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Two-dimensional steady state hydrodynamic analysis of gas—solids flow in vertical pneumatic conveying systems
Powder Technology, 48 (1986) 67 - 74
67
Two-Dimensional Steady State Hydrodynamic Analysis of Gas-Solids Flow in Vertical Pneumatic Conveying Systems M. A. ADEWUMI The Pennsylvania State University, University Park, PA 16802 (U.S.A.) and H. ARASTOOPOUR
Illinois Institute of Technology, Chicago, IL 60616 (U.S.A.) (Received March 18, 1986)
Presented at the 1st World Ongress on Rwticle Technology, held in Nuremberg (F.R.G.) April, I986
SUMMARY
Pneumatic conveying, as a mode of solid particles transportation, is widely used in the petroleum, chemical, petrochemical, gas processing and food processing industries. Within the last decade, there has been increasing demand for an optimized utilization of this technology, particularly in the energy conversion processes. Such optimized u tilization of this technology requiresgood understanding of the hydrodynamic behavior of gas-solids mixture flow in pipes and the capability to predict such behavior. Empirical correlations have been developed in the literature for predicting some of the more commonly sought design variables such as pressure drop. Most of such correlations were developed using one-dimensionally based experimental data. Apart from being limited to the data-base used in developing them, they ignore the effect of radial non-uniformity of the basic variables such as solid velocity, void fraction, etc. More recent experimentations confirm this non-radial uniformity of such important design variables as solid velocity and solid concentmtion. We present a two-dimensional steady state two-phase hydrodynamic model to describe upward co-current pneumatic conveying of solid particles in vertical pipe. The model incorporates viscous dissipation terms in both the gas phase and the particulate phase. Numerical solution was obtained using the numerical method of lines. The predicted system variables and their distributions under different operating conditions agree with the observed behavior of the system reported in
the literature. The predicted solid velocity profiles using this model agree reasonably well with available experimental data.
INTRODUCTION
Pneumatic conveying, as a mode of solid particles transportation, has various applications in various industries, ranging from the chemical, petrochemical and food processing industries to the petroleum industry. Circulation loops of catalyst crackers in refineries, long-distance transportation of coal using pneumatic pipelines, coal feeding in solid fuel processing such as coal gasification, and pneumatic drilling of oil wells are a few representative examples of the processes where the technology of pneumatic conveying of solid particles is used. A more efficient utilization of this mode of solid transportation demands that one possesses the capability to predict some very important design parameters such as minimum transport velocity of the gas/air, the power requirement, choking velocity, etc. Several articles have appeared in the literature reporting experimental measurements and/or empirical correlations of various design variables needed in gas-solid pneumatic conveying. While the literature on this is voluminous, the capability for predicting the physical behavior of the system and the associated phenomena is rather inadequate. This calls for a more basic understanding and description of this system. This assertion has been attested to by recent state-of-the-art reviews of the work in this field by three of 0 Elsevier Sequoia/Printed in The Netherlands
68
the foremost investigators in this field (Knowlton [ 11, Leung [ 21 and Arastoopour [3]). Most of the experimental studies have been designed to measure pressure drop in the system and are also one-dimensionally based, thereby ignoring the effect of radial nonuniformity of the basic variables such as solid velocity profile, void fraction, etc. The predictive models for describing gassolids system variables that have been developed in the literature fall into two general categories, namely the empirical approach and the fundamental hydrodynamic approach. In the former approach, using experimental data generated and/or collected, empirical correlations are developed for predicting the important design variables and parameters, such as pressure drop, choking velocity, etc. The common characteristic of these correlations is that they do not usually apply beyond the range of the data-base used in developing them. As for the latter approach, few attempts have been made in the literature to apply generalized multiphase flow equations to gassolids flow systems. Pioneering efforts at developing such models to describe gas-solids systems were made by Jackson [ 4, 51, Anderson and Jackson [6], and Soo [ 7,8]. Kliegel and Nickerson [9] were among the first to analyze some form of multiphase flow equations for gas-solids system. They basically used the method of characteristics to solve the equations for gas-particles mixture in axially symmetric nozzles. Other pioneering efforts, especially in the application of multiphase flow equations to dilute gassolid pneumatic conveying, include those of Arastoopour and Gidaspow [ 10 - 121, Arastoopour et al. [ 131, Shih et al. [ 141, and Nakamura and Capes [ 15,161. As most often done in the literature on this topic, most of these earlier models are onedimensional steady state models, except for that of Shih et al. [14], which is a twodimensional steady state model for horizontal pipe. While the one-dimensional models may have been quite successful in pressure drop predictions (e.g., Arastoopouret a2. [ 10 - 13]), experimental evidences have suggested that the assumption of non-radial variation of these basic variables may not be true. Measuring gas and solid velocities at various radial locations in a vertical pneumatic conveying
system, both Doig [17] and Reddy [18] found significant radial non-uniformity. In fact, both their gas and solid velocity profiles seem to follow power-law radial variation. Furthermore, Doig’s observation of non-uniformity in radial distribution of solid concentration agrees with a similar observation earlier by Soo [ 191. Doig also notices that the particle velocity profile attains ‘fully developed flow’ away from the pipe entrance. A more recent study by Lee and Durst [20] confirms this non-uniform particle velocity profile across the pipe crosssection. More recently however, Adewumi and Arastoopour [21] presented a pseudotwo-dimensional model for describing pneumatic conveying of solid particles in vertical pipes. While their model neglects the radial components of the velocity vectors, it accounts for the radial variation of the axial components of the velocity vectors as well as the other system variables. Their model agrees reasonably well with Reddy’s experimental data on solid velocity profile. However, neglecting the radial velocities may overshadow some important effects such as localized internal circulation of solids, which of course may become appreciable as the system approaches choking condition and also in the entrance region where feed conditions have the most impact. From the foregoing discussion, it seems that a good knowledge of the local particle velocity distribution is an integral part of the basic understanding of gas-solids flow behavior in pneumatic conveying. Such phenomena as choking and clumping that have been experimentally observed might have their roots in the two-dimensional effects described.
MODELDEVELOPMENT
Volume-averaging technique was used to derive the general multiphase flow equations for solid pneumatic conveying (see Adewumi [22]). The basic assumptions made in the development of the two-dimensional steady state model are summarized below: (1) The system is isothermal and there is no interphase mass transfer. (2) The particle size is uniform and the particles have the same density.
69
(3) Solid particles of uniform size and density are defined as a particulate phase, where by each phase is assumed to form a continuum. (4) The particulate phase is assumed to be incompressible, while the gas phase obeys the ideal gas law. (5) Pressure gradient is assumed to exist in the gas phase only. (6) Flow is assumed to be steady. The resulting conservative and additional constitutive equations that define our model are presented below. The primitive variables are V,, V,, the axial components of the gas and particulate phases velocity vectors, respectively, Us and US, the corresponding radial components, (xg and ps, the gas void fraction and density and the pressure P, which is coupled to pg via the equation of state.
Particulate phase :
aw - ~,hw4i az
+--
I
a
r ar Ir(l
-
~&?9~*21
=F,
(6)
where FDZ and FD, are the drag forces in the axial and radial directions, respectively, and FGg, FGS are the gas and particulate phases gravitational forces, respectively. The conatitutive equations Assumptions made yield the following equations of state for the gas and particulate phases. Gas phase equation of state: Pg = VP I
(7)
Mg
rl=RT
Continuity equations Gas phase:
I
Particulate phase equation of state:
(1)
ps = pso = constant
(3)
The gravity forces are defined as
Particulate phase:
Fcig = ‘I[gP&
(9)
(10) @,)P,, g The drag forces are defined as below. The basic assumption is that the functional form of the drag force expressions is the same in both the axial and radial directions. Drag force in the axial direction:
FGS = (I-
Momentum equations in Z-direction Gas phase :
F DZ=
3 cnz(l--
4
@&s-2’67( v, -v,)l
v, - v,(ps
d,
Drag force in the radial direction: Particulate phase:
FDr=-
(11)
3 c,~l-a,)~,-2’67(Ug-Us)lUg-U,Ipg 4
d,
where CC, a =--
r
ar
r(l--a,)2
1
+ F,z-FGS
(1.0 + 0.15ReZ0S687) ReZ < 1000
(4) c DZ=
I0.44
Momentum equations in r-direction Gas phase :
aw,P,v,u,) az =-_
+ +
(13)
$mgPgug2)
cD, =
(1.0 + 0.15Re,0.687)
Re, < 1000 Re,>
i+P -_
at- -
ReZ > 1000
FD,
(5)
1000 (14)
70
Boundary conditions At r = 0 (the pipe axis), we assume symmetry and hence the following:
ReZ
and Re, =
u, = 0 44
-&Id, k
(16)
Chamcteristic analysis of the model The set of quasi-linear partial differential equations describing our model can be re-written in the form
:+A($
=C(U)
whose characteristic equation is IM-A(
=0
The details of evaluating these characteristics have been worked out by Adewumi [22]. The six resulting characteristics are
u, = 0
av,_0 -ib
av,
-=
o
ar
aal3 -=
ar
0
At r = R (at the pipe wall): u, = 0 u, = 0 V, = 0 or very small number
NUMERICAL SCHEME
Since all these characteristics are real, then this model is well posed as an initial-value problem.
Initiul conditions These are basically the inlet conditions into the pipe, but they serve as our initial conditions, since we march in Z-co-ordinate. At 2 = 0, we specify V,, V,, cygand P, based on the feed conditions and the specified solid loading. Parabolic distribution is assumed for the radial velocities at the pipe inlet. These profiles are such that they give zero value at the wall and the pipe axis.
Numerical method of lines is used to solve the quasi-linear set of partial differential equations constituting this model. This method basically transforms a set of partial differential equations (and the associated initial and boundary conditions) into a system of coupled ordinary differential equations. This is achieved by diseretizing the partial derivatives in one independent variable (in the case of two independent variables) while leaving the other continuous. The resulting system of ordinary differential equations is usually large and stiff and hence must be solved using robust stiff ODE integrator. The numerical method of lines is implemented in FORSIM package [23]. FORSIM, in addition, provides options for various ODE integrators and the GEARHINDMARSH method, which uses variable step implicit backward difference method, is used.
PARAMETRIC STUDY
Parametric study was conducted on a system similar to the experimental system of Reddy [ 181. Uniform size and density
71
particles 2.70 X 10m4 m in diameter and 2590 kg/m3 in density were used. The value of ‘solid viscosity’ used in our model was calculated based on the experimental data of Arastoopour and Cutchin [ 241 and Arastoopour et al. [25] for flow of coarse particles relative to the mixture of gas and fine particles. The estimated value of solid viscosity is 0.1 Pa.s [ 221. The input needed in this model includes the inlet values of the primary variables V,, V,, cyg,P, U, and U, at the pipe inlet. Since these were not explicitly measured by Reddy, we specified continuous functions that satisfy Reddy’s gas and solid mass balances and ensure ‘continuity’ from initial to boundary conditions. This helps to remove the initial-boundary conditions singularity at the pipe wail and thus reduces, considerably, the CPU time required to solve the problem. We assumed uniform pressure of 1.2549 X lo5 Pa, 2 = 0. Figure 1 shows the solid velocity profile at two axial locations in the pipe. It also gives the solid velocity profiles at two inlet gas velocities of 8.7 and 14.7 m/s. At any fixed inlet gas velocity, the profiles show evidence of developing flow as it moves along the pipe. Near the pipe inlet (0.0244 m above the inlet), the solid velocity profile is almost flat except I
-
vao
= 14.7 m/r
- -
VP0 = 5.7 m/r
=O.lm
0
s2.7 dip P, Ir.
_
x lo-‘m
=1.22m ‘2590
near the wall, whereas at a higher axial location (1.22 m above the pipe entrance), a more pronounced non-uniform profile, tending towards parabolic, is observed. At the inlet, due to the entrance effect and possible flow instability, the radial velocity would be relatively large. However, because of the presence of the wall, the particles hit the wall and are deflected back into the fastflowing gas stream at the core and are accelerated. Thus, as they move up the pipe, the axial velocity becomes very dominant and the radial velocity becomes low for continuity to be satisfied. This eventually gives rise to ‘developed’ flow, analogous to that which obtains in single-phase flow, where the radial velocity is practically zero. The solid velocity is generally higher for the higher inlet gas velocity. This is due to the larger amount of drag exerted on the solid by the gas, since drag is proportional to the square of the relative velocity between the gas and the solid particles. In the region close to the wall, the gas velocity is lower than that in the core of the pipe. The solid velocity is correspondingly lower and the area swept along by the gas becomes smaller for lower gas velocities, thus giving rise to the thicker solid boundary layer zone. Figure 2 shows the solid velocity profile along the axis of the pipe. It shows rapid acceleration near the pipe entrance and a lower one away from the pipe entrance. As the particles are injected into a high-
kg/m3
/LQ
=O.l Pa-s ‘l.SX IO-5Pa-r
P,
* 1.25481
IO5 Pa
D
so. I m ‘2.7~ IO-‘m
PI
* 2590
ps
so.1 Pa-r * I.8 I 10-5 PO-,
voo = 14.7 m/r
* I.22 m
2
I.”
“..a
I
RADIAL
DISTANCE,
(DIMENSIONLESS)
Fig. 1. Effect of gas velocity on solid velocity profile (at varying Z), predicted by the two-dimensional model.
IO-
I
o".O
kg/m3
I
0.5 AXIAL DISTANCE,
I
I.0 m
Fig. 2. Effect of gas velocity on solid velocity profile, predicted by the two-dimensional model.
72
velocity gas stream, a tremendous amount of drag is exerted on them since the relative velocity between the gas and solid is very high near the pipe entrance. Since the drag is proportional to the square of the relative velocity, the drag is proportionately high. Despite this, gravity will try to decelerate these particles but the drag force is dominant in this zone, usually referred to as the acceleration zone. As they move away from the entrance region, their velocity is closer to that of the conveying gas and drag is therefore considerably reduced to a magnitude of the same order as the decelerating gravitational force. As they move farther up the pipe, the drag is balanced by the solid weight and thus a constant velocity, called the equilibrium velocity, is attained. This figure clearly shows this tendency, even though the pipe length is not long enough for the particles to reach this ultimate velocity. Also, the fire shows the solid velocity profile at two different inlet gas velocities (8.7 and 14.7 m/s). At higher inlet gas velocity, the solid velocity is consistently higher. This is due to the larger drag exerted on the particles by the faster flowing gas. Moreover, the profile for the lower gas velocity shows the tendency to reach its equilibrium velocity at a shorter length. This is due to the fact that, since the solid particles in both cases have the same terminal velocity, their equilibrium velocities are only dependent on the conveying gas velocity. The solid for the lower gas velocity has lower equilibrium velocity and thus
should attain this within a shorter pipe length. Figure 3 shows the pressure drop profile along the pipe axis. It shows rapid pressure drop in the acceleration region, up to about 0.6 m from the pipe inlet, after which the pressure gradient is almost constant. In the acceleration region, both the drag force and the momentum outflow contributions to the pressure drop are high, thus accounting for the higher pressure gradient in this zone. As the solid particles reach their equilibrium values, the drag force will be constant and is balanced by the solid weight, the solid velocity is essentially constant and the acceleration contribution becomes negligible. The pressure gradient is thus essentially equal to the weight of the solids. Although this tendency is portrayed in Fig. 3, the pipe is too short to fully observe the pressure gradient constancy. On the same figure, we show the effect of two different inlet gas velocities on the pressure drop profile, with the higher gas velocity giving rise to higher pressure drop throughout the system. The higher the gas velocity, the higher the drag on the solid, the higher the acceleration of the solid and the solid and gas frictional forces. All these account for contributions to the higher pressure drop at higher gas velocities. Figure 4 is a plot of the solid radial velocity as a function of the radial position. The magnitude of the radial velocities imposed 0.16
I 0
C.,2-
pg
-1.6x
P,
‘I.2546
v,,
so.1
so.1 82.7
I
I
m x 10-4,
P,
=1.22m ‘2590kgh3
p*
=O.l
Pa-r
IO-Spa-r x lOsPa m/s
aoo no.904 0 0.0
I
I
0.5 AXIAL
1.0 DISTANCE.
m
Fig. 3. Effect of gas velocity pressure drop profile, predicted by the two-dimensional model.
0.0
0.2
0.4 RADIAL
DISTANCE.
0.6
0.6
1.0
IOIMENSIONLESSI
Fig. 4. Solid radial velocity profile (at varying Z), predicted by the two-dimensional steady state model.
73
at the pipe inlet are relatively small compared with their axial counterparts. The imposed distribution is parabolic, with vanishing values at the centre and at the wall, in order to satisfy symmetry at the centre and the existence of the wall, respectively. Nevertheless, some observations of their distributions away from the pipe inlet deserve mentioning. Generally, these velocities remain small compared with their axial counterparts. This is expected, since the flow develops as the fluid moves away from the entrance, as indicated by Fig. 1. Around the pipe entrance, the particles close to the wall move slowly towards the centre of the pipe; in other words, they are entrained by the high gas velocity zone at the central portion of the pipe. Farther up in the pipe, these particles are replaced by faster-moving particles, thus causing the radial movement of the fastermoving particles towards the pipe wall (see Fig. 4). This is probably the main cause of some kind of local circulation or vortex motion calculated in our gas-solids system.
time the particles enter the test section. The inlet conditions to the test section, such as pressure, gas and solid velocities, and void fraction, were not reported by Reddy and these variables were not measured at the pipe wall. These are generally needed to impose both the initial and boundary conditions on our equations. We specified an inlet void fraction of 0.984. Using this, the gas flow rate (measured) and a solid loading of 0.4 (as specified by Reddy) we estimated the inlet velocity of the particulate phase. Figure 5 shows the comparison of the predicted solid velocity profile with Reddy’s experimental data. The predicted solid velocity profile compares reasonably well with the experimental data. Apparent discrepancies between the experimental data and predicted results in the wall region depict the effect of the no-slip boundary condition imposed on the solid velocity at the wall. This may not be appropriate. Improved measurement of the system variables at the wall will help the specification of the boundary conditions and should improve the performance of this model.
COMPARISON WITH REDDY’S EXPERIMENTAL DATA ACKNOWLEDGEMENT
An attempt was made to compare the predicted numerical results with Reddy’s [18] experimental data. The test section of Reddy’s set-up consists of a 0.1 m I.D., 1.2 m long vertical column. This is preceded by a calming section 22 times the pipe diameter to ensure uniformly stable motion by the
PARTICLE
SIZE.
LOADING
NUMBER=SS.OOO
EXPERIMENTAL
PREDICTED
o.ov 0.0
dpn270p
RATIO -0.40
REYNOLD’S REOOY’S
The authors would like to thank the Gas Research Institute, Basic Research Division, for partial financial support under Contract No. 5084260-1071 and the Institute of Gas Technology for initial financial support and its research facilities.
DATA
RESULTS
I
I
I
1.0
2.0
3.0
DISTANCE
FROM THE
I
I
4.0
WALL, cm
Fig. 5. Comparison of calculated solid velocity profile with Reddy’s [ 181 experimental data.
74 LIST OF SYMBOLS CD, CDZ
4 FD, FDZ F Gtz FGS iif
Pg r R Re,
Rez
drag coefficient in r-direction drag coefficient in Zdirection particle diameter drag force in rdirection drag force in Z-direction gas phase gravitational force particulate phase gravitational force gravitational acceleration gas molecular weight pressure radial co-ordinate gas constant particulate Reynolds number in rdirection particulate Reynolds number in Z-direction absolute temperature radial component of gas velocity radial component of solid velocity axial component of gas velocity axial component of solid velocity axial co-ordinate
Greek symbols gas volume fraction % h characteristics of the system of equations gas viscosity Elg particulate phase viscosity PS gas density ps solid density Ps
Subscripts gas phase quantities g S 0
particulate phase quantities inlet conditions
REFERENCES T. M. Knowlton, Solids Transport-state-of-theArt and Research Needs, paper presented at NSF Workshop on Fluidization and Fluid-Particles Systems - Research Needs and Priorities, RPI, New York, 1979. L. S. Leung, in J. R. Grace and J. M. Matsen
3 4 5
10 11 12 13 14 15 16
17 18 19 20 21
22 23 24 25
(eds.), Fiuidization, Plenum Press, New York, 1980. H. Arastoopour, in N. P. Cheremisinoff (ed.), Encyclopedia of Fluid Mechanics, vol. 4, Gulf, Houston, TS, 1986, section 2. R. Jackson, Trans. Inst. Chem. Engrs., 41 (1963) 13. R. Jackson, in J. R. Davidson and D. Harrison (eds.), Fluid Mechanical Theory in Fluidization, Academic Press, New York, 1971. T. B. Anderson and R. Jackson, Ind. and Eng. Chem. Fund., 6 (1967) 527. S. L. Soo, The Physics of Fluids, 20 (1977) 568. S. L. Soo, Multiphase Transport, 1 (1979) 291. J. R. Kiiegel and G. R. Nickerson, Flow of GasParticle Mixtures in Axially Symmetric Nozzles, Paper presented at the American Rocket Society Propellants, Combustion and Liquid Rockets Conference, Palm Beach, Florida (26 - 28 April, 1961). H. Arastoopour and D. Gidaspow, Znd. and Eng. Chem. Fund., 18 (1979) 123. H. Arastoopour and D. Gidaspow, Chem. Eng. Sci., 34 (1979) 1063. H. Arastoopour and D. Gidaspow, Powder Technol., 22 (1979) 77. H. Arastoopour, S. C. Lin and S. A. Weil, AZChE J., 28 (1982) 467. Y. T. Shih, H. Arastoopour and S. A. Weil, Znd. and Eng. Chem. Fund., 21 (1982) 37. K. Nakamura and C. E. Capes, The Can. J. Chem. Eng., 51 (1973) 29. K. Nakamura and C. E. Capes, in D. L. Keairns (ed.), Fluidisation Technology, Vol. 2, Hemisphere, Washington, DC, 1976, pp. 159 - 184. I. D. Doig, Ph.D. Thesis, Univ. New South Wales, Australia (1965). K. V. S. Reddy, Ph.D. Thesis, Univ. Waterloo (1967). S. L. Soo, Znd. and Eng. Chem. Fund., 1 (1962) 33. S. L. Lee and F. Durst, Znt. J. Multiphase Flow, 8 (1982) i25. M. A. Adewumi and H. Arastoopour, PseudoTwo-Dimensional Steady State Two-Phase Flow Model for Gas-Solids Vertical Pneumatic Conveying Systems, Paper presented at the 1985 annual meeting of the American Institute of Chemical Engineers, Nov. 10 - 15, 1985 in Chicago, IL. M. A. Adewumi, Ph.D. Thesis, Iilinois Institute of Technology, Chicago (1985). M. B. Carber, Atomic Energy of Canada Limited Report AECL-4844 (1974). H. Arastoopour and J. H. Cutchin III, Chem. Eng. sci., 40 (1985) 1135. H. Arastoopour, C. H. Wang and S. A. Weil, Chem. Eng. Sci., 37 (1982) 1379.
67
Two-Dimensional Steady State Hydrodynamic Analysis of Gas-Solids Flow in Vertical Pneumatic Conveying Systems M. A. ADEWUMI The Pennsylvania State University, University Park, PA 16802 (U.S.A.) and H. ARASTOOPOUR
Illinois Institute of Technology, Chicago, IL 60616 (U.S.A.) (Received March 18, 1986)
Presented at the 1st World Ongress on Rwticle Technology, held in Nuremberg (F.R.G.) April, I986
SUMMARY
Pneumatic conveying, as a mode of solid particles transportation, is widely used in the petroleum, chemical, petrochemical, gas processing and food processing industries. Within the last decade, there has been increasing demand for an optimized utilization of this technology, particularly in the energy conversion processes. Such optimized u tilization of this technology requiresgood understanding of the hydrodynamic behavior of gas-solids mixture flow in pipes and the capability to predict such behavior. Empirical correlations have been developed in the literature for predicting some of the more commonly sought design variables such as pressure drop. Most of such correlations were developed using one-dimensionally based experimental data. Apart from being limited to the data-base used in developing them, they ignore the effect of radial non-uniformity of the basic variables such as solid velocity, void fraction, etc. More recent experimentations confirm this non-radial uniformity of such important design variables as solid velocity and solid concentmtion. We present a two-dimensional steady state two-phase hydrodynamic model to describe upward co-current pneumatic conveying of solid particles in vertical pipe. The model incorporates viscous dissipation terms in both the gas phase and the particulate phase. Numerical solution was obtained using the numerical method of lines. The predicted system variables and their distributions under different operating conditions agree with the observed behavior of the system reported in
the literature. The predicted solid velocity profiles using this model agree reasonably well with available experimental data.
INTRODUCTION
Pneumatic conveying, as a mode of solid particles transportation, has various applications in various industries, ranging from the chemical, petrochemical and food processing industries to the petroleum industry. Circulation loops of catalyst crackers in refineries, long-distance transportation of coal using pneumatic pipelines, coal feeding in solid fuel processing such as coal gasification, and pneumatic drilling of oil wells are a few representative examples of the processes where the technology of pneumatic conveying of solid particles is used. A more efficient utilization of this mode of solid transportation demands that one possesses the capability to predict some very important design parameters such as minimum transport velocity of the gas/air, the power requirement, choking velocity, etc. Several articles have appeared in the literature reporting experimental measurements and/or empirical correlations of various design variables needed in gas-solid pneumatic conveying. While the literature on this is voluminous, the capability for predicting the physical behavior of the system and the associated phenomena is rather inadequate. This calls for a more basic understanding and description of this system. This assertion has been attested to by recent state-of-the-art reviews of the work in this field by three of 0 Elsevier Sequoia/Printed in The Netherlands
68
the foremost investigators in this field (Knowlton [ 11, Leung [ 21 and Arastoopour [3]). Most of the experimental studies have been designed to measure pressure drop in the system and are also one-dimensionally based, thereby ignoring the effect of radial nonuniformity of the basic variables such as solid velocity profile, void fraction, etc. The predictive models for describing gassolids system variables that have been developed in the literature fall into two general categories, namely the empirical approach and the fundamental hydrodynamic approach. In the former approach, using experimental data generated and/or collected, empirical correlations are developed for predicting the important design variables and parameters, such as pressure drop, choking velocity, etc. The common characteristic of these correlations is that they do not usually apply beyond the range of the data-base used in developing them. As for the latter approach, few attempts have been made in the literature to apply generalized multiphase flow equations to gassolids flow systems. Pioneering efforts at developing such models to describe gas-solids systems were made by Jackson [ 4, 51, Anderson and Jackson [6], and Soo [ 7,8]. Kliegel and Nickerson [9] were among the first to analyze some form of multiphase flow equations for gas-solids system. They basically used the method of characteristics to solve the equations for gas-particles mixture in axially symmetric nozzles. Other pioneering efforts, especially in the application of multiphase flow equations to dilute gassolid pneumatic conveying, include those of Arastoopour and Gidaspow [ 10 - 121, Arastoopour et al. [ 131, Shih et al. [ 141, and Nakamura and Capes [ 15,161. As most often done in the literature on this topic, most of these earlier models are onedimensional steady state models, except for that of Shih et al. [14], which is a twodimensional steady state model for horizontal pipe. While the one-dimensional models may have been quite successful in pressure drop predictions (e.g., Arastoopouret a2. [ 10 - 13]), experimental evidences have suggested that the assumption of non-radial variation of these basic variables may not be true. Measuring gas and solid velocities at various radial locations in a vertical pneumatic conveying
system, both Doig [17] and Reddy [18] found significant radial non-uniformity. In fact, both their gas and solid velocity profiles seem to follow power-law radial variation. Furthermore, Doig’s observation of non-uniformity in radial distribution of solid concentration agrees with a similar observation earlier by Soo [ 191. Doig also notices that the particle velocity profile attains ‘fully developed flow’ away from the pipe entrance. A more recent study by Lee and Durst [20] confirms this non-uniform particle velocity profile across the pipe crosssection. More recently however, Adewumi and Arastoopour [21] presented a pseudotwo-dimensional model for describing pneumatic conveying of solid particles in vertical pipes. While their model neglects the radial components of the velocity vectors, it accounts for the radial variation of the axial components of the velocity vectors as well as the other system variables. Their model agrees reasonably well with Reddy’s experimental data on solid velocity profile. However, neglecting the radial velocities may overshadow some important effects such as localized internal circulation of solids, which of course may become appreciable as the system approaches choking condition and also in the entrance region where feed conditions have the most impact. From the foregoing discussion, it seems that a good knowledge of the local particle velocity distribution is an integral part of the basic understanding of gas-solids flow behavior in pneumatic conveying. Such phenomena as choking and clumping that have been experimentally observed might have their roots in the two-dimensional effects described.
MODELDEVELOPMENT
Volume-averaging technique was used to derive the general multiphase flow equations for solid pneumatic conveying (see Adewumi [22]). The basic assumptions made in the development of the two-dimensional steady state model are summarized below: (1) The system is isothermal and there is no interphase mass transfer. (2) The particle size is uniform and the particles have the same density.
69
(3) Solid particles of uniform size and density are defined as a particulate phase, where by each phase is assumed to form a continuum. (4) The particulate phase is assumed to be incompressible, while the gas phase obeys the ideal gas law. (5) Pressure gradient is assumed to exist in the gas phase only. (6) Flow is assumed to be steady. The resulting conservative and additional constitutive equations that define our model are presented below. The primitive variables are V,, V,, the axial components of the gas and particulate phases velocity vectors, respectively, Us and US, the corresponding radial components, (xg and ps, the gas void fraction and density and the pressure P, which is coupled to pg via the equation of state.
Particulate phase :
aw - ~,hw4i az
+--
I
a
r ar Ir(l
-
~&?9~*21
=F,
(6)
where FDZ and FD, are the drag forces in the axial and radial directions, respectively, and FGg, FGS are the gas and particulate phases gravitational forces, respectively. The conatitutive equations Assumptions made yield the following equations of state for the gas and particulate phases. Gas phase equation of state: Pg = VP I
(7)
Mg
rl=RT
Continuity equations Gas phase:
I
Particulate phase equation of state:
(1)
ps = pso = constant
(3)
The gravity forces are defined as
Particulate phase:
Fcig = ‘I[gP&
(9)
(10) @,)P,, g The drag forces are defined as below. The basic assumption is that the functional form of the drag force expressions is the same in both the axial and radial directions. Drag force in the axial direction:
FGS = (I-
Momentum equations in Z-direction Gas phase :
F DZ=
3 cnz(l--
4
@&s-2’67( v, -v,)l
v, - v,(ps
d,
Drag force in the radial direction: Particulate phase:
FDr=-
(11)
3 c,~l-a,)~,-2’67(Ug-Us)lUg-U,Ipg 4
d,
where CC, a =--
r
ar
r(l--a,)2
1
+ F,z-FGS
(1.0 + 0.15ReZ0S687) ReZ < 1000
(4) c DZ=
I0.44
Momentum equations in r-direction Gas phase :
aw,P,v,u,) az =-_
+ +
(13)
$mgPgug2)
cD, =
(1.0 + 0.15Re,0.687)
Re, < 1000 Re,>
i+P -_
at- -
ReZ > 1000
FD,
(5)
1000 (14)
70
Boundary conditions At r = 0 (the pipe axis), we assume symmetry and hence the following:
ReZ
and Re, =
u, = 0 44
-&Id, k
(16)
Chamcteristic analysis of the model The set of quasi-linear partial differential equations describing our model can be re-written in the form
:+A($
=C(U)
whose characteristic equation is IM-A(
=0
The details of evaluating these characteristics have been worked out by Adewumi [22]. The six resulting characteristics are
u, = 0
av,_0 -ib
av,
-=
o
ar
aal3 -=
ar
0
At r = R (at the pipe wall): u, = 0 u, = 0 V, = 0 or very small number
NUMERICAL SCHEME
Since all these characteristics are real, then this model is well posed as an initial-value problem.
Initiul conditions These are basically the inlet conditions into the pipe, but they serve as our initial conditions, since we march in Z-co-ordinate. At 2 = 0, we specify V,, V,, cygand P, based on the feed conditions and the specified solid loading. Parabolic distribution is assumed for the radial velocities at the pipe inlet. These profiles are such that they give zero value at the wall and the pipe axis.
Numerical method of lines is used to solve the quasi-linear set of partial differential equations constituting this model. This method basically transforms a set of partial differential equations (and the associated initial and boundary conditions) into a system of coupled ordinary differential equations. This is achieved by diseretizing the partial derivatives in one independent variable (in the case of two independent variables) while leaving the other continuous. The resulting system of ordinary differential equations is usually large and stiff and hence must be solved using robust stiff ODE integrator. The numerical method of lines is implemented in FORSIM package [23]. FORSIM, in addition, provides options for various ODE integrators and the GEARHINDMARSH method, which uses variable step implicit backward difference method, is used.
PARAMETRIC STUDY
Parametric study was conducted on a system similar to the experimental system of Reddy [ 181. Uniform size and density
71
particles 2.70 X 10m4 m in diameter and 2590 kg/m3 in density were used. The value of ‘solid viscosity’ used in our model was calculated based on the experimental data of Arastoopour and Cutchin [ 241 and Arastoopour et al. [25] for flow of coarse particles relative to the mixture of gas and fine particles. The estimated value of solid viscosity is 0.1 Pa.s [ 221. The input needed in this model includes the inlet values of the primary variables V,, V,, cyg,P, U, and U, at the pipe inlet. Since these were not explicitly measured by Reddy, we specified continuous functions that satisfy Reddy’s gas and solid mass balances and ensure ‘continuity’ from initial to boundary conditions. This helps to remove the initial-boundary conditions singularity at the pipe wail and thus reduces, considerably, the CPU time required to solve the problem. We assumed uniform pressure of 1.2549 X lo5 Pa, 2 = 0. Figure 1 shows the solid velocity profile at two axial locations in the pipe. It also gives the solid velocity profiles at two inlet gas velocities of 8.7 and 14.7 m/s. At any fixed inlet gas velocity, the profiles show evidence of developing flow as it moves along the pipe. Near the pipe inlet (0.0244 m above the inlet), the solid velocity profile is almost flat except I
-
vao
= 14.7 m/r
- -
VP0 = 5.7 m/r
=O.lm
0
s2.7 dip P, Ir.
_
x lo-‘m
=1.22m ‘2590
near the wall, whereas at a higher axial location (1.22 m above the pipe entrance), a more pronounced non-uniform profile, tending towards parabolic, is observed. At the inlet, due to the entrance effect and possible flow instability, the radial velocity would be relatively large. However, because of the presence of the wall, the particles hit the wall and are deflected back into the fastflowing gas stream at the core and are accelerated. Thus, as they move up the pipe, the axial velocity becomes very dominant and the radial velocity becomes low for continuity to be satisfied. This eventually gives rise to ‘developed’ flow, analogous to that which obtains in single-phase flow, where the radial velocity is practically zero. The solid velocity is generally higher for the higher inlet gas velocity. This is due to the larger amount of drag exerted on the solid by the gas, since drag is proportional to the square of the relative velocity between the gas and the solid particles. In the region close to the wall, the gas velocity is lower than that in the core of the pipe. The solid velocity is correspondingly lower and the area swept along by the gas becomes smaller for lower gas velocities, thus giving rise to the thicker solid boundary layer zone. Figure 2 shows the solid velocity profile along the axis of the pipe. It shows rapid acceleration near the pipe entrance and a lower one away from the pipe entrance. As the particles are injected into a high-
kg/m3
/LQ
=O.l Pa-s ‘l.SX IO-5Pa-r
P,
* 1.25481
IO5 Pa
D
so. I m ‘2.7~ IO-‘m
PI
* 2590
ps
so.1 Pa-r * I.8 I 10-5 PO-,
voo = 14.7 m/r
* I.22 m
2
I.”
“..a
I
RADIAL
DISTANCE,
(DIMENSIONLESS)
Fig. 1. Effect of gas velocity on solid velocity profile (at varying Z), predicted by the two-dimensional model.
IO-
I
o".O
kg/m3
I
0.5 AXIAL DISTANCE,
I
I.0 m
Fig. 2. Effect of gas velocity on solid velocity profile, predicted by the two-dimensional model.
72
velocity gas stream, a tremendous amount of drag is exerted on them since the relative velocity between the gas and solid is very high near the pipe entrance. Since the drag is proportional to the square of the relative velocity, the drag is proportionately high. Despite this, gravity will try to decelerate these particles but the drag force is dominant in this zone, usually referred to as the acceleration zone. As they move away from the entrance region, their velocity is closer to that of the conveying gas and drag is therefore considerably reduced to a magnitude of the same order as the decelerating gravitational force. As they move farther up the pipe, the drag is balanced by the solid weight and thus a constant velocity, called the equilibrium velocity, is attained. This figure clearly shows this tendency, even though the pipe length is not long enough for the particles to reach this ultimate velocity. Also, the fire shows the solid velocity profile at two different inlet gas velocities (8.7 and 14.7 m/s). At higher inlet gas velocity, the solid velocity is consistently higher. This is due to the larger drag exerted on the particles by the faster flowing gas. Moreover, the profile for the lower gas velocity shows the tendency to reach its equilibrium velocity at a shorter length. This is due to the fact that, since the solid particles in both cases have the same terminal velocity, their equilibrium velocities are only dependent on the conveying gas velocity. The solid for the lower gas velocity has lower equilibrium velocity and thus
should attain this within a shorter pipe length. Figure 3 shows the pressure drop profile along the pipe axis. It shows rapid pressure drop in the acceleration region, up to about 0.6 m from the pipe inlet, after which the pressure gradient is almost constant. In the acceleration region, both the drag force and the momentum outflow contributions to the pressure drop are high, thus accounting for the higher pressure gradient in this zone. As the solid particles reach their equilibrium values, the drag force will be constant and is balanced by the solid weight, the solid velocity is essentially constant and the acceleration contribution becomes negligible. The pressure gradient is thus essentially equal to the weight of the solids. Although this tendency is portrayed in Fig. 3, the pipe is too short to fully observe the pressure gradient constancy. On the same figure, we show the effect of two different inlet gas velocities on the pressure drop profile, with the higher gas velocity giving rise to higher pressure drop throughout the system. The higher the gas velocity, the higher the drag on the solid, the higher the acceleration of the solid and the solid and gas frictional forces. All these account for contributions to the higher pressure drop at higher gas velocities. Figure 4 is a plot of the solid radial velocity as a function of the radial position. The magnitude of the radial velocities imposed 0.16
I 0
C.,2-
pg
-1.6x
P,
‘I.2546
v,,
so.1
so.1 82.7
I
I
m x 10-4,
P,
=1.22m ‘2590kgh3
p*
=O.l
Pa-r
IO-Spa-r x lOsPa m/s
aoo no.904 0 0.0
I
I
0.5 AXIAL
1.0 DISTANCE.
m
Fig. 3. Effect of gas velocity pressure drop profile, predicted by the two-dimensional model.
0.0
0.2
0.4 RADIAL
DISTANCE.
0.6
0.6
1.0
IOIMENSIONLESSI
Fig. 4. Solid radial velocity profile (at varying Z), predicted by the two-dimensional steady state model.
73
at the pipe inlet are relatively small compared with their axial counterparts. The imposed distribution is parabolic, with vanishing values at the centre and at the wall, in order to satisfy symmetry at the centre and the existence of the wall, respectively. Nevertheless, some observations of their distributions away from the pipe inlet deserve mentioning. Generally, these velocities remain small compared with their axial counterparts. This is expected, since the flow develops as the fluid moves away from the entrance, as indicated by Fig. 1. Around the pipe entrance, the particles close to the wall move slowly towards the centre of the pipe; in other words, they are entrained by the high gas velocity zone at the central portion of the pipe. Farther up in the pipe, these particles are replaced by faster-moving particles, thus causing the radial movement of the fastermoving particles towards the pipe wall (see Fig. 4). This is probably the main cause of some kind of local circulation or vortex motion calculated in our gas-solids system.
time the particles enter the test section. The inlet conditions to the test section, such as pressure, gas and solid velocities, and void fraction, were not reported by Reddy and these variables were not measured at the pipe wall. These are generally needed to impose both the initial and boundary conditions on our equations. We specified an inlet void fraction of 0.984. Using this, the gas flow rate (measured) and a solid loading of 0.4 (as specified by Reddy) we estimated the inlet velocity of the particulate phase. Figure 5 shows the comparison of the predicted solid velocity profile with Reddy’s experimental data. The predicted solid velocity profile compares reasonably well with the experimental data. Apparent discrepancies between the experimental data and predicted results in the wall region depict the effect of the no-slip boundary condition imposed on the solid velocity at the wall. This may not be appropriate. Improved measurement of the system variables at the wall will help the specification of the boundary conditions and should improve the performance of this model.
COMPARISON WITH REDDY’S EXPERIMENTAL DATA ACKNOWLEDGEMENT
An attempt was made to compare the predicted numerical results with Reddy’s [18] experimental data. The test section of Reddy’s set-up consists of a 0.1 m I.D., 1.2 m long vertical column. This is preceded by a calming section 22 times the pipe diameter to ensure uniformly stable motion by the
PARTICLE
SIZE.
LOADING
NUMBER=SS.OOO
EXPERIMENTAL
PREDICTED
o.ov 0.0
dpn270p
RATIO -0.40
REYNOLD’S REOOY’S
The authors would like to thank the Gas Research Institute, Basic Research Division, for partial financial support under Contract No. 5084260-1071 and the Institute of Gas Technology for initial financial support and its research facilities.
DATA
RESULTS
I
I
I
1.0
2.0
3.0
DISTANCE
FROM THE
I
I
4.0
WALL, cm
Fig. 5. Comparison of calculated solid velocity profile with Reddy’s [ 181 experimental data.
74 LIST OF SYMBOLS CD, CDZ
4 FD, FDZ F Gtz FGS iif
Pg r R Re,
Rez
drag coefficient in r-direction drag coefficient in Zdirection particle diameter drag force in rdirection drag force in Z-direction gas phase gravitational force particulate phase gravitational force gravitational acceleration gas molecular weight pressure radial co-ordinate gas constant particulate Reynolds number in rdirection particulate Reynolds number in Z-direction absolute temperature radial component of gas velocity radial component of solid velocity axial component of gas velocity axial component of solid velocity axial co-ordinate
Greek symbols gas volume fraction % h characteristics of the system of equations gas viscosity Elg particulate phase viscosity PS gas density ps solid density Ps
Subscripts gas phase quantities g S 0
particulate phase quantities inlet conditions
REFERENCES T. M. Knowlton, Solids Transport-state-of-theArt and Research Needs, paper presented at NSF Workshop on Fluidization and Fluid-Particles Systems - Research Needs and Priorities, RPI, New York, 1979. L. S. Leung, in J. R. Grace and J. M. Matsen
3 4 5
10 11 12 13 14 15 16
17 18 19 20 21
22 23 24 25
(eds.), Fiuidization, Plenum Press, New York, 1980. H. Arastoopour, in N. P. Cheremisinoff (ed.), Encyclopedia of Fluid Mechanics, vol. 4, Gulf, Houston, TS, 1986, section 2. R. Jackson, Trans. Inst. Chem. Engrs., 41 (1963) 13. R. Jackson, in J. R. Davidson and D. Harrison (eds.), Fluid Mechanical Theory in Fluidization, Academic Press, New York, 1971. T. B. Anderson and R. Jackson, Ind. and Eng. Chem. Fund., 6 (1967) 527. S. L. Soo, The Physics of Fluids, 20 (1977) 568. S. L. Soo, Multiphase Transport, 1 (1979) 291. J. R. Kiiegel and G. R. Nickerson, Flow of GasParticle Mixtures in Axially Symmetric Nozzles, Paper presented at the American Rocket Society Propellants, Combustion and Liquid Rockets Conference, Palm Beach, Florida (26 - 28 April, 1961). H. Arastoopour and D. Gidaspow, Znd. and Eng. Chem. Fund., 18 (1979) 123. H. Arastoopour and D. Gidaspow, Chem. Eng. Sci., 34 (1979) 1063. H. Arastoopour and D. Gidaspow, Powder Technol., 22 (1979) 77. H. Arastoopour, S. C. Lin and S. A. Weil, AZChE J., 28 (1982) 467. Y. T. Shih, H. Arastoopour and S. A. Weil, Znd. and Eng. Chem. Fund., 21 (1982) 37. K. Nakamura and C. E. Capes, The Can. J. Chem. Eng., 51 (1973) 29. K. Nakamura and C. E. Capes, in D. L. Keairns (ed.), Fluidisation Technology, Vol. 2, Hemisphere, Washington, DC, 1976, pp. 159 - 184. I. D. Doig, Ph.D. Thesis, Univ. New South Wales, Australia (1965). K. V. S. Reddy, Ph.D. Thesis, Univ. Waterloo (1967). S. L. Soo, Znd. and Eng. Chem. Fund., 1 (1962) 33. S. L. Lee and F. Durst, Znt. J. Multiphase Flow, 8 (1982) i25. M. A. Adewumi and H. Arastoopour, PseudoTwo-Dimensional Steady State Two-Phase Flow Model for Gas-Solids Vertical Pneumatic Conveying Systems, Paper presented at the 1985 annual meeting of the American Institute of Chemical Engineers, Nov. 10 - 15, 1985 in Chicago, IL. M. A. Adewumi, Ph.D. Thesis, Iilinois Institute of Technology, Chicago (1985). M. B. Carber, Atomic Energy of Canada Limited Report AECL-4844 (1974). H. Arastoopour and J. H. Cutchin III, Chem. Eng. sci., 40 (1985) 1135. H. Arastoopour, C. H. Wang and S. A. Weil, Chem. Eng. Sci., 37 (1982) 1379.