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Compressibility of the Fluid EPJ Web of Conferences 67, 02048 (2014) DOI: 10.1051/epjconf/20146702048 C Owned by the authors, published by EDP Sciences, 2014 Compressibility of the fluid Jana Jablonská1,a 1 Department of Hydromechanics and Hydraulic Equipment, Faculty of Mechanical Engineering, VŠB-Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech republic Abstract. The presence of air in the liquid causes the dynamic system behaviour. When solve to issue of the dynamics we often meet problems of cavitation. Cavitation is an undesirable phenomenon, since it causes a disruption of the surrounding material and material destruction. Cavitation is accompanied by loud sound effects and reduces the efficiency of such pumps, etc. Therefore, it is desirable to model systems in which the cavitation might occur. A typical example is a solution of water hammer. 1 Introduction Fluid density after compression is given by m m In solving dynamics of hydraulic systems we often 0 4 4 (4) encounter the problem with cavitation. Cavitation is an V0 V 1 p 1 p undesirable phenomenon that we are trying to prevent. Cavitation depends on the liquid used in the hydraulic Thereof system. Liquids - hydraulic fluids are typical of its air content. The air in the fluid occurs in two forms - C p S D1 T (5) dissolved and undissolved. Dissolved air according to the 0 E K U literature [4] ia controled by Henry's law, its release is complex and difficult to describe action. If we consider the fluid flow with undissolved air it is a flow of mixture - water – air. 3 Determining the modulus of elasticity of liquid 2 Compressibility of the fluid The modulus of elasticity of water is generally in the literature determines the value of 2.1·109 Pa under normal Compressibility of the fluid is character to shrink in conditions, which corresponds to theory 1 (see Figure 3). volume when increasing the external pressure. It is The modulus of elasticity is dependent on pressure, expressed by coefficient of compressibility [2, 7] temperature (see Figure 1) and a significant effect on it has a content of undissolved (free) air in the liquid. The 1 1 C V S V 1 = - D T 4 (1) undissolved air greatly reduces the modulus of elasticity K V E p U V p T konst in the liquid. You can think of it as the air in the form of bubbles in the liquid. [2, 3, 7, 8, 9] where V is original volume, V is change of volume Liquid containing undissolved air (free) is no longer caused by change in pressure p at constant temperature. homogeneous environment and is a suspension of gas Therefore we can write (air) in the liquid (water) Module compressibility of air under normal conditions is 1.4·106 Pa [2, 4], it is three V V V0 , p p0 p (2) orders of magnitude less than the value for water. From equation 1 and 2 can be deduced 4 V0 V V V 1 p (3) a Jana Jablonská: [email protected] This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20146702048 EPJ Web of Conferences velocity of pressure waves in the range 4 1 a s 20 800 m s [2, 5]. Table 1. Measured and calculated values. Experimenttal Literature values values Run time of wave 0.467 - - T s Act. Speed of 4 1 249.59 20 800 soundn a s m s Modulus compressibility 62.172 · 106 0.399 · 106 638.7 · 106 K Pa Fig. 1. Basic properties of water depending on temperature [3] 3.2 Determining moduluss of elasticity from RLC 3.1 Determining modulus of elasticity from the resistances experiment Another possibility is to deterrmine the modulus of liquid One way of precise identification modulus of liqquid elasticity with the inclusionn of pipes ´wall elasticity elasticity (suspension water - air) in the system is (without considering air) [7]. measurement. The graph of pressure course in the 1 1 d 1 0.029 hydraulic shock can be deducted value of the period tp = K K E 4 s 2.14109 7004106 40.005 (see Figure 2). [1] voda (9) K 114130435Pa 3.3 Determining modulus of elasticity with air – theory 2 If from formula (1) generally applies that dpp K V (10) dV For volume of mixure must then apply Fig. 2. Record of pressure during hydraulic shock (valve at tthe (11) beginning of pipeline) V Vk V g From period can be determined the time course of wave T Where Vk is liquid volume, Vg is gas volume (air) aat t 0.934 T p 0.467 s (6) reference (atmospheric) pressure. 2 2 If the volume fraction of gas p (air) in a liquid is defined as Spread velocity of shock wave to be calculated from length of piping l 58.28m (pump – closing valve) and V g ; 4 p Vg p Vk (12) from the periood Vk 24l 2458.28 Than volume fraction dependds on the pressure and from a 249.59m4 s 1 (7) S,exp T 0.467 equation can be deduced 1 Module of liquid compressibility C p S n D ref T (13) p D p p T K E ref U a ; K a2 4 s,exp s,exp (8) K 249.592 4998 62.172 4106 Pa To determine the theoretical modulus of liquid elastiicity for the tube with fluid in the literature indicates sppread 02048-p.2 EFM 2013 After substituting equation (12) and (13) to equation (11) 4 Possible solutions applies Based on the evaluation, depending on the density from C 1 S pressure and determination of modulus of liquid elasticity D C p S n T 4 D ref T I defined these alternative solutions: V Vk p Vk Vk D1 D T T D p p T (14) E E ref U U 1. Theory 1 V Vk 1 p C p S C p 98000 S = 4 D1 rel T 998 4 D1 T ref E K U E 2.14109 U After substituting equation (14) to equation (10) C 1 S D C p S n T dp 2. From experiment K V D1 D ref T T k D T D E pref p U T dV E U (15) C p S C p 98000 S = 4 D1 rel T 99844 D1 T dp ref E K U E 62172146 U K V 1 k p dV After derivation can be deduced for modulus of liqquid 3. From RLC resistancees elasticity form C p S C p 98000 S 4 D rel T 4 D T 1 = ref D1 T 998 1 C p S n E K U E 114130435U 1 4D ref T D T E pref p U 4 K K ref 1 (16) p n 4. Theory 2 4 ref 1 K ref n1 n 4 p p n C S ref D T D T Where pref p0 is atmospheric pressure, is relative D T D T gas content at atmospheric pressure, n is gas specific heat D T ratio. D T D p T = 4 1 ref D 1 T D C p S n T D 1 4 D o T T D T D E po p U T K D ref 1 T D p n T 1 4 K o D refef n1 T D 4 T E n po p n U After substitution C S D T D T Fig. 3. Modulus of elasticity depending on pressure and voluume D T fraction D T On Figure 3 is outlined the theoretical value of water D T D T modulus is called a theory 1 . In the above equations is D p T = 980 4 1 marked as Kref and for calculation is specified value D 1 T 9 . D C S 1.4 T Kref = 2.1·10 Pa From Figure 8 it is evident that peak 4 D 98000 T D 1 0.001 D T T pressure in the hydraulic shock to move around 200 kPa D E 98000 p U T ( 2.14109 relative pressure), while the Figure 3 shows that when D 1 T D 98000 1.4 T the pressure is to change modulus of liquid elasticity is 1 0.0014 2.14109 D 1.41 T very significant. Therefore, this parameter must be E 1.4 4 98000 p 1.4 U included in the calculations. The value of elastic modulus specified from experiments and designated by the RLC resistance in the range (200 ÷ 400) kPa (absolute pressure) corresponds approximately to the curve with consideration of air volume fraction = 0.01 [3, 7, 8]. 02048-p.3 EPJ Web of Conferences 5. From literature [5] dR 2 pvap tt p t (18) p b = dt 3 0.0009999994 p 0.099249 l Mathematical model iis defined for multiphase mixture consisted from waterr and vapour eventually air. For multiphase flow simulattion the Mixture model is used. This model is advisable, when the velocity of individual phase translation differs. Model provides phase changeover, for this occasion the volumetric fractions of phases are defined. Continuity equation forr the mixture has the form u m m m, jj 0 (19) t x j where um, j is the mass-averagged velocity and m is the mixture density defined by expression.
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