Reducing Environmental Hazards of Prismatic Storage Tanks under

Vibrations

OLENA SIERIKOVA *

Applied Mechanics and Environmental Protection Technologies Department

National University of Civil Defence of Ukraine

vul. Chernishevska, 94, Kharkiv, Ukraine, 61023

UKRAINE

ELENA STRELNIKOVA, DENYS KRIUTCHENKO, VASIL GNITKO

Department of Hydroaeromechanics of Power Machines

A.M. Pidhorny Institute for Mechanical Engineering Problems NAS of Ukraine

vul. Pozharskoho, 2/10, Kharkiv, Ukraine 61046

UKRAINE

Abstract: - Regular operation, pre-repair and repair work on tanks, as well as outflows of oil products and other

flammable liquids under the influence of seismic loads, fires and explosions on tanks are the source of

technogenic impact on the environment. Therefore, the influence treatment of the fluctuations and vibrations on

the storage tanks for environmentally hazardous liquids and the assessment of reducing the load on nature is a

very relevant scientific and practical issue to improve the environmental safety of areas adjacent to the

tanks. This paper treats free and forced liquid vibrations in prismatic tanks filled with an incompressible ideal

liquid. The developed method allows us to estimate the level of the free surface elevation in prismatic tanks

under suddenly enclosed loadings. The proposed method makes it possible to determine a suitable place with a

proper height for installation of the baffles in tanks by using numerical simulation and thus shortening the

expensive field experiments. The proposed approach could be applied to various environmentally hazardous

liquids. This will increase the environmental safety level of the territories adjacent to stationary tanks with

environmentally hazardous liquid. It will also be possible to prevent emergencies.

Key-Words: -

Dynamical Systems, Distributed Systems, Applied Systems Theory, oil products, flammable liquids,

environmental safety, seismic loads, storage tanks

Received: March 24, 2022. Revised: October 21, 2022. Accepted: November 16, 2022. Published: December 31, 2022.

1 Introduction

Prismatic tanks are widely used in various fields of

modern technology and construction as tanks,

containers, locks of hydraulic structures. They are

also used to store the variety of liquids: drinking and

firefighting water, oil, liquefied natural gas, wine,

etc.

On large farms, liquid fertilizers, fuel for

agricultural machinery, manure, silage, and so on

are stored in such tanks. These substances often

store centrally in large quantities, which increases

the risk of environmental damage. Leakage of stored

substances could be associated with insufficient

tightness of storage tanks, accidents or operators

unintentional actions or maintenance personnel, as

well as seismic loads. The environmental hazard of

substances used in agriculture is mainly caused by

the high concentration of nutrients, as well as the

possible presence of pathogens, veterinary drugs

[1,2], or toxicity to aquatic organisms (for example

gasoline, certain pesticides). Agricultural waste

could also have a high biological oxygen demand,

where the biological oxygen demand of bovine

slurry could be up to 50 times higher than domestic

wastewater. The result may be the death of aquatic

organisms [3].

Reservoirs for storing oil products are

environmentally dangerous sources of technogenic

impact on the environment, acting as objects of

uncontrolled emissions of steam-air or steam-gas-air

mixtures and oil products spills, followed by fires

and explosions. The environmental significance of

storage largely depends on its potential to pollute

the environment and on the physical and chemical

properties of the stored substances. Oil tanks are

used on the farm to store gasoline and diesel fuel.

Properly designed oil storage facilities must prevent

leakage and potential contamination of soil, surface

or groundwater. Soil contamination includes the

hazardous waste, oil spills, sludge from the

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treatment process, and coke dust. Soil contamination

reduces the fertility of the soil and introduces

foreign particles, which may affect the growth and

quality of crops. One of the environmental impacts

that may arise out of the implementation and

operation of the storage tank is oil spill and

evaporation of products that pollute the area. Each

activity involved in the operation of a tank farm has

a potential spill risk and evaporation of different

types of gases causing pollution [4].

2 Problem Formulation

The analysis of the environmental impact sources

during the tanks operation indicates that steel

vertical tanks, even during normal operation, are

environmentally hazardous. Fires and explosions on

tanks from combustible substances and flammable

liquids often occur during cleaning and preparation

for repairs, as well as during the repair work itself.

Regular operation, pre-repair and repair work on

tanks with oil products, as well as outflows of oil

products under the influence of seismic loads are the

source of technogenic impact on the environment

due to the occurrence of environmentally hazardous

situations, accompanied by explosions and fires, and

posing the real threat to the population life and

health. Therefore, the influence treatment of the

fluctuations and vibrations on the storage tanks for

environmentally hazardous liquids and the

assessment of reducing the load on nature is a very

relevant scientific and practical issue to improve the

environmental safety of areas adjacent to the tanks

[5,6].

In previous studies [7], the authors have

considered seismic loads on cylindrical reservoirs

[8,9], the use of composite materials with

nanoinclusions as reservoir materials, as well as the

effect of flooding and flooded soils on technogenic

objects and on the seismic activity increasing in

[10,11]. This paper considers prismatic storage

tanks for environmentally hazardous liquids. The

environment inside the tank has significantly

affected their dynamic characteristics (frequencies

and vibration modes) and stress-strain state. If a

tank structure resting on the ground experiences

dynamic or kinematic effects as a result of

earthquakes, the impulsive wave pressure

component and other perturbations, then its

response (displacements, forces, frequencies and

vibration modes) depends significantly on the

degree of vessel filling, the elastic properties of the

foundations, and also stiffness inertial

characteristics of the structure itself.

Failures of these tanks, following destructive

earthquakes or explosives, may lead to

environmental catastrophes, loss of valuable

contents, and disruption of fire-fighting efforts.

Liquid sloshing is also a reason for damage of roofs

and upper walls of storage tanks. Inadequately

designed tanks and liquid storages were damaged in

past extensive earthquakes.

Comprehensive reviews of the phenomenon of

sloshing [9], including analytical predictions and

experimental observations were done in the different

works [7,12]. It has been proved that exact solutions

for the linear liquid sloshing are limited by tanks

geometry with straight walls, as rectangular and

cylindrical containers.

Furthermore, it is difficult or impossible to

obtain analytical solutions for reservoirs and tanks

with complex geometric shapes. Therefore, many

numerical methods have been implemented to solve

linear boundary value problems of liquid sloshing.

Diverse simplified theoretical studies have been

provided [12,13], and other numerical methods have

been developed. Some of these researches have been

used as the basis for current design standards.

Raynovskyy and Tymokha [14] have developed the

analytical linear multimodal method to analyze 2D

liquid sloshing in horizontal cylindrical tanks. The

liquid volume method has been developed at

[12,13]. Dynamic analysis of fluid-filled shell

structures using compatible finite element and

boundary element methods has been performed at

[15,16].

Many slosh suppression devices have been

proposed to damp fluid motion, reduce structural

loads caused by fluid spillage, and prevent

instability [17]. These devices are rigid or elastic

with different sizes and orientations, annular

partitions, and different plates that partially cover

the free surface [18]. The suppression systems

design requires the quantitative knowledge of the

slosh characteristics.

In practice, the baffles effect could be seen after

the baffles installation. However, such experimental

research is too expensive. Therefore, the computing

technologies development for qualitative numerical

modeling is a very relevant issue.

The paper treats the free and forced liquid

vibrations in prismatic tanks filled with the

incompressible ideal liquid. Reservoirs with both

horizontal and vertical baffles have been considered.

The changing dynamical reservoir behavior with the

internal baffle has been established. The harmonic

and impulse loads have been supposed to be applied

to the considered structures.

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3 Problem Solution

Issues concerned with liquid vibrations in rigid

prismatic and cylindrical tanks have been

considered in the paper. The considered reservoirs

have been presented on Figure 1. The horizontal and

vertical baffles have been installed into cylindrical

tanks. At the first stage, reservoirs have their own

modes and frequencies. Their own modes have been

used as basic systems for the forced vibration

problems under harmonic and impulse loads.

It has been supposed the liquid in the containers

as the ideal and incompressible one, and the fluid

motion caused by shell vibrations as irrotational.

Then its relative velocity V has a potential

 

zyxt ,,,

, so that

Vzyx 















;;

. (1)

The moistened shell surface has been designated

by S1, and the free surface by S0. Suppose the

Cartesian coordinate system 0xyz has been

connected with considered containers, the liquid

free surface S0 coincides with the plane z = 0 at the

rest state. If the liquid-filled shell is under the force

acting in the horizontal plane, then the coordinate

axes with orts may be chosen so that acceleration as

has been considered as

as = as(t) i, (2)

where the factor as(t) depends only on time t, and

i is the unit vector along Ox.

Firstly it has been obtained the relation between

the velocity potential, accelerations due to driving

forces, and the liquid pressure. Equation (2) has

been acquired in the following form:

 

 

tax ss a

(3)

Note that for the gravitational acceleration there

have been obtained form

 

gzg

where g is the gravity acceleration, z is the

vertical coordinate of a point in the liquid.

a) b)

Figure 1: Partially filled with the liquid reservoirs

Motion equations for the ideal liquid could be

represented in the vector form as follows:

    

pgztxasw

where w is the fluid flow acceleration,  is the

liquid density, and p is the fluid pressure. When

const

the following relation is valid:

    





 p

gztxas

. (4)

Therefore, the liquid particle's acceleration

under gravitational forces and horizontal excitations

always has the potential (an analog of the Prandtl’s

potential). The driving forces are usually considered

starting from the rest state. Thus, according to

Kelvin's theorem, if the liquid motion starts from

rest, then its relative velocity V has the potential.

Formula has been obtained for the pressure

(Bernoulli’s equation) considering the gravity and

horizontal accelerations. Using equation (4) and

assuming that the flow is irrotational, Bernoulli

equation has been received in the following form:

 

















 2

gzxta

pp s

, (5)

where p0 is for atmospheric pressure.

If small liquid oscillations have been considered

(the linearized formulation has been studied), then

2

, and there have been obtained formula

 

















 gzxta

pp s0

. (6)

Pressure p on the shell walls in the absence of the

horizontal volume force has been determined from

the linearized Bernoulli equation by the formula

pgz

p



















, (7)

Formulas (5)-(7) shows that to receive the liquid

pressure it is necessary to obtain the liquid potential

Ф. It has been assumed the flow to be inviscid and

incompressible, the irrotational fluid motion in the

3D reservoir described by the Laplace equation for

the velocity potential 

2

. (8)

The mixed boundary value problem for the

Laplace equation has been formulated to determine

this potential. The non-penetration condition on the

wetted tank surfaces S1 has been applied [16]







On the free surface (z = 0), the following

dynamic and kinematic boundary conditions must

be satisfied:

0; 0

0











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where the unknown function

 

tyx ,,

describes the free surface shape and position.

Therefore, for the velocity potential there have

been obtained the following boundary-value

problem

2

;







;

0; 0

0











. (9)

where

pp 

has been received from equation

(6) at

 

yxtz ,,

To calculate the liquid vibrations in the presence

of baffles, consider at first the cylindrical shell with

a ring baffle, Fig. 1b). Let wetted surface be

bafbot12111SSSSS 

The multi-domain (boundary super-elements)

method has been used. Note, it has introduced the

"artificial" interface surface Sint [13], and divided

the region filled with the liquid on the two parts

21;

, bounded by surfaces S11, Sbaf, Sint , S bot and

S12, Sbaf, Sint, S0, respectively. On the interface

surface Sint, the following boundary conditions have

been set [19,20]:

2int1int

2int1int ;



 











SS nn

(10)

The zero eigenvalue obviously exists for problem

(9), but it has been excluded with the help of the

following orthogonality condition:







 dS

. (11)

So, equations (9)-(11) for calculating the

unknown functions

 

tyx ,,

and

 

zyxt ,,,

have been obtained.

Consider the cylindrical quarter tank, Fig.

1d). Suppose R is tank radius, and H is for filling

level. Using cylindrical coordinate system there

have been obtained the following boundary value

problem

2

,0,0





















 rzr hzRr

0; 0

0











(12)

with orthogonality condition (11) and

pp 

obtained from (7) at

 

yxtz ,,

For the prismatic tank depicted in Fig. 1a) there

could be solved the boundary value problem

(9) with the performance of equation (11).

3.1 Mode decomposition method

For all presented boundary value problems let

consider the auxiliary boundary value problems

have been received

2





















 g

.(13)

Differentiate the second correlation in (13) by

respect to t and substitute received equality for

t



into the first relation. Further the auxiliary function

 as

   

zyxezyxt ti ,,,,,  

has been presented.

It has been come to the eigenvalue problem

2







g















 dS

(14)

Suppose the solutions of eigenvalue problem

(14) are their own modes



with corresponding

own frequencies



. Consider now the potential Ф

in the next form:





 M

kkk



. (15)

As the equation for the free surface there have

been gained the following expression:







 M

. (16)

3.2 Reducing to systems of boundary integral

equations

For the prismatic reservoirs, the boundary value

problem (9), (11) has been solved analytically using

the Fourier method of variable separation.

Shell of revolution with and without baffles

Consider the boundary value problem (9),(11) for

the cylindrical shell without baffles.

It has been supposed the structure under

consideration is a shell of revolution, in cylindrical

coordinates system (r, z, ) there have been gained

   

.,...2,1,,...1,0,cos,,,  kzrzr kk

(17)

Here  is a harmonica number, indexes k is for

mode numbers, corresponding to . Thus,

frequencies and modes of free vibrations are

considered separately for different values of .

Dropping indices k, the main relation for

determining functions k could be written in the

form [19,20]

 

SS 00

2









  nn

.(18)

Here S = S1  S0; both points P and P0 belong to

the surface S. By P – P0 there have been denoted

the Cartesian distance between points P and P0.

With boundary conditions (9), there have been

gained the system of integral equations in the form

[6,7]

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









































































111

001

011

0000

111

SSS

(19)

Here for convenience it has been denoted by 0

values of the potential on the free surface and by 1

its values on the shell walls.

There have been searched solutions in form (17)

for the system (19).

The following integral operators have been

presented below:

111 ),(

PPrn

 





;

000 

 















000

;

PPn

S



 

;



000

.(20)

Then the boundary value problem (9) takes the

form

1



 CB

;

10 SP 

;

01 2



 F

;

00 SP 

After excluding function 1 from these relations

the following eigenvalue problem relative to 0

have been obtained

.;0)()(

FBDAECDA 

 

Its solution gives natural modes and frequencies

of liquid sloshing in rigid tank. Evaluation of

integral operators in (21) has been carried out by the

method proposed in [9,19,20].

Having defined the basic functions k, substitute

them in expressions (16) for velocity potential and

(17) for the free surface elevation. Then substitute

the received relations in the boundary condition on

the free surface that correspond to application of the

dynamical condition [20,21].

 







sxtag

As in the cylindrical system of coordinates there

 cosx

, there could be interest only in the first

harmonica, i.e. in formula (17) it has been only

considered =1. Furthermore, there could be the

following equation on the surface S0

 







  

dgd s

kkk



Due to validity of relation (15) on the surface S0

the equality given above takes the form

 

  

tadd s

kkkk

kkk



. (21)

Accomplishing the dot product of equality (21)

 

l,1

and having used orthogonality of own

modes, it has been received the system of ordinary

differential equations of the second order

   

 

MkFFtadd

kks

kk ,1;

2









.(22)

Suppose that before applying the horizontal

loading, the tank was at the rest state. Then it must

be solved system (22) under zero initial conditions.

The analogical procedure for the shell of

revolution with ring baffles has been proposed and

described in detail in authors’ papers [16,18,21]. It

should be noted that the system of second order

differential equations has the form (22), but with

other modes and frequencies.

According to [22-25] there have been assumed

for the shell of revolution with two vertical baffles

could be sought the basic function in the next form:

   

.,...2,1,,...1,0,2cos,,,  kzrzr kk

(23)

Using this correlation boundary conditions have

been satisfied on the vertical baffles. The system of

singular integral equations for k(r,z) acquires the

form analogical to system (19). Let  be a generator

of the surface S1. Reducing integrals in this system

to one-dimensional ones the system of one-

dimensional integrals that has the same form as in

[18] has been obtained. For one-dimensional

integrals in (20) there have been gained formulas

 



















dzrzzzdS

PPn )(),()(

;

 





dPPdS

PP 0

),()(

Here kernels are differed from those obtained in

[16,19] and are presented as follows:

         

;

0









































  zr nk

nkk

zzrr

zz EFE

   

;

,0k

PP 



 F

   

 









222 sin14cos1611 dkkE

;

   













022 sin1

4cos

;

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 

;2; 0

2rrbzzrra 

k

2

. (24)

Note, that the system of second order differential

equations has the form (22), but with other modes

and frequencies, has been obtained by using the

solution of the system of integral equation (19) but

with kernels (24).

3.3 Liquid free vibrations

The prismatic reservoir

 

Hzbybaxa  0,,

, filled with the

ideal incompressible liquid has been considered.

The own modes and frequencies of fluid vibrations

in the prismatic reservoir have been determined by

the Fourier method of separation of variables. As

first, the solution of Laplace equation with non-

penetration boundary conditions (first and second

equalities in (9)) has been found

cos

cosch 





























 b

zA klklklkl

Satisfying the condition on the free surface (third

equation in (9)), there have been obtained

Hklklkl 



 chsh

From this the expression for the natural

frequencies of the liquid in the prismatic tank have

been found

 

Hg klklkl  tanh

The hereinafter the expressions for the first 8

own modes of fluid vibrations (they also are the

system of basis functions for solving the problem of

forced vibrations) have been presented

 

sin)0cos(

101 y

xCx 



 

),0cos(

sin

210 yx

Cx 





 

sin

311 y

Cx 



 

),0cos(cos

420 yx

Cx 



 

),cos(0cos

502 y

xCx 



 

sincos

621 y

Cx 



 

,cos

sin

712 y

Cx 



 

Cx 

 coscos

822

(25)

Table 1 shows the numerical values of

frequencies ij and the frequency parameter ij for

the prismatic reservoir in the form of a cube with a

= b = H = 1m.

Table 1. Natural frequencies of fluid oscillations

in the prismatic tank

ij

ij

1.77245385

4.05116419

1.77245385

4.05116419

2.50662827

5.71001255

3.54490770

5.89216585

3.54490770

5.89216585

3.96332729

6.23315169

3.96332729

6.23315169

5.01325655

7.01253864

For receiving the expression for function  the

following formula (15) for the velocity potential has

been used. Dependence

 

jinn ,

has been shown in

Table 1, the functions n have been determined by

the formulas

 

   

jinnyx

ab ij

n,;,

cosh

1





, (26)

where ij has been found from (25). So the free

surface elevation has been expressed by formula

(16), where the basic function n have been defined

in (26)

The vibration modes of the free surface have

been shown on Figure 2.

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS

DOI: 10.37394/23201.2022.21.27

Olena Sierikova , Elena Strelnikova,

Denys Kriutchenko, Vasil Gnitko

E-ISSN: 2224-266X

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Fig.2. Modes of free surface vibrations in the

prismatic tank

Thus, the basic system for the liquid forced

vibration in rigid prismatic tanks has been built.

3.4 Forced vibrations of the liquid under

harmonic loads

Suppose that at the initial time the liquid in the

prismatic tank is at rest. The tank subjected to

periodic loads cost, that applied in the horizontal

direction (parallel to Ox axis, Fig. 1). The system of

differential equations of the fluid motion has been

obtained, starting from the next boundary condition

on the free surface:

   

0, 





Hz

stxayxg

. (27)

Substituting expressions (24), (25) into relation

(27) leads to

       

0cos











  tx

zyx

tdgHyxtd

kkk



. (28)

Then the following system of differential

equations has been gained accomplishing dot

product of correlation (28) by



and using the

orthogonality of sloshing modes:

   

0cos

11 



 ttdtd



;

   

0cos

22 



 ttdtd



;

   

0cos

121

33 



 ttdtd



(29)

Solution of system (29) is following:

 

tttd 1

1coscos

8





;

 

;coscos

2tttd 





(30)

 

.coscos

121

3tttd 





Time history of the free surface elevation in the

point  with coordinates

1,1,1  zyx

during 10

sec at  =1.1Hz has been shown on Figure 3.

Figure 4 shows the influence of the load’s

frequency  on the free surface level elevation.

Figure 3. Time history

of free surface via

elevation

Figure 4. Free surface

elevation loads

frequencies

It would be noted that convergence here is

achieved at M=3.

The peaks in the graph correspond to the

frequencies 10 and 20 (table 1). These frequencies

are the most dangerous ones, for example, during

transportation of the tank under consideration. The

function describing the free surface elevation of the

liquid has been obtained as

   







 M

Hyx

where the coefficients

 

tdk

have been

determined by formulas (30).

3.5 Forced vibrations of the liquid under

impulse loads

It has been considered the rigid prismatic tank filled

with the liquid. The tank parameters are the

following: H = b = 1m and a = 1m or a = 2m. The

pressure p on tank walls from formula (7) has been

determined. Here

 

tas

is the function characterizing

external influence (a horizontal seism or an

impulse).

The load is suddenly applied to the lateral

surface of the prismatic tank as(t) = Q0a(t), where

Q0 = 1 МPа is the distributed pressure, and

 











.,0

,,1

(31)

It has been supposed before applying the

horizontal impulse the tank was at the state of rest.

Then there have been solved systems (30) under

zero initial conditions. The operational method [20]

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS

DOI: 10.37394/23201.2022.21.27

Olena Sierikova , Elena Strelnikova,

Denys Kriutchenko, Vasil Gnitko

E-ISSN: 2224-266X

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Volume 21, 2022

has been applied here for receiving the solution of

system (30).

The following values for coefficients

 

Mktdk,1, 

have been obtained:



































TtTtt

Ttt

)(cos

)cos(

0)cos(

)(

2222

On Figure 5 the free surface elevation in the

point x=1, y=1, z=1 depending on time has been

shown for T=1.5 sec.

a) b)

Figure 5. Time-histories of the free surface

elevation

Figures 5 a) and b) correspond to a=1m and

a=2m, respectively, and this is the side of the

prismatic tank along which the load has been

applied. Increasing this side leads to alignment and

mitigation of sloshing amplitudes

4 Conclusion

The developed method allows us to estimate the

level of the free surface elevation in prismatic tanks

under suddenly enclosed loadings. The free and

forced liquid vibrations in prismatic tanks of equal

heights have been considered. The benchmark tests

have been considered that validated the obtained

results. The effects of the baffle installation and

their influence on changing the elevation of the free

surface have been taken into account. The

elaborated approach allows us to carry out the

numerical simulation of baffled tanks with baffles of

different sizes and with different positions in the

tank. This gives the possibility of governing the

baffle radius and its position within the tank. It is

very topical, because practically, the effect of

baffles could be seen usually only after the baffle

has been installed. The proposed method makes it

possible to determine a suitable place with a proper

height for installation of the baffles in tanks by

using numerical simulation and thus shortening the

expensive field experiments. It would be noted that

only the ideal incompressible liquid is under

consideration. The proposed approach could be

applied to various environmentally hazardous

liquids. The proposed approach will be easily

generalized to elastic tanks with elastic baffles.

Thus, the future research concerned with free and

forced liquid vibrations in elastic tanks with elastic

baffles. The geometry of the tank also could be

easily changed, so the results will be obtained for

conical, spherical and compound shells with and

without baffles. It will allow giving

recommendations about installation of protective

elements (covers, partitions). This will increase the

environmental safety level of the territories adjacent

to stationary tanks with environmentally hazardous

liquid. It will also be possible to prevent

emergencies.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Olena Sierikova: conceptualisation, data curation,

formal analysis, methodology.

Elena Strelnikova carried out the simulation and the

optimization.

Denys Kriutchenko: visualization, data curation.

Vasil Gnitko: carried out the simulation and the

optimization.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS

DOI: 10.37394/23201.2022.21.27

Olena Sierikova , Elena Strelnikova,

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E-ISSN: 2224-266X

257

Volume 21, 2022