FREIA Report 2020/01 July 9, 2020 Department of Physics and Astronomy Uppsala University Benchmarking a Cryogenic Code for the FREIA Helium Liquefier Elias Waagaard Supervisor: Volker Ziemann Subject reader: Roger Ruber Bachelor Thesis, 15 credits Uppsala University Uppsala, Sweden Department of Physics and Astronomy Uppsala University Box 516 SE-75120 Uppsala Sweden Papers in the FREIA Report Series are published on internet in PDF format. Download from http://uu.diva-portal.org Abstract The thermodynamics inside the helium liquifier in the FREIA laboratory still contains many unknowns. The purpose of this project is to develop a theoretical model and im- plement it in MATLAB, with the help of the CoolProp library. This theoretical model of the FREIA liquefaction cycle aims at finding the unknown parameters not specified in the manual of the manufacturer, starting from the principle of enthalpy conservation. Inspiration was taken from the classical liquefaction cycles of Linde-Hampson, Claude and Collins. We developed a linear mathematical model for cycle components such as turboexpanders and heat exchangers, and a non-linear model for the liquefaction in the phase separator. Liquefaction yields of 10% and 6% were obtained in our model simula- tions, with and without liquid nitrogen pre-cooling respectively - similar to those in the FREIA liquefier within one percentage point. The sensors placed in FREIA showed simi- lar pressure and temperature values, even though not every point could be verified due to the lack of sensors. We observed an increase of more than 50% in yield after adjustments of the heat exchanger design in the model, especially the first one. This constitutes a guideline for possible future improvements of the liquefier. Sammanfattning Termodynamiken bakom heliumf¨orv¨atskaren i FREIA-laboratoriet inneh˚allerfortfarande m˚angaok¨anda aspekter. Detta kandidatarbete syftar till att utveckla en teoretisk modell och implementera den i MATLAB med hj¨alp av biblioteket CoolProp. Denna modell av FREIA:s f¨orv¨atskningscykel syftar till att hitta de ok¨anda parametrar som inte specifi- cerats av tillverkaren, och baserar sig p˚aprincipen om entalpins bevarande. Inspiration togs fr˚ande klassiska f¨orv¨atskningscyklerna Linde-Hampson, Claude och Collins. Vi ut- vecklade en linj¨ar matematisk modell f¨or cykelkomponenter s˚asomexpansionsturbiner och v¨armev¨axlare, och en icke-linj¨ar modell f¨or sj¨alva f¨orv¨atskningen i fasseparatorn. En f¨orv¨atskningsverkningsgrad p˚a10% och 6% uppn˚addesi v˚aramodellsimuleringar, med respektive utan f¨orkylning med flytande kv¨ave - liknande verkningsgraderna i FREIA- f¨orv¨atskaren inom en procentenhet. Sensorerna placerade i FREIA visade p˚aliknande tryck och temperaturer, ¨aven om bristen p˚asensorer gjorde att vi inte kunde bekr¨afta varje punkt. Vi observerade en ¨okning p˚amer ¨an 50% i verkningsgrad efter att ha juste- rat v¨armev¨axlardesignen n˚agot,speciellt f¨or den f¨orsta. Detta kan utg¨ora riktlinjer f¨or var man fram¨over kan f¨orb¨attra den faktiska f¨orv¨atskaren. Contents 1 Introduction 1 2 Fundamental Thermodynamics 2 3 Cycle Components 6 3.1 Compressor . .6 3.2 Heat Exchanger . .6 3.2.1 Counterflow Two-fluid Heat Exchanger . .7 3.2.2 Counterflow Three-fluid Heat Exchanger . 11 3.3 Turboexpander . 15 3.4 Joule-Thomson Valve . 17 4 Classical Liquefaction Cycles 18 4.1 Linde-Hampson Cycle . 18 4.2 Claude Cycle . 20 4.3 Collins Cycle . 23 5 The FREIA Helium Liquefier 25 5.1 Simulations and Results of the FREIA Model . 28 6 Discussion 33 7 Future Prospects 35 8 Conclusion 35 9 Popular Science Summary 36 10 Acknowledgements 37 References 38 11 Appendices 39 11.1 FREIA LN2 Enthalpy Model Function . 39 11.2 FREIA LN2 Enthalpy Model Tester Script . 44 1 Introduction Cryogenics, the study of material behaviour at low temperatures, has a long tradition in Upp- sala. An important step to better understand the concept of temperature was taken there in 1742, when Anders Celsius launched his centigrade thermometer scale. This scale was based on phase transitions: the freezing and boiling point of water at a particular atmospheric pres- sure. This standardized temperature measurement around the world until this day, although he initially proposed that water should boil at 0◦C and freeze at 100◦C. This scale was reversed after his death and then accepted worldwide [1]. Today, the FREIA Laboratory at the Uppsala University carries out experiments for accelerator physics and instrumentation at low temper- atures with liquid nitrogen and liquid helium as coolants. However, lighter gases such as these are not trivial to liquefy. Carl von Linde and William Hampson independently succeeded with the liquefaction of air in 1895 using the cycle that was later named after them [2]. It would take until 1908 for helium to be liquefied for the first time, achieved by Heike Kamerlingh Onnes with a pre-cooled Linde-Hampson cycle [3]. Nowadays, cryogenics play a fundamental role to supply coolants to a range of scientific experi- ments, medical applications and industrial machinery. For instance, liquid helium is often used for the cooling of superconducting magnets in Functional Magnetic Resonance Imaging (fMRI) machines [4]. The need for low-temperature equipment in industry is also growing rapidly. The industrial gas market size is estimated to grow from USD 80 Billion in 2015 to reach more than USD 100 Billion by 2020, dominated by large international corporations such as the Linde Group and Air Liquide [5]. Only the system of superconducting magnets at the Large Hadron Collider at CERN use a total of 120 metric tonnes of helium to be cooled down to the working temperature of 1.8 K [6]. Due to the abundant applications and few deposits on Earth, the supply of liquid helium is limited and causes very fluctuating market prices. As pointed out by Sophia Hayes in Physics Today, the market price of liquid helium was USD 5 per litre in 2010, but this had almost quadrupled by 2016 [7]. This volatility of the global helium supply stresses even further the need for robustly designed and leakproof helium liquefaction systems. Moreover, simulations help to optimize and guide towards improving the efficiency where the information provided by the manufacturer is not sufficient. In the FREIA Laboratory, the helium liquefier Linde L140 built by the Linde Group delivers over 140 l/h of liquid helium into a 2000 l storage dewar, and includes liquid nitrogen pre-cooling and a helium recovery system [8]. However, the process of helium liquefaction still contains many unknown parameters at several points in the cycle, where sensors are missing. These unknowns include gas flows, temperatures and pressures. The purpose of this project is to construct a theoretical model of the liquefier, starting from drawings of the vendor. This theoretical model is based on balancing the flow of enthalpies through the system at thermodynamic equilibrium, and is implemented in a model simulation in MATLAB. The system parameters from the FREIA control system, such as temperatures and pressures, are compared with output parameter values of the simulation. For higher temperatures and lower pressures, helium behaves like an ideal gas for which calcu- lations are relatively simple. However, in many instances, we need to consider the behaviour of real gases, whose thermodynamic relations are far more complex. For this project, we use the open-source library CoolProp that contains tables for many thermodynamic quantities, in- cluding real gases to a very good approximation [9]. Thus, our model is applicable also for real systems. 1 In this report, we first describe some essential terminology in thermodynamics in Section 2, followed by mathematical models for the individual cycle components in Section 3. These com- ponents will then be put together in liquefaction cycle models that are simulated in MATLAB. We will gradually increase the complexity of the models, starting from the historically impor- tant liquefaction cycles of Linde-Hampson, Claude and Collins in Section 4, to finally simulate the FREIA liquefaction cycle in Section 5. The MATLAB code to simulate the FREIA lique- faction cycle can be found in Appendices 11.1 and 11.2. A popular science summary can be found in Section 9. 2 Fundamental Thermodynamics To start with, several recurrent thermodynamic concepts need to be defined. We will often come back to the first law of thermodynamics, which states the conservation of the internal energy U, or the energy associated to the random motion of molecules in a system. According to the first law, an infinitesimal change dU in the internal energy is dU = δQ + dW; (1) where δQ is the infinitesimal heat of the process and dW is the infinitesimal work. In other words, the sum of all incoming energies in a system equals the outgoing energies or what is accumulated [2]. To describe a system, we use principal properties such as pressure P , volume V , molar quantity n, and temperature T . P and T do not depend directly on the size or the amount of material in the system, and are called intensive quantities. P is the force exerted per unit area of the system, and T is a quantitative expression of the heat in the system. For such a system, we express the infinitesimal work as dW = −P dV . In this report, we shall always give P in terms of absolute pressure, in unit [bar(a)]. Conversely, V and n are extensive quantities and depend on the size of the system, or how much is contained in it. The thermodynamic relation between these quantities is called an equation of state. In classical thermodynamics, the equation of state for an ideal gas is PV = nRT; (2) where n is the amount of substance of the gas (in moles) and R = 8:314 [J/(mol K)] is the ideal gas constant.