quantitative evaluation of solute dispersion in …15th international conference on fluid control,...

15th International Conference on Fluid Control, Measurements and Visualization 27-30 May 2019, Naples, Italy

Paper ID:146 1

Quantitative Evaluation of Solute Dispersion

in Irregular-Shaped Micro-Channels

Fumina Togi1,*, Tetsumi Kubota1, Kizuki Toyama1, Shusaku Harada1

1Faculity of Engineering, Hokkaido University, Japan *[email protected]

Abstract Mass transport in liquid-filled micro-channels is important for various engineering processes. For instance, solute dispersion in natural flow channels such as rock cracks or gaps in soils is related to resource and environmental engineering as mineral resources extraction or pollutant dispersion in underground. In general, natural flow channels have a complicated shape and the solute dispersion in these channels is affected by geometric characteristics of the channel. The purpose of this study is to examine the effect of channel geometry on the transport of soluble substances and to understand the dispersion process in irregular-shaped micro-channels. We conducted experiments using various micro-channels and reveal the effect of the irregularity of wall surface on solute transport. The experimental results using rough-walled channels with various heights and wavelengths show that the dispersion coefficient depends on the wall roughness complicatedly. From the experimental results, a comprehensive dispersion model is proposed as a function of Péclet number taking into account of the effect of channel geometry by reference to previous studies. The proposed model describes well the dispersion coefficient obtained from the experiments with various-shaped channels over the wide range of Péclet number. Keywords: solute dispersion, micro channel, wall roughness

1 Introduction

It is well-known that there are many kinds of natural micro-channels such as void networks in soils or fractures in bedrocks. In general, natural channels have irregular shapes with the roughness of the wall surface. These complicatedness of the geometric characteristics prevents from predicting the mass transport in the channels quantitatively.

Many researchers have been studied solute dispersion in liquid-filled micro-channels. The most significant dispersion process in micro-channels is known as Taylor dispersion, which is a laminar dispersion caused by a combination of velocity difference of solvent and solute molecular diffusion [1]. This kind of dispersion process has been examined using various-shaped micro-channels. For example, Doshi et al. [2] investigated the effect of sidewall on the dispersion process in open and close channels. Chatwin et al. [3] and Fukushima et al. [4] studied the influences of the cross-sectional aspect ratio of channels on the longitudinal laminar dispersion in rectangular and elliptical channels. Detwiler et al. [5] experimentally studied the relationship between Péclet number and dispersion coefficient in rough-walled open channels. However, the comprehensive models on the laminar dispersion in micro-channels involving the effect of these channel geometries have not been fully established yet.

The purpose of this study is to examine the effect of the channel geometry on the solute transport and to establish an inclusive model of the laminar dispersion in micro-channels. We conducted experiments using in various-shaped channels, e.g., the rectangular channels with flat wall surface (rectangular channel), the channels with regularly-roughed sidewall (regularly-roughed channel) and the channels with irregularly-roughed wall having various roughness (irregularly-roughed channel).

2 Experimental methodology

We performed image analysis in order to quantify the concentration field of soluble substances in micro-channels. The method of image analysis is based on absorption photometry which is a typical method for measurement of solute concentration in chemical analysis. In this study, we applied this method to quantify two-dimensional and heterogeneous solute concentration field in micro-channels. The quantification of solute concentration is based on


Paper ID:146 2

Lambert-Beer law as follows.

110

0

logI

CaI

ε

= −

(1)

where I0 and I1 is the intensity of the light before and after passing through the solution, ε is the molar absorption coefficient, C is the solute concentration and a is the thickness of the channel. In the experiment, we used copper sulfate dissolved glycerin solution and adopted near-infrared lights with about 800 nm wavelength which is absorbed by cupric ion most. Figure 1 shows the relationship between concentration of cupric ion and the light intensity passing through a micro-channel. By using this relation, we quantify the local concentration field from the brightness of each pixel in the images taken in the experiment.

Fig.1 Relation between concentration of copper sulfate solution and light intensity

3 Experimental setup

Figure 2 shows a schematic diagram of the experimental system. A micro-channel was set between CCD camera and near-infrared light source. Pure glycerin solution was filled in the micro-channel before the experiment. Then copper sulfate dissolved glycerin solution was provided into a micro-channel with constant volumetric flow rate Q by a syringe pump. We took images every 60 seconds and quantified concentration by image analysis explained above.

We prepared various-shaped micro-channels, i.e., the rectangular channels with flat walls (rectangular channel), the channels with regularly-roughed wall (regularly-roughed channel) and with irregularly-roughed wall (irregularly-roughed channel) on one side. The design of irregularly-roughed wall was done based on the following equation [6];

1

1

( ) sinx

x x

x

Nx

k kN

xkh x m n

W

π+

− +

= +

(2)

Fig.2 Schematic diagram of experimental system


Paper ID:146 3

where h is the height of the wall roughness, W is the channel length, Nx is the maximum wave number and mkx and nkx are random numbers. In order to characterize the wall roughness, we define two parameters as the amplitude σ and the wavelength T. The amplitude σ is defined as the standard deviation of the height of roughed-wall. On the other hand, T characterizes the minimum wavelength of the roughed-wall calculated from the maximum wave number as T=W/Nx. The size of each channel is W=100 mm in length (x direction), a = 0.2 mm in height (y direction) and b = 2 mm in average width (z direction). In every experiment, we controlled Péclet number Pe=v0a/D (v0 (=Q/ab) : average velocity, D : molecular diffusion coefficient) by changing the volumetric flow rate Q.

4 Results and discussion

Figure 3 shows images of quantified concentration field in a rectangular channel and a regularly-roughed channel for Péclet number Pe ~ 250. As shown in the figure, the solute concentration field in the rectangular channel shows almost parabolic profile. On the other hand, the solute concentration field in the roughed channel indicates more sharp profile. Although the average flow rate of solution in these channels are similar, solute in the roughed channel moves faster than that in the rectangular channel. Obviously it is caused by the effect of roughed wall. It is found from the concentration field in the roughed channel that dead water region exists near the roughed wall and mass transfer in the roughed channel is much enhanced compared with the smooth rectangular channel. This is because the solute migrates at the center of the channel selectively due to the existence of dead water region generated in the vicinity of the sidewall in roughed channels.

We also examined the effect of the irregularity of wall surface on solute transport. Figure 4 shows the experimental results of quantified concentration field in various channels with irregular-shaped wall for Pe ~ 250. These results clearly show the difference of solute transport between various-shaped channels. From the experimental results using roughed channels with various heights and wavelengths, it is found that the solute dispersion greatly depends on the characteristics of the wall roughness complicatedly. The solute transport is enhanced in cases of the channels having the larger height σ and the shorter wavelength T, since large dead water region is generated at the vicinity of the sidewall.

Figure 5 shows the profile of cross-sectional average concentration in x direction (streamwise direction) in the irregularly-roughed channel for Pe = 250. The plots in Fig.5 indicate the average concentration at 80, 100 and 120 minutes from the beginning of experiment. The smooth lines are fitting results by the following equation [1].

t=60min

t=80min

t=100min

(a) rectangular chanel (b) regularly-roughed channel

Fig. 3 Quantified concentration field in micro-channels for Pe~250 (b = 2mm)

t=20min

t=40min

t=60min

(a) σ = 0.125mm, T = 2mm (b) σ = 0.25mm, T = 2mm, (c) σ = 0.25mm, T = 12mm

Fig. 4 Quantified concentration fields in irregularly-roughed channels for Pe~250


Paper ID:146 4

Fig. 5 Profile of average concentration over cross-section in irregularly-roughed channel

Table 1 Dispersion coefficient obtained from fitting of experimental results

geometric condition dispersion coefficient Dx amplitude σ wavelength T

(a) 0.125mm 2mm 3.5×10-5 cm2/s (b) 0.25mm 2mm 9.7×10-4 cm2/s (c) 0.25mm 12mm 3.3×10-5 cm2/s

( )

−−=

−−2

1

2

1

0

0 2

1erf

2

1

2

1tDtvx

C

Cx

m (3)

By means of the fitting results, we can calculate dispersion coefficient Dx in respective channels. Table 1 shows the dispersion coefficient in irregularly-roughed channels. From Table 1, the dispersion coefficient is larger in case of the channels with the larger amplitude σ and the shorter wavelength T of the wall roughness. These results correspond to the observation of quantified concentration images shown in Fig.4.

Figure 6 shows the relationship between the dispersion coefficient and Péclet number Pe. The plots are obtained from the experiments. Each line shows the model results calculated from the following equation [5].

2Taylormacro1 PePe

D

Dx αα ++= (4)

−×=

Γ

119512.7Taylorα (5)

where Γ (= b/a) is the cross-sectional aspect ratio of the channel. αmacro and αTaylor are the coefficients which describe the effect of wall roughness and the effect of cross-sectional shape of the channel, respectively. Eq.(5) indicates the dispersion model for a rectangular channel obtained from previous study [3].

At lower Péclet number, the solute dispersion is dominated by molecular diffusion and Dx/D approaches to unity. On the other hand, at higher Péclet number, the advection effect is significant and the dispersion coefficient is determined only by the cross-sectional shape of the channel.

● : Experiment ― : Theory ●：80min ●：100min ●：120min


Paper ID:146 5

Fig. 6 Relationship between Péclet number Pe and non-dimensional dispersion coefficient Dx/D

The wall roughness affects the solute dispersion only at moderate Péclet number. If we can relate the coefficient αmacro to the characteristics of wall roughness (e.g. amplitude σ and wavelength T), Eq.(5) would be a comprehensive dispersion model as a function of Péclet number involving the effect of channel geometry. Solid lines in Fig.6 are the calculation results of Eq.(5) with various αmacro. The calculation results describe well the experimentally-obtained dispersion coefficient in various-shaped channel over the wide range of Péclet number. In case of smooth rectangular channels, the experimental results are close to the model results with αmacro = 0, especially under high Pe conditions (Pe ~ 1000). On the other hand, the dispersion coefficients in small rough-walled channels (σ = 0.125mm) are close to the model with αmacro = 1. Furthermore, in case of large rough-walled channels (σ = 0.25mm), the dispersion coefficients are close to the model results with αmacro = 16. From these results, one of the possible formulation of αmacro can be given as follows.

4

macro 16

=

b

σα (6)

The physical meaning of the above model is still not clear and it should be more complicated functions of the characteristics of the wall roughness. In this study, we partly evaluated the effect of wall roughness on solute dispersion under limited conditions.

5 Conclusion

We examined the effect of wall roughness on solute dispersion by experiments using various-shaped micro-channels. From the quantification of solute concentration fields, it is found that mass transport complicatedly depends on the characteristics of the wall roughness. The dispersion coefficient in micro-channel is determined not only by Péclet number but also the cross-sectional shape and wall roughness of the channel.

References

[1] Taylor, G., Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube, Proc. R. Soc. Lon. Ser. A, Math. Phys. Eng. Sci., 219-1137 (1953), 186-203.

[2] Doshi, M. R., Pankaj, M. D., William, N. G., Three Dimensional Laminar Dispersion in Open and Closed Rectangular Conduits, Chem. Eng. Sci., 33-7 (1978), 795-804.

[3] Chatwin, P. C., Paul, J. S., The Effect of Aspect Ratio on Longitudinal Diffusivity in Rectangular Channels, J. Fluid Mech., 120 (1982), 347-358.

[4] Fukushima, K., Hayakawa, N., Laminar Dispersion in an Elliptical Pipe and in a Rectangular Pipe, J. Japan Soc. Fluid Mech., 2 (1983), 34-42.

● : σ = 0.25mm● : σ = 0.125mm● : σ = 0mm

αmacro= 16

(σ = 0.25mm)

αmacro= 1

(σ = 0.125mm)

αmacro= 0

(rectangular channel)


Paper ID:146 6

[5] Detwiler, R. L., Rajaram, H., Glass, R. J., Solute Transport in Variable-Aperture Fractures: An Investigation of the Relative Importance of Taylor Dispersion and Macrodispersion, Water Resour. Res., 36-7 (2000), 1611-1625.

[6] Yamamoto, Y., Hisataka, F., Harada, S., Numerical Simulation of Concentration Interface in Stratified Suspension: Continuum-Particle Transition, Int. J. Multiphase Flow, 73 (2015), 71-79.

quantitative evaluation of solute dispersion in …15th international conference on fluid control,...

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