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Role of diffusion on molecular tagging velocimetry technique for rarefied gas flow analysis

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Abstract

The molecular tagging velocimetry (MTV) is a well-suited technique for velocity field measurement in gas flows. Typically, a line is tagged by a laser beam within the gas flow seeded with light emitting acetone molecules. Positions of the luminescent molecules are then observed at successive times and the velocity field is deduced from the analysis of the tagged line displacement and deformation. However, the displacement evolution is expected to be affected by molecular diffusion, when the gas is rarefied. Therefore, there is no direct and simple relationship between the velocity field and the measured displacement of the initial tagged line. This paper addresses the study of tracer molecules diffusion through a background gas flowing in a channel delimited by planar walls. Tracer and background species are supposed to be governed by a system of coupled Boltzmann equations, numerically solved by the direct simulation Monte Carlo (DSMC) method. Simulations confirm that the diffusion of tracer species becomes significant as the degree of rarefaction of the gas flow increases. It is shown that a simple advection–diffusion equation provides an accurate description of tracer molecules behavior, in spite of the non-equilibrium state of the background gas. A simple reconstruction algorithm based on the advection–diffusion equation has been developed to obtain the velocity profile from the displacement field. This reconstruction algorithm has been numerically tested on DSMC generated data. Results help estimating an upper bound on the flow rarefaction degree, above which MTV measurements might become problematic.

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Notes

  1. The standard deviation of the yellow dots displacements with respect to the fitting line amounts to 19 μm, or 3 % of the maximum observed displacement along the channel centerline.

References

  • Arkilic EB, Breuer KS, Schmidt MA (2001) Mass flow and tangential momentum accommodation in silicon micromachined channels. J Fluid Mech 437:29–43

    Article  MATH  Google Scholar 

  • Bird GA (1994) Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press, Oxford

    Google Scholar 

  • Bruno D, Catalfamo C, Laricchiuta A, Giordano D, Capitelli M (2006) Convergence of Chapman–Enskog calculation of transport coefficients of magnetized Argon plasma. Phys Plasmas 13(7):072307

    Article  Google Scholar 

  • Cattafesta LN, Sheplak M (2011) Actuators for active flow control. Annu Rev Fluid Mech 43:247–272

    Article  Google Scholar 

  • Cercignani C (1988) The Boltzmann equation and its applications. Springer, Berlin

    Book  MATH  Google Scholar 

  • Colin S (2005) Rarefaction and compressibility effects on steady and transient gas flows in microchannels. Microfluid Nanofluid 1:268–279

    Article  Google Scholar 

  • Colin S (2012) Gas microflows in the slip flow regime: a critical review on convective heat transfer. J Heat Transf Trans ASME 134:020908

    Article  Google Scholar 

  • Colin S, Lalonde P, Caen R (2004) Validation of a second-order slip flow model in rectangular microchannels. Heat Transf Eng 25:23–30

    Article  Google Scholar 

  • Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford

    Google Scholar 

  • Dongari N, Sharma A, Durst F (2009) Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes. Microfluid Nanofluid 6(5):679–692

    Article  Google Scholar 

  • ElBaz A, Pitz R (2012) N\(_2\)O molecular tagging velocimetry. Appl Phys B Lasers Opt 106(4):961–969. doi:10.1007/s00340-012-4872-5

    Article  Google Scholar 

  • Elsnab JR, Maynes D, Klewicki JC, Ameel TA (2010) Mean flow structure in high aspect ratio microchannel flows. Exp Therm Fluid Sci 34:1077–1088

    Article  Google Scholar 

  • Ewart T, Perrier P, Graur I, Meolans JG (2006) Mass flow rate measurements in gas micro flows. Exp Fluids 41:487–498

    Article  Google Scholar 

  • Ferziger JH, Kaper HG (1972) Mathematical theory of transport processes in gases. North-Holland, Amsterdam

    Google Scholar 

  • Gendrich CP, Koochesfahani MM, Nocera DG (1997) Molecular tagging velocimetry and other novel applications of a new phosphorescent supramolecule. Exp Fluids 23:361–372

    Article  Google Scholar 

  • Hammer P, Pouya S, Naguib A, Koochesfahani M (2013) A multi-time-delay approach for correction of the inherent error in single-component molecular tagging velocimetry. Meas Sci Technol 24:105302

    Article  Google Scholar 

  • Hu H, Koochesfahani MM (2006) Molecular tagging techniques for micro-flow and micro-scale heat transfer studies. In: Proceedings of FEDSM09. ASME, FEDSM2009-78059

  • Ismailov M, Schock H, Fedewa A (2006) Gaseous flow measurements in an internal combustion engine assembly using molecular tagging velocimetry. Exp Fluids 41:57–65

    Article  Google Scholar 

  • Kandlikar SG, Colin S, Peles Y, Garimella S, Pease RF et al (2013) Heat transfer in microchannels—2012 status and research needs. J Heat Transf Trans ASME 135(9):091001. doi:10.1115/1.4024354

    Article  Google Scholar 

  • Kaskan WE, Duncan ABF (1950) Mean lifetime of the fluorescence of acetone and biacetyl vapors. J Chem Phys 18(4):427–431

    Article  Google Scholar 

  • Koochesfahani MM (1999) Molecular tagging velocimetry (MTV): progress and applications. In: 30th AIAA fluid dynamics conference, Norfolk, VA, AIAA99-3786

  • Koochesfahani MM, Nocera DG (2007) Molecular tagging velocimetry. In: Tropea C, Yarin AL, Foss JF (eds) Handbook of experimental fluid dynamics, chap. 5.4. Springer, Berlin, pp 362–382

    Google Scholar 

  • Kovach KM, LaBarbera MA, Moyer MC, Cmolik BL, van Lunteren E et al (2015) In vitro evaluation and in vivo demonstration of a biomimetic, hemocompatible, microfluidic artificial lung. Lab Chip 15:1366–1375. doi:10.1039/C4LC01284D

    Article  Google Scholar 

  • Lempert WR, Boehm M, Jiang N, Gimelshein S, Levin D (2003) Comparison of molecular tagging velocimetry data and direct simulation Monte Carlo simulations in supersonic micro jet flows. Exp Fluids 34:403–411

    Article  Google Scholar 

  • Lempert WR, Ronney P, Magee K, Gee KR, Haugland RP (1995) Flow tagging velocimetry in incompressible flow using photo-activated nonintrusive tracking of molecular motion (PHANTOMM). Exp Fluids 18:249–257

    Article  Google Scholar 

  • Louisos W, Hitt DL (2005) Influence of wall heat transfer on supersonic MicroNozzle performance. J Spacecr Rockets 49:1123–1131

    Google Scholar 

  • Lu CJ, Steinecker WH, Tian WC, Oborny MC, Nichols JM et al (2005) First-generation hybrid MEMS gas chromatograph. Lab Chip 5:1123–1131

    Article  Google Scholar 

  • Matsuda Y, Misaki R, Yamaguchi H, Niimi T (2011a) Pressure-sensitive channel chip for visualization measurement of micro gas flows. Microfluid Nanofluid 11:507–510

    Article  Google Scholar 

  • Matsuda Y, Uchida T, Suzuki S, Misaki R, Yamaguchi H et al (2011b) Pressure-sensitive molecular film for investigation of micro gas flows. Microfluid Nanofluid 10:165–171

    Article  Google Scholar 

  • Maurer J, Tabeling P, Joseph P, Willaime H (2003) Second-order slip laws in microchannels for helium and nitrogen. Phys Fluids 15:2613–2621

    Article  Google Scholar 

  • Morini GL, Yang Y, Chalabi H, Lorenzini M (2011) A critical review of the measurement techniques for the analysis of gas microflows through microchannels. Exp Therm Fluid Sci 35:849–865

    Article  Google Scholar 

  • Niu C, Hao Y z, Li D, Lu D (2014) Second-order gas-permeability correlation of shale during slip flow. SPE J 19:786–792

    Article  Google Scholar 

  • Perrier P, Graur IA, Ewart T, Meolans JG (2011) Mass flow rate measurements in microtubes: from hydrodynamic to near free molecular regime. Phys Fluids 23:042004

    Article  Google Scholar 

  • Pitakarnnop J, Varoutis S, Valougeorgis D, Geoffroy S, Baldas L et al (2010) A novel experimental setup for gas microflows. Microfluid Nanofluid 8:57–72

    Article  Google Scholar 

  • Pitz RW, Lahr MD, Douglas ZW, Wehrmeyer JA, Hu S et al (2005) Hydroxyl tagging velocimetry in a supersonic flow over a cavity. Appl Opt 44:6692–6700

    Article  Google Scholar 

  • Resibois P, de Leener M (1977) Classical kinetic theory of fluids. Wiley, New York

    Google Scholar 

  • Samouda F, Barrot C, Colin S, Baldas L, Laurien N (2012a) Analysis of gaseous flows in microchannels by molecular tagging velocimetry. In: Proceedings of the ASME 2012 10th international conference on nanochannels, microchannels and minichannels (ICNMM2012). ASME, pp 221–228. ISBN 978-0-7918-4479-3

  • Samouda F, Brandner JJ, Barrot C, Colin S (2012b) Velocity field measurements in gas phase internal flows by molecular tagging velocimetry. J Phys. In: Conference series—proceedings of 1st European conference on gas MicroFlows (GASMEMS2012), vol 362, p 012026

  • Samouda F, Colin S, Barrot C, Baldas L, Brandner JJ (2015) Micro molecular tagging velocimetry for analysis of gas flows in mini and micro systems. Microsyst Technol 21:527–537

    Article  Google Scholar 

  • Schembri F, Bodiguel H, Colin A (2015) Velocimetry in microchannels using photobleached molecular tracers: a tool to discriminate solvent velocity in flows of suspensions. Soft Matter 11:169–178

    Article  Google Scholar 

  • Seungdo A, Gupta NK, Gianchandani YB (2014) A Si-micromachined 162-stage two-part Knudsen pump for on-chip vacuum. J Microelectromech Syst 23:406–416

    Article  Google Scholar 

  • Sharipov F (2011) Data on the velocity slip and temperature jump on a gas–solid interface. J Phys Chem Ref Data 40:023101

    Article  Google Scholar 

  • Stier B, Koochesfahani MM (1999) Molecular tagging velocimetry (MTV) measurements in gas phase flows. Exp Fluids 26:297–304

    Article  Google Scholar 

  • Sugii Y, Okamoto K (2006) Velocity measurement of gas flow using micro PIV technique in polymer electrolyte fuel cell. In: Proceedings of 4th international conference on nanochannels, microchannels and minichannels. ASME, pp 533–538

  • Thompson BR, Maynes D, Webb BW (2005) Characterization of the hydrodynamically developing flow in a microtube using MTV. J Fluids Eng 127:1003–1012

    Article  Google Scholar 

  • Yang Y, Gerken I, Brandner JJ, Morini GL (2014) Design and experimental investigation of a gas-to-gas counter flow micro heat exchanger. Exp Heat Transf 27:340–359

    Article  Google Scholar 

  • Yoon SY, Ross JW, Mench MM, Sharp KV (2006) Gas-phase particle image velocimetry (PIV) for application to the design of fuel cell reactant flow channels. J Power Sour 160:1017–1025

    Article  Google Scholar 

  • Zhang WM, Meng G, Wei X (2012) A review on slip models for gas microflows. Microfluid Nanofluid 13:845–882

    Article  Google Scholar 

Download references

Acknowledgments

This research obtained financial support from the European Community Seventh Framework Program (FP7/2007-2013) under Grant Agreement No. 215504, from the Fédération de Recherche Fermat, FR 3089, and from the Project 30176ZE of the PHC GALILEE 2014 Program. The latter is supported by the Ministère des Affaires Etrangères et du Développement International (MAEDI) and the Ministère de l’Enseignement Supérieur et de la Recherche (MENESR).

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Correspondence to Aldo Frezzotti.

Appendix: Advection–diffusion approximation

Appendix: Advection–diffusion approximation

Assuming that the tracer species is instantaneously added in negligible amount to the undisturbed steady flow of the background gas allows rewriting Eq. (1) as follows:

$${\varvec{v}}\circ \frac{\partial f_1}{\partial {\varvec{r}}}=Q_{11}(f_1,f_1)$$
(34)
$$\frac{\partial f_2}{\partial t}+{\varvec{v}}\circ \frac{\partial f_2}{\partial {\varvec{r}}}=Q_{21}(f_2,f_1).$$
(35)

The linear Eq. (35) describes the unsteady diffusion of tracer molecules through a non-equilibrium background by neglecting tracer molecules interaction (\(Q_{22}(f_2,f_2)=0\)) and their effects on the background gas flow described by the non-linear Boltzmann equation (34).

As shown below, the kinetic equations, either in the general form expressed by Eq. (1) or in the limit case described by Eqs. (34, 35), can be effectively used to simulate tagged molecules diffusion. However, their mathematical form is too complicated to formulate a method for the reconstruction of the velocity field in the background gas from the observed displacement of tagged molecules.

A simpler description of tagged molecules transport is provided by the diffusion equation (Ferziger and Kaper 1972; Crank 1975):

$$\frac{\partial \rho _2}{\partial t} + \frac{\partial }{\partial {\varvec{r}}}\circ \left( \rho _2{\varvec{u}}^\mathrm{hyd}\right) =\frac{\partial }{\partial {\varvec{r}}}\circ \left[ \rho \frac{m_1m_2}{(\rho /n)^2}{\mathscr {D}}_{12}\frac{\partial }{\partial {\varvec{r}}}\left( \frac{\rho _2}{\rho }\right) \right]$$
(36)

where the binary diffusion coefficient \({\mathscr {D}}_{12}\) takes the following form:

$$\begin{aligned} {\mathscr {D}}_{12}&=\frac{3}{16nm_{12}}\frac{\sqrt{2\pi m_{12} k_B T}}{\pi \sigma _{12}^2},\\ m_{12}&=\frac{m_1m_2}{m_1+m_2} \end{aligned}$$
(37)

in the first approximation of the diffusion coefficient of a binary mixture of hard sphere molecules. Equation (36) can be derived from Eq. (1) assuming that the scale of spatial gradients is much larger than the mean free path (Ferziger and Kaper 1972) and the contributions of pressure and temperature gradients to the diffusion driving force can been neglected. The contribution of the external force field \({\varvec{F}}\) to the diffusion driving force is automatically cancelled, since both species are subject to the same force (Ferziger and Kaper 1972). In the considered geometry, u hyd reduces to its axial component \(u_x\) which can be assumed to depend only on y and z in the region accessible to tagged molecules during their luminescence life time. Hence, the velocity field divergence can be neglected and, for small χ 2 values Eq. (36) takes the form:

$$\frac{\partial n_2}{\partial t} + u_x(y,z)\frac{\partial n_2}{\partial x}={\mathscr {D}}_{12}\nabla ^2 n_2 - \frac{1}{\tau _p} n_2 .$$
(38)

The diffusion coefficient \({\mathscr {D}}_{12}\) is assumed to be constant because of the small variation of temperature and density in the flowfield. The additional source term \(-\frac{1}{\tau _p} n_2\) has been added at r. h. s. of the above equation to take into account the decay of tagged molecules number as a result of phosphorescence intensity decrease, being \(\tau _p\) the phosphorescence lifetime of acetone molecules (Kaskan and Duncan 1950).

However, the source term can be eliminated by the following rescaling:

$$n_2(x,y,z,t)=N_0\exp {\left( -\frac{t}{\tau _p}\right) }p(x,y,z,t)$$
(39)

where \(N_0\) is the total number of tagged molecules initially created and the new unknown probability density p(xyzt) obeys the equation:

$$\frac{\partial p}{\partial t}+u_x(y,z)\frac{\partial p}{\partial x}={\mathscr {D}}_{12}\nabla ^2 p .$$
(40)

Since the tagged molecules displacements are small when compared with the channel length \(L_x\), the x-coordinate domain is considered unbounded. Accordingly, p is defined in the domain \(\varOmega =\{(x,y,z)\in {\mathscr {R}}_3: -\infty <x<+\infty , -L_y/2<y<L_y/2, -L_z/2<z<L_z/2\}\). Under the assumption that collisions with channel walls do not cause tagged molecules absorption by promoting a non-radiative deexcitation, the following boundary conditions can be assigned at walls:

$$\begin{aligned} \frac{\partial p}{\partial z}&=0,\;\; z=\pm L_z/2 \\ \frac{\partial p}{\partial y}&=0,\;\; y=\pm L_y/2 . \end{aligned}$$
(41)

The initial probability distribution is assigned as \(p(x,y,z,0)=p_0(x,y,z)\). The shape of \(p_0(x,y,z)\) is related to the way the gas is illuminated by the laser beam. In the following developments, it is assumed that a thin cylindrical beam produces a y-independent initial state of the form:

$$p_0(x,z)= {\left\{ \begin{array}{ll} \frac{1}{V_0} &\quad (x,y,z)\in {\mathscr {C}}_0 \\ 0&\quad (x,y,z)\notin {\mathscr {C}}_0 \end{array}\right. }$$
(42)

where \({\mathscr {C}}_0\) is the set \({\mathscr {C}}_0=\{(x,y,z)\in {\mathscr {R}}_3: x^2+z^2<r_0^{2},-L_y/2<y<L_y/2\}\) and \(V_0=\pi r_0^{2}L_y\) its volume.

The relationship between the velocity field \(u_x\) and the displacement \(s_x(y,t)\) can be easily obtained from Eq. (40). Since the displacement is obtained as a function of y, it is useful to introduce the probability \(P_{xz}(x,y,z,t)\) that a molecule has a position (xz) on a plane at fixed y:

$$\begin{aligned} P_{xz}(x,y,z,t)&=\frac{p(x,y,z,t)}{P_y(y,t)}, \\ P_y(y,t)&=\int p(x,y,z,t)\,{\hbox {d}}x\,{\hbox {d}}z . \end{aligned}$$
(43)

The displacement is now obtained as:

$$\begin{aligned} s_x(y,t)&=\int x P_{xz}(x,y,z,t)\,{\hbox {d}}x \,{\hbox {d}}z \\& = \frac{1}{P_y(y,t)} \int x p(x,y,z,t)\,{\hbox {d}}x \,{\hbox {d}}z . \end{aligned}$$
(44)

It is shown that \(P_y(y,t)\) is a constant because of the symmetry of the initial state. Integrating Eq. (40) over x and z while taking into account the boundary conditions (41) leads to the following for \(P_y(y,t)\)

$$\frac{\partial P_y}{\partial t}={\mathscr {D}}_{12}\frac{\partial ^2 P_y}{\partial y^2}$$
(45)

where \(-\frac{L_y}{2}<y<\frac{L_y}{2}\) and \(\frac{\partial P_y}{\partial y}=0\) at \(y=\pm \frac{L_y}{2}\). Since the illumination can be assumed to be uniform along the beam, as confirmed by experiments, the initial state does not depend on y. Hence \(P_y(y,0)=P_0\), being \(P_0\) a constant. As a matter of fact, it can be immediately seen that the function \(P_y(y,t)=P_0\) is a solution of Eq. (45) and, because of the uniqueness theorem, it is the only solution with the prescribed initial state.

The evolution equation for \(s_x(y,t)\) is readily obtained by multiplying Eq. (40) by x and integrating over x and z:

$$\frac{\partial s_x}{\partial t}=\overline{u}_x(y,t)+{\mathscr {D}}_{12}\frac{\partial ^2 s_x}{\partial y^2},$$
(46)

with initial state \(s_x(y,0)=0\) and boundary conditions \(\frac{\partial s_x}{\partial y}=0\) at \(y=\pm \frac{L_y}{2}\). The velocity \(\overline{u}_x(y,t)\) is defined as:

$$\overline{u}_x(y,t)=\frac{\int u_x(y,z)p(x,y,z,t)\,{\hbox {d}}x\,{\hbox {d}}z}{P_0}.$$
(47)

As is clear, tagged molecules displacement evolve under the action of the average velocity \(\overline{u}_x(y,t)\), which is determined by the gas velocity field \(u_x(y,z)\) and the spatial molecules distribution described by p(xyzt). If \(u_x(y,z)\approx u_x(y)\), as it happens in the central part of the channel when \(L_z\gg L_y\), then \(\overline{u}_x(y,t)\approx u_x(y)\) as long as tagged particles positions remain confined in the region where \(\partial u_x/\partial z\approx 0\). The above considerations impose a limit to the measurement duration which should not exceed a limit time t l , after which the tagged molecules z coordinates variance would be larger than channel width. A rough estimation of t l is given by the following expression:

$$t_l=\frac{L_z^{2}}{2{\mathscr {D}}_{12}}.$$
(48)

For \(t<t_l\), the approximation \(\overline{u}_x(y,t)\approx u_x(y)\) holds and Eq. (46) takes the form:

$$\frac{\partial s_x}{\partial t}=u_x(y)+{\mathscr {D}}_{12}\frac{\partial ^2 s_x}{\partial y^2}.$$
(49)

The solution of Eq. (49), with initial state and boundary conditions stated above, can be given in closed form as (Crank 1975):

$$\begin{aligned} s_x(y,t)&=(W_0,u_x)t \,W_0+\sum _{k=1}^{\infty }\tau _{k}(W_{k},u_x)\\&\quad (1-e^{-t/\tau _{k}})W_{k}(y). \end{aligned}$$
(50)

In Eq. (50), the symbol \((\,,\,)\) denotes the scalar product of any two functions in \((-\frac{L_y}{2},+\frac{L_y}{2})\), defined as

$$(f,g)=\int _{-\frac{L_y}{2}}^{+\frac{L_y}{2}} f(y)g(y)\,{\hbox {d}}y.$$
(51)

The functions

$$W_{k}(y)= {\left\{ \begin{array}{ll} \frac{1}{\sqrt{L_y}}, & \quad k=0\\ \sqrt{\frac{2}{L_y}}\cos \left( \frac{2\pi k}{L_y}y\right) , & \quad k=1,\dots ,\infty \end{array}\right. }$$
(52)

obey the conditions \((W_{k},W_{l})=\delta _{kl}\), being \(\delta _{kl}\) the Kronecker’s delta. The time constants \(\tau _{k}=\frac{1}{{\mathscr {D}}_{12}}\left( \frac{L_y}{2\pi k}\right) ^2\) characterize the exponential time evolution of the amplitudes associated to the spatial modes W k .

The direct linear relationship between the velocity field u x and the average displacement s x can be seen more clearly by recasting Eq. (50) in the form:

$$s_x(y,t)=\int _{-\frac{L_y}{2}}^{+\frac{L_y}{2}} G(y,y^{\prime }|t)u_x(y^{\prime })\,{\hbox {d}}y^{\prime },$$
(53)

where the Green function \(G(y,y^{\prime }|t)\) takes the form:

$$\begin{aligned} G(y,y^{\prime }|t)=W_{0}(y)W_{0}(y^{\prime })t+\sum _{k=1}^{\infty } \tau _{k}(1-e^{-t/\tau _{k}})W_{k}(y)W_{k}(y^{\prime }) \end{aligned}$$
(54)

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Frezzotti, A., Si Hadj Mohand, H., Barrot, C. et al. Role of diffusion on molecular tagging velocimetry technique for rarefied gas flow analysis. Microfluid Nanofluid 19, 1335–1348 (2015). https://doi.org/10.1007/s10404-015-1649-2

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