Pressure drop in pipeline

Pressure Drop in Vertical Tubes in Transport of Solids by Gases Measurements of total pressure drop and static pressure drop in the transport of solid particles through vertical 0.267- and 0.532-inch inside diameter glass tubes by an air stream are reported. Closely sized sand (28- t o 35-, 35- to48-,48-t060-, and60- to80- meshTyler),and bothground and spherical silica-alumina cracking catalyst were used. Solids circulation rates ranged from 2 to 54 pounds per second per square foot a t various constant air rates from 0.9 to 3 pounds per second per square foot, equivalent t o 12 to 40 feet per second. Direct measurements of the dispersed solids density were made; these permitted the average particle velocity, slip velocity, and solids static pressure drop t o be calculated. The total pressure drop

I E of the inherent advantages of a fluidized system is the 0” ease with which solids can be added to or removed from the reaction zone by pneumatic transport. A means of estimating the pressure drop in the ducts carrying the gas-solid mixture is essential in designing such a system. This becomes particularly important in designing small equipment such as pilot units where the risers (as the vertical ducts are callpd in fluid catalytic cracking units) may be as small as 0.5 inch inside diameter. In pipes this size, friction loss betiveen the solid particles and the pipe walls is an unknoivn and conceivably a large part of the total pressure drop. At the other extreme, in the 6-foot inaid? diameter risers found in commercial fluid catalytic cracking units, the pressure drop is usually assumed to be entirely due to the static head of the gas-solid mixture.

During operation of a full scale fluid cat cracker, the riser pressure drop serves as a guide in estimating catalyst circulation iates. In the operation of a pilot unit, where inore exact measurements can be made, it is found that n-hen a constant carriei aii rate is maintained in the spent catalyqt riser, a linear calibratioii can be made between riser pressure drop and 8o11ds circulation rate for a given cracking catalyst. The iiser pressure diop also is used to hold the catalyst circulation rate a t a desired value by means of a differential pressure 1 ecorder-conti oller n-hich actuates the spent catalyst slide valve.

I n systems where all or part of the reaction tabes place in the riser, that is, when the reactant gas is used as a carrier, it uould be desirable to estimate the riser solids concentration and velocitv so as to provide a measure of the degrce of contact betn-een the gas and solid. This again is directly related to the static head component of the total riser pressure drop.

A knowledge of the solids flow rate, gas velocity, and total pressure drop in a vertical riser offers no clew as to the relative values of the static and fiiction components of the total pressure diop. If a section where acceleration of the particles has ceased is considered, the mass-average solids velocity or the concentration of solids in the riser also must be known before the static component can be calculated. A t constant solids f l o ~ rate (in weight per unit of time) and carrier gas velocitv, a coarse heavy material will move a t a slower velocity than a fine low density powder. Consequently with the heavy material the static head rvill be higher than in the case of the light material; the friction

1 Present address, Atlantic Refining Company, Philadelphia, Pa.

can be regarded as the sum of the drop due t o the carrier gas alone plus a solids pressure drop. The residual obtained by subtracting the solids static head from the total solids pressure drop was treated as an apparent solids friction drop, which conformed to the Fanning equation when solids velocity and dispersed solids density were used. However, later measurements showed tha t the major portion of the apparent friction was due to particle acceleration in the test section. The mathematical treatment of the accelerating period is presented. Drag correlation for free-falling particles in still fluids relates relative velocity of gas and solids to force required t o overcome inertia, gravity, and solids friction.

0. H. HARIU‘ A N D M. C. MOESTAD UNIVERSITY OF P E N N S Y L V A N I A . P H I L A D E L P H I A , P A .

loss d l certainly differ due to the difference in velocity of the two materials

The object of this investigation was to study the effect of the variables mentioned in the preceding paragraph on the pressure drop in two sizes of vertical glass tube.

LITERATURE

Little can be found in the literature on the subject of riser pressure drops in fluidized systems. Daniels (4 ) , in describing the hydraulics of a fluid catalrtic cracking unit, mentions that riser gas velocities of 15 to 30 feet per second are requiird and that pressure drop in the riser.. is estimated commonly by assuining i t equal to the static head requirement and disregarding an> fluid friction. He calculates the density of the gas-solid mixture by multiplying the flow density (total flow of solid plus gas in pounds per hour divided bv total volume in cubic feet per houi) by an assumed slip factor of 2.0, equivalent to taking the velocitv of the particles of solid as one half the velocity of the gas. For example, with a carrier gas velocitv of 20 feet per second, a slip factor oi 2 would requiie the dip velocity t o be 10 feet per second. In viex of the fact that the average free-falling velocity of the catalyst particles in the carrier gas is of the order of 1 foot per second, the slip factor should be about 20 19 or onlv slightly over 1.0 unless there is corisidei able friction betireen the particles and the riser lvall.

Dalla Valle (3) presents a summary of most of the literature pertaining to the theory of particle transport prior to 1940. The emphasis up to that time n as on the pneumatic handling of giain He developed equations for the air 7-elocity required to transport particles. The banie result can be arrived a t by use of the inore general and fundaniental correlation of drag coefficient against Reynolds number, the data for which are suin- marized by Lapple and Shepherd (8). Cramp ($) presents an equation, which appears fundamentally sound, for calculating the total pressure drop in a pneumatic conveyer. Methods developed by Hudson (5, 6) for the design of pneumatic con- veyers are strictly empirical. Jennings (7’) shows a method of calculating the accelerating distance in vertical transport. Chat- ley (I), in calculating the poiver requirements of a grain con- vever, recognizes that an additional term should be added for friction between the solids and the pipe wall, but states that no information is available for estimating this.

1148

lune 1949 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY 1149

In a paper recently published, Vogt and White ( I d ) report measurements of pressure drop in vertical and horizontal pipe carrying suspensions of solids in air. They derive a correlation in terms of the ratio of pressure drop with the suspension flowing to that obtained with the carrier gas alone, the weight ratio of solids flow to gas flow-, and properties of the solid and gas. Their analysis is difficult of interpretation for vertical risers because the allowance for static head is not clearly indicated. No measurements of particle velocity were made. For reasons shown above the present authors believe it is desirable to have a method of estimating both solids static head and solids friction pressure drop. Both these components of pressure drop are taken into account in the theoretical analysis which follows.

THEORY

An analysis of measurements of all the pressure drops en- countered in a transport system consisting of a horizontal pipe, a bend, and a vertical section would be a complex problem. Particle velocities probably would be different in the horizontal and vertical sections. The nature of the friction loss between particles and the pipe wall varies between the horizontal section, the bend, and the vertical pipe. I n the first, the weight of the particles probably concentrates most of the friction on the lower half of the pipe; in the second case centrifugal force has a large effect on the friction; and in the vertical section friction losses are due to the particles striking the walls and other particles. Static head is important in the vertical tube but is zero in the horizontal. Kinetic energy changes are important: initial acceleration a t the point of entry of the solids requires energy which must be supplied by the carrier gas; at the bend, and for some distance above it, the particles must be accelerated upward, resulting in an energy transfer from the gas to the solids over and above tha t required for solids friction and potential energy change; i f the over-all pressure drop is an appreciable fraction of the initial static pressure, acceleration of both the gas and particles, due to the increase in specific volume of the gas, must be accounted for in the energy balance.

Consider the ideal vertical riser in which uniform spheres are moving upward at a constant mass-average velocity of us feet per second, carried by a gas moving a t ug. To support the particles, the gas must slip past them at a relative velocity (slip velocity) of Au, equal to (u, - us), and exert a force, F p pounds, on each particle. The drag correlation, as shown below, relates Au to F,. This force is equal to the weight of the particle whether i t is standing still or moving a t a constant velocity if there are no external forces acting on the particle, such as friction against the tube walls. Then 4u will be the free-falling velocity of the spheres. I n the presence of external retarding forces on the particles, brought about by collisions with the walls and between particles, a somewhat higher slip velocity is required. I n the range of slip velocities covered in most of the present experimental work, F , varies approximately as ( A U ) ~ . ~ .

The mass rate of flow of solids per unit cross section of riser can be designated G, pounds per (square foot) (second). Because of the slippage, the true dispersed solids density, p d s (in pounds of solids per cubic foot), is higher than the flow density, which has been defined above as total weight flowing per hour divided by total volume of flow per hour. The following rigorous relation is used in the derivation below:

Ga = UsPds (1)

Tn the experimental work, us was calculated from measurements of G, and P d s .

Over L vertical feet of riser, the pressure drop in the carrier gas due to supporting the weight of the dispersed solids can be regarded as a solids static head, equal to L feet of a fluid of density pds. Changing to a unit of pressure commonly used in fluidized systems, the solids static head in inches of water is

APz = Pds L x 12/62.3 = 0.1925pd,L, (2)

Ap,/L = 0.1925 Gg/u8 (3)

01'

Expressing this as pressure drop per foot per unit of mass velocity,

Apn/LG, = 0 .1925 /~ , (4) An assumption is now made that the Fanning friction equation

(IO) will apply to the energy lost by the particles in striking the riser wall and in impacts between particles. Then the solids friction loss,

Ap.f8 = 2 f,Lu3pdS(O.1925)/gD, ( 5 )

( 6 )

(7)

Zxpressing pds in terms of us

AptJ = 0.1925 X 2f,Lu,2 G,lgD;lc,, or

Ap/,/LG, = 0.1925 X 2 fszcS/gDr

A manometer connected across length L will read 4 p , the sum of a pressure drop due to gas friction against the walls, (Ap~u) , plus the solids friction loss plus the solids static head, or

AP Ap/o f A ~ j a + A T J ~ (8 )

The static head of the gas phase will not be indicated if the manometer lines are filled with a gas of the same density. If the volumetric concentration of particles is low, the true velocity of the gas will not differ appreciably from the superficial velocity. The surface area of the riser in contact with the gas is affected only to a slight degree by the presence of the particles. Con- sequently, i t is reasonable to assume that the gas friction loss is the same as if the solids were absent.

The total solids pressure drop, from Equation 8, is

Aps Ap - Ap~p/o = AP, 4- Apfa (9)

(10)

Combining Equations 4 and 7 ,

Apa/LGs = 0.1925 [I/'us 4- (2fa/gDr) X ( % ) I Equation 10 is one of the basic equations used in the analysis

of the experimental data. It is in accordance with earlier ob- servations that A p / L is a linear function of G,, with its origin at Apfg /L and G, = 0, if us and fa can be considered constant. Here Ap,/LG, represents the slope of a plot of A p / L against G,. The present experimental data verify this linear relation (Figures 6 and 7 ) . An interesting feature of Equation 10 is that it is entirely independent of the gas velocity and the gas and solid properties, except for the possible effect of these variables on the Fanning friction factor for the solids, fs. -4 second relation between A..o,/LG, and u8 is necessary to solve

for these values at a given set of operating conditions. This is derived from the drag correlation (8, IZ), which relates the relative velocity of the gas and solids, Au, to the force exerted on the particles by the gas. By this correlation, the force in pounds on a particle is

F , = Cp,( A ~ ) ~ . 4 , / 2 9 = 7rCpo( Au)'D;/8 g (11)

where C is a function of the Reynolds number, Dp( Au)p , /~ , . The number of particles in L feet of riser is

The total force on all the particles is

F = (TCPA AUI2Dg/8 9 ) X (6pdsLArl~D3,pp) (13)

The total force on the particles divided by the area of the tube and L equals the pressure drop per foot in pounds/square foot, or

F I L A , = 3 CP& AU)'Pds/4 gDpP, (14)

Converting to inches of water and substituting G8/uI for p d s ,

1150 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 41, No. 6

I p , / L = 0.1925 X 3 C p , ( AZL)~G,/-I gDnpP1/* or

Ap,;LG, = (0.0045/Dp) ( p o / p p ) I ( Au)'/u81 C' (15)

( l t i j

Simult'aneous solution of Equations 10 and I 6 anioiiii ts to a force-balance in which t,he force required to support the particles and overcome solids friction, as specified by 1;quatioii 10, is balanced by the force exerted by the gas oii the partic:lw as s h o ~ i i by Equation 16. To relate A u to ul, t'he gas velocity must be specified. For a single smoot,h sphere, C will vary with Au as showi by the curve oi Figure 10, ivhrrc C is plotted against Re.

A graphical represent'atiori of Equations 10 aiid 16 aids in explaining their significance. I n Figure 2, lines AC and .2D are calculated from Equation 10, assuming j equal to 0.004 and using the two riser diameters used in the clxperimxital ~ v o ~ k . Line B R is the static head component of ICquatiori 10 arid is independent of riser diameter. Ap,/LC, approac1it.s infinit?, as 7 1 r approaches zero, as Ap,/LG, is proportional to 1. /1/~. A s ug increases, ap,/LG, reaches a minimum aiid t,heri bibcomes greater when the solids friction loss increases iiiore rapidly t hari the static head decreases. With increasing diamctcr the minimum moves to the right. Line EG is calculated from Ikpatioii 16 for 0.00165 foot dianieter spheres with a density of 165 pounds per cubic foot (the size and density of one of the sands used in the expwinient) carried by a 25 foot per second air stream at atinos- pheric conditions. The curve for free-falling sphcws in Figure 10 is assumed to hold. Point F is the force balancc point for the larger riser, determining immediately the values oi' arid Au, as well as ap,/LG,, which is the suni of ap,/LG, arid ~ p j , / l , G . . Thr: ordinate a t point H is Ap,/LG,, t,he static hc,ad component, wht:reas the solids friction drop per unit inass velocity Apja//,G.>, is represented by the distance H F . Comparison of thc iriter- sections G arid P shows the effecl oi decreasing thc size of thv riser a t a constant air velocity. The solids velocity, u . , tlr- creases slightly in going from F to G, resulting iii a soinr:n-hat higher static head; the friction drop practically doubles. If the riser dianieter is increased to the point, that solids friction is negligible, the line for total pressure drop such as A 1) or ,IC becomes practically coincident with line AB. drops to J and the slip velocity, 25 - 11.9 = 13.1 feet pw sc:corid, corresponds to the free-falling velocity of the spheres. This citn be considered to be approximately t,rue for large risers.

An expression for the mechanical efficiency of this ideal riser now can be dcveloped. The powcr output is cyii:iI to thcl P t lT2

at v-hic'l lift work is done on the solids, or

EIp. output = G,A,L/SSO (17)

Tho 1m\ver input is that lost by the air aiid cdii be ~~o~)r i~sci i t td I)?

Then Au = uo - us.

The in1

Hp. input = &pug AT/(0.1925) X (550) (18)

R-liwe Ap = Spja + Apj\ + A p z in inches of wat,er. The mechanical efficiency, hp. outlhp. in, is a maximuin when riser diameter is so largc as to niake air arid solid friction lo rrcgligihlr, so tha't A p = A p = pd.L(O.1 923). Thc~l~c~fol*i?

Fcff,",ex = G,,I J,/,; /,* L u g . 1 j . (19)

.k:ff.,,,,, = U9/ i / [ , ( 20 1

Thus, the iiiaximum mecliariical efficiency i i i 111 accclcrating section of riser is simply the invrrrc. o which is defined as

Slip factor = ~ i , / i c ~ (21 1

The nicc1iaiiic:iil ene loss is due to turbulriiw r:tt.at,eti by the gas slipping past the particles, arid appears as heat. Tlie efTicic,iicy ~ v i l l be lowered by air and solids friction losws.

It must be remembered that this analysis is concerned oi i lv

with a vertical section and one in which acceleration of gm rtnt l solids is negligible.

P A R T I C L E ACCELERATION

Variatioris iri the valurs of fs calculated from the expoi iiiiriii ~ i l data by Equation 10 indicated the possibility that the p a r t w h ~ ~ had not reached an equilibrium velocity before entering thtl t section, but were still accelerating. Therefore, the residual solid5 piessure drop, after subtracting the measured static hcad, woulti be due not only to the foice everted by the gas on the partic I( to overcome solids iiictiorial forces, but also to the force ieq to accelerate the particle.;. In an effort to distinguish bct the frictional and acceltliatioii components, the following niai 1 1 ( - matical treatment was derived.

Applied to ono pa1 ticlr, thc i i c ' t upwxd acceleration ( * A I I i l l

t ~ p r cscled as

~vl i tw the first term oii the right t h upmard acceleration duc, I ( J

the diag force of the cariicr gas and the second and third ter~ii. a i r the domnnard accelerations tluv to gravity and the frictioiial force pLlr particlc, respectiwly. If fs and the empirical rclaLioii twtweeii (Au) and C' arc. kiion 11, 101 a specified gas velocit\ / io, a can b(> calculated for all valuc.5 o r 11, from zero to the poilit u-licire a = 0. The lattcr c .or r iy)ond~ to the equilibrium solitli vrlocj ty.

Thr time-velocity and dislaiiuc-vt,loc.ity relations caii be 011-

t a i n d by giaphicnl intcyqi atioii of the L01101\ ing tv, o equatioiis:

i2:3i

I-k~uatiori 23 is iritegratcd by plotting I ,'CL (as ordinatt.) agaiii-~ l i s . T l i e time rcquircd to w : t ( > l i

velocity u2 from ul is thc area under thc curve. Similarly, l:qii:i-

tion 24 is integrated by plot,tiiig u,/a 8s ordinate against us. Finally, Equation 10 can bo expressed differentially t o iricliitlr~

t l i ~ prttssure drop rryuircid lo :tc:c:i~lc,rate the particles:

0, is calculated by TGquation 22.

(25)

Hy t,his equation, i," - ($1) cari t jc piottod as a function of i i i . ii'

the varhtion of u with I,&, is kiiovn by Equation 2 2 . Sin(:(> t l i c , relat,ion botumm us and /, (:an be derived from Equation 24, (2) can t,hrii be plotted as a function of distance, 1,. 11 i \

apparent that when the acceleration approaches zero, us ai)- proachcs a constant equilibrium value and Equation 25 bwoiii(+ identical with Equation 10.

Equation 25 shows that the p urc drop per foot can be aik cutiaf.inely high value a t a point ere us is small because both

t I1 ti s t at>icI h car1 (k) arid the acceleration pressure drop

f i i ~ largo. Th(~or(.t,ically, ~l.rcrc, I L ~ is zero, --I ih (2) is infinite,

Portunatrly this oarinoh occur actually, sirice for a finit,c G, i u i d

%<TO u s , the solids corici~iitr~atioii pds would be iiifinitc:, w h i r ~ l ~

is physically impossible. Siiicc the '- term applies to liorimiit t i l

as \vel1 as vertical ti~aiispoi~i, this analysis helps in explairiiiig the high pressure drops and the turbulent conditions normally t:ncountered a t the poiiit, wherc the solids are introduced into t,hc, carrier stream. drop might be decreased hy mcchmically imparting to t'he solitlh

((2

g21s

It also indicates that the acceleration p

June 1949 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 1151

a velocity in the direction of the gas stream before the two are eter 7 served as a measure of mixed. the solids flow rate,

Two brass Parker three-wai- plug valves, I and L, and :t copper tube by-pass, K (approximately the same inside diameter as J). were used to trap a sample of solids in thta riser for a measurement of pda The sample was removed for weighing by: breaking a close- fitting rubber sleeve connectioi~ between I and P-4; stopping the solids flow through B . connecting the short nipple ex- tending up from Z (containing the solids) to a small cyclone separator fitted with a weighing

4 Storage section Dimension, Inclirs flask and an internal steel wool Filter 10 x 10 AIR filter (not shown in Figure I ) ,

and turning valve I so that the Cyclone A Inner tube 1 5 (11)) Outer tube 2.5 ( 1 . T ) ) Figure Apparatus carrier air forced the sanlplr~

into the flask. Storage tube Combined length, D and r Lowei tube, inside diam. 1 82 Pressure taps

P-1 to P-2 6 P-1 to P-3 23 Inside diainetei

This might, for example, be done by a curved nozzle.

APPARATUS

The apparatus shown in Figure 1 was designed to introduce continuously into a metered air stream an adjustable flow of solid particles, with facilities for mcasuring the solid circulation rate, the pressure drop across a length of vertical riser carrying the gas-solid suspension, and the weight of solids contained at any instant in a known volume of riser. Dimensions of the apparatus are given in Table I.

Table I . Dimensions of Apparatus

60

PROCEDURE

0 126

16 The materials shown in Table I1 were sized with Tyler standard

screens in a Ro-Tap machine, allowing 20 minutes for a small batch. The sands were further screened by hand until i t ap-

Dimensions, Inches peared that the amount passing the smaller screen was negligiblL

J, I.D. 0 267 0 532 During operation thv I to L 41 5 53 9 1 to P-4 5 9 cracking catalysts wen

found to contain a small amount of fines; thc

P-4 to P.5 32 0 40.1 Taps, i d. 0 125 0.12: Plug valve poits, .I and L 0 25 0 5 Straightening section befoie \al\ e I 6 7

ground catalyst contained more than the micro- spheroidal, as evidenced

The fluidized bed storage section consists of glass tubes F and D by the amount of dust rising to the filter H. connected by a valve, E, a conical brass screen with a cone- Densities of the sand shaped copper seat. The lower end of D is fitted to a brass tube

standpipe, C, at the bottom of which is located slide valve B used particles were measured by water displacement iii

to control the solid circulation rate. The slide valve was con- structed from a 0.5-inch gate valve by replacing the gate with a

EI 50-ml. pycnometer. machined plunger fitting into a cylinder. The flow of solids was Since the catalyst particles controlled by varying the opening between the end of the plunger

and the apex of a triangular hole in the wall of the cylinder. A stream of air for fluidizing the bed of solids in D was intro are porous, and a measuie-

l o l 5 *' 3G 35 ment of their absolutcl duced through nozzle F-2. -4 second stream entered F-I, at fi densities and por0sitit.i rate adjusted to maintain a constant positive pressure drop across

the slide valve as indicated by manometer 8. Manometer 6 ua, Ft./Sec. indicated the pressure drop through a fixed length of the fluidized Figure 2. Plot of Equations ,o was scope considered of this work, beyond v a ~ u ~ , thv

quoted by Webb ( I S ) weie and 16 bed, between taps P-1 and P-2, and served as a guide in setting the air rate through F-2 Starting with an unfluidized bed, as F-1 and F-2 were increased, manometer 6 showed an increasing A6 is static head component of Equa- reading until at the fluidization point it reached a maximum and t ion 10: AC and AD are plots for two wed in calculatillg indicated the true bed density. At higher fluidizing air rates the riser diameters, assuming fs = 0.004. particle density. The,

EG is plot of Equation 16 for mater ia l density reading decreased due to expansion of the bed. A wi th gas velocity of 25 ft./sec. average particle diametei , Manometer 7 measured the pressure between P-1

and a point above the bed, P-3. It is in effect a level meter since it measures the weight of solids above point P-I. Small bleed streams of air were added through bubblers to pressure t a s P-1, P-2, and P-3 to prevent their blockage by soEd particles.

Standpipe Length Inside dianirtcr 1 . 2 6

--______ H . Riser section Tube 1 Tube 2

Table I I . Solids Properties .I and B = Otta \%a sand C and D = Sea sand Two calibrated orifices were used at A to cover

the range of carrier velocities desired in the two sizes E = Micros heroidal oraching. catalyst F = Grouiicr cracking ratalyst

of glass riser J. During a run, the carrier air picked h ~ ~ ~ ~ ~ i ~ l A B C D h I up the solid particles falling through the slide valve Tyler scregll sise, mesh and transported them through riser J i n t o cyclone G. The solids fell through tube F and valve E (open) back into the fluidized bed in D whereas the carrier D ~ , arithmetic av. diameter, 0.00165 0.00117 o OOOYO o 00070 o O O O ~ B o ooo41

Manometers 4 and 5 measured the static pressure and pressure drop in the riser 3.

The solid circulation rate was measured by K'* for 'Irc 1 0 2 7 2 6 6 5 8 4 3 4 2 2 2 2 closing valve E which caused a fluidized bed of ut, calculated free-fall velocity 12 8 9 2 7 2 5 2 1 0 I 0

particles to build up in tube F. The rate remained in ft*'sec.d unchanged because the total weight of solids above the slide valve remained constant, ensuring a C K2 = D p p g / p g so Re = K2( Au), PO = 0.075, PO = 0 018 X 0 000672. constant slide valve Ap. As the level dropped For spherical particles; for rough particles, ut is lower.

35 48 B O 100 LOO 35 28 48 BO 80 200 200

1007 thru 1004 on

air plus blped air and fluidizing air left the system through the heavy cloth filter on the top of cone H . density, 1b leu ft. 165 165 165 169 615 61

feet

808 574 141 343 176 170 Ki, for airb

Calculated, assuming absolute density = 2 33 g./cc and poie 5olunie = 0.6 cc b Used in calculating drag; Ki = 222 D p p p / p g ; PO = 0.075 Ib /cu f t .

in D, the rate of change in the reading of manom-

1152 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 41, No. 6

D,, was taken as the arithmeticmean of the tmo screen openings. Photo- micrographs of the six materials are shown in Figure 13. Each scale division

I corresponds to 100 microns.

I T ith each mate-

of the orifice meter Figure 3. Measurement of Flow 2-that is, constant

Rate of Solids carrier air rate- a series of runs was made covei-

ing"as wide a range of solids circulation rates as possible. Two i o four carrier air rates were used with each material in each ot the two risers. Tables 111 and I T show the range of variables covered

The procedure employed in a run n a s as follows: after the material was charged to the unit, F-2 vias adjusted to fluidize The bed in D without excessive slugging. The carrier rate was adjusted to the desired pressure drop on meter 2 and slide valve

0 I 2 3 4 rial, a t one setting 4 0

Time, Min. 4p Across Valve, in. H g

Figure 4. Gas Leakage through Slide Valve

B n a y opened to start the circulation of solids. Valves I and L n-ere in the position shown in Figure 1, 60 the gas-solid mixture flowed through riser J . The air rate to F-1 was adjusted to maintain a desired constant pressure drop across the slide valve as shown by manometer 8. The air rate to F-2 was finally adjusted to as near the minimum fluidization point as possible to reduce slugging and fluctuations in the level meter 7. Readings n-ere taken of manometers 2 through 8.

~~ ~~

Table I I I . Smoothed Data and Calculated Results for 0.267-Inch I.D. Riser

&?/ A p , )LS< Solid See. In. HzO Lb.,"r A 2 1 . 9

21 .9 21.9 21.9 21.9

Av. 21.9 3 5 . 3 35.4 3 5 . 5 3 5 . 6

AV. 3 5 . 5 40 .9 4 1 . 1 4 1 . 3 4 1 . 5

sv. 41.2 B 2 1 . 7

2 1 . 7 21 .7 2 1 . 7 21 .7

Av. 21.7 3 0 . 7 3 0 . 7 30 .7 30 .7 30 .7

d v . 30 .7 4 1 . 2 4 1 . 3 4 1 . 5 4 1 . 6

Av. 4 1 . 4 C 1 3 . 3

1 3 . 3 1 3 . 3 1 3 . 3

Av. 1 3 . 3 2 0 . 3 2 0 . 3 2 0 . 3 2 0 . 3 2 0 . 3

.&v. 2 0 . 3 2 9 . 1 2 9 . 1 2 9 . 1 2 9 . 1 2 9 . 1

Av. 29.1

2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 . . .

3 . 0 4 . 0 5.0 6 . 0

3 . 0 4 . 0 5 . 0 6 .O

2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

3 . 0 4 . 0 B.0 6 . 0

2 . 0 3 . 0 4 . 0 B.0

2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

. . .

. j .

. . .

. . I

. . .

. . i

. . .

. . .

1 5 . 7 29 .4 4 3 . 0 5 6 . 5 7 0 . 2

1 9 . 5 3 1 . 7 4 4 . 0 5 6 . 1

. .

. . 1 3 . 0 2 4 . 5 3 6 . 0 4 7 . 5

1 8 . 9 32.9 4 7 . 0

( 4 . 5

1 0 . 5 2 2 . 6 3 4 . 6 4 6 . 8 5 8 . 8

. .

01.0

. .

. . 12.4 2 2 . 2 3 2 . 1 4 2 . 0

2 0 . 2 3 1 . 4 4 3 . 0 54 .4

1 8 . 3 3 2 . 8 4 7 . 2 61.5 7 5 . 8

1 1 . 1 22 .9 3 4 . 8 4 6 . 7 5 8 . 5

. .

. .

. .

. .

M S , Grams 1 . 1 0 1 . 7 6 2 .41 3 . 0 5 3 . 7 1

0 .76 1 . 0 4 1 . 3 1 1 . 5 8

0 . 4 2 0 . 6 8 0 . 9 4 1 . 2 0

0 . 7 8 1 . 2 6 1 . 7 5 2 . 2 3 2 . 7 1

0 . 3 7 0 . 6 5 0 .94 1 . 2 2 1 . 5 0

0 . 2 7 0 . 4 5 0 .62 0 . 8 0

1 . 4 0 2 .18 2 . 9 6 3 , 7 3

0 . 8 2 1 . 3 5 1 . 8 8 2 . 4 1 2 . 9 4

0 . 3 1 0 . a 0 . 8 7 1 . 1 4 1 . 4 1

. .

. .

. .

. .

. .

. .

. .

. .

. .

G., Lb. 1 (Sq. Ft.) u8, ~ p ~ ,

Cu. Ft. (See.) Ft.,'Sec. In. H20 1 . 8 3 1 1 . 3 6 2 0 . 9 4 2 . 9 3 2 1 . 1 7 . 2 1 . 5 1 4 . 0 2 3 0 . 9 7 . 7 2 . 0 6 5 . 0 8 4 0 . 5 8 . 0 2 . 6 1 6 . 1 8 5 0 . 4 8 . 2 3 . 1 7 . . . . . 7 . 5 I .

1 . 2 7 14.0 11 0 0 . 6 5 1 . 7 3 2 2 . 8 1 3 . 2 0 . 8 9 2 .18 3 1 . 6 1 4 . 5 1 . 1 2 2 .63 4 0 . 3 15 3 1 . 3 5 . . . . 1 3 5 I .

0 .70 9 . 3 1 3 . 3 0 . 3 6 1 . 1 3 17 .6 1 5 . 6 0 . 6 8 1 . 5 7 2 5 . 8 1 6 . 4 0 . 8 1 2 . 0 0 3 4 . 1 1 7 . 1 1 . 0 3 . . . . l 5 , 6 . .

1 . 3 0 1 3 . 5 1 0 . 4 0 . 6 7 2 . 1 0 23 .6 1 1 . 2 1 .08 2.92 33 .7 11 5 1 . 5 0 3 . 7 2 43 .7 1 1 . 7 1 . 9 1 4 .52 5 3 . 3 1 1 . 8 2 . 3 2

1 1 . 3 0 . 6 2 7 . 5 12 1 o 32 1 . 0 8 1 6 . 2 15 .0 0 . 5 6 1 . 5 7 2 4 . 8 15 .8 0 . 8 1 2 . 0 3 3 3 . 5 1 6 . 5 1 . 0 4 2 . 5 0 4 2 . 1 1 6 . 8 1 . 2 8

. 1 6 . 0 0 .45 8 . 9 20 0 . 2 3 0 . 7 5 1 5 . 9 21.2 0 . 3 9 1 . 0 3 2 3 . 0 2 2 . 3 0 . 5 3 1 . 3 3 3 0 . 7 2 2 . 6 0 . 6 8

. . . . 21.5 , .

2 . 3 1 1 4 . 5 6 . 2 1 . 2 0 3 . 6 4 2 2 . 6 6 . 2 1 . 3 7 4 . 9 4 3 0 . 8 6 . 2 2 . 5 4 6 . 2 1 3 9 . 0 6 . 3 3 . 2 0

6 . 2 1 . 3 7 1 3 . 1 9 . 6 0 . 7 1 2.2,s 2 3 . 5 1 0 . 4 1 . 1 6 3 . 1 4 3 3 . 8 1 0 . 8 1 . 6 1 4 .02 4 4 . 1 1 1 . 0 2 . 0 6 4 . 9 0 5 4 . 3 1 1 . 1 2 .52

10.6 . . 0 . 5 2 8 . 0 l 5 , 3 0 . 2 7 0 . 9 8 1 6 . 4 1 6 . 7 0 .50 1 . 4 5 2 4 . 9 1 7 . 1 0 . 7 6 1 . 9 0 3 3 . 4 1 7 . 6 0 . 9 8 2 . 3 5 4 2 . 0 1 7 . 8 1 . 2 1

. . . . 1 6 . 9

.lPJ,/ua In. HzO

0 .54 . . . . . I

. .

. . 1 . 3 7 . . . . . . . .

1 . 8 2 , . . . . . . . .

0 . 5 4 . . . . . . . . . .

1 . O G . . . . . . . . . .

1 . 8 3 . . . . . . . . .

0 . 2 0 . . . I . . . .

0 . 4 6 . . . . . . . . . .

0 . 9 4

Apparent Frictiona .lPaB APfa AU,

In. H2O in. H;O fs Ft./Sec. U ~ , ' V . ~ C Re - \pdLGa 1 . 4 6 0 . 5 2 0,0052 1 5 . 7 3 . 5 4 . . . . . , 0 0487 2 . 4 6 0 . 9 5 0 , 0 0 4 4 1 4 . 7 3 . 0 5 . , , . , , 0.0439 3 . 4 6 1 . 4 0 0.0041 1 4 . 3 2 .85 . . . . . . 0.0420 4 . 4 6 1 . 8 5 0.0040 1 3 . 9 2 . 7 5 , . . . . , 0.0413 5 . 4 6 2 . 2 9 0.0039 1 3 . 7 2 .69 . . . . , . 0.0406

, , , 1 4 . 4 _ _ 1 . 2 7 147 0 .0435 1 . 6 3 0 . 9 8 0 .0043 2 4 . 3 3 . 2 1 . . . . . . 0.0436 2 . 6 3 1 . 7 4 0,0040 2 2 . 2 2 . 6 8 . . . . , , 0,0432 3 . 6 3 2 . 5 1 0 , 0 0 3 8 2 1 . 0 2 . 4 5 . . , , . , 0.0430 4 .63 3 . 2 8 0.0037 2 0 . 3 2 . 3 3 , . . . . . 0,0430

2 2 . 0 . . 0 . 9 8 224 0 ,0432 1.18 0 . 8 2 0 ,0046 2 7 . 6 3 07 . . , . . . 0.0475 2 . 1 8 1 . 6 0 0 , 0 0 4 1 2 5 . 5 2 . 6 4 . . . . . . 0.0464 3.18 2 . 3 7 0.0039 2 4 . 9 2 . 5 2 . . , . . . 0.0462 4 .18 3 . 1 5 0.0037 2 4 . 4 2 .42 . . . . . . 0.0460 . . . . , . , 2 5 . 6 . . 0 . 8 9 262 0.0466

1 . 4 6 0 . 7 9 0.0039 11.3 2 . 0 9 . . , . . . 0.0405 2 . 4 6 1 . 3 8 0 ,0038 1 0 . 5 1 . 9 4 . . , . . , 0.0390 3 .46 1 . 0 6 0 ,0036 1 0 . 2 1 . 8 9 . . . . . . 0.0384 4 . 4 6 2 . 8 5 0 . 0 0 3 5 10.0 1 . 8 6 . , . . . , 0,0383 5 . 4 6 3 . 1 4 0 , 0 0 3 5 9 . 9 1 . 8 4 . . . , , . 0,0383

. . . . . . . 10 .4 . . 2.34 74 0.0389 0 . 9 4 0 . w 0.0047 . . . . . 0.0470 1.94 i . i s o.0040 1 5 . 7 2 . 0 4 . , , . . , 0.0448 2 . 9 4 2 . 1 3 0 , 0 0 3 8 1 4 . 9 1 . 9 4 , . , . , , 0.0445 3 . 9 4 2 . 9 0 0 , 0 0 3 7 1 4 . 2 1 . 8 6 . . . . . . 0.0441 4 . 9 4 3.66 0 , 0 0 3 6 1 3 . 9 1 . 8 3 . , , , . . 0,0440

1 4 . 7 , , 1 . 9 1 107 0.0449 1 . 1 7 0 94 0.0037 2 1 . 2 2 . 0 8 . . , . . . 0.0492 2 . 1 7 1 . 7 8 0 , 0 0 3 7 2 0 . 1 1 . 9 4 . , , , , , 0.0511 3 . 1 7 2 . 6 4 0 ,0036 1 9 . 3 1 . 8 6 . . , . . . 0.0517 4.17 3.49 0 ,0036 1 9 . 0 1 . 8 4 . , , . . . 0.0519

. . . . . , , 1 9 . 9 . . 1 . 5 7 145 0.0508 1 . 8 0 0 .60 0 ,0046 7 . 1 1 . 8 7 . , , . , , 0.0405 2 . 8 0 0 . 9 3 0 , 0 0 4 6 7 . 1 1 . 8 7 . . , . . . 0.0464 3 . 8 0 1 . 2 6 0 ,0046 7 . 1 1 . 8 7 . . . . . . 0.0462 4 . 8 0 1 .60 0 .0045 7 . 0 1 . 9 0 . . . , . . 0.0461

. . . . . . . . . . . . 2.54 40 0 0463 1 . 5 4 0 . 8 3 0 .0048 1 0 . 7 2 . 1 2 . . . . . . 0.0440 2 . 5 4 1.38 0 ,0039 9 . 9 1 . 9 5 . , . . . . 0.0405 3 . 5 4 1 . 9 3 0 ,0037 9 . 5 1.88 . . , , . , 0.0392 4 . 5 4 2 . 4 8 0 ,0036 9 . 3 1 . 8 4 . . . . . . 0 , 0 3 8 5 5 . 5 4 3 . 0 2 0 . 0 0 3 3 9 . 2 1 . 8 3 . , . . .. 0.0382

. . . I . . . 9 . 7 . . 2 . 0 0 54 0.0401 1 . 0 6 0 . 7 9 0 .0046 13 .8 1 . 9 0 . 2 . 0 6 1 . 5 6 0.0040 1 2 . 4 1 . 7 4 . 3 . 0 6 2 . 3 1 0 . 0 0 3 8 1 2 . 0 1 . 7 0 . 4 .06 3 . 0 8 0 ,0037 11.5 1 . 6 5 . 5 . 0 6 3 . 8 5 0.0036 11.3 1 . 6 3 .

, 0.04Y6 . 0.0470 . 0.0460 . 0,0456 . 0.0461

12 .2 . . 2 . 3 2 68 0 ,0467

a Includes pressure drop due t o acceleration of the particles. (Concluded on p a g e 1163)


---- CALCULATED the slide valve, A p , L - OBSERVEJ

The solids circulation rate, ws, was then measured by closing valve E and reading manometer 7 a t precisely 15-second or 30- second intervals. These level readings were plotted during the run against time, as shown in Figure 3. The best line drawn through the points gave the average rate of change of manometer 7 in centimeters per minute. Manometer 7 was pre- viously calibrated for each of the materials by weighing the solids removed through the slide valve and noting the change in the meter reading. The calibrations checked the calculated values within 1 %. Figure 3 shows three circulation rate determinations. The slope of each line times the calibration factor gives the solids rate in pounds per hour.

After the solids circulation rate was determined] valve E was opened, allowing the solids to flow back into D. With no changes made in the settings, valves I and L were turned simultaneously to by-pass the flow through K and isolate a sample of solids in J . After removing the sample as described above, it was weighed to the nearest 0.01 gram.

It was found that there was an appreciable leakage of air down through the slide valve, adding an unknown amount to the measured carrier gas. A method was devised to calibrate the leak; a carbon dioxide stream was substituted for the air a t F-1, the aeration point above the slide valve. Orsat analyses of the riser gas while solids were flowing through the valve gave a measure of the amount of leakage as a function of slide valve A p . With all four sands (A through D) the leakage was found to depend only on

These measurements are shown in Figure 4.

50 100 200 400

opening. This is because at the aeration rates used, the standpipe remained unfluidized and the main resistance to downward flow of aeration gas from point F-1 was in the

Standard Cu. Ft./Hr. (32') 2-inch depth of particles. With the finer and less dense cracking catalysts no consistent cali-

bration was found. A slight flow of aeration gas in F-1 would fluidize the standpipe and then the leakage, as measured by the carbon dioxide method, appeared to vary with valve pressure drop, the valve opening, and the fluidized density of the material in the standpipe. At a constant valve pressure drop the leakage was approximately proportional to the solids circulation rate. Use of this correction, as shown in Tables I11 and IV, introduced an uncertainty which could not be evaluated.

The friction drop due to air flow alone was measured a t various air rates for both glass tubes used as risers; these data are presented in Figure 5 . Some of the points were taken over a period

Figure 5. Pressure Drop in Absence of Solids

Fut/ Solid Sec.

39.9 39.9 40.0 40.1

Av. 40.0 D 12.2

12.2 12.2 12.2

Av. 12.2

19.7 19.7 19 .7 19.7

Av. 19.7

28.7 28.7 28.7 28.7

Av. 28.7 E . .

. . Av. 19.3 '

.4v. 27, 8

F . . . . . . Ax-. 19.6C

45.. 28.0C A 17.8 (GO, 17.8 car- 17 .8 rier) 17.8

Av. 17.8

Table Ill. Smoothed Data and Calculated Results for 0.267-Inch I.D. Riser (Concluded)

A P , w e , Av.Is, In. Hz0 Lb./Hr. Gram::

3 . 0 4 . 0 6 . 0 6.0

2 . 0 3 . 0 4 . 0 5 .0

2 . 0 3.0 4 . 0 5 . 0

2 . 0 3 . 0 4 . 0 5 . 0

1 . 0 1.5 2 .0 2.25

. . .

. . .

. . .

. . .

. . . 1.25 1 .5 2 . 0 2.25

1 . o 1 .5 2 . 0 2 .5

1.25 1 .5 2 . 0 2.25

2 .5 3 .0 4 . 0 4.75

. . .

. . .

. . .

. . .

12.9 21.2 29.5 37.8

20.2 33.2 46.3 59.4

18.5 31.2 44.0 57.0

11 6 22.3 33 .1 43.9

4 .6 10.5 16.7 15.8

. .

. .

. .

. .

. . 2 . 6 4 . 5 8 .3

10.2

6 . 3 12 ,8 19.4 26.0

4 .0 6 . 8

12 ,6 15.3

23.5 30 .5 45.0 55 .8

. .

. .

. .

. .

0.26 0.44 0.63 0 .81

1.38 2.17 2.96 3.75

0.77 1.26 1 .76 2.25

. .

. .

. . 0.32 0.60 0.88 1.16 . .

0.15 0.31 0.48 0.56

0.03 0.06 0.14 0.18

0.16 0.32 0.49 0.65

0.06 0.10 0.18 0.22

1.77 2.20 3.03 3.67

. .

. .

. .

. .

..

Ct.7 Cu. Ft

0 .43 0.73 1.05 1.35

. . 2.30 3.62 4.94 6.25

1.28 2.10 2.93 3.75

0.53 1.00 1.47 1.93

0.25 0.52 0.80 0.93

0 . 0 5 0.10 0.23 0.30

0.27 0.53 0.82 1.08

0.10 0.17 0.30 0.37

2.95 3.67 5.05 6.11

. .

. .

. .

. .

. .

. .

..

. .

Gs, Lb./ Apparent Frictiona (SS. Ft.) us A P Z , Anfa A P S A p j s Au ,

(Sec.) Ft./S'ec. In. Hz0 In . Hi0 In . Hi0 in. HzO f. Ft./Sec. 9.3

15.1 21.2 27 .1 . .

14.5 23.8 33 .1 42.5

13.2 22.3 31.5 40.8

8 .3 16.0 23.7 31.4

3 . 3 7 .5

12.0 14.2

1.9 3 . 2 5.9 7.3

4.5 9 . 2

13.9 18.6

2 . 9 4 9 9 .0

11 .o

16.8 21.9 32.2 40.0

. .

. I

. .

. *

. .

. .

. .

. .

21.6 20.8 20.2 20 .1 20.7

6 .3 6 .6 6 . 7 6 .8 6.6

10.3 10.6 10.8 10.9 10.7 15.6 16.0 16.1 16.3 16.0 13 14.4 15.0 15.3 14 .9 . .

28' 24 25 16.7 17.3 17.0 17.2 17.1

29 29 30 30 29 5

5.7 6 0 6.4 6.5 6 .1

0.22 0.38 0.54 0.65

1.18 1.86 2.54 3 .21

0.66 1 .os 1.51 1.93

0.27 0.51 0 .76 0.99

0.13 0.27 0 .41 0.48

. .

. .

. .

, .

. . 0.03 0.05 0.12 0.16

. . 0.14 0.27 0 . 4 2 0.56

0.05 0.09 0.15 0.19

1.52 1.88 2.60 3.14

. .

. .

. .

1.68 . . . . . . . .

0.19 . . . . . . . .

0 .44 . . . . . . . .

0.92 . . . . . . . .

0.38 0.40 0.42 0.44

0.82 0.83 0.85 0.86

0.38 0.40 0.42 0.44

0.82 0.83 0.86 0.86

0.35

t .

. .

. .

. .

. . . . ..

. .

1.32 2.32 3.32 4.32

1.81 2.81 3.81 4 .81

1.56 2 56 3.56 4.56

1.08 2.08 3.08 4.08

0.62 1.10 1.58 1.81

0.43 0.67 1.15 1.39

0.62 1.10 1.58 2.06

. .

. .

. .

. .

. .

. .

. . 0.43 0.67 1.15 1.39

2.15 2.65 3.66 4.40

. I

. .

1 .10 0.0038 18.3 1.94 0.0043 19.1 2.78 0.0045 15.8 3.63 0.0046 20.0 . . . . . . 19.3

0.63 0,0048 5 .5

1.27 0.0040 a . a 1.60 0,0038 5 .4

0.95 0,0042 f.2

. . . . . 5 . 6

0 . 9 0 0.0046 9.4 1.48 0.0044 9 . 1 2.05 0,0042 8 .5 2.63 0.0041 8 . 8

. . . . . 9 . 0 0 .81 0,0043 13 .1 1.57 0.0043 12.7 2.32 0,0042 12.6 3.09 0.0042 12.4

. . t . . 12.7 0.49 . . . . b 0.83 0.0052 . . 1.17 0,0045 , . 1.33 0,0044 , .

4.4 0.40 0.0068 b 0.62 0.0054 . . 1.03 0,0048 . . 1.23 0,0047 . . . . . . . . . .

0 .48 0,0044 b 0.83 0.0037 . . 1.16 0.0034 . . 1.50 0,0032 . . . . . . . . 2 .5

0.38 0.0038 b 0.58 0.0035 . . 1.00 0,0032 . . 1.20 0,0031 . .

. . . . . .. 0.63 0.0046 12.1 0 .77 0,0041 11.8 1.05 0,0036 11.4 1.26 0.0034 11.3

. . . 11.7

ua/uz 1.85 1 .92 1.98 1.99

1.53 1.85 1.82 1.75

1.51 1.88 1.82 1.81

1.84 1.75 1.78 1 .76

, .

. .

. .

. . b . . . . . .

1 . 3 b

. .

. . . .

. . b . . . . . *

1.15 b

. .

. . . .

. . 3.12 2.98 2.80 2.72

C Re . . . . . . . . . . . . . . . . . , . . . . , 1.40 108 . . . . . . . . . , . . . . . . . . . . . . , 3.2 23 . . . . . . . . , . . . . . . . . . . . . . . . 1.95 35

. . . . I .

. . , . . .

. . . . , . . . . . . . 1.65 55 . . . . . . . . . . . . . . . . . . I . . . . . 7 .0 9.7 . . , . . . , . . . . . . . . . . , . . . . . .

b b

. . . . . , . . . . . . . . . . , . . . . . .

20 5 .5 . . . . . . . . . . . , . . . . . . , ~ . , . .

b b

. . . . . .

. . . . . .

. . . . . .

. . . . . . 1.05, 21 .6

b Not calculated due to the l o a accuracy of us and u~). 6 Estimated because slide-valve leakage could not be measured accurately with materials E and F; leakage estimatpd a t 1 cu. f t . / l O lb. wg.

Aps/LGs 0.0531 0 . C576 0.0587 0.0897 0,0573

0.0441 0.0431 0,0424 0.0441 0,0443 0.0430 0.0424 0.0418 0.0429

0.0487 0.0481 0.0481 0,0481 0.048f

0.055 0.049 0.048. 0,051 0.088. 0,078. 0.065. 0.063 0.073:

0.045. 0.043 0.042 0.043 0.056. 0.051 0.048 0.047 0.051 0.0479- 0.0453 0.0452 0.0412 0.0442

0 , 0468

. . . .

. . . .

Vol. 41, No. 6 1154 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

Table IV. Smoothed Data and Calculated Results for 0.532-Inch I.D. Riser

1 7 . 1 0 . 8 0 1 7 . 1 1 . 1 0 1 7 . 0 1 . 4 0 1 7 . 0 1 .70 1 7 . 1 .. 29.5 0.80 2 9 . 5 1 . 1 0 2 9 . 4 1 . 4 0 29 .3 1 .70 29 .4 . . 40.5 1 .40 40 .4 1 . 7 0 4 0 . 3 2 .00 40.4 . .

1 9 . 0 2 . 5 6 3 0 . 5 ?.SO 4 2 . 2 0.05 53 .9 6.30

14.0 0 .73 31 .6 1 .38 4 9 . 0 2 40 66 .8 3 . 2 0

24 .0 0 , 7 9 3 9 . 5 1 .32 55 .0 1.8:;

. . , .

. . . .

0.82 3 . 4 1 4 . 2 1 . 2 2 3 . 4 8 4 . 5 1 69 7 . 5 5 4 . 7 2 . 0 2 9 . 6 8 4 . 8

4 . 6 0 , 2 3 4 2.51 10 .7 0 .50 5.66 11 .3 0 . 7 7 8 . 8 0 11.4 1.03 11.90 11 .6

11.3 0.253 4 . 3 1 17.0

0 5<J '1.87 16.6 0 .42 7 .10 16.7

16 .7 1 6 . 6 0 .50 1 7 . z 1 . 0 8 0 3 4 f i : j . O Y 8 . 9 1 6 . 5 0 . 8 0 35 r? 2 . 1 8 0 . 7 0 6 . 3 7 9 . 1 1 6 . 5 1 10 54 .0 3 .28 1 .05 9 . 7 0 u 2 1 6 . 4 1.40 72.5 4 .?8 i . ~ i i M.o:! ~ 4 . 3 1 0 . 8 . . . . , .

40.0 1 . 1 0 7 6 0 . 2 4 39 .9 L 4 0 2 2 . 0 0 . 6 0 3 9 . 8 1 . 7 0 36 .6 0.95 39.9 . . . , . .

9.1 U.077 1 3 : 17.: 0 .192 3 . 9 5 20.6 0 . 3 0 3 6 J 7 21 .5

2 1 . l t 1 6 . 1 0.50 2 0 . 2 1 00 0.3Y 3.63 1 1 . 0 16 .1 0 . 8 0 3 8 . 0 2 . 0 ; ( ] . t i t i 6 83 10 .4 1 6 . 1 1.10 5 6 . 4 3 . 1 1 1 . 0 0 10 .11 10.1 1 6 . 0 1 4 0 7 4 . 6 4 . 1 7 1 R1 13.29 9 . 9 1 6 . 1 , . . 10 1 28.6 0 . 8 0 1 6 . 2 0 .49 0 I d 7 2 . 9 1 1 8 . 5 28 .5 1.10 3 5 . 6 1 . 0 7 0 34c3 6 . 3 8 18.6 28.4 1 . 4 0 53,O 1.6.5 0 . iXO 9.88 18 .6 28.5 . . 1 8 . 6 1 5 . 8 0 50 2 2 . 0 1,O.j 0 . 2 3 7 3 . 9 3 1 1 . 7 1 5 . 8 0 .50 40.; ; 2.10 0.674 7 27 1 0 . 8 1 5 . 7 1 10 5 9 . 4 L1.j 1 . 0 1 10.67 10 .6 1.5.7 1.40 78 .0 4 20 I 3.5 1 4 . 0 0 1 0 . 1 1.5.8 1 . 1 0 . 8 28 .2 0 . 8 0 1 8 . 0 0 . 4 9 0.1.57 3 . 2 3 20 .6 2 8 . 2 1 . 1 0 37.0 1 . 0 1 0 , 3 2 4 6 . 6 5 2 0 . 6 2 8 . 1 1 . 4 0 06 .0 1 . 6 2 0 . 4 9 10.07 2 0 . 6 28.2 . . . ' 20.6

0.40 1 6 . 0 0 . 6 2 O,l9!) 2 .88 1 4 . 5 0.60 3 2 . 5 1 .40 0.450 2.84 1 3 . 0

d 0.80 48.8 2 .28 0.716 8.75 12 .2 14 .2 .. . . 1 2 . 6 b

0 . 4 0 9 . 3 0 . 2 3 0.050 1 .67 21 0 . 6 0 21 .0 0 .67 0.183 3 .77 2 0 . 6

d 0 .80 3 3 . 0 0 .91 0 292 5 . 9 2 2 0 . 3 2 0 . 2 , . . . 20.6

0 . 4 0 11 .8 0 .30 0 OQ6 2 . 1 2 22 0 .60 23.7 0 . 6 1 0 195 1.25 2 1 . 8

d 0.80 35 .5 0 .92 0.29: 6 .37 21 .6 20.2 . . . . 2 1 . 8

0 . 5 3 0 . 7 9 1 . 0 4 1.30

0 . 1 3 0 . 3 2

0 . 6 6

0 .16

0 .38

0 .22 0 .43 0 .65 0.91

0 . 0 3 0 . 1 2 0 . 2 0

0 . 2 1 0 .43 0 .65 0 .56

0 . 1 0 0 . 2 2 0 .34

0 . 2 2 0 . 4 3 0,6,5 0 .87

0 . 1 0 0 . 2 1 0.31

0 . 1 3 0 .29 0 . 4 6

0 .05 0 .12 0 .19

0.05 0 . 1 3 0.20

. .

0 . s o

0 .27

. .

. .

. ,

. .

, .

. .

. .

, .

. .

lncludes pressure drop due t o acceleratiori of the particles. b Using average of the last two nuinhers. e N o t calculated due t o low accuracy of u a and up. 1 Estimatod. by assuining R -lide-val\.i: leakage of 1 CII. f t . / t O lb. ?cy

0 . 2 3 . . . . . . . .

0 . . . . . . . . .

0 . 9 6 , . . . . .

0.21 . . . . . .

0 .95 . . . .

. . 0.20

, . . . . . . .

0.23 , , . . , .

0.20 . . . . . . . .

0 , 3 3 , . , .

. . 0.16

. .

. .

. . 0 . 2 9

. .

. .

. . 0.29

I . . .

0 . 5 7 0 . 8 7 1 , 1 7 1.47 . .

r) 2.1

n 84 0 . 34

1 .14

0 .44 0 . 7 4 1 . 0 4

0 . 2 9 0 .89 0 .89 1 , 19

0 . l . j 0 43 0 , 7 5

0 . 3 0

0 . 9 0 1 . 2 0

(1 27 I). 57 0.87

0 . 80 0, 80 0 . 9 0 1 .20

0 .27 0 . 5 7 0 87

0.24 0 .44 0 .B4

0 .11 0 . 3 1 0 . 8 1

0 11 0 . 3 1 0 .5l

. .

. .

. .

n.60

. .

. .

( .

0 . 0 4 0.0031 1 2 . 9 0 . 0 8 0,0036 1 2 . 6 0.13 0,0041 12 3 0 . 1 7 0.0041 12 .2

12 3 0 . 0 9 0.0037 1 8 . 8 0 . 2 2 0.0038 18 2 0.34 0.0038 1 8 . 0 0 .48 0,0039 17 .7

18 .2 0.28 0,0042 23 .2 0 . 4 7 0.0044 23 .7 0.66 0,0045 2 3 . 7

2 3 . 7 0 07. 0,0028 7 . 7 0 . 1 4 0.0027 7 . 4 0.21 0.0020 7 . 3 0 . 2 8 0.0025 7 . 1

7 . 1 0 .10 0 , 0 0 4 7 29.3 0.33 0.0045 I%:+ 0.55 0.0043 1 8 . 3

0 . 0 9 0 .0025 5 . 1 0.17 O.O02(j 5 7 0 . 2 3 0,0027 t i .0 0 .34 0.002R 6 . 1 -.. 0 . 1 7 n.0033 i o . L 0 . 3 3 0.0033 Q 9 0 . 3 3 0.0032 9 . 8

. . . . 9 . 9 0 . 0 8 0.002 4 . 1 0 .17 0.0024 5 . 0 0.25 0.0024 . 5 . 1 0 . 3 3 0,0025 3 . 3

5 0 0 17 0 0028 7 f> 0 . 3 6 0 0028 7 7 0 56 0.0030 7 . 5

. . . . . 7 0 0 .11 0.0029 0 . 1 5 0,0022 c 0 . 1 8 0.0019

. . . . . . l . ( j

0 06 0.0018 0 19 0.0027 0.32 0,0030 . ,

. . . . . , . . 0 06 0 002 e 0 18 0.0021 0 31 0 0025

U"/W*

4 . 1 1 3.81 3 .63 3 . 5 3

2 . 7 5 2.61 2.58 2 , .i:i

2.38 2.42 2.12

1 .87 1 .82 1 , 7 0 1.77

2 . 2 8 1 . 9 4 1 .8,5

1 . f 7 1 .<55 1 iio 1 . 0 2

1 . .X 1 . 3 1 .5:3

. .

, .

. .

. .

I . :i i I , 48 I .18 l , , j l

1 . 3 7 1 .38 1 , 36

. .

. . , .

1 ,1:3

c

. .

. . , .

. .

c y R e . . . . , . . . . . . . . . , . . . , . . I I I 128

. . . . . . . . . , . , , . , . . . . . . 0 . 7 9 18fi

. . . , , . . . . , . . . . . . 0.75 242

. . . . . . . . . .

, . . . 2 , f i 3 24

. . , . .

. . . . . . . . .

1 . I(< I

. . . . . 3 . J 6 32

. . . . , , . . . .

2 28 , j5 . . . . . . . . . . . . . . , , . . . . . . 3 . 7 6 22 . . . . . . . . .

, . . . 3.52 3:i . o . . . , . . , . . . . . . . . 20 :3 . : . . . . . . . . . . . , , . . , . .

. . . . . . . . . . . . . . . . .

Lips 1 L 0, 0.0500 0.0476 0,0462 0.0435 0.0473 0 ,0287 0.0286 0 ,0286 0.0286 0,0286 0.0307 0.0312 0.0314 0,0311 0.0281 0 . 0 2 7 7 0.027,i 0,0274 0.0277 0.0332 0.0341 0.0342 0.0338 0.0247 0.0262 0.026G 0.0270 0.0264 0 .0288 0.0268 0.026.4 0.0273 0,02-L2 0.025:i 0.0258 0.0261 0,0254 0,0306 0,0284 0.0277 0.0289 0.025 0.023 0.022 0.0233 0 .020 0.025 0,026 0.023 0.016 0,022 0.024 0.021

of a month, during Tvhich time a iiuniber of run6 were made with mid . The constancy o€ the data indicates no change in thc ioughness of the tube walls. Figure 5 also shows the pressure drops calculated by the Fanning equation using the f line for smooth tubing ( I O ) . The vertical line on the curve for tube 1 is a t a Reynolds number of 2100, a t the transition point between \iscous and turbulent flow. The range of air flows used in tubc 2 as well into the turbulent flow region.

one series of runs also was made in tubc 1 with material A\,

using rarbon dioxide as a carrier gas. Arialysk oi the trends of the apparent solids friction factor

j h indicated the possibility that particle acceleration was con- tributing to the component of pressure drop originally believed to he entirely solids friction. .\ pressure tap therefore, was in- italled on the larger glass riser midway between the original. tapr fo pcrmit Ap measurements across the individual halves. Meas- urements were made a t one air velocity with sand A and two velocities with sand B over the iiiaxiniuin range of solids f l o ~ iatcs. The air velocities were the same as those used in the original measurements. The ratio of Aps in the lower half to that in the upper half remained constant for all values of G, a t R given carrier velocity Data are summarized in Table V.

DATA AND CALCULATION OF RESULTS

.ill oi the measurements 01 ut6, solids flow rate in pounds p ( > ~ hour, and MS, the weight in grams of solids trapped between the two by-pass valves in the riser, wcw plottcd againit ~ p , the. observed riser pressure drop for each series of runs. -\ wriw consisted of measurements a t a constant carrier rate ~ i t h one inatciial axid riser. Examples of these plots are Figures 6, 7 , 8, and 9. In no case mas there a deviation trom a straight-line ivlatiou. In almost every plot the straight lineis extrapoltttd to the eupcrimental value of air friction drop with no solids R o n ta1rt.n from Figure 5 . The value used for the air rate was thv \urn of tiicl inctcrcd carrier air and the slide valve leakage. Thr ail ltiakagr was ralculated by multiplying the observed value for carbon clioxide a t the experimental slide valve Ap, by 1.24- that is, 1/4%% The curvc for uo = 35.4 feet per wcond in Figure 8 shows the greatest deviation from straight-line eutra- polation to A p f o ~ It is included to illustrate one possible sourcc of wror in the data. I n thi3 series, the deviation is believed due to a turbulence that was observed directly above the 1owc.r bypass valve I, but below the lower pressure tap, causing a highvr concentration of solids at that point than elseyhere. Thvit '

June 1949 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY 1155

determine us at the two values of np,/LG, in the lowrr and upper halves. The contribution of acceleration to the apparent j 8

shown in Table IV was approximated by equating __ to

w 29.4 15 8 28.2 a, from Equation 25. The term a/u, equals du/dL, or appioxi-

Inately (u, upper - us lower)/L, where IJ is one half the length ~ P S (upper)

of the test section. This treatment is not completely rigorous because i t assumes the average velocity in each half occurs at thr A P ~ L b V . 1 mid-point, but it serves the purpose of illustrating the large effect on pressure drop of what would appear to be a small amount of A P ~ L (av.)

Table V. Measurements of Pressure Drop in Lower and Upper Halves of 0.5-Inch Riser

____ ~p~ (lower) a 1.47 1 73 2.00

2fsua hIateria1 A L) D QD,

i p S / L (lower) 1.19 1 27 1.34

A P S / L (upper) 0.81 0 . 7 3 0.66

A ~ J L G s (av.) b Lower half Upper half

0.0286 0,0254 0.0289 acceleration.

0,0232 0.0185 0.0191 0.0340 0.0322 0.0387

DISCUSSION 20.6

10.1 19.3 22 1

The data in Tables I11 and IV are smoothed values from lineai ug (av.)a Lower UpperC 12.5 11 0 0024 6 oo28 plots of w,, solids flow rate, and M,, the weight of particles trapped

.r8 due to acceleration 0.0026 0.0018 0,0018 in a known volume of riser, against Ap, observed riser pressure is apparentb

Friction, fa 0.0012 0 0006 Oo10 drop. A total of 224 measurements of ws and 256 of M, was used a Measured.

11 3 10 8 10.3

0 0038

i n the original plots. In Tables I11 and IV the following relations are shown:

At a constant carrier gas velocity, ug, the solids velocity, ti,, and slip velocity Au, can be considered practically constant a t all values of wa.

With one material in one riser, increasing uo results in a lower Iispersed solids density, higher apparent solids friction drop, Apf,,, for the same value of G,.

For one riser, the major effect of decreasing the particle sizc and density is to decrease the Sl1P velocltY.

As the components of ApB other than static head brconie qiI1nller, the slip velocity decreases and approaches the free-falling velocity of the particles. The slip velocities in the larger tube at

b From Table IV. c Calriilated by Equation 16.

11 ns a virud evidence of wine tuibulence a t this point in all the lulls, klut in thls partlcular st111es conditions nere evidently such 515 to inalce the error a maximum.

The slopes of the lines in Eigurc 7 show that at the lowest gas yclocit), 17.1 feet per second, the, value of Ip , /LG, (propor- tlonal to the inverse of the slope on this Plot) is the highest of the three vrlocities used. It is less a t ug = 29.5 feet per second and slightly higher again a t un = 40 4 feet, per second This is in accordancr ni th thc theoretical lines AF'C iii

30

Itisel static pressure was a linear function of Ap 70

60

and static pressure drop, Apz , and

The slip velocity, Au, increases a t the same time.

Flgulc 2

also. rill of the original data were smoothed in this manner, so the data arid ealculations shown in Tables I11 and IV are based on interpolation a t even values of i 50

-IQ. Values of G, and ,ws were calculated directly 2 40

Gl/pdJ. Both Apjg and uB are based on the sum of the g 30

tions 2 and 9 were usrd to calculate Apr and Ap8. The difference between Ap, and Apz was originally considered as solids friction, and is given in Tables I11 arid IV as apparent Apf.. This value was used in ISquation 5 to calculate the apparent values of fa A p , In. He0 A p , In. Hz0

,hewn in the tables. C and Re were calculated from F~~~~~ 6. pressure D~~~ Figure 7. Pressure Drop the average values of A~JLG,, uB, and (Au) for each against Solids Flow Rate a t series. For convenience in calculating C and Rr the Constant Carrier Air Rates Constant Carrier Air Rates expression (0.0O45p,/Dppp) in Equation 16 was grouped for Tube 2 as l /K? , and D p p p / p o of the Reynolds number func-

I1 for each material, based on air at 80" F. and 0.3 inch .of mercury gage pressure. The carrier air temperature

sure from 0.1 to 0.8 inch of mercury, but the average values were considered sufficiently accurate for 3 20 G and Re. However, ug was based on the measured temperature and pressure. Experimental values of C 15 -- against Re are shown in Figure 10.

from tos and M,, and u j from the equation us = -I

iwtered carrier gas and the slide valve leakage. Equa- 20

10

0 1 2 3 4 5 6 7 0 05 I O 15 20 25

against Solids FloC Rate a t

for Tube 1

35

30

~ tion as Kz. Values of K1 and Kz are shown in Table MATERIAL A, TUBE NO I VOL* I 32x1OiFTJ ---- -- 1

I 7 varied from 74" to 87" F. and the riser static pres- 25 L " 0 1

6

" 5 E !?

T I O p 2 To determine a n order of magnitude of the ac-

celeration effect on the apparent solids friction, the measurements of Ap in the upper and lower halves of the larger iiser were used to calculate the mass-average velocity in

I

the two halves, as shown in Table V. The measured ratio 0 .{-IO 15 2 0 2 5 of Ap8 in the lower half to that in the upper half was con- A,, In. Hz0 Ap, In. Ha0

Figure 9. Measurements verted to Ap,/LG, for the two sections, based on the over- F,gure all Value shown in Table Iv for the corresponding sand Density of Dispersed Solids in of Density of Dispersed and air velocity. Equation 16 was used graphically to Tube 1 Solids in Tube 2

Measurements of

1156 I N D U S T R I A L A N D E N G I N E E R I N G

LEGEND POINT MATERIAL

I

IO' IO0 IO' IOP I oa IO'

Figure 10

the loivest air velocity are in fair agreement with the calculated free-falling velocities of spheres of the same diameter, shown in Table 11. I t should be remembered that a nonspherical particle will have a lover free-falling velocity than a sphere of the same volume.

The solids velocities reported in Tables I11 and I\- represent integrated averages during an accelerating period; these were calculated by dividing the mass velocit,y by the average dispersed solids density.

The data for cracking catalysts arc of liniited value, as is evident in the tables. The low fluidized density of these materials, compared to t,hat for the sands (30 pounds per cubic foot, as compared with 80 to 90 pounds per cubic foot) limited the available slide valve pressure drop and therefore the maximum solids flow rate a t maxiinuin valve opening. Solids vcloc- ities were high, causing l o r values of pds. Since the latter value appeared most subject t o experimental error, values of us calculated froin p d s are not very accurate. The estimated value of ug is also questionable, since consistent measuremcnts of the amount of gas leaking t,hrough the slide valve could not be made when the solids in the standpipe w ~ r c in a fluidized condition. However, the data do indicate t,hat with these low-density fine particles, the relative velocity of the carrier gas and solids is small.

In view of the data in Table IT, shon-ing the large effect of solids acceleration on the apparent solids friction drop, t,he small variation of f8 calculated by assuming no accelerattion, as shown in Tables I11 and IV, a t first appears unexpected. HoiTever, it can be shown that a const,ant fractional increase in 2 ( s per unit of length will result in a constant additive increase in .fa for all values of ut. The second and third serics of Table V, using the smallest sand, do show t,he same fractional increase (about 15bz,) in us a t velocities of about 10 and 20 feet per second. The first series, with the largest sand, shows only a slightly greater fractional increase (about 2lyG). There is some indication, therefore, that the fractional increase of us in all runs does not vary greatly. Due to the acceleration effect, the present data offer no rigorous justification for the use of the Fanning equation for correlating the actual solids friction drop. However, after allowing for the acceleration effect in the three series of runs in Table V, the actual frict,ion factor appears t o be practically constant a t 0.001 within the accuracy of the experimental data. The Fanning equation, when used with the solids velocity and dispersed density, is simply a mathematical expression for the state- ment that the particles lose a constant fraction of their kinetic

energy, foot pounds per pound, due to friction bebeen the particles and the wall of the riser, and between the particles themselves. Finally, its form in Equation 10 lends itself to

2g

C H E M I S T R Y Vol. 41, No. 6

relatively simple mathematical treatmcnt. For these reasons its uie is recommended.

Because of the approximations necessary in the analysis of t!ie results, measurements of the accelerating effect were limited in the present work to the runs shown in Table Ti. In a con- tinuation of this problem, one of the phases that \vi11 be studied is the pressure drop in vertical glass and steel pipes in a section where solids accclera- tion is negligiblc. With more accurate determinations of the friction loss, it,s mechanism and the variables which affect it will be better undcr- stood.

The experimental values of the drag coefficient, C, plotted against the Reynolds number based on the slip velocity show remarkably close agreement with the line for free-falling spheres in still fluids, Figure 10. The average deviation of the data for the four sands froin the line for free-falling spheres is represented by the dashed line, with the average C I CO (where CO is the drag

coefficient for free-falling spheres and C the obscrvrd value, both at the same Reynolds number) equal to 1.47. This deviation factor is believed to be due to a combination of the follo-ing: shape of the particles; surface roughness of the particles; the arbitrary expression of the mean particle diameter as the arith- nietic mean of the smallest and largest scieen openings used: the velocity gradient and turbulence of the carrier gas; a hindered settling behavior due to the effect of the particles on the flow pattern of the fluid; and the use of an average solids velocity for cases in w!iic!i particle accelrration was actually orcurring. Of thcsc, an estimated correction for the departure of the particles from a spherical shape brings the C/CO ratio to a more uniform value and nearer unity as shown in Table VI.

Table V I . Values of C/CO and Shape Factor tor Four Sands

h B C D

1 . 2 2 1 . 6 5 1.60 1 . 4 6

1.16 1 . 4 0 1 .25 1 . 2 5

1 OJ 1.20 1 .28 1.17

Thc shapc factors were assigned by Leva (9) on the basis CJf

the photoinicrographs shoxn in Figure 11. The only evidence regarding the velocity gradient and turbulence of the carrier gas is as follows: in two series of runs (materials C and D in the smaller tube) the carrier gas flow, in the absence of solids, was mcll within the viscous region. I\-evertheless, the relation between C and Re did not differ from that in cases n-here the gas flow was definitely turbulent,. This raises a question as to the legitimacy of assuming that the pressure drop of the carrier gas, in the presence of solids, is the same as if the solids w x e absent. The only answer is that deducting this pressure drop from the tota,l pressure drop, as is done in these calculations, results in values of the drag coefficient, C, that are consistent and also quit,e near those for free-falling spheres. The present' lyorlr does not provide information regarding any of the other four items.

The carbon dioxide run checks alniost exactly the data for air in Figure 10, showing that the difference in viscosity and density is accounted for in the correlation. The points for the cracking catalysts E and F are only approximations due to the uncertainty of the calculated slip velocity. Since the slip velocity of a material of this size and density is so low, an error in calculating it has a relatively minor effect on the particle velocity if the carrier velocity is in the usual operating range of 20 to 30 feet per second, typical in t,he riser of a catalytic cracking unit. Recausc


A = 25- t o 28-mesh Ottawa sand B = 35- t o 48-mesh Ottawa sand C = 48- t o 60-mesh sea sand

D = 60- t o 80-mesh sea sand f = 100- t o 200-mesh rnicrospheroidal catalyst F = 100- t o 100-mesh ground catalyst

Figure 11. Material Used

(I scale division = 100 microns)

of the agreement found between thc free-fall line of Figure 10 and the data for sand, where the slip velocities were measurable, it is reasonable to assume that the free-fall line can safely be used for the 100 to 200 mesh particles.

' E F F E C T O F A RANGE O F P A R T I C L E S I Z E S

When the material being transported covers a wide range of particle sizes, as is desirable in fluidized systems, some average diameter must be used in calculating the mass-average slip velocity. If it is assumed that interferencc between particles is so great that all of them are moving a t the same velocity, the diameter of a particle having an average cross-sectional area would be the logical choice, since the force on each particle, by the drag correlation, is proportional to the product of. ( A u ) ~ and A p . However, i t is more likely that a velocity gradient exists, whereby the largest particles are moving at the lowest velocity (with the highest slip). In this case, the average diameter to use would be somewhere between the cross-sectional average diameter and the diameter of the largest particle. By way of example, the effect of diameter on the free-falling velocity of cracking catalyst in air at 900" F. and 10 pounds per square inch gage is as follows: 40 mesh (420 microns), 4.8 feet per second; 200 mesh (74 microns), 0.28 feet per second; 5 microns, 0.001 feet per second. For large risers, where the wall friction is small, the slip velocity can be assumed to equal the free-falling velocity. Some sort of average free-falling velocity presumably would apply for the usual case of a mixture of sizes, but the present work with materials of narrow size range provides no information on this point.

S A M P L E CALCU L A T l ONS

The method of calculation of the values listed in Tables IV and V will be illustrated for material A in the 0.532-inch riser

with a carrier air velocity of 29.5 feet per second. Equations 22 through 25 will be applied also to calculate the change of solids velocity and pressure drop with vertical distance during acceleration of the solids from zero velocity to the equilibrium value.

At 1.10 inch H20 total Ap,

ws .If8 = 1.56 grams from Figure 9 Cross sectional area of riser = 0.00155 square foot G, = 31'6 = 5.66 pounds/(square foot)(second) \701ume of riser between by-pass valves = 6.86 X 10-3 cubic foot

'Oo0 = 0.501 pound/cubic foot Pds = Bverage us = G8/pda = 5.66/0.501 = 11.3 feet/second Distance between pressure taps = 40.1 inches Solids static head = A p , = 40.1 X pds/62.3 = 40.1 X 0.501,'

Air friction ressure drop a t 29.4 feet/second = A p f , = 0.56

A. Table IV:

= 31.6 pounds/hour from Figure 7

0.00155 X 3600

454 X 6.86

62.3 = 0.32 inch of H20

inch of Hs8 from Figure 7 Solids pressure drop Aps = A p - A p f s = 1.10 - 0.56 = 0.54

inch of HzO Apparent solids friction drop = A p B - Apz = 0.54 - 0.32 =

0.22 inch of H20

By Equation 7,

APlQD? = 0.22 X 32.2 X 0.0444 = o.oo38 = 0.385LG,u8 0.385 X 3.34 X 5.66 X 11.3

slip velocity = Au = ug - us = 29.5 - 11.3 = 18.2 feet/second

By Equation 16,

c = APs D P P P Ua LG, 0 . 0 0 4 5 ~ ~ ( A u ) ~

1158 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY Vol. 41, No. 6

- D B p p __ = 808 for material A in air ( K , in Table 11)

Re = n__ D (Au), ,

0 .0045~~

0 5 4 X 808 X 11.3 (J-L -___~ 3.34 X 5.66 X (18.2)2 0‘79

MU

’2’ = 10.2 for material A in air ( K 2 in Tablc 11)

Re = 10.2 X 18.2 = 186

These values of C arid RP fall evartlp on the frw-fall iiI1cl of Figure 10. R. Acceleration measureinellti, Tahlr V:

Po

In measuring the effect of solids acceleration 011 the pressure drop across the two halves of the riser, Ape (Iowcr)//Ip, (upper) was found to be constant at 1.47 over a range of solids flow rates for material -4 at an average ug of 29.4 (Table V). This ratio and the over-all value of Ap,/LG, of 0.0286 from Table I T T were used to calculate apg/LG, for the lower and upper sections at 0.0232 and 0.0340, respectively. By Equation 16, Aps/LG8 was plotted against us, using the free-fall line of Figure 10 to relaw C to ( A u ) since the average C and Re fell on this line. This plot showed us to be 10.3 at Ap,/LG, = 0.0340 and 12.5 at’ 0.0232. It was assumed that these velocities occurred at the mid-points of the two halves.

The acceleration component of Equation 25

Solving for the contribution of acceleration to , f a ,

For the same conditions as above (material A, 0.532-i tirh risw, average ? i o = 29.4 feet /second)

-AB- = 0.206 recipi,oca~ tcct 4P,D,

aPSUlxlej* = 0.001

‘1s = 0.0454 ,eciproc:al feet D,.

u !viis calculated over a range of ing to a = 0.

from tcs = 0 to us corlespi)iiti- Examples arc given below:

0 2 9 . 4 300 0 .65 116 0 84 0 2 3 . 4 239 0 . 7 2 81 ..? 1 .6 1 7 , .?I

14 15.4 167 0 , 8 7 1 2 , a 8 . 9 1 . 1 I l e = Kz(Au,) = 10.2 ( A L L ) .

b From the free-fall line i n Figiirc 10, sirlcr ri ir? ohaer ied ai’elaae R e a n d C fit thiq l ine.

Values of a 5 0 obtained arc plottcd agaiiist n, in Figurca I2 To calculate, 1, as R function of u,, the (the curvcd solid line).

folloiring integration m-aa perfozined

(24)

This can bc don? gi,aphically by plotting exact values of u,/a against I / , FTowrver, littlc (war is ~ntroduced by using the

US, Ft./Sec.

Figure 12. Plot of Acceleration Equations

(Sample calculation)

dashed straight line in Figure 12, rh ich permits a mathc.niat,ical integrat,ion. The straight line must coincide as nearly as possi1,lc

with the curve a t low values of (1, where u. is a largc: t 1 u 1 1 d ) c ~ l ~ .

At high values of a, tho diffcrencc bct,wcen the two lincs l i a h a niinor effect on t’he integrat,ion brcauao here u / n is small. ( t i t !

straight line,

(2 ( i j ( I

1 1 , (1 = a;, - x ) i

wlierc U A is the: j/-int,c, pl, = N!: ur = cquilibriuni u s = 14.2s. Substituting t,liis expi io11 for (I in l?qiiat:ion 24 and it i tvgi~ati i~g ht,tn-cen = 0 nnct w1 = i i r rewilts i l l

- ‘ 1 > ( N i l ) I& l i e

(1 l J g - I / ,$ 1, = ~ ii, 111

1i:rainples of L cdculatcttl i t r 1 his mtiiiii(~r tire t is follons:

I& = 0 2 1 0 I , = 0 0 .03 1 .29 7 . 5 2 feet

This luiiction is shown in Liiguro 12 as thc curved lint: I,! ‘rll(~J- retically, the eyuilibrium velocity is never reached. H ~ ~ Y C Y W , SSTl of the final velocity is attained in this casc a t 7 . 5 f e ~ t abovct the t,hcoretical aero I c ~ ~ e l . In the prcsent, casc, in which t11(% vertical section is preceded by n bend, the zero levcl c:annoi. I)c ascribed any physical significance until more is learned n twut llrc niechanism of flow of the solids in curvc~I sections.

L = Distance, Ft.

Figure 13. Pressure Drop and Solids Velocity against Distance

during Acceleration

(Sample calculation)

June 1949 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY 1159

1Squation 25 was used to calculate the differential pressure drop as a function of us.

-1 constant value of 0.001 was again assumed for f 8 . At us = 10, x, (2) = 0.1925 0.100 + 0.0014 X 10 4- ( a&) = 0.0361

Similarly, other values are as follows:

us = 2 5 12 14 14.1

1 - - ( d p / d L ) = 0.312 0.124 0.02A2 0.0181 0.0178 G,

This relation is plottrd also in Figure 12 as the curve marked d l ) .

The values o€ as (dp/clL) and us are rcplotted in Figure 13

itgairiat L as abscissa. I n this form, the area under the dp 4 s c'urvc' (line A) represents total - between any two values of G,

/.. A considerable distance must be traveled before the accelerating ctffect becomes insignificant.

This theoretical approach was substantiated by the measurements of Ap in the two halves of the riser (Table V). The dis- tancc between the outer pressure taps, length ab, was positioned by t i ial-and-error so that the integrated average -'- " that is,

the area under the curve between the abscissa values of a and b -corresponded to the measured over-all value of 0.0286. The abscissa at d' then corresponds to the center pressure tap. The ordinates of cd and ef are the measured values of Ap,/LG, in the lower and upper halves, respectively. The agreement between these measured values and the integrated average values for the two halves (obtained by measuring areas) is excellent. L\lso, the integrated average solids velocity from curve B in Figure 13 over the length ab agrees with the measured average us of 11.3. These two agreements confirm the above method of iriterpretation of the observed data.

Curve A, Figure 13, also indicates that the distance corresponding to a hypothetical zero velocity (u, = 0, L = 0) was immediately above the lower by-pass valve, which was 9 inches below the bottom pressure tap. This probably means that con- stdwable interference to solids flow occurred due to the combined ;tction of the bend and the by-pass valve.

The pressure drop required to accelerate the solids cannot be calculated by multiplying the increase in kinetic energy of the particles (foot pounds per pound) by the solid-to-gas flow ratio (pounds solids per pound of gas) to yield foot pounds per pound of gas, because such a calculation assumes 100% energy transfer troni the gas to the solids. Actually, the efficiency of energy tiansfer is at all times equal to us/uB. For any slip velocity, there exists a definite force F , exerted by the gas on an individual particle. I n one second the mechanical energy lost by the gas = F p X u,, while the mechanical work done on the particle is only F,, x 2L".

1

LG,

APPLICATION

111 a fluid catalytic cracking unit spent catalyst riser, 6 feet in iriside diameter, air at 900' F., a n average pressure of 10 pounds J ) C ~ square inch gage, and a linear velocity of 25 feet per second is used to transport catalyst from the reactor slide valve to the rrgenerator. A differential pressure recorder connected to taps across the upper half of the 100-foot vertical duct (where solids ncceleration is negligible) shows a reading of 0.25 inch of water per foot. With the same air flow and no catalyst circulation, the meter indicates a zero pressure drop.

1. What is the catalyst flow rate if the effective average particle diameter is (a) 50 microns; ( b ) 100 microns? Assume th&t the solids friction drop is negligible.

2. Show that the solids friction drop is a negligible fraction of the total pressure drop. The value for f s for cracking catalyst in a brick-lined pipe is not known, but use values of 0.001 and 0.004 to cover the possible range.

SOLUTION

1. Keglecting solids friction, Equation 10 becomes

Apa 0.1925 or G - APsua 8 - - 0.1925 L mi = - US

It has been shown that if the solids friction is negligible, the slip velocity, Au, will be equal to the free falling velocity of the particles.

(a) D, = 50 microns = 1.64 X feet.

For particles of this diameter, free-fall is in the viscous region where C = 24/Re (Figure 10). With no solids friction, the forcr on one particle will be equal to its weight or (?/6)D;(pp - p q ) . Substituting these values of C and F in Equation 11 results 111 Stokes' law

With p p equal to 61 pounds per cubic foot (Table 11) and pu = 0.032 X 0.000672 pound per (foot)(second)

Au = 0.14 foot per second

U a = ug - 471 = 25.0 - 0.1 = 24.9

From Rquation 26

0, = 0'2:,:2y'9 = 32.4 pounds per (square foot)(secontl)

32.4 X 3600 X 0.785 X 36 2000

- = The catalyst flow in tons/hour =

1650. ( b ) D, = 100 microns = 3.28 X feet

In air a t these conditions, a particle this size still conforms to Stokes' law during frec fall, so

AIL = (100/50)2 X 0.14 = 0.56 foot per second

(This corresponds to Re = 0.42, which is well in the viscous range, Figure 10, showing that Stokes' law is applicable.)

U . = 25.0 - 0.6 = 24.4

0.25 X 24.4 = 31.7 0.1925 G, = ,

In tous per hour, catalyst flow = 31.7 X 1.8 X 28.3 = 1615. Thus, the value used for the particle diameter is of minor im-

portance in this calculation when the air velocity is high compared to the slip velocity.

2. Effect of solids friction:

0.192511/u8 + (2f,lgD,)u,l (10) LG,= If the solids friction is appreciable, the slip velocity will be higher than that calculated for free-fall. Arbitrarily using a value of 24 for ul, the relative values of static and friction pressure drops can be calculated over'a range offs

fs = 0.001 0.004

0.0416 0.0416 1 Relative static Ap = U, Sf&, 0.0003 0.0011

Total 0.0419 0.0427 Relative friction Ap = -- gDr

Friction Apasaper centof thetotal 0.7% 2.6% This shows that for all practical purposes, the solids friction

drop can be neglected in this articular problem and Equation 26 can be used directly with a s i p velocity equal to the free-falling velocity.

1160 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y VoL 41, No. 6

This problem illustrates how, without knowing exactly what effective particle diameter to use for a mixture of sizes and the value of the solids friction factor, f8, a relation between solids flow rate and pressure drop can be estimated. This value of pressure drop holds only in that part of the vertical riser where particle acceleration is negligible. It should be emphasized that in a complete system the pressure drops due to initial particle acceleration, bends, and acceleration after bends can be a major portion of the total, indicating the need for considerable experimental work on these phases of the problem.

CONCLUSION

It has been shoivn that in a vertical riser through which solid particles are transported by a stream of gas, the total pressure drop can be considered the sum of that due to the gas flow alone, as though no solids n-ere present, and a solids pressure drop, The latter consists of a solids static head, a solids friction loss due to contact beheen the particles and the pipe wall, and, for a considerable distance above the point where the solids start to move in a vertical direction, an acceleration pressure drop.

The solids static head can bc expressed as the vertical distance times the dispersed solids density; the latter is related to the solids mass and linear velocities. For the friction loss component, the Fanning equation, used with the solids linear velocity and dispersed solids density, is recommended. Due to a relatively large amount of acceleration pressure drop in the present experiments, the solids friction loss could not be measured in- dependently. However, after allowing for the acceleration effect the order of magnitude of the Fanning friction factor for sands in glass tubing is about 0.001.

The pressure drop that is required to produce acceleration of the particles in the lower section of a vertical riser was shown by measurements and by a mathematical analysis to be a significant portion of the total. An extremely high pressure drop is found to be possible at the point of introduction of the solids into the gas stream. Equations were developed for calculating the variation of pressure drop and solids velocity with distance in the accelerating section.

At any time, the solids velocity corresponds to a state of dy- namic equilibrium, in which the rctarding forces of gravity, friction, and inertia must be exactly equaled by the force exerted by the gas OII the particles. The latter force was shown to be related to the slip velocity by thc drag correlation for free-falling particles in still fluids. This is a completely unexpected situation, since the turbulent flow conditions existing in the transport of solid particles by a gas bear no resemblance to those for the free fall of individual particles in a still fluid. Deviation from the relation for free-falling spheres is believed mostly due to the ir- regular shape of the particles; this can be corrected for by an appropriate shape factor.

To extend the theory of particle transport, further experimental M-ork is needed. The following are some of the many interesting problems that have come to mind during the course of the present work:

Effect of a mixture of particle sizes Horizontal and inclined pipes Friction losses in commercial piping Losses in bends Effect of a bend on the acceleration drop immediately following

Losses a t the point of entrance of the solids into the gas stream it

N O M E N C L A T U R E

a = acceleration of particles, feet per second per second A , = projected area of a particle, square feet A , = cross-sectional area of the riser, square feet C = drag coefficient = 2Fpg/p,( A U ) ~ A ~ , dimensionless CO = drag coefficient for free-falling spheres from Figure 10,

dimensionless

D, = diameter of a part cle, feet D, = diameter of the riser, feet F = force, pounds F, = force on one particle, pounds fs = Fanning friction factor for the solid particles, dimension-

G, = mass velocity of solids, pounds per second per square foot g = acceleration of gravity, 32.2 feet per second per second K1 = grouping of the constant and solid and air properties in the

drag equation (Equation 16), equal to D,p,/0.0O45pv, inches of water times square seconds per pound

KZ = grouping of the solids and air properties in Reynolds number, = Dppy/,uo, seconds per foot

L = vertical distance in riser, feet Ma = observed weight of solids between the riser by-,pass valves,

grams A’ = number of solid particles in L feet of riser Re = Reynolds number based on the particle diameter and slip

velocity = Dp( AIL j po /pg , dimensionless u0 = gas vclocit,y, feet per second tia = mass-average velocity of solids particles, feet per second ut = terminal free-falling velocity of a part,icle, feet, per second ws = flow rate of solids, pounds per hour A p = total observed pressure drop across L feet, inches of water Apf, = pressure drop in L feet due to gas friction, inches of water Ap/,= pressure drop in L feet due to solids friction, inches of

Ap, = total solids pressure drop in L feet = ApJ“a + Ap,,

ApZ = solids &tic pressure drop in L feet, inches of water Au = slip velocity = tio - 2 1 , feet per second X = particle area-volume shape factor, dimensionless po = gas viscosity, pounds per foot second pda = dispersed solids densit,?, pounds of dispersed solids per

po = gas density, pounds per cuhic foot, p p = particle density, poufids per cubic foot,

less

wat,er

inches of water

cubic foot

LITERATURE C I T E D

(1) Chatiey, H., Engineering, 149, 230 (1940). (2) Cramp, W,, Chemistru &: Industru, 44, 207 (1925). (3) Dalla Valle, J. M., “Micromeritics,” 1st ed., New York, Pitman

(4) Daniels, L. S., Petroleum Re.fi?zer, 25, 435 (1946). (5) Hudson, W. G., Chem. & X e t . Eng., 51, 147 (1944). (6) Hudson, W. G., “Conveyors and Related Equipment,” X e F

(7) Jennings, M., Engineering, 150, 361 (1940). (8) Lapple, C. E., and Shepherd, C. B., IBD. ENG. CHEM., 32, 606

(9) Leva, bl., personal communication.

Publishing Corp., 1940.

York, John Wiley & Sons, 1944.

(1940).

(10) Perry, J. H., “Chemical Engineeis’ Handbook,” 2nd ed., p. 807,

(11) Ib id . , p. 1852. (12) Vogt, E. G., and White, R. R. , IND. ENG. CHEV., 40, 1731 (19481. (13) Webb, G . M., Petroleum Processing, 2, 397 (1947), RECEIVED February 2 5 , 1919.

New York, McGraw-Hill Book Co.. 1941.

COURTESY S T A N D A R D O l L C O W P A N Y (NEW J E R S E Y )

Fluid Catalyst being Drained from Petro- leum Cracker during Plant Shutdown

Pressure drop in pipeline

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