Ind. Eng. Chem. Fundam. 1982,21, 352-360
352
PI, Pz
= constants in eq 22 = mean ionic activity coefficient A+", Lo = limiting (zero concentration) ionic conductance
u*
(A/ cm2)(V/ cm) (g-equiv/mL) pure solvent and solution respectively, CP t = molar extinction coefficient Literature Cited Po, h1 = viscosities of
Astarita, G. "Mass Transfer wlth Chemical Reaction", 1st ed.; Elsevier: New York, 1967; Chapters 1 and 3. Barron, C. H.; O'Hern, H. A. Chem. Eng. Sci. 1986, 2 7 , 397. Bengtsson. S.; Bjerle, I. Chem. Eng. Sci. 1975, 3 0 , 1429. Chen, T. I.; Barron, C. H. Ind. Eng. Chem. Fundam. 1972, 11, 466. Chhabria, M. C.; Sharma, M. M. Chem. M g . Sci. 1974, 2 9 , 993. Danckwerts, P. V. Trans. Faraday SOC. 1951, 4 7 , 1014. Danckwerts, P. V. "Gas-Liquid Reactions", 1st ed.; McGraw-Hill: New York, 1970; Chapters 3, 5,and 10. De Waai, K. J. A,; Okeson, J. C. Chem. f n g . Sci. 1966, 2 7 , 559. Fuller, E. C.; Christ, R. H. J . Am. Chem. SOC. 1930, 5 2 , 2743. Garner, F. H.; Lane, J. J. Trans. Inst. Chem. Eng. 1959, 3 7 , 162. Glasstone, S . "Thermodynamics for Chemists"; Von Nostrand: New York (Second East-West Reprint), 1965; Chapter 16, p 402. Kronig, R.; Brink, J. C. Appl. Sci. Res. 1950, A 2 , 142. Linek, V. Chem. Eng. Sci. 1968, 2 7 , 777. Linek, V.; Mayrhoferovi, J. Chem. Eng. Sci. 1970, 2 5 , 787. Linek, V.; Mayrhoferovg, J. Collect. Czech. Chem. Commun. 1970, 3 5 , 680. Linek, V.; Tvrdik, J. Biotechnol. Bioeng. 1971, 13, 353. Linek, V. Chem. Eng. Sci. 1971, 2 6 , 491. Linek, V. Chem. Eng. Sci. 1972, 2 7 , 627. Linek, V.; Vacek, V. Chem. Eng. Sci. 1981, 3 6 , 1747.
Miura, K.; Miura, T. Heat Transfer Jap. Res. 1974, 3 , 75. Nyvit, V.; Kastanek, F. Collect. Czech. Chem. Commun. 1975, 4 0 , 1853. Onken, V.; Schalk, W. Ger. Chem. Eng. 1978, 1 , 191. Perry, R. H.; Chilton, C. H.; Kirkpatrick. S. D. "Chemical Engineers' Handbook", 4th ed.; McGraw-Hill: New York, 1963; Chapter 3. Ratcliff. G. A.; Holdcraft, J. G. Trans. Inst. Chem. Eng. 1963, 4 7 , 315. Rea, R. C.; Sherwood, T. K. "The Properties of Gases and Liquids", 2nd ed.; McGraw-Hill: New York, 1966, Chapter 11, pp 561-563. Reith, T. Ph.D. Thesis, Technical University, Delft, 1966. Sawicki, J. E.; Barron. C. H. Chem. Eng. J . 1973, 5 , 153. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. "Mass Transfer", 1st ed.; McGraw-Hill: New York, 1975; Chapter 8, pp 371. Sivaji, K. Ph.D. Thesis, I.I.T. Bombay, 1977. Srikrishna, M. Ph.D. Thesis, I.I.T., Bombay. 1968. Srlkrishna, M.; Murthy, G. S . R. N. Indian Chem. Eng. 1971, 8 , T-1. Srivastava, R. D.; McMillan, A. F.; Harris, I. J. Can. J . Chem. Eng. 1968. 46. 181. Sussman, S.; Portnoy, I. L. Anal. Chem. 1952, 2 4 , 1652. Treybai, R . E. "Mass Transfer Operations", 2nd ed.; McGraw-Hill: New York, 1968; Chapter 3, p 51. Trushanov, N. N.; Tsirlin, A. M.; Nikitenko, A. M.; Khodov, G. Ya. J . Appl. Chem. USSR 1975, 4 8 , 300. Tsirlin, A. M.; Trushanov, V. N.; Khadov, G. Ya. J. Appl. Chem. USSR 1973, 4 6 , 2307. Wesselingh, J. A.; Van't Hoog, A. C. Trans. Inst. Chem. Eng. 1970, 4 8 , T69. Westerlerp, K. R.; Van Dierendonck, L. I.; De Kraa. J. A. Chem. Eng. Sci. 1963, 18, 157. Wilke, C. R.; Chang, P. AIChE J . 1955, 7 , 264. Yagi, S.; Inoue, H. Chem. Eng. Sci. 1962, 17. 411.
Receiued for review February 2, 1981 Accepted July 14, 1982
Semicompartmental Approach to Fluidized Bed Reactor Modeling. Application to Catalytic Reactors K. Vlswanathan Department of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi- I 100 16, India
A new comprehensive model for fluidized bed catalytic reactors is presented for first-order reactions. The bed
is divided into four phases, namely, bubble, cloud, wake, and emulsion. In accordance with the mixing patterns in the phases, the emulsion phase is divided into a number of compartments, and gas in other phases is assumed to be in plug flow. The compartment height is equated to a representative average bubble size in that compartment that would give true exchange coefficients and conversion levels as the actual bed. This semicompartmental approach, besides being more realistic, makes analytical approach possible even for varying bubble size with the height of the bed thus making computations much simpler than other models. Predictions of concentration profiles, conversion levels, and bed expansion show good agreement with published experimental observations.
Introduction An extensive review of available models for catalytic reactions in fluidized beds has been made by Horio and Wen (1977). The model of Fryer and Potter (1972) appears to be a good description in view of its capability to predict the observed concentration minima within the bed. Their model assumes concentration in all the phases to vary differentially which in turn means plug flow of gas in all the phases. On the other hand, the model of Kat0 and Wen (1969) is based on the division of the bed into many compartments, the number of compartments being the same for all the phases. The gas in each phase in each compartment was assumed to be well mixed. Since residence time distribution for gas in different phases is different, it seems implausible to consider the same number of compartments for all the phases. Though each bubble, cloud, and wake may be well mixed, the
concentration in each of them varies differentially as they rise through the bed, thereby implying plug flow of gas in bubble, cloud, and wake phases. On the other hand, some sort of circulation is created in the emulsion due to the flow of bubbles. Hence compartmentation of only emulsion seems logical. Since the axial mixing length is considered to be the same order of magnitude as the bubble size (Horio and Wen, 1977), emulsion may be compartmentalized with the height of each compartment equal to the representative average bubble size in that compartment. However, this may not truly represent mixing in the emulsion, but since it facilitates the definition of average bubble size and hence the height of a compartment, this is used. This method of analysis only would be realistic for fluidized beds of large diameters and small heights where the number of compartments may not be large enough so
0198-4313/82/1021-0352$01.25/00 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 353
Emulsion
Phase
u:,
104
Ce,
Reactant concentration, dimensionleri
bub 5
Gas velocity
5.8
3t
Solids v e l o c i t y
4
60-
U0=104
Fractional volume
Porosity
Gas exchange c o efticients
'
1
Em'
~
Fluidizing gas
ub
Emf
velocity
41
Figure 1. Representation of a fluidized bed reactor according to semicompartmental model.
as to approximate to plug flow in the emulsion. Fluidized bed coal combustors fall in this category. However, in the present study the model is developed for catalytic reactors and tested with Fryer and Potter's (1976) data on ozone decomposition on sand catalyst. An additional advantage of the present model is that the solutions are attainable analytically even for the case of varying bubble size with height, whereas the model proposed by Fryer and Potter (1972) can be solved only numerically as the coefficients in the resulting third-order differential equation for bubble concentration are very strong functions of height. Features of the Model The fluidized bed is divided into four phases as shown in Figure 1. The hydrodynamic features are essentially the same as in the Kunii and Levenspiel (1969) and Fryer and Potter (1972) models. The equations are modified incorporating the fact that solids move up in wakes and down in clouds, whereas gas moves up both in clouds and wakes. The emulsion phase is divided into a number of compartments with all bubbles of equal size in each compartment. Further, the size of each compartment is made equal to the representative average bubble size in that compartment and is estimated by the method described in the Appendix. Bubble Size. The variation of bubble size with the height of the bed is assumed to be given by Mori and Wen's (1975) correlation as db = d b m - ( d b m - dbO) exp(-ph) (1) where dbm = 0.65[A,(uo - Umf)]o'4 and dbo = 0.35[A,(uo ~ ~ ~ ) for / nperforated ~ ] ~ 'distributor ~ plates, or dbo = 0.00376(u0- umf)2for porous distributor plates. Compartment Size. It is estimated by the following equation derived in the Appendix.
I
I
5
15
10 h
20
25
crn
Figure 2. Predicted variation of 6, ub, and 6ub with the height of the bed.
ue can be obtained by solving the following equations (Viswanathan and Rao, 1980) simultaneously. ub = UO - UeEmf 0.71(gdbJ1" (3)
+
6 6 =
= 3uf/(ub - uf) U O - Uetmf
(4)
ub[l
+ ( a + 6)tmfI - U e t m f [ l + a + PI
(5)
Uetmf
= umf- a6ubtmf/(l- 6 - as)
(6)
Typical calculated values of 6, ub, and bubble flow 6ub are shown in Figure 2. The critical inlet superficial gas velocity, ucr, at which backflow of gas starts in emulsion can be obtained from eq 5 and 6 as ucr _ - [I + ( a + P ) t m f l ( 1 - 6 - a S ) / ( a t m f ) Umf
(7)
All the variables given above would vary for all the compartments. The subscript i has not been used with them just for the purpose of clarity. Bed Expansion. Since the emulsion is considered to be at minimum fluidization conditions, i.e., its porosity remains constant at emf, the bed expansion is only due to bubbles. If Lmfand Lf represent the length of the bed at minimum fluidization and the expanded bed length, respectively, one obtains Ahi = Ahm,i/(1- S i ) (8) N i=l
Ahm,i = Lmf
(9)
From eq 8 and 9 it follows that N
C(1- 6i)Ahi = Lmf
i=l
Estimation of Ahi for the last compartment, i.e., AhN is deferred to a later stage. Other Hydrodynamic Equations. Bubble rise velocity ub, cloud size P, bubble fraction 6, and emulsion gas velocity
(10)
The compartment height Ahi and the bubble fraction Si are evaluated using eq 2 and 5 up to the point where the left-hand side of eq 10 just exceeds Lmf. Estimation for the Last Compartment, A h p From eq 10, one obtains
354
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 N-1
L,f -
(1 - 6i)Ahi
The average bubble size in the last compartment is given by dbrn
where dbN-land dbNare evaluated using eq 1 applied at hN-l and hW To obtain dbN and AhN, eq 11and 12 have to be solved iteratively. Starting with an assumed value for 6 N ( = AhN can be calculated from eq 11 and dbN from eq 12. dN can then be calculated from eq 5 and checked with the assumed value. In fact, 6 N = 6N-l itself is seen to be within 2% of the exact 6 N value for the data considered. Gas Exchange Coefficients. The gas exchange coefficient from bubble to cloud is given by Davidson and Harrison (1963) as
The gas in cloud and wake are assumed to be intimately mixed (see Figure 1). There is an exchange of solids from cloud-wake to emulsion and the coefficient for solids transfer is derived by Yoshida and Kunii (1968) and given in Kunii and Levenspiel (1969) as
where
In a modified form, eq 22 becomes
For backflow of gas in emulsion, Cei on the right-hand side of eq 21-23 should be replaced by C~-l.The procedure of solution is given only for upflow. Since the bubble size is a constant within each compartment given by eq 2 or 12, all the coefficients in eq 20,21, and 23 are constant and also since Cei, is constant in each compartment, the differential-difference equations given above can be solved analytically. Equations 20 and 21 yield [bo + b1D + b2D2]Cb = Cei
(24)
cc = Cb + a$Cb However, by assuming solids to enter the wake from the cloud at a velocity us + Ub, a slightly different equation is obtained, which is 3(1 - c m f ) U m f ( l - 6) [Kcelbs = dbtmf(l- 6 - Ly6)(1 - Uf/Ub)
(14)
The gas exchange coefficient from cloud-wake to emulsion including the transfer of gas in the interstices of solids moving from wake to emulsion then becomes
The solution of eq 24 is C. Cb = bo From eq 25 and 26 Cei C, = - + (1
(25)
+ dlemlx+ d2emzX
(26)
+ uOml)dlemlx+ (1 + uOm2)d2em2x
where -bl f (b12- 4b0b2)”~
(28) 2bz It should be mentioned at this stage that dl, d2,m,, m2, and all the coefficients would vary with the compartment. Subscript i is not used with them for purposes of clarity. Substituting for Cc from eq 27 in eq 23, one obtains m1,2 =
where the first term in eq 15 representing the diffusive transfer was given by Kunii and Levenspiel (1969). Solids Fraction. Let Yb, yc, and -ye represent the ratio of volume of solids to volume of bubbles in bubble, cloud-wake, and emulsion phases respectively. Assuming Y b to be zero, one obtains Yc
= (a + P)(1 - emf)
(16) (17)
Material Balance Equations Since the cloud and wake are assumed to be intimately mixed and both travel at the same velocity, they can be lumped together in reactant gas material balance equations. The reactant gas material balance equations for bubble and cloud-wake phases then become
(27)
b0
On rearranging the above equation one obtains
c . =Cei-1 + f1d1 + f2d2 et
l+g
where Kce a2m1
f l = -(1
+ aoml)[emlx~ -
I
emlx~-l
Ind. Eng.
Chem. Fundam., Vol. 21, No. 4,
cc= Cb + uo(mldlemlx+ rn2d2emzx)
1
(32)
The constants dl and d2 can be estimated from eq 31 and 32 with three boundary conditions, Cb(Xi-l), CC(xi-J, and Cei-l. Downflow of Gas in Emulsion. The starting equations are eq 21 and 23 with Cei on the right-hand side replaced by Cei-l. Following the same procedure as outlined above for upflow of gas in emulsion, the final solution equations for downflow are
(34) C, = Co + ao(mldlemlx+ m2d2emzx)
355
NM “N-
Substituting for Cei from eq 29 in eq 26 and 27, Cb and C, can be expressed in terms of known Cei-l as
1982
(35)
where f l , f 2 , and g are as given by eq 30. It is to be noted that the equations for upflow and downflow do not coincide for u, equal to zero. This is so because by definition of compartmentation the concentration of emulsion gas inside a compartment equals that at its exit. For upflow it is Cei whereas for downflow it is Cei-l (see Figure 3).
UPFLOW
BACKFLOW
Figure 3. Representation of concentration in emulsion.
dilution of the bubbles is to the fullest extent and the rest of the bubbles having clouds through which this emulsion gas flows also are of such small size that the exchange coefficient Kbcmay be assumed to be infinite with the result that the assumed boundary condition is reasonable. An added advantage of the assumed boundary condition is that it facilitates in obtaining an approximate analytical solution. This dilution of bubbles was recently considered by Viswanathan and Rao (1980). It should be remarked, however, that this dilution is at most only a few percent (about 1% for the data considered here) of the total gas flow. Mathematically, this boundary condition is given by
Boundary Conditions and Calculational Procedure (a) Upflow of Gas in Emulsion. The boundary conditions for any compartment are the outlet concentrations of the previous compartment. Starting boundary conditions are: at h = 0, i = 1
It should be noted that here the value of Ca is not known. One more boundary condition is required to solve the equations. It can only be said that the emulsion gas at the top of the bed comes out of bubbles and cloud-wakes. Fryer and Potter (1972) assumed the gas supply to emulsion to be from clouds alone so that
Cb(0) = Cc(0) = CeO = 1
CeN = Cc(1)
(36)
The calculation procedure involves the following steps. Step 0. Initialize i = 0. Step 1. i = i + 1. Step 2. Evaluate dbifrom eq 2 or 12 and all the required coefficients at dbi. Step 3. Evaluate dl and d2 simultaneously from eq 31 and 32 (eq 34 and 35 for backflow of gas in emulsion). Step 4. Establish the profiles cb(x) and C,(x). Step 5. Evaluate Cb(Xi), C,(xi),and Cei. Step 6. Use Cb(Xi), C,(xi),and Cei as boundary conditions for the next compartment and go to step 1. (b) Backflow of Gas in Emulsion. It is assumed that the downflowing emulsion gas mixes with the entering fresh gas and rises in the form of bubbles and cloud-wakes. This appears reasonable in view of the following reasons. Since a distribution of bubble sizes at any time at the distributor may be expected, it may be postulated that the down-flowing emulsion gas reverses its direction to flow up through only such bubbles which have superficial flow less than up Hence bubbles from size 0 to some value db* (
