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Ind. Eng. Chem. Res. 2011, 50, 1150–1157
Laminar Flow in Helical Coils: A Parametric Study Ritu Gupta, R. K. Wanchoo,* and T. R. M. Jafar Ali UniVersity Institute of Chemical Engineering and Technology, Panjab UniVersity, India -160 014
In the present study, experimental observations on pressure drop measurements for fully developed, incompressible Newtonian fluid flowing through helical coils of constant circular cross section under laminar flow conditions are reported. To elucidate the effect of coil pitch and coil diameter on the friction factor, 5 types of coils with 12 different combinations of varying radii, such that 8.3 e p/dt e 66.7 and 11.7 e Dc/dt e 105.48, have been used. The experimental observations indicated that the Germano number (NGn) successfully signifies the combined effect of various coil parameters on the pressure drop. Hence, the data corresponding to low and high Germano numbers (NGn e 70 and NGn > 70) have been treated separately, to yield suitable correlations for the laminar flow region. The correlations developed explain the present experimental data to within (10% (in the range of 1 < NGn < 130) and the data available in the literature to within (15%. 1. Introduction Helical coils provide large heat-transfer area per unit volume, high heat- and mass-transfer flux, and small residence time distribution. The industrial processes involving curved tubes include heat exchangers, ultrafiltration, rectification and absorption, nanofluids, chemical reactors, nuclear reactors, and piping systems. It can be seen that curved geometries have wide range of applications, varying from human organs (lungs, blood vessels, catheter, etc.) to industrial devices (cooling coils, mixers, microdevices, etc.).1-5 For the last few decades, flow through coiled tubes has been the focus of attention for many researchers, because of the complex fluid dynamics that occur in such systems. In a recent review, Vashisth et al.6 gave an excellent account of industrial applications and contributions made by various investigators, involving heat, mass, and momentum transfer in curved geometries. As the fluid enters a curved section, centrifugal force acts outward from the center of curvature on the fluid elements. The centrifugal force is dependent on the local axial velocity of the fluid particle. Because of no-slip conditions at the wall, the axial velocity in the core region is greater than the velocity near the wall. Pressure gradients parallel to the axis of the symmetry are uniform along lines normal to that of the symmetry axis. To maintain the momentum balance between the centrifugal force and the pressure gradient, slowermoving fluid elements move toward the inner wall of the curved tube, leading to the development of secondary flow in the flowing fluid. Flow pattern inside the coiled tube is affected by the coil curvature.7,8 The critical velocity of the fluid is affected by changes in the coil curvature. An analytical solution for fully developed laminar flow in a curved circular tube was first developed by Dean.9,10 He developed a series solution as a perturbation of the Poiseuille flow in a straight pipe for low Dean number values (NDe < 17). Dravid et al.11 have shown that the secondary flow pattern can change substantially in form as some of the parameters, such as NDe change. In the presence of a hydrodynamic boundary layer, the fluid particles near the wall have lower axial velocity and, hence, experience lower centrifugal action than the fluid particles flowing in the tube core, characterized by higher axial velocities. As a consequence, the fluid from the tube core region is pushed toward the outer wall of the tube, where it bifurcates and drives the fluid near * To whom correspondence should be addressed. Tel.: +91 172 2534933. Fax: +91 172 2779173. E-mail:
[email protected].
the wall toward the inner wall of the tube, thus forming a pair of recirculating counter-rotating vortices.12 In the case of helical coils, an increase in coil elevation per revolution causes additional torsional or rotational force. The combined effect of torsional and centrifugal force results in a specific flow field. Ito13 performed experiments on smooth curved pipes with curvature ratios from 1:16.4 to 1:64.8, to determine the friction factors for laminar and turbulent flow conditions, and the following was proposed:
[
NDe fc ) 21.5 fs (1.56 + log10 NDe)5.73
]
when flow is laminar and 13.5 < NDe < 2000 and
[ Rr ]
fc
1/2
2 -1/4
[ ( Rr ) ]
) 0.029 + 0.034 NRe
( Rr )
for0.034 < NRe
2
< 300.
for 0.034 < NRe(r/R)2 < 300. For NRe((r)/(R))2 values below 0.034, the friction factor was equivalent to that of a straight pipe. Barua14 theoretically analyzed the motion of flow in a stationary curved pipe for large Dean numbers (30 < NDe < 2000) and proposed the following: fc ) 0.509 + 0.0918√NDe fs Nunge and Lin,15 in their studies, reported on the friction factor between straight tubes and highly curved tubes and concluded that, at high NDe, the ratio of friction factors decreased with increasing curvature. These results are in contradiction with the results of Austin and Seader.16Tarbell and Samuels17 developed the following friction factor correlation that is based on the Reynolds number and the curvature ratio, instead of NDe.
[
]
fc 7.964 × 10-3 ) 1 + 8.279 × 10-4 + × (R/r) fs 2 NRe - 2.096 × 10-7NRe This correlation is supposed to be valid for NDe ) 20-500. Van Dyke18 observed that, for zero-curvature pipe, with large NDe, the pressure drop was proportional to the NDe1/4. However, this observation is not consistent with the results of earlier
10.1021/ie101752z 2011 American Chemical Society Published on Web 12/06/2010
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Table 1. Friction Factor Correlations under Laminar flow Conditions in Curved Tubes authors 10
correlation
( )
2 4
Dean
fc NDe ) 1.03058 fs 288
White28
fc 11.6 0.45 )1 - 1 NDe fs ) 1 (for NDe < 11.6)
Adler19
fc ) 0.1064√NDe fs
Ito13
fc NDe ) 21.5 5.73 fs 1.56 + log ( 10 NDe)
Ito13
geometry
( )
2 2
+ 0.1195
NDe 288
[ ( ) ]
1/0.45
[
fc
2 -0.2
remarks
torus
analytical
small d/D, NDe < 20
circular tube
empirical
D/d ) 5.15, 50, 2050
experimental and theoretical
large NDe
empirical
13.5 < NDe < 2000
circular tube
empirical
NRe(d/D)2 > 6
helical
empirical
2000 < NRe < 9000, 0.037 < d/D < 0.097
]
Dd ) 0.079[N (Dd ) ]
method
Re
Kubair and Varrirer29
fc ) 0.7716 exp 3.553
Barua14
fc ) 0.509 + 0.0918√NDe fs
torus
theoretical
large NDe
Mori and Nakayama30
0.1080√NDe fc ) fs 1 - (3.253/ √NDe)
circular tube
theoretical
experimentally verified, 13.5 < NDe < 2000
Srinivasan et al.31
fc )
helical
empirical
0.0097 < d/D < 0.135 30 < NDe < 300 30 < NDe < 300 30 < NDe < NRe crit(d/D)1/2 NRe < NRe crit
( Dd )]N
[
-0.5 Re
32 NRe
( ) ( ) ( )
-0.5
D d
fc ) 1.8 NRe
fc ) 1.084 NRe Ito32
-0.6
D d
fc ) 5.22 NRe
fc ) 0.1033√NDe fs
D d
-0.2
{[ ( )] ( ) } 1.729 NDe
1+
0.5
-
1.729 NDe
0.5
theoretical
-3
Collin and Dennis33
fc ) 0.38 + 0.1028√NDe fs
torus
numerical
large NDe
Mishra and Gupta34
fc ) 1 + 0.033(log10 NHe)4 fs
helical
empirical
1 < NHe < 3000
helically coiled tube
correlation
where Hart et al.35
p )] [ (2πD
NHe ) NDe 1 +
{ [
fc ) f 1 +
1.5 0.090NDe 70 + NDe
]}
where f)
0.07725
[log10(NRe /7)]2
Hasson36
fc ) 0.556 + 0.0968√NDe fs
Mujawar and Rao37
fc 0.36 ) 0.26NDe fs
2 1/2
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investigators,19,20 who have indicated that pressure drop is proportional to NDe1/2. For large NDe, further investigation is required. Based on the studies on laminar flow in helical coils with a finite pitch,21-25 it is observed that, for a wide range of NDe values, the hydrodynamics of helical coils is governed by three parameters, namely, the Dean number (NDe), the Germano number (NGn), and the curvature ratio (dt/Dc). The flowgoverning parameters have been defined as NDe ) NRe
NGn ) NRe × δ ) NRe
{
() dt Dc
0.5
π2(Dc /dt) [π(Dc /dt)]2 + (p/dt)2
Table 2. Helical Coil Geometrical Parameters Tube Diameter [mm] inside outside outside diameter diameter, diameter, tube length, of coil support, coil pitch, coil type dt dto Lc [m] D [mm] p [mm] A B C D E
4.50 6.50 7.94 9.53 12.01
10.50 11.50 13.00 16.00 18.00
3.98 5.00 5.00 5.00 5.00
114.30 266.85 323.10 462.00 655.00
40 60 100 200 300
Table 3. Geometric Details of Different Helical Coils Used in the Present Study
}
According to Liu and Masliyah,26 for high NDe, a flow pattern parameter γ (which is defined as γ ) NGn/NDe3/2) was used for the transition from two-vortex flow to one-vortex flow. Consequently, a helical flow is governed by NDe, dt/Dc, and NGn or γ. The effect of torsion is evident only through the use of NGn or γ. When the curvature ratio (dt/Dc) is negligibly small, the flow in a helical coil can be explained by NDe and NGn or γ. When both dt/Dc and NGn or γ are negligible, the flow in a helical pipe is governed by NDe alone. However, for low NDe (NDe < 20), Liu and Masliyah26 used another flow transition parameter, γ* (which is defined as γ* ) NGn/NDe2). Ali27 developed empirical correlations between the pressure drop and the flow rate for helical coils, using characteristic dimensionless groups such as the Euler number (NEu), Reynolds number (NRe), and a new geometrical number that is a function of the equivalent diameter of the coil, length of the coiled tube, and the inside diameter of the tube. The author further reports that there could be four regions of flow: a laminar flow range, a turbulent flow range, and two ranges of transitional flow. For each region, correlations have been proposed using three characteristic dimensionless groups. Available friction factor correlations for laminar flow through helical coils are given in Table 1. Mostly, these correlations lack universal character, because of the fact that each correlation is restricted to certain geometrical or flow conditions. Disagreement regarding the relevant dimensionless groups and the form of dependency has often occurred between investigators. Table 1 showed that the correlations are applicable over a limited range of NDe values and, by and large, do not satisfy the boundary conditions (i.e., fc f fs for NDe f 0). Also, the effect of various critical parameters that affect the hydrodynamics of flow through coils has not been included, which probably could be the reason for the lack of universality in these correlations. 2. Experimental Section The helical coils were made from flexible poly(vinyl chloride) (PVC) tubing with different inside diameters. Helical coils with different pitches were constructed by winding the PVC tubing around cylindrical coil supports that had different diameters (see Table 2). To evaluate the effect of various parameters, experimental measurements were made under varying geometric and flow conditions. Five types of coils with 12 different combinations of varying radii of curvature yielded coil curvature values (δ) in the range of 9.44 × 103-81.53 × 103 and coil torsion values (τ) in the range of 0.47 × 103-18.53 × 103, as given in Table 3. The torsion parameter (β) lies in the range of 3.32 × 10-3 to 46.33 × 10-3, which is much smaller than the threshold
coil
Lc/dt
Dc/dt
p/dt
E D C A A A A B A B B A
416.67 524.93 629.92 883.33 883.33 883.33 883.33 766.92 883.33 766.92 766.92 883.33
11.66 14.49 17.04 28.02 28.38 29.53 34.95 43.38 43.97 51.94 102.77 105.48
8.33 10.49 12.60 8.89 13.33 22.22 44.44 15.39 66.67 15.39 15.39 22.22
δ τ β NRe crit [× 103] [× 103] [× 103] (from eq 1) 81.53 65.55 55.60 35.33 34.46 32.03 24.59 22.76 18.45 19.08 9.71 9.44
18.53 15.10 13.09 3.57 5.15 7.67 9.95 2.57 8.90 1.80 0.47 0.63
45.90 41.71 39.26 13.41 19.62 30.30 44.86 12.04 46.33 9.21 3.32 4.61
9479.91 8718.76 8204.59 6860.48 6830.18 6737.08 6362.68 5925.96 5900.29 5596.54 4585.75 4553.61
limit of β ) 0.5 that was suggested by Yamamoto et al.38 Therefore, in the present investigation, the effect of torsion on the flow was neglected. However, it was assumed that the torroidal approximation was valid and the focus was on the effect of curvature on the fluid flow. Furthermore, Table 3 shows that the ratio of dt/Dc closely approximates the coil curvature (δ). Hence, for the present study, the coil geometry was assumed to be characterized by the ratio of coil diameter to the tube diameter. The critical values of the Reynolds number (NRe crit) for each coil was determined using the correlation of Srinivasan et al.:31
[ ()]
NRe crit ) 2100 1 + 12
dt Dc
0.28
(1)
Complete details of coil geometrical parameters and the corresponding critical Reynolds numbers used in the study are given in Tables 2 and 3. The schematic of the experimental setup is given in Figure 1. To minimize the end effects on pressure drop measurements, the pressure taps were located away from the pipe ends by a distance of more than 100 pipe diameters. The pressure drop for each run was measured using carbon tetrachloride (CCl4) or mercury as the manometer fluid. The inner radius of the tubes was evaluated by weighing the water inside a known length of helical coil. The measurements were accurate to within (0.5%. In all experimental runs, water was used as the working fluid and was introduced to the coil from the overhead tank under ambient conditions. The water flow rate was measured using a calibrated rotameter. The setup was first calibrated by obtaining the pressure drop data on straight tubes. The measurements from straight tubes (fs) were in close agreement with the following resistance correlations39 for smooth straight pipes: For laminar flow:
fs )
16 NRe
(2)
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Figure 1. Experimental setup.
For turbulent flow:
0.125 NRe0.32 (for 3000 < NRe 70
a
b
for a
for b
goodness of fit, R2
root-mean-square deviation, rmsd
variance
0.903 0.525
0.227 0.516
( 0.042 ( 0.036
( 0.015 ( 0.018
0.98 0.98
4.65 × 10-5 6.23 × 10-5
6.86 × 10-7 1.39 × 10-6
For the same NRe value, the friction factor decreases marginally as the p/dt value increases. From similar results on other coils with different geometrical combinations, it can be inferred that both Dc/dt and p/dt play a significant role in increasing or decreasing the pressure drop in the coils. The effect of these parameters also is consistent with the results of other investigators (Mishra and Gupta34). 3.1. Correlation of Data. Figure 2 shows that the coil friction factor (fc) seems to have a reasonable dependence on Dc/dt and p/dt. Since neither NRe or NDe alone cannot elucidate the combined effect of Dc/dt and p/dt, it was considered worthwhile to use the Germano number (NGn) as the correlating parameter. The Germano number is defined as NGn ) NReδ and represents the combined effect of the variation in coil diameter and coil pitch. The ratio of the friction factors for a helical coil (fc) to that for a straight pipe (fs) is shown in Figure 3, as a function of NGn. As can be seen from Figure 3, the ratio fc/fs has a reasonable dependence on NGn. Furthermore, an inflection point at NGn ≈ 70 is observed. This behavior can be attributed to the fact that, at high NDe, the effect of torsion becomes apparent andm hencem the flow pattern changes from two-vortex flow to one-vortex flow, as observed by Liu et al.25 Therefore, it is evident that no single correlation can represent the data
over the entire range of laminar flow. An attempt was made to represent the present friction factor data, in the form b fc ) fs(1 + aNGn )
(9)
where fs ) 16/NRe and a and b are two adjustable constants. Using the Nelder-Mead standard routine for nonlinear parametric estimation, the friction factor data for helical coils, under laminar flow conditions, were fitted to the model equation (eq 9). The model parameters thus obtained (a and b) are given in Table 4. Present data were compared to a set of nine most widely referred to correlations (see Table 1). Figure 4 shows a parity plot between the experimentally measured values of the friction factor (fc) and the calculated values based on present model (eq 9) and the models available in the literature. The present model (eq 9) predicts the observed coil friction factor values to within (10%. However, model equations available in the literature predict the present data to within (30%. Statistical comparison, in terms of the absolute average error parameter (%AE ) 100/ N| ∑[fpred - fexp)/fexp] |, where N is the number of data points), is given in Table 5. This table shows that our correlation (eq 9) explains the present data regarding the friction factor on helical
Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011
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Figure 4. Parity between fc(calculated, based on eq 9) and fc(experimental). Table 5. Statistical Comparison of Different Correlations for Prediction of the Coil Friction Factor to the Present Experimental Data investigator 19
Adler White28 Mori and Nakiyama30 Srinivasan et al.31 Ito32 Collin and Dennis33 Mishra and Gupta34 Hasson36 Mujawar and Rao37 present model (eq 9)
absolute average error, AE [%] 5.09 8.01 11.83 11.50 8.80 5.87 8.33 7.15 5.51 2.62
coils over a wide range of coil diameters and coil pitches under laminar flow conditions to a reasonable accuracy, compared to other correlations. The friction factor data from Cioncolini and Santini12 on helical coils, with coil diameter to tube diameter ratios in the range of 6.9-369, were compared with the predictions of the present model (eq 9). Figure 5 shows a parity plot between the literature data of Cioncolini and Santini12 and the predicted friction factor. The present model (eq 9) predicts the data of Cioncolini and Santini,12 to within a reasonable accuracy of (15%, in comparison to other model equations, where the accuracy of predictions is within (30%. Statistical comparison, in terms of the parameter %AE, is given in Table 6. The proposed model equation (eq 9) predicts the available
Table 6. Statistical Comparison of the Present Model with Other Models, Using the Data of Cioncolini and Santini12 on the Friction Factor for Flow through Helical Coils investigator 19
Adler White28 Mori and Nakiyama30 Srinivasan et al.31 Ito32 Collin and Dennis33 Mishra and Gupta34 Hasson36 Mujawar and Rao37 present model (eq 9)
absolute average error, AE [%] 7.26 27.05 31.20 8.31 27.56 23.07 26.93 26.10 12.02 3.28
literature data reasonably well, with an absolute average error of %AE ) 3.28%, compared to the correlations available in the literature. 4. Conclusion (1) The parametric study on laminar flow through helical coils clearly indicates a significant dependence of the coil friction factor (fc) on coil geometrical parameters, such as Dc/dt and p/dt. (2) In the present study, the combined effect of coil geometrical parameters on the coil friction factor is wellrepresented by the Germano number (NGn), in comparison to the conventional Dean number (NDe), in the range of 1 < NGn < 130.
Figure 5. Parity between fc(calculated) and fc(experimental) (using data taken from Cioncolini and Santini12).
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(3) Based on the present experimental data obtained on coils with varying geometrical parameters (8.3 e p/dt e 66.7 and 11.7 e Dc/dt e 105.48), the coil friction factor (fc) is very wellrepresented, in terms of NGn, by the following correlations: 0.227 fc ) fs(1 + 0.903NGn )
(for NGn e 70)
fc ) fs(1 +
(for NGn > 70)
0.516 0.525NGn )
These values are accurate to within (10%, with fs ) 16/NRe. (4) These correlations also predict the friction factor data on coils available in the literature reasonably well, to within (15%. (5) The available correlations for fc explained the present experimental data within an absolute average error (%AE) ranging from 5.09% to 11.83%. Furthermore, these models predicted the available literature data on helical coils with an absolute average error of %AE ) 7.26%-31.20%. (6) The present correlation (eq 9) gave excellent predictions of the present data and the data available in the literature, with %AE values of 2.62% and 3.28%, respectively. It can be safely concluded that the helical coil friction factor data (fc), under laminar flow conditions, is very well-represented in terms of the Germano number (NGn), in comparison to the Dean number (NDe), in the range of 1 < NGn < 130. Notation ∆m ) mass of water collected [kg] ∆P ) pressure drop across coil [Pa] ∆t ) collection time [s] m ˙ ) mass flow rate of water [kg/s] a, b ) constants of eq 9 D, Dc ) coil diameter [m] d, dt ) tube diameter [m] dto ) outer diameter of tube [m] f ) friction factor g ) acceleration due to gravity [m/s2] h ) manometer reading [m] Lc ) coil length [m] N ) number of data points NDe ) Dean number NGn ) Germano number NHe ) Helical coil number [-] NRe ) Reynolds number NRe crit ) critical Reynolds number p ) coil pitch [m] R ) coil radius [m] r ) tube radius [m] rmsd ) root-mean-square deviation V ) velocity of water [m/s] Greek Symbols δ ) coil curvature; δ ) π2(Dc/dt)]/{[π(Dc/dt)]2 + (p/dt)2} τ ) coil torsion; τ ) π(p/dt)/{[π(Dc/dt)]2 + (p/dt)2]} β ) torsion parameter; β ) τ/(2δ)1/2 γ ) flow pattern parameter γ* ) flow transition parameter µ ) viscosity of water [Pa s] F ) density of coil fluid (water) [kg/m3] Subscripts c ) coil exp ) experimental m ) manometric fluid pred ) predicted s ) straight tube t ) tube
AbbreViations %AE ) absolute average error; %AE ) (100/N)| ∑[(fpred fexp)/fexp] |
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ReceiVed for reView August 24, 2010 ReVised manuscript receiVed November 16, 2010 Accepted November 19, 2010 IE101752Z
