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Modelling of vertical pneumatic-conveying hydrodynamics
Modelling of vertical pneumatic-conveying hydrodynamics K. N. Theologos Department
and N. C. Markatos
of Chemical
Engineering,
National
Technical
University
of Athens,
Athens,
Greece
A finite-domain procedure for simulating the hydrodynamics of vertical pneumatic conveying is presented. A parametric study considering six difSerent spec@ations for particles and 41 difSerent operating conditions has been carried out. Predictions, which have been obtained using two dtflerent empirical correlations for the description of the interphase momentum transfer and three deferent empirical correlations for the description of the solid-to-wall friction, have been compared with experimental data from the literature. It is concluded that in dilute transport situations realistic predictions of pressure drop, particle holdup, and of choking behavior can be made. Finally, a calibration procedure is proposedforjtting the calculated to the experimental data of vertical pneumatic conveying.
Keywords: mathematical
modelling, vertical pneumatic conveying, two-phase flow, interphase drag force
1. Introduction Pneumatic conveying, or pneumatic transport, is commonly used in industry for loading and unloading of dry bulk materials, whereas physical operations such as drying can be carried out in flowing gas-solid Chemical reactions such as catalytic suspensions. cracking of hydrocarbons can also be completed in riser reactors, where the catalyst is conveyed by the hydrocarbon vapors. In these operations, it is important to determine pressure drop and choking velocity and also to predict the solids concentration and the particle holdup along the conveying line. In this study, the one-dimensional differential equations describing the above process have been solved by using a finite-domain procedure. These equations have been solved in conjunction with empirical correlations, describing the drag force between the particles and the fluid and the pressure drop due to wall friction with the fluid and with the particles. A parametric study considering six different specifications for particles and 41 different operating conditions has been carried out. Predictions have been obtained two different empirical equations for the using description of the drag force between the particles and the fluid and three different equations for the description
Address reprint requests to Professor Markatos at the Computational Fluid Dynamics Unit, Section 11, Dept. of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Str., 157 73 Zografou, Athens, Greece. Received
306
4 November
Appl.
Math.
1992; accepted
Modelling,
25 October
1994,
1993
Vol. 18, June
of the wall solid friction factor. The results obtained have been compared with the experiments of Hariu and Molstad.’ Predictions have also been made near the choking conditions, and results have been compared with Zenz’s’ experimental data.
2. The mathematical
model
Vertical pneumatic conveying of solids by gases can be described in many situations by means of onedimensional, isothermal, steady-state mass and momentum balances. The one-dimensional assumption is a good approximation for the operation of a conveying line, but it cannot take into account phenomena such as backmixing of solids due to distributor geometry. The isothermal assumption is a good approximation for dispersed flow due to adequate heat transfer and low rates of heat generation by friction. Three dimensional, nonisothermal, and transient cases could also be handled by similar modelling techniques, as described here. However, the simplified version, apart from its significant practical relevance, allows for quick and inexpensive testing and validation of physical concepts and of modelling philosophy and techniques.
2.1
The dlflerential
equations
One-dimensional, isothermal, and steady-state, vertical pneumatic conveying can be described by four differential equations, i.e., the continuity and the
0
1994 Butterworth-Heinemann
Modeling momentum
equations
of vertical pneumatic-conveying
hydrodynamics: 2.3
for each phase.3
Gas continuity
;
(EWJ = 0
(1)
Particles continuity
;
[(l - &)WJ = 0
(2)
Gas momentum
k kpswsws)
The frictional force between the wall and the gas-solid suspension is assumed to be separated into two terms due, separately, to the fluid alone (f,,) and to the effect of solid particles (f,,). The frictional pressure drop due to conveying the fluid alone in a pipe of length L and diameter D is defined by Fanning’s equation: = Fg
Particles momentum
;
= -(l -
- E);
+fpg
f, = 0.316Re-
+fpw
(1 - 4Pp - Pg)S
(4)
where w is the velocity, p the density for the phase considered, and the subscripts g and p refer to the gaseous and particulate phases, respectively. g is the gravitational acceleration, E is the void fraction, P is the pressure (assumed the same for both phases) and the forces, J are described below.
The major empirical input in the present modelling technique is the determination of the fluid-particle interphase momentum transfer. In vertical pneumatic conveying a drag force (&J is exerted on the particles from the gas, while an opposite force (f,, = -fPy) is exerted on the gas by the particles. The drag force controls the slip velocity between the two phases, the acceleration zone of the particulate phase, and the concentration of particles all over the conveying line. The drag force on a single spherical particle of diameter d, in an infinite fluid is usually expressed in terms of a drag coefficient C, as F, = 0.5C&d;/4)p,(w,
- wp)’
(5)
To take into account the effect of concentration of particles in sedimentation or fluidization, Richardson and Zaki4 showed that the drag force per unit volume of particles in mixtures of fluid and spherical particles can be expressed as F, = 3/4C,
PAWS- WA2 E-2,65 (6) 4
The drag coefficient C, can be related to the particle Reynolds number by means of the relations: C, = 24/Re,(l
‘j4
(10)
The wall friction due to solid particles is defined differently by various authors. Most of them define a solid friction factor, following Fanning’s equation, based on the particle velocity and the dispersed solid density:i5 APF,
=
f,PAl - E)WiL
(11)
20
Yang5 correlated the experimental data for vertical pneumatic conveying obtained by Hariu and Molstad’ and derived the following empirical equation for the prediction of the solid friction factor: fps3/(1 - E) = 0.0206[(1 - &)Re,/Re,]-0.869
The drag force
2.2
(9)
20
Assuming a smooth pipe, the friction factor f, can be obtained by the empirical Blassius formula:
4PpWpWpl
C(1 -
fsP,W,2L
(3)
+fgp +fgW
and N. C. Markatos
The wall friction
Ap = --E g
K. N. Theologos
+ Re,0.687)
and pLsis the viscosity
where Re, is the Reynolds number based on the particle terminal (free-fall) velocity. The same author correlated additional experimental data and derived an updated equation: &/(l
- E) =
0.0126[(1 - &)Re,/Re,]-0.979
V,/v > 1.5
0.0410[(1 - &)Re,/Re,]- ‘.021 V,/l: < 1.5 (13)
where V, is the gas superficial velocity particle terminal (free-fall) velocity. 2.4
and
r/; is the
Numerical solution
The above set of equations has been solved using a numerical technique. In order to solve the equations, the entire domain of interest is discretized into a grid of finite control volumes. The differential equations are integrated over these volumes in order to derive finitedomain equations. These equations are solved in an iterative manner and yield values of the dependent variables at the centers of the control volumes. The finite-domain equations are solved by using a developed computer program, which is similar to the one described in Ref. 3. The relevant numerical details have been published extensively (e.g., Refs. 6 and 7) and are not repeated here.
(7)
3. Results and discussion
where Re, = (dpPg/~JI~g
(12)
- wPl of the gas.
(8)
To evaluate the performance of the proposed model a parametric study and comparisons with experimental data have been carried out. The vertical lift line studied
Appl.
Math.
Modelling,
1994,
Vol. 18, June
307
Modelling
of vertical pneumatic-conveying
hydrodynamics:
here is the same as the one used by Hariu and Molstad.’ A vertical line of 13.5-mm internal diameter and 1.368-m height has been considered. To facilitate the validation, an appropriate mesh has been used to calculate values at the same locations where the experimental ones had been measured. The results have been checked for grid effects and were found grid independent for a grid consisting of 10 control volumes. This grid independency is presented for a typical run in Figure 1. Convergence was easy to obtain, and a typical computer run was accomplished within 150 s CPU time on an Acorn Archimedes 440 personal workstation. Six different specifications for particles to be conveyed by air have been considered, as shown in Table 1. Predictions have been made for 41 different operating conditions, the ones used by Hariu and Molstad. For the present predictions at the inlet boundary of the lift line, the gas phase is considered to enter the tube with a
7.00
I
1
6.00 5.00
0
N%=S
0
N%=lO
4.00 3.00 Z.(x) 1.00
oxxJ ’ 0.00
1.oo
0.40
0 20
Dinwwiwlcss Figure
1 (a).
twigill
Grid independency
of solid-phase
velocity
results.
K. N. Theologos
and N. C. Markatos
velocity equal to the experimental superficial velocity, whereas the solid phase is considered to enter the tube with zero momentum. The latter assumption is made because of a lack of any experimental data about the solid velocity or the solid volume fraction at the inlet of the lift line. Near the inlet boundary, the velocity and the volume fraction of the solid phase were calculated, assuming only interphase transfer of momentum due to the frictional force exerted from the gas to the particles. In every case, for a given gas superficial velocity, V,, and solid mass flow rate, F,, the following variables have been calculated all over the lift line: gas and particle velocities, wg and wp, bed voidage, E, and pressure, P. Also, the total pressure drop, AP, and the particles holdup, M,, have been calculated in every case. 3.1
Typical predictions
The results presented in Figures 2, 3, and 4 were computed assuming zero pressure drop due to solid friction; the drag force was calculated by equation (5). The values used for the production of these figures are presented in Appendix A. Figure 2 shows the distributions of solid volume fraction and of phase velocities along the height of the lift line, for conveying material A by air. As shown in this figure, the gas velocity decreases as the solids accelerate; the solid volume fraction decreases, due to the increase in solid velocity at a constant solid mass flow rate, as expected. Figure 3 shows the effect of solid mass flow rate on the pressure drop along the lift line for a given gas superficial velocity. Predictions are presented for material A. As expected, the pressure drop increases as the solid mass flow rate increases, because more solids have to be accelerated and conveyed by the fluid. This prediction is consistent with experimental data obtained by Hariu and Molstad.
NZ=S
0 0
N%=lO
0
NZ=20
14.0 _
9.OL3
pN%=.50
O.(X)
020
0.40
0 60
Dimensionless
Figure results.
Table
1 (b).
1.
Grid independency
Particle
Particle diameter Particle density
(pm)
of solid-phase
volume
fraction
A
0
C
D
E
F
503 2643
357 2643
274 2643
213 2707
110 977
110 977
3.9
2.8
2.2
1.6
0.3
0.3
(m/set)
Appl.
0.4
0.2
(kg/m3) Particle free-fall
308
1.oo
properties
Material
velocity
0 no
height
Math.
Dimensionless
Figure
Modelling,
1994,
Vol. 18, June
2.
Solid volume
fraction
0.0
(I.8
length
and phase velocities.
Model/kg
0.0
0 2
of vertical pneumatic-conveying
0 4
(1.6
Dimensionless
Figure
3.
0 x
10
length
Effect of solid flow rate on pressure
drop.
Figure 4 shows the effect of the gas superficial velocity on the solid phase velocity. Predictions are presented for material A, for three different gas superficial velocities. For each gas superficial velocity, solid velocities have been calculated assuming different solid flow rates. As is shown, the solid mass flow rate has a minor effect on the profile of the solid velocity, which is affected primarily by the gas superficial velocity.
3.2
A parametric study
A parametric study concerning appropriate empirical correlation
hydrodynamics:
K. N. Theologos
and N. C. Markatos
the drag force and the wall friction factor has been carried out. Predictions have been obtained using two different empirical equations for the description of the drag force between the particles and the fluid and three different equations for the description of the wall solid friction factor. The parametric study runs (six in total) carried out are shown in Table 2. The operating conditions and the predictions obtained for each of the six runs for each of the 41 different operating conditions are presented in Appendix B. For each run the total pressure drop, AP, and the particles holdup, M,, were calculated. Then, a percent error between the predicted and the experimental value was calculated. The maximum and mean errors for the run that gave best fit to the experimental data are presented in Table 3. The predictions obtained are reasonable, with the exception of the calculated pressure drop for particles E and F and the solid holdup for material B. The best predictions were obtained assuming that there was no pressure drop due to solid friction (runs 1 and 4). This assumption is reasonable here, because dilute phase transport is considered. For material A the best predictions were obtained for a solid friction factor calculated by equation (12). The parametric study showed that the use of equation (6) instead of equation (5) in the calculation of the interphase friction had a minor effect on the prediction of the total pressure drop, AP, and the particles holdup, M,. This is because dilute phase transport is considered and the values of the void fraction, E, are close to unity. In vertical pneumatic conveying, the total pressure drop consists of three individual contributions due to acceleration, gravity, and wall friction:
the selection of the for the prediction of
AP, = AP, + AP, + AP,
(14)
To determine the acceleration term, it is important to know the momentum of the solid phase at the inlet of
Table
2.
Parameter
study runs
fp = 0
8
Fo, equation Fo, equation
3.x0
(5) (6)
fp, equation
1 4
(12)
f,, equation
2 5
(13)
3 6
2 A 3.30 :’ :: p 2.80 _;; z ,” 230
Table
3.
1.80
Material
0.80
I-
0.2
0.4 Dimensionless
Figure 4. velocity.
Effect
of gas superficial
0.6
08
1.0
length
velocity
on the
solid
phase
A B C D E F
Deviation
Best fit run 2 4 4 1 1 I,4
Appl.
Math.
between
experimental
Maximum BP error (%)
Mean AP error (%)
12.7 14.4 11.2 12.6 30.7 27.6
6.4 6.4 6.5 8.1 22 17.9
Modelling,
1994,
data and predictions Maximum M, error (%)
Mean A& error (%)
14.8 25 2.2 11.4 10.8 10
5.1 15.5 7.0 2.3 5.3 9.5
Vol. 18, June
309
Modeling
of vertical pneumatic-conveying
hydrodynamics:
the lift line. For a known solid mass flow rate, the initial velocity or the initial volume fraction is necessary for the determination of the momentum at the inlet of the lift line. Harm and Molstad calculated an average solid volume fraction by measuring the total particle holdup in the lift line. For a unique specification of pressure drop in vertical pneumatic conveying the inlet void fraction (or solid velocity) has to be measured as mentioned by Arastoopour and Gidaspow’ and as proven by Figures 5 and 6. These figures show very good agreement between predictions and experiment, and even though errors occurred in the prediction of both pressure drop and particles holdup, the predicted linear dependence between these variables accurately fits the experimental values.
Chocking behavior: Comparison with experiment
3.3
For a fixed solid mass flow rate, reducing the gas flow rate has two major consequences. One consequence is that the frictional resistance of the flowing mixture is reduced. Also, gas and solid both rise more slowly, the voidage decreases, and the static head rises. At high gas superficial velocities, when decreasing the gas velocity the change in the frictional resistance predominates and the pressure drop decreases. A further lowering of gas flow rate causes a rapid rise in solid inventory and static head that produces an increase in pressure drop.’ This competition of forces that change with flow rate in the opposite direction results in the occurrence of a minimum in the pressure drop curve, which is known as chocking behavior. 7.00 ,
I 0
0.00
1
I
150
200
250
RW
350
400
450
500
Pressure do-op (Nim2)
Figure 5.
Particle holdup vs. pressure drop (material A)
4.50
K. I\ 5
‘heologos
and N. C. Markatos
10
c
z_
”
E?g
22 2
t-2
a
1 100
10
Superficial gas velocity (Wee)
Figure
7.
Choking
behavior.
This behavior is predicted by the present modelling technique, and it is presented in Figure 7. In this figure the effect of gas superficial velocity on the total pressure drop is presented. Predicted values were obtained using Zenz’s’ experimental conditions. The calculated pressure drops are in very good agreement with the experimental data, even though neither an initial nor an average solid concentration is reported.
3.4
A calibration procedure
The total pressure drop is not the only parameter that has to be determined in vertical pneumatic conveying. In many operations, such as gas-oil catalytic cracking in riser reactors, the particle (i.e., catalyst) concentration has to be determined accurately, as it controls the rate of reaction. Unfortunately, there are no experimental measurements of the local voidage in vertical pneumatic conveying, but only average calculations through measurements of the total particle holdup in the lift line. The comparison between the experimental and calculated values reported in section 3.2 above showed that the use of empirical correlations for the determination of the solid to wall friction caused deviations between calculated and experimental values. In order to calculate accurately the total pressure drop and the particle holdup the following calibration procedure is proposed. First a total pressure drop, APJe = 0, is calculated assuming zero friction between the particles and the wall. Then, for the same operating conditions, the pressure drop APJp,eq,13, is calculated using equation 13 for the determination of the solid-to-wall friction factor. Assuming that the error in the calculation of pressure drop is caused only by the determination of the solid-to-wall friction, the friction factor, f,, is corrected by multiplication with the following expression:
ii~~~~
AP,,,
- AP,
=o
APfp,eq.i3 - AP,* = o cl
0.50 -0.00 -I 100
150
2w
250
I 350
3ul
Pressure drop (N/m21
Figure 6.
310
Particle holdup vs. pressure drop (material
Appl.
Math.
Modelling,
1994,
D)
Vol. 18, June
value of the pressure where AP,,, is the experimental drop. This technique was applied at 11 different operating conditions for material A and not only gave accurate predictions for the pressure drop, but also corrected the particle holdup. Predictions, which gave a maximum
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
dP
200
3(x1
4,X,
pressure
Experimcnlal
SW drop
0
1
2
3
Experimental
(Sll”2~
Figure friction
8. factor
Predicted
4 particle
5
6
7
holdup
(Kr)
vs. experimental
values
using
a calibrated
fD.
error of 2% for the prediction of pressure drop and 4% for the prediction of the particle holdup, are presented in Figure 8. The above-mentioned procedure shows that in the case where experimental data of total pressure drop for vertical pneumatic conveying are available, a fine tuning of a hydrodynamic model can be carried out through a calibration of the friction factor between the particles and the wall, f,.
particle diameter force exerted on the gas by the particles force exerted on the gas by the wall j: P9 force exerted on the particles by the gas f PW force exerted on the particles by the wall acceleration due to gravity pipe length i pressure (assumed the same for both phases) P (d,p,lp,) I wg - wp I Rep ~~,P,lP,) I: Ret gas superficial velocity v, particle terminal (free-fall) velocity v gas velocity W9 particles velocity WP axial distance Z voidage (i.e., gas volume fraction) & gas viscosity p9 gas density ps particles density PP
6. Acknowledgement 4. Conclusion The results presented, and the many more obtained but not reported here due to space restrictions, lead to the following conclusions: Pressure drop and particle holdup in dilute-phase vertical pneumatic conveying can be predicted using numerical techniques. Realistic predictions of practical accuracy for dilute transport can be obtained using standard drag correlations for a single spherical particle in an infinite fluid and ignoring the solid-wall frictional force. The proposed modelling technique can realistically predict the choking behavior in vertical pneumatic conveying. For accurate predictions of pressure drop and particle holdup in vertical pneumatic conveying, the initial solid velocity (or volume fraction) has to be measured. A calibration procedure of the hydrodynamic model with reference to experimental data can be carried out through calibration of the solid-to-wall friction factor,
I 2 3
4 5 6
8
Nomenclature
CD
D
7. References
7
f P’ I.
This research was performed partially in the framework of the JOU-2-CT92-122 EEC contract. The financial support of the EEC (Directorate General XII) is gratefully acknowledged. The computations were performed using the PHOENICS computer program of CHAM, Ltd., Wimbledon, London, UK.
drag coefficient pipe diameter
9
Harm, 0. H., Molstad, M. C. Pressure drop in vertical tubes in transport of solids by gases. Ind. Eng. Chem. 1949,41(6), 1148-1160 Zenz, F. A. Two-phase fluid-solid flow. Ind. Eng. Chem. 1949, 41 (12) 2801-2806 Markatos, N. C. and Shinghal, A. K. Numerical analysis of one-dimensional, two phase flow in a vertical cylindrical passage. Ado. Eng. Sofrware 1982, 4 (5), 99-106 Richardson, J. F., Zaki, W. N. Trans. Insl. Chem. Eng. 1954, 32, 35-53 Yang, W. C. A correlation for solid friction factor in vertical pneumatic conveying lines. AIChE J. 1978, 24 (3), 548-552 Markatos, N. C. and Moult, A. The computation of steady and unsteady turbulent, chemically-reacting flows in axisymmetrical domains. Trans. Instn. Chem. Engrs. 1979, 51, 156162 Markatos, N. C. Modelling of two-phase transient flow and combustion of granular propellants. Int. J. Multiphase Flow 1986, 12 (6), 913-933 Arastoopour, H., Gidaspow, D. Vertical pneumatic conveying using four hydrodynamic models. Ind. Eng. Chem. Fundum. 1979, 18 (2), 123-130 Kunii, D., Levenspiel, 0. Fluidizution Engineering. John Wiley & Sons, New York, 1969.
Appendix A: Predictions of pressure, phase velocities, and phase volume fractions in a vertical lift line for the experimental conditions used by Hariu and Molstad (material A, tube 2) Predictions vs. experimental Dimensions
data (Hariu and Molstad-1949)
of apparatus
Riser i.d. (mm) Riser height (m)
13.5 1.368
Appl.
Math.
Modelling,
1994,
Vol.
18,
June
311
Modelling Solid
of vertical pneumatic-conveying
K. N. Theologos
and N. C. Markatos
properties
Material Particle Particle Particle
hydrodynamics:
diameter (pm) density (kg/m3) free-fall velocity
Experiment
(m/set)
B
C
D
503 2643 3.9
357 2643 2.8
274 2643 2.2
213 2707 1.6
E
F
110 977 0.3
110 977 0.3
1 Material A Gas superficial velocity Solid mass flow rate
Z (m) Dimens. length P (N/m2) Wl (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
5.2 m/set 2.39 g Jsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.1 249 5.24 0.9923 7.7E-3 4.42
0.2 204 5.23 0.9945 5.5E-3 4.09
0.3 173 5.23 0.9950 5.OE-3 3.95
0.4 146 5.22 0.9953 4.7E-3 3.87
0.6 120 5.22 0.9954 4.6E-3 3.83
0.7 96 5.22 0.9955 4.5E-3 3.80
0.8 71 5.22 0.9956 4.4E-3 3.78
0.9 47 5.22 0.9957 4.3E-3 3.74
2 Material A Gas superficial velocity Solid mass flow rate
Z (ml Dimens. length P (N/m2) Wl (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
A
5.2 m/set 3.84 g lsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 359 5.26 0.9879 1.2E-3 4.42
0.188 290 5.24 0.9913 8.7E-3 4.08
0.313 245 5.24 0.9922 7.8E-3 3.94
0.438 205 5.24 0.9926 7.4E-3 3.87
0.563 169 5.24 0.9928 7.2E-3 3.83
0.688 133 5.24 0.9929 7.1 E-3 3.81
0.813 99 5.24 0.9930 7.OE-3 3.78
0.938 65 5.23 0.9932 6.8E-3 3.73
3 Material A Gas superficial velocity Solid mass flow rate
5.2 m/set 5.32 g lsec
Z (ml
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m2) AP (N/m2) W2 (m/set)
0.063 468
0.188 376 -92 1.18 1.2E-2 4.08
0.313 316 -152 1.32 1 .l E-2 3.94
0.438 265 -203 1.39 1 .OE-2 3.86
0.563 217 -251 1.43 9.8E-3 3.82
0.688 171 -297 1.46 9.7E-3 3.79
0.813 126 -342 1.48 9.5E-3 3.77
0.938 82 -386 1.52 9.3E-3 3.73
0 0.85 1.6E-2 4.44
R2 (vol/vol) Wslip (m/set)
Experiment
4
Z (m) Dimens. length P (N/m2) AP (N/m2) W2 (m/set) R2 (vol/vol) Wslip (m/set)
312
Appl.
Material A Gas superficial velocity Solid mass flow rate
5.2 m/set 6.79 g/set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 576 0 0.87 2.1 E-2 4.44
0.188 461 -115 1.20 1.5E-2 4.08
0.313 387 -189 1.34 1.3E-2 3.93
0.438 324 -252 1.41 1.3E-2 3.86
0.563 265 -311 1.45 1.2E-2 3.81
0.688 208 -368 1.48 1.2E-2 3.78
0.813 153 -423 1.50 1.2E-2 3.76
0.938 99 -477 1.54 1.2E-2 3.72
Math. Modelling,
1994, Vol. 18, June
Modeling Experiment
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
5 Material A Gas superficial velocity Solid mass flow rate
9.0 m/set 1.76 g/set
Z (ml
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m2) AP (N/m*) W2 (m/set)
0.063 282 0 1.67
0.188 233 -49 2.52
0.313 199 -83 2.96
1.9E-3 6.50
1.6E-3 6.05
0.438 168 -114 3.24 1.4E-3 5.77
0.563 140 -142 3.43 1.4E-3 5.58
0.688 113 -169 3.57 1.3E-3 5.44
0.813 86 -196 3.67 1.3E-3 5.34
0.938 60 -222 3.75 1.2E-2 5.26
2.8E-3 7.35
R2 (vol/vol) Wslip (m/set)
Experiment
6 Material A Gas superficial velocity Solid mass flow rate
9.0 m/set 3.98 g/set
2 (m)
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m*) AP (N/m*) W2 (m/set)
0.063 429 1.68
0.188 340 -89 2.53
6.3E-3 7.38
4.2E-3 6.51
0.313 285 -144 2.95 3.6E-3 6.08
0.438 238 -191 3.22 3.3E-3 5.81
0.563 196 -233 3.40 3.1 E-3 5.63
0.688 156 -273 3.53 3.OE-3 5.50
0.813 118 -312 3.63 2.9E-3 5.40
0.938 80 -349 3.71 2.8E-3 5.32
0
R2 (vol/vol) Wslip (m/set)
Experiment
7 Material A Gas superficial velocity Solid mass flow rate
Z Cm) Dimens. length P (N/m*) AP (N/m*) W2 (m/set) R2 (vol/vol) Wslip (m/set)
Experiment
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 577 0 1.69 9.7E-3 7.40
0.188 449 -129 2.53 6.5E-3 6.53
0.313 372 -205 2.96 5.5E-3 6.10
0.438 309 -268 3.22 5.1 E-3 5.83
0.563 252 -325 3.40 4.8E-3 5.65
0.688 200 -377 3.52 4.6E-3 5.52
0.813 149 -428 3.61 4.5E-3 5.43
0.938 101 -476 3.69 4.4E-3 5.35
8 Material A Gas superficial velocity Solid mass flow rate 0.068
Z (ml Dimens. length P (N/m*) AP (N/m2) W2 (m/set) R2 (vol/vol) Wslip (m/set) Experiment
0.063 725 0 1.68 1.3E-3 7.34
P
(N/m*)
Wl (m/set) W2 (m/set) RI (vol/vol) R2 (vol/vol) Wslip (m/set)
8.9 m/set 8.42 glsec
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.188 556 -169 2.51 8.9E-3 6.47
0.313 459 -266 2.93 7.6E-3 6.05
0.438 379 -346 3.18 7.OE-3 5.78
0.563 309 -416 3.35 6.6E-3 5.61
0.688 244 -481 3.47 6.4E-3 5.49
0.813 182 -543 3.56 6.3E-3 5.40
0.938 122 -603 3.63 6.1 E-3 5.33
9 Material A Gas superficial velocity Solid mass flow rate
Z (m)
9.0 m/set 6.17 g/set
12.3 3.02
m/set g Jsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
526 12.44 2.28 0.9965 3.5E-3 10.16
430 12.43 3.49 0.9977 2.3E-3 8.95
364 12.42 4.10 0.9980 2.OE-3 8.32
307 12.42 4.49 0.9982 1.8E-3 7.93
254 12.42 4.76 0.9983 1.7E-3 7.66
203 12.42 4.95 0.9984 1.6E-3 7.47
154 12.42 5.10 0.9984 1.6E-3 7.32
106 12.42 5.21 0.9985 1.5E-3 7.21
Appl.
Math.
Modelling,
1994,
Vol. 18, June
313
Modelling
of vertical pneumatic-conveying
Experiment
10 Material A Gas superficial velocity Solid mass flow rate
Z (m) P
(N/m*)
WI (m/set) W2 (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
K. N. Theologos
and N. C. Markatos
12.3 m/set 4.98 g /set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
677 12.37 2.27 0.9942 5.8E-3 10.10
539 12.35 3.45 0.9962 3.8E-3 8.90
452 12.34 4.05 0.9967 3.3E-3 8.29
378 12.34 4.43 0.9970 3.OE-3 7.91
310 12.33 4.69 0.9972 2.8E-3 7.65
247 12.33 4.87 0.9973 2.7E-3 7.46
186 12.33 5.01 0.9974 2.6E-3 7.32
127 12.33 5.12 0.9974 2.6E-3 7.22
11 Material A Gas superficial velocity Solid mass flow rate
(N/m*)
WI (m/set) W2 (m/set) RI (vol/vol) R2 (vol/vol) Wslip (m/set)
12.3 m/set 6.93 g/set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
837 12.4 2.28 0.9920 8.OE-3 10.12
655 12.4 3.45 0.9947 5.3E-3 8.92
545 12.4 4.04 0.9955 4.5E-3 8.32
454 12.4 4.41 0.9958 4.2E-3 7.94
371 12.4 4.66 0.9961 3.9E-3 7.69
294 12.4 4.84 0.9962 3.8E-3 7.51
220 12.4 4.98 0.9963 3.7E-3 7.37
149 12.3 5.08 0.9964 3.6E-3 7.26
Z (m) P
hydrodynamics:
5.30
4.50 9.OE-3
4.30
"g=12.3m,rec S.OE-3
^ 3.80 d < 5 x 3.3" .=: :: z s 2.30 % z w 2 30
1.80 &OE-3 1.30
0.0
0.2
0.4
0.6
Dimensionless
0.8
1.0
0.80 0.0
3.OE-3 0.2
0.4 Dimensionless
length
0.6
0.8
1.0
length
Figure A.1. Effect of solid flow rate on pressure drop.
Figure A.2. Effect of gas superficial velocity on the solid velocity.
314
Vol. 18, June
Appl.
Math.
Modelling,
1994,
0.0
0.2
0.4
Dimensionless
0.6
0.8
I.0
length
Figure A.3. Solid volume fraction and phase velocities.
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
Appendix B: Comparison of predictions of pressure drop and solid holdup in a vertical lift line with experimental data (Hariu and Molstad-1949) Exp. Run Run Run Run Run Run
: Experimental 1: Predicted 2:Predicted 3:Predicted 4:Predicted 5:Predicted 6:Predicted
value value (f, value (f, value (f, value (f, value (f, value (f,
= 0, F, (equation (equation 12), F, (equation 13), F, = 0, F, (equation (equation 12), F, (equation 13), F,
5)) (equation (equation 6)) (equation (equation
5)) 5)) 6)) 6))
AP (N/m') Solid
5.2
Al 2 3 4 5 6 7 8 9 10 11 B
c
D
E
F
u9 (Wsec)
12 13 14 15 16 17 18
9.0
12.3
5.0
12.2
19 20 21 22 23 24 25
4.9
26 27 28 29 30 31 32
4.8
33 34 35 36 37 38
4.3
39 40 41
6.2
8.7
8.6
6.2
(kg:Zec)
Exp.
Run
(1)
Run
MS (2)
Run
(3)
Exp.
Run
(1)
(9r) Run
(2)
Run
(3)
2.39
199
3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93
274 349 423 199 274 349 423 349 423 498
166 233 301 368 183 247 309 372 323 378 438
183 264 345 426 200 291 383 475 371 462 561
237 347 459 569 206 300 404 486 387 478 587
2.56 3.80 5.05 6.30 0.73 1.56 2.40 3.20 0.79 1.32 1.85
2.00 3.18 4.35 5.51 0.57 1.27 1.96 2.71 0.66 1.10 1.53
2.18 3.48 4.78 6.07 0.65 1.48 2.31 3.20 0.80 1.34 1.88
2.71 4.30 5.88 7.44 0.67 1.52 2.36 3.22 0.85 1.40 1.96
2.17 4.47 6.80 9.14 0.95 2.77 4.61
124 199 274 349 274 348 423
124 205 286 366 254 309 362
140 242 346 451 270 367 465
144 245 349 451 282 392 500
1.08 2.18 3.28 4.38 0.24 0.60 0.95
1.22 2.58 3.89 5.19 0.18 0.52 0.87
1.33 2.84 4.31 5.77 0.21 0.63 1.07
1.35 2.86 4.32 5.77 0.23 0.68 1.14
2.55 4.79 7.11 9.39 2.04 4.47 6.93
124 199 274 349 199 274 349
125 193 263 333 178 245 310
145 236 331 425 210 321 437
149 240 334 428 223 343 466
1.00 2.05 3.11 4.17 0.49 1.07 1.65
1.22 2.29 3.38 4.45 0.48 1.06 1.64
1.33 2.51 3.73 4.93 0.56 1.25 1.94
1.35 2.53 3.75 4.94 0.59 1.30 2.01
2.77 5.10 4.78 9.83 2.27 4.66 7.06
124 199 274 349 199 274 349
122 187 253 319 181 243 305
148 240 336 433 225 345 470
154 248 344 439 246 381 516
1.05 2.10 3.15 4.20 0.49 1.01 1.52
1.17 2.15 3.15 4.12 0.49 1.01 1.53
1.28 2.37 3.48 4.57 0.58 1.20 1.83
1.30 2.40 3.51 4.60 0.61 1.26 1.91
2.02 4.10 6.15 1.17 2.65 4.16
100 149 199 100 149 199
77 117 158 88 113 138
108 191 279 123 206 298
116 203 290 147 251 363
0.62 1.40 2.23 0.25 0.57 0.91
0.66 1.33 1.99 0.26 0.59 0.93
0.69 1.39 2.09 0.28 0.64 1.02
0.69 1.40 2.10 0.29 0.67 1.05
1.49 2.99 4.49
100 149 199
94 119 144
140 226 320
169 277 388
0.30 0.61 0.92
0.33 0.67 1.00
0.36 0.73 1.10
0.37 0.75 1.13
Ug: Superficial gas velocity (mlsec) Fs:Solidmass flow rate(kg/set) AP: Pressuredrop (N/m*) MS: Solidholdup (gr)
Appl.
Math.
Modelling,
1994,
Vol. 18, June
315
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
AP error(%) ug (m/set)
Solid Al
5.2 2 3 4 5 6 7 8 9 10 11
9.0
12.3
MS error(%)
(kg:Zec)
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93 Maximum error Errorsmean value
-16.6% -15.0% -13.8% -13.0% -8.0% -9.9% -11.5% -12.1% -7.4% -10.6% -12.0% 16.6% 11.8%
-8.0% -3.6% -1.1% 0.7% 0.5% 6.2% 9.7% 12.3% 6.3% 9.2% 12.7% 12.7% 6.4%
19.1% 26.6% 31.5% 34.5% 3.5% 9.5% 15.8% 14.9% 10.9% 13.0% 17.9% 34.5% 17.9%
-21.9% -16.3% -13.9% -12.5% -21.9% -18.6% -18.3% -15.3% -16.5% -16.7% -17.3% 21.9% 17.2%
-14.8% -8.4% -5.3% -3.7% -11.0% -5.1% -3.7% 0.0% 1.3% 1.5% 1.6% 14.8% 5.1%
5.9% 13.2% 16.4% 18.1% -8.2% -2.6% -1.7% 0.6% 7.6% 6.1% 5.9% 18.1% 7.8%
AP error(%) ug (m/set)
Solid B
12 13 14 15 16 17 18
5.0
12.2
(kg:Zec)
Run(l)
2.17 4.47 6.80 9.14 0.95 2.77 4.61 Maximum error Errorsmean value
0.0% 3.0% 4.4% 4.9% -7.3% -11.2% -14.4% 14.4% 6.5%
MS error(%)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
12.9% 21.6% 26.3% 29.2% -1.5% 5.5% 9.9% 29.2% 15.3%
16.1% 23.1% 27.4% 29.2% 2.9% 12.6% 18.2% 29.2% 18.5%
13.0% 18.3% 18.6% 18.5% -25.0% -13.3% -8.4% 25.0% 16.5%
23.1% 30.3% 31.4% 31.7% -12.5% 5.0% 12.6% 31.7% 21.0%
25.0% 31.2% 31.7% 31.7% -4.2% 13.3% 20.0% 31.7% 22.4%
MS error(%)
AP error(%) ug (m/set)
Solid
c
316
19 20 21 22 23 24 25
4.9
8.7
Appl.
Math.
(kg:Zec)
Run (1)
2.55 4.79 7.11 9.39 20.4 4.47 6.93 Maximum error Errorsmean value
0.8% -3.0% -4.0% -4.6% -10.6% -10.6% -11.2% 11.2% 6.4%
Modelling,
1994,
Vol. 18, June
Run (2) 16.9%
18.6% 20.8% 21.8% 5.5% 17.2% 25.2% 25.2% 18.0%
Run (3) 20.2% 20.6% 21.9% 22.6% 12.1% 25.2% 33.5% 33.5% 22.3%
Run(l) 22.0% 11.7% 8.7% 6.7% -2.0% -0.9% -0.6% 22.0% 7.5%
Run (2)
Run (3)
33.0% 22.4% 19.9% 18.2% 14.3% 16.8% 17.6% 33.0% 20.3%
35.0% 23.4% 20.6% 18.5% 20.4% 21.5% 21.8% 35.0% 23.0%
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
MS error (%)
AP error (%)
Solid D
26 27 28 29 30 31 32
ug
(m/s=) 4.8
8.6
Fs (kg/=) 2.77 5.10 4.78 9.83 2.27 4.66 7.06 Maximum error Errors mean value
Run (1) -1.6% -6.0% -7.7% -8.6% -9.0% -11.3% -12.6% 12.6% 8.1%
Run (2) 19.4%
20.6% 22.6% 24.1% 13.1% 25.9% 34.7% 34.7% 22.9%
Run (3)
Run (1)
Run (2)
Run (3)
24.2% 24.6% 25.5% 25.8% 23.6% 39.1% 47.9% 47.9% 30.1%
11.4% 2.4% 0.0% -1.9% 0.0% 0.0% 0.7% 11.4% 2.3%
21.9% 12.9% 10.5% 8.8% 18.4% 18.8% 20.4% 21.9% 15.9%
23.8% 14.3% 11.4% 9.5% 24.5% 24.8% 25.7% 25.7% 19.1%
MS error (%)
AP error (%)
Solid E
33 34 35 36 37 38
ug (m/set) 4.3
6.2
(kg::ec) 2.02 4.10 6.15 1 .17 2.65 4.16 Maximum error Errors mean value
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
-23.0% -21.5% -20.6% -12.0% - 24.2% - 30.7% 30.7% 22.0%
8.0% 28.2% 40.2% 23.0% 38.3% 49.7% 49.7% 31.2%
16.0% 36.2% 45.7% 47.0% 68.5% 82.4% 82.4% 49.3%
6.5% -5.0% -10.8% 4.0% 3.5% 2.2% 10.8% 5.3%
11.3% -0.7% -6.3% 12.0% 12.3% 12.1% 12.3% 9.1%
AP error (%)
Solid F
39 40 41
ug
and N. C. Markatos
Run (3) 11.3% 0.0% -5.8% 16.0% 17.5% 15.4% 17.5% 11 .O%
MS error (%)
Fs
Wsec)
Wg/sec)
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
6.2
1.49 2.99 4.49 Maximum error Errors mean value
-6.0% -20.1% -27.6% 27.6% 17.9%
40.0% 51.7% 60.8% 60.8% 50.8%
69.0% 85.9% 95.0% 95.0% 83.3%
10.0% 9.8% 8.7% 10.0% 9.5%
20.0% 19.7% 19.6% 20.0% 19.7%
23.3% 23.0% 22.8% 23.3% 23.0%
Appl.
Math.
Modelling,
1994,
Vol. 18, June
317
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
AP (N/m2) Solid
ug Wsec)
Al
5.2 2 3 4 5 6 7 8 9 10 11
B
c
D
E
F
9.0
12.3
12 13 14 15 16 17 18
5.0
12.2
19 20 21 22 23 24 25
4.9
26 27 28 29 30 31 32
4.8
8.7
8.6
33 34 35 36 37 38
4.3
39 40 41
6.2
6.2
Ug: Superficial
gas velocity
Ms (gr)
(kg:Zec)
Exp.
Run (4)
Run (5)
Run (6)
Exp.
Run (4)
Run (5)
Run (6)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93
199 274 349 423 199 274 349 423 349 423 498
164 230 294 357 182 247 309 370 323 378 438
181 260 337 414 200 290 382 474 371 462 561
245 352 459 564 207 309 410 509 395 499 612
2.56 3.80 5.05 6.30 0.73 1.56 2.40 3.20 0.79 1.32 1.85
1.97 3.11 4.22 5.31 0.56 1.27 1.95 2.68 0.66 1.10 1.52
2.15 3.40 4.63 5.83 0.65 1.47 2.29 3.16 0.80 1.34 1.87
2.85 4.34 5.78 7.18 0.67 1.51 2.32 3.18 0.84 1.39 1.93
2.17 4.47 6.80 9.14 0.95 2.77 4.61
124 199 274 349 274 348 423
127 204 283 361 254 309 362
140 240 342 444 270 367 465
145 249 353 457 281 400 518
1.08 2.18 3.28 4.38 0.24 0.60 0.95
1.21 2.55 3.83 5.09 0.18 0.52 0.87
1.32 2.80 4.23 5.64 0.21 0.63 1.07
1.31 2.76 4.14 5.49 0.27 0.66 1.11
2.55 4.79 7.11 9.39 2.04 4.47 6.93
124 199 274 349 199 274 349
124 192 262 329 179 244 310
145 234 328 422 210 321 437
153 248 346 443 229 360 494
1 .oo
2.05 3.11 4.17 0.49 1.07 1.65
1.22 2.27 3.35 4.39 0.48 1.06 1.64
1.33 2.49 3.69 4.86 0.56 1.24 1.94
1.31 2.44 3.59 4.72 0.57 1.26 1.95
2.77 5.10 4.78 9.83 2.27 4.66 7.06
124 199 274 349 199 274 349
121 186 252 316 181 243 305
148 239 334 429 225 345 470
159 258 359 459 254 400 548
1.05 2.10 3.15 4.20 0.49 1.01 1.52
1.17 2.15 3.13 4.09 0.49 1.01 1.53
1.27 2.36 3.45 4.53 0.57 1.20 1.82
1.25 2.30 3.35 4.38 0.59 1.21 1.83
2.02 4.10 6.15 1.17 2.65 4.16
100 149 199 100 149 199
77 117 157 88 113 138
108 190 276 123 205 297
124 217 309 153 266 384
0.62 1.40 2.23 0.25 0.57 0.91
0.66 1.33 1.98 0.26 0.59 0.93
0.68 1.39 2.08 0.28 0.64 1.01
0.64 1.30 1.94 0.27 0.62 0.97
1.49 2.99 4.49
100 149 199
94 119 144
140 224 317
177 292 409
0.30 0.61 0.92
0.33 0.67 1.00
0.36 0.73 1.09
0.35 0.70 1.05
Fs: Solid mess flow rate (kg/set)
(m/set)
AP: Pressure drop (N/m*)
MS: Solid holdup (gr)
MS error (%)
AP error (%) ug Solid
Wsec)
Al
5.2 2 3 4 5 6 7 8 9 10 11
318
9.0
12.3
Appl.
Math.
(kg:Eec)
Run (4)
Run (5)
Run (6)
Run (4)
Run (5)
Run (6)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93 Maximum error Errorsmean value
-17.6% -16.1% -15.8% -15.6% -8.5% -9.9% -11.5% -12.5% -7.4% -10.6% -12.0% 17.6% 12.5%
-9.0% -5.1% -3.4% -2.1% 0.5% 5.8% 9.5% 12.1% 6.3% 9.2% 12.7% 12.7% 6.9%
23.1% 28.5% 31.5% 33.3% 4.0% 12.8% 17.5% 20.3% 13.2% 18.0% 22.9% 33.3% 20.5%
-23.0% -18.2% -16.4% -15.7% -23.3% -18.6% -18.8% -16.3% -16.5% -16.7% -17.8% 23.3% 18.3%
-16.0% -10.5% -8.3% -7.5% -11.0% -5.8% -4.6% -1.3% 1.3% 1.5% 1.1% 16.0% 6.2%
11.3% 14.2% 14.5% 14.0% -8.2% -3.2% -3.3% -0.6% 6.3% 5.3% 4.3% 14.5% 7.8%
Modelling,
1994,
Vol. 18, June
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
MS error(%)
AP error(%) Solid B
12 13 14 15 16 17 18
ug
Wsec) 5.0
12.2
(kg:Zec)
Calc. (4)
2.17 4.47 6.80 9.14 0.95 2.77 4.61 Maximum error Errorsmean value
2.4% 2.5% 3.3% 3.4% -7.3% -11.2% -14.4% 14.4% 6.4%
Calc. (5) 12.9% 20.6% 24.8% 27.2% -1.5% 5.5% 9.9% 27.2% 14.6%
Calc. (6) 16.9%
25.1% 28.8% 30.9% 2.6% 14.9% 22.5% 30.9% 20.3%
Calc. (4)
Calc. (5)
12.0% 17.0% 16.8% 16.2% -25.0% -13.3% -8.4% 25.0% 15.5%
22.2% 28.4% 29.0% 28.8% 12.5% 5.0% 12.6% 29.0% 19.8%
AP error(%) Solid
c
19 20 21 22 23 24 25
ug
Wsec)
(kg;Zec)
Calc. (4)
Calc. (5)
4.9
2.55 4.79 7.11 9.39 2.04 4.47 6.93 Maximum error Errorsmean value
0.0% -3.5% -4.4% -5.7% -10.1% -10.9% -11.2% 11.2% 6.5%
16.9% 17.6% 19.7% 20.9% 5.5% 17.2% 25.2% 25.2% 17.6%
8.7
Solid D
26 27 28 29 30 31 32
(m/set) 4.8
8.6
(kg:lec) 2.77 5.10 4.78 9.83 2.27 4.66 7.06 Maximum error Errorsmean value
Calc. (4) -2.4% -6.5% -8.0% -9.5% -9.0% -11.3% -12.6% 12.6% 8.5%
Calc. (5) 19.4% 20.1% 21.9% 22.9% 13.1% 25.9% 34.7% 34.7% 22.6%
Calc. (6) 23.4% 24.6% 26.3% 26.9% 15.1% 31.4% 41.5% 41.5% 27.0%
Calc. (4) 22.0% 10.7% 7.7% 5.3% -2.0% -0.9% -0.6% 22.0% 7.0%
Solid E
33 34 35 36 37 38
(m/set)
(kg;Zec)
Calc. (4)
4.3
2.02 4.10 6.15 1.17 2.65 4.16 Maximum error Errorsmean value
-23.0% -21.5% -21.1% -12.0% -24.2% -30.7% 30.7% 22.1%
6.2
Calc. (5) 8.0% 27.5% 38.7% 23.0% 37.6% 49.2% 49.2% 30.7%
Appl.
21.3% 26.6% 26.2% 25.3% 12.5% 10.0% 16.8% 26.6% 19.8%
Calc. (6)
Calc. (5) 33.0% 21.5% 18.6% 16.5% 14.3% 15.9% 17.6% 33.0% 19.6%
31.0% 19.0% 15.4% 13.2% 16.3% 17.8% 18.2% 31.0% 18.7%
MS error(%) Calc. (6) 28.2% 29.6% 31.0% 31.5% 27.6% 46.0% 57.0% 57.0% 35.9%
Calc. (4)
Calc. (5)
Calc. (6)
11.4% 2.4% -0.6% -2.6% 0.0% 0.0% 0.7% 11.4% 2.5%
21.0% 12.4% 9.5% 7.9% 16.3% 18.8% 19.7% 21.0% 15.1%
19.0% 9.5% 6.3% 4.3% 20.4% 19.8% 20.4% 20.4% 14.3%
AP error(%)
ug
_
Calc. (6)
MS error(%)
BP error(%)
ug
and N. C. Markatos
MS error(%) Calc. (6) 24.0% 45.6% 55.3% 53.0% 78.5% 93.0% 93.0% 58.2%
Math.
Calc. (4) 6.5% -5.0% -11.2% 4.0% 3.5% 2.2% 11.2% 5.4%
Modelling,
1994,
Calc. (5) 9.7% -0.7% -6.7% 12.0% 12.3% 11.0% 12.3% 8.7%
Vol. 18, June
Calc. (6) 3.2% -7.1% -13.0% 8.0% 8.8% 6.6% 13.0% 7.8%
319
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos MS error (%)
AP error (%) ug (m/set)
Solid F
39 40 41
(kg:Zec)
6.2
1.49 2.99 4.49 Maximum error Errors mean value
Calc. (4)
Calc. (5)
Calc. (6)
Calc. (4)
Calc. (5)
Calc. (6)
-6.0% -20.1% -27.6% 30.7% 19.0%
40.0% 50.3% 59.3% 59.3% 45.1%
77.0% 96.0% 105.5% 105.5% 84.2%
10.0% 9.8% 8.7% 11.2% 8.5%
20.0% 19.7% 18.5% 20.0% 16.7%
16.7% 14.8% 14.1% 16.7% 13.3%
7.00 G.00 ~5.00 9 ~4.00 B $
3.lxl
ti a 2.OU l.Ctl
+
0.00
150
7.00
250
300 t’~easule
Figure
350 dlotl
400
450
503
200
250
3w t’lessure
Particle holdup vs. pressure drop: run I.
B.I.
I50
(N/1112)
Figure
8.2.
350
4w
450
500
drot, (N/1112)
Particle holdup vs. pressure drop: run 4.
4.50 4.M)
4.00 Material
D 3.50
3.50 g
gj 3.00
3.00
9 -a 2.50
2 ‘c) 2.50
2 2 2.00 .L! a?G 1.50
B s 2.00 .o a!i 1.50
1.00
t.co
0.50
0.50 0.00
0.00 100
150
200 Prersul-e
Figure
8.3.
320
Appl.
250
300
Modelling,
loo
150
1994,
200 Pteswte
drot, (N/1112)
Particle holdup vs. pressure drop: run 2.
Math.
350
Vol. 18, June
Figure
8.4.
25U
300
dl-Ott (N/1112)
Particle holdup vs. pressure drop: run 4.
350
and N. C. Markatos
of Chemical
Engineering,
National
Technical
University
of Athens,
Athens,
Greece
A finite-domain procedure for simulating the hydrodynamics of vertical pneumatic conveying is presented. A parametric study considering six difSerent spec@ations for particles and 41 difSerent operating conditions has been carried out. Predictions, which have been obtained using two dtflerent empirical correlations for the description of the interphase momentum transfer and three deferent empirical correlations for the description of the solid-to-wall friction, have been compared with experimental data from the literature. It is concluded that in dilute transport situations realistic predictions of pressure drop, particle holdup, and of choking behavior can be made. Finally, a calibration procedure is proposedforjtting the calculated to the experimental data of vertical pneumatic conveying.
Keywords: mathematical
modelling, vertical pneumatic conveying, two-phase flow, interphase drag force
1. Introduction Pneumatic conveying, or pneumatic transport, is commonly used in industry for loading and unloading of dry bulk materials, whereas physical operations such as drying can be carried out in flowing gas-solid Chemical reactions such as catalytic suspensions. cracking of hydrocarbons can also be completed in riser reactors, where the catalyst is conveyed by the hydrocarbon vapors. In these operations, it is important to determine pressure drop and choking velocity and also to predict the solids concentration and the particle holdup along the conveying line. In this study, the one-dimensional differential equations describing the above process have been solved by using a finite-domain procedure. These equations have been solved in conjunction with empirical correlations, describing the drag force between the particles and the fluid and the pressure drop due to wall friction with the fluid and with the particles. A parametric study considering six different specifications for particles and 41 different operating conditions has been carried out. Predictions have been obtained two different empirical equations for the using description of the drag force between the particles and the fluid and three different equations for the description
Address reprint requests to Professor Markatos at the Computational Fluid Dynamics Unit, Section 11, Dept. of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Str., 157 73 Zografou, Athens, Greece. Received
306
4 November
Appl.
Math.
1992; accepted
Modelling,
25 October
1994,
1993
Vol. 18, June
of the wall solid friction factor. The results obtained have been compared with the experiments of Hariu and Molstad.’ Predictions have also been made near the choking conditions, and results have been compared with Zenz’s’ experimental data.
2. The mathematical
model
Vertical pneumatic conveying of solids by gases can be described in many situations by means of onedimensional, isothermal, steady-state mass and momentum balances. The one-dimensional assumption is a good approximation for the operation of a conveying line, but it cannot take into account phenomena such as backmixing of solids due to distributor geometry. The isothermal assumption is a good approximation for dispersed flow due to adequate heat transfer and low rates of heat generation by friction. Three dimensional, nonisothermal, and transient cases could also be handled by similar modelling techniques, as described here. However, the simplified version, apart from its significant practical relevance, allows for quick and inexpensive testing and validation of physical concepts and of modelling philosophy and techniques.
2.1
The dlflerential
equations
One-dimensional, isothermal, and steady-state, vertical pneumatic conveying can be described by four differential equations, i.e., the continuity and the
0
1994 Butterworth-Heinemann
Modeling momentum
equations
of vertical pneumatic-conveying
hydrodynamics: 2.3
for each phase.3
Gas continuity
;
(EWJ = 0
(1)
Particles continuity
;
[(l - &)WJ = 0
(2)
Gas momentum
k kpswsws)
The frictional force between the wall and the gas-solid suspension is assumed to be separated into two terms due, separately, to the fluid alone (f,,) and to the effect of solid particles (f,,). The frictional pressure drop due to conveying the fluid alone in a pipe of length L and diameter D is defined by Fanning’s equation: = Fg
Particles momentum
;
= -(l -
- E);
+fpg
f, = 0.316Re-
+fpw
(1 - 4Pp - Pg)S
(4)
where w is the velocity, p the density for the phase considered, and the subscripts g and p refer to the gaseous and particulate phases, respectively. g is the gravitational acceleration, E is the void fraction, P is the pressure (assumed the same for both phases) and the forces, J are described below.
The major empirical input in the present modelling technique is the determination of the fluid-particle interphase momentum transfer. In vertical pneumatic conveying a drag force (&J is exerted on the particles from the gas, while an opposite force (f,, = -fPy) is exerted on the gas by the particles. The drag force controls the slip velocity between the two phases, the acceleration zone of the particulate phase, and the concentration of particles all over the conveying line. The drag force on a single spherical particle of diameter d, in an infinite fluid is usually expressed in terms of a drag coefficient C, as F, = 0.5C&d;/4)p,(w,
- wp)’
(5)
To take into account the effect of concentration of particles in sedimentation or fluidization, Richardson and Zaki4 showed that the drag force per unit volume of particles in mixtures of fluid and spherical particles can be expressed as F, = 3/4C,
PAWS- WA2 E-2,65 (6) 4
The drag coefficient C, can be related to the particle Reynolds number by means of the relations: C, = 24/Re,(l
‘j4
(10)
The wall friction due to solid particles is defined differently by various authors. Most of them define a solid friction factor, following Fanning’s equation, based on the particle velocity and the dispersed solid density:i5 APF,
=
f,PAl - E)WiL
(11)
20
Yang5 correlated the experimental data for vertical pneumatic conveying obtained by Hariu and Molstad’ and derived the following empirical equation for the prediction of the solid friction factor: fps3/(1 - E) = 0.0206[(1 - &)Re,/Re,]-0.869
The drag force
2.2
(9)
20
Assuming a smooth pipe, the friction factor f, can be obtained by the empirical Blassius formula:
4PpWpWpl
C(1 -
fsP,W,2L
(3)
+fgp +fgW
and N. C. Markatos
The wall friction
Ap = --E g
K. N. Theologos
+ Re,0.687)
and pLsis the viscosity
where Re, is the Reynolds number based on the particle terminal (free-fall) velocity. The same author correlated additional experimental data and derived an updated equation: &/(l
- E) =
0.0126[(1 - &)Re,/Re,]-0.979
V,/v > 1.5
0.0410[(1 - &)Re,/Re,]- ‘.021 V,/l: < 1.5 (13)
where V, is the gas superficial velocity particle terminal (free-fall) velocity. 2.4
and
r/; is the
Numerical solution
The above set of equations has been solved using a numerical technique. In order to solve the equations, the entire domain of interest is discretized into a grid of finite control volumes. The differential equations are integrated over these volumes in order to derive finitedomain equations. These equations are solved in an iterative manner and yield values of the dependent variables at the centers of the control volumes. The finite-domain equations are solved by using a developed computer program, which is similar to the one described in Ref. 3. The relevant numerical details have been published extensively (e.g., Refs. 6 and 7) and are not repeated here.
(7)
3. Results and discussion
where Re, = (dpPg/~JI~g
(12)
- wPl of the gas.
(8)
To evaluate the performance of the proposed model a parametric study and comparisons with experimental data have been carried out. The vertical lift line studied
Appl.
Math.
Modelling,
1994,
Vol. 18, June
307
Modelling
of vertical pneumatic-conveying
hydrodynamics:
here is the same as the one used by Hariu and Molstad.’ A vertical line of 13.5-mm internal diameter and 1.368-m height has been considered. To facilitate the validation, an appropriate mesh has been used to calculate values at the same locations where the experimental ones had been measured. The results have been checked for grid effects and were found grid independent for a grid consisting of 10 control volumes. This grid independency is presented for a typical run in Figure 1. Convergence was easy to obtain, and a typical computer run was accomplished within 150 s CPU time on an Acorn Archimedes 440 personal workstation. Six different specifications for particles to be conveyed by air have been considered, as shown in Table 1. Predictions have been made for 41 different operating conditions, the ones used by Hariu and Molstad. For the present predictions at the inlet boundary of the lift line, the gas phase is considered to enter the tube with a
7.00
I
1
6.00 5.00
0
N%=S
0
N%=lO
4.00 3.00 Z.(x) 1.00
oxxJ ’ 0.00
1.oo
0.40
0 20
Dinwwiwlcss Figure
1 (a).
twigill
Grid independency
of solid-phase
velocity
results.
K. N. Theologos
and N. C. Markatos
velocity equal to the experimental superficial velocity, whereas the solid phase is considered to enter the tube with zero momentum. The latter assumption is made because of a lack of any experimental data about the solid velocity or the solid volume fraction at the inlet of the lift line. Near the inlet boundary, the velocity and the volume fraction of the solid phase were calculated, assuming only interphase transfer of momentum due to the frictional force exerted from the gas to the particles. In every case, for a given gas superficial velocity, V,, and solid mass flow rate, F,, the following variables have been calculated all over the lift line: gas and particle velocities, wg and wp, bed voidage, E, and pressure, P. Also, the total pressure drop, AP, and the particles holdup, M,, have been calculated in every case. 3.1
Typical predictions
The results presented in Figures 2, 3, and 4 were computed assuming zero pressure drop due to solid friction; the drag force was calculated by equation (5). The values used for the production of these figures are presented in Appendix A. Figure 2 shows the distributions of solid volume fraction and of phase velocities along the height of the lift line, for conveying material A by air. As shown in this figure, the gas velocity decreases as the solids accelerate; the solid volume fraction decreases, due to the increase in solid velocity at a constant solid mass flow rate, as expected. Figure 3 shows the effect of solid mass flow rate on the pressure drop along the lift line for a given gas superficial velocity. Predictions are presented for material A. As expected, the pressure drop increases as the solid mass flow rate increases, because more solids have to be accelerated and conveyed by the fluid. This prediction is consistent with experimental data obtained by Hariu and Molstad.
NZ=S
0 0
N%=lO
0
NZ=20
14.0 _
9.OL3
pN%=.50
O.(X)
020
0.40
0 60
Dimensionless
Figure results.
Table
1 (b).
1.
Grid independency
Particle
Particle diameter Particle density
(pm)
of solid-phase
volume
fraction
A
0
C
D
E
F
503 2643
357 2643
274 2643
213 2707
110 977
110 977
3.9
2.8
2.2
1.6
0.3
0.3
(m/set)
Appl.
0.4
0.2
(kg/m3) Particle free-fall
308
1.oo
properties
Material
velocity
0 no
height
Math.
Dimensionless
Figure
Modelling,
1994,
Vol. 18, June
2.
Solid volume
fraction
0.0
(I.8
length
and phase velocities.
Model/kg
0.0
0 2
of vertical pneumatic-conveying
0 4
(1.6
Dimensionless
Figure
3.
0 x
10
length
Effect of solid flow rate on pressure
drop.
Figure 4 shows the effect of the gas superficial velocity on the solid phase velocity. Predictions are presented for material A, for three different gas superficial velocities. For each gas superficial velocity, solid velocities have been calculated assuming different solid flow rates. As is shown, the solid mass flow rate has a minor effect on the profile of the solid velocity, which is affected primarily by the gas superficial velocity.
3.2
A parametric study
A parametric study concerning appropriate empirical correlation
hydrodynamics:
K. N. Theologos
and N. C. Markatos
the drag force and the wall friction factor has been carried out. Predictions have been obtained using two different empirical equations for the description of the drag force between the particles and the fluid and three different equations for the description of the wall solid friction factor. The parametric study runs (six in total) carried out are shown in Table 2. The operating conditions and the predictions obtained for each of the six runs for each of the 41 different operating conditions are presented in Appendix B. For each run the total pressure drop, AP, and the particles holdup, M,, were calculated. Then, a percent error between the predicted and the experimental value was calculated. The maximum and mean errors for the run that gave best fit to the experimental data are presented in Table 3. The predictions obtained are reasonable, with the exception of the calculated pressure drop for particles E and F and the solid holdup for material B. The best predictions were obtained assuming that there was no pressure drop due to solid friction (runs 1 and 4). This assumption is reasonable here, because dilute phase transport is considered. For material A the best predictions were obtained for a solid friction factor calculated by equation (12). The parametric study showed that the use of equation (6) instead of equation (5) in the calculation of the interphase friction had a minor effect on the prediction of the total pressure drop, AP, and the particles holdup, M,. This is because dilute phase transport is considered and the values of the void fraction, E, are close to unity. In vertical pneumatic conveying, the total pressure drop consists of three individual contributions due to acceleration, gravity, and wall friction:
the selection of the for the prediction of
AP, = AP, + AP, + AP,
(14)
To determine the acceleration term, it is important to know the momentum of the solid phase at the inlet of
Table
2.
Parameter
study runs
fp = 0
8
Fo, equation Fo, equation
3.x0
(5) (6)
fp, equation
1 4
(12)
f,, equation
2 5
(13)
3 6
2 A 3.30 :’ :: p 2.80 _;; z ,” 230
Table
3.
1.80
Material
0.80
I-
0.2
0.4 Dimensionless
Figure 4. velocity.
Effect
of gas superficial
0.6
08
1.0
length
velocity
on the
solid
phase
A B C D E F
Deviation
Best fit run 2 4 4 1 1 I,4
Appl.
Math.
between
experimental
Maximum BP error (%)
Mean AP error (%)
12.7 14.4 11.2 12.6 30.7 27.6
6.4 6.4 6.5 8.1 22 17.9
Modelling,
1994,
data and predictions Maximum M, error (%)
Mean A& error (%)
14.8 25 2.2 11.4 10.8 10
5.1 15.5 7.0 2.3 5.3 9.5
Vol. 18, June
309
Modeling
of vertical pneumatic-conveying
hydrodynamics:
the lift line. For a known solid mass flow rate, the initial velocity or the initial volume fraction is necessary for the determination of the momentum at the inlet of the lift line. Harm and Molstad calculated an average solid volume fraction by measuring the total particle holdup in the lift line. For a unique specification of pressure drop in vertical pneumatic conveying the inlet void fraction (or solid velocity) has to be measured as mentioned by Arastoopour and Gidaspow’ and as proven by Figures 5 and 6. These figures show very good agreement between predictions and experiment, and even though errors occurred in the prediction of both pressure drop and particles holdup, the predicted linear dependence between these variables accurately fits the experimental values.
Chocking behavior: Comparison with experiment
3.3
For a fixed solid mass flow rate, reducing the gas flow rate has two major consequences. One consequence is that the frictional resistance of the flowing mixture is reduced. Also, gas and solid both rise more slowly, the voidage decreases, and the static head rises. At high gas superficial velocities, when decreasing the gas velocity the change in the frictional resistance predominates and the pressure drop decreases. A further lowering of gas flow rate causes a rapid rise in solid inventory and static head that produces an increase in pressure drop.’ This competition of forces that change with flow rate in the opposite direction results in the occurrence of a minimum in the pressure drop curve, which is known as chocking behavior. 7.00 ,
I 0
0.00
1
I
150
200
250
RW
350
400
450
500
Pressure do-op (Nim2)
Figure 5.
Particle holdup vs. pressure drop (material A)
4.50
K. I\ 5
‘heologos
and N. C. Markatos
10
c
z_
”
E?g
22 2
t-2
a
1 100
10
Superficial gas velocity (Wee)
Figure
7.
Choking
behavior.
This behavior is predicted by the present modelling technique, and it is presented in Figure 7. In this figure the effect of gas superficial velocity on the total pressure drop is presented. Predicted values were obtained using Zenz’s’ experimental conditions. The calculated pressure drops are in very good agreement with the experimental data, even though neither an initial nor an average solid concentration is reported.
3.4
A calibration procedure
The total pressure drop is not the only parameter that has to be determined in vertical pneumatic conveying. In many operations, such as gas-oil catalytic cracking in riser reactors, the particle (i.e., catalyst) concentration has to be determined accurately, as it controls the rate of reaction. Unfortunately, there are no experimental measurements of the local voidage in vertical pneumatic conveying, but only average calculations through measurements of the total particle holdup in the lift line. The comparison between the experimental and calculated values reported in section 3.2 above showed that the use of empirical correlations for the determination of the solid to wall friction caused deviations between calculated and experimental values. In order to calculate accurately the total pressure drop and the particle holdup the following calibration procedure is proposed. First a total pressure drop, APJe = 0, is calculated assuming zero friction between the particles and the wall. Then, for the same operating conditions, the pressure drop APJp,eq,13, is calculated using equation 13 for the determination of the solid-to-wall friction factor. Assuming that the error in the calculation of pressure drop is caused only by the determination of the solid-to-wall friction, the friction factor, f,, is corrected by multiplication with the following expression:
ii~~~~
AP,,,
- AP,
=o
APfp,eq.i3 - AP,* = o cl
0.50 -0.00 -I 100
150
2w
250
I 350
3ul
Pressure drop (N/m21
Figure 6.
310
Particle holdup vs. pressure drop (material
Appl.
Math.
Modelling,
1994,
D)
Vol. 18, June
value of the pressure where AP,,, is the experimental drop. This technique was applied at 11 different operating conditions for material A and not only gave accurate predictions for the pressure drop, but also corrected the particle holdup. Predictions, which gave a maximum
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
dP
200
3(x1
4,X,
pressure
Experimcnlal
SW drop
0
1
2
3
Experimental
(Sll”2~
Figure friction
8. factor
Predicted
4 particle
5
6
7
holdup
(Kr)
vs. experimental
values
using
a calibrated
fD.
error of 2% for the prediction of pressure drop and 4% for the prediction of the particle holdup, are presented in Figure 8. The above-mentioned procedure shows that in the case where experimental data of total pressure drop for vertical pneumatic conveying are available, a fine tuning of a hydrodynamic model can be carried out through a calibration of the friction factor between the particles and the wall, f,.
particle diameter force exerted on the gas by the particles force exerted on the gas by the wall j: P9 force exerted on the particles by the gas f PW force exerted on the particles by the wall acceleration due to gravity pipe length i pressure (assumed the same for both phases) P (d,p,lp,) I wg - wp I Rep ~~,P,lP,) I: Ret gas superficial velocity v, particle terminal (free-fall) velocity v gas velocity W9 particles velocity WP axial distance Z voidage (i.e., gas volume fraction) & gas viscosity p9 gas density ps particles density PP
6. Acknowledgement 4. Conclusion The results presented, and the many more obtained but not reported here due to space restrictions, lead to the following conclusions: Pressure drop and particle holdup in dilute-phase vertical pneumatic conveying can be predicted using numerical techniques. Realistic predictions of practical accuracy for dilute transport can be obtained using standard drag correlations for a single spherical particle in an infinite fluid and ignoring the solid-wall frictional force. The proposed modelling technique can realistically predict the choking behavior in vertical pneumatic conveying. For accurate predictions of pressure drop and particle holdup in vertical pneumatic conveying, the initial solid velocity (or volume fraction) has to be measured. A calibration procedure of the hydrodynamic model with reference to experimental data can be carried out through calibration of the solid-to-wall friction factor,
I 2 3
4 5 6
8
Nomenclature
CD
D
7. References
7
f P’ I.
This research was performed partially in the framework of the JOU-2-CT92-122 EEC contract. The financial support of the EEC (Directorate General XII) is gratefully acknowledged. The computations were performed using the PHOENICS computer program of CHAM, Ltd., Wimbledon, London, UK.
drag coefficient pipe diameter
9
Harm, 0. H., Molstad, M. C. Pressure drop in vertical tubes in transport of solids by gases. Ind. Eng. Chem. 1949,41(6), 1148-1160 Zenz, F. A. Two-phase fluid-solid flow. Ind. Eng. Chem. 1949, 41 (12) 2801-2806 Markatos, N. C. and Shinghal, A. K. Numerical analysis of one-dimensional, two phase flow in a vertical cylindrical passage. Ado. Eng. Sofrware 1982, 4 (5), 99-106 Richardson, J. F., Zaki, W. N. Trans. Insl. Chem. Eng. 1954, 32, 35-53 Yang, W. C. A correlation for solid friction factor in vertical pneumatic conveying lines. AIChE J. 1978, 24 (3), 548-552 Markatos, N. C. and Moult, A. The computation of steady and unsteady turbulent, chemically-reacting flows in axisymmetrical domains. Trans. Instn. Chem. Engrs. 1979, 51, 156162 Markatos, N. C. Modelling of two-phase transient flow and combustion of granular propellants. Int. J. Multiphase Flow 1986, 12 (6), 913-933 Arastoopour, H., Gidaspow, D. Vertical pneumatic conveying using four hydrodynamic models. Ind. Eng. Chem. Fundum. 1979, 18 (2), 123-130 Kunii, D., Levenspiel, 0. Fluidizution Engineering. John Wiley & Sons, New York, 1969.
Appendix A: Predictions of pressure, phase velocities, and phase volume fractions in a vertical lift line for the experimental conditions used by Hariu and Molstad (material A, tube 2) Predictions vs. experimental Dimensions
data (Hariu and Molstad-1949)
of apparatus
Riser i.d. (mm) Riser height (m)
13.5 1.368
Appl.
Math.
Modelling,
1994,
Vol.
18,
June
311
Modelling Solid
of vertical pneumatic-conveying
K. N. Theologos
and N. C. Markatos
properties
Material Particle Particle Particle
hydrodynamics:
diameter (pm) density (kg/m3) free-fall velocity
Experiment
(m/set)
B
C
D
503 2643 3.9
357 2643 2.8
274 2643 2.2
213 2707 1.6
E
F
110 977 0.3
110 977 0.3
1 Material A Gas superficial velocity Solid mass flow rate
Z (m) Dimens. length P (N/m2) Wl (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
5.2 m/set 2.39 g Jsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.1 249 5.24 0.9923 7.7E-3 4.42
0.2 204 5.23 0.9945 5.5E-3 4.09
0.3 173 5.23 0.9950 5.OE-3 3.95
0.4 146 5.22 0.9953 4.7E-3 3.87
0.6 120 5.22 0.9954 4.6E-3 3.83
0.7 96 5.22 0.9955 4.5E-3 3.80
0.8 71 5.22 0.9956 4.4E-3 3.78
0.9 47 5.22 0.9957 4.3E-3 3.74
2 Material A Gas superficial velocity Solid mass flow rate
Z (ml Dimens. length P (N/m2) Wl (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
A
5.2 m/set 3.84 g lsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 359 5.26 0.9879 1.2E-3 4.42
0.188 290 5.24 0.9913 8.7E-3 4.08
0.313 245 5.24 0.9922 7.8E-3 3.94
0.438 205 5.24 0.9926 7.4E-3 3.87
0.563 169 5.24 0.9928 7.2E-3 3.83
0.688 133 5.24 0.9929 7.1 E-3 3.81
0.813 99 5.24 0.9930 7.OE-3 3.78
0.938 65 5.23 0.9932 6.8E-3 3.73
3 Material A Gas superficial velocity Solid mass flow rate
5.2 m/set 5.32 g lsec
Z (ml
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m2) AP (N/m2) W2 (m/set)
0.063 468
0.188 376 -92 1.18 1.2E-2 4.08
0.313 316 -152 1.32 1 .l E-2 3.94
0.438 265 -203 1.39 1 .OE-2 3.86
0.563 217 -251 1.43 9.8E-3 3.82
0.688 171 -297 1.46 9.7E-3 3.79
0.813 126 -342 1.48 9.5E-3 3.77
0.938 82 -386 1.52 9.3E-3 3.73
0 0.85 1.6E-2 4.44
R2 (vol/vol) Wslip (m/set)
Experiment
4
Z (m) Dimens. length P (N/m2) AP (N/m2) W2 (m/set) R2 (vol/vol) Wslip (m/set)
312
Appl.
Material A Gas superficial velocity Solid mass flow rate
5.2 m/set 6.79 g/set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 576 0 0.87 2.1 E-2 4.44
0.188 461 -115 1.20 1.5E-2 4.08
0.313 387 -189 1.34 1.3E-2 3.93
0.438 324 -252 1.41 1.3E-2 3.86
0.563 265 -311 1.45 1.2E-2 3.81
0.688 208 -368 1.48 1.2E-2 3.78
0.813 153 -423 1.50 1.2E-2 3.76
0.938 99 -477 1.54 1.2E-2 3.72
Math. Modelling,
1994, Vol. 18, June
Modeling Experiment
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
5 Material A Gas superficial velocity Solid mass flow rate
9.0 m/set 1.76 g/set
Z (ml
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m2) AP (N/m*) W2 (m/set)
0.063 282 0 1.67
0.188 233 -49 2.52
0.313 199 -83 2.96
1.9E-3 6.50
1.6E-3 6.05
0.438 168 -114 3.24 1.4E-3 5.77
0.563 140 -142 3.43 1.4E-3 5.58
0.688 113 -169 3.57 1.3E-3 5.44
0.813 86 -196 3.67 1.3E-3 5.34
0.938 60 -222 3.75 1.2E-2 5.26
2.8E-3 7.35
R2 (vol/vol) Wslip (m/set)
Experiment
6 Material A Gas superficial velocity Solid mass flow rate
9.0 m/set 3.98 g/set
2 (m)
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
Dimens. length P (N/m*) AP (N/m*) W2 (m/set)
0.063 429 1.68
0.188 340 -89 2.53
6.3E-3 7.38
4.2E-3 6.51
0.313 285 -144 2.95 3.6E-3 6.08
0.438 238 -191 3.22 3.3E-3 5.81
0.563 196 -233 3.40 3.1 E-3 5.63
0.688 156 -273 3.53 3.OE-3 5.50
0.813 118 -312 3.63 2.9E-3 5.40
0.938 80 -349 3.71 2.8E-3 5.32
0
R2 (vol/vol) Wslip (m/set)
Experiment
7 Material A Gas superficial velocity Solid mass flow rate
Z Cm) Dimens. length P (N/m*) AP (N/m*) W2 (m/set) R2 (vol/vol) Wslip (m/set)
Experiment
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.063 577 0 1.69 9.7E-3 7.40
0.188 449 -129 2.53 6.5E-3 6.53
0.313 372 -205 2.96 5.5E-3 6.10
0.438 309 -268 3.22 5.1 E-3 5.83
0.563 252 -325 3.40 4.8E-3 5.65
0.688 200 -377 3.52 4.6E-3 5.52
0.813 149 -428 3.61 4.5E-3 5.43
0.938 101 -476 3.69 4.4E-3 5.35
8 Material A Gas superficial velocity Solid mass flow rate 0.068
Z (ml Dimens. length P (N/m*) AP (N/m2) W2 (m/set) R2 (vol/vol) Wslip (m/set) Experiment
0.063 725 0 1.68 1.3E-3 7.34
P
(N/m*)
Wl (m/set) W2 (m/set) RI (vol/vol) R2 (vol/vol) Wslip (m/set)
8.9 m/set 8.42 glsec
0.205
0.342
0.479
0.616
0.752
0.889
1.026
0.188 556 -169 2.51 8.9E-3 6.47
0.313 459 -266 2.93 7.6E-3 6.05
0.438 379 -346 3.18 7.OE-3 5.78
0.563 309 -416 3.35 6.6E-3 5.61
0.688 244 -481 3.47 6.4E-3 5.49
0.813 182 -543 3.56 6.3E-3 5.40
0.938 122 -603 3.63 6.1 E-3 5.33
9 Material A Gas superficial velocity Solid mass flow rate
Z (m)
9.0 m/set 6.17 g/set
12.3 3.02
m/set g Jsec
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
526 12.44 2.28 0.9965 3.5E-3 10.16
430 12.43 3.49 0.9977 2.3E-3 8.95
364 12.42 4.10 0.9980 2.OE-3 8.32
307 12.42 4.49 0.9982 1.8E-3 7.93
254 12.42 4.76 0.9983 1.7E-3 7.66
203 12.42 4.95 0.9984 1.6E-3 7.47
154 12.42 5.10 0.9984 1.6E-3 7.32
106 12.42 5.21 0.9985 1.5E-3 7.21
Appl.
Math.
Modelling,
1994,
Vol. 18, June
313
Modelling
of vertical pneumatic-conveying
Experiment
10 Material A Gas superficial velocity Solid mass flow rate
Z (m) P
(N/m*)
WI (m/set) W2 (m/set) Rl (vol/vol) R2 (vol/vol) Wslip (m/set)
Experiment
K. N. Theologos
and N. C. Markatos
12.3 m/set 4.98 g /set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
677 12.37 2.27 0.9942 5.8E-3 10.10
539 12.35 3.45 0.9962 3.8E-3 8.90
452 12.34 4.05 0.9967 3.3E-3 8.29
378 12.34 4.43 0.9970 3.OE-3 7.91
310 12.33 4.69 0.9972 2.8E-3 7.65
247 12.33 4.87 0.9973 2.7E-3 7.46
186 12.33 5.01 0.9974 2.6E-3 7.32
127 12.33 5.12 0.9974 2.6E-3 7.22
11 Material A Gas superficial velocity Solid mass flow rate
(N/m*)
WI (m/set) W2 (m/set) RI (vol/vol) R2 (vol/vol) Wslip (m/set)
12.3 m/set 6.93 g/set
0.068
0.205
0.342
0.479
0.616
0.752
0.889
1.026
837 12.4 2.28 0.9920 8.OE-3 10.12
655 12.4 3.45 0.9947 5.3E-3 8.92
545 12.4 4.04 0.9955 4.5E-3 8.32
454 12.4 4.41 0.9958 4.2E-3 7.94
371 12.4 4.66 0.9961 3.9E-3 7.69
294 12.4 4.84 0.9962 3.8E-3 7.51
220 12.4 4.98 0.9963 3.7E-3 7.37
149 12.3 5.08 0.9964 3.6E-3 7.26
Z (m) P
hydrodynamics:
5.30
4.50 9.OE-3
4.30
"g=12.3m,rec S.OE-3
^ 3.80 d < 5 x 3.3" .=: :: z s 2.30 % z w 2 30
1.80 &OE-3 1.30
0.0
0.2
0.4
0.6
Dimensionless
0.8
1.0
0.80 0.0
3.OE-3 0.2
0.4 Dimensionless
length
0.6
0.8
1.0
length
Figure A.1. Effect of solid flow rate on pressure drop.
Figure A.2. Effect of gas superficial velocity on the solid velocity.
314
Vol. 18, June
Appl.
Math.
Modelling,
1994,
0.0
0.2
0.4
Dimensionless
0.6
0.8
I.0
length
Figure A.3. Solid volume fraction and phase velocities.
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
Appendix B: Comparison of predictions of pressure drop and solid holdup in a vertical lift line with experimental data (Hariu and Molstad-1949) Exp. Run Run Run Run Run Run
: Experimental 1: Predicted 2:Predicted 3:Predicted 4:Predicted 5:Predicted 6:Predicted
value value (f, value (f, value (f, value (f, value (f, value (f,
= 0, F, (equation (equation 12), F, (equation 13), F, = 0, F, (equation (equation 12), F, (equation 13), F,
5)) (equation (equation 6)) (equation (equation
5)) 5)) 6)) 6))
AP (N/m') Solid
5.2
Al 2 3 4 5 6 7 8 9 10 11 B
c
D
E
F
u9 (Wsec)
12 13 14 15 16 17 18
9.0
12.3
5.0
12.2
19 20 21 22 23 24 25
4.9
26 27 28 29 30 31 32
4.8
33 34 35 36 37 38
4.3
39 40 41
6.2
8.7
8.6
6.2
(kg:Zec)
Exp.
Run
(1)
Run
MS (2)
Run
(3)
Exp.
Run
(1)
(9r) Run
(2)
Run
(3)
2.39
199
3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93
274 349 423 199 274 349 423 349 423 498
166 233 301 368 183 247 309 372 323 378 438
183 264 345 426 200 291 383 475 371 462 561
237 347 459 569 206 300 404 486 387 478 587
2.56 3.80 5.05 6.30 0.73 1.56 2.40 3.20 0.79 1.32 1.85
2.00 3.18 4.35 5.51 0.57 1.27 1.96 2.71 0.66 1.10 1.53
2.18 3.48 4.78 6.07 0.65 1.48 2.31 3.20 0.80 1.34 1.88
2.71 4.30 5.88 7.44 0.67 1.52 2.36 3.22 0.85 1.40 1.96
2.17 4.47 6.80 9.14 0.95 2.77 4.61
124 199 274 349 274 348 423
124 205 286 366 254 309 362
140 242 346 451 270 367 465
144 245 349 451 282 392 500
1.08 2.18 3.28 4.38 0.24 0.60 0.95
1.22 2.58 3.89 5.19 0.18 0.52 0.87
1.33 2.84 4.31 5.77 0.21 0.63 1.07
1.35 2.86 4.32 5.77 0.23 0.68 1.14
2.55 4.79 7.11 9.39 2.04 4.47 6.93
124 199 274 349 199 274 349
125 193 263 333 178 245 310
145 236 331 425 210 321 437
149 240 334 428 223 343 466
1.00 2.05 3.11 4.17 0.49 1.07 1.65
1.22 2.29 3.38 4.45 0.48 1.06 1.64
1.33 2.51 3.73 4.93 0.56 1.25 1.94
1.35 2.53 3.75 4.94 0.59 1.30 2.01
2.77 5.10 4.78 9.83 2.27 4.66 7.06
124 199 274 349 199 274 349
122 187 253 319 181 243 305
148 240 336 433 225 345 470
154 248 344 439 246 381 516
1.05 2.10 3.15 4.20 0.49 1.01 1.52
1.17 2.15 3.15 4.12 0.49 1.01 1.53
1.28 2.37 3.48 4.57 0.58 1.20 1.83
1.30 2.40 3.51 4.60 0.61 1.26 1.91
2.02 4.10 6.15 1.17 2.65 4.16
100 149 199 100 149 199
77 117 158 88 113 138
108 191 279 123 206 298
116 203 290 147 251 363
0.62 1.40 2.23 0.25 0.57 0.91
0.66 1.33 1.99 0.26 0.59 0.93
0.69 1.39 2.09 0.28 0.64 1.02
0.69 1.40 2.10 0.29 0.67 1.05
1.49 2.99 4.49
100 149 199
94 119 144
140 226 320
169 277 388
0.30 0.61 0.92
0.33 0.67 1.00
0.36 0.73 1.10
0.37 0.75 1.13
Ug: Superficial gas velocity (mlsec) Fs:Solidmass flow rate(kg/set) AP: Pressuredrop (N/m*) MS: Solidholdup (gr)
Appl.
Math.
Modelling,
1994,
Vol. 18, June
315
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
AP error(%) ug (m/set)
Solid Al
5.2 2 3 4 5 6 7 8 9 10 11
9.0
12.3
MS error(%)
(kg:Zec)
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93 Maximum error Errorsmean value
-16.6% -15.0% -13.8% -13.0% -8.0% -9.9% -11.5% -12.1% -7.4% -10.6% -12.0% 16.6% 11.8%
-8.0% -3.6% -1.1% 0.7% 0.5% 6.2% 9.7% 12.3% 6.3% 9.2% 12.7% 12.7% 6.4%
19.1% 26.6% 31.5% 34.5% 3.5% 9.5% 15.8% 14.9% 10.9% 13.0% 17.9% 34.5% 17.9%
-21.9% -16.3% -13.9% -12.5% -21.9% -18.6% -18.3% -15.3% -16.5% -16.7% -17.3% 21.9% 17.2%
-14.8% -8.4% -5.3% -3.7% -11.0% -5.1% -3.7% 0.0% 1.3% 1.5% 1.6% 14.8% 5.1%
5.9% 13.2% 16.4% 18.1% -8.2% -2.6% -1.7% 0.6% 7.6% 6.1% 5.9% 18.1% 7.8%
AP error(%) ug (m/set)
Solid B
12 13 14 15 16 17 18
5.0
12.2
(kg:Zec)
Run(l)
2.17 4.47 6.80 9.14 0.95 2.77 4.61 Maximum error Errorsmean value
0.0% 3.0% 4.4% 4.9% -7.3% -11.2% -14.4% 14.4% 6.5%
MS error(%)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
12.9% 21.6% 26.3% 29.2% -1.5% 5.5% 9.9% 29.2% 15.3%
16.1% 23.1% 27.4% 29.2% 2.9% 12.6% 18.2% 29.2% 18.5%
13.0% 18.3% 18.6% 18.5% -25.0% -13.3% -8.4% 25.0% 16.5%
23.1% 30.3% 31.4% 31.7% -12.5% 5.0% 12.6% 31.7% 21.0%
25.0% 31.2% 31.7% 31.7% -4.2% 13.3% 20.0% 31.7% 22.4%
MS error(%)
AP error(%) ug (m/set)
Solid
c
316
19 20 21 22 23 24 25
4.9
8.7
Appl.
Math.
(kg:Zec)
Run (1)
2.55 4.79 7.11 9.39 20.4 4.47 6.93 Maximum error Errorsmean value
0.8% -3.0% -4.0% -4.6% -10.6% -10.6% -11.2% 11.2% 6.4%
Modelling,
1994,
Vol. 18, June
Run (2) 16.9%
18.6% 20.8% 21.8% 5.5% 17.2% 25.2% 25.2% 18.0%
Run (3) 20.2% 20.6% 21.9% 22.6% 12.1% 25.2% 33.5% 33.5% 22.3%
Run(l) 22.0% 11.7% 8.7% 6.7% -2.0% -0.9% -0.6% 22.0% 7.5%
Run (2)
Run (3)
33.0% 22.4% 19.9% 18.2% 14.3% 16.8% 17.6% 33.0% 20.3%
35.0% 23.4% 20.6% 18.5% 20.4% 21.5% 21.8% 35.0% 23.0%
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
MS error (%)
AP error (%)
Solid D
26 27 28 29 30 31 32
ug
(m/s=) 4.8
8.6
Fs (kg/=) 2.77 5.10 4.78 9.83 2.27 4.66 7.06 Maximum error Errors mean value
Run (1) -1.6% -6.0% -7.7% -8.6% -9.0% -11.3% -12.6% 12.6% 8.1%
Run (2) 19.4%
20.6% 22.6% 24.1% 13.1% 25.9% 34.7% 34.7% 22.9%
Run (3)
Run (1)
Run (2)
Run (3)
24.2% 24.6% 25.5% 25.8% 23.6% 39.1% 47.9% 47.9% 30.1%
11.4% 2.4% 0.0% -1.9% 0.0% 0.0% 0.7% 11.4% 2.3%
21.9% 12.9% 10.5% 8.8% 18.4% 18.8% 20.4% 21.9% 15.9%
23.8% 14.3% 11.4% 9.5% 24.5% 24.8% 25.7% 25.7% 19.1%
MS error (%)
AP error (%)
Solid E
33 34 35 36 37 38
ug (m/set) 4.3
6.2
(kg::ec) 2.02 4.10 6.15 1 .17 2.65 4.16 Maximum error Errors mean value
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
-23.0% -21.5% -20.6% -12.0% - 24.2% - 30.7% 30.7% 22.0%
8.0% 28.2% 40.2% 23.0% 38.3% 49.7% 49.7% 31.2%
16.0% 36.2% 45.7% 47.0% 68.5% 82.4% 82.4% 49.3%
6.5% -5.0% -10.8% 4.0% 3.5% 2.2% 10.8% 5.3%
11.3% -0.7% -6.3% 12.0% 12.3% 12.1% 12.3% 9.1%
AP error (%)
Solid F
39 40 41
ug
and N. C. Markatos
Run (3) 11.3% 0.0% -5.8% 16.0% 17.5% 15.4% 17.5% 11 .O%
MS error (%)
Fs
Wsec)
Wg/sec)
Run (1)
Run (2)
Run (3)
Run (1)
Run (2)
Run (3)
6.2
1.49 2.99 4.49 Maximum error Errors mean value
-6.0% -20.1% -27.6% 27.6% 17.9%
40.0% 51.7% 60.8% 60.8% 50.8%
69.0% 85.9% 95.0% 95.0% 83.3%
10.0% 9.8% 8.7% 10.0% 9.5%
20.0% 19.7% 19.6% 20.0% 19.7%
23.3% 23.0% 22.8% 23.3% 23.0%
Appl.
Math.
Modelling,
1994,
Vol. 18, June
317
Modelling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos
AP (N/m2) Solid
ug Wsec)
Al
5.2 2 3 4 5 6 7 8 9 10 11
B
c
D
E
F
9.0
12.3
12 13 14 15 16 17 18
5.0
12.2
19 20 21 22 23 24 25
4.9
26 27 28 29 30 31 32
4.8
8.7
8.6
33 34 35 36 37 38
4.3
39 40 41
6.2
6.2
Ug: Superficial
gas velocity
Ms (gr)
(kg:Zec)
Exp.
Run (4)
Run (5)
Run (6)
Exp.
Run (4)
Run (5)
Run (6)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93
199 274 349 423 199 274 349 423 349 423 498
164 230 294 357 182 247 309 370 323 378 438
181 260 337 414 200 290 382 474 371 462 561
245 352 459 564 207 309 410 509 395 499 612
2.56 3.80 5.05 6.30 0.73 1.56 2.40 3.20 0.79 1.32 1.85
1.97 3.11 4.22 5.31 0.56 1.27 1.95 2.68 0.66 1.10 1.52
2.15 3.40 4.63 5.83 0.65 1.47 2.29 3.16 0.80 1.34 1.87
2.85 4.34 5.78 7.18 0.67 1.51 2.32 3.18 0.84 1.39 1.93
2.17 4.47 6.80 9.14 0.95 2.77 4.61
124 199 274 349 274 348 423
127 204 283 361 254 309 362
140 240 342 444 270 367 465
145 249 353 457 281 400 518
1.08 2.18 3.28 4.38 0.24 0.60 0.95
1.21 2.55 3.83 5.09 0.18 0.52 0.87
1.32 2.80 4.23 5.64 0.21 0.63 1.07
1.31 2.76 4.14 5.49 0.27 0.66 1.11
2.55 4.79 7.11 9.39 2.04 4.47 6.93
124 199 274 349 199 274 349
124 192 262 329 179 244 310
145 234 328 422 210 321 437
153 248 346 443 229 360 494
1 .oo
2.05 3.11 4.17 0.49 1.07 1.65
1.22 2.27 3.35 4.39 0.48 1.06 1.64
1.33 2.49 3.69 4.86 0.56 1.24 1.94
1.31 2.44 3.59 4.72 0.57 1.26 1.95
2.77 5.10 4.78 9.83 2.27 4.66 7.06
124 199 274 349 199 274 349
121 186 252 316 181 243 305
148 239 334 429 225 345 470
159 258 359 459 254 400 548
1.05 2.10 3.15 4.20 0.49 1.01 1.52
1.17 2.15 3.13 4.09 0.49 1.01 1.53
1.27 2.36 3.45 4.53 0.57 1.20 1.82
1.25 2.30 3.35 4.38 0.59 1.21 1.83
2.02 4.10 6.15 1.17 2.65 4.16
100 149 199 100 149 199
77 117 157 88 113 138
108 190 276 123 205 297
124 217 309 153 266 384
0.62 1.40 2.23 0.25 0.57 0.91
0.66 1.33 1.98 0.26 0.59 0.93
0.68 1.39 2.08 0.28 0.64 1.01
0.64 1.30 1.94 0.27 0.62 0.97
1.49 2.99 4.49
100 149 199
94 119 144
140 224 317
177 292 409
0.30 0.61 0.92
0.33 0.67 1.00
0.36 0.73 1.09
0.35 0.70 1.05
Fs: Solid mess flow rate (kg/set)
(m/set)
AP: Pressure drop (N/m*)
MS: Solid holdup (gr)
MS error (%)
AP error (%) ug Solid
Wsec)
Al
5.2 2 3 4 5 6 7 8 9 10 11
318
9.0
12.3
Appl.
Math.
(kg:Eec)
Run (4)
Run (5)
Run (6)
Run (4)
Run (5)
Run (6)
2.39 3.84 5.32 6.79 1.76 3.98 6.17 8.42 3.02 4.98 6.93 Maximum error Errorsmean value
-17.6% -16.1% -15.8% -15.6% -8.5% -9.9% -11.5% -12.5% -7.4% -10.6% -12.0% 17.6% 12.5%
-9.0% -5.1% -3.4% -2.1% 0.5% 5.8% 9.5% 12.1% 6.3% 9.2% 12.7% 12.7% 6.9%
23.1% 28.5% 31.5% 33.3% 4.0% 12.8% 17.5% 20.3% 13.2% 18.0% 22.9% 33.3% 20.5%
-23.0% -18.2% -16.4% -15.7% -23.3% -18.6% -18.8% -16.3% -16.5% -16.7% -17.8% 23.3% 18.3%
-16.0% -10.5% -8.3% -7.5% -11.0% -5.8% -4.6% -1.3% 1.3% 1.5% 1.1% 16.0% 6.2%
11.3% 14.2% 14.5% 14.0% -8.2% -3.2% -3.3% -0.6% 6.3% 5.3% 4.3% 14.5% 7.8%
Modelling,
1994,
Vol. 18, June
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
MS error(%)
AP error(%) Solid B
12 13 14 15 16 17 18
ug
Wsec) 5.0
12.2
(kg:Zec)
Calc. (4)
2.17 4.47 6.80 9.14 0.95 2.77 4.61 Maximum error Errorsmean value
2.4% 2.5% 3.3% 3.4% -7.3% -11.2% -14.4% 14.4% 6.4%
Calc. (5) 12.9% 20.6% 24.8% 27.2% -1.5% 5.5% 9.9% 27.2% 14.6%
Calc. (6) 16.9%
25.1% 28.8% 30.9% 2.6% 14.9% 22.5% 30.9% 20.3%
Calc. (4)
Calc. (5)
12.0% 17.0% 16.8% 16.2% -25.0% -13.3% -8.4% 25.0% 15.5%
22.2% 28.4% 29.0% 28.8% 12.5% 5.0% 12.6% 29.0% 19.8%
AP error(%) Solid
c
19 20 21 22 23 24 25
ug
Wsec)
(kg;Zec)
Calc. (4)
Calc. (5)
4.9
2.55 4.79 7.11 9.39 2.04 4.47 6.93 Maximum error Errorsmean value
0.0% -3.5% -4.4% -5.7% -10.1% -10.9% -11.2% 11.2% 6.5%
16.9% 17.6% 19.7% 20.9% 5.5% 17.2% 25.2% 25.2% 17.6%
8.7
Solid D
26 27 28 29 30 31 32
(m/set) 4.8
8.6
(kg:lec) 2.77 5.10 4.78 9.83 2.27 4.66 7.06 Maximum error Errorsmean value
Calc. (4) -2.4% -6.5% -8.0% -9.5% -9.0% -11.3% -12.6% 12.6% 8.5%
Calc. (5) 19.4% 20.1% 21.9% 22.9% 13.1% 25.9% 34.7% 34.7% 22.6%
Calc. (6) 23.4% 24.6% 26.3% 26.9% 15.1% 31.4% 41.5% 41.5% 27.0%
Calc. (4) 22.0% 10.7% 7.7% 5.3% -2.0% -0.9% -0.6% 22.0% 7.0%
Solid E
33 34 35 36 37 38
(m/set)
(kg;Zec)
Calc. (4)
4.3
2.02 4.10 6.15 1.17 2.65 4.16 Maximum error Errorsmean value
-23.0% -21.5% -21.1% -12.0% -24.2% -30.7% 30.7% 22.1%
6.2
Calc. (5) 8.0% 27.5% 38.7% 23.0% 37.6% 49.2% 49.2% 30.7%
Appl.
21.3% 26.6% 26.2% 25.3% 12.5% 10.0% 16.8% 26.6% 19.8%
Calc. (6)
Calc. (5) 33.0% 21.5% 18.6% 16.5% 14.3% 15.9% 17.6% 33.0% 19.6%
31.0% 19.0% 15.4% 13.2% 16.3% 17.8% 18.2% 31.0% 18.7%
MS error(%) Calc. (6) 28.2% 29.6% 31.0% 31.5% 27.6% 46.0% 57.0% 57.0% 35.9%
Calc. (4)
Calc. (5)
Calc. (6)
11.4% 2.4% -0.6% -2.6% 0.0% 0.0% 0.7% 11.4% 2.5%
21.0% 12.4% 9.5% 7.9% 16.3% 18.8% 19.7% 21.0% 15.1%
19.0% 9.5% 6.3% 4.3% 20.4% 19.8% 20.4% 20.4% 14.3%
AP error(%)
ug
_
Calc. (6)
MS error(%)
BP error(%)
ug
and N. C. Markatos
MS error(%) Calc. (6) 24.0% 45.6% 55.3% 53.0% 78.5% 93.0% 93.0% 58.2%
Math.
Calc. (4) 6.5% -5.0% -11.2% 4.0% 3.5% 2.2% 11.2% 5.4%
Modelling,
1994,
Calc. (5) 9.7% -0.7% -6.7% 12.0% 12.3% 11.0% 12.3% 8.7%
Vol. 18, June
Calc. (6) 3.2% -7.1% -13.0% 8.0% 8.8% 6.6% 13.0% 7.8%
319
Modeling
of vertical pneumatic-conveying
hydrodynamics:
K. N. Theologos
and N. C. Markatos MS error (%)
AP error (%) ug (m/set)
Solid F
39 40 41
(kg:Zec)
6.2
1.49 2.99 4.49 Maximum error Errors mean value
Calc. (4)
Calc. (5)
Calc. (6)
Calc. (4)
Calc. (5)
Calc. (6)
-6.0% -20.1% -27.6% 30.7% 19.0%
40.0% 50.3% 59.3% 59.3% 45.1%
77.0% 96.0% 105.5% 105.5% 84.2%
10.0% 9.8% 8.7% 11.2% 8.5%
20.0% 19.7% 18.5% 20.0% 16.7%
16.7% 14.8% 14.1% 16.7% 13.3%
7.00 G.00 ~5.00 9 ~4.00 B $
3.lxl
ti a 2.OU l.Ctl
+
0.00
150
7.00
250
300 t’~easule
Figure
350 dlotl
400
450
503
200
250
3w t’lessure
Particle holdup vs. pressure drop: run I.
B.I.
I50
(N/1112)
Figure
8.2.
350
4w
450
500
drot, (N/1112)
Particle holdup vs. pressure drop: run 4.
4.50 4.M)
4.00 Material
D 3.50
3.50 g
gj 3.00
3.00
9 -a 2.50
2 ‘c) 2.50
2 2 2.00 .L! a?G 1.50
B s 2.00 .o a!i 1.50
1.00
t.co
0.50
0.50 0.00
0.00 100
150
200 Prersul-e
Figure
8.3.
320
Appl.
250
300
Modelling,
loo
150
1994,
200 Pteswte
drot, (N/1112)
Particle holdup vs. pressure drop: run 2.
Math.
350
Vol. 18, June
Figure
8.4.
25U
300
dl-Ott (N/1112)
Particle holdup vs. pressure drop: run 4.
350