-transfer coefficients between the surface of the cathode and the fluidizing medium were obtained from
k=-
1
nFCA
where lz = mass-transfer coefficient (m/s), F = Faraday constant (C/g-equiv), n = number of gram-equivalence, C = concentration of iodine in the aqueous solution (mol/mol of water), A = area of the cathode (m2),and i = diffusional current (PA). The procedure was repeated after changing the superficial velocity of the fluidizing medium. The temperature of the fluidizing bed was maintained at 10.5 f 0.5 "C.
Results and Discussion The relationship between the mass-transfer coefficient, estimated from the diffusional current method, and void fraction of the bed is illustrated in Figure 5 . Note that the maximum values of mass-transfer coefficients exist in the range of the void fraction from 0.57 to 0.85. This feature is closely related to the characteristics of fluidized particles, specifically, the partition function of particle velocities discussed earlier. It is similar to those obtained by other investigators (Itoh et al., 1973; Morooka et al., 1980). Correlations for the wall-to-bed mass-transfer coefficients obtained in the present work, as well as those obtained by other investigators in the form of eq 1, are illustrated in Figure 6, in which the correlating equations proposed by others are represented by the straight lines. Deviations of these correlating equations are as much as
346 Ind. Eng. Chem. Res., Vol. 26, No. 2, 1987 Table 11. Sample Calculations 1. given conditions glass beads, D, = 0.0458 cm superficial velocity U, = 3.34 cm/s
2. estimation of the parameters Cr2, = 2.60 cm/s"
k
d,,='c S ~ ' S C=~e(/ ~2uz - -
u,,)Sc213 = e(
$)Sc2I3(
-
2uz -
Uf
- uzc
)
=
ufo
-
= 0.0782
uz,
2
X
3'34 = 0.0640' 3.34 - 2.60
This value is obtained from the previous paper (Yutani and Ototake, 1980); U,, is plotted as a function of Ufin Figure 3.
Note that
uz= -um
Table 111. Three Tswical Calculated Results Based on the Present conditions U,, cmls t k , cmfs Re 4.34 0.947 1.97 290.0 1.94 171.8 3.92 0.919 2.11 124.3 3.62 0.897 2.19 95.3 3.34 0.876 2.23 69.5 0.848 3.00 2.31 50.3 2.64 0.815 2.30 38.7 2.24 0.787 2.27 25.9 1.19 0.740 2.25 15.4 1.52 0.691 2.22 12.7 1.25 0.652 8.04 2.04 0.92 0.595 5.46 1.78 0.548 0.70
I
'
I
I
'
I
ErMrimantal Dala
'
I
'
I
1
Data calcd results
E J ~ 0.0568 0.0619 0.0712 0.0782 0.0861 0.0974 0.106 0.120 0.139 0.158 0.180 0.194
U,,, cmls
Re'
3.90 3.29 2.94 2.60 2.15 1.75 1.47 1.13 0.87 0.76 0.64 0.58
319.0 198.1 147.6 116.4 89.1 67.1 53.1 36.5 22.0 17.6 10.5 6.4
e J,'
0.0516 0.0536 0.0600 0.0640 0.0672 0.0730 0.0773 0.0853 0.0974 0.114 0.138 0.166
I 3
1 u
n
h v)
1
--
,
Re 0 6
0 8
07
09
[-I Figure 5. Relationship between the mass-transfer coefficients, k , and the void fraction of a liquid-solids fluidized bed, e , obtained experimentally.
i _ i I _ . -
2
3
Figure 6. Correlations of mass-transfer coefficients in terms of the Reynolds number, eq 1, for the glass beads-water system.
number, Re', as shown in Figure 8. It can be expressed as tJD'
*30-50% in the lower range of the Reynolds number; nevertheless, the flow in this range is not completely laminar. Such deviations may be due to the fact that the effects of fluctuating velocities of particles on the masstransfer coefficients are not explicitly taken into account in these correlations. To take such effects into account, a correlation is proposed here on the basis of eq 14 and 15. The resultant correlation is depicted in Figure 7; sample calculations are provided in Tables I1 and 111. Note that the deviations of the data from the proposed correlation are much smaller than those of the previously available correlations. Plotting the experimental data logarithmically according to eq 14 and 15 has yielded a relationship between the modified mass factor, tJD', and the modified Reynolds
,
:-I
= 0.4Re'-04
0.5 =S R e ' < lo3
(19)
where JD'
= S~'SC'/~
(20)
Conclusion The accuracy of the present correlation is within f 510% in the low range of the modified Reynolds number, Re ', indicating that it is more precise than other correlations. However, in the high range of the modified Reynolds number, Re ', the present correlation appears to be only as accurate as the available correlations based on eq 1. This implies that the effects of fluctuating velocities of particles on the mass- or heat-transfer coefficient are rather small in this region; probably these coefficients are strongly affected by the turbulence induced in the fluid phase itself.
Ind. Eng. Chem. Res. 1987,26, 347-356
br Morooko
16'1
03
1
1
,
il980)
et 01 I
t
'
!
IO
1
1
I
, L
Ir, Qeor ?e [-
i
Figure 7. Typical example of comparison between eJD vs. Re and cJD'
VS.
Re'.
10
347
Sc = Schmidt number = k / v S t = Stanton number, k / U or k / U , St' = modified Stanton number U = average quasi-steady-state velocity of particles corresponding to the average void fraction t during bed expansion U , = superficial velocity of the fluidizing medium U, = initial superficial velocity of the fluidizing medium Uc = minimum value of U U, = average velocity of particles relative to the upward superficial velocity of the fluidizing medium U,, = minimum value of U z j Uzj= instantaneous velocity of an individual particle relative to the upward superficialvelocity of the fluidizing medium ,r = velocity defined by eq 6 u:) = average value of uz; U , = superficial velocity of fluid at a new steady state after the stepwise disturbance W = number of ways WA = weight of particles in compartment A AWA = weight of particles which have passed into compartment B from compartment A Greek Symbols a2(U,.) = variance
of the instantaneous particle velocity, UZj, derined by eq 8 a*( UZj&= variance of the instantaneous particle velocity relative to the average velocity given by eq 10 a(Uzj)r. = standard deviation of Uzj t = void fraction of the bed
Literature Cited
Figure 8. Correlations between the modified Reynolds number, Re', and the modified e J D factor, JD', for the glass beads-water system.
Nomenclature
D, = particle diameter JD = mass-transfer factor JD' = modified mass-transfer factor k = mass-transfer coefficient L = length of the electrode N = total number of particles N . = number of particles in the jth state of particles P(UJ = partition function of particle velocities Re = Reynolds number R' = modified Reynolds number
Carbin, D. C.; Gabe, D. R. Electrochim. Acta 1974, 19, 645. Itoh, R.; Komazawa, Y.; Omodaka, K.; Yamamoto, M. Kagaku Kogaku 1973,37, 497. Jagannadharaju,G. J. V.; Venkata Rao, C. Indian J . Technol. 1965, 3,201. King, D. G.; Smith, J. W. Can. J . Chem. Eng. 1967, 45, 329. Le Goff, P.; Vregnes, F.; Coeuret, F.; Bordet, J. Ind. Eng. Chem. 1969, 61(10),8. Morooka, S.; Kusakabe, K.; Kato, Y. Int. Chem. Eng. 1980,20,433. Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill: New York, 1965; Chapter 7. Storck, A.; Vergnes, F.; Le Goff, P. Powder Technol. 1975,12, 215. Yutani, N.; Ototake, N. Kagaku Kogaku Ronbunshu 1980, 6, 570. Yutani, N.; Ototake, N.; Too, J. R.; Fan, L. T. Chem. Eng. Sci. 1982, 37, 1079.
Received for review August 16, 1984 Revised manuscript received March 19, 1986 Accepted August 25, 1986
Simplified Model Predictive Control Gomatam R. Arulalan and Pradeep B. Deshpande* Chemical Engineering Department, University of Louisville, Louisville, Kentucky 40292
A new methodology for multivariable control is presented. T h e method utilizes easily available step responses for implementation. It is easy to understand and implement. It requires a digital computer for implementation, but the memory requirements are modest. Interaction compensation is inherent as in other predictive control methods, and t h e robustness issues can be addressed in t h e design stage. Simulation results are presented which show t h e excellent capability of t h e method. The PID controller has been a work horse in process industries for over 40 years. The classical methods of Ziegler-Nichols and Cohen-Coon are industry standards. PID-type controllers are routinely used in SISO (singleinput single-output) applications with good results, but success with this type of controller for multivariable systems has been limited. It is now recognized that the limitations of the PID-type controller can be traced to its characteristics. The PID controller came into existence on the basis of hardware 0888-5885/87/2626-0347$01.50/0
realizability; bellows, orifices in pneumatics and potentiometers, operational amplifiers, resistors, and capacitors in the electronic world could be used to construct it. However, with the evolution of digital control computers, much better designs can be produced without any consideration for hardward realizability. This, in part, has spurred research and development to evolve better control strategies for process systems. The demands of the times is the second source of incentives. Processes today are much more complex, re0 1987 American Chemical Society
