Pneumatic transport of solids in an inclined geometry

Pneumatic Transport of Solids in an Inclined Geometry G . E. KLINZING, N . D. ROHATGI, C. A . MYLER, S. DHODAPKAR, A . ZALTASH Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, PA, 15261, USA and M. P. MATHUR U.S. Department of Energy, Pittsburgh Energy Technology Center, Pittsburgh, PA Flow studies were conducted in 0.0266-m and 0.0504-m glass pipes held at various angles of inclination. Measurements in these experimental setups included particle velocities, solid mass flow rates, and pressure drops in both the upper and the lower halves of the pipe. Visual observations of the flow patterns were made through the glass section. Particles used in this system included glass particles of 67-, 450-, and 900-pm diameter, as well as iron oxide of 400-pm diameter. Mass flows from the two halves of the pipe were obtained by splitting the flow with a knife-edged separator. Experiments were also performed in a 0.0095-m transfer line in an inclined loop. The angle of inclination of the test loop was varied from 0 to 90 degrees from the horizontal. The effect of angle of loop inclination, tube diameter, and particle characteristics on basic flow parameters were studied. Des etudes ont CtC menCes dans des tubes de verre de 0,0266 et 0,0504 m a diffkrents angles d’inclinaison. Les mesures prises dans ces montages expkrimentaux comprennent les vitesses de particules, les dCbits massiques de solides et les pertes de charge dans la moitiC infkrieure et supCrieure du tube. Des observations visuelles des conditions d’koulement ont CtC effectukes a travers la section de verre. On a utilisC des particules de verre de 67, 450 et 900pm de diamktre ainsi que des particules d’oxyde de fer de 400pm. Les Ccoulements massiques des deux sections du tube ont CtC obtenus en dtdoublant I’Ccoulement a I’aide d’un siparateur biseautk. Des expkriences ont Cgalement CtC rCalisCes dans une conduite de transfert de 0,0095 m dans une boucle inclinie. L’angle d’inclinaison de la boucle d’essai varie d e 0 B 90 degrCs par rapport a I’horizontal. On a ktudiC I’effet de I’angle d’inclinaison de la boucle, du diamktre du tube et des caracteristiques des particules sur les paramktres de base de I’tcoulement. Keywords: inclined pneumatic transport, pneumatic conveying, solids transport, gas-solids flow. T he growing international awareness of the energy crisis has compelled engineers to explore new energy sources. In the development of energy sources for the future, the use of fossil fuel ranks high as an alternative. Many fossil fuel energy processes, such as coal combustion, coal gasification, and power generation, depend on the movement of solids by pneumatic transport. Much work has been done on gas-solid transport systems, but many of the studies have been performed only on horizontal and vertical systems. Despite many frequent references in the literature to the advisability of using diagonals rather than vertical runs wherever possible in piping systems ( Zens, 1960), research and development in the area of inclined flow are very limited. The central element of this study is to obtain fundamental experimental data concerning different flow configurations of pneumatic-transport systems in order to place the modeling and scale-up procedures on a firmer basis. The information of particle velocities, voidages, visual behaviors, and pressure drops was closely monitered in these systems. Visually at high gas velocities, the particles appear to be in suspension and are distributred more or less uniformly across the pipe cross-section. Then, as the air velocity is reduced, there is a tendency for particles to congregate in the lower half of the pipe. With further reduction in air velocity, deposition occurs, in some cases uniformly and in others in preferred areas of the pipe, leading to dune formation (Owen, 1969). Numerous forces exist in the pneumatic-transport systems. The direction and magnitude of these forces significantly affect the flow behavior. The most general form o f a force balance in pneumatic transport is for inclined flow. In the transport of solids by a gas stream, a force balance on the particle and the fluid yields the following (Myler, 1985; Klinzing, 1981): - dFgSsin Am8 dUJ dt 8 dFaddiriona, . . . . . . . . . . . . . . . . . . . (1) Amg aP = dF, - dFJj: - - Pf ax - dFRf sin 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) Substituting the expressions for the forces of gravity, drag, and friction into Equations (1) and ( 2 ) and further simplification gives the following: dU 3CD,~-~.’pf(Uf - U p ) 2 dt 4 ( P p - pfWp THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOLUME 67. APRIL, 1989 2f,Ui -~ Dt (3) 237 ;' (4) As can be seen from Equations (3) and (4), the dra coefficient for a single particle has been corrected by E -4. B, as suggested by Wen and Yu (1966). It should be noted also that the additional forces were neglected. They include the lift forces, the cohesive forces, and the forces due to electrostatics. For steady flow conditions, the pressure drop can be obtained by adding the reduced forms of Equations (3) and (4) as a sum of the individual contributions from various forces in the system: Figure 1 - Variable-incline/high-pressurecoal flow test loop. Figure 2 - Schematic of the 0.0266-m-diameter system The form of the solids friction term in Equations (1) and (3) is not explicitly stated. The friction factor is a function of velocity, gas density, particle shape, and other factors as well as pipe inclination. A point should be made about Equations (1) through (4); that is, these equations are written for an axial direction only. Additionally, the solution presented in Equation (5) is for steady, axial flow. It was born out in the experiments that such conditions are rarely, if ever, completely met. As an approximation, however, the equations are supported by experimental data. As pointed out by Yang (1974), the solids friction, particle velocity and particle concentration are coupled. Given the differences in particle velocity and concentration profiles between inclinations, particle friction almost necessarily must be different. This difference can be incorporated into the friction factor to allow differences due to pipe inclination. The most troublesome term in this balance is the solids frictional contribution. A number of investigators have developed expressions based on experimental data for this frictional term (Yang, 1974; Konno and Saito, 1969; Capes and Nakamura, 1973). The experimental determination of the particle velocity, solids mass flow rate, and pressure drop permits a more comprehensive evaluation of these expressions. Another important factor in all pneumatic transport systems is the transport line voidage given by: E = l - 4 ws .................... 07.upP p One sees from Equation (6) that a knowledge of the solids flow rate and particle velocity will yield the voidage. Experimental Setups For an enhanced understanding of the mechanism of the gas-solid flow and for a more comprehensive study of the pneumatic systems, a number of flow experiments were conducted using various pipe sizes oriented at different angles of inclination. Three experimental setups were used: 0.0095-m, 0.0266-m, and 0.0504-m gas-solid systems. 238 0 . 0 0 9 5 ~TRANSPORT SYSTEMS A flow diagram of the variable inclined flow test loop is shown in Figure 1. In this test loop, located in the Coal Utilization Division of the Pittsburgh Energy Technology Center, a Petrocarb Injector System is used to transfer pulverized coal through a 0.0095-m-diameter pipe. This setup is completely pneumatic. The Petrocarb coal injection system consists essentially of a primary injector, which is designed to operate continuously 24 hours a day. It has proven capability of delivering pulverized coal at constant, controlled, and reproducible flow rates for long periods of time, the maximum flow rate being 0.13 kg/s, into a receiver operating at a maximum total pressure of 113 kPa. The test loop is mounted on a rigid support frame and is connected to the Petrocarb Injection System by flexible stainless-steel lines. The support can be raised to any desired inclined position, defined by angle 8 with respect to the horizontal. Pressure transducers are used to measure pressure drops in the line. Two sets of Auburn monitors are installed in the loop to measure particle velocity and voidage in the transport lines. The position of the Auburn monitor sets can be changed along the loop to determine the change in particle velocity and voidage with distance from the solid injection point. All data acquisition is done by a computer. The following test conditions were used: a. Transport gas: nitro en at nominal flow rates of 8.50, 16.99, and 25.49 m /hr. 8 b. Solids: Pittsburgh S e a m coal with weight average particle sizes of 47 pm and 61 pm. c. Angle of inclination of the test loop: 0, 30, 60, and 87.5 degrees from horizontal. d. Injection tank pressure: 345 and 552 kPa. e. Duration of each test: 20 minutes. f. Solids flow rates ranging from 0 to 0.1 kg/s. 0 . 0 2 6 6 - ~DIAMETER SYSTEM This setup, located at the University of Pittsburgh, is shown in Figure 2 (Myler, 1987). Air was metered into the system THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL, 1989 TABLE1 Properties of Particles Used Material pp (kg/m3) 2470 2395 2464 5004 Glass Glass Glass Iron Ore Dp(pm) Shape U,(m/s) 67 450 900 400 Sphere Sphere Crushed Flake 0.46 3.97 7.45 5.87 Nylon Screws 0.0266 m r- 1 4 m I 1 - Screw Terminal Figure 4a view). / L---_ Insulator - Schematic of the 0.0504-m-diameter system (side L. Aluminum Figure 3 - Split electrostatic ring probe. through a rotameter by two valves, one before and the other just after the rotameter. This arrangement allowed the rotameter calibration pressure to be maintained for various gas flow rates. Because of the limitations of the air supply, the maximum flow rate obtainable was approximately 50.5 m3/hr. Following the rotameter was a Plexiglas column used to humidify the air. The humidifier could provide relative humidities greater than 75 % in order to reduce the elecrostatic charges. Solids were fed into the system using a volumetric screw feeder. Solids entered near the bottom of the test unit through a modified tee fitting. Two different tee fittings were used. The first one used in the 0 and 30 degrees inclinations was made by a 0.0508-m Excelon pipe, which was tapered to 0.025 m by 0.076 m. This pipe was molded to a 0.0266-m section of PVC pipe to form a tee fitting. The other, used in 45 degrees inclination, was made by putting cascadetype baffling below the inlet of a 0.0266-m tee fitting. The fitting was modified to allow the gas an opportunity to accelerate the particles to their steady velocities. The resulting gas-solid mixture flowed through the test unit. The test unit consisted of an entrance copper section, a main glass test section and a PVC flow splitter 0.0266 m in diameter. The chosen layout insured that all two-phase flow mixture sections were of the orientation studied, thereby reducing the effects of other orientations on the result obtained. Piping for different configurations was changed according to the desired orientation. The entrance copper section was provided to ensure that the solid particles were fully accelerated when they entered the test section. The entrance section was 2.4-m, 2.12-m, and 1.54-m long for 0-, 30-, and 45-degrees inclinations, respectively. The entrance section was made of copper to reduce electrostatic charges produced in the system, by grounding it. The main glass test section was used to observe the fully developed flow of solids. The length of this section was 2.1-m, 1.9177-m, and 1.848-m for 0, 30, and 45 degrees, respectively. Two split electrostatic aluminum ring probes were mounted in this section for determining particle velocities in the upper and lower halves of the pipe. A fiber insulator was used to separate the two halves, as shown in Figure 4b - Schematic of the 0.0504-m-diameter system (top view). Figure 3. The voidage was calculated from the particle velocity and the solids flow rate measurements using Equation (6). Pressure taps for pressure drop measurements were made at each end of the glass test section. Following the test section was a 0.9-m long section of PVC pipe, which was designed to split the flow into an upper and lower fraction. The upper and lower streams entered separate 0.0285-m3 collection tanks. Air from these gas-solid mixtures passed through filter bags located at the top of each collection vessel leaving the solids in the tanks. Particles of two different materials, glass and iron ore, were used. Their properties are given in Table 1. 0 . 0 5 0 4 - ~DIAMETER SYSTEM This test loop is shown in Figures 4a and 4b (Myler, 1987). Air was supplied to the unit by a Roots blower. The blower could deliver 424.75 m3/hr of air at 41.37 kPa. Air was sent through a humidifier unit. The air exiting the humidifier unit flowed to a tee fitting, where a bypass gate valve allowed the control of air discharge to the test unit. The air exiting the gas delivery unit passed through an Elster Turbine meter. Four different pipe orientations were studied: 90, 60, 30 and 0 degrees from the horizontal. Results and Discussions As mentioned earlier, flow studies were performed in 0.0095-m, 0.0266-m, and 0.0504-m systems held at different inclinations. The test sections in the last two setups were made of glass to allow the observation of the flow patterns for each experimental run. Measurements in these two experimental setups included particle velocities, solids mass flow rates, and pressure drops in both the upper and lower halves of the pipe. THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL, 1989 239 I 1 I I 1 I I -c cn 0 '2 0.8 0 .-0E 0 I I 1 I 1 I LEGEND - 2 1.0- -5 i./ P c A = 0 Degrees = 30 Degrees 0 = 45 Degrees v 1.0 A A g 0.8 8 0.6- Z? 0.40.2- 9 \ \ = A Uppor at Ip = 21.41 kebrn' V owo or at lP= 21.41 0 Uppor at = 40.40 0 Lowor at Ip = 40.49 A Uppor at jp = 59.02 0 Lowar at la = 59.82 0.4 0 Uppor at ;1 = 249.77 at ~p = ~ 9 . 7 7 + owo or I 10 I 0 I 1 I 20 1 30 U, ( m / 4 Figure 5 - U,,/Ug vs. U8 for 450-pm glass beads in 0.0266-mdiameter system at 30" from horizontal. 1.2 A - 2 1.0 -Ec 2 cn 0 0 C ;. 0.8 1.0- = = o = I I I E - 5 0.4- - - - -27 0.6-0 A LEBEND A Uppor at Ip = 53.93 kglmn' Lowor at lP= 53.93 Uppor at jP = 82.71 0 Lowor at Jp = 82.71 Uppor at lP= 120.40 0 Lowor at jp = 120.40 Uppor-at iP= isg.82 Lowor at lP= 180.82 \ \ V 9 0.4 0 - A 0.2 0.2 + I I 10 1 I 20 I I 30 I - - 0 -s5 0.6 0 = I LEGEND 450-pm Glass QOGpmGlass 67-pm Qlase w p m iron ore *g 0.8.-E 0 I V 0 I - I 40 U, (m/r) Figure 6 - U p / U8 vs. U, for 450-pm glass beads in 0.0504-mdiameter system at 60" from horizontal. I 0 I 10 1 I 20 I 30 PARTICLE VELOCITY Figure 8 - Average U p fU, vs. U, for 67-, 450-, and 900-pm glass, and 400-pm iron ore, in 0.0266-m-diameter system at 30" from horizontal when W,= 0.0232 kg/s. The particle velocity was determined by cross-correlating the two signals from two probes set at a fixed separation along the pipe. To confirm the particle velocity measurements obtained by the cross-correlation technique, two simultaneously actuated valves were used to determine mean void fraction. The average particle velocity could then be calculated using Equation (6). These values agreed well with those obtained from cross-correlation technique (Myler, 1987). Figures 5 and 6 show the ratio of particle velocity to gas velocity as a function of gas velocity for 450-pm glass beads in 0.0266-m and 0.0504-m-diameter pipes at 30 and 60 degrees, respectively. As can be seen, this ratio levels off to a constant value at high gas velocities for all the solids mass flow rates studied. At lower gas velocity, this ratio drops sharply from this constant value. These figures show that the constant Up/ U, ratio increases with increasing pipe diameter. It should be noted that they are also at different orientations. However, the effect of inclination angle will be discussed later. As mentioned earlier, particle velocities were measured in both the lower and upper halves of the pipe. At high gas velocity, the velocities are almost the same, but at lower gas velocity, the upper-half particle velocity is generally higher than the lower one. A comparison of Up/ Ugratio for different orientations at constant mass flow rate is shown in Figure 7. As can be seen, the ratio increases slightly with the decreasing angle of inclination. In the 0.0095-m transport system, the particle velocity remained fairly constant also with respect to angle of loop inclination. Figure 8 shows the effect of particle characteristics on the U p / U , ratio. This ratio increases with decreasing particle size. However, for the experiments with 0.0095 m diameter pipe transfer line, the effect of average coal particle size on the particle velocity was small owing to the narrow 240 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL, 1989 I . I I I 1 I - V . 800 I LEGEND A Uppar at Jp = 21.41 k g l r d Lower at ~p = 21.41 Upper i t jP = 40.49 0 Lower at Jp = 40.49 Uppor i t jP = SB.BZ 0 Lowaratlp = 59.92 Uppor at jP = z40.n Lowor at Jp = 240.77 Air only 0 A 700 - + 600 1 I I LEQEND A = ODegrees = 30 Degrees 0 = 45 Degrees 0 = Alr only v - 6 -500E 3' 2 P -I 2 0 10 20 ! :zi Figure 9 - A P / L vs. UR for 450-pm glass beads in 0.0266-mdiameter system at 30" from horizontal. 200 100 I 10 I A V - 0 A + 0I 20 I I Figure 11 - A P / L vs. U,for 450-pm glass beads in 0.0266-mdiameter system at O", 30", and 45" from horizontal when W, = 0.0332 kg/s. A 1 I II LEQEND Uppar at jp = 53.83 kdr.mY Lowarrtjp = 53.83 Upporcltjp = 82.71 0 Lower at jp = 82.71 Upporatjp = 1 2 0 4 0 Lowor at jp = 120.40 W Upporat jp = 189.82 Lowor atjp = 189.82 0 Air only 0 I I I I LEQEND = 450.pm Glass V = W&pm Qlars O= 67-pm Qlars O = m p r n Iron Ore .= Alr only 1 I I 20 10 U, (m/s) Figure 10 - A P / L vs. U, for 450-pmglass beads in 0.0504-mdiameter system at 60" from horizontal. Figure 12 - A P / L vs. U, for 67-, 450-, and 900-pm glass, and 400-pm iron ore in 0.0266-m-diameter system at 30" from horizontal when W, = 0.0232 kg/s. THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL, 1989 241 LEQEND A = I t Wa = 0.0110 kqlr V = at Wa = 0.0125 0 = at Wa = 0.0333 0 = I t WE = 0.1386 - I 2 1.0C 0 .-u) C 0) E 0.8.-0 - Y 3 r" 0 . 6 ' 0.4 I I 10 0 U, (m/s) I 1 20 I I 30 Figure 13 - W,, / W,,vs. U, for 450-pm glass beads in 0.0266-mdiameter system at 30" from horizontal. difference between the coal sizes (47-pm and 61-pm). The U p / U g ratio for the 400-pm iron ore is much higher than that for the 450-pm glass beads, which is conjectured to be due to the shape of these particles. PRESSURE DROP Pressure drop measurements in the 0.0266-m and 0.0504-m setups were made along the top and the bottom section of the pipe. The main purpose of this measurement was to quantify the lift forces present in the systems studied. This proved to be an inclusive method of measurement for lift, as it was found the pressure drop across both halves was almost equal (Figures 9 and 10). As can be seen from these figures, the pressure drop increases with solids flow rate but decreases with increasing pipe diameter. The minimum point shifts to higher gas velocity with increasing solids flux. The pressure drop increases with angle of inclination (see Figure 1 1). The gas velocity at minimum pressure drop also increases with the angle of inclination. This difference seems to be due to the gravity effect ( g sin 0 ) and the solids friction factor. The solids friction factor was obtained using the following equation: A= ( A P \ / L ) D, 2 u; Pp ........................ - - 0.2- -1 t 1 0 I I 1 I 20 10 I 1 30 U, (m/sl Figure 14 - W,,/W,, vs. Ug 450-pm glass beads in 0.0266-mdiameter system at O", 30", and 45" from horizontal when W,= 0.0332 kgls. for the solids friction factor was fitted into the Konno-Saito type correlation V; = C/Fr,). It was found that the value of C goes through a maximum at 30 degrees. Another approach which one could follow, is a modification of the basic force balance (Equation 3) where the solids friction contribution is broken down into various interaction terms encountered in gas-solid systems. This type of approach, although addressing some of the weaknesses of the force balance of Equation (3), requires much larger data base to study the individual interaction terms separately. The pressure drop increases with particle size (see Figure 12). The same trend was observed in the 0.0095-m setup for the coal particles studied. The gravity force contribution to the total pressure drop did not vary much with particle sizes due to the narrow size range of coal particles studied. The gas frictional term could not be measured independently in the presence of particles, hence it could safely be assumed to be constant. Thus, the difference observed may be attributed to the solid friction factor. As mentioned earlier, the particle velocity in the 0.0094-m system was not affected by the coal particle size studied. Thus the solids friction factor must account for the change in pressure drop. SOLIDSMASSFLOWMEASUREMENTS (7) (1-6) Measurements of the solids mass flow rates in both the upper and lower halves of the 0.0266-m and 0.0504-m pipe where A P , is defined as: indicate that a uniform suspended flow regime with particles uniformly distributed over the cross-section of the pipe AP AP\ - - - ~g sin 0 [pfe + p,, ( l - e ) ] depends on gas velocity, particle characteristics, and system L L characteristics (i.e., orientation and pipe size). At high gas - -APU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @I velocity, even though the distribution seems uniform visually, more solids were at the bottom than at the top. In some cases, L however, the opposite was observed owing to electrostatics. At any given solids mass flow rate, as the gas velocity is It should be noted that all the terms in the right-hand side decreased, more solids were contained in the lower half than of Equation (8) are known experimentally. The data obtained 242 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL, 1989 -t 1.4 1.2 - - Particle velocities were also measured in the lower and upper halves of the pipe. These measurements were almost equal at high gas velocities. Then, as the gas velocity decreases, the solids velocities were no longer equal, with the top velocity generally higher. The ratio of U p / U g showed a slight increase with the decreasing angle of inclination. This trend could be due to the variation of gravity and solids friction terms. This ratio also increases with decreasing particle size. Measurements of solids mass flow rate were made in both the upper and lower halves of the pipe. In general, the ratio of these measurements (upper/lower ratio) decreased with decrease in gas velocity to a point and then increased. this increase may be due to the increase in particle velocity in the top half of the pipe. LEQEND A = +m Qlrrr = OOOpm Qlrar 0 = 67rmQlrra 0 = lOOpmImnOn v 1.0- .-0 u) c E .- 0.8- - w Y 5 \ 0.6- 3 0.4 -- Acknowledgements 0.2 c I 0 I 1 I 10 1 20 1 1 30 U, ( m i 4 The authors wish to acknowledge the National Science Foundation and the Coal Utilization Division of the Pittsburgh Energy Technology Center for support to undertake this study. Disclaimer Figure 15 - W,,,/W,, vs. Ux for 67-, 450-, and 900-pm glass, and 400-pn iron ore, in 0.0266-m-diameter system at 30” from horizontal when W, = 0.0232 kg/s. References in this paper to any specific commercial product, process, or service is to facilitate understanding and does not necessarily imply its endorsement or favoring by the United States Department of Energy. in the upper half of the pipe. There was, however, no distinct interface between the two. As the gas velocity is further reduced, a clear interface between the upper and lower halves was developed. This interface never reached the center of the pipe. Figure 13 shows the unique relationship of solids mass flow ratio of upper to lower halves as a function of gas velocity at constant solids flow rate. As mentioned earlier, this ratio decreases with gas velocity at a given solids flow rate. At sufficiently low velocities, an increase is seen in the ratio, which is caused by the decrease in particle velocity in the lower dense flow with increase in velocity of the upper dilute flow. A comparison of this mass flow ratio at different inclinations srudied reveals that the mass flow ratio for the 45-degree incline is lower than for the 0-degree incline but higher than for the 30-degree incline (see Figure 14). The profile in the horizontal system is much steeper than the other two orientations. Figure 15 shows the effect of particle characteristics on this ratio. The most interesting trend observed is the mass ratio of 67-pm glass beads, which was lying in between the ratio 450-pm and 900-pm glass. This variation could be due to the cluster formation effect in the case of 67-pm glass particles. Nomenclature = = = = = = = = = = = = = = = = = = = = = = = = Conclusions Experimental measurements of particle velocity, pressure drop, and mass flow rate of solids were made in three different setups with different pipe sizes. In the 0.0266-m and 0.0504-m systems, these parameters were measured in both the upper and lower halves of the pipe. Pressure-drop measurements were made along the pipe at the top and the bottom of the pipe. Under most conditions, these measurements were equal. The pressure drop increases with the angle of inclination and the particle size. The increase in pressure drop with angle 0 from the horizontal could be attributed to the changes in static head and the solids friction factor. single-particle drag coefficient particle diameter (m) tube diameter (m) gas friction factor particle Froude number solids friction factor drag force (N) gas gravitational term (N) solids gravitational term (N) frictional term due to the gas phase (N) frictional term due to the presence of solids (N) gravitational acceleration (,m/s2) solids mass flux (kg/s . m-) mass of gas in differential element x (kg) mass of solids in differential element x (kg) pressure gradient (Pa/m) pressure drop per unit length (Palm) air only pressure drop per unit length (Pa/m) actual gas velocity (m/s) superficial gas velocity (m/s) particle velocity (m/sec) total solids mass flow rate (kg/s) mass flow rate of solids in lower half of the pipe (kg/s) mass flow rate of solids in upper half of the pipe (kg/s) Greek letters E 0 PJ PP void fraction angle of inclination from the horizontal (degrees) gas density (kg/m3) = solids density (kg/m’) = = = References Capes, C. E. and K. Nakamura, “Vertical Pneumatic Conveying: An Experimental Study with Particles in the Intermediate and Turbulent Flow Regimes”, Can. J. Chem. Eng. 51, 31-38 (1973). T H E CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOLUME 67. APRIL, 1989 243 Klinzing, G. E., “Gas-Solid Transport”, McGraw-Hill Book Co., New York (1981). Kono, H. and S . Saito, “Pneumatic Conveying of Solids Through Straight Pipes”, Chem. Eng. Japan 2, 211-217 (1969). Myler, C. A., “Gas-Solid Transport in a 0.0508-m Pipe at Various Inclinations With and Without Electrostatics”, M.S. Thesis, University of Pittsburgh (1985). Myler, C. A ., “Use of a Thermodynamic Analogy for Pneumatic Transport in Horizontal Pipes”, Ph.D. Thesis, University of Pittsburgh (1987). Owen, P. R . , “Pneumatic Transport”, J. Fluid Mech. 39, 407-432 ( 1969). 244 Wen, C. Y. and Y. H. Yu, “Mechanics of Fluidization”, Chem. Eng. Prog. Symp. Series 62, No. 62, 1 0 0 - 1 1 1 (1966). Yang, W. C., “Correlations for Solid Friction Factors in Vertical and Horizontal Pneumatic Conveyings”, AIChE J. 20, 605-607 (1974). Zenz, F. A. and D. F. Othmer, “Fluidization and Fluid-Particle Systems”, Reinhold Publishing Corporation, New York (1960). Manuscript received January 15, 1988; revised manuscript received September 7, 1988; accepted for publication September 14, 1988. THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 67, APRIL. 1989