Flat-bottom Cylindrical Tanks Research Articles

Development and application of 3D-PTV measurements to lab-scale stirred vessel flows

In this work, guidelines for reliable 3D-PTV measurements of fluid flows in stirred vessels at the lab-scale have been developed. The flow of water at Re=12,000 in a flat bottom cylindrical tank (T=180 mm) stirred with a 6 blades Rushton turbine (c=D=T/3) has been measured at different camera frame rates (125 – 3600 fps) and tracer concentrations (0.001 – 0.010 px−2). The best compromise between the number of measured data, tracking efficiency and CPU time (206 frame-1, 39% efficiency, 0.026 min frame-1) has been obtained at 125 fps and 0.002 px−2. These results can be expressed in terms of the normalized parameters φ∼0.5 and p∼2 and scaled to different experimental conditions. The Savitzky–Golay filter, used to enhance the measurements signal-to-noise ratio, has been optimized by testing different values of the polynomial order (0–3) and filter width (321 data points). A 2nd order, 11-points filter gave the best results, based on considerations regarding the reduced Chi-squared and velocity distributions. With the best conditions and filter, the uncertainty in the measured tracer positions was in the order of 255 μm. Finally, un unbiased distribution of the flow decorrelation time has been determined from the velocity autocorrelation functions along the trajectories longer than 6 impeller revolutions. This method could be used to compare the macro-mixing performance in different flow systems.

Fluid Pressures on Unanchored Rigid Flat-Bottom Cylindrical Tanks Under Action of Uplifting Acceleration

To protect flat-bottom cylindrical tanks against severe damage from uplift motion, accurate evaluation of accompanying fluid pressures is indispensable. This paper presents a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank in the same manner as the procedure outlined and discussed previously by the authors (Taniguchi, T., and Ando, Y., 2010, “Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration,” ASME J. Pressure Vessel Technol., 132(1), p. 011801). With perfect fluid and velocity potential assumed, the Laplace equation in cylindrical coordinates gives a continuity equation, while fluid velocity imparted by the displacement (and its time derivatives) of the shell and bottom plate of the tank defines boundary conditions. The velocity potential is solved with the Fourier–Bessel expansion, and its derivative, with respect to time, gives the fluid pressure at an arbitrary point inside the tank. In practice, designers have to calculate the fluid pressure on the tank whose perimeter of the bottom plate lifts off the ground like a crescent in plan view. However, the asymmetric boundary condition given by the fluid velocity imparted by the deformation of the crescent-like uplift region at the bottom cannot be expressed properly in cylindrical coordinates. This paper examines applicability of a slice model, which is a rigid rectangular tank with a unit depth vertically sliced out of a rigid flat-bottom cylindrical tank with a certain deviation from (in parallel to) the center line of the tank. A mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with Fourier series. It well converges with a few first terms of the Fourier series and accurately calculates the values of the fluid pressure on the tank. In addition, the slice model approximates well the values of the fluid pressure on the shell of a rigid flat-bottom cylindrical tank for any points deviated from the center line. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. The proposed mathematical and graphical methods are cost effective and aid in the design of the flat-bottom cylindrical tanks that allow the uplifting of the bottom plate.

Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration

Although uplift motion of flat-bottom cylindrical shell tanks has been considered to contribute toward various damages to the tanks, the mechanics were not fully understood. As well as uplift displacement of the tanks, fluid pressure accompanying the uplift motion of the tanks may play an important role in the cause of the damage. An accurate estimate of the fluid pressure induced by the uplift motion of the tanks is indispensable in protecting the tanks against destructive earthquakes. As a first step of a series of research, this study mathematically derives the fluid pressure on a rigid rectangular tank with a unit depth accompanying angular acceleration, which acts on a pivoting bottom edge. The rectangular tank employed herein is equivalent to a thin slice of the central vertical cross section of a rigid flat-bottom cylindrical shell tank. Assuming a perfect fluid and velocity potential, a continuity equation is given by the Laplace equation in Cartesian coordinates. The fluid velocities accompanying the motions of the walls and bottom plate constitute the boundary conditions. Since this problem is set as a parabolic partial differential equation of the Neumann problem, the velocity potential is solved with the Fourier-cosine expansion. The derivative of the velocity potential with respect to time gives the fluid pressure at an arbitrary point inside the tank. A mathematical solution for evaluating the fluid pressure accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with the Fourier series. The proposed mathematical solution well converges with a few first terms of the Fourier series. Values of the fluid pressure computed by the explicit finite element (FE) analysis well agrees with those by the proposed mathematical solution. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. Consequently, the mathematical solution given by the Fourier series converges easily and provides accurate evaluation of the fluid pressures on a rigid rectangular tank accompanying the angular acceleration acting on the pivoting bottom edge. Irregularity in the fluid pressure distribution increases as the tank becomes taller.

Fluid pressure on unanchored rigid flat-bottom cylindrical tanks due to uplift motion and its approximation

As well as the uplift displacement of the tanks, the fluid pressure accompanying the uplift motion of the unanchored tanks has been considered to contribute toward their various damages. Its accurate estimate is indispensable to secure the safety of the tanks at an earthquake event. Assuming the ideal fluid and velocity potential, a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration that acts on a pivoting bottom edge is derived. In actual case scenarios, practitioners should calculate the fluid pressure accompanying the rocking motion of the tank which possesses a crescent-like uplift region in a plan view of the bottom plate. To satisfy this demand, this paper also proposes an approximation method featuring a rectangular tank with a unit depth off and parallel with the center line of the rigid flat-bottom cylindrical tank, denoted a slice model. The mathematical solution is given by the explicit function of aspect of the tank but is a Fourier series. It well converges with a few first terms of the Fourier series. A comparison with the FE analysis with a three-dimensional half tank model reveals the analytical accuracy for evaluating values of the fluid pressure on the rigid flat-bottom cylindrical tank. In addition, the slice model reasonably approximates the values of the fluid pressure on the shell off and parallel with the center line of the rigid flat-bottom cylindrical tank. For designer’s convenience, diagrams that depict the normalized fluid pressure are also presented. The curved component of the fluid pressure distribution increases as the tank becomes taller or the azimuth angle approaches to 0 or 180 degrees.

平底円筒貯槽の浮上り挙動への陽解法の適用性に関する基礎的研究

The uplift of the flat-bottom cylindrical shell tank has been considered to contribute to the various damage of the tanks. The motion of the free surface and the fluid pressure characterize dynamical effects of the fluid on the tank induced by the uplift motion. Therefore, to properly evaluate the seismic-resistance performance of the tank, account must be made of these behaviors mathematically and numerically. The present paper examines the applicability of the explicit FE-Analysis to numerically compute these behaviors. As we shortly see, the explicit FE-Analysis accurately predicts the motion of the free surface observed during the free rocking test of the tank and the fluid pressure on the tank mathematically derived based on Laplace equation and associate boundary conditions.

Seismic design rules for flat bottom cylindrical liquid storage tanks

A set of design rules have been developed that permit a detailed assessment to be made of large flat bottomed cylindrical bulk storage tanks subject to seismic loading. It is proposed that these should form an Appendix to the new British Standard BS7777. Equations and appropriate coefficients are developed to assess the stresses in the tanks shell due to impulsive and convective liquid mass effects, including hoop and axial stresses. Appropriate stress limits are proposed for both OBE and SSE events. Lift-off of the bottom due to rocking is considered, including limits on annular plate width and local peak foundation loads. A worked example is included showing use of the equations that are proposed for BS7777.