Hassan Saghi, Dezhi Ning, Shunqi Pan,3 and Reza Saghi

Received:03 September 2021/Accepted:24 October 2021

?Harbin Engineering University and Springer-Verlag GmbH Germany,part of Springer Nature 2022

Abstract A dual-baffled rectangular tank with different configurations is proposed to reduce the sloshing effect, and design optimization is conducted through numerical simulations with open-source software, namely OpenFOAM, based on the computational fluid dynamic model.A series of physical experiments in the dual-baffled rectangular tank is performed for model validation and design optimization with the measured water surface elevation distributions along the tank. The optimization uses the calculated maximum horizontal force exerted on the tank and entropy generation (EG) as the criterion. Results show that the dual-baffle configuration positioned at the tank center is more effective in reducing the sloshing than that of the single baffle when the relative baffle height and initial water depth ratio(Hb/Hw,where Hband Hw represent baffle height and static water depth, respectively) are larger than 0.5. However, such an effect then diminishes when the ratio is larger than 0.75. The effect of the dual-baffle configuration on the sway motion under the condition of different motion amplitudes and frequencies is also evaluated. The results show that the reduction in the maximum horizontal force is almost the same for dual- and single-baffled configurations and reaches the minimum when the sway motion amplitude is near 0.03 m. The dual-baffled configuration for the angular frequency of the sway motion is more effective than the single-baffled in reducing the sloshing at the low angular frequencies but is only less effective at high angular frequencies. Furthermore, the optimal baffle inclination angle is 85° when the inclined straight and curved baffles are used,and curved baffles can successfully decrease the horizontal force exerted on the tank and EG.

Keywords Rectangular storage tank;Dual-baffled rectangular tank;Sloshing phenomenon;Optimization;Horizontal force;Entropy generation

1 Introduction

Fluid sloshing is a free surface oscillation of the con‐tained fluid due to the impulsive loads in a partially filled tank(Abramson 1996).This phenomenon is one of the ma‐jor concerns in the design of dynamic systems, such as aerospace vehicles, road tankers, seagoing vessels, lique‐fied natural gas carriers, elevated water towers, and cylin‐drical petroleum tanks. Numerical and experimental stud‐ies on sloshing have been performed by many researchers.The fluid was considered to be inviscid (Frandsen 2004;Ketabdari and Saghi 2012a; 2012b; Ketabdari and Saghi 2013a; 2013b; 2013c; Ketabdari and Saghi 2015; Saghi 2016; Saghi and Ketabdari 2012) and viscous (Chen and Nokes 2005; Wu et al. 2012) in the numerical modeling.Therefore, Laplace and Reynolds-averaged Navier–Stokes(RANS) equations were used as governing equations. The shape of the storage tank is another important concern in the modeling of sloshing. The performance of different geometric shapes of storage tanks, such as trapezoidal(Ketabdari and Saghi 2015), circular conical (Gavrilyuk et al.2005),elliptical(Hashemian and Aghabeigi 2012),rect‐angular (Saghi and Lakzian 2017; Saghi et al. 2020a; Sa‐ghi et al.2021),cylindrical(Papaspyrou et al.2004;Shek‐ari et al. 2009), and spherical (Yue 2008; Curadelli et al.2010), were investigated. For example, Saghi and Lakzian(2017) numerically studied the sloshing phenomenon in rectangular tanks with different dimensions to find the op‐timum aspect ratio of the tank based on some criteria, in‐cluding entropy generation. The baffles have been widely used in the storage tanks to alleviate the sloshing-induced forces on storage tanks and damp out the sloshing. Some researchers used the baffles with different locations and shapes as a promising way to reduce the sloshing impact.Therefore, Ning et al. (2012) exerted the coupled vertical and horizontal motions on a vertical baffled rectangular storage tank. The sloshing phenomenon in the work of Jung et al. (2012) was modeled in a rectangular tank with different baffle heights and water depths. They found that the free surface oscillation was minimized when the ratio of baffle height to water depth was around 0.9. Nayak and Biswal Kishore (2015) numerically simulated the seismic response of rectangular liquid tanks with a bottom-mount‐ed submerged block. Their results showed that the slosh‐ing in the system is highly sensitive to the frequency of tank motion. Some researchers also used a single vertical baffle. For example, Golla et al. (2021) experimentally studied the effect of three types of vertical baffles, includ‐ing surface-piercing bottom-mounted, flush-mounted, and immersed bottom-mounted baffle configurations, on the sloshing noise in a rectangular tank in different sloshing re‐gimes and various fill levels. Yu et al. (2020) conducted some experimental studies to find the optimum number and position of the vertical baffles to suppress the sloshing impact. Xue et al. (2017) experimentally investigated the effectiveness of different types of baffles in suppressing pressure under a wide range of frequencies. They mea‐sured the dynamic impact pressure alongside the central line of the tank wall for tanks without baffles and with dif‐ferent kinds of baffles, including perforated vertical baf‐fles,surface-piercing bottom-mounted vertical baffles,ver‐tical baffle flushing with a free surface,and immersed bot‐tom-mounted vertical baffles. Some researchers also re‐cently investigated the dual baffle. For instance, Cho and Kim (2016) also studied the effect of dual vertical baffles on sloshing reduction in a swaying rectangular tank. They showed that the submergence depth and baffle location are important parameters affecting the performance of the dual vertical baffles. Saghi et al. (2020b) evaluated the effec‐tiveness of the oblique dual baffle on the sloshing impact arising from sway motion on the rectangular storage tank.They also examined the effect of baffle geometric parame‐ters, including baffle position and orientation, and showed that the suggested optimum baffle can reduce sloshing loads by up to 15%. Ma et al. (2021) numerically studied suppressing the sloshing with single and dual vertical baf‐fles. They showed that the spacing distance between the baffles is an important parameter; thus, the peak impact pressure from the sloshing phenomenon can be inhibited more than approximately three times low, thus producing dual baffles with a poor arrangement.

EG has been presented as an optimization tool by Bejan(1979; 1987) and was employed by other researchers in the last decade. For instance, Saghi and Lakzian (2017)and Saghi (2018) considered the EG criterion to optimize the 2D rectangular and elliptical storage tanks. They also applied EG in the dam-break flow with obstacles and found the optimum shape of the obstacle (Saghi and Lak‐zian 2019). EG was considered in this study as a criterion to compare different scenarios of the dual baffle. There‐fore, the efficiency of the dual baffle is increased by de‐creasing EG.Therefore, the optimum dual baffle is related to the minimum EG. Along these lines, some parameters of dual baffles, including position and length, have been considered. Meanwhile, the researchers did not use the curved dual baffle.

A similar study was only conducted by Kamath et al.(2021)to evaluate the effects of the baffle on the sloshing.Different from their work for the effects of the single baf‐fle on the sloshing in roll motion,the single-and dual-baf‐fle effects on sloshing in sway motion were studied in the present paper, in which the baffle curvature and EG were considered.Therefore,a series of experimental and numer‐ical tests are performed in this study to examine the effects of the curvature of the dual baffle in rectangular storage tanks on reducing the sloshing. The numerical model based on OpenFOAM is used to solve the unsteady incom‐pressible RANS equations by using an interFoam solver and applied to the various dual-baffled configurations for the de‐sign optimization using some criteria including the total EG.

2 Experimental setup used to validate the numerical model

The experiments of a tank sloshing with single and dual baffles as the passive measures were conducted at Hakim Sabzevari University. Figure 1(a) shows the shake table used in the experiments, which is 1.35 m in length and 0.80 m in width. The thickness of the steel plate of the shake table body is 8 mm.An electric motor with 4 250.4 W power and 1450 r/min angular velocity is used in the pow‐er point system(Figure 1(b)).The shake table can generate a harmonic oscillation with a frequency below 10 Hz and 0–0.07 m amplitude. The liquid tank is 0.45 m in length,0.15 m in width, and 0.30 m in height (Figure 1(c)). The storage tank comprises Plexiglas and is fixed on the shake table. The thickness of the tank wall is 0.015 m; thus, the tank is rigid. The baffles comprise a glass with 0.003 m thickness.The width of the baffles is the same as the inner width of the tank(0.15 m).The height of the baffles is also variable (Hb= 0.02, 0.04, and 0.06 m).A digital camera is placed in front of the shake table to record the free surface oscillation. The “Aoao Video to Picture Converter” and“Plot digitizer”software are used to extract the required in‐formation from the photos taken.

Figure 1 Experimental setup(Hakim Sabzevari University)

3 Numerical Model description and bound‐ary conditions

A 2D numerical study on the single and dual-baffled rectangular storage tanks was conducted in this paper to optimize against the sloshing phenomenon.The coordinate system of the tank without baffle and a pressure probe namedP, which is in the middle of the tank, is shown in Figure 2(a).For comparison,the single baffle positioned at the tank center was also considered, as shown in Figure 2(b).The tank length is divided into two and three equal parts by the single and dual baffles,respectively.The variableHdenotes the tank depth,Hwis the water depth,Hbis the baffle height,t2is the baffle thickness, andLis the length of the tank in the figure.

The fluid was considered viscous, incompressible, and laminar in the present study.Therefore, the Navier–Stokes equations can be used as the governing equations as fol‐lows(Versteeg and Malalasekera 2007):

whereUis the velocity vector,tis the time,vis the kine‐matic viscosity of the fluid,Pis the dynamic pressure,andgis the gravitational acceleration.

The volume of fluid (VOF) method was used in this paper to describe scalar corresponding phase fields.The parameter α defined as Equation (3) is updated at each time step by using Equation (4). This equation was solved via the multidimensional universal limiter for explicit solution method (Rudman 1997; Saghi and Ketabdari 2014; Ketabdari and Saghi 2013d; Ketabdari and Saghi 2012c):

Figure 2 Coordinate system of the baffled rectangular storage tank

whereαis the quantity of water per unit of volume.Open‐FOAM in version 7.0 is adopted in the present study to model the related sloshing phenomenon in the rectangular storage tank.First,the“interFoam”solver,which takes ad‐vantage of handling dynamic meshes for moving surfaces,is chosen to model the sway motion. In addition, “inter‐Foam” can solve the three-dimensional Navier–Stokes equation for two incompressible phases using finite vol‐ume discretization and the VOF method. The solver algo‐rithm called the pressure implicit method for the pressurelinked equation is a combination of pressure implicit with operator splitting and semi-implicit method for pressurelinked equation algorithms. Thus, “interFoam” is used in the present study due to its various advantages (Open‐FOAM 2019). Time derivative terms were discretized by using the first-order implicit Euler discretization scheme.The second-order centered Gauss linear scheme was also used to discretize the gradient parameters. The skewnesscorrected centered Gauss linear correction was employed for the Laplace derivative terms, and the upwind scheme was utilized to handle the divergence terms. The domain has been bounded by walls in this study.Hence,the veloci‐ty field, namely the movingWallVelocity boundary condi‐tion, was employed for these boundaries. By contrast, the fixedFluxPressure and zeroGradient boundary conditions were respectively implemented for the pressure field and the phase zone(OpenFOAM 2019).

The EG analysis is also adopted to investigate the irre‐versible aspects of the second law of thermodynamics and the effects of the new design on loss reduction. The fluid friction and heat transfer dissipation are both irreversible processes.The irreversibility for Newtonian fluids is intro‐duced as local EG by Bejan(1979):

whereSis the entropy generation rate,kis the fluid ther‐mal conductivity,Tiis the fluid temperature,μis the fluid dynamic viscosity,andφis viscous dissipation function.

The temperature is considered ambient (i.e., 293 K) in this study. The temperature gradient is often negligible in rectangular storage tanks. Thus, the isothermal flow is fo‐cused on the sloshing phenomenon; only the EG created by the fluid friction is considered. The parameters of the viscous dissipation (φ) and the total EG (Sgen) are respec‐tively defined as follows:whereu,v,andware the velocity components,Δx,Δy,and Δzare the mesh size, andNx,Ny,andNzare the number of the meshes in thex-,y-,andz-directions,respectively.

The accumulated total EG (Sgen,cum) at timeTcis evaluat‐ed as:

4 Some tests to validate the numerical model

The dependence of the numerical results on the mesh size is first examined. Thus, a tank withH= 0.3 m,L=0.45 m,W=0.15 m,andHw=0.06 m(Figure 2(a))under‐goes harmonic sway motion with 0.04 m amplitude and 5 rad/s angular frequency.The horizontal force exerted on the tank was estimated at different mesh sizes, and the re‐sults are shown in Figure 3. These results indicate a mesh size independence for the mesh sizes smaller than 0.005 m.The sensitivity of time-step integration was estimated based on Courant number 0.5.

Figure 3 Horizontal force on the tank to evaluate the mesh size independence

In this step, different cases, including the simple tank(Figure 2(a)), single-baffled tank (Figure 2(b)), and dualbaffled tank(Figure 2(c)),were modeled under a harmonic sway motion, and the numerical model was validated by comparing with the experimental results obtained in the present study. First, a sway motion with 0.04 m amplitude and 5 rad/s angular frequency was exerted on the simple tank,and the distribution of the surface elevation along the tank was then numerically and experimentally calculated as shown in Figures 4 and 5.The water depth in this test is 0.06 m. The results indicate a good consistency between the results.

As another test, a sway motion with 0.04 m amplitude and 5 rad/s angular frequency was exerted on the singlebaffled tank, and the results are shown in Figures 6 and 7.In this test,Hb= 0.12 m,t2= 0.003 m, andHw= 0.12 m.The results show that air is trapped in the rolled free sur‐face at the tip of the baffle, which can be easily captured by the numerical model.

Figure 4 Snapshot of the free surface in a simple rectangular tank

Figure 5 Comparison of the free surface oscillation in a simple rectangular tank for the experimental and numerical results at t=2.0 s and t=5.0 s

Figure 6 Snapshot of the free surface in a single-baffled rectangular tank

Figure 7 Comparison of the free surface oscillation in a single-baffled rectangular tank for the experimental and numerical results at t=2.0 s and t=5.0 s

Finally, a harmonic sway motion of 0.04 m amplitude and 4.6 rad/s was exerted on the tank, and the distribution of the surface elevation along the tank from experimental and numerical results att= 2 and 5 s are respectively shown in Figures. 8 and 9. In this test,Hb= 0.12 m,t2=0.003 m, andHw= 0.12m. The figures reveal the presence of similar phenomena between experimental and numeri‐cal simulations at different instants.

Asway motion of amplitudea=0.001 m andw=5.83 rad/s angular frequency was exerted on the tank withH=0.33 m,L= 0.48 m length (Figure 2(a)), andHw= 0.09 m to con‐firm that the model encapsulates effective sloshing pres‐sure magnitudes applied on tank walls. The pressure at pointP(Figure 2(a)) was calculated, and the results were compared with those of Xue et al. (2019) in Figure 10.A good agreement was found between the results.

5 Results and discussions

5.1 Effects of dual-baffle height

First,the effects of dual-baffle height were evaluated on the results of horizontal wave force on the tank. The tank parameters are the same as the tank with the single baffle(Figure 2(b)). A sway motion with 0.04 m amplitude and 5 rad/s angular frequency was exerted on the tank. The static water depth was selected as 0.06 m. The dual-baffle height and thickness were considered asHb= 0.01, 0.03,and 0.06 m andt2= 0.003 m, respectively. The single baf‐fle with the same heights positioned at the tank center,named as a middle baffle, was also modeled separately to compare their effects on reducing the sloshing phenome‐non. The related results are shown in Figure 11. This fig‐ure reveals that the difference among results with and with‐out baffles is quite small for the baffle heightHb=0.01 m,as shown in Figure 11(a). Thus, the baffle effect of slosh‐ing reduction can be neglected for small baffle heights.The results of horizontal wave forces on the tank become smaller than that without baffle and tend to stabilize with the further increase in baffle height,especially for the dualbaffle case,as shown in Figure 11(c).

The maximum horizontal force exerted on the tank(Fxmax)and the decrement percentage of maximum horizontal force (DF) were calculated for different baffle heights based on the obtained results, and the findings are shown in Figure 12.The parameter DF is calculated as follows:

Figure 8 Comparison of the free surface in the rectangular tank between experimental results and numerical results at different time instants

Figure 9 Comparison of the free surface oscillation for experimental and numerical results

Figure 10 Comparison of the pressure data between the results of the current paper and those of Xue et al.(2019)

whereFxmaxbis the maximum horizontal force exerted on the simple tank (without baffle). TheFxmaxis calculated based on the horizontal force exerted on the tank perimeter during the simulation. The horizontal force exerted on the tank is estimated as follows:

wherenxis the component of the unit vector perpendicular to the tank surface in thex-direction,pis the dynamic pres‐sure on the tank perimeter,andAis the element area on the wet tank surface.

Figure 11 Effect of baffle height on the horizontal force exerted on the tank

Figure 12 The effects of the baffle height on the maximum horizontal force exerted on the tank

The results in Figure 12(b) indicate that the parameter DF of the dual-baffled tank(Figure 2(c))forHb/Hw>0.5 is more than that of the single-baffled tank (Figure 2(b)).Therefore, the baffle effect on reducing the sloshing is ob‐served for the dual baffle compared with the single baffle positioned at the tank center.However,the effects of a sin‐gle baffle are more substantial than those of a dual baffle when the relative baffle heightHb/Hw＜0.5.Finally,the re‐sults show that the effects of single and dual baffles are nearly constant when the relative baffle heightHb/Hw>0.75.

The EG was estimated for the storage tanks with differ‐ent dual-baffle heights based on Eq. (9), and some results are shown in Figure 13(a).The variation of the total cumu‐lative EG with baffle height is also shown in Figure 13(b).The baffle effect is also constant as the relative baffle heightHb/Hw>0.75,which is in accordance with Figure 12.Therefore,an adaptable baffle length can be selected based on practical engineering.

5.2 Effects of sway motion parameters

The effects of sway motion amplitude on the sloshing reduction with single and dual baffles were evaluated in this subsection.The different motion amplitudes(a=0.01,0.02,0.03,and 0.04 m)and 1.36 rad/s were exerted on the rectangular storage tank with the static water depth of 0.06 m. The baffle height and thickness wereHb= 0.06 m andt2=0.003 m,respectively.The horizontal force exerted on the tank was then estimated for different sway motion amplitudes.The parameter DF was calculated for different sway motion amplitudes (a) based on the obtained results,and the findings are shown in Figure 14(a). DF decreases with the motion amplitude whena＜0.03 m,which means that the effect of sloshing reduction is apparent. However,the variation trend is the opposite; that is, DF starts to in‐crease slowly whena>0.03 m.Furthermore,EG was esti‐mated for the dual baffle and different sway motion ampli‐tudes, as shown in Figure 14(b). The results show that the EG increases with the increment of sway motion amplitude.

The effects of sway motion frequency on DF were eval‐uated.The sway motion with 0.04 m amplitude and differ‐ent frequencies were exerted on the rectangular storage tank with a static water depth of 0.06 m. The horizontal force exerted on the tank was estimated for different sway motion frequencies. The decrement percentage of maxi‐mum horizontal force (DF) was calculated for different sway motion frequencies (f) based on the obtained results,and the findings are shown in Figure 14(c).The results al‐so reveal that the effect of dual baffle relative to middle single baffle is observed in the low angular frequencies and is only slightly evident in the high angular frequencies.

6 Optimization of inclined and curved dualbaffle configurations

This section aims to examine the effects of different de‐signs of the dual-baffled water tanks on sloshing.Three de‐signs include one inclined straight and two curved baffle designs, as shown in Figure 15. The dual straight and curved baffles (i.e., inner and outer curved baffles) in‐clined with different inclination angles are designed with the following equations:

Figure 13 Sgen,cum for the storage tanks with different dual baffle heights

Figure 14 The effects of sway motion frequency on DF

Figure 15 Sketch of the dual baffle

In these experiments, the sway motion with 0.04 m amplitude and 1.36 Hz frequency is applied on a rectan‐gular storage tank with the initial water depths of 0.06 m and 0.18 m. The dual-baffle height (Hb) varies between 0.01 and 0.06 m, and the thickness (t2) is maintained at 0.003 m.

The maximum horizontal force and EG were estimated for different inclination angles (θ= 65°, 75°, 85°, 95°,105°) for the inclined straight dual-baffle configuration shown in Figure 15(a), and the results are shown in Fig‐ure 16.The value of EG and maximum horizontal force in the rectangular storage tank with 80°and85° inclination angles is the smallest compared with the other inclination angles forHw=0.06 m and 0.18 m,respectively.

The two configurations in Figures 15(b) and 15(c) are used in the rectangular storage tanks(case 1)to evaluate the efforts of the curved baffle configurations on the sloshing reduction. The water depth is set to 0.18 m.The sway motion used in this set of tests is similar to that previously used (0.04 m amplitude and 1.36 rad/s frequency). The results of horizontal force exerted on the tank and EG are estimated and compared with the in‐clined dual baffle, as shown in Figure 17. The results in Figure 17(a) indicate that the maximum horizontal force exerted on the tank for inclined dual baffle is 24.66 N,while those for the curved dual baffles shown in Figures 15(b) and 15(c) are 22.13 and 20.61 N, respectively.Thus, using the curved dual baffle can further decrease the maximum horizontal force exerted on the tank by around 10%–16% in this study. Figure 17(b) shows that using the curved dual baffle can increaseEGby around 10% and decrease EG by around 20% for the two cas‐es. Therefore, the curved dual baffle shown in Figure 15(c)is suggested as the optimum dual baffle.

The distributions of EG in the tanks with inclined and curved dual baffles at an inclination angle of 85°(Figure 15)are shown in Figure 18 at the selected time steps. The re‐sults show that the distribution of EG in the rectangular storage tank with curved dual baffles is less than that with inclined dual baffles.

Figure 16 Variation of the maximum horizontal force and EG with inclination angle at Hw=0.06 m and Hw=0.18 m

Figure 17 Comparison of inclined and curved dual baffles with a sidewall angle of 85°

Figure 18 Snapshots of the EG distribution at different times and for the optimum rectangular storage tanks

7 Conclusions

A series of numerical and experimental tests were per‐formed in this study to model the sloshing phenomenon in the dual-baffled rectangular storage tanks. The dual-baf‐fled rectangular storage tank was optimized based on some criteria,which comprise the maximum horizontal force ex‐erted on the tank and EG. The results show that the dualbaffled tank is more effective than the single-baffled one in sloshing reduction when the relative baffle heightHb/Hw>0.5, while such an effect is constant when the relative baf‐fle heightHb/Hw> 0.75, where dual- and single-baffled configurations have nearly the same effect in sloshing re‐duction. The results also show that the EG decreases with the increment of the baffle height. Furthermore, the decre‐ment percentage of the maximum horizontal force (DF) is the same for single and dual-baffled tanks when the sway motion amplitude isa＜ 0.03 m, where the parameter DF reaches the minimum, during the evaluation of the effects of sway motion amplitude and frequency on the sloshing.The dual-baffle effect on reducing the sloshing is more sub‐stantial compared with the single-baffle in the low angular frequencies and is only minimal in the high angular frequen‐cies considering the angular frequency of the sway motion.

The inclined and curved dual baffled rectangular tanks with different sidewall angles were also considered in this study for optimization. The results show that the horizon‐tal force is minimum for the sidewall angle of 80°at the cases of shallow water depth. However, the horizontal force was found to be minimum at the sidewall angle of 85°with the increase in the water depth.The EG is always minimum at the sidewall angle of 85°.Therefore,the side‐wall angle of 85° is selected as the optimum sidewall an‐gle. Finally, the curved dual baffle with different curve equations was investigated.The results show that the opti‐mum curved dual baffle decreases the horizontal force ex‐erted on the tank by around 10%–15% considering in‐clined dual baffle.Furthermore,using this curved dual baf‐fle decreases EG by around 20%.Therefore,the curved du‐al baffle is suggested as the optimum dual baffle.