STUDY OF INHERENT FREQUENCY OF HELMHOLTZ RESONATOR

Transcription

1 005 WJTA Amerian Waterjet Conferene August -3, 005! Houston, Texas Paper 6B-4 STUDY OF INHERENT FREQUENCY OF HELMHOLT RESONATOR Gong Weili An Liqian Cui Longlian Xie Guixin Shool of Mehanis, Arhiteture & Civil Engineering China University of Mining and Tehnology Beijing, P.R. China ABSTRACT Self-resonating water jet has been the subjet of interest for researhers. The inherent frequeny of Helmholtz Resonator is one of the important parameters for design of self-resonating water jet nozzle devie. A new parametri model for predition of the inherent frequeny of Helmholtz resonator used in the field of water jet tehnology was proposed in this paper. Development of the model was based on the assumptions that the length and the diameter of Helmholtz resonating hamber and the length of the straight pipe segment of water jet nozzle are in the same order of magnitude. In omparison with the existing parametri model, the assumptions are more reasonable and the physial model on whih the new parametri model was developed is in good agreement with the real onfiguration of Helmholtz resonator found in self-resonating water jet. As a result, the new model was expeted to be more aurate in predition of the inherent frequeny of Helmholtz resonator. Organized and Sponsored by the Water Jet Tehnology Assoiation

2 . INTRODUCTION Compared with ontinuous high pressure water jet, pulsed high water jet has higher jetting effiieny. The priniple of self-resonating was widely used for generating the pulsed water jet. The inherent frequeny of the Helmholtz resonator is the key parameter for the self-resonating pulsed water jet tehnology. Fluid self-resonane ours under the ondition that the indution frequeny (pressure agitation frequeny) is onsistent with the inherent frequeny of Helmholtz resonator. Therefore, determination of the inherent frequeny of Helmholtz resonator is the interested subjet in this field. The widely used mathematial model for predition of the natural frequeny of Helmholtz resonator was given by []: f D πd L (.) Where f is the inherent frequeny of Helmholtz resonator, A is the water nozzle diameter, L is the length of inlet nozzle s straight pipe segment, D is the diameter of the inlet nozzle, D is the diameter of the resonating hamber, and is the overall veloity of agitation wave in elasti tube. The definition of is given by: (.) a 0 Where a is the wave veloity of in water and 0 denote the wave veloity of in elasti tube wall. The physial model from whih Eq. (.) was developed is shown in Fig.. This model development was based on the assumption that self-resonating hamber is a onentrated Parameter, the length of the hamber is muh less than its diameter and an therefore be negleted. This assumption, however, does not agree well with the typial struture of self-resonator used in self-resonating water jet tehnologies, onsequently the model is expeted to have some deviation in preditions. Therefore, development of a new predition model with higher predition auray is required and provides the motivation for the study presented in this paper.

3 . SOLUTION OF GOVERNING EQUATIONS OF TRANSIENT FLOW. Governing equations of transient flow in tube The segment of a round-setioned tube with A being its setional area was hosen as ontrol volume (see Fig. ). The length of ontrol volume is x, the absissa of inlet plane is x, the absissa of outlet plane is x x. It is apparent that x and x are independent of time t, ρ is the density of water, and u is the axial veloity of the fluid. In the ase of flow in the tube with round ross setion, u y u z 0, u x uu(x), the ross setional area is a funtion of x and t, i. e., AA(x,t), owing to the elastiity of the tube wall. The ontinuity equation for the elasti tube an therefore be written as: ρa ρau 0 (.) It was assumed that flow was ontinuous, inompressible, isothermal, fritionless, one dimensional, without body fores, and flowing within the tube of round ross setion, Navier-Stokes equation an be written as: u u u p ρ (.) In the ase of transient flow and fluid resonane, inertia fore beomes the major fator and the visosity of the fluid an be negleted. Therefore, Eq. (.) and Eq. (.) are the governing equations for the study of transient flow in the tube.. Linearization of the governing equations The veloity of the wave in water represents the influene of water ompressibility upon the propagating veloity of the agitation wave, whih is defined as: a dp dρ (.3)

4 The veloity of the wave in elasti tube wall resents the influene of the elastiity of tube wall upon the propagating veloity of the agitation wave, whih is defined as: 0 A dp ρ da (.4) Aording Eq. (.3), Eq. (.4) and Eq. (.), the ontinuity equation (.) an be transformed into: A p q 0 (.5) Where qρau is the mass flow rate. Let q denote integral of the first term of Eq. (.5) along the tube segment L. When L is small, the first term an be onsidered to be onstant. Then one obtains: A p A p q L dx L 0 (.6) The fluid apaitane per unit tube length is defined as: q A C (.7) p The final form of the linearized ontinuity equation is given by: p C q (.8) Multiply the ontinuity equation (.) by u, and the Navier-Stokes equation (.) by ρa, then added together, Eq. (.) beomes: q qu p A (.9)

5 Assuming that the flow veloity of the fluid is muh less than the propagating veloity of the agitation wave, the seond term of migration aeleration of qu/ in Eq. (.9) an be negleted. Hene the Navier-Stokes equation (.9) is linearized as: q l p (.0) Where l is the fluid indutane defined by: l p q A (.) Where p is the pressure drop along the tube segment L. The ideal fluid one-dimensional linearization governing equations for transient flow in the tube are ahieved by a sequene of mathematial manipulations, as given by Eq. (.8) and Eq. (.0)..3 Solution of governing equations with fluid impedane method Aording to the assumptions, the pressure p an be expressed as: pp(x,t) and mass flow rate as: qq(x,t). The orresponding Laplae transformations an be given as: P(x,s)L[p(x,t)], and Q(x,s)L[q(x,t)]. The initial onditions are: ( x, 0) 0 p( x,0) 0 q (.) Alternating Eq. (.8) and Eq. (.0) by Laplae transformation yields: ( x, s) P Q ( x, s) lsq CsP ( x, s) ( x, s) 0 0 (.3) Eq. (.3) is the linearized and Laplae-transformed governing equations. The fluid resistane is defined as: R p/q. Its Laplae transformation of P/Q is noted as the fluid impedane. The fluid apaitane C representing fluid ompressibility and tube wall s elastiity, as well as the fluid indutane l aused by the unsteady flow, will exhibit reatane, as the fluid flowing through the pipe system. Laplae transformation of expression (.7) (fluid

6 apaitane) leads to CQ/sP. Similarly, the apaitive reatane is defined as P/Q/S. And by Laplae transformation of Eq. (.) we obtain lp/sq. Thus the definition of indutive reatane is: l P/Qls. Under the boundary onditions of x0, P P(0,s), Q Q(0,s): solution to the governing equation (.3) is: P Q ( s, x) ( x, s) P h sx P sh Q sh sx sx sx Qh (.4) Where /A is the harateristi impedane of the tube. At the end of the tube (xl), the pressure and flow rate are: ( s L), Q Q( s, ) P P, L (.5) Substituting Eq. (.5) into Eq. (.4), and letting sjω, by algebrai manipulations we obtain: P P os Q P Q j sin Q j sin os (.6) And further we an obtain the solution to the governing equation for one dimensional transient flow in the tube by fluid impedane method : jtg L tg ω (.7) Where P /Q, P /Q, are the fluid impedane at the beginning and the end of the tube respetively, ω is the irular frequeny.

7 .4 The inherent frequeny of Helmholtz resonator The typial struture of Helmholtz resonator found in self-resonating water jet tehnologies indiates that the length and diameter of resonating hamber and length of the straight pipe segment of the jet nozzle are same order of magnitude. The physial model of Helmholtz resonator is shown in Fig. 3. The straight pipe segment stands for the water jet nozzle s straight segment. The length of straight segment is L, the diameter D, the harateristi impedane. and are the fluid impedane at the beginning and the end of straight segment respetively. The length of Helmholtz resonator is L, the diameter D, the harateristi impedane. 3 and 4 are the fluid impedane at the beginning and the end of the hamber respetively. 3 Applying Eq. (.7) to the straight pipe segment, then we obtain: j tg j tg (.8) Applying Eq. (.7) to the Helmholtz hamber in the same way, we obtain: j tg 4 3 j 4tg (.9) The onditions under whih resonane takes plae in the pipe system an be desribed by: 0 and 4, whih was substituted into Eq. (.8) and Eq. (.9) respetively, and then we obtain: j tg (.0) 3 jtg (.)

8 Where ωπf, f is the inherent frequeny of Helmholtz resonator. Sine Eq. (.0) and Eq. (.) are related by: 3, the following relation was ahievable: πfl tg tg πfl D D (.) Where /A and /A are the harateristi impedane of the straight pipe segment and Helmholtz resonating hamber, and the orresponding ross setional area are and A π D A π D 4. Sine L and L are same order of magnitude, and muh less than the overall veloity of agitation wave, Eq. (.) an be simplified into: 4 πfl tg πfl tg πf L L (.3) Substituting Eq. (.3) into Eq. (.), the parametri model of the inherent frequeny of Helmholtz resonator is obtained as: f D π λ D L (.4) Where the penalty parameter λ is the ratio of the straight pipe segment to the diameter of the water jet nozzle, i.e. λl /D. For self-resonating water jet, Strouhal number is given by: fd (.5) u S d Where f is the dominant frequeny of the separated vorties, u is the water jetting veloity. Mah number is defined as Mu/. Combining Eq. (.4) and Eq. (.5) and eliminating f, we an therefore obtain the relationship between the strutural parameters and the Strouhal number and Mah number, given by: D D (.6) π λs d M D L

9 3. CONCLUSIONS A new parametri model for prediting the natural frequeny of Helmholtz resonator was developed, and the relationship between the strutural parameters and Strouhal number and Mah number was obtained. The assumptions on whih the model development based agree with Helmholtz resonator used in self-resonating water jet, and is expeted to be more aurate in preditions. The obtained results would be of help to understanding the mehanism of self-resonating water jet. 4. ACKNOWLEDGMENTS Thanks should go to Proessor F. Mei at University of Siene and Tehnology Beijing, where the first author of this paper had been working as a PhD andidate under her aademi guide and kind help in this researh work. 5. REFERENCES. S. honghou, Theory and Tehnology of Water Jet, Petroleum University Publishing House, Beijing. 998, L. haorong, Appliation of fluid transient flow theory to hydrauli oal and rok fragmentation, High Pressure Water Jet. No., (979), G. Weili, Study of self-resonating water jet & water jet ultra-fine omminution tehnology, PhD Dissertation. Shool of Resoures Engineering, University of Siene & Tehnology Beijing, 999, NOMENCLATURE f -Inherent frequeny of Helmholtz resonator A -Water nozzle diameter L -Llength of inlet nozzle s straight pipe segment D -Diameter of the inlet nozzle D -Diameter of the resonator hamber -Overall veloity of agitation wave in elasti tube a-the wave veloity of in water 0 -the wave veloity of in elasti tube wall ρ-density of water u-axial veloity of the fluid. q-mass flow rate

10 C-Fluid apaitane per unit tube length l-fluid indutane R-Fluid resistane -Fluid impedane. -Charateristi impedane of the tube ω-cirular frequeny. λ-ratio of the straight pipe segment to the diameter of the water jet nozzle S d - Strouhal number M-Mah number

11 7. GRAPHICS 0 V A Helmholtz resonating hamber Straight segment L Fig. The existing physial model of Helmholtz resonator y (ρ ua) x x x x (ρua) x x x Fig.. Control volume in the tube with round ross setion 3 4 P Q D P Q D Helmholtz resonating hamber Straight segment L L Fig. 3 Physial model of Helmholtz resonator