Correlation for Predicting Choking Flowrates in Vertical Pneumatic

Nomenclature

a = interfacial area per unit volume, ft’ /ftJ A , B = indication of two different liquids C, = specific heat, Btuilb, OF C,? = specific heat of liquid A , Btuilb, C, = specific heat of the liquid mixture, Btuilb, O

O

F

F

C, = specific heat of water, Btuilb, F D = impeller diameter, f t d o = diameter of drop, ft h = heat transfer coefficient, Btu/ (min-ft’--”F) k = constant dependent on the properties of both liquids L = characteristic length, f t m = mass flow rate, lb,/min N = impeller speed, rps NRe = Reynolds number, pD‘N/@ N w e = Weber number, PN’ D3/umterfaclal d = volume _ _ fraction of the dispersed phase, Vd,,/ O

Vtotai

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= = pu = u =

PA, P B

inlet temperature of liquid A , F outlet temperature of liquid A , F average temperature, F inlet temperature of water, F outlet temperature of water, ’F volume fraction of two-phase mixture volume of the heat exchange vessel, ft’ viscosity of the fluid, lb,/ft-see viscosities of liquids A and E , lb,/ft-see viscosity of the liquid mixture, lb,/ft-sec w,, = viscosity of water, lb,/ft-see p = density of the fluid, lb,/ft’ O

O

pm

densities of liquids A and B , lb,/ft’ density of the liquid mixture, lb,/ft’ density of water, lb,/ftJ surface tension, lb,/sec‘

literature Cited

Bodman, S.W., Cortez, D. H., Ind. Eng. Chem Process Des. Develop., 6 ( l ) ,127-33 (1967). Chapman, F. S.,Holland, F. A., “Liquid Mixing Processing in Stirred Tanks,” 1st ed., Reinhold, New York, 1966, p 227-8. Hinze, J., AIChE J., 1, 289-95 (1955). Misek, T., “Hydrodynamicke Chovani Michanych Kapalinovych Extraktoru,” Kandidatska Disertacni Prace, V.S.C.H.T. (Hydrodynamic Behavior of Agitated Liquid Extractors.) Diss. Inst. Chem. Technol., Prague, 1960. Olney, R. B., Carlson, G. J., Chem. Eng. Progr., 43, 47380 (1947). Perry, J. H., “Chemical Engineer’s Handbook,” 4th ed., McGraw-Hill, New York, 1963, sec 21, p 14. Rodger, W. A,, Trice, V. G., Rushton, J. H., Chem. Eng. Progr., 52, 515-20 (1956). Sterbacek, Z., Tausk, P., Int. Ser. Monogr. Chem. Eng., 5,45-8 (1965). Treybal, R. E., AIChE J., 4, 202 (1958). Vermeulen, T., Williams, G. L., Langlois, G. E., Chem. Eng. Progr., 51, 85-94 (1955). RECEIVED for review May 14, 1970 ACCEPTED November 12, 1970

Correlation for Predicting Choking Flowrates in Vertical Pneumatic Conveying 1. S. Leung, Robert J. Wiles’, and Donald J. Nicklin Department of Chemical Engineering, University of Queensland, St. Lucia, Brisbane, Australia A correlation i s derived to predict choking flowrates of solids in vertical pneumatic conveying by assuming: That choking occurs in such a narrow band of voidages that voidage can be taken as constant; and that a t this voidage the slip velocity equals the free fall velocity. This simple theory correlates to within +70% the data reported by previous workers. Choking flowrates predicted by earlier correlations are severalfold different from recently reported observations. The present method i s applicable to mixtures of particles as well as to particles of uniform size distribution.

I

n vertical pneumatic transport, the particles are often carried up the column as an apparently evenly dispersed suspension with low volumetric concentration (generally less than 5%). If, however, the air velocity is gradually reduced a t the same mass flowrate of solids, the in-line solids concentration increases. A point will be reached where the air velocity is insufficient to support the particles as a uniform suspension. The entire suspension collapses and is then transported up the column in slug flow. The point of choking is the transition point from upflow of

’ To whom correspondence should be addressed.

solids as a thin suspension (often referred to as leanphase flow) to slugging flow (sometimes referred to as dense-phase flow). This phenomenon was described in detail by Zenz and Othmer (1960). If the air velocity is further reduced below the choking point with the same mass flow of solids, slug or bubble flow will continue until the slip velocity-Le., the average air velocity minus the average solids velocity-is below that a t incipient fluidization for the same air-solids system. Fluidized flow-i.e., slug or bubble flow-is no longer possible, and the solids in the column form a packed bed. The transition from fluidized flow to a packed bed in Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971

183


vertical pneumatic conveying is entirely analogous to the transition from fluidized flow to packed-bed flow in vertical hydraulic conveying. The latter transition we have discussed elsewhere (Leung et al., 1969), and a method for predicting the transition in hydraulic conveying was described. The same method may be used for predicting this transition in pneumatic conveying. Vertical pneumatic conveying is generally carried out in the lean-phase flow regime. Slug flow (or dense-phase flow) is often to be avoided because of the erratic nature of the flow, the pressure fluctuations, high pressure drops, and pipeline vibrations. Both pneumatic transport processes are used, however, and a recent study by Sandy et al. (1970) has shown that in an application for the vertical transport of 70- to SO-mesh fused alumina particles, the power requirement for dilute-phase conveying was less than that for dense-phase conveying by a factor of 10. However, the gas requirement for the dilute-phase transport was roughly 20 times greater than that for densephase transport. The economic advantage then lies with dilute-phase conveying as long as air can be used as the conveying gas (at no charge). In lean-phase flow it is desirable to operate a t as low an air flowrate as possible from energy requirements, pipe erosion, and particle attrition considerations. It is, therefore, important when designing vertical transport systems to predict the flow conditions a t choking. Little satisfactory information, however, is available in the literature for predicting the choking point. Zenz and Othmer (1960) correlated the results on choking for 18 runs with a plot of R (=G,/upf) vs. u ’ l gdp: where

R G,

= is termed the “loading ratio” = superficial mass flow of solids a t choking, lb/ft2

se c u = superficial fluid velocity a t choking, ft/sec pi = fluid density, lb/ft’ p, = solid density, lb/ftJ d = particle diameter, ft g = acceleration owing to gravity, (ft sec-’)/sec

The correlation of Zenz is of doubtful value as it does not take into account the terminal falling velocity of the particles, ut, and the quantity u‘lgdp; is not dimensionless. The invalidity of the correlation can readily be demonstrated. For G, = 0, the transition from lean-phase t o slugging fluidization-Le., choking in a fluidized bedoccurs a t a voidage of close to unity (Ormiston, 1966; Zenz and Othmer, 1960) and the superficial gas velocity is close to ut. The fluid velocity in the correlation of Zenz may be replaced by u t which is then evaluated using Stokes’ Law and Newton’s Law Stokes’ Law Regime

Ut

=

d2 (Ps - P 1811

Newton’s Law Regime u: = 3 d ( P s - P

i k

Clearly the magnitude of the right-hand side of Equations l and 2 depends particularly on the properties of the gasisolid system. For the same value of [GIup,],[u’igdp:] may vary manyfold for different systems. When G/upi takes finite values, we have that u = AG, + Bu, where A and B are functions of the voidage, and there is no obvious reason for the Zenz correlation to apply under such conditions. That it does in certain cases is probably fortuitous. Leva (1959) compared the results of Culgan (1952), Lewis et al. (1949), and Zenz (1949) by plotting R vs. u’igdp;. His Figure 6-4 shows clearly that the correlation is of little general assistance in predicting choking flowrates. Doig and Roper (1963a) correlated the Tesults on choking and saltation by plotting log [ u / ( g d ) ‘1 vs. log R , and suggested the following empirical equations for predicting choking:

+

‘1

log [ u / (gd)’ = 0.030 u t 0.25 log for u, less than 10 ft/sec

0

O

i k

A

log [ u /(gd)’

0

‘1 =

Ut ~

-2 28

R

+ 0.25 log R

(3a)

(3b)

for ut greater than 10 ft/sec

0.1

5

184

‘

1 IO

I

I

I 50

I

I

I

I IC

Ind. Eng. Chern. Process Des. Develop., Vol. 10, No. 2, 1971

where u t , the terminal falling velocity, is expressed in ft per sec. Doig and Roper recognized the importance of including u, in their correlation. However, the empirical equation was obtained from the limited results on choking, and Figure 1 shows that Equation 3a does not correlate the recent results of Ormiston (1966). In a recent study, Ormiston investigated the transition from lean-phase flow to slugging in vertical pneumatic conveying. In particular, the solid flowrate, the air flowrate, and the voidage in the riser were measured in 1- and 1.5-in. diam clear plastic risers. The measured voidage of the lean-phase a t the onset of choking, t c , ranged from 0.931 to 0.987. No general method for predicting the choking point was proposed by Ormiston. Partly from lack of information on how the choking flowrates can be predicted, the design of a vertical pneu-

matic conveying system is based on actual operating experience and “rules of thumb” such as those given by the Engineering Equipment Users’ Association Handbook (1963) (Figure 2 ) . I n the present work the reported results on choking are analyzed and a method for correlating these results is described. T h e correlation is extended to include nonuniformly sized particles. Method

The present method for calculating flowrates a t the onset of choking is based on the following two observations of reported results:


The voidage a t the onset of choking, c L , for various systems lies within a narrow range. The slip velocity, u , ~(= V f - V,) a t the onset of choking is equal to ur, the terminal falling velocity of a single particle. Table I summarizes results on choking in the literature, with the exception that the recent data of Jones et al. (1967b) are not included. All the reported e< (with the

exception of five runs) fall within the range of 0.93 to 0.99. As choking may be approached from lean-phase flow and from dense-phase flow, i t is important to point out that c C is here defined as the voidage in the leanphase situation a t the onset of choking. Table I shows five values of choking voidage reported by Lewis et al. (1949) below 0.93. As they were not primarily interested in measuring ec, it is likely that some of their voidage measurements were made during the transition from leanphase flow to slug flow. The careful measurements of Ormiston were made in the lean-phase flow regime prior to the transition. For the present correlation, we shall assume that an average voidage a t the onset of choking of cc

= 0.97

At a voidage range of 0.93 to 0.99, the slip velocity approaches ut according to the correlation of Richardson and Zaki (1954). Our present assumption that us; = u, has also been confirmed by the experimental results of Jung (1958)-reported by Doig and Roper (1963b)-Jones

Table I. Summary of literature Values for Voidage at Choking Superficial gas velocity, ft/sec

Zenz (1949)

8.85 11.2 10.3 4.92 10.0 11.2 12.8 14.8 16.7 11.7 13.4 15.6 20.2 26.2 33.6 11.2 13.0 15.0 20.2 5.75 7.8 9.6 Lewis et al. 4.0 (1949) 4.3 5.5 5.8 6.6 6.9 5.2

Superficiol solid velocity, ft/sec

0

0 0 0 0.057 0.119 0.206 0.311 0.426 0.0081 0.030 0.058 0.118 0.193 0.291 0.043 0.112 0.192 0.416 0.0175 0.063 0.108 0.0131 0.026 0.058 0.0832 0.122 0.148 0.0131

Voidage at transition

0.945 0.987 0.983 0.978 0.9426 0.9459 0.9450 0.9452 0.9450 0.9871 0.9873 0.9872 0.9871 0.9873 0.9871 0.9826 0.9825 0.9837 0.9831 0.9797 0.9831 0.9770 0.9935 0.987 0.980 0.969 0.966 0.963 0.991

Conduit/ particle system identification“

A/ a Alb Alc A/ d Ala Ala AI a A! a AI a Aib Alb Alb Aib Alb AI b Alc Aic Alc AI c AI d A! d Aid Bie Ble Ble Ble Ble Bie B! f

“ A = 1.75 in., B = 1.25 in., C = 4.92 in., D = 1.0 in. (also 1.5) a = Rape seed. density, 68.0 lblft’ ; a v diam, 1670 microns; uniform, almost spherical; terminal veloc, 8.85 ft/sec b = Sand, density, 165 lblft”; av diam, 930 microns; uniform, sharply angular; terminal veloc, 11.2 ft/sec c = Glass beads, density, 155 lb/ft’; av diam, 588 microns; uniform, spherical; terminal veloc, 10.3 ft/see d = Salt crystals, density, 131 lb/ftJ; av diam, 167 microns; broad size distribution, granular; terminal veloc, 4.92 ftlsec e to g = Glass beads, density, 155 lbift’; uniform, spherical e = av diam, 40 microns; terminal veloc, 0.40 ftlsec f = av diam, 100 microns; terminal veloc, 1.80 ft/sec

Superficial gas velocity, ft/sec

Voidage Superficial solid velocity, ft/sec

at

transition

Conduit/ particle system identification‘

0.977 0.0259 Blf 0.961 0.0482 Blf 0.959 0.061 Bl f 0.955 0.0964 BI f 0.951 0.135 Bl f B/f 0.943 0.171 B; f 0.941 0.193 0.938 0.208 BI f 0.960 0.012 Bl g 0.908 0.0318 Big 0.870 0.0773 Big 0.891 0.122 Bl g 0.897 0.135 Big 0.900 0.173 Big > 0.96 3.3 t o 13.1 Gunther Clh all cases (1957) actual velocities, not superficial D/i 0.063 0.978 Ormiston 6.85 D/i 0.126 0.975 (1966) 7.74 D/i 0.254 0.950 11.5 Dlj 0.113 0.976 8.85 0.971 10.5 0.170 Dlj 7.57 Dik 0.034 0.981 0.128 0.966 9.40 D/k 0.256 0.948 10.4 Dik 11.35 Dik 0.442 0.931 D! 1 0.086 0.973 9.15 0.125 10.4 0.963 Di1 Dil 0.251 0.950 11.5 g = av diam, 280 microns; terminal veloc, 7.0 ft/sec h = Wheat, density, 80.5 lb/ftJ; a v diam, 2600 microns; uniform, rounded; termmal veloc, 31 ft/sec i to 1 = Sand, density, 164 t o 168 lblft’ ; fairly uniform, angular shape i = av diam, 120 microns; terminal veloc, 3.6 ftisec j = av diam, 151 microns; terminal veloc, 4.59 ftfsec k = av diam, 225 microns; terminal veloc, 5.74 ftisec 1 = av diam, 265 microns; terminal veloc, 6.23 ft 1sec Terminal velocities for i to 1 calculated by Ormiston (1966) using shape factor of 0.5.

Lewis et al. (1949)

5.6 5.7 5.7 7.0 7.0 7.5 8.0 8.0 7.6 8.2 8.3 8.9 9.3 9.7 ?

Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971

185

Awrage Particfe

Gas Density

Diameter

p

f

TerminaI Settling Velocity of SingIe Particle U

Solid Density f3

S

v =u p

t

voidage at Choking Point

vs = us/ I - € us/=v- vs Downloaded by UNIV OF CALIFORNIA SAN DIEGO on September 14, 2015 | http://pubs.acs.org Publication Date: April 1, 1971 | doi: 10.1021/i260038a008

d

%

size is somewhat speculative, since the evidence for assuming a critical voidage of about 0.97 is limited to only a few points. For vertical pneumatic conveying of nonuniformly sized particles, it is important to recognize that the size distribution of the particles in the riser will be different from the size distribution of the feed particles. The difference arises as the smaller particles (with lower u t ) tend to rise faster than the larger particles. Thus the particles in the pipe will contain a higher fraction of the larger particles compared with the feed solids, and this may partially explain the observation that mixed-size particles behave similarly to a uniform material of higher mean diameter. Assuming the voidage at the onset of choking for mixed solids is also 0.97 and the slip velocity for each size fraction is equal to ut for that fraction, the following continuity equation may be written: UsXzf

= (1 -

t J XLt

(Uicc

- Ut,)

(7)

where

Gas Superficial Velocity 11

x,/ = volume fraction of particles in the feed with terminal velocity of ut, xtr = volume fraction of particles in the riser with terminal velocity of ut,

Solid Swrficial Velocity Us

A second continuity equation may be written for the whole solids stream,

Figure 2. Terms used in pneumatic conveying

et al. (1966), Konno and Saito (1969), Ormiston (1966), and Zenz (1949). I n particular, Konno and Saito show that the approximation is valid over a wide range of solid velocities. Reports that the slip velocity may be much higher than ur are explained by the agglomeration of particles owing to electrostatic effects (Doig and Roper, 196313). I t is now possible to estimate the solid flowrate at the onset of choking from the above two assumptions. Referring to Figure 2,

v,

us = (1- €4 ZXLt

- Ut&)

(8) Provided the size distribution of the feed solids is known, the critical air rate may be calculated from the above two equations a t any given solid flowrate. A specimen calculation is given in the following section. Specimen Calculation of Critical Solid Flow. For Uniformly Sized Particles. The problem is to specify the minimum air flowrate and line size for transporting 36,000 lb per hr of a uniformly sized solid material up a vertical riser. The density of solids is 100 lb per ft3 and the terminal (Uicc

Average Solids Velocity, = U / tc - U t (4) Volumetric solid flowrate per unit cross-sectional area (or superficial solid velocity). us

= (u/tc- uO(1 -

tc)

(5)

Equation 5 is similar to an equation first proposed by Zenz (1949) and is also analogous to an equation derived by Ormiston (1966) from a material balance and the Richardson-Zaki relationship. Taking c e as 0.97, Equation 5 may be rewritten as

u = 32.3

+ 0.97

(6) The critical volumetric air rate (defined as the air rate a t the onset of choking) may be calculated from Equation 6 at a given solid flowrate provided ut is known. A generalized plot of u vs. u, for various values of u! is given in Figure 3. Solids of Mixed Sizes. For pneumatic conveying, it is generally recognized (Zenz and Othmer, 1960) that the critical air flowrate a t a given solid rate for mixed-size solids is higher than that for a uniform-sized material of the same mean diameter. Thus a mixed-size material behaves similarly to a uniform-sized one of a higher average diameter. The present method for predicting critical flowrates (now extended to nonuniformly sized particles) predicts this trend. I t should be emphasized however, that the use of Equation 6 for solids of mixed 186

US

Ut


Superficial Fluid

Velocity

U

ft/sec

Figure 3. Superficial solid velocity as a function of superficial gas velocity Plotted for c r = 0.97


falling velocity of particles is 5 ft per sec. The corresponding volumetric solid flow is 0.1 ft3 per sec, Assume that the pressure in the riser is atmospheric in which case the terminal falling velocity in the riser is equal to 5 ft per sec. Note: The terminal velocity of Figure 2 refers to the conditions in the pipe. This may be very different from atmospheric pressure. The following calculation procedure may be adopted: Assume a pipe size. Calculate the superficial solid velocity from the pipe size and the specified volumetric solid flow. The superficial air velocity is computed from Equation 6 or Figure 3. Calculate the air flowrate. Repeat for a different pipe size. The results of the calculation are given in Table 11. The air flowrate should be multiplied by a safety factor of two to allow for errors of the correlation. The pressure drop in the system may be estimated in the usual way (Chen 1963; Engineering Equipment Users' Handbook, 1963; Jones et al., 1967a), and the economic combination of pipe size and energy requirements may be computed. For Nonuniformly Sized Particles. The problem is to specify the air flow and line size if the particles in the previous example consist of three size fractions as follows: W t fraction

ut for fraction, ft/sec

0.2 0.5 0.3

1 5 20

For the solution, the following calculation procedure may be adopted: Carry out the first two steps of the previous (uniformly sized particles) calculation; substitute us into Equation 7 for each fraction of solids and solve for t h e superficial air velocity-e.g.,

0.2 U , = (1 - 0.97) [ ( ~ / 0 . 9 7-) 11 xi+ 0.5 U , = (1 - 0.97) [ ( ~ / 0 . 9 7-) 51 ~ ? t 0.3 U , = (1 - 0.97) [ ( ~ / 0 . 9 7 ) 201 (1 -

Table II. Air Requirements for Choking Conditions in Different-Sized Pipes" Internal diameter of pipe, in.

Superficial solid velocity, ft/sec

2

4.59

3 4 6 8

2.04 1.15 0.505 0.287

153 (off graph) 70 42 23 14.2

Specimen calculation, uniform-sized particles.

-

xX)

The three equations contain three unknowns, x l r , x a , and u. I t should be pointed out that when expressions for x,. and x?, are substituted into the third of the three equations, a cubic results and with mixed solids of more than three fractions the solution becomes progressively more difficult. Repeat for a different line size. Table I11 gives the calculated air flow and line size combinations. Again it is recommended that the air flowrate should be multiplied by a safety factor of two and the most economic combination may be computed in the usual way. Table I11 when compared with Table I1 confirms that the air requirements for mixed-sized particles are the greater (for the same average diameter). The results of Table I11 confirm the size distribution of the particles in the pipe is different (with higher average diameter) than that in the feed particles. Discussion

Validity of Present Correlation. The present method for predicting critical solid flowrates depends on the validity of the two assumptions, viz

(i) usl = ut

(ii) t c = 0.97 Assumption: u , ~= u t . The actual slip velocity at the onset of choking may be estimated by the RichardsonZaki (1954) equation.

3.34 3.43 4.28 4.5 4.95 O

From Figure 4.

Table 111. Air Requirements for Choking Conditions in Different-Sized Lines' Internal diameter of pipe, in.

Superficial solid velocity, ft/sec

Superficial air velocity, ft/sec

Proportions of different sizes in situ

157

0.19 0.49 0.32 0.17 0.44 0.39 0.16 0.44 0.40 0.12 0.35 0.53 0.08 0.25 0.67

2

4.59

3

2.04

79.5

4

1.15

47.5

6

0.505

28.5

8

0.287

23.5

Volumetric air flow, ft'/sec

3.42 3.90 4.15

5.16 8.2

Specimen calculation, nonuniformly sized particles.

(9) where n is a function of d / D and the Reynolds number, p I u t d / p . For fully developed turbulent flow as is general in most pneumatic conveying systems, n = 2.4. The error involved in the assumption of us!= uiis estimated below by substituting the extreme value of tc-i.e., 0.931-into Equation 9. Thus we obtain U,,/Ut

xi!

Volumetric air flow, ft /sec

Superficial air velocityb, ft/sec

US^

=

tn -

= 0.905

~t

According to the Richardson-Zaki equation, our assumption overestimates the slip velocity by not more than 10%. I t is possible, however, that the slip velocity is significantly different from ut as a result of agglomeration owing to electrostatic effects. The present method should be used with caution for systems where electrostatic effects are known to be significant. Assumption: tc = 0.97. While the error in assuming u , ~ = ut is likely to be small, the use of tc = 0.97 in the present correlation will involve much higher errors. Taking an actual c C of 0.987 (the maximum value reported by Ormiston), the calculated critical solid rate from Equation 5 will be some 57% lower than the rate calculated from t C = 0.97. At the other extreme, the solid rate calculated for tc = 0.931 (the lowest voidage reported by Ormiston) will be some 130% higher than that calculated from t , = 0.97. I n this case, however, the error is somewhat compensated by the overestimate of ut and the combined error may be somewhat lower. The value of = 0.97 for the present work is chosen as an arbitrary mean but the results in Table I do suggest that for most systems Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971

187

35


30

3

I

j

I

I

-

I

I

I

I

/

1

/

I

/

/

A

,I

,’,

I I

I

l

l

I

the voidage a t the onset of choking is very close to 0.97. Figure 4 compares the critical air flowrates reported in the literature with values calculated by the present method. Disregarding the data points of Jones, agreement is within ~ 7 0 % .Even if all the points are considered, about 75% of them are seen to lie within ~ 7 0 % .This is reasonable, especially when all other methods presented in the literature for predicting choking flowrates are unsatisfactory. A safety factor would still need to be applied. The “choking air velocities” from the recent data of Jones et al. (196713) are consistently higher than the predicted values from the present correlation. Jones et al. studied pressure drop in vertical pneumatic conveying in 0.305-, 0.402-, and 0.870-in. id steel tubes. They were not primarily concerned with the onset of choking, and their choking data (1967b) are not included in Jones et al. (1967a). At a given mass flowrate of solid they measured the pressure drop a t reducing air flow. The choking flowrates were not reported as such, but are assumed to be the air flowrates a t which a drastic change in pressure drop was observed. However, this flowrate can be considerably higher than the true critical choking gas flowrate. When operating a t a superficial velocity near u t , a sudden reduction in superficial air velocity, u , would result in a substantial reduction in solid velocityi.e., a sharp increase in solid hold-up and pressure drop although u is still higher than the critical choking velocity. Hence the sharp increase in pressure drop noted in the results of Jones et al. is likely to correspond to increase of solid hold-up a t u greater than the true critical velocity. This point was recognized by Ormiston who emphasized the considerable care which had to be taken in decreasing 188


the air flow and allowing the system to settle down at each setting of the air flow, an indication of this being the steadying of the pressure drop. Once the critical air flow had been reached, Ormiston increased it again slightly to reestablish steady transport. Measurements of all variables including the voidage were then taken on the dilutephase flow side of the transition, but the voidage was measured on the slugging side of the transition as well. T o reduce electrostatic charging effects, Ormiston took the precaution of humidifying his air supply t o 75% relative humidity. I t is of interest that Jones et al. used a much taller riser (21 ft overall) than either Zenz (1949, 1960) (4-ft test section) or Lewis et al. (1949) (10 ft overall). This could perhaps be a contributing factor in causing their choking air velocities to be higher. The possibility that full acceleration was not achieved in the experiments of both Zenz and Lewis e t al. has been suggested previously by Leva (1959). The test section employed by Ormiston in obtaining his previously unpublished results was 5 ft long from the solids inlet point to the bottom of two plug valves, the latter being 9 ft apart. The total riser height was 18 ft. General Discussion. The present method for predicting flowrates a t the onset of choking is an approximate one and much further work will be required to extend and improve the correlation. The accuracy of the correlation will be considerably improved if the exact value of t t for a given system is known. Zenz and Othmer suggested that te may be a function of particle diameter and the ratio of particle t o fluid density. They suggest that the mechanism is one of wake capture, with the particle, once caught in the wake of another particle when the voidage becomes sufficiently low, causing a “chain reaction” since the drag on the first particle in the turbulent wake is not as high as that outside of the influence of the wake. This suggestion needs investigation, but it is difficult to see why the critical voidage, e,, should not then be a constant for a particular gas-particle system, independent of the flowrates of gas and solid. The results of Ormiston would tend to suggest that a constant te is not the case, although we may assume this for a design correlation because the range of values is narrow. The results of most other workers cannot be used to check this point as they did not measure the voidage directly although the results of Lewis et al. do suggest the same tendency for t c to depend on solid flowrate. No correlation has yet been proposed, however, for predicting tc from the physical properties of a system. I t should be stressed, perhaps, that we have considered the transition from lean-phase flow to slugging flow as a result of the inherent instability of the dispersion. We have not considered, for instance, the transition or instability as a result of the characteristics of the blower or compressor in the pneumatic conveying system. With blowers characterized by reducing delivery a t increasing delivery pressure (centrifugal fans have such a characteristic), the transition to the slug-flow regime may be triggered off by a sudden increase in solid flowrate (or a sudden reduction of gas flowrate). This causes an increase of pressure drop and hence, the blower delivery pressure. The blower delivery falls, causing a further increase in solids concentration in the pipe with a corresponding further increase of pressure drop, and so on. The instability may well lead to choking depending on the characteristics

of the blower and the operating conditions prior to the initial perturbation.


Conclusions

Vf = average interstitial gas velocity = U / C , ft/sec V , = average solids velocity = us/(1 - e ) , ft/sec x,f = volume fraction of particles in the feed solids stream xLt = volume fraction of particles in the riser

By assuming a constant voidage a t the onset of choking in vertical pneumatic conveying, an equation (Equation 6) is presented which successfully correlates the reported measurements of choking flowrates to within =k70%. This correlation may be used for predicting choking flowrates for the vertical conveying of uniform and nonuniform particles.

GREEKLETTERS t = void fraction a t any condition = critical void fraction (at choking) p = gas viscosity, lb/sec ft p f = density of gas, lb/ft3 p 3 = density of solids, lb/ft’

Appendix

literature Cited

The data collected in Table I in general require no explanation, except for two points. The first four sets of results of Zenz (for zero solid velocity), unlike the remainder attributed to that worker, do not come from Table 10.3 of Zenz and Othmer (1960). They are taken from Table 7.1 of the same reference. For these low voidages in a fluidized bed i t was assumed that u equals ut,and the terminal velocities were calculated in the usual way, by means of the C D N ~vs. , NReplot. In the work of Lewis et al. (1949), the solid rate was fixed and the gas rate decreased gradually until the slugging point was reached. The calculated terminal (free falling) velocities for the Lewis data are those given in the original reference. The voidage values given in Table I were not stated explicitly by Lewis but have been calculated by the present authors from their data by dividing the measured mass of solid per unit volume of riser by the absolute solid density. The superficial solid velocity was simply found from the stated values of solid mass feed rate and the absolute solid density.

Chen, Ja-Min, Ph.D. Thesis, University of Michigan, Ann Arbor, Mich., 1963. Culgan, J. M., Ph.D. Thesis, Georgia Institute of Technology, Atlanta, Ga. 1952. Doig, I. D., Roper, G. H., Aust. Chem. Eng., 4 (l),9 (1963a). Doig, I. D., Roper, G. H., ibid., 4 (4), 9 (1963b). Engineering Equipment Users’ Association, “Pneumatic Handling of Powdered Materials,” pp 52-64, Constable, London, 1963. Gunther, W., Dissertation, Techn. Hochschule, Karlsruhe, West Germany, 1957. Jones, J. H., Braun, W. G., Daubert, T. E., Allendorf, H. D., AIChE J . , 12, 1070 (1966). Jones, J. H., Braun, W. G., Daubert, T. E., Allendorf, H . D., ibid., 13, 608 (1967a). Jones, J. H., Braun, W. G., Daubert, T. E., Allendorf, H. D., Document 9358, American Documentation Institute, Library of Congress, Washington, 196713. Jung, R., Forsch. Ingenieur., 24, 50 (1958). Konno, H., Saito, S., J . Chem. Eng. Japan, 2, 211 (1969). Leung, L. S., Wiles, R. J., Nicklin, D. J., Trans. Inst. Chem. Eng., 47, T271 (1969). Leva, M., “Fluidization,” pp 135-47, McGraw-Hill, New York, N. Y., 1959. Lewis, W. K., Gilliland, E. R., Bauer, W. C., Ind. Eng. Chem., 41, 1104 (1949). Ormiston, R. M., Ph.D. Thesis, Cambridge University, England, 1966. Richardson, J. F., Zaki, W. N., Trans. Inst. Chem. Eng., 32, 35 (1954). Sandy, C. W., Daubert, T. E., Jones, J. H., Chem. Eng. Progr. Symp. Ser, 66 (105), 133 (1970). Zenz, F. A., Ind. Eng. Chem., 41, 2801 (1949). Zenz, F. A., Othmer, D. F., “Fluidization and Fluid Particle Systems,” pp 253-8, 318-32, Reinhold, New York, N.Y., 1960.

Nomenclature

a‘ = average diameter of particle, pm CO = drag coefficient on a single particle D = internal diameter of riser, in. g = gravitational acceleration, (ft/sec)/sec G, = mass flux of solids, lb/hr ft2 Gf = mass flux of gas, lb/hr ft’ N K e = Reynolds number n = index of the Richardson-Zaki equation R = loading ratio = Gs/Gf u = superficial gas velocity, ft/sec u, = superficial solids velocity, ft/sec ud = slip velocity = V, - V,, ft/sec ut = terminal velocity of a single particle in an infinite medium, ft/sec ut, = terminal velocity of component i of a solids mixture, ft/sec

RECEIVED for review May 8, 1970 ACCEPTED November 5, 1970


189

Correlation for Predicting Choking Flowrates in Vertical Pneumatic

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