Methods for transfer matrix evaluation applied to thermoacoustics

Transcription

1 Proceeding of the Acoutic 2012 Nante Conference April 2012, Nante, France Method for tranfer matrix evaluation applied to thermoacoutic F. Bannart, G. Penelet, P. Lotton and J.-P. Dalmont Laboratoire d acoutique de l univerité du Maine, Bât. IAM - UFR Science Avenue Olivier Meiaen Le Man Cedex 9 flavio.bannart@univ-leman.fr 3029

2 23-27 April 2012, Nante, France Proceeding of the Acoutic 2012 Nante Conference The deign of a thermoacoutic (TA) prime-mover partly relie on the knoledge of it onet condition, i.e. the reonance frequency of the elf-utained ocillation and the minimum heat poer upply hich i neceary for the phenomenon outbreak. The onet condition can be calculated once the tranfer matrix of the TA core i determined, hichever by analytical modeling or acoutic meaurement. The latter, hoever, conit of an intereting option to avoid thermophyical or geometrical conideration of complex tructure, mainly ith repect to the TA regenerator, a the TA core i treated a a black box. To experimental method are conidered in thi ord: the To-Load Method and the Impedance Method. Thee method are here preented, dicued and compared on everal apect, taking into account imulated and experimental reult for a tack and a regenerator both ubmitted to a teep temperature gradient. Regarding thee apect of invetigation, the To-Load Method a proved to be the bet for the tack, meanhile the Impedance Method for the regenerator. In the econd cae, important experimental difficultie arie due to the regenerator intrinic propertie. Therefore, the Impedance Method a conceived toard the invetigation into thi pecific problematic and ucceeded on it purpoe. 1 Introduction Deigning a thermoacoutic (TA) machine lean on the knoledge of it onet condition, i.e. heating poer upply Q and acoutic frequency f, hich generate the required thermoacoutic intability in it core to make it ork. Such condition can be calculated if the Tranfer Matrix (T) of the TA core i available. The TA core i defined a the region here the temperature gradient i not null. It T matrix can be obtained experimentally and, aiming at thi goal, to method have been invetigated: the To-Load Method and the ne Impedance Method. In both cae the TA core i treated a a black box. Thu, by mean of acoutical meaurement only, it T matrix can be obtained a an acoutical to-port ytem, no mattering ho complex are the geometrical or thermophyical propertie of the TA core. Therefore, it i avoided the need of analytical modeling. The thermoacoutic phenomenon take place ithin the porou material, hich play the main role inide the TA core. Depending on it average pore ize and poroity, the material may be claified a a tack or regenerator, determining o it application, either tanding-ave TA engine or traveling-ave one, repectively [3]. For the firt cae the average pore diameter i required to reach a fe thicknee of the fluid thermal boundary layer performed during the acoutic ocillation nearby the olid all, in order to etablih an eentially irreverible heat tranfer in beteen them [5]. For the regenerator, meanhile, the quet i for a reverible thermal interaction [4]. That demand pore a mall a poible, regarding other combined contraint of the regenerator, a it poroity and axial length, hich mut allo the ound ave pa through. The To Load Method [1] conit of a procedure of preure meaurement here a claical 4 microphone method i applied tice, each time for one acoutic load condition, in order to obtain the 4 coefficient of the T matrix from meaured tranfer function beteen the microphone. The Impedance Method, on the other hand, conit of obtaining T from the meaurement of the Impedance Matrix of the TA core by imply applying the proper analytical expreion that interrelate them. The impedance meaurement, for intance, are made by mean of a pecific device called Acoutic Impedance Senor [2]. The TA core here conidered, hoever, i an aymmetrical ytem; and thi apect preuppoe to impedance meaurement to fulfill the matrix. It can be applied a to acoutic load technique, a in the previou method, but, in our cae, thee to meaurement are differentiated on the experimental configuration by taking each one in a different axial ene, tated a direct ene and inverted one. Independently on the method to be applied, in order to characterize the TA core thermal apect related to T, everal thermal gradient condition hall be impoed along the TA core longitudinal axi, hich may be provided by a ytem of arbitrary heat poer upply Q combined ith cooling heat exchanger. Once the teady tate regime i reached for each Q, a correponding T ( Q, f ) can be calculated afterard, therefore etablihing in the final reulting T the coupled thermal and acoutical behavior of the TA core, implicitly. A tack made of ceramic and a regenerator contituted of a grid of teel pile are teted ith both method. The experimental reult obtained are compared mutually and alo ith repect to theoretical reult ith imulated noie and bia. Beide, to foam are alo teted ith the Impedance Method: one made of a Ni-Cr alloy and the other made of Reticulated Vitreou Carbon (RVC). 2 The To-Load Method A chematic draing of the experimental apparatu ued for the determination of the T matrix of the TA core i hon in the Figure (1a). The TA core under tudy i hon in Figure (1b). Figure 1: Experimental apparatu - from [1]. A temperature gradient i applied along the tack/regenerator by mean of 3 heat exchanger: a hot 3030

3 Proceeding of the Acoutic 2012 Nante Conference April 2012, Nante, France exchanger (d) hich upplie a heat poer Q to the ytem, a cold exchanger (b) hich enure that the left ide of the tack i maintained at room temperature, and a econd cold exchanger (f) i ued to control the temperature ditribution along the thermal buffer tube (e). In order to characterize the propagation of acoutic ave through the TA core, to traight duct are connected to it end. The duct on the left i connected to an electrodynamic loudpeaker, hile the duct on the right i either open to free pace or cloed ith a rigid all. To pair of microphone are fluh mounted along each duct at both ide of the TA core. Auming that an harmonic plane ave i propagating in the device at the angular frequency ω, the acoutic preure p ( x, and the acoutic volume velocity U ( x, are ritten a belo: a) Ceramic Stack. jϖt ζ ( x, = R[ ζ ( x) e ], (1) here ζ repreent either p or U, the accent indicate the complex amplitude and R [] denote the real part of a complex number. With thi aumption, the complete apparatu can be ued to meaure the complex amplitude of acoutic preure p ( x l,r ) and the acoutic volume velocitie U ( x l,r ) at the left ide and right ide of the to-port, repectively, and thu to deduce the T matrix uing the folloing equation: p ( xr ) p ( xl ) = T U ( xr ) U ( xl ) T = T pp up T T pu uu p ( xl ). (2) U ( xl ) The meaurement method ued here to obtain T i a claical to-load, four-microphone method [6,7]. The baic principle of thi method conit of calculating p ( x l,r ) and U ( x l,r ) from the meaurement of acoutic preure at poition x i [i=1,2,3,4, ee Fig.1(a)]. A et of to meaurement i hoever required to obtain the four coefficient T pp, T pu, T up, T uu of the T matrix from the meaured acoutic preure p ( x i ), and thi i realized here by uing to different load, the firt ith the open duct, and the econd ith the duct cloed by a rigid all. The T matrix meaurement are taken in the frequency range from 50z to 200z, and they follo 28 different value of heat poer upply, here Q varie from 0W to 81W ith a contant increment of 3W. 2.1 Experimental Reult The Figure (2) preent the behavior of the T pp coefficient, choen a an arbitrary example, from 50 z to 200 z and for 3 different value of heat poer upply Q : 0 W, 36 W and 72 W, arbitrarily choen a ell. b) Grid of Inox Pile Figure 2: Module (olid line) and phae (dotted line) of Tpp in the frequency domain for Q = 0W (o), Q = 36W (quare) and Q = 72W (diamond) To-Load Method. When there i no heat upply into the acoutic ytem, it i expected to have an amplitude of the average T determinant being approximately 1, ith average phae cloe to 0 rad. Thee condition characterize a reciprocal ytem and tate for a good meaurement quality. The reciprocity for the Ceramic Stack reulted in the folloing value: Amplitude average: 1.010; tandard deviation: 3.3e-3. Phae average(rad): 1.9e-3; tandard deviation: 2.4e-3. Therefore, the T matrix may be conidered reciprocal, and thee reult reciprocity prove a quite good repeatability ith repect to a previou ork [1], contributing o to the conitency of the To-Load Method, for thi kind of material (tack). oever, thee reciprocity apect reulted inufficient good for the Grid of Steel Pile: Amplitude average: ; tandard deviation: Phae average(rad): 1.9e-3; tandard deviation: Even though very good on the average, the tandard deviation reulted poor. It i a conequence of a high level of noie in T, in pite of applying the ame ell controlled experimental procedure. Thi effect i confirmed in a theoretical enitivity analyi for a regenerator, here arbitrary noie and bia are introduced to imulated tranfer function. When applying the ame analyi to the tack, the imulation reulted in lo enitivity to both parameter, coherently ith the experimental behavior. Thi contributed hence toard the confidence on thi analyi application to other poible olution for thi problem, before proceeding ith another experimentation. That ha leaded to the folloing method. 3031

4 23-27 April 2012, Nante, France 3 The Impedance Method The Impedance Method ha been developed a an attempt to experimentally obtain the T matrix of a regenerator TA core. Thi choice came out from our effort to overcome the limitation of the previou method, hich indeed became better undertood after a theoretical enitivity analyi. ence, before tarting ith the econd method engagement, it a done a enitivity analyi for the Impedance Method a ell, at the ame arbitrary level of imulated noie and bia previouly tated for the firt method cae, in order to evaluate the perpective prediction. 3.1 Method Definition The T coefficient, tated jut a in the definition (2), are calculated from the to-port equation that relate them to the coefficient, the latter hich are actually meaured uing the Acoutic Impedance Senor. The matrix i defined a folloing: p p U = U = U U. (3) The coefficient are obtained from experimental meaurement of ound preure uing a pecific device coupled to the TA core under tet. Thi device i called Impedance Senor and it comprie three microphone, a piezo-electric buzzer and the houing itelf. The figure 1 ho the etup cheme of uch Impedance Senor hen coupled to the TA core and the adaptive part. The firt meaurement i made in the direct ene, hile the econd meaurement i made in the inverted one, a hon in the repective Figure 3 and 4. Mic 1 PE Buzzer Mic 2 Ac. Imp.Senor Ref. Plane Adapt. Part Figure 4: Impedance Senor coupled ith the TA core Invere Sene. 3.2 Method Application In order to take into account the microphone poition and the Reference Plane, the 11 and coefficient are generally expreed a folloing [2]: 11 Proceeding of the Acoutic 2012 Nante Conference Regenerator or Stack Qh TA Core ( K) β =, (4) 1 ( δ K) 1+ δ ) =. (5) 31( 11 δk T Mic 3 Mic 1 PE Buzzer Mic 2 Ref. Plane Regenerator or Stack Qh For 22 and 12 are applied the ame equation (4) and (5), in correpondence. The term are tranfer function that relate the ound preure meaurement after the calibration: Mic 3 p 2 2 = and p1 1 p =, (6) p1 1 Ac. Imp.Senor Adapt. Part TA Core Figure 3: Impedance Senor coupled ith the TA core Direct Sene. The piezo-electric buzzer i loaded by a mall cavity on it rear face and it radiate toard a duct on it front face. The microphone 1 meaure the preure p1( x, inide the mall cavity, hile the microphone 2 meaure the preure p 2( x, in the duct, before the entrance of the Adaptive Part. From p 1 and p 2, and taking into account geometrical conideration, the acoutic impedance 11 can be calculated at the Reference Plane. On the oppoite extremity, encloing the ytem under tet, the tranfer impedance can be calculated hen conidering alo the preure meaurement at the poition, here the microphone 3 i placed. Therefore, the ytem to be meaured i defined a the region beteen and. A et of 2 meaurement i required to obtain the 4 coefficient of the matrix. In one of thee it i realized inverion of TA core. here the term are the microphone correponding enitivitie. The term β, δ, K and K T are parameter related to the geometry and the calibration of the device (ee ref. [2] for more detail). The meaurement of demand a previou calibration for each experimentation. All meaurement folloed the pattern adopted into the previou method, ith an input poer upplied ranging beteen 0W and 81W ith a contant increment of 3W 3.3 Calibration The procedure i, at firt, cloing the device ith a rigid plate that enure a good approximation to a infinite impedance. Thi rigid plate, neverthele, ha an orifice hich encloe the third microphone. The tranfer function and 31 are meaured and here labeled ith the ubcript cal ever ince, to be ued later for the all et of remaining meaurement. A a conequence of uch configuration, it reult that the impedance at the Reference Plane i the ame a the 3032

5 Proceeding of the Acoutic 2012 Nante Conference tranfer impedance to the microphone 3, due to the fact that the poition i coincident. Further development lead to: K δ cal = and KT = 31 cal (7) Therefore, a a final reult of the calibration, the parameter K and K T are obtained directly from the meaurement of the tranfer function and both ratio of enitivity 2 1 and 3 1 are implicitly taken into account. Once uch parameter are available and the calibration tranfer function are regitered, one can realize the folloing tep, hich concern the impedance meaurement itelf. 3.4 Impedance Meaurement The parameter K and K T can be replaced by (7) into the equation (4) and (5). Therefore, the calibration i conidered in the determination of. Thee equation provide a imple manner to calculate the matrix directly from the tranfer function, from the geometrical parameter, and from the ga thermophyical propertie. The ample here teted are exactly the ame of the previou method. Beide, to other ample are teted for the Impedance Method: a foam made of a Ni-Cr alloy and another made of RVC. 3.5 Experimental Reult The Figure (5) preent the behavior of the T pp coefficient in the ame ay a in the Figure (2), for the ame 3 different value of heat poer upply Q. a) Ceramic Stack b) Grid of Inox Pile Figure 5: Module (olid line) and phae (dotted line) of Tpp in the frequency domain for Q = 0W (o), Q = 36W (quare) and Q = 72W (diamond) Impedance Method. In analogy ith previou method, the reciprocitie are here preented, for Q = 0W either ay. For the Ceramic Stack it follo: Amplitude average: ; tandard deviation: 4.5e-3. Phae average(rad): -1.4e-3; tandard deviation: 3.6e-3. Thee reult approache the reciprocity almot likely the To-Load Method, but not a much a it. The enitivity to bia reulted lightly bigger in the enitivity imulation, coherently. For the Grid of Steel Pile regenerator good reult are achieved, a een belo: Amplitude average: ; tandard deviation: Phae average(rad): 9.0e-3; tandard deviation: 7.2e-3. With repect to the other ample, the value of reciprocity are preented too. Ni-Cr alloy: Amplitude average: ; tandard deviation: Phae average(rad): ; tandard deviation: RVC: Amplitude average: ; tandard deviation: Phae average(rad): ; tandard deviation: Under an overvie, thee reult indicate that the Impedance Method eam to be robut no matter the material to be characterized inide the TA core, excepting for the foam of Ni-Cr alloy, hich call for additional invetigation. 4 Concluion and Perpective We have preented a ne method for the tranfer matrix meaurement of a thermoacoutic core, either regenerator or tack. We have hoed that thi ne method, contrarily to the To-Load Method, i accomplihing independently of the material under tet, primarily if dealing ith one that reemble a regenerator. Future ork ill be devoted to confirm the validation of the Impedance Method, and to the exploration of it reult on the goal of the dimenioning of thermoacoutic machine. Reference April 2012, Nante, France [1] M. Guedra, G. Penelet, P. Lotton, J.-P. Dalmont, Theoretical prediction of the onet of thermoacoutic intability from the experimental tranfer matrix of a thermoacoutic core, J. Acout. Soc. Am. 130(1), (2011). [2] C.A. Macaluo, J.P. Dalmont, Trumpet ith nearperfect harmonicity: Deign and acoutic reult,, J. Acout. Soc. Am. 129(1), (2011). [3] G.W. Sift, Thermoacoutic A unifying perpective for ome engine and refrigerator, J. Acout. Soc. Am. 113, (2003). [4] M.E.. Tijani, Loudpeaker-driven thermo-acoitic refrigeration PhD Thei, Techniche Univeriteit Eindhoven (2001) [5] G.W. Sift, Thermoacoutic engine, J. Acout. Soc. Am. 84(4), (1988). 3033

6 23-27 April 2012, Nante, France Proceeding of the Acoutic 2012 Nante Conference [6]. Böden, M. Abom, Influence of error on the tomicrophone method for meauring acoutic propertie in duct, J. Acout. Soc. Am. 79(2), (1986). [7] M.L. Munjal, Acoutic Duct and Muffler With Application to Exhaut and Ventilation Sytem Deign, Wiley-intercience, Ne York, 352 page (1987) 3034