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A universal model to calculate cyclone pressure drop
Powder Technology 171 (2007) 184 – 191 www.elsevier.com/locate/powtec
A universal model to calculate cyclone pressure drop Jianyi Chen a,⁎, Mingxian Shi b a
Institute of Mechanical and Electrical Engineering, China University of Petroleum, Dongying, Shandong, China 257061 b School of Petrochemical Engineering, China University of Petroleum, Beijing, China 102200 Received 8 February 2006; received in revised form 4 September 2006; accepted 27 September 2006; Available online 7 October 2006
Abstract The definition and composition of the pressure drop over a tangential inlet, reverse flow cyclone have been analyzed. It is assumed that two factors mainly contribute to the pressure drop, i.e., the local loss and the loss along the distance. The former includes the expansion loss at the cyclone inlet and the contraction loss at the entrance of the outlet tube (or vortex finder). The latter consists of the swirling loss resulting from friction at the cyclone walls and the dissipation of gas dynamic energy in the outlet tube. By use of the measured results of the flow field in cyclones, the calculation methods for each loss have been developed. And a universal model to predict the cyclone pressure drop is thus obtained simply by summing each loss. A detailed comparison between the calculated and experimental results shows that this accurate model is suitable either for pure or for dust laden gases at normal or high temperatures and can meet the requirement of most cyclone designs. © 2006 Elsevier B.V. All rights reserved. Keywords: Cyclone separator; Pressure drop; Calculation; Model
1. Introduction Cyclone separators have been the most common devices for the removal of dispersed particles from their carrying gases because of their simple structure, low cost and ease of operation. The pressure drop is one of the two important performance parameters. Many researchers [1–7] have developed different procedures to estimate the pressure drop of cyclones. Most of the procedures, however, are empirical and suitable only for pure gases, and not very satisfactory in generality. Additionally, the pressure drop of a cyclone under the condition of dust-laden gases or high temperature gases is of importance, and moreover it is much different from that under the condition of normal and pure gases. Briggs [8] found that the pressure drop decreases with an increase of the dust loading. Muschelknautz [6], Yuu [9], Hoffmann [10] and Luo [11] not only verified Briggs' findings but also discovered other interesting phenomena. For example, Muschelknautz [6] and Luo [11] found that the pressure drop falls with the rise of the dust loading until one turning point is reached at which the pressure drop begins to increase. As for the ⁎ Corresponding author. Tel.: +86 546 8393034; fax: +86 546 8393914. E-mail address: [email protected] (J. Chen). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.09.014
determination of the pressure drop of cyclones working at elevated temperatures, only very little experimental data have been reported, which are insufficient for a correct evaluation of the temperature effect on the pressure drop. Bohnet [12] observed that either the pressure drop or the drag coefficient decreases with an increase of gas temperature. Xu [13] and Chen [14] also confirmed such an observation. The prediction of the pressure drop over a cyclone dealing with high temperature or dust-laden gases, however, has not been well solved. The present paper analyses the definition and composition of the pressure drop over a tangential inlet, reverse flow cyclone, and then establishes a universal model for its prediction by employing the measured flow fields in cyclones. It is shown that the present model is good in accuracy and suitable for the prediction of the pressure drop over cyclones operating with pure or dust-laden gases at normal or high temperatures. 2. Definition and composition of the pressure drop 2.1. Definition of the pressure drop Fig. 1 is a schematic diagram of a most commonly used, tangential inlet and reverse flow cyclone. Generally, the
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
185
Fig. 3. The axial velocity near the cyclone outlet.
flow rectifier. Hoffmann [10] observed that the static pressure at the outlet wall is close to the static pressure that would be measured after an ideal rectifier. Therefore, the static pressure at the wall of the outlet tube minus the static pressure at the inlet gives the true dissipative loss or the pressure drop of a cyclone. The latter three ways have been widely used in research and engineering practices. And the pressure drop in this paper is defined as Eq. (1) and taken in a way similar to that of Meissner and Löffler. Fig. 1. Sketch of a reverse flow cyclone separator.
2.2. Composition of the pressure drop pressure drop over a cyclone is the difference of static pressure between the inlet and the outlet, which can be written as: DP ¼ Psi −Pso
ð1Þ
The static pressure at the inlet cross-section is uniformly distributed because there is no swirling motion. It can be easily measured with a pressure tapping in the wall. But the static pressure at the outlet wall is quite different from its crosssectional average due to the strong swirling flow. The dynamic pressure stored in the swirling motion can be significant. The determination of the static pressure downstream of a cyclone, hence the pressure drop, becomes more complicated and difficult. In the past, this problem has been approached in several ways. Stairmand [4] presumably measured the static pressure at the outlet wall immediately downstream of a cyclone as he ignored the influence of the swirl. Shepherd and Lapple [1] discharged the gas directly from the cyclone to atmosphere (taking the downstream pressure as the ambient pressure). Meissner and Löffler [15] measured the static pressure after a
Fig. 2. (a). Slot inlet. (b). Volute inlet.
The pressure drop over a cyclone consists of a local loss and a frictional loss (or a loss along the distance). The local loss includes an expansion loss at the cyclone inlet, ΔP1, and a contraction loss at the entrance of the outlet tube, ΔP2. The frictional loss includes a swirling loss due to the friction between the gas flow and the cyclone wall, ΔP3, and a dissipation loss of the gas dynamic energy in the outlet, ΔP4. Therefore, the pressure drop can be expressed as: DP ¼ DP1 þ DP2 þ DP3 þ DP4
ð2Þ
3. Method for pure gases 3.1. The expansion loss at the cyclone inlet—ΔP1 The gas flow will expand both radially and axially after entering a cyclone, resulting in a local expansion loss. For a slot inlet, the radial expansion loss is a function of the variable b / (R − re). But for a volute inlet, the radial expansion loss is a function of the variable b / (R + b − c~b − re), where c~ = c / b and
Fig. 4. The tangential and axial velocity in the cyclone outlet.
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Table 1 Comparison between measured and calculated drag coefficients under normal temperature and pure gas conditions ~ Model D/m a/D b/D KA dr S/D Stairmand
0.305 0.305 0.305 0.610 1.524 4.572 0.335 0.287 0.287 0.300 0.400 0.400 0.400 0.400 0.400 0.600 0.600 0.600 0.600 0.600 0.800 0.800 0.800 0.800 0.800 1.200 1.200 1.200 1.200
Stern Lapple PV
0.50 0.35 0.50 0.25 0.60 0.67 0.61 0.53 0.53 0.56 0.49 0.49 0.56 0.56 0.64 0.56 0.56 0.56 0.49 0.64 0.56 0.56 0.56 0.49 0.64 0.48 0.55 0.55 0.55
0.22 0.31 0.20 0.13 0.25 0.13 0.32 0.23 0.11 0.25 0.22 0.22 0.26 0.26 0.29 0.26 0.26 0.26 0.22 0.29 0.25 0.25 0.25 0.22 0.29 0.22 0.25 0.25 0.25
7.14 7.14 7.85 25.12 5.23 8.83 4.05 6.31 13.87 5.50 7.20 7.20 5.48 5.48 4.18 5.45 5.45 5.45 7.33 4.26 5.51 5.51 5.51 7.20 4.26 7.48 5.65 5.65 5.65
0.45 0.39 0.50 0.50 0.50 0.33 0.56 0.52 0.52 0.32 0.32 0.25 0.32 0.44 0.32 0.32 0.24 0.44 0.32 0.44 0.45 0.31 0.25 0.31 0.31 0.25 0.25 0.31 0.44
0.50 0.35 0.50 1.50 1.50 0.40 0.91 1.60 1.60 0.56 0.49 0.49 0.56 0.56 0.64 0.56 0.56 0.56 0.49 0.64 0.56 0.56 0.56 0.49 0.64 0.48 0.55 0.55 0.55
H1 / D
H/D
dc / D
2.5 2.5 1.5 2.0 2.0 1.1 1.4 2.1 2.1 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6
5.0 5.0 4.0 4.0 4.0 2.0 2.7 4.3 4.3 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8
0.375 0.375 0.375 0.375 0.375 0.375 0.400 0.250 0.250 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400
(1) (2) (3) (4) (5) (6) (7) Note: ξ(0) p —this method; ξp —Shepherd and Lapple [1]; ξp —First [2]; ξp —Alexander [3]; ξp —Stairmand [4]; ξp —Barth [5]; ξp —Muschelknautz [6]; ξp — Casal [7].
c is the width of the inlet cutting into the cyclone body, see Fig. 2. And the radial expansion loss can be expressed as [16]: DP1r ¼
1−
b R þ b−e c b−re
2
qg Vi2 2
The axial expansion loss, however, is not as easy to determine. For the sake of simplicity, a correction coefficient, ki, is employed to account for the contribution of the axial expansion loss. ki is less than unity and its usual value is 0.3. As a result, the expansion loss at the cyclone inlet, ΔP1, can be expressed as follows: DP1 ¼
1−ki
b R þ b−ec b−re
2
qg Vi2 2
A volute inlet changes to a slot inlet when c~ = 1. For common designs, c~ is not less than 1 / 3. With the sizes being nondimensionalized by the cyclone barrel diameter D, it is obtained that: DP1 ¼
1−
2ki be 1 þ 1:33 e b− der
2
qg Vi2 2
ð3Þ
3.2. The contraction loss at the entrance of the outlet tube —ΔP2 The contraction loss at the entrance of the outlet tube (or vortex finder) occurs because of an abrupt reduction of the flow area when gases enter the outlet tube from the separation space of a cyclone. According to the axial velocity profile in the separation space [17], the axial flow can be divided into an upward flow and a downward flow, as shown in Fig. 3. And the flow areas of the upward and downward flow are roughly 1 / 3 and 2 / 3 of the cross-sectional area of the cyclone barrel, respectively. pffiffiffi It means that their interfacial radius approximates 3R=3, or the averaged axial velocity of the upward flow is equal to 3Vi / KA. Consequently, ΔP2 can be expressed as [16] 2 2 1−3 de r qg V 2 2 qg 3Vi i DP2 ¼ 0:5 1−3 der ¼ 4:5 2 KA 2 KA2
ð4Þ
3.3. The swirling loss—ΔP3 The friction between the gas flow and the cyclone wall due to the gas viscosity will result in a loss of swirling energy, which is defined as the swirling loss of a cyclone here. Because of this viscous friction, the tangential flow in a cyclone is a combination
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187
Vi/m/s
ξm
ξ(0) p
ξ(1) p
ξ(2) p
ξ(3) p
ξ(4) p
ξ(5) p
ξ(6) p
ξ(7) p
9.28 9.28 5.05 15.07 16.07 16.15 16.07 14.97 28.42 20.00 17.62 17.43 18.89 16.66 14.48 14.98 13.15 14.95 18.04 15.27 17.90 16.48 15.58 17.14 16.93 17.01 21.90 13.11 18.00
8.70 9.07 5.68 3.38 10.60 11.40 7.25 7.19 3.68 20.30 17.90 26.89 21.74 11.77 27.28 22.56 37.78 13.15 18.06 14.70 13.76 24.35 38.67 20.02 30.20 30.85 43.18 25.18 13.16 σ
8.04 9.97 5.90 2.98 8.68 13.49 8.45 5.49 3.65 20.85 16.97 26.86 22.22 11.62 28.56 22.85 41.30 12.31 17.98 15.22 13.02 24.76 39.71 19.55 32.43 31.40 43.73 25.25 13.98 3 7.3 × 10−
8.69 11.60 6.40 2.00 9.60 12.80 9.76 7.33 3.33 23.01 17.58 29.06 23.10 11.73 30.28 23.08 40.69 11.85 17.16 15.24 11.51 23.42 36.77 17.92 30.29 26.85 35.56 23.12 11.48 2 2.1 × 10−
7.08 9.44 6.18 1.89 9.07 19.57 11.91 6.65 3.02 22.69 17.34 28.66 22.78 11.57 29.86 22.76 40.12 11.68 16.92 15.03 11.35 23.09 36.25 17.67 29.87 26.48 35.06 22.80 11.32 2 5.0 × 10−
5.84 6.73 5.23 2.65 6.00 14.26 5.62 4.89 3.55 17.01 13.23 23.90 17.07 7.92 22.73 17.03 36.49 7.98 12.94 9.45 7.78 17.39 31.61 13.51 22.85 21.74 30.28 17.22 7.84 2 8.0 × 10−
7.35 10.16 5.26 1.77 9.55 19.96 7.81 5.92 2.69 20.86 16.88 29.63 22.18 10.29 29.08 23.76 45.55 10.95 17.67 14.09 11.02 25.54 43.32 19.55 33.04 34.79 46.07 27.44 11.72 2 4.2 × 10−
6.47 8.93 5.41 2.90 8.85 16.79 9.37 6.74 3.62 22.05 16.44 29.49 22.14 10.12 29.66 22.10 43.72 10.22 16.01 13.11 9.91 22.51 38.68 16.80 29.75 26.87 37.20 22.23 9.92 2 2.9 × 10−
7.10 11.00 5.71 2.04 8.39 11.09 10.45 6.41 2.77 22.00 15.50 28.21 22.11 10.71 32.11 22.11 45.83 10.82 15.05 14.56 10.50 22.45 39.84 15.83 32.03 25.38 37.96 22.04 10.43 2 3.3 × 10−
5.66 8.27 4.14 2.51 6.40 9.56 6.54 4.70 2.82 25.71 15.97 39.61 25.88 8.41 42.81 25.84 75.40 8.53 15.33 12.58 8.18 26.53 61.99 16.51 42.82 34.16 58.13 25.93 8.15 1 1.3 × 10−
of a quasi-free vortex and a quasi-forced vortex, instead of a combination of a free vortex and a forced vortex. Referring to the theories by Barth [5] and Muschelknautz [6], the swirling loss can be calculated by the following equation:
DP3 ¼
f0 Fs qg ðVhw Vhe Þ1:5 2 ð0:9Qi Þ
cyclone hopper, and the swirling velocity is only somewhat slightly less than in the separation space [17,18]. It means that there is also friction at the wall of the cyclone hopper, which will lead to an additional pressure loss. This friction loss is accounted for simply by adding the surface area of the cyclone hopper into Fs in this paper, and thus
ð5Þ Fs ¼
~ ~ where Qi = abVi = πD2Vi / 4KA, Vθw = V θwVi and Vθe = V θeVi. Qi stands for the inlet gas flow rate. For a quasi-free vortex flow, e e −n Vehe ¼ Vehw re −n e ¼ V hw d r :
ð6Þ
p þ ðD þ dc Þ 2
DP3 ¼
qg Vi2 2
ð7Þ
Where Fs is the total area of the contact surfaces between the gas flow and the cyclone wall. Usually Fs is the sum of the top cover area, the surface area of the cyclone barrel and cone, and the external surface area of the outlet tube. However, experimental results showed that there also exists a swirling motion in the
H22
ðD−dc Þ2 þ pDb Hb þ 4
ð8Þ
Fs can also be non-dimensionalized as
Eq. (5) is transformed into: 3 4KA f0 FsVe hw 0:9pD2 de 1:5n r
p 2 2 D −De þ pDe S þ pDH1 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fes ¼
and
4Fs e 2r þ 4 der Se þ 4 He1 ¼ 1− d pD2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 e eb Heb þ 1 þ d c 4 He2 þ 1− dec þ 4 D 3 −1:5n DP3 ¼ 1:11f0 KA FesVehw de r
qg Vi2 2
ð9Þ
Where f0 is the friction coefficient between a pure gas flow and a cyclone wall. For a steel cyclone f0 = 0.005 and for
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cyclones having a different roughness of wall surface the value of f0 can be determined as specified in reference [6]. 3.4. The dissipation loss of the gas dynamic energy in the outlet—ΔP4 In the outlet tube, the gas tangential velocity is still very high and the tangential gas flow is also a combination of a quasi-free vortex and a quasi-forced vortex. The swirl flow does not decay significantly in the outlet tube. The axial velocity, however, is totally different from that in the separation space. In the vicinity of the swirl center the axial velocity is very low, but it becomes quite high in the annular region near the outlet wall, see Fig. 4 [18]. Most of the gas flows axially through this region. It is natural to divide the gas flow in the outlet tube into two regions: a core region, in which the axial velocity is assumed as zero, and an annular region in which the axial velocity is uniformly distributed. The tangential flow conforms to the law of a quasiforced vortex in the core region and of a quasi-free vortex in the annular region, respectively. Assuming that rc is the radius of the boundary between the core and the annular region, the axial velocity in the annular region is then as follows: Vze ¼
4Qi D2 1 ¼ Vi ¼ Vi 2 2 2 2 2 pðDe −Dc Þ KA ðDe −Dc Þ KA ðder − rec2 Þ
pressure drop over a cyclone in most cases) that it can be omitted for the sake of simplicity. Therefore the above expression can be shortened as n¼
þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi V¯ h ¼ Vhc Vhe ¼ Vehw ðerc ree Þ0:5n Vi ¼ Vehw ðe rc der Þ0:5n Vi
ð12Þ
Consequently, the pressure loss due to the dynamic dissipation or ΔP4 is " 2 # qg ¯ 2 qg Vi2 1 Ve hw 2 DP4 ¼ ðVh þ Vze Þ ¼ þ 2 2 2 ðe rc der Þn KA2 ð e d r −erc2 Þ2
2ki e b 1 þ 1:33eb−e dr
2 þ
3 −1:5n þ1:11f0 KAFesVehw der
4:5ð1−3 der Þ KA2 2 1 Ve þ hw n þ 2 2 e ðe rc der Þ KA ðd r −erc2 Þ2
3
ð15Þ
e 2hw 1 V þ ðe rc der Þn KA2 ðde2r −e r 2c Þ2
ð16Þ
1:11KA−0:21de r Re0:06 pffiffiffiffiffiffiffiffiffiffi 1 þ f0 Fes KAder 0:16
ð17Þ
Where Re is the cyclone outlet Reynolds number and can be defined as Re ¼
qg Ve De qg Vi D ¼ lg lg KAder
ð18Þ
And the swirl exponent n can also be correlated with Re by the following equation " # S−a −0:5 0:12 n ¼ 1− exp − 0:26Re 1þ ð19Þ b
j j
3.5. Calculation and comparison ð13Þ
Now the pressure drop over a cyclone, ΔP, can be easily obtained by summation of ΔP1 ∼ ΔP4. But it is more often to use the drag coefficient, ξ, which is derived as n ¼ 1−
−1:5n
þ1:11f0 KAFesVeh w de r
In addition, the dimensionless tangential velocity at the cyclone wall is Vehw ¼
According to the transportation law of the moment-ofmomentum of gas rotation, the mean tangential velocity in the outlet tube can be approximately expressed as
2
2 rec ¼ 0:38 der þ 0:5 de r
ð10Þ
ð11Þ
2ki eb e der 1 þ 1:33 b−
Now the pressure drop or the drag coefficient over a cyclone can be determined by Eq. (15) if the dimensions of and the tangential velocity in a cyclone are known. Take a Model PV cyclone as an example. The measured results of its flow field showed that rc is normally 0.5–0.6 times the radius of the outlet tube [17,18]. Furthermore, when nondimensionalized, rc can be correlated with a dimensionless diameter of the cyclone outlet, d˜r, as follows
The non-dimensional tangential velocity at the radius, rc, is Vehc ¼ Vehw er c−n
1−
ð14Þ
A detailed calculation showed that the second term in Eq. (14), i.e., the contraction loss, is so small (not more than 1% the
Table 1 lists the measured and calculated drag coefficients of dozens of cyclones operated under ambient temperature and pure gas conditions. The calculation is accomplished by using Eqs. (15)–(19) and assuming ki = 0.3 and f0 = 0.005. For Model Stairmand, Stern and Lapple cyclones, the measured results are cited from the corresponding references, while for Model PV cyclones the measurements are taken elsewhere by the authors. All the test gases were ambient air. The calculated drag coefficients by other researchers are also listed for comparison. The accuracy of the present method is satisfactory except for few cases. In average, the mean square deviation between the measured and calculated results of the present method is less than that of other methods by one or two orders of magnitude.
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Table 2 Comparison between measured and calculated drag coefficients at different gas temperatures ~ D/m KA dr Vi/m/s T/K ρg /kg/m3 ξm ξ(0) ξ(1) p p
ξ(2) p
ξ(3) p
ξ(4) p
ξ(5) p
ξ(6) p
ξ(7) p
0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300
22.69 22.69 22.69 22.69 22.69 22.69 22.69 22.69
17.01 17.01 17.01 17.01 17.01 17.01 17.01 17.01
20.86 18.55 16.59 16.66 14.85 14.85 14.85 14.85
22.05 22.05 22.05 22.05 22.05 22.05 22.05 22.05
22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00
25.71 25.71 25.71 25.71 25.71 25.71 25.71 25.71
5.50 5.50 5.50 5.50 5.50 5.50 5.50 5.50
0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315
20.00 19.62 20.11 16.14 36.29 28.24 19.82 16.56
303 470 685 676 973 973 973 973
1.08 0.69 0.49 0.49 0.34 0.34 0.34 0.34
20.30 19.30 18.40 18.40 17.40 16.60 17.10 17.20
Listed in Table 2 is a comparison between the measured and calculated drag coefficients of a Model PV cyclone operated at different temperatures. The values of ki and f0 used in Table 2 are assumed to be the same as in Table 1. The gas is normal pressure air. Obviously the calculated results agree very well with the measured ones. In fact, the rise of gas temperature will lead to a decrease of gas density and an increase of gas viscosity, both of which contribute to a weaker swirling motion and consequently result in a reduction of the swirling loss in the separation space and the dissipation loss in the outlet. It is indicated that the drag coefficient and the temperature can be correlated with simply through a Reynolds number, Re, which has an influence on the swirl exponent and tangential velocity at the cyclone wall. 4. Method for dust-laden gases The calculation of the pressure drop over a cyclone for dust-laden gases is more complicated and has not been well solved, though Briggs [10], Muschelknautz [6], Luo [11] and Bohnet [12] developed some empirical or semi-empirical
20.85 18.52 17.00 16.50 16.98 16.40 15.63 15.27
methods.In the case of dust-laden gases, it is also reasonable to keep the assumption that the pressure drop over a cyclone still consists of the four losses defined in Section 3, but their calculation methods are somehow different. Let Ci stand for the inlet dust loading. Checking Eqs. (3), (4), (9) and (13), one observes that under the dust-laden gas condition, the gas density ρg shall be replaced by (ρg + Ci) in Eq. (3) and [ρg + (1 − η)Ci] in Eqs. (4), (9) and (13), where η is the separation efficiency of a cyclone. A low dust loading generally means a much smaller value of (1 − η)Ci relative to ρg. Even at high dust loadings (1 − η)Ci is also much less than ρg because a high dust loading will lead to a high separation efficiency η, usually approaching unity. Therefore, it is not necessary to use [ρg + (1 − η)Ci] instead of ρg in the calculations of ΔP2, ΔP3 and ΔP4 by Eqs. (4), (9) and (13). However, two variables, f0 and ~ V θw, bear relation to the dust loading Ci and hence need to be redetermined while the remainders can be kept unchanged. For ~ ′ stand for the friction coefficient dust-laden gases, let f and Vθw and non-dimensional tangential velocity at the cyclone wall, respectively. Considering that the contraction loss ΔP2 can be omitted and ρg is replaced by (ρg + Ci) in Eq. (3), the drag
Table 3 Comparison between measured and calculated drag coefficients at different dust loadings ~ dr Vi/m/s T/K ρg/kg/m3 Ci/kg/m3 ξm D/m KA 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400
5.48 5.48 5.48 5.48 5.48 5.48 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18
0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.440 0.440 0.440 0.440 0.440
20.00 20.00 20.00 20.00 20.00 20.00 15.20 15.20 15.20 15.20 15.20 15.20 22.80 22.80 22.80 22.80 22.80
293 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293
1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20
0.002 0.010 0.050 0.200 0.500 1.000 0.010 0.050 0.200 0.500 1.000 2.000 0.010 0.050 0.200 0.500 1.000
23.01 23.01 23.01 23.01 23.01 23.01 23.01 23.01
22.33 21.25 19.88 19.17 19.17 19.54 25.52 24.87 23.44 22.46 22.79 24.09 14.47 13.60 13.02 12.73 13.74 σ
ξ(0) p
ξ(6) p
ξ(8) p
ξ(9) p
ξ(10) p
20.85 20.35 19.86 19.61 19.68 19.97 25.31 24.68 24.26 24.18 24.34 24.85 13.61 13.27 13.13 13.24 13.53 3 1.6 × 10−
21.43 20.77 19.49 17.68 16.13 14.85 28.46 26.97 24.81 22.91 21.30 19.63 13.23 12.43 11.28 10.26 9.40 2 1.9 × 10−
22.32 22.30 22.27 22.20 22.13 22.05 26.66 26.62 26.54 26.45 26.35 26.21 15.03 15.00 14.96 14.91 14.85 2 1.4 × 10−
22.32 22.31 22.26 22.16 22.04 21.89 26.66 26.60 26.49 26.34 26.16 25.88 15.03 15.00 14.93 14.85 14.75 2 1.3 × 10−
21.50 20.03 16.92 14.13 15.16 20.41 23.94 20.23 16.89 18.12 24.39 40.19 13.49 11.40 9.52 10.21 13.75 2 5.2 × 10−
(6) (8) (9) (10) Note:ξ(0) p —this method; ξp —Muschelknautz [6]; ξp —Briggs [8]; ξp —Smolik [21]; ξp —Baskakov [22].
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J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
coefficient of a cyclone for dust-laden gases, ξc, can be ~ determined by Eq. (20) if f0 and Vθw in Eqs. (9) and (13) are ~ substituted by the corresponding variables f and Vθw ′ . nc ¼
Ci 1þ qg þ
!
2ki be 1− 1 þ 1:33 e b − der
V2 1 Ve hw nþ 2 2 e ðe rc d r Þ KA ðder −e r 2c Þ2
2
V3 −1:5n þ1:11fKAFesVehw de r
ð20Þ
According to Muschelknautz [6] qffiffiffiffiffiffiffiffiffiffiffiffi f ¼ f0 1 þ 3 Ci =qg
ð21Þ
Based on the author's measurements [19], together with Ogawa's research [20], the following equation is recommended ~′ to calculate V θw , that is V¼ Vehw
Vehw 1 þ 0:35ðCi =qg Þ0:27
ð22Þ
Table 3 lists the measured and calculated drag coefficients of three Model PV cyclones at different inlet dust loadings. Here the assumption of ki = 0.3 and f0 = 0.005 still holds. The test gas is the ambient air and the test dust is talc powder of 325 mesh. The inlet dust loading ranges from 0.002 to 2 kg of dust per m3 of gas. The calculated results of other published methods are also listed in Table 3 for comparison. It is clear that the accuracy of the present method is very good. 5. Conclusions The pressure drop over a cyclone consists of the expansion loss at the cyclone inlet, the contraction loss at the entrance of the outlet tube, the swirling loss due to the friction at the cyclone wall, and the dissipation loss of gas dynamic energy in the outlet tube. The latter two parts are most important and the second part can be omitted due to its minor contribution to the pressure drop. The swirling loss depends not only on the friction at the cyclone wall but also on the distribution of the gas tangential velocity. The gas flow in the outlet tube can be divided into two regions: a core region, in which the axial velocity is assumed as zero, and an annular region in which the axial velocity is uniformly distributed. The dissipation loss in the outlet tube is equal to the sum of the axial dynamic energy and the mean tangential dynamic energy. The influence of gas temperature on the pressure drop is attributed to the change of the intensity of gas swirling motion. The rise of gas temperature leads to a decrease of gas density and an increase of gas viscosity, both resulting in a weaker swirling flow and a reduction in the swirling loss and dissipation loss. In the case of dust-laden gases, though an increase of the dust loading will lead to a greater expansion loss at the cyclone inlet,
the swirling loss decreases due to a weaker swirling flow. Consequently the pressure drop decreases at the beginning and then turns to increase with the dust loading. The present method is suitable for the prediction of the pressure drop of cyclones operating with pure or dust-laden gases at normal or high temperatures. And a detailed comparison between the experimental results and calculations have shown that the present method is higher in accuracy and better in general use than other methods, and can meet the design requirement of most commonly used, reverse flow cyclones. Nomenclature a Cyclone inlet height, a = ãD, m ~ b Cyclone inlet width, b = b D, m Ci Inlet dust loading, kg/m3 ~ De Cyclone outlet diameter, De = d rD, m D Cyclone barrel diameter, m ~ Db Cyclone hopper diameter, Db = D bD, m ~ dc Diameter of dust exit, dc = d cD, m ~ Fs Area of contact surface, Fs = F sπD2 / 4, m2 f0 Friction coefficient ~ S Insert depth of cyclone outlet, S = S D, m ~ H Cyclone body height, H = H D, m ~ H1 Cyclone barrel height, H1 = H 1D, m ~ ~ ~ H2 Cyclone cone height, H2 = H 2D = (H − H 1)D, m ~ Hb Hopper height, Hb = H bD, m KA Inlet area ratio, KA = πD2 / 4ab ki Correction coefficient of expansion loss n Swirl exponent Psi Static pressure at the cyclone inlet, Pa Pso Static pressure at the cyclone outlet, Pa Qi Inlet gas flow rate, m3/s ~ re Radius of cyclone outlet, re = d r R, m rc Radius of the core flow, rc = r~cR, m R Radius of cyclone barrel, m ~ Re Reynolds number, Re = ρgDVi / μgKAd r T Gas temperature, K Ve Mean axial velocity in cyclone outlet, m/s Vi Inlet velocity, m/s ~ Vθ Gas tangential velocity, Vθ = V θVi, m/s ~ Vθw Tangential velocity at radius R, Vθw = V θwVi, m/s ~ Vθe Gas tangential velocity at re,Vθe = V θeVi, m/s Vz Gas axial velocity, m/s Vze Axial velocity in outlet annular region, m/s V¯θ Mean tangential velocity in cyclone outlet, m/s ΔP Cyclone pressure drop, Pa ΔP1 Expansion loss at the cyclone inlet, Pa ΔP2 Contraction loss at the cyclone outlet, Pa ΔP3 Swirling loss, Pa ΔP4 Dissipation loss in the cyclone outlet, Pa μg Gas dynamic viscosity, Pa·s ξ Drag coefficient of a cyclone ξm Measured drag coefficient of a cyclone ξp Calculated drag coefficient of a cyclone ρg Gas density, kg/m3 σ Mean squared deviation, σ = Σ(ξp − ξm)2 / N
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
Acknowledgements This work was jointly sponsored by Program for New Century Excellent Talents in University (NCET-04-0108) and National Basic Research Program of China (2005CB221201). References [1] C.B. Shepherd, C.E. Lapple, Flow pattern and pressure drop in cyclone dust collectors, Ind. Eng. Chem. 31 (8) (1939) 972–984. [2] M.W. First, Fundamental Factors in the Design of Cyclone Dust Collectors: (doctoral thesis), Harvard University, Cambridge, Mass., 1950. [3] R. McK Alexander, Fundamentals of cyclone design and operation, Proceedings of the Australian Institute of Mining Metals, Nos. 152–153, (1949) 203–228. [4] C.J. Stairmand, Pressure drop in cyclone separators, Engineering 168 (1949) 409–412. [5] W. Barth, L. Leineweber, Beurteilung und Auslegung von Zyklonabscheidern, Staub 24 (2) (1964) 41–55. [6] E. Muschelknautz, Auslegung von Zyklonabscheidern in der technischen Praxis, Staub Reinhalt. Luft 30 (5) (1970) 187–195. [7] J. Casal, J.M. Martinez-Benet, A better way to calculate cyclone pressure drop, Chem. Eng. 1 (1983) 99–100. [8] L.W. Briggs, Effect of dust concentration on cyclone performance, Trans. Am. Inst. Chem. Eng. 42 (1946) 511–526. [9] S. Yuu, T. Jotaki, Y. Tomita, et al., The reduction of pressure drop due to dust loading in a conventional cyclone, Chem. Eng. Sci. 33 (1) (1978) 1573–1580. [10] A.C. Hoffmann, A.V. Santen, R.W.K. Allen, et al., Effects of geometry and solid loading on the performance of gas cyclones, Powder Technol. 70 (1992) 83–91. [11] X.L. Luo, J.Y. Chen, M.X. Shi, Research on the effect of the particle concentration in gas upon the performance of cyclone separators, J. Eng. Thermphys. 13 (3) (1992) 282–285 (in Chinese).
191
[12] M. Bohnet, T. Lorenz, Separation efficiency and pressure drop of cyclones at high temperatures. In: Gas Cleaning at High Temperatures, R. Clift, J.P.K. Seville, (Eds.), Blackie Academic and Professional, Chapman and Hall, Glasgow, UK,17–31, 1993. [13] S.S. Xu, J.Y. Xu, C.K. Xu, Study of the influence of temperature and pressure on high temperature dust separation properties of cyclone separators, Power Eng. 17 (2) (1997) 52–58 (in Chinese). [14] J.Y. Chen, M.X. Shi, Experimental research on cyclone performance at high temperatures, Proc. 9th World Filtration Congress (CD), New Orleans, U.S.A., Apr. 2004. [15] P. Meissner, F. Löffler, Zur Berechnung des Stromungsfeldes im Zyklonabscheider, Chemie-Ing.-Techn. 50 (1978) 471. [16] R.D. Blevins, Applied Fluid Dynamics Handbook, Van Nostrand Reinhold Company Inc., New York, USA, 1984, pp. 71–79. [17] X.L. Wu, Z.L. Ji, Y.H. Tian, et al., Experimental study on flow field of PV type cyclone separator, Acta Pet. Sin., Pet. Process. Sect. 13 (3) (1997) 93–99 (in Chinese). [18] L.Y. Hu, M.X. Shi, Three dimensional time averaged flow structure in cyclone separator with volute inlet, J. Chem. Ind. Eng., China 54 (4) (2003) 549–556 (in Chinese). [19] J.Y. Chen, X.L.. Luo, M.X. Shi, Pressure drop calculation of PV type cyclone under dust-bearing condition, Pet.-Chem. Equipment Technol. 18 (4) (1997) 1–3 (in Chinese). [20] A. Ogawa, O. Seito, H. Nagabayashi, Distribution of the tangential velocities on the dust laden gas flow in the cylindrical cyclone dust collector, Part. Sci. Technol. 6 (1988) 17–28. [21] J. Smolik, Air Pollution Abatement, Part I, Scriptum No.401-2099, Techn. Univ. of Prague, 1975. [22] A.P. Baskakov, V.N. Dolgov, Y.M. Goldvin, Aerodynamics and heat transfer in cyclones with particle-laden gas flow, Exp. Therm. Fluid Sci. 3 (1990) 597–602.
A universal model to calculate cyclone pressure drop Jianyi Chen a,⁎, Mingxian Shi b a
Institute of Mechanical and Electrical Engineering, China University of Petroleum, Dongying, Shandong, China 257061 b School of Petrochemical Engineering, China University of Petroleum, Beijing, China 102200 Received 8 February 2006; received in revised form 4 September 2006; accepted 27 September 2006; Available online 7 October 2006
Abstract The definition and composition of the pressure drop over a tangential inlet, reverse flow cyclone have been analyzed. It is assumed that two factors mainly contribute to the pressure drop, i.e., the local loss and the loss along the distance. The former includes the expansion loss at the cyclone inlet and the contraction loss at the entrance of the outlet tube (or vortex finder). The latter consists of the swirling loss resulting from friction at the cyclone walls and the dissipation of gas dynamic energy in the outlet tube. By use of the measured results of the flow field in cyclones, the calculation methods for each loss have been developed. And a universal model to predict the cyclone pressure drop is thus obtained simply by summing each loss. A detailed comparison between the calculated and experimental results shows that this accurate model is suitable either for pure or for dust laden gases at normal or high temperatures and can meet the requirement of most cyclone designs. © 2006 Elsevier B.V. All rights reserved. Keywords: Cyclone separator; Pressure drop; Calculation; Model
1. Introduction Cyclone separators have been the most common devices for the removal of dispersed particles from their carrying gases because of their simple structure, low cost and ease of operation. The pressure drop is one of the two important performance parameters. Many researchers [1–7] have developed different procedures to estimate the pressure drop of cyclones. Most of the procedures, however, are empirical and suitable only for pure gases, and not very satisfactory in generality. Additionally, the pressure drop of a cyclone under the condition of dust-laden gases or high temperature gases is of importance, and moreover it is much different from that under the condition of normal and pure gases. Briggs [8] found that the pressure drop decreases with an increase of the dust loading. Muschelknautz [6], Yuu [9], Hoffmann [10] and Luo [11] not only verified Briggs' findings but also discovered other interesting phenomena. For example, Muschelknautz [6] and Luo [11] found that the pressure drop falls with the rise of the dust loading until one turning point is reached at which the pressure drop begins to increase. As for the ⁎ Corresponding author. Tel.: +86 546 8393034; fax: +86 546 8393914. E-mail address: [email protected] (J. Chen). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.09.014
determination of the pressure drop of cyclones working at elevated temperatures, only very little experimental data have been reported, which are insufficient for a correct evaluation of the temperature effect on the pressure drop. Bohnet [12] observed that either the pressure drop or the drag coefficient decreases with an increase of gas temperature. Xu [13] and Chen [14] also confirmed such an observation. The prediction of the pressure drop over a cyclone dealing with high temperature or dust-laden gases, however, has not been well solved. The present paper analyses the definition and composition of the pressure drop over a tangential inlet, reverse flow cyclone, and then establishes a universal model for its prediction by employing the measured flow fields in cyclones. It is shown that the present model is good in accuracy and suitable for the prediction of the pressure drop over cyclones operating with pure or dust-laden gases at normal or high temperatures. 2. Definition and composition of the pressure drop 2.1. Definition of the pressure drop Fig. 1 is a schematic diagram of a most commonly used, tangential inlet and reverse flow cyclone. Generally, the
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
185
Fig. 3. The axial velocity near the cyclone outlet.
flow rectifier. Hoffmann [10] observed that the static pressure at the outlet wall is close to the static pressure that would be measured after an ideal rectifier. Therefore, the static pressure at the wall of the outlet tube minus the static pressure at the inlet gives the true dissipative loss or the pressure drop of a cyclone. The latter three ways have been widely used in research and engineering practices. And the pressure drop in this paper is defined as Eq. (1) and taken in a way similar to that of Meissner and Löffler. Fig. 1. Sketch of a reverse flow cyclone separator.
2.2. Composition of the pressure drop pressure drop over a cyclone is the difference of static pressure between the inlet and the outlet, which can be written as: DP ¼ Psi −Pso
ð1Þ
The static pressure at the inlet cross-section is uniformly distributed because there is no swirling motion. It can be easily measured with a pressure tapping in the wall. But the static pressure at the outlet wall is quite different from its crosssectional average due to the strong swirling flow. The dynamic pressure stored in the swirling motion can be significant. The determination of the static pressure downstream of a cyclone, hence the pressure drop, becomes more complicated and difficult. In the past, this problem has been approached in several ways. Stairmand [4] presumably measured the static pressure at the outlet wall immediately downstream of a cyclone as he ignored the influence of the swirl. Shepherd and Lapple [1] discharged the gas directly from the cyclone to atmosphere (taking the downstream pressure as the ambient pressure). Meissner and Löffler [15] measured the static pressure after a
Fig. 2. (a). Slot inlet. (b). Volute inlet.
The pressure drop over a cyclone consists of a local loss and a frictional loss (or a loss along the distance). The local loss includes an expansion loss at the cyclone inlet, ΔP1, and a contraction loss at the entrance of the outlet tube, ΔP2. The frictional loss includes a swirling loss due to the friction between the gas flow and the cyclone wall, ΔP3, and a dissipation loss of the gas dynamic energy in the outlet, ΔP4. Therefore, the pressure drop can be expressed as: DP ¼ DP1 þ DP2 þ DP3 þ DP4
ð2Þ
3. Method for pure gases 3.1. The expansion loss at the cyclone inlet—ΔP1 The gas flow will expand both radially and axially after entering a cyclone, resulting in a local expansion loss. For a slot inlet, the radial expansion loss is a function of the variable b / (R − re). But for a volute inlet, the radial expansion loss is a function of the variable b / (R + b − c~b − re), where c~ = c / b and
Fig. 4. The tangential and axial velocity in the cyclone outlet.
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J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
Table 1 Comparison between measured and calculated drag coefficients under normal temperature and pure gas conditions ~ Model D/m a/D b/D KA dr S/D Stairmand
0.305 0.305 0.305 0.610 1.524 4.572 0.335 0.287 0.287 0.300 0.400 0.400 0.400 0.400 0.400 0.600 0.600 0.600 0.600 0.600 0.800 0.800 0.800 0.800 0.800 1.200 1.200 1.200 1.200
Stern Lapple PV
0.50 0.35 0.50 0.25 0.60 0.67 0.61 0.53 0.53 0.56 0.49 0.49 0.56 0.56 0.64 0.56 0.56 0.56 0.49 0.64 0.56 0.56 0.56 0.49 0.64 0.48 0.55 0.55 0.55
0.22 0.31 0.20 0.13 0.25 0.13 0.32 0.23 0.11 0.25 0.22 0.22 0.26 0.26 0.29 0.26 0.26 0.26 0.22 0.29 0.25 0.25 0.25 0.22 0.29 0.22 0.25 0.25 0.25
7.14 7.14 7.85 25.12 5.23 8.83 4.05 6.31 13.87 5.50 7.20 7.20 5.48 5.48 4.18 5.45 5.45 5.45 7.33 4.26 5.51 5.51 5.51 7.20 4.26 7.48 5.65 5.65 5.65
0.45 0.39 0.50 0.50 0.50 0.33 0.56 0.52 0.52 0.32 0.32 0.25 0.32 0.44 0.32 0.32 0.24 0.44 0.32 0.44 0.45 0.31 0.25 0.31 0.31 0.25 0.25 0.31 0.44
0.50 0.35 0.50 1.50 1.50 0.40 0.91 1.60 1.60 0.56 0.49 0.49 0.56 0.56 0.64 0.56 0.56 0.56 0.49 0.64 0.56 0.56 0.56 0.49 0.64 0.48 0.55 0.55 0.55
H1 / D
H/D
dc / D
2.5 2.5 1.5 2.0 2.0 1.1 1.4 2.1 2.1 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6
5.0 5.0 4.0 4.0 4.0 2.0 2.7 4.3 4.3 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8
0.375 0.375 0.375 0.375 0.375 0.375 0.400 0.250 0.250 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400
(1) (2) (3) (4) (5) (6) (7) Note: ξ(0) p —this method; ξp —Shepherd and Lapple [1]; ξp —First [2]; ξp —Alexander [3]; ξp —Stairmand [4]; ξp —Barth [5]; ξp —Muschelknautz [6]; ξp — Casal [7].
c is the width of the inlet cutting into the cyclone body, see Fig. 2. And the radial expansion loss can be expressed as [16]: DP1r ¼
1−
b R þ b−e c b−re
2
qg Vi2 2
The axial expansion loss, however, is not as easy to determine. For the sake of simplicity, a correction coefficient, ki, is employed to account for the contribution of the axial expansion loss. ki is less than unity and its usual value is 0.3. As a result, the expansion loss at the cyclone inlet, ΔP1, can be expressed as follows: DP1 ¼
1−ki
b R þ b−ec b−re
2
qg Vi2 2
A volute inlet changes to a slot inlet when c~ = 1. For common designs, c~ is not less than 1 / 3. With the sizes being nondimensionalized by the cyclone barrel diameter D, it is obtained that: DP1 ¼
1−
2ki be 1 þ 1:33 e b− der
2
qg Vi2 2
ð3Þ
3.2. The contraction loss at the entrance of the outlet tube —ΔP2 The contraction loss at the entrance of the outlet tube (or vortex finder) occurs because of an abrupt reduction of the flow area when gases enter the outlet tube from the separation space of a cyclone. According to the axial velocity profile in the separation space [17], the axial flow can be divided into an upward flow and a downward flow, as shown in Fig. 3. And the flow areas of the upward and downward flow are roughly 1 / 3 and 2 / 3 of the cross-sectional area of the cyclone barrel, respectively. pffiffiffi It means that their interfacial radius approximates 3R=3, or the averaged axial velocity of the upward flow is equal to 3Vi / KA. Consequently, ΔP2 can be expressed as [16] 2 2 1−3 de r qg V 2 2 qg 3Vi i DP2 ¼ 0:5 1−3 der ¼ 4:5 2 KA 2 KA2
ð4Þ
3.3. The swirling loss—ΔP3 The friction between the gas flow and the cyclone wall due to the gas viscosity will result in a loss of swirling energy, which is defined as the swirling loss of a cyclone here. Because of this viscous friction, the tangential flow in a cyclone is a combination
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
187
Vi/m/s
ξm
ξ(0) p
ξ(1) p
ξ(2) p
ξ(3) p
ξ(4) p
ξ(5) p
ξ(6) p
ξ(7) p
9.28 9.28 5.05 15.07 16.07 16.15 16.07 14.97 28.42 20.00 17.62 17.43 18.89 16.66 14.48 14.98 13.15 14.95 18.04 15.27 17.90 16.48 15.58 17.14 16.93 17.01 21.90 13.11 18.00
8.70 9.07 5.68 3.38 10.60 11.40 7.25 7.19 3.68 20.30 17.90 26.89 21.74 11.77 27.28 22.56 37.78 13.15 18.06 14.70 13.76 24.35 38.67 20.02 30.20 30.85 43.18 25.18 13.16 σ
8.04 9.97 5.90 2.98 8.68 13.49 8.45 5.49 3.65 20.85 16.97 26.86 22.22 11.62 28.56 22.85 41.30 12.31 17.98 15.22 13.02 24.76 39.71 19.55 32.43 31.40 43.73 25.25 13.98 3 7.3 × 10−
8.69 11.60 6.40 2.00 9.60 12.80 9.76 7.33 3.33 23.01 17.58 29.06 23.10 11.73 30.28 23.08 40.69 11.85 17.16 15.24 11.51 23.42 36.77 17.92 30.29 26.85 35.56 23.12 11.48 2 2.1 × 10−
7.08 9.44 6.18 1.89 9.07 19.57 11.91 6.65 3.02 22.69 17.34 28.66 22.78 11.57 29.86 22.76 40.12 11.68 16.92 15.03 11.35 23.09 36.25 17.67 29.87 26.48 35.06 22.80 11.32 2 5.0 × 10−
5.84 6.73 5.23 2.65 6.00 14.26 5.62 4.89 3.55 17.01 13.23 23.90 17.07 7.92 22.73 17.03 36.49 7.98 12.94 9.45 7.78 17.39 31.61 13.51 22.85 21.74 30.28 17.22 7.84 2 8.0 × 10−
7.35 10.16 5.26 1.77 9.55 19.96 7.81 5.92 2.69 20.86 16.88 29.63 22.18 10.29 29.08 23.76 45.55 10.95 17.67 14.09 11.02 25.54 43.32 19.55 33.04 34.79 46.07 27.44 11.72 2 4.2 × 10−
6.47 8.93 5.41 2.90 8.85 16.79 9.37 6.74 3.62 22.05 16.44 29.49 22.14 10.12 29.66 22.10 43.72 10.22 16.01 13.11 9.91 22.51 38.68 16.80 29.75 26.87 37.20 22.23 9.92 2 2.9 × 10−
7.10 11.00 5.71 2.04 8.39 11.09 10.45 6.41 2.77 22.00 15.50 28.21 22.11 10.71 32.11 22.11 45.83 10.82 15.05 14.56 10.50 22.45 39.84 15.83 32.03 25.38 37.96 22.04 10.43 2 3.3 × 10−
5.66 8.27 4.14 2.51 6.40 9.56 6.54 4.70 2.82 25.71 15.97 39.61 25.88 8.41 42.81 25.84 75.40 8.53 15.33 12.58 8.18 26.53 61.99 16.51 42.82 34.16 58.13 25.93 8.15 1 1.3 × 10−
of a quasi-free vortex and a quasi-forced vortex, instead of a combination of a free vortex and a forced vortex. Referring to the theories by Barth [5] and Muschelknautz [6], the swirling loss can be calculated by the following equation:
DP3 ¼
f0 Fs qg ðVhw Vhe Þ1:5 2 ð0:9Qi Þ
cyclone hopper, and the swirling velocity is only somewhat slightly less than in the separation space [17,18]. It means that there is also friction at the wall of the cyclone hopper, which will lead to an additional pressure loss. This friction loss is accounted for simply by adding the surface area of the cyclone hopper into Fs in this paper, and thus
ð5Þ Fs ¼
~ ~ where Qi = abVi = πD2Vi / 4KA, Vθw = V θwVi and Vθe = V θeVi. Qi stands for the inlet gas flow rate. For a quasi-free vortex flow, e e −n Vehe ¼ Vehw re −n e ¼ V hw d r :
ð6Þ
p þ ðD þ dc Þ 2
DP3 ¼
qg Vi2 2
ð7Þ
Where Fs is the total area of the contact surfaces between the gas flow and the cyclone wall. Usually Fs is the sum of the top cover area, the surface area of the cyclone barrel and cone, and the external surface area of the outlet tube. However, experimental results showed that there also exists a swirling motion in the
H22
ðD−dc Þ2 þ pDb Hb þ 4
ð8Þ
Fs can also be non-dimensionalized as
Eq. (5) is transformed into: 3 4KA f0 FsVe hw 0:9pD2 de 1:5n r
p 2 2 D −De þ pDe S þ pDH1 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fes ¼
and
4Fs e 2r þ 4 der Se þ 4 He1 ¼ 1− d pD2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 e eb Heb þ 1 þ d c 4 He2 þ 1− dec þ 4 D 3 −1:5n DP3 ¼ 1:11f0 KA FesVehw de r
qg Vi2 2
ð9Þ
Where f0 is the friction coefficient between a pure gas flow and a cyclone wall. For a steel cyclone f0 = 0.005 and for
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cyclones having a different roughness of wall surface the value of f0 can be determined as specified in reference [6]. 3.4. The dissipation loss of the gas dynamic energy in the outlet—ΔP4 In the outlet tube, the gas tangential velocity is still very high and the tangential gas flow is also a combination of a quasi-free vortex and a quasi-forced vortex. The swirl flow does not decay significantly in the outlet tube. The axial velocity, however, is totally different from that in the separation space. In the vicinity of the swirl center the axial velocity is very low, but it becomes quite high in the annular region near the outlet wall, see Fig. 4 [18]. Most of the gas flows axially through this region. It is natural to divide the gas flow in the outlet tube into two regions: a core region, in which the axial velocity is assumed as zero, and an annular region in which the axial velocity is uniformly distributed. The tangential flow conforms to the law of a quasiforced vortex in the core region and of a quasi-free vortex in the annular region, respectively. Assuming that rc is the radius of the boundary between the core and the annular region, the axial velocity in the annular region is then as follows: Vze ¼
4Qi D2 1 ¼ Vi ¼ Vi 2 2 2 2 2 pðDe −Dc Þ KA ðDe −Dc Þ KA ðder − rec2 Þ
pressure drop over a cyclone in most cases) that it can be omitted for the sake of simplicity. Therefore the above expression can be shortened as n¼
þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi V¯ h ¼ Vhc Vhe ¼ Vehw ðerc ree Þ0:5n Vi ¼ Vehw ðe rc der Þ0:5n Vi
ð12Þ
Consequently, the pressure loss due to the dynamic dissipation or ΔP4 is " 2 # qg ¯ 2 qg Vi2 1 Ve hw 2 DP4 ¼ ðVh þ Vze Þ ¼ þ 2 2 2 ðe rc der Þn KA2 ð e d r −erc2 Þ2
2ki e b 1 þ 1:33eb−e dr
2 þ
3 −1:5n þ1:11f0 KAFesVehw der
4:5ð1−3 der Þ KA2 2 1 Ve þ hw n þ 2 2 e ðe rc der Þ KA ðd r −erc2 Þ2
3
ð15Þ
e 2hw 1 V þ ðe rc der Þn KA2 ðde2r −e r 2c Þ2
ð16Þ
1:11KA−0:21de r Re0:06 pffiffiffiffiffiffiffiffiffiffi 1 þ f0 Fes KAder 0:16
ð17Þ
Where Re is the cyclone outlet Reynolds number and can be defined as Re ¼
qg Ve De qg Vi D ¼ lg lg KAder
ð18Þ
And the swirl exponent n can also be correlated with Re by the following equation " # S−a −0:5 0:12 n ¼ 1− exp − 0:26Re 1þ ð19Þ b
j j
3.5. Calculation and comparison ð13Þ
Now the pressure drop over a cyclone, ΔP, can be easily obtained by summation of ΔP1 ∼ ΔP4. But it is more often to use the drag coefficient, ξ, which is derived as n ¼ 1−
−1:5n
þ1:11f0 KAFesVeh w de r
In addition, the dimensionless tangential velocity at the cyclone wall is Vehw ¼
According to the transportation law of the moment-ofmomentum of gas rotation, the mean tangential velocity in the outlet tube can be approximately expressed as
2
2 rec ¼ 0:38 der þ 0:5 de r
ð10Þ
ð11Þ
2ki eb e der 1 þ 1:33 b−
Now the pressure drop or the drag coefficient over a cyclone can be determined by Eq. (15) if the dimensions of and the tangential velocity in a cyclone are known. Take a Model PV cyclone as an example. The measured results of its flow field showed that rc is normally 0.5–0.6 times the radius of the outlet tube [17,18]. Furthermore, when nondimensionalized, rc can be correlated with a dimensionless diameter of the cyclone outlet, d˜r, as follows
The non-dimensional tangential velocity at the radius, rc, is Vehc ¼ Vehw er c−n
1−
ð14Þ
A detailed calculation showed that the second term in Eq. (14), i.e., the contraction loss, is so small (not more than 1% the
Table 1 lists the measured and calculated drag coefficients of dozens of cyclones operated under ambient temperature and pure gas conditions. The calculation is accomplished by using Eqs. (15)–(19) and assuming ki = 0.3 and f0 = 0.005. For Model Stairmand, Stern and Lapple cyclones, the measured results are cited from the corresponding references, while for Model PV cyclones the measurements are taken elsewhere by the authors. All the test gases were ambient air. The calculated drag coefficients by other researchers are also listed for comparison. The accuracy of the present method is satisfactory except for few cases. In average, the mean square deviation between the measured and calculated results of the present method is less than that of other methods by one or two orders of magnitude.
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
189
Table 2 Comparison between measured and calculated drag coefficients at different gas temperatures ~ D/m KA dr Vi/m/s T/K ρg /kg/m3 ξm ξ(0) ξ(1) p p
ξ(2) p
ξ(3) p
ξ(4) p
ξ(5) p
ξ(6) p
ξ(7) p
0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300
22.69 22.69 22.69 22.69 22.69 22.69 22.69 22.69
17.01 17.01 17.01 17.01 17.01 17.01 17.01 17.01
20.86 18.55 16.59 16.66 14.85 14.85 14.85 14.85
22.05 22.05 22.05 22.05 22.05 22.05 22.05 22.05
22.00 22.00 22.00 22.00 22.00 22.00 22.00 22.00
25.71 25.71 25.71 25.71 25.71 25.71 25.71 25.71
5.50 5.50 5.50 5.50 5.50 5.50 5.50 5.50
0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315
20.00 19.62 20.11 16.14 36.29 28.24 19.82 16.56
303 470 685 676 973 973 973 973
1.08 0.69 0.49 0.49 0.34 0.34 0.34 0.34
20.30 19.30 18.40 18.40 17.40 16.60 17.10 17.20
Listed in Table 2 is a comparison between the measured and calculated drag coefficients of a Model PV cyclone operated at different temperatures. The values of ki and f0 used in Table 2 are assumed to be the same as in Table 1. The gas is normal pressure air. Obviously the calculated results agree very well with the measured ones. In fact, the rise of gas temperature will lead to a decrease of gas density and an increase of gas viscosity, both of which contribute to a weaker swirling motion and consequently result in a reduction of the swirling loss in the separation space and the dissipation loss in the outlet. It is indicated that the drag coefficient and the temperature can be correlated with simply through a Reynolds number, Re, which has an influence on the swirl exponent and tangential velocity at the cyclone wall. 4. Method for dust-laden gases The calculation of the pressure drop over a cyclone for dust-laden gases is more complicated and has not been well solved, though Briggs [10], Muschelknautz [6], Luo [11] and Bohnet [12] developed some empirical or semi-empirical
20.85 18.52 17.00 16.50 16.98 16.40 15.63 15.27
methods.In the case of dust-laden gases, it is also reasonable to keep the assumption that the pressure drop over a cyclone still consists of the four losses defined in Section 3, but their calculation methods are somehow different. Let Ci stand for the inlet dust loading. Checking Eqs. (3), (4), (9) and (13), one observes that under the dust-laden gas condition, the gas density ρg shall be replaced by (ρg + Ci) in Eq. (3) and [ρg + (1 − η)Ci] in Eqs. (4), (9) and (13), where η is the separation efficiency of a cyclone. A low dust loading generally means a much smaller value of (1 − η)Ci relative to ρg. Even at high dust loadings (1 − η)Ci is also much less than ρg because a high dust loading will lead to a high separation efficiency η, usually approaching unity. Therefore, it is not necessary to use [ρg + (1 − η)Ci] instead of ρg in the calculations of ΔP2, ΔP3 and ΔP4 by Eqs. (4), (9) and (13). However, two variables, f0 and ~ V θw, bear relation to the dust loading Ci and hence need to be redetermined while the remainders can be kept unchanged. For ~ ′ stand for the friction coefficient dust-laden gases, let f and Vθw and non-dimensional tangential velocity at the cyclone wall, respectively. Considering that the contraction loss ΔP2 can be omitted and ρg is replaced by (ρg + Ci) in Eq. (3), the drag
Table 3 Comparison between measured and calculated drag coefficients at different dust loadings ~ dr Vi/m/s T/K ρg/kg/m3 Ci/kg/m3 ξm D/m KA 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400 0.400
5.48 5.48 5.48 5.48 5.48 5.48 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18
0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.315 0.440 0.440 0.440 0.440 0.440
20.00 20.00 20.00 20.00 20.00 20.00 15.20 15.20 15.20 15.20 15.20 15.20 22.80 22.80 22.80 22.80 22.80
293 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293 293
1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20
0.002 0.010 0.050 0.200 0.500 1.000 0.010 0.050 0.200 0.500 1.000 2.000 0.010 0.050 0.200 0.500 1.000
23.01 23.01 23.01 23.01 23.01 23.01 23.01 23.01
22.33 21.25 19.88 19.17 19.17 19.54 25.52 24.87 23.44 22.46 22.79 24.09 14.47 13.60 13.02 12.73 13.74 σ
ξ(0) p
ξ(6) p
ξ(8) p
ξ(9) p
ξ(10) p
20.85 20.35 19.86 19.61 19.68 19.97 25.31 24.68 24.26 24.18 24.34 24.85 13.61 13.27 13.13 13.24 13.53 3 1.6 × 10−
21.43 20.77 19.49 17.68 16.13 14.85 28.46 26.97 24.81 22.91 21.30 19.63 13.23 12.43 11.28 10.26 9.40 2 1.9 × 10−
22.32 22.30 22.27 22.20 22.13 22.05 26.66 26.62 26.54 26.45 26.35 26.21 15.03 15.00 14.96 14.91 14.85 2 1.4 × 10−
22.32 22.31 22.26 22.16 22.04 21.89 26.66 26.60 26.49 26.34 26.16 25.88 15.03 15.00 14.93 14.85 14.75 2 1.3 × 10−
21.50 20.03 16.92 14.13 15.16 20.41 23.94 20.23 16.89 18.12 24.39 40.19 13.49 11.40 9.52 10.21 13.75 2 5.2 × 10−
(6) (8) (9) (10) Note:ξ(0) p —this method; ξp —Muschelknautz [6]; ξp —Briggs [8]; ξp —Smolik [21]; ξp —Baskakov [22].
190
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
coefficient of a cyclone for dust-laden gases, ξc, can be ~ determined by Eq. (20) if f0 and Vθw in Eqs. (9) and (13) are ~ substituted by the corresponding variables f and Vθw ′ . nc ¼
Ci 1þ qg þ
!
2ki be 1− 1 þ 1:33 e b − der
V2 1 Ve hw nþ 2 2 e ðe rc d r Þ KA ðder −e r 2c Þ2
2
V3 −1:5n þ1:11fKAFesVehw de r
ð20Þ
According to Muschelknautz [6] qffiffiffiffiffiffiffiffiffiffiffiffi f ¼ f0 1 þ 3 Ci =qg
ð21Þ
Based on the author's measurements [19], together with Ogawa's research [20], the following equation is recommended ~′ to calculate V θw , that is V¼ Vehw
Vehw 1 þ 0:35ðCi =qg Þ0:27
ð22Þ
Table 3 lists the measured and calculated drag coefficients of three Model PV cyclones at different inlet dust loadings. Here the assumption of ki = 0.3 and f0 = 0.005 still holds. The test gas is the ambient air and the test dust is talc powder of 325 mesh. The inlet dust loading ranges from 0.002 to 2 kg of dust per m3 of gas. The calculated results of other published methods are also listed in Table 3 for comparison. It is clear that the accuracy of the present method is very good. 5. Conclusions The pressure drop over a cyclone consists of the expansion loss at the cyclone inlet, the contraction loss at the entrance of the outlet tube, the swirling loss due to the friction at the cyclone wall, and the dissipation loss of gas dynamic energy in the outlet tube. The latter two parts are most important and the second part can be omitted due to its minor contribution to the pressure drop. The swirling loss depends not only on the friction at the cyclone wall but also on the distribution of the gas tangential velocity. The gas flow in the outlet tube can be divided into two regions: a core region, in which the axial velocity is assumed as zero, and an annular region in which the axial velocity is uniformly distributed. The dissipation loss in the outlet tube is equal to the sum of the axial dynamic energy and the mean tangential dynamic energy. The influence of gas temperature on the pressure drop is attributed to the change of the intensity of gas swirling motion. The rise of gas temperature leads to a decrease of gas density and an increase of gas viscosity, both resulting in a weaker swirling flow and a reduction in the swirling loss and dissipation loss. In the case of dust-laden gases, though an increase of the dust loading will lead to a greater expansion loss at the cyclone inlet,
the swirling loss decreases due to a weaker swirling flow. Consequently the pressure drop decreases at the beginning and then turns to increase with the dust loading. The present method is suitable for the prediction of the pressure drop of cyclones operating with pure or dust-laden gases at normal or high temperatures. And a detailed comparison between the experimental results and calculations have shown that the present method is higher in accuracy and better in general use than other methods, and can meet the design requirement of most commonly used, reverse flow cyclones. Nomenclature a Cyclone inlet height, a = ãD, m ~ b Cyclone inlet width, b = b D, m Ci Inlet dust loading, kg/m3 ~ De Cyclone outlet diameter, De = d rD, m D Cyclone barrel diameter, m ~ Db Cyclone hopper diameter, Db = D bD, m ~ dc Diameter of dust exit, dc = d cD, m ~ Fs Area of contact surface, Fs = F sπD2 / 4, m2 f0 Friction coefficient ~ S Insert depth of cyclone outlet, S = S D, m ~ H Cyclone body height, H = H D, m ~ H1 Cyclone barrel height, H1 = H 1D, m ~ ~ ~ H2 Cyclone cone height, H2 = H 2D = (H − H 1)D, m ~ Hb Hopper height, Hb = H bD, m KA Inlet area ratio, KA = πD2 / 4ab ki Correction coefficient of expansion loss n Swirl exponent Psi Static pressure at the cyclone inlet, Pa Pso Static pressure at the cyclone outlet, Pa Qi Inlet gas flow rate, m3/s ~ re Radius of cyclone outlet, re = d r R, m rc Radius of the core flow, rc = r~cR, m R Radius of cyclone barrel, m ~ Re Reynolds number, Re = ρgDVi / μgKAd r T Gas temperature, K Ve Mean axial velocity in cyclone outlet, m/s Vi Inlet velocity, m/s ~ Vθ Gas tangential velocity, Vθ = V θVi, m/s ~ Vθw Tangential velocity at radius R, Vθw = V θwVi, m/s ~ Vθe Gas tangential velocity at re,Vθe = V θeVi, m/s Vz Gas axial velocity, m/s Vze Axial velocity in outlet annular region, m/s V¯θ Mean tangential velocity in cyclone outlet, m/s ΔP Cyclone pressure drop, Pa ΔP1 Expansion loss at the cyclone inlet, Pa ΔP2 Contraction loss at the cyclone outlet, Pa ΔP3 Swirling loss, Pa ΔP4 Dissipation loss in the cyclone outlet, Pa μg Gas dynamic viscosity, Pa·s ξ Drag coefficient of a cyclone ξm Measured drag coefficient of a cyclone ξp Calculated drag coefficient of a cyclone ρg Gas density, kg/m3 σ Mean squared deviation, σ = Σ(ξp − ξm)2 / N
J. Chen, M. Shi / Powder Technology 171 (2007) 184–191
Acknowledgements This work was jointly sponsored by Program for New Century Excellent Talents in University (NCET-04-0108) and National Basic Research Program of China (2005CB221201). References [1] C.B. Shepherd, C.E. Lapple, Flow pattern and pressure drop in cyclone dust collectors, Ind. Eng. Chem. 31 (8) (1939) 972–984. [2] M.W. First, Fundamental Factors in the Design of Cyclone Dust Collectors: (doctoral thesis), Harvard University, Cambridge, Mass., 1950. [3] R. McK Alexander, Fundamentals of cyclone design and operation, Proceedings of the Australian Institute of Mining Metals, Nos. 152–153, (1949) 203–228. [4] C.J. Stairmand, Pressure drop in cyclone separators, Engineering 168 (1949) 409–412. [5] W. Barth, L. Leineweber, Beurteilung und Auslegung von Zyklonabscheidern, Staub 24 (2) (1964) 41–55. [6] E. Muschelknautz, Auslegung von Zyklonabscheidern in der technischen Praxis, Staub Reinhalt. Luft 30 (5) (1970) 187–195. [7] J. Casal, J.M. Martinez-Benet, A better way to calculate cyclone pressure drop, Chem. Eng. 1 (1983) 99–100. [8] L.W. Briggs, Effect of dust concentration on cyclone performance, Trans. Am. Inst. Chem. Eng. 42 (1946) 511–526. [9] S. Yuu, T. Jotaki, Y. Tomita, et al., The reduction of pressure drop due to dust loading in a conventional cyclone, Chem. Eng. Sci. 33 (1) (1978) 1573–1580. [10] A.C. Hoffmann, A.V. Santen, R.W.K. Allen, et al., Effects of geometry and solid loading on the performance of gas cyclones, Powder Technol. 70 (1992) 83–91. [11] X.L. Luo, J.Y. Chen, M.X. Shi, Research on the effect of the particle concentration in gas upon the performance of cyclone separators, J. Eng. Thermphys. 13 (3) (1992) 282–285 (in Chinese).
191
[12] M. Bohnet, T. Lorenz, Separation efficiency and pressure drop of cyclones at high temperatures. In: Gas Cleaning at High Temperatures, R. Clift, J.P.K. Seville, (Eds.), Blackie Academic and Professional, Chapman and Hall, Glasgow, UK,17–31, 1993. [13] S.S. Xu, J.Y. Xu, C.K. Xu, Study of the influence of temperature and pressure on high temperature dust separation properties of cyclone separators, Power Eng. 17 (2) (1997) 52–58 (in Chinese). [14] J.Y. Chen, M.X. Shi, Experimental research on cyclone performance at high temperatures, Proc. 9th World Filtration Congress (CD), New Orleans, U.S.A., Apr. 2004. [15] P. Meissner, F. Löffler, Zur Berechnung des Stromungsfeldes im Zyklonabscheider, Chemie-Ing.-Techn. 50 (1978) 471. [16] R.D. Blevins, Applied Fluid Dynamics Handbook, Van Nostrand Reinhold Company Inc., New York, USA, 1984, pp. 71–79. [17] X.L. Wu, Z.L. Ji, Y.H. Tian, et al., Experimental study on flow field of PV type cyclone separator, Acta Pet. Sin., Pet. Process. Sect. 13 (3) (1997) 93–99 (in Chinese). [18] L.Y. Hu, M.X. Shi, Three dimensional time averaged flow structure in cyclone separator with volute inlet, J. Chem. Ind. Eng., China 54 (4) (2003) 549–556 (in Chinese). [19] J.Y. Chen, X.L.. Luo, M.X. Shi, Pressure drop calculation of PV type cyclone under dust-bearing condition, Pet.-Chem. Equipment Technol. 18 (4) (1997) 1–3 (in Chinese). [20] A. Ogawa, O. Seito, H. Nagabayashi, Distribution of the tangential velocities on the dust laden gas flow in the cylindrical cyclone dust collector, Part. Sci. Technol. 6 (1988) 17–28. [21] J. Smolik, Air Pollution Abatement, Part I, Scriptum No.401-2099, Techn. Univ. of Prague, 1975. [22] A.P. Baskakov, V.N. Dolgov, Y.M. Goldvin, Aerodynamics and heat transfer in cyclones with particle-laden gas flow, Exp. Therm. Fluid Sci. 3 (1990) 597–602.