0.8
nounced in the case of the beds made up of large-sized particles and iron shot, than in small-sized beds. Ostergaard’s correlation fits the data of the small-sized particles (below 0.33 cm) and satisfactorily in low ranges of gas velocities, but failed in the case of large-sized particles (more than 0.33 cm), and a t high gas flow rates. Though the particle size in the case of iron, is 0.3 cm, Ostergaard’s correlation failed to predict satisfactorily in water and kerosine. This may be owing to the high inertia of the iron particles. The correlation developed by the authors fits the data for all the systems satisfactorily by means of two straight lines.
0.7
0-6 n
tX
d
0.5
c3
0.4
0.3 03
Nomenclature
0.4
0.5
Figure 20.
6
0.7
0.6
0.0
(GALC) (exptl) vs. t (calcd)
Comparisons of bed porosity dato predicted through Equation 2 with experimental
0 %?-In. sand-air-water
@Rockwool shot-air-water ARockwool shot-air-kerosine
and the terminal velocities of the particles, whereas Ostergaard’s method involves trial and error procedure and cannot be applied beyond a certain range of particle size. Perhaps the limitation of the above Equations 2 and 3 is that they fail to predict bed porosities for zero gas velocity condition. Bed porosities corresponding to zero gas velocities can be estimated from the equations of Richardson and Zaki (1954). I n gas-liquid fluidized beds, porosity reduction is observed with beds of small particles. As particle size is increased, the bed porosity reduction is decreased. Also, bubble coalescence is more pronounced in the case of small-sized beds. Disintegration of bubbles is more pro-
Ki? = constant in Equation 1 m,n = exponents in Equation 1 U, = superficial gas velocity (based on empty cross section of tube), cm/sec Ui‘= superficial liquid velocity (based on empty cross section of the tube), cm/sec Ut = terminal velocity of the particle, cm/sec € = bed porosity pi = viscosity of liquid, poise u = surface tension of the liquid with air, dynes/ cm literature Cited
Brown, G. G., “Unit Operations,” Wiley, New York, N. Y., 1950. Ostergaard, K., Chem. Erg. Sci., 20, 165 (1965). Ostergaard, K., Theisen, P. I., ibid., 21, 413 (1966). Richardson, J. F., Zaki, W. N., Trans. Znst. Chem. Eng., 32, 35 (1954). Stewart, P. S . B., Davidson, J. F., Chem. Eng. Sci., 19, 319 (1964). Turner, R., “Fluidization Papers Joint Symp., London,” 1963. RECEIVED for review January 15, 1970 ACCEPTED December 22, 1970
Pressure Drop in Archimedian Spiral Tubes Shaukat Ali and C. V. Seshadri’ Department of Chemical Engineering, Indian Institute of Technology, Kanpur, Zndia
S p i r a l flow geometries have the advantages of compactness and higher heat transfer rates when used in heat exchangers. These, however, are offset by the disadvantages of greater pressure drops and the difficulty of removing This paper presents a study Of pressure drop in isothermal, steady Newtonian flow in an Archimedian tube spiral. Archimedian spirals are described by the equation:
r = (p/2~)0
(1)
where r and 0 are polar coordinates and p is the pitch, a constant for a particular coil. The spiral geometry used in this study is shown in Figure 1. To whom correspondence should be addressed. 328 Ind. Eng. Chem.
Process Des. Develop., Vol. 10, No. 3, 1971
Previous Work
Srinivasan et al. (1968) have reviewed the little work that exists on spiral coils. Friction Factors. Kubair and Kuloc?r (1966) have proposed the following equations for the Fanning friction factor, fs,: fx
= 12.74 [df/LDa,]O’N~~O5, 300
10,000):
4.0
3.9
4.1
4.2
log NRc
3. Critical Reynolds numbers prediction (for coils changing and /?,ni , p, and d, constant)
with R,, Coil no.
Symbol
Coil no.
Symbol
A V
2
X
5
3
0
6
Table 1. Critical Reynolds Numbers
These lines are shown in Figure 2, with the data points for 17 coils. The dotted portions of the lines are extrapolations of Equations 10 and 11 and represent the data in the extrapolated region within the accuracy of the predictions. The maximum deviation in the laminar flow region is 9 . 6 5 and that in the turbulent flow region is 8.57. The critical Reynolds number is also a function of the geometrical parameters of a spiral. Secondary flows tend to stabilize laminar flow; however, in a spiral, forward and reverse transitions between laminar and turbulent flow take place. The initial straight-tube turbulence in the feed line is damped out in the innermost turns and as the intensity of secondary circulation decreases in the outer turns, forward transition takes place. On the other hand. when inlet to the coil is from outside, turbulence in the outermost turn gets damped out in the inner turns as the intensity of secondary circulation increases and reverse transition takes place. T o test the uniqueness of the hydrodynamics, experiments were carried out feeding the spiral from the outermost turn also. There was no change in the pressure drop measurements, indicating that the transition lengths for the forward and reverse transition were the same, whether the spiral was fed on the inside or the outside. T o locate the transition point accurately, the method of White (1929) and Adler (1934), originally suggested for curved pipes. was used. This consists of plotting the ratio f.t / vs. N & where f\ is the friction factor for straight tubes. The ratio f a c / f c is equivalent t o the ratio (AP)$ , and ( A I ' ) , are the pressure drops in spiral coil and straight tube, respectively, a t the same Reynolds number. Figure 3 shows the graphs obtained for the spirals with R,,, changing, other parameters being the same. The friction factors for straight tubes were calculated from Drew (1932) : f k
fs
= 0.00140
+ 0.125 NR:"~',N R >~ 3000
Coil no.
(NRe)crit
I
( N R e ) c r ~ tII
Obsd
Eq 4
Eq 5
E q 14
Obsd
E q 15
6607 6760 6680 6530 6530 6530 6600 6820 5250 7850
6828 6940 6985 6940 6622 6496 6793 6940 6830 7160
6521 6781 6891 6521 6521 6521 6917 6568 6766 7352
6548 6705 6770 6548 6548 6548 6865 6867 6778 7016
10,470 10,000 10,350
10,480 10.480 10,480 11,314 9,444 8,981 9,723 11,975 10,117 10,445
1
2 3 8 9 10 13 14 16
17
9.440 9,000 10,000 10,960 11,600 10,350
The unforeseen conclusion from these graphs was that there are two critical Reynolds numbers in a particular spiral, one where turbulence has set in only in the outer turns, and the other, presumably, when the inertia forces, even in the innermost turns, are sufficient to overcome the damping effect of the secondary flows. Hence, the first break in Figure 3 corresponds to laminar flow everywhere except in a small region in the outer turns and the second break corresponds t o the flow becoming turbulent everywhere. Figure 3 shows that the first critical Reynolds number, ( N R ~I varies ) ~ ~with ~ ~R,, while the second critical Reyn11 remains unaffected. Similarly, olds number, (NReIcnt only (NRe)cnt II varied with R,, and both critical Reynolds numbers were proportional to pitch and diameter. Using the following criteria, we have:
Rmax Rmm P
-
-
+
3
OJ) 039
( N ~ e ) c r i It
--
(NReIcnt 11
Rmax
or Rmn
+
2100 2100 9
(NRe)crlt I
or
(NRejcnt 11
+
2100
(13)
The following equations were obtained by data analysis:
(12) Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
331
Table I presents the predictions of Equations 4, 5 , 14, 15 and experimental data. The agreement of values from Equations 4 and 5 is quite good. Equation 15 cannot be compared with other equations because this has not been reported before.
AP = pressure drop, g c m - ’ s e - * ( L P ) $= pressure drop in straight tube, g cm-’ sec-’ ( A P ) ~=? pressure drop in spiral coil, g cm-’ sec-’ r = polar coordinate R,,, = maximum radius of coil, cm R,,, = minimum radius of coil, cm Y = average velocity of flow, cm/sec Greek letters = viscosity of fluid, g cm-’ sec-’ = density of fluid, g cm-’ 0 = angle, radians
j~
Nomenclature
p
a,b,c,d = empirical constants d , = inside tube diameter, cm average coil diameter, cm maximum coil diameter. cm minimum coil diameter, cm Fanning friction factor for straight tube as defined in Equation 6 = Fanning friction factor for spiral coil as defined in Equation 6 empirical constant length of coil, cm additional straight lengths. cm number of coil turns Reynolds number as defined in Equation 7 (NRe )crlt = critical Reynolds number (NRe)cnt I = first critical Reynolds number (NRe)cr,t 11 = second critical Reynolds number p = pitch of coil, cm fqc
literature Cited
Adler, M., 2. Angeu. Math. Mech., 14, 257 (1934). Ali, S., M. Technol. Thesis, Indian Institute of Technology, Kanpur, India, 1969. Drew, T. B., Koo, E . C., McAdams, W. H., Trans. Amer. Inst. Chem. Eng., 28, 56 (1932). Kubair, V., Kuloor, N. R., Indian J . Technol., 4, 3 (1966). Kubair, V., Varrier, C. B. S., Trans. Indian Inst Chem. Eng., 14, 93 (1961/62). Srinivasan, P. S., Nandapurkar, S. S., Holland, F . A., Chem. Eng. (London), 208, CE113 (1968). White, C. M., Proc. Roy. Soc., Ser. A , 123, 645 (1929).
RECEIVED for review April 7, 1970 ACCEPTED December 17, 1970
Segregation Models of Solid Mixtures Composed of Different Densities and Particle Sizes Tatsuo Tanaka
Department of Chemical Process Engineering, Hokkaido Uniuersity, Sapporo, Japan A new model describing the mechanism of segregation of particles differing in size and density was proposed and the calculated results based on this model agreed fairly well with experimental data. This model was further developed to give a nomograph allowing adjustment of some operating variables to minimize the overall degree of segregation when particles were subject to heap process under gravity.
Segregation of solid mixtures is sometimes most troublesome to industries dealing with particulate matter-e.g., in filling and emptying bins and hoppers. Mechanisms of segregation of particles differing in size in emptying a hopper were discussed previously (Shinohara et al., 19701, and the authors referred t o hypothetical hopper or screen models. That work stated that the small particles tend t o pass through the interstices of the large particles when they are flowing. The theoretical analysis based on this model was quite successful in the interpretation of segregation of binary mixtures discharging from hoppers. Since 332
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
Brown (1939) pointed out that segregation is a surface phenomenon, some experimental work has been presented by Matthee (1968) and Lawrence and Beddow (1969) using mixtures of different particle sizes. However, no theoretical treatment has appeared for a mixture of particles with different densities. According to experience with a centrally filling hopper, particles form a conical surface as the bin is filling and the segregation apparently occurs in the thinner surface layers moving on the heap. The heavy particles tend to remain near the central portion while the lighter parti-
