- Email: [email protected]
An investigation of the flow structure through abrupt enlargement of circular pipe
Powder Technology 104 Ž1999. 296–303 www.elsevier.comrlocaterpowtec
An investigation of the flow structure through abrupt enlargement of circular pipe Predrag Marjanovic a
a,)
, Avi Levy b, David J. Mason
a
Centre for Industrial Bulk Solids Handling, Department of Physical Sciences, Glasgow Caledonian UniÕersity, Cowcaddens Road, Glasgow G4 0BA, UK b Department of Mechanical Engineering, Ben-Gurion UniÕersity of NegeÕ, P.O. Box 653, Beer-SheÕa 84105, Israel
Abstract There are many examples of flow of gas, with or without solids, where an abrupt change of cross-sectional area has to be implemented. A typical example is long distance pneumatic conveying where single bore pipeline normally requires high pressure drop which consequently results in high gas and solids velocities towards the end of pipeline. In those cases, stepped pipeline is used in order to reduce both gas and solids velocity whenever they reach unacceptable level. The flow structure through an abrupt enlargement of a conduit can be very complex, specially in the case of gas–solids flow. Both velocity and pressure fields are very important for the analysis and design of the overall flow characteristics. The compressible flow of gas through an abrupt enlargement was modelled using both a 1-D analytical model and 3-D numerical model. A summary of the analytical 1-D model of compressible flow through an abrupt enlargement and the results obtained for the flow through stepped pipe Ž81 to 105 mm. for several flow conditions are presented in this paper. Those results have been compared with the 3-D numerical model which solves basic conservation equations and very good agreement was found. Finally the same 3-D numerical model was applied to several regimes of gas–solids flow in order to investigate the structure of gas and solids velocity field, as well as pressure distribution along the pipe. Special attention has been paid to the pressure recovery downstream from the enlargement due to its important role in the design of stepped pipeline pneumatic conveying system. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Pneumatic conveying; Pipe enlargement; Pressure drop; Flow structure; Stepped pipe
1. Introduction Flow through confined conduits, when it is possible to form a cross-section which is approximately perpendicular to the streamlines, is a typical example where velocity component in one direction is predominant over the others. At the same time, as non-uniformity of a flow parameter profile is not too great it is possible to introduce, using 1-D approach, the average values of all flow parameters and then substitute the real parameter profiles in a cross-section with the constant ones. This 1-D analysis gives some valuable results, sometimes even highly accurate. However, a 1-D character of a flow in conduit can be disrupted in different ways. The most frequent cause is the change of either the direction of the flow or the cross-sec-
) Corresponding author. Tel.: q44-141-331-3741; fax: q44-141-3313448; E-mail: [email protected]
tional area of conduit. In both cases it is very likely that detachment from the wall will occur and turbulent eddies will form. Typical examples of such flows are stepped pneumatic conveying pipes and bends. 1-D analysis takes the presence of the areas with disrupted 1-D character globally, by introducing some additional terms into the basic equations which are highly dependent upon the 1-D character of the flow. These are the conservation of momentum and energy; the equations of continuity and state are not influenced due to the way in which they had been established. Using this approach the analytical model for single phase compressible flow through abrupt enlargement has already been developed by Marjanovic and Djordjevic w1x. The first logic step in the analysis of this flow structure is the analysis of gas only flow. In most practical applications of the flow in pipes, 1-D model of flow gives sufficient accuracy of the results, providing that few assumptions which have to be made are correctly introduced.
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 0 7 - 2
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
The next step is to add the interactive influence of solid particles which significantly complicates the flow structure.
297
area, A o is the outlet pipe area., Mi is Mach number at the step inlet, and g is the ratio of specific heats Žg s c prc v .. 3. Gas–solids flow
2. Gas only flow The 1-D model of subsonic compressible flow of gas through abrupt enlargements and contractions was analysed and solved analytically for both isothermal and adiabatic conditions; detailed analysis and results are given in Ref. w1x. The basic assumption used relates to the accurate evaluation of the reaction force term in the momentum equation, i.e., the force which the fluid exerts on the pipe walls. For the flow through an abrupt enlargement ŽFig. 1., the reaction force consists mainly of the pressure force exerted on the annular area. Gas pressure over the annular area is approximately uniform and equal to the pressure in upstream station ‘i’. Therefore, the following set of equations was used to define gas flow through an abrupt enlargement Žsubscript ‘i’ relates to the inlet and ‘o’ to the outlet of the enlargement.. Conservation of mass: m ˙ g s r i u i A i s ro u o A o
Ž 1.
Conservation of momentum: ypi Ž A o y A i . s p i A i y po A o q m ˙ g Ž ui y uo . Ideal gas law: pi po s sR r i Ti ro To
Ž 2.
Ž 3.
The results for two major flow parameters Žvelocity and pressure. are presented as: uo ui
1 s
Ž g q 1 . Mi2
Ž g Mi2 q A .
(
pi
s 1q
gy1 2
Mi2 Ž 1 y Fu2 .
E Ž ri ´ i fi .
q = Ž ri ´ i™ u i f i . s = Ž Gf i ´ i= Ž f i . . q Sf i Ž 6 . Et where t is the evolution time, ´ i , r i and u i are the volume fraction, the density and the velocity of phase i, f i is the conserved property of phase i such as enthalpy, momentum per unit mass, etc., and Gf i and Sf i are the diffusion coefficient and source term of f i . Since the steady state solution was considered in this paper, the transient term in Eq. Ž6. was set to zero. Only basic substitutions necessary to derive the mass, momentum and energy equations from the general equation are presented in Table 1. Details of the source terms used to model mass, momentum and energy exchange between phases, and each phase and the pipe wall are given by Levy et al. w3x.
4. Results
y Mi4 q 2 Ž g A y g y 1 . Mi2 q A2 s Fu po
A 3-D model based upon the concept of interdispersed continua was used to model this flow through an abrupt enlargement. This model solves the conservation equations for mass and momentum for the gas and solid phases by using a finite-volume numerical method. The 3-D model employed in this study is incorporated in the PHOENICS software by CHAM, UK. The general models and their solutions are described by Spalding w2x. This model uses a finite-volume formulation of the conservation equations for mass and momentum for two phases. PHOENICS provides solutions to the discretised version of a set of differential equations having the following general form for the i th phase of a multi-phase flow:
1 Fu A
Ž 4. Ž 5.
where A s Ž A orA i . ) 1 is area ratio Ž A i is the inlet pipe
The major difference between the outcome of 1-D analytical approach and the numerical study is that the former gives the value of all flow parameters at the inlet of the enlargement and at the downstream section where the gas flow is reattached to the pipe wall Žsee Fig. 1., whereas the latter gives 3-D results. In order to compare those two sets of results, the numerical study 3-D results were translated to 1-D results using the standard averaging transformation:
fs
Fig. 1. Flow through an abrupt enlargement.
Ýf x y A x y Ý Ax y
Ž 7.
Parameter f is any flow variable Že.g., gas velocity and pressure across the pipe cross-sectional area.. However, the numerical study does give 3-D distribution of velocity and, therefore, offers more detailed ‘insight’ into the structure of gas–solids flow. That beneficiary information is also presented in this paper.
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
298
Table 1 The meaning of terms in the general single-phase conservation equation for particular variables Conservation equation
Variable f i
Diffusion coefficient, Gf i
Source term, Sf i
Mass
1 w MrM x
0
Ø Mass transfer between the i th phase and the other phases w MrL3 T x
Momentum
u w MLrTrM x
m w MrLT x
Ø Ø Ø Ø Ø
Pressure gradient Body forces Žgravity, buoyancy, etc.. Other viscous terms Žthat cannot be represented in the general form. Momentum transfer between the i th phase and the other phases Momentum transfer due to mass transfer w MLrL3 T 2 x
Energy
H s h q 1r2 u 2 w ErM x
krc p s mrPr w MrLT x
Ø Ø Ø Ø
Volumetric rate of heat generation Ždue to burning, external heating. Work done Žmechanical, gravitational, etc.. Energy transfer between the i th phase and the other phases Energy transfer due to mass transfer w ML2rL3 T 3 x
4.1. Gas and material properties The following gas and material properties were chosen to be analysed with 1-D analytical and 3-D numerical approaches. 4.1.1. Gas properties Ø Gas Ø Characteristic gas constant Ø Ratio of specific heats
Air R s 287 Jrkg K g s c prc v s 1.4
4.1.2. Material properties Ø Material Ø Particle diameter Ø Particle density
Polyethylene pellets d s 3 mm rs s 880 kgrm3
4.1.3. Pipe geometry and flow conditions The following pipe geometry Žsee Fig. 1. and flow conditions were used in this study: Ø Pipeline bores
Ø Pipeline lengths Ø Temperature Ø Solids loading ratio
Ø Gas flow rate
stepped from D i s 81 mm to Do s 105 mm Žarea ratio A s A orA i s 1.68. Li s Lo s 6 m T s 293 K s constant Five values were used in the analysis: SLR s m ˙ srm˙ g s 2, 4, 6, 8, 10 Five values were used in the analysis: m ˙ g s 0.08, 0.106, 0.12, 0.13, 0.14 kgrs
4.2. Gas flow analysis The individual global results obtained for single phase flow using two approaches and their comparison are pre-
sented in Table 2. Pressure ratio is defined as the ratio of pressures po Žat the step outlet, but at the point where the ‘dead zone’ after the step disappears. and p i Žat the step inlet.. As expected, due to relatively low gas velocity, both before and after the step, the pressure ratio is very close to unity. Nevertheless, the agreement between two sets of results is remarkably good. 4.3. Gas–solids flow analysis The calculation domain for the numerical study of this flow problem Ž6 m upstream and 6 m downstream of the enlargement. was divided into 120 axial slices, with each slice containing 25 control volumes. Using body fitted coordinates, the control volume 5 = 5 = 120 rectangular grid was generated for 3-D flow model. The axial length of the control volumes linearly decreased towards the abrupt enlargement. The body fitted routines distorted the grid in order to achieve a circular cross-section and the pipeline geometry shown in Fig. 1. The structure of gas–solids flow through an abrupt enlargement is much more complex than gas only flow due to several characteristic features. First, there are two different velocity fields — gas and solids. Second, neither gas nor solids velocity profiles are axisymmetric due to the influence of solids volume fraction distribution across the pipe Žsolids volume fraction increases towards the bottom of a pipe.. In the ideal case both gas and solids horizontal velocity profile are close to symmetric, but the vertical ones are distorted, unless both gas and solids velocities are extremely high, possibly beyond practical pneumatic conveying applications. Finally, the transition from the steady velocity profiles upstream of the step to the steady velocity profiles downstream of the step is not only influenced by the pipe geometry but also by the exchange of momentum between two phases. An illustration of the above phenomenon is given with horizontal and vertical velocity profiles in Fig. 2 for one
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
299
Table 2 Comparison between the results obtained using 1-D analytical and 3-D numerical approaches Flow regime no.
Gas flow rate, m ˙ g Žkgrs.
Gas velocity at step inlet, u z1,i Žmrs.
Mach number at step inlet, Mi
Gas velocity ratio, u z1,oru z1,i
1-D analysis approach, p1,o rp1,i
3-D numerical approach, p1,orp1,i
1 2 3 4 5
0.08 0.106 0.12 0.13 0.14
13.28 17.60 19.92 21.59 23.25
0.0387 0.0513 0.0581 0.0629 0.0678
0.5949 0.5948 0.5947 0.5946 0.5945
1.00051 1.00089 1.00114 1.00134 1.00155
1.00051 1.00091 1.00117 1.00138 1.00161
set of flow conditions Žgas flow rate m ˙ g s 0.106 kgrs and SLR s 6.. Various symbols represent calculated numerical results; the velocity profile lines are drawn simply by using ‘smoothed line’ option in a spreadsheet program as no further detailed analysis of these profiles was pursued. Both profiles confirm the features explained above. They also indicate the downstream pipe section where the momentum exchange from solids to gas phase has been
Pressure ratio
Relative error Ž%.
0.0003 0.0020 0.0034 0.0045 0.0058
finished, i.e., when solids velocity equates gas velocity. This has been also confirmed in the following analysis of pressure distribution along the pipe. The gas and solids velocity results for each of 25 control volumes in one cross-sectional area of pipe were averaged Žusing Eq. Ž7.. and these results are presented in Fig. 3 for 7 m pipe length Ž1 m upstream and 6 m downstream from the step.. These results were obtained
Fig. 2. Horizontal and vertical gas and solids velocity profiles before and after the pipe enlargement Žthe step is located at z s 6 m..
300
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
Fig. 3. Average gas and solids velocity distribution for the flow through an abrupt enlargement for different flow regimes.
for SLR s 4 and 10 and five different gas flow rates, as indicated in Fig. 3. Although this transformation to 1-D presentation inevitably distorted the real flow structure image, specially gas velocity and immediately after the step, it is still possible to estimate the length of pipe over which the momentum exchange from solids to gas phase is taking place. This length increases with the increase of both gas flow rate and solids loading ratio. More detailed analysis on this length is given below when pressure distribution is scrutinised. The initial numerical calculation of pressure distribution along the pipe was carried out with the ambient conditions at pipe outlet. However, as the pressure recovery at the step was the major aim of this investigation, those results were modified in order to get identical pressure at the step inlet for each given gas flow rate. The pressure at the step inlet obtained for lowest gas flow rate was chosen as the reference value for all flow regimes. Gas pressure data which were calculated along the pipeline for all other flow regimes were then adjusted by adding Žor subtracting. the
difference between the calculated value for given flow regime and the chosen reference value. In that way the pressure gradients remained unchanged, the comparison of pressure recovery data was much easier and the graphical presentation of the results is much clearer. The numerical results for gas pressure distribution along the same pipe section, modified as explained above, and for the same combination of flow conditions are presented in Fig. 4. The major features of the structure of gas–solids flow through an abrupt enlargement are as follows. Ø Pressure gradient for the fully developed flow both before after the step is directly proportional to both gas flow rate and solids loading ratio. Ø The ratio of pressure gradients before and after the step for the same flow conditions can be estimated for turbulent flow regime and for gas only flow as Ž DorDi . 4.75 s 3.3. Depending on gas flow rate, this ratio varies between 2.8 and 4.0 Žfor SLR s 4. and between 3.8 and 5.7 Žfor SLR s 10.. Qualitatively, this agrees with the expected results.
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
301
Fig. 4. Pressure distribution for the flow through an abrupt enlargement for different flow regimes.
Ø Gas pressure rises after the step as the kinetic energy transforms into the potential energy. This increase is higher than the theoretical value for gas only flow due to addi-
tional momentum transfer from solid particles which are flowing at higher velocity along the transition zone after the step.
Fig. 5. Ratio of pressures at the outlet Žfully developed gas–solids flow. and inlet of an abrupt enlargement for different flow regimes.
302
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
Fig. 6. Transition length from an abrupt enlargement to the point where gas–solids flow is fully developed.
Ø Gas pressure reaches its maximum at the point when gas and solids velocity equalise, i.e., where momentum transfer from solids to gas phase finishes. This is the point when the average gas velocity reaches its minimum value. However, solid particles decelerate further downstream and gas–solids flow fully develops when slip velocity reaches its constant value. The ratio of pressures at the point when gas–solids flow fully develops Ž pout . and at the inlet to the enlargement Ž p in . is presented in Fig. 5. The calculated data are represented by symbols; the straight lines are determined by a simple linear regression for illustrative purposes. Ø Consequently, two transition lengths can be identified: Ža. From the step to the point where gas pressure reaches its maximum and average gas velocity its minimum; this value varies from about 1 m Žat low gas flow rate and SLR. to about 3 m Žat high gas flow rate., i.e., non-dimensional length Ž LtrrDo . varies from 10 to 30. Žb. From the step to the point where both pressure gradient and slip velocity reach constant values; this length is obviously larger than the previous one and it varies from about 2 m Žat low gas flow rate and SLR. to about 3.5 m Žat high gas flow rate., i.e., non-dimensional length Ž LtrrDo . varies from 19 to 39 Žsee Fig. 6.. Similarly as in Fig. 5, the calculated data are represented by symbols and the straight lines are determined by a simple linear regression for illustrative purposes. Further investigation, analytical, numerical and experimental, on the conditions and flow structure which develop when gas–solids mixture flow through an abrupt enlargement is required due to its importance for the design of stepped pneumatic conveying pipeline. The flow conditions at the step have been attracting considerable interest both from the researchers and designers. We hope that
further analysis of the presented approach will contribute to reliable determination of the conditions at which pipeline blockage at the step can be avoided.
5. Conclusions The results presented in this paper are the initial considerations and analysis of the flow structure which develops when gas–solids mixture flows through an abrupt enlargement. They include the analysis of both velocity and pressure fields which are highly critical for a reliable stepped pipeline design of a pneumatic conveying system. The results indicate possible influences of various flow parameters which had been analysed in this study Žgas flow rate and solids loading ratio. on the pressure recovery after an abrupt enlargement. More work is planned to widen the range of flow parameters in the analysis, to pay more attention on the transverse velocity profiles, to model all those results, apply them to the design of stepped pipelines and finally to verify them by structured pneumatic conveying test trials.
6. List of symbols A d D L m ˙ M p R S
Pipe area Particle diameter Pipe bore Pipe length Mass flow rate Mach number Pressure Characteristic gas constant Source term
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
T Temperature u Velocity ´ Volume fraction f Variable g Ratio of specific heats Žs c prc v . G Diffusion coefficient r Density Subscripts i inlet to abrupt enlargement o outlet to abrupt enlargement 1 phase 1 Žgas. 2 phase 2 Žsolids.
303
References w1x P. Marjanovic, V. Djordjevic, On the compressible flow losses through abrupt enlargements and contractions, J. Fluids Eng., Trans. ASME 116 Ž1994. 756–762. w2x D.B. Spalding, Numerical computation of multi-phase fluid flow and heat transfer, in: C. Taylor, K. Morgan ŽEds.., Recent Advances in Numerical Methods, Vol. 1, Chap. 5, Pineridge Press, Swansea, UK, 1980, pp. 139–167. w3x A. Levy, T. Mooney, P. Marjanovic, D.J. Mason, A comparison of analytical and numerical models for gas–solid flow through straight pipe of different inclinations with experimental data, Powder Technol. 93 Ž1997. 253–260.
An investigation of the flow structure through abrupt enlargement of circular pipe Predrag Marjanovic a
a,)
, Avi Levy b, David J. Mason
a
Centre for Industrial Bulk Solids Handling, Department of Physical Sciences, Glasgow Caledonian UniÕersity, Cowcaddens Road, Glasgow G4 0BA, UK b Department of Mechanical Engineering, Ben-Gurion UniÕersity of NegeÕ, P.O. Box 653, Beer-SheÕa 84105, Israel
Abstract There are many examples of flow of gas, with or without solids, where an abrupt change of cross-sectional area has to be implemented. A typical example is long distance pneumatic conveying where single bore pipeline normally requires high pressure drop which consequently results in high gas and solids velocities towards the end of pipeline. In those cases, stepped pipeline is used in order to reduce both gas and solids velocity whenever they reach unacceptable level. The flow structure through an abrupt enlargement of a conduit can be very complex, specially in the case of gas–solids flow. Both velocity and pressure fields are very important for the analysis and design of the overall flow characteristics. The compressible flow of gas through an abrupt enlargement was modelled using both a 1-D analytical model and 3-D numerical model. A summary of the analytical 1-D model of compressible flow through an abrupt enlargement and the results obtained for the flow through stepped pipe Ž81 to 105 mm. for several flow conditions are presented in this paper. Those results have been compared with the 3-D numerical model which solves basic conservation equations and very good agreement was found. Finally the same 3-D numerical model was applied to several regimes of gas–solids flow in order to investigate the structure of gas and solids velocity field, as well as pressure distribution along the pipe. Special attention has been paid to the pressure recovery downstream from the enlargement due to its important role in the design of stepped pipeline pneumatic conveying system. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Pneumatic conveying; Pipe enlargement; Pressure drop; Flow structure; Stepped pipe
1. Introduction Flow through confined conduits, when it is possible to form a cross-section which is approximately perpendicular to the streamlines, is a typical example where velocity component in one direction is predominant over the others. At the same time, as non-uniformity of a flow parameter profile is not too great it is possible to introduce, using 1-D approach, the average values of all flow parameters and then substitute the real parameter profiles in a cross-section with the constant ones. This 1-D analysis gives some valuable results, sometimes even highly accurate. However, a 1-D character of a flow in conduit can be disrupted in different ways. The most frequent cause is the change of either the direction of the flow or the cross-sec-
) Corresponding author. Tel.: q44-141-331-3741; fax: q44-141-3313448; E-mail: [email protected]
tional area of conduit. In both cases it is very likely that detachment from the wall will occur and turbulent eddies will form. Typical examples of such flows are stepped pneumatic conveying pipes and bends. 1-D analysis takes the presence of the areas with disrupted 1-D character globally, by introducing some additional terms into the basic equations which are highly dependent upon the 1-D character of the flow. These are the conservation of momentum and energy; the equations of continuity and state are not influenced due to the way in which they had been established. Using this approach the analytical model for single phase compressible flow through abrupt enlargement has already been developed by Marjanovic and Djordjevic w1x. The first logic step in the analysis of this flow structure is the analysis of gas only flow. In most practical applications of the flow in pipes, 1-D model of flow gives sufficient accuracy of the results, providing that few assumptions which have to be made are correctly introduced.
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 9 . 0 0 1 0 7 - 2
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
The next step is to add the interactive influence of solid particles which significantly complicates the flow structure.
297
area, A o is the outlet pipe area., Mi is Mach number at the step inlet, and g is the ratio of specific heats Žg s c prc v .. 3. Gas–solids flow
2. Gas only flow The 1-D model of subsonic compressible flow of gas through abrupt enlargements and contractions was analysed and solved analytically for both isothermal and adiabatic conditions; detailed analysis and results are given in Ref. w1x. The basic assumption used relates to the accurate evaluation of the reaction force term in the momentum equation, i.e., the force which the fluid exerts on the pipe walls. For the flow through an abrupt enlargement ŽFig. 1., the reaction force consists mainly of the pressure force exerted on the annular area. Gas pressure over the annular area is approximately uniform and equal to the pressure in upstream station ‘i’. Therefore, the following set of equations was used to define gas flow through an abrupt enlargement Žsubscript ‘i’ relates to the inlet and ‘o’ to the outlet of the enlargement.. Conservation of mass: m ˙ g s r i u i A i s ro u o A o
Ž 1.
Conservation of momentum: ypi Ž A o y A i . s p i A i y po A o q m ˙ g Ž ui y uo . Ideal gas law: pi po s sR r i Ti ro To
Ž 2.
Ž 3.
The results for two major flow parameters Žvelocity and pressure. are presented as: uo ui
1 s
Ž g q 1 . Mi2
Ž g Mi2 q A .
(
pi
s 1q
gy1 2
Mi2 Ž 1 y Fu2 .
E Ž ri ´ i fi .
q = Ž ri ´ i™ u i f i . s = Ž Gf i ´ i= Ž f i . . q Sf i Ž 6 . Et where t is the evolution time, ´ i , r i and u i are the volume fraction, the density and the velocity of phase i, f i is the conserved property of phase i such as enthalpy, momentum per unit mass, etc., and Gf i and Sf i are the diffusion coefficient and source term of f i . Since the steady state solution was considered in this paper, the transient term in Eq. Ž6. was set to zero. Only basic substitutions necessary to derive the mass, momentum and energy equations from the general equation are presented in Table 1. Details of the source terms used to model mass, momentum and energy exchange between phases, and each phase and the pipe wall are given by Levy et al. w3x.
4. Results
y Mi4 q 2 Ž g A y g y 1 . Mi2 q A2 s Fu po
A 3-D model based upon the concept of interdispersed continua was used to model this flow through an abrupt enlargement. This model solves the conservation equations for mass and momentum for the gas and solid phases by using a finite-volume numerical method. The 3-D model employed in this study is incorporated in the PHOENICS software by CHAM, UK. The general models and their solutions are described by Spalding w2x. This model uses a finite-volume formulation of the conservation equations for mass and momentum for two phases. PHOENICS provides solutions to the discretised version of a set of differential equations having the following general form for the i th phase of a multi-phase flow:
1 Fu A
Ž 4. Ž 5.
where A s Ž A orA i . ) 1 is area ratio Ž A i is the inlet pipe
The major difference between the outcome of 1-D analytical approach and the numerical study is that the former gives the value of all flow parameters at the inlet of the enlargement and at the downstream section where the gas flow is reattached to the pipe wall Žsee Fig. 1., whereas the latter gives 3-D results. In order to compare those two sets of results, the numerical study 3-D results were translated to 1-D results using the standard averaging transformation:
fs
Fig. 1. Flow through an abrupt enlargement.
Ýf x y A x y Ý Ax y
Ž 7.
Parameter f is any flow variable Že.g., gas velocity and pressure across the pipe cross-sectional area.. However, the numerical study does give 3-D distribution of velocity and, therefore, offers more detailed ‘insight’ into the structure of gas–solids flow. That beneficiary information is also presented in this paper.
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
298
Table 1 The meaning of terms in the general single-phase conservation equation for particular variables Conservation equation
Variable f i
Diffusion coefficient, Gf i
Source term, Sf i
Mass
1 w MrM x
0
Ø Mass transfer between the i th phase and the other phases w MrL3 T x
Momentum
u w MLrTrM x
m w MrLT x
Ø Ø Ø Ø Ø
Pressure gradient Body forces Žgravity, buoyancy, etc.. Other viscous terms Žthat cannot be represented in the general form. Momentum transfer between the i th phase and the other phases Momentum transfer due to mass transfer w MLrL3 T 2 x
Energy
H s h q 1r2 u 2 w ErM x
krc p s mrPr w MrLT x
Ø Ø Ø Ø
Volumetric rate of heat generation Ždue to burning, external heating. Work done Žmechanical, gravitational, etc.. Energy transfer between the i th phase and the other phases Energy transfer due to mass transfer w ML2rL3 T 3 x
4.1. Gas and material properties The following gas and material properties were chosen to be analysed with 1-D analytical and 3-D numerical approaches. 4.1.1. Gas properties Ø Gas Ø Characteristic gas constant Ø Ratio of specific heats
Air R s 287 Jrkg K g s c prc v s 1.4
4.1.2. Material properties Ø Material Ø Particle diameter Ø Particle density
Polyethylene pellets d s 3 mm rs s 880 kgrm3
4.1.3. Pipe geometry and flow conditions The following pipe geometry Žsee Fig. 1. and flow conditions were used in this study: Ø Pipeline bores
Ø Pipeline lengths Ø Temperature Ø Solids loading ratio
Ø Gas flow rate
stepped from D i s 81 mm to Do s 105 mm Žarea ratio A s A orA i s 1.68. Li s Lo s 6 m T s 293 K s constant Five values were used in the analysis: SLR s m ˙ srm˙ g s 2, 4, 6, 8, 10 Five values were used in the analysis: m ˙ g s 0.08, 0.106, 0.12, 0.13, 0.14 kgrs
4.2. Gas flow analysis The individual global results obtained for single phase flow using two approaches and their comparison are pre-
sented in Table 2. Pressure ratio is defined as the ratio of pressures po Žat the step outlet, but at the point where the ‘dead zone’ after the step disappears. and p i Žat the step inlet.. As expected, due to relatively low gas velocity, both before and after the step, the pressure ratio is very close to unity. Nevertheless, the agreement between two sets of results is remarkably good. 4.3. Gas–solids flow analysis The calculation domain for the numerical study of this flow problem Ž6 m upstream and 6 m downstream of the enlargement. was divided into 120 axial slices, with each slice containing 25 control volumes. Using body fitted coordinates, the control volume 5 = 5 = 120 rectangular grid was generated for 3-D flow model. The axial length of the control volumes linearly decreased towards the abrupt enlargement. The body fitted routines distorted the grid in order to achieve a circular cross-section and the pipeline geometry shown in Fig. 1. The structure of gas–solids flow through an abrupt enlargement is much more complex than gas only flow due to several characteristic features. First, there are two different velocity fields — gas and solids. Second, neither gas nor solids velocity profiles are axisymmetric due to the influence of solids volume fraction distribution across the pipe Žsolids volume fraction increases towards the bottom of a pipe.. In the ideal case both gas and solids horizontal velocity profile are close to symmetric, but the vertical ones are distorted, unless both gas and solids velocities are extremely high, possibly beyond practical pneumatic conveying applications. Finally, the transition from the steady velocity profiles upstream of the step to the steady velocity profiles downstream of the step is not only influenced by the pipe geometry but also by the exchange of momentum between two phases. An illustration of the above phenomenon is given with horizontal and vertical velocity profiles in Fig. 2 for one
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
299
Table 2 Comparison between the results obtained using 1-D analytical and 3-D numerical approaches Flow regime no.
Gas flow rate, m ˙ g Žkgrs.
Gas velocity at step inlet, u z1,i Žmrs.
Mach number at step inlet, Mi
Gas velocity ratio, u z1,oru z1,i
1-D analysis approach, p1,o rp1,i
3-D numerical approach, p1,orp1,i
1 2 3 4 5
0.08 0.106 0.12 0.13 0.14
13.28 17.60 19.92 21.59 23.25
0.0387 0.0513 0.0581 0.0629 0.0678
0.5949 0.5948 0.5947 0.5946 0.5945
1.00051 1.00089 1.00114 1.00134 1.00155
1.00051 1.00091 1.00117 1.00138 1.00161
set of flow conditions Žgas flow rate m ˙ g s 0.106 kgrs and SLR s 6.. Various symbols represent calculated numerical results; the velocity profile lines are drawn simply by using ‘smoothed line’ option in a spreadsheet program as no further detailed analysis of these profiles was pursued. Both profiles confirm the features explained above. They also indicate the downstream pipe section where the momentum exchange from solids to gas phase has been
Pressure ratio
Relative error Ž%.
0.0003 0.0020 0.0034 0.0045 0.0058
finished, i.e., when solids velocity equates gas velocity. This has been also confirmed in the following analysis of pressure distribution along the pipe. The gas and solids velocity results for each of 25 control volumes in one cross-sectional area of pipe were averaged Žusing Eq. Ž7.. and these results are presented in Fig. 3 for 7 m pipe length Ž1 m upstream and 6 m downstream from the step.. These results were obtained
Fig. 2. Horizontal and vertical gas and solids velocity profiles before and after the pipe enlargement Žthe step is located at z s 6 m..
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Fig. 3. Average gas and solids velocity distribution for the flow through an abrupt enlargement for different flow regimes.
for SLR s 4 and 10 and five different gas flow rates, as indicated in Fig. 3. Although this transformation to 1-D presentation inevitably distorted the real flow structure image, specially gas velocity and immediately after the step, it is still possible to estimate the length of pipe over which the momentum exchange from solids to gas phase is taking place. This length increases with the increase of both gas flow rate and solids loading ratio. More detailed analysis on this length is given below when pressure distribution is scrutinised. The initial numerical calculation of pressure distribution along the pipe was carried out with the ambient conditions at pipe outlet. However, as the pressure recovery at the step was the major aim of this investigation, those results were modified in order to get identical pressure at the step inlet for each given gas flow rate. The pressure at the step inlet obtained for lowest gas flow rate was chosen as the reference value for all flow regimes. Gas pressure data which were calculated along the pipeline for all other flow regimes were then adjusted by adding Žor subtracting. the
difference between the calculated value for given flow regime and the chosen reference value. In that way the pressure gradients remained unchanged, the comparison of pressure recovery data was much easier and the graphical presentation of the results is much clearer. The numerical results for gas pressure distribution along the same pipe section, modified as explained above, and for the same combination of flow conditions are presented in Fig. 4. The major features of the structure of gas–solids flow through an abrupt enlargement are as follows. Ø Pressure gradient for the fully developed flow both before after the step is directly proportional to both gas flow rate and solids loading ratio. Ø The ratio of pressure gradients before and after the step for the same flow conditions can be estimated for turbulent flow regime and for gas only flow as Ž DorDi . 4.75 s 3.3. Depending on gas flow rate, this ratio varies between 2.8 and 4.0 Žfor SLR s 4. and between 3.8 and 5.7 Žfor SLR s 10.. Qualitatively, this agrees with the expected results.
P. MarjanoÕic et al.r Powder Technology 104 (1999) 296–303
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Fig. 4. Pressure distribution for the flow through an abrupt enlargement for different flow regimes.
Ø Gas pressure rises after the step as the kinetic energy transforms into the potential energy. This increase is higher than the theoretical value for gas only flow due to addi-
tional momentum transfer from solid particles which are flowing at higher velocity along the transition zone after the step.
Fig. 5. Ratio of pressures at the outlet Žfully developed gas–solids flow. and inlet of an abrupt enlargement for different flow regimes.
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Fig. 6. Transition length from an abrupt enlargement to the point where gas–solids flow is fully developed.
Ø Gas pressure reaches its maximum at the point when gas and solids velocity equalise, i.e., where momentum transfer from solids to gas phase finishes. This is the point when the average gas velocity reaches its minimum value. However, solid particles decelerate further downstream and gas–solids flow fully develops when slip velocity reaches its constant value. The ratio of pressures at the point when gas–solids flow fully develops Ž pout . and at the inlet to the enlargement Ž p in . is presented in Fig. 5. The calculated data are represented by symbols; the straight lines are determined by a simple linear regression for illustrative purposes. Ø Consequently, two transition lengths can be identified: Ža. From the step to the point where gas pressure reaches its maximum and average gas velocity its minimum; this value varies from about 1 m Žat low gas flow rate and SLR. to about 3 m Žat high gas flow rate., i.e., non-dimensional length Ž LtrrDo . varies from 10 to 30. Žb. From the step to the point where both pressure gradient and slip velocity reach constant values; this length is obviously larger than the previous one and it varies from about 2 m Žat low gas flow rate and SLR. to about 3.5 m Žat high gas flow rate., i.e., non-dimensional length Ž LtrrDo . varies from 19 to 39 Žsee Fig. 6.. Similarly as in Fig. 5, the calculated data are represented by symbols and the straight lines are determined by a simple linear regression for illustrative purposes. Further investigation, analytical, numerical and experimental, on the conditions and flow structure which develop when gas–solids mixture flow through an abrupt enlargement is required due to its importance for the design of stepped pneumatic conveying pipeline. The flow conditions at the step have been attracting considerable interest both from the researchers and designers. We hope that
further analysis of the presented approach will contribute to reliable determination of the conditions at which pipeline blockage at the step can be avoided.
5. Conclusions The results presented in this paper are the initial considerations and analysis of the flow structure which develops when gas–solids mixture flows through an abrupt enlargement. They include the analysis of both velocity and pressure fields which are highly critical for a reliable stepped pipeline design of a pneumatic conveying system. The results indicate possible influences of various flow parameters which had been analysed in this study Žgas flow rate and solids loading ratio. on the pressure recovery after an abrupt enlargement. More work is planned to widen the range of flow parameters in the analysis, to pay more attention on the transverse velocity profiles, to model all those results, apply them to the design of stepped pipelines and finally to verify them by structured pneumatic conveying test trials.
6. List of symbols A d D L m ˙ M p R S
Pipe area Particle diameter Pipe bore Pipe length Mass flow rate Mach number Pressure Characteristic gas constant Source term
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T Temperature u Velocity ´ Volume fraction f Variable g Ratio of specific heats Žs c prc v . G Diffusion coefficient r Density Subscripts i inlet to abrupt enlargement o outlet to abrupt enlargement 1 phase 1 Žgas. 2 phase 2 Žsolids.
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References w1x P. Marjanovic, V. Djordjevic, On the compressible flow losses through abrupt enlargements and contractions, J. Fluids Eng., Trans. ASME 116 Ž1994. 756–762. w2x D.B. Spalding, Numerical computation of multi-phase fluid flow and heat transfer, in: C. Taylor, K. Morgan ŽEds.., Recent Advances in Numerical Methods, Vol. 1, Chap. 5, Pineridge Press, Swansea, UK, 1980, pp. 139–167. w3x A. Levy, T. Mooney, P. Marjanovic, D.J. Mason, A comparison of analytical and numerical models for gas–solid flow through straight pipe of different inclinations with experimental data, Powder Technol. 93 Ž1997. 253–260.