Axisymmetric Tanks Research Articles

Vibration analysis of liquid in an axisymmetric tank covered by a diaphragm

Vibration analysis of liquid in an axisymmetric tank covered by a diaphragm

A NURBS-based isogeometric boundary element method for analysis of liquid sloshing in axisymmetric tanks with various porous baffles

An isogeometric boundary element method (IGA-BEM) based on the non-uniform rational B-splines (NURBS) is firstly performed to investigate the liquid sloshing in axisymmetric tanks with the porous baffles. The proposed method can completely maintain the advantages of the BEM that only the boundary of a domain requires discretization. By applying the NURBS basis functions it can exactly describe the geometry of the boundary. Meanwhile, it can also be obtained better solution field approximation at the domain boundary. Furthermore, as to the axisymmetric geometries of the containers considered in this paper, the 3-D liquid sloshing problems can be effectively reduced to 2-D ones on half of the cross-sections of the containers, which can significantly increase the computational efficiency. Meanwhile, the zoning method is employed in this paper to treat the arbitrary mounted porous baffles, and the Laplace equation is utilized as the governing equation of the potential flow model by assuming the fluid motion to be inviscid, irrotational and incompressible. Additionally, the weighted residual method together with the Green’s theorem is applied to develop the BEM integral equation. The natural sloshing frequencies and dynamic sloshing forces solved by the proposed method are compared with the available literatures and the traditional boundary element method (BEM). Good agreements are observed in the comparisons between numerical results and those of the existing literatures. And higher accuracy and convergence can be achieved by the proposed IGA-BEM method with significantly fewer nodes than the traditional BEM. Moreover, spherical tanks with the coaxial hemispherical, wall-mounted conical or surface-piercing cylindrical porous baffle, ellipsoidal tanks with spheroidal or surface-piercing cylindrical porous baffle, and the toroidal tank with tubular porous baffle are considered to investigate the effects of the porous-effect parameter, radius, length, height, horizontal and vertical semi-axes of the porous baffle on the sloshing characteristics (i.e. dynamic sloshing forces and surface elevations). The results show that the surface-piercing cylindrical porous baffle offers more noticeable suppression on sloshing response than the hemispherical and spheroidal porous baffles. Changing the radius of the tubular porous baffle has almost negligible effect on the sloshing force acting on the toroidal tank. The excitation frequency corresponding to the maximal value of sloshing force can be altered evidently by changing the porous-effect parameter of the porous baffle. In addition, choosing reasonable porous-effect parameter, radius, horizontal semi-axes and relatively larger length, height as well as vertical semi-axes for the porous baffles yields considerable suppression on the sloshing response.

Transient performance of a thermocline energy storage system using the two-energy equation model

This work investigates the thermal behavior of the charging cycle of a thermal energy storage system for a concentrated solar power plant filled with solid porous material. A transient model that describes turbulent flow in a hybrid medium (porous/clear) with both forced and natural convection was used. The mean flow macroscopic equations are developed based on the concept of double-decomposition. Governing equations were discretized using the SIMPLE method and the system of algebraic equations was relaxed by the SIP procedure. The k-ε turbulence model was used for modeling turbulence. The two-energy equation model was used to evaluate heat transfer between the solid and fluid phases. Simulations are based on an axisymmetric tank with external convection, hot fluid inlet at the top and distributor regions at the top and bottom of the tank. Effects of Reynolds number (Ret), porosity (ϕ) and permeability (K) were investigated. Additionally, storage (ηst) and charging (ηchg) efficiencies were compared to quantify the effectiveness of the charge. Consequently, a new thermal charge efficiency (ηth) was proposed to evaluate the charging cycle. Finally, it was found that the most efficient charging parameters were the ones with higher porosity and lower permeabilities. Moreover, increasing Ret values increased the thermal charge efficiency up to a maximum and thereafter presented a stabilized trend with further increasing of Ret values.

A Numerical Study on Influent Flow Rate Variations in a Secondary Settling Tank

The secondary settling tank is an essential unit for the biochemical treatment of domestic sewage, and its operational effect influences the quality of effluent. Under the influence of the confluence of rainwater and sewage, wastewater use habits, etc., the inflow of the secondary sedimentation tank changes over time. In this paper, OpenFOAM, an open-source computational fluid dynamics package, was used to study the dynamic behaviors of solid–liquid two-phase flow in the tank under influent flow rate variations. A coupled method including a mixture model, drift equation and a dynamic boundary method is proposed. Numerical investigations were carried out for a 2D axisymmetric sedimentation tank using 12 cases. With increasing influent flow rate, sludge accumulates continuously in the bottom left side of the tank, sludge hopper, and inlet; the sludge blanket thickness near the right end of the tank increases continuously; and the sludge concentration in the tank approximately linearly increases with time, with a low slope. The developed framework is generic and is, therefore, expected to be applicable for modelling sludge sedimentation processes.

Boundary element analysis of liquid sloshing characteristics in axisymmetric tanks with various porous baffles

This paper aims to investigate the effects of the porous baffles on the suppression of sloshing for the tanks with axisymmetric geometries under lateral excitation. Based on the assumptions of inviscid, irrotational, incompressible liquid and small amplitude sloshing, an axisymmetric boundary element method (BEM) for 3D Laplace equation is derived by using the Green's theorem together with the weighted residual method. And a zoning method is employed to model fluid domain in the tanks with complex porous baffles. Meanwhile, the porous baffles are treated motioning together with the tanks, and the velocity across the porous baffle is assumed to be linearly proportional to the pressure gradient between each side of the porous baffle. And the mechanism of suppressing the sloshing response is mainly the energy dissipation of the fluid passing through the porous baffle. Moreover, the linear free surface boundary conditions are also used to solve the governing equations. Compared with other numerical methods, the most prominent advantage of the BEM in solving axisymmetric potential problem is that only the boundaries of half the cross-section instead of the entire problem domain should be discretized, which can cut down large amount of memory and time costs. The present method is verified by comparing the numerical results with the existing literatures, and excellent agreements are obtained. Meanwhile, the proposed models are applied to investigate the effects of the porous baffles on sloshing response in circular cylindrical, annular cylindrical and conical tanks. The effects of the porous baffle length, porous-effect parameter, installation angle and baffle height on the sloshing force, natural frequency and surface elevation are studied. Additionally, some typical sloshing pressure distributions, velocity potential contours and velocity fields are plotted. The results show that swirls at the tips of the baffles can be observed in many cases, and the top-mounted porous baffle makes more significant suppression effects on sloshing response than that of bottom-mounted porous baffle, while increasing the number of ring porous baffles can achieve better restraint effects on sloshing response. And increasing the baffle length of the horizontal wall-mounted ring porous baffle can significantly decrease the sloshing frequencies, as well as the first non-dimensional natural frequency decreases with decrease in porous-effect parameter of the coaxial porous baffle. In addition, remarkable effects on sloshing can be obtained when reasonable designed by selecting the optimal porous-effect parameter, installation angle and baffle height. And this paper can be a useful guide for the seismic design and analysis of many actual liquid storage tanks (such as the Advanced Passive PWR, large water cooling tower, etc.).

Нормальные колебания жидкости, вытекающей из вращающегося бака

The purpose of the study was to analyze oscillations of liquid which partially or completely fills the fuel tank rotating around the longitudinal axis of the launch vehicle. The relevance of the problem is substantiated by the importance of assessing the effect of the intake, measuring, damping devices and other inner tank devices on the liquid fuel oscillations. The study introduces the formulation of problems on the motion of an incompressible rotating liquid flowing from an axisymmetric tank of arbitrary shape through the intake device, and gives a detailed study conducted for the fuel compartment in the form of a cylinder as the most common one. Finally, we give the solutions of the problems on normal, i.e. natural, oscillations of the liquid with boundary conditions on a free surface and a surface with resistance — the intake drain surface.

Investigation of thermal energy storage systems in concentrated solar power

This study presents computational fluid dynamics (CFD) modelling results obtained for the first cycle on a sensible thermal energy storage system based on rocks and pebbles freely poured into a mild steel tank. This study is primarily focused on proposing a plan to find an optimized particle size for the packed bed storage using various heat transfer fluids namely water, therminol-vp1, and air. The computational domain consists of an axisymmetric tank of cylindrical shape filled with a porous bed coupled with the wall. The governing equations have been solved for incompressible turbulent flow and fully developed forced convection, based on the two-phase transient model equation (SSTKW) to calculate the temperature of the fluid and solid phases. The porosity has been considered as 0.28 for the packed bed while the thermodynamic properties of both phases are temperature-dependent. The influence of the thermal dispersion within the porous bed, as well as the effective conductivity between the beads, were considered. The heat transfer coefficient was calculated according to the correlation for forced convection within porous media. Experimental temperature profile at the inlet of the bed is applied as a boundary condition in the simulation. Simulation results showed a good agreement with experimental data.

Sloshing in vertical and horizontal axisymmetric tanks

Sloshing in vertical and horizontal axisymmetric tanks

Overhead water tank shapes with depth-independent sloshing frequencies for use as TLDs in buildings

Sloshing water in the overhead water tank of a multi-storeyed building may be utilized to act as a tuned liquid damper for vibration control under wind and earthquake excitation. In conventional rectangular or circular water tanks, tuning presents difficulties as the sloshing frequency varies significantly with change in the depth of water in the tank. To address this issue, in this paper, we find shapes of tanks wherein the sloshing frequency is essentially independent of water depth over a large and useful range of water levels. Both two-dimensional as well as axisymmetric (three-dimensional) tank shapes are found. We use a direct boundary element method to find the sloshing frequencies in each case. In each case, a tentative simple analytical form for the tank shape is chosen with three free parameters, and these parameters are adjusted to obtain shapes where the first lateral sloshing frequency has negligible variation with water depth. For axisymmetric tanks, the circumferential (azimuthal) variation in field variables is restricted to the first harmonic, in the interest of lower computational effort. For both planar and axisymmetric cases, the working range of water depths is taken to be from 0.2 to 2 times the tank width. In both cases, the variation in first lateral sloshing mode frequency is found to be under 0.2% over the working range. In comparison, for constant width tanks such as the rectangular or circular ones, over the same range of water depths, the corresponding variation is more than 60 times greater.

Slosh Damping Caused by Friction Work Due to Contact Angle Hysteresis

The damping ratio of low-gravity sloshing induced by the friction work due to contact angle hysteresis is calculated using a newly developed semianalytical sloshing analysis method for arbitrary axisymmetric tanks. The damping ratio is estimated for the first mode for the case with small contact angle and its variation, as a fundamental study for propellant sloshing. First, the relation between the friction force and the change in the contact angle caused by contact angle hysteresis is derived from the equilibrium among the friction force and the three surface tensions. Next, the virtual work done by the friction force is calculated to determine the nonlinear damping term to be introduced into the modal equation for sloshing. Physical considerations are made for the dependencies of the damping ratio on the Bond number and filling level. Although the damping term arises from the second-order small term in the friction force, the damping ratio can reach the same order as the viscous damping ratio for low Bond numbers because, due to the hysteresis nonlinearity, the friction damping ratio increases with decreasing Bond number more markedly than the viscous ratio.

A comparison between CFD simulation and experimental investigation of a packed-bed thermal energy storage system

This work presents the comparison between CFD and experimental results obtained on a sensible thermal energy storage system based on alumina beads freely poured into a carbon steel tank. Experimental investigations of charging and discharging phases were carried out at a constant mass flow rate using air as heat transfer fluid. The experimental set-up was instrumented with several thermocouples to detect axial and radial temperature distribution as well as reservoir wall temperature. The experimental results were compared with those obtained from CFD simulations carried out with the FLUENT software. The computational domain consists of an axisymmetric tank of cylindrical shape filled with a porous bed coupled with the wall. The governing equations are solved for incompressible turbulent flow and fully developed forced convection, based on the two-phase transient model equation (LTNE-local thermal non-equilibrium) to calculate the temperature of fluid and solid phases. The porosity of the bed is considered variable in the radial direction, while the thermodynamic properties of both phases are temperature-dependent.The influence of the thermal dispersion within the porous bed, as well as the effective conductivity between the beads was considered. The heat transfer coefficient was calculated according to correlation for forced convection within porous media. Numerical results show a good agreement with experimental ones if thermal properties are considered temperature-dependent and the experimental temperature profile at the inlet of the bed is applied as a boundary condition in the simulations.

The shape of the fundamental sloshing mode in axisymmetric containers

We numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model and reduces the problem to an appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method for making images of an oscillating fluid.

A scaled boundary finite element approach for sloshing analysis of liquid storage tanks

The scaled boundary finite element method (SBFEM) is a promising alternative for sloshing analysis of liquid storage tanks. For linear sloshing problems associated with liquid storage tanks, in most cases, like the upright tanks of arbitrary uniform cross section, the horizontal tanks of arbitrary uniform cross-section and the axisymmetric tanks of arbitrary meridian shape, by employing SBFEM the sloshing analysis can be greatly simplified. Only one dimensional discretization is needed, thus the computational effort is reduced to a great extent, whereas high accuracy of the results is ensured. Numerical examples validate the efficiency and accuracy of the proposed approach.

Thermal modeling of hydrogen storage by absorption in a magnesium hydrides tank

This paper summarizes numerical results of hydrogen absorption simulated in an axisymmetric tank geometry containing magnesium hydride heated to 300 °C and at moderate storage pressure 1 MPa. The governing equations are solved with a fully implicit finite volume numerical scheme used by a commercial software FLUENT. The effect of the different kinetic reaction equations modeling hydrogen absorption was studied by the introduction of a specific subroutine at each time step in order to consider which one will provide results close to available experimental results. Spatial and temporal profiles of temperature and concentration in hydride bed are plotted. Results show that suitable method for our two-dimensional study is a CV-2D technique because it generates the smallest error especially during the beginning of the reaction. Also, its computational time is the shortest one compared to the other methods.

Mechanical Models of Low-Gravity Sloshing Taking Into Account Viscous Damping

Mechanical models of damped low-gravity sloshing are developed using a proposed analytical method for arbitrary axisymmetric tanks. It is shown that (a) the complex amplitudes of the force and moment caused by the conventional mechanical model do not coincide with the complex amplitudes of the force and moment calculated from the modal equation of sloshing and (b) these differences arise not only from the damping ratio but also from the surface tension although the surface tension does not cause energy dissipation. A mechanical model for correcting these differences is developed. The mass of this correction model is found to be equal to the mass of the liquid that fills the domain bounded by the meniscus and the plane that includes the contact line of the meniscus with the tank wall. With decreasing Bond number, the correction model mass as well as the damping ratio increase and, therefore, the correction becomes important. The force and moment caused by the conventional uncorrected mechanical model have phase lag with respect to the force and moment calculated from the modal equation of sloshing near the resonant frequency. Therefore, the correction is important for the dynamics and control analysis of a space vehicle.

Dynamic Stability of Plane Free Surface of Liquid in Axisymmetric Tanks

When liquid filled containers are excited vertically, it is known that, for some combinations of frequency and amplitude, the free surface undergoes unbounded motion leading to instability, called parametric instability or parametric resonance, while for other combinations the free surface remains plane. In this paper, the stability of the plane free surface is investigated theoretically when the vessel is a vertical axisymmetric container. The effect of coupled horizontal excitation on the stability is examined. The dynamics of sloshing flows under specified excitations are simulated numerically using fully nonlinear finite element method based on non-linear potential flow theory. A mixed Eulerian-Lagrangian technique combined with 4th-order Runge-Kutta method is employed to advance the solution in time. A regridding technique based on cubic spline is applied to the free surface for every finite time step to avoid possible numerical instabilities.

Studying the coupled eigenoscillations of an axisymmetric tower-elevated tank system by the multimodal method

Employing the virtual work variational principle and the linear multimodal method for the liquid sloshing in an axisymmetric tank, we study coupled eigenoscillations of a tower and an elevated tank partially filled by a liquid. An emphasis is placed on the case of an upright circular cylindrical tank. Theoretical results are compared with known experimental data.

Viscous Damping Ratio of Low-Gravity Sloshing in Arbitrary Axisymmetric Tanks

A review of earlier studies in the area of sloshing shows that, although there are extensive studies on the main flow analysis, theoretical investigations of viscous damping are scarce, especially for arbitrary axisymmetric tanks. In this paper, an analytical method for calculating the viscous damping ratio of low-gravity sloshing in arbitrary axisymmetric tanks is presented, and its dependence on the Bond number and the filling level is examined. Using two systems of spherical coordinates, an analytical expression of bidirectional flow in the boundary layer is derived. By substituting this expression into the virtual work expression of the Navier–Stokes equations’ viscous terms, the modal damping term is determined. It is shown that as the Bond number decreases the damping ratio increases, due to prolongation of the natural period rather than to increase in the damping coefficient. A notable finding is that for high filling levels the damping coefficient decreases with decreasing Bond number, unlike the damping ratio. The reason for this is explained by introducing a correlation parameter whose Bond number dependence is similar to that of the damping coefficient. The analysis is verified by comparing with earlier experimental results.

A successive boundary element model for investigation of sloshing frequencies in axisymmetric multi baffled containers

This study presents a developed successive Boundary Element Method to determine the symmetric and antisymmetric sloshing natural frequencies and mode shapes for multi baffled axisymmetric containers with arbitrary geometries. The developed fluid model is based on the Laplace equation and Green's theorem. The governing equations of fluid dynamic and free surface boundary condition are also applied to proposed model. A zoning method is presented to model arbitrary arrangement of baffles in multi baffled axisymmetric tanks. The influence of each zone on neighboring zones is applied by introducing interface influence matrix which correlates the velocity potential of interfaces to their flux. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. The proposed method has a considerable effect on decreasing computational cost and a good accuracy in determining the sloshing natural frequencies. The obtained results for different types of container based on the application of the presented study are validated in comparison with the literature and very good agreement is achieved. Finally, the effect of baffle parameters on the sloshing natural frequencies was investigated and some conclusions are outlined.

Theoretical Determination of Modal Damping Ratio of Sloshing Using a Variational Method

This paper investigates a variational method for theoretically determining the first damping ratio of sloshing in cylindrical and arbitrary axisymmetric tanks. In this method, a virtual work expression for the viscous terms in the Navier–Stokes equations is transformed into the first-mode damping term, thereby extending Hamilton’s principle for nonviscous sloshing to a variational principle of viscous sloshing. By applying the Galerkin method to the variational principle, a computationally efficient analysis is conducted. For arbitrary axisymmetric tanks, a method for reducing the influence of the change in the fluid velocity boundary condition on the damping estimation is investigated. The proposed method provides theoretical foundations for past empirical results for cylindrical and spherical tanks.