Flow through triple helical microchannel

Flow through helical tubes and channels have been examined in different contexts, for facilitating heat and mass transfer at low Reynolds number flow, for generating plug flow to minimize reactor volume for many reactions. The curvature and torsion of the helices have been shown to engender secondary flow in addition to the primary axial flow, which enhances passive in-plane mixing between different fluid streams. Most of these studies, however, involve a single spiral with circular cross-section, which in essence is symmetric. It is not known, however, how the coupled effect of asymmetry of cross-section and the curvature and torsion of channel would affect the flow profile inside such tubes or channels. In this context, we have presented here the analysis of fluid flow at low Reynolds number inside a novel triple helical channel that consists of three helical flow paths joined along their contour length forming a single channel. We have carried out both microparticle image velocimetry (micro-PIV) and 3D simulation in FLUENT of flow of a Newtonian fluid through such channels. Our analysis shows that whereas in conventional single helices, the secondary flow is characterized by two counter-rotating vortices, in the case of triple helical channels, number of such vortices increases with the helix angle. Such flow profile is expected to enhance possibility of mixing between the liquids, yet diminish the pressure drop.

Figure 1

(a) Side view of a single helical microchannel with helix angle $\theta$ and helix radius $a$. (b) Side view of a triple helical channel. $\lambda$ defines the pitch or the wavelength of the channel. (c) The front view of the inlet of triple helical channel. (d) 3D view of triple helical channel.

Figure 2

(a) Triple helical microchannel with $\theta ={16}^0$. (b, c) Images taken by the micro-PIV at two different time steps at Reynolds number, $\mathrm{Re}=7$. Images (b) and (c) were captured at two different time steps, 10 μs apart.

Figure 3

Determination of radial velocity at any constituent lobe of the triple helical channel. (a–d) Cross-sectional images of different locations on the axial length of the channel between $-\lambda /6$ to $\lambda /6$. The arrow represents the radial direction through which $v_x$ is estimated at these different locations. (e) The $v_x$ data for all these different locations are compiled together to obtain the radial velocity at a particular cross-section of the channel.

Figure 4

(a, b) Schematic representation of computational grid generation at the inlet and cross-section parallel to the main channel axis of triple helical channel. (c, d) Expanded view at different positions of the cross-section. (e) Plot of area-weighted average axial velocity, $\overline{v_a}$ as a function of number of nodes $N_g$ in the triple helical channel shows that the grid convergence is achieved at $12\times {10}^5$. The dashed line shows the inlet average velocity to the channel.

Figure 5

(a, b) Contour profiles of axial velocity $v_a$ single helical channel and triple helical channel $\theta =9^0-34^0$ for $\mathrm{Re}=1-500$. For lower helix angles, spatial variation in $v_a$ occurs as $\theta$ is increased. Eventually, with increase in $\mathrm{Re}$ and $\theta$, the maximum $v_a$ is shifted toward the wall of the triple helix.

Figure 6

The axial velocity $v_z$ from experiment (dotted lines) and simulation (solid lines) are plotted along a radial direction for $\theta =16^{\mathrm{o}}{,34}^{\circ }$ and $\mathrm{Re}=7$. Axial velocities are recorded at a typical cross-section as represented by the dashed line in image (c).

Figure 7

(a) Contour profiles of radial velocity, $v_r$ triple helical channel with $\theta =9^0\text{−}34^0$ for $\mathrm{Re}=1-500$. For lower helix angles, spatial variation in $v_r$ occurs as $\theta$ is increased. Eventually, with increase in $\mathrm{Re}$ and $\theta$, the maximum negative $v_r$ is shifted toward the wall of the triple helix. (b) Maximum radial velocity, $v_r{|}_{\max }$ as obtained from different sets of simulations of flow in triple helical channel fall on a single master line when plotted against the dimensionless quantity $\mathrm{Re}{\left(\kappa {\tau }^2{a}^3\right)}^{0.25}$. Symbols $•,◇,*,\mathrm{\Delta },\square$ and $◯$ represent the data obtained for triple helical channel at $\mathrm{Re}=1$,10,50,100, 150, and 500, respectively.

Figure 8

Radial velocity $v_r$ from experiment (symbols) and simulation (solid lines) are plotted against the degree of rotation $\beta$ (as shown in image) in a particular circular lobe. The plots (a) and (d) correspond respectively to $\theta =16^{\circ }$ and $22^{\circ }$ and Reynolds number $\mathrm{Re}=7$; and plots (b) and (c) represent, respectively, $\theta =22^{\circ },34^{\circ }$, and $\mathrm{Re}=0.7$. The data sets 1–3 represent three different radial locations as shown by the dashed lines as shown in the representative image of the cross-section. For any helix angle $\theta$ and $\mathrm{Re}$, spatial variation of $v_r$ is such that positive and negative poles are generated in a single lobe in triple helix.

Figure 9

(a, b) Contour profiles of tangential velocity $v_t$ in single and triple helical channels ($\theta =9^{\mathrm{o}}-34^{\mathrm{o}}$) for $\mathrm{Re}=1-500$, respectively. Positive and negative tangential velocity implies flow in clockwise and anticlockwise directions, respectively.

Figure 10

A line is drawn through the location of maximum tangential velocity (a–c) at any lobe of a triple helical channel and the center of the channel cross-section. Tangential velocity are plotted along this line at different Reynolds number: $\mathrm{Re}=1$, $100$, and $500$ and for different helix angle $\theta$. Symbols $\circ ,\square ,*$, and $\mathrm{\Delta }$ represent $\theta =3^{\circ }$, $9^{\circ }$, ${22}^{\circ }$, and ${34}^{\circ }$, respectively. (d) The maximum tangential velocity, $v_t{|}_{\max }$ for both triple and single helical channel are found to scale linearly with $\mathrm{Re}{\left(\kappa {\tau }^2{a}^3\right)}^{0.25}$. Symbols $•,◇,*,\mathrm{\Delta },\square$, and $\circ$ represent the data obtained for triple helical channel at $\mathrm{Re}=1,10,50,100,150$, and 500, respectively, and ♦, $▴$, and $\blacksquare$ represent that for single helical channels at $\mathrm{Re}=10$, $100$, and $150$, respectively.

Figure 11

(a) Plots of vorticity streamlines of triple helical channel with $\theta ={16}^{\circ }-34^{\mathrm{o}}$ for $\mathrm{Re}=1-500$. Number of vortices increases and the vortices shift towards the wall with increase in $\theta$ and $\mathrm{Re}$. (b) Maximum vorticity plotted against $\mathrm{Re}{\left(\kappa {\tau }^2{a}^3\right)}^{0.1}$ show that the data from different simulations fall on a single straight line. Symbols $•,◇,*,\mathrm{\Delta },\square$, and ○ represent the data obtained for triple helical channels at $\mathrm{Re}=1$, $10$, $50$, $100$, $150$, and $500$, respectively, and ♦, $▴$, and $\blacksquare$ represent that for single helical channels at $\mathrm{Re}=10$, $100$, and $150$, respectively.

Figure 12

(a, b) Pressure drop, $\mathrm{\Delta }p$ data estimated for triple helical channels of axial length: 2 mm and hydraulic diameter: $\sim 55\mu \mathrm{m}$ are plotted against torsion, $\tau$ and curvature, $\kappa$ for a range of Reynolds number, $\mathrm{Re}:1-500$. (c) The $\mathrm{\Delta }p$ data from different simulations were used for calculating the friction factor, $f$, all of which were found to scale linearly with dimensionless number: $f=17.2\mathrm{R}{\mathrm{e}}^{-1}\exp \left(\kappa /\tau \right)$. In this figure, we have plotted the $f$ calculated for a single helical channel, which also shows that $f$ scales linearly as $f=16.5\mathrm{R}{\mathrm{e}}^{-1}\exp \left(\kappa /\tau \right)$.