IMECE PULSE TUBE CRYOCOOLER DYNAMICS AND SIMULATION

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1 Proceedings of of the the 2011 ASME ASME 2011 International Mechanical Engineering Congress and & Exposition IMECE2011 November November 11-17, 11-17, 2011, Denver, Colorado, USA IMECE PULSE TUBE CRYOCOOLER DYNAMICS AND SIMULATION Christopher Davis, M.S. Mechanical and Aerospace Engineering Department University of California Davis Davis, CA, USA Nesrin Sarigul-Klijn, Ph.D. Mechanical and Aerospace Engineering Department University of California Davis Davis, CA, USA ABSTRACT The Pulse Tube cryocooler is a relatively new device with a promising future in space grade cryogenics. Unlike many active cryocoolers, the pulse tube is a regenerative type cooler with no cold end moving parts. The lack of cold end moving parts permits it to be used in application with mechanically sensitive or high fidelity components. The pulse tube currently lacks adequate scientific literature and understandings that other space qualified cryocoolers have. A systematic approach to understand, model, and study is provided in this work. Further, the importance of a standardized convention of the dynamically complex pulse tube is elucidated to clarify conflicting accounts found in literature. Flow solutions of the pulse tube demands great conceptual understanding and are carried out though numerical simulation. A 2-D pulse tube is modeled in detail assuming axisymmetric conditions. It is found that flow initialization periods are substantially long in comparison to cycle time due to the great inertial resistance found in pulse tubes along with the large volume associated to the reservoir. During this period, the pulse tube section temperature profile is found to significantly rise prior to a refrigeration permitting profile. It is also found that porous medium (inertial) resistance coefficients can be utilized to sufficiently model long spans of inertance tubing without the need for increased computationally-demanding mesh size. INTRODUCTION Pulse Tube (P-T) cryocoolers and their thermalacoustic off shoots are well suited for a variety of applications including space, high fidelity sensors, and liquefaction; they have even been used in the processing of ice cream. [1] The unique design and operation of the P-T makes it superior to similar application cryocoolers with regards to efficiency and reliability in an optimized configuration. One of the most important parameters to control for optimal configuration is the phase shift between pressure and flow. By properly setting the phase difference between these two parameters entropy flow is maximized in the pulse tube section, while entropy losses are minimized in porous sections like the regenerator. Unfortunately, the P-T is relatively young in the cryocooler field, and does not have sufficient public documentation to reach a mastering level like that of the Stirling cooler. Much work that has been done to design and optimize the P-T cooler is corporate based and remains proprietary. [2] A large majority of the work that is available lacks cohesion with convention and notation. Additionally, high computational demands inhibit detailed P-T studies. Many non-commercial entities do not have access to machines capable of running studies in a timely fashion. A computational study of the P-T is carried out in consideration of flow stability and optimization. Previous multidimensional CFD studies of the P-T lack much practical geometric considerations for optimal flow conditions. Additionally, the type of P-T considered, commonly consists of a narrow inertance tube nearly an order of magnitude longer than the pulse tube section itself. In computational modeling, these extra nodes can multiply the time required to obtain useful solutions. An alternative means of obtaining a solution is suggested by substitute of a much shorter inertance tube section and assigned an equivalent inertial resistance coefficient. NOMENCLATURE A Area cp Specific heat capacity at constant pressure e0 Total energy h Enthalpy H Total enthalpy HHX Hot end heat exchanger P, p Pressure Q Heat R Real number axis S,s Entropy 1 Copyright 2011 by ASME

2 T Temperature t Time W Work θ,φ,φ Pressure-flow phase angle Subscripts Amp - Amplitude Avg - Average C Cold end of pulse tube section H Hot end of pulse tube section In Indicating an inward flow Irr irreversible BACKGROUND AND CONVENTION P-Ts are regenerative coolers similar to the Stirling type cooler but without cold end moving parts. Simply put, the working fluid inside acts more like an electrical 'AC' system rather than a continuous flow-loop DC system. Gas particles at one end of the device typically never reach the other end. The single 'AC' driving force located at the ambient temperature end (shown left) of the P-T isolates any mechanical or electromagnetic vibrations from the cold end and thus, the object being cooled is not subject to those vibrations. This attribute makes the P-T ideal for sensitive equipment such as infrared and other optoelectronics sensor systems. Having essentially only half of the mechanical parts that the Stirling cooler does, the reliability of the P-T is greatly increased since there is considerably less parts vulnerable to fatigue and failure. Over its existence, the P-T has developed many modification and additions to improve the thermal cycle and efficiency. Some of the most mentionable are the orifice, inertance tube, and buffer volume or reservoir. These additions are incorporated into the design at the 'hot' end of the pulse tube. Below, Figure 1 shows a simplified version of a pulse tube cryocooler. Starting from the left, the P-T is driven by a pressure oscillator, shown here as a piston. Next is the regenerator, a heat storage matrix which temporarily stores heat of compression during the cycle. The regenerator is followed on the right by the cold heat exchanger (CHX) and is labeled in the Tcold section. Separating the cold and hot end is the pulse tube section. At the end of the pulse tube, the hot heat exchanger (HHX) removes heat into the ambient environment. Figure 1: Simple Pulse Tube The working gas within the pulse tube is in a closed system and considered compressible. Pressure has amplitude variations as it passes through porous mediums such as the regenerator but, typically does not experience considerable phase variations throughout the P-T. Flow rates of the gas within the device are phase shifted off pressure. The amount of lag or lead the flow rate has with pressure is a spatial variable within the P-T. Pulse Tube Cycle The P-T's thermal cycle is one of the most unintuitive refrigeration cycle, yet still works off the principle of compression- heat rejection- expansion- and heat lift. The complication of understanding the cycle comes with the 'where' and 'when' aspects. To begin, consider the cycle process shown in Figure 2. On top is a plot of ideal phase between pressure and volume flow, negative number indicating flow is lagging pressure. During stead state, a near-linear temperature profile exists within the pulse tube section. The cycle begins at (1) with the compressor in the fully expanded state. This will be referred to as the phasor-cycle zero degree point with associated volume flow. The compressor begins to move toward the regenerator, compressing the gas in the pulse tube section following adiabatic relations. (2) The gas in the compressor region is moved through the regenerator which temporarily stores some of its heat. This gas enters the cold end of the pulse tube at a constant design temperature, T C. Once this gas is in the pulse tube, its temperature also increases according to the adiabatic relations. Gas at the 'hot' end of the pulse tube is now hotter than the ambient temperature. Flow at the hot end eventually begins to pass through the heat exchanger. As the compressor reaches the equivalent Top Dead Center position, pressure reaches a max and flow continues to flow into the pulse tube section due to inertial and impedance (lag) effects. During state (3), the compressor begins to move back to the original position, dropping pressure and pulling gas back into the compressor section. When this begins, gas at the hot end is still moving away from the compressor and passing through the heat exchanger. Gas in the pulse tube section again follows adiabatic relations and cools. (4) After the effective expansion and cooling takes place, the gas at the cold end of the pulse tube is now below original conditions. The gas shifts back into the regenerator, passing through the cold heat exchanger picking up heat from the object being cooled. In an ideally designed P-T, the gas leaving the cold heat exchanger is the same temperature as it was when it left the regenerator, and also, gas is only passed through the cold heat exchanger till the original temperature profile is met. This prevents 'hot' gas from being passed through the heat exchanger and reducing its cycle average cooling. State (5) indicates the equivalent Bottom Dead Center condition. Gas at the hot end, lagging more than the cold end, continues to flow back into the pulse tube till the original temperature profile is met. This completes the thermodynamic cycle of the P-T which then begins again at state (1). 2 Copyright 2011 by ASME

3 gas must be forced though. This causes great losses; from a design standpoint, it is important to minimize these losses. The goal is to transmit the maximum amount of acoustic power through the regenerator with the least amount of losses. This is done when pressure and flow rate are in phase with each other. However, compliance effects in the regenerator cause the flow to lead pressure by about 60 o. To offset this, phase lagging devices force flow rate to lag pressure by about 30 o or half of the effect of the regenerator. This shifts the zero-phase point to the middle of the regenerator and results with the maximum amount of power to be transmitted through the regenerator with least amount of losses. Depending on design, the following key points will have a typical pressure-flow rate phase relation of: regenerator- warm end: +30 o, center: 0 o, cold end: -30 o ; pulse tube hot end: +50 o. [3] Figure 2: Pulse Tube Cycles 1-5 THEORY AND MODELING A '1-5' cycle model shown in Figure (2) is convenient to obtain a quick cut-away understanding of how the P-T operates. However, complete and withstandable proof of not only how but also why the P-T cycle works must be demonstrated using the laws of thermodynamics and conservation. A purely thermodynamic explanation for the cycle can be done so with the first two laws of thermodynamics which are given by: It is important to remember for convention purposes, that the pressure cycle should begins with the piston in the equivalent BDC position as shown in the cycle figure; this is the low point of the pressure wave and degrees away from the velocity's magnitude minimum.[3] (1) (2) Figure 3: Phasor Diagram The last term on the Right Hand Side (RHS) identifies any change over time to the system of the respective quantity, S irr is the entropy generated from irreversibility's, subscripts 'e' and 'i' indicate quantities exiting (exit) and entering (in) the system. The work term on the RHS of the first law, being the work that the gas does on its surroundings, is zero from the assumption that the gas is contained within a rigid device. The quantities above can be expressed as a cycle average value by integrating in the manner Phase Shifting Effects It is frequently noted that for ideal conditions, phase effects must cause flow rate to lag pressure by about thirty degrees, [2-5]. However, what is not so readily available is the overall profile of phase shift between flow rate and pressure, and a clean explanation for why it exists. In order to begin, one must recognize that the regenerator is a porous median that the Terms such as enthalpy and entropy are state quantities, meaning their value is only dependent on the current state of the system. Because of this, their 'zero' value may be taken at (3) 3 Copyright 2011 by ASME

4 the average system condition, that is, if the system were to stop oscillating and all components were to reach steady state. The resulting thermodynamic relations become: Here, the 'in' and 'exit' values are combined to identify net flow, also, the temperature T is taken as the average value. The last term on the RHS naturally becomes zero when the system is in cyclic steady state resulting in the following forms (4) (5) cos(φ) is the angle between the two phasors, Fig. 3. Substituting into equation (6), the maximum (ideal) heat lift is the product of pressure and volume flow. If the tube is assumed to be perfectly insulated a corollary to equation (11) is that enthalpy flow must be constant from one end of the tube to the other. Computational Modeling and Simulation Any real system will consist of entropy losses that the above solution cannot explain. To obtain a useful solution that can be applied to physical designs, a more complete set of equations must be used. A Navier-Stokes approach, including conservation of mass, momentum, and energy, is used in numerical form to obtain flow and thermal solutions for a P-T design. (6) Now, if a piston was to appear at any given point in the pulse tube, the theoretical work done or potential for work to be done on that piston could be found by combining equations (6) and (7) that yield: (7) (8) The ideal cycle however, is adiabatic and reversible [2] driving the entropy term to zero. The cycle averaged work term can be rewritten as the product of dynamic pressure and volumetric flow also known as work flow or acoustic power. This equation states that the enthalpy flow in the ideal system is directly related the flow's power. The values for averaged pressure and volume flow rate can be taken individually over the integral of one cycle as show above. Conveniently, both terms are sinusoidal and the average or quadratic mean (RMS) is well known as: (9) (10) Where P amp is the peak amplitude. Taking the product of these two phasors leads to: (11) Figure 4: Joule-Thomson Effect of Various Gases in Cryogenic Regime. Three-dimensional solutions to the full form Navier-Stokes equations can be extremely complex, unstable and untimely. Prior to solving, it is prudent to simplify the set of equations as much as possibly while also maintaining sufficient accuracy. Assumptions with regards to the working fluid's properties along with symmetry are very useful in developing a P-T solution. Helium is almost always the working fluid in P-T coolers because of its low boiling point and ideal gas properties. [3] The Joule-Thomson effect is a good gage to determine how closely a gas matches ideal gas properties within the P-T for adiabatic expansion; a value of zero indicates ideal gas properties. Another convenient feature of the P-T is that nearly always designed axially-symmetric. Additionally, flow and refrigeration stability requirements demand that the flow through the pulse tube section remain laminar and close to plug flow. This is why flow straightening devices are often incorporated at both ends of the pulse tube section.[2] For these reasons, 2-D models can be used to closely approximate full three-dimensional models while significantly reducing computation time. 4 Copyright 2011 by ASME

5 Current Models A numerical 2-D simulation is carried out to demonstrate the performance of the P-T cooler and also comparison of P-T designs. Three models are considered, first being a recreation of available literature designs [4, 5], second being our proposed geometric improvement, and lastly our proposed model to improve flow characteristics and decrease computation time. All three model's dimensions are along the same order as shown below. The models below are presented in order and denoted as: (A) 'literature' model, (B) modified' model, and (C) 'inertance tube alternative' model. Figure 5: Literature Model (A) Mesh is use to create an unstructured mesh. The literature model, (A) shown in Figure (5),consists of sharp, 90 o edges and abrupt radial changes between sections. For comparison, the literature model consists of 4026 nodes. Model (B) is shown in Figure (6) with sloped walls connecting sections of changing diameter. This model contains a refined mesh of 11,389 nodes. The inertance tube has been extended from that of model (A) for increased inertial effects. Finally, model (C) is presented to suggest a means to quickly model inertial effects with reduced calculation time, and without the need to redesign the geometry between trials. The model still contains a small inertance tube section, allowing flow to develop in a fashion similar to that of a fully sized inertance tube. It also permits the region to be defined as a porous medium and associated inertial resistance coefficients assigned to it. Model (C) shown in Figure (7) was developed with 23,692 nodes to achieve desired accuracy. All models were tested at an operating frequency of 34Hz using a Fluent User Defined Function for controlling the piston head velocity. The non-heat-exchanging wall of the P-T were modeled as aluminum, 'well insulated,' and adiabatic. The walls and 'fins' within the sections of heat transfer, (after cooler, cold & hot heat exchanger) where modeled as aluminum with thermal properties resembling a well-designed heat exchanger surface. Both the after cooler and hot heat exchanger surfaces were set at constant ambient temperature of 240 K. The walls of the inertance tube were also set to ambient temperature to resemble actual operational condition. The cold heat exchanger was left adiabatic to observe minimum cold finger temperature. DISCUSSION OF FINDINGS All models produce favorable temperature profiles within the pulse tube section with varying degrees of success. Model (A) has considerable fluid dynamic losses due to the rough geometric design. Figure (8) [(HHX reversed flow)] shows flow at the pulse tube hot end and hot heat exchanger experience circulation and reversed boundary layers. A recurring trend for this geometric model is the circulation of flow at locations transitioning from a section of smaller diameter to larger. The height of circulation is proportional to the jump in radius. Figure 6: Modified Model (B) Mesh Design Figure 7: Inertance Alternative (C) Mesh Design The models are created in ICEM CFD and imported into Ansys FLUENT for simulation. The geometries designed are 2-D axis-symmetric cross sections of the P-T. A block method Figure 8: (A) Velocity vectors of HHX during compression 5 Copyright 2011 by ASME

6 The modified mesh (B) shown in Figure (9) takes into consideration fluid dynamics to minimize losses due to swirling and abrupt radius change. The heat fins implemented in the after cooler and hot/cold heat exchanger further help to straighten the flow. exists between heat fins. However, the magnitude of circulating gas is nearly insignificant. The temperature profile of the modified mesh is shown in Figure (10). A much more distinct profile is seen compared to the literature model. Temperature peaks are observed at the hot end, while more favorable phase conditions produce a much lower averaged cold temperature. For this given configuration and inertance tube design, hot gas from the inertance tube is shown to inject into the core stream of the pulse tube section like plume flow. An alternative method of producing inertial effects in the inertance tube section is utilized to test viability. This method has the added benefit of allowing for the rapid transition of P-T 'designs' without requiring reediting the actual mesh. Model (C), having the greatest inertial resistance, also produces the least penetration depth of hot gas from the inertance tube into the pulse tube section. The temperature profile is the most linear and has the best temperature separating effects of all three designs. Figure 9: (B) Velocity vectors of CHX during compression Figure 9 shows the modified geometry of the cold heat exchanger passing flow from the regenerator into the pulse tube. The flow remains well straightened as it passes through the cold heat exchanger. The modified hot heat exchanger sees a boundary layer phenomenon similar to that observed in model A(and for the same reason), with a strong core stream of velocity and opposing velocity around it. Some circulation still Figure 10: Pulse tube thermal contours. Top: (A)Literature Model, Middle: (B)Modified Model, Bottom: (C)Alternative Inertance linear and has the best temperature separating effects of all three designs. 6 Copyright 2011 by ASME

7 Figure 11: Area Average Velocity Comparisons at Hot Heat Exchanger (HHX) and Cold Heat Exchanger (CHX) Figure 12: (C) Pulse tube temperature profile with time 7 Copyright 2011 by ASME

8 CONCLUSION Without question, the P-T presents a great challenge to fully understand and accurately model. Consideration must be taken to produce multi-dimensional geometries that are conducive to the expected flow. The 'boxed' off geometry presented in literature model (A) is hardly conducive of laminar, planar flow required by the P-T. Many losses from swirling and streaming are eliminated in the modified geometry presented above. This design is much more conducive to smooth planar flow, and thus transmission of PV (acoustic) power. Velocity profiles found in the literature model have high peaks of magnitude in the center core stream of flow and very low values of flow near the walls due to the conflicting jumps in radius. The literature model is shown to have the greatest averaged velocity followed by the modified model, and lastly, the alternative inertance tube model. All velocity profiles have a varying degree of phase shift from pressure yet all models agree with the suggested pressurevelocity phasor diagrams with pressure crossing t 0 at a minimal value, (like an inverted cosine) and velocity follow shortly after with a positive velocity during compression. It is understood that by increasing inertial resistance, the thermal barrier in the pulse tubes shifts towards the hot end and thus allows the temperature profile to reach much lower temperatures at the cold end. As the inertial effect decreases, the hot gas from both the hot end and the inertance tube, have a greater penetration depth into the pulse tube. The unimpeded hot core stream injects into the pulse tube at a high temperature while cooler, oncoming gas, flows near the walls around this stream. The purpose of the pulse tube is to sustain a thermal barrier, but when significant amounts of core flow penetrate the pulse tube and mix with cooler gas, the pulse tube no longer serves as a useful component. The high inertial resistance of the alternative inertance tube modeling (C) produces the most ideal profile and also reaches the lowest temperature. The thermal barrier can easily be seen within the modified mesh (center). Here, the hot gas from the inertance tube disrupts the gradual temperature profile as seen in that of (C), although cooling is still achieved. Configurations producing flow seen in (B) have reduced stability due to the sharp temperature barrier. A further concern for flow stability is heeded when applying to three dimensional flow assumptions since it is much more prone to swirling and mixing. Finally, when comparing contours of (B) and (C), it is obvious that the literature model is not conducive to flow fields required by the P-T for refrigeration. ACKNOWLEDGEMENTS The first author, Christopher Davis would like to acknowledge the support received as the recipient of 2009 N&M Sarigul-Klijn Flight Research/Space Engineering award endowed on 17 th December 2003 in commemoration of the 100 th anniversary of powered flight with a purpose of encouraging students to conduct research in flight related areas. His winning proposal was titled High Performance Space Grade Cryo-Coolers. REFERENCES [1] Ken Brown. Chilling at Ben & Jerry's : Cleaner, Greener [Internet]. The Wall Street Journal, April 15, m (Accessed on December 22, 2010) [2] Ray Radebaugh. Development of the Pulse Tube Refrigerator as an Efficient and Reliable Cooler. In: Proc institute of refrigeration, London, [3] C. Davis. An Exploration of Pulse Tube Dynamics and Computational Modeling. Master s Thesis, University of California, Davis, March [4] Jeesung (Jeff) Cha. CFD Simulation of Multi- Dimensional Effects in Inertance Tube Pulse Tube Cryocoolers. Master s Thesis Georgia Institute of Technology, May [5] Y.P. Banjare, R.K. Sahoo, S.K. Sarangi. CFD Simulation of Orifice Pulse Tube Refrigerator. Department of Mechanical Engineer, NIT Rourkela, Orissa, [6] S.W.K. Yuan, D.G. T. Curran, and J.S. Cha. A NON- TUBE INERTANCE DEVICE FOR PULSE TUBE CRYOCOOLERS. The Aerospace Corporation, El Segundo, CA, October [7] C. Davis M. Langston, R. Friend, J. Garcia, C. Lin, A. Jordan, W. Shaw, M.Sarigul-Klijn, N. Sarigul-Klijn. "Universal Long Duration Tug Concept. AIAA SPACE 2008 Conference & Exposition , 2008 [8] Ansys Fluent. Ansys Fluent 12.0 Tutorial Guide. Ansys INC 2009 [9] W.R. Smith. One-Dimensional Models for Heat and Mass Transfer in Pulse-Tube Refrigerators. Cryogenics 41, pp , [10] E.W. Lemmon, M.O. McLinden and D.G. Friend. Thermophysical Properties of Fluid Systems. NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P.J. Linstrom and W.G. Mallard, National Institute of Standards and Technology, Gaithersburg MD, 20899, retrieved June 26, Copyright 2011 by ASME