Pei, D. C. T., Reddy, K. V. S., Canadian Congress of Applied Mechanics, Quebec, May 1967, Paper F20. Perry, J. H., et al., “Chemical Engineers’ Handbook,” 4th ed., p. 5-62, McGraw-Hill, New York, 1963. Peskin, R. L., et al., U. S. Atomic Energy Comm., Quart. Repts. 63-1, 63-2, Contract .4T(30-1) 2930 (1963). Peskin, R. L., Dwyer, H. A , A . S . M . E . Paper 65-WA/FE24 (1965). Soo, S. L., Regalbuto, J. .4., Can. J . Chem. Eng. 38, 160 (1960). Soo, S. L., IND.ENG.CHEM.FUNDAMENTALS 1, 33 (1962). Soo, S. L., Ihrig, H. K., El Kouk, .4. F., Annual Meeting, A.S.M.E., Paper 59-A-59, 1959.
Soo, S. L., Trezek, G. J., Dimick, R. C., Hohnstreiter, G. F . , I N D .ENC.CHEM.FUNDAMENTALS 3, 98 (1964). Thomas. D. G., i4.Z.Ch.E.J. 10 (1964). Yannopoulos, N. J., Themelis, N. J., Gauvin. \V. H., Can. J . Chem. Eng. 44, 231 (1966). Zahm, A. F., J . Frankltn Znst. 217, 153 (1934). Zenz, F. A., Othmer, D. F., “Fluidization and Fluid-Particle Systems,” Reinhold, New York, 1960.
RECEIVED for review August 7, 1967 ACCEPTED March 4, 1968
TURBULENT FLOW OF GAS=SOLIDS SUSPENSIONS H. E. McCARTHY
E. I . du Pont de Nemours G3 Co., Wilmington, Del. 79898 J . H. O L S O N University of Delaware, Newark, Del.
7977 7
lo6
0.75-2.75 2.0
>lo6
...
106
1.2
x
105
0.5
Theoretically, Pigford and Thomas (1965) and Murray (1965) have shown that axial density variations should grow in a fluidized bed but should damp out in the radial direction. No data have been presented upon these fluctuation estimations for gas-solids flow at high Reynolds numbers. Experimentally, a number of workers have measured the time-averaged, radial concentration distribution of solids (Table I I ) . I t is concluded that if the Reynolds number is greater than lo5, the particle density is less than 4 grams per cc., and the particle size is less than 200 microns, homogeneous flow is achieved. The effect of solids upon turbulent dissipation has been investigated theoretically by Saffman (1962), who analyzed the effect of coarse and fine dust upon the stability of laminar flow. His work shows that the stability of laminar flow is increased (higher critical N R e ) by the presence of large solid particles. This development admits the possibility of reduced dissipation for gas-solids flow at higher N R e . Experimental data upon turbulent dissipation in gas-solids flow are indirect and contradictory. I t is assumed that the turbulent dissipation is related to the turbulent eddy diffusivity in the central core of pipe flow through Reynolds analogy. The effects of solids upon the eddy diffusivity may be classified as follows: T h e intensity of fluid turbulence may be altered by the solids. I t is usually assumed that the scale of turbulence is reduced. The correlation length of the gas eddies may be increased. The solids are expected to have longer correlation lengths than the fluid because of their higher momentum. Thus in liquids one expects a significant transport of fluid attached to the boundary layer of the solid; in gases the boundary laver transport is less significant. This effect should increase the eddy dkusivity. There is the usual “detour” contribution for eddv diffusion around the spheres. This effect should decrease‘ the eddy diffusivity.
-
Since one may use these possibilities to predict either an increase or a decrease in the eddy diffusivity, it is not surprising that these experiments are less than conclusive. So0 et al. (1964) found no change in the eddy diffusivity of helium injected into the central core of a gas-solids flow system. Kada and Hanratty (1960) repeated the experimental technique in a 472
l&EC FUNDAMENTALS
50 35 50 200 7200
Comment
Large axial variations in solids density Homogeneous flow Homogeneous flow
T y k e of Flow
= 10% density variations
increase near wall Homogeneous Homogeneous Homogeneous Homogeneous
Reynolds number in which the data were affected by a von Karman vortex street, and Bobkowicz and Gauvin (1965) found a n increase in eddy diffusivity but a decrease in frictional loss for liquid-solids flow. Thus it is concluded that the Reynolds analogy approach is not informative. Julian and Dukler (1965) proposed an eddy viscosity model based upon the Reynolds analogy for two-phase, homogeneous gas-solids flow. Using the logarithmic velocity profile, their model predicts monotonic increase in eddy viscosity with solids loading. Their model is suspect because: I t was tested with data for pneumatic conveying in which the flow is clearly inhomogeneous, and the frictional losses should always increase in gas-solids systems, which is contrary to experimental evidence. I t is concluded that homogeneous piston flow of solids is approximately achieved on gas-solids flow a t high Reynolds numbers and low solids-gas weight ratios. There is some disagreement upon how closely the flow approaches the homogeneous limit. In addition, there is little experimental or theoretical information upon the contribution of solids to energy dissipation in two-phase gas-solids flow. Solids Acceleration
I n this work the acceleration of the solid particles is calculated using standard drag coefficient and the use of this correlation involves the following separate problems : Does the standard drag coefficient hold for particles in accelerating flow? In dilute systems, can one consider the drag acting on one particle and then sum over particles? Measurement of drag coefficients in accelerating fields has been attempted by many individuals, and their efforts have been summarized by Rudinger (1963). These experiments are difficult and show contrary trends. For example, Ingebo (1950) found a correlation which lies below the terminal velocity drag coefficient curve. Rudinger’s (1963) data crossed both Ingebo’s curve and the standard drag curve. Finally, Crowe (1962) obtained a curve slightly higher than the standard curve. The scatter in these data suggests the use of the standard drag coefficient (Zahm, 1927) in accelerating flows.
The drag coefficient for a swarm of particles is less than that for a single sphere (Richardson and Zaki, 1954). However, Richardson’s correction for swarms is unimportant when, as in this work, the volume fraction of spheres is less than 0.001. Accordingly, the acceleration force upon solids ie found by a summation of the drag upon individual particles. Thus the drag is given as
p
dVa g, dt
MsVx dVo
C b r ( V g - V 2 AP
sc dz
2 8,
Ms = - - = -- -
Table 111.
r l n m A
B C
Position
Equipment Dimensions
r”l r-7
f7PlP-l PIP-! D E
Length, Pt.
Tolcronce, Ft.
(1)
which gives the solids velocity as
[The virtual mass (Hinze, 1961) is negligible in gas-solidsflow, since ‘/BM,p,/p, M , is nearly identical to M,.] The use of Equations 1 and 2 is further supported by the work of Torobin and Gauvin (1959, 1960, 1961a, b) in their review of the effect of wake interaction and free stream turhulence upon drag coefficients. They concluded that these contrihutions tend to cancel and therefore suggest the use of the total terminal drag coefficient rather than the summation of drag contributions in accelerating flow, I n any event, a large relative error on the drag coefficient makes only a small contrihution to the calculations reported in this work. The objectives of this work are to measure the friction factor for two-phase flow and to relate these ohservations to the solids, density, velocity profile, homogeneity of flow, and drag coeficient for gas-solids flow. The over-all aim is to present a model for two-phase flow which yields a useful representation of the experimental data from other established information.
+
Experimental Equipment and Procedures
The pertinent dimensions for equipment used (Figure 1) are given in Table 111. Equipment specifications can be obtained from the authors. The three pipe sections were joined by rubber-cushioned flanges and gasketed usipv Teflon amliets drilled to the exact inside diameter of the p centered using a Teflon tween pipe sections. Th,
a sandblast through a hypodermic needle to eliminate chipping on the inside of the pipe. T o achieve a leak-free seal a t the pressure taps, aluminum fixtures were machined to fit the clamps fastened around the pipe. The inside of these plugs was recessed, and a ruhher insert was installed to hear against the pipe. The pressure transmitters were then calibrated. A pressure transducer and recorder were calibrated for the maximum pressure for each run. Air was supplied by a 550-cu. foot per minute compressor. Air flaw was metered using a calibrated orifice assembly. The operating pressure was fixed using a pressure regulator and was adjusted for each run. During each run an air bleed, prior to the pressure regulator, was maintained to eliminate pulsations of the compressor. Solids were fed using a specially built feeder, which could be pressurized. Solids were metered by a screw with a variable-speed drive into a side stream which was taken after the orifice assembly. A valve was used to regulate pressure drop across the feeder supply and discharge to fluidize the feed stream. For free-flowing material, a spring-loaded flapper valve was constructed to shut off feed when the screw was not turning. The pressure drop across the flowmeter and seven of the pressure transmitters was recorded using a pneumatic recorder. The discharge of the pipe was maintained a t amospheric pressure; therefore, the flow was controlled by regulating the upstream pressure. The recorder was set to measure a AP of 5 inches of water full scale, so it had to be reset for each transmitter and each run. After the recorder was calibrated, “no load” data were collected. The hydrostatic pressure of the flowine stream was taken usins a Dressure transducer.
VOL. 7
NO, 3
AUGUST 1968
473
~~
~~
Solids Used Spec$cations 201 0
Table IV.
Solid Type Lucite
Manufacturer Du Pont
Glass bead
Potter Bros.
Size “N”
Calcium carbonate (calcite)
C. A . Wagner
Camel white
Particle Size 40 mesh 577, 60 mesh 11% - 80mesh 47, - 100 mesh +270 mesh -200 mesh Size distribution shown on Figure 2
100%
Acrylic resin, lot 890653
-
Carthy, 1966). For the present, a continuum approach is used and component equations are developed from the equations for a continuum. T h e method differs from that of other workers who have attempted to add component equations to obtain a n equation for the two-phase system. T h e applicability of the model developed here and a comparison with the equation of other workers are presented below. PRESSURE DROP FOR A GAS-SOLIDS MIXTURE.This paper defines a mass average velocity according to Truesdell and Toupin (1960). We limit this discussion to homogeneous flow-i.e., no concentration gradients in a differential control volume. The fundamental premise is that the total mass flow is equal to the sum of the component mass flows as n
2
I
I
1
I
I
3
4
5
6
7 8 9 1 0
l
l
1
PARTICLE SIZE IMICROICSI
Figure 2.
PmVz
I 20
Particle size distribution of calcium carbonate
collected in a 30-sq. foot Dacron filter bag and weighed, and the feed rate was determined. A second “no load” run was then made. This procedure was carried out for each of three air flow and four feed rates for the glass beads and the Lucite. Only one condition was possible for the calcium carbonate because of the difficulty encountered in feeding. Data on the three types of solids used are shown in Table I V and Figure 2. The particle size distribution for the glass beads and Lucite was determined using Tyler screens and an Alpine laboratory jet sieve. A Model B Coulter counter was used to determine the size distribution of the calcium carbonate. The particle size was checked periodically during the runs; no significant size reduction occurred. Photographs using a narrow depth of field camera were taken during the runs with solids to determine the particle velocity. A two-flash system with a time delay was used. A Strobotach flash was triggered by a camera shutter and in turn triggered an Edgerton time-delay flash. A phototube pickup was triggered by both flashes and the delay time measured using an oscilloscope. The Strobotach flash was 1 microsecond long, while the Edgerton flash was 0.5 microsecond in duration; hence the two flashes could be distinguished on the photograph. Pictures were taken using the glass beads and calcium carbonate. To focus the camera a point was established by focusing on the front of a 10/32 screw. The pipe center was established and distance measured by turns o i the screw. In the picture the screw was in focus while the first threads were out of focus; thus the depth of field of the camera was less than l / l c inch. The flash units were set up at the bottom of the pipe and pictures taken a t right angles in the front of the pipe. Velocities were measured in the center section of the pipe to eliminate distortion due to curvature. The distance the particles traveled was measured using a micrometer-operated comparator. The micrometer could be read to 0.001 inch. Magnification was calculated by using the ruler photographed on the pipe.
P*Vi
=
(3)
i=l
where i refers to the ith component and p m and mass average. Thus the mass average velocity is defined by:
(4) and the mass average density is equal to the sum of the component densities. I t is clear that in a system where there is no mass exchange between the phases, the continuity equation holds for the mass average and each component. Adopting the continuum approach, one can derive the timeaveraged equation of motion as shown Bird et d. (1962). For one-dimensional flow with frictional losses represented as shear stress a t the wall the equation is: Pm
DVZ
gc Dt
dP dz
2 rw r
where
and
For steady-state flow, Equation 3 becomes
pml ,d- vz -- - - dP -ZdZ
dz
2 rm r
Substituting in Equation 5 in the derivative one obtains
From the continuity equation, one obtains Theory
Mathematical Expressions for Pressure Drop of a GasSolids Mixture and Acceleration of Solids. Basic equations for momentum and energy are developed elsewhere (Mc474
l&EC FUNDAMENTALS
Vzrefer to the
(5)
I n the system under study in this work, the gas can be considered as ideal. For temperature calculations, it can be assumed that the gas temperature and solids temperature are equal (Appendix B). I n the system where gas stagnation temperature and the solids initial temperature are equal, an enthalpy balance gives
Applying the continuity equation and the ideal gas laws and assuming constant temperature, one obtains:
which is a n established solution (Lapple, 1943). I n the case of adiabatic flow, one obtains
Differentiating Equation 9 and making appropriate substitutions, one obtains:
This same problem has been considered by various workers. So0 (1960, 1961, 1962), Peskin andDwyer (1965), and Marble
(1962, 1964) have considered the flow of a gas-solid mixture by taking a linear summation of the momentum and obtaining a n equation of the type
for one-dimensional steady-state flow. The difference in Equations 12 and 13 arises in consideration of the momentum of the mixture. I t can be shown (Truesdell and Toupin, 1960) that the momentum of the mixture is given by
This equation is developed and discussed more fully by McCarthy (1966). One can now see that the two equations, 12 and 13, can be made equivalent by defining the pressure
Pcrn)Gij =
2 P("Gz, + p(')[V,(') - V i ] [ V j ( ' )- V,]
(15) Results
5 3 1
where s = number of components and n = total number of components, in Cartesian notation. The interaction terms could also be included in the wall stress tensor by redefining this quantity; however, these definitions are artifacts, since the ternis arise from momentum consideration. If the slip velocity (V,-V,) is negligibly small, Equations 12 and 13 reduce to the same equation. Equation 14 states that the momentum of the mixture is the linear sum of the momentum of the gas and solids. I n a system where gas and solids velocities are equal, the continuum model yields
This equation can be treated as a compressible fluid flow equation and an analytical solution can be developed. The detailed derivations are shown ir. Appendix A. Equation 8 reduces to
Gm dVz gc dz where G ?,
dP
2 jGmVz
dz
Dgc
g,, f,and D are constants.
The above equations are implicit in pressure and need only the initial or final pressure, mass flow, and stagnation temperature to solve for pressure drop. One often wants to solve for mass flow for a given AI'; rearrangement of Equations 17 and 18 results in a more direct calculation for mass flow. Equation 16 was solved by Lapple (1943) for isothermal and adiabatic flow. His solutions are in terms of velocity and require a cumbersome graphical solution for pressure. I n his analysis, Lapple included terms accounting for an inlet to the pipe by taking an isentropic expansion to the inlet velocity. The maximum (choked) flow may be obtained from the above equations by taking dGJdP = 0, as was done by Lapple. The isentropic expansion to the inlet could be easily added to the above equation, allowing a direct calculation of flow rate from stagnation conditions. Peskin and Dwyer (1965) considered the flow as incompressible and solved for pressure drop from this model. They accounted for the change in pressure drop due to solids as a change in the particle drag coefficient. Acceleration effects, which are slow enough not to increase the slip velocity, can readily be accounted for by using either Equation 17 or 18. I n most particle flow regions, the temperature change of the mixture is small, and Equation 17 gives a good approximation of the pressure drop.
(16)
The analysis of the gas-solids flow in this work comprises: Experimental determination of the pressure drop along the pipe. Photographic examination of the solid particulate velocities. Calculations based on mathematical models. Accordingly, the organization of the results displays both experimental and computational data and compares the two where possible. Pressures were recorded experimentally and compared with predicted values from the summation model and the continuum model. Particulate velocities were measured photographically and compared to velocities predicted from the acceleration model Equation 2. These comparisons were then used to illustrate the applicability of the models presented in this paper. Gas Friction Factor. T h e friction factor for single-phase gas flow as a function of pressure drop and flow rate was determined to check the experimental procedure. T h e friction factor was computed to match the measured pressure drops a t flow rates measured using the orifice meter assembly. T h e friction factor was calculated using the one-dimensional continuum equation, A-1 1. VOL. 7
NO. 3
AUGUST
1960
475
The friction factor obtained for gas flow shows that the experimental procedures gave expected results for a smooth pipe. These data lend confidence to the results obtained for twophase flow. Gas-Solids Friction Factor. If the slip velocity (velocity of gas velocity of solids) is small, the continuum and momentum summation mathematical models are essentially equivalent, If the acceleration is slow also, the choice of drag coefficients is not critical. Figures 5 and 6 show model prediction for solids and gas velocity, using the standard drag coefficients along the pipe length. Figure 5 shows the predicted behavior of an experimental run using glass beads a t solids and gas flow conditions which match a particular experiment. Figure 5 shows that the solids velocity approximates the gas velocity for the sections from 4 to 12 feet. I n the first 4 feet the gas velocity is fairly constant, and the solids are accelerated to approximate this velocity. I n the length beyond 12 feet, the gas velocity increases as the exit of the pipe is approached; here, both the solids and the gas are accelerating rapidly, and the slip velocity increases. Figure 6 shows a similar calculation for the 3micron C a C 0 3 particles; it appears that the solids velocity is essentially identical to the gas velocity for all points in the pipe. T h e glass beads represent the larger, more dense particles, while the C a C 0 3 is the smallest employed. The slip velocity is less than 10% of the gas velocity in the 4- to 12-foot region for all materials considered in this work. T h e section between 4 and 12 feet was thus chosen for calculation of the friction factor in two-phase flow. The twophase friction factor was calculated using Equation 18 and properties of the gas-solids mixture. The computed data are shown in Figure 7 with the best fit quadratic curve. The calculated curve was:
,010
,009 ,008
,007
-
,006 ,005
LT
,004
0 V
s z
P
4
,003 95% CONFIDENCE LIMITS’
LL
,002
.oo
I
I
1
I
I
I
2
3
4
5
6
7
I
I
I
0 9 1 0
REYNOLDS NUMBER ( x i o - 5 )
Figure 3.
Gas friction factor for 1 -inch glass pipe
T h e best fit line of friction factor as a function of Reynolds B In (ATRe) was computed number in the form of In ( F ) = A using regression analysis. The best fit line calculated was
+
- 0.211 In ( N R e )
In ( F ) = -2.95
f 2 phase
(22a)
~
f
or
F
= 0.055/(NRe)1.21106
= 1.011
gas
+ 0.82M’,/W0 + 0.479(JVs/W,)2
The friction factor correlation obtained in this work in view of the data scatter can be represented by
(22b)
Figure 3 shows the regression line plotted with the literature data taken from Lapple (1956) for smooth pipes. The data plotted in Figure 4 show that the data in general fell within 5% of the literature line. The lines corresponding to 1 2 0 are shown on Figure 3.
f 2 phase
+
-
~-
- 1.0 0.8Mfs/W, 0.5(W,/WO)* (24) gas This correlation is the best representation of the data obtained but is valid only in the region tested. The rate of de-
f
t
f .0035
t5%
.. .0030
I
-5%
I
!
I
3x105
2x105
I
4x105
NRE
Figure 4. 476
l&EC FUNDAMENTALS
(23)
Friction factor for 1 -inch glass pipe
I
5x10’
1
900
r
r t
850 8OOb
I
7501
I
8501
I
800
./
700
650
c
600
5501
3
7/
GAS VELOCITY
500
350 ~ , A S VELOCITY OLIDS
'250 OOi
/ 0 5 200 2 0
2
4
6
8
IO
I2
14
16
18
20
200;
'
2
I
I
I
4
6
8
LENGTH ( F T I
Figure 5.
I
I
,
f
,
12 '
14
16
18
20
LENGTH ( F T I
Figure 6.
Gas and solids velocity
Sample 91 6-1 2
Gas and solids velocity Sample 1029-2 3-micron C a C 0 8
64-micron gloss beads W,/W, = 0.168
crease in the friction factor decreases with increasing weight ratio. This is consistent Lvith the observation that if the converse were true, a point of zero friction loss would be approached, Lvhich is obviously impossible. This correlation predicts a minimum a t FVs/ Tf', = 0.8; ho\vever, the standard deviation estimate is 1 0 . 4 , so this minimum should be used with caution. At higher loading \z here particle-particle and particle-wall impacts become important, one might expect an increase in friction [actor. So0 and Trezek (1966a) show a correlation of friction factor us. Reynolds number us. mass ratio of solids, which gives qualitative agreement with the present correlation in the region studied here. So0 (1966a) also shows a n increase in friction factor a t mass ratios greater than 1. The difference between the present correlation and that of So0 is probably due to the use of the summation model by Soo. So0 and Trezek (1966b) also show a friction factor correlation. Their data Mere obtained in the acceleration zone and hence are difficult to compare \+ith the present data. This correlation shows a friction factor as a function of Z / L which is due to the model used in this work. Solids Velocity Profile. Solids velocity was measured by measuring the distance on a photograph (Figure 8) taken with a narrow depth of field camera. T h e focal plane was moved through the flow field. Velocities were measured a t different positions on each photograph as well as on different photographs. These data (Table V) show less than a 10% variation with position from r / r , = 0 to 0.8. If we approximate the scale of turbulence by scale = ,SU/U,,, we find that the scale is 0.06 to 0.13. Laufer (1954) found that the turbulence scale is 0.1 to 0.5 in the region of the photographs. This scale of turbulence can more than account for variations seen. T h e inertia of the solids probably reduces the turbulence scale.
IO
W,/W,
= 0.041
1.10
1.05
I .oo
0 95
v) Y
2 090 212
0.85
0.8C
0.75
0.7C
Figure 7. Two-phase friction factor-gas weight flow ratio VOL. 7
NO. 3
friction factor vs.
AUGUST 1968
477
W J w, 0.035
Material
50-micron glass
0.030 0.049 0.002
Table V. Position, R/Ra 0 0.5
0.0 0 0.5 0.8 0 0.5 0.0 0 0.5
0.167
0.0 0 0.5 0.0
Velocities Aucragcd Velocity, F!. Sac. C"
537.5 591.5 503.3 471.6 551.1 476.6 408.7 494.6 428.4 553.9 539.1 519.5 379.7 332.1 420.0
31.0 3.6 41.9 20.5 47.9 50.0 24.6 79 .o
26.2
55.27 20.0 32.2 36.0 22.0 23.0
Celculatcd Vs, Ft./Sac.
526.0 480.0 436.0 506.5 370.0
575.0
Scale 0.05 0.01
0.06 0.06 0.09 0.10 0.06 0.08 0.06 0.02 0.05
0.06 0.05 0.02 0.06 0.06 0.03 0.13 0.06 0.10
.
;..,.
Figure 8.
. . . .
.
j.
1.
Photographs for measuring particle velocities
The gas velocity shows only a 10% variation from the average velocity in the turbulent core (Laufer, 1954). So0 et al. (1964) and Peskin and Dwyer (1965) have shown that the gas velocity profile maintains this shape in the presence of solids. Their data, however, show a 20 to 25% radial 478
I&EC FUNDAMENTALS
variation from the average velocity for the solids. Data obtained in this work show only a 10 to 15% deviation from this average velocity from the center line to R/R,, = 0.8. The data reported in the literature show a 35 to SO% lower velocity a t R/Ro = 0.8. Since our measurements appear to he valid to =!=lo%, one can conclude that the reported velocity profile was not observed. Thus from our data we conclude that in the central core, the solids and gas velocities are essentially Constant. Solids Velocity. The measured particulate velocities are also compared to the computed velocities a t the point where the photographs were taken (Table V). T h e solids velocity, computed using the standard drag coefficient, agrees with measured velocities. I n the case of CaCOa, the average measured velocity is 579 feet per second and the computed velocity is 575 feet per second. Since this is also the gas velocity (the slip velocity is essentially zero for any reasonable estimation of drag coefiicient), this datum shows the validity of the velocity measurements. The computed velocities are well within the accuracy (*IO%) of the experimental measurements. The particle velocities were measured 14.75 feet downstream from the pipe inlet, and this is a region where acceleration does occur. More rapid acceleration occurs a t the pipe discharge, hut in this work the velocities were not measured a t this location. Figure 9 shows the effect of increasing the drag coefficient by 20 and 50%. The velocities predicted here show more deviation from the measured data. This increase in drag coefficient also results in deviation from the measured pressuresfor example, increasing the drag coefficient 20% decreases the predicted discharge pressure for sample 916-12 from 1.0 to -0.7 p.s.i.g. The velocities calculated using the standard drag coefficient correlations deviate less than 10% from the measured value, and a change in drag coefficientscauses larger deviations. The velocities were measured in a portion of the pipe where there was a fairly rapid acceleration (Figure 5). I t therefore is concluded that in the accelerating flows where the velocities were measured, the agreement between predicted and measured velocities lends validity to the use of the standard drag coefficient correlation. I n measuring solids velocity a t higher loading, one might he able to determine how the standard drag coefficients are altered when particle-wake interactions occur. A fuller definition of the effect of an accelerat-
@ STANDARD
50% INCREASE x 20% INCREASE
30r 200
i
100
0
1
I
I
ing stream on drag force could be obtained near the pipe discharge. Pressure Profiles. Figures 10 through 14 compare measured pressure data with computed pressures using both Equations 12 and 13 for the entire pipe length. We define Equation 12 as the continuum model and Equation 13 as the summation model. First, compare the computed data using the continuum model with the experimental data (Figures 10 to 14). As the = 0.090 (Figure IO) solids loading is increased from to Ws,’FVo = 0.419 (Figure 13) the deviation betwren the calculated and predicted curve is slight. There is no trend noted as the solids loading is increased; in fact, the deviations that are present are due to the experimental error in the correlation curve. Figures 10 to 14 also compare the summation and the continuum model. I n all the figures except 14, the summation model gives pressures which deviate markedly from experimental data. The deviation increases with increased solids loading. Figure 14 shows data for the C a C 0 8 run, in which the solids velocity and the gas velocity were essentially equal, and shows agreement between both mathematical models and the data. One would expect agreement for this run, since both models are identical if the solids and gas velocities are equal, These data show that the use of a mathematical model in which a linear momentum summation is assumed is not valid. The pressures calculated deviate markedly from experimental data and the deviation increases as solids loading is increased. So0 and Trezek (1966b) show similar curves; however, in their work the friction factors were not determined in a portion
I
I
I
I
I
1
ws/Mio
DATA CALCULATED 0 CALCULATED (Summation) w
x
(
-
0;5
2.5
4.5
I
6,5
.t
6.5
VOL. 7
I
10.5
I
12.5
t
I
14.5
16.5
I
\
J
16.5 20.5
NO. 3 A U G U S T 1 9 6 8
479
\
2 -
*
I -
o-
I
I
,
I
I
I
I
I
t
LENGTH
Figure 13.
[FT)
Pressure vs. pipe length
Sample 929-22 W,/W, = 0.41 9 64-micron glass beads
“I
13
OJ
0.5
1
2.5
1
4,5
I
6.5
I
8.5
I
10.5
I
12.5
1
14.5
1
16.5
I
18.5
{
20.5
LENGTH ( F T )
Figure 12.
Pressure vs. pipe length
Sample 923-1 2 W,/WQ = 0.341 64-micron glass beads 480
I&EC FUNDAMENTALS
LENGTH ( F T )
Figure 14.
Pressure vs. pipe length Sample 1029-22 W,/Wo = 0.041 3-micron CaC03
of the pipes with low acceleration and applied in the acceleration region. I t is thus hard to obtain an accurate comparison with the present data.
We can define pa V , = Go, which is constant, and p I = pa for single phase. Because ofvery small viscosity change,
DG
Conclusions
A two-phase friction factor correlation for turbulent flow < lo6)of a gas-solidsflow in the dilute (0 < T.1’8/W, (lo5 < XT, < 0.6) region has been obtained. This correlation (Equation 24) shows that the friction factor decreases when solids are added to the system. The correlation obtained is limited to the region studied and must be considered a first-order approximation to a more complex function. I n the experimental work, it was noted that the pressure drop did not increase when solids were added to the gas stream, even though the density was increased. Since it has been shown that the structure of turbulence in the central core is not affected by the solids, the reduction in friction factor appears to be the result of turbulent damping in the “buffer” zone due to the inertia of the solids. This effect was not a function of particle size or density; the reduction depended only on the ratio of solids flow to gas flow. Particle velocities measured in this \vork do not show any significant variation with respect to radial position in the region 0 ,, + PI
for
[(l
-:)>I.
w
This development gives two equations capable of solving a compressible fluid flow problem for the above conditions. Both equations involve an implicit solution but can easily be programmed for a computer. These equations can be modi, and V,; however, V, = fied for solid-gas flow by using G V g . T h e temperature can be calculated from Equation 11 by defining a mixture specific heat as (A-12)
(A-13 )
Residence time in these tests is 0.05 second; therefore, conduction presents no problem in this system. Now consider heat transfer from the surface. The limiting Nuslett number for a sphere in stagnant air is 2 (McAdams, 1954). For the 50-micron particle, this leads to h, = 183
T - To T I - To
= ___
where T i s temperature a t time t Tois the initial temperature constant T I is the surface temperature and is constant so
To
= 0
7a
=1
B.t.u. sq. ft. hr. O F.
which would allow for rapid heat transfer from the surface. To determine the time required for heat transfer, one can calculate a character time T T by considering heat transfer from the surface.
=
7
a2
t = 0.00217 second
qp =
I n determining the degree to which the temperature of a solid will follow the gas temperature, one must consider the conduction of heat from the center of the particle to the surface and the heat transfer from the surface. Conduction of heat to the surface is considered first.
at 0.95 - = 0.4
So for a 50-micron glass particle
Appendix 8. Temperature of Solids in Accelerating Gas Stream
Define a dimensionless temperature =
7 =
(aDp2)(hm)(AT) 2aDpK,AT for one particle
Total Q = nq = ppCp(2aDsK/mCP)(AT)
By looking a t acceleration due only to Stokes law drag, as was done by Marble (1962, 1964), a characteristic acceleration time, ( r P )can , be defined.
For air r T / r p = 1.94, so the time required to reach temperature equilibrium is twice as long as required for acceleration. As shown in Figure 7, the acceleration in this system is rapid, so one can say that the temperature of the solids follows the temperature of the gas.
where a = K / p C, By making a transformation = rr
Equation B-2 will become
Nomenclature
au _ -- a-d2u
br2
Make Equation B-3 dimensionless by letting ta
e = - - az ,
r
R=a
where a = sphere radius Thus
d2u _ -- _ du
be
bR2
This equation can be solved by separation of variables as shown by Carslaw and Jaeger (1959) to give 482
I&EC FUNDAMENTALS
(B-10) (B-11)
Because only radial flow is present
at
(B-7)
Define a characteristic time to reach a A T = E as
r p = m/6aap
U
(B-6)
A = area, sq. ft. B = parameter group = 2 g,C,T, C = specific heat of solid, ft./" R. C, = specific heat of gas, ft./" R. C, = drag coefficient for particles = diameter, ft. f = friction factor for pipe G = mass flow rate, lb./sq. ft. sec. F = drag force, lb. gc = Newton's conversion factor, ft/sec.2 K = ratio of specific heats, (C,/C,) L = length of pipe, ft. M = mass (slugs) P = pressure q = heat flux, B.t.u./sec. Q = total heat flux, B.t.u./sec. r = radial coordinate, ft.
D
= universal gas constant, ft./” R. = time, sec. T = temperature, R.
R t
”
U = transformed variable V = velocity, ft./sec. W = weight flow, lb./sec. i = not defined See Eq. A-8 SUBSCRIPTS g = gas p = particle s = solid z = mixture m = mixture o = initial or stagnation
GREEKLETTERS a
p
POL?, G d B 2fL = system parameter = -
= system parameter = -
D
~ C C P P O
‘k = system parameter = ___
R G ~ B
p a = density of solids in two-phase system p p = particle density pf
= fluid density
6
density of gas in two-phase system viscosity, lb. sec./sq. ft. = volume fraction of Darticles
u
=
7
= dimensionless temp.
pu = I.I =
turbulent correlation function
literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,’’ Second printing, Wiley, New York, 1962. Bobkowicz, A. J., Gauvin, W. H., Can. J . Chem. Eng. 43, 87 (April 1965). Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed., pp. 233-4, Oxford Press, New York, 1959. Crowe, C. T., Ph.D. thesis, University of Michigan, 1962. DePew, C. A., Farbar, L., J . Heat Transfer 85,164 (1963). Goto, Keishi, Iinoya, Koichi, Chem. Eng. Japan 2, 144-5 (1964). Hinkle, B. L., Ph.D. thesis, Georgia Institute of Technology, 1953. Hinze, J. O., Appl. Sci. Res. A-11, 33 (1961). Ingebo, R. E., NACA Techn. Note 3762 (September, 1950). Julian, F. M., Dukler, A. E., A.Z.Ch.E. J . 11, 855-8 (September 1965). Kada, Hisao, Hanratty, T. J., A.Z.Ch.E. J . 6, 624 (December, 1960).
Lapple, C. E., “Fluid and Particle Mechanics,” University of Delaware, Newark, Del., March 1956. L a d e . C. E.. Trans. A.Z.Ch.E. 39. 345-432 11943). ~, Lazer,’John,’NACA Refit. 1174, (f954). McAdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill, New York, 1954. McCarthv. H. E.. Master’s thesis. Universitv of Delaware. 1966. Marble, F: E., “Dynamics of a Gas Containing Small Solid Particles,” Fifth AGARD Combustion and Propulsion Colloquium, Brunaschweig, April 1962. Marble, F. E., Phys. Fluids 7, No. 8, 1270 (August 1964). Mehta, N. C., Smith, J. M., Comings, E. W., Znd. Eng. Chem. 49,986-92 (1957). Murray, J. D., J . Fluid Mech. 21, Part 3, 465-93 (March 1965). Orr, Clyde, Jr., “Particulate Technology,” pp. 124-78, Macmillan, New York, 1966. Peskin, R. L., Dwyer, H. A., “A Study of the Mechanics of Turbulence Gas-Solid Shear Flows,” ASME Winter Annual Meeting, Chicago, Ill., Nov. 7-14, 1965. Pigford, R. L., Baron, Thomas, IND.ENG.CHEM.FUNDAMENTALS, 4, 81 (1965). Richardson, J. F., Zaki, W.N., Trans. Inst. Chem. Engrs. 32, No. 1, 35 (1954). Rudinger, G., Reprinted from the Multiphase Symposium, ASME, 1963. Saffman, P. G., J . FluidMech.Vo1. 13, Part 1, 120-28 (May 1962). Savins, J. G., SOC.Petrol. Eng. J . 4,203 (September, 1964). Soo, S. L., A.Z.Ch.E. J . 7, 384 (1961). Soo, S. L., “Fluid Dynamics of Multiphase Systems,” Blaisdell, rCaltham, Mass., 1967. Soo, S. L., Proceedings of Symposium on Interaction between Fluids and Particles, 3rd Congress of European Federation of Chemical Engineering, pp. 50-63, Institute of Chemical Enaineers, London, 1962. Soo, S. L., Ihrig, H. K., Jr., Elkouh, A. F., J . Basic Eng. 82, 60921 (September 1960). Soo, S.L., Regalbato, J. A., Can. J . Chem. Eng. 38, 160-6 (October 1960). Soo, S.’L., Trezek, G. L., IND.ENG. CHEM. FUNDAMENTALS 5, 388 (1966a). Soo, S. L., Trezek, G. L., “Proceedings of Heat Transfer and Fluid Mechanics Institute,” p. 148, Stanford University Press, 1966b. Soo, S. L., Trezek, G. L., Dimick, R. C., Hohnstreiter, G. F., IND.ENG.CHEM.FUNDAMENTALS 3, 98, (1964). Sproull, PV. T., Nature 190, 976 (June 10, 1961). Thomas, D. G., A.Z.Ch.E. J . I O , No. 3, 303-8 (1964). Torobin, L. B., Gauvin, I$’, H., A.Z.Ch.E. J . 7 , No. 3, 406-10 (September 196la). Torobin, L. B., Gauvin, LV. H., A.Z.Ch.E. J . 7, No. 4, 615-19 (December 1961b). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng. 37,1959 (December 1959), 38, 189 (December 1960). Truesdell, C., Toupin, R., “Classical Field Theories, Encyclopedia of Physics,” Sect. 157, Springer-Verlag, Berlin, West Germany, 1960. Zahni, A. F., NACA Rept. 253 (1927). ~~
RECEIVED for review April 14, 1967 ACCEPTED January 28, 1968
VOL. 7
NO. 3
AUGUST 1968
483
