Optimization of Baffled Rectangular and Prismatic Storage Tank Against the Sloshing Phenomenon

The fluid motion in partially filled tanks with internal baffles has wide engineering applications. The installation of baffles is expected to reduce the effect of sloshing as well as the consequent environmental damages. In the present study, a series of experimental tests are performed to investigate the sloshing phenomenon in a baffled rectangular storage tank. In addition, the sloshing phenomenon is also modeled by using OpenFoam. Based on the experimental and numerical studies, optimization of the geometric parameters of the tank is performed based on some criteria such as tank area, entropy generation, and the horizontal force exerted on the tank area due to the sloshing phenomenon. The optimization is also carried out based on the entropy generation minimization analysis. It is noted that the optimum baffle height is in the range of hb/hw=0.5–0.75 in the present study (where hb and hw are the baffle height and water depth, respectively). Based on the results, the optimal design of the tank is achieved with RA= 0.9–1.0 (where RA=L/W, L and W are the length and width of the tank, respectively). The results also show that the increase of hb can lead to a decrease of the maximum pressure and horizontal force exerted on the tank. It is also noted that the horizontal force exerted on the tank firstly continues to increase as the sway motion amplitude increases. However, as the normalized motion amplitude parameter, a/L (The parameter a is the motion amplitude), exceeds 0.067, the effect of motion amplitude on the force is not obvious. The same optimization is also performed in the multiple-variable-baffled tank and prismatic storage tank.


Introduction
A rigid tank partially filled with liquid fluid is involved in various engineering projects, such as liquefied natural gas carriers, road tankers, seagoing vessels, aerospace vehicles, elevated water towers and petroleum cylindrical tanks. Sloshing phenomenon is one of the major concerns in the design of liquid storage tanks, which has been studied extensively using both numerical and experimental methods. If the liquid fluid is assumed to be inviscid, its sloshing in the tank can be simulated using potential flow theory, such as Ketabdari and Saghi (2012, 2013a, 2013b, 2013c, Ketabdari et al. (2015), , and Saghi (2016). To overcome the inability of potential flow theory, the Reynolds-averaged Navier−Stokes (RANS) equations were used as the governing equations in recent numerical studies (Chen and Nokes, 2005). The shape of the storage tank is an important factor that affects sloshing. The performance of different geometric shapes of storage tanks such as rectangular (Huang et al., 2010), elliptical (Hasheminejad and Aghabeigi, 2012), cylindrical (Papaspyrou et al., 2004), circular conical (Gavrilyuk et al., 2005), spherical (Yue, 2008) and trapezoidal tanks (Saghi, 2016) were investigated by researchers. Installing baffles properly was reported to be an efficient way to reduce sloshing. Special attention was paid to the effect of vertical baffles on the natural frequencies of the tank. Ning et al. (2012) used the higher-order boundary element method (HOBEM) to investigate the effect of vertical baffles on water sloshing in a 2D rectangular storage tank under the coupled horizontal and vertical motion. Jung et al. (2012) numerically compared the surface elevation in a baffled rectangular tank with different ratios of baffle height to initial water depth. They found that the minimum elevation of the free surface occurs when the ratio is around 0.9.
Upper mounted baffles were also used by some other researchers such as Goudarzi and Farshadmanesh (2015), who numerically analyzed the effects of dimensions and locations of upper mounted baffles on water sloshing in tanks and reported the maximum waver height can be reduced by 50%. Kolaei et al. (2015) reported top-mounted baffles suppress sloshing better than bottom-mounted and centermounted partial baffles in the cylindrical storage. However, the center-mounted baffle is more effective when the water depth is near 50% of tank height. The bottom-mounted baffle is effective only at very low filling-depth.
The effectiveness of horizontal baffles to reduce the sloshing impacts in the storage tanks has also investigated. Sanapala et al. (2018) numerically simulated the sloshing due to vertical harmonic and seismic excitations in a rectangular water tank with two horizontal baffles fixed to the sidewalls. The results showed that better sloshing suppression is sensitive to the baffle location rather than baffle width. Panigrahy et al. (2009) reported that the ring baffles are more effective in reducing sloshing in comparison with the vertical or horizontal baffles. Biswal et al. (2003) found that an annular baffle in a cylindrical storage tank is more effective when it is installed near the free surface, and the flexibility of the baffle is not an important parameter. Wang et al. (2019) reported that, under pitching excitations, the free surface oscillation reduces when a rigid annular baffle in a circular cylinder tank moves close to the free surface. Using experimental method, Akyıldız et al. (2013) found that ring baffles are very effective in reducing the sloshing loads on a cylindrical tank with roll motion. Wang et al. (2016) installed the T-shaped baffles consisting of bottommounted baffle and surface-piercing baffle in horizontal elliptical tanks to reduce the sloshing effect. They found that the bottom-mounted baffle possesses little effect on the motion mode in high frequencies and the length of the horizontal baffle has large effect on suppressing the sloshing near the free surface. Cho et al. (2017) numerically studied the sloshing reduction in a swaying rectangular tank by installing a horizontal porous baffle. They showed that horizontal porous baffles installed at the side walls are more effective to suppress the sloshing in comparison with the one installed at the center. Kuzniatsova and Shimanovsky (2016) provided the guidelines on the optimal design of the hole size and hole area on the performance of perforated baffles in reducing the liquid oscillation. Floating baffles are another kind of baffles used by some researchers to reduce the sloshing effect in liquid storage tanks (Koh et al., 2013). Yu et al. (2019) reported that the perforated floating plates with the median solidity ratio is the most optimal one in suppressing the sloshing. some researchers such as Hwang et al. (2016) used the elastic baffles as the sloshing damping device, or elastically mounted rigid baffles (Kim et al., 2018).
In the last decade, the entropy generation minimization analysis has been widely used as an effective tool for the purpose of optimization. This method was firstly used by Bejan (1979Bejan ( , 1987, and then was employed by other researchers Lakzian, 2017, Saghi, 2018;Saghi and Lakzian, 2019). However, in contrast to the numerous studies of the entropy generation of internal flows, the entropy generation has rarely been used for the optimization related to the external flows such as the sloshing phenomenon. So, in the current study, the novel points of this paper is to use some criteria consisting of entropy generation for optimizing baffled and prismatic tanks against sloshing phenomenon.

Experimental setup
A 135 cm long and 80 cm wide shake table (see Fig. 1a) is used to conduct a series of experiments of the water sloshing phenomenon in a rectangular tank at Hakim Sabzevari University (see Fig. 1b). The shake table body is made of steel plate with 8 mm thickness, equipped with a hydraulic power point system with an electro motor with 5.7 Hp power, and operated at 1 450 rpm angular velocity. The shake table can generate a harmonic oscillation with the amplitude ranging from 0 to 0.07 m, and frequency below 10 Hz. A transparent 0.45 long, 0.15 m wide and 0.3 m high rectangular tank filled with dyed water is fixed at the center of the oscillation platform (see Fig. 1b). The tank walls and the baffles are made of 0.3 cm thick Plexiglas sheets, so that the tank can be considered rigid. During the experiment, the platform oscillates harmonically along the length direction Hassan SAGHI et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 664-676 665 of the water tank. The "Aoao Video to Picture Converter" and "Plot digitizer" softwares are used to extract the freesurface-shape from the images recorded by a digital camera placed in front of the shake table.

Numerical method in OpenFoam (OF)
The water is considered as viscous, incompressible and the flow inside the tank is laminar. Therefore, the Navier−Stokes equations are used as governing equations as follows (OpenFoam, 2019): (1) where is the fluid density; the velocity vector; the time; the gravitational acceleration; and is the dynamic pressure. In the modelling, the air and water are separated by an interface. The volume of fluid (VOF) method has been used to describe the phase movement. In this method, the parameter α is defined as the quantity of water per unit of volume in each cell so that (Rudman, 1997): Based on this parameter, the fluid density and the dynamic viscosity of the cell are computed as follows : The solver of Multidimensional Universal Limiter for Explicit Solution (MULES) is used to solve the above equation and calculate α in all the cells at each time step.
Horizontal force exerting on the tank is estimated as follows: where n x is the component of the unit vector perpendicular on the tank surface in the x direction, and is the element area on the wet tank surface (A).

InterFoam solver in OpenFoam
In this paper, OpenFoam (OF) in version 7.0 is used. Among the different solvers integrated in OF, the in-terDyMFoam solver takes the advantages of handling dynamic meshes for moving surfaces. In addition, interDyM-Foam can solve the three-dimensional Navier−Stokes equation for two phases using a finite volume discretization and VOF method. The solver algorithm used by InterFoam/In-terDyMFoam is called PIMPLE, and is a combination of PISO and SIMPLE algorithms. Owing to its various advantages, interDyMFoam is used in the present study (Open-Foam, 2019). The maximum courant number of 0.5 is also considered as the criterion to adjust the time step.

Entropy generation (EG)
The irreversible aspects of the second law of thermodynamics were used as entropy generation (EG) analysis to investigate the effects of the new design on losses reduction. The irreversibility for Newtonian fluids is introduced as local EG by Bejan (1979): where T em is the fluid temperature, k is the fluid thermal conductivity, μ is the fluid dynamic viscosity, and is EG rate . In this study, the temperature is considered as the ambient temperature (293 °K). The temperature gradient often is negligible in the rectangular storage tanks. Thus, the present study focuses on the isothermal flow in the sloshing phenomenon; only the EG created by the fluid friction is considered. The viscous dissipation coefficient is expressed as follows: Thus, the total EG can be integrated as follows: The accumulated EG (S gen-,cum ) and average EG at time t are evaluated as:

Mesh size independency
The dependence of the numerical results on the mesh size is then examined. To achieve this, a rectangular tank with 0.45 m in length, 0.3 m in height and 0.15 m in width was used. The water depth in the tank was 0.06 m. The tank undergoes harmonic sway motion with amplitude of 0.04 m and angular frequency of 5 rad/s. The horizontal forces exerted on the side wall of the tank based on meshes with different precisions are compared with each other. The results for different mesh sizes are shown in Fig. 2.
Based on the results, there is no considerable change in the results for the mesh sizes smaller than dx = dy = dz= 0.001 m. It means that the results are independent of the mesh sizes smaller than dx = dy = dz = 0.001 m. Therefore, these mesh sizes are considered in all the subsequent calculations.

Model validation
The numerical model was validated by comparing numerical results with the experimental data. A 0.45 m long, 0.3 m high and 0.15 m wide rectangular tank is filled with water to 0.06 m in depth. The tank undergoes harmonic sway motion with a 0.04 m in amplitude and a 5 rad/s in angular frequency. Experimental and numerical results of the surface elevation distribution along the tank at different time are shown in Fig. 3.
Then, the free surface elevations along the tank were extracted from the photos shown in Fig. 3 to make quantitative comparison in Fig. 4. It can be seen that there is a good agreement between the numerical and experimental results.
In another test, a 0.12 m high, 0.15 m wide and 0.003 m thick baffle is vertically installed at the middle of the tank. Then, the tank undergoes harmonic sway motion with 0.04 m in amplitude and 5 rad/s in angular frequency. Experi- Fig. 2. Horizontal force on the tank to evaluate the mesh size independency. Fig. 3. Snapshot of the free surface in the rectangular tank.
Hassan SAGHI et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 664-676 667 mental and numerical results of the surface elevation distribution along the tank at different time are shown in Fig. 5. From the figures, it can be seen that air is trapped in the rolled free-surface at the tip of the baffle and the vortex is formed at the right side of the baffle at t=4−12 s. Especially, the vortex becomes the deepest at t=12 s and can be easily captured in the numerical figure. Violent sloshing phenomena can be observed numerically and experimentally. Then, the free surface elevations along the tank were extracted from the photos shown in Fig. 5 to make the comparison more explicitly, and the results are given in Fig. 6. It can be seen that there is a good agreement between the numerical and experimental results in the case of the baffled tank.
In the above model validation, the surface-elevation distributions are compared with those obtained from the experiments. It will be more persuasive if the comparison for the hydrodynamic pressure is provided. So, a two-dimensional numerical modelling carried out by Xue et al. (2019) was used here. The tank has a size of 0.48 m in length, 0.33 m in height, and the water depth of 0.09 m. A sway motion of amplitude a=0.1 cm and 5.83 rad/s angular frequency were exerted on the tank. The pressure on the corner of the right wall of the tank was estimated, and the results are shown in Fig. 7. It can be seen that there is a good agreement between the results.  Fig. 9.
Owing to the strong motion of the tank, both the pressure and horizontal wave force are not steady with apparently strong transient effects for the case of short baffles. However, the results reach the steady state and are decreased to the minimum when the baffle height increases to 0.06 m, which is also due to the baffle inducing the viscous effects and change of nature frequency of the tank.
In this step, the maximum pressure (P max ), the decrement percentage of maximum pressure (D P ), the maximum horizontal force (F max ), and the decrement percentage of maximum horizontal force (D F ) were calculated for different baffle heights as follows: where F max-wb is the maximum horizontal force exerting on The results show that an increase of h b can lead to a decrease of the maximum pressure and horizontal force exerted on the tank due to the sloshing reduction with the baffle height increase. It can also be seen that the pressure and force decrement for the baffles h b /h w in the range of 0.5−0.75 is nearly constant. It means that the increment of baffle height does not affect considerably the sloshing phenomenon. Therefore, the optimum baffle height is sugges-ted in the range of h b /h w =0.5−0.75 by combining the economic effects, i.e., large cost for long baffle.
Herein, the time sequence of velocity vectors for a sway motion with 0.04 m and 1.33 s period, and for h b /h w =1/6 and 2/3 is illustrated in Fig. 12.
The results show that by increasing the baffle height, there are some vortices behind the baffle after the flow being separated from the baffle tip. After a while, the vortex becomes larger as shown in Fig. 12b (t=15 s), which can also induce more viscous effects and result in smaller sloshing force.

Effect of sway motion amplitude
The effect of motion amplitude on the sloshing process is evaluated. The size of the tank keeps the same as that in    Fig. 13. For the harmonic motion of the tank, its period is fixed at 1.36 s, while its amplitude increases from 0.001 m to 0.04 m. The parameter D F was calculated for different sway motion amplitudes shown in Fig. 14.
The results show that as the motion amplitude increases from 0.01 m to 0.03 m, the horizontal force exerted on the tank continues to decrease, which is in accordance with the phenomenon of relative force difference between that with and without baffle decreasing with the motion amplitude shown in Fig. 13. While, as the motion amplitude exceeds 0.03 m, the effect of motion amplitude on the horizontal wave force is not obvious for the proposed baffle.   m 2 . The water depth in the tank was 0.06 m. The tank undergoes harmonic sway motion with 0.04 m in amplitude and 5 rad/s in angular frequency. The n-th order natural frequency of a rectangular container is estimated as (Jung et al., 2015): ω = 5 rad/s) Based on Eq. (13), the first natural frequencies of the tank with different lengths (L=0.15−0.45 m) are in the range of 15−25 rad/s. So, the adopted angular frequency ( is far from the natural frequencies of the tank to prevent the resonance phenomenon. To optimize the rectangular storage tank, the horizontal force exerted on the tank is calculated for different tank areas (A T ) and the results with respect to R A (= L/W) are shown in Fig. 15a. Entropy Generation (EG) analysis is also performed for the optimal design of the rectangular storage tank. The parameter is estimated by using Eq. (9), and the results of with respect to R A are shown in Fig. 15b.
The results show the horizontal force exerted on the tank perimeter (F xmax ) and the cumulative entropy generation (S gen,cum ) achieve the minimum in the range of R A = 0.9−1.0, i.e., the tank shape being similar to the square. Therefore, R A = 0.9−1.0 can be suggested for the optimum design of the rectangular storage tank.

Effect of baffle on the optimum storage tank
A rectangular storage tank with an optimum R A , i.e., R A = 0.92 in Fig. 15, and equipped with a baffle of 0.06 m in height is considered. The condition without a baffle has also been considered for the purpose of comparison. The comparison is shown in Fig. 16. Initially, the difference of EG between that with and without baffle is quite small. The baffle effect is quite obvious and the related EG value is quite small as the time is larger than 11 s. The increase of EG values is dramatic over time in the tank without baffle. This phenomenon may be related to the violent free surface  motions experienced in a containment system without baffles.
The distributions of entropy generation in tanks with and without baffles are shown in Fig. 17 at selected time steps. It can be seen that the distribution of EG in a rectangular storage tank with baffle is less obvious than that without baffle at t=5 s and 10 s, but quite oppositely at t=15 s, 20 s and 25 s, which is consistent with those in Fig. 16.

Evaluation of multiple-variable-baffled tank
Multiple-baffle has been employed by some researchers to mitigate the sloshing effect on the tank. For instance, Chu et al. (2018) experimentally studied the sloshing phenomenon in a rectangular water tank with multiple bottom-moun-ted baffles. Multiple-baffle is also used by some researchers such as Shekari (2020) in flexible cylindrical container to scrutinize the sloshing response subjected to lateral excitations. But the baffle height is constant in the previous researches. So, in this section, we evaluate the effects of the baffle height in a multiple-variable-baffled tank. A sway motion with 0.04 m in amplitude and 1.36 Hz in frequencywas exerted on a rectangular storage tank with 0.45 m in length, 0.15 m in width, and 0.3 m in height. The water depth is considered as 0.12 m. Five variable baffles with different heights were considered and positioned in the tank in the present study, in which the '3-9-12-9-3' type baffles are shown in Fig. 8. The horizontal forces exerted on the tank were estimated for different type baffle arrangements, and the results are given in Fig. 18.
The main aim of this section is to compare of the effect of the baffles at different locations. Firstly, the horizontal forces exerted on the tanks with the types '3-6-9-6-3' and '3-6-12-6-3' are compared. As shown in Fig. 19a, by increment of 33% of the height of the baffle B 1 (Only one baffle), F x is decreased around 30%. Then, the baffle '3-9-12-9-3' is considered and compared with the type '3-6-12-6-3'. It can be seen that F x decreases around 20% by increment of 33% of the height of the baffle B 2 (two baffles) (see Fig. 19b). Finally, the results in Fig. 19c show that by increment of 33% of the height of the baffle B 3 (two baffles), F x decreases around 10%. As the baffle B 1 is one and the  Hassan SAGHI et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 664-676 baffles B 2 and B 3 are two, the effect of B 1 on sloshing reduction is three times that of B 2 and six times that of B 3, respectively.

Optimization of prismatic storage tank
In this section, the prismatic storage tanks with different parameter a (see Fig. 20) are considered so that the heights of the tanks are equal to that of the optimum rectan-gular storage tank, i.e., the aspect ratio (R A =L * /W=0.92). Moreover, the volume of the tanks is the same with each other. Based on these, the dimensions of the prismatic tanks can be determined and summarized in Table 1. As both water volumes are equal and the water depth in the rectangular storage tank is 0.06 m, the water depth (h w ) in the prismatic tank can also be calculated.
In Table 1, a is the horizontal length of the truncated area, L* is the total length of the tank (L*=L+2a), L is the length of the bottom, W is the width, h w is the water depth, and A T is the tank area.
Then, a sway motion with 2 cm in amplitude and 4.6 rad/s in angular frequency was exerted on the prismatic storage tank. The horizontal force and EG were calculated. In order to find the optimum shape for the prismatic storage tank, the cumulative entropy generation (S gen,cum ) and maximum horizontal force exerted on the tank were calculated as shown in Fig. 21. From the figure, it can be seen that the parameter a of the optimum chambered storage tank can be found in the range of a=0.025−0.03. Therefore, the dimensionless length of the horizontal length of the truncated area  18. Schematic sketch of the '3-9-12-9-3' type baffles (unit: cm).  is suggested as a/L in the range of 0.13−0.15.

Conclusion and Discussion
In this paper, a series of experimental tests have been performed to investigate the sloshing in rectangular storage tanks. The sloshing phenomenon has also been modeled by using OpenFoam. In addition, efforts have also been made on the optimal design of a rectangular tank based on some criteria associated with the horizontal force exerted on the tank perimeter as well as the entropy generation analysis. The results reveal that the optimum baffle height is in the range of (h b /h w =0.5−0.75) in which h b and h w are the baffle height and the depth of the filled water, respectively. Based on the results, the optimal design of the tank is achieved when R A = 0.9−1.0. An increase of h b can lead to a decrease of the maximum pressure and horizontal force exerted on the perimeter of the tank. The horizontal force exerted on the tank perimeter firstly continues to increase as the sway motion amplitude increases. As the normalized motion amplitude parameter a/L exceeds 0.067, the effect of motion amplitude on the force is not obvious. In a multiplebaffled tank, the baffle position at the middle possesses the most efficient sloshing reduction in comparison with the other positions. Besides the rectangular tank, the proposed optimization method also can be adopted in the complicated tank, such as prismatic tank.
It should be noted that the pressure and velocity measurements were not considered in the experiments because the touch sensors in the lab were not sensitive and accurate enough under such violent sloshing conditions, which was the shortcoming in the present study. The same experiments will be performed to measure the velocity field by using the high-precision optical sensors in the coming future.