The Regenerator-Displacer to be Modeled
Contents
1. N cycle is the same as the 1 cycle In other words any guessed initial conditions must be washed out by performing the transient analysis of many crank rotations with only predictions of the final cycle of interest to the analyst e Having to work in the time domain for each cycle means needing to take at least one hundred time steps per cycle since otherwise important events would be missed during each cycle Combining this with the above requirement means that from thousands to tens of thousands of time steps might be required to achieve predictions for one design point e Large gradients within the regenerator dictate a minimum spatial resolution as well and this compounds the above computational cost e The copious heat transfer area between the material and the fluid must be taken into account This could actually help simplify the model if it is large enough but otherwise it can present another numerical challenge splitting the analytic hairs between the temperature of the local fluid and the co located regenerator material It might seem adequate to find methods that can achieve answers for one design point in a few hours of computation However rarely would such information be of value since such models themselves must be iterated to 1 calibrate them against available test data 2 test sensitivities and uncertainties and 3 perform sizing studies For example the engineer might want to find the optimum regenera2. based correction factors applied to a 1D model It is even possible to correct the simulations of a 1D model based on a few solutions of a slower and therefore less frequently exercised 3D model Axial Resolution For the current design an axial resolution of 30 is adequate 40 or 50 subdivisions do not appreciably affect the answers However this number should be revisited with any significant change to either the design or the operating conditions All of the models are built to enable easy changes to the resolution to facilitate such investigations with those variations being easiest to apply within the FloCAD and text file versions Infinite Convective Heat Transfer within the Regenerator The heat transfer area between the fluid and the metal in the regenerator is huge and the conduction convection lengths are incredibly short the convective heat transfer coefficients are enormous despite the laminar flow Under most circumstances the temperature difference between the lead shot and the helium next to it is less than a degree much less than the temperature difference axially Temperature profiles of helium solid red versus lead blue dashed are presented at 4 locations through the regenerator 300 280 260 240 220 200 180 160 Temperature K 140 120 100 80 O 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 36C Crank Angle degrees The only exceptions to the above observ
3. be worth the bother but this hasn t been verified 0 Enlarged as they are in the junction case above as needed to conserve total volume 11 Comments in the text file version reveal these CAPILs and the HTRLMP call to the junction in the middle Comments in the include insert file reveal the one way conductors used to model advection In the FIoCAD model use the Advection tab in the R D cylinder to define the helium flow However since that option by default would cause lead to flow instead of helium a parallel and again with identically named nodes to force an automatic merge cylinder of helium would have to be constructed
4. lead shot In this system the details of the compressor and heat exchanger are neglected by idealizing the inlet and the outlet to be represented by constant temperatures and pressures The dynamic seal is also assumed to be perfect The no load adiabatic condition is sought and parasitic heat leaks through the canister wall are neglected In the following table appropriate SINDA FLUINT register names that are used in the subsequently described model are listed Note that the units in the model may differ from those reported in the table and this is noted in the Register Name column if so Parameter Speed Source Pressure Source Temperature Sink Pressure Conductivity Effective conductivity Density Specific heat Bead diameter Regenerator permeability Regenerator porosity Displacer diameter Displacer length Displacer stroke Gap at zero degrees hot end Gap at 180 degrees cold end Valve opening max Value Units GM Cycle 75 rpm 400 psia 300 K 200 psia Register Name rpm Phi in Pa Thi Plo in Pa Lead Shot Properties assumed constant for now 34 W m k 0 34 W m K 11350 kg m 130 J kg K 1 45 inch Regenerator Displacer 23e10 m 40 6 cm 15 cm 2 5 cm 1 25 cm 0 2 cm Valves 2 5 cm Klead KeffLead Rlead CpLead Dbead in cm Plead PorDisp Ddisp in m Ldisp in m DispDisp in m GapHot in m GapCold in m AvMax in m Expanding the model to inclu
5. of these alternatives might even be needed for other regenerators or for other design scenarios even different RPM Therefore any conclusions implied by the presentation of the final model are not firm they are suggestions only Regenerator Displacers Stirling cycle and Gifford McMahon GM cycle engines the displacer is the piston on the cold heat input end of the device GM cycles also use regenerators as do many Stirling cycles at least those intent on achieving the highest possible thermodynamic efficiency In fact regenerators are also key features in pulse tube cryocoolers a derivative of a Stirling cycle that lacks a displacer Regenerators are porous bodies usually cylindrical in shape through which fluid passes back and forth in a periodic motion They are often made either of packed but not sintered beads or screens stacked perpendicular to the flow direction Regenerators are hot on one end and cold on the other An ideal regenerator maintains that temperature gradient heating fluid as it enters one direction and cooling it when it flows in the other direction In one variation of the GM cycle the displacer is also the regenerator the regenerator displacer is a cylinder containing a porous material This porous cylinder is driven back and forth between the cold end say of a cryocooler perhaps used in a vacuum pump or sensor cooling application and the hot end where heat is rejected by exhausting out the
6. step was taken such that these massless nodes then disappear by automatic merging operations overwhelmed by the mass containing nodes of the R D cylinder before the model is launched in SINDA FLUINT Since the pipe flow passages and cylinder are defined geometrically it is relatively easy to change the resolution within Thermal Desktop FIOCAD d Technically Thermal Desktop plus FloCAD RadCAD was not used in this model although radiation is often important in most realistic cryogenic designs AutoCAD 2008 Thermal Desktop 5 1 Patch o C Documents and SettingssBrentMy Models Gifford McMahon FloCAD regenerator displacer dwg IR Model 4 d 300 6 300 6 6 4 25 1 eof 203 6 Ie 9 130 8 106 5 82 24 7 98 SO 98 Temperature LK Time 15 2 sec Summary of Alternatives Investigated Roads not Taken The presented thermal fluid model successfully captures the significant effects of a regenerator displacer and does so quickly However it is not the only possible way of modeling such a device and it is unlikely to represent the best way of modeling significant variations of this device e g different material pressure levels rpm etc Therefore a discussion of alternatives and of the sensitivities of the assumptions applied should be valuable to those engineers using these models as a starting point In other words if the available models are applied to new si
7. 30 tanks finite control volumes and 31 STUBEs zero inertia flow paths The resolution has been picked to match that of the thermal model each fluid control volume corresponds to a co located thermal mass The heat transfer between the helium and the lead shot must be estimated as well as the pressure drop Built in correlations were applied to this very low Reynolds number This conclusion has good implications for the use of the SINDA FLUINT Advanced Design Modules when applied to regenerators incremental changes starting from a prior solution should converge quickly to the new solution This FPROP DATA description f6018NL helium inc is available from C amp R as are many others Many common fluids are kept updated on www crtech com situation by treating the STUBEs as effective ducts The flow area AF was set to the frontal area times the porosity then the total bead surface area per axial length was calculated as the cross sectional wetted perimeter PWdisp setting the hydraulic diameter DH value accordingly as 4 AF PWaisp The actual heat transfer and pressure drop of course will be influenced by the Biot number of the lead beads the tortuosity of the channels and the interruptions of the boundary layers etc Such factors should ultimately be calibrated using test data and SINDA FLUINT offers many scaling factors e g UAM for tie scaling FCLM for laminar friction scaling etc and even automated methods for performing suc
8. RadCAD for radiation and FIOCAD for fluid flow offers all of the features listed above for Sinaps but it goes a step further it provided geometric based calculations it can help build the thermal fluid network and provide network level inputs based on top level inputs Plain and post processed 3D drawings of the R D model in Thermal Desktop FIoCAD shown below make a sharp contrast with the abstract Sinaps diagrams the cylindrical R D is in fact constructed as a cylinder on the screen In other words the size and shape of the R D was defined geometrically so the program was able to generate the nodes and conductors representing the lead shot automatically The fluid model was drawn to the right in the figures as a FloCAD pipe The length of this pipe is defined geometrically and its member tanks and STUBEs are generated according to the current discretization level axial resolution which is 30 in this case The diameter and flow area of this pipe were set manually whereas in most models they are calculated by FIoCAD In fact in most models the wall represented by the thermal structure is pointed to from the FIoCAD pipe and heat transfer calculations are performed automatically However FloCAD lacks provisions for flowing through porous media so instead a fake arithmetic or massless wall of nodes was constructed in the FIoCAD pipe These nodes were then numbered identically with the corresponding nodes in the regen submodel This
9. Regenerator Displacer Modeling in SINDA FLUINT February 20 2008 Overview This document has two purposes 1 To provide guidelines for regenerator modeling using SINDA FLUINT 2 To demonstrate three completely different methods of creating the same SINDA FLUINT model any of which could be used as starting points for a new regenerator model since they have all been created parametrically a text based input file b Sinaps diagram and c Thermal Desktop FloCAD CAD drawing A hybrid regenerator displacer device used in a single stage Gifford McMahon GM cryocooler will be modeled although the results are applicable to stationary regenerators as well The models are available for inspection www crtech com so the descriptions are not exhaustive The organization of this document is as follows First the concept of a regenerator displacer R D will be described along with details of the particular R D to be modeled Second a summary of the corresponding SINDA FLUINT model will be presented the full user s manual is available at www crtech com for reference along with key results Next a summary of the application of each of the modeling environments text Sinaps FloCAD to this problem will be provided Significantly variations of this model that were explored but not presented will be also briefly described Perhaps this information will help future users avoid dead ends However it is more likely that some
10. Top Hot side 370 FS a eT Ie IG Bottom Cold side Pressure psia 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Crank Angle degrees This rapid equalization is caused by the aforementioned fact that the valve openings register AvMax are rather large A noticeable jump in flow through the regenerator occurs at these points which is not performance enhancing The good news is that if more realistic valve responses were modeled the resulting model would execute even faster since SINDA FLUINT would not have to resolve via automatically chosen time steps the artificial valve transients that resulted in the baseline model It is also possible that further simplifications could be made in the model that were not feasible because of the extreme case that this severe valve transient event represented It was for that reason to err on the side of a cautiously conservative modeling approach that this unrealistic valve throat area was not reduced Text Input File It is challenging to say anything positive about a text input file which is based on methods dating back to the 1960s and which has not been in significant use for at least the last decade After all text based files are better read and written by computer programs as a way of communicating with each other which is how Sinaps and Thermal Desktop both communicate with SINDA FLUINT though not exclusively Nonetheless the text file is presented
11. and closing AORI value was controlled as a truncated sinusoid For example the HP valve starts to open at zero degrees reaches full open position at 45 degrees and closes at 90 degrees This was modeled as an orifice opening that moves as a sinusoid from 0 to radians A sine based expression could have been input for the valve throat AORI values but instead a schedule of valve position versus cycle fraction was specified as an array ARRAY DATA with the values cyclically interpolated using the D11CYL utility in FLOGIC 0 This seemingly complex method was chosen such that future analysts using these models as starting points can more easily apply any arbitrarily complicated description of valve motion with time The resulting profiles are as follows HP Valve LP Valve 2 4 2 0 1 8 1 6 1 4 1 2 1 0 0 8 Valve Opening cm2 0 6 20 40 60 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Crank Angle degrees To avoid the need to fully close the orifice a very small opening was allowed instead The flow through this leak is so small as to not be of any significance to the answer As will be discussed later the valve throat areas chosen were too large causing pressures to equalize very quickly and a sudden surge to occur in regenerator flow rate each time a valve started to open It was decided to purposely leave this mistake in the model in order to stress the methodology investigation
12. ate could vary during the cycle Unfortunately this assumption was not appropriate for this system However the use of junctions instead of tanks might be applicable in other cases although it is difficult to suggest exactly in which cases this statement might be true Fortunately it is trivial to switch between tanks and junctions in the pipe or HX macro and if the results don t change significantly to take advantage of the speed improvements that result One dimensional response The validity of the 1D assumption itself can be questioned For most stationary regenerators a 1D assumption is very good In fact the intent of stacked screens is to make the regenerator respond as a 1D device For regenerator displacers the 1D assumption can be questioned because of the seals fluid will enter and exit the regenerator at the sides taking a short cut along the sides of Inspection of the expressions used for these volumes will reveal this term 0 5 PorDisp Ldisp and that it was zeroed out in the final model the piston and therefore making the fluid velocities at least 2D field While a 2D or even 3D flow field can be analyzed in the above approach the generation of the fluid model is cumbersome and the solution speeds can be expected to drop significantly by at least an order of magnitude At some point such a scientific approach might yield to an engineering approach take into account the lack of true 1D behavior via test
13. ation that of fluid and metal temperatures being the same at any location are 1 when the valves crack open and the flow rate pulses and 2 within the first millimeter or two of the current entrance The first temporal case would be eliminated by a more realistic valve model but also does not last for very long in any case it represents a short fraction of the total cycle as can be seen in the above plot The second spatial case for inlet variations only affects a very small portion of regenerator and again not for the entire cycle Why is this observation important If the flow is truly 1D and if like the use of junctions described above the flow rate can be assumed to be spatially but not temporally constant through the regenerator then a very significant approximation can be made The fluid model can be replaced by a simple superposition of an advection term to the thermal model if the distinction between helium and lead temperatures is negligible i e if the convective heat transfer within the regenerator is infinite In other words the motion of helium through the regenerator can be removed from the helium model and relocated to the regen thermal model as an advection material flow term with the nodes in the thermal model now simultaneously representing both the helium and the lead at the same geometric location Specifically if the flow rate and entrance temperatures can be predicted using a much reduced fluid s
14. because 1 the entire model can be read at least by those engineers which retain that skill and because 2 the text file contains some of the variations that have been commented out Sinaps Diagram based Model Sinaps allows the user to communicate with the SINDA FLUINT software by diagramming the thermal and fluid network on the screen This diagram is shown post processed colored below and zoomed in with no post processing Sinaps also provides forms for providing and reviewing editing inputs including a tree like hierarchical model browser It also allows material properties to be defined and stored in a database for various analysis cases to be defined within one model whereas a text file can contain only one case In this model a Sinaps pipe is used to represent the helium flow within the regenerator with nodes representing the lead shot shown to the right in the figures below The pressurization sources and valves are shown at the top of the diagrams Finally Sinaps allows plots of results to be launched from within the diagram as well as from other locations such as from the Register edit table High pressure Low pressure Time 159999971385771 source sink Lead top Low pressure HELIUM 200 EGEN 17 REGEN 1 oi HEHEHTEHERHEI NUBE L F 7 I Helium top FloCAD Geometry based Model Thermal Desktop with companion modules
15. de temperature varying and even anisotropic e g stacked screens properties is trivial but distracting 2 This is estimated as 1 of the total taking into account that the beads contact poorly and at small points with large constriction resistances Ideally this value should be measured empirically Valve opening min 0 0025 AvMin in m Valve cam schedule inlet 0 90 degrees sinusoid assumed Valve cam schedule exhaust 150 300 degrees sinusoid assumed Thermohydraulic Modeling Challenges Regenerator modeling presents various challenges to off the shelf general purpose thermohydraulic solutions such as SINDA FLUINT SINDA FLUINT is capable of solving a wide variety of fluid phenomena with as few or as many simplifying assumptions as the analyst wishes to apply However these solutions occur either in the time domain as transients or as time independent steady state solutions There is no option for solving in phase space Therefore a regenerator poses the following problems e Atransient solution is intrinsically required no time independent state exists which is of any use to the designer The fact that the diffusivity of the regenerator material is of critical importance to the designer provides a validation of the need to work in the time domain e Furthermore a cyclically converged solution is required meaning that several at least two or three but potentially hundreds of cycles must be analyzed before the
16. g the factor of 0 9999 is used to make sure that both the start and end of the last cycle 0 and 360 degree positions are captured since otherwise the start point might be missed due to numerical round off An optional text based user file USER1 provides a summary of key temperatures for review The column titles were printed in OPERATIONS Results Discussion A key output is the cycle averaged temperature of tank helium 2 the helium temperature at the bottom cold side of the regenerator The figure below shows a result of about 55K This is optimistic for a single stage device but keep in mind that the cold end has been assumed adiabatic and that not even systematic heat leaks through the canister have been included so 55K is a reasonable result given the assumptions Note that the temperatures of the lead shot are less variable than the helium next to them as is expected Helium hot Helium cold Middle helium Top of Lead Middle lead Bottom of lead Temperature K 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Crank Angle degrees The jumps visible in the green trace above are caused by the opening of each valve as can be also seen in the pressure plot below At those times pressures in the R D matrix are equalizing with either the supply pressure from 0 to 10 degrees or the exhaust pressure from 150 to 160 degrees
17. h a match between test and predictions However the best guess methodology described above yielded a good match for permeability and pore size data that was available for similarly sized beads albeit sintered Therefore the equilavent duct approach was deemed close enough for the purposes of this model At the bottom of the regenerator a single tank 2 exists to represent the helium at the cold end The volume of this tank is cycled sinusoidally using a VDOT rate of change of volume expression in time where TIMEN is the current problem time to represent the motion of the displacer The volume of the tank 1 at the top hot end of the R D is similarly cycled though opposite in phase Unfortunately the depiction in Sinaps and FloCAD is upside down from that of Welch s Figure 2 Any comments or naming schemes therefore refer to the hot end as the top and the cold end as the bottom Hot Side 90 Cold Side Volume cc 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Crank Angle degrees Because it must accommodate various mechanical objects this top hot side tank has a larger dead space smallest volume than that of cold side which is as small as feasible The hot side tank is also connected to the high pressure HP source and low pressure LP sink by the valves as described in the next subsection Valve Model The valves were modeled as orifices whose opening
18. low pressure port In a paper by Kimo Welch http www genvactech com Vac Technologies Vac Tech Article 1 htm the action of a single stage R D is nicely summarized in the figure below Cooling f Station Displacer Piston Regenerator bed in piston 2 A 4A Gas Seal Lil Valve starts Valve Fully i to open E VU VAL C17 E fi 1 HPHL 2 HP HL HP HL He He He x Primary X Expans ion olume E et 2 E r 1 T ter E x p 5 HP HL HP HL 7 HP HL 8 HP HL He He He He Figure 2 Displacer Operated in Forward Direction Kimo M Welch In order to investigate modeling techniques a SINDA FLUINT model of the above regenerator displacer was constructed In order to demonstrate the various tools with which such a model can be built executed and interpreted three variations of this model are available 1 An ASCII text file 2 A Sinaps model 3 A Thermal Desktop FIOCAD model These models and the associated documentation which included recommendations for methodology are available by clicking URL The starting point for the model to be developed is represented by diagram 2 in the above figure corresponding to a crank angle of zero degrees with the cold volume at its smallest the warm volume at its largest and the source HP valve just starting to open The Regenerator Displacer to be Modeled A single stage helium GM cycle will be investigated and the R D consists of
19. ly thermal initial conditions can be guessed and isothermal is a frequent choice Therefore there are not many software options for setting more complex initial conditions other than imported them from prior analyses However in this case significant computational savings can be achieved by starting from a better guess of the initial temperature profile within the regenerator As few as 5 cycles can then be integrated corresponding to just a few seconds of real time to achieve the desired result whereas with more lazy initial conditions as many as a few hundred cycles might be required For the regenerator a simple linear temperature profile suffices as an initial condition The profile will transition from the hot temperature 7hi 300K based on the temperature of the heat exchanger outlet to the cold temperature 7 0 50K guessed then corrected based on earlier runs A little Fortran based user logic placed in OPERATIONS is the best way to accomplish this An alternative is to apply results from an earlier run using the various restart options available in SINDA FLUINT and Thermal Desktop Even if the earlier run is different the temperature profile is likely to be close enough to accelerate convergence Regenerator Fluid Model Real gas helium is used based on data from NIST s REFPROP program The regenerator fluid model consists of an HX duct macro or the equivalent FloCAD or Sinaps Pipe This macro or pipe consists of
20. of a linear 1D string of diffusion nodes representing a porous cylindrical section that has been subdivided into 30 axial The chosen design is generic to avoid intellectual property concerns Because test data is therefore unavailable minimum required was defined analytically any measure that decreased accuracy including any simplifying assumptions and that significantly affected the results was discarded Conversely any simplification that made no significant effect on results was retained Please keep in mind that these conclusions are very likely to be specific to the specific design investigated and that some amount of methodology exploration will be needed when these models are applied to any new designs or even to the same design if exposed to a different set of boundary conditions sections 29 axial linear conductors between these 30 nodes represent the weak conduction per KeffLead in the packed shot Because the thermal load on the cold head has been neglected as has the canister itself no further thermal structure is needed The lead thermal properties are approximate and can be easily replaced or updated to include temperature dependencies with more accurate data Setting the initial temperatures of this stack of regenerator nodes was critically important in reducing run times Recall that many transient cycles must be integrated before initial conditions are washed out and cyclic convergence is achieved Normal
21. s in order to avoid an overly simplified model that might not be as easily adapted to different designs or conditions Execution and Output Control To prevent the software from taking a time step greater than 1 of the period 1 rps in seconds where rps is rom 60 the output interval is set to this value OUTPTF 0 01 rps However as will be described shortly output is later disabled for all but the last cycle The register cycles contains the number of cycles to run for convergence even though only the results of the last engine cycle are needed Cycles has been set to 20 which is conservatively high as few as 5 cycles are often adequate with good initial conditions see Thermal Model above on Page 6 To test model variations such as new valve schedules set cycles 1 To verify convergence a plot of key variables can be made over the last cycle If the starting point at zero degrees crank angle is the same as the ending point at 360 degrees for all key variables then the cycle can be considered converged and the influence of any guessed initial conditions has been eliminated To produce binary and text output at only the 20 cycle and not before the output operations in OUTPUT CALLS in submodel helium are bracketed by the conditional phrase if timen ge 0 9999 timend 1 rps then where TIMEND is the problem end time and is equal to cycles rps with 1 rps equal to the cycle period in seconds The tolerancin
22. tor design including valve timing perhaps itself based on a computationally intensive criterion such as cool down time from room temperature Speed will always be important Because of the above challenges an investigation was undertaken to discover what were the minimum required modeling techniques that could be applied to a specific problem a single stage regenerator displacer in a simplified GM cycle To keep the problem tenable a 1D response gradients in the axial direction only was assumed adequate although implications of extension to a 2D or 3D model are summarized where applicable The following section describes the baseline model A section at the end of this document describes possible variations A SINDA FLUINT Model Given the above challenges it is gratifying that a model could be developed which executes fast enough between about a minute on a PC that it can itself be exercised repeatedly as necessary for parametric explorations The model is available for inspection as described later Therefore only a summary of important topics will be documented here The model uses standard SI units m Pa sec kg J etc In FloCAD the drawing itself can be in different units but the underlying SINDA FLUINT model will be generated in either standard SI or US Customary English units Two submodels were used a thermal submodel regen and a fluid submodel helium Thermal Model The thermal model consists simply
23. tuations some of the underlying decisions may have to be revisited Fluidic Options Tubes For helium at the relatively low pressures being considered the inertia of the fluid is negligible If this were not true e g a long thin regenerator using a liquid or other dense fluid tubes might be required instead of STUBE connectors For this model tubes only add solution cost but don t affect the answers FTIEs For similar reasons axial conduction within the fluid itself is negligible therefore axial FTIEs are not needed For short fat regenerators using a liquid or other conductive fluid this might not be true For this model FTIEs only add relatively minor solution cost but don t affect the answers Junctions A significant speed up can be gained by using volume less junctions instead of tanks If junctions are used to represent the helium with the regenerator then half that volume should be added to each of the end volumes so the effects of the total volume are not lost The elimination of volume within the regenerator is a significant assumption since it not only neglects the mass changes of the helium within the regenerator due to temperature or pressure it also makes the implicit assumption that the mass flow rate is a constant in space but not in time In other words if junctions were used the mass flow rate into the regenerator at the current entrance would exactly equal the mass flow rate at the exit though the flow r
24. ubmodel consisting of end tanks and a pair of CAPIL connectors to represent the flow resistance of the regenerator then those factors can be applied to a superimposed set of one way SINDA conductors The speed up of the resulting model is tremendous almost an order of magnitude even with greatly increased axial resolution Unfortunately in the particular case investigated the infinite convective heat transfer assumption was not appropriate a full fluidic solution was required However this conclusion may not be the true for other designs or operating points and the tremendous speed up that results is worth the attempt Perhaps correction factors can be applied to yield the same result as the more complete solution in the same way a faster 1D model can be corrected based on a few 3D solutions As was mentioned above SINDA is replete with opportunities for applying correction factors and it features methods for calculating them based on available test data Or this fast but approximate advection model might be used to explore a wider design space parametrically with the more complete solution used to perform final verifications and fine tuning For the extra careful analyst the thermal mass of those nodes can be increased slightly to cover the gain of helium mass whose Cp is significant even if its density is not This gain would have to assume either a constant or a temperature dependent mass of helium It is unlikely to
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