# Design Guidelines for Thermally Driven Micropumps of Different Architectures Based on Target Applications via Kinetic Modeling and Simulations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Pump Designs, Manufacturing Materials, and Fabrication Process

- Pump A consists of an array of multiple parallel narrow microchannels in one single pumping stage to achieve high mass flow rate ($\dot{m}$) performance. The layout area is $a\times a$ with $n$ parallel narrow microchannels of diameter $d$ and length $L$.
- Pump B consists of a multistage system where each stage is formed by one single narrow pumping microchannel followed by one wide channel (where the reduced counter thermal transpiration flow will appear) to achieve high pressure difference ($\Delta P$) performance. The layout area is $\left(W\times W\right)+\left(b\times b\right)$. The diameter of the narrow and wide channels are $d$ and $D$, respectively, and the length of both channels is $L$.
- Pump C combines the two previous designs. More specifically, it consists of a multistage system and each stage is formed by an array of $n$ parallel narrow pumping microchannels, followed by one wide channel where the reduced counter thermal transpiration flow will appear. This design provides high $\Delta P$ and $\dot{m}$ performances, due to the multi-stage cascade system and to the multiple narrow microchannels per stage, respectively. The layout area is $\left(W\times W\right)+\left(c\times c\right)$. The diameter of the narrow and wide channels are $d$ and $D$, respectively, while the length of all channels is $L$.

## 3. Kinetic Modeling

## 4. Results and Discussion

^{2}. A schematic view of the corresponding layouts to be examined is shown in Figure 3. As the microchannel diameter is reduced, the number n of parallel microchannels in the layout is increased, keeping the same area ratio between the flow and the layout cross section areas. In this way, the comparison of the performance characteristics of the different layouts always involves the same cross section flow area. Additional details of the layout geometry are provided in Table 1.

#### 4.1. Pump A: One Pumping Stage with Multiple Parallel Microchannels

#### 4.2. Pump B: Multistage Pumping with One Narrow and One Large Channel Per Stage

#### 4.3. Pump C: Multistage Pumping with Multiple Parallel Microchannels Per Stage

## 5. Concluding Remarks

^{2}, are presently under construction. Since the theoretical pressure differences cover several orders of magnitude, the device could decrease the pressure of a system from atmospheric pressure ${P}_{in}=100$ kPa down to 1 or 2 Pa, with associated mass flow rates higher than ${10}^{-10}$ kg/s. These theoretical performances shall be corrected to deal with specific operational constraints such as leakages, issues in thermal management of the device, etc.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Tabulated Results of the Kinetic Coefficients

**Table A1.**Kinetic coefficients ${G}_{P}$ and ${G}_{T}$ for the pressure- and temperature-driven flows, respectively, in terms of the gas rarefaction parameter $\delta $.

δ | 0.0005 | 0.001 | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |

G_{P} | 1.5023 | 1.5008 | 1.4904 | 1.4800 | 1.4636 | 1.4514 | 1.4418 | 1.4339 |

G_{T} | 0.7502 | 0.7486 | 0.7366 | 0.7243 | 0.7042 | 0.6884 | 0.6752 | 0.6637 |

δ | 0.06 | 0.07 | 0.08 | 0.09 | 0.1 | 0.2 | 0.3 | 0.4 |

G_{P} | 1.4273 | 1.4217 | 1.4168 | 1.4127 | 1.4101 | 1.3911 | 1.3876 | 1.3920 |

G_{T} | 0.6536 | 0.6444 | 0.6359 | 0.6281 | 0.6210 | 0.5675 | 0.5303 | 0.5015 |

δ | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.2 | 1.4 |

G_{P} | 1.4011 | 1.4130 | 1.4270 | 1.4425 | 1.4592 | 1.4758 | 1.5158 | 1.5550 |

G_{T} | 0.4779 | 0.4576 | 0.4367 | 0.4237 | 0.4092 | 0.3959 | 0.3721 | 0.3514 |

δ | 1.6 | 1.8 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 |

G_{P} | 1.5956 | 1.6373 | 1.6799 | 1.9014 | 2.1315 | 2.3666 | 2.6049 | 2.8455 |

G_{T} | 0.3330 | 0.3165 | 0.3016 | 0.2439 | 0.2042 | 0.1752 | 0.1531 | 0.1359 |

δ | 8.0 | 9.0 | 10.0 | 20.0 | 30.0 | 40.0 | 50.0 | |

G_{P} | 3.0878 | 3.3314 | 3.5749 | 6.0492 | 8.5392 | 11.0360 | 13.4950 | |

G_{T} | 0.1220 | 0.1106 | 0.1014 | 0.05426 | 0.03685 | 0.02785 | 0.02212 |

## References

- Schomburg, W.K.; Vollmer, J.; Bustgens, B.; Fahrenberg, J.; Hein, H.; Menz, W. Microfluidic components in LIGA technique. J. Micromech. Microeng.
**1994**, 4, 186–191. [Google Scholar] [CrossRef] - Laser, D.J.; Santiago, J.G. A review of micropumps. J. Micromech. Microeng.
**2004**, 14, 35–64. [Google Scholar] [CrossRef] - Maxwell, J.C. On stresses in rarefied gases arising from inequalities of temperature. Proc. R. Soc. Lond.
**1878**, 27, 304–308. [Google Scholar] - Reynolds, O. XVIII. On certain dimensional properties of matter in the gaseous state. Philos. Trans. R. Soc. Lond. Ser. A
**1879**, 170, 727–845. [Google Scholar] - Knudsen, M. Eine Revision der gleichgewichtsbedingung der gase. Thermische Molekularströmung. Ann. Phys.
**1909**, 336, 205–229. [Google Scholar] [CrossRef] - Knudsen, M. Thermischer molekulardruck der gase in Röhren. Ann. Phys.
**1910**, 338, 1435–1448. [Google Scholar] [CrossRef] - Sone, Y.; Waniguchi, Y.; Aoki, K. One-way flow of a rarefied gas induced in a channel with a periodic temperature distribution. Phys. Fluids
**1996**, 8, 2227–2235. [Google Scholar] [CrossRef] - Sone, Y. Molecular Gas Dynamics: Theory, Techniques, and Applications; Birkhäuser: Basel, Switzerland, 2007; ISBN 9780817643454. [Google Scholar]
- Aoki, K.; Mieussens, L.; Takata, S.; Degond, P.; Nishioka, M.; Takata, S. Numerical Simulation of a Knudsen Pump Using the Effect of Curvature of the Channel; Global Science Press: Hong Kong, China, 2007. [Google Scholar]
- Aoki, K.; Takata, S.; Kugimoto, K. Diffusion approximation for the knudsen compressor composed of circular tubes. AIP Conf. Proc.
**2008**, 953–958. [Google Scholar] [CrossRef] - Colin, S. Single-Phase Gas Flow in Microchannels. In Heat Transfer and Fluid Flow in Minichannels and Microchannels; Elsevier: Amsterdam, The Netherlands, 2014; pp. 11–102. [Google Scholar]
- Aoki, K.; Degond, P.; Takata, S.; Yoshida, H. Diffusion models for Knudsen compressors. Phys. Fluids
**2007**, 19, 117103. [Google Scholar] [CrossRef][Green Version] - Li, X.; Oehrlein, G.S.; Schaepkens, M.; Ellefson, R.E.; Frees, L.C. Spatially resolved mass spectrometric sampling of inductively coupled plasmas using a movable sampling orifice. J. Vac. Sci. Technol. A
**2003**, 21, 1971–1977. [Google Scholar] [CrossRef] - Hamad, F.; Khulbe, K.C.; Matsuura, T. Comparison of gas separation performance and morphology of homogeneous and composite PPO membranes. J. Membr. Sci.
**2005**, 256, 29–37. [Google Scholar] [CrossRef] - McNamara, S.; Gianchandani, Y.B. On-chip vacuum generated by a micromachined Knudsen pump. J. Microelectromech. Syst.
**2005**, 14, 741–746. [Google Scholar] [CrossRef] - Gupta, N.K.; Gianchandani, Y.B. Porous ceramics for multistage Knudsen micropumps-modeling approach and experimental evaluation. J. Micromech. Microeng.
**2011**, 21, 095029. [Google Scholar] [CrossRef] - Vargo, S.E. Initial results from the first MEMS fabricated thermal transpiration-driven vacuum pump. AIP Conf. Proc.
**2001**, 585, 502–509. [Google Scholar][Green Version] - Gupta, N.K.; An, S.; Gianchandani, Y.B. A Si-micromachined 48-stage Knudsen pump for on-chip vacuum. J. Micromech. Microeng.
**2012**, 22, 105026. [Google Scholar] [CrossRef] - An, S.; Gupta, N.K.; Gianchandani, Y.B. A Si-Micromachined 162-stage two-part knudsen pump for on-chip vacuum. J. Microelectromech. Syst.
**2014**, 23, 406–416. [Google Scholar] [CrossRef] - Qin, Y.; An, S.; Gianchandani, Y.B. Arrayed architectures for multi-stage Si-micromachined high-flow Knudsen pumps. J. Micromech. Microeng.
**2015**, 25, 115026. [Google Scholar] [CrossRef][Green Version] - Qin, Y.; Gianchandani, Y.B. A fully electronic microfabricated gas chromatograph with complementary capacitive detectors for indoor pollutants. Microsyst. Nanoeng.
**2016**, 2, 15049. [Google Scholar] [CrossRef] - Cheng, Q.; Qin, Y.; Gianchandani, Y.B. A Bidirectional Knudsen Pump with Superior Thermal Management for Micro-Gas Chromatography Applications. In Proceedings of the IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NV, USA, 22–26 January 2017; pp. 167–170. [Google Scholar]
- Courson, R.; Cargou, S.; Conédéra, V.; Fouet, M.; Blatché, M.-C.; Serpentini, C.L.; Gué, A.-M. Low-cost multilevel microchannel lab on chip: DF-1000 series dry film photoresist as a promising enabler. RSC Adv.
**2014**, 4, 54847–54853. [Google Scholar] [CrossRef][Green Version] - Sharipov, F. Rarefied gas flow through a long tube at any temperature ratio. J. Vac. Sci. Technol. A
**1996**, 14, 2627–2635. [Google Scholar] [CrossRef] - Sharipov, F.; Seleznev, V. Data on Internal Rarefied Gas Flows. J. Phys. Chem. Ref. Data
**1998**, 27, 657–706. [Google Scholar] [CrossRef] - Sharipov, F. Non-isothermal gas flow through rectangular microchannels. J. Micromech. Microeng.
**1999**, 9, 394–401. [Google Scholar] [CrossRef] - Sharipov, F.; Bertoldo, G. Rarefied gas flow through a long tube of variable radius. J. Vac. Sci. Technol. A
**2005**, 23, 531–533. [Google Scholar] [CrossRef] - Graur, I.; Sharipov, F. Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section. Microfluid. Nanofluid.
**2009**, 6, 267–275. [Google Scholar] [CrossRef] - Ritos, K.; Lihnaropoulos, Y.; Naris, S.; Valougeorgis, D. Pressure- and temperature-driven flow through triangular and trapezoidal microchannels. Heat Transf. Eng.
**2011**, 32, 1101–1107. [Google Scholar] [CrossRef] - Tatsios, G.; Lopez Quesada, G.; Rojas-Cardenas, M.; Baldas, L.; Colin, S.; Valougeorgis, D. Computational investigation and parametrization of the pumping effect in temperature-driven flows through long tapered channels. Microfluid. Nanofluid.
**2017**, 21, 99. [Google Scholar] [CrossRef] - Pantazis, S.; Valougeorgis, D.; Sharipov, F. End corrections for rarefied gas flows through circular tubes of finite length. Vacuum
**2014**, 101, 306–312. [Google Scholar] [CrossRef] - Valougeorgis, D.; Vasileiadis, N.; Titarev, V. Validity range of linear kinetic modeling in rarefied pressure driven single gas flows through circular capillaries. Eur. J. Mech. B Fluids
**2017**, 64, 2–7. [Google Scholar] [CrossRef] - Naris, S.; Valougeorgis, D.; Kalempa, D.; Sharipov, F. Flow of gaseous mixtures through rectangular microchannels driven by pressure, temperature, and concentration gradients. Phys. Fluids
**2005**, 17, 100607. [Google Scholar] [CrossRef] - Rojas Cardenas, M.; Graur, I.; Perrier, P.; Meolans, J.G. Thermal transpiration flow: A circular cross-section microtube submitted to a temperature gradient. Phys. Fluids
**2011**, 23, 031702. [Google Scholar] [CrossRef] - Rojas-Cárdenas, M.; Perrier, P.; Graur, I.; Méolans, J.G. Time-dependent experimental analysis of a thermal transpiration rarefied gas flow. Phys. Fluids
**2013**, 25, 72001. [Google Scholar] [CrossRef] - Sharipov, F. Data on the velocity slip and temperature jump on a gas-solid interface. J. Phys. Chem. Ref. Data
**2011**, 40, 023101. [Google Scholar] [CrossRef]

**Figure 1.**Representative view of single-stage pump A, and of two consecutives stages of multistage pumps B and C. Gray arrows denote the pumping flow direction, the hot and cold regions being at the top and the bottom of each stage, respectively.

**Figure 2.**Schematic of fabrication process with the superposition of dry film (DF) photoresist layers and the use of lamination (grey cylinders) and lithography techniques. Typical thicknesses of the DF layers are 5, 25, 50, and 100 µm, while the thickness of the glass wafer is 500 µm. In the cooled reservoirs the glass substrate can be replaced with a silicon substrate to improve temperature uniformity.

**Figure 3.**View of layouts with increasing the number n of microchannels and decreasing the channel diameter d, keeping the same ratio between the channels and the overall cross sections.

**Figure 4.**Ratios of total maximum mass flow rates ${\dot{m}}_{1}/{\dot{m}}_{n}$ and associated maximum pressure differences $\Delta {P}_{n}/\Delta {P}_{1}$ for various one-stage layouts (numbers of microchannels $n=4,25,100,400$ and corresponding diameters $d=50,20,10,5$ μm) compared to the reference layout ($n=1$, $d=100$ μm).

**Figure 5.**(

**a**) Pressure difference and (

**b**) mass flow rate versus inlet pressure for the one-stage pump A with $d=5,10,20$ μm (mass flow rates are given for a single channel of the pump). The rarefaction parameter is ranging from 0.03 to 130, and it increases with the inlet pressure and the diameter, providing the maximum pressure difference at $\delta =3-4$, in the transition regime.

**Figure 6.**Maximum pressure difference (corresponding to zero net mass flow rate) of pump B with single narrow and wide microchannels of diameters $d=10$ μm and $D=100$ μm, respectively, in each stage, at various inlet pressures, versus the number of stages with (

**a**) $N\le 1000$ and (

**b**) $N\le 100$. The range of the rarefaction parameter in the narrow channel is (

**a**) $\delta \approx 0.6-56$ and (

**b**) $\delta \approx 0.6-12$ for ${P}_{in}=1$ kPa, and (

**a**) $\delta \approx 60-86$ and (

**b**) $\delta \approx 60-62$ for ${P}_{in}=100$ kPa. Always, $\delta $ is increasing with the inlet pressure and the number of stages.

**Figure 7.**Maximum mass flow rate (corresponding to zero pressure difference) of pump B with single narrow and wide microchannels of diameters $d=10$ μm and $D=100$ μm, respectively, in each stage, at various inlet pressures, versus the number of stages.

**Figure 8.**Inlet pressure evolution of a system connected to pump B as a function of its number of stages, considering narrow diameter channels $d=5,10,20$ μm and a constant outlet pressure ${P}_{out}=100$ kPa (red solid symbols represent the pressure and stage number, where the maximum slope, corresponding to the peak values of Figure 5a, is observed).

**Figure 9.**Performance characteristic curves of a Knudsen pump, based on pump C, for a number of stages $N=1,5,10,20$ with inlet pressures ${P}_{in}=1,5,20,100$ kPa when narrow microchannels diameter $d=10$ μm and wide channel diameter $D=100$ μm.

**Figure 10.**Performance characteristic curves of a Knudsen pump, based on pump C, for a number of stages $N=40,100,200,500,1000$ with inlet pressures ${P}_{in}=1,5,20,100$ kPa when narrow microchannels diameter $d=10$ μm and wide channel diameter $D=100$ μm.

Total Layout Area a × a (μm × μm) | Microchannel Diameter d (μm) | Number n of Parallel Microchannels |
---|---|---|

200 × 200 | 50 | 4 |

200 × 200 | 20 | 25 |

200 × 200 | 10 | 100 |

200 × 200 | 5 | 400 |

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**MDPI and ACS Style**

López Quesada, G.; Tatsios, G.; Valougeorgis, D.; Rojas-Cárdenas, M.; Baldas, L.; Barrot, C.; Colin, S. Design Guidelines for Thermally Driven Micropumps of Different Architectures Based on Target Applications via Kinetic Modeling and Simulations. *Micromachines* **2019**, *10*, 249.
https://doi.org/10.3390/mi10040249

**AMA Style**

López Quesada G, Tatsios G, Valougeorgis D, Rojas-Cárdenas M, Baldas L, Barrot C, Colin S. Design Guidelines for Thermally Driven Micropumps of Different Architectures Based on Target Applications via Kinetic Modeling and Simulations. *Micromachines*. 2019; 10(4):249.
https://doi.org/10.3390/mi10040249

**Chicago/Turabian Style**

López Quesada, Guillermo, Giorgos Tatsios, Dimitris Valougeorgis, Marcos Rojas-Cárdenas, Lucien Baldas, Christine Barrot, and Stéphane Colin. 2019. "Design Guidelines for Thermally Driven Micropumps of Different Architectures Based on Target Applications via Kinetic Modeling and Simulations" *Micromachines* 10, no. 4: 249.
https://doi.org/10.3390/mi10040249