Dimensionless Analysis for Regenerator Design

Transcription

1 Dimensionless Analysis for Regenerator Design Jinglei Si, Jon Pfotenauer, and Greg Nellis University of Wisonsin-Madison Madison, WI ABSTRACT Regenerative eat exangers represent a ruial omponent in te design of single and multi-stage ryooolers. Bot te eat transfer and fluid dynamis tat our in te regenerator influene its performane signifiantly and an only be modeled adequately by solving te oupled mass, energy, and momentum onservation equations. Beause of te inerent sopistiation required, numerial solutions desribing regenerator performane require substantial omputational time in order to onverge wi limits teir utility wit respet to optimization and design. Tis report investigates te extrapolation of te performane of a regenerator from a known base ase to onditions removed from te base ase using a simple, analytial model tat is based on running a few simulations in REGENv3.2 at onditions near te base ase and orrelating te results in terms of te key dimensionless parameters. Te analytial model an be used for aelerated parametri study and optimization of a pulse tube ryoooler; te proess of fitting an extrapolating funtion to te REGENv3.2 results must be aomplised periodially as te optimization proess moves away from te base ase. In tis paper, te orrelating funtions are disussed and te metodology is demonstrated. Te base ase performane is determined using REGENv3.2 and te extrapolated performane, using te orrelating funtions, is ompared to te performane at te same ondition predited diretly by REGENv3.2. Correlations are developed to predit te mass flow rate and pase at te warm end of te regenerator, as well as te eat exanger ineffetiveness and pressure drop. INTRODUCTION Te regenerator-design ode developed at NIST[1] in its most reent version, REGENv3.2, provides a powerful tool for te user to investigate te influene of geometry, material seletion, frequeny, temperature, pressure ratio, and te pase between flow and pressure on regenerator performane. Wile te broad array of allowable input parameters enables a wide variety of questions to be investigated, te multipliity of oies an be intimidating to te new user, and te optimal approa for utilizing REGENv3.2 for regenerator design is not immediately obvious. Also, te program is omputationally intensive to run, partiularly for low temperature simulations. A large number of iterations (around 10,000), time steps per yle (around 250), and spatial mes points (around 200) are required in order to guarantee te aurate results [2]. Tese numerial parameters ause REGENv3.2 to require approximately 24 ours to simulate a single ondition wen run on a typial personal omputer. Te intent of tis investigation is to identify orrelating funtions tat are based on te key dimensionless parameters and an be used to extrapolate te performane of a regenerator over a wide range of operating onditions based Cryooolers 14, edited by S.D. Miller and R.G. Ross, Jr. International Cryoooler Conferene, In., Boulder, CO,

2 420 REGENERATOR MODELING AND PERFORMANCE INVESTIGATION on a minimum number of REGENv3.2 simulations. Tese orrelating funtions an subsequently be integrated wit models of te oter omponents in a system and used to alulate te regenerator performane witin an extensive optimization and design proess witout aving to repeat te detailed investigation wit REGEN at every operating ondition. As te optimization proess moves away from te base ase tat is used to generate te orrelating funtions, te extrapolation will be less aurate and, eventually, te proess must be repeated using a new base ase. FEATURES OF REGENV3.2 Te performane of te regenerator is ritial to te overall performane of a pulse tube system beause te dominant loss meanisms in te yle our in te regenerator. Te input parameters required by REGENv3.2 inlude te operating parameters (e.g., average pressure, pressure ratio, frequeny, et.) and te geometry (e.g., diameter, lengt, matrix type, et.). Witin a system simulation, REGENv3.2 is most often used to move from te old end to te warm end of te regenerator. Tat is, te amplitude of te mass flow rate at te old end ( ) and its pase relative to te pressure variation (θ) are typially inputs to te simulation wile te amplitude of te mass flow rate at te ot end of te regenerator ( m! ) and its pase angle relative to te pressure variation (θ) are typially outputs. Additionally, REGENv3.2 predits te average pressure drop aross te regenerator ( P ) and te average rate of entalpy flow troug te regenerator (EHTFLX); te EHTFLX parameter araterizes te net termal performane of te regenerator and inludes te loss of refrigeration aused by te ineffetiveness of te regenerator as well as eat ondution troug te matrix. Te remainder of tis paper disusses te orrelating funtions tat ave been developed to allow te predition of tese key outputs (, θ, P, and EHTFLX) over a wide range of operating onditions. Te mass flow rate and its pase at te ot end are predited based on a simple, pasor analysis of te regenerator. Te entalpy flux and pressure drop are predited based on preparing loally valid and well-beaved orrelations for te ineffetiveness and frition fator of te regenerator. HOT END MASS FLOW RATE AND PHASE Te mass flow rate at te ot end and its pase must be estimated before te termal loss or pressure drop as tis quantity feeds te orrelating funtions for te oters. Te ot end mass flow rate, m!, and its pase wit respet to pressure, θ, are predited approximately using a pasor analysis, as disussed in [3]. Te pasor model neglets flow-indued pressure loss troug te regenerator and onsiders only te effet of mass storage in te omponent. In tis limit, te pressure is spatially uniform and an be represented by te average pressure ( P ) and dynami pressure amplitude ( P " ): were te pressure amplitude an be expressed as: P= P+ P " sin( ωt) (1) ( PR 1) P " = ( PR + 1) P (2) and ω is te angular frequeny and PR is te pressure ratio, defined by te ratio of te maximum pressure to te minimum pressure during te yle. Te derivative of pressure wit respet to time (t) is given by: dp π = P " ωos( ω t ) = P " ωsin ω t + (3) dt 2

3 421 DIMENSIONLESS ANALYSIS FOR REGENERATOR DESIGN T Regenerator T Pressure Dead volume V Mass P m Figure 1. Semati of te regenerator Figure 1 illustrates a semati of te regenerator and sows tat an instantaneous mass balane is: were m is te mass of gas stored in te regenerator. dm = + (4) dt Te gas stored in te regenerator is assumed to obey te ideal gas law; te mass of te gas is terefore: PV m = (5) RT were V is te void volume of te regenerator and T is te mass average temperature of te regenerator, wi is assumed to be onstant. If a linear temperature distribution is assumed, ten T is [3]: Substituting Eq. (5) into Eq. (4) yields: T T T = (6) T ln( ) T PV d dm RT V dp m m m dt dt RT dt! = +! = +! = + (7) Note tat te mass storage term in Eq. (7) is proportional to te time derivative of pressure; aording to Eqs. (1) and (3), te pasor representing mass storage in te regenerator must be oriented at 90 o relative to te real axis. Te pasor relation expressed by Eq. (7) is sown in Fig. 2 were te magnitude of te mass storage term is V dp ωpv " = RT dt RT (8) In Fig. 2, te mass flow rates are represented as pasors; te magnitude of te pasor represents te amplitude of te mass flow variation wile te angle between te pasor and te real axis represents te pase between te mass flow rate and te pressure. A mass flow pasor tat lies on te real axis is in pase wit te pressure variation. Mass flow pasors tat lie in te 1 st quadrant lead te pressure variation in time wile tose in te 4 t quadrant lag te pressure variation.

4 422 REGENERATOR MODELING AND PERFORMANCE INVESTIGATION θ θ θ V dp RT dt real axis Pressure Figure 2. Pasor relationsip for te regenerator In order to explore te limits of te orrelating funtions disussed in tis paper, te funtion is generated at a partiular base ase and ten used over a range of operating onditions. Te result predited by te orrelating funtion is ompared wit te result obtained from running REGENv3.2 diretly. No base ase is needed to aomplis te pasor analysis; owever, Table 1 provides te range of onditions used to evaluate its auray. Figs. 3(a) and 3(b) sow te ot end mass flow rate and pase angle, respetively, predited using pasor analysis as a funtion of te same quantity predited diretly using REGEN. Notie tat under most operating onditions te pasor analysis agrees wit REGEN to witin ± 15%. Te larger deviations our wen te pressure drop aross te regenerator, wi was negleted in te pasor analysis, beomes large. Tis ondition is not typial of a well-defined regenerator. Table 1. Te range of te geometry and operating onditions used to generate Fig. 3 Parameter Range Mean pressure (MPa), P 2 to 3 MPa Pressure ratio, PR 1.1 to 1.3 Mass flow rate at te old end, 0.1 to 0.5 kg/s Pase at te old end, θ -10 to -80 Porosity, φ 0.5 to 0.8 Frequeny, f 30 to 60 Hz Lengt, L r 2 to 13 m Area, A r 100 to 500 m 2 Matrix type Sreen Matrix material Stainless steel më (from pasor analysis) % -15% Units: kg/s më (from REGEN) (a) % -15% Units: Degree Pase Angle (from REGEN) Figure 3. Comparison of (a) te ma plitude, and (b) te pase angle at te ot end of te regenerator predited by te pasor analysis as a a funtion of te same quantity predited diretly from REGENv3.2 Pase Angle (from pasor anal.) (b)

5 DIMENSIONLESS ANALYSIS FOR REGENERATOR DESIGN 423 Te average mass flow rate witin te regenerator an be estimated using te pasor analysis and provides an input to te orrelating funtions for te termal and pressure loss. Te mass flow rate at any point along te regenerator an be represented as a pasor and te amplitude of te average mass flow rate witin te regenerator is obtained by integrating tis pasor in pase spae from te old end to te ot end of te regenerator: 1 θ avg = θ θ θ dθ (9) were te amplitude of te mass flow rate is a funtion of pase angle (Fig. 2): m osθ =! (10) osθ Substituting Eq. (10) into Eq. (9) leads to: arrying out te integration leads to: 1 θ osθ avg = dθ θ θ (11) θ osθ mos m =! θ! avg ( ln(seθ tan θ ) ln(seθ tan )) θ θ + + θ (12) Tis average mass flow rate provides a good referene tat is used in te subsequent setions. avg REGENERATOR THERMAL LOSS Te output EHTFLX is te total rate of termal loss predited by REGENv3.2 and inludes te eat loss aused by te ineffetiveness of te regenerator and te eat ondution troug te matrix. Tis termal loss must be deduted from te available aousti power in te pulse tube wen alulating te net ooling power. Te termal performane of te regenerator is omplex and influene by fators su as te property variation of te matrix and te working fluid, dispersion and ondution, pressurization losses, et. Te value EHTFLX learly annot be predited in any way oter tan te use of a sopistiated numerial model tat solves te oupled mass momentum and energy equations. However, over a small region of operating onditions it is expeted tat te dimensionless termal performane of a regenerator an be orrelated using a small set of dimensionless numbers. Te dimensionless parameters annot be alulated preisely but an be determined approximately using te known inputs; tis idea forms te basis of te orrelating funtion for te regenerator termal loss. Te dimensionless termal loss is te ineffetiveness, wi is defined as te ratio of te termal loss to te total eat transferred in te matrix (i.e., te energy required bring te average mass flow from T to T ): EHTFLX ineff = (13) avg p ( T T ) were p is te eat apaity of te working gas, T is te temperature of te warm end and T is te temperature of te old end. Note tat p, as well as te oter properties required to implement te orrelating funtions, are alulated at te mass average temperature, Eq. (6), and mean pressure. Te independent dimensionless numbers tat orrelate te ineffetiveness are expeted to be te apaity ratio (CR) and te number of transfer units (NTU). Te apaity ratio is te ratio of te apaity of te flow troug te regenerator to te apaity of te regenerator matrix. Te number of transfer units reflets te ratio of te total ondutane of te matrix to te apaity of te flow troug te regenerator. As alulated below, tese definitions are only approximately orret sine te properties and oter arateristis are only estimates evaluated at te mean temperature and some average operating ondition:

6 424 REGENERATOR MODELING AND PERFORMANCE INVESTIGATION avg f CR = L A ρ (1 φ ) r r m m knual r r 1 NTU = d d avg p p (14) (15) were ρ m is te matrix density, m is te matrix eat apaity, φ is te matrix porosity, L r, and A r are te lengt and area of te regenerator, respetively, k is te ondutivity of te gas (evaluated at te average temperature), d is te ydrauli diameter of te matrix and Nu is te Nusselt number araterizing eat transfer between te matrix and te gas. Te Nusselt number is alulated aording to te Reynolds number and Prandtl number aording to [4]: Nu = 0.68Re Pr (16) mavg d Re =! µ Ar φ (17) were Re is te Reynolds number, and Pr is te Prandtl number, µ is te dynami visosity at te average temperature. Note tat CR and NTU, as defined by Eqs. (14) and (15), an be alulated for an arbitrary set of operating onditions and geometry witout running REGENv3.2. Te form of te orrelating funtion tat relates te dependent parameter ineff to te independent parameters NTU and CR is: b+ NTU ineff = exp( a + ) (18) CR NTU were a, b, and are undetermined oeffiients based on running REGENv3.2 in a ontrolled fasion for ases tat are small perturbations relative to te base ase. Beause tere are 3 unknown oeffiients, tree ases must be run using REGENv3.2 to determine te oeffiients, a, b and. Tese ases inlude te base ase as well as two additional ases. Table 2 gives te geometries and te operating onditions used in te base ase. As an example of tis proedure, a first set of values for NTU and CR are determined from te geometry and onditions defined in Table 2 for te base ase via Eq.s (14) and (15) and te ineffetiveness for te same set of parameters is determined by REGENv3.2. A seond set of NTU and CR values are osen (somewat arbitrarily) by dereasing NTU by 20% wile inreasing CR by 40%. Eq.s (14) and (15) are ten used to bak out values for area A r and lengt L r. Tese new values along wit te oter parameters from Table 2 are used as input to REGENv3.2 to alulate te assoiated ineffetiveness for tis seond ase. A similar metod wit a tird ase were now NTU is inreased by 20% ompared to te base ase, and CR is dereased by 40% ompared to te base ase, provides a tird set of A r and L r values from wi a tird value of ineffetiveness is obtained via REGENv3.2. Finally, te tree oeffiients in Eq. (18) an be determined from tese tree sets of NTU, CR, and ineffetiveness values. Table 2. Te geometry and operating onditions used in te base ase Parameter Range Mean pressure (MPa), P 2.5 MPa Pressure ratio, PR 1.2 Mass flow rate at te old end, kg/s Pase at te old end, θ -45 Porosity, φ Frequeny, f 45 Hz Lengt, L r 2 m Area, A r 300 m 2 Matrix type Sreen Matrix material Stainless steel

7 DIMENSIONLESS ANALYSIS FOR REGENERATOR DESIGN ineff (from te orrelation) % -15% ineff (from RGEN) Figure 4 Verify te auray of te orrelation wen te operating onditions ange Additional ases ave also been run to verify te auray of te orrelation wit te oeffiients given by te base ase and te two additional ases. Te ranges witin wi te parameters listed in Table 1 ave been varied to test te orrelation are also defined in Table 1.. Te value of EHTFLX for all tese ases is determined by REGENv3.2, and ten onverted to te assoiated value of ineffetiveness via Eq. (13). Te values of te ineffetiveness determined in tis way define te x-axis in Fig. 4. Te ineffetiveness values defining te y axis, are determined via Eq. (18) from teir respetive vales of NTU and CR and te previously determined oeffiients a, b, and.. Te grap demonstrates tat te orrelation an be used to predit te ineffetiveness to witin 15% of te REGENv3.2 value for 95% of te ases. REGENERATOR PRESSURE DROP Te pressure drop aross te regenerator an be orrelated most onveniently in terms of a frition fator (f), wi is a dimensionless parameter defined aording to: 2 Pd ρ φ A r f = (19) 2L r avg were ρ is te gas density at te average temperature and pressure. Te frition fator is assumed a funtion of Reynolds number, wi was defined in Eq. (17). Te orrelating funtion tat relates te frition fator to te Reynolds number is: e f = d + (20) Re g were d, e and g are undetermined oeffiients. In a similar fasion as wit te ineffetiveness alulation, tree ases must be run using REGENv3.2 to determine te oeffiients, a, b and.in Eq. (20). Again, Table 2 provides te geometries and operating onditions used in te base ase. REGENv3.2 is used to determine te pressure drop for ea of te tree ases, and Eq. (19) is used to alulate te assoiated frition fator. Wit te known frition fators and Reynolds numbers, te oeffiients d, e and g are determined by te tree speifi versions of Eq. (20). Additional ases ave also been run to verify te auray of te orrelation as ompared to te preise solution from REGENv3.2.. Te ranges witin wi te parameters listed in Table 1 ave been varied to test te orrelation are also defined in Table 1. Te omparison is sown in Fig. 5 were te pressure drop defining te x-axis is determined by REGENv3.2, wile tat defining te y-axis is determined from Eq.s (20) and (19) for te predetermined values of d, e, and g.. Te grap demonstrates tat te orrelation predits te frition fator to witin 15% of

8 426 REGENERATOR MODELING AND PERFORMANCE INVESTIGATION Pressure Drop (from predition), Pa % -15% Pressure Drop (from REGEN), Pa Figure 5 Pressure drop from REGEN vs. pressure drop from te predition of te orrelations te value provided by REGENv3.2. Te large disagreement ours wen te porosity is different tan te base ase. DISCUSSION AND CONCLUSIONS REGENv3.2 provides a powerful and aurate numerial model for regenerator design. However, in some ases an exessive amount of time is required to obtain useful results. An alternative metod, utilizing a ombination of pasor analysis and dimensionless orrelations an be used to greatly sorten te alulation time for regenerator designs. Te pasor analysis is used to alulate te mass flow rate and pase wit respet to pressure at te warm end, and predits tese to witin 15% of te values preisely determined by REGENv3.2. Correlations ave also been developed to determine te ineffetiveness and pressure drop losses of te regenerator. Te orrelations require running tree known ases using REGENv3.2 to determine te oeffiients for eiter te ineffetiveness or te pressure drop. One te oeffiients are determined, te orrelations an be used to estimate te ineffetiveness or pressure drop to witin 15% of te values preisely determined by REGENv3.2. Furtermore, te orrelations are useful over a fairly large range in a variety of parameters inluding te average pressure, pressure ratio, frequeny, porosity, and old end mass flow rate, pase angle, and temperature. From te above disussion, we may onlude tat te pasor analysis an be used as an aurate meanism to alulate te mass flow rate and pase at te warm end of te regenerator. Te dimensionless orrelations an also be used to alulate te termal loss and pressure drop aross te regenerator. One te orrelations are known, tey an also be used for design alulations. Te primary advantage of tis metod is tat it will sorten te required alulating time for design studies as ompared wit tose required by te full use of REGENv3.2, and yet maintain an aeptable auray. ACKNOWLEDGMENT Tis work was supported by a subontrat (NNJ04JA20C) from Atlas Sientifi. REFERENCES 1. Gary, J., O Gallager,A. Radebaug, R. and Marquardt, E., REGEN3.2 User Manual, NIST, (2001).

9 DIMENSIONLESS ANALYSIS FOR REGENERATOR DESIGN Pfotenauer, J. M. Si, J. L. and Nellis, G. F., A Parametri Optimization of a Single Stage Regenerator Using REGEN 3.2, Cryooolers 13, Springer Siene & Business Media, New York (2004), pp Potratz, S., Design and test of a ig apaity pulse-tube, 2005, p Akermann, R.A. Cryogeni regenerative eat exangers, Plenum press, New York, 1997, p. 124.