Slip velocities in pneumatic transport part I

Powder

Technology,

47 (1986) 167 - 177

167

Slip Velocities in Pneumatic Transport Part1 S RAVI SANKAR* Unwersrty

and T N SMlTH

of Adelaide,

S A 5001 (Au&alto)

SUMMARY

Results of some measurements of slap velocrty between sohd and gas an vertacal pneumatic transport are presented. Tests cover particles of sand, glass and steel shot up to 700 p m durmeter and transport tubes of 12.7 to 38.1 mm m durmeter. Slip velocity measurements were made at sohd volumetric concentrataons up to 10%. The countercurrent flow arrangement, whrchpermrtted easy and accurate determmatlon of sokd velocrty from pressure drop data at hagh concentratrots and low solid velocatles, as described Results indicate that the slrp velocrty 18 a strong function of concentrataon and increases with mcreastng concentmtron These trends confirm the conclusion of Matsen that slip velocrty should increase with rncreasrng concentratron m order to explain chokmg phenomenon observed m gas-solrd flows through the pipes. Extenswn of Zakr’s correlation, whuzh predicts decreasmg slap velocities with ancreasing concen tratron to transport sztuatwn, IStherefore not apphcable Shp velocltzes are severalfold hagher than the correspondmg single particle termmal velocttles, mdlcatzng the posslbihty of formatron of clusters

INTRODUCTION

Slip velocity, the velocity of the carrymg fluid relative to the moving solid, is the prune factor m the design of pneumatic transport systems. Its value represents the smallest fluid velocity at which any transport is feasible. When a substantially greater fluid *Present address Department of Chemical Englneermg, Umveraty of Alberta, Edmonton, Alta , T6G 2G6 (Canada) 0032~6910/86/$3

50

velocity is chosen for practical transport of sohd at a specified rate through a vertical conduit, the slip velocity may be used to calculate the sohd m the conduit. These m turn may be used to estimate the contributions of sohd friction and sohd weight to the pressure gradient which must be applied to the motive fluid m order to mamtam transport. Such calculations are complicated by doubt over the proper value of slip velocity. It 1s easy to estimate the termmal settlmg velocity of a smgle particle of the sohd material, but it I not permissible to identify this result with the shp velocity m vertical transport. The value may be close at low solid loadmgs m conduits of large diameter, but very substantial differences arise from the effects of proximity of particles and their mteractions with one another and with the wall of the conduit. Some correlations of slip velocity with sohd properties, such as that of Yang [ 11, are available but should be used with cucumspection [ 21. Until these have been developed and substantiated by successful application to a wide range of variables, the product designer must resort to particular data reported m sources hke the E.E.U.A. Handbook [3] in order to estimate safe transport velocities and operating pressures. Such data cover only certam sohd materials and are somewhat problematical in apphcation, smce they are expressed variously as mmimum fluid velocities, maxmum sohd volume fractions or maxmum ratios of solid to gas flows. None of these ISsufficient to provide a specification of operatmg conditions. Moreover, the range of variables to which they may be applied rehably is short. The limitation to range of conditions is a reflection of the dependence of slip velocity on pertment variables and of a degree of mter0 Elsevler Sequola/Prmted m The Netherlands

168

action between these vanables. In addition to snnple particle properties such as size and density, factors such as sohd volume fraction, diameter of conduit and velocity of fluld must be expected to have some influence. Dehneatlon of the effects of these vanables on shp velocity 1sof evident value to design. Explanation of the effects and the govemmg mechanisms ISof interest not only from the sclentlfic point of view but also m the development of policies and procedures for the operation and control of transport systems.

COUNTERCURRENT

/ I

PRESSURE TAPPINGS

r

j-l------

I

,

f

I

1

I

r

‘F

,

J ---

a

EXPERIMENTS

The fust phase of the expenmental programme, whose prune objective IS to determine slip velocities at large concentrations and low transport velocltles, 1spresented m this paper Smce the present&on of prelimmary fmdmgs of this programme [4], more data covermg a wide range of parameters have been obtamed These expenmen@ are designed to mvestigate the effect of concentration on the slip velocity and also to study the mfluence of pertment vmables such as the tube diameter particle size and density. Briefly, expenments were cmed out using an apparatus m which sohd flows downward agamst a rising stream of air Although this type of arrangement does not represent actual transport conditions, it corresponds to the region between batch fluldizatlon and cocurrent transport. The following factors prompted the choice of such an arrangement (1) It 1s of direct interest for some operations such as heat transfer, drymg and reaction. (u) It facilitates investigation of lower limit to transport velocity. (m) It allows accurate and easy determmatlon of slip velocltles, especially at large solid loadings, which would otherwise be difficult unth cocurrent transport.

Appam tus

r-

The apparatus used m the countercurrent expernnents 1sdepicted m Fig. 1. Sohds fall from an open contamer at a rate fixed by the size of the orifice selected for the test. To ensure even distribution of solids, the arrangement shown m Fig 2 was used. The stream of sohds was dlstnbuted to a width corre-

ROTA METER

MANOMETERS

AIR SUPPLY

I Fig 1 Countercurrent

RECEIVER

flow apparatus

SPlid Feed Container

\

I

kor’f’ce

i

Pipe section having same diameter of the test Sectloll

0 4 concentric Pipe section to contain spilling from overflow

Sieve

converging cone

*4' 1 =t 1-I

Fig 2 Arrangement

Test section

for even dlstrlbutlon

of sol&

spondmg to that of the test conduit by a sieve urlth appropriately chosen mesh. From the sieve, the solids fall through a distance calculated to allow them to accelerate to near equlhbrium transport velocity before they pass mto the test conduit. Air from the blower, metered through a rotameter, enters the closed recelvmg vessel at the bottom of the tube and passes through a convergmg section mto the tube. From the top of the

169 TABLE 1 Detruls of materials used No

Matenal

Den&y Ws)

Particle diameter Wm)

Shape

1 2 3 4 5 6

Glass beads Sand Glass beads Steel shot Steel shot Steel shot

2 2 2 7 7 7

96 173 644 179 375 637

Spherical Spherical Spherical Spherical Spherical

47 63 47 62 62 62

tube, the an flow is diffused to the atmosphere through a &vergmg section. The test conduit was 3 m long and was provided with pressure tappings at 0.5 m mtervals. Two ‘U’ tube manometers were connected to pressure tappmgs 1 m apart, at the top and bottom ends of the test section. Butanol (density 0.808 g/cm3) was used as the manometnc fluid for good sensitivity. The leads of the manometer were provided with needle valves to dampen any high-frequency oscillations m pressure differential which are characteristic of two-phase flows. Experrmen tal procedure With a fixed flow of solid through the tube, pressure drop readmgs at the top and bottom ends of the tube were recorded at several gas flow rates. The an flow was raised in mcrements from zero to the pomt at which solid flow becomes unstable or rs arrested, which is characterised by large fluctuations m pressure drop readings. The procedure was repeated with different solid flow rates fixed by selected orifice sizes at the sohd feed contamer. Range of varuables rnvesttgated SIX different materials with mean particle sizes ranging from 96 to 644 pm and densities rangmg from 2.5 to 7.6 g/cm3 were mvestigated m four test conduits with diameters rangmg from 12.7 to 38.1 mm. Details of these variables are presented m Table 1 and Table 2. The volume to surface mean diameters of particles presented in the above tables were obtained from the sieve analysis of materials usmg standard BSI sieves. Detarls of the analysis of each material are given elsewhere [5]. The materials chosen were found to be

Termmal velocity (m/s) 0 522 1 230 4 740 2 785 -5945 9 379

TABLE 2 Test section details No

Tube diameter (mm)

Tube material

1 2 3 4

12 19 25 38

Steel Steel Steel Steel

7 1 4 1

closely sized. Expenments were conducted at several sohd flow rates fixed by the feed container orifice diameter, which varied from 6 to 25 mm Cahbration of the sohd flow rate with different onfrce sizes for all types of sohd material used IS presented in Table 3. The feed rate from the contamer was found to be constant over the penod of the test.

ANALYSIS OF DATA

It is desired that solids reach eqmhbrmm velocity by the tune they enter the test section, as steady state conditions are of mterest. Details of the mmimum fall distance required to reach equilibrium velocity m still air, for all particles mvestigated, are presented in Table 4. The calculated droppmg d&ance for the heaviest and largest particle used m the tests to reach its terminal velocity m still an was calculated to be 9.4 m. This requnement 1s further reduced with mcreasmg upward gas velocity. Furthermore, the drstances calculated are overestrmated m the sense that it IS impossible to achieve a distribution of particles where particles are dropped individually. In reahty, a maximum

170

TABLE3 Callbratlon of mass flow rate of solids No

1

2 3 4 5 6

Sohd material

96 w

Mass flow rate (g/s) Orifice size (mm)

Glass

173 /.en Sand 644 m Glass 179 m Steel 375 /lm Steel 637 m Steel

6

8

10

12

15

19

25

8 74 7 71 4 76 26 63 21 97 18 07

16 9 15 21 10 71 53 1 45 44 39 18

27 66 24 53 18 33 84 47 73 46 64 58

49 39 46 6 33 87 158 9 140 7 127 1

83 77 77 54 61 83 2714 245 8 226 3

139 9 133 4 104 2 464 2 425 5 386 8

290 47 284 0 227 8 974 3 919 3 854 7

TABLE4 Droppingdistancesm an No

1

2 3 4 5 6

Particle size (cun) 96

173 644 179 375 637

Density (g/cm3)

Termmal velocity (m/s)

DroPPlng dIstancea (m)

2 47 2 63 2 47 7 62 7 62 7 62

0 52 1 23 4 74 2 79 5 95 9 38

0 05 0 21 2 81 1 03 4 64 10 01

droppmg distance of about 1 m was found to be quite adequate m the majority of the runs. The above-mentioned steady state condition 1srepresented by correspondence of pressure gradients at the top and bottom ends of the test conduit This was reahsed in a substantial number of observations However, at large solid feed rates correspondmg to low gas flows, the pressure drop at the bottom end of the tube was found to be higher than the pressure drop at the top end of the tube, mdicatmg mcomplete acceleration of solids. This results from the fall of a group of particles whose dropping distance is larger than that of an individual particle. However, such observations with large differences in pressure gradients at the top and bottom ends of the tube were excluded from the analysis. Observations with pressure drop readings differmg by more than 10% at both ends of the tube were not recorded. A pan of sohd volumetric concentration and slip velocity was obtained at each observation.

Concentration-slip veloctty calculatrons Solid volumetric concentration was derived from the measured pressure drop. For steady state conditions, the total pressure drop (Ap)$ can be expressed as follows (Ap)t = (A&

+ (AP), + (Ap)rs + (A&

(1)

Since the density of solid is quite large (by a factor of 1000) m comparison with the density of an, the static head of au (Ap)zs can be ignored. The solid-wall frrction component (Ap)fs can also be ignored, as its magnitude is insignificant in comparison with the solid static head (Ap),,, especially at larger volumetric concentrations. In addition, sohd velocities are always much less than particle termmal velocity, due to countercurrent arrangement The maxmum solid velocity encountered was about 10 m/s. Such low solid velocities justify neglect of the sohdwall friction component. The gas-wall friction component was expervnentally determmed with only air flowing through the test

171

section and was found to be negligible in comparison with the sohd static head term. Following the above arguments, the total pressure drop can be approxunated to sohd static head component without mcurrmg much error. (AP)~ = (APL,

(2)

The pressure drop due to the solid hold-up is expressed as follows: (AP), = cp,gL

X 0

(3)

A

m B

From eqns. (2) and (3), the solid volumetrrc concentratron can now be derived from the pressure drop as follows. c= -UP),, (4) P&L From known quantltles of sohd and an volumetrrc fluxes through the tube, sohd and an velocltles were denved as follows: vg=

-

XX

xx +

+

2+ 'hbe

12 7mm ($) 96p Glass beads

9

2

C&CENTikNl &)

19

12

Fig 3 Effect of concentration

%

l-c

(6) The total upward volumetric flux of air
(7)

The positive sign on the sohd velocity u appropriate, smce the sohds are flowmg in a dvection opposite to the air flow.

RESULTS

Effect of concentratzon The concentration and slip velocity values derived from the above-mentioned procedure were analysed for a sohd matenal m a test conduit. Shp velocity normahsed with respect to particle terminal velocity was plotted agamst sohd volumetric concentration to determme the relatlonshlp between them. Results of 96 E.trnglass beads m a 12.7 mm tube are presented in Fig. 3. The following observations can be made from thusplot.

(1) Slip velocity is a unique function of solid volumetnc concentration. Higher mass flow rates of solids result m extension of the concentration range. (ri) While at very low concentrations slip velocity 1salmost equal to calculated partrcle terminal velocity, its value 1slarger than the terminal velocity at higher concentrations. (m) The rate of change of slip velocity with concentration decreases with increasing concentration. The above trends are typical of the results obtained wrth other particles in all the tubes mvestrgated. These results are presented elsewhere [5]. Effect of partde properties In order to assess the mfluence of particle properties such as density and size on the shp velocity-concentratron relationship, slip velocities normahsed with respect to the corresponding particle termmal velocltles were plotted agamst sohd volumetric concentration. The results of srx different solid matenals m a 12.7 mm diameter tube are presented in Fig. 4. The following observatrons can be made from this graph (1) At a gwen concentration, smaller partrcles have larger dnnensronless slip velocities.

Matenal

L 0

12 7mm

(#) Tube

96p Glass beads

A

1731.‘Sand

0

644,~ Glass beads

+

179p Steel shot

X

375~ Steel shot

0

637~1 Steel shot

0

0

0 0 0

0

0

0

0

”

$3 O 0

0

0

O-

@%

O 08

0

0

0

0

0

:a

O0

00 ^ 00

96w

Glassbeads

2

10

Fig 5 Effect of tube diameter

(n) The degree of dependence of slip velocity decreases with increasing particle size (m) The effect of particle density appears to be of little significance in comparison with the effect of particle size. This was concluded from the observation that the results of 173 pm sand (p, = 2 6 g/cm3) and 179 E.crnsteel shot (p, = 7 6 g/cm3) tend to fall on the same line. Similar observations can be made with 644 pm glass beads and 637 pm steel shot. Plots with other tube sizes are presented elsewhere [ 51. All these plots show similar trends

Effect of tube daameter Plottmg dimensionless shp velocity agamst concentration for a given material m different tube sizes, the following observations can be made Results of 96 pm glass beads in four different tubes are presented m Fig. 5. (1) At a given solids concentration dimensionless slip velocity mcreases with decreasmg tube size. (n) The degree of dependence of slip velocity on concentration decreases with mcreasmg tube diameter. Similar trends were observed with other particles These plots are presented elsewhere [ 51

Summary of results The trend of entire experimental data from the tests with six different particles m four different tubes can be summarised as follows. (1) Dimensionless shp velocity is a unique function of concentration for a given sohd material and tube size. (ii) Slip velocity mcreases with increasing concentration and its magnitude is larger than the correspondmg smgle particle termmal velocity (m) The degree of dependence of slip velocity on concentration decreases with increasing tube size and particle size. (iv) The influence of particle density on the shp velocity-concentration relationship is less significant than that of particle size Comparison with extstmg data Slip velocities related exphcitly to volume fraction are rarely reported in the literature. In general, data are presented m terms of loading ratio or gas to sohd velocity ratio. Lack of mformation about the correspondmg solid and gas mass flow rates limits the derivation of the variables of interest. In addition, the malority of the data published are confined to very low concentrations at high transport velocities, where solid-wall friction effects dommate. Matsen [6] presents the following correlation based on the elutriation

173

data of Wen and Hashmger [ 71, with 70 pm glass beads m a 100 mm dra. bed. c, -= ct 4 _

c < 0.0003

1

c > 0.0003

= 10.8~0.293

vt

The above correlation suggests a hnear relationship between the logarrthmlc values of volumetric concentration of sohds and the dunenslonless slip velocity Analysis of the present expelvnental data suggests that such a linear relationship does mdeed exrst. Figure 6 represents the data obtamed with 96 I.crnglass beads m a 12 7 mm tube. Observations wrth other particles m all the tubes mvestlgated gave smnlar results. The values of A and I3 obtamed by lmear regressron are presented in Table 5 for all the expenments. The correlation coefflclents are also presented. These values are greater than 0.95 for the majority of the data, mdlcatmg a good fit However, It

TABLE

127mm(#) 'hbe 96~~ Glass

-P 5

-2

beads

[CONCENTRATlON~’ -15

-0

5

log,o

Fig 6 Concentration-shp scale

velocity

map on a log-log

5

Parameter

estimates

of correlation

2

= A@

h No

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Tube size

Particle size

Particle density

(mm)

&ml

(g/cm31

12 19 25 38 12 19 25 38 12 19 25 38 12 19 25 38 12 19 25 38 12 19 25 38

96 96 96 96 173 173 173 173 644 644 644 644 179 179 179 179 375 375 375 375 637 637 637 637

2 2 2 2 2 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 7 7 7 7

7 1 4 1 7 1 4 1 7 1 4 1 7 1 4 1 7 1 4 1 7 1 4 1

47 47 47 47 63 63 63 63 47 47 47 47 62 62 62 62 62 62 62 62 62 62 62 62

A

39 24 13 10 33 12 6 7 7 3 1 1 20 14 7 5 9 3 3 3 5 2 1 1

B

00 05 28 45 37 64 81 55 38 23 88 29 31 60 43 82 07 40 20 31 60 48 76 79

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Regression coefficient

544 482 356 305 642 426 297 320 374 232 160 058 476 467 349 286 359 212 215 231 281 164 113 126

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

988 995 992 974 992 977 989 987 993 967 918 622 992 991 971 980 989 964 959 960 963 953 925 881

174 1s interesting

to note that the values of A and B varied, dependmg on the system properties, z e , tube diameter and particle properties These two parameters are correlated to three dimensionless groups which characterise the system, namely particle Reynolds number (Be,), particle to tube diameter ratio d,/d,, particle to air density ratio ps/ps The entire data of countercurrent experiments can be represented by the followmg correlation.

A = 93.67(Re,)-

B

=l.O75(Re,)-

1

08

0

Authors

data

0

Matsen’s

data

/”

/” ,’

o.994(G.) lfi14(Ps)O-‘~ o*s(dP)o-*16(P$i3

(8)

d,wg Re, = Fig 8 Correlation of parameter B

Erg The values of the parameters A and B predicted by the above correlation are plotted agamst the observed values (Fig. 7 and Fig. B), and the correspondence 1s reasonably good For comparison, the values of A and B reported by Matsen [6] are also presented. Since Matsen’s correlation was based on Wen’s experiments with 71 pm glass spheres m a

0

Authors

0

Matsen s data

data

“/

,/”

“‘/ii

O

Fig 7 Correlation of parameter A

101.1 mm tube, the parameters A and B were estimated from these system properties using the above correlation. The agreement between the estimated and reported values is good, considering the fact that the tube size used m Wen and Hashmger’s work was larger than the tube diameters used in the present study by at least a factor of 3. Yerushalmi and Cankurt [B] report increasing slip velocities with mcreasmg concentration, based on their expernnents with 50 pm spherical catalyst particles m a 152 mm tube. However, the concentration-&p velocity relationship was not unique, but depended on the mass flow rate of solids. The sohd volumetric concentration and slip velocity were inferred from the pressure gradient measured over the middle of the test section. Unfortunately, no mention was made as to whether steady state conditions prevailed m such measurements. If the acceleration of sohds m the test section is incomplete, the concentrations and slip velocities may have been overestimated. This m turn explams the mtroduction of solid flux as a parameter m the concentration-slip velocity relationship. Also, the size range of catalyst particles used m their tests was reported to be 0 - 130 ,um. Information on the distribution of the particle size was not presented, except for the mean particle size. The reported shp velocity at neghgible sohd

175

volumetric concentration (C < 0.1%) is about 13 tunes larger than single particle termmal velocity, when a correspondence 18expected mstead. Yousfi and Gau [9] present the following correlation for slip velocity, based on their experiments with 20 E.trnand 183 pm particles m 38 mm and 50 mm diameter tubes The range of sohd volumetric concentrations reported is 0.5% to 22%. -18

(PelPs)

The above expression suggests that shp velocity is a strong function of concentration and that it increases with mcreasing concentration. Since this correlation involves gas Reynolds number as a parameter, comparison with the correlation proposed (eqn. 8)) is not feasible. However, the following observations reported by Yousfi and Gau [lo] agree with the trends observed m the present work: (1) While the dimensionless shp velocity ranged from 8 to 40 m the case of 55 pm catalyst particles, its value varied from 40 to 300 in the case of 20 pm particles. (n) For large 290 pm polystyrene particles, dimensionless slip velocity was much smaller than that observed with fme particles. Its value ranged from 1 to 4 (m) The effect of concentration on shp velocity was less significant in the case of large particles The large magnitudes of dunenslonless shp velocity reported in the case of fine particles need some attention. In fact, dimensionless shp velocities as high as 300 were reported, while the value predicted by the correlation (eqn. 8)) is only about 37 at the maximum solid concentration studied (22%). This discrepancy is possibly due to electrostatic charging and adhesion of particles These effects are known to be enhanced with decreasmg particle size Nevertheless, the correlation proposed (eqn. 8)) is not recommended for such fine particle systems until additional data are available m support of it. Some remarks on the correkataon On closer exammation of the correlation (eqn. B)), the powers on particle Reynolds number and diameter ratio groups are almost the same m magnitude, but opposite m sign. Approximating the magnitudes of the powers

to the same value, a new dimensionless group is realised.

This new dimensionless group is the tube Reynolds number correspondmg to particle termmal velocity. The significance of this group IS that it represents the ideal chokmg flow gas Reynolds number. The correlation presented earlier can be rewritten m terms of this new group as follows 2 =A$’ vt

(9) B = 1.075(Ret)-0445

dtw,

Ret = -

k.

The above correlation suggests that the higher the choking Reynolds number the smaller the magnitude of dimensionless slip velocity A and the degree of dependence on concentration B. The contribution of density ratio is also significant. This might appear to contradict the observed correspondence of the slip velocity-concentration relationship between particles of almost the same size with large differences in densities. The following reasoning should clarify the matter. Particle Reynolds numbers of the materials used m this study range from 3 to 400. In this intermediate region, the experimental study of Jones et al [ 111 suggests that the termmal velocity of a particle is proportional to

Ut a

Since the density ratio for gas sohd systems is very large, the above expression can be approximated to

(10)

176

On exammmg eqns. (10) and (9), the effect of particle density on parameters A and B 1s made clear. Although higher particle density results m larger values of the density ratio term, a correspondmg mcrease m particle terminal velocity results m little change m the value of parameter A. Slmllarly, parameter B suffers little change. Although eqn. (9) IS attractive m terms of fewer dimensionless groups, eqn. (8) should be preferred until additional data are acquired to establish that the correspondence of powers on particle Reynolds number and tube to particle size ratio is not fortuitous. Proposed correla tton The form of the correlation (eqn. 8)) to predict slip velocity at any particular sohd concentration is not completely satisfactory m the sense that it can not be extended to very low concentrations, and requires the specification of a lower limit to concentration below which slip velocity is approximately equal to particle termmal velocity. Therefore, other forms of correlation which extend to zero concentration are considered. One such form is as follows.

(11) Unfortunately, the degree of fit for the above type of correlation is very poor. The left-hand term m the above equation is very sensitive at low concentrations and for large particles where u,/ut term is closer to unity. As a consequence, the correlation (eqn. 8)) presented earher is preferred, but with the followmg modification:

concentration c,,, is larger than for small particles. This, mdeed, 1s experimentally observed fact. While for 96 pm glass spheres m a 12.7 mm tube slip velocity started to mcrease even from concentrations as low as O.l%, its value remamed approximately equal to termmal velocity even up to 1% concentration m the case of 644 pm glass beads m a 38.1 mm tube.

CONCLUSIONS

(1) The influence of sohd volumetric concentration on slip velocity m the case of gassolid flows m pipes is drfferent from that of fluidization and sedimentation phenomena. Use of Zaki’s correlation, which predicts decreasing slip velocity with concentration, is mappropriate for gas-solid flows m pipes. (2) The expenments carried out, covering a wide range of particle and tube sizes, confirm the conclusion of Matsen [6] that slip velocity should increase with concentration m order to explam the choking phenomenon observed m gas-solid flows. (3) Additional investigations are necessary to explam the slip velocities bemg larger than the correspondmg particle termmal velocities. One possible explanation is that the higher slip velocities are due to formation of clusters whose effective terminal velocities are larger. Explanation of possible mechanisms of the formation of such clusters would be of great value.

LIST OF SYMBOLS

(12)

A B c Gl-l,,

where cmin = A-1’B The above correlation suggests that the lower hmit of concentration cmin depends upon the values of parameters A and B, which m turn depend on the system properties. The expressions for these parameters suggest that their values decrease with mcreasmg particle size and tube diameter. In other words, for large particles, the value of mmmum

R% R% Ret

parameter m eqn. (8) parameter m eqn. (8) volume fraction of sohds lower hmit of volume fraction of sohds particle diameter, m tube diameter, m acceleration due to gravity, m/s2 length of test section between pressure taps, m particle Reynolds number tube Reynolds number tube Reynolds number at particle termmal velocity

177

%

4 % vt

gas velocity, m/s

REFERENCES

shp velocity, m/s solid velocity, m/s particle terminal velocity, m/s

Greek symbols total pressure drop, Pa pressure drop due to static head of gas, Pa pressure drop due to gas-wall frrction, Pa pressure drop due to solid-wall friction, Pa pressure drop due to static head of solids, Pa gas density, kg/m3 solid density, kg/m3 volumetric flux of gas m the tube, m/s volumetric flux of sohds m the tube, m/s gas vrscoslty, kg/(m-s)

4

5 6 7 8 9 10

11

W C Wang,AZChE J, 21 (1975) 1013 L S Leung and R J Wdes, Znd Eng Chem Process Des Dev , 15 (1976) 552 Engmeermg Eqmpment Users Assoclatlon, Pneumatu Handlmg of Powdered Mate&s, Constable, London, 1963 S Raw Sankar and T N Smith, Proc ZOth $ Australian Chemical Engmeerrng Conference, The Institute of Engmeers Austraha, Sydney, 1982, pp 299 - 303 S Raw Sankar, Ph D Thesis, Umv of Adelaide (1984) J M Matsen, Powder Technol, 32 (1982) 21 C Y Wen and R F Hashmger, AZChE J, 6 (1960) 220 J Yerushalml and N T Cankurt, Powder Technol, 24 (1979) 187 Y Yousfi and G Gau, Chem Eng Scz, 29 (1974) 1939 Y Yousfl and G Gau, Chem Eng Set , 29 (1974) 1947 J H Jones, W G Braun, T E Daubert and H D Allendorf, AZChE J , 12 (1966) 1070