Deign Guideline For a Theroacoutic Refrigerator Ra C. Dhuley, M.D. Atrey Deartent of Mechanical Engineering Indian Intitute of Technology Bobay, Powai Mubai-400 076 Develoent of refrigerator baed on Theroacoutic technology i a novel olution to the reent day need of cooling without cauing environental hazard. With added advantage of abence of oving art and circulating refrigerant, thee device can attain very low teerature aintaining a coact ize. The reent theoretical work i baed on theory of linear theroacoutic[1]. Under the hort tack and invicid aution, an algorith for deign of a tanding wave theroacoutic refrigerator, with ain focu on the tack, i decribed. A tack i deigned for a given cooling requireent of the refrigerator and certain choen oeration araeter. Key Word: deign, theroacoutic, refrigeration Noenclature reure (N -2 ) Q c cooling ower (W) T teerature (K) W acoutic ower (W) f frequency (Hz) COP coefficient of erforance ρ denity (kg -3 ) L length of reonator () a ound eed ( -1 ) L length of tack () λ wavelength () A cro ection area of tack ( 2 ) k wave nuber ( -1 ) l half late thickne () ω angular frequency (rad -1 ) y 0 half late acing () γ ratio of ecific heat П late erieter () δ k theral enetration deth () x tack centre oition () β theral exanion coefficient (k -1 ) K theral conductivity (W -1 k -1 ) Subcrit C iobaric ecific heat (J kg -1 k -1 ) ean Γ noralized teerature a alitude gradient 1 local alitude ecific entroy (J kg -1 k -1 ) er late
INTRODUCTION Theroacoutic i the interaction between heat and ound. Of the everal alication[1,2] of theroacoutic in energy converion, the theroacoutic refrigerator i a well known device. A theroacoutic refrigerator ue the energy of ound or reure wave to bring about refrigeration. The ain coonent of a theroacoutic refrigerator are the reonator, the tack, the acoutic driver and the heat exchanger a hown below in Figure 1. OPERATING AND DESIGN PARAMETERS[6] The ai of the reent analyi i to develo a deign rocedure for a theroacoutic refrigerator and to deterine the deign araeter baed on certain given oerating araeter. Variou oerating and deign araeter are given in Table 1. The roertie of working ga like denity, theral conductivity, ratio of iobaric to iochoric ecific heat and eed of ound in ga lay an iortant art in thi analyi. The working ga hould have a low boiling oint, high ound eed and hould be inert to the coonent of the refrigerator. Table 1 Oerating and Deign araeter Figure 1 Scheatic of theroacoutic refrigerator The tanding reure wave generated by acoutic driver in the reonator caue to-and-fro otion of the ga fro one end of the tack to the other thereby roviding a ean of tranorting heat. The ga near the reure node get cooled due to rarefaction and ick u heat fro the tack aking one end of tack cold. Thi ga ove toward reure antinode and get heated u due to coreion. It loe heat to the tack and ake the other end of tack hot. Thu, a teerature gradient i et u along the tack length. With heat exchanger at the two end of the tack, thi teerature gradient can be ued to roduce refrigeration. A detailed decrition of theroacoutic cooling cycle ha been given by Arnott et. al[3]. Braun et. al[4] have develoed a deign otiization rogra baed on the theroacoutic iulation rogra known a DELTAE develoed by Swift[1]. A deign otiization rocedure baed on noralization of araeter ha been given by Wetzel et. al[5] and Tijani et. al[6] Oerating araeter ean reure, ean teerature, T frequency, f reure alitude, a ASSUMPTIONS[1,6] Deign Paraeter reonator length, L tack length, L tack cro ection area, A late thickne, 2l late acing, 2y 0 tack centre oition, x Following are the aution ade in the reent analyi 1. The length of the tack i uch aller a coared to the wavelength of the tanding wave L <<λ (Short tack aroxiation). It can be aued that the local reure and velocity alitude of the ocillating ga olecule i ore or le ae over the entire length of the tack. 2. The flow of the ga in the tack i aued to be invicid. Friction at the inner wall of the reonator i alo neglected.
3. The teerature difference acro the tack end i aued to be all a coared to the ean teerature T <<T. It can be aued that the thero-hyical roertie of the ga do not vary ignificantly over the tack length and hence can be aued contant. 4. The whole analyi i carried out at teady tate oeration of the refrigerator. The ean teerature of the ga i T and the teerature gradient acro the tack reain contant with tie. 5. Conductivity of late aterial i neglected. GOVERNING EQUATIONS[1] The equation governing the entire analyi i the general heat tranfer equation exreed in ter of entroy tranort. Neglecting vicoity of the ga, it can be written a where k=2π/λ. On account of the oroity B of the tack, eqn.3 get odified to, a u1 in( kx) ab (4) The critical teerature gradient i an iortant araeter governing the refrigeration action of a theroacoutic device. It i given by, T crit ( 1) kb cot( kx) (5) where β i the theral exanion coefficient of the ga. For ideal ga, it i equal to the invere of the abolute teerature. Of utot concern in the deign rocedure are the cooling ower roduced by the tack, the work inut needed to roduce thi cooling ower and thu the COP. Thee are given by the following exreion :- T( v ) ( KT ) t (1) Q k T ( 1) 1u1 4 (6) where i ecific entroy. The equation tate that the rate of change of aount of heat at a certain oint deend on convective flow of heat by ean of entroy and conduction of heat. Variou other exreion[1] relevant to reent analyi uch a thoe for the cooling ower and the acoutic ower can be derived fro eq(1). Referring to Figure 2, the acoutic driver roduce a region of axiu reure variation at x=0 i.e a reure antinode and velocity node. Subequently, following equation can be written down for local reure and velocity alitude in the reonator :- 1 a co( kx) a u1 in( kx) a (2) (3) W 2 2 k LT ( 1 ) ( 1) 4 C T a COP tan( kx) BL ( 1) (7) (8) where, δ k i the theral enetration deth of the ga and i an iortant araeter governing the late acing of the tack. It i defined a the ditance through which heat diffue in the ga in tie 1/ω and i given by k 2K C (9) Π i the erieter of the cro ection of the late. Auing the thickne of the late to
be very all, the erieter i equal to twice the late width (Π=2w). Γ i the ratio of actual teerature gradient to the critical teerature gradient of the tack. region within δ k fro the late and hence the late acing i choen to be 2δ k [8]. Hence, 2y 0 =0.212. DESIGN PROCEDURE Thi ection decribe the deign rocedure of a refrigerator roducing a cooling ower of 4 W and a cold end teerature of 210 K. The oerating araeter choen are hown in Table 2[6]. Table 2 Oerating and working ga araeter Oerating araeter Heliu roertie =10 bar ρ =1.9244 kg/ 3 T =250 K a=935 / f=400 Hz k=0.14 W/K a =0.2 bar T h =283 K γ=1.67 Deterining the deign araeter Reonator length The allet oible length of the reonator which will roduce a tanding wave i equal to quarter of the wavelength. A tanding wave can be generated by keeing one end of reonator cloed (driver end) and iulating the other end a oen. Thi can be done by attaching a ufficiently large buffer[6,7] of ga at T at thi end. For thi ot fundaental cae we get L=58.8 c. Plate acing The theral enetration deth of the ga at the given frequency i found out to be 0.106. Figure 3 how variation of heat flux with ditance fro the late. It can be een that alot all the tranort of heat take lace in Figure 3 Heat tranort v. ditance fro late. Plate thickne The late thickne can be found out fro the exreion 1 2 2 ( 1) l y0 B (10) where B i the oroity of the tack. The oroity i defined a the ratio of oen area in the cro ection to the total cro ection area of the tack. The oroity i choen uch a not to diturb the acoutic tanding wave ignificantly and i norally taken in the range 0.7-0.8[5,6]. Chooing B=0.75, we get 2l=0.07. Stack length, centre oition and area of cro ection A grahical aroach ha been ileented to deterine the tack length, centre oition and the cro ection area. It i aued that the tack i ade of n arallel late over one another and the cro ection of the tack i a quare of width w. In thi cae, the erieter of the late cro ection becoe 4nl+4(n-1)y 0. Knowing the fact that total cooling ower roduced by
the tack i n tie that roduced by a ingle late, it can be written Q [ ( (11) c n n l y0) y0] [ T 1] k u 1 1 L T crit obtained. The cro ection area of thi tack coe out to be 7.06 c 2. Thi tack will roduce a cooling ower of 4 W at COP cloe to 2. Putting the known quantitie in eq(11), an equation in three variable x, L and n i obtained, 2 0.282n 0.212 n 509.54 0 tan( kx) in(2 kx)(0.221 1) L (12) Siilarly, fro eq(8) we get, tan( kx) COP 0.75 L (13) Figure 5 Variation of nuber of late with tack length. COP and n are lotted with L for different oition of tack centre in the reonator. Theoretically, every oint on uch a curve will give u a tack roducing a cooling ower of 4 W and a cold end teerature 210 K. Figure 4 and Figure 5 how variation of COP and n reectively with L at different tack centre location. Figure 4 Variation of COP with tack length For intance, if L =0.1 i choen at x =0.1, a tack with 95 arallel late i RESULTS AND DISCUSSIONS The deign araeter obtained fro the analyi are hown in Table 3 Paraeter Table 3 Reult Value L 588 L 100 A 706 2 2l 0.07 2y 0 0.212 100 x The tack length (L =10 c) i very all a coared to the wavelength of the acoutic field (λ=235.2 c). Hence, the aution of hort tack i erfectly valid. At the tack centre location (x =10 c) the critical teerature gradient i 1235.13 K/. The actual teerature gradient acro the tack end i 750 K/. Clearly, Γ=0.607<1 which i riary neceity[1] for the device to function a a refrigerator, i et. Another erfectly valid choice of tack roducing the ae cooling ower at ae
cold end teerature i L =8.5 c and x =8c. In thi cae, the tack will conit of 102 late and a cro ection of 8.2 c 2. The correonding COP i 1.9. Tijani et al.[6] obtained a tack with cro ection 11.8 c 2 and COP of 1.5 with ae tack length and et of oeration araeter. The difference in the cro ection can be accounted for the author in [6] had taken vicoity into account and hence needed ore nuber of tack late to roduce 4 W ower. Referring to Figure 4, high value of COP (~3) are exected fro the analyi. But, it hould be noted that thi i COP of the tack and not of the refrigerator. Even when the vicoity i neglected, ore ower ha to be inut to account for loe at heat exchanger, driver, etc. (though not a art of thi analyi). cooling alication, International Journal of Refrigeration, 32, (2007), 1059-1071. 5. Wetzel M., Heran C., Deign Otiization of theroacoutic refrigerator, International Journal of Refrigeration, 20, (1997), 3-21. 6. Tijani M.E.H., Zeeger J.C.H., dewaele A.T.A.M.., Deign of theroacoutic refrigerator, Cryognenic, 42, (2002), 49-57 7. Tijani M.E.H., Zeeger J.C.H., dewaele A.T.A.M.., Contruction and working of a theroacoutic refrigerator, Cryognenic, 42, (2002), 59-66 8. Tijani M.E.H., Zeeger J.C.H., dewaele A.T.A.M., Otial tack acing for theroacoutic Refrigeration, J Acout Soc. A, 112(1), (2002), 128-133 CONCLUSIONS A ile deign rocedure for a tanding wave theroacoutic refrigerator ha been decribed. With a choice of oerating araeter and heliu a working ga, grahical aroach ha been ued to deterine the geoetrical araeter of the tack. A refrigerator roducing a cooling ower of 4W at cold end teerature of 210K at teady tate i deigned theoretically uing thi deign rocedure. Reference 1. Swift G.W., Theroacoutic engine, J Acout Soc. A, 84, (1998), 1146-1180 2. Swift G.W., Theroacoutic- A unifying erective of oe engine and refrigerator, Acoutical Society of Aerica Publication (2002). 3. Arnott W, Raet R, Ba H.E, Theroacoutic Engine, Ultraonic Syoiu, (1991), 995-1003. 4. Paek I.,Braun J.E., Mongeau L., Evaluation of tanding-wave theroacoutic cycle for