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Impact of collisions between fine and coarse particles on the terminal velocity of coarse particles
Powder Technology 363 (2020) 181–186
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Short communication
Impact of collisions between fine and coarse particles on the terminal velocity of coarse particles Robert Zarzycki, Renata Włodarczyk ⁎, Rafał Kobyłecki, Zbigniew Bis Department of Advanced Energy Technologies, Faculty of Infrastructure and Environment, Czestochowa University of Technology, Poland
a r t i c l e
i n f o
Article history: Received 25 August 2019 Received in revised form 19 December 2019 Accepted 6 January 2020 Available online 09 January 2020 Keywords: Coarse particles Suspension of fine particles in a gas stream Double collisions Terminal velocity of coarse particles
a b s t r a c t The paper presents an analysis of the effect of collisions between coarse and fine particles in the vertical flow of a stream of fine particles suspended in gas on the exchange of momentum in these conditions. It was found that the phenomenon of double collisions of fine particles with the surface of coarse particles can be observed during the flow of a stream of fine particles suspended in gas around moving and stationary coarse particles. This phenomenon changes the nature of the relationship between the force of mutual interactions and mass flow rate of fine particles. The proposed corrective relationship significantly improved the consistency of the value of coarse particle terminal velocity depending on mass flow rate of fine particles suspended in the gas flowing around coarse particles. The range of importance of the relationships concerning the terminal velocity of coarse particles was determined. © 2020 Published by Elsevier B.V.
1. Introduction In practice, in flow systems based on multisolids pneumatic transport, especially in boilers with circulating fluidized bed (CFB), the materials used are loose, being a more or less complex polydisperse mixture. Particles of bulk material, especially those representing a mixture of different (in terms of size and/or density) loose materials transported by gas upwards, move at significantly different velocities, as demonstrated e.g. in [1,4–12]. At higher concentrations, these velocity differences lead to mutual collisions between the particles or their clusters, strands, etc. Due to momentum exchange between the colliding particles an additional force of mutual interactions appear [4,8,10,11] and change the overall balance of forces, which consequently results in the change of the movement conditions of both types of colliding particles. The relationships developed in [4,10], concerning the force of mutual interactions between the colliding fine and coarse particles, allowed for a qualitative and satisfactory quantitative description of the changes (decrease in value) of the velocity of coarse particle transport in the stream of fine particles in the gas. This phenomenon was also observed and described for flows with distinct non-homogeneity of the flow structure (multi-solid dense pneumatic transport or multi-solid circulating fluidized beds) [8,11].
⁎ Corresponding author. E-mail address: [email protected] (R. Włodarczyk).
https://doi.org/10.1016/j.powtec.2020.01.018 0032-5910/© 2020 Published by Elsevier B.V.
Most of the developed relationships have concerned the flow conditions in which the flow of suspension of fine particles occurred in gas with relatively low concentration [2,10]. Multiple observations and studies of polydisperse material flow in laboratory model rigs and industrial fluidized bed boiler systems carried out by the authors indicated a real possibility of the flow of a suspension of fine particles with significantly high concentrations (cf. Fig. 1). The current paper is focused on the verification of the force of mutual interactions between fine and coarse particles under conditions of the flow of suspension of fine particles in gas with higher concentrations.
2. Interaction force between fine and coarse particles in vertical flow In the studies [4,10] the authors provided a detailed analysis of the exchange of momentum between colliding fine particles (A) and coarse particles (B) in the flow. Using the results of the analyses presented in [4,10], the equation for the vertical component of the force of interactions between colliding coarse and fine particles can be written as:
ð F AB Þz ¼
2 π 2 dA σ ρ jw j2z dB þ 1 ð1 þ eÞ A A 3 1AB ! 4 dB ρ d 2 A A3 þ 1 ρB dB
ð1Þ
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If particles B are stationary w1B = 0 (e.g. suspended solids), then the relationship (5) takes the form:
ð F AB Þz ¼
2 π 2 dA Gs dB þ 1 ðw1A Þz 4 dB
ð6Þ
The above expression for the mutual interaction force is also valid in the case of core/annulus type flows regardless of the location of the coarse particle (core or annulus). This means that local aerodynamic conditions prevailing in the core and in the annulus (i.e. velocity and concentration of fines) in the immediate vicinity of coarse particle are important for the determination of the interaction force. The value of the interaction force depends on the difference between the velocities of coarse and fine particles while its orientation and sense corresponds to the relative velocity. In case the coarse particle is in the core, the senses of the relative velocity vector and the vector of the mutual interaction force are consistent with each other and opposed to the sense of the vector of the gravity force – such orientation facilitates the carryover of coarse particles. In case the coarse particle is found in the annulus region and the sense of the relative velocity vector is consistent with that of the gravity force (i.e. the clusters of fines fall faster than coarse particle) then the mutual interaction force is consistent with the sense of the gravity force vector and thus the movement of coarse particle downwards is accelerated. It should be pointed out that if the velocities of fine and coarse particles become even then the value of the mutual interaction force approaches zero. 3. Terminal velocity of coarse particles in gas-solid suspension
Fig. 1. Flow of gas and quartz sand particles around a freely moving isolated spherical coarse particle (a) or group of spherical coarse particles (b). Coarse particle diameter dpc = 8 mm, quartz sand particle size dpf = 260 μm, gas velocity U = 5 m/s. The solid mass flow rates Gsf = 3 kg/m2 s (case a) and Gsf = 5 kg/m2 s (case b).
The case, where coarse particles remain stationary and are suspended in a vertical upward flowing gas transporting fine particles, satisfies the condition for the determination of the terminal velocity for the coarse particles since according to the definition the particle terminal velocity is the lowest velocity of the gas flow at which the phenomenon of the ‘hovering’ of the isolated particles in the stream of the upward flowing fluid is observed. The terminal velocity is also considered to be the velocity of the beginning of pneumatic transport. As defined above, the condition of the ‘hovering’ of the isolated coarse particles is that U = Utc,
For the case
mA ≅ 0 (dA b bdB) and e = 1 the expression (1) takes the mB
form: 2 π 2 dA ð F AB Þz ¼ dB þ 1 σ A ρA jw1AB j2z 4 dB
ð2Þ
2 π 2 dA ðw1B Þ 2 dB þ 1 σ A ρA ðw1A Þ2z 1− 4 dB ðw1A Þz
ð F d Þz þ F g þ F fc z ¼ 0
ð3Þ
where the subscript ‘z’ refers to a vertical component of any given value. For circulating fluidized bed and pneumatic transport, the following relationship is valid: σ A ρA ðw1A Þz ¼ Gs
dition of the hovering of coarse particles in the stream of vertically and upwards flowing suspension of gas and fine particles may be determined from the force balance, according to the following relationship:
ð4Þ
ð5Þ
2 U tc
ρg 3 cDc 4 ρpc dpc
2 dpf 1þ U−U tf dpc 3 Gs þ −g ¼ 0 dpc 2 ρpc
ð8Þ
Using the expressions for the characteristic numbers Re and Ar, Eq. (8) can be transformed into the form suitable for the calculation of the terminal velocity of coarse particles:
From Eq. (3) the following relationship may be obtained: 2 π ðw1B Þ 2 2 dA ð F AB Þz ¼ Gs dB þ 1 ðw1A Þz 1− 4 dB ðw1A Þz
ð7Þ
The implementation of the commonly known expressions for the gravity force and the gas drag force and putting dB = dpc, dA = dpf, (w1A)z = wfp and the expression for the interaction force (6) Eq. (7) can be rearranged and written as:
or
ð F AB Þz ¼
dwcp c ¼ 0 and wp = 0. In such case, the condt
Retc
2 2 30;5 0 0 Gs dpc Retf 4 Arc 7 Gs 6 Gs ¼4 2 þ2 2 þ 5 − 3 cD c cD c cDc cDc dpf
ð9Þ
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where: Retf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpf U tf 4ðArÞ f dpc U tc ; Retc ¼ ¼ ; Retf ¼ 3 cD f ν ν
0
Gs ¼
2 dpf Gsdpc 1þ ρg ν g dpc
ð10Þ
ð11Þ
The drag coefficients for fine and coarse particles, cD f and cD c, can be expressed as functions of the Reynolds number in the following form: cDf ¼
dpf U 24 ≤1000 ; Re f ¼ 1 þ 0:15 Re0:687 f ν Re f
ð12Þ
cDc ¼
24 dpc U ≤1000 ; Rec ¼ 1 þ 0:15 Re0:687 c Rec ν
ð13Þ
For Re f N 1000 and for Re c N 1000 the drag coefficients c Df = cDc = 0.44. The dimentionless Eq. (9) was used in the studies of [3,4,10] for the determination of the terminal velocity of coarse particles in the flow of gas and fine particles and for the comparison of the calculated values with the experimental data. The degree of compliance of the solution of Eq. (9) with authors' own measurement results is shown in Fig. 2: the agreement is quite satisfactory just for a limited range of solids mass flow rates (roughly up to 5 kg/m2 s). In this study, we decided to remove these restrictions and present a correction for the determination of the mutual interaction force by taking into consideration the role of double collisions which may occur during two-phase gas-solid flows around movable or stationary elements, as shown in Figs. 1 and 3. The mechanism of the double collisions is shown in Fig. 4. They result from the interactions between the incoming fine particles and those rebounding from the surfaces of coarse particles or stationary cylindrical elements (cf. Fig. 3). Consequently, the number of particles that hit (impact) the surface of the stationary (Fig. 3) or moving (Fig. 1) elements is multiplicated since the fines that have been rebounced from the coarse elements (e.g. coarse particles) interact with the incoming fine particles and hit the surface of the coarse particles again. The real stream of particles that interact with the surface of the element can thus be expressed by a simple function: Gsreal ¼ k Gs f Gs f
ð14Þ
The coefficient k(Gsf) is related to the mass flow rate of fine particles since the phenomenon of double collisions clearly increases with the increase of the Gsf (cf. Figs. 1 and 3).
Fig. 3. Flow of gas and quartz sand particles around a stationary 20 mm cylinder, sand particle diameter dpf = 260 μm, gas velocity U = 4 m/s, Gsf = 5,7 kg/m2 s. The noticeable cluster of particles near the cylinder on the windward side represents the zone of double collisions.
In order to verify the above assumptions and to determine a functional form for the k(Gsf) the investigations were carried out at a test stand schematically shown in Fig. 5. The setup allows to measure the sum of the aerodynamic force and the force that is the result of momentum exchange between fine particles and the surface of a coarse ball suspended on a strain gauge arm. As seen in Fig. 5 the flow of fine particles was organized in a CFB loop consisting of the riser, the cyclone and the recirculation system with control and measurement of the solids mass flow rate. At the beginning of the experiments coarse particles of various diameters were placed at the end of the strain gauge arm and the arm was then introduced into the flow of fine particles. In each case the strain gauge system enabled to compensate the weigh of the coarse particle and thus to carry out direct measurement of the sum of aerodynamic force and mutual interaction force between the coarse particle and the fines. The value of the mutual interaction force was determined as the difference between the sum of those two forces and the aerodynamic force measured for the air flow with the same gas velocity but without the fines. The problem associated with the compensation of the interaction force between the fines and the strain gauge arm was solved by an additional ‘corrective’ measurement, carried out for the flow of gas and fines with no coarse particle. Fig. 6 shows an example relationship between the sum of the aerodynamic force and mutual interaction force and the mass flow rate of fine quartz sand particles with a mean diameter of dpf = 67 μm. The coarse body was a spherical coal particle (dpc = 8 mm) and the investigations were carried out at the experimental setup shown in Fig. 5. The results indicate the validity of the assumption of double collisions. The analysis of the significant number of authors' experimental data sets allowed to choose the following relationships for k(Gsf): 2 !2 3 Gs f 4 5 k Gs f ¼ 2− exp − Gs f
Fig. 2. Terminal velocity of spherical coarse particles (dpc = 8 mm) versus the mass flow rate of fine quartz sand particles (dpf = 67 μm).
ð15Þ
The continuous red line in Fig. 6 was drawn based on the Eqs. (6) and (15). The dashed lines determine the boundary values of the mutual interaction force (k = 1 in case the double collisions are not taken into account and k = 2 if the double collisions occur). The use of the relationship (9) supplemented with formula (15) brings about significant improvement in the compliance between the calculated and measured values of the terminal velocity of coarse particles, as shown in Fig. 7. It may be concluded that at high solids mass flow rates the terminal velocity of coarse particles asymptotically approaches the value of the terminal velocity of fine particles, Utf, which
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Fig. 4. The structure of the flow of gas and fine inert particles around the coarse particles for various mass fluxes of fine solids.
is represented by the red line in Fig. 7. This fact is confirmed by the analysis of the relationship (5) – in case the velocities of coarse and fine particles are even then the value of the mutual interaction force becomes 0 (the force disappears).
4. Scope of applicability The question of the scope of application of the corrective relationship (15) remains open. Observation of the flow around a stationary cylinder allowed to find out that for high values of the solids mass flow rates a stationary cluster of particles is formed around the cylinder. The particles are ‘glued’ to the coarse object from the front side (inflow of the fines) as shown in Fig. 8 and also schematically drawn in Fig. 4. Therefore, it can be assumed that there is another limit value of the mass flow rate of fine particles, Gs f ∗∗ (cf. Fig. 4), beyond which the
Fig. 5. Schematic of the test rig for the measurement of the interaction force between coarse spherical particle in the flow of fine particles suspended in gas (1 = coarse particle, 2 = tensometric balance, 3 = A/C converter, 4 = computer, 5 = measurement of the volumetric flow of fine particles, 6 = valves, 7 = primary cyclone, 8 = secondary cyclone, 9 = fabric filter, 10 = ID fan).
double collisions disappear and the character of the flow changes radically around a stationary (Fig. 8) or a freely moving coarse particle (Fig. 9). For the quantitative presentation of the phenomena described above, the expression for the free path of spherical particles in gas flow (analogy to the movement of gas molecules in the kinetic theory) is proposed as: 1 l ¼ pffiffiffi 2 2πdpf n f
ð16Þ
The increase of the mass flow rate of fine particles, Gsf, in the vertically and upward flowing gas in a channel with limited transverse dimensions naturally leads to the reduction of the distance between the solids. Interaction of such gas-solids mixture with the surface of a stationary or moving element immediately brings about the increase of the concentration of fine particles at the surface of those elements and thus leads to double impacts described above. The double impacts are thus the logical consequence of the ‘multiplication’ of particle concentration in this zone. The increase of numerical concentration, nf, in the zone directly adjacent to the coarse elements causes a rapid decrease of the free path length (Eq. (16)) and explains the phenomenon of
Fig. 6. The sum of the aerodynamic drag force (Fa) and Fcollision (the interaction force between fine quartz sand particles of dpf = 67 μm and spherical coarse particle of dpc = 8 mm) versus the mass flow rate of fine particles. The continuous line represents the approximation with Eqs. (6) and (15); Gs⁎f = 9 kg/(m2 s).
R. Zarzycki et al. / Powder Technology 363 (2020) 181–186
185
inflowing and rebounded fine particles become obvious. If the average length of the free path and the size of the fine particles are equal then the particles rebounded from the surface are not able to move away since they are ‘pressed’ back to the surface of the coarse particle/element and have to remain there. Such case is presented in Fig. 8 where a cluster of particles attached to the surface of a large body is seen. The criterion for the determination of such state can be obtained from the relation (16) by introducing the volume of fine particles to the numerator and denominator (assuming the particles are spherical and the relationship between the volumetric concentration and 3
number of particles is: σ f ¼ n f
πdpf ). Thus we obtain: 6
3
πdpf l ¼ pffiffiffi 2 6 2πdpf σ f
ð17Þ
which can be easily transformed into the dimensionless form:
Fig. 7. Terminal velocity of spherical coarse particles (dpc = 8 mm) versus the mass flow rate of fine quartz sand particles (dpf = 67 μm).
l 1 ¼ pffiffiffi dpf 6 2σ f
The condition for the certainty (probability equal to 1) of double collisions between the inflowing and rebounded fine particles occurs when l/dpf = 1. The critical value of the concentration of fine particles in the gas can be thus derived from Eq. (18): 1 σ f ≥ pffiffiffi 6 2
Fig. 8. The flow around a stationary 20 mm cylinder immersed into the circulating fluidized bed of quartz sand, dpf = 260 μm, U = 5 m/s, Gsf = 45 kg/(m2 s).
double collisions. However, further increase of the numerical concentration of fine particles (the increase is proportional to the solid mass flow rate, Gsf) brings about that at some point the length of the free path becomes similar to the size of particles and the collisions between the
ð18Þ
ð19Þ
The increase of the solids mass flow rate beyond the value determined by Eq. (19) leads to the change of the flow structre around the elements, as shown in Figs. 8–9, and thus the relationship (15) must be corrected at least in terms of the shape and mass of coarse particles. The critical concentration calculated from Eq. (19) is σf∗∗ = 0.118. Above that value, the loss of stability and homogeneity of the gas-solids suspension flow is likely to occur, leading to the formation of strands and clusters, well known from the visualization and observation of such flow structures, as shown e.g. in Fig. 8. Under such conditions, large particles ‘get stuck’ in the clusters and the behaviour of those particles depends on the movement conditions of the clusters. The description of the aerodynamics of such gas and bulk material flow was also reported by Wirth [11,12]. In conclusion, we also would like to emphasize that attention should be put during analysis of the solution of relationship (9) for high mass flow rates of fine particles, Gsf; as shown in Fig. 7 the asymptote of the solution is the value of the terminal velocity of fine particles, Uf t (red line). This fact is confirmed by the visualization shown in Fig. 9. 5. Summary The preliminary analyses and studies presented and discussed in the current paper allow the formulation of the following brief conclusions:
Fig. 9. The flow around an 8 mm spherical particle (white spot) freely moving in the circulating fluidized bed of quartz sand, dpf = 260 μm, U = 5 m/s, Gsf = 58 kg/(m2 s).
1. The phenomenon of double collisions can be observed during the flow of a stream of fine particles suspended in vertically flowing gas around moving or stationary coarse particles. The intensity of collisions and interactions between the particles increases with the increase of the mass flow rate of fine particles. 2. Double collisions increase the number of fine particles which collide with the surfaces of coarse particles. 3. The proposed correction (Eq. (15)) was positively verified and its implementation into the force balance equation significantly improved the calculation of the terminal velocity of coarse particles versus the mass flow rate of fine particles. 4. At high values of the solid mass flow rate of fine particles, Gsf, the terminal velocity of coarse particles is similar to the terminal velocity of fine particles.
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R. Zarzycki et al. / Powder Technology 363 (2020) 181–186
List of symbols cD c
drag coefficient for coarse particles, −
cD f dA dB dpc dpf e g l nf
drag coefficient for fine particles, − mean size of particles A, m mean size of particles B, m diameter of coarse particles, m diameter of fine particles, m restitution coefficient, − gravity acceleration, m/s2 free path length for fine particles between collisions, − numerical concentration of fine particles (number of particles per unit volume), − relative velocity of colliding particles, m/s velocity of coarse particle, m/s velocity of fine particle, m/s gravity force acting on coarse particles, N vertical component of the gas drag force acting on coarse particle, N vertical component of the interaction force between coarse and fine particles, N mass flow rate of fine particles, kg/(m2s) critical mass flow rate of fine particles, kg/(m2s) gas velocity, m/s terminal velocity of coarse particle, m/s terminal velocity of fine particles, m/s density of particle A, kg/m3 density of particle B, kg/m3 density of gas, kg/m3 density of coarse particle, kg/m3 concentration of fine particles A, − concentration of fine particles, − critical concentration of fine particles, − – kinetic viscosity, m2/s
w1AB wcp wfp Fg (Fd)z (Ffc)z Gs Gsf∗ U Utc Utf ρA ρB ρg ρpc σA σf σf∗∗ ν
Funding This research was funded by grants BSPB-400-301/19.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] J. Adhnez, L.F. de Diego, P. Gayan, Transport velocities of coal and sand particles, Powder Technol. 77 (1993) 61–68. [2] M. Benali, K. Shakourzadeh-Bolouri, J.F. Large, Powder Technol. 66 (1991) 285–292. [3] Z. Bis, Fluidized Beds, in: M. Lackner, F. Winter, A.K. Agarwal (Eds.), Handbook of Combustion, 4, Wiley-Vch 2010, pp. 399–433. [4] Z. Bis, Fluidyzacja cyrkulacyjna mieszanin polidyspersyjnych, monograph No. 63, Częstochowa Univ. of Technology, 1999. [5] P. Buffière, R. Moletta, Collison frequency and collisional particle pressure in threephase fluidized beds, Chem. Eng. Sci. 55 (22) (2000) 5555–5563. [6] C. Hu, K. Luo, S. Wang, L. Junjie, J. Fan, The effects of collisional parameters on the hydrodynamics and heat transfer in spouted bed: a CFD-DEM study, Powder Technol. 353 (2019) 132–144. [7] C.J. Meyer, D.A. Deglon, Particle collision modeling – a review, Minerals Eng. 24 (8) (2011) 719–730. [8] S. Satija, L.S. Fan, Terminal velocity of dense particles in the multisolid pneumatic transport bed, Chem. Eng. Sci. 40 (2) (1985) 259–267. [9] L. Wang, B. Wu, Z. Wu, R. Li, X. Feng, Experimental determination of the coefficient of restitution of particle-particle collision for frozen maize grains, Powder Technol. 338 (2018) 263–273. [10] K.K. Win, W. Nowak, H. Matsuda, M. Hasatani, Z. Bis, J. Krzywanski, W. Gajewski, Transport velocity of coarse particles in multi-solid fluidized bed, J. of Chem. Eng. of Japan 28 (/5) (1995) 535–540. [11] K.E. Wirth, Influence of the particle size distribution on the operation behavior of circulating fluidized beds, Proc. of the 2nd Int. Conf. on Multiphase Flow, Kyoto, 1995. [12] K.E. Wirth, Fluid mechanics of circulating fluidized beds, Chem. Eng. Technol. 14 (1991) 29–38.
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Short communication
Impact of collisions between fine and coarse particles on the terminal velocity of coarse particles Robert Zarzycki, Renata Włodarczyk ⁎, Rafał Kobyłecki, Zbigniew Bis Department of Advanced Energy Technologies, Faculty of Infrastructure and Environment, Czestochowa University of Technology, Poland
a r t i c l e
i n f o
Article history: Received 25 August 2019 Received in revised form 19 December 2019 Accepted 6 January 2020 Available online 09 January 2020 Keywords: Coarse particles Suspension of fine particles in a gas stream Double collisions Terminal velocity of coarse particles
a b s t r a c t The paper presents an analysis of the effect of collisions between coarse and fine particles in the vertical flow of a stream of fine particles suspended in gas on the exchange of momentum in these conditions. It was found that the phenomenon of double collisions of fine particles with the surface of coarse particles can be observed during the flow of a stream of fine particles suspended in gas around moving and stationary coarse particles. This phenomenon changes the nature of the relationship between the force of mutual interactions and mass flow rate of fine particles. The proposed corrective relationship significantly improved the consistency of the value of coarse particle terminal velocity depending on mass flow rate of fine particles suspended in the gas flowing around coarse particles. The range of importance of the relationships concerning the terminal velocity of coarse particles was determined. © 2020 Published by Elsevier B.V.
1. Introduction In practice, in flow systems based on multisolids pneumatic transport, especially in boilers with circulating fluidized bed (CFB), the materials used are loose, being a more or less complex polydisperse mixture. Particles of bulk material, especially those representing a mixture of different (in terms of size and/or density) loose materials transported by gas upwards, move at significantly different velocities, as demonstrated e.g. in [1,4–12]. At higher concentrations, these velocity differences lead to mutual collisions between the particles or their clusters, strands, etc. Due to momentum exchange between the colliding particles an additional force of mutual interactions appear [4,8,10,11] and change the overall balance of forces, which consequently results in the change of the movement conditions of both types of colliding particles. The relationships developed in [4,10], concerning the force of mutual interactions between the colliding fine and coarse particles, allowed for a qualitative and satisfactory quantitative description of the changes (decrease in value) of the velocity of coarse particle transport in the stream of fine particles in the gas. This phenomenon was also observed and described for flows with distinct non-homogeneity of the flow structure (multi-solid dense pneumatic transport or multi-solid circulating fluidized beds) [8,11].
⁎ Corresponding author. E-mail address: [email protected] (R. Włodarczyk).
https://doi.org/10.1016/j.powtec.2020.01.018 0032-5910/© 2020 Published by Elsevier B.V.
Most of the developed relationships have concerned the flow conditions in which the flow of suspension of fine particles occurred in gas with relatively low concentration [2,10]. Multiple observations and studies of polydisperse material flow in laboratory model rigs and industrial fluidized bed boiler systems carried out by the authors indicated a real possibility of the flow of a suspension of fine particles with significantly high concentrations (cf. Fig. 1). The current paper is focused on the verification of the force of mutual interactions between fine and coarse particles under conditions of the flow of suspension of fine particles in gas with higher concentrations.
2. Interaction force between fine and coarse particles in vertical flow In the studies [4,10] the authors provided a detailed analysis of the exchange of momentum between colliding fine particles (A) and coarse particles (B) in the flow. Using the results of the analyses presented in [4,10], the equation for the vertical component of the force of interactions between colliding coarse and fine particles can be written as:
ð F AB Þz ¼
2 π 2 dA σ ρ jw j2z dB þ 1 ð1 þ eÞ A A 3 1AB ! 4 dB ρ d 2 A A3 þ 1 ρB dB
ð1Þ
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R. Zarzycki et al. / Powder Technology 363 (2020) 181–186
If particles B are stationary w1B = 0 (e.g. suspended solids), then the relationship (5) takes the form:
ð F AB Þz ¼
2 π 2 dA Gs dB þ 1 ðw1A Þz 4 dB
ð6Þ
The above expression for the mutual interaction force is also valid in the case of core/annulus type flows regardless of the location of the coarse particle (core or annulus). This means that local aerodynamic conditions prevailing in the core and in the annulus (i.e. velocity and concentration of fines) in the immediate vicinity of coarse particle are important for the determination of the interaction force. The value of the interaction force depends on the difference between the velocities of coarse and fine particles while its orientation and sense corresponds to the relative velocity. In case the coarse particle is in the core, the senses of the relative velocity vector and the vector of the mutual interaction force are consistent with each other and opposed to the sense of the vector of the gravity force – such orientation facilitates the carryover of coarse particles. In case the coarse particle is found in the annulus region and the sense of the relative velocity vector is consistent with that of the gravity force (i.e. the clusters of fines fall faster than coarse particle) then the mutual interaction force is consistent with the sense of the gravity force vector and thus the movement of coarse particle downwards is accelerated. It should be pointed out that if the velocities of fine and coarse particles become even then the value of the mutual interaction force approaches zero. 3. Terminal velocity of coarse particles in gas-solid suspension
Fig. 1. Flow of gas and quartz sand particles around a freely moving isolated spherical coarse particle (a) or group of spherical coarse particles (b). Coarse particle diameter dpc = 8 mm, quartz sand particle size dpf = 260 μm, gas velocity U = 5 m/s. The solid mass flow rates Gsf = 3 kg/m2 s (case a) and Gsf = 5 kg/m2 s (case b).
The case, where coarse particles remain stationary and are suspended in a vertical upward flowing gas transporting fine particles, satisfies the condition for the determination of the terminal velocity for the coarse particles since according to the definition the particle terminal velocity is the lowest velocity of the gas flow at which the phenomenon of the ‘hovering’ of the isolated particles in the stream of the upward flowing fluid is observed. The terminal velocity is also considered to be the velocity of the beginning of pneumatic transport. As defined above, the condition of the ‘hovering’ of the isolated coarse particles is that U = Utc,
For the case
mA ≅ 0 (dA b bdB) and e = 1 the expression (1) takes the mB
form: 2 π 2 dA ð F AB Þz ¼ dB þ 1 σ A ρA jw1AB j2z 4 dB
ð2Þ
2 π 2 dA ðw1B Þ 2 dB þ 1 σ A ρA ðw1A Þ2z 1− 4 dB ðw1A Þz
ð F d Þz þ F g þ F fc z ¼ 0
ð3Þ
where the subscript ‘z’ refers to a vertical component of any given value. For circulating fluidized bed and pneumatic transport, the following relationship is valid: σ A ρA ðw1A Þz ¼ Gs
dition of the hovering of coarse particles in the stream of vertically and upwards flowing suspension of gas and fine particles may be determined from the force balance, according to the following relationship:
ð4Þ
ð5Þ
2 U tc
ρg 3 cDc 4 ρpc dpc
2 dpf 1þ U−U tf dpc 3 Gs þ −g ¼ 0 dpc 2 ρpc
ð8Þ
Using the expressions for the characteristic numbers Re and Ar, Eq. (8) can be transformed into the form suitable for the calculation of the terminal velocity of coarse particles:
From Eq. (3) the following relationship may be obtained: 2 π ðw1B Þ 2 2 dA ð F AB Þz ¼ Gs dB þ 1 ðw1A Þz 1− 4 dB ðw1A Þz
ð7Þ
The implementation of the commonly known expressions for the gravity force and the gas drag force and putting dB = dpc, dA = dpf, (w1A)z = wfp and the expression for the interaction force (6) Eq. (7) can be rearranged and written as:
or
ð F AB Þz ¼
dwcp c ¼ 0 and wp = 0. In such case, the condt
Retc
2 2 30;5 0 0 Gs dpc Retf 4 Arc 7 Gs 6 Gs ¼4 2 þ2 2 þ 5 − 3 cD c cD c cDc cDc dpf
ð9Þ
R. Zarzycki et al. / Powder Technology 363 (2020) 181–186
183
where: Retf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpf U tf 4ðArÞ f dpc U tc ; Retc ¼ ¼ ; Retf ¼ 3 cD f ν ν
0
Gs ¼
2 dpf Gsdpc 1þ ρg ν g dpc
ð10Þ
ð11Þ
The drag coefficients for fine and coarse particles, cD f and cD c, can be expressed as functions of the Reynolds number in the following form: cDf ¼
dpf U 24 ≤1000 ; Re f ¼ 1 þ 0:15 Re0:687 f ν Re f
ð12Þ
cDc ¼
24 dpc U ≤1000 ; Rec ¼ 1 þ 0:15 Re0:687 c Rec ν
ð13Þ
For Re f N 1000 and for Re c N 1000 the drag coefficients c Df = cDc = 0.44. The dimentionless Eq. (9) was used in the studies of [3,4,10] for the determination of the terminal velocity of coarse particles in the flow of gas and fine particles and for the comparison of the calculated values with the experimental data. The degree of compliance of the solution of Eq. (9) with authors' own measurement results is shown in Fig. 2: the agreement is quite satisfactory just for a limited range of solids mass flow rates (roughly up to 5 kg/m2 s). In this study, we decided to remove these restrictions and present a correction for the determination of the mutual interaction force by taking into consideration the role of double collisions which may occur during two-phase gas-solid flows around movable or stationary elements, as shown in Figs. 1 and 3. The mechanism of the double collisions is shown in Fig. 4. They result from the interactions between the incoming fine particles and those rebounding from the surfaces of coarse particles or stationary cylindrical elements (cf. Fig. 3). Consequently, the number of particles that hit (impact) the surface of the stationary (Fig. 3) or moving (Fig. 1) elements is multiplicated since the fines that have been rebounced from the coarse elements (e.g. coarse particles) interact with the incoming fine particles and hit the surface of the coarse particles again. The real stream of particles that interact with the surface of the element can thus be expressed by a simple function: Gsreal ¼ k Gs f Gs f
ð14Þ
The coefficient k(Gsf) is related to the mass flow rate of fine particles since the phenomenon of double collisions clearly increases with the increase of the Gsf (cf. Figs. 1 and 3).
Fig. 3. Flow of gas and quartz sand particles around a stationary 20 mm cylinder, sand particle diameter dpf = 260 μm, gas velocity U = 4 m/s, Gsf = 5,7 kg/m2 s. The noticeable cluster of particles near the cylinder on the windward side represents the zone of double collisions.
In order to verify the above assumptions and to determine a functional form for the k(Gsf) the investigations were carried out at a test stand schematically shown in Fig. 5. The setup allows to measure the sum of the aerodynamic force and the force that is the result of momentum exchange between fine particles and the surface of a coarse ball suspended on a strain gauge arm. As seen in Fig. 5 the flow of fine particles was organized in a CFB loop consisting of the riser, the cyclone and the recirculation system with control and measurement of the solids mass flow rate. At the beginning of the experiments coarse particles of various diameters were placed at the end of the strain gauge arm and the arm was then introduced into the flow of fine particles. In each case the strain gauge system enabled to compensate the weigh of the coarse particle and thus to carry out direct measurement of the sum of aerodynamic force and mutual interaction force between the coarse particle and the fines. The value of the mutual interaction force was determined as the difference between the sum of those two forces and the aerodynamic force measured for the air flow with the same gas velocity but without the fines. The problem associated with the compensation of the interaction force between the fines and the strain gauge arm was solved by an additional ‘corrective’ measurement, carried out for the flow of gas and fines with no coarse particle. Fig. 6 shows an example relationship between the sum of the aerodynamic force and mutual interaction force and the mass flow rate of fine quartz sand particles with a mean diameter of dpf = 67 μm. The coarse body was a spherical coal particle (dpc = 8 mm) and the investigations were carried out at the experimental setup shown in Fig. 5. The results indicate the validity of the assumption of double collisions. The analysis of the significant number of authors' experimental data sets allowed to choose the following relationships for k(Gsf): 2 !2 3 Gs f 4 5 k Gs f ¼ 2− exp − Gs f
Fig. 2. Terminal velocity of spherical coarse particles (dpc = 8 mm) versus the mass flow rate of fine quartz sand particles (dpf = 67 μm).
ð15Þ
The continuous red line in Fig. 6 was drawn based on the Eqs. (6) and (15). The dashed lines determine the boundary values of the mutual interaction force (k = 1 in case the double collisions are not taken into account and k = 2 if the double collisions occur). The use of the relationship (9) supplemented with formula (15) brings about significant improvement in the compliance between the calculated and measured values of the terminal velocity of coarse particles, as shown in Fig. 7. It may be concluded that at high solids mass flow rates the terminal velocity of coarse particles asymptotically approaches the value of the terminal velocity of fine particles, Utf, which
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Fig. 4. The structure of the flow of gas and fine inert particles around the coarse particles for various mass fluxes of fine solids.
is represented by the red line in Fig. 7. This fact is confirmed by the analysis of the relationship (5) – in case the velocities of coarse and fine particles are even then the value of the mutual interaction force becomes 0 (the force disappears).
4. Scope of applicability The question of the scope of application of the corrective relationship (15) remains open. Observation of the flow around a stationary cylinder allowed to find out that for high values of the solids mass flow rates a stationary cluster of particles is formed around the cylinder. The particles are ‘glued’ to the coarse object from the front side (inflow of the fines) as shown in Fig. 8 and also schematically drawn in Fig. 4. Therefore, it can be assumed that there is another limit value of the mass flow rate of fine particles, Gs f ∗∗ (cf. Fig. 4), beyond which the
Fig. 5. Schematic of the test rig for the measurement of the interaction force between coarse spherical particle in the flow of fine particles suspended in gas (1 = coarse particle, 2 = tensometric balance, 3 = A/C converter, 4 = computer, 5 = measurement of the volumetric flow of fine particles, 6 = valves, 7 = primary cyclone, 8 = secondary cyclone, 9 = fabric filter, 10 = ID fan).
double collisions disappear and the character of the flow changes radically around a stationary (Fig. 8) or a freely moving coarse particle (Fig. 9). For the quantitative presentation of the phenomena described above, the expression for the free path of spherical particles in gas flow (analogy to the movement of gas molecules in the kinetic theory) is proposed as: 1 l ¼ pffiffiffi 2 2πdpf n f
ð16Þ
The increase of the mass flow rate of fine particles, Gsf, in the vertically and upward flowing gas in a channel with limited transverse dimensions naturally leads to the reduction of the distance between the solids. Interaction of such gas-solids mixture with the surface of a stationary or moving element immediately brings about the increase of the concentration of fine particles at the surface of those elements and thus leads to double impacts described above. The double impacts are thus the logical consequence of the ‘multiplication’ of particle concentration in this zone. The increase of numerical concentration, nf, in the zone directly adjacent to the coarse elements causes a rapid decrease of the free path length (Eq. (16)) and explains the phenomenon of
Fig. 6. The sum of the aerodynamic drag force (Fa) and Fcollision (the interaction force between fine quartz sand particles of dpf = 67 μm and spherical coarse particle of dpc = 8 mm) versus the mass flow rate of fine particles. The continuous line represents the approximation with Eqs. (6) and (15); Gs⁎f = 9 kg/(m2 s).
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inflowing and rebounded fine particles become obvious. If the average length of the free path and the size of the fine particles are equal then the particles rebounded from the surface are not able to move away since they are ‘pressed’ back to the surface of the coarse particle/element and have to remain there. Such case is presented in Fig. 8 where a cluster of particles attached to the surface of a large body is seen. The criterion for the determination of such state can be obtained from the relation (16) by introducing the volume of fine particles to the numerator and denominator (assuming the particles are spherical and the relationship between the volumetric concentration and 3
number of particles is: σ f ¼ n f
πdpf ). Thus we obtain: 6
3
πdpf l ¼ pffiffiffi 2 6 2πdpf σ f
ð17Þ
which can be easily transformed into the dimensionless form:
Fig. 7. Terminal velocity of spherical coarse particles (dpc = 8 mm) versus the mass flow rate of fine quartz sand particles (dpf = 67 μm).
l 1 ¼ pffiffiffi dpf 6 2σ f
The condition for the certainty (probability equal to 1) of double collisions between the inflowing and rebounded fine particles occurs when l/dpf = 1. The critical value of the concentration of fine particles in the gas can be thus derived from Eq. (18): 1 σ f ≥ pffiffiffi 6 2
Fig. 8. The flow around a stationary 20 mm cylinder immersed into the circulating fluidized bed of quartz sand, dpf = 260 μm, U = 5 m/s, Gsf = 45 kg/(m2 s).
double collisions. However, further increase of the numerical concentration of fine particles (the increase is proportional to the solid mass flow rate, Gsf) brings about that at some point the length of the free path becomes similar to the size of particles and the collisions between the
ð18Þ
ð19Þ
The increase of the solids mass flow rate beyond the value determined by Eq. (19) leads to the change of the flow structre around the elements, as shown in Figs. 8–9, and thus the relationship (15) must be corrected at least in terms of the shape and mass of coarse particles. The critical concentration calculated from Eq. (19) is σf∗∗ = 0.118. Above that value, the loss of stability and homogeneity of the gas-solids suspension flow is likely to occur, leading to the formation of strands and clusters, well known from the visualization and observation of such flow structures, as shown e.g. in Fig. 8. Under such conditions, large particles ‘get stuck’ in the clusters and the behaviour of those particles depends on the movement conditions of the clusters. The description of the aerodynamics of such gas and bulk material flow was also reported by Wirth [11,12]. In conclusion, we also would like to emphasize that attention should be put during analysis of the solution of relationship (9) for high mass flow rates of fine particles, Gsf; as shown in Fig. 7 the asymptote of the solution is the value of the terminal velocity of fine particles, Uf t (red line). This fact is confirmed by the visualization shown in Fig. 9. 5. Summary The preliminary analyses and studies presented and discussed in the current paper allow the formulation of the following brief conclusions:
Fig. 9. The flow around an 8 mm spherical particle (white spot) freely moving in the circulating fluidized bed of quartz sand, dpf = 260 μm, U = 5 m/s, Gsf = 58 kg/(m2 s).
1. The phenomenon of double collisions can be observed during the flow of a stream of fine particles suspended in vertically flowing gas around moving or stationary coarse particles. The intensity of collisions and interactions between the particles increases with the increase of the mass flow rate of fine particles. 2. Double collisions increase the number of fine particles which collide with the surfaces of coarse particles. 3. The proposed correction (Eq. (15)) was positively verified and its implementation into the force balance equation significantly improved the calculation of the terminal velocity of coarse particles versus the mass flow rate of fine particles. 4. At high values of the solid mass flow rate of fine particles, Gsf, the terminal velocity of coarse particles is similar to the terminal velocity of fine particles.
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List of symbols cD c
drag coefficient for coarse particles, −
cD f dA dB dpc dpf e g l nf
drag coefficient for fine particles, − mean size of particles A, m mean size of particles B, m diameter of coarse particles, m diameter of fine particles, m restitution coefficient, − gravity acceleration, m/s2 free path length for fine particles between collisions, − numerical concentration of fine particles (number of particles per unit volume), − relative velocity of colliding particles, m/s velocity of coarse particle, m/s velocity of fine particle, m/s gravity force acting on coarse particles, N vertical component of the gas drag force acting on coarse particle, N vertical component of the interaction force between coarse and fine particles, N mass flow rate of fine particles, kg/(m2s) critical mass flow rate of fine particles, kg/(m2s) gas velocity, m/s terminal velocity of coarse particle, m/s terminal velocity of fine particles, m/s density of particle A, kg/m3 density of particle B, kg/m3 density of gas, kg/m3 density of coarse particle, kg/m3 concentration of fine particles A, − concentration of fine particles, − critical concentration of fine particles, − – kinetic viscosity, m2/s
w1AB wcp wfp Fg (Fd)z (Ffc)z Gs Gsf∗ U Utc Utf ρA ρB ρg ρpc σA σf σf∗∗ ν
Funding This research was funded by grants BSPB-400-301/19.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] J. Adhnez, L.F. de Diego, P. Gayan, Transport velocities of coal and sand particles, Powder Technol. 77 (1993) 61–68. [2] M. Benali, K. Shakourzadeh-Bolouri, J.F. Large, Powder Technol. 66 (1991) 285–292. [3] Z. Bis, Fluidized Beds, in: M. Lackner, F. Winter, A.K. Agarwal (Eds.), Handbook of Combustion, 4, Wiley-Vch 2010, pp. 399–433. [4] Z. Bis, Fluidyzacja cyrkulacyjna mieszanin polidyspersyjnych, monograph No. 63, Częstochowa Univ. of Technology, 1999. [5] P. Buffière, R. Moletta, Collison frequency and collisional particle pressure in threephase fluidized beds, Chem. Eng. Sci. 55 (22) (2000) 5555–5563. [6] C. Hu, K. Luo, S. Wang, L. Junjie, J. Fan, The effects of collisional parameters on the hydrodynamics and heat transfer in spouted bed: a CFD-DEM study, Powder Technol. 353 (2019) 132–144. [7] C.J. Meyer, D.A. Deglon, Particle collision modeling – a review, Minerals Eng. 24 (8) (2011) 719–730. [8] S. Satija, L.S. Fan, Terminal velocity of dense particles in the multisolid pneumatic transport bed, Chem. Eng. Sci. 40 (2) (1985) 259–267. [9] L. Wang, B. Wu, Z. Wu, R. Li, X. Feng, Experimental determination of the coefficient of restitution of particle-particle collision for frozen maize grains, Powder Technol. 338 (2018) 263–273. [10] K.K. Win, W. Nowak, H. Matsuda, M. Hasatani, Z. Bis, J. Krzywanski, W. Gajewski, Transport velocity of coarse particles in multi-solid fluidized bed, J. of Chem. Eng. of Japan 28 (/5) (1995) 535–540. [11] K.E. Wirth, Influence of the particle size distribution on the operation behavior of circulating fluidized beds, Proc. of the 2nd Int. Conf. on Multiphase Flow, Kyoto, 1995. [12] K.E. Wirth, Fluid mechanics of circulating fluidized beds, Chem. Eng. Technol. 14 (1991) 29–38.