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Computational investigation and parametrization of the pumping effect in temperature-driven flows through long tapered channels

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Abstract

The temperature-driven rarefied gas flow and the associated pumping effects through long channels with linearly diverging or converging cross sections are computationally investigated. The implemented kinetic modeling is well known and relies on the infinite capillary methodology coupled with the mass conservation principle along the channel. The net mass flow rate and the induced pressure difference between the channel inlet and outlet are parametrized in terms of the geometrical and operational data including the channel inclination and the inlet pressure. Specific attention is given to the diode effect. The investigated flow setups include (a) the maximum pressure difference scenario with zero net mass flow rate (maximum pumping effect), (b) the maximum net mass flow rate scenario with equal inlet and outlet pressures and (c) all intermediate flow cases where both the net mass flow rate and the pressure difference are different than zero. In the first limit case, the pressure difference is always increased with the channel inclination and, depending on the inlet pressure, it may be larger for either the diverging or converging channel. In the second limit case, the mass flow rate is always decreased when the channel inclination is increased and it is always higher for the diverging channel. In both limit cases, optimum operation scenarios, in terms of the diode effect and the overall performance, are extracted. For intermediate cases, the characteristic curves of the net mass flow rate versus the pressure difference have been developed, indicating that the mass flow rate is inversely proportional to the pressure difference. The results strongly depend on the channel inclination. The present work may support decision making on the suitability of tapered channel flow to meet certain pumping specifications and the design of cascade-type thermally driven micropumps.

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Acknowledgements

This project has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie Sklodowska-Curie Grant Agreement No. 643095.

Author information

Correspondence to D. Valougeorgis.

Appendix: Computation of reduced flow rates in pressure- and temperature-driven flow between parallel plates

Appendix: Computation of reduced flow rates in pressure- and temperature-driven flow between parallel plates

The linearized Shakhov model equation for the pressure- and temperature-driven flow between two plates is given in dimensionless form by (Sharipov 2002):

$$c_{y} \frac{{\partial \varphi_{j} }}{\partial y} + \delta \varphi_{j} = \delta \left[ {u_{j} + \frac{2}{15}q_{j} \left( {c_{y}^{2} - \frac{1}{2}} \right)} \right] - S_{j}^{\varphi }$$
(11)
$$c_{y} \frac{{\partial \psi_{j} }}{\partial y} + \delta \psi_{j} = \delta \frac{4}{15}q_{j} - S_{j}^{\psi }$$
(12)

Here, the subscripts j = PT denote the pressure- and temperature-driven cases, respectively, y is the space variable normal to the two plates, c y is the molecular velocity in the y-direction, φ j (yc y ) and ψ j (yc y ) are the reduced linearized distribution functions, δ is the gas rarefaction parameter, and u j and q j are the axial components of the bulk velocity and heat flux given by

$$u_{j} = \frac{1}{\sqrt \pi }\int\limits_{ - \infty }^{\infty } {\varphi_{j} e^{{ - c_{y}^{2} }} dc_{y} } \quad {\text{and}}\quad q_{j} = \frac{1}{\sqrt \pi }\int\limits_{ - \infty }^{\infty } {\left[ {\psi_{j} + \left( {c_{y}^{2} - \frac{1}{2}} \right)\varphi_{j} } \right]e^{{ - c_{y}^{2} }} dc_{y} } ,$$
(13)

while the source terms are

$$S_{P}^{\varphi } = \frac{1}{2},S_{T}^{\varphi } = \frac{1}{2}\left( {c_{y}^{2} - \frac{1}{2}} \right),S_{P}^{\psi } = 0 \quad {\text{and}} \quad S_{T}^{\psi } = 1.$$
(14)

The associated boundary conditions can be written as

$$\varphi_{j} \left( {1/2,c_{y} } \right) = 0,c_{y} < 0,\varphi_{j} \left( { - 1/2,c_{y} } \right) = 0,c_{y} > 0$$
(15)
$$\psi_{j} \left( {1/2,c_{y} } \right) = 0,c_{y} < 0,\psi_{j} \left( { - 1/2,c_{y} } \right) = 0,c_{y} > 0$$
(16)

The system of Eqs. (11) and (12) coupled with the expressions (13) and (14) subject to the boundary conditions (15)–(16) is solved numerically to yield the dimensionless flow rates, also known as kinetic coefficients,

$$G_{P} = - 2\int\limits_{ - 1/2}^{1/2} {u_{p} dy} \quad {\text{and}}\quad G_{T} = 2\int\limits_{ - 1/2}^{1/2} {u_{T} dy} .$$
(17)

Tabulated values of the flow rates G P and G T are provided in Table 1 for δ ∊ [10−2, 50], and they are used in the solution of Eq. (9). For values of δ > 50 the analytical slip expressions \(G_{P}^{slip} = \frac{\delta }{6} + \sigma_{P}\) and \(G_{T}^{slip} = \frac{{\sigma_{T} }}{\delta }\) where σ P  = 1.018 and σ T  = 1.175 are applied (Sharipov and Seleznev 1998).

Table 1 Kinetic coefficients G P and G T in terms of the gas rarefaction parameter δ

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Tatsios, G., Lopez Quesada, G., Rojas-Cardenas, M. et al. Computational investigation and parametrization of the pumping effect in temperature-driven flows through long tapered channels. Microfluid Nanofluid 21, 99 (2017). https://doi.org/10.1007/s10404-017-1932-5

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Keywords

  • Diverging and converging channels
  • Diodicity
  • Rarefied gas dynamics
  • Thermal transpiration
  • Linear kinetic modeling
  • Knudsen pump