The Thermoacoustic Stirling Heat Engine(TASHE) designed by B ack- haus, a device without moving parts which operates at a frequency of 85 Hz with an average pressure of 3 MPa that is capable of using susta inable energies, is applied to run an Inertance Pulse Tube Refrigerator(IP TR) with 1 W cooling power at 90 K. The coupling of these devices ca used to eliminate all moving parts as well as miniaturizing the refrigerat or to use for cooling superconducting magnets for MRI systems. A new meth od for the design of the IPTR performed by using numerical simulation of REGEN3.3. Moreover, to have a better vision of the overall conﬁg uration of IPTR and verify the Results of REGEN3.3, DeltaEC is used as an aux- iliary software. Fortunately, both software results matched perfect ly, and the performance of the IPTR was acoustically and thermodynamica lly ideal. Keywords: Thermoacoustics,InertancePulseTubeRefrigerator,REGEN3.3, DeltaEC, Sustainable Energy. 1 Introduction Since the introduction of thermoacoustic technology, cooling below 120 K tem- perature without moving parts hasn’t been a dream. There are many appl i- cations in which cryogenic refrigerators are part of the system like liq uefaction of oxygen for storage at hospitals and home use, storage of biological cells and specimens, etc [1]. These cryogenic refrigerators utilize thermo dynamic cycle concepts that can divide into recuperative (steady ﬂow) and regener ative (oscil- lating ﬂow) [2]. The Stirling, Giﬀord-MacMaihan, and Pulse-Tube are d iﬀerent types of the regenerative cryogenic refrigerator. These refrigerators usually are run by compressors that impose moving parts to the system and decreas e reli- ability. However, the introduction of the Thermoacoustic Stirling Heat Engine (TASHE) by Backhaus and Swift [3] let to run a regenerative cryogenic ref rig- erator with a signiﬁcant advantage; the system wasn’t more than some welde d tubes and heat exchangers. The coupling of the Oriﬁce Pulse Tube Ref rig- erator(OPTR) with the TASHE makes it a suitable match to eliminate eve ry ✯Sharif University of Technology. Azadi Ave, Tehran,Iran. Mahdi.lavari@alum.sharif.ir . 1 moving part to encourage more reliability, less complicity of constru ction, fewer frequency interferences, etc. The TASHE also can be supplied wit h various sources of sustainable energy like wasted heated from power plants or sol ar en- ergy. But, the TASHE-OPTR solution mentioned above doesn’t apply to t he cases that the refrigerator needs to miniaturize for superconductin g applica- tions. So as well as eliminating moving parts, it needs to pay attent ion to the decreasing size of the refrigerator. TheIPTRworksthermodynamicallyintheStirlingcyclewasﬁrstl ypatented byRobertStirling[4]in1816astheStirlingheatengine. Hershelintr oducedthis cycle as a refrigerator in 1834 [5]. In 1861, Hershel’s concept became practical by Alexander Kirk [6].In 1946, Philips Company [7] liqueﬁed air by using t he Reverse Stirling Engine. In 1963, Giﬀord and Longsworth [8] caused major advance by introducing a refrigeration device that worked without di splacer at the cold part of the Stirling refrigerator that nowadays is called the B asic Pulse Tube Refrigerator(BPTR). In 1984, an oriﬁce was placed inside the pulse t ube by Mikulin et al. [9].A year later, Radebaugh et al. [10] moved this oriﬁce out of pulse tube at its warm end caused the oriﬁce to work as a ﬂow impedance and, as a result, a temperature of 60 K achieved by using Helium gas. This pulse tube arrangement consists of the regenerator, pulse tube, oriﬁc e and the reservoir alongside the heat exchangers (see Figure 1) that now referre d to as the Oriﬁce Pulse Tube Refrigerator(OPTR). RegeneratorPulse Tube ReservoirCold Heat ExchangerHot Heat Exchanger Ambient Heat Exchanger Oriﬁce Figure 1: General arrangement of an OPTR Later studies on developing OPTR led to the discovery of the Inertan ce tube. In 1996, Godshalk et al. [11] obtained a beneﬁcial phase shift caused by the Inertance tube. This component was simply a long and narrow tub e that produced favorable impedance modiﬁcation in the OPTR. In 1998, Roach and Kashani [12] examined a comparison between the refrigerators with an oriﬁce and Inertance tubes. The results indicated that the Inertan ce tubes yield dramatic improvements over the use of oriﬁce. In 2000, Marquardt an d Radebaugh [13] practiced an Inertance tube in the pulse tube refrigerat or to achieve a cooling power of 19 W at 90 K. In 2008, Vanapalli [14] analyzed operation and various aspects of miniaturization of the IPTR. He increased the frequency of the system to decrease the whole volume of the IPTR and studied the eﬀects of the diﬀerent geometrical and acoustical paramete rs on the eﬃciency of the IPTR to minimize various losses. The regenerator geome try was the stainless steel screen with the cold end at 80 K and the warm end at 300 K. The average pressure, pressure ratio at the cold end and the freq uency of operation were 3.5 MPa, 1.3 and 120 Hz respectively. In 2017, Srikanth and et al. [15] designed an IPTR that delivered 1 W cooling at 80 K and 100 Hz Frequency. 2 Figure 2: The Thermoacoustic Stirling Heat Engine(TASHE) [22] Although the TASHE and the IPTR organized on the same phenomenon, called thermoacoustic technology, the TASHE had elapsed through diﬀerent histories. To study thermoacoustic technology, knowledge of the hist ory of the TASHE is inevitable. Thermoacoustics is the conversion between heat e nergy and acoustic energy. Thermoacoustics can be divided into the thermoacou s- tic engine (acoustic power generator) and thermoacoustic refrigerator (ac oustic power consumer). According to this classiﬁcation, the TASHE is a ther moa- coustic engine and the IPTR is a thermoacoustic refrigerator. Thermoacou stic systems acoustically consist of standing-wave systems and travelin g-wave sys- tems. from the acoustical aspect, 90◦diﬀerences between pressure phase and velocity phase is called standing-wave and traveling-wave means pr essure oscil- lation is in phase with velocity oscillation. To follow thermoacoustic history, Byron Higgins [16] in 1777 accidentally discovered that acoustic oscillation occurred in certain positions of the ﬂame in the tube. Moreover, in certain lengths of the fuel supply line, oscillations occurred and could be heard. Sondhauss tube, a tube with one end close d and the other open, was the ﬁrst thermoacoustic eﬀect studied by Sondh auss in 1850 [17]. In 1959, Rijke [18] installed a well-heated screen in a tube and observed severe acoustic oscillation. This Sondhauss and Rijke tubes regarded as the ancestors of standing-wave and traveling-wave thermoacoustic mac hines, respectively [19]. Further development achieved by Carter and et al. [20] in 1962. They placed the stack in Sondhauss tube to increase the area for heat transfer. Their thermoacoustic engine produced 27 W acoustic Power f rom 600 W power input.In 1979, Ceperley [21] oﬀered a type of thermoacoustic engin e was diﬀerent from the Sondhauss tube. Miraculously he observed the m otion and pressure cycles of the gas in the regenerator of a normal Stirling engi ne are identical with those occurring in the regenerator of a traveling- wave heat engine. It is now known as a traveling-wave Stirling engine [1]. Ce perley’s con- cept wasn’t practical until 1999, Backhaus and Swift designed and fabricate d a novel traveling-wave thermoacoustic engine, which is known as a T hermoa- coustic Stirling Heat Engine(TASHE). The TASHE achieved 42% of Carnot eﬃciency which is comparable to that of a traditional engine [3]. The worl d’s ﬁrst TASHE shown in Figure 2 worked at above 80 Hz frequency [22]. About 10 years before the introduction of the TASHE, in 1990, Swift and Radebaugh [1] developed a pulse tube refrigerator run by a thermoacousti c standing-wave engine. This thermoacoustically driven pulse tube r efrigerator (TADPTR) led to a refrigeration temperature of 90 K with an operating fre - 3 Figure 3: Prototype of acoustic liquiﬁer(TASHE-OPTRs) [23] quency of 40 Hz. In 2000, Los Alamos National Laboratory(LANL) and PRAX- IER revised the TASHE as the driver, instead of an original standing-wave thermoacoustic engine that used in TADPTR [24]. This thermoacoustic sys - tem worked with helium at an average pressure of 450 psi, with oscillation s up to 45 psi in amplitude at a frequency of 40 Hz [25]. The major thermoacous- tic subsystems were an engine to generate high-intensity acoustic p ower from high-temperature heat, a wave tube to transport the acoustic power f rom the engine to the refrigerators and to determine the 40 Hz operating freque ncy and three refrigerators to generate useful cryogenic refrigeration while consuming the acoustic power. These three pulse tube refrigerators were des igned with regenerators of 20 cm diameter and pulse tubes of 10 cm diameter [26]. Modifying the frequency of the world’s ﬁrst TASHE to 40 Hz in TASHE- OPTR, provoked an increase in the volume of the whole conﬁguration and made it a bulky system. the discrepancy of these two types of TASHE size i s obvious by comparison between Figure 2 and Figure 3. The frequency of the world’ s ﬁrst TASHE is changed to provide a good match with a typical Pulse Tube Re- frigerator worked with 40 Hz frequency. Until recently. The poor heat t ransfer and high pressure-drop losses in the regenerative heat exchanger at fre quencies above 60 Hz performed an obstacle to gain eﬃcient operation of regenerative refrigerators. In 2006, Radebaugh and O’Gallaher [27] noticed that the right combination of frequency and pressure, along with optimized regenerat or geom- etry, can lead to eﬃcient regenerator operation at a frequency much abov e 60 Hz. By raising the frequency of pulse tube, the rational result is th e miniatur- ization of the whole system and improving the intensity of the output because of the increasing of oscillating mass ﬂow. As a gap found in the literature review, the design of the Inertance Pul se Tube Refrigerator run by the ﬁrst world’s TASHE (with operation freque ncy above 80 Hz) can be a good candidate to gain advantages like miniaturization of the system along with no moving parts, mechanical vibration-free, using of sustainable energies, etc. The approach for modeling and analysis in th is paper 4 VresevoirLITLPTLR DRDPT RegeneratorPulse TubeInertance Tube ReservoirCold Heat ExchangerHot Heat Exchanger Ambient Heat Exchanger Figure 4: General arrangement of a typical inline IPTR mostly conducted by numerical simulation. So 1-D models, DeltaEC by LANL and REGEN3.3, have been used to design a unique miniaturized refrige rator system driven by the ﬁrst world’s TASHE. The system that is discus sed is referred to as the Inertance Pulse Tube Refrigerator(IPTR) and pro duces a cooling power of 1.0 W at 90 K and 85 Hz. 2 The Inertance Pulse Tube Refrigerator The IPTR is a Stirling-type refrigerator that works with an oscillatin g pressure where the amplitude of oscillation is typically about 10% to 30% of the average pressure. The average pressure is in the range of 1 to 3 MPa and freque ncies are in the range of 20 to 60 Hz [1]. Helium is normally used as the working ﬂuid because of its ideal gas properties, high thermal conductivity, and high ratio of speciﬁc heats. In an ideal IPTR with an isothermal regenerator and an adiabatic pulse tube, the process is reversible. Consequently, the COP of the r efrigerators has a maximum called the COP of Carnot of the refrigerator. It is given by Equati on 1 [28] but the COP of the cooler reduced by the dissipated energy. COPCarnot=Th Tc−Th(1) HereThis the temperature of the hot end of the regenerator, Tctemperature of the cold end of the regenerator. Since the design of ideally operating I PTR is determined by deeply understanding the function and operation p rinciple of its components. The IPTR includes four major components consists of t he regenerator heat exchanger, pulse tube, inertance tube, and reservoi r. These components presented in Figure 4. 2.1 Regenerator The IPTR is a regenerative refrigerator. This classiﬁcation is becau se of the existence of at least one regenerator heat exchanger that operates with the oscillating ﬂow. The regenerator consists of a pile of stainless-stee l screens. The regenerator is sandwiched by cold and hot heat exchangers, therefore, mai ntains 5 a smooth temperature proﬁle between the two adjacent heat exchangers . The function of the regenerator is to act as a heat storage tank. While hot gas passed through the porous medium of the regenerator, heat absorbed by the heat capacity of the matrix for a half cycle. In backward mode, the cold gas ﬂowing in the opposite direction picked up the heat from the mat rix and returned the matrix to its original temperature before the cycle repeated. As a result, these heat exchanges transmit acoustic power or PV power of the driver, here the TASHE, to the cold end of the regenerator of the IPTR wi th a minimum of loss. The time-averaged acoustic power [29] transmits throu gh the regenerator is given by: /angbracketleft˙W/angbracketrightt=/angbracketleftP˙V/angbracketrightt= (1/2)|P||˙V|cosθ. (2) Where the bold symbols represent time-varying or phasor variables, s uch as pressure P, volume ﬂow rate ˙V. The phase angle θis the phase between pressure and ﬂow. Most losses of the IPTR happen in the regenerator. These losses consist of thermal ineﬀectiveness (enthalpy ﬂow), pressure drop and therm al conduction. These losses are functions of the dimensions and operation parameters of t he regenerator. Vanapalli [14] analyzed the regenerator losses and determine t he variationoflosseswithspeciﬁcareaandlengthoftheregenerator. Hedisc overed that to have lower fractional void volume losses, the speciﬁc area and t he length should be smaller which, is conﬂicting with the pressure drop and h eat transfer losses (speciﬁc area should be larger for lower losses) and conduction l oss (length should be larger for lower loss). The optimum speciﬁc area is a tradeoﬀ b etween pressure drop, enthalpy ﬂow, conduction, and void volume losses. Rade baugh and O’Gallagher [27] also suggested much smaller hydraulic diameters rege n- erator material and smaller regenerator volumes at high pressures requir ed for the high-frequency operation of the regenerator and reduction of losses. 2.2 Pulse Tube The IPTR operates ideally with adiabatic compression and expansion in th e pulse tube, therefore it should insulate the processes at its two ends. Besides, it must be long enough that gas ﬂowing from the warm end traverses only partw ay through the pulse tube before the ﬂow reversed, so the gas parcels th at ﬂow in from one side of the pulse tube never reach another side. With these c riteria, gas in the middle portion of the pulse-tube never leaves the pulse-t ube and forms a temperature gradient that insulates the two ends. This middle por tion gases act like a displacer in Stirling-type refrigerator. Turbulence in the pulse-tube must be avoided due to its eﬀect on the insulation of the two ends. 2.3 Inertance Tube The inertance eﬀect mechanism has developed to cause a beneﬁcial ph ase shift between the ﬂow and the pressure at the warm end of the pulse tube. T hese mechanisms have led to improved eﬃciencies in the pulse tube r efrigerators. An Inertance tube can be placed at the warm end of the pulse tube to have a proper phase of mass ﬂow with respect to the pressure in the regene rator. 6 RegeneratorPulse TubeImaginary axis Real axis PVahxфh фc фpt ∙ Ambient Heat Exchanger(ahx)Cold Heat Exchanger(chx)Hot Heat Exchanger(hhx) RTambientPVpt∙ γRTcPVchx∙ RTcPVrg∙ RTrgPVhhx∙ RTW Pmhhx∙ mw∙ m c∙ m chx∙ m ahx∙ m inertance tube∙ Figure 5: The phasor diagram of an IPTR Radebaugh [30] pointed out the correct phasing occurs when the mass ﬂow or thevolumeﬂowthroughtheregeneratorisapproximatelyinphasewithp ressure. The phasor diagram (see Figure 5) of the regenerator and pulse tube indi- cates they have a similar eﬀect on the phase angle between pressure an d ﬂow. When gas ﬂows due to rising pressure from the driver to reservoir v olume, so it passes through the regenerator, pulse tube. Both the regenerator and the pulse tube caused the ﬂow phase to lead the pressure phase similarly. Rade baugh and et al. [29] learned to achieve optimum phase angle of ﬂow lagging pressure by about 30◦at the cold end means that the Inertance tube will need to lag the ﬂow at its inlet by about 60◦to accommodate the change in phase through the pulse tube. 2.4 Reservoir Volume The function of reservoir volume is to be large enough that the eﬀect of pr essure can be omitted despite oscillating ﬂow occurs. 3 The TASHE The TASHE is shown in Figure 6, is an example of a traveling-wave refrige rator. The TASHE provides one way to achieve the required acoustic work to dr ive the IPTR. The TASHE produces acoustic power while the gas in the regenerat or ex- periences thermal expansion when the pressure is high and thermal c ontraction when the pressure is low. Therefore each cycle occurred generate s the acoustic waves and makes them more strength. Like the IPTR, if the regenerator of the TASHE is in a suitable acoustic network, the ﬂow along the regenerator ’s temperature gradient is in phase with the pressure, so good thermal contact between gas and regenerator is required to cause the thermal expansion an d contraction steps to be in phase with the oscillating pressure. 7 Figure 6: Schematic cross-section of the Backhaus-Swift thermoacousti c-Stirling engine. (a) is the arrangement of TASHE that consists of the looped acoustical network shown in (b) and a standing-wave resonator [31]. The ﬁrst’s world TASHE (see Figure 6) is a 30 bar helium ﬁlled the syste m, oscillating at 85 Hz. The resonator was essentially half-wavelength, wit h pres- sure oscillations 180◦out of phase at the right end of the fat portion on the right and in the small torus containing the heat exchangers on the left. T he highest velocity occurred in the center of the resonator, at the smal l end of the long cone. The highest pressure amplitude occurred in the ambi ent heat exchanger [22]. 4 Numerical Simulation of the IPTR The approaches for design and analysis of the IPTR mostly conducted by nu- merical simulation. Base on this fact, the modeling of IPTR investigat ed by REGEN3.3 and DeltaEC. DeltaEC has a one-dimensional approach and treats a thermoacoustic model as a series of standardized segments. DeltaEC p erforms simultaneous integrations along x of the conservation equations of momentum , continuity, and energy. Its equations are modiﬁed by using sinusoid al parame- ters and takes advantage of practical parameters to include losses like pr essure drops. REGEN3.3 integrates with respect to both x and t but makes no res tric- tive assumption that the time dependence must be periodic. 4.1 DeltaEC In 1868, Kirchhoﬀ calculated acoustic attenuation in a duct due to oscillator y heat transfer between the solid isothermal duct wall and the gas sustain ing the sound wave [20]. In 1969-1983, Rott et al. [19] established a sound theoretical foundation applicable to both thermoacoustic prime mover and heat pumps . In 2001, Swift wrote a book and presented a simpliﬁed form of governing equat ion 8 for acoustics waves under “Rott’s acoustic approximation” [22]. The main idea was to apply sinusoidal terms of pressure, temperature, and density to the con- servation law of mass, momentum, and energy and neglecting the second-or der involving the product of two small value terms in equations. These equations represent a set of linear and time-dependent equations that can found in swift’s book. Ward and Swift [32] developed a program for thermoacoustic compu- tation base on numerical integration of Rott’s acoustic approximation. This software is now named Design Environment for Low-amplitude thermoacous tic Engines Conversion(DeltaEC) [33]. DeltaEC deals with one spatial dimension sequence of thermoacoustic el e- ments called segments. It integrates momentum and continuity equati ons that include additional eﬀects such as dissipation of acoustic power along th e sides of the geometry chosen by the user such as ducts, compliances, transduc ers, and thermoacoustic stacks and regenerators. It uses diﬀerent equations in diﬀerent segments to suit local circumstances but typical types of equations that used as momentum and continuity equations [34] are given by: dp1 dx=−iωρm AU1, (3) dU1 dx=−iωA ρma2p1. (4) WhereAis an cross-sectional area, p1pressure amplitude, U1volume ﬂow rate,asound speed, ρmmean density, ωangular frequency. The Equation 3 is derived from the momentum equation of ﬂuid mechanics and the Equation 4 is derived from the continuity equation of ﬂuid mechanics. DeltaEC assuming that oscillatory ﬂow losses can be calculated by using the steady-ﬂow Moody friction factor at each instant of time during the osc illatory ﬂow, AturbulencealgorithmcanbeenabledinDUCTandCONE,whichspec ify the relative roughness (roughness height divided by pipe diameter) . DeltaEC uses a shooting method, by guessing and adjusting any unkno wn boundary conditions at the initial segment to achieve desired boundary condi- tions elsewhere. The user of DeltaEC enjoys considerable freedom i n choosing which variables used as boundary conditions and which computed as part of the solution. With its multi-parameter shooting method to satisfy a variety of mixed boundary conditions, DeltaEC gives the user considerable fre edom in choosing which variables are computed as solutions. So it is obvious that any corrupt and impractical inputs as boundary condition caused outputs conv erges to wrong and impractical answers. In this paper, the literature revi ew used to consider practical inputs for DeltaEC. 4.2 REGEN3.3 The ﬁrst version of REGEN was distributed by the National Institute of Stan- dards and Technology(NIST) as REGEN3.1 in 1989 [35]. This software was revised and distributed as REGEN3.2 and REGEN3.3. The numerical approx - imation of REGEN is a discretization of the diﬀerential equations for th e con- servation of mass, momentum, and energy in the gas and regenerator matrix. The result is a nonlinear system of equations for the temperature, pr essure and mass ﬂux at all the mesh points simultaneously is solved by a Newton iter ation. 9 The program ﬁnds the solution to the conservation equations that satisfy t he input conditions. In REGEN3.3, losses associated with regenerator ineﬀectiveness (ent halpy ﬂow), conduction through the matrix, and pressure drop calculated by t he model. The optimization process through the use of REGEN3.3 conducted by altering the oscillation frequency and temperatures of both ends of the re- generator to set an optimum speciﬁc area of the regenerator. 5 Design Process This section describes a systematic design procedure of an IPTR ope rating around 90 K temperature with 1 W cooling. The design method starts with deﬁning inputs and requirements. The primary parameters of inter est for an IPTR are cooling power, operating temperature, average and dynamic pr essure, and frequency. Because the TASHE, as the driver, imposed ﬁxed input s on the IPTR, the measures of merit should be considered to choose a suitabl e segment from the DeltaEC model of the TASHE from reference [34]. 5.1 Input from the TASHE The TASHE consists of various segments that shown in Figure 7 from the DeltaEC model of the TASHE with diﬀerent values of acoustic power and ph ase angle between ﬂow and pressure. The TASHE-OPTR designed by LANL and PRAXIER used the resonator segment of the TASHE as output for the OPTR shown in Figure 8a and Figure 8b. This segment is a standing-wave segmen t because the phase relation between ﬂow and pressure is above 60◦. As a result, the phase relation needed to be adjusted to pursue a favorable input for the OPTR. This reason indicated that adjustment of the ﬁrst world’s TASHE w as necessary to ﬁt correctly with the eﬃcient OPTR to minimize loss es. To join an IPTR with the ﬁrst world’s TASHE, some measures of merit evaluated fr om the literature review to make a suitable eﬀect on the design of the IP TR. They listed as follows: Figure 7: DeltaEC model of the TASHE 1. Flow phase should lead pressure phase more as possible at the cold end of 10 (a) The schematic of the TASHE-OPTR [25] (b) The TASHE-OPTR [26] Figure 8: The TASHE-OPTR regenerator to have pressure and ﬂow in phase in the midpoint of regen- erator [14]. 2. More acoustic power means more refrigeration power. 3. Higher pressure ratio is a better choice but this should not exceed 1.5 because above this value, the equations can’t assume linear anymore and the low-amplitude approximation is not valid [22]. 4. The inline arrangement of the IPTR (see Figure 4) is the most eﬃcient because it decreases void volume losses [2]. According to these measures of merit, Segment 9 selected as input fr om the DeltaEC model of the TASHE. Table 1 exhibits its parameters. The pres sure ratio, acoustic power and phase relation between ﬂows with pressure p erfectly match with desired values. 11 Table 1: Segment 9 of DeltaEC model of the TASHE Segment 9 parameters values Frequency(Hz) 85 Mean pressure(MPa) 3.1 Temperature hot end of regenerator(K) 325 Amplitude of pressure(MPa) 0.3 Phase of pressure(◦) 0 Amplitude of volume ﬂow (m3/s) 0.016 Phase of volume(◦) 32 Acoustic power(W) 2009 Pressure ratio 1.21 Phase volume ﬂow leads pressure(◦) 32 5.2 Numerical Simulation After deﬁning inputs and requirements, the approximate conﬁgurati on of the IPTR should be estimated before running the simulation code to des ign the IPTR. Since there are too many possible combinations in the selecti on of geo- metric parameters such as diameter and length of the regenerator and the p ulse tube, using all of the possible combinations in simulation code is not p ractical. As a starting point, three typical IPTR design parameters that their f requency is within the range 60-120 degree and mean pressure are between 25-35 bars were selected from the literature review shown in Table 2. Table 2: Three typical design data of the IPTR Design Parameter Ref. [30] Ref. [15] Red. [14] Frequency(Hz) 60 100 120 Mean pressure(MPa) 2.5 3.5 3.5 Temperature hot end of regenerator(K) 300 300 300 Temperature cold end of regenerator(K) 80 80 80 Speciﬁc area of regenerator(m2/kg) 0.052 0.058 0.054 Length of regenerator(mm) 52 35 30 Diameter of regenerator(mm) 25.6 7 9.5 COP 0.117 0.1044 0.078 Phase that ﬂow lead pressure at cold end(◦) -39 -30 -30 Pressure ratio 1.3 1.3 1.3 Refrigeration power(W) 25 1 6.1 As shown in Table 2, ﬂow phase angle lead pressure phase angle about 30◦ and it totally agrees with Radebaugh [29]. Hence -30◦phase angle at the cold end of the regenerator used as a ﬁxed parameter in the design of the IPTR. Another parameter is the speciﬁc area of the regenerator which is the rat io of the gas cross-sectional area to mass ﬂow at the cold end. Vanapalli [14] investigated that the value speciﬁc area in miniaturization of the IPT R played a big role in the eﬃciency of overall IPTR. As shown in Table 2, 0.054 m2/kg,the 12 average of values in Table 2, used as an initial value for the speciﬁc area val ue in REGEN3.3. The pressure ratio (the ratio of the maximum pressure to the minimum pressure) is directly related to the intensity of the ac oustic wave. unfortunately, the TASHE imposed 1.21 as the pressure ratio on the IPTR. After specifying deﬁned cooling power (1 W), operating temperatu re (90 K), average pressure (3.1 MPa), dynamic pressure (0.3 MPa), and frequenc y (85 Hz). A model of the IPTR with input from the TASHE developed by DeltaEC along with a series of run carried out by REGEN3.3 to determine the siz e of the regenerator and maximizing the COP of the regenerator. The Figure 9 represents segments of the IPTR that modeled by DeltaEC. Segments in Figure 9 from left to right are the regenerator (No.4), pulse tube (No.9), inertanc e tube (No.11) and reservoir volume (No.12). Figure 9: The IPTR modeled by DeltaEC 6 Result To design a regenerator that can perform 1 W of cooling at 90 K, REGEN3.3 performed using an iterative optimization process. Initial guesses made with respect to the regenerator length, regenerator inner diameter and an opt imal phase shift (between pressure and velocity) at the cold end of the r egenerator. Simultaneously, initial guesses for conﬁguration of the IPTR obtained fr om ref- erences were used to set up a series of run on DeltaEC. The eﬀorts we re made to bring DeltaEC and REGEN3.3 outputs closer together as same as maximum COP was the target. 6.1 REGEN3.3 Many runs were applied by REGEN3.3 to conﬁgure regenerator until maximu m COP reached. Results from the ﬁnal run result shown in Table 3. 13 Table 3: Regenerator parameters derived from REGEN3.3 Parameter Values Frequency(Hz) 85 Mean pressure(MPa) 3.1 Refrigeration power (W) 1 Temperature of hot end of regenerator(K) 325 Temperature of cold end of regenerator(K) 90 Phase that ﬂow lead pressure at cold end(◦) -28 Acoustic power at hot end of regenerator (W) -1917 Acoustic power at hot end of regenerator (W) 549 COP 0.121 Material of regenerator Stainless steel Porosity of regenerator 0.6939 Hydraulic diameter of regenerator( µm) 44.88 Working ﬂuid Helium Diameter of regenerator(m) 0.08 Length of regenerator(m) 0.05 6.2 DeltaEC Diﬀerent conﬁgurations of the IPTR with input from the TASHE tested by DeltaEC by changing diﬀerent segments of DeltaEC. The result can foun d in Table 4. The primary 3-D model of the designed IPTR depicted in Figure 10. Table 4: The IPTR parameters derived from DeltaEC Design Parameter Values Frequency(Hz) 85 Mean pressure(MPa) 3.1 Refrigeration power(W) 1 Temperature hot end of regenerator(K) 325 Temperature cold end of regenerator(K) 90 Acoustic power at hot end of regenerator (W) -1959 Acoustic power at cold end of regenerator (W) 519 Diameter of regenerator(m) 0.08 Length of regenerator(m) 0.05 Diameter of pulse tube(m) 0.04 Length of pulse tube(m) 0.055 Diameter of inertance tube(m) 0.006 Length of inertance tube(m) 0.05 Volume of reservoir(m3) 0.0011 7 Performance of the IPTR The regenerator works isothermally and the pulse tube works adiabaticall y in an ideal IPTR. Figure 11 originated by DeltaEC, represents the temper ature gradient over the IPTR length. As shown in Figure 11, temperature proﬁl e 14 Hot Heat Exchanger(hhx)Regenerator Cold Heat Exchanger(chx)Pulse Tube Ambient Heat Exchanger(ahx)Inertance Tube Reservoir Input from TASHEFigure 10: The 3-D model of the IPTR dropped (from Segment 1 to segment 4) from 325 K with a linear slope to 90 k over regenerator length. This linear slope indicates that the regene rator is isothermal. It means that heat exchange between gas parcels and porous medi a intheregeneratorisperfectandatanysurfaceofit, thetemperaturei sconstant. The temperature proﬁle (from segment 5 to segment 9) displays tempe rature raised from 90 K to 300K with a nearly nonlinear slope due to the adiabatic background of the pulse tube. The rest segments include the inertan ce tube and the reservoir experienced 300 K (ambient) temperature. Figure 11: Temperature gradient over length of the IPTR from DeltaEC To evaluate the IPTR from the acoustical viewpoint, the amplitude of pr es- sure over the IPTR length proves the nature of sinusoidal waves ins ide the refrigerator. Figure 12 attests that the amplitude of pressure along IP TR fol- lows one-quarter of the cosine function. It seems the IPTR perfect ly managed from an acoustical perspective. 15 Figure 12: Amplitude of pressure over length of the IPTR from DeltaEC Figure 13 is the amount of acoustic power over the length of the IPTR. Figure 13 explains that the regenerator is the most important component of the IPTR due to its massive consumption of acoustic power (from segmen t 1 to segment 4). So by optimizing the regenerator, the acoustic power that r emains at the cold end of the regenerator can be increased for a ﬁxed input acousti c power. It is noteworthy to pay attention that the left acoustic power consumed by the inertance tube (from segment 10 to segment 11) to provide a suit able phase shift in the IPTR. Figure 13: Acoustic power over length of the IPTR from DeltaEC 8 Veriﬁcation of the IPTR Model To judge and further verify the validity of the design made by REGEN3.3 and DeltaEC, their results compared. The acoustic power at the cold and hot end of the regenerator is evaluated by both software is shown on Table 5. the smal lest diﬀerences between outputs of REGEN3.3 and DeltaEC represents that two models perfectly matched with each other. REGEN3.3 and DeltaEC calcu late equations with diﬀerent assumptions. For REGEN3.3, losses associated w ith regenerator ineﬀectiveness (enthalpy ﬂow), conduction through the matrix, and pressure drop are computed by the model and conduction loss through the tube wall containing the regenerator material is calculated separately. However, DeltaEC utilized some practical factors to calculate losses. Therefor e, these 16 percentages of discrepancy mean that the dominant losses of both models s et with minimum values. Table 5: The acoustic power at both end of regenerator Parameter REGEN3.3 DeltaEC Diﬀerence Acoustic powers at hot end(W) -1917 -1959 2.1% Acoustic powers at cold end(W) 549 519 5.7% Another important graph between outputs of REGEN3.3 to verify the valid- ity and convergent of the results on the model is the conservation of ene rgy in theregenerator. Thetotalenergyshouldbeequaltothesumoftheheatcon duc- tion and the enthalpy ﬂow. From Figure 14, the convergence of the numeric al solution can be perceived as well as satisfying the ﬁrst law of thermo dynamics. The energy ﬂow at any cross-section of the regenerator must be equal to th e sum of enthalpy ﬂow and heat conduction. So, Figure 14 conﬁrms that the conservation of energy along regenerator length is constant. Figure 14: The conservation of energy in the regenerator 9 Conclusion This paper dealt with investigating a method to design an Inertance P ulse Tube Refrigerator to be driven by the TASHE. The literature review showe d that high-frequency operation of the IPTR with acceptable eﬃciency is av ailable. Two separate models by DeltaEC and REGEN3.3 was considered to design th e IPTR. Inputs, constraints, and measures of merit derived from the l iterature review and the DeltaEC model of the TASHE. Both REGEN3.3 and DeltaEC models approximately shared similar results due to the diminish ing of losses on both models. The results of Models indicated that the IPTR with 1 W cooling at 85 Hz and 90 K was available with reasonable eﬃciency. Performance study of the IPTR exhibited that the pressure amplitude was cosine funct ion and the regenerate and pulse tube worked isothermally and adiabatically, respe ctively. Finally, the veriﬁcation of models evaluated by the conversion of ener gy and comparison between the two model results. 17