Isentropic Sound Waves Propagation in a Tube Filled with a Porous Media

Transcription

1 INTERNATIONAL JOURNAL OF ECHANICS Ientropic Sound Wave Propagation in a Tube Filled with a Porou edia H.. Duwairi Abtract A rigid frame, cylindrical capillary theory of ound propagation in porou media that include the nonlinear effect of the Forchheimer type i laid out by uing variational olution. It i hown that the five main parameter governing the propagation of ound wave in a fluid contained in rigid cylindrical tube filled with a aturated porou media are hear wave number, = R ρ ω / μ, reduced frequency parameter, k = wr a, poroity,, Darcy number, Da = R K, and Forchheimer number, C. The manner in which the flow = C F influence the attenuation and the phae velocitie of the forward and backward propagating ientropic acoutic wave i deduced. It i found that the incluion of the olid matrix increae wave attenuation and phae velocitie for both forward and backward ound wave, while increaing the poroity and the reduced frequency number decreaed attenuation and phae velocitie. The effect of the teady flow i found to decreae the attenuation and phae velocitie for forward ound wave and enhance them for the backward ound wave. T Key-Word - fluid flow, porou medium, ound wave I. INTRODUCTION he acoutic problem cover a wide range of practical problem. If the acoutic improvement are retricted to interior pace (building hall, theater, dwelling, factorie, vehicle cabin, etc.), uually mineral wool or open pore foam can be ued to olve the problem. For outdoor problem, for intant, acoutic noie barrier againt traffic noie, the aborption i provided by granular material uch a porou concrete or imilar material, a they behave better with bad weather and other atmopheric phenomena, a well a they can be cleaned (with a preurized water) without looing their acoutic propertie. In porou material uch a fibrou and granular, the aborption proce of the acoutic wave take place through vicoity and thermal loe of the acoutic energy inide the micro tube forming the material. The problem of a anucript received ay 9, 7: Revied verion received July, 7. Thi work i done during a abbatical leave year granted from the Univerity of Jordan to the author Dr. Hamzeh Duwairi at the German Jordanian Univerity H.. Duwairi, echanical Engineering Department, Faculty of Engineering & Technology, The Univerity of Jordan, 9 Amman, Jordan. (phone: ; fax: 9--; duwairi@ ju.edu.jo). propagation of ound wave in fluid contained in a plain medium i a claical one, to which famou name are connected like Helmholtz [], Kirchhoff [] and Rayleigh []. Since then many paper have been written on the ubject; often in relation to the tudied dealing with the dynamic repone of preure tranmiion line. A variational treatment of the problem of ound tranmiion in narrow tube i decribed by Cumming [] a an alternative to the more uual analytical procedure which i limited to mathematically tractable geometrie. A firt approximation to the effect of mean flow on ound propagation through cylindrical capillary tube i achieved by Peat []. A ound tranmiion in narrow pipe with uperimpoed uniform mean flow and acoutic modeling of automobile catalytic converter i done by Dokumaci []. A numerical tudy on the propagation of ound through capillary tube with mean flow i achieved alo by Jeong and Ih [7] and finally an approximate diperion equation for ound wave in a narrow pipe with ambient gradient i done by Dokumaci []. The problem of ound wave propagation in a tationary or flowing fluid in a porou medium i not addreed yet. An attempt i made in thi article to develop a implified nonlinear theory that predict the propagation characteritic of a tationary or flowing fluid in aturated porou media. Thi theory i an extenion of the claical plain medium theory, uing a modification to Darcy law due to the Forchheimer effect. Analytical expreion for the propagation contant are obtained from variational olution. Comparion with previou work in the limit of plain medium how an excellent agreement. II. PROBLE FORULATION Conider a rigid tube filled with a aturated porou material, the fluid i aumed to be a tationary or movable inide the tube. The x-coordinate i meaured along the tube and the r- coordinate i meaured normal to the axial direction. Under the boundary layer approximation the baic equation which govern acoutic wave propagation in a rigid tube filled with a porou media are the continuity and momentum equation: ρ ρ ρ ( v v u + u + v + ρ + + ) = () t x r r r x u u u p ρ + u + v = t x r x () μ u K CF ρ u / K u + μ r u + r r Iue, Volume, 7

2 p = () r Where u,v are the velocity component in the axial and normal direction; repectively. ρ and p are the fluid denity and preure, μ i the abolute vicoity, K i the permeability and i the poroity of the porou medium, i the Forchheimer coefficient. Further implification of C F the governing equation reult from the aumption that the radial velocity component, v, i zero. Since one i dealing only with capillary tube the radial velocity might be expected to be negligible, if not identically zero. The effect of thi i to decouple the continuity equation () from the momentum equation (). Next, it i aumed that the flow through the capillary duct i a uperpoition of a fully developed laminar, incompreible, axial teady flow and a mall harmonic acoutic diturbance of frequencyω. The teady flow i taken to have contant denity ρ and a peed of ound a uch that the fluid variable can be expanded in the form: ( ( ) ) Γξ iω ρ = ρ + αρ η e e t, ( ( ) ( ) ) Γξ iω u = a η + αu η e e t (-) Γξ iω v aαv( η) e e t Γξ iωt p = ρa γ ) p ( ξ ) + αp( η) e e (-7) =, ( ( ) Where α << and γ i the ratio of pecific heat. It i een that the teady flow variable p and ach number together with acoutic variable ρ, u, v and p are dimenionle. Now introduce the following variable in the tranformation: ξ = ωx / a, η = r / R () R i the radiu of the capillary duct. The axial acoutic wave motion ha been aumed to have complex propagation contant Γ which can be expanded a: Γ = Γ + i Γ (9) Where Γ repreent the wave attenuation per unit ditance and Γ repreent the phae hift over the ame ditance. The aumed form of the variable, equation (-7) are ubtituted into the governing equation () and ()-() and term of imilar order in α equated. It i found that for zeroth order, the teady flow olution, the equation of continuity and radial momentum are identically atified, while the axial momentum equation () become: dp d d = η Da γ dξ ϕη dη dη CF k () Here = R ρ ω / μ i the hear wave number, k = ωr / a i the reduced frequency parameter, Da = R / K i the Darcy number and C i the Forchheimer number. Thi i the claical F equation of Hagen-Poieuilli flow, the olution of which, with no-lip boundary condition, give a parabolic velocity profile: dp η ( = = η ) γ ξ () d Where i the mean ach number of the teady flow. The linearized acoutic equation follow from equating term of firt order in α in the governing equation, and are: i ρ + Γu + Γ( η ) ρ = () INTERNATIONAL JOURNAL OF ECHANICS Iue, Volume, 7 iu Γ [ + ( η u] = ( Γ / γ ) p + () d u (/ ) + (/ η) du dη ( Da / ) u (CF Da / k) u dη Where p i the preure, i the poroity of the porou medium. Note that the, Da and C will reflect the effect of F the porou matrix ize on the acoutic problem under conideration. The cae of = or Da = correpond to the plain medium without the preence of the olid matrix and any value of < < or Da > repreent a porou medium with different pore pace. For the cae of = and Da =, the governing equation () and () reduce to thoe obtained by Peat [] for the cae of a pure plain medium. In the limit of zero teady flow, =, thee equation are found to reduce to thoe for the reduced frequency olution of Tijdeman [9]. It will be aumed hat the tube are rigid which implie the nolip boundary condition of the fluid velocity at wall: u = at η = () The olution of equation ()-(7) i greatly implified if one aume that the acoutic diturbance occur ientropically, ince then: p = ργ () III. VARIATIONAL SOLUTIONS Now the problem reduce to that of olving the continuity and axial momentum equation for the velocity component and the preure p, which i contant over a radial cro ection. A variational olution with the following form of axial acoutic velocity variation i ought: u = C( η ) where C i contant () Equation () correpond to the Euler-Lagrange equation: f d u = (7) u dη ( du / dη) du iu η Γ Γ ( / ) η + + ( ) puη + ( η dη γ () f = dη Da C Da + u η + ( η k Here C, and thu for a given form of trial olution, = C F the bet approximation to equation () correpond to the minimum of the functional: F = du iu η Γ (/ ) η + + ( ) puη + dη γ Γ Da C Da ( η + u η + ( η k (9) dη Thu, the aumed form of trial olution for u, equation (), i ubtituted into thi expreion and the minimum i found by etting F C =, which reult in an expreion for the contant C; o that: pγ C = γ i Γ Da C F Da k ()

3 The known from of trial olution i now inerted into the continuity equation () with ubtitution for the denity from the ientropic relation () and integrated over the domain; thi lead to: i i Da C Da ( ) Γ Γ + k () i Da i C Dai = k With a olution of the propagation contant of the form: i i Da C Da ϕ k Γ = ± ( ) () i i Da C Da k i Da i C Da ( ) k ( ) IV. RESULTS AND DISCUSSION In the limit of zero teady flow =, comparion of variational olution with exact olution a given by Peat [] in the limit of plain medium for = and Da = i hown in table. Figure i a plot of the modulu of wave attenuation per unit ditance, Γ, and phae hift, Γ, for varying hear wave number and ach number =,.,.,. and for elected value of Da =, C =., =. and k =.π. It i clear that a the ach number i increaed the attenuation and phae velocitie are decreaed for the forward ound wave and increaed for the backward ound wave propagated in a porou media; thi i due unfavorable colliion with the olid matrix for the forward ound wave and favorable fluid flow velocitie on the propagation of the backward ound wave. Figure. how the effect of increaing Darcy number Da =,.,,, for elected value of =., C =., =. and k =. π, it i clear that a the Darcy number i increaed the attenuation and phae velocitie are increaed for both the forward and backward ound wave; thi i due to favorable olid matrix effect on damping the propagated ound wave. Figure. how the effect of poroity =.,.,.7,.,. 9 on attenuation and phae velocitie for elected value of Da =, C =. =. and k =.π, it i found that the increaing of the poroity decreae the attenuation and phae velocitie for both the forward and backward propagated ound wave; thi i due to the mall effect of the olid matrix a moving toward the plain media limit. Figure. how the effect of Forchheimer term C =.,,,, on the attenuation and phae velocitie for elected value of Iue, Volume, 7 INTERNATIONAL JOURNAL OF ECHANICS Da =, =., =. and k =. π, it i found that a the Forchheimer term i increaed the attenuation and phae velocitie are decreaed for the forward and backward ound wave; thi i due to favorable damping effect of the fluid inide the large ued pore of the olid matrix. / Finally, figure. how the effect of increaing the reduced frequency parameter k =.π,.π,. π,.π,.π,. π on the attenuation and phae velocitie for elected value of Da =, =., =. and C =, it i found that a the reduced frequency i increaed both the attenuation and phae velocitie for the forward and backward ound wave are decreaed; thi i due to higher frequency of the impacted ound wave on the olid matrix, it i important to note that the ame effect i noticed for ound wave propagated in a plain medium. V. CONCLUSIONS In thi work the acoutic problem for a tationary or flowing fluid in a circular tube filled with a porou media i invetigated; the following concluion remark are obtained: - It i found that the effect of increaing Darcy number or Forchheimer number i to increae the attenuation and phae velocitie for both forward and backward ound wave; thi i due to favorable role of olid matrix in damping ound wave. - It i found that the effect of increaing poroity or reduced frequency parameter i to decreae attenuation and phae velocitie for both forward and backward ound wave; thi i due to abence of favorable role of porou matrix and high incident ound wave trength repectively. Γ' 7

4 INTERNATIONAL JOURNAL OF ECHANICS Γ" Phae Shift Forward wave Phae Shift Forward wave Γ' Γ" Phae Shift Backward wave Phae Shift Backward wave Fig. The Effect of ach number Fig. The Effect of Darcy number 7 Iue, Volume, 7

5 INTERNATIONAL JOURNAL OF ECHANICS Phae Shift Forward wave Phae Shift Forward wave Phae Shift Backward wave Phae Shift Backward wave Fig. The Effect of Poroity Fig. The Effect of Forchheimer number Iue, Volume, 7 7

6 INTERNATIONAL JOURNAL OF ECHANICS Shear wave number, Preent Peat [] Phae Shift Forward wave Phae Shift Backward wave Fig. Effect of reduced frequency number REFERENCES [] H. V. Helmholz, Verhandlung der Naturhitorih- edizinichen Verein zu Heidel-berg, Bd III, vol.,. [] G. Kirchhoff, Uber den Einflu der Waermeleitung in einem Ga auf den Schallbewegung, Proggendorfer Annalen, vol., pp. 77-9,. [] Lord Rayleigh, Theory of ound, volume II, London: The acmillan Company econd edition, pp. 9-, 9. [] A. Cumming, Sound propagation in narrow tube of arbitrary cro-ection, Journal of Sound and Vibration, vol., no., pp. 7-, 99. [] K. S. Peat, A firt approximation to the effect of mean flow on ound flow on ound propagation through cylindrical capillary tube, Journal of Sound and Vibration, vol. 7, no., pp. 7-9, 99. [] E. Dokumaci, Sound tranmiion in narrow pipe with uperimpoed uniform mean flow and acoutic modeling of automobile catalytic converter, Journal of Sound and Vibration, vol., no., pp. 799-, 99. [7] A.-W. Jeong and J.-G. Ih, A numerical tudy on the propagation of ound through capillary tube with mean flow, Journal of Sound and Vibration, vol. 9, no., pp. 7-79, 99. [] E. Dokumaci, An approximate diperion equation for ound wave in a narrow pipe with ambient gradient, Journal of Sound and Vibration, vol., no., pp. 7-,. [9] H. Tijdeman, On the propagation of ound wave in cylindrical tube, Journal of Sound and Vibration, vol., pp. -, 97. Table : Comparion of variational olution with Peat [] Iue, Volume, 7