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Effect of mechanical properties of powder on pneumatic conveying in inclined pipe
Powder Technology 122 Ž2002. 150–155 www.elsevier.comrlocaterpowtec
Effect of mechanical properties of powder on pneumatic conveying in inclined pipe M. Hirota) , Y. Sogo, T. Marutani, M. Suzuki Department of Chemical Engineering, Himeji Institute of Technology, 2167 Shosha, Himeji, Hyogo, 671-2201 Japan Received 2 June 2000; received in revised form 13 September 2000; accepted 1 December 2000
Abstract Pneumatic conveying of fine powder has merits, such as no dust pollution and wide flexibility of pipeline layout. Thus, pneumatic conveying is widely used in industry. However, there is no information about the relation between the pressure drop for pneumatic conveying of fine powder and the mechanical properties of powder. We explained that the pressure drop of pneumatic conveying of powder in a horizontal pipe could be estimated from the dynamic friction coefficient of the powder in the previous paper. However, the relation between the pressure drop of pneumatic conveying in an inclined pipe and the mechanical properties of the powder is not cleared yet. The effect of mechanical properties and the angle of an inclined pipe on the pressure drop for pneumatic conveying of fine powder was examined and compared with the calculated results by our model. Based on these results, it is cleared that the pressure drop for pneumatic conveying of fine powder can be estimated from the dynamic friction coefficient of the powder and the inclined angle of the pipe. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Pneumatic conveying; Inclined pipe; Mechanical property; Dense phase; Dynamic friction
1. Introduction Nowadays, the pneumatic conveying of powder is widely used in many kinds of powder processes because of merits like almost no dust pollution and wide flexibility of pipeline layout. Widely used too are the high-density and low-velocity conveying of powder due to merits like the low energy loss, the prevention of the attrition of pipes and that it inhibits to make the fines of powder w1–3x. We hereby test the high-density and low-velocity pneumatic conveying of powder using a progressive cavity pump ŽMohno pump. for the feeder and discuss the effect of mechanical properties of powder on the conveying properties w4–6x. Based on the results, the pressure drop on the pneumatic conveying using horizontal pipe can be estimated from the mechanical properties of the powder w6x. In a real powder plant, inclined or vertical pipes are widely used; however, almost none studied the conveying properties in the pipes.
)
Corresponding author. Tel.rfax: q81-792-67-4842. E-mail address: [email protected] ŽM. Hirota..
In this paper, the effect of mechanical properties of powder and the inclined angle of pipe on the pressure drop was studied using three different kinds of test powders.
2. Experimental The schematic diagram of our experimental apparatus is shown in Fig. 1. The experimental procedure is almost the same as that in our previous paper w6x, so only the scheme of the procedure is shown in here. The test powder feeds to the transport pipe Ž6. by a progressive cavity pump Ž4. ŽHeishin Mohno pump.. The pressure in the pipe is measured by two pressure gauges Ž7., the mass flow rate of the powder and the volume flow rate of multiphase flow were measured by load cell Ž9. and orifice flow meter Ž10., respectively. As the progressive cavity pump was used only for the feeding in this experiment, the flow rate and the solid–air mixing ratio can be controlled in the limited region; a the second flow of air was fed into the transport pipe from the air inlet Ž5. located close to the outlet of the pump. A clear polyvinyl chloride pipe with 0.016-m inner diameter was
0032-5910r02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 1 . 0 0 4 1 1 - 9
M. Hirota et al.r Powder Technology 122 (2002) 150–155
151
Fig. 1. Schematic diagram of apparatus.
used as transport pipe and the tests were performed on angles 08, 208, 308, 458, 608 and 908 of an inclined pipe. Three kinds of test powder, namely, flyash, silica and soft flour, were used for our experiments. Data for 50% particle diameter, particle density and particle–particle dynamic friction factor Žcalled dynamic internal friction factor. are shown in Table 1. The dynamic internal friction factor is used for the calculation of a theoretical pressure drop per unit length of pipe. In here, the dynamic internal friction factor of a test powder is obtained from the angle of yield locus measured by powder shear test; the detail of the shear test was reported in the previous papers w7,8x.
3. Calculation method Tsuji w9x calculated the pressure drop of powder plug conveying in a horizontal pipe by the following equation: Dp L
s 2 m W Ž 1 y ´ . rp g q 4 m W Ž K W q 1 . Ccos f cos Ž g q f W . q C W 4
1
D Ž 1.
.
powder plug is conveyed in a pipe, part of the powder stays on the bottom of the pipe and the powder plug is moving on the retention layer. Furthermore, it seems that the particles in a moving plug are rotating and the dynamic friction condition is shown in a plug conveying. These phenomena relate to the pressure drop of the pipe. Therefore, we used the powder properties of the critical state line ŽCSL., obtained from particle–particle shear tests in this paper. Generally, CSL of the powder layer shows almost straight line that coincides with the origin on a s – t plane Žsee Fig. 2.. Thus, the particle–particle cohesion, C, and particle–wall cohesion, C W , are zero and Eq. Ž1. becomes: Dp s 2 mdi Ž 1 y ´ . rp g . Ž 2. L Morikawa w10x proposed that the pressure drop could be calculated using the following equation Žthe calculated value shows half the value of Eq. Ž1..: Dp s m W Ž 1 y ´ . rp g . Ž 3. L On the other hand, the friction factor, l, for solid–air two-phase flow can be calculated by the following equa-
In this calculation, the values of powder properties in Eq. Ž1. have to be a dynamic value because the plug does not remain stationary, but is moving in a pipe. When Table 1 Properties of sample powders Sample powder
50% Particle diameter wmmx
Particle density wkgrm3 x
Coefficient of dynamic internal friction w – x
Silica Flyash Soft flour
3.2 13.2 26.0
2650 2320 1400
0.84 0.60 0.76 Fig. 2. CSL of flyash.
M. Hirota et al.r Powder Technology 122 (2002) 150–155
152
The following equation can be obtained:
l s
mdi
2C1 Fr
,
Ž 6.
where C1 is a constant, i.e., 1, using Eq. Ž2., or 2, using Eq. Ž3.. Based on Eq. Ž6., the friction factor, l, for powder plug conveying in a horizontal pipe can be obtained from the dynamic internal friction factor only. On the other hand, it seems that the pressure drop of the powder conveying in an inclined pipe can be calculated as the summation of the pressure drops by the potential energy and the particle friction; thus, Eq. Ž2. and Eq. Ž3. can be rewritten as follows: Dp
ž / L
Fig. 3. Relation between Fr and l r mdi .
tion Žwhich neglects the pressure drop by gas because of high powder density in the plug flow.:
ls
2 DD p L rp Ž 1 y ´ . u 2
.
Ž 4.
Froude number, Fr, is defined as follows: Fr s
u
gD
Ž 7.
where the first term of the right side in the parenthesis shows the difference of potential energy, and the second term indicates the pressure drop caused by friction.The following equation can be derived from ŽEqs. Ž4., Ž5. and Ž7.:
ls
2 Ž sin u q C1 mdi cos u . Fr
Ž 8.
or it could be simplified as:
2
.
s Ž 1 y ´ . rp g Ž sin u q C1 mdi cos u . ,
Ž 5.
l Fr 2
s sin u q C1 mdi cos u .
Fig. 4. Effect of inclined angle on friction factor, l.
Ž 9.
M. Hirota et al.r Powder Technology 122 (2002) 150–155
153
Fig. 5. Effect of inclined angle on friction factor, l.
In this paper, we compare the experimental results with the calculated ones by Eqs. Ž8. and Ž9..
4. Results and discussion Fig. 3 shows the relation between lrmdi and Fr for the powder conveying in a horizontal pipe. The velocity, u, in Eq. Ž4. was obtained from volumetric flow rate of solid–air two-phase flow and the rate of solid volume flow, assum-
ing an air flow rate equal to a particle flow rate same as that in our previous paper w6x. In this figure, most of the experimental plots lay between the calculated lines of Eq. Ž6., whose C1 was substituted either with 1 or 2. The internal dynamic friction factor was used for the calculation. The reason why there is internal dynamic friction using for the calculation is that the powder is staying on the bottom of the pipe and the particles in a moving plug are rotating. These results were presented in the previous paper w6x; thus, the detail of this phenomenon is not shown here.
Fig. 6. Effect of powder properties on relation between lr2 Žsin u q C1 mdi cos u . and Fr.
154
M. Hirota et al.r Powder Technology 122 (2002) 150–155
powder, influences wall-friction factor, which we will also discuss. Eq. Ž8. can be transformed to the following equation:
l
1 s
2 mdi Ž C1cos u q sin u .
Fig. 7. Effect on inclines angle on relation between l r2 Žsin u qC1 mdi cos u . and Fr.
In the next stage, we compared the calculated results by Eq. Ž8. for the powder conveying in an inclined pipe with our experimental results. Fig. 4 shows the relation between friction factor, l, and Froude number, Fr, for a soft flour conveying in a inclined pipe. All experimental results lay between two calculated lines obtained by Eq. Ž8., where C1 s 1 or 2. Fig. 5 shows the comparison between calculated and experimental results for flyash conveying in an inclined pipe. In this case, all experimental results also lay between two calculated lines obtained by Eq. Ž8.. In the case of silica powder, the results are not shown in this paper, but similar results can be found. From these results, it is clear that a pressure drop coefficient for the powder conveying in an inclined pipe can be obtained from a dynamic internal friction factor and an inclined angle of pipe. Therefore, only the dynamic internal friction factor, among the various properties of
Fr
.
Ž 10 .
Based on this equation, it should be considered that there is no effect on powder properties of the plot of the experimental results on lr2 mdi Žsin u q C1cos u . y Fr coordinates. Fig. 6 shows the experimental results of each inclined angle of pipe. Usually, C1 s 1 or 2 in Eq. Ž10., but the plots in this figure are calculated using C1 s 1.5. From this figure, it is clear that all experimental data of each angle of an inclined pipe lay almost on a straight line. Thus, the effect of the powder’s mechanical properties cannot be found in the results. Fig. 7 shows the superimposed results of each inclined angle of pipe in Fig. 6. It is clear from this figure that the effect of the angle of an inclined pipe on the results cannot be recognized. Therefore, in the case of the powder conveying in an inclined pipe, pressure drop coefficient for the powder conveying in an inclined pipe can be estimated from dynamic internal friction factor and inclined angle of pipe. Fig. 8 shows the effect of the angle of an inclined pipe on the left side l Frr2 of Eq. Ž9.. Most of the experimental results lay between the calculated lines of Eq. Ž9. that was substituted 1 or 2 for C1. The value of l Frr2 takes a maximum value at 30–458 of inclined angle of pipe.
5. Conclusion We performed the experiment of the high-density and low-velocity powder conveying in a pipe and concluded the following results from the discussion about the effect of mechanical powder properties and the angle of an inclined pipe on a pressure drop coefficient. In the range of the formation of stable plug, the pressure drop coefficient in an inclined pipe can be estimated
Fig. 8. Effect of inclined angle on l Frr2
M. Hirota et al.r Powder Technology 122 (2002) 150–155
by Eq. Ž8. from dynamic internal friction factor and the inclined angle of pipe. Therefore, the value of l Frr2 change due to powderfriction factor; however, the value takes a maximum value at 30–458 of inclined angle of pipe. Nomenclature C Particle–particle cohesion ŽPa. Constant in Eqs. Ž6. – Ž10. Ž – . C1 CW Particle–wall cohesion ŽPa. D Internal diameter of pipe Žm. Fr Froude number Ž – . Gravitational acceleration Žmrs 2 . g Fr constant in Tsuji’s equation Ž – . KW L Distance between pressure gauges Žm. u Flow velocity Žmrs. Dp Pressure difference between pressure gauges ŽPa. ´ Void fraction Ž – . f Angle of internal friction Ž8. fW Angle of wall friction Ž8. g Defined angle in Tsuji’s equation Ž8. l Friction factor Ž – . mdi Dynamic internal friction factor Ž – . Dynamic wall-friction factor Ž – . mW u Inclined angle of pipe Ž8. rp Particle density Žkgrm3 .
s t
155
Normal stress ŽPa. Shear stress ŽPa.
Acknowledgements The authors are grateful to Heishin Soubi for providing a progressive cavity pump and to Nisshin Flour Milling and Nihon Taika Genryou for providing test powders throughout this research project.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
T. Kano, J. Soc. Powder Technol., Jpn. 19 Ž1982. 740. Y. Morikawa, Micromeritics, Jpn. 34 Ž1990. 66. N. Uchida, Powder Sci. Eng., Jpn. 24 Ž10. Ž1992. 44. T. Oshima, H. Eitoku, M. Hirota, M. Suzuki, J. Soc. Powder Technol., Jpn. 30 Ž1993. 646. H. Eitoku, M. Hirota, M. Suzuki, T. Oshima, Y. Iwata, T. Matsumoto, J. Soc. Powder Technol., Jpn. 32 Ž1995. 606. M. Hirota, T. Matsumoto, M. Adomi, Y. Kita, M. Suzuki, Adv. Powder Technol. 9 Ž1998. 71. M. Hirota, T. Kobayasgi, O. Sano, T. Oshima, J. Soc. Powder Technol., Jpn. 21 Ž1984. 137. M. Hirota, Oshima, Powder Technol. 53 Ž1987. 49. H. Tsuji, Kuuki yusou no kiso. Youkendo, Tokyo, 1984, p. 114. Y. Morikawa, Micromeritics, Jpn. 32 Ž1988. 38.
Effect of mechanical properties of powder on pneumatic conveying in inclined pipe M. Hirota) , Y. Sogo, T. Marutani, M. Suzuki Department of Chemical Engineering, Himeji Institute of Technology, 2167 Shosha, Himeji, Hyogo, 671-2201 Japan Received 2 June 2000; received in revised form 13 September 2000; accepted 1 December 2000
Abstract Pneumatic conveying of fine powder has merits, such as no dust pollution and wide flexibility of pipeline layout. Thus, pneumatic conveying is widely used in industry. However, there is no information about the relation between the pressure drop for pneumatic conveying of fine powder and the mechanical properties of powder. We explained that the pressure drop of pneumatic conveying of powder in a horizontal pipe could be estimated from the dynamic friction coefficient of the powder in the previous paper. However, the relation between the pressure drop of pneumatic conveying in an inclined pipe and the mechanical properties of the powder is not cleared yet. The effect of mechanical properties and the angle of an inclined pipe on the pressure drop for pneumatic conveying of fine powder was examined and compared with the calculated results by our model. Based on these results, it is cleared that the pressure drop for pneumatic conveying of fine powder can be estimated from the dynamic friction coefficient of the powder and the inclined angle of the pipe. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Pneumatic conveying; Inclined pipe; Mechanical property; Dense phase; Dynamic friction
1. Introduction Nowadays, the pneumatic conveying of powder is widely used in many kinds of powder processes because of merits like almost no dust pollution and wide flexibility of pipeline layout. Widely used too are the high-density and low-velocity conveying of powder due to merits like the low energy loss, the prevention of the attrition of pipes and that it inhibits to make the fines of powder w1–3x. We hereby test the high-density and low-velocity pneumatic conveying of powder using a progressive cavity pump ŽMohno pump. for the feeder and discuss the effect of mechanical properties of powder on the conveying properties w4–6x. Based on the results, the pressure drop on the pneumatic conveying using horizontal pipe can be estimated from the mechanical properties of the powder w6x. In a real powder plant, inclined or vertical pipes are widely used; however, almost none studied the conveying properties in the pipes.
)
Corresponding author. Tel.rfax: q81-792-67-4842. E-mail address: [email protected] ŽM. Hirota..
In this paper, the effect of mechanical properties of powder and the inclined angle of pipe on the pressure drop was studied using three different kinds of test powders.
2. Experimental The schematic diagram of our experimental apparatus is shown in Fig. 1. The experimental procedure is almost the same as that in our previous paper w6x, so only the scheme of the procedure is shown in here. The test powder feeds to the transport pipe Ž6. by a progressive cavity pump Ž4. ŽHeishin Mohno pump.. The pressure in the pipe is measured by two pressure gauges Ž7., the mass flow rate of the powder and the volume flow rate of multiphase flow were measured by load cell Ž9. and orifice flow meter Ž10., respectively. As the progressive cavity pump was used only for the feeding in this experiment, the flow rate and the solid–air mixing ratio can be controlled in the limited region; a the second flow of air was fed into the transport pipe from the air inlet Ž5. located close to the outlet of the pump. A clear polyvinyl chloride pipe with 0.016-m inner diameter was
0032-5910r02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 0 1 . 0 0 4 1 1 - 9
M. Hirota et al.r Powder Technology 122 (2002) 150–155
151
Fig. 1. Schematic diagram of apparatus.
used as transport pipe and the tests were performed on angles 08, 208, 308, 458, 608 and 908 of an inclined pipe. Three kinds of test powder, namely, flyash, silica and soft flour, were used for our experiments. Data for 50% particle diameter, particle density and particle–particle dynamic friction factor Žcalled dynamic internal friction factor. are shown in Table 1. The dynamic internal friction factor is used for the calculation of a theoretical pressure drop per unit length of pipe. In here, the dynamic internal friction factor of a test powder is obtained from the angle of yield locus measured by powder shear test; the detail of the shear test was reported in the previous papers w7,8x.
3. Calculation method Tsuji w9x calculated the pressure drop of powder plug conveying in a horizontal pipe by the following equation: Dp L
s 2 m W Ž 1 y ´ . rp g q 4 m W Ž K W q 1 . Ccos f cos Ž g q f W . q C W 4
1
D Ž 1.
.
powder plug is conveyed in a pipe, part of the powder stays on the bottom of the pipe and the powder plug is moving on the retention layer. Furthermore, it seems that the particles in a moving plug are rotating and the dynamic friction condition is shown in a plug conveying. These phenomena relate to the pressure drop of the pipe. Therefore, we used the powder properties of the critical state line ŽCSL., obtained from particle–particle shear tests in this paper. Generally, CSL of the powder layer shows almost straight line that coincides with the origin on a s – t plane Žsee Fig. 2.. Thus, the particle–particle cohesion, C, and particle–wall cohesion, C W , are zero and Eq. Ž1. becomes: Dp s 2 mdi Ž 1 y ´ . rp g . Ž 2. L Morikawa w10x proposed that the pressure drop could be calculated using the following equation Žthe calculated value shows half the value of Eq. Ž1..: Dp s m W Ž 1 y ´ . rp g . Ž 3. L On the other hand, the friction factor, l, for solid–air two-phase flow can be calculated by the following equa-
In this calculation, the values of powder properties in Eq. Ž1. have to be a dynamic value because the plug does not remain stationary, but is moving in a pipe. When Table 1 Properties of sample powders Sample powder
50% Particle diameter wmmx
Particle density wkgrm3 x
Coefficient of dynamic internal friction w – x
Silica Flyash Soft flour
3.2 13.2 26.0
2650 2320 1400
0.84 0.60 0.76 Fig. 2. CSL of flyash.
M. Hirota et al.r Powder Technology 122 (2002) 150–155
152
The following equation can be obtained:
l s
mdi
2C1 Fr
,
Ž 6.
where C1 is a constant, i.e., 1, using Eq. Ž2., or 2, using Eq. Ž3.. Based on Eq. Ž6., the friction factor, l, for powder plug conveying in a horizontal pipe can be obtained from the dynamic internal friction factor only. On the other hand, it seems that the pressure drop of the powder conveying in an inclined pipe can be calculated as the summation of the pressure drops by the potential energy and the particle friction; thus, Eq. Ž2. and Eq. Ž3. can be rewritten as follows: Dp
ž / L
Fig. 3. Relation between Fr and l r mdi .
tion Žwhich neglects the pressure drop by gas because of high powder density in the plug flow.:
ls
2 DD p L rp Ž 1 y ´ . u 2
.
Ž 4.
Froude number, Fr, is defined as follows: Fr s
u
gD
Ž 7.
where the first term of the right side in the parenthesis shows the difference of potential energy, and the second term indicates the pressure drop caused by friction.The following equation can be derived from ŽEqs. Ž4., Ž5. and Ž7.:
ls
2 Ž sin u q C1 mdi cos u . Fr
Ž 8.
or it could be simplified as:
2
.
s Ž 1 y ´ . rp g Ž sin u q C1 mdi cos u . ,
Ž 5.
l Fr 2
s sin u q C1 mdi cos u .
Fig. 4. Effect of inclined angle on friction factor, l.
Ž 9.
M. Hirota et al.r Powder Technology 122 (2002) 150–155
153
Fig. 5. Effect of inclined angle on friction factor, l.
In this paper, we compare the experimental results with the calculated ones by Eqs. Ž8. and Ž9..
4. Results and discussion Fig. 3 shows the relation between lrmdi and Fr for the powder conveying in a horizontal pipe. The velocity, u, in Eq. Ž4. was obtained from volumetric flow rate of solid–air two-phase flow and the rate of solid volume flow, assum-
ing an air flow rate equal to a particle flow rate same as that in our previous paper w6x. In this figure, most of the experimental plots lay between the calculated lines of Eq. Ž6., whose C1 was substituted either with 1 or 2. The internal dynamic friction factor was used for the calculation. The reason why there is internal dynamic friction using for the calculation is that the powder is staying on the bottom of the pipe and the particles in a moving plug are rotating. These results were presented in the previous paper w6x; thus, the detail of this phenomenon is not shown here.
Fig. 6. Effect of powder properties on relation between lr2 Žsin u q C1 mdi cos u . and Fr.
154
M. Hirota et al.r Powder Technology 122 (2002) 150–155
powder, influences wall-friction factor, which we will also discuss. Eq. Ž8. can be transformed to the following equation:
l
1 s
2 mdi Ž C1cos u q sin u .
Fig. 7. Effect on inclines angle on relation between l r2 Žsin u qC1 mdi cos u . and Fr.
In the next stage, we compared the calculated results by Eq. Ž8. for the powder conveying in an inclined pipe with our experimental results. Fig. 4 shows the relation between friction factor, l, and Froude number, Fr, for a soft flour conveying in a inclined pipe. All experimental results lay between two calculated lines obtained by Eq. Ž8., where C1 s 1 or 2. Fig. 5 shows the comparison between calculated and experimental results for flyash conveying in an inclined pipe. In this case, all experimental results also lay between two calculated lines obtained by Eq. Ž8.. In the case of silica powder, the results are not shown in this paper, but similar results can be found. From these results, it is clear that a pressure drop coefficient for the powder conveying in an inclined pipe can be obtained from a dynamic internal friction factor and an inclined angle of pipe. Therefore, only the dynamic internal friction factor, among the various properties of
Fr
.
Ž 10 .
Based on this equation, it should be considered that there is no effect on powder properties of the plot of the experimental results on lr2 mdi Žsin u q C1cos u . y Fr coordinates. Fig. 6 shows the experimental results of each inclined angle of pipe. Usually, C1 s 1 or 2 in Eq. Ž10., but the plots in this figure are calculated using C1 s 1.5. From this figure, it is clear that all experimental data of each angle of an inclined pipe lay almost on a straight line. Thus, the effect of the powder’s mechanical properties cannot be found in the results. Fig. 7 shows the superimposed results of each inclined angle of pipe in Fig. 6. It is clear from this figure that the effect of the angle of an inclined pipe on the results cannot be recognized. Therefore, in the case of the powder conveying in an inclined pipe, pressure drop coefficient for the powder conveying in an inclined pipe can be estimated from dynamic internal friction factor and inclined angle of pipe. Fig. 8 shows the effect of the angle of an inclined pipe on the left side l Frr2 of Eq. Ž9.. Most of the experimental results lay between the calculated lines of Eq. Ž9. that was substituted 1 or 2 for C1. The value of l Frr2 takes a maximum value at 30–458 of inclined angle of pipe.
5. Conclusion We performed the experiment of the high-density and low-velocity powder conveying in a pipe and concluded the following results from the discussion about the effect of mechanical powder properties and the angle of an inclined pipe on a pressure drop coefficient. In the range of the formation of stable plug, the pressure drop coefficient in an inclined pipe can be estimated
Fig. 8. Effect of inclined angle on l Frr2
M. Hirota et al.r Powder Technology 122 (2002) 150–155
by Eq. Ž8. from dynamic internal friction factor and the inclined angle of pipe. Therefore, the value of l Frr2 change due to powderfriction factor; however, the value takes a maximum value at 30–458 of inclined angle of pipe. Nomenclature C Particle–particle cohesion ŽPa. Constant in Eqs. Ž6. – Ž10. Ž – . C1 CW Particle–wall cohesion ŽPa. D Internal diameter of pipe Žm. Fr Froude number Ž – . Gravitational acceleration Žmrs 2 . g Fr constant in Tsuji’s equation Ž – . KW L Distance between pressure gauges Žm. u Flow velocity Žmrs. Dp Pressure difference between pressure gauges ŽPa. ´ Void fraction Ž – . f Angle of internal friction Ž8. fW Angle of wall friction Ž8. g Defined angle in Tsuji’s equation Ž8. l Friction factor Ž – . mdi Dynamic internal friction factor Ž – . Dynamic wall-friction factor Ž – . mW u Inclined angle of pipe Ž8. rp Particle density Žkgrm3 .
s t
155
Normal stress ŽPa. Shear stress ŽPa.
Acknowledgements The authors are grateful to Heishin Soubi for providing a progressive cavity pump and to Nisshin Flour Milling and Nihon Taika Genryou for providing test powders throughout this research project.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
T. Kano, J. Soc. Powder Technol., Jpn. 19 Ž1982. 740. Y. Morikawa, Micromeritics, Jpn. 34 Ž1990. 66. N. Uchida, Powder Sci. Eng., Jpn. 24 Ž10. Ž1992. 44. T. Oshima, H. Eitoku, M. Hirota, M. Suzuki, J. Soc. Powder Technol., Jpn. 30 Ž1993. 646. H. Eitoku, M. Hirota, M. Suzuki, T. Oshima, Y. Iwata, T. Matsumoto, J. Soc. Powder Technol., Jpn. 32 Ž1995. 606. M. Hirota, T. Matsumoto, M. Adomi, Y. Kita, M. Suzuki, Adv. Powder Technol. 9 Ž1998. 71. M. Hirota, T. Kobayasgi, O. Sano, T. Oshima, J. Soc. Powder Technol., Jpn. 21 Ž1984. 137. M. Hirota, Oshima, Powder Technol. 53 Ž1987. 49. H. Tsuji, Kuuki yusou no kiso. Youkendo, Tokyo, 1984, p. 114. Y. Morikawa, Micromeritics, Jpn. 32 Ž1988. 38.