Department of mechanics, Faculty of engineering Hamadan branch, Islamic azad university, Hamadan, Iran. Abstract

Transcription

1 194 Ciência enatura, Santa Maria, v. 37 Part 1 015, p ISSN ipressa: ISSN on-line: X Vibration Siulation of the cylindrical reservoir shell containing fluid vortex with the help of Vib-Shape software Hossein Moradi 1, Farzan Barati,* 1 Departent of echanics, Haadan branch, Islaic azad university, Haadan, Iran Departent of echanics, Faculty of engineering Haadan branch, Islaic azad university, Haadan, Iran Abstract This paper addresses the siulation of the vibrational behavior of a flat-bottoed cylindrical reservoir in different levels of fluid vortex with the help of Vib-Shape software. The reservoir is equipped with a ixer, which creates fluid vortex and required loading. The copliance of the shell vibration ode shape of the entioned odel obtained fro the collected data and siulation in the aforeentioned software is copared with the ode shapes derived fro the Donnell s theory. The results are in the good agreeent with the governing theories of the proble. Keywords: Vibration, cylindrical reservoir, ixer, vortex fluid, ode shape.

2 195 1 Introduction T he dynaic behavior of the cylindrical reservoirs filled with fluid has a lot of applications in the industries where the cylindrical reservoirs are used for aintenance or storage of the liquids. When the vortex and rotation of the fluid is considered with the variable level of the fluid siultaneously, the effect of inertial coupling between the fluid and wall otion of the reservoir on the dynaic behavior and stability of the reservoir becoe ore proinent. Although nuerous papers have been published about cylindrical reservoirs epty or fluid-filled, or suberged in the fluid, this type of siulation about cylindrical reservoirs with variable fluid level, which contains vortexes due to the ixer, have not been carried out as required. The ai of this research is to siulate the vibration ode shape of the cylindrical reservoir resulting fro the existence of vortex fluid. In addition, the copliance with theoretical results of Donnell s theory is checked. The governing theory - Donnell s theory of shells The equation of the otion in the linear vibration of the circular cylindrical shell can be obtained by eliinating the nonlinear ters in the nonlinear equation of the otion. The equations of the otion in the Donnell s theory are as follows: u v 4 w w w vr w k R R x 4 4 x x 1v E Where: w R t k 1 1h R (1 c) X,r and θrepresents a cylindrical coordinate syste, R indicates the radius of the shell, ν is the Poisson's ratio and E is the Young's odulus. The answer is in the following for: 1 u C cos x cos n cos t v C sin x sin n cos t w C sin x cos n cos t 3 ) a( ) b( ) c) Where the λ=π/l, =1,, and n=0,1,,. According to the figures 1 and, and n are the nubers of axial and circuferential half-waves, respectively. Also, ω is the free angular frequency and t is the tie. )1 a( u 1 v u 1 v v w 1 v u R R vr R x x x E t Figure1. The nuber of circuferential waves (n) deterining the ode shapes. )1 b( v 1 v v 1 v u w 1 v v R R R x x E t Figure. The nuber of axial half-waves () deterining the ode shapes.

3 196 Equations ( a,b,c) describe the odes shape Ci, i=1,, 3 are the constant coefficients related to the doain. For every asyetric ode shape (ie, for n> 0) a secondary ode exists, which resebles the priary ode, but is perpendicular to it. As a result, the sine and cosine functions in the angular coordinate (θ) are transfored by π/n (according to the figure 3). This secondary ode causes the axiu vibrational doain accrues in the place, where the first ode has one node. Figure 3. Identical asyetric odes couple; integrated solid line: cosine ode; dashed line: sine ode; Tried line: nondefored shell 3 Description of the reservoir A thin-walled cylindrical shell (closed botto and open end) with the radius of , length of 345, wall thickness of 3.7 and the fluid density ofρf=1000 kg/3, is considered. The cylindrical shell is ade fro S.S with the echanical properties of ρsh= 700 kg/3, υ = 0.3 and E = 00 Gpa. direction. So, the resulting Flow Rangeis in the turbulent/transition range. Figure 5. The ixing propeller and it s location The reservoir pressure is atospheric pressure. The fluid in the reservoir is nonviscous and incopressible. In addition, the vibration of the plate of the reservoir botto due to the ixer support is neglected. Also, the distortion of the plate and the shell, and their effect on the vibrational behavior of the reservoir is ignored. 4 Experient schedule The vibration test of the entioned reservoir in the cited condition is perfored in three following cases: 1- The reservoir filled with the fluid up to 30% of the height and the rotational speed of 000 RPM. - The reservoir filled with the fluid up to 50% of the height and the rotational speed of 000 RPM. 3- The reservoir filled with the fluid up to 75% of the height and the rotational speed of 000 RPM. 5 Location of the sensor Figure 4. The cylindrical reservoir with a ixer The ixer propeller (below figure) is located in the center of the cylindrical shell and due to the structure, the Pitched Blade Turbine have puping action in the axial/radial 80 points are deterined for easuring the radial otion of the cylindrical shell under the radial loading of the fluid vortex and the resulting flow, which have been shown on the reservoir. During the test, the aount of the vibration and radial otion of these points is easuredbyan acceleration transducer in each of the three deterined cases.

4 197 6 Data processing and odal identification The data for the vibrational behavior of the cylindrical shell is captured with Easy-Viber Vibration Data Collector and then exained in the Vib-Shape and Spectra-Pro environent. The siulated odel of the reservoir is as the figure 6. In the right figure, Corner Point Nubers, and in the left one, the Corner Point Directions are presented. 80 points were specified for easuring the radial otion of the cylindrical shell under the radial loading of the fluid vortex and the resulting flow. The easuring unit for the radial otion is μ. The iage of the Vib-Shape environent, which shows the aniation of the shell in a oent, is presented for each of the three experients. Figure 8. Case - The reservoir filled with the fluid up to 50% of the height and the rotational speed of 000 rp. Figure 9. Case 3- The reservoir filled with the fluid up to 75% of the height and the rotational speed of 000 rp. 7 The obtained results due to the vibrational ode shape of the cylindrical reservoir Figure 6. The siulated odel of the reservoir. right figure, Corner Point Nubers, and in the left one, the Corner Point Directions. The iage of the Vib-Shape environent, which shows the aniation of the shell in a oent for Deviation Ani. Max condition are presented for each case in the figures 7,8 and 9. The following results are obtained in the three case of the experients: 1- In the case 1, the obtained ode shape has sufficient syetry and relatively good copliance with the results of Donnell s theory (according to figures 3 and 5). The vibration ode Figure 7. Case 1-The reservoir filled with the fluid up to 30% of the height and the rotational speed of 000 rp. - In the case, the obtained ode shape has relatively good syetry and fairly close copliance with the results

5 198 of Donnell s theory (according to figures 3 and 6). The vibration ode 3- In the case 3, the obtained ode shape has sufficient syetry and relatively good copliance with the results of Donnell s theory (according to figures 3 and 7). The vibration ode 4- Increasing the fluid level in the reservoir in the constant ixing rotation, the vibration value is increased in soe points copared to the lower fluid level (Except soe points). So, it can be concluded that the obtained results have adequate accuracy and are consistent with the Donnell s theory of shells. Aabili,Marco, (008), Nonlinear vibrations and stability of shell and plates, New York, Cabridge University Press. Askhari E., and Daneshand F.,(009), Coupled Vibration of a Partially Fluid-Filled Cylindrical Container with a Cylindrical Internal Body, Journal of Fluids and Structures,Vol. 5, Gonçalves Paulo B., N. del Prado Zenón J. G.,(010), Nonlinear Vibrations of Axially Loaded Cylindrical Shells Partially Filled With Fluid,Journal of Brazilian Conference on Dynaic, 9 th, Leissa, Arthur W.,(1973), Vibration of Shell,Washington D.C,National Aeronautics and Space adinistration. Michael P. Païdoussis, (014), Fluid-Structure InteractionsSlender Structuresand Axial Flow, Second edition, Volue 1, Elsevier Acadeic Press. Michael P. Païdoussis, (004), Fluid-Structure InteractionsSlender Structuresand Axial Flow, Second edition, Volue, Elsevier Acadeic Press. Strutt H. and Runzi D., (000), Handbook of EKATO ixing technology, Schopfhei, Gerany,EKATO Ruhr-Und Mischtechnik GbH ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 8 Suggestions Due to extensive use of storage and pressurized reservoirs in various industries, particularly oil, gas and petrocheical industry, and also according to the key role of these equipents in ters of production, process and safety issues in the cases dealing with the high process teperatures and pressures, it is suggested that the vibration behavior of these equipents would be studied in the case of ore weld seas, also investigating the vibration ode shapes in this case is suggested. References