7 American WJTA Conference and Expo August 9-, 7 Houston, Texas Paper ANALYSIS OF FLOW INSIDE THE FOCUSING TUBE OF THE ABRASIVE WATER JET CUTTING HEAD Viém Mádr, Jana Viiamsoá, Libor M. Haáč VŠB Technica Uniersity of Ostraa Ostraa, Czech Repubic ABSTRACT To make the anaysis of fow in the focusing tube of abrasie water jet cutting head, the cacuation of intera of the exit eocities of water jet from the focusing tube into the free space is necessary. The cacuation of ongitudina eocity in the axis of water jet at the exit from the focusing tube, the cacuation of expansion eocity of water jet at its diergence in the air and the cacuation of mean eocity of water jet at the end of the focusing tube are performed. For the cacuations, it was necessary to determine a radius of jet at its exit from the focusing tube. The anaysis of ongitudina eocity in the focusing tube of abrasie water jet cutting head seres to the determination of a negatie pressure in the mixing chamber of the cutting head. The aue of a negatie pressure is one of the boundary conditions of mathematica modeing of water jet fow in the cutting head. Organized and Sponsored by the WaterJet Technoogy Association
. INTRODUCTION The anaysis of a process of mixing of the iquid with air and partices of materias inside the cutting head is of importance to improement in the efficiency of injection abrasie iquid jet. The mathematica description of iquid jet fow through the mixing chamber and the focusing tube can be done on the basis of knowedge of mass or oume concentration of air in the jet and the determination of infuence of aeration of the jet on its eocity profie or on the mean eocity of the jet. The anaysis of a ongitudina eocity in the focusing tube of abrasie water jet cutting head is of importance not ony to the mathematica description of the phenomenon, but aso to the determination of pressure of air inside the mixing chamber of equipment concerned. At each point of stream, a fuctuating quantity, e.g. eocity, can be decomposed into two components, the mean (time-smoothed) eocity and the fuctuation eocity the magnitude and direction of which changes ery quicky in time. The mean eocity is the aerage aue of instantaneous eocity in the time intera, which is great enough with regard to the time of turbuent osciations in fuctuation eocity. In the corresponding time intera, the mean aue of fuctuation eocity is equa to zero. Simiar considerations hod aso true for other fuctuating quantities, such as pressure, density, and others. Athough the time mean aues of fuctuating components of quantities are equa to zero, the mean time aue changes aong the cross sectiona area of fow of tubes. If we take, in the course of dynamica cacuations, their mean aue aong the cross sectiona area of fow into account, then the turbuent fow may be considered to be one-dimensiona steady fow of iquid and reationships deried for this kind of fow can be used.. THE INTERVAL OF EXIT VELOCITIES OF WATER JET FROM THE FOCUSING TUBE TO THE FREE SPACE In the cutting head, the water jet used for cutting of materias passes, after exiting the nozze, through the mixing chamber and the focusing tube (Figure ). As known, a pure water jet or abrasie water jet is used for materia cutting. When an abrasie is used, it is mixed with the water jet in the mixing chamber. In the foowing considerations and cacuations, we sha be concerned ony with a pure water jet passing the system mixing chamber focusing tube. Cacuations are executed on the foowing conditions: the iquid is incompressibe, the iquid remains in the jet; it does not accumuate in the mixing chamber, the jet is not aerated. In the course of cacuations, we take the turbuent fow of water in the jet as a steady fow with the fow fied characterised by the mean time aues of fuctuating quantities (Koář et. a. 983). At the beginning of cacuations, we sha determine the intera of mean eocities at which the iquid can moe at the exit from the focusing tube. From the equation of continuity it foows that
= S S d π = d π, () where S is the cross-section of the nozze at the entry into the mixing chamber, S is the exit cross-section of the focusing tube, d is the diameter of the nozze, d is the diameter of the focusing tube and is the mean eocity of water jet at the entry into the mixing chamber. The aue of eocity is the maximum if the jet does not extend radiay from the axis, which is, with reference to the existence of radia components of eocity, impossibe. The aue of eocity is minimum if the jet expands in the radia direction in reation to the axis of fow so that its cross-section at the exit from the focusing tube wi be identica with the cross-section of fow at the exit of focusing tube. Specific cacuations are performed for cutting head gien by the foowing parameters and aues of quantities characterising the condition of water: - water nozze at the entry into the mixing chamber: (index ), d =.5 mm (r =.5 mm), - mixing chamber: (index s), s = 3 mm, d s = 7.4 mm, - focusing tube: (index ), = 76 mm, d =. mm (r =.5 mm), where is a reeant ength and d is a reeant diameter of respectie parts of the cutting head. The temperature t, water density ρ, pressure p and mean eocity of water jet at the entry into the mixing chamber are expected to hae the foowing aues: t = C, ρ = 998 kg.m -3, p = 4 MPa, = 65 m.s -. For the cutting head set ike that, the mean exit eocity of the iquid jet outfowing the focusing tube is aid 39,65 m.s. 3. THE VALUE OF LONGITUDINAL VELOCITY IN THE AXIS OF WATER JET AT THE EXIT FROM THE FOCUSING TUBE The water jet moing in the air begins expanding after oercoming a certain distance from the nozze, caed initia section. Aong the initia section the eocity in the jet axis is uniform. Behind this section, the eocity begins to diminish as a resut of jet expansion. The ength of initia section p is gien by the reationship of A. J. Mioič [] p = 45d. () The ongitudina maximum eocity aong the axis of water jet moing in the air is, within its compact part behind the initia section, gien by reationship (Agroskin et. a. 955)
45d =, (3) where is the distance measured aong the jet axis from the end of initia section. For the cutting head with the aboe-presented parameters, we sha get the ength of the initia section p = 36 mm and the eocity within the section s = 65 m.s - after inserting into these reationships. At the distance k = 53 mm from the end of the water nozze, the eocity of water begins to decrease according to reationship (3). Hence we sha determine the aue of eocity in the axis of jet at the end of the focusing tube; the eocity reaches the aue = 65 m.s -. 4. THE EXPANSION VELOCITY OF WATER JET AT ITS DIVERGENCE IN THE AIR In the radiay expanding part of water jet behind the initia section, a decrease in the eocity of water jet occurs. For the cacuation of expansion eocity e in the direction perpendicuar to the axis of water jet, we sha use a reationship deried by Haáč (Haáč et. a. 999a) = p p ρ e u at ρ, (4) where p is the pressure of iquid before the nozze, ρ is the density of iquid under norma conditions, u is the eocity of water jet at the entry into the mixing chamber taken approximatey as equa to. In a case of the set cutting head, we sha obtain for the aue of temperature t = C and the density of water ρ = 998 kg.m -3 the cacuated aue of expansion eocity is approximatey e = 65 m.s -. The expansion of the jet at the nozze outet causes a sma increase of water jet diameter. Neertheess, the expansion attenuates ery quicky. 5. RADIUS OF JET AT THE OUTLET FROM THE FOCUSING TUBE We sha determine the radius of jet y as a sum of its radius r at the entry into the mixing chamber and the distance y, i.e. the distance traeed by the margina part of jet in the radia direction perpendicuary to the axis of the tube per time needed by the jet to trae the distance from the entry into the mixing chamber to the exit from the focusing tube y = r + y = r + et, (5) where t is the time needed by the jet to trae the tota interna path in the cutting head.
The tota interna path is taken as a sum of the ength of mixing chamber s and the ength of focusing tube. We sha diide the interna path into two sections, i.e. the initia section and the section in which the eocity of jet diminishes. We sha designate: the mean eocity of jet aong the initia section, the mean eocity of jet aong the section with the decreasing eocity. In a case of the initia section, the eocity aong the path is uniform. With the other section, we sha determine the mean aue of eocity from equation (3) by repacing distance by ariabe x. By subsequent integration aong the ength of section we sha obtain 45 d = 45d n, (6) dx = x x p x p where x = p. For the set parameters of cutting head, the foowing then hods true: = 65 m.s, = 43 m.s. The presented time is gien by a sum of times needed for coering the two aboe-mentioned sections, i.e. k t = t + t p = +. (7) For the set cutting head we sha obtain amost t =.87* -4 s. After putting down into reationship (5) we sha obtain the aue of radius of water jet y at the exit from the focusing chamber. For the set aues of cutting head it has the aue about y = 5 mm. This resut ceary presents that the cassica theories presented for ow eocity jets are unusabe for high-eocity water jets. Therefore we used the theory presented in Haáč et. a. (999a) and cacuated the radius of the jet at the end of the focusing tube, i.e. in the distance = 89 mm from the water nozze outet. The radius is then y =.34 mm. The width of the gap r = r y between the focusing tube and the water jet at the exit from cutting head is then r =.86 mm.
6. THE MEAN VALUE OF THE VELOCITY OF WATER JET AT THE END OF THE FOCUSING TUBE The equation of continuity wi be written for the cross-section of jet at its entry into the mixing chamber and the exit cross-section of jet at its exit from the focusing tube as foows S S y =, (8) where S =.49 mm is the cross-section of jet at the entry into the mixing chamber, = 65 m.s - is the mean eocity of water jet at the entry into the mixing chamber, S y =.33 mm is the cross-section of water jet at the exit from the focusing tube, is the mean aue of eocity of jet in its exit cross-section. With the set cutting head, the mean eocity of water jet at the exit from the focusing tube is expected to be = 97 m.s - according to the equation (8). The theoretica mode pubished in Haáč et. a. (999b) yieds for the ength of the jet core with the assigned aues of the nozze diameter and water pressure approximatey k = 6.5 mm. The aerage eocity at the water nozze outet is just = 646 m.s -. The axis eocity at the end of the focusing tube is cose to the aue o = 57 m.s - and the jet diameter is about r =.34 mm. Those parameters yied the jet cross-section neary S y =.33 mm by the end of the focusing tube and the aerage outet eocity of the water jet is amost = 96 m.s -. Simutaneousy, the aerage eocity determined from the eocity profie of the same mode and the effectie aue of the jet cross-section is cose to the aue = 94 m.s -, i.e. amost the same as the one determined from the equation of continuity. 7. CONCLUSION The anaysis of ongitudina eocity in the focusing tube of abrasie water jet cutting head consists in the determination of minimum eocity, at which the jet can exit from the focusing tube into free space, in the determination of initia section of the jet and its eocity at the end of the focusing tube, in the determination of aerage eocity aong the whoe interna path of the jet, in the determination of expansion eocity of the jet at its diergence in the air and in the cacuation of mean aue of eocity of water at the end of the focusing tube. The performed anaysis of ongitudina eocity in the focusing tube of abrasie water jet cutting head is necessary for the determination of negatie pressure in the mixing chamber of cutting head, which is one of boundary conditions of mathematica modeing the fow of water jet through the cutting head. 8. ACKNOWLEDGEMENTS The authors thank the Ministry of Industry and Trade (project H-PK/) and the Grant Agency of the Czech Repubic (project 5/6/56) for support proided to research.
9. REFERENCES Agroskin, I.I., Dmitrije, G.T., Pikao, F.I. (955): Hydrauics I. SNTL, Praha, 4 p. (in Czech) Haáč, L., Haáčoá, I., Mádr, V. (999a): Veocity profie of the supersonic iquid jet. Transactions of the VŠB Technica Uniersity of Ostraa, Mining and Geoogica Series, No., Vo. XLV, pp. 77-83 (in Czech) Haáč, L.M., Haáčoá, I.M., Mádr, V. (999b): Quick Method for Determination of the Veocity Profie of the Axia Symmetrica Supersonic Liquid Jet. Proceedings of the th American Waterjet Conference, M. Hashish (ed.), WJTA, Houston, Texas, pp. 89-99 Koář, V., Patočka, C., Bém, J. (983): Hydrauics. SNTL, Praha, 48 p. (in Czech). GRAPHICS air and materia partices inet iquidnozze thimbe iquid infow focussing tube mixed jet generation iquid, air and materia partices outfow mixing process acceeration process iquidnozze iquid jet generation Figure. A diagrammatica section through the cutting head with the representation of processes eading to the generation of abrasie iquid jet at its exit.