Modeling the Mean Residence Time of Liquid ... - ACS Publications

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Modeling the Mean Residence Time of Liquid Phase in the Gas− Liquid Cyclone Junwei Yang,† Xupeng Zhang,‡ Guoping Shen,† Jiazhi Xiao,*,† and Youhai Jin† †

State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Qingdao 266580, China Shandong Shtar Science & Technology Petrochemical Co., Ltd., Rizhao 276806, China

‡

S Supporting Information *

ABSTRACT: Residence time is a key parameter for the cyclones with mass transfer or reaction process. A model for the cylinder-on-cone cyclone was presented to calculate the mean residence time of the liquid phase, based on a force balance on the wall film. This model was validated using an experiment, which measured the residence time of the liquid phase by the hold-up method. The results demonstrated that the predicted residence times are in good overall agreement with the measured values. Furthermore, the liquid residence time decreases obviously with increasing droplets loading, and decreases less with increasing inlet velocity. The liquid residence time increases with the cylinder diameter. The wall film velocity, however, decreases with increasing cylinder diameter, because the perimeter of liquid wall film is directly proportional to the cylinder diameter.

1. INTRODUCTION Cyclone separation is an efficient separation technology for the multiphase flow, which has been widely used in the separation of heterogeneous systems. The heat transfer, mass transfer, and residence time characteristics of a cyclone give it unique advantages in some of the drying,1 gas absorption,2 or reaction processes.3 Residence time becomes a key parameter if there is heat and mass transfer or reaction process in cyclones.4−7 Because there are usually two phases in a cyclone, the residence time in cyclones can be classified into two categories: the continuous phase (gas or liquid) and the dispersed phase (particles or droplets) residence times. Many studies were conducted for the residence time of continuous phase by experiment and numerical simulation. Lede et al.8 measured the residence time distribution of the gas phase in a cyclone reactor using the tracer method. On the basis of the experimental results, a general model was proposed, which described the cyclone as a plug flow reactor followed by a more or less bypassed stirred volume. The model predicted results were in good agreement with the measured results. With the development of computational fluid dynamics (CFD), the CFD method was also successfully used to simulate the residence time distribution of the continuous phase in cyclones.9−12 The residence time distribution of the continuous phase was computed by solving the transient scalar transport equation. The measurement of the particles’ residence time in a cyclone is more difficult. Lede et al.13 measured the residence time of solid particles in a cyclone reactor using four different methods (photocell method, piezoelectric method, photographic method, hold-up method), and the four methods have led to similar results. Kang et al.14 measured the residence time distribution of solid particles in three different cyclones using the KCl-coated tracer. The results showed that solid backmixing is significant, in contrast to Lede et al.,13 who concluded that the behavior of the particles is close to plug flow. Li et al.15 studied the solid particle residence time in a cyclone using the hold-up method. In addition, there were also many reports © XXXX American Chemical Society

using the numerical simulation to study the residence time of the solid particles in cyclones.16−20 The solid particles were modeled using the Euler−Lagrange model or Eulerian two-fluid model. However, for the gas−liquid cyclone, the incoming liquid droplets are centrifuged to the wall of the cyclone, forming a liquid wall film because of the droplet coalescence. The velocity of liquid film at the wall is zero, while there is a slip velocity between the solid particles and the wall of cyclone. Therefore, the friction exerted on solid particles at the wall is of a different nature than that exerted on liquid droplets, even if they are all dispersed phases. It is necessary to study the liquid residence time in the gas−liquid cyclone. Currently, there are few reports on the flow characteristic and residence time of the liquid phase in cyclones. Hreiz et al.21 conducted the hydrodynamics and velocity measurements in a GLCC (gas−liquid cylindrical cyclone). The gas−liquid swirl flow in a GLCC was characterized qualitatively by flow visualizations. Hoffmann et al.22 presented a model to estimate the residence time of the liquid phase in the cyclone with vane-type inlet; no experimental verification was given. This work presented a model to estimate the residence time of the liquid phase in the cylinder-on-cone cyclone, on the basis of a force balance on the wall film. Furthermore, this model was experimentally validated with the residence time of the liquid phase measured by the hold-up method. The effects of the droplet loading, gas inlet velocity, and cyclone diameter on the residence time of the liquid phase were investigated. Received: July 16, 2015 Revised: September 11, 2015 Accepted: October 12, 2015

A

DOI: 10.1021/acs.iecr.5b02617 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

2. MODELING THE FLOW CHARACTERISTICS OF WALL FILM Hoffmann et al.22 presented a model, on the basis of a force balance on the wall film, to estimate the flow characteristics of the wall film in the “straight-through” cyclone with a vane-type inlet. In this paper, the flow characteristics of the wall film in the cylinder-on-cone cyclone were modeled by a similar method. Furthermore, the model was validated with experimental data. According to the observation during the experiment, the liquid phase flow in the cylinder-on-cone cyclone can be classified into three flow patterns (Figure 1): spiral strip, spiral

Figure 2. Force analysis of the cyclone: (a) cylinder section, (b) cone section.

across dz, τg,i and τl,w are the gas/liquid and wall/liquid shear stresses in the actual flow direction, respectively, α is the angle of the gas and liquid wall film flows relative to the horizontal, ρg, and ρl are the gas and liquid densities, Si, Sw are the perimeters of gas flow and the liquid wall film, and g is the acceleration of gravity. Eliminating dp between eqs 1 and 2 and simplifying gives

Figure 1. Flow patterns of liquid phase: (a) spiral strip, (b) spiral band, (c) spiral film.

τl,w sin αSw Al wall shear stress

band, and spiral film. At low liquid flow rates, the incoming liquid droplets are centrifuged to the wall of the cyclone, and form a spiral strip because of the droplet coalescence. The liquid phase spirals down the wall of the cyclone with the effect of gravity and shear stresses. The liquid is maintained on the walls of the cyclone by the centrifugal forces. The spiral strip is gradually transformed into a spiral band, which is similar to the ribbon flow pattern reported by Hreiz et al.,23 with the increasing liquid flow rate. For higher liquid flow rates, the flow pattern of the liquid wall film transits from spiral band to spiral film, which is similar to the annular flow reported by Hreiz et al.23 The model, in this paper, is mainly available for the spiral film flow. Thus, the assumptions of this model are as follows: (1) all of the droplets entering the cyclone flow down as a uniformly thick film. (2) The gas and liquid phases spiral down the wall of cyclone at a constant angle to the horizontal, and the “Dean Effect”24 was ignored. (3) The incoming droplets are spun to the wall immediately, and the residence time from the inlet to the wall of a cyclone can be ignored. (4) The model is for a steady state only. 2.1. Cylinder Section. On the basis of the assumptions above, a force balance on the liquid phase in the axial z direction (Figure 2a) can be written as follows:

interfacial shear stress

(3)

where Ag = π[(D − 2δ) − de ]/4, δ is the film thickness, and de is the diameter of gas outlet, which is approximately equal to the diameter of upward gas core. Al = π[D2 − (D − 2δ)2]/4, A is the fraction of the total cross-sectional area, and A = πD2/4. Furthermore, the wetted wall perimeter, Sw = πD, and the interface perimeter, Si = π(D − 2δ). The wall shear stress in the film flow direction is defined as 1 τl,w = ρl v l 2fl,w (4) 2 Here, f l,w is the wall friction factor. There are many correlations for the wall friction factor in the gas/liquid pipe flow. However, there is little correlation for the wall friction factor for a spiral film with centrifugal force. Many reported correlations were used during the model development in this work, among which the Ouyang−Aziz correlation25 and Swamee−Jain correlation26 are more accurate for the prediction of residence time; even then, the Swamee−Jain correlation is developed for the friction factor in turbulent pipe flows. Furthermore, the Swamee−Jain correlation can account for the effect of wall roughness on the friction factor. The correlation of Ouyang−Aziz is 2

−Al dp − τl,w sin αSw dz + τg,i sin αSi dz + ρl Al g dz = 0 (1)

Here, the first term is the force caused by the axial pressure drop, the second and third terms are the wall/liquid and gas/ liquid shear stress components in the axial z direction, respectively, and the fourth term is the gravity of liquid. A force balance on the gas phase gives −Ag dp − τg,i sin αSi dz + ρg Ag g dz = 0

⎛1 1 ⎞⎟ − τg, i sin αSi⎜⎜ + − (ρl − ρg )g = 0 Ag ⎟⎠ ⎝ Al gravity

fl,w =

0.0926 1.629 ⎛ vs,g ⎞ ⎜ ⎟ ⎜ ⎟ Rel 0.516 ⎝ vs,l ⎠

2

(5)

where vs,g is the superficial gas velocity, vs,g = mg/(ρgA), vs,l is the superficial liquid velocity, vs,l = ml/(ρlA), and Rel is the liquid film Reynolds number ρ v lDe Rel = l μl (6)

(2)

where Ag, Al are the horizontal cross-sectional areas of gas and liquid wall film, respectively, dp is the axial pressure difference B


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Industrial & Engineering Chemistry Research Here, De is the film hydraulic diameter, De = 4Al/Sw, and the mean liquid velocity in the actual flow direction vl can be written as ml vl = ρl Al sin α (7)

Here, the first term is the force caused by the pressure drop, the second and third terms are the wall/liquid and gas/liquid shear stresses components in the wall direction, respectively, and the fourth term is the gravity component of liquid. A force balance on the gas phase gives

where ml and mg are the liquid and gas mass flow rates, respectively. The correlation of Swamee−Jain is

(14)

fl,w =

−A mg dp − τg,i sin αSmi dz + ρg A mg g cos β dz = 0

where Amg, Aml are the average cross-sectional areas of gas and liquid wall film in the cone section, respectively, Smi, Smw are the average perimeters of gas flow and liquid wall film, and β is the angle of the wall relative to the vertical. Eliminating dp between eqs 13 and 14 and simplifying gives

0.25 ⎡ ⎢⎣log10

(

e 3.7D

+

5.74 Re l 0.9

⎤2 ⎥⎦

)

(8)

−

where e is the absolute wall roughness, e = 0.046 mm (smooth wall). The liquid film Reynolds number Rel is calculated as eq 6. Similar to the wall shear stress, the interfacial shear stress in the gas flow direction is defined as τg,i =

1 f ρ (vg − v l)2 2i g

fg , i

= 1 + 300

δ D

(9)

(10)

(11)

vl =

where Reg, the gas Reynolds number, is defined as

Reg =

ρg vgDg μg

2

2

4ml 2

πρl [D − (D − 2δ)2 ]sin α

(16)

The mean residence time of the liquid phase over the cyclone cylinder can be obtained:

(12)

Here, Dg is the diameter of gas flow, Dg = D − 2δ, and vg is the gas velocity in the actual flow direction. It should be noted that the flow field of the gas phase in the cyclone is different from that in the annular flow. The gas velocity of annular flow in the pipe is uniform; thus, vg can be given as the mean gas velocity.29 However, the gas velocity distribution in the cyclone is more complex. It is well-known that the tangential gas velocity profile in cyclones resembles a Rankine vortex, and is approximately equal to the inlet velocity at the near wall region.22 The axial gas velocity is far less than the tangential velocity, and gradually decreases along the axial direction. According to the definition of interfacial shear stress, vg should be the gas velocity near the gas/liquid interface. Thus, the gas velocity vg was approximately given as the inlet gas velocity. 2.2. Cone Section. Because the diameter of the cone section gradually reduces, an equivalent diameter Dm was used to simplify the calculation, Dm = (D + Dc)/2. Here, Dc is the diameter of underflow outlet. Then, a force balance on the liquid phase in the wall direction (Figure 2b) can be written as follows:

θ1L =

H1 −

b 2

v l sin α

(17)

Here, H1 is the height of cyclone cylinder, and b is the height of the inlet. (3) With substution of the shear stresses, τg,i and τl,w, into the force balance equation, eq 15, an implicit equation, containing the film thickness, δ, was obtained. The implicit equation can be solved iteratively for the film thickness by trial and error. (4) The mean liquid velocity in the actual flow direction can be calculated: vml =

4ml πρl [Dm 2 − (Dm − 2δ)2 ]sin α

(18)

The mean residence time of the liquid phase over the cone section can be obtained θ2L =

H2 vml sin α cos β

(19)

Here, H2 is the height of the cone section. To reduce the deviation caused by the simplification of the average diameter, the cone section can also be solved by a segment method. The segment method is to divide the cone section into many segments at the axial direction, and the residence times in each segment can be calculated as the method mentioned above. (5)

−A ml dp − τl,w sin αSmw dz + τg,i sin αSmi dz + ρl A ml g cos β dz = 0

(15)

where Amg = π[(Dm − 2δ) − de ]/4, Aml = π[Dm − (Dm − 2δ)2]/4, and Am is the fraction of the total cross-sectional area, Am = πDm2/4. Furthermore, the wetted wall perimeter Smw = πDm, and the interface perimeter Smi = π(Dm − 2δ). The wall and interfacial shear stresses are calculated using the same method as that of the cylinder section. 2.3. Solution of Model Equations. The steps to calculate the residence time of the liquid phase in cyclones are as follows: (1) With a substitution of the shear stresses, τl,w and τg,i, from eqs 4 and 9, into the force balance equation, eq 3, an implicit equation, containing the film thickness, δ, was obtained. The implicit equation can be solved iteratively for the film thickness by trial and error. (2) Then, the mean liquid velocity in the actual flow direction can be calculated 2

For the turbulent gas flow, the interfacial friction factor, fg,i, can be calculated using the Blasius’ correlation:28 fg , i = 0.079Reg −0.25

A ml

⎛ 1 1 ⎞⎟ + τg,i sin αSmi⎜⎜ + A mg ⎟⎠ ⎝ A ml

+ (ρl − ρg )g cos β = 0

The ripples and waves at the liquid film surface significantly increase the interfacial friction. To account for the effect of ripples and waves on the interfacial friction, the correlation by Wailis27 was used: fi

τl,w sin αSmw

(13) C


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Figure 3. Schematic of the experimental system: (a) sketch of the experimental setup, and (b) schematic of the cyclone geometry. (Units = millimeters.)

as C0 = Ql/(Ql + Qg). The measured residence time of the liquid phase can be calculated as

The total residence time in the cyclone is the sum of residence times in cylinder and cone sections, θ = θ1L + θ2L. As a result of wall friction and the effects of gravity, the angle at which the liquid wall film flows relative to the horizontal will increase somewhat over the length of the cyclone. Therefore, this angle was just taken as an approximation. In general, for the volute or tangential inlet cyclones, an average value of 45−55° was suggested22 on the basis of observation.

θobs =

Vs Ql

(20)

where θobs is the measured residence time of liquid phase, Vs is the liquid holdup, and Ql is the liquid flow at the cyclone inlet. The separation efficiency of gas−liquid cyclone used in this work is greater than 99% at the investigated operating conditions, which has little effect on the test of residence time. The investigated operating conditions are given in the Supporting Information.

3. EXPERIMENTAL SECTION The model verification was conducted on a cold model experiment, and the sketch of the experimental equipment was shown in Figure 3. Air was drawn into the cyclone using an induced draft fan. The total air flow into the cyclone was controlled through a main valve, and measured using a Pitot tube. Water was injected by an atomizing nozzle, and the volume flow rate is measured by a turbine flowmeter with accuracy grade ±0.5%. First of all, the valves, V1 and V2, were opened. The atomized droplets and carrier gas were drawn into the cyclone by a draft fan. The incoming liquid droplets were centrifuged to the wall of the cyclone, formed a liquid wall film spiraling down along the wall, and flowed into the tank through the valve V1. Air flowed out through the vortex finder. When the operation reached a steady state, the liquid holdup, Qs, was measured when the valves V1 and V2 were shut off simultaneously. In order to ensure the repeatability of testing, each condition was tested more than three times. The cyclone was made of transparent plexiglass with volute inlet. The cyclone was 300 mm in height and 150 mm in diameter. The slot inlet was 70 mm in height and 35 mm in width, and the length of the inlet pipe is 1.0 m, so that the gas− liquid flow pattern in the inlet pipe was fully developed. The angle of the cone wall relative to the vertical was 16.7°. The gas velocity at the cyclone inlet was controlled in the range 10−20 m/s; the flow of water was 0.2−1.0 m 3 /h, and the corresponding droplets flow fraction of inlet, C0, ranged from 0.1 to 1.0 v%. The definition of droplet flow fraction was given

4. RESULTS AND DISCUSSION 4.1. Model Verification. To verify the accuracy of this model, liquid residence time was calculated for the different investigated operating conditions, and the wall friction factors were calculated using the correlations of Ouyang−Aziz and Swamee−Jain, respectively. In the verification experiments, the angle of the liquid film flow relative to the horizontal is given as 45° on the basis of observation. The comparison results were presented in Figure 4, and the predicted residence times are in good overall agreement with the measured values. The mean relative deviations are less than 15%, and the mean relative deviations obtained from Ouyang−Aziz and Swamee−Jain correlations are 14% and 12%, respectively. Most of the results obtained from Ouyang−Aziz correlation are greater than that of Swamee−Jain correlation. Furthermore, the predicted values of two models are both greater than the measured values at the liquid film Reynolds number Rel = 3100. The forces on the liquid wall film in the vertical direction include the wall shear stress, interfacial shear stress, and gravity. The wall shear stress is the largest and most important force, which is the main factor influencing the residence time of the liquid wall film. As shown in Figure 5, the wall friction factors calculated using the correlations of Ouyang−Aziz and Swamee−Jain decrease with the increase of liquid film Reynolds number, and the wall friction factors obtained from Swamee− D


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Figure 6. Relationship between Reynolds number and mean residence time of liquid wall film (vin = 18m/s).

Figure 4. Measured versus calculated mean residence time of liquid phase.

of the vortex core in a GLCC. For the film flow in the cyclone of this work, the centrifugal force and interfacial shear stress on the wall film may result in a later transition from laminar to turbulent flow. Thus, further study for the effect of the centrifugal force on the flow pattern of film flow is needed. 4.2. Effect of the Droplets Loading. The effect of the droplets loading on the residence time of the liquid phase was investigated by modeling and experiment. Figure 7 showed the

Figure 5. Comparison of the wall friction factors obtained from different correlations.

Jain correlation are less than that of the Ouyang−Aziz correlation; however, the difference becomes narrowing with the increase of liquid film Reynolds number. That may explain the difference of residence times obtained from the correlations of Ouyang−Aziz and Swamee−Jain. Furthermore, at Rel ≈ 3000 the results predicted by both models tend to a constant with the increase of liquid film Reynolds number, because the correlations of Ouyang−Aziz and Swamee−Jain were developed for the pipe flow or falling film flow without centrifugal force, which evolves from laminar to turbulent flow at Rel = 2500−3000.30 As shown in Figure 6, this characteristic leads the predicted residence times to a constant with the increase of liquid film Reynolds number. However, the measured residence times still decrease with the increase of liquid film Reynolds number at Rel = 3100. Thus, the predicted residence times are all greater than the measured values at Rel = 3100. Wang et al.31 reported that the thickness distribution of the rotary falling film is more uniform than that of the free falling film. It can be inferred that the fluctuation of liquid film with centrifugal force is suppressed. Hreiz et al.21 also reported that, at high Re, at the center of the flow, centrifugal effects damp turbulence and lead to a laminarization

Figure 7. Effect of the droplet loading on the liquid residence time (vin = 18 m/s).

relationship between the droplet loading and the residence times of the liquid phase at vin = 18m/s. The results indicate that the liquid residence times decrease with increasing droplet loading of inlet gas. The predicted residence times are in agreement with the measured values at vin = 18 m/s. Moreover, the effects of the droplet loading on the thickness and velocity of liquid wall film at vin = 18 m/s were shown in Figure 8. The correlation of Ouyang−Aziz was used to calculate the wall friction factor. Results indicate that the thickness and velocity of liquid wall film increase with the droplet loading of the inlet gas. The increasing rate of film thickness is greater than that of the film velocity. 4.3. Effect of the Inlet Velocity. Figure 9 showed the relationship between the measured residence time of the liquid E


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Figure 8. Effect of the droplet loading on the thickness and velocity of liquid wall film (vin = 18 m/s).

Figure 10. Ratio of forces to resultant force (vin = 18 m/s).

were investigated by modeling. The correlation of Swamee− Jain was used to calculate the wall friction factor. As shown in Figure 11, at H1 = 300 mm, Ql = 0.95 m3/h, the liquid

Figure 9. Effect of the inlet velocity on the mean residence time of liquid phase. Figure 11. Effects of the cylinder diameter on the liquid residence time and film velocity (H1 = 300 mm, Ql = 0.95 m3/h).

phase and the incoming liquid flow at different inlet velocities. The results indicate that the residence time decreases less with increasing inlet velocity at the same liquid flow. At Ql = 0.53 m3/h, the residence times with vin = 10 and 18 m/s are 1.02 and 0.95 s, respectively; the difference is only 0.07 s, as the inlet velocity affects the liquid residence time by changing the interfacial shear force. It can be inferred that the gas shear stress has less impact on the liquid residence time under the experimental conditions. The liquid wall film is mainly affected by two forces in the vertical downward direction: the interfacial shear stress and gravity. To analyze the effect of vertical downward forces on the film flow, the ratios of gravity and interfacial shear stress to vertical downward resultant force in eq 3 were calculated at vin = 18m/s, respectively. As shown in Figure 10, the ratio of gravity to resultant force is 80−90%, and the ratio of interfacial shear stress is only 10−20%. Therefore, the main reason for the lesser effect of inlet velocity on the liquid residence time is that the interfacial shear stress is far less than the gravity on liquid wall film. 4.4. Effect of the Cylinder Diameter. The effects of the cylinder diameter on the liquid residence time and film velocity

residence time increases with the cylinder diameter. The wall film velocity, however, decreases with increasing cylinder diameter, because the perimeter of the liquid wall film is directly proportional to the cylinder diameter.

5. CONCLUSIONS Residence time is a key parameter for the cyclones accompanied by heat and mass transfer or reaction process. This work presented a model to estimate the liquid residence time in the cylinder-on-cone cyclone, based on a force balance on the wall film. Furthermore, this model was validated using an experiment, which measured the residence time of liquid phase by hold-up method. The results demonstrated that the predicted residence times are in good overall agreement with the measured values. The predicted values are greater than the measured values at the liquid film Reynolds number Rel = 3100, because the centrifugal force and interfacial shear stress on the wall film may result in a later transition from laminar to turbulent flow. F


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Industrial & Engineering Chemistry Research Vs liquid holdup, m3

The effects of the droplet loading and inlet velocity on the flow characteristics of wall film were investigated by modeling and experiment. The results indicate that the liquid residence time decreases with increasing droplet loading of inlet gas. The thickness and velocity of liquid wall film increase with the droplet loading. The liquid residence time decreases less with increasing inlet velocity, because the interfacial shear stress is far less than the gravity on liquid wall film. Furthermore, the liquid residence time increases with the cylinder diameter. The wall film velocity, however, decreases with increasing cylinder diameter, because the perimeter of the liquid wall film is directly proportional to the cylinder diameter.

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Greek Letters

α angle of the gas and liquid wall film flows relative to the horizontal β angle of the wall relative to the vertical δ film thickness, mm θobs measured residence time of liquid phase, s θ1L residence time of liquid phase over cyclone cylinder, s θ2L residence time of liquid phase over cyclone cone, s μg gas viscosity, Pa s μl liquid viscosity, Pa s ρg gas density, kg/m3 ρl liquid density, kg/m3 τg,i gas/liquid shear stress in the actual flow direction, N τl,w wall/liquid shear stress in the actual flow direction, N

ASSOCIATED CONTENT

S Supporting Information *

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b02617. Investigated operating conditions (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Tel: +86(532)8698 1812. Fax: +86(532)8698 1787. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge the financial support from the China Postdoctoral Science Foundation (2014M551984), Shandong Postdoctoral Innovation Project Foundation (201402040), and Qingdao Postdoctoral Applied Research Project (2014).

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NOMENCLATURE Ag horizontal cross-sectional areas of gas, m2 Al horizontal cross-sectional areas of liquid wall film, m2 b cyclone inlet width, m C0 droplets flow fraction of inlet, v%, C0 = Ql/(Ql + Qg) D cyclone barrel diameter, m De film hydraulic diameter, m de diameter of gas outlet, m Dg diameter of gas flow, m Dm equivalent diameter of cone section, m e absolute wall roughness, mm fg,i interfacial friction factor f l,w wall friction factor g acceleration of gravity, m/s2 H1 height of cyclone cylinder, m H2 height of cyclone cone, m ml liquid mass flow rate, kg/s mg gas mass flow rate, kg/s p static pressure, Pa Ql liquid flow at the cyclone inlet, m3/s Rel liquid film Reynolds number Reg gas Reynolds number Si wetted wall perimeter, m Sw interface perimeter, m vin air velocity at the inlet, m/s vl mean liquid velocity in the actual flow direction, m/s vg gas velocity in the actual flow direction, m/s vs,g superficial gas velocity, m/s vs,l superficial liquid velocity, m/s G


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