Journal of Membrane Science 288 (2007) 321–335
Dean vortices applied to membrane process Part II: Numerical approach R. Moll a , D. Veyret b , F. Charbit a , P. Moulin a,∗ a
Laboratoire en Proc´ed´es Propres et Environnement (LPPE-CNRS-UMR 6181), Universit´e Paul C´ezanne Aix Marseille III, Europˆole de l’Arbois, BP 80, Bat. Laennec, Hall C, 13545 Aix en Provence Cedex 04, France b Laboratoire IUSTI-CNRS UMR 6595, Technopˆ ole de Chˆateau Gombert, 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France Received 2 February 2006; received in revised form 2 November 2006; accepted 21 November 2006 Available online 26 November 2006
Abstract This work is devoted to the numerical study of the flow inside helical hollow fibers. For a wide range of Reynolds numbers and of helix shapes, corresponding to woven or non-woven, permeable or non-permeable fibers, the corresponding three-dimensional velocity field is calculated in order to study the velocity profiles and the local wall shear stress. The influence of permeation is not qualitatively marked within the ranges of conditions that reproduce the classical experimental ultrafiltration conditions. The permeate flux density depends on the permeability of the membrane and is constant on the perimeter of a cross-section of the tube. For all the permeability situations considered, the Dean flow depends on three parameters: the Reynolds number, the curvature and the helix torsion. The effect of each of these parameters depends largely on the relative values of the two others. The Dean number makes it possible to predict the mean shear stress, as well as its distribution along the perimeter of a cross-section. For woven fibers, as their diameter is small, the pitch has a significant influence on the curvature. © 2006 Elsevier B.V. All rights reserved. Keywords: Dean vortices; Numerical simulation; Woven hollow fibers; Pitch effect; Diameter effect; Permeation effect
1. Introduction Despite the apparent simplicity of the Dean flow, its numerical or analytical resolution [1,2] as well as the understanding of its behavior is still the subject of research works. The numerical resolution of the flow has been achieved several times for a torus [3,4]. It has been shown that the pressure distribution depends on the Dean number, that the pressure increases slightly from the inside towards the outside of the tube and that the convective terms are not important for a Dean number lower than 16 [5]. The analytical resolution used by Dean [6] was based on a method called the perturbation method. Akiyama and Cheng [7] showed that this method was not valid: they obtained friction coefficients in agreement with the experiment and observed the influence of the inlet flow on the heat transfer. Ranade and Ulbrecht [8] used laser Doppler anemometry to measure the velocity profile of Newtonian and non-Newtonian
∗
Corresponding author. Tel.: +33 4 42 90 85 01; fax: +33 4 42 90 85 15. E-mail address:
[email protected] (P. Moulin).
0376-7388/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2006.11.037
liquids in torus and obtained a good agreement with his numerical predictions for the axial flow but not for the secondary flow. As the flow in a helical tube is extremely complex, Germano [9,10] set up an orthogonal coordinate system along a generating line and developed the equations allowing the resolution of the flow for a helical tube. In particular, they introduced the torsion in these equations. It was thought previously that only the curvature was at the origin of the secondary flow and so the torsion effect was not taken into account. The torsion can cause a 90◦ translation of the secondary flow [11] and lead to an asymmetrical flow [12], which was not predicted by the Dean number [13]. In a turbulent regime the torsion does not modify the axial velocity profile but modifies the intensity and distribution of the radial velocity [14]. Silva et al. [15] studied rectangular and elliptical tubes (in which a stronger secondary flow is generated) in terms of pressure drop, velocity profile and heat transfer, calculated by numerical simulation. Before the transition to a turbulent regime, there is a region of vortex instability. In this region, each vortex separates into two and then four vortices.
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Steady, fully-developed laminar flows in helical pipes with a finite pitch were numerically studied by Liu and Masliyah [16]. They found out that the effect of torsion can be neglected for curvature ratio less than 0.01 (for moderate Dean number) and that for flow with high Dean number and curvature ratio, increasing the torsion changes the relative position of the secondary flow vortices. Mizushima et al. [17] studied the phenomenon of coalescence and division of the Dean vortices. They showed that their division is generated by a phenomenon of resonance interactions. They determined the domain of stability of the Dean vortices. Daskopoulos and Lenhoff [18] studied analytically the bifurcation structure. Despite symmetrical perturbations, the two patterns resulting from the bifurcation (one or two pairs of vortices) remained stable. Yamamoto et al. [19] studied the stability of the flow in a helical tube. They showed that up to a certain value of the torsion the critical Dean number decreases (transition to a turbulent regime), and then starts increasing when this value is exceeded. The shape of the sections can also have a strong influence on the flow: Bolinder [20] studied the flow in helical square-section tubes. They showed that the transition from the first stability region with two vortices to the second region with four vortices occurs farther than in circular-section tubes. The purpose of this work is the numerical simulation of the flow and shear stress profiles for different values of the coiled diameter, of the pitch and of the Reynolds number. First non-permeable hollow fibers are studied. Then the influence of permeation on the flow and on the shear stress is studied. 2. Method FIDAP [21] was used to simulate the flow. The simulated flow corresponded to a single phase, incompressible fluid with constant Newtonian rheological properties. The membrane walls were considered to be rigid and steady, isothermal flow is considered. The Navier-Stokes and continuity equations in a fixed Cartesian coordinate system (O, x, y, z) are: 2 ∂u ∂u ∂u ∂P ∂ u ∂2 u ∂2 u ρ u +v +w + 2 + 2 =− +μ ∂x ∂y ∂z ∂x ∂x2 ∂y ∂z (1)
∂v ∂v ∂v ρ u +v +w ∂x ∂y ∂z
∂P =− +μ ∂y
∂2 v ∂2 v ∂2 v + 2 + 2 ∂x2 ∂y ∂z
(2)
∂w ∂w ∂w ρ u +v +w ∂x ∂y ∂z
∂u ∂v ∂w + + =0 ∂x ∂y ∂z
∂P =− +μ ∂z
∂2 w ∂2 w ∂2 w + + ∂x2 ∂y2 ∂z2 (3)
(4)
Since the regime is considered as being steady, this is a system of local equations that will have to be spatially integrated.
Fig. 1. Boundary conditions.
In terms of boundary conditions, it is supposed that the region of the tube upstream of the simulated domain is straight and that the tube outlet is free (Fig. 1). Thus, the selected boundary conditions are: (i) at the inlet of the membrane: a parabolic velocity profile is specified; (ii) at the outlet of the membrane: a free flow under atmospheric pressure is assumed; (iii) at the wall of the tube: - In the case of non-permeable fibers, the velocity of the fluid in contact with the wall is zero. - In the case of permeable fibers, the Forcheimer-Brinkman model is used to simulate a flow through the porous membrane. For a fluid with a constant viscosity, a form of the Darcy model can be found: μ ∂P + =0 Kx ∂x
(5)
μ ∂P + =0 Ky ∂y
(6)
μ ∂P + =0 Kz ∂z
(7)
where Kx , Ky , Kz are the permeabilities in the three space directions. The porous medium is defined by its porosity and the Darcy permeability coefficients that characterize its capacity to let the fluid go through. The permeate flow is considered free after its passage through the porous membrane and the pressure outside the membrane is equal to the atmospheric pressure. The basis of the finite element-based algorithm is the use of a projection method to approximate the solution of the NavierStokes equations. The simulations were performed on a SUN Enterprise 4000 (six processors) requiring 1–2 h of computational time. Between 25 and 50 iterations (depending on the Reynolds number and geometry parameters) were required for the problem to converge, each of these iterations was done in 150 s. Table 1 illustrates the refinement of the mesh based on the accuracy of the solution obtained for the velocity. Both graded and uniform meshes were examined: during the first simulations
R. Moll et al. / Journal of Membrane Science 288 (2007) 321–335 Table 1 Grid accuracy
323
Table 3 Characteristics of the geometries
Number of nodes per section × length (eight-node brick elements)
Type of mesh
u0,0
244 × 360 392 × 360 788 × 360 1126 × 720
Graded Graded Graded Graded
0.6348 0.6480 0.6657 0.6662
0.6114 0.6115 0.6120 0.6120
1.565 1.566 1.570 1.571
244 × 360 392 × 360 788 × 360 1126 × 720
Uniform Uniform Uniform Uniform
0.6288 0.6399 0.6537 0.6602
0.6113 0.6115 0.6117 0.6118
1.561 1.563 1.565 1.568
rmax
θ max
it was observed that the velocity gradients were stronger near the wall, so the mesh was refined there. Comparison of the velocity at the center of the pipe and the location of the maximum velocity shown in Table 1 is presented for four different meshes in both non-uniform and uniform distribution. Depending on the grid refinement and type, the difference ranges within 0.1–3.0%. The graded mesh gave more accurate results then the uniform. As the velocities and the coordinates of each of the mesh nodes were known, it was thus possible to calculate the wall shear stress. The stress tensor was calculated in helical coordinates [22]. A comparison with the results obtained by Liu and Masliyah [16] (Table 2 and Fig. 2) shows a good agreement. 3. Results and discussion 3.1. Description of the flow The Dean flow depends on both the tube geometry and the flow velocity in the tube. In order to get a comprehensive view, two tubes with different geometries were considered (Fig. 3). The characteristics of these two geometries are detailed in Table 3. The definitions of the different symbols are given in Appendix A. The V01 tube has the same characteristics as the one used by Moulin [23] during laser visualization experiments. The other tube was equivalent to a woven hollow fiber. Figs. 4 and 5 show the variations in the velocity profiles in the two tubes when the Reynolds number increases. For low Reynolds numbers (about 100), the profile of the axial velocity is quasi-parabolic and there is only one pair of vortices in the secondary flow. When the Reynolds number increases, the velocity profile varies. The representation of the axial velocity (i.e. orthogonal to the section) reveals that little by little the flow gets farther from the center of the tube to get closer to the Table 2 Comparison with Liu and Masliyah [16] (Re = 1000, De = 100, l = 0.01) η
γ
Liu u0,0
Liu umax
Present work u0,0
Present work umax
0.02 0.185 0.2 1
0.02 0.185 0.2 1
0.6288 0.8929 0.9290 0.9975
0.8733 0.9796 0.9812 0.9977
0.6223 0.8896 0.9243 0.9956
0.8670 0.9702 0.9730 0.9961
Name
di (mm)
dc (mm)
p (mm)
ε
λ
V01 T01
3.2 0.7
20.00 1.22
30 40
0.2610 0.0104
0.124 0.109
extrados of the bend and follows the helix torsion. As the flow rate increases, the variations in the velocity field are even more pronounced: the parabolic shape spreads out along the external wall of the bend where the velocities become maximum, and then turns into the well-known bean-like shape [6,8]. This is even more marked for higher Reynolds numbers (above 1500) for which the velocity gradients are higher. The radial velocity profile (i.e. orthogonal to the axis) varies as well: for low Reynolds numbers the two vortices are symmetrical with regard to the diameter, and the radial velocity is maximum at the center of the section. A translation of the centers of the vortices towards the intrados occurs when the circulation flow rate increases. The region where the radial velocity is maximum lies between the centers of the vortices and the wall. A similar behavior has been observed by several authors [12,19,24,25]. The velocity profile follows the distribution of the centrifugal forces. For a hollow fiber configuration, the flow generally remains laminar; that is the reason why turbulence is not simulated in this work. In helical tubes however, it is acknowledged that beyond critical Reynolds number, an unstable region appears which contains one or two more pairs of vortices. Mishra and Gupta [26] used an empirical relation to predict the critical Reynolds number in a helical tube: 0,32 di Rec = 2.104 (8) dc Two was the highest number of vortices detected in any of the cases studied here. 3.2. Development of the flow Austin and Seader [4] estimated the length of the flow development in a coiled tube with a parabolic inlet flow rate by the relation: di 1/3 θA = 87.3 4De (9) dc By extrapolating the Austin relation, as Mishra and Gupta did [26], the flow is fully developed for an angle of 216◦ for Re = 320 and for an angle of 371◦ for Re = 1627. This was confirmed by comparing the changes in the velocity vectors from a section to another. 3.3. Wall shear We studied the influence of the flow rate and then that of the coil diameter and of the pitch, for a constant Reynolds number and then for a constant modified Dean number.
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Fig. 2. Axial velocity profiles (di = 1 mm, dc = 4 mm, p = 2 mm): (a) results obtained by Liu and Masliyah [16]; (b) present study; (i) Re = 10; (ii) Re = 20; (iii) Re = 100; (iv) Re = 201; (v) Re = 1003.
3.3.1. Influence of the flow rate For a given flow rate, the helical tube was compared with a straight tube with the same internal diameter. For the straight tube, the wall shear stress was calculated using the relation: 8μv τstraight = (10) di
For the helical tube, we calculated the stress tensor and then the ratio of the local wall shear stress to the straight wall shear stress. The coordinate system used to display the shear stress is shown in Fig. 6. Figs. 7 and 8 show the variation of the shear stress ratios as a function of the angular position for the V01 and T01 tubes, respectively. The wall shear stress is strong at the
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325
Fig. 3. Tube representation: V01 and T01.
extrados of the bend (angle between 90◦ and 270◦ ) and weak at the intrados. As the flow rate increases, the difference between the wall shear stress in a helical tube and that in a straight tube increases. For helical tubes, locally, the difference between the shear stress at the intrados and extrados is also greater when the
Fig. 5. Influence of flow rate on the velocity profile. Left: axial velocity; right: radial velocity (T01, θ = 524◦ ).
flow rate is higher. The shear stress ratio can be slightly lower than 1, due to the fact that the flow rate near the intrados in a helical tube is smaller than that near the wall in a straight tube. The variation is less significant for T01 than for V01: T01 is less curved and for a same flow rate, the Dean number and the shear stress are lower. However, the increase and distribution of the local shear stress vary in the same way for both helixes. It will
Fig. 4. Influence of flow rate on the velocity profile. Left: axial velocity; right: radial velocity (V01, θ = 524◦ ).
Fig. 6. Intrados and extrados positions for the shear stress results.
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Fig. 9. Variation of the dimensionless curvature vs. coiled diameter (di = 1 mm, p = 40 mm, Re = 1000). Fig. 7. Variation of the wall shear stress vs. angle in a section for different Reynolds number (V01, θ = 524◦ ).
be shown later that the wall shear stress depends mainly on the Dean number. 3.3.2. Influence of the coiled diameter Table 4 summarizes all the simulations presented in this paragraph. The coiled diameter of a helical tube represents the inclination of the bend. The effect of the variation of the coiled diameter depends on the initial configuration of the tube. In the case of a torus, the curvature decreases when the coiled diameter increases. In the case of helical tubes, when the coiled diameter increases, the curvature increases up to a maximum value, above which it starts decreasing. In the following, the influence of the coiled diameter on the flow is examined using different helical configurations: first, the Reynolds number is kept identical for all the tubes, then the Dean number. 3.3.2.1. Constant Reynolds number. The dimensionless curvature increases with the coil diameter up to dc = 14.4 mm, then decreases (Fig. 9). For S01 to S09 tubes, the stronger the curvature, the higher the axial and radial velocities are near the tube wall. The local shear stress was determined for each one of these tubes. Fig. 10a shows the local stress for the tubes in which the curvature increases when the coiled diameter increases and Fig. 10b the local stress for the tubes in which the coiled diameter is equal to or more than 14.4 mm. The symmetry of the stresses on the perimeter of a cross-section of the tube is not strictly identical. For the smallest coiled diameters (1.8 mm) the curve representing the shear stress is quasi-symmetrical with respect to the 150◦ . For greater coiled diameters (from 5 mm)
Fig. 8. Variation of the wall shear stress vs. angle in a section for different Reynolds number (T01, θ = 524◦ ).
the symmetry was with respect to the 180◦ . Within the interval corresponding to the intrados (0–45◦ and 315–360◦ ) the shear stress remained close to the one calculated for a straight tube, and outside this region the closer to the extrados of the tube, the stronger the difference between the geometries. 3.3.2.2. Constant modified Dean number. Several tubes were used with the same modified Dean number of 138. For each tube, the reference was a straight tube through which the same fluid flows, with the same Reynolds number. For a constant modified Dean number, the more curved the tube, the lower the Reynolds number. For De = 138, there was a two-fold increase of the flow rate from S01 to S07. According to Mishra and Gupta [26], for a same modified Dean number, the ratio of the mean shear stress in a helical tube to that in a straight tube with the same flow rate is the same whatever the geometry of the tube. In Fig. 11, it can be seen that the stress shapes are identical and that there is a weak rotation effect similar to the one observed on the axial velocity profile. Although the stress ratios are the same, the Reynolds
Fig. 10. (a and b) Variation of wall shear stress ratio vs. angle in a section for different coiled diameter (di = 1 mm, p = 40 mm, θ = 524◦ , Re = 1000).
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Table 4 Simulations to study the effect of the coiled diameter Name S01 S02 S03 S04 S05 S06 S07 S08 S09 S10 S11 S01 S12 S06b S07b S08b S09b
di (mm) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
dc (mm) 1.8 2.6 4.0 5.0 6.0 7.2 14.4 28.8 57.6 3.0 10.0 1.8 3.6 7.2 14.4 28.8 57.6
p (mm) 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
Re 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1323 962 752 699 810 1073
De
ε
λ
104 124 149 163 174 183 197 170 128 132 195 138 138 138 138 138 138
2.18 × 10−2
0.154 0.151 0.143 0.136 0.129 0.119 0.069 0.026 0.007 0.149 0.097 0.154 0.145 0.119 0.0689 0.0257 0.00732
3.08 × 10−2 4.49 × 10−2 5.34 × 10−2 6.06 × 10−2 6.73 × 10−2 7.79 × 10−2 5.81 × 10−2 3.31 × 10−2 3.51 × 10−2 7.63 × 10−2 2.2 × 10−2 4.1 × 10−2 6.7 × 10−2 7.8 × 10−2 5.8 × 10−2 3.3 × 10−2
wall or extrados. This entails a higher and less homogeneous wall shear stress. The radial velocity increases proportionally to the curvature and the vortices get closer to the internal tube wall. The increase in the curvature has the same effect as an increase in the flow rate. If two helical tubes with different geometries have the same (mean) wall shear stress, they will not have the same Reynolds number or the same modified Dean number, as represented in Fig. 13. The modified Dean number depends on the curvature and on the Reynolds number. The simulations carried out for constant Dean numbers show that other parameters (among which the torsion) also influence the flow, but to a lesser extent. Fig. 11. Variation of wall shear stress ratio vs. angle in a section for different coiled diameter (di = 1 mm, p = 40 mm, θ = 524◦ , De = 128).
number – and thus the mean shear stress – is different for each tube. The variation of the mean shear stress as a function of the coiled diameter is represented in Fig. 12. For a constant Dean number, the less curved the tube, the higher the shear stress. To conclude on the influence of the coiled diameter, it can be said that for a given internal diameter, circulation velocity and pitch there is a coiled diameter for which the modified Dean number is maximized. As the curvature increases, the axial flow spreads out and the velocity gets higher closer to the external
Fig. 12. Variation of the average wall shear stress vs. coiled diameter for a constant modified Dean number (di = 1 mm, p = 40 mm, θ = 524◦ , De = 138).
3.3.3. Influence of the pitch The influence of the pitch on the flow has not very often been taken into account, although it can considerably decrease the curvature. In a helical tube, the modified Dean number is defined by: di De = Re (11) dc (1 + (p/πdc )2 )
Fig. 13. Variation of the Reynolds number and of the modified Dean number vs. coiled diameter for a constant wall shear stress τ w = 16 Pa (di = 1 mm, p = 40 mm, θ = 524◦ ).
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R. Moll et al. / Journal of Membrane Science 288 (2007) 321–335 Table 5 Simulations to study the pitch effect
Fig. 14. (a and b) Variation of ratio R2 vs. ratio p/dc (Re = 1000).
Let us consider now a torus with the same internal diameter and coil diameter as this helical tube and in which the flow rate is the same. The Dean number in this tube is: di DeTeq = Re (12) dc The influence of the pitch on the Dean number can be estimated by the ratio: R2 =
De 1 = DeTeq 1 + (p/πdc )2
(13)
The variation of this ratio R2 with the pitch is represented in Fig. 14a and b. There are two types of situations: if the pitch is smaller than or nearly the same as the coiled diameter, then the decrease in the curvature and Dean number is slight (Fig. 14a) whereas it becomes stronger for a pitch higher than the coiled diameter (Fig. 14b). Fig. 15 shows the variation of the torsion as a function of the pitch for a given coiled diameter. For all the configurations studied, an increasing pitch led principally to a decrease in the curvature and torsion. We therefore tried to deter-
Fig. 15. Variation of the torsion vs. pitch (di = 1 mm, dc = 1.8 mm, Re = 1000).
Name
di (mm)
dc (mm)
p (mm)
Re
De
ε
λ
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P01b P03b P04b P06b P17
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 14.4 14.4 14.4 14.4 14.4 14.4 14.4 14.4 1.8 1.8 1.8 1.8 1.8
30 40 50 60 70 80 90 120 40 50 60 70 80 90 100 120 30 50 60 80 100
1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1647 1973 2299 3279
138 104 84 70 60 53 47 42 197 177 159 143 130 118 109 93 138 138 138 138 138
0.038 0.022 0.014 0.010 0.007 0.006 0.004 0.004 0.078 0.063 0.050 0.041 0.034 0.028 0.024 0.017 0.038 0.014 0.010 0.007 0.004
0.202 0.154 0.124 0.104 0.089 0.078 0.070 0.063 0.069 0.069 0.067 0.063 0.060 0.056 0.052 0.046 0.202 0.124 0.104 0.089 0.063
mine what influence the pitch has on the flow. The simulations used are summarized in Table 5. 3.3.3.1. Constant Reynolds number. Fig. 16 shows the ratio of the local wall shear stress for tubes that differ by their pitch (small coil diameter). The mean shear stress decreases from 13 to 8 Pa for a pitch increasing from p = 30 to 200 mm. The greater the curvature, the greater the difference between the shear stress at the extrados and that at the intrados. 3.3.3.2. Constant modified Dean number. The wall shear stress ratio is represented in Fig. 17 for the tubes considered (De = 138). This ratio is the same whatever the tube. However, for a constant Dean number, the greater the pitch, the greater the mean wall shear stress (the circulation velocity increases with the pitch). To conclude on the pitch effect, it can be said that when the pitch increases, the curvature and the Dean flow intensity decrease. For a constant modified Dean number, the velocity profile is the same everywhere. The pitch variation has a stronger
Fig. 16. Variation of wall shear stress ratio vs. angle in a section for different pitchs (di = 1 mm, dc = 1.8 mm, θ = 524◦ , Re = 1000).
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Table 6 Geometry parameters
Fig. 17. Variation of wall shear stress ratio vs. angle in a section for different pitchs (di = 1 mm, dc = 1.8 mm, θ = 524◦ , De = 138).
influence on the torsion than the coiled diameter variation, but it does not seem to have any significant influence on the flow. A variation of the curvature due to a change in the coiled diameter or the pitch seems to have a similar effect on the torsion. 3.4. Prediction of the wall shear stress Curved tubes are very often used in industry (heat transfer, piping). It is necessary to be able to predict certain properties of the flow in such tubes, such as the pressure drop, directly linked to the shear stress: L P = 4(τw ) (14) di with L is the length of the tube (m); di is the internal diameter of the tube (m); τ w is the wall shear stress (Pa); P is the pressure drop (Pa). To predict pressure drops, most of the existing correlations are only suitable for given parameter ranges: Reynolds number, ratio of the internal diameter to the coiled diameter, ratio of the pitch to the coiled diameter. Certain predictions are based on the modified Dean number only, others also involve geometrical ratios corresponding to the curvature or the torsion. Predicting a property of the flow using a correlation is useful only if it is simple and it covers a wide enough range of geometrical and hydrodynamical parameters. The most often used correlation is that of Mishra and Gupta [26] which is expressed in the laminar regime under the form: f/2helical 4 = 1 + 0.033(logDe ) f/2straight
Parameter
Range
di (mm) dc (mm) p (mm) Re De
0.2–2 0.36–110 0–200 10–3000 0.4–730
such tubes cannot be predicted by the correlation. The other correlations that take into account the effect of the pitch also apply to limited geometry regions and do not apply to woven hollow fibers. A campaign of 500 simulations was carried out on non-permeable tubes, in which only the mean value of the wall shear stress for a fully developed flow was considered. More than half of the simulations involved tubes which were outside the validity range studied by Mishra. Table 6 presents the range considered in these simulations. The friction coefficient and the shear stress being directly proportional, only the ratio of the mean shear stress in a helical tube to that in a straight tube was considered. Fig. 18 shows the variation of the mean wall shear stress as a function of the modified Dean number, for two series of tubes (tubes with a geometry within the validity range and tubes with a geometry outside this validity range). The respective variations of the shear stress as a function of the modified Dean number were not discernable. The differences did not exceed 1 or 2%. The values obtained were always lower than the values predicted using Mishra’s correlation, but remained within the predicted error margin (6%). For low Dean numbers, there was a threshold where the ratio of the mean shear stress in a helical tube to that in a straight tube was equal to 1. The end of the threshold corresponds to the critical Dean number (Dec = 13), that is to say to the number above which improvements, for example increase in the pressure drops or in the heat transfer, can be observed. Castelain et al. [27], Mallubhotla and Belfort [28] and Moulin et al. [29] observed experimentally a critical Dean number of about 20. Soeberg [5] obtained a similar result for the resolution in a torus (Dec = 16). We carried out many more simulations
(15)
where f/2 is the friction coefficient defined by: f τw = 2 2 ρv
(16)
where v is the mean circulation velocity (m s−1 ); ρ is the fluid density (kg m−3 ). The validity range for the relation is: 0.003
