12
393 7 Thermoplastic Pressure Vessel Design 7.1 Thermoplastic Thin-Walled Pressure Vessels The term thin-walled or thin-shelled pressure vessel describes a hollow cylinder in which the circumferential stress (frequently called hoop stress) in the wall is assumed to be constant throughout the thickness of the wall when the cylinder is subjected to an internal or external fluid pressure. Thermoplastic materials have been used for the fabrication of many pressure vessel devices, such as toilet flush valves, spray paint containers, butane lighters, irrigation sprayers and valves, brake master cylinders, radiator end cores, garden hoses, tubing, end connections, pumps, and so forth. Figures 7-1 and 7-2 illustrate some of these pressure vessel applications. Figure 7-1 Thermoplastics pressure vessel applications (Courtesy: Du Pont) Water spray gun Figure 7-2 Thermoplastic pressure vessel applications

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Page 1: Plastic Vessel Pressure Design

393

7 Thermoplastic Pressure Vessel Design

7.1 Thermoplastic Thin-Walled Pressure

Vessels

The term thin-walled or thin-shelled pressure vessel describes a hollow cylinder in which the circumferential stress (frequently called hoop stress) in the wall is assumed to be constant throughout the thickness of the wall when the cylinder is subjected to an internal or external fluid pressure. Thermoplastic materials have been used for the fabrication of many pressure vessel devices, such as toilet flush valves, spray paint containers, butane lighters, irrigation sprayers andvalves, brake master cylinders, radiator end cores, garden hoses, tubing, end connections, pumps, and so forth. Figures 7-1 and 7-2 illustrate some of these pressure vessel applications.

Figure 7-1 Thermoplastics pressure vessel applications (Courtesy: Du Pont)

Water spray gun

Figure 7-2 Thermoplastic pressure vessel applications

Page 2: Plastic Vessel Pressure Design

394 7 Thermoplastic Pressure Vessel Design

7.2 Thin-Walled Cylinder Basic Principles

The circumferential stress (a) in a thin-walled cylinder subjected to an internal pressure (P) per unit area is found by applying an equation of equilibrium tothe forces acting on the half cylinder shown in Figure 7-3. The length is uniform,wall thickness is (t), and inside radius is (r).

2 2 dP r aσ× = ∫ (7-1)

But under the assumed conditions, da a tσ σ σ= =∫ .

Therefore, Barlow’s Equation is:

P r

×= (7-2)

Spherical Closed End Thin-Walled Pressure Vessels

To calculate the stress of a cylindrical pressure vessel with a spherical base, under uniform internal pressure, using Figure 7-4 as a model, the maximum stress equation should be applied:

Max. 2

P r

×=

× (7-3)

Flat Closed End Thin-Walled Pressure Vessel

To find the maximum stress of a cylindrical pressure vessel with a circular fl at bottom base, under uniform internal pressure and using Figure 7-5 as a model,the following equations should be applied:

Center defl ection:4

23

3(1 )

16

P r

E tδ υ

⎛ ⎞× ×= −⎜ ⎟× ×⎝ ⎠

(7-4)

Maximum moment:2

Max. 8

P rM

×= (7-5)

Maximum stress:2

Max. 2 2

6 0.75M P r

t tσ

× × ×= = (7-6)

Example 7-1

The shank/riser toilet flush valve shown in Figure 7-6 needs to withstand a 2,000 psi burst pressure and 120 psi continuous internal pressure for 10 years. The burst pressure would be the controlling factor for the design, ratherthan the continuous pressure. The material selected for this application is acetal homopolymer with a tensile strength of 10,000 psi. Calculate the wall thickness of the shank/riser by using Barlow’s equation (Equation 7-2).

Barlow’s EquationP r

×= or

2,0000.20

10,000

P r rt r

σ

× ×= = =

The stress for the shank/riser wall thickness at 120 psi pressure over ten years is calculated by using the isochronous creep stress/time long-term pipe test

da dar

P2r x P

t

Figure 7-3 Thin-walled cylinder mathematical model

tr

Pr

Figure 7-4 Spherical closed end thin-walled pressure vessel

P Pt

r

Figure 7-5 Flat closed end thin-walled pressure vessel

Page 3: Plastic Vessel Pressure Design

7.2 Thin- Walled Cylinder Basic Principles

data shown in Figure 7-7. For 10 ycars the tensile stress is 1,750 psi. The wall thickness can be calculated as t, = (120 x r) 1 1,750 = 0.068 r. The burst pressure wall thickness t , = 0.20 r is the control bctor for dimensioning, because the wall: thickness for burst pressure requires 0.20 10.068 = 2.94 times the wall thickness calculated to retain the 110 psi internal pressure over 10 ycars.

The shanklriser outside diameter at the base (left side) is the root diameter of the threads, and its wall thickness is t = 0.20 x 0.453 = 0.0906 in. Because the toy (right side) outside diameter is smaller, its wall thickness t = 0.20 x 0.375 = 0.075 in, without considering the reinforcement caused by the perpendicular wall of the toilet valve housing. These two shanklriser inside diameters form a tapered wall, which not only provides the most efficient design, but aIso helps during the part ejection from the long core of the mold. To improve the molding process efficiency, the long core requires the surface to be hardened to 60 R,, the surface to be coated with a low coefficient of friction mold release and polished in the longitudinal direction (a RMS). The core also requires an independent water cooling system. The shanklriser is to he gated at the flange width (thicker wall sectinn equal to 0.1 56 i n ) using an insulated runnerless mold with three cavities.

Figure?-6 Acetal homopolymer shank/riser toilet flush valve

Internal pressure at 73" f.

Time, [hours)

10.000

e 2 - -

Figure 7-7 Fsochronous acetal homopolymer pipe burst data (Courtesy: Du Pont)

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Page 4: Plastic Vessel Pressure Design

396 7 Thermoplastic Pressure Vessel Design

7.3 Thick-Walled Pressure Vessels

For the pressure vessels with a relatively large wall thickness, the mathematical model shown in Figure 7-8 should be applied. If the variation in the stress from the inner surface to the outer surface is relatively large, the value of the stress found from Barlow’s Equation 7-2 is not a satisfactory measure of the signifi cant circumferential stress in thick-walled pressure vessels.

Barlow’s Equation applies only to thin-walled cylinders; this fact will be evident from a consideration of the equilibrium of the forces acting on the thin-walled cylinder shown in Figure 7-8. In addition, a satisfactory solution for the thick-walled cylinder problem requires the determination not only of the circumferential stress at any point in the cylinder, but of both of the principal stresses whose vectors lie in the plane of the paper, namely, the circumferential or tangential stress σt and the radial stress σr as shown in Figure 7-9.

7.3.1 Lame’s Equation for Thick-Walled Cylinders

Lame’s equation is used to calculate the maximum tangential and radial stresses of thick-walled cylinders subjected to internal and external pressures.

Figure 7-9 represents a relatively long open-ended thick-walled cylinder subjectedto internal and external fl uid pressures P1 and P2, respectively. We will useFigure 7-9 to find the circumferential stress σt and the radial stress σr at a point at any distance from the central axis of the. From the conditions of symmetry,it is known that there is no shearing stress on the planes on which σt and σr act and therefore they are principal stresses.

Finding both σt and σr requires that sections be passed through the body so that the portion of the body isolated by the sections will be acted on by forces that involve the two stresses. Such a portion is obtained by first passing two concentric sections through the body and thus isolating a thick-walled cylinder;this element with the forces acting on it is shown in Figure 7-10. A diametral plane is then passed through the element, isolating one half the element that may be expressed by the two stresses (one at each side of the half cylinder), as shown in Figure 7-11.

By applying one of the equations of equilibrium to the forces in Figure 7-11 wefind that the algebraic sum of the vertical components of the forces is equal tozero and the following equation is obtained:

t r r rd 2 d 2 d 2 d dσ ρ σ ρ ρ σ ρ σ= − − (7-7)

The term 2 dρ dσr is negligibly small. The tensile stress becomes:

t r r(d /d )σ σ ρ σ ρ= − − (7-8)

If the stresses σr and σt in Figure 7-11 are assumed to be positive, that is, if both stresses are assumed to be tensile stresses, the previous equation is:

t r r(d /d )σ σ ρ σ ρ= + (7-9)

A rational assumption concerning the strains in a thick-walled cylinder is that longitudinal strains of the fibers are equal. This means that transverse (parallel) sections that are plane before the fl uid pressures P1 and P2 are applied remain plane and parallel after the pressures are applied. This will be true at least for a

r1

r2

PP

da

da

P x 2r1

Figure 7-8 Thick-walled cylinder mathematical model

r2

P2 P1

r1

dt

r r+ d

r

Figure 7-9 Radial and tangential stresses caused by in/out pressures

d

r ad

r r+ d( ) ad

P2

P1

Figure 7-10

d

t dt d

r 2

r r+ d( ) 2( + )d

P2P1

Figure 7-11

Page 5: Plastic Vessel Pressure Design

397

cylinder with open ends, and it will also be nearly true for a closed cylinder at sections well removed from the ends of the cylinder.

The relation between the longitudinal strain εl of any longitudinal fi ber and the stresses acting on the fiber in an open ended thick-walled cylinder is:

1 r t( / ) ( / )E Eε υ σ υ σ= − (7-10)

and according to the above assumption,εl is constant. In addition, Poisson’s ratioν and the modulus of elasticity E are constants of the material, resulting in:

t r Constant 2σ σ κ− = =

The constant is denoted by 2 κ for convenience. The previous two equations givetwo relations between σr and σt. From these two equations we obtain:

r r2 2 (d /d )κ σ ρ σ ρ= − − (7-11)

But the right hand side of this equation, when multiplied by ρ, becomes the derivative, with respect to ρ, of –(ρ2 σr), and therefore the equation may be written:

2rd( )/d 2ρ σ ρ κ ρ= − (7-12)

The integration of this equation gives:

2 2rρ σ κ ρ β= − + (7-13)

where β is a constant of integration. Therefore:

2r ( / )σ β ρ κ= − (7-14)

and from the equation σt – σr = Constant = 2 κ

2t ( / )σ β ρ κ= + (7-15)

The values of the constants β and κ are found by substituting the values σr and ρ that were obtained from the physical conditions or assumptions stated. Forexample, assuming the cylinder to be subjected to both internal and external pressures P1 and P2, respectively, we observe that

σr = P1 where ρ = r1 and σt = P2 where ρ = r2

and from the Equation 7-14, 2r ( / )σ β ρ κ= − we obtain:

21 1( / )P rκ β= − + and 2

2 2( / )P rκ β= − +

from which κ and β are:

2 21 1 2 2

2 22 1

P r P r

r rκ

−=

− (7-16)

2 21 2

1 22 22 1

( )r r

P Pr r

β×

= −−

(7-17)

7.3 Thick-Walled Pressure Vessels

Page 6: Plastic Vessel Pressure Design

398 7 Thermoplastic Pressure Vessel Design

The substitution of these values of κ and β in Equations 7-14 and 7-15:

2 2 2 2 21 1 2 2 1 2 1 2

t 2 22 1

( / ) ( )P r P r r r P P

r r

ρσ

− + × −=

− (7-18)

2 2 2 2 22 2 1 1 1 2 1 2

r 2 22 1

( / ) ( )P r P r r r P P

r r

ρσ

− + × −=

− (7-19)

It is evident from Equation 7-18 that the maximum value of σt occurs at the inner surface where ρ has its minimum value r1. The maximum value of σr willalways be the larger of the two pressures P1 and P2.

7.3.2 Maximum Stresses with Internal and External Pressures

By setting ρ = r1 in Equations 7-18 and 7-19, the maximum stresses (at the inner surface) are:

Maximum tensile stress 2 2 2

1 1 2 2 2t 2 2

2 1

( ) 2P r r P r

r rσ

+ −=

− (7-20)

(Lame’s Equation)

Maximum radial stress r 1Pσ = if 1 2P P>

7.3.3 Maximum Stresses for Internal Pressure Only

The maximum radial stress value σr is equal to P1; these maximum stresses are shown in Figure 7-12.

If the internal pressure is P1 and the external pressure is zero (P2 = 0), themaximum stress equations are reduced to:

2 21 1 2

t 2 2 22 1

1P r r

r rσ

ρ

⎛ ⎞×= +⎜ ⎟− ⎝ ⎠

(7-21)

2 21 1 2

r 2 2 22 1

1P r r

r rσ

ρ

⎛ ⎞×= −⎜ ⎟− ⎝ ⎠

(7-22)

These equations show that the maximum values of σt occur at the inner surface,when ρ = r1. The new equation is:

Maximum tensile stress 2 2

1 1 2t 2 2

2 1

( )P r r

r rσ

+=

− (7-23)

Table 7-1 provides more equations for calculating different pressure vessel types of loading.

P = 02

r 1= P

Max. t

Figure 7-12 Maximum stress, internal pressure, mathematical model

Page 7: Plastic Vessel Pressure Design

399

Table 7-1 Cylindrical Pressure Vessel Equations

Vessels geometry Type of loading Cylindrical pressure vessel equations

Thin-walled cylindery

T

R2

2

1

P2

P2

P1

Uniform internalorexternal pressure

1 21 2

P R

×=

×1 2

2P R

×= 2

2 2 1( )R

RE

σ υ σΔ = − ×

Y2 2

2 Y 21 4

TP

R R

E T

σ

σ

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎛ ⎞+⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2 11 1

2 1

2R R

PR R

σ−

= ×+

Thin-walled cylinder with spherical bottom base

R1

T

P1

Uniform internal pressure Maximum stress at middle of spherical base

1 1Max.2

P R

×=

×

Thin-walled cylinder with circular fl at bottom base

R1

T

P1

Uniform internal pressure Maximum stress at middle of circular fl at base

421 1

3

3(1 )

16

P R

E Tδ υ

⎛ ⎞× ×= −⎜ ⎟× ×⎝ ⎠

21 1

Max. 8

P RM

×=

21 1

2 2

0.756Max.

P RM

T Tσ

× ××= =

Thick-walled cylinder

R2 R1

2

13

P2

P2

P1

Uniform internal pressureP2 = 0 1 0σ =

2 22 1

2 1 2 22 1

Max.R R

PR R

σ+

=−

22

3 1 2 22 1

Max.R

PR R

σ =−

2 21 2 1

1 1 2 22 1

R R RR P

E R RυΔ

⎛ ⎞+= +⎜ ⎟−⎝ ⎠

22 1

2 1 2 22 1

2R RR P

E R RΔ

⎛ ⎞×= ⎜ ⎟−⎝ ⎠

Uniform external pressureP1 = 0 1 0σ =

22

2 2 2 22 1

2Max.

RP

R Rσ

×=

−2

3Max.

Max.2

σσ =

21 1

1 2 2 22 1

2R RR P

E R RΔ

⎛ ⎞×= ⎜ ⎟−⎝ ⎠

2 22 2 1

2 2 2 22 1

R R RR P

E R RυΔ

⎛ ⎞+= −⎜ ⎟−⎝ ⎠

R2 = External radius, R1 = Internal radius, T = Thickness, υ = Poisson ratio,δ = Defl ection, P2 = External pressure, P1 = Internal pressure, E = Flexural modulus,σ = Stress, σY = Yield stress, M = Moment of force

7.3 Thick-Walled Pressure Vessels

Page 8: Plastic Vessel Pressure Design

400 7 Thermoplastic Pressure Vessel Design

7.4 Designing Cylinders for Cost Reduction

The typical design of spherical closed end pressure vessels uses smooth, thick-walled cylinders where the hoop stress is double the axial stress. The use ofthinner walls and circumferential ribs to reinforce the vessel in the hoop directionimproves the process efficiency, lowers part weight, cycle time, and manufacturingcosts. A design comparison is shown in Figures 7-13 and 7-14.

Figure 7-13 shows a typical pressure vessel design with 0.250 in wall thickness.The recommended ribbed design is shown in Figures 7-14 and 7-15. The wall thickness was reduced to 0.125 in and ribs were added to provide the same strength of the thick-walled cylinder, in both directions, hoop and axial.

Fillet radius and draft angles on the ribs reduce stress concentration and simplify part ejection from the mold. The stresses caused by the spherical closed end cylinder wall were checked using standard pressure vessel equations. The stress level was negligible for both designs.

The estimated cost savings of the ribbed design is 30%. Because of the reduction in the molding cycle from 85 s for the 0.250 in, thick-walled design to 45 s for the recommended 0.125 in walls and circumferential ribbed design.

7.5 Thermoplastic Pressure Vessels Design

Guidelines

When good design, proper resin selection, mold design, quality injectionmolding, and testing are employed, thermoplastic pressure vessels will provide satisfactory and safe service performance to the end users. Because thermoplasticshave nonlinear stress-strain relationships over a large range of strains, the usual analytical techniques applied to pressure vessels may not be accurate enough to predict failures. The design parameters for a thermoplastic pressure vessel should be very conservative and the injection molding process should be set up to comply with the “A-1 Quality Control” requirements.

7.5.1 Preliminary Pressure Vessel Design

A cylinder is considered thin-walled when the ratio between the wall thickness to inside radius is 0.50 or less; in this case, Barlow’s Equation (Eq. 7-2) should be used. Lame’s Equation (Eq. 7-20) should be used for thick-walled cylinders having a ratio greater than 0.50. Other sound engineering techniques areavailable to the designer (e.g., finite element analysis, more sophisticatedequations, etc.) to determine the wall thicknesses and dimensions of the load-bearing members.

When designing a pressure vessel cylinder with a snap-fit end cap, the internal pressure will deform the wall of the vessel more than its cap. This reduces snap-fi ttightness and causes leakage. It is recommended to redesign the exterior snap-fit cap and add an “O” ring to eliminate the leakage. Figure 7-16 shows a poor product design, operational problems, and design recommendations.

When designing a pressurized cylinder with a bolted end cap and a top-seated “O”ring, a very high load is required to compress the “O” ring axially, consequently deformation and creep of the seal flanges occur. This effect worsens when the distance between the “O” ring and the bolts is increased. To compensate for this

Figure 7-13 Thick wall cylinder (Courtesy: Du Pont)

Figure 7-14 Recommended design (Courtesy: Du Pont)

0.062 R. Draft angle0.250

0.1250.375

Figure 7-15 Recommended design, ribbed cross section detail

Page 9: Plastic Vessel Pressure Design

401

effect, redesign the end cap and the cylinder wall to move the “O” ring to seal bycompressing radially.Additional benefits are gained by reducing creep. Stiffeningthe flange with ribs or a metal ring under the bolts can help. Figure 7-17 shows a poor design, operational problems, and design recommendations.

When designing a pressurized cylinder with self-tapping screw end caps and a top axial seated “O” ring, the end cap requires snap-fi t fingers with the “O”ring moved to seal radially. With the low internal pressure required for this valve application, the use of snap-fits can make the assembly faster and moreeconomical. Figure 7-18 shows a poor design while Figure 7-19 shows the design recommendations.

Carefully study the thermoplastic material properties for the cylinder.

Specify the sensitive areas of the pressure vessel, such as type, size, number, and location of the gate, sharp corners, weld lines, and ribs.

The design pressure of the cylinder should be lower than 150 psi, or 15% of the maximum required bursting pressure.

Creep stress of the resin must be used based on best available data.

Poor design

Internalpressure

Operational problems

Pressureexpands wall

(leakage)

Internalpressure

Pressure & "O" ringeliminates leakage

Design recommendations

Internalpressure

"O" ring

Figure 7-16 Pressure vessel with snap-fit end cap design (Courtesy: Du Pont)

Poor design

Axial"O" ring Axial bolts compression

forces apart joining walls

Operational problems

Best seal, radial "O"ring in compression

Design recommendation

Radial"O" ring

Figure 7-17 Pressure vessel with bolted end cap design

Valve assembly top view

Axial "O" ring

Self-tap screw

Valve cross section view

Figure 7-18 Poor valve design,“O” ring with self-tapping screw end cap (Courtesy: Du Pont)

Valve assembly top view

Valve cross section view

Radial"O" ring

Valve base withsix lock pockets

Valve end cap withsix lock springs

Figure 7-19 Valve with six snap-fit end caps,design recommendations

7.5 Thermoplastic Pressure Vessels Design Guidelines

Page 10: Plastic Vessel Pressure Design

402 7 Thermoplastic Pressure Vessel Design

7.6 Testing Prototype Thermoplastic Pressure

Vessels

Build a cylinder prototype tool and mold several prototype samples.

For each pressure vessel to be tested, use the same thermal and moistureenvironment of the product in its end use operation. All cycling and burst tests will be conducted in this environment.

The designer must decide whether a cycling pressure test is needed. If the pressurevessel end use involves frequent pressurization and depressurization, a cycling fatigue test is recommended.

The cylinder pressure should be tested starting from atmospheric to design pressure and back to atmospheric 100,000 times or less, if a lower service life is acceptable.

If a cycling test was done on the prototype pressure vessel, a burst test must be done with the same specimen used in the cycling test. Otherwise, use any good sample conditioned to resemble the operating environment for the burst test.

Burst pressure and the mode of failure should be carefully recorded.

7.6.1 Redesign and Retesting the Pressure Vessels

If needed, modify the pressure vessel design according to the outcome of the previous tests.

Conduct a burst test with one of the modifi ed pressure vessels to confi rm the improvements made by the redesign modifi cations.

Build a production tool and mold commercial quality cylinders.

Retest the pressure vessel periodically to confirm if the quality of the injection molding process meets the quality control requirements established for the production pressure vessels.

7.7 Pressure Vessel Regulations

Pressure vessels are regulated by industry codes to establish design safetyguidelines for dimensioning and testing vessels of various materials and end use applications. One of the most prominent groups that regulate pressure vessel design is the American Society of Mechanical Engineers (ASME). The ASME Boiler and Pressure Vessel Codes have become an American National Standard (accepted by the American National Standards Institute) and are mandatory bylaw in several states.

Injection molded thermoplastic pressure vessels do not always fall under the jurisdiction of the ASME code because of the small size of the pressure vessels andthe type of end use applications. However, the National Sanitation Foundation is very active in regulating the design, testing, and manufacture of injection molded thermoplastic pressure vessels.

Page 11: Plastic Vessel Pressure Design

403

7.7.1 ASME Pressure Vessel Code

This regulation is applicable for materials of flexural modulus as low as1.0 × 106 psi. Some thermoplastic materials meet this criterion. Creep ofthermoplastic materials is not considered by the code, but must be taken into account in designing thermoplastic pressure vessels.

Tensile Strength

The materials considered by this code are the materials that have tensile strengthsfrom 12,000 psi to 25,000 psi.

Design Pressure

Sets the maximum design pressure lower than 150 psi, or 15% of the bursting pressure.

Design Temperature

Temperature is set at 150 °F and the code requires the burst pressure test to be done at this temperature.

Operating Pressure

Should be less than, or equal to the design pressure.

Bursting Pressure

Bursting pressure is the hydrostatic pressure at which a prototype pressurevessel bursts.

Loadings

Several types of pressure loadings are considered; the most important being internal and external pressure. The other types are mechanical impact loads,reactions of supporting lugs, rings, and so forth.

Stress Caused by Combined Loads

Stresses are analyzed using the membrane stresses produced by bending and shearing loadings. The maximum membrane stress during normal operation of the pressure vessel must not exceed 15% of the maximum membrane stress at the bursting pressure.

Proof of Design Adequacy

A prototype pressure vessel must be subjected to 100,000 pressure cycles, rangingfrom atmospheric to the design pressure. After this test, the same pressure vessel should burst at a pressure no less than six times the specified maximum design pressure. The test fluid should have a minimum temperature of 150 °F.

Pressure Relief Devices

It is require that all pressure vessels be provided with protection against over pressures.

7.7 Pressure Vessel Regulations

Page 12: Plastic Vessel Pressure Design

404 7 Thermoplastic Pressure Vessel Design

Three types of pressure relief devices are acceptable:

Direct spring loaded safety relief valves

Pilot operated valves

Rupture disks

Set Pressures

A single pressure relief device will be set at the design pressure.

Set Pressure Tolerances

For safety and safety relief valves:

±2 psi when the operating pressure is lower than 70 psi

±3 psi when the operating pressure is higher than 70 psi

Rupture discs ±5 psi for all pressures

Permissible Over Pressures

Single relief devices should withstand 110% of the design pressure.

Testing Requirements

Apply cyclic pressure and adhere to burst test conditions:

Test fluids should be water or other liquids

Temperature of test fluid should be 150 °F

Cycle pressure from atmospheric to design pressure and back, 100,000times

Following the cyclic test, the pressure vessel should burst. The minimum burst pressures to be six times the design pressure.