Flow Variation of Duckbill Valve Jet

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AN ARTICLE IN JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004

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4. element mesh for the initially closed duckbill valve; two rows of elements are used for both the bill and the cuff. The interior of the DBV is circular upstream; at the other end, the bill is rather flat; there is a uniform portion for both the bill and the cuff. In the transition between these two ends, the cross section in the DBV interior can be well approximated by an ellipse. The half width in the minor-axis central plane decreases from R p ~ at the cuff ! to zero at the bill; similarly the half width in the symmetry plane Fig. 4. Computed deformation of duckbill valve: ~ a ! deformed shape; ~ b ! longitudinal section; and ~ c ! pressure variation ~ upstream pressure p 0 5 5.0 kPa) 974 / JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 MEASURIT www.measurit.com

7. longitudinal directions, resulting in this large normal strain. The solution also shows that the change in rubber thickness at the bill is less than 2%; this implies the synthetic material is tough enough to sustain the hydraulic pressure. Fig. 5 ~ b ! shows the computed longitudinal strain e 33 for the entire valve; it is seen that the strain distribution is quite even, except at the transition between the saddle and the cuff of the valve, the anchor point to hold the valve back against the internal pressure. The change in rubber thickness in the longitudinal direction at this position is over 5%. Fig. 5 ~ c ! shows the computed circumferential strain e 22 for the DBV. As expected, the maximum strain ~ and stress ! occurs at the middle part of the valve and then gradually decreases to the minimum value at the edge of the valve. At the critical region, the deformation is over 6% in the circumferential direction. It is in- teresting to note that the valve sustains more circumferential strain than the normal strain as a result of increasing internal pressure. The convergence of the solution has been checked by doubling the mesh resolution at the bill. The computed maximum valve deflection is changed by around 0.5%. Hydraulic Characteristics of Duckbill Valve In order to derive the hydraulic characteristics of the DBV, the computation is performed for a range of pressures, p 0 5 1.25– 10.0 kPa; this corresponds to a flow range of Q 5 0.0139– 0.1432 m 3 / s. Fig. 6 shows the deformed bill which is similar in shape for all flows. Given the equilibrium deformed shape for each pressure, the flow and specific energy can be ob- tained from Eqs. ~ 1 ! and ~ 2 ! . The most important hydraulic char- acteristic of this flow device is the relationship between pressure and flow, the so called ‘‘head–discharge’’ relationship. Fig. 7 ~ a ! shows the computed variation of pressure with flow; it is seen that the relationship is very linear for all the thicknesses tested. On the Fig. 6. ~ Color ! Computed valve opening shape ( d p 5 305 mm) Fig. 7. Computed hydraulic characteristics of duckbill valve for different rubber thicknesses: ~ a ! pressure-discharge variation and ~ b ! opening area as function of flowrate JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 977 MEASURIT www.measurit.com

8. other hand, Fig. 7 ~ b ! shows that the valve opening area ~ at the bill ! a j varies nonlinearly with the discharge. Given the linearity of the head–discharge relation, all the computed results can be nondimensionalized by the values p 0.3 , Q 0.3 corresponding to an area ratio of a j / a p 5 0.3. The material properties are then ac- counted for in the scaling, and Figs. 8 ~ a and b ! show that all the numerical results collapse onto unified normalized plots for both the head–discharge and area opening relations. The best-fit rela- tions are p p 0.3 5 0.993 Q Q 0.3 (9) a j a p 5 0.286 S p p 0.3 D 0.61 (10) It should be noted that the choice of a j / a p 5 0.3 is arbitrary and will not affect the normalized head–discharge relation. The nu- merical predictions are very close to the theoretical proportional- Fig. 8. Normalized relations of computed hydraulic characteristics: ~ a ! pressure-discharge relation; ~ b ! valve opening ratio versus pressure; and ~ c ! energy head 978 / JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 MEASURIT www.measurit.com

10. Urmston Road Sewage Outfall, Hong Kong ~ Horton et al. 1995 ! , with diameter d p 5 0.305 m and thickness t 0 5 1.57 cm. A steel plate was inserted into the valve to provide support, and a vertical loading was incrementally applied onto the rubber valve mem- brane. The experimental measurements of stress and strain ~ not shown ! demonstrate a very linear relation; the data indicate a Young’s modulus of E 5 7.32 MPa, which is consistent with typi- cal values for elastomers ~ Lindley 1970 ! . Fig. 9 ~ a ! shows the comparison of the best fit of the pressure- flow numerical results @ Fig. 7 ~ a !# with the experimental data of the 305 mm valve. It is seen that the experimental data support the linear P – Q relation except at large flows, when the pressure measurement is probably affected by the size of the experimental setup; the high flow data also deviate somewhat from energy con- servation checks. On the other hand, the comparison of the best fit numerical result with measured jet velocity @ Fig. 9 ~ b !# shows ex- cellent agreement ~ the jet velocity for the highest flow was out- side measurement range ! . Given the fact that synthetic fabric sup- port was not modeled, the agreement is quite satisfactory. In Fig. 10, the normalized valve opening area and head–discharge rela- tions are compared with the collective data of tests on different valve sizes. It is quite remarkable that the theoretical relation derived in this study is supported by all the data; this appears to suggest that the DBV valve deformation is chiefly dependent on the mechanics of the rubber deformation, and only secondarily on minor synthetic support ~ present in all the commercial valves ! and upstream connection ~ e.g., the 8 in. valve has a flange connec- tion ! . Concluding Remarks A nonlinear large deformation finite element analysis of a DBV has been performed. The deformed valve is computed iteratively from sequential standard large deformation analysis of the DBV subjected to an internal pressure loading obtained by assuming a one-dimensional energy conserving flow in the interior of the valve. The calculations show that the maximum stress occurs around the two sides of the saddle of the DBV; maximum strains are on the order of 5%. The most significant results are the pre- diction of a linear pressure–discharge relation, and a nonlinear variation of jet velocity/valve opening area with discharge. The Fig. 10. Comparison of predicted duckbill valve hydraulic characteristics with experimental data of duckbill valve: ~ a ! normalized valve opening area and ~ b ! normalized head–discharge relation 980 / JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 MEASURIT www.measurit.com

5. containing the major axis increases from R p to l /2 5 p R p /2 at the bill ~ Fig. 2 ! . The deformation of the elastomer for a given upstream pres- sure can be obtained by incremental solutions from zero pressure up to the desired value ~ Lo 1988, 1992 ! . Static equilibrium must hold at any instant of time for the elastomer solution domain, and the governing equation is πs 1 b 5 0 (4) subject to the inner surface boundary condition of zero displace- ment at the inflow pipe ~ the ‘‘cuff’’ ! and the fluid pressure load u 5 0 inflow pipe (5) s ï n 52 p ~ x ! n inner surface (6) Fig. 5. ~ Color ! Computed strain for whole duckbill valve ~ upstream pressure p 0 5 0.5E4 Pa) : ~ a ! principal normal strain ( e 11 ); ~ b ! principal longitudinal strain ( e 33 ) ; and ~ c ! principal circumferential strain ( e 22 ) JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 975 MEASURIT www.measurit.com

1. Flow Variation of Duckbill Valve Jet in Relation to Large Elastic Deformation Joseph H. W. Lee 1 ;S.H.Lo 2 ; and K. L. Lee 3 Abstract: Duckbill-shaped elastomer valves are often installed on wastewater effluent diffusers, stormwater outfalls, and industrial flow systems to prevent backflow and sediment/salt water intrusion. Unlike fixed diameter nozzles, the flow from a duckbill valve ~ DBV ! depends both on the driving pressure and the size of the valve opening. A nonlinear large deformation finite element analysis of a prototype DBV is reported herein. The elastomer is modeled as a hyperelastic incompressible solid, and the flow inside the DBV, shaped like a converging nozzle, is treated as energy conserving. The deformed valve is computed iteratively from sequential standard large deformation analysis of the internal flow and pressure loading. The calculations show that the valve opening is lip shaped, and the maximum stress occurs around the two sides of the saddle of the DBV; maximum strains are on the order of 5%. In contrast to the traditional square-root head–discharge dependence, a linear pressure–discharge relation is predicted for a range of elastomer thickness; the jet velocity/valve opening area varies nonlinearly with discharge. The normalized predictions of valve discharge flow and opening area as a function of the driving pressure are in excellent agreement with experimental data. DOI: 10.1061/ ~ ASCE ! 0733-9399 ~ 2004 ! 130:8 ~ 971 ! CE Database subject headings: Elastomer; Finite element method; Deformation; Fluid-structure interaction; Elasticity; Water flow. Introduction Duckbill-shaped elastomer valves are often installed on wastewa- ter effluent diffusers, stormwater outfalls, and industrial flow sys- tems to essentially permit flow only in one direction. A ‘‘duck- bill’’ valve ~ DBV ! is manufactured of neoprene flexible elastomer material reinforced with synthetic fabric @ Figs. 1 ~ a and b !# , much like a car tire; it resembles a short piece of rubber hose that has been flattened at one end ~ the ‘‘bill’’ ! . At the other end the valve is typically clamped onto an existing nozzle port ~ the ‘‘cuff’’ ! ; the transition between the bill and the cuff is termed the ‘‘saddle.’’ These valves can be easily mounted onto discharge ports, and have been increasingly used on submarine wastewater outfalls in many countries ~ e.g., Roberts et al. 1984; Water Research Center 1990 ! . These valves can prevent sea water, sediment, and aquatic life from entering submarine outfalls, and achieve optimal mixing of the underwater effluent discharge with the receiving water. The design implications and practical advantages of DBVs have been discussed ~ Duer 1998 ! . A DBV is a flow sensitive variable-area valve. At no flow condition, the flaps of the valve remain closed. As the flowrate increases, pressure is exerted on the flaps of the valve, and the valve opens more. The flow from a DBV differs from a conven- tional round nozzle or pipe discharge significantly. For flow from a round pipe of fixed diameter, the jet discharge velocity varies linearly with the flow which has a square-root power dependence on the driving pressure. For a DBV, however, experiments have shown that the jet velocity varies nonlinearly with the flowrate; reasonably high velocities can be maintained even at very small flowrates ~ Abromaitis and Raftis 1995; Horton et al. 1995 ! . The flow however varies approximately linearly with the driving pres- sure. Despite the use of DBVs by practitioners, to the writers’ knowledge a large deformation analysis of such valves has hith- erto not been reported. The flow from a DBV depends both on the driving pressure and the size of the valve opening; the pressure is, however, determined by the flow. A full three dimensional ~ 3D ! large deformation analysis of this fluid–structure interaction is necessary for an understanding of the hydraulic characteristics of this flow device. In this paper, a nonlinear finite element analysis of the large deformation of a DBV is presented. First, the theoretical formu- lation of the large deformation of the DBV, modeled as a hyper- elastic incompressible solid, is given. The iterative solution of the valve deformation and the flow field is described. Second, details of the finite element implementation ~ via the nonlinear finite ele- ment program ABAQUS ! and the computed deformation of a typi- cal DBV ~ for different rubber thicknesses ! are presented. Finally, the derived hydraulic characteristics of a DBV are compared with experimental data and discussed. Theory Consider a DBV attached to a pipe of diameter d p 5 2 R p and cross-sectional area a p 5 p d p 2 /4. Fig. 2 shows a schematic dia- gram of the centerline section of such a valve ~ in the plane of the major axis ! . Without loss of generality we assume the flow 1 Professor, Dept. of Civil Engineering, The Univ. of Hong Kong, Pokfulam Rd., Hong Kong, China. E-mail: hreclhw@hkucc.hku.hk 2 Reader, Dept. of Civil Engineering, The Univ. of Hong Kong, Pokfulam Rd., Hong Kong, China. E-mail: hreclsh@hkucc.hku.hk 3 Research Assistant, Dept. of Civil Engineering, The Univ. of Hong Kong, Hong Kong, China. Note. Associate Editor: Michelle H. Teng. Discussion open until Janu- ary 1, 2005. Separate discussions must be submitted for individual pa- pers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November 15, 2001; approved on January 12, 2004. This paper is part of the Journal of Engineering Mechanics , Vol. 130, No. 8, August 1, 2004. ©ASCE, ISSN 0733-9399/2004/8-971–981/$18.00. JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 971 MEASURIT www.measurit.com

11. normalized predictions of pressure and valve opening area are in excellent agreement with experimental data. Acknowledgments The research was supported in part by a grant from the Hong Kong Research Grants Council. The assistance of the Hong Kong Drainage Services Department in providing the 305 mm DBV is gratefully acknowledged. References ABAQUS solution manual . ~ 1998 ! . Hibbitt, Karlsson, and Sorensen, Inc. Abromaitis, A. T., and Raftis, S. G. ~ 1995 ! . ‘‘Development and evaluation of a combination check valve/flow sensitive variable orifice nozzle for use on effluent diffuser lines.’’ Proc. WEFTEC 95 , Vol. 1, Part 2, Miami Beach, Fla, 701–709. Duer, M. J. ~ 1998 ! . ‘‘Use of variable orifice duckbill valves for hydraulic and dilution optimization of multiport diffusers.’’ Water Sci. Technol., 38 ~ 10 ! , 277–284. Horton, P. R., Hranisavljevic, D., Wilson, J. R., and Miller, B. M. ~ 1995 ! . ‘‘Northwest New Territories Sewage Outfall, Urmston Road, Hong Kong: tideflex check valve testing.’’ Rep. No. 95/19 , Australian Water and Coastal Studies Pty Ltd., Water Research Laboratory, Univ. of New South Wales, Sydney, Australia. Lee, J. H. W., Karandikar, J., and Horton, P. R. ~ 1998 ! . ‘‘Hydraulics of duck-bill elastomer check valves.’’ J. Hydraul. Eng., 124 ~ 4 ! , 394 – 405. Lindley, P. B. ~ 1970 ! . Engineering design with natural rubber , The Natu- ral Rubber Producers Research Association, London. Lo, S. H. ~ 1986 ! . ‘‘Treatment of large deformation problem in elastoplas- ticity by the finite element method.’’ Doc-Ing thesis, L’Ecole Nation- ale des Ponts et Chaussees, Paris. Lo, S. H. ~ 1988 ! . ‘‘Stress evaluation algorithm for rate constitutive equa- tions in finite deformation analysis.’’ Int. J. Numer. Methods Eng., 26 ~ 1 ! , 121–141. Lo, S. H. ~ 1992 ! . ‘‘Geometrically nonlinear formulation of 3D finite strain beam element with large rotations.’’ Comput. Struct., 44 ~ 1/2 ! , 147–157. McKenzie, L. C., Horton, P. R., and Wilson, J. R. ~ 1996 ! . ‘‘Burwood Beach Ocean Outfall—testing of 8 inch check valve.’’ Rep. No. 96/18 , Water Research Laboratory, Univ. of New South Wales, Sydney, Aus- tralia. Roberts, D. G. M., Flint, G. R., and Moore, K. H. ~ 1984 ! . ‘‘Weymouth and Portland marine treatment scheme: tunnel outfall and marine treatment works.’’ Proc. Inst. Civ. Eng., Part I, 76 ~ 1 ! , 117–143. Utah Water Research Laboratory. ~ 1994 ! . ‘‘Pressure and performance tests of four inch and six-inch tideflex check/diffuser valves.’’ Rep. No. USU-361 , Utah State Univ., Logan, Utah. Water Research Centre. ~ 1990 ! . Design guide for marine treatment schemes , Vol. 4, WRc, Swindon, United Kingdom, 495–501. JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 981 MEASURIT www.measurit.com

3. ably uniform at these two sections. Once a j is computed, the total energy head E and flow Q can be determined iteratively from the above equations. As a j 2 ! a p 2 , E ' p 0 / g can be first assumed. This gives V j 5 A 2 gE and hence the flow Q 5 V j a j . With this first good estimate of Q , V p 5 Q / a p can be substituted into Eq. ~ 2 ! to obtain a revised energy head E and flow Q . Typically the calcu- lation converges in 2–3 iterations. In addition, the longitudinal pressure variation is given by V ~ x ! 2 2 g 1 p ~ x ! g 5 E (3) where p ( x ) 5 pressure at any section x between the inflow and DBV bill; and V ( x ) 5 Q / A ( x ) and A ( x ) 5 cross-sectional average velocity and internal cross-sectional area, respectively. Note that as a first approximation, the use of Eq. ~ 3 ! to determine the pres- sure distribution on the inner surface of the valve tacitly assumes a flat velocity profile and neglects the normal pressure variation due to streamline curvature and elevation changes. Large Deformation Finite Element Analysis For strains up to about 10%, an elastomer behaves like a perfectly elastic incompressible solid ~ Lindley 1970 ! with Poisson’s ratio l 5 0.5; material volume is conserved, such that an elongation of the rubber is accompanied by decrease of thickness. It can be shown that the DBV under consideration satisfies these approxi- mations. In the finite element analysis we assume an isotropic material with l 5 0.499; no other synthetic support fabric is con- sidered. Computations are performed on a 12 in. ( d p 5 305 mm) diameter DBV with thickness 1.57 cm; the Young’s modulus of elasticity is ~ obtained from our experiments ! E 5 7.32 MPa. For a given upstream pressure the nonlinear analysis is per- formed using the ABAQUS code ~ Hibbitt et al. 1998 ! . In the finite element analysis, the unknown DBV body configuration is de- fined by means of interpolation functions in terms of a number of control nodal points. The valve is modeled by 224 continuum stress/displacement 20-node brick elements; the total number of nodes for the quadratic elements is 2,086. Fig. 3 shows the finite Fig. 2. Schematic diagram for duckbill valve as converging nozzle Fig. 3. Finite element mesh of duckbill valve ~ diameter 5 30.5 cm; thickness 1.57 cm ! JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 973 MEASURIT www.measurit.com

2. through the valve is an incompressible fluid like water, with con- stant density r and specific weight g 5 r g , where g 5 acceleration due to gravity. The flow issuing from the valve discharges as a submerged jet into a similar fluid. The flow cross section changes from a circle of radius R p at the inflow ~ at the ‘‘cuff’’ ! to the unknown lip-shaped cross section at the exit ~ the ‘‘bill’’ ! . At the no flow ~ or zero excess pressure ! condition, the flaps of the valve remain closed. As the driving pressure at the inflow section D p 0 ~ or excess pressure head D p 0 / r g ) increases, a corresponding un- known normal stress load is exerted on the internal surface of the flaps of the valve; the valve opening area a j and hence the flow Q increases. As the flow accelerates inside the DBV, the excess pres- sure D p ( x ) @ henceforth referred to simply as p ( x ) ] decreases from p 0 at the inflow to zero at the exit. The velocity varies from V p at the inflow pipe to V j 5 Q / a j at the discharge. In view of the converging flow geometry, and confirmed by experiments, the flow is energy conserving to first approximation—i.e., the DBV acts like a smooth nozzle with negligible energy loss. Our objec- tive is to predict, for a given upstream pressure, the 3D shape of the DBV ~ and hence the pressure distribution ! , the flow discharge, and other related hydraulic characteristics. Energy Conservation of Duckbill Valve Flow The flow within the streamline-shaped DBV is assumed to be energy conserving and one dimensional. Application of mass con- tinuity and the integral mechanical energy equation between a section upstream ~ 1 ! in the inflow pipe and the bill opening, sec- tion ~ 2 ! , can be written as Q 5 V p a p 5 V j a j (1) V p 2 2 g 1 p 0 g 5 V j 2 2 g 5 E (2) where p 0 / g 5 h 5 driving pressure head; and V p 2 /2 g , V j 2 /2 g 5 velocity head ~ kinetic energy per unit weight of fluid ! at the inflow and outflow, respectively. Note that the flow is reason- Fig. 1. ~ a ! Duckbill-shaped elastomer valve; and ~ b ! definition diagram of duckbill valve characteristics 972 / JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 MEASURIT www.measurit.com

9. ity coefficient of unity. Note that for ( a j / a p ) 2 ! 1, we would have Q ' A 2 gha j . Since Q } h , we must have a j } A h , and it can be shown that p p 0.3 5 Q Q 0.3 (11) a j a p 5 0.3 S p p 0.3 D 0.5 (12) which is in close agreement with the best fit equations. This gives additional support to the accuracy of the computations. Similarly the relation between total energy head E 5 p 0 / g 1 V p 2 /2 g and flow can be obtained @ Fig. 8 ~ c !# ; as expected the relation is very simi- lar to that of pressure discharge. Comparison with Experimental Data Experiments on the hydraulics of duckbill elastomer check valves have been carried out by different investigators. These include the tests of the Utah Water Research Laboratory ~ 1994 ! for different types of a commercial valve of nominal port diameter d p 5 100 and 150 mm ~ 4 and 6 in. ! , and those of the Univ. of New South Wales’ ~ UNSW ! Water Research Laboratory on a 305 mm ~ 12 in. ! valve ~ Horton et al. 1995 ! , and on a 200 mm ~ 8 in. ! rubber check valve ~ McKenzie et al. 1996 ! . In each test a steady water flow is discharged through a submerged DBV ~ fitted onto an inflow round pipe ! in a receiving water tank. For each discharge Q , the centerline ~ maximum ! velocity V j was measured by an acoustic doppler velocimeter; the pressure head loss across the valve D p / g was measured by standard piezometers attached to the upstream pipe and the downstream tank. The valve opening area a j was also measured by two means: ~ 1 ! as inferred from the velocity, a j 5 Q / V j and ~ 2 ! by a point gage traversing across the valve. The approach velocity in the port was determined from the measured discharge and the area of the approach pipe V p 5 Q / a p . Addi- tional details on the experiments and the results can be found in the original reports ~ Lee et al. 1998 ! . Stress–strain measurements were independently made by the authors for a 12 in. Tideflex duckbill check valve used for the Fig. 9. Comparison of computed duckbill valve hydraulic characteristics ( d p 5 305 mm, E 5 7.32 MPa, t 0 5 1.57 cm) with experimental data: ~ a ! pressure and ~ b ! jet velocity as function of discharge JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 / 979 MEASURIT www.measurit.com

6. where s 5 Cauchy stress tensor; b 5 body force vector ~ 5 0 for this problem ! ; and u , n 5 displacement and surface normal vectors, re- spectively. The virtual work equation of Eq. ~ 4 ! taking into account the surface boundary conditions is given by G ~ u ! 5 E V s ij S ]h i ] x j D d V 2 E ]V t i h i da 5 0 (7) where t 52 p ( x ) n and h 5 arbitrary C 1 functions over V . Given the elastic material constitutive relations, the current configuration V and the stress state s are functions of displace- ment u . Through a finite element discretization, V , s , and n can all be expressed in terms of unknown values at nodal points. As a result, Eq. ~ 7 ! is a nonlinear differential equation in displacement vector u . The tangent stiffness matrix, and hence the incremental equation corresponding to a single load step, can be obtained through a linearization process ~ Lo 1986 ! . The deformed shape under a given pressure load is to be determined in a sequence of load steps, in which the equilibrium state of each load step is obtained by iteration. The foreseeable difficulties are how pres- sure load should be applied and whether convergence can be eas- ily achieved for the duckbill valve flow which can cause a dis- placement as large as 30% of the duckbill valve opening, with an associated internal strain of 5–10%. We are also concerned about the stability of the fluid–structure interaction; fortunately, as will be shown later, the use of carefully selected load steps with a simple geometrical update of the Bernoulli flow proved to be adequate and stable for the solution process. For a given upstream pressure p 0 , the 3D large deformation of the valve can be obtained from an iterative solution of a static finite element analysis of the valve deformation as follows: 1. A uniform initial pressure equal to the prescribed upstream inflow pressure p 0 is assumed. A standard 3D finite element large deformation analysis is then performed and for a typi- cal run loading is applied in about ten increments. The cross- sectional area A ( x ) and the DBV opening area a j can be obtained from the nodal coordinates of the solution. 2. The pressure p 0 will push open the duckbill causing the water to flow. The flow Q can be obtained from Eqs. ~ 1 ! and ~ 2 ! as outlined above, and the change in fluid pressure within the DBV p ( x ) from Eq. ~ 3 ! . 3. With the updated internal pressure distribution, the standard analysis is repeated, and a new DBV shape is determined. 4. Steps 2 and 3 are repeated until the computed Q converges. Typically, six iterations are required for convergence. The solu- tion for a range of upstream driving pressure and several rubber thicknesses t 0 have been obtained. All calculations were per- formed on an IBM9076 SP2 scalable 48-node parallel computer; a solution takes around 10 CPU min. Finite Element Solution Fig. 4 ~ a ! shows the computed configuration of the deformed DBV for an upstream pressure of p 0 5 5.0 kPa ~ thickness t 0 5 1.57 cm) . It is seen that the DBV opening is lip shaped, and that the outer surface of the valve is saddle shaped. The longitu- dinal section shows that the bill portion is relatively flat @ Fig. 4 ~ b !# . In accordance with the change in velocity ~ by continuity ! , the pressure drops sharply to a small value near the exit; the pressure gradient at the inner surface of the bill is relatively small @ Fig. 4 ~ c !# . The computed flow for this pressure is Q 5 0.0705 m 3 / s, with a relative valve opening area of a j / a p 5 0.29 and exit jet velocity of V j 5 3.32 m/s, respectively. The maximum deflection of the duckbill valve opening is 0.0421 m. Fig. 5 ~ a ! shows the computed principal normal strain e 11 for the entire DBV. It is seen that the most critical part is at the two sides of the saddle of the valve. This is because the elements in these areas are deformed steeply in both the circumferential and Fig. 5. Ñ Continued ! 976 / JOURNAL OF ENGINEERING MECHANICS © ASCE / AUGUST 2004 MEASURIT www.measurit.com

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