Ind. Eng. Chem. Fundam. 1981, 20, 186-195
180
Preliminary investigations (Horvlth and Pedersen, 1977) suggest that the method developed here can be used for the quantitative treatment of axial dispersion of tracers subject to chemical transformation in automated analyzers with segmented flow. a = tracer concentration in sequential sampling A = transfer area in liquid slugs; cf. eq l a c = concentration dt = tube diameter df = film thickness h = mass transfer coefficient i, j, k , m, n = segment number 1,1, = liquid and air segment lengths m = moment p = sample number in sequential sampling p = normalized first moment s = number of wash segments between sample inputs
S = cross sectional area of annular liquid film St = Stanton number for mass transfer t = time u = average velocity of bulk liquid slug V = volume
= dimensionless bulk concentration
y = dimensionless film concentration
Greek Symbols CY = l/dr B = lldt y = surface tension c = 1/(1 + la) 7
Superscripts * = convoluted tracer profile 0 = tracer profile for sequential samples Literature Cited
Nomenclature
1:
j = order of the moment
Begg, R. D. Anal. Chem. 1972, 44, 631. Brethetton, F. P. J . FkrMMech. 1961, 70, 166. Carslaw, H. S.; Jaeger, J. C. “Operatknal Methods In Applied MWmaW, 2nd ed.; Oxford Unlvedty: Cambrldgs, 1948; pp 299-303. Concus, P. J. J. phys. Chem. 1970, 74, 1818. Duda, J. L.; Vrentas, J. S. J. F M M e c h . 1971a, 45,247. Duda, J. L.; Vrentas, J. S. J. FbidMech. 197lb. 45,261. Falrbrother, F.: Stubbs, A. E. J . Chem. Soc. 19821, 1 , 527. Gross, J. F.; Aroesty, J. -1972, 8 , 225. HorvHth, Cs; Pedecsen, H. In Advances in Automated Analysis, Technicon International Congress 1976”, Vd. 1; Mediad: Tarrytown, NY, 1977: pp 86-95. HorvHth. Cs;Sardl, A.: Solomon, B. A. physkl. (2”.phys. 1978a, 4, 125. HorvHth, Cs; Solomon, 8. A.; Ewsser, J-M. Ind. Eng. Chem. Fundam. 197Sb, 12, 431. Ldn, L. P.; Narayan, S.; Dellenbach, R.; HOrvBth, Cs. CUn. Chem., 1976, 22, 1017. L&, L. P.; Sansur, M.;Snyder, L. R.; ~orvHth,Cs. Clin. Chem., 1977, 23, 1556. Ray, W. H., Jr. J . Mecromd. Scl. Rev. 1972, C8, 1. Skeggs, L. T., Jr. Am. J. CNn. PaW. 1957, 28, 311. Snyder, L. R. J. chrometq. 1976, 125, 287. Snyder, L. R.; Adler, H. J. Anal. Chem. 1976a, 48, 1017. Snyder, L. R.; Adler, H. J. Anal. Chem. 1976b, 48,1922. Taylor, G. I. J. FbidMech. 1961, 70, 161. Theirs, R. E.: Reed, A. H.; Delander, K. C h . Chem. 1971, 17, 42. Vrentas, J. S.; Duda, J. L.; Lehmkuhl, G. D. Id.Eng. Chem. Fundam. 1978, 17, 39. Wallis, Q. B. “OneDlmenslonalTwo-Phase Flow”; McGrawHIII: New York, 1969; pp 282-314.
Receiued for review December 11, 1978 Resubmitted June 23, 1980 Accepted May 4,1981
= ut11
a2 = variance 1.1 = viscosity Subscripts b = bulk f = film
This work was supported by Grants No. GM 20993 and CA 28037 from the National Institute of Health, US. Public Health Service,
DHEW.
Mass and Momentum Transfer to Newtonian and Non-Newtonian Fluids in Fixed and Fluidized Beds S. Kumar and S. N. Upadhyay’ Lbpatiment of Chemical Engineering d Technobgy, InstlMe of Technology, Banaras Hindu Unkerslty, Varanasi22 7005, Mia
Mass transfer and pressure drop measurements during flow of Newtonian and nonHewtonian fluids through fixed and fluidized beds of uniformly sized cylindrical pellets and spheres have been made. Demlneralized water and 1.O % aqueous carboxymethylcellulose (CMC)solution have been used as the fluids. Measurements have covered a particle Reynolds number range from 0.0387 to 6000 and Schmidt numbers from 816 to 71 871. It has been shown that with appropriate choice of 8 viscosity, the results for both Newtonian and non-Newtonian fluids can be expressed by corretatlons based on the capillary tube bundle model.
Introduction
Heat, mass, and momentum transfer characteristics of Newtonian fluids flowing through fixed and fluidized particulate systems have been extensively studied and the
* To whom corrmpondence &odd be addressed at the Department of Energy Engineering, College of Engineering, University of Illinois at Chicago Circle, Box 4348,Chicago,IL 60680. 01964313/81/1020-0186$01.25/0
available information is adequately reviewed by many workers (Barker, 1965; Kunii and Levenspiel, 1969; Leva, 1959; Pandey et al., 1978; Upadhyay and Tripathi, 1975a). The study of flow of non+htc”an fluids through P a ticulate systems has gained momentum recently due to its applications in many areas (Savins, 1969). For example, a knowledge of non-Newtonian flow mechanism through porous media is needed in diversified fields such as ceramic engineering, filtration of polymer solutions and slurries, 0 1981 Amerlcan Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 187
flow through ion-exchange beds, ground water hydrology, petroleum reservoir engineering and secondary oil recovery, polymer proceasing, etc. The production of microbiological mass in fermenters and activated sludge units and extraction of metals from leached pulp by ion exchange without prior filtration are examples where the fluid is usually non-Newtonian in nature and the particles are in the fluidized state. In spite of these applications the flow of non-Newtonian fluids through fluidized beds is much less studied (Brea et al., 1976; Mishra et al., 1975; Singh et al., 1976; Yu et al., 1968). From the review of the available published literature it is clear that earlier packed bed pressure drop studies are limited to small Reynolds numbers and mostly to particles of spherical shape. Fluidized bed studies are limited only to spherical particles and no reported information is available on heat and mass transfer to non-Newtonian fluids flowing through fixed and fluidized beds. The aim of the present work was to study the influence of non-Newtonian properties upon mass transfer and pressure drop during flow through fEed and fluidized beds of cylindrical and spherical particles. The measurements have been made over a wide range of flow conditions using 1.0% aqueous CMC solution and demineralized water. Theoretical Background Momentum Transfer. The capillary tube bundle approach together with the mean hydraulic diameter concept or Darcy's permeability concept have been successfully extended to non-Newtonian fluids (Brea et al., 1976; Mishra et al., 1975; Savins, 1969). For a purely viscous fluid obeying the power-law equation
For non-Newtonian fluids the laminar and turbulent contributions have been assumed additive and the Ergun equation (Ergun, 1952) fm
+
=150 1.75 NR'em
has been frequently used to describe the friction factorReynolds number relation in the entire range @rea et al., 1976; Mishra et al., 1975). Equations similar to eq 11, but with different numerical coefficients, have also been used. The laminar and turbulent contribution Coefficients in these have ranged from 150 to 180 and 1.7 to 1.75, respectively (Brea et al., 1976; Hanna et al., 1977). Quation 11, however, is the most favored relation. For upward flow of a fluid through a particulate bed, the particles become fluidized when the drag force by the upward moving fluid is sufficient to support the weight per unit area of particles in the bed. Thus at minimum fluidization the force balance gives
AP = (1 - d ( P , - P f W A
(12)
and
= K(+)"
(1) flowing through a packed bed of spheres, the equivalent , capillary model shows that the average shear stress T ~ and average shear rate can be related as (Mishra et al., 1975) 7
For n' = 1,eq 9 reduces to the usual Newtonian modified Reynolds number
where 2tDp De = 3(1 - t)
-)
K' = K( 3 n f + 1 4n
(3) n'
(4)
and n'= n (5) For Newtonian fluids, where n = 1 and K' = p , eq 2 reduces to
From eq 12 and 13
From eq 11 and 12, for fmed bed of c = td, an equation for predicting minimum fluidization velocity can be developed. This equation in its final form is fmfNR,em2/(2-n') = 150 NR,edn'/(2-n') + 1.75 NR,d2/(2-n') (15) This equation is similar to one obtained by Brea et al. (1976). Mishra et al. (1975) defined a frictional velocity as
and a frictional Reynolds number as Dp"'(Us*)2-"'pt
Comparing eq 2 and 6, an effective viscosity, peM, for non-Newtonian fluids can be defined as
From eq 16 and 17 they obtained
(7) The modified friction factor, f,, and modified Reynolds number, NRPem, can be written as
where NR'emf
and
=
Dpn'(Us)'-"'pf K'(l - t)[12(1 - t ) / ~ ~ ] " ' - '
(19)
Using eq 11 and 18 they obtained an equation for predicting the m i n i u m fluidization velocity. Their equation
188 Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981
t.'f r
;r - -
19
14
20
Figure 1. Experimental setup: 1, demineralized water line; 2, water reservoir; 3,centrifugal pump; 4,constant-temperature reservoir; 6,cooling coil; 6,temperature controller; 7, thermometer; 8, two-stage pump; 9, constant-level overhead tank; 10, surge tank; 11, rotameters; 12, constant-temperature bath; 13,monometer; 14,differential manomet.@r;15, calming section; 16, test column; 17,inert bed; 18, active bed; 19, weighing vessel; 20, dial balance; 21, fluid collecting tank.
differs from eq 15 only in the rearrangement of dimensionless groups and numerical coefficients. Their coefficient of the last term, 1.19, is erroneous; it should be 7.143. The corrected form of their equation is NR,emt/(2-n') + 85.716NR,&"/(2-"') - 7.143N*,,2/(2-"') 0 (20) For beds of nonapherid particles, the particle diameter Dp in all the above equations is replaced by D'&, where Db is the volume-based particle diameter and 4, is the shape factor, accounting for the departure from spherical shape. Mass Transfer. The material balance equation on the bulk stream for the mass transfer in beds is written as N A = V(C2- C,) (21) and the dissolution rate equation as N A = AkcACh where
PERSPEX PIPE
I 1
(22)
The relevant dimensionless groups are
and
Another dimensionless group which accounts for nonNewtonian behavior is (Pigford, 1955)
These dimensionless groups along with Reynolds number, NRie can be related in the conventional way as N S h = f(6, NR'e, NSc) (28) and Jd = f(6, NR'e) (29)
40
RESSURE TAP--__--
t
PERFORATED PLATE
45
--CALMING SECTION FLUID I N 0
'-81 Ill
Pp4
iE;1
-
"b" FLUID IN
Figure 2. Test columns: all dimensions are in centimeters,
Experimental Program Experimental Setup and Procedure. A schematic diagram of the experimental setup used is shown in Figure 1. A variable speed pump was used to pump the fluid from the thermostated reservoir to the test section. Flow rate was metered with a calibrated rotameter to the bottom of the test section containing the bed. After passing through the test section, the fluid was either collected in a reservoir or discharged to the drain. For making measurements at very low flow rates, the gravity-induced flow from an overhead reservoir through a constant head tank was employed. The fluid was maintained at a constant temperature by means of a cooling coil and a temperature controller and was kept adequately mixed by agitation. Mass transfer measurements with spheres were made in a test section consisting of 12.6-cm diameter Perspex pipe. The dimensions are shown in Figure 2. The weight loss of a soluble solute sphere placed amidst inert spheres of identical dimensions was measured at various flow rates. Such test spheres were placed along the axis of the bed at about four sphere diameters below the top end of the bed. The bed rested on a stainless steel wire mesh screen placed between the bottom flanges. Pressure taps in the top and bottom flanges attached to the test section were provided for measuring the pressure drop across the bed. Measurements with pellets were made in test sections consisting of 8.0 and 5.6 cm diameter glass pipes. The
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, I981 189 Table I. Characteristics of the Particles surface area, A,, cmz
volume, V,, cm3
3.030 2.017 1.735 1.354 0.8624 0.6410 0.5193
28.85 12.79 9.457 5.762 2.337 1.291 0.8472
14.57 4.297 2.735 1.300 0.3358 0.1379 0.0733
3.050 2.020 1.690 1.265
29.24 12.82 8.976 5.029
diameter, d , cm
thickness h , cm
eq diam,= D,, cm
eq diam,b
density, p s ,
D,, CM
sphericity, +s
g/cm3
3.030 2.017 1.735 1.354 0.8624 0.6410 0.5193
1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.478 2.478 2.478 2.478 2.316 2.316 2.316
3.05 2.02 1.690 1.265
1.000 1.000 1.000 1.000
1.221 1.212 1.175 1.122
Glass Pellets 0.6283 1.203 0.2374 0.8563 0.1325 0.6845
1.063 0.7683 0.6326
0.7807 0.8050 0.8542
2.375 2.375 2.375
Benzoic Acid Pellets 0.4872 1.143 0.2069 0.8278 0.0596 0.5388
0.9761 0.7337 0.4845
0.7298 0.7854 0.8085
1.338 1.251 1.251
Glass Spheres
a
1.283 0.885 0.640
0.486 0.386 0.412
4.545 2.304 1.472
1.283 0.878 0.554
0.3770 0.3415 0.2470
4.103 2.154 0.9124
D,= (A,/n)‘/*.
3.030 2.017 1.735 1.354 0.8624 0.6410 0.5193
Benzoic Acid Spheres 14.86 3.05 4.316 2.02 2.527 1.690 1.060 1.265
D’,= ( ~ V , / I I ) ” ~ .
details of the 8.0 cm diameter test section are shown in Figure 2. The dimensions and other arrangements for 5.6 cm diameter test section were identical with those used earlier (Upadhyay and Tripathi, 1975b). In each case, the weight loss of an active bed of benzoic acid pellets sandwiched between beds of inert glass pellets of identical dimensions was measured at various flow rates. For making the mass transfer runs, the test sample (weighed to the nearest 0.05 mg) was inserted and the fluid flow was set at the desired rate. Measurements of the flow rate, inlet and outlet fluid temperatures, bed height, and pressure drop across the bed were taken during the runs. Depending upon the flow rate, a run lasted for 10 to 30 min. After the runs made with aqueous CMC solution the test sample was taken out, washed with a saturated solution of benzoic acid, and was placed in a desiccator for drying to constant weight. The weight loss so obtained was used to calculate the mass transfer rate. In a separate set of blank runs, the loss in weight of the test sample during charging into and removal from the test section and during washing was determined. At least ten such measurements were made for each particle size and the mean of these was used as a correction. The correction was found to be a function of particle size. All the main test run readings for a particular particle size were corrected by subtracting the appropriate correction. Those readings where the correction was more than 20% of the total weight loss were rejected. The fluidized bed studies were made with 8.0 and 5.6-cm diameter columns using the same flow loop. The top retaining screen and the top layer of inert particles were removed to allow free expansion of the bed. Measurements of pressure drop, flow rate, bed height, and inlet and outlet temperatures were made during the runs. In a separate set of runs the pressure drop measurements with inert particles (glass beads, pellets, and spheres) were also made under fixed and fluidized-bed conditions. The measured pressure drops were corrected for the empty column pressure drop obtained under a separate set of blank runs. Packing Materials. The particles studied in this work were uniformly sized spheres and pellets of benzoic acid
and glass. The characteristics of various packing materials are given in Table I. Benzoic acid pellets and spheres were made from CP grade benzoic acid obtained from M/S Sarabhai Merck (Baroda, India). The spheres were made either by casting molten benzoic acid in spherical moulds of suitable dimensions or by coating benzoic acid on the surface of g h spheres. Coating was done by dipping the glass spheres, held in special tongs, in molten benzoic acid. The coated layer thus obtained was quite strong and smooth and was about 1.5 mm thick; the test spheres were of true spherical shape. Casting was done in brass moulds which were machined from brass rods as described elsewhere (Steinberger and Treybal, 1960; Tripathi et al., 1971). The cast spheres were lapped to true spherical shape. Diameter measurements were made on all the test spheres at a minimum of six radial positions and the arithmetic mean of these was used for evaluating the surface area and volume. Benzoic acid pellets of uniform sizes were obtained by pelletizing powdered benzoic acid in a single-punch pelleting machine. All the benzoic acid pellets and spheres were washed with water to remove surface dust and were dried in a desiccator before being used. Glass beads, pellets, and spheres of appropriate dimensions were obtained from M/S Banaras Bead Manufacturing Co. (Varanasi, India). The diameter and thickness measurements were made on 50 randomly chosen samples for each particle size. Densities of the pellets and beads were determined by liquid displacement while the densities of spheres were calculated from the mean volume and mass. The liquid used was distilled water for glass beads and pellets and a saturated aqueous solution of benzoic acid for benzoic acid pellets. Fluids. The fluids used in all the fixed and fluidized bed experiments were demineralized water and 1% aqueous CMC solution. The latter was prepared from Celpro Grade CMC obtained from M/SCellulose Products Ltd. (Ahemadabad, India). The aqueous solution was prepared by dissolving a known weight of CMC in a known quantity of water. Special care was taken to avoid the
190
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 1
I
I
look
Dc l a 6 c m
1
0
N-o!
q-
10
Y
m-
*
0.
a
1
0.1 0.1
FLOW LINE
1
IOL 1 .O% A Q .
CMC'S O L U T I O N
-
FLOW LINE
$9
10 0.1 1
-50.0
4
5
10
100
f
100
1000 10
[I 2Us( 1-b/€z#8 Db
]
1000
(6' )
Figure 4. Shear stress-shear rate data for flow of demineralized water through beds of pellets. SYMBOL D i , c m SHAP
1 1 , I
o.;;g;,l;,]
0.6326 P
P
a
. (6)
0 0.1
0.01
0.1
1
10
U,(cm/s)
Figure 3. Typical AP-U, plots for fixed and fluidized beds: P, pellets, S, spheres.
formation of "cat's eyes" during solution preparation. Once prepared, the CMC solution was reused till ita benzoic acid concentration was about 60% of the saturation solubility. All the measurements with aqueous CMC solution were made at a carefully controlled temperature of 25 f 0.2 "C. No such control of temperature was employed in case of runs made with water. The rheological constants of aqueous CMC solution were obtained using a capillary tube viscometer with a diameter of 0.17 cm and an L I D ratio of 400. The wall shear stresses, D A P / 4 L , calculated from the measured and corrected pressure differences across the capillary tube, were plotted against wall shear rates, (8U,/D), on logarithmic coordinates. The rheological parameters were determined using the power law equation (DAP/4L) = K'(8U,/D)n'
(30)
The value of K'was found to be 0.4493 (g/cm s2-"'); that of n 'was 0.8538. These were used to calculate the effective viscosity of the aqueous CMC solution for fixed and fluidized beds. No detectable difference was observed in the values of K' and n ' determined for different batches of CMC solution; no changes were observed due to use in the mass transfer and pressure drop runs. Viscosity of demineralized water was taken to be that of pure water according to Perry (1963). The solubility of benzoic acid in water was taken from Kumar et al. (1977; 1978);values for aqueous CMC solution at 25 "C were determined experimentally (Kumar et al., 1978). The diffusion coefficients of benzoic acid in water were those used earlier (Kumar et al., 1977). Diffusion coefficients in aqueous CMC solution at 25 "C were determined experimentally using the laminar flow dissolution technique (Kumar and Upadhyay, 1980). Results and Discussion Pressure Drop Data. The pressure drop investigations were carried out with six sizes of cylindrical pellets and ten sizes of spheres using test columns with diameters of
5.6 and 12.6 cm, and using demineralized water and 1% aqueous CMC solution as test fluids. The fluidized state could be achieved only with particles of sizes smaller than 1 cm. Pressure dropsuperficial flow velocity plots for 1% aqueous CMC solution flowing through beds of pellets ( D p = 0.6326 cm) and spheres ( D p= 0.5193 cm) contained in a 12.6-cm diameter column are shown as typical examples in Figure 3. Similar plots were also obtained for other systems. These plots are similar for both fluids. It is observed that for the beds of spheres the transition from fixed to fluidized state is gradual and smooth. The pressure drop increases continuously in the fixed bed region and remains constant in the fluidized state. The beds of pellets behave similarly in the fixed state. In the fluidized state, however, these beds behave differently and at the transition point the pressure drop first increases and then decreases before becoming constant. Many earlier workers have observed similar behavior (Bradshaw, 1961; Bradshaw and Bennett, 1961; Kunii and Levenspiel, 1969; Leva, 1959). The maximum in the AP-Us plot is attributed to reorientation of the particles before the bed fluidizes completely. For a bed of nonspherical particles this reorientation causes an increase in the bed height followed by a decrease, as well as a change in the flow cross section. Hence a hump in the plot is obtained. For spherical particles, the reorientation causes little or no change in the flow cross section and bed height; hence a smooth transition is observed from the fixed to the fluidized state. In order to verify the validity of shear stress and shear rate relationships discussed earlier, shear stressahear rate plots were prepared for all the systems. Typical examples of such plots for water and 1% aqueous CMC solutions flowing through fixed and fluidized beds are shown in Figures 4 and 5. The solid lines in these figures are the capillary-tube flow lines for the corresponding fluids. The effect of the state of bed is clearly evident. For both fluids, the fixed and fluidized states are marked by an abrupt change at the onset of fluidization. These plots also help to distinguish the flow mechanism. For example, from Figure 4a, which shows the shear stress-shear rate plot for beds of pellets through which water was flowing, it is clear that at low flow rates the shear stressahear rate plot for packed beds follows the line for capillary-tube flow and the effect of particle size is absent. As the flow rate is increased the fixed-bed data starts deviating from the capillary-tube flow line and the effect of particle size comes into the picture. The data for different particle sizes follow different lines. Although less pronounced, this effect
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981
and fluidized bed data do not follow the same trend at high velocities. Similar behavior has been reported by Mishra et al. (1975). The plot of friction factor vs. the modified Reynolds number for all the spherical particles studied is shown in Figure 6. Similar plots for pellets are shown in Figures 7 and 8. The Ergun and BlakeKozeny equations are also shown in all these figures as continuous and dotted lines. The average deviations of the data for pellets and spheres from the Ergun equation are *15.0% and *19.0%, resp. Use of equivalent particle diameter based on the surface area of the pellets in the friction factor and Reynolds number was unsuccessful in correlating the data with the Ergun equation. Such a particle diameter is often used in correlating the particle-fluid heat and mass transfer data of fixed and fluidized beds. This is probably due to the fact that, unlike heat and mass transfer, the friction factor depends upon the form drag; hence a volume-based diameter is more appropriate. Bed Expansion Data. In the fluidized state the void fraction, e, varies with superficial velocity, U,.Figure 9 shows a few typical logarithmic plots of void fraction vs. flow velocity. For both water and 1% aqueous CMC solution, these plots seem at obey a relation of the form u, = em (31) A similar relation for Newtonian fluids was proposed by Richardson and Zaki (1954) where m is a function of Reynolds number and (Dp/D,) ratio. The slopes, m,for beds fluidized by 1%aqueous CMC solution are always higher than those of the corresponding plots for water fluidized beds, thus indicating an influence of non-Newtonian behavior. This can be approximately demonstrated as follows. For the laminar flow regime eq 11 gives
cm 9. cm 1D0627 12.6 0 . 7 6 8 3 12.6
SYMBOL D'
C
08624
v
05193
VV I
]
l,lIl.1ll
10
,
04845
5.6
l,l,ll~llll1 100 200
(8-1 )
Figure 6. Shear stress-shear rate data for flow of 1.0% aqueous CMC solution through fiied and fluidized beds; symbols 0 , 0 , and v are for fluidized beds.
persists in the fluidized state. The point at which the separation from the capillary tube flow line occurs is taken as the point below which the flow is laminar and above which it becomes turbulent. In this region inertial effects predominate over viscous ones and separation of data for various particle sizes is obtained. For CMC solution, for which the shear stress-shear rate plots are shown in Figure 5, most of the fEed bed data fall along the capillary-tube flow line. The few deviations are probably due to experimental errors involved in the pressure drop measurements. The close agreement between the fixed-bed data and capillary-tube flow line confirms the validity of the shear stress and shear rate expressions based on the capillary tube bundle model. Here also, data in the turbulent region show an effect of particle size and deviate from the capillary-tube flow b e . The abrupt change in the fluidized region for beds involving both fluids can be described in the following manner. As fluidization starts, the bed expands and the cross section available for the flow increases. At the same time the tortuosity factor, 25/12, obtained for fixed beds also changes due to the change in the effective bed height and the onset of continuous particle movement. Due to the increased flow cross section the average velocity in the bed decreases causing a decrease in the shear rate with increasing shear stress. Because of this decrease, the fixed
0.01
0.1
1
which reduces to (33) for Newtonian fluids. From eq 32 it is clear that for pseudo-plastic fluids (n'< 11,the slopes of the V,-t curves
10
%em
101
(0'
100
%em)
Figure 6. Friction factor-Reynolds number data for fixed and fluidized beds of spheres.
1000
10000
192 Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 40OOt,
,
I
PARTICLES PELLETS
WATER 1 om A a CMC O, cm
v
DCc m j
0
10627 07883 06326 0 9977 6 1
126 126 126 5 6
0
04 48 84 45 5 0
5 6 6 5
I
A
e
07337
g5 g6
’
1 07337P 04845.P 08624s 06410,s
1
56
56 m l 2 6 A 126 T
10 20 30 EXPERIMENTAL (fdm,(g/cm.s2)
01
1
10
NRem
100 (Or
1000
10000
NFikn)
Figure 7. Friction factor-Reynolds number data for fixed and fluidized beds of pellets. __
- -
40
Figure 10. Comparison of experimental and calculated shear stresses at minimum fluidization: P, pellets; S, spheres.
-7
T--
SYMBOL D~ cm o 3050 c 2020 1690
1265
A
0 0
#
A 0
1
T 0
,
,
I
2
3
4
1.0627,P 12.6 0.7683.P 12.61 0.6326,P 12.61 0.9761.P 5.6 0.7337.P 5.6 0.4845.P 5.6 0.8824,s 12.6 0.6413 12.6 0.5193,s 12.6
I
4
5
1
6
7
EXPERIMENTAL U,,(cm/sl
Figure 11. Comparison of experimental and calculated minimum fluidizing velocities: P, pellets; S, spheres. Figure 8. Friction factor-Reynolds number data for fixed and fluidized beds of pellets (measurements taken during mass transfer runs).
JI
(BFPELLETS
(AbSPHERES 0’, 1 2 . 6 c m
P
SYMBOL 0
0
0.2
t
Dp, c m 1:0627 12.6 0.7683 12.6 06326 126
0.641 0.5193
0.2
0
B
,
8
Figure 9. Typical bed voidage-velocity data: (A) and (B),demineralized water; (C) and (D), 1.0% aqueous CMC Solution.
will be greater than those for Newtonian systems. Brea et al. (1976) also observed a similar influence of nonNewtonian properties on U,- e plots. When fluidization is initiated, the pressure drop across the bed remains constant and equals the net weight of
particles in the bed. Under, this situation a comparison of the experimental shear stress with that calculated from eq 14 provides an alternative check of the power-law based average shear stress used for particulate systems. The experimental minimum fluidization velocities were obtained from void fraction-superficial velocity plots, and the corresponding shear stresses were calculated from the measured pressure drops. A comparison of the calculated and experimental shear stress values is shown in Figure 10. It is observed that the two values are in excellent agreement. In designing fluidization equipment, the most important parameter to be considered is the minimum fluidization velocity. Equation 20 was used to calculate the minimum fluidization velocity for the various particlefluid systems. The average minimum fluidization void fraction for beds containing spheres was taken as 0.4 and for pellets as 0.42. These values are the averages of corresponding measured values and are typical void fractions for loosely packed beds. The predicted Umfis compared with measured values in Figure 11. The fractional deviations are of the same order of magnitude as those for the pressure drop from the Ergun equation. Mass Transfer Data. Fixed and fluidized bed mass transfer measurements were made with four sizes of pellets and four sizes of spheres in test columns with diameters of 5.6,8.0, and 12.6 cm. Measurements with spheres were made only in the 12.6-cm diameter column employing both coated and cast spheres. Under the operating conditions used, the fluidized state could be achieved only with the pellets in the 5.6-cm diameter column. Measured weight losses and estimated inlet and outlet concentrations were used to calculate the mass transfer coefficients. The equivalent particle diameter, D p , for spheres was the average measured diameter; for pellets it was the equivalent diameter based on surface area. The equivalent diameter
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981 193 I
I
PARTICLES: SPHERES, BEDS: FIXED, Dc: 12.6 SYMBOL
a n
0
3.050 2.020 1.680 1.285 3.030 2.017
A 0
V
N
cm
Do, cm NATURE CAST I+ 8'
"
COATED
I
'9
0.1
loc
--
. -I: -
e
o
E
.*
$ 8%
1.89,.
8
0.0 15
'
I
(0)-SPHERE 8
\
I
1
I I PARTICLES: PELLETS, BEDS: FIXED h FLUIDIZED SYMBOL Dp, cm Oc, cm
0
1,1431 1.1036 0.8278 1.1431 0.8278
A
0 V A 10 0.1
1
8.0 8.0 8.0 5.6 5.6
10
0.1 0.6308
0.03 0.1
1
10
100
1000
NRe
ve-
Figure 14. Mass transfer data for demineralized water flowing through fmed and fluidized beds: eJ,-N, plot. Solid curves, eq 34 and 35; dashed curves, eq 36. 10
1
DEMINERALIZED WATER 10 1000
-
,
11111111111
,
l ~ l ~ l l l l l l l l ~ l s l l l l l l lI
a
-
100 =
--- V0
0
8. O
-
-0
10
I
V
1.354
l ~ 1 I 1 1 l 1 l l /
,
1
vv o* 60
-
I
I11I1111111
' 8
FIXED FLUIDIZED
-
V
V 0 A
,
0.1
Ec
(E)
EMINERALIZED WATER FLUIDIZED Cm Dc, C m 1,14318.0
v 1 8 1 8 1 1 1 1 1 1 1
'I
I ~ l ~ I l I l I L
71.0% AQ. CMC SOLUTION Dc: 5 . 6 cm -FIXED FLUIDIZED $,cm 1 0 0 1.143 - v 0.82 0 0.53
l 1 1 , 1 1 l / l L
A I
5,
-
1
0
0
A
A
v
oV cm 1.1431 0.8278 0.6338
1.1036 80 0.8278 8.0 1.14315.6 0.8278 5.6
1 , 1 1 1 1 1 / 1 1 1
, !
, , , / .
,
0.1
0
0
Figure 16. Mass transfer data for 1.0% aqueous CMC solution flowing through fixed and fluidized beds: eJd-Nh' plot. Solid curves, eq 34 and 35; dashed curves, eq 36.
spectively, for all the spheres and pellet sizes studied. Experimental data for the two fluids fall on separate parallel curves. These plots do not show any effect of particle size and column diameter. The fluidized bed Sherwood numbers for both water and 1% aqueous CMC solution are independent of Reynolds number and are lese than the fixed-bed values. The slight scatter of the data points observed in both fixed and fluidized-bed regions are due to the differences in void fractions for beds of different particle sizes. Mass transfer results for water and 1%aqueous CMC solution are shown separately as plots of cJd vs. Nb in Figures 14 and 15, along with the lines representing dd = 1.1068Nb4*72 N b < 10 (34) d d 0.4598N~4~4069 N b > 10 (35) and d d = 0.765Nb4'82 + 0.365Nb-0.ses (36) These expressions were developed earlier (Dwivedi and
194
Ind. Eng. Chem. Fundam., Vol. 20, No. 3, 1981
-E
PARTICLES: SPHERES, BEDS: FIXED, D, SYMBOL D D , cm NATURE
F
a
3050 2020
A
2017
a
1732
0
.
10
12 6 'cm
CAST ', 1
F
...........
I,
-.o---
PARTICLES: P ~ L L E T S BEDS: , ~ I X E DLL F L U I D I Z ~ D D~ : 5 . 6 cm
r.. .H V
1'
....
r
V
0.5388
o_/
__-
08270
Figure 17. Mass transfer data for 1.0% aqueous CMC solution flowing through h e d and fluidized beds: J v d & - i V ~plot. , ~ Dashed curves, eq 37; solid curves, eq 38. 01
10
100
1000
NR e
Figure 16. Mass transfer data for demineralized water flowing through fixed and fluidized beds: eNsh.Nh-1/3-NR,plot. Dashed curves, eq 37; solid curves, eq 38.
Upadhyay, 1977) for correlating fixed and fluidized bed maSS transfer data. The average deviations of the new data from eq 34-36 are f 18.89%, f 9.04%, and ilO.l%, respectively. A comparison of the experimental results with the asymptotic equations of Karabelas et al. (1971) eNshNec-'/3= [(0.12(NG,Ns31/4)6 + (1.05( N g k )1/3)6] 1/6Ns-1/3 (37) for NReC 10, 123 C NsC C 42000 ~.iVsi-,IVs,-'/~ = [(1.05N~e'/~)~ + (0.525Nbo.56)6]1/6 (38) for 0.001 C NReC 1,42000 C Nh C 70600 and also for 1 < NReC 2500,123 < Nk C 70600 is shown in Figure 16 and 17. The expressions were modified earlier (Kumar et al., 1977) to include the void fraction effect. From these figures it is clear that eq 37 and 38 hold for both the fixed and the fluidized-bed data. The average deviations of the data from eq 37 and 38 are f13.5% and f9.84%, respectively. Although Pigford's 6 factor has not been included in the figures comparing the results for aqueous CMC solution with eq 34-39, its effect was negligible because it was nearly equal to unity (1.0141). It would be appropriate to emphasize here that the above findings are based only on one non-Newtonian fluid. Thus the results reported have somewhat limited applicability and do not provide a rigorous test of the effective viscosity approach. It would be useful therefore to acquire more experimental data with non-Newtonian fluids of widely varying flow parameters. Conclusions The capillary tube bundle model of a particulate bed can be successfully combined with the rheological data from a capillary tube viscometer to relate flow rate-pressure drop results for both fixed and fluidized beds. The Ergun equation can be successfully used to correlate the data of power-law fluids. The non-Newtonian mass transfer characteristics of fured and fluidized beds are similar to those of Newtonian fluids. Using an effective non-Newtonian viscosity based on the capillary tube bundle approach, the resulta for both Newtonian and power-law type, pseudoplastic, non-Newtonian fluids can be correlated by the same equations. However, more data with fluids of widely varying K and
n are needed to test such an approach rigorously. Acknowledgment The authors are grateful to the CSIR, New Delhi, India, for the financial support to one of them (S.K.) in the form of a Senior Research Fellowship. This material is based upon a thesis by Surendra Kumar for the doctoral degree at the Banaras Hindu University. Nomenclature A = mass transfer area, L2 A, = surface area of the particle, L2 C1= inlet concentration, MIL3 C2 = outlet concentration, MIL3 C, = saturation solubility, MIL3 ACh = log mean concentration difference, MIL3 d = nominal diameter, L D, D, = tube diameter, L De = equivalent diameter, L = surface area based equivalent particle diameter, L 2 t = volume based equivalent particle diameter, L D, = molecular diffusivity, L 2 / T f, = modified friction factor fd = modified friction factor at the onset of fluidization h = thickness, L Jd = mass transfer factor k, = mass transfer coefficient, L I T K = consistency index, MILP-" K' = generalized consistency index, M/LP-" L = bed length, L L d = bed length at the onset of fluidization, L m = constant n = flow behavior index n' = generalized flow behavior index N = mass transfer flux, M/L2T N G=~G r d o f number, D,3gPt(p, - pt)/F2 or D,3gPt(p, - pd/r% NRe= Reynolds number = (D USflp) NRle _. - = Reynolds number for non-Newtonian fluids, (DpUSfI M d N*bf = frictional Reynolds number defined by eq 17 Nbm = modified Reynolds number = D,U&f/r(l- 4 Nn,..- = modified Revnolds number for non-Newtonianfluids. D , u d ~ L a(1 f f- 4 N= modified Reynolds number NRem at the onset of fluidization NRI~+= modified Reynolds number NR" at the onset of fluidization N; =Schmidt number p/pfDmor p~eff/p& Nsh = Sherwood number, ,%@,/Dm AP = pressure drop, M / L Us= superficial fluid velocity, L / T V = volumetric flow rate, L 3 / T V, = volume of particle, L3 + = shear rate, 1/T
Znd. Eng. Chem. Fundam. 1081, 20, 195-204
6 = Pigford‘s non-Newtonian factor e = void fraction emf = void fraction at the onset of fluidization pi = density of fluid, MIL3 ps = density of solid, M L3 7 = shear stress, M I L T, = average shear stress, M / L P p = viscosity, MILT p , = ~ effective viscosity for non-Newtonian fluid, MILT = shape factor
4
Literature Cited Barker. J. J. Ind. Eng. Chem. 1985, 57(4), 43-51; 1965(5), 33-39. Bradshaw, R. D. R.D. Thesis, Purdue University, Lafayette, IN, 1981. B r a W W , R. D.; Bermett, C. 0. AIChEJ. 1981, 7, 48-52. B r a , F. M.; Edwards, M. F.; Wllklnson, W. L. Chem. Eng. Scl. 1978, 31, 329-338. Dwhredl, P. N.; Upaclhyay, S. N. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 157-185. Ergun, S. Chem. Eng. Frog. 1952, 48, 89-94. Henna, M. R.; Kozidtl. W.; Tlu, C. Chem. Eng. J . 1977, 13, 93-99. Karabeias, A. J.; Wagner, T. H.; Hanrany, T. J. Chem. €ng. Scl. 1971, 26, 1581-1589. Kumar, S. Ph.D. Thesls In Chemlcal Engineering, Banaras Hindu Unhrerslty, Indla, 1978. Kumar, S.; Upadhyay, S. N. Ind. Eng. Chem. Fundam. 1980, 19, 75-79.
105
Kumar, S.; Upadhyay, S. N.; Mathur, V. K. I d . Eng. Chem. process Des. Dev. 1077. 16. 1-8. -Kumar, S.; Upadhyay, S . N.; Mathur, V. K. J. Chem. Eng. Data 1978, 23, 139-141. Kunll, D.; Levensplel, 0. “FluidizationEnglneerlng”, Wlley: New York, 1989. Leva, M. “Fluldlzatlon”. McOraw-Hm: New York, 1959. Mlshra, P.: Slngh, D.; Mlshra, I. M. Chem. Eng. Scl. 1975, 30, 397-405. McCune, L. K.; Wilheim, R. H. Ind. Eng. Chem. 1949, 41, 1124-1134. Pandey, D. K.; Upadhyay. S. N.; Oupta, S. N.; Mlshra. P. J . Sci. Ind. Res. 1978, 37, 224-249. Peny, R. H. “Chemlcai Engineers’ Handbook”, 4th ed.; Mc(jraw-Hiii: New York, 1983. Pigford, R. L. Chem. Eng. Symp. Ser. No. 17, 1955, 51, 79-92. Rlchardson, J. F.; Zaki, W. N. Trans. Inst. Chem. Eng. 1954, 32, 35-53. Richardson, J. F.; Zakl, W. N. Chem. €ng. Scl. 1954, 3 , 85-73. Savins, J. G. Ind. Eng. Chem. 1989, 61, 18-47. Slngh, D.; Prasad, B.; Mlshra, P. Indian J. Techno/. 1978, 14, 591-595. Stelnberger, R. L.; Treybai, R. E. AIChE J . 1980, 6, 227-232. Tripathi, G.; Slngh, S. K.; Upadhyay, S. N. Indhn J. Techno/. 1971, 9 , 285-291, Upadhyay, S. N.; Trlpathl, Q. Indhn J. Techno/. 1972, 10, 381-388. Upadhyay, S. N.; Trlpathi, G. J . Sci. Ind. Res. 1975a, 34, 10-35. Upadhyay, S. N.; Trlpathl, G. J . Chem. Eng. Data 1975b, 20, 20-28. Wamsley, W. W.; Johanson, L. H. Chem. Eng. Prog. 1954, 50, 347-355. Yu, Y. H.; Wen, C. Y.; Belle, R. C. Can. J . Chem. Eng. 1988, 46, 149-154.
- - - . -.
Received for review February 7 , 1980 Accepted February 25,1981
A “Transformation”Method for Calculating the Research and Motor Octane Numbers of Gasoline Blends Michael ti. Rusln‘ American Petroleum Institute, 2101 L Street, Northwest, Washington, DC 20037
Harold S. Chung’ and John F. Marshall W l l Research and Devebpment Corporatbn, Central Research Divisbn Laboratory. Prlnceton, New Jersey 08540
To overcome the conceptual and practical difficulties inherent in current methods for estimating the Research and Motor Octanes of unleaded and leaded gasoline blends, a new, general (non-refinery-speclfic) procedure has been constructed. It incorporates hypotheses on how blending nonllnearitles are produced by processes occurring inside a knock test engine. Three effects are considered: interaction of major chemical classes of hydrocarbons, variability of conditions inside the engine when rating different fuels, and sulfur/lead antagonism. Measured component properties rather than “blending values” are employed in the calculations. Specifically, the method requires measured Research and Motor Octanes of the components, their concentrations, and contents of olefins, aromatics, and paraffins (saturates). Based on data derived from 564 gasolines of widely varying composition, the standard errors of prediction for the model were found to be approximately four-tenths of an octane number.
I. Introduction The understanding of the nature of physical property blending represents an important class of problems in the petroleum industry. Practical questions in actual operation are involved. One example is the description of the process of combining refinery streams of different composition, volatility, octane number, etc. to formulate gasolines meeting the unleaded, regular, and premium (or “super unleaded”) grade specifications. Another is the extension *Corresponding author; present address: Mobil Research and Development Corporation, Field Research Laboratory, P.O. Box 900, Dallas, TX 75221. This research was conducted while the author was employed by the Mobil Research and Development Corporation in Princeton, NJ.
of such a methodology, say, by adaptation to linear programming models, to the broader tasks of refinery design and optimization. Often, the properties under consideration are empirically defined and, as such, are not expected to blend linearly. Octane numbers, as measured by the standard ASTM D2699 and D2700 methods, are a case in point. The Research and Motor Octanes of a gasoline blend are not accurately estimated by the volumetric averaging of the corresponding component octanes. Nonlinear models are needed for improved prediction. In a previous paper (Ruin, 1975) we examined the mathematical structure of these models and showed that, in order to be consistent with certain “common sense” physical mixing behavior, they must all be of the form consisting of three steps: (a) transformation of component properties according to a
0196-4313~a111020-0195501.2510 0 1981 American Chemical Socletv
