Pressure Drop in Vertical Tubes in Transport of Solids by Gases can be regarded as t h e sum of t h e drop due t o t h e carrier gas alone plus a solids pressure drop. T h e residual obtained by subtracting t h e solids static head from t h e total solids pressure drop was treated as an apparent solids friction drop, which conformed t o t h e Fanning equation when solids velocity and dispersed solids density were used. However, later measurements showed t h a t t h e m a j o r portion of t h e apparent friction was due t o particle acceleration in t h e test section. T h e mathematical treatm e n t of t h e accelerating period is presented. Drag correlation for free-falling particles i n still fluids relates relative velocity of gas and solids t o force required t o overcome inertia, gravity, and solids friction.
M e a s u r e m e n t s of total pressure drop and static pressure drop in t h e transport of solid particles through vertical 0.267- and 0.532-inch inside diameter glass tubes by a n air stream are reported. Closely sized sand (28- t o 35-, 35to48-,48-t060-, and60- to80- meshTyler),and bothground and spherical silica-alumina cracking catalyst were used. Solids circulation rates ranged f r o m 2 t o 54 pounds per second per square foot a t various constant air rates f r o m 0.9 t o 3 pounds per second per square foot, equivalent t o 12 t o 40 feet per second. Direct measurements of t h e dispersed solids density were made; these permitted t h e average particle velocity, slip velocity, and solids static pressure drop t o be calculated. T h e t o t a l pressure drop
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 .
0”
E of the inherent advantages of a fluidized system is the 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 a n 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 t o 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. I n 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 t o 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
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.
I
1
Present address, Atlantic Refining Company, Philadelphia, Pa.
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 d i p velocity t o be 10 feet per second. I n 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 suinmarized 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 conveyers are strictly empirical. Jennings (7’) shows a method of calculating the accelerating distance in vertical transport. Chatley (I), in calculating the poiver requirements of a grain convever, 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
INDUSTRIAL A N D ENGINEERING CHEMISTRY
lune 1949
I n 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 t o t h a t obtained with the carrier gas alone, the weight ratio of solids flow t o 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
UsPds
(1)
Tn the experimental work, uswas calculated from measurements of
G, and
Pds
x
L
12/62.3 = 0.1925pd,L,
(2)
01'
Ap,/L
=
(3)
0.1925 Gg/u8
Expressing this as pressure drop per foot per unit of mass velocity,
Apn/LG, = 0 . 1 9 2 5 / ~ ,
(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, (5)
Ap.f8 = 2 f,Lu3pdS(O.1925)/gD,
Zxpressing pds in terms of us
An analysis of measurements of all the pressure drops encountered 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, t h e 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 t o 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 t h a t required for solids friction and potential energy change; if the over-all pressure drop is a n 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 t o (u,- us),and exert a force, F p pounds, on each particle. The drag correlation, as shown below, relates Au t o 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 a r e 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 =
APz =
1149
Pds.
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 t o L feet of a fluid of density pds. Changing t o a unit of pressure commonly used in fluidized systems, the solids static head in inches of water is
AptJ = 0.1925 X 2f,Lu,2 G,lgD;lc,, or
Ap/,/LG, = 0.1925 X 2 fszcS/gDr
(6)
(7)
A manometer connected across length L will read 4 p , the sum of a pressure drop due to gas friction against the walls, ( A p ~ u ) , plus the solids friction loss plus the solids static head, or AP
Ap/o
f A ~ j a+
(8)
ATJ~
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 t o a slight degree by the presence of the particles. Consequently, i t is reasonable t o assume t h a t the gas friction loss is the same as if the solids were absent. The total solids pressure drop, from Equation 8, is Aps
A p - Ap~p/o= AP,
4-
(9)
Apfa
Combining Equations 4 and 7 ,
Apa/LGs = 0.1925 [I/'us 4- (2fa/gDr) X
(%)I
(10)
Equation 10 is one of the basic equations used in the analysis of the experimental data. It is in accordance with earlier observations t h a t A p / L is a linear function of G,, with its origin at A p f g / 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 t h a t 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. -4second relation between A..o,/LG, and u8is necessary t o 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 , / 92 =
7rCpo(
Au)'D;/8 g
(11)
where C is a function of the Reynolds number, Dp(A u ) p , / ~ , . The number of particles in L feet of riser is
The total force on all the particles is
F = ( T C P AAUI2Dg/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, Converting to inches of water and substituting G8/uIfor p d s ,
(14)
INDUSTRIAL AND ENGINEERING CHEMISTRY
1150 Ip,/L
=
0.1925 X 3
Cp,(
AZL)~G,/-I gDnpP1/*or
Ap,;LG, = (0.0045/Dp)( p o / p p ) I ( Au)'/u81 C'
(15) (ltij
Simult'aneous solution of Equations 10 and I 6 anioiiii ts to a force-balance in which t,he force required t o 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. Then A u = uo - us. 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 1 6 aids in explaining their significance. I n Figure 2, lines A C 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 71r 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 a t atinospheric 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 iritersections G arid P shows the effecl o i decreasing thc size of thv riser a t a constant air velocity. The solids velocity, u . , tlrcreases 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) o r ,IC becomes practically coincident with line A B . The in1 drops to J and the slip velocity, 25 - 11.9 = 13.1feet pw sc:corid, corresponds to the free-falling velocity of the spheres. This citn be considered t o 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 l T 2 at v-hic'l lift work is done on the solids, or EIp. output = G,A,L/SSO
(17)
T h o 1m\ver input is t h a t lost by the air aiid cdii be ~ ~ o ~ ) r i ~ s cI)? iittd
Hp. input
=
&pug AT/(0.1925) X (550)
(18)
+
+
R-liwe Ap = Spja Apj\ A p z in inches of wat,er. T h e mechanical efficiency, hp. outlhp. in, is a maximuin when riser diameter is so largc as t o 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)
It must be remembered that this analysis is concerned o i i l v with a vertical section and one in which acceleration of gm rtntl solids is negligible. PARTICLE 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 t o the force ieq to accelerate the particle.;. In an effort to distinguish bct the frictional and acceltliatioii components, the following niai 11( matical treatment was derived. Applied to ono pa1 ticlr, thc i i c ' t u p w x d acceleration ( * A I I i l l t ~ pcscled r as
~ v l i t wthe 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 ( A u ) and C' arc. kiion 11, 101 a specified gas velocit\ / i o ,a can b(>calculated for all valuc.5 o r 11, from zero to the poilit u-licire a = 0. T h e lattcr c . o r r i y ) o n d ~to the equilibrium solitli vrlocj ty. T h r time-velocity and dislaiiuc-vt,loc.ity relations caii be 011t a i n d by giaphicnl intcyqi atioii of the L01101\ ing tv,o equatioiis: i2:3i
I-k~uatiori23 is iritegratcd by plotting I ,'CL (as ordinatt.) agaiii-~ 0,is calculated by TGquation 22. T l i e time rcquircd to w : t ( > l i velocity u2 from ul is thc area under thc curve. Similarly, l:qii:ition 24 is integrated by plot,tiiig u,/a 8s ordinate against u s . Finally, Equation 10 can bo expressed differentially t o iricliitlr~ t l i ~prttssure drop rryuircid lo :tc:c:i~lc,ratethe particles: lis.
(25)
Hy t,his equation,
= U9/i/[,
( 20 1
~i,/ic~
-
cari tjc piottod as a function of
iii.
ii'
(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
largo.
f i i ~
(k)
arid the acceleration pressure drop
Th(~or(.t,ically, ~l.rcrc,I L ~is zero,
ih(2)
--I
((2
is infinite,
Portunatrly this oarinoh occur actually, sirice for a finit,c G, i u i d u s , the solids corici~iitr~atioiipds would be iiifinitc:, w h i r ~ l ~
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,"($1)
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,
t I1 ti s t at>icIh car1
.k:ff.,,,,,
Vol. 41, No. 6
%
i t r 1 his mtiiiii(~rtire tis follons: 10
0.03
1 .29
7 . 5 2 feet
This luiiction is shown in Liiguro 12 as thc curved lint: I,! ‘rll(~Jretically, 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 ~ abovct t the t,hcoretical aero I c ~ ~ e l I. n 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 t w u t llrc niechanism of flow of the solids in curvc~Isections.
0.0454 ,eciproc:al feet
u !viis calculated over a range of from tcs = 0 ing t o a = 0. Examples arc given below:
0 0 14
111 lJg
(1
to u s corlespi)iiti-
0.65 116 0 84 0.72 81 ..? 1.6 1 7 , .?I 167 0,87 1 2 ,a 8.9 1.1 I l e = Kz(Au,) = 10.2 ( A L L ) . F r o m the free-fall line i n Figiirc 10, sirlcr riir? o h a e r i e d ai’elaae R e a n d C 300
239
15.4
b
fit thiq line.
Values of a 5 0 obtained arc plottcd agaiiist n, in Figurca I 2 (the curvcd solid line). To calculate, 1, as R function of u,,the 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
L = Distance, Ft.
Figure 13. Pressure Drop and Solids Velocity against Distance during Acceleration (Sample calculation)
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
June 1949
1Squation 25 was used to calculate the differential pressure drop as a function of us.
-1constant value of 0.001 was again assumed for f 8 .
x, (2)
=
(
0.1925 0.100
+ 0.0014 X
At us = 10,
1. What is the catalyst flow rate if the effective average particle diameter is (a) 50 microns; ( b ) 100 microns? Assume th&tthe solids friction drop is negligible. 2. Show t h a t 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 t o cover the possible range.
10 4SOLUTION
a&)
= 0.0361
1.
Keglecting solids friction, Equation 10 becomes
us = 2 - - ( d p / d L ) = 0.312
G,
12
14.1
0.124
0.02A2 0.0181
14
0.0178
This relation is plottrd also in Figure 12 as the curve marked dl).
The values o€
as(dp/clL) and
itgairiat L as abscissa.
us are rcplotted in Figure 13
I n this form, the area under the d p
c'urvc' (line A) represents total
between G,
4
s
8
-
-
US
5
1
-
0.1925 or G
Apa
mi = -
Similarly, other values are as follows:
1
1159
APsua 0.1925 L
It has been shown t h a t if the solids friction is negligible, the slip velocity, Au, will be equal t o 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
any two values of
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 distancc between the outer pressure taps, length ab, was positioned by /..
so t h a t the integrated average -" '- that is, LG, the area under the curve between the abscissa values of a and t i ial-and-error
b -corresponded t o the measured over-all value of 0.0286. The abscissa at d' then corresponds t o 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 t o 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 t h a t constdwable interference t o 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) t o yield foot pounds per pound of gas, because such a calculation assumes 100% energy transfer troni the gas t o the solids. Actually, the efficiency of energy tiansfer is at all times equal t o 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". 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 t o 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.
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 Ua =
ug -
471 =
25.0 - 0.1 = 24.9
From Rquation 26
0,=
0'2:,:2y'9
= 32.4 pounds per (square foot)(secontl)
The catalyst flow in tons/hour
=
32.4 X 3600 X 0.785 X 362000
=
1650. ( b ) D, = 100 microns = 3.28 X
feet
I n 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 t o 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 G, = , 0.1925 I n 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 importance in this calculation when the air velocity is high compared to the slip velocity. 2. Effect of solids friction:
LG,=
0.192511/u8
+ (2f,lgD,)u,l
(10)
If the solids friction is appreciable, the slip velocity will be higher than t h a t 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 f s = 0.001 0.004 1 Relative static A p = U,
Relative friction Ap
Sf&, = -gDr
Total
Friction A p a s a p e r centof thetotal
0.0416 0.0003
0.0416 0.0011
0.0419
0.0427
0.7%
2.6%
This shows t h a t 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 t o the free-falling velocity.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1160
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 t h a t part of the vertical riser where particle acceleration is negligible. It should be emphasized t h a t in a complete system the pressure drops due t o 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 t h a t 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 b e h e e n the particles and the pipe wall, and, for a considerable distance above the point where the solids start t o 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 independently. 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 t h a t 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 dynamic 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 irregular shape of the particles; this can be corrected for by a n 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:
D, D, F
VoL 41, No. 6
= diameter of a part cle, feet
diameter of the riser, feet force, pounds F, force on one particle, pounds fs Fanning friction factor for the solid particles, dimensionless 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 t h e drag equation (Equation 16), equal t o D,p,/0.0O45pv, inches of water times square seconds per pound K Z = 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(AILj p o / p g , 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 w s = flow rate of solids, pounds per hour A p = total observed pressure drop across L feet, inches of water A p f , = pressure drop in L feet due to gas friction, inches of water Ap/,= pressure drop in L feet due t o solids friction, inches of wat,er A p , = total solids pressure drop in L feet = ApJ“a Ap,, inches of water 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 cubic foot po = gas density, pounds per cuhic foot, pp = particle density, poufids per cubic foot, = = = =
+
LITERATURE CITED
(1) Chatiey, H., Engineering, 149, 230 (1940). Chemistru &: Industru, 44, 207 (1925). (2) Cramp, W,,
(3) Dalla Valle, J. M., “Micromeritics,” 1st ed., New York, Pitman
Publishing Corp., 1940. (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 York, John Wiley & Sons, 1944. (7) Jennings, M., Engineering, 150, 361 (1940). (8) Lapple, C. E., and Shepherd, C. B., IBD.E N GCHEM., . 32, 606 (1940). (9) Leva, bl.,personal communication. (10) Perry, J. H., “Chemical Engineeis’ Handbook,” 2 n d ed., p. 807, New York, McGraw-Hill Book Co.. 1941. (11) I b i d . , 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.
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 it Losses a t the point of entrance of the solids into the gas stream NOMENCLATURE
a
A, A, C CO
= acceleration of particles, feet per second per second = projected area of a particle, square feet = cross-sectional area of the riser, square feet = drag coefficient = 2Fpg/p,( A U ) ~ A dimensionless ~, = drag coefficient for free-falling spheres from Figure
dimensionless
COURTESY
10,
STANDARD O l L COWPANY
(NEW J E R S E Y )
Fluid Catalyst being Drained f r o m Petrol e u m Cracker during Plant Shutdown
