Pressure Drop in Vertical Solid-Gas Suspension Flow Javaid I. Khan and David C. Pei* Chemical Engineering Department, Cniversity of Waterloo, Waterloo, Ontario
There has been a well-recognized need in the literature to improve the reliability of existing correlations for predicting the pressure drop in dilute phase vertical solid-gas suspension flow. To accomplish this, most of the available data in literature were collected and experiments conducted to extend the range of parameters. On the basis of these data, a comparative study of the existing correlations was made and a more reliable predictive correlation i s proposed.
T h e r e are a number of industrial problems which involve solid-gas suspension flow such as pneumatic conveying, combustion of solid fuels: catalytic cracking, cooling systems for nuclear reactors, etc. I n all cases, the knowledge of pressure losses is probably the most fundamental parameter for design purposes. Approximately fourteen both dimensional (Belden a d Kassel, 1949; Cramp and Priestley, 1924a, 1924b, 1 9 2 4 ~ ; Jones and A\llendorf, 1967; Leung, et al., 1971; Razumov, 1962; Reddy, 1967; Sproule, 1961) and dimensionless (Bootliroyd 1971; Farbar, 1949; Ghosh and Preni Chand, 1968; Hariu and Molstad, 1939; JIetha, ef al., 1957; T'ogt and White, 1938) correlations are available in the literature for predicting pressure drops in vertical dilute phase solid-gas suspension flon-s. Generally in developing such expressions the iiiveatigators have used their own experimental data covering only a limited i'aiige of variables in their respective systems. As a result, there are serious disagreemelit. among them aiid colisiderable aniouiit of uncertainty over the accuracj- or validity of these correlations. It is with this in mind t'hat a comparative study was made on all the available data iri literature having almost ,similar experimental conditions (Cliandock, 1970; Farbar, 1949; Hariu aiid Xolstad, 1949; Jones and -,lllendorf, 1967; Reddy, 1967: Vogt and Khite, 1948) as n-ell as the data collected iii this investigation whirli extends certairi range of variables. In developing a more reliable correlation, the following two criteria were used : (i) the correlatioii should be dimensionless in format and (ii) it should he based 011 readily available physical properties of the system. literature Review
I n view of the limited space only a brief review of the important discrepancies among all the investigat'ors are cited below. (i) Buoyancy Effects. Some researchers (Boothroyd, 1966; Ghosh aiid Prem Chand, 1968; Razumov, 1962; Vogt and Khit'e, 1948) have shon-n t h a t the density ratio or the buoyancy effect is a n important correiat'ing factor, whereas the others (Belden and Kassel, 1949; Cramp and Priestley! 1924a, 192413, 1 9 2 4 ~ ;Farbar, 1949; Hariu and Xolstad, 1949; l f e t h a , e! al., 1957; Sproule, 1961) have not shon-11 such a depeiidenq-. Probably they thought that the loading ratio itself takes care of the density ratio. (ii) Diameter Ratio. I t has been reported by some n-orkers (Cramp and Priestlej-, 1924a, 1924b, 1 9 2 4 ~ ; Khan, 1972; Metha, et al., 1957; Razumov, 1962; T'ogt and White, 1948) that the diameter ratio (particle to pipe) 428
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
does affect pressure drop. This was found untrue by the others (Belden and Iiassel, 1949; Boothroyd, 1966; Farbar, 1949; Ghosh arid Prem Chand, 1968; Hariu and Molstad, 1949; Jones and Xllendorf, 1967; Leung, et al., 1971; Sproule, 1961). Boothroyd (1966) rioted that a diameter ratio below the value of 0.007 does effect the pressure but' not beyond that. (iii) Loading Ratio. It is agreed by almost all investigators t h a t the loading ratio definitely contributes to the pressure losses, though some have shown (Richardson and McLeiinan, 1960; Tchen, 1947) a decrease i n pressure drop by addition of solids. Thomas (1962) attributes the decreaRe in pressure drop over a certain range of particle loading to the non-Sewtonian behavior of the suspension. (iv) Reynolds and Froude Numbers. Generally it' is agreed bb- all workers t h a t there is a n increase in pressure losses with the increase of Reynolds number in vertical pneumatic conveying. The ratio of the inertia forces to the gravitational forces (usually known as Froude number) as a correlating parameter for pressure gradients was used only by Ghosh and Preni Chand (1968) aiid Stemerding (1962). Duckworth 11971), on the ot,her hand, suggested that the influence of Reynolds number is insignificant and only the Froude iiuniber is of considerable importance. He remarked that the inclusion of t,he Reynolds number as a significant variable probably st,ems from the failure in studying the separate effects of the Reynolds aiid Froude numbers. (v) Shape Factor of Particles. The cont,ribution or the effects of the shape of the solid particles on the pressure gradients were first. thoroughly studied by Jones and *%llendorf (1966, 1967). This was supported by Leung, et al. (1971), and Razumov (1962). However, the majority have found that these effects are negligible. (vi) Ratio of t h e D r a g Coefficient of t h e Solid Particle to the Friction of the Pipe. The dependence of pressure drop on the ratio of drag coefficient of a solid particle to the friction of the pipe was first observed by Ghosh and Prem Chand (1968). No such parameter has been taken into consideration by other workers. (vii) Pressure Drop Data. There are abundance of experimental data on pressure drop in the literature o n vertical solid-gas suspension flow under a variety of experimental conditions. Cramp and Priestley (1924a, 1924b, 1924c) made the pressure drop measurements a t pressures less t h a n t,he at,mospheric pressure. Beldeii a n d Icassel (1949) took the pressure gradient' readings which include the acceleration effects. Boothroyd's data are limited only to particles less than 30 p in size. Ghosh and Prem
Table I Range
Variable
Density of the fluid, lb/fta Density of the solids, lb/ft3 Diameter of the particles, ft Diameter of the pipes, ft Fluid velocity, ft/sec Froude number Loading ratio Reynolds number Shape factor
0,060474-0.091312 40.0-476.0 0.000132-0.0191 0,041-0.33 8,(5-160 9.4-1 21 .8 0.4-28.8 3500-100.000 0.41-1.0
-" 0
e 0
ONE DATA POINT TWO DATA POINTS THREE DATA POINTS FOUR OR MORE DATA POINTS
/' /'
;
4
Figure 1 . Sketch of the equipment: ( 1 ) air blower housing, (2) pressure relief valve, (3) air bleed valve, (4) air tank, (5) flexible hose, (6) rotameter, (7) mixing tee, (8) calming section, (9) pressure drop measurement section, ( 1 0 ) expansion section, ( 1 1 ) cyclone separator, ( 1 2) solids control valve, ( 1 3) solids feeding line, (14) solids feed tank, ( 1 5) flow observation section, ( 1 6) solids control valve, ( 1 7) solids sampling valve
PREDICTED A Ps / A Pf
Figure 2. Analysis of the proposed correlation
Chand (1968) proposed a correlation applicable to vertical solid-gas suspensioii flow, but the data used were collected in a horizoiital test section. Lloreover, some of the workers (Capes, 1971; Lapidus and Elgin, 1961; Razumov, 1962; Zeiiz, 1949) took the pressure drop data in the dense phase flow. Therefore, all the data have been excluded for correlation purpose arid only the data (Chandock, 1970; Farbar, 1949; Hariu aiid l\lolst.ad, 1949; Jones and Aillendorf, 1967; Khan, 1972; Redd)-, 1967; Vogt and K h i t e , 1948) in the dilute phase f l o ~under almost identical experimental conditions were used. 111summary (i) it is possible to have a correlation limited to steady-st'ate, dilute-phase vertical solid-gas suspension flow aiid (ii) further experimentation is necessary t o expand tlie range of variables, Le., shape factor, density ratio, diameter ratio, etc. Experimental Equipment and Measurements
The equipment used for the measurement of pressure drop is schematically shown i n Figure 1. .iir was blown by a blower which passes through ,I storage tank aiid is metered by a calibrated rotameter. 'She solid particles (wheat,, oats, aiid barley) are fed by gravity from a solids regulatory valve. 1he suspension then passes upward through a vertical 10 ern i.d. liipe. -111 expansion :section and a pair of cyclones are used to recover t'he solids whereas the air is vented out. The detail description of each component is given by Khan (1972). 7 7
Pressure drop measurements \yere made Lvith a micromanometer (Casella-London). The readings were taken with a n accuracy of +0.01 mm of water. The "size" of a particle is probably the representative dimelisions that best describes the degree of coiiimunicatioii mmetric particle the of the liarticle. For a spherically diameter is that dimension and thus ,*ize. The diameter of a particle deviating from spherical symmetry ma)- be as one dimensional distance bet\veen two points on the surface of tlie particle passing through its center of gravity. At least 200 measureiiieiits n-ere made for each solid used aiid tlie surface mean diameter satisfying the foregoing definition was then calculated. The particle shape is considered iii coiinection with tlie coilcept of size (Hariu and Nolstad, 1949). Shape factor which corrects for both the difference in dimensionality of the original measurements arid noiisphericitj- of shape was calculated as 6 = V ZAYidi3n-liere V i q the volume of the liquid displaced by S iparticles haviug diameter d. Data Processing and Discussions
The total data including those reported in the literature (Chandock, 1970; Farbar, 1949; Hariu aiid Xolstad, 1949; Jones and Illendorf, 1967; Khan, 1972; Reddy, 1967; Vogt and Khite, 1948) on pressure drop for the steady-state dilutephase vertical transport' of solids in pipes are about 1200 Ind. Eng. Chem. Process Des. Develop., Vol. 12,
No. 4, 1973 429
Table II Investigator
Suggested Correlation
Belden and Kassel (1949)
SLPd+ 2Uf(0.049G + 0.22Gr)
Boothroyd (1966)
APT/APf = e ( ~ s / ~ f ) ~ % / f )
Lf
gD t (Re)O ,
Cramp and Priestley (1924a, 1924b, 1924c)
APa/APf
Farbar (1949)
e tan LY
70
70
RMSD
DMAX
71
745
58
99
205
650
79
32 1
Ghosh and Prem Chand (1968)
213
1503
Hariu and Malstad (1949)
106
189
Jones and Allendorf (1967)
37
93
Metha and Smith (1957)
83
162
Razumov (1962)
143
721
Stemerding (1962)
183
1013
Vogt and White (1945)
148
131
=
-
Proposed correlation
1.0
--
.
.e
Cd 1 A
I
1
I
d
?
Re/Fr I
a"
I
IO I ,5
1.0
I
20
I
I
I
l
30
40
e/+
.. ... ..-.. :..-.** . .*.* .
9
-
~
50
.y
Figure 3. Effect of the various groups on the performance of the correlation
points. These data are tabulated in Appendix I (available as supplementary material) lvhich covers a fairly large range of variables as listed in Table I. The root-mean-square deviation (RSAID) and the maximum deviation in percentage (DLIAX) were computed for all correlations and the individual results are reported in Table 11. 430 Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 4, 1973
23.5
51
It is also evident that with the possible exception of Jones and Xllendorf's (1967) correlation, all the others have failed in predicting the pressure drops for vertical transport of solids in pipes with reasonable accuracy. The expression proposed by Jones and ;illendorf gave 14y0 R l I S D for their own data and 37y0 for all the available data. Their data further suggested that the diameter ratio could be a correlating parameter, but they hold the opinion that this is not so. Xoreover Leung, et al. (1971), pointed out that the correlation suggested by Jones and hllendorf (1967) is somewhat in error. They modified it by incorporating voidage fraction into the expression. The modified correlation, however, gave 23y0 RUSD for Jones and Xllendorf's own data. T o conclude, the foregoing comparative study has clearly demonstrated that the pressure drop for the transport of solids in vertical pipes will depend upon the following seven parameters: (i) loading ratio; (ii) density ratio; (iii) diameter ratio; (iv) ratio of the inertia forces to the viscous forces, known as Reynolds number; (v) ratio of the inertia forces to the gravitational forces, known as Froude number; (vi) the ratio of the drag coefficient of the solids and the friction of the pipe; (vii) shape factor of the solids used. The data which are multivariant were then treated statistically using .kitken's method by pivotal condensation (1937). The correlation so obtained is
The RMSD was calculated for this correlation and was found to be 23.5y0 with maximum deviation of 5170, A plot of experimentally observed values against the predicted values for this correlation is presented in Figure 2. I n order to study the contribution of various groups taken into account a t their different values in this correlation,
( 4 P s ~ A P f ) c a l o d / ( ~ ~ s / L ~ P f ) o bvalues sd were plotted against these parameters and are presented in Figures 3. It can be observed t h a t the proposed correlation correlates remarkably well for all parameters.
Conclusions
Aicomparative study of all the existing correlat'ions in the literature 011 dilute-phase solid-gas suspension flow has indicated the need t o improve the reliability and range of these expressions. It was found t h a t the pressure loss in vertical dilute-phase solid-gas suspension flow can be correlated successfully within 2357, RNSD of all the available data in the literature. Nomenclature
T h e fuiidamental dimensions are presented by I.' = force, L = length, Jf = mass, 7' = time. ectioiial area of the lift line, L z Cd = drag coefficient of the particle., 4 3 g [ ( P , - Pf), Pf ] iD,,i t 7 5 zdimen?ionless ), D , = diameter of t'he particles, L D t = diameter of the lift line, L F r = Froude number. 17r:gDt, dimeiisioiiless f = Faiiniiig frictioii factor, dimeiisionle;s j5 = solids friction -factor, tlimensioiileis Gf = fluid mass flow rate, .llL-?T-l G , = solids mass flow rate, J I L - T - 1 G,* = qolids mass flow rate (t'onslhr cm2), AIL-T 1 L, = length of entrance zone, L L f = length of the lift line. L A P f = pyessure c~ropdue tb fluit1 alone. F L - ~ A P s = pressure drop due to the presence of solids, F L P 2 APt = total pressure drop, FL-* A p t * = total pressure drop (mi),FL-2 Re = Reynolds number, dimensionless L-f = velocity of the fluid media, LT-1 rf* = velocity of t'he fluid media (m 'sec), LT-' C , = velocity of the solids, LT-1 = velocit'y of the solids (nivsec), LT-l C3 = slip velocity, LT-1
r,*
GREEKLETTER^ function of air flow rate on velocity, dimeiisioiiless loading ratio, tlinieiisionless 6 voidage fractioii, dimeiiiionless Q = shape factor of the particles, climeiisioiiless A = friction of the pipe given by Ulassius equation, dimeiisioiiless (Y
0
= = =
A,
1 p ps pf
7
friction factor of the solids, dimeiisioiiless average segregation factor, tlimeriiioiilcss = viscosity of the fluid, JfL-1T-1 = density of the solid, = density of the fluid, .11LP3 = radial variation of solid velocity, dimensionless =
=
literature Cited
Aitken, A . C., Proc. Roy. SOC.,57, 172 11937). Belden, 1). H., Kassel, L., I t i d . Eng. Chem., 41, 1174 (1949). Boothrovd. K. G.. Trans. Irist. Chem. Etior.. 44. TY06 11966). Boothro-d; lt. G:, Priezcmatraiisporf, 1, "1 '(1971). Capes, C. E., Cau. J . Chem. Eiig., 49, 182 (1971). Chandock, S. S., Ph.L). Thesis, University of Waterloo, 1970. Cramp, W.: Priestley, A , , Erlgiiieer, No. 64 (Jan 11, 1924a). Cramp, W., Priestlev, A , , Etigiiieer, No. 64 (Jan 18, 1924b). Cramp, W., Priestley, A,, Erigzneer, No. 89 (Jan 25, 1 9 2 4 ~ ) . Ihckworth, 11. A , , Pneumatrarzsport, 1, 12 (1971). Farbar, L., Iiid. Etig. Chenz., 4, 6, 1184 (1949). Ghosh, I). P., Prem Chand, J., Agr. Eng. K c s . , 13, 1, 29 (1968). Hariu, 0. H., Molstad, 11.C., I n d . Elig. C h e n ~ . ,41, 6, 1148 ( 1949). Jones, H. J., Allendorf, 11. H., AIChE J . , 12, 6, 1070 (1966). Jones, H. J., Allendorf, 11. H., AIChE J., 13, 3, 608 (1967). Khan, J. I., 1I.A.Sc. Thesis, University of Waterloo, 1972. Lapidus, L., Elgin, J. C., AIChE J . , 3, 1, 63 (1931). Leung, L. S., Wiles, It. J., Sicklin, I). J., Ptieic?rzatrarisport, 1, B7 (1971). JIetha, S . C., Smith, J. 1L,Coming, E. W., I t i d . Etig. Chem., 49, 6 , 986 (1%3). Razumov, I. 11.R . , Iiit. Chrm. Erig., 2 , 4, 339 (1962). Reddy, K. 5'. S.,Ph.D. Thesis, University of Waterloo, 1967. Richardson, J. F., JIcLennan, AI.,Zratis. Inst. Chcm. Erigr., 38, 32 11960). Stemerding, S., Cheni. Eng. Sci., 17, 599 (1962). Tchen, C. 11.,Ph.D. Thesis, Delft, 1947. Sproule, W.T., A-atzire ( L o d o n ) , 190 (1961). Thomas, 11. G., "Symposium on International ItePearch on Thermodynamics and Transport Properties," AG511E(196'2). Vogt, E. G., White, 11. R., I d . Erig. Chenz., 40,
