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Satterfield, C. N., AIChEJ., 21, 209 (1975). Satterfield, C.N., Ma, Y. H.,Sherwood, T. K., Int. Chem. Eng. Symp. Ser., 28, 22 (!968). Satterfieid, C. N., Ozel. F., AIChEJ., 19, 1259 (1973). Satterfield,C.N.. Pelossof, A. A., Sherwood, T. K., AIChEJ., 15, 226 (1969). Schwartz, J. G., Weger, E., Dudukovic. M. P.. AIChEJ., 22, 894 (1976). Sedriks, W., Kenney, C. N., Adv. Chem. Ser., No. 109, 251 (1972). Sedriks, W., Kenney, C. N., Chem. Eng. Sci., 28, 559 (1973). Sherwood, T. K., Farkas, E. J., Chem. Eng. Sci., 21, 573 (1966). Sherwood, T. K.. Pigford, R. L., Wilke, C. R., “Mass Transfer”, p 242, McGraw-Hill, New York, N.Y., 1975.
Van Krevelen, D. W., Krekels, J. T. C., Red. Trav. Chim., 67, 512 (1948). Wilson, E. J., Geankopiis, C. J., Ind. Eng. Chem. Fundam., 5, 9 (1966).
Recieued f o r review August 18, 1977 Accepted February 10,1978 We acknowledge the Fellowship Grant from the Government of Japan and the financial assistance of the National Science Foundation, Grant ENG 76-01153. The Girdler Chemical Company, Inc., provided the alumina for catalyst preparation and Dow Chemical Company supplied a-methylstyrene.
Analysis of Gas Flow in Fluidized Bed Reactors Dragomir B. Bukur Faculty of Technology, University of Novi Sad, 21000 Novi Sad, Yugoslavia
Gas flow distribution and the cross-sectional areas occupied by the bubble and particulate phases in a fluidized bed are examined for the models of Partridge and Rowe, Kunii and Levenspiel, and Fryer and Potter. It has been shown that for these models there exists a critical value of fluidizing gas velocity, defined as the gas velocity at which the area of one of the phases becomes equal to the total cross-sectional area of the bed. For a given set of model parameters the value of the critical fluidizing gas velocity can be predicted. Existence of such a critical velocity imposes restrictions on the applicability of these models. Some modifications of the Partridge-Rowe model are proposed in order to extend the region of validity of this model while retaining its basic features.
Introduction In recent years a number of two-phase models have been proposed for prediction of reactant conversion in fluidized bed catalytic reactors, but only a few of them take into account the bubble clouds and/or the bubble wakes (Grace, 1971). The models of Partridge and Rowe’(1966b), Kunii and Levenspiel (1968b3,and Fryer and Potter (1972) are typical representatives of the latter class of reactor models. Some investigators have pointed out certain difficulties in application of the Partridge-Rowe (1966a,b) model. Potter (1971) examined the gas flow distribution for this model and found that at sufficiently high fluidizing gas velocity, the downward flow of the dense phase gas may occur, while the original model assumes cocurrent upward flow of the two gas phases. Recently, Chavarie and Grace (1975a) found, for their experimental data, that the area of the bubble-cloud phase calculated according to model assumptions exceeds the total cross-sectional area of bed. The countercurrent backmixing models of Kunii and Levenspiel (1968a) and Fryer and Potter (1972) assume the upward flow of the bubbles, and the bubble clouds and wakes and the downward flow of the dense phase gas. These models have been used in the analysis of various kinetic and diffusional processes in fluidized beds. No restrictions on their use have been reported yet. In this paper the gas flow distribution and phase areas, for the above mentioned models, are examined in some detail. It is found that there exist two critical values of the fluidizing gas velocity for each of these models. One of them is defined as the gas velocity where there is no gas movement in the dense phase, and the other one as the gas velocity where the area of one of the phases becomes equal to the total cross-sectional area of the bed. The values of these critical velocities are given in terms of model parameters, which are usually known or can be estimated from available experimental results and/or existing theoretical expressions. Existence of the critical fluidizing gas velocity, where the area occupied by one of the phases 0019-7874/78/1017-0120$01.00/0
becomes equal to the total cross-sectional area of the bed, restricts the use of these models to values of fluidizing gas velocity which are smaller than this critical value. Partridge-Rowe Model In the present analysis a simplified version of the Partridge-Rowe (1966a) model will be considered. It will be assumed that all bubbles are of constant size throughout the bed and that the density changes due to pressure drop in the bed can be neglected. The total gas flow is divided into two streams. One stream represents the flow in form of gas bubbles and associated clouds, and the remainder of flow is accounted for by percolation of the interstitial gas through the dense phase. This statement can be written as G = Gc
+ GI
(1)
According to Partridge and Rowe (1966a,b),the volumetric flow rate of bubbles is the same as in the simple two-phase theory of fluidization (Toomey and Johnstone, 1952), i.e., GB = ( U - U,f)A
(2)
The volumetric flow rate of bubble-cloud units can be obtained from geometric considerations. If we denote AB as the area of the bed occupied by bubbles, then the cross-sectional area occupied by bubbles and clouds would be
(3) where f c is the ratio of the cloud volume (or cloud “overlap” region) to the bubble volume. Since the clouds contain particles with an assumed voidage of tmf, the area available for gas flow in the bubble-cloud phase is ACG= AB(^
+ dc)
(4)
The bubbles and the clouds are rising together with the absolute velocity U B ,hence 0 1978 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
and
UBACG= G c Combining eq 4 , 5 , and 6 one obtains
(6)
GC = G B ( + ~ cmffc)
(7)
From eq 1, 2, and 7 the superficial gas velocity of the interstitial gas in the dense phase U I ,can be written as
UI =
u - (U - u m f ) ( 1 +
cmj‘c)
(8)
From the last equation it follows that the downflow of the interstitial gas will occur ( U I < 0), if the superficial gas velocity U exceeds the critical value given by
This result was first obtained by Potter (1971), who pointed out that for U > U,,, the model of Partridge and Rowe (1966b) for prediction of reactant conversion in a fluidized bed reactor would not be valid and that the boundary conditions appropriate to the countercurrent backmixing model would be required. However, it will be shown that the above condition also implies that the area occupied by the bubble-cloud phase is greater than the total cross-sectional area of the bed, which represents an inherent physical incompatibility and restricts the use of this model to the values of superficial gas velocity which are smaller than Ucr. Partridge and Rowe (1966a) used Murray’s (1965) theory to estimate values of fc. They obtained that the ratio of the cloud to the bubble volume can be approximated by the expression 1
fc
=a-1
(10)
where a is the ratio of bubble to remote interstitial gas velocity (a > 1 when cloud formation occurs)
121
Modifications of Partridge-Rowe Model Chavarie and Grace (1975a,b) concluded that the simple two-phase theory overestimates the visible bubble flow rate Gg. Instead of eq 2 they used the measured visible bubble flow rates and found that A c < A for the range of experimental conditions investigated (2.38Umf< U < 4.28Umf).They also considered another modification of the Partridge-Rowe model, by taking into account the bubble wakes and making allowance for a possible gas downflow in the dense phase. With this modification, the model becomes similar to the countercurrent back-mixing model of Fryer and Potter (1972) which will be discussed later in this paper. Let us examine some of the assumptions of the original Partridge-Rowe model and see whether it could be improved without changing its basic characteristics. From eq 2 and 5 one obtains
where U g is given by eq 12. Since in principle U can assume any value, it follows that for
u > umf + UBm
(16)
we would have AB > A , or cg > 1, where tg is the volume fraction of bubbles in the bed. This represents another mechanical incompatibility of the Partridge-Rowe model. A simple way to remove this incompatibility is to choose the value of the absolute bubble velocity which would be compatible with the condition 0 < EB < 1. A commonly accepted relationship proposed by Davidson and Harrison (1963) uB=
u- umf+
uBw
(17)
would be sufficient for this purpose. With this expression for U B ,eq 13 becomes (18) For this case, it can be easily shown that A c
> A , for U >
(U c r ) ~ where , ( Ucr)2is given by
and U Bcan be calculated from the experimental correlation of Rowe and Partridge (1965)
ugw= K
a
(12)
Substituting eq 10 into eq 3, using eq 5 with Ug = Ugm,and eq 11 one obtains
From eq 13 it follows that A c > A for U > (Ucr)l,where (Ucr)l is given by eq 9, which can also be written as (Ucr)l= umf
) + ,a - 1
(1
(14)
Hence, the model is valid only for U < Ucr.The value of U,, decreases with the value of a , which in turn depends on an equivalent bubble diameter. Thus, in the case of small bubbles it is more likely to have U > Uc,and A c > A . In a recent paper Chavarie and Grace (1975a) published results on ozone decomposition in a two-dimensional fluidized bed reactor. In the analysis of their experimental data they used the PartridgeRowe (1966b) model which allows for bubble size variations within the bed and found that A c values computed from eq 13 were larger than the total bed cross-sectional area, particularly a t lower bed levels (i.e., where the bubbles are small).
The critical value of fluidizing gas velocity computed from eq 19 is higher than the value obtained from eq 14. However, eq 14 still represents the criterion for reversal of flow in the dense phase. Hence, for ( Ucr)l < U < ( Ucr)2there would be the downward flow of the interstitial gas in the dense phase, and the boundary conditions appropriate to the countercurrent backmixing model (Fryer and Potter, 1972) would be required. Further improvement of the Partridge-Rowe model can be achieved using a modification proposed by Bukur et al. (1977). If one assumes that the gas flow rate through the dense phase remains the same as at incipiency of fluidization, Le., GI = UmfA,one obtains from eq 1 and 7
Then, from eq 5,13,17,and 20, the area of the bubble-cloud phase is
It follows that the area occupied by the bubble-cloud phase will be greater than the total cross-sectional area of the bed
122
for U
Ind. Eng. Chem. Fundam., Vol. 17,
No. 2,
1978
> ( Ucr)s,where ( U c r )is~given by
Table I. Influence of a on the Values of Critical Fluidizing Gas Velocities (Ucr)llumf
(UCA2IUnlf
(UcrMUmf
3
3 5
5 13
31
5 7
9 13
41 85
91 183
01
The value of the critical superficial gas velocity computed from eq 22 is much higher than the values which are obtained from eq 14 and 19, as can be seen from Table I. (The values in Table I were obtained assuming tmf = 0.50.) I t should be noted that in this case the downward flow of the dense phase gas is not possible. Hence, this modification of the Partridge-Rowe model considerably extends the region of applicability of the original model while retaining its basic features. Countercurrent Backmixing Model Several investigators (Stephens et al., 1967; Van Deemter, 1967; Kunii and Levenspiel, 1968a) have independently developed backmixing models. The basic characteristic of these models is the downward flow of the dense phase gas. In recent years this model has been refined by Latham and Potter (1970) and Fryer and Potter (1972). According to these authors, the bubbles move upward carrying clouds and wakes. The area occupied by this ascending material is
A1 = AB
+ Ac + A w = A t ~ ( +1 f,+ fw)
(23)
where f wis the ratio of wake to bubble volume. The area occupied by the dense phase is
A2 = A
- Ai
(24)
Since there is no net flow of solids through the bed, the upward flow of solids in the bubble wakes must be balanced by the downward flow of solids in the clouds and in the dense phase. If, in addition, one assumes that the gas velocity relative to the descending particles is constant and equal to the gas velocity relative to the particles a t incipient fluidization, Latham and Potter (1970)showed that the downward flow of the dense phase gas will occur for U > U,,, where U,, is given by -u c r - [I - t ~ ( + 1 f w ) l [ 1 + cmf(fw + fell (25) umf
tmffw
and the superficial gas velocity of the bubbles is given by
where tg = UGB/UB, and ug is defined by eq 17. Hence, eq 26 is quadratic in UGB.Usually, one introduces a simplifying assumption that the volume of the cloud region can be neglected (Kunii and Levenspiel, 1968a; Fryer and Potter, 1972),i.e., fc = 0, in which case the expression for UGB becomes (27) Hence
Substitution of eq 28 into eq 23 gives
From eq 29 it follows that the area occupied by bubbles and wakes will be larger than the total cross-sectional area of the bed for U > U,,*, where U,,* is given by
uc,*= UB- - u m f fw
(30)
2
13
If it would happen that U,,* < U,,, where U,, is given by eq 25, with fc = 0, then there would be no downflow of the dense phase gas. However, in the cases where fc sz 0, the value of Ucr* will be greater than U,,, and the countercurrent backmixing model would be valid for
u,, < u s u,,*
(31)
The obtained result imposes a restriction on the use of the countercurrent backmixing model, which was believed to be particularly useful at high gas velocities. Some of the results reported by Fryer and Potter (1972) were obtained for values U > U,,*. From the data in their Figure 5a, Uc,* = 44.8 cm/s, and they used the values of superficial gas velocity up to U = 50 cm/s. In case of small bubbles, one cannot neglect the volume of clouds, and the value of U,,* would be smaller than the one calculated from eq 30, but the exact expression for Uc,* cannot be obtained in a closed form. Conclusions The analysis of gas flow distribution and phase areas shows that the Partridge-Rowe (1966a,b) model is valid only for the values of fluidizing gas velocity which are smaller than the corresponding critical values. One of the critical values refers to the situation where the downflow of the dense phase gas occurs (eq 9), and the other one is defined as the gas velocity where the area of bubble-cloud phase becomes equal to the total cross-sectional area of the bed (eq 9,19 and 22). Some modifications of the Partridge-Rowe model are proposed in order to extend the region of applicability of this model. The modification which uses Davidson and Harrison’s (1963) expression for the absolute bubble velocity (eq 17), and where it is assumed that the gas flow rate through the dense phase is the same as at incipiency of fluidization was found to be particularly useful. For the countercurrent backmixing models of Kunii and Levenspiel (1968a) and Fryer and Potter (1972) it was also found that there exists a critical velocity, such that for this value of fluidizing gas velocity the area of the ascending phase (bubbles and bubble wakes) is equal to the total area of the bed. The value of this critical velocity would be even smaller for a model which takes into account the bubble clouds as well. Nomenclature A = cross-sectional area of bed AB = cross-sectional area occupied by bubbles A c = cross-sectional area occupied by bubbles and clouds ACG = area available for gas flow in the bubble-cloud phase A w = cross-sectional area occupied by bubble wakes AI = cross-sectional area occupied by ascending material A2 = cross-sectional area occupied by the dense phase dg = bubble diameter f c = ratio of the cloud volume to the bubble volume f w = ratio of the wake volume to the bubble volume G = volumetric flow rate of gas G B = volumetric flow rate of bubbles G c = volumetric flow rate of bubbles and clouds G I = volumetric flow rate of the interstitial gas
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
K = a constant in eq 12 U = superficial fluidizing gas velocity U B = absolute bubble rise velocity UB- = rising velocity of an isolated bubble U,, = superficial critical fluidizing velocity UGB = superficial bubble gas velocity U I = superficial interstitial gas velocity Umf = superficial minimum fluidization velocity Greek Letters a = ratio of bubble velocity to interstitial gas velocity a t minimum fluidization CB = fraction of bed occupied by bubbles emf = voidage a t incipient fluidization Literature Cited Bukur, D. B., Amundson, N. R., "Some Model Studies of Fluidized Bed Reactws", p IO, Chemical Engineering Department, University of Minnesota, 1974. Bukur. D., Caram, H. S.,Amundson, N. R., "Chemical Reactor Theory-A Review", L. Lapidus and N. R. Amundson, Ed., p 692, Prentice-Hall, Englewood Cliffs, N.J., 1977.
123
Chavarie, C., Grace, J. R., lnd. Eng. Chem. Fundam., 14, 79 (1975a). Chavarie, C., Grace, J. R., lnd. Eng. Chem. Fundam., 14, 86 (1975b). Davidson, J. F., Harrison, D., "Fluidized Particles", p 100, Cambridge University Press, 1963. Fryer, C., Potter, 0. E., lnd. Eng. Chem. Fundam., 11,338 (1972). Grace, J. R., A.l.Ch.E. Symp. Ser., 67, No. 116. 159 (1971). Kunii, D., Levenspiel, O., lnd. Eng. Chem. Fundam., 7, 446 (1968a). Kunii. D.. Levenspiel, O., lnd. Eng. Chem. Fundam., 7, 481 (1968b). Latham, R., Potter, 0. E., Chem. Eng. J., 1, 152 (1970). Murray, J. D., J. Fluid Mech., 21, 465 (1965). Partridge, 6. A., Rowe, P. N.. Trans. lnst. Chem. Eng., 44, T349, (1966a). Partridge, B. A., Rowe, P. N., Trans. lnst. Chem. Eng., 44, T335 (1966b). Potter, 0. E., "Fluidization", J. F. Davidsonand D. Harrison, Ed., p 332, Academic Press, London, 1971. Rowe, P. N., Partridge, B. A,, Trans. lnst. Chem. Eng., 43, 157 (1965). Stephens, G. K., Sinclair, R. J., Potter, 0. E., Powder Techno/., 1, 157 (1967). Toomey, R. D.,Johnstone, H. P., Chem. Eng. Prog., 48 (9, 220 (1952). Van Deemter, J. J., "Proceedings of the International Symposium on Fluidization", A. A. H. Drinkenburg, Ed., p 334, Netherlands University Press, Amsterdam, 1967.
Received for reoiew M a y 2, 1977 Accepted January 26, 1978
EXPERIMENTAL TECHNIQUES
A High-Frequency Rheometer Syamal K. Poddar, Paul Klelnsmlth, and Willlam C. Forsman* Department of Chemical and Biochemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19 174
A rheometer has been developed to study the high-frequency, linear viscoelastic behavior of liquids (solvents or
solutions) with viscosities varying over a wide range. This instrument could be of particular interest in the study of the viscoelastic effects associated with the presence of short-range interactions such as intramolecular hydrogen bonding or chain solvation in polymer solutions. The apparatus consists of a glass rod bonded to a quartz crystal of the same diameter which is driven in torsional oscillation by a pulse modulated sinusoidal emf. The wave packets are reflected by the free end of the rod, return up the rod, and are sensed by the quartz crystal. The primary data are the differences in phase shift and attenuation of the reflected waves in the rod assembly when the rod is partially immersed in a viscoelastic solution and when in air. The paper deals with the development of the equipment, its calibration (which also tests the basic assumptions underliningthe operation of the equipment and its mathematical description), and testing the equipment with distilled water, glycerol, and a glycerol-water mixture. Equations have been developed for determining the complex dynamic modulus G* of liquids.
Introduction The instrument described here is a modification of a design introduced by McSkimin (1952) that utilizes a quartz crystal vibrating in a torsional mode, which in turn excites traveling torsional wave in a glass rod that serves as the viscometer probe. Recent modifications of this apparatus have been reported by Barlow and Erginsav (1972), Barlow and Lamb (1959), and Barlow et al. (1964, 1969), who focused on improving the stability and accuracy of the equipment and extending the frequency to the megahertz range. A similar device was described by Glover et al. (1968). An alternative design described by Mason (1947) which uses vibrations excited in a totally immersed crystal is potentially operated in the same frequency range and more accurate for determining the dynamic viscosity of relatively nonviscous fluids. Unfortunately, this instrument uses water'soluble crystals and is thus unsuitable for measurements on aqueous solutions. 0019-7874/78/1017-0123$01.00/0
Analysis of the operational characteristics of a rheometer employing traveling torsional waves was given in McSkimin's (1952) paper and discussed by Barlow (1959) and Mewis (1967). The analysis given here adopts a different point of view and offers a detailed method for processing of experimental data. In addition, it was our intent to explore the suitability of this instrument for measuring high-frequency viscoelastic behavior of solutions (or simple liquids) having dynamic viscosities in the centipoise range. Finally, we should like to add that the calibration technique we describe for this equipment is new and offers proof of the adequacy of the basic physical assumptions and the mathematical description (McSkimin, 1952; Barlow, 1959; Mewis, 1967). Description A component diagram is given in Figure 1, and a photograph of the complete experimental setup with all the electrical
0 1978 American Chemical Society
