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Time-frequency analysis of the vortex motion in a cylindrical cyclone separator
Chemical Engineering Journal 373 (2019) 1120–1131
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Time-frequency analysis of the vortex motion in a cylindrical cyclone separator
T
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Zhuwei Gaoa,b, Juan Wanga,b, , Jiangyun Wanga,b, Yu Maoa a b
State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, PR China Beijing Key Laboratory of Process Fluid Filtration and Separation, Beijing 102249, PR China
H I GH L IG H T S
flow in cyclone is double vortex motion, not single vortex motion. • The total frequency is neither a fixed value nor many decayed values. • The to three frequencies, the flow can be divided to three regions. • According motion frequency of vortex has self-preservation. • The • The motion frequency presents quantum characteristics.
A R T I C LE I N FO
A B S T R A C T
Keywords: Cyclone separator Numerical simulation Double vortex motion Time-frequency analysis Vortex energy transfer Quantum characteristic
This study aims to describe the double vortex motion via time-frequency analysis in cyclone separator, and propose a new cognition of the quantum characteristic of vortex energy transfer. A cylindrical cyclone separator with a large length-to-diameter ratio was designed to eliminate cone and dust hopper effects. Velocity was measured with a Phase Doppler Particle Analyzer (PDPA) and numerical simulations were performed via Reynolds stress model. The model was validated by comparison between numerical and experimental results. The results indicate that the total frequency distribution of fluctuating tangential velocity is neither a fixed value nor many decayed values in the cyclone separator. The total frequency has three specific values, according to which the flow in the cyclone separator can be divided to three regions (top, middle, and bottom). The flow performances differ in the three regions and have different spectrum forms. The fluid flow develops from single vortex motion dominated by a concentrated frequency to double vortex motion dominated by two concentrated frequencies. The energy is transferred from the high-frequency vortex to another low-frequency vortex. In this process, the motion frequency of vortex has self-preservation. The motion frequency of each vortex remains unchanged until the vortex disappears and the energy dissipates; the motion frequency in the whole cyclone is discontinuous, which encompasses the quantum characteristic.
1. Introduction The cyclone separator is a piece of industrial equipment that is utilized for gas-solid separation. It has simple structure, convenient operation and high separation efficiency which make it popular within in petroleum, chemical, coal power generation, and other industrial fields [1]. The flow field in the cyclone separator is complex and turbulent. The flow is unsteady, which causes velocity [2] and pressure [3] fluctuations which in turn vibrate the cyclone shell [4]. The dynamic flow performance in the cyclone separator thus has a far-reaching significance for engineering.
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As per the real-world industrial process, the working condition of cyclone separator is dynamic and continuous [5]. Some scholars have used the spectrum method [6] to analyze the dynamic continuous flow field as a transition from the time domain to the frequency domain [7]. Vortex motion has garnered a great deal of research attention in particular [8]. Table 1 briefly describes several notable publications regarding the dynamic properties of the vortex motion in cyclone separators. The extant research mostly centers on the so-called Precessing Vortex Core (PVC) phenomenon or the instantaneous tangential velocity. These works have provided a comprehensive and workable understanding of the vortex motion in cyclone separators.
Corresponding author at: State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, PR China. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.cej.2019.05.054 Received 20 November 2018; Received in revised form 19 March 2019; Accepted 9 May 2019 Available online 10 May 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.
Chemical Engineering Journal 373 (2019) 1120–1131
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ε φ ω γ μ μt ρ
Nomenclature a b C d dp De D DH e g fg L Le I p r R S t ttes u, v u¯ u′ x, y, z
inlet height (mm) inlet weight (mm) model constant wall thickness (mm) diameter of tracer particle (mm) vortex finder diameter (mm) cylinder diameter (mm) hydraulic diameter (mm) eccentric distance (mm) gravitational acceleration (m·s−2) gas flow rate (kg·s−1) length of inlet (mm) length of vortex finder (mm) turbulence intensity pressure (Pa) radial coordinate (mm) radius of the cylinder (mm) symmetric strain rate tensor (s−1) time (s) residence time (s) velocity (m/s) mean velocity (m/s) fluctuating velocity (m/s) coordinate (m)
turbulent dissipation rate (m2·s−3) the phase angle (°) angular frequency of the turbulence components wave length (nm) dynamic viscosity (kg·m−1s−1) eddy viscosity (kg·m−1s−1) density (kg·m−3)
Subscripts i, j, k t max p g
directions in the Cartesian coordinate system tangential maximum value tracer particle gas
Abbreviations CFD DPT HWA LDV LES PIV PVC RSM
computational fluid dynamic dynamic pressure transducer hot-wire anemometer laser-Doppler velocimetry large eddy simulation particle image velocimetry precessing vortex core Reynolds stress model
Greek symbols δij
Kronecker symbol
Table 1 Previous study on the dynamic property of the vortex. Year
Auther
Method/Model
Cone
Comments
1999 2000 2002
Hoekatra [9] Derksen [10] Solero [11]
LDV, RSM LES LDV
Yes No Yes
2003
Otermair [12]
LES
Yes
2005 2005
Derksen [13] Peng [14]
LES DPT
Yes Yes
2007
Wu [15]
RSM, PIV
Yes
2010
Wang [16]
HWA
Yes
2010
Gao [17]
DPT
Yes
2011 2013 2014
Gronald [18] Shukla [19] Cai [20]
LES RSM, LES HWA, DPT
Yes Yes Yes
2016
Gu [21]
DPT, LES
Yes
The forced core region of the flow is dominated by the so-called precessing vortex core. The core of the main vortex is observed to move about the geometrical axis of the cyclone in a quasi-periodic manner. A low frequency periodic oscillation is super-imposed to the chaotic turbulence spectrum. There is a frequency of rotation from a spectral analysis of the fluctuations. And this frequency is proportional to the volumetric gas flow. The position of the axis of the vortex changes with height. The maximum deviation of the vortex axis from the cyclone axis is at the bottom of the bin. The instantaneous velocity signal is quasi-periodical and exhibits a peak in the spectrum. The frequency with which the vortex core rotates varies with the gas flow rate and was found to be about the same as the frequency with which the gas rotates higher in the separator. The processing vortex core (PVC) phenomenon existed in all axial positions of the cyclone separator. The motion frequency of vortex core center is not the same with the frequency of the pulsation velocity where the PVC phenomenon exists. The real tangential velocity has a basic frequency near the center region, and the amplitude of real tangential velocity near the center region is bigger than that near the wall. The frequency of the inner vortex is different from that of the outer vortex. The inner vortex flow fluctuates stronger and faster than its outer partner. There is no apparent frequency in the dust hopper, while a shift towards low frequency in the drop tube. Due to the precession of the vortex core, there are high velocity fluctuation levels near the geometrical center. The flow is instability, which not only causes noise, but also causes velocity and the pressure fluctuation. The fluctuation will lead to vibration of the cyclone shell. There are two dominant frequencies of the pressure fluctuation in the gas flow. The second main frequency is not an integer multiple of the primary main frequency.
energy transfer quantum characteristics in the cyclone separator. We used a cylindrical cyclone separator with a large length-to-diameter ratio to observe the vortex motion of swirling fluid. We used a Reynolds stress model (RSM) to simulate the gas flow and the spectrum method to illustrate velocity fluctuations. The flow field simulation results were compared against experimental data to confirm the accuracy of the tangential velocity, fluctuation, and spectrum data. Our goal is to comprehensively understand the relationship between global frequency and vortex motion in the cyclone separator.
Previous researchers [9–11,13–14,16,22–24] have suggested that the flow in cyclone reflected in the spectrum has a main frequency. Recent studies, however, have shown that many monitoring points present two main frequencies [17,21]. Interestingly, the second main frequency is not an integer multiple of the primary main frequency. Most scholars consider the energy in the cyclone separator to decay gradually, but do not precisely observe how the energy transfers in the separator. This study was conducted to observe double vortex motion via timefrequency analysis. We also established a new conception of vortex 1121
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2. Experimental setup
Table 2 The physical dimensions of the cyclone.
2.1. Cyclone geometry in detail A cylindrical cyclone was designed as the basis for our vortex motion experiments. Fig. 1 shows the structure diagram of the cyclone. The cylindrical cyclone separator has no influence of the cone, which has a large length-to-diameter ratio [25]. This structure results in lower pressure drop and lower separation efficiency inside the cylindrical cyclone compared to the traditional cyclone. The cylindrical cyclone can thus be applied in the vertical multi-pipe cyclone separation system. The structure is mainly built into the secondary or tertiary cyclone to separate finer solid particles. The entrance structure of the cylindrical cyclone is a single inlet volute with 140-mm diameter. The horizontal plane across the bottom of the entrance is set here as the reference plane and the positive direction is upward. The geometric center of the vortex finder is set as the origin. The basic dimensions of the cyclone are listed in Table 2 as they correspond to the structure as shown in Fig. 1.
vg
(C1 + C3 γ )2 + (C2 ω + C3 γ )2 (C1 + C3 γ )2 + (ω + C3 γ )2
φ = arctan ωmax =
ω (C1 + C3 γ )2 (C2 − 1) (C1 + C3 γ )2 + (C2 ω + C3 γ )(ω + C3 γ )
0.38vRe 0.56D−1,
ω ≤ ωmax
Dimensions (mm)
Diameter of the cyclone body Height of the cylindrical part Vortex finder diameter Height of the Inlet Width of the Inlet Vortex finder length Length of the Inlet Length of the outlet Eccentric distance Wall thickness
D H De a b S L Le e d
140 1919 50 76 36 76 150 924 16 5
C1 = Re =
36μp (2ρp + ρg ug D μg
ρg ) dp2 , γ=
, C2 =
3ρg 2ρp + ρg
, C3 =
18 (2ρp + ρg ) dp
ρg μg π
,
πω 2
vp is the velocity of the trace particle; vg is the velocity of the gas; φ is the phase angle; and ω is the angular frequency of the turbulence components. vp/vg and φ/ω are 0.996 and 2.5 μs, respectively. So propylene glycol particles are satisfactory tracer particle for the PDPA measurement system.
Fig. 2 shows the schematic diagram of the experimental system. A Phase Doppler Particle Analyzer (PDPA) (Denmark Dantec Co.) was used to measure the flow field of the cyclone separator. The test vessel is a cylindrical plexiglass tube. The experiment was conducted under atmospheric pressure and ambient temperature conditions. The gas (air) was first drawn into the cyclone from the inlet, then formed a rotating flow, then escaped through the air outlet. To measure the air volume, a pitot tube was set between the air outlet of the separator and the filter. Twelve monitoring sections were recorded during the experiment: z = −10, −100, −300, −500, −700, −900, −1100, −1300, −1500, −1700, −1900 and −1910 mm. Tracer particles were released into the cyclone during the test as the PDPA measured their velocity based on the Doppler Effect. A test window was set up for each measurement point and placed under 2-mm thick optical glass to reduce the interference of the light path. The tracer particle selection has an important influence on the system’s measurement accuracy. Tracer particles need strong optical behavior and stable chemical properties, but also must flow well. We selected propylene glycol particles with 2-μm diameter produced by a LZL-type particle generator and assessed their behavior by Eqs. (1) to (3) [26].
=
Symbol
where
2.2. Experimental system
vp
Parameters
3. Numerical methods 3.1. Turbulence model In addition to the above considerations, it is important to ensure a manageable computational cost [27–29]. The Reynolds stress model (RSM) and the large eddy simulation (LES) model are widely used in the numerical simulation of cyclone separator [30]. The LES model requires a large number of meshes which require larger amounts of computation and longer monitoring time than the RSM. Previous researchers [31–33] have considered the RSM model superior for predicting the behavior of swirling flow in cyclone separators, so we used the RSM model to analyze the cyclone flow field in this study. The index of gas compressibility is the Mach number. The gas is generally considered incompressible when its Mach number is less than 0.3. When the Mach number is greater than 0.3, the compressibility of the gas gradually increases and the influence of compressibility must be considered in the calculation. Generally, the velocity of the flow field inside the cyclone separator is less than 80 m/s. Ambient air is considered to fall into the range of incompressible fluid, so the flow in the cyclone separator can be considered to be incompressible and viscous. For incompressible fluid with constant temperature in the cyclone separator, who has no energy transfer with the environment, the
(1)
(2) (3)
Fig. 1. The structure diagram of cyclone separator. 1122
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Tracking particle
Gas
Gas z=-10 mm z=-100 mm
r/R=1 r/R=0.86 r/R=0.71
z=-300 mm
r/R=0.57 r/R=0.29 r/R=0.14 r/R=0
Fan
Pitot tube
z=-500 mm
Filter z=-700 mm z=-900 mm z=-1100 mm z=-1300 mm
PDPA
Data collect box
z=-1500 mm z=-1700 mm z=-1900 mm z=-1910 mm
PDPA measurement
Cyclone
Computer
Experimental apparatus
Fig. 2. Schematic diagram of the experimental system.
Table 3 Numerical schemes for this study.
Table 4 Numerical simulation conditions.
Numerical setting
Scheme
Name
Condition
Parameter
Number
Pressure discretization Pressure velocity coupling Momentum discretization Turbulent kinetic energy Turbulent dissipation rate Reynolds stress
PRESTO! SIMPLEC QUICK Second-order upwind Second-order upwind First-order upwind
Medium
Air
Boundary
Initialization
Inlet Outlet Wall Inlet
Calculation
Transient
Density Viscosity Velocity Pressure Standard & no slip Hydraulic diameter Turbulence intensity Time step size
1.225 kg/m3 1.789 × 10−4 Pa·s 15 m/s 101.325 kPa — 48.857 4.835% 1 × 10−4 s
Table 5 Selection of the time step. Parameters
symbol
Number
Air density Gas flow rate Cyclone volume Residence time Time step size
ρ fg V tres t
1.225 kg/m3 0.060 kg/s 0.033 m3 0.674 s 1 × 10−4 s
The velocity components can be simplified as the mean u¯i and fluctuating ui′ (i = 1, 2, 3) components, which are related by: Fig. 3. The computational grid of the cyclone.
ui = u¯i + ui′
where the Reynolds stress term − ρui′¯u′j includes the turbulence closure, which must be modelled to solve Eq. (5). In the RSM, the transport equations can be written as follows:
governing equations are as follows.
∂ (ui ) =0 ∂x i ρ
∂ (uj ui ) ∂p ∂ui −ρ =− +ρ ∂x i ∂x j ∂t
(6)
(4)
∂ (ui′¯u′j ) ∂x j
+
∂uj ⎞ ∂ ⎛ ∂ui + )⎟ + ρgj ⎜μ ( ∂x i ⎠ ∂x j ⎝ ∂x j
ρ (5)
∂ (ui′¯u′ j ) ∂t
+ρ
∂ (uk ui′¯u′j ) ∂xk
= Dij + Pij + Πij + εij + Sij
The turbulent diffusion term is: 1123
(7)
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Fig. 4. The distributions of tangential velocity, z = −100 mm.
Dij = −
∂ ¯ ⎞⎤ ∂ ⎡ ρui′ u¯ ′j u′k + (p′¯u′j )δik + (p′¯ui′ )δjk − μ ⎛ ui′ u′j ∂ ∂xk ⎢ xk ⎠⎥ ⎝ ⎦ ⎣ ⎜
Fig. 6. Distribution of fluctuating tangential velocity at z = −100 mm.
The dissipation term is:
⎟
(8)
εij = −2μ
The stress production term is:
∂uj ∂u + u′j ¯uk′ i ⎞ Pij = −ρ ⎛ui′¯uk′ ∂xk ∂xk ⎠ ⎝ ⎜
(11)
The source term is: Sij In above formulas, δij is the Kronecker factor, μ is the molecular viscosity.
⎟
(9)
The pressure strain term is:
¯ ∂u ∂u j⎞ Πij = p ⎛⎜ i + ⎟ ∂ ∂ x x j i ⎝ ⎠
∂u¯i′ ∂u¯i′ ∂xk ∂xk
(10)
Fig. 5. Comparison of numerical results and experimental data. 1124
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time is about 0.674 s in this study. Therefore, the time step of 1 × 10−4 s is an acceptable value to simulate the transient flow in the cyclone separator. 3.3. Grid system and independence The cyclone computational grid was established in the ANSYS ICEM software. The whole computational domain was divided into structured hexahedron grids. The volute inlet in the cyclone needs special treatment to prevent an excessively sharp grid from forming [35]. We adopted a finer grid near the wall to simulate the turbulent flow [36,37]. A grid domain containing 426,810 cells was adapted to our simulation as shown in Fig. 3. We tested three grid domains in our preliminary computation containing 234,610, 426,810, and 585,410 hexahedral cells, respectively. The differences in cyclone pressure drop between the 234,610 and 585,410 cells were about 8.1%, and the differences in cyclone pressure drops between the 426,810 and 585,410 cells were less than 1.8%. The relative error in tangential velocity at the position of z = −100 mm between 234,610 and 585,410 cells was about 7.9%, and the relative error in tangential velocity between 426,810 and 585,410 cells was less than 1.7% (Fig. 4). The grid domains of 426,810 and 585,410 cells produced according results, suggesting that computed results are independent of the mesh size.
Fig. 7. Spectrum of fluctuating tangential velocity at z = −100 mm.
4. Results and discussion 4.1. Model validation We compared the numerical tangential velocity and axial velocity results against the experimental data to validate our models, as shown in Fig. 5. Both are in close agreement, which indicates that the models presented here have good prediction accuracy for the cyclone separator flow field. 4.2. Spectral analysis of dynamic flow 4.2.1. Tangential velocity fluctuation The flow in the cyclone separator is a dynamic continuous process. The velocity components are decomposed into the mean u¯i and fluctuating ui′ (i = 1, 2, 3) parts which have the following relationship:
Fig. 8. Total frequency distribution in the cyclone separator.
3.2. Simulation conditions 3.2.1. Numerical schemes We used the commercial CFD code Fluent 16.0 for the purposes of this study. Calculations were performed via segregated (pressure-based) solver method. The phase-coupled SIMPLEC algorithm was implemented to manage the coupling between velocity and pressure. The pressure gradient term was in PRESTO! (pressure staggering option) interpolation format. The numerical settings are summarized in Table 3.
ui = u¯i + ui′
(12)
When the flow is fully developed, the mean velocity is a fixed value, so fluctuations in velocity is mainly reflected in the fluctuating velocity. The distribution of fluctuating tangential velocity in monitoring point at z = −100 mm is shown as an example in Fig. 6; the fluctuating tangential velocity changes with time from −7 to 7 m/s. 4.2.2. Spectrogram of fluctuating tangential velocity The interaction of irregular gas turbulence pulsations and regular vortex core oscillations causes fluctuations in the cyclone. The tangential velocity also changes over time. We used Matlab software to compile an algorithm which manages this instantaneous signal. We also used a Fast Fourier Transform (FFT) [38] algorithm to complete the transformation from the time domain to the frequency domain, then analyzed the frequency, amplitude, and other relevant information in the spectrum. The transformation equations are:
3.2.2. Boundary conditions The velocity inlet condition was imposed at the inlet boundary surface as per the actual operating conditions of the system as shown in Fig. 3. The outlet boundary surface was placed under pressure outlet conditions to reflect the fully developed pipe flow. The axial gradient of variables at the outlet boundary surface was zero. The fluid flow was modeled using the standard wall function and no-slip boundary condition near the wall. Detailed simulation conditions, including the air density and dynamic viscosity, are shown in Table 4.
N /2 + 1
N /2 + 1
F (k ) =
3.2.3. Time step The time step is a critical aspect in any transient flow field analysis. In unsteady simulation, the time step should be less than the average residence time [34]. The average residence time (tres) in the cyclone separator is determined by the cyclone volume (V) and the gas flow rate (fg); it is calculated as tres = ρV/fg. As shown in Table 5, the residence
∑ n=1
f (2n) e−i2πnk (N /2) + e−i2π / N
∑
f (2n + 1) e−i2πnk (N /2)
n=1
(13) Fig. 7 shows the spectrum of fluctuating tangential velocity at the monitoring point r/R = 0.14 in the position z = −100 mm. The main frequency of the spectrum at this point is 49.5 Hz. In the spectrum, 1125
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Fig. 9. Spectrum of fluctuating tangential velocity at r/R = 0.14.
high-frequency and low-amplitude sections reflect the flow turbulence, which is irregular and random. The low-frequency and high-amplitude section has obvious periodicity reflective of the vortex core oscillation.
three specific values: 49.5, 36.5, and 8.5 Hz.
4.2.3. Total frequency in the cyclone separator The main frequency of vortex motion is determined by the coupling effect of flow parameters and geometric parameters in the cyclone separator. We analyzed all the monitoring points to determine the total frequency in the system as shown in Fig. 8. The total frequency distribution is neither a main frequency nor many decayed values throughout the whole separator. The main frequency distribution has
The main frequency observed during signal analysis and processing corresponds to vortex motion. The spectrum of the fluctuating tangential velocity also indirectly reflects changes in internal energy in the cyclone separator. We analyzed the axial distribution spectrum to observe the energy transfer of the internal flow as shown in Fig. 9. During the initial movement of the fluid, the frequency exerts obvious single dominant characteristics. The main frequency generally
4.3. Vortex self-preservation
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Fig. 10. Fluctuation and spectrum of fluctuating tangential velocity at z = −300 mm.
Fig. 11. Fluctuation and spectrum of fluctuating tangential velocity at z = −1100 mm.
Fig. 12. Fluctuation contours of tangential velocity at z = −300 mm.
this time, but the energy gradually turned to the new vortex (corresponding frequency of 36.5 Hz). The new secondary vortex developed gradually until a relationship formed between the primary and secondary vortexes. The secondary vortex corresponding to 36.5 Hz
corresponds to a vortex motion with relative periodic regularity. With sufficient development of gas flow, the vortex began to divide and another new vortex formed characterized by double vortex motion. The initial vortex (corresponding frequency of 49.5 Hz) still existed during 1127
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Fig. 13. Fluctuation contours of tangential velocity at z = −1100 mm.
Fig. 14. Fluctuation and spectrum of fluctuating tangential velocity at z = −700 mm.
The flow field distribution of single vortex motion is illustrated as a single main frequency. We zeroed in on two selections for subsequent analysis. Figs. 10 and 11 show the fluctuation and spectrum of fluctuating tangential velocity of r/R = 0.14 at positions z = -300 mm and z = -1100 mm. The frequencies of z = -300 mm and z = -1100 mm are 49.5 Hz and 36.5 Hz, so the period of vortex motion is:
became the main vortex and the primary vortex corresponding to 49.5 Hz became the auxiliary vortex. The vortex energy completely transformed during this process, however, the motion frequency of the vortex did not change as the vortex decayed. In other words, in the process of energy transfer, the motion frequency of the vortex tended toward self-preservation. 4.4. Single vortex motion and double vortex motion As per the above analysis, we know that energy was transferred from one vortex to another vortex. The flow changed from a single vortex motion dominated by a concentrated frequency to double vortex motion dominated by two concentrated frequencies. We next sought to verify its correctness, the key was focused on the correctness of double vortex motion.
T1 =
1 1 = = 0.0202s f1 49.5Hz
(14)
T2 =
1 1 = = 0.0274s f2 36.5Hz
(15)
We next divided the period into many equal parts and selected a set of data every 0.0040 s. Fig. 12 and Fig. 13 show fluctuation contours of tangential velocity at the positions of z = −300 mm and z = −1100 mm. As shown in Fig. 12, the tangential velocity distribution is similar to the condition of the initial position (t = 5.0000 s) after 0.0200 s. This time is very close to 0.202 s, which was the period of vortex motion in the above analysis at the position of z = −300 mm. Similarly, this time of the position of z = −1100 mm was about 0.0280 s (corresponding to 0.0274 s), where its motion was similar to the initial position (Fig. 13).
4.4.1. Single vortex motion The flow field distribution of single vortex motion was illustrated by single main frequency. Two selections were selected for the analysis. Figs. 10 and 11 shows the fluctuation and spectrum of fluctuating tangential velocity of r/R = 0.14 in the position of z = −300 mm and z = −1100 mm. 1128
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Fig. 15. Fluctuation contours of tangential velocity at z = −700 mm.
In effect, the flow of the two sections was mainly a single vortex motion.
4.5. Analysis of vortex motion The frequency corresponds to the internal position in the cyclone separator as per a detailed global analysis. Fig. 16 shows the global frequency distribution in the cyclone. There were three values of main frequency in the cyclone, as mentioned above. The internal flow field can be divided into three regions (top, middle, and bottom) according to these three main frequency values.
4.4.2. Double vortex motion Not all the flow we observed was characterized by single vortex motion. We analyzed the monitoring point at the axial position of z = −700 mm as per its fluctuation and fluctuating tangential velocity spectrum as shown in Fig. 14. The tangential velocity distribution at z = −700 mm was not similar to the distribution of z = −300 mm and z = −1100 mm, which had two main frequencies. Fig. 15 shows fluctuation contours of tangential velocity at the position of z = −700 mm. The distribution of tangential velocity was not close to 0.0202 s (corresponding to 49.5 Hz) nor was it close to 0.0274 s (corresponding to 36.5 Hz) similar to the initial position (t = 5.0000 s). It was about 0.0240 s, and the flow was similar to the initial flow. However, it is worth noting that the period of motion did not correspond to the two main frequencies. We found that the cross section of z = −700 mm was not a single vortex motion, but a collection of two or more complex vortex motions. This phenomenon can be explained by signal analysis. A single signal corresponds to a vortex motion which has a main frequency. The new signal period of the superposition of two signals falls between the two original signal periods. Based on the distribution of the flow field and the principle of the superposition of the signal, the results of spectrum diagram indicate that the flow of the z = −700 mm cross section does indeed have double vortex motion. The flow develops from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies as the fluid moves throughout the cyclone separator. The energy is transmitted from a vortex with a higher frequency (with gradually decreasing amplitude) to a vortex with a lower frequency (with gradually increasing amplitude). The vortex motion throughout this process also tends toward self-preservation. We found that the motion frequencies of the vortexes did not change until the vortex disappeared and the energy dissipated. As the energy dissipated, the high-frequency vortex disappeared and a new low-frequency vortex was generated. The flow then changed from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies. The global frequency of the flow was discontinuous in the whole cyclone separator, which represents the quantum characteristic.
(1) Top region (0 < z < 3.57D) The range of the top region in our system was between z = 0 and −500 mm. The gas from the inlet section flowed into the internal space of cyclone and developed under the effect of centrifugal force. The main frequencies were approximately the same along the radial direction. The flow in this region was a single vortex motion with a main frequency of 49.50 Hz. The single vortex motion was relatively regular, and the vortex structure was not broken in this region. Most of the energy of the vortex motion was not consumed in this region, but transferred to the downward region. The downward flow brought the energy to the middle region, while the upward flow brought the energy to the outlet region of the separator. The radial centripetal flow kept the main frequencies of the axial sections basically consistent. (2) Middle region (3.57D < z < 10.71D) At the axial position of z = −500 to −1500 mm, the frequency changed along the radial direction and was no longer a fixed value as in the top region. In other words, the flow of this region had changed. There were two main frequencies in this region: 49.5 Hz and 36.5 Hz. The amplitude of the vortex with 49.5 Hz decreased continuously, indicating that the energy of this vortex also decreased. The amplitude of the vortex with 36.5 Hz increased continuously, indicating that the energy of this new vortex gradually increased. In other words, the energy of the vortex of 49.5 Hz was gradually transmitted to the vortex of 36.5 Hz. The frequency of two vortexes was unchanged throughout this process because of the frequency of the vortex was self-sustaining. The flow changed from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies, during which time the energy was transferred and attenuated. 1129
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of the two vortexes was also unchanged, i.e., was self-sustaining during the energy transfer process. We found that the energy in the cyclone separator was transmitted from the higher-frequency vortex (gradually decreasing amplitude) to the low-frequency vortex (gradually increasing amplitude). In the process of this motion, the vortex was self-sustaining and its motion frequency did not change. The global frequency of the vortex was not continuously distributed in the whole space, but rather jumped from one frequency to another showing quantum characteristics. 5. Conclusions A cylindrical cyclone with a large length-to-diameter ratio was used in this study to assess various aspects of vortex performance. By numerical simulation and PDPA experiments, we observed the double vortex motion as per using time-frequency analysis in the cyclone separator and established a new conception of the quantum characteristic of vortex energy transfer. We analyzed the motion of vortexes in the flow process as per their motion frequency distribution. Our main conclusions can be summarized as follows. (1) The total frequency distribution is neither a main frequency nor many decayed values in the whole cyclone separator. The main frequency distribution has three specific values. (2) The flow develops from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies as fluid moves through the cyclone separator. (3) According to the vortex motion and main frequency distribution, the internal flow field of the cyclone separator can be divided into three regions: a. The top region (0 < z < 2.57D), which has a stable main frequency of 49.5 Hz and consumes relatively little energy. b. The middle region (3.57D < z < 10.71D), which features the coexistence of multi-master frequencies with 49.5, 36.5, and 8.5 Hz. Energy is transferred and decayed between vortexes in this region. c. The bottom region (z > 10.71D), where the main frequencies are relatively low and the flow is a combination of small-scale vortices (4) The motion frequency of the vortex tends toward self-preservation. The motion frequency of any one vortex does not change until the vortex disappears. The energy is transmitted from the vortex with higher frequency (gradually decreasing amplitude) to a vortex with lower frequency (gradually increasing amplitude). Therefore, the global frequency of vortex motion is discontinuous in the whole cyclone separator representative of the quantum characteristic. Acknowledgements The authors gratefully acknowledge the support from the National Basic Research Program of China (No. 21106181) and the Science Foundation of China University of Petroleum, Beijing (No. 2462015YQ0303). Fig. 16. Global frequency distribution in cyclone separator.
References (3) Bottom region (z > 10.71D)
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The range of this region was from z = −1500 mm to the bottom of the separator. The distributions of main frequencies changed again in this section. The main frequency values were almost constant in the radial direction and were relatively low. The flow was relatively stable in this region; it was a combination flow of small-scale vortices. The vortex with 49.5 Hz disappeared completely while the vortex with 36.5 Hz was retained accompanied by a new vortex of 8.5 Hz. Similar to the middle region, the energy transferred from the high-value vortex with 36.5 Hz to the low-value vortex with 8.5 Hz. The motion frequency 1130
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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej
Time-frequency analysis of the vortex motion in a cylindrical cyclone separator
T
⁎
Zhuwei Gaoa,b, Juan Wanga,b, , Jiangyun Wanga,b, Yu Maoa a b
State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, PR China Beijing Key Laboratory of Process Fluid Filtration and Separation, Beijing 102249, PR China
H I GH L IG H T S
flow in cyclone is double vortex motion, not single vortex motion. • The total frequency is neither a fixed value nor many decayed values. • The to three frequencies, the flow can be divided to three regions. • According motion frequency of vortex has self-preservation. • The • The motion frequency presents quantum characteristics.
A R T I C LE I N FO
A B S T R A C T
Keywords: Cyclone separator Numerical simulation Double vortex motion Time-frequency analysis Vortex energy transfer Quantum characteristic
This study aims to describe the double vortex motion via time-frequency analysis in cyclone separator, and propose a new cognition of the quantum characteristic of vortex energy transfer. A cylindrical cyclone separator with a large length-to-diameter ratio was designed to eliminate cone and dust hopper effects. Velocity was measured with a Phase Doppler Particle Analyzer (PDPA) and numerical simulations were performed via Reynolds stress model. The model was validated by comparison between numerical and experimental results. The results indicate that the total frequency distribution of fluctuating tangential velocity is neither a fixed value nor many decayed values in the cyclone separator. The total frequency has three specific values, according to which the flow in the cyclone separator can be divided to three regions (top, middle, and bottom). The flow performances differ in the three regions and have different spectrum forms. The fluid flow develops from single vortex motion dominated by a concentrated frequency to double vortex motion dominated by two concentrated frequencies. The energy is transferred from the high-frequency vortex to another low-frequency vortex. In this process, the motion frequency of vortex has self-preservation. The motion frequency of each vortex remains unchanged until the vortex disappears and the energy dissipates; the motion frequency in the whole cyclone is discontinuous, which encompasses the quantum characteristic.
1. Introduction The cyclone separator is a piece of industrial equipment that is utilized for gas-solid separation. It has simple structure, convenient operation and high separation efficiency which make it popular within in petroleum, chemical, coal power generation, and other industrial fields [1]. The flow field in the cyclone separator is complex and turbulent. The flow is unsteady, which causes velocity [2] and pressure [3] fluctuations which in turn vibrate the cyclone shell [4]. The dynamic flow performance in the cyclone separator thus has a far-reaching significance for engineering.
⁎
As per the real-world industrial process, the working condition of cyclone separator is dynamic and continuous [5]. Some scholars have used the spectrum method [6] to analyze the dynamic continuous flow field as a transition from the time domain to the frequency domain [7]. Vortex motion has garnered a great deal of research attention in particular [8]. Table 1 briefly describes several notable publications regarding the dynamic properties of the vortex motion in cyclone separators. The extant research mostly centers on the so-called Precessing Vortex Core (PVC) phenomenon or the instantaneous tangential velocity. These works have provided a comprehensive and workable understanding of the vortex motion in cyclone separators.
Corresponding author at: State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, PR China. E-mail address: [email protected] (J. Wang).
https://doi.org/10.1016/j.cej.2019.05.054 Received 20 November 2018; Received in revised form 19 March 2019; Accepted 9 May 2019 Available online 10 May 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.
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ε φ ω γ μ μt ρ
Nomenclature a b C d dp De D DH e g fg L Le I p r R S t ttes u, v u¯ u′ x, y, z
inlet height (mm) inlet weight (mm) model constant wall thickness (mm) diameter of tracer particle (mm) vortex finder diameter (mm) cylinder diameter (mm) hydraulic diameter (mm) eccentric distance (mm) gravitational acceleration (m·s−2) gas flow rate (kg·s−1) length of inlet (mm) length of vortex finder (mm) turbulence intensity pressure (Pa) radial coordinate (mm) radius of the cylinder (mm) symmetric strain rate tensor (s−1) time (s) residence time (s) velocity (m/s) mean velocity (m/s) fluctuating velocity (m/s) coordinate (m)
turbulent dissipation rate (m2·s−3) the phase angle (°) angular frequency of the turbulence components wave length (nm) dynamic viscosity (kg·m−1s−1) eddy viscosity (kg·m−1s−1) density (kg·m−3)
Subscripts i, j, k t max p g
directions in the Cartesian coordinate system tangential maximum value tracer particle gas
Abbreviations CFD DPT HWA LDV LES PIV PVC RSM
computational fluid dynamic dynamic pressure transducer hot-wire anemometer laser-Doppler velocimetry large eddy simulation particle image velocimetry precessing vortex core Reynolds stress model
Greek symbols δij
Kronecker symbol
Table 1 Previous study on the dynamic property of the vortex. Year
Auther
Method/Model
Cone
Comments
1999 2000 2002
Hoekatra [9] Derksen [10] Solero [11]
LDV, RSM LES LDV
Yes No Yes
2003
Otermair [12]
LES
Yes
2005 2005
Derksen [13] Peng [14]
LES DPT
Yes Yes
2007
Wu [15]
RSM, PIV
Yes
2010
Wang [16]
HWA
Yes
2010
Gao [17]
DPT
Yes
2011 2013 2014
Gronald [18] Shukla [19] Cai [20]
LES RSM, LES HWA, DPT
Yes Yes Yes
2016
Gu [21]
DPT, LES
Yes
The forced core region of the flow is dominated by the so-called precessing vortex core. The core of the main vortex is observed to move about the geometrical axis of the cyclone in a quasi-periodic manner. A low frequency periodic oscillation is super-imposed to the chaotic turbulence spectrum. There is a frequency of rotation from a spectral analysis of the fluctuations. And this frequency is proportional to the volumetric gas flow. The position of the axis of the vortex changes with height. The maximum deviation of the vortex axis from the cyclone axis is at the bottom of the bin. The instantaneous velocity signal is quasi-periodical and exhibits a peak in the spectrum. The frequency with which the vortex core rotates varies with the gas flow rate and was found to be about the same as the frequency with which the gas rotates higher in the separator. The processing vortex core (PVC) phenomenon existed in all axial positions of the cyclone separator. The motion frequency of vortex core center is not the same with the frequency of the pulsation velocity where the PVC phenomenon exists. The real tangential velocity has a basic frequency near the center region, and the amplitude of real tangential velocity near the center region is bigger than that near the wall. The frequency of the inner vortex is different from that of the outer vortex. The inner vortex flow fluctuates stronger and faster than its outer partner. There is no apparent frequency in the dust hopper, while a shift towards low frequency in the drop tube. Due to the precession of the vortex core, there are high velocity fluctuation levels near the geometrical center. The flow is instability, which not only causes noise, but also causes velocity and the pressure fluctuation. The fluctuation will lead to vibration of the cyclone shell. There are two dominant frequencies of the pressure fluctuation in the gas flow. The second main frequency is not an integer multiple of the primary main frequency.
energy transfer quantum characteristics in the cyclone separator. We used a cylindrical cyclone separator with a large length-to-diameter ratio to observe the vortex motion of swirling fluid. We used a Reynolds stress model (RSM) to simulate the gas flow and the spectrum method to illustrate velocity fluctuations. The flow field simulation results were compared against experimental data to confirm the accuracy of the tangential velocity, fluctuation, and spectrum data. Our goal is to comprehensively understand the relationship between global frequency and vortex motion in the cyclone separator.
Previous researchers [9–11,13–14,16,22–24] have suggested that the flow in cyclone reflected in the spectrum has a main frequency. Recent studies, however, have shown that many monitoring points present two main frequencies [17,21]. Interestingly, the second main frequency is not an integer multiple of the primary main frequency. Most scholars consider the energy in the cyclone separator to decay gradually, but do not precisely observe how the energy transfers in the separator. This study was conducted to observe double vortex motion via timefrequency analysis. We also established a new conception of vortex 1121
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2. Experimental setup
Table 2 The physical dimensions of the cyclone.
2.1. Cyclone geometry in detail A cylindrical cyclone was designed as the basis for our vortex motion experiments. Fig. 1 shows the structure diagram of the cyclone. The cylindrical cyclone separator has no influence of the cone, which has a large length-to-diameter ratio [25]. This structure results in lower pressure drop and lower separation efficiency inside the cylindrical cyclone compared to the traditional cyclone. The cylindrical cyclone can thus be applied in the vertical multi-pipe cyclone separation system. The structure is mainly built into the secondary or tertiary cyclone to separate finer solid particles. The entrance structure of the cylindrical cyclone is a single inlet volute with 140-mm diameter. The horizontal plane across the bottom of the entrance is set here as the reference plane and the positive direction is upward. The geometric center of the vortex finder is set as the origin. The basic dimensions of the cyclone are listed in Table 2 as they correspond to the structure as shown in Fig. 1.
vg
(C1 + C3 γ )2 + (C2 ω + C3 γ )2 (C1 + C3 γ )2 + (ω + C3 γ )2
φ = arctan ωmax =
ω (C1 + C3 γ )2 (C2 − 1) (C1 + C3 γ )2 + (C2 ω + C3 γ )(ω + C3 γ )
0.38vRe 0.56D−1,
ω ≤ ωmax
Dimensions (mm)
Diameter of the cyclone body Height of the cylindrical part Vortex finder diameter Height of the Inlet Width of the Inlet Vortex finder length Length of the Inlet Length of the outlet Eccentric distance Wall thickness
D H De a b S L Le e d
140 1919 50 76 36 76 150 924 16 5
C1 = Re =
36μp (2ρp + ρg ug D μg
ρg ) dp2 , γ=
, C2 =
3ρg 2ρp + ρg
, C3 =
18 (2ρp + ρg ) dp
ρg μg π
,
πω 2
vp is the velocity of the trace particle; vg is the velocity of the gas; φ is the phase angle; and ω is the angular frequency of the turbulence components. vp/vg and φ/ω are 0.996 and 2.5 μs, respectively. So propylene glycol particles are satisfactory tracer particle for the PDPA measurement system.
Fig. 2 shows the schematic diagram of the experimental system. A Phase Doppler Particle Analyzer (PDPA) (Denmark Dantec Co.) was used to measure the flow field of the cyclone separator. The test vessel is a cylindrical plexiglass tube. The experiment was conducted under atmospheric pressure and ambient temperature conditions. The gas (air) was first drawn into the cyclone from the inlet, then formed a rotating flow, then escaped through the air outlet. To measure the air volume, a pitot tube was set between the air outlet of the separator and the filter. Twelve monitoring sections were recorded during the experiment: z = −10, −100, −300, −500, −700, −900, −1100, −1300, −1500, −1700, −1900 and −1910 mm. Tracer particles were released into the cyclone during the test as the PDPA measured their velocity based on the Doppler Effect. A test window was set up for each measurement point and placed under 2-mm thick optical glass to reduce the interference of the light path. The tracer particle selection has an important influence on the system’s measurement accuracy. Tracer particles need strong optical behavior and stable chemical properties, but also must flow well. We selected propylene glycol particles with 2-μm diameter produced by a LZL-type particle generator and assessed their behavior by Eqs. (1) to (3) [26].
=
Symbol
where
2.2. Experimental system
vp
Parameters
3. Numerical methods 3.1. Turbulence model In addition to the above considerations, it is important to ensure a manageable computational cost [27–29]. The Reynolds stress model (RSM) and the large eddy simulation (LES) model are widely used in the numerical simulation of cyclone separator [30]. The LES model requires a large number of meshes which require larger amounts of computation and longer monitoring time than the RSM. Previous researchers [31–33] have considered the RSM model superior for predicting the behavior of swirling flow in cyclone separators, so we used the RSM model to analyze the cyclone flow field in this study. The index of gas compressibility is the Mach number. The gas is generally considered incompressible when its Mach number is less than 0.3. When the Mach number is greater than 0.3, the compressibility of the gas gradually increases and the influence of compressibility must be considered in the calculation. Generally, the velocity of the flow field inside the cyclone separator is less than 80 m/s. Ambient air is considered to fall into the range of incompressible fluid, so the flow in the cyclone separator can be considered to be incompressible and viscous. For incompressible fluid with constant temperature in the cyclone separator, who has no energy transfer with the environment, the
(1)
(2) (3)
Fig. 1. The structure diagram of cyclone separator. 1122
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Tracking particle
Gas
Gas z=-10 mm z=-100 mm
r/R=1 r/R=0.86 r/R=0.71
z=-300 mm
r/R=0.57 r/R=0.29 r/R=0.14 r/R=0
Fan
Pitot tube
z=-500 mm
Filter z=-700 mm z=-900 mm z=-1100 mm z=-1300 mm
PDPA
Data collect box
z=-1500 mm z=-1700 mm z=-1900 mm z=-1910 mm
PDPA measurement
Cyclone
Computer
Experimental apparatus
Fig. 2. Schematic diagram of the experimental system.
Table 3 Numerical schemes for this study.
Table 4 Numerical simulation conditions.
Numerical setting
Scheme
Name
Condition
Parameter
Number
Pressure discretization Pressure velocity coupling Momentum discretization Turbulent kinetic energy Turbulent dissipation rate Reynolds stress
PRESTO! SIMPLEC QUICK Second-order upwind Second-order upwind First-order upwind
Medium
Air
Boundary
Initialization
Inlet Outlet Wall Inlet
Calculation
Transient
Density Viscosity Velocity Pressure Standard & no slip Hydraulic diameter Turbulence intensity Time step size
1.225 kg/m3 1.789 × 10−4 Pa·s 15 m/s 101.325 kPa — 48.857 4.835% 1 × 10−4 s
Table 5 Selection of the time step. Parameters
symbol
Number
Air density Gas flow rate Cyclone volume Residence time Time step size
ρ fg V tres t
1.225 kg/m3 0.060 kg/s 0.033 m3 0.674 s 1 × 10−4 s
The velocity components can be simplified as the mean u¯i and fluctuating ui′ (i = 1, 2, 3) components, which are related by: Fig. 3. The computational grid of the cyclone.
ui = u¯i + ui′
where the Reynolds stress term − ρui′¯u′j includes the turbulence closure, which must be modelled to solve Eq. (5). In the RSM, the transport equations can be written as follows:
governing equations are as follows.
∂ (ui ) =0 ∂x i ρ
∂ (uj ui ) ∂p ∂ui −ρ =− +ρ ∂x i ∂x j ∂t
(6)
(4)
∂ (ui′¯u′j ) ∂x j
+
∂uj ⎞ ∂ ⎛ ∂ui + )⎟ + ρgj ⎜μ ( ∂x i ⎠ ∂x j ⎝ ∂x j
ρ (5)
∂ (ui′¯u′ j ) ∂t
+ρ
∂ (uk ui′¯u′j ) ∂xk
= Dij + Pij + Πij + εij + Sij
The turbulent diffusion term is: 1123
(7)
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Fig. 4. The distributions of tangential velocity, z = −100 mm.
Dij = −
∂ ¯ ⎞⎤ ∂ ⎡ ρui′ u¯ ′j u′k + (p′¯u′j )δik + (p′¯ui′ )δjk − μ ⎛ ui′ u′j ∂ ∂xk ⎢ xk ⎠⎥ ⎝ ⎦ ⎣ ⎜
Fig. 6. Distribution of fluctuating tangential velocity at z = −100 mm.
The dissipation term is:
⎟
(8)
εij = −2μ
The stress production term is:
∂uj ∂u + u′j ¯uk′ i ⎞ Pij = −ρ ⎛ui′¯uk′ ∂xk ∂xk ⎠ ⎝ ⎜
(11)
The source term is: Sij In above formulas, δij is the Kronecker factor, μ is the molecular viscosity.
⎟
(9)
The pressure strain term is:
¯ ∂u ∂u j⎞ Πij = p ⎛⎜ i + ⎟ ∂ ∂ x x j i ⎝ ⎠
∂u¯i′ ∂u¯i′ ∂xk ∂xk
(10)
Fig. 5. Comparison of numerical results and experimental data. 1124
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time is about 0.674 s in this study. Therefore, the time step of 1 × 10−4 s is an acceptable value to simulate the transient flow in the cyclone separator. 3.3. Grid system and independence The cyclone computational grid was established in the ANSYS ICEM software. The whole computational domain was divided into structured hexahedron grids. The volute inlet in the cyclone needs special treatment to prevent an excessively sharp grid from forming [35]. We adopted a finer grid near the wall to simulate the turbulent flow [36,37]. A grid domain containing 426,810 cells was adapted to our simulation as shown in Fig. 3. We tested three grid domains in our preliminary computation containing 234,610, 426,810, and 585,410 hexahedral cells, respectively. The differences in cyclone pressure drop between the 234,610 and 585,410 cells were about 8.1%, and the differences in cyclone pressure drops between the 426,810 and 585,410 cells were less than 1.8%. The relative error in tangential velocity at the position of z = −100 mm between 234,610 and 585,410 cells was about 7.9%, and the relative error in tangential velocity between 426,810 and 585,410 cells was less than 1.7% (Fig. 4). The grid domains of 426,810 and 585,410 cells produced according results, suggesting that computed results are independent of the mesh size.
Fig. 7. Spectrum of fluctuating tangential velocity at z = −100 mm.
4. Results and discussion 4.1. Model validation We compared the numerical tangential velocity and axial velocity results against the experimental data to validate our models, as shown in Fig. 5. Both are in close agreement, which indicates that the models presented here have good prediction accuracy for the cyclone separator flow field. 4.2. Spectral analysis of dynamic flow 4.2.1. Tangential velocity fluctuation The flow in the cyclone separator is a dynamic continuous process. The velocity components are decomposed into the mean u¯i and fluctuating ui′ (i = 1, 2, 3) parts which have the following relationship:
Fig. 8. Total frequency distribution in the cyclone separator.
3.2. Simulation conditions 3.2.1. Numerical schemes We used the commercial CFD code Fluent 16.0 for the purposes of this study. Calculations were performed via segregated (pressure-based) solver method. The phase-coupled SIMPLEC algorithm was implemented to manage the coupling between velocity and pressure. The pressure gradient term was in PRESTO! (pressure staggering option) interpolation format. The numerical settings are summarized in Table 3.
ui = u¯i + ui′
(12)
When the flow is fully developed, the mean velocity is a fixed value, so fluctuations in velocity is mainly reflected in the fluctuating velocity. The distribution of fluctuating tangential velocity in monitoring point at z = −100 mm is shown as an example in Fig. 6; the fluctuating tangential velocity changes with time from −7 to 7 m/s. 4.2.2. Spectrogram of fluctuating tangential velocity The interaction of irregular gas turbulence pulsations and regular vortex core oscillations causes fluctuations in the cyclone. The tangential velocity also changes over time. We used Matlab software to compile an algorithm which manages this instantaneous signal. We also used a Fast Fourier Transform (FFT) [38] algorithm to complete the transformation from the time domain to the frequency domain, then analyzed the frequency, amplitude, and other relevant information in the spectrum. The transformation equations are:
3.2.2. Boundary conditions The velocity inlet condition was imposed at the inlet boundary surface as per the actual operating conditions of the system as shown in Fig. 3. The outlet boundary surface was placed under pressure outlet conditions to reflect the fully developed pipe flow. The axial gradient of variables at the outlet boundary surface was zero. The fluid flow was modeled using the standard wall function and no-slip boundary condition near the wall. Detailed simulation conditions, including the air density and dynamic viscosity, are shown in Table 4.
N /2 + 1
N /2 + 1
F (k ) =
3.2.3. Time step The time step is a critical aspect in any transient flow field analysis. In unsteady simulation, the time step should be less than the average residence time [34]. The average residence time (tres) in the cyclone separator is determined by the cyclone volume (V) and the gas flow rate (fg); it is calculated as tres = ρV/fg. As shown in Table 5, the residence
∑ n=1
f (2n) e−i2πnk (N /2) + e−i2π / N
∑
f (2n + 1) e−i2πnk (N /2)
n=1
(13) Fig. 7 shows the spectrum of fluctuating tangential velocity at the monitoring point r/R = 0.14 in the position z = −100 mm. The main frequency of the spectrum at this point is 49.5 Hz. In the spectrum, 1125
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Fig. 9. Spectrum of fluctuating tangential velocity at r/R = 0.14.
high-frequency and low-amplitude sections reflect the flow turbulence, which is irregular and random. The low-frequency and high-amplitude section has obvious periodicity reflective of the vortex core oscillation.
three specific values: 49.5, 36.5, and 8.5 Hz.
4.2.3. Total frequency in the cyclone separator The main frequency of vortex motion is determined by the coupling effect of flow parameters and geometric parameters in the cyclone separator. We analyzed all the monitoring points to determine the total frequency in the system as shown in Fig. 8. The total frequency distribution is neither a main frequency nor many decayed values throughout the whole separator. The main frequency distribution has
The main frequency observed during signal analysis and processing corresponds to vortex motion. The spectrum of the fluctuating tangential velocity also indirectly reflects changes in internal energy in the cyclone separator. We analyzed the axial distribution spectrum to observe the energy transfer of the internal flow as shown in Fig. 9. During the initial movement of the fluid, the frequency exerts obvious single dominant characteristics. The main frequency generally
4.3. Vortex self-preservation
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Fig. 10. Fluctuation and spectrum of fluctuating tangential velocity at z = −300 mm.
Fig. 11. Fluctuation and spectrum of fluctuating tangential velocity at z = −1100 mm.
Fig. 12. Fluctuation contours of tangential velocity at z = −300 mm.
this time, but the energy gradually turned to the new vortex (corresponding frequency of 36.5 Hz). The new secondary vortex developed gradually until a relationship formed between the primary and secondary vortexes. The secondary vortex corresponding to 36.5 Hz
corresponds to a vortex motion with relative periodic regularity. With sufficient development of gas flow, the vortex began to divide and another new vortex formed characterized by double vortex motion. The initial vortex (corresponding frequency of 49.5 Hz) still existed during 1127
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Fig. 13. Fluctuation contours of tangential velocity at z = −1100 mm.
Fig. 14. Fluctuation and spectrum of fluctuating tangential velocity at z = −700 mm.
The flow field distribution of single vortex motion is illustrated as a single main frequency. We zeroed in on two selections for subsequent analysis. Figs. 10 and 11 show the fluctuation and spectrum of fluctuating tangential velocity of r/R = 0.14 at positions z = -300 mm and z = -1100 mm. The frequencies of z = -300 mm and z = -1100 mm are 49.5 Hz and 36.5 Hz, so the period of vortex motion is:
became the main vortex and the primary vortex corresponding to 49.5 Hz became the auxiliary vortex. The vortex energy completely transformed during this process, however, the motion frequency of the vortex did not change as the vortex decayed. In other words, in the process of energy transfer, the motion frequency of the vortex tended toward self-preservation. 4.4. Single vortex motion and double vortex motion As per the above analysis, we know that energy was transferred from one vortex to another vortex. The flow changed from a single vortex motion dominated by a concentrated frequency to double vortex motion dominated by two concentrated frequencies. We next sought to verify its correctness, the key was focused on the correctness of double vortex motion.
T1 =
1 1 = = 0.0202s f1 49.5Hz
(14)
T2 =
1 1 = = 0.0274s f2 36.5Hz
(15)
We next divided the period into many equal parts and selected a set of data every 0.0040 s. Fig. 12 and Fig. 13 show fluctuation contours of tangential velocity at the positions of z = −300 mm and z = −1100 mm. As shown in Fig. 12, the tangential velocity distribution is similar to the condition of the initial position (t = 5.0000 s) after 0.0200 s. This time is very close to 0.202 s, which was the period of vortex motion in the above analysis at the position of z = −300 mm. Similarly, this time of the position of z = −1100 mm was about 0.0280 s (corresponding to 0.0274 s), where its motion was similar to the initial position (Fig. 13).
4.4.1. Single vortex motion The flow field distribution of single vortex motion was illustrated by single main frequency. Two selections were selected for the analysis. Figs. 10 and 11 shows the fluctuation and spectrum of fluctuating tangential velocity of r/R = 0.14 in the position of z = −300 mm and z = −1100 mm. 1128
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Fig. 15. Fluctuation contours of tangential velocity at z = −700 mm.
In effect, the flow of the two sections was mainly a single vortex motion.
4.5. Analysis of vortex motion The frequency corresponds to the internal position in the cyclone separator as per a detailed global analysis. Fig. 16 shows the global frequency distribution in the cyclone. There were three values of main frequency in the cyclone, as mentioned above. The internal flow field can be divided into three regions (top, middle, and bottom) according to these three main frequency values.
4.4.2. Double vortex motion Not all the flow we observed was characterized by single vortex motion. We analyzed the monitoring point at the axial position of z = −700 mm as per its fluctuation and fluctuating tangential velocity spectrum as shown in Fig. 14. The tangential velocity distribution at z = −700 mm was not similar to the distribution of z = −300 mm and z = −1100 mm, which had two main frequencies. Fig. 15 shows fluctuation contours of tangential velocity at the position of z = −700 mm. The distribution of tangential velocity was not close to 0.0202 s (corresponding to 49.5 Hz) nor was it close to 0.0274 s (corresponding to 36.5 Hz) similar to the initial position (t = 5.0000 s). It was about 0.0240 s, and the flow was similar to the initial flow. However, it is worth noting that the period of motion did not correspond to the two main frequencies. We found that the cross section of z = −700 mm was not a single vortex motion, but a collection of two or more complex vortex motions. This phenomenon can be explained by signal analysis. A single signal corresponds to a vortex motion which has a main frequency. The new signal period of the superposition of two signals falls between the two original signal periods. Based on the distribution of the flow field and the principle of the superposition of the signal, the results of spectrum diagram indicate that the flow of the z = −700 mm cross section does indeed have double vortex motion. The flow develops from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies as the fluid moves throughout the cyclone separator. The energy is transmitted from a vortex with a higher frequency (with gradually decreasing amplitude) to a vortex with a lower frequency (with gradually increasing amplitude). The vortex motion throughout this process also tends toward self-preservation. We found that the motion frequencies of the vortexes did not change until the vortex disappeared and the energy dissipated. As the energy dissipated, the high-frequency vortex disappeared and a new low-frequency vortex was generated. The flow then changed from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies. The global frequency of the flow was discontinuous in the whole cyclone separator, which represents the quantum characteristic.
(1) Top region (0 < z < 3.57D) The range of the top region in our system was between z = 0 and −500 mm. The gas from the inlet section flowed into the internal space of cyclone and developed under the effect of centrifugal force. The main frequencies were approximately the same along the radial direction. The flow in this region was a single vortex motion with a main frequency of 49.50 Hz. The single vortex motion was relatively regular, and the vortex structure was not broken in this region. Most of the energy of the vortex motion was not consumed in this region, but transferred to the downward region. The downward flow brought the energy to the middle region, while the upward flow brought the energy to the outlet region of the separator. The radial centripetal flow kept the main frequencies of the axial sections basically consistent. (2) Middle region (3.57D < z < 10.71D) At the axial position of z = −500 to −1500 mm, the frequency changed along the radial direction and was no longer a fixed value as in the top region. In other words, the flow of this region had changed. There were two main frequencies in this region: 49.5 Hz and 36.5 Hz. The amplitude of the vortex with 49.5 Hz decreased continuously, indicating that the energy of this vortex also decreased. The amplitude of the vortex with 36.5 Hz increased continuously, indicating that the energy of this new vortex gradually increased. In other words, the energy of the vortex of 49.5 Hz was gradually transmitted to the vortex of 36.5 Hz. The frequency of two vortexes was unchanged throughout this process because of the frequency of the vortex was self-sustaining. The flow changed from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies, during which time the energy was transferred and attenuated. 1129
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of the two vortexes was also unchanged, i.e., was self-sustaining during the energy transfer process. We found that the energy in the cyclone separator was transmitted from the higher-frequency vortex (gradually decreasing amplitude) to the low-frequency vortex (gradually increasing amplitude). In the process of this motion, the vortex was self-sustaining and its motion frequency did not change. The global frequency of the vortex was not continuously distributed in the whole space, but rather jumped from one frequency to another showing quantum characteristics. 5. Conclusions A cylindrical cyclone with a large length-to-diameter ratio was used in this study to assess various aspects of vortex performance. By numerical simulation and PDPA experiments, we observed the double vortex motion as per using time-frequency analysis in the cyclone separator and established a new conception of the quantum characteristic of vortex energy transfer. We analyzed the motion of vortexes in the flow process as per their motion frequency distribution. Our main conclusions can be summarized as follows. (1) The total frequency distribution is neither a main frequency nor many decayed values in the whole cyclone separator. The main frequency distribution has three specific values. (2) The flow develops from a single vortex motion dominated by a concentrated frequency to a double vortex motion dominated by two concentrated frequencies as fluid moves through the cyclone separator. (3) According to the vortex motion and main frequency distribution, the internal flow field of the cyclone separator can be divided into three regions: a. The top region (0 < z < 2.57D), which has a stable main frequency of 49.5 Hz and consumes relatively little energy. b. The middle region (3.57D < z < 10.71D), which features the coexistence of multi-master frequencies with 49.5, 36.5, and 8.5 Hz. Energy is transferred and decayed between vortexes in this region. c. The bottom region (z > 10.71D), where the main frequencies are relatively low and the flow is a combination of small-scale vortices (4) The motion frequency of the vortex tends toward self-preservation. The motion frequency of any one vortex does not change until the vortex disappears. The energy is transmitted from the vortex with higher frequency (gradually decreasing amplitude) to a vortex with lower frequency (gradually increasing amplitude). Therefore, the global frequency of vortex motion is discontinuous in the whole cyclone separator representative of the quantum characteristic. Acknowledgements The authors gratefully acknowledge the support from the National Basic Research Program of China (No. 21106181) and the Science Foundation of China University of Petroleum, Beijing (No. 2462015YQ0303). Fig. 16. Global frequency distribution in cyclone separator.
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