# Design and modelling of a Knudsen micropump for integration in a Pulsed tube Cryo-cooler

Thanasis Basdanis, Phassawat Leelaburanathanakul, Thierry Camps, Marcos

Rojas-Cardenas, Christine Barrot-Lattes, Lucien Baldas, Jürgen Brandner,
Stéphane Colin

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Thanasis Basdanis, Phassawat Leelaburanathanakul, Thierry Camps, Marcos Rojas-Cardenas, Christine Barrot-Lattes, et al.. Design and modelling of a Knudsen micropump for integration in a Pulsed tube Cryo-cooler. 9th Micro and Nano Flows Conference, University of Edinburgh, Sep 2025, Edinburgh, United Kingdom. hal-05499544

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Submitted on 8 Feb 2026

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# Design and modelling of a Knudsen micropump for integration in a Pulsed tube Cryo-cooler

Thanasis BASDANIS¹*, Phassawat LEELABURANATHANAKUL¹, Thierry CAMPS², Marcos ROJAS-CARDENAS¹, Christine BARROT¹, Lucien BALDAS¹, Jürgen J. BRANDNER³, Stéphane COLIN¹

¹Institut Clément Ader (ICA), Université de Toulouse, CNRS, INSA Toulouse, ISAE-SUPAERO, IMT Mines Albi, Toulouse, France

²CNRS, LAAS, 7 Avenue du Colonel Roche, Toulouse, F-31400, France

³Institute of Microstructure Technology (IMT), Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany

* Corresponding author: Tel.: +33(0) 767944648; Email: basdanis@insa-toulouse.fr

## Abstract

Miniaturized cryogenic cooling systems are important for scientific and industrial applications. Pulsed tube cryocoolers (PTCC) can be used for this purpose. However, the mechanical vibrations generated by conventional compressors should be further reduced. This work focuses on the design and modelling of a Knudsen pump that can be integrated into a PTCC to serve as a vibration free compressor. Knudsen pumps utilize thermal transpiration flow driven by temperature gradients along microchannels, eliminating the need for moving parts. However, for high-performance operation at atmospheric or higher pressures, submicron channels are required. Bessel beam technology has been used for the first time to create nanochannels in the microfabrication of Knudsen pumps, further expanding the limits of fabrication. Through numerical simulations of gas flow and heat transfer, a single-stage, parallel-channel design has been developed and optimized taking into account the microfabrication constraints. The resulting design consists of 300,000 nanochannels with a radius of 0.3 µm. By creating a mean temperature difference of 47 °C, a maximum mass flow of $\dot{m}_{max} = 2.2 \times 10^{-10}$ kg/s and a maximum pressure difference of $\Delta P_{max} = 1872$ Pa was achieved.

**Keywords:** Microfluidics, Knudsen pump, Heat transfer, Rarefied gas flows, Thermal transpiration

## 1. Introduction

Knudsen pumps should be designed to operate under rarefied conditions [1]. For operation at atmospheric or higher pressures, such as in pulsed tube cryocoolers (PTCC), channels of diameter 1 µm or smaller are needed, posing significant modelling and fabrication challenges. A popular modern technique for fabricating channels in Knudsen pumps is deep reactive-ion etching (DRIE) [2]. However, DRIE is not suitable for production of channels with diameters of less than few microns and high length-to-radius ratios. An alternative solution proposed in this context is the Bessel beam laser processing. The precise opening of submicron diameter channels with high writing speeds [3] makes this technology suitable for the present work.

The Knudsen pump should consist of multiple parallel channels to increase the net mass flow and multiple stages connected in series to amplify the pressure difference. Additionally, channel dimensions must be optimized based on the rarefaction conditions that enhance the performance of the Knudsen pump under the working pressures of the PTCC, while thermal management strategies should be refined to maximize temperature gradients along the channels. In the present work, a single-stage parallel-channel Knudsen pump is proposed, based on the characteristics and limitations of Bessel beam laser processing. The design is optimized thanks to numerical gas flow simulations using the kinetic theory of gases, as well as heat transfer simulations conducted in ANSYS.
# 2. Design of the Knudsen pump

## 2.1 Gas flow modelling

To optimize the radius of the channels, thermal transpiration flow through long circular channel is modelled using the Shakhov kinetic model based on the kinetic theory of gases. Rarefied conditions, under which the maximum pressure difference is obtained, are selected as optimal, i.e., the rarefaction parameter is around $δ_m = 3.5$. Thus, for operation under atmospheric conditions, a radius of $R = 0.3$ µm is proposed.

## 2.2 Channel array design

A 2×2 cm² glass plate with a thickness of 500 µm is used and the central 1×1 cm² area is designated for the fabrication of the nanochannels along the thickness of the plate. Considering constraints related to heat management and structural integrity of the glass plate, a spacing of 20 µm between the channels has been selected, resulting in 300,000 nanochannels with a radius of $R = 0.3$ µm arranged in hexagonal pattern.

## 2.3 Heat transfer modelling

Heat transfer simulations using ANSYS Mechanical are performed and presented in Fig. 1 to determine the optimal design and positioning of the heaters, heat sink, as well as to select the most suitable materials. Three metal layers of titanium, titanium-tungsten and gold are applied to heat up one side of the glass. On the opposite side, silicon pillars are designed on a silicon plate, and the silicon plate is anodically bonded to the glass plate. The silicon pillars are positioned around every channel on the bottom of the glass plate to operate as heat sinks and maximize temperature gradients along the channels.

# 3 Results

Based on the heat transfer simulations, a Joule heating of 10 W creates a mean temperature difference of 47 °C across the 300,000 nanochannels. By demonstrating that using only the mean temperature difference is sufficient to accurately predict gas flow in multiple parallel channels with varying temperature differences, the proposed single-stage Knudsen pump design achieves a maximum mass flow rate of $\dot{m}_{max} = 2.2 \times 10^{-10}$ kg/s and a maximum pressure difference of $\Delta P_{max} = 1872$ Pa.

Although current technological limitations make it challenging to meet the requirements of a PTCC, ongoing efforts in this direction may further facilitate this integration. However, the present work is an important step towards the development of Knudsen pumps for atmospheric or over-atmospheric applications.

<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Thermal simulation of the single-stage Knudsen pump on top of a vacuum fitting: a) top view of temperature field, b) cross-sectional view AA', c) detailed view of glass top beneath the heaters, d) detailed view of glass bottom above the pillars.</figcaption></figure>

# Acknowledgements

This work was funded by the French National Research Agency (ANR) and the German Research Agency (DFG) under the project "ANR-2022-CE92-0017-01".

# References

1. X. Wang, T. Su, W. Zhang, Z. Zhang, S. Zhang, Knudsen pumps: a review, *Microsyst. Nanoeng.* 6 (1) (2020) 26.

2. T. Byambadorj, X. Zhao, Y. Qin, Y.B. Gianchandani, Monolithic SOI through-wafer Knudsen pumps with mechanically robust Si channels, *Sens. Actuators A Phys.* 365 (2024) 114825.

3. R. Meyer, et al., Extremely high-aspect-ratio ultrafast Bessel beam generation and stealth dicing of multi-millimeter thick glass, *Appl. Phys. Lett.* 114 (20) (2019) 201105.