Analytical and finite element modeling of pressure vessels for seawater reverse osmosis desalination plants

Desalination 397 (2016) 126–139

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Analytical and finite element modeling of pressure vessels for seawater reverse osmosis desalination plants A.M. Kamal, T.A. El-Sayed ⁎, A.M.A. El-Butch, S.H. Farghaly Department of Mechanical Design, Faculty of Engineering, Helwan University, P.O. Box 11718, Mataria, Cairo, Egypt

H I G H L I G H T S • • • •

The pressure vessel for SWRO has been modeled using analytical solution and finite element modeling. Comparison between stainless steel and fiber reinforced composite materials has been done. The design parameters of pressure vessel have been optimized. The results of both analytical solution and finite element modeling have showed good agreement.

a r t i c l e

i n f o

Article history: Received 24 October 2015 Received in revised form 14 June 2016 Accepted 16 June 2016 Available online xxxx Keywords: Pressure vessel Sea water desalination plant Composite materials Analytical and FEM Membrane unit

a b s t r a c t A pressure vessel (PV) which contains the membrane elements of seawater reverse osmosis (SWRO) desalination has been modeled using analytical solution and finite element modeling (FEM) to optimize the PV design parameters. Two types of PV materials have been compared namely; stainless steel and fiber reinforced composite materials. Von-Mises yield criterion and Tsai-Wu failure criterion are used for the design of stainless steel and composite PVs respectively. E-glass/epoxy and carbon/epoxy composite materials are considered in this work. In addition, hybrid composite materials are introduced for layers through the vessel thickness. The results have shown that the optimum lay-up is achieved using the angle-ply [± ϴ]ns at winding angle of 54° for E-glass/ epoxy and 55° for carbon/epoxy PVs while for hybrid composite PVs the optimum lay-up is [90G/±50C/90G]ns. Also, the results have shown that the composite PVs have lighter weight than the stainless steel PVs. The carbon/epoxy PVs introduce the optimum weight savings but in terms of the total PVs cost, the hybrid composite PVs can be used. © 2016 Elsevier B.V. All rights reserved.

(continued)

Nomenclature Symbol

Meaning

Units

a, b, c, d A, B, D Ai cθ Di E E1 E2 E3 f1, f2 f11, f12, f22, f66 Fa

Laminate compliance matrices Laminate stiffness matrices Flow area Cosine of winding angle Inner diameter of PV Young's tensile modulus Longitudinal tensile modulus Transverse tensile modulus (in 2-direction) Transverse tensile modulus (in 3-direction) Tsai-Wu coefficients Tsai-Wu coefficients Side thrust

MPa−1 MPa mm2 – mm MPa MPa MPa MPa MPa−1 MPa−2 N

Abbreviations: PV, pressure vessel; RO, reverse osmosis; SWRO, seawater reverse osmosis; FEM, finite element modeling. ⁎ Corresponding author. E-mail address: [email protected] (T.A. El-Sayed).

http://dx.doi.org/10.1016/j.desal.2016.06.015 0011-9164/© 2016 Elsevier B.V. All rights reserved.

Symbol

Meaning

Units

Ftu Fty F1c F1t F2c F2t F6 G12 G13 G23 h hk hk-1 i k L Mx,y Mx, My, Ms n

Tensile ultimate strength Tensile yield strength Longitudinal compressive strength Longitudinal tensile strength Transverse compressive strength Transverse tensile strength In-plane shear strength Shear modulus in the 1–2 plane Shear modulus in the 1–3 plane Shear modulus in the 2–3 plane Thickness of pressure vessel Coordinate of layer k upper surface Coordinate of layer k lower surface Number of layers through vessel thickness Layer order Length of PV Moments matrix per unit length Bending and twisting moments per unit length Number of basic laminates

MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa mm mm mm – – mm N N –

A.M. Kamal et al. / Desalination 397 (2016) 126–139

1. Introduction

(continued) Symbol

Meaning

Units

nc ng Nx,y Nx, Ny, Ns p Qx,y Q1,2 Q11, Q12, Q22, Q66 sθ Sf Sfmin Sfk ΔSf t T Vf Wcomp Wst z α β γs γos γ4 γ5 γ6 ϵ ϵx ϵy ϵx, y ϵo x;y ϵox, ϵoy ϵu1t ϵu2t ϵ1 ϵ2 ϵ1,2 ϵ3 ϴ κx, y κx, κy, κs ρc ρcomp ρg ρst ν ν12

Number of carbon/epoxy layers Number of E-glass/epoxy layers Forces matrix per unit length Axial, hoop and shear forces per unit length Internal fluid pressure Transformed lamina stiffness matrix w. r. t. global axes Lamina stiffness matrix w. r. t. material axes Components of Q1,2

– – N/mm N/mm MPa MPa MPa MPa

Sine of winding angle Safety factor of PV Minimum safety factor Safety factor of layer k Safety factor error Ply (layer) thickness Transformation matrix Fiber volume fraction Weight of composite PV Weight of stainless steel PV Coordinate of layer midplane w. r. t. reference plane Coefficient of S2fk Coefficient of Sfk In-plane shear strain w. r. t. global axes In-plane shear strain of reference plane Shear strain in the 2–3 plane Shear strain in the 1–3 plane Shear strain in the 1–2 plane Normal strain Axial strain (in x-direction) Hoop strain (in y-direction) Strains matrix w. r. t. global axes Reference plane strains matrix Axial and hoop strains of reference plane Ultimate longitudinal tensile strain Ultimate transverse tensile strain Longitudinal tensile strain (in 1-direction) Transverse tensile strain (in 2-direction) Strains matrix w. r. t. material axes Transverse tensile strain (in 3-direction) Fiber orientation (winding angle) Reference plane curvatures matrix Components of κx,y Density of carbon/epoxy material Density of composite material Density of E-glass/epoxy material Density of stainless steel Poisson's ratio Poisson's ratio at load in 1-direction and strain in 2-direction Poisson's ratio at load in 1-direction and strain in 3-direction Poisson's ratio at load in 2-direction and strain in 1-direction Poisson's ratio at load in 2-direction and strain in 3-direction Poisson's ratio at load in 3-direction and strain in 1-direction Poisson's ratio at load in 3-direction and strain in 2-direction Normal stress Maximum normal stress Minimum normal stress Axial stress (in x-direction) Hoop stress (in y-direction) Radial stress (in z-direction) Stresses matrix w. r. t. global axes Longitudinal tensile stress (in 1-direction) Transverse tensile stress (in 2-direction) Stresses matrix w. r. t. material axes Transverse tensile stress (in 3-direction) Shear stress Maximum shear stress Shear stress in the x-y plane Shear stress in the 2–3 plane Shear stress in the 1–3 plane Shear stress in the 1–2 plane

– – – – – mm – – kg kg mm – – – –

ν13 ν21 ν23 ν31 ν32 σ σmax σmin σx σy σz σx,y σ1 σ2 σ1 ,2 σ3 τ τmax τs τ4 τ5 τ6

127

– – – – – – – – – – – – – – Deg. – – kg/m3 kg/m3 kg/m3 kg/m3 – – – – – – – MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa

Reverse osmosis (RO) is considered one of the most efficient methods for seawater desalination and many researches have been devoted to optimize this process. The main components of a seawater reverse osmosis (SWRO) desalination plant are seawater intake, pretreatment, reverse osmosis system and post-treatment is shown in Fig. 1 [1]. Spiral wound membrane elements (from 1 to 6 elements) are installed in a PV which is usually fabricated from stainless steel or fiber-reinforced composite materials [2], see Fig. 2. Much research have been developed to optimize the design parameters of composite PVs such as properties of fiber and matrix, fiber winding angle, fiber concentration and number of layers through the vessel thickness. Krikanov [3] proposed a method to design hybrid laminated composite PVs under different loading conditions. The vessel consists of a hoop layer reinforcing the cylindrical part and a helical layer reinforcing the vessel domes. He concluded that when the vessel axial strain is constrained, transformation between the helical layer and the hoop layer should be done with a suitable angle and when the hoop strain is constrained, the hoop layer material should be replaced with higher stiffness material. Xu et al. [4] proposed a 3D parametric finite element model to estimate the failure evolution and strength of composite hydrogen storage vessels. They established a solution algorithm using ANSYS finite element software to examine the damage progress and the failure properties of composite structures with increasing internal pressure using different failure criteria. Their results show a good agreement between the theoretical failure pressure and the experimental burst pressure especially for Tsai-Wu failure criterion. Son and Chang [5] calculated the stress distributions in the composite layers of a hydrogen PV by using three different modeling techniques which are laminate-based, full ply-based and hybrid (combining a laminate-based modeling for the dome part with a ply-based modeling for the cylinder part). The models were created using ABAQUS 6.9-1commercial finite element software. The results showed that the PV failure occurred by the transverse tensile stress at the border under the test pressure of (105 MPa) which is 1.5 of the operating pressure (70 MPa). The finite element results were verified using experimental results of a PV prototype. The results showed that complex geometries can be simply modeled using the laminate-based modeling technique and a reasonable overall stress distribution can be obtained. Moreover, the full ply-based modeling technique can be used to obtain more accurate stress distributions results in both the metal liner and the composite layers. Onder et al. [6] studied the effect of winding angle of composite PVs on the burst pressure using an analytical solution and finite element method. They performed an experimental method to verify the optimum winding angle for symmetric and antisymmetric shells. The analytical and experimental results were in good agreement for some orientations. Parnas and Katirci [7] used the classical lamination theory and generalized plane strain model to derive an analytical solution to evaluate the behavior of fiber reinforced composite PVs. This solution includes the effect of the internal pressure, axial force and body force due to rotation. In addition, the effect of temperature and moisture variation is considered. Xie et al. [8] introduced a technique for the optimization of fiber orientations and weighting factors in hybrid-fiber multilayersandwich cylindrical shells subjected to external pressure. The results of the examples presented in their work showed that the maximum critical pressure can be obtained when using higher modulus fiber in the hoop direction and lower modulus fiber in the longitudinal direction. Mian et al. [9] developed an optimization method for composite PV lay-ups using finite element analysis and calculated the relative weight saving compared with the reference aluminum PV. Three composite materials namely; S-glass/epoxy, Kevlar/epoxy and carbon/epoxy are

128

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Fig. 1. Schematic diagram of a typical SWRO desalination plant [1].

used. They also established a MATLAB code to verify their results. They concluded that the optimum lay-up is achieved at the angle-ply [± ϴ]ns and at a winding angle of 54° for all composite materials. Wang et al. [10] studied experimentally the properties of PVs made of carbon and aramid fibers hybrid composites with epoxy resin at different ply types and hybrid ratios. They concluded that the vessels with longitudinally unique fiber type have high strength and positive hybrid effect while the vessels with longitudinally different fiber types have low strength and negative hybrid effect. Assam et al. [11] studied the influence of several design parameters on the performance of filament-wound composite PVs. These PVs consists of inner liner and composite layers. The material properties and the thickness of the pressure vessel components are investigated. Their results show that for large number of layers the effect of liner on the internal pressure is very limited. In addition, they showed that increasing both the number of layers and the layer thickness results in gain in the operating efficiency and the ultimate failure pressure. Taghavian et al. [12] presented a new method for designing composite PVs under different loading and constraints conditions using lattice structures. They developed an analytical approach and verified the results experimentally. They concluded that significant material saving can be achieved by using of lattice structures for hoop strain suppressing compared with that of addition of extra plies while for axial strain suppressing, no considerable material saving is recognized. M. Sabour and Foghani [13] proposed a method to design a metalcomposite PV using a combination of finite element method and fuzzy decision making. They used three types of fibers which are carbon, glass and Kevlar and optimized the vessel weight for each type. The results showed that PVs with carbon fibers have minimum weight and high strength compared with other PVs. They also concluded that a

uniform stress distribution between the steel liner and the composite layers is obtained when using carbon fibers materials. Daniel and Ishai [14] presented an example of a PV subjected to internal pressure and external torque. They compared three types of composite materials with different lay-ups using an analytical approach. Then, they determined the optimum lay-up for each material which achieves the minimum vessel thickness. Finally, they calculated the relative weight savings for the optimum PV compared with an aluminum reference PV. The present research focused on optimization of the SWRO (PV) design parameters using both analytical solution and FEM. Comparison between stainless steel and composite material PVs is evaluated with respect to material saving, specific strength and cost. Four different sizes of PVs are used which are the standard sizes available in the international markets. For composite material PVs, E-glass/epoxy, carbon/ epoxy and hybrid materials are selected with five symmetric laminate lay-ups. 2. Analytical solution A thin-walled cylindrical PV subjected to internal pressure is investigated using linear elasticity analysis. Two types of PV materials have been compared namely; stainless steel and fiber reinforced composite materials, to optimize the PV design parameters. The optimization process is carried out by using a computational program established by MATLAB (R2012b). 2.1. Analysis of stainless steel PV The SWRO stainless steel PV is considered as an open pipe subjected to internal pressure and side thrust exerted at both sides of the pipe due

Fig. 2. Spiral wound element PV assembly [2].

A.M. Kamal et al. / Desalination 397 (2016) 126–139

to the effect of end caps on the vessel, see Fig. 3. The value of axial forces, Fa in N, is calculated, neglecting the areas of feed, reject and product ports, as follows: F a ¼ pAi

ð1Þ

π 2 D 4 i

ð2Þ

; Ai ¼

129

Table 1 Properties of stainless steel 316L [16]. Properties

Values

Density, ρst, kg/m3 Young's modulus, E, GPa Poisson's ratio, ν Tensile yield strength, Fty, MPa Tensile ultimate strength, Ftu, MPa

7750 193 0.31 207 586

where, is the internal fluid pressure in MPa is the inner diameter of PV in mm is the flow area in mm2. Four industrial sizes of PVs are selected to be 2.5, 4, 8 and 16 in. respectively according to the available sizes in the international markets. The vessel length, L, is 0.4 m for 2.5 in. diameter and 1 m for the other sizes which determined to be suitable for one membrane element. The value of internal pressure is selected to be 8 MPa (80 bar) which suitable for most SWRO desalination processes. The longitudinal and hoop stresses, σx and σy in MPa, acting on an element of the cylindrical shell along the axial and hoop directions (x and y), see Fig. 3, are obtained as follows: p Di Ai

σx ¼

pDi 4h

ð3Þ

σy ¼

pDi 2h

ð4Þ

where, h is the thickness of PV in mm. For thin-walled PV, the radial stresses can be neglected, σz = 0, (plane stress condition). The principal stresses for this state of stress are σ max ¼

σx þ σy þ τ max 2

ð5Þ

σ min ¼

σx þ σy −τ max 2

ð6Þ

τ max ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ −σ 2 x y þ τ2 ; τ ¼ 0 2

ð7Þ

The stainless steel material is considered to be isotropic and linear elastic with the following stress-strain relation (Hook's law): σ ¼ Eϵ

ð8Þ

Many stainless steel grades have been used for SWRO (PV) such as austenitic grade 316L, highly alloyed austenitic grade 904L, duplex grade 2205 and super duplex 2507 [15]. For cost considerations, the selected stainless steel grade in this work is 316L with the properties at room temperature shown in Table 1 [16].

Fig. 3. PV loading.

Using the von-Mises yield criterion for the design of stainless steel PV as follows:  σ 2max þ σ 2min −σ max σ min ¼

F ty Sfall

2 ð9Þ

Where, Sfall is the allowable safety factor (Sfall = 2). From the above analysis, the stainless steel PV thickness can be calculated as follows: h¼

pffiffiffi 3 pDi S 4 F ty fall

ð10Þ

Then, the weight of the stainless steel PV, can be obtained as follows: W st ¼

h i π 2 ρst ðDi þ 2hÞ −D2i L 4

ð11Þ

where Wst is the weight in kg and ρst is density in kg/m3. 2.2. Analysis of composite PV The SWRO multi-layered filament wound composite PV for the tube of membrane unit is analyzed at the same loading conditions (internal pressure and side thrust) of the stainless steel PV, Fig. 3. The same internal diameters and lengths are used. Two types of composite materials are used namely; E-glass/epoxy and carbon/epoxy (AS4/3501-6) and their properties are shown in Table 2 [14]. The PV wall thickness is determined for each vessel size at different types of symmetric laminate lay-ups, see Table 3, with the variation of fiber orientation or winding angle (angle between fiber direction and axis of the vessel, ϴ in degree, see Fig. 4). Hybrid composite materials are also used with the same lay-ups by introducing layers with different composite materials as shown in Table 3. For example the lay-up [+ϴC/−ϴG]ns means that one layer of carbon/epoxy material at positive winding angle and the other layer of E-glass/epoxy material at negative winding angle. The subscripts can be defined as follows:

Table 2 Properties of composite materials [14].

Properties

E-glass/epoxy

Carbon/epoxy (AS4/3501-6)

Fiber volume fraction, Vf Density, ρ, kg/m3 Longitudinal modulus, E1, GPa Transverse modulus, E2, GPa In-plane shear modulus, G12, GPa Major Poisson's ratio, ν12 Minor Poisson's ratio, ν21 Longitudinal tensile strength, F1t, MPa Transverse tensile strength, F2t, MPa In-plane shear strength, F6, MPa Ultimate longitudinal tensile strain, ϵu1t Ultimate transverse tensile strain, ϵu2t Longitudinal compressive strength, F1c, MPa Transverse compressive strength, F2c, MPa

0.55 2100 39 8.6 3.8 0.28 0.06 1080 39 89 0.028 0.05 620 128

0.63 1580 142 10.3 7.2 0.27 0.02 2280 57 71 0.015 0.006 1440 228

130

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Table 3 Types of symmetric laminate lay-ups. Lay-up

[±ϴ]ns

[0/±ϴ]ns

[90/±ϴ]ns

[0/±ϴ/90]ns

[90/±ϴ/90]ns

Material

E-glass/epoxy Carbon/epoxy [+ϴC/−ϴG]ns [±ϴC/±ϴG]ns

E-glass/epoxy Carbon/epoxy [0C/±ϴG]ns [0G/+ϴG/−ϴC]ns

E-glass/epoxy Carbon/epoxy [90C/±ϴG]ns [90G/+ϴG/−ϴC]ns

E-glass/epoxy Carbon/epoxy [0C/±ϴG/90C]ns [0G/±ϴC/90G]ns

E-glass/epoxy Carbon/epoxy [90C/±ϴG/90C]ns [90G/±ϴC/90G]ns

n is the multiple of plies or number of basic laminates and s is the symmetric stacking sequence. The analysis procedure of multidirectional laminates is considered using the classical lamination theory with its assumptions [14,17] and applied for the basic laminate unit for each lay-up (n = 1). The unidirectional composite material of each lamina is considered to be orthotropic and linear elastic with the following stress-strain relations (referred to the principal material axes 1, 2, 3) [14]: 2

1 6 E 1 6 6 8 9 6 − ν 12 6 ϵ1 > > E1 > > 6 > > > 6 ν 13 ϵ2 > > > > > 6 < = 6− ϵ3 E1 ¼6 6 γ4 > > > > 6 0 > > > γ > > > 6 > : 5> ; 6 6 γ6 6 0 6 6 4 0

ν 21 E2 1 E2 ν 23 − E2 −

ν31 E3 ν32 − E3 1 E3 −

3 0

0

0

0

0

0 0

0

0

1 G23

0

0

0

1 G13

0

0

0

0

0 7 7 7 8 9 0 7 7> σ 1 > 7> > > > 7> > > > σ2 > > 0 7 7< σ 3 = 7 7> τ4 > > > 0 7 > τ5 > > 7> > > > 7> 7: τ6 ; 7 0 7 7 1 5

σ3 = τ4 = τ5 =0, γ4 =γ5 =0, ϵ ≠ 0 (eliminated from Eq. (14)) 8 9 2 Q 11 < σ1 = or σ 2 ¼ 4 Q 12 : ; τ6 0

ð12Þ

From the symmetry of the compliance matrix, then ð13Þ

Q 12 Q 22 0

38 9 0 < ϵ1 = 0 5 ϵ2 or in shortened form σ1;2 ¼ Q 1;2 ϵ 1;2 ð15Þ : ; γ6 Q 66

And the components of lamina stiffness Q1 , 2 can be determined as follows: Q 11 ¼

G12

ν 12 ν 21 ν 13 ν 31 ν 23 ν 32 ¼ ; ¼ ; ¼ E1 E2 E 1 E3 E2 E3

Where,

E1 ν 21 E1 E2 ;Q ¼ ;Q ¼ ; Q ¼ G12 ð16Þ 1−ν12 ν 21 12 1−ν 12 ν 21 22 1−ν 12 ν 21 66

The transformed lamina stiffness Qx , y referred to global axes (x, y) can be determines as follows: 9 9 8 8 < σ1 = < σx = ¼ T σy or in shortened form σ1;2 ¼ Tσx;y σ ; : 2; : τ6 τs

ð17Þ

8 9 ϵ > < 1 > = ϵ2 ¼T 1 > : γ > ; 6 2

ð18Þ

8 9 ϵ > < x > = ϵy or in shortened form ϵ1;2 ¼ Tϵ x;y 1 > : γ > ; s 2

Where the transformation matrix Where Ei is the Young's modulus along axis i, Gij is the shear modulus in direction j on the plane whose normal is in direction i, νij is Poisson's ratio for the i-j plane, σ1 , σ2 & σ3 are the normal stresses in directions 1, 2 & 3 and τ4 , τ5 & τ6 are the shear stresses in 2–3, 1–3 and 1–2 planes respectively. In the current applications, composite materials are used in the form of thin laminates loaded in the plane of the laminate. The stress-strain relations for a thin lamina (plane stress condition) can be expressed as followings: 2

1 8 9 6 E 1 < ϵ1 = 6 6 ν 12 ϵ2 ¼ 6 − 6 : ; 6 E1 γ6 4 0

ν 21 E2 1 E2

−

0

3 0 78 9 7 < σ1 = 7 7 0 7 σ2 7 : τ6 ; 1 5 G12

ð14Þ

2

cθ 2 T ¼ 4 sθ 2 −cθ sθ

sθ 2 cθ 2 cθ sθ

3 2cθ sθ −2cθ sθ 5; cθ ¼ cosθ; sθ ¼ sinθ cθ 2 −sθ 2

σx;y ¼ Q x;y ϵ x;y

ð19Þ ð20Þ

Using Eqs. (15) to (20) thus, 2 Q 11 Q x;y ¼ T −1 4 Q 12 0

Q 12 Q 22 0

3 0 0 5T 2Q 66

ð21Þ

After dividing the third column of, Qx ,y, by 2, the laminate stiffness matrices can be expressed as follows: i

The extensional stiffness matrix A ¼ ∑ Q kx;y ðhk −hk−1 Þ;

ð22Þ

The coupling stiffness matrix B ¼

  1 i 2 2 ∑ Q kx;y hk −hk−1 2 k¼1

ð23Þ

The bending stiffness matrix D ¼

  1 i 3 3 ∑ Q k h −hk−1 3 k¼1 x;y k

ð24Þ

k¼1

Where, hk and hk-1 are the z-coordinates of the upper and lower surfaces of layer k measured from the laminate reference plane and i is the number of layers (plies) in the basic laminate unit, see Fig. 5. Then, the laminate compliance matrices can be obtained as follows:

Fig. 4. Fiber orientation in composite PV.

 n  o a ¼ A−1 − −A−1 B d BA−1 ;

ð25Þ

A.M. Kamal et al. / Desalination 397 (2016) 126–139

131

The shear force and moments per unit length Ns =Mx = My = Ms = 0

The reference plane strains matrix ϵ o x;y

8 o9 < ϵx = ¼ ϵoy ¼ aN x;y þ bMx;y : o; γs ð33Þ

The reference plane curvatures matrix κx;y Fig. 5. Layers coordinate notation.

ð34Þ

  b ¼ −A−1 B d;

ð26Þ

  c ¼ −d BA−1 ;

ð27Þ

 n o −1 d ¼ D− BA−1 B

ð28Þ

The mechanical loading matrices are

The forces matrix

N x;y

8 9 < Nx = ¼ Ny ; : ; Ns

The moments matrix M x;y

ð29Þ

9 8 < Mx = ¼ My ; : Ms

ð30Þ

pDi in N=mm; Where; the longitudinal force per unit length Nx ¼ σ x h ¼ 4

ð31Þ

the hoop force per unit length N y ¼ σ y h ¼

pDi 2

8 9 < κx = ¼ κy ¼ cN x;y þ dM x;y : ; κs

in N=mm:

ð32Þ

Then, the layer strains referred to global axes (x, y) can be given as 8 9 8 o9 8 9 < ϵ x = < ϵx = < κx = o ϵy ¼ ϵy þ z κy : ; : o; : ; γs κs γs

ð35Þ

Where, z = coordinate of layer mid-plane measured from the laminate reference plane, see Fig. 5. Thus the layer strains referred to the principal material axes 1, 2 can be written as 9 9 8 8 ϵ ϵ > > = = < 1 > < x > ϵy ϵ2 ¼T 1 1 > > > ; : γ ; : γ > 2 6 2 s

ð36Þ

And the layer stresses referred to the principal material axes 1, 2 are 9 2 8 Q 11 < σ1 = ¼ 4 Q 12 σ : 2; 0 τ6

Q 12 Q 22 0

38 9 0 < ϵ1 = 0 5 ϵ2 : ; γ6 Q 66

ð37Þ

The design of composite PV is based on the Tsai-Wu failure criterion with the following form f 1 σ 1 þ f 2 σ 2 þ f 11 σ 21 þ f 22 σ 22 þ f 66 τ26 þ 2f 12 σ 1 σ 2 ¼ 1

Fig. 6. Finite element model for stainless steel PVs. (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in.

ð38Þ

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Table 4 FEM information.

Table 5 Element size in (PVs) FEM.

Geometry

Surface body

Stiffness behavior Thickness mode Analysis type Coordinate system

Flexible Manual 3-D Global Cartesian (x, y, z)

Mesh Physics preference Element type Smoothing Mapped face meshing method Body sizing type Behavior

Mechanical Linear quadrilateral (SHELL181) Medium Quadrilaterals Element size Soft

Solution Physics Analysis Solver Environment temperature

Structural Static structural Mechanical APDL 22 °C

f 11

1 1 − F 1t F 1c 1 ¼ F 1t F 1c

1 1 − F 2t F 2c 1 ; f 66 ¼ 2 F6

; f2 ¼ ; f 22 ¼

1 F 2t F 2c

; f 12 ¼ −

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 11 f 22 ð39Þ

where f1 & f2 are in MPa−1, f11 ,f22 , f66 & f12 are in MPa−2. Then, this failure criterion is applied for each layer at failure stresses Sfk σ1, Sfk σ2and Sfk τ6 where, Sfk is the safety factor of layer k. Substitution of these failure stresses in Eq. (38) yields αS2fk

Element size mm

Total number of elements

Stainless steel

Composite

2.5 4 8 16

2.5 4 8 16

12,960 20,240 10,160 5120

11.357 16.037 9.095 5.288

52.167 118.436 111.759 110.386

Solving Eq. (40), the layer safety factor is

Sfk ¼

−β þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2 þ 4α 2α

ð42Þ

Using the first-ply failure (FPF) approach, the laminate safety factor equals to the minimum value of layer safety factor, Sfmin. The allowable PV thickness, ha in mm, can be determined as followings:

Where, the Tsai-Wu coefficients can be calculated as followings: f1 ¼

Solving time (sec)

PV size in

þ βSfk −1 ¼ 0

ð40Þ

ha ¼

Sfall ho Sfmin

ð43Þ

where; ho is the thickness of the basic laminate unit in mm ðho ¼ i  t Þ

ð44Þ t is the ply thickness = 0.2 mm, Sfall = 2 (the same value for stainless steel PV). The number of basic laminates, n can be calculated and approximated to the higher integer number as follows: n¼

Sfall ha ¼ ho Sfmin

ð45Þ

Then, the PV thickness is

where; α ¼ f 11 σ 21 þ f 22 σ 22 þ f 66 τ 26 þ 2f 12 σ 1 σ 2 ; β ¼ f 1 σ 1 þ f 2 σ 2 ð41Þ

h ¼ n  ho

Fig. 7. Variation of safety factor versus element size for stainless steel PVs. (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in.

ð46Þ

A.M. Kamal et al. / Desalination 397 (2016) 126–139

133

And the PV safety factor is Sf ¼

h S ha fall

ð47Þ

The weight of composite PV, Wcomp in kg, is W comp ¼

h i π 2 ρcomp ðDi þ 2hÞ −D2i L 4

ð48Þ

where, ρcomp. is the density of composite material in kg/m3. In case of hybrid composite material, the weight of PV can be determined as follows: W comp ¼

i π ρg þ ρc h 2 Di þ 2h −D2i L 4 2

at ng ¼ nc

ð49Þ

3.2. Modeling of composite PV The finite element model for composite PVs for different lay-ups given in Table 3 has been performed at the optimum lay-ups resulting from the analytical solution. The same FEM information of stainless steel PVs is used. The selected element size and total number of elements for both stainless steel and composite PVs are shown in Table 5. Also, the table shows the solving time of both stainless steel and composite (for example, E-glass/epoxy [±ϴ]ns lay-up) PVs.

where, ρg. ρc. ng. nc.

Fig. 8. Distribution of the von-Mises stress for 8 in. PV.

is the density of E-glass/epoxy material, is the density of carbon/epoxy material, is the number of E-glass/epoxy layers, is the number of carbon/epoxy layers,

and

W comp

i π 2ρg þ ρc h 2 Di þ 2h −D2i L ¼ 3 4

4. Results and discussion at ng ¼ 2nc

ð50Þ

3. Finite element modeling The finite element modeling technique has been established for the SWRO (PV) at the same sizes, materials and loading conditions that used in the analytical solution to verify the results. ANSYS Workbench version 15 mechanical module is used in case of stainless steel PV and ANSYS Composite PrepPost (ACP) module is used in case of composite PV. 3.1. Modeling of stainless steel PV The finite element model for stainless steel PVs of different sizes are shown in Fig. 6. A DELL-Laptop-Core i5 has CPU speed of 2.5 GHz and RAM of 4 GB with Windows 7-64 bit operating system is used. The FEM information used for all vessel sizes is shown in Table 4. 3.1.1. Mesh convergence study Since the computed results of the FEM are affected by mesh size [18–21], a mesh convergence study has been performed to determine the element size for various vessel sizes. The variation of PV safety factor with element size for all vessel sizes is shown in Fig. 7. As shown from this figure, the safety factor decreases with decreasing of the element size and close to the allowable value (Sfall = 2) for all vessel sizes. It is noticed that the solving time increases with decreasing of the element size, so that a suitable element size is selected for each vessel size.

4.1. Stainless steel PV The wall thickness and weight of stainless steel PVs of different sizes for both analytical solution and FEM are shown in Table 6. It is clear that the safety factor obtained from the FEM is slightly larger than that obtained from the analytical solution for the same vessel wall thickness and size. As could be observed from Table 6, the FEM results are in good agreement with the analytical solution within an acceptable range of the safety factor error i.e. b 0.4%. The distributions of the equivalent (von-Mises) stress and the safety factor for 8 in. PV are shown in Figs. 8 and 9 respectively. From these figures, the maximum stress is 103.11 MPa and the minimum safety factor is 2.0076. 4.2. Composite PV The analysis of composite PVs is performed for all vessel sizes with the different materials and lay-ups shown in Table 3. The variation of vessel thickness with fiber orientation is plotted for each vessel and the optimum winding angle at minimum thickness is determined. Then, the weight saving of each composite PV compared with the stainless steel one is calculated. FEM results are compared with that of the analytical solution and the safety factor error is determined for each vessel.

Table 6 Wall thickness and weight of stainless steel PVs. PV size inch

h mm

Weight kg

Analytical solution Sf

FEM Sf

ΔSf %

2.5 4 8 16

2.125 3.401 6.801 13.602

1.359 8.697 34.787 139.150

2 2 2 2

2.0073 2.0079 2.0076 2.0076

0.365 0.395 0.380 0.380

Fig. 9. Distribution of the safety factor for 8 in. PV.

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Fig. 10. Variation of allowable thickness versus fiber orientation for (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in. PVs for [±ϴ]ns lay-up.

4.2.1. Analytical solution The variation of the allowable PV thickness with fiber orientation (angle, ϴ) of different vessel sizes for [± ϴ]ns lay-up is shown in Fig. 10. From this figure, it is clear that the allowable thickness depends on fiber orientation and material type. The carbon/epoxy vessel has allowable thickness lower than the E-glass/epoxy vessel at the same winding angle for all vessel sizes. This is due to the high stiffness and strength of the carbon fibers compared with the E-glass fibers. It is also observed that the minimum allowable thickness is achieved at the optimum fiber orientation of 54° for E-glass/epoxy vessel and 55° for carbon/epoxy vessel for all vessel sizes. These results show good agreement with the previous works [6,7,9,14]. In case of hybrid composite materials, the allowable thickness values are between that of E-glass/epoxy and carbon/epoxy composite

materials depending on the amount of each material and the arrangement of layers through the vessel wall thickness. The number of E-glass/epoxy layers is taken to be the same as carbon/epoxy layers. The minimum allowable thickness is achieved at the optimum fiber orientation of 56° for [+ϴC/−ϴG]ns and 53° for [±ϴC/±ϴG]ns lay-ups for all vessel sizes. The difference between vessel thicknesses for hybrid lay-ups decreases with the variation of fiber orientation far from the optimum angle for all vessel sizes due to the little effect of carbon fibers stiffness and strength at the extreme values of winding angle (same thicknesses at ϴ = 0° and at ϴ = 90°) as shown in Fig. 10. For example, the minimum allowable thickness for [+ ϴC/− ϴG]ns lay-up is 8.177 mm while for [± ϴC/± ϴG]ns lay-up is 4.592 mm for 8 in. PV. This is due to the existence of carbon fibers in helical direction for both positive and negative values of ϴ in [± ϴC/± ϴG]ns lay-up

Fig. 11. Variation of allowable thickness versus fiber orientation for (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in. PVs for [0/±ϴ]ns lay-up.

A.M. Kamal et al. / Desalination 397 (2016) 126–139

135

Fig. 12. Variation of allowable thickness versus fiber orientation for (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in. PVs for [90/±ϴ]ns lay-up.

(balanced laminate) which increase the strength and stiffness of the whole vessel while for [+ ϴC/− ϴG]ns lay-up (unbalanced laminate), the carbon fibers are existent in helical direction for positive values of ϴ only. The variation of the allowable PV thickness with fiber orientation of different vessel sizes for [0/± ϴ]ns lay-up is shown in Fig. 11. It is observed from this figure that the minimum allowable thickness is achieved at the optimum fiber orientation of 90° for E-glass/epoxy and carbon/epoxy vessels for all vessel sizes. For hybrid composite vessels, the number of E-glass/epoxy layers is taken to be twice the number of carbon/epoxy layers. The minimum allowable thickness is achieved at the optimum fiber orientation of 90° for [0C/± ϴG]ns and 79° for [0G/+ ϴG/− ϴC]ns lay-ups for all vessel sizes. For example, the

minimum allowable thickness for [0C/±ϴG]ns lay-up is 9.541 mm while for [0G/+ ϴG/− ϴC]ns lay-up is 7.814 mm for 8 in. PV. This is due to the existence of carbon fibers in helical direction of negative values of ϴ for [0G/+ ϴG/− ϴC]ns lay-up which increase the strength and stiffness of the whole vessel than that of the existence of carbon fibers in longitudinal direction only (angle, 0°) for [0C/±ϴG]ns lay-up. The variation of the allowable PV thickness with fiber orientation of different vessel sizes for [90/±ϴ]ns lay-up is shown in Fig. 12. It can be noticed from this figure that the minimum allowable thickness is achieved at the optimum fiber orientation of 49° for E-glass/epoxy and 46° for carbon/epoxy vessels for all vessel sizes. For hybrid composite vessels, the number of E-glass/epoxy layers is taken to be twice the number of carbon/epoxy layers. The minimum allowable thickness is

Fig. 13. Variation of allowable thickness versus fiber orientation for (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in. PVs for [0/±ϴ/90]ns lay-up.

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Fig. 14. Variation of allowable thickness versus fiber orientation for (a) 2.5 in., (b) 4 in., (c) 8 in. and (d) 16 in. PVs for [90/±ϴ/90]ns lay-up.

achieved at the optimum fiber orientation of 26° for [90C/±ϴG]ns and 51° for [90G/+ ϴG/− ϴC]ns lay-ups for all vessel sizes. For example, the minimum allowable thickness for [90C/± ϴG]ns lay-up is 6.852 mm while for [90G/+ ϴG/− ϴC]ns lay-up is 8.794 mm for 8 in. PV. This is due to the existence of carbon fibers in hoop direction (angle, 90°) for [90C/± ϴG]ns lay-up which increase the strength and stiffness of the whole vessel than that of the existence of carbon fibers in helical direction of negative values of ϴ for [90G/+ϴG/− ϴC]ns layup. This leads to that the carbon fibers are more effective in hoop direction which has higher stresses than other directions. These results show good agreement with the previous work [8]. The variation of the allowable PV thickness with fiber orientation of different vessel sizes for [0/±ϴ/90]ns lay-up is shown in Fig. 13. From this figure, it is obvious that the minimum allowable thickness is achieved at the optimum fiber orientation of 90° for E-glass/epoxy and 71° for carbon/epoxy vessels for all vessel sizes. For hybrid composite vessels, the number of E-glass/epoxy layers is taken to be the same as

carbon/epoxy layers. The minimum allowable thickness is achieved at the optimum fiber orientation of 90° for [0C/± ϴG/90C]ns and 61° for [0G/±ϴC/90G]ns lay-ups for all vessel sizes. For example, the minimum allowable thickness for [0C/±ϴG/90C]ns lay-up is 4.958 mm while for [0G/±ϴC/90G]ns lay-up is 5.572 mm for 8 in. PV. This is due to the existence of carbon fibers in both hoop and longitudinal directions for [0C/± ϴG/90C]ns lay-up while they existent in helical direction only for [0G/±ϴC/90G]ns lay-up. The variation of the allowable PV thickness with fiber orientation of different vessel sizes for [90/±ϴ/90]ns lay-up is shown in Fig. 14. This figure demonstrates that the minimum allowable thickness is achieved at the optimum fiber orientation of 44° for E-glass/epoxy and 38° for carbon/epoxy vessels for all vessel sizes. In case of hybrid composite vessels, the number of E-glass/epoxy layers is taken to be the same as carbon/epoxy layers. The minimum allowable thickness is achieved at the optimum fiber orientation of 0° for [90C/± ϴG/90C]ns and 50° for [90G/± ϴC/90G]ns lay-ups for all vessel sizes. For example, the

Table 7 Wall thickness, weight and safety factor for 2.5 in. PVs. Analytical solution Lay-up

[±ϴ]ns

[0/±ϴ]ns

[90/± ϴ]ns

[0/±ϴ/90]ns

[90/± ϴ/90]ns

Material

E-glass/epoxy Carbon/epoxy [+ϴC/−ϴG]ns [±ϴC/±ϴG]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG]ns [0G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG]ns [90G/+ϴG/ϴC]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG/90C]ns [0G/±ϴC/90G]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG/90C]ns [90G/±ϴC/90G]ns

Optimum lay-up

[±54]4s [±55]2s [+56C/−56G]4s [±53C/±53G]s [0/±90]4s [0/±90]s [0C/±90G]3s [0G/+79G/−79C]3s [90/±49]3s [90/±46]s [90C/±26G]2s [90G/+51G/−51C]3s [0/±90/90]3s [0/±71/90]s [0C/±90G/90C]s [0G/±61C/90G]2s [90/±44/90]3s [90/±38/90]s [90C/±0G/90C]2s [90G/±50C/90G]s

n

4 2 4 1 4 1 3 3 3 1 2 3 3 1 1 2 3 1 2 1

FEM

ha mm

h mm

Sf

2.923 0.847 2.555 1.435 3.921 0.942 2.981 2.442 3.315 0.894 2.141 2.748 3.659 0.918 1.549 1.797 3.465 0.912 1.735 1.324

3.2 1.6 3.2 1.6 4.8 1.2 3.6 3.6 3.6 1.2 2.4 3.6 4.8 1.6 1.6 3.2 4.8 1.6 3.2 1.6

2.1898 3.7790 2.5047 2.2301 2.4483 2.5485 2.4149 2.9484 2.1720 2.6837 2.2417 2.6200 2.6234 3.4845 2.0655 3.5607 2.7704 3.5081 3.6892 2.4171

ΔSf %

Weight kg

Weight saving %

h mm

Sf

0.563 0.207 0.494 0.241 0.865 0.154 0.585 0.585 0.638 0.154 0.383 0.585 0.865 0.207 0.241 0.494 0.865 0.207 0.494 0.241

58.537 84.776 63.671 82.271 36.314 88.652 56.948 56.948 53.074 88.652 71.812 56.948 36.314 84.776 82.271 63.671 36.314 84.776 63.672 82.271

3.2 1.6 3.2 1.6 4.8 1.2 3.6 3.6 3.6 1.2 2.4 3.6 4.8 1.6 1.6 3.2 4.8 1.6 3.2 1.6

2.1895 3.7801 2.4498 2.2294 2.4501 2.5502 2.4169 2.9102 2.1722 2.6842 2.2427 2.5744 2.6253 3.4869 2.0666 3.5639 2.7709 3.5093 3.6883 2.4135

−0.012 0.029 −2.192 −0.031 0.072 0.068 0.081 −1.296 0.008 0.020 0.045 −1.739 0.072 0.068 0.053 0.089 0.019 0.034 −0.024 −0.150

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Table 8 Wall thickness, weight and safety factor for 4 in. PVs. Analytical Solution Lay-up

[±ϴ]ns

[0/±ϴ]ns

[90/± ϴ]ns

[0/±ϴ/90]ns

[90/± ϴ/90]ns

Material

E-glass/epoxy Carbon/epoxy [+ϴC/−ϴG]ns [±ϴC/±ϴG]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG]ns [0G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG]ns [90G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG/90C]ns [0G/±ϴC/90G]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG/90C]ns [90G/±ϴC/90G]ns

Optimum lay-up

[±54]6s [±55]2s [+56C/−56G]6s [±53C/±53G]2s [0/±90]6s [0/±90]2s [0C/±90G]4s [0G/+79G/−79C]4s [90/±49]5s [90/±46]2s [90C/±26G]3s [90G/+51G/−51C]4s [0/±90/90]4s [0/±71/90]s [0C/±90G/90C]2s [0G/±61C/90G]2s [90/±44/90]4s [90/±38/90]s [90C/±0G/90C]2s [90G/±50C/90G]2s

n

6 2 6 2 6 2 4 4 5 2 3 4 4 1 2 2 4 1 2 2

FEM

ha mm

h mm

Sf

4.676 1.355 4.088 2.296 6.274 1.507 4.770 3.907 5.304 1.431 3.426 4.397 5.855 1.469 2.479 2.876 5.544 1.459 2.776 2.118

4.8 1.6 4.8 3.2 7.2 2.4 4.8 4.8 6 2.4 3.6 4.8 6.4 1.6 3.2 3.2 6.4 1.6 3.2 3.2

2.0529 2.3620 2.3481 2.7876 2.2953 3.1856 2.0124 2.4570 2.2625 3.3546 2.1016 2.1833 2.1862 2.1778 2.5819 2.2254 2.3086 2.1926 2.3057 3.0214

minimum allowable thickness for [90C/±ϴG/90C]ns lay-up is 5.551 mm while for [90G/±ϴC/90G]ns lay-up is 4.236 mm for 8 in. PV. This is due to the existence of carbon fibers in hoop direction only for [90C/±ϴG/ 90C]ns lay-up while they existent in helical direction for [90G/± ϴC/ 90G]ns which increase the strength and stiffness of the whole vessel. From the previous analysis of the different types of hybrid composite lay-ups, it can be concluded that the existence of carbon fibers in helical direction of both positive and negative values of ϴ is more effective in increasing the strength and stiffness of the whole vessel than that of the existence of them in hoop direction only but less effective than both hoop and longitudinal directions. Also, the existence of carbon fibers in helical direction of positive or negative values of ϴ is more effective than the longitudinal direction only but less effective than hoop direction only. The minimum allowable thickness, thickness, safety factor, weight and relative weight saving of 2.5 in., 4 in., 8 in. and 16 in. composite PVs at the optimum lay-ups are shown in Tables 7–10 respectively.

ΔSf %

Weight kg

Weight saving %

h mm

Sf

3.371 0.820 2.953 1.939 5.170 1.239 3.093 3.093 4.261 1.239 2.293 3.093 4.562 0.820 1.939 1.939 4.562 0.820 1.939 1.939

61.241 90.572 66.040 77.701 40.551 85.748 64.441 64.441 51.005 85.748 73.631 64.441 47.545 90.572 77.701 77.701 47.545 90.572 77.701 77.701

4.8 1.6 4.8 3.2 7.2 2.4 4.8 4.8 6 2.4 3.6 4.8 6.4 1.6 3.2 3.2 6.4 1.6 3.2 3.2

2.0527 2.3626 2.2991 2.7867 2.2967 3.1877 2.0141 2.4280 2.2626 3.3554 2.1026 2.1488 2.1875 2.1790 2.5838 2.2275 2.3091 2.1933 2.3051 3.0167

−0.009 0.027 −2.088 −0.033 0.061 0.066 0.082 −1.180 0.003 0.025 0.049 −1.581 0.060 0.053 0.075 0.092 0.020 0.033 −0.026 −0.156

From these tables, it is obvious that the lowest value of the minimum allowable thickness is achieved at the [±54]ns lay-up of the E-glass/epoxy PVs and at the [±55]ns lay-up of the carbon/epoxy PVs while for hybrid composite PVs, it is achieved at the [90G/±50C/90G]ns lay-up for all vessel sizes. The relative weight saving for each composite PV compared with the stainless steel PV of the same size can be calculated as followings: Weight saving ¼

W st− W comp W st

ð51Þ

It is clear from Tables 7–10 that the carbon/epoxy PVs have weight savings higher than that of the E-glass/epoxy PVs, good agreement with the previous work [14], and the hybrid composite PVs have values between them for all lay-ups and vessel sizes. For example, the optimum lay-up, [±55]4s, for carbon/epoxy 8 in. vessel has weight saving of 90.572% while [± 53C/±53G]3s lay-up has weight saving of

Table 9 Wall thickness, weight and safety factor for 8 in. PVs. Analytical Solution Lay-up

[±ϴ]ns

[0/±ϴ]ns

[90/± ϴ]ns

[0/±ϴ/90]ns

[90/± ϴ/90]ns

Material

E-glass/epoxy Carbon/epoxy [+ϴC/−ϴG]ns [±ϴC/±ϴG]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG]ns [0G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG]ns [90G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG/90C]ns [0G/±ϴC/90G]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG/90C]ns [90G/±ϴC/90G]ns

Optimum lay-up

[±54]12s [±55]4s [+56C/−56G]11s [±53C/±53G]3s [0/±90]11s [0/±90]3s [0C/±90G]8s [0G/+79G/−79C]7s [90/±49]9s [90/±46]3s [90C/±26G]6s [90G/+51G/−51C]8s [0/±90/90]8s [0/±71/90]2s [0C/±90G/90C]4s [0G/±61C/90G]4s [90/±44/90]7s [90/±38/90]2s [90C/±0G/90C]4s [90G/±50C/90G]3s

n

12 4 11 3 11 3 8 7 9 3 6 8 8 2 4 4 7 2 4 3

ha mm

h mm

9.353 2.710 8.177 4.592 12.547 3.014 9.541 7.814 10.608 2.862 6.852 8.794 11.710 2.939 4.958 5.752 11.089 2.919 5.551 4.236

9.6 3.2 8.8 4.8 13.2 3.6 9.6 8.4 10.8 3.6 7.2 9.6 12.8 3.2 6.4 6.4 11.2 3.2 6.4 4.8

FEM ΔSf %

Sf

Weight kg

Weight saving %

h mm

Sf

2.0529 2.3620 2.1524 2.0907 2.1040 2.3892 2.0124 2.1499 2.0363 2.5159 2.1016 2.1833 2.1862 2.1778 2.5819 2.2254 2.0201 2.1926 2.3057 2.2660

13.483 3.280 10.788 5.774 18.853 3.697 12.370 10.763 15.254 3.697 9.173 12.370 18.248 3.280 7.757 7.757 15.848 3.280 7.757 5.774

61.242 90.572 68.987 83.403 45.806 89.373 64.441 69.061 56.151 89.373 73.631 64.441 47.545 90.572 77.701 77.701 54.442 90.572 77.701 83.403

9.6 3.2 8.8 4.8 13.2 3.6 9.6 8.4 10.8 3.6 7.2 9.6 12.8 3.2 6.4 6.4 11 3.2 6.4 4.8

2.0527 2.3626 2.1100 2.0900 2.1056 2.3905 2.0141 2.1254 2.0363 2.5165 2.1026 2.1452 2.1875 2.1791 2.5838 2.2275 2.0205 2.1933 2.3052 2.2625

−0.009 0.027 −1.972 −0.034 0.075 0.055 0.082 −1.138 0.001 0.023 0.049 −1.745 0.060 0.058 0.075 0.092 0.022 0.033 −0.022 −0.157

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Table 10 Wall thickness, weight and safety factor for 16 in. PVs. Analytical Solution Lay-up

[±ϴ]ns

[0/±ϴ]ns

[90/± ϴ]ns

[0/±ϴ/90]ns

[90/± ϴ/90]ns

Material

E-glass/epoxy Carbon/epoxy [+ϴC/−ϴG]ns [±ϴC/±ϴG]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG]ns [0G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG]ns [90G/+ϴG/−ϴC]ns E-glass/epoxy Carbon/epoxy [0C/±ϴG/90C]ns [0G/±ϴC/90G]ns E-glass/epoxy Carbon/epoxy [90C/±ϴG/90C]ns [90G/±ϴC/90G]ns

Optimum lay-up

[±54]24s [±55]7s [+56C/−56G]21s [±53C/±53G]6s [0/±90]21s [0/±90]6s [0C/±90G]16s [0G/+79G/−79C]14s [90/±49]18s [90/±46]5s [90C/±26G]12s [90G/+51G/−51C]15s [0/±90/90]15s [0/±71/90]4s [0C/±90G/90C]7s [0G/±61C/90G]8s [90/±44/90]14s [90/±38/90]4s [90C/±0G/90C]7s [90G/±50C/90G]6s

n

24 7 21 6 21 6 16 14 18 5 12 15 15 4 7 8 14 4 7 6

ha mm

h mm

18.705 5.419 16.353 9.183 25.095 6.027 19.081 15.629 21.215 5.724 13.704 17.588 23.420 5.877 9.915 11.503 22.178 5.838 11.103 8.473

19.2 5.6 16.8 9.6 25.2 7.2 19.2 16.8 21.6 6 14.4 18 24 6.4 11.2 12.8 22.4 6.4 11.2 9.6

83.403%. Then, the hybrid composite material which saves about 50% carbon fibers has weight saving close to the carbon/epoxy composite material with deference of 7.169%. Also, the optimum lay-up, [0/± 90]3s, for carbon/epoxy 8 in. vessel has weight saving of 89.373% while [0G/+79G/−79C]7s lay-up has weight saving of 69.061%. Then, the hybrid composite material which saves about 66.6% carbon fibers has weight saving deference of 20.312% from the carbon/epoxy composite material. Thus, the low cost of PVs can be achieved by using of the hybrid composite material for the same lay-up and vessel size. 4.2.2. FEM solution The PV thickness and safety factor obtained from the FEM for 2.5 in., 4 in., 8 in. and 16 in. composite PVs at the optimum lay-ups are shown in Tables 7–10 respectively. The same vessel wall thickness is obtained as in the analytical solution with some safety factor error for each vessel. As could be observed from these tables, the safety factors obtained from the finite element modeling are close to that obtained from the analytical solution within an acceptable range i.e. b0.2% for most lay-ups except that for hybrid lay-ups which have carbon fibers in one helical direction only, the safety factor error increases to about 2.2%.

FEM ΔSf %

Sf

Weight kg

Weight saving %

h mm

Sf

2.0529 2.0667 2.0546 2.0907 2.0084 2.3892 2.0124 2.1499 2.0363 2.0966 2.1016 2.0469 2.0495 2.1778 2.2591 2.2254 2.0201 2.1926 2.0175 2.2660

53.932 11.457 41.115 23.094 71.784 14.787 49.480 43.051 61.016 12.287 36.692 46.257 68.175 13.119 27.047 31.029 63.394 13.119 27.047 23.094

61.242 91.766 70.453 83.403 48.413 89.373 64.441 69.061 56.151 91.170 73.631 66.757 51.006 90.572 80.563 77.701 54.442 90.572 80.563 83.403

19.2 5.6 16.8 9.6 25.2 7.2 19.2 16.8 21.6 6 14.4 18 24 6.4 11.2 12.8 22.4 6.4 11.2 9.6

2.0525 2.0672 2.0156 2.0899 2.0099 2.3908 2.0143 2.1242 2.0362 2.0971 2.1026 2.0121 2.0510 2.1793 2.2608 2.2272 2.0204 2.1933 2.0170 2.2625

−0.019 0.023 −1.899 −0.039 0.075 0.067 0.092 −1.194 −0.004 0.024 0.049 −1.698 0.071 0.067 0.074 0.079 0.017 0.033 −0.025 −0.157

The distribution of the safety factor for [±54]12s lay-up 8 in. E-glass/ epoxy PV is shown in Fig. 15. 5. Conclusion The SWRO pressure vessels (PVs) have been modeled using analytical solution and finite element modeling (FEM) for stainless steel and fiber reinforced composite materials to optimize the PV design parameters. The used PV sizes are 2.5, 4, 8 and 16 in. according to the standard sizes in the international markets. For stainless steel PVs, the same wall thickness is obtained from both analytical solution and FEM for each vessel size with acceptable safety factor error. For composite PVs, the analytical solution results show that the allowable PV thickness depends on fiber orientation and material type for a certain lay-up. The carbon/epoxy vessels have allowable thickness lower than the E-glass/epoxy vessels at the same winding angle for all vessel sizes. This is due to the high stiffness and strength of the carbon fibers compared with the E-glass fibers. In case of hybrid composite materials, the allowable thickness values are between that of E-glass/epoxy and carbon/epoxy materials depending on the amount of each material

Fig. 15. Distribution of the safety factor for [±54]12s lay-up 8 in. E-glass/epoxy PV.

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and the arrangement of layers through the vessel thickness. By comparing the different types of lay-ups, the optimum lay-ups are [±54]ns layup for the E-glass/epoxy PVs, [±55]ns lay-up for the carbon/epoxy PVs and [90G/±50C/90G]ns lay-up for the hybrid composite PVs for all vessel sizes. Thus, for hybrid composite PVs, it is recommended to use carbon fibers in helical direction of both positive and negative values of winding angle to increase the strength and stiffness of the whole vessel. The composite PVs show better weight savings compared with the stainless steel PVs. The carbon/epoxy PVs have weight savings higher than that of the E-glass/epoxy PVs and the hybrid composite PVs have values between them for all lay-ups and vessel sizes. Thus, the low cost PVs can be achieved by using of the hybrid composite material for the same lay-up and vessel size. Finally, the FEM results show good agreement with the analytical solution results. References [1] Y.M. Kim, S.J. Kim, Y.S. Kim, S. Lee, I.S. Kim, J.H. Kim, Overview of systems engineering approaches for a large-scale seawater desalination plant with a reverse osmosis network, Desalination 238 (2009) 312–332. [2] M.C. Porter, Handbook of Industrial Membrane Technology, Noyes, Park Ridge, N.J., 1990 [3] A.A. Krikanov, Composite pressure vessels with higher stiffness, Compos. Struct. 48 (2000) 119–127. [4] P. Xu, J.Y. Zheng, P.F. Liu, Finite element analysis of burst pressure of composite hydrogen storage vessels, Mater. Des. 30 (2009) 2295–2301. [5] D.-S. Son, S.-H. Chang, Evaluation of modeling techniques for a type III hydrogen pressure vessel (70 MPa) made of an aluminum liner and a thick carbon/epoxy composite for fuel cell vehicles, Int. J. Hydrog. Energy 37 (2012) 2353–2369. [6] A. Onder, O. Sayman, T. Dogan, N. Tarakcioglu, Burst failure load of composite pressure vessels, Compos. Struct. 89 (2009) 159–166.

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