Processing equipment design

Exercise No. 6: Calculation of conical cover of vessel loaded with

internal or external overpressure

Lecturer: Pavel Hoffman

http://fsinet.fsid.cvut.cz/cz/U218/peoples/hoffman/index.htm

e-mail: pavel.hoffman@fs.cvut.cz

2 PED-ex.6

Calculation of conical cover of vessel loaded with internal or external overpressure 100 kPa (according ČSN 690010, part 4.6)

Sketch of cylindrical vessel with conical cover and neck 1 – cone 2 – cylinder

H a1

f D1

s2

s1

f Di

f De

3 PED-ex.6

Given data:

• External diameter of cylindrical part De = 1500 mm

• Realized wall thickness of cylindrical part s2 = 7 mm

• Internal diameter of cylindrical part Di = D = 1486 mm

• Calculating internal/external overpressure pi,e = 100 kPa

• Allowed stress of material of cylind. part 1 = D1 = 140 MPa

• Allowed stress of material of conical part 2 = D2 = 140 MPa

• Modulus of elasticity for both materials E = 206 GPa

• Poisson constant for both materials = 0,3

• Sum of all allowances (corrosion etc.) c1 = c2 = 1 mm

• Neck = tube f 159 x 4.5 d1 = 150 mm

• Height of conical cover (for a1 = 70°) H 240 mm

• Coefficient of longitudinal weld in cone φW = 0,7

Note: Following calculations are valid for angle a1 70 °. We will perform calculations for 2 angels: a1 = 45° and a1 = 70°.

PED-ex.6 4

Internal overpressure 100 kPa – half apex angle α1 = 45°

Calculated length of transition part (it corresponds to the reach of stress peak)

)(*cos

*7,01

1

1cs

Da

a

mma 9,87)15,8(*45cos

1486*7,0

1

Calculated diameter of conical shell

DC = D – 1,4 * a1* sina1

DC = 1486 – 1.4 * 87.9 * sin45 = 1399.0 mm

Conical shell wall thickness is estimated on the basis of the relationship in ČSN as s1 = 8.5 mm

sC = sR / cos α1 = (7 – 1) / cos 45 = 6 / 0.707 8.5 mm

Lt = 0.55*√(D*s) L = 1.65*√(D*s)

PED-ex.6 5

Determining whether the calculation formulas given in CSN can be used for our case:

The minimum calculated wall thickness of conical shell

mmp

pDs

PD

CCR 01.1

45cos

1*

100.07.0*140*2

100.0*1399

cos

1*

**2

*

1

a

Maximum internal pressure that can withstand the conical shell

MPa

csD

csp

C

PD 74.0

)15.8(45cos

1399

)15.8(*7.0*140*2

)(cos

)(***2

1

1

1

a

PED-ex.6 6

Internal overpressure 100 kPa – half apex angle α1 = 70°

Conical shell wall thickness is estimated on the basis of the relationship in ČSN again as s1 = 8.5 mm

Calculated length of transition part (it corresponds to the reach of stress peak)

)(*cos

*7.0 1

1

1 csD

a a

mma 3.126)15.8(*70cos

1486*7.01

Calculated diameter of conical shell

DC = D – 1.4 * a1* sina1

DC = 1486 – 1.4 * 126.3 * sin70 = 1320 mm

for angle is longer reach of stress peak

for angle is calculated ø of cone

( x for a1 = 45° it was 87.9 mm)

( x 1399 mm)

PED-ex.6 7

Determining whether the calculation formulas given in CSN can be used for our case:

The minimum calculated wall thickness of conical shell

mmp

pDs

PD

CCR 97.1

70cos

1*

100.07.0*140*2

100,0*1320

cos

1*

**2

*

1

a

Maximum internal pressure that can withstand the conical shell

MPa

csD

csp

C

PD 36.0

)15.8(70cos

1399

)15.8(*7.0*140*2

)(cos

)(***2

1

1

1

a

( x 1.01 mm)

( x 0.74 MPa)

PED-ex.6 8

External overpressure 100 kPa – half apex angle α1 = 45°

Preliminary determination of wall thickness s1

The wall thickness is specified according equations valid for cylinder (part 4.5) multiplied by value 1/cosa1.

In our previous example of the cylindrical vessel loaded with external pressure we specified the cylinder thickness without allowances:

mmDp

DKMaxsD

R 80.4*2

**1.1;10** 2

22

Then is the preliminary wall thickness of the conical part

mms

s RR 8.6

45cos

8.4

cos 1

21

a

K1

K3

K2

= the real situation in which the heater has worked

PED-ex.6 9

Preliminary designed wall thickness together with all allowances is

s1 = sR1 + c = 6.8 + 1 8.0 mm Calculating diameter of a smooth conical shell without torus in connection with the cylinder (used in next calculations)

DC = D – 1.4 * a1* sina1

where is

)(*cos

*7.0 1

1

1 csD

a a

mma 9.84)10.8(*45cos

1486*7.01

DC = 1486 – 1.4 * 84.9 * sin45 = 1402 mm

Calculated length of transition part (it corresponds to the reach of stress peak)

( x for internal pressure and 45° it was 87.9 mm)

( x 1399 mm)

PED-ex.6 10

Checking of the preliminary specified wall thickness for the external overpressure Allowed external overpressure

2

1

E

P

P

p

p

pp

Allowed external overpressure in plastic state (region) (similar relationship to internal pressure = achieving of allowable stress in the shell)

Allowed external overpressure in elastic state (region) (stability loss in the elastic region, it is for σ < σD)

MPa

csD

csp

C

DP 985.0

)10.8(45cos

1402

)10.8(*140*2

)(cos

)(**2

1

1

1

a

)(

)(**2

csD

csp D

P

EEE

E

U

ED

cs

D

cs

l

D

Bn

Ep

)(*100*

)(*100**

*

*10*8.20 1

2

1

1

6

For cylinder:

D DE

l lE

D DK / cos a1

PED-ex.6 11

nU = 2,4 ... safety coefficient against a stability loss (see cylinder) where effective dimensions of a conical shell are:

mmdD

lE 7.94445sin*2

1501486

sin*2 1

1

a

21

;EEE

DDMaxD

mmn

dDD

U

E 8.115645cos*4.2

1501486

cos* 1

11

a

1

1

11

1

2 **)(*31.0cos

aa

tgcs

dDdD

DDE

mmtgDE .565145*10.8

1501486*)1501486(*31.0

45cos

14862

effective length of conical shell

effective diameter of conical shell

D – higher diameter d1 - lower diameter

(for a1 = 0 DE1 = (D+d1)/nu D; DE2 = D)

d1

D

lE α1 H

R-r1

PED-ex.6 12

DE = 1156.8 mm

)(*100**45,9;0.1

1

12111cs

D

l

DBBMinB E

E

E

Parameter B1 is specified from equation

0.15)10.8(*100

8.1156*

7.944

8.1156*45.912

B

B1 = 1.0

8.1156

)10.8(*100*

8.1156

)10.8(*100*

7.944

8.1156*

0,1*4.2

10*206*10*8.20236

Ep

MPap E 622.0

Allowed external overpressure in elastic state (region) is

like for the cylinder

PED-ex.6 13

Then is the maximal allowed external overpressure for the preliminary specified cone wall thickness s1 = 8.0 mm

MPaMPap 100.0526.0

622.0

985.01

985.0

2

Such conical shell is too overdesigned. Therefore we will repeat

these calculations for a newly estimated cone wall thickness.

Note: When calculating the cylindrical part of the vessel to the external overpressure the calculated wall thickness was sR = 4.8 mm, it means that the realized wall thickness (with allowances for corrosion etc.) was about 6 mm.

PED-ex.6 14

b) New estimation of the preliminary designed wall cone thickness together with all allowances

s1 = sR1 + c = 4 + 1 5 mm Calculating diameter of a smooth conical shell without torus in connection with the cylinder (used in next calculations)

DC = D – 1.4 * a1* sina1 where is

)(*cos

*7.0 1

1

1 csD

a a

mma 1.64)15(*45cos

1486*7.01

Calculated length of transition part (it corresponds to the reach of stress peak)

(results of the last iteration of the cone wall thickness calculation)

(thinner wall shorter length of transition part)

( x 8 mm)

( x 84.9 mm)

PED-ex.6 15

DC = 1486 – 1.4 * 64.1 * sin45 = 1422.5 mm

Checking of the preliminary specified wall thickness for the external overpressure

Allowed external overpressure

2

1

E

P

P

p

p

pp

Allowed external overpressure in plastic state (region)

EEE

E

U

ED

cs

D

cs

l

D

Bn

Ep

)(*100*

)(*100**

*

*10*8.20 1

2

1

1

6

where effective dimensions of a conical shell are

mmdD

lE 7.94445sin*2

1501486

sin*2 1

1

a

21;

EEEDDMaxD

Allowed external overpressure in elastic state (region)

MPa

csD

csp

C

DP 556.0

)15(45cos

5.1422

)15(*140*2

)(cos

)(**2

1

1

1

a

( x 1156.8 mm)

( x 0.985 MPa)

PED-ex.6 16

mmn

dDD

U

E 8.115645cos*4.2

1501486

cos* 1

11

a

1

1

11

1

2 **)(*31.0cos

aa

tgcs

dDdD

DDE

mmtgDE 9.920545*15

1501486*)1501486(*31.0

45cos

14862

DE = 1156.8 mm

)(*100**45.9;0.1

1

12111cs

D

l

DBBMinB E

E

E

7.19)15(*100

8.1156*

7.944

8.1156*45.912

B

B1 = 1.0

( x 1402 mm)

PED-ex.6 17

8.1156

)15(*100*

8.1156

)15(*100*

7.944

8.1156*

0.1*4.2

10*206*10*8.20236

Ep

MPap E 154.0

Then is the maximal allowed external overpressure for this specified cone wall thickness s1 = 5 mm

MPaMPap 100.0148.0

154.0

556.01

556.0

2

Note: For s1 = 4.5 mm is the maximal external pressure p = 0.107 MPa

= 107 kPa. The value is practically equal to the given external overpressure. For the cylinder was the wall thicknes s2 = 4.8 mm.

( x 0.622 MPa)

( x for s1 = 8 mm it was 0.526 MPa)

PED-ex.6 18

External overpressure 100 kPa – half apex angle α1 = 70 °.

Preliminary specification of wall thickness s1 The wall thickness is specified according equations valid for cylinder (part 4.5) multiplied by value 1/cosa1.

In our previous example of the cylindrical vessel loaded with external pressure we specified the cylinder thickness without allowances:

mmDp

DKMaxsD

R 80.4*2

**1.1;10** 2

22

Then is the preliminary wall thickness of the conical part

mms

s RR 04.14

70cos

8.4

cos 1

21

a

For α1 = 45 ° it was sR1 = 6.8 mm disadvantage of flat cones

PED-ex.6 19

Preliminary designed wall thickness together with all allowances is

s1 = sR1 + c = 14.04 + 1 15.0 mm Calculating diameter of a smooth conical shell without torus in connection with the cylinder (used in next calculations)

DK = D – 1.4 * a1* sina1

where is

)(*cos

*7.0 1

1

1 csD

a a

mma 6.172)10.15(*70cos

1486*7.01

DC = 1486 – 1.4 * 172.6 * sin70 = 1259 mm

(x for 45° and s1 = 8 mm was a1 = 84.6 mm)

(x 1402 mm)

Calculated length of transition part (it corresponds to the reach of stress peak)

PED-ex.6 20

2

1

E

P

P

p

p

pp

Allowed external overpressure in plastic state (region) (similar relationship to internal pressure = achieving of allowable stress in the shell)

MPa

csD

csp

C

DP 061.1

)10.15(70cos

1259

)10.15(*140*2

)(cos

)(**2

1

1

1

a

Allowed external overpressure in elastic state (region) (stability loss in the elastic region, it is for σ < σD)

EEE

E

U

ED

cs

D

cs

l

D

Bn

Ep

)(*100*

)(*100**

*

*10*8.20 1

2

1

1

6

(x for 45° and estimation 8 mm it was 0.556 MPa)

Checking of the preliminary specified wall thickness for the external overpressure Allowed external overpressure

PED-ex.6 21

nU = 2,4 ... safety coefficient against a stability loss (see cylinder) where effective dimensions of a conical shell are:

mmdD

lE 9.71070sin*2

1501486

sin*2 1

1

a

21

;EEE

DDMaxD

mmn

dDD

U

E 239270cos*4.2

1501486

cos* 1

11

a

1

1

11

1

2 **)(*31.0cos

aa

tgcs

dDdD

DDE

mmtgDE 1289070*10.15

1501486*)1501486(*31.0

70cos

14862

D – higher diameter d1 - lower diameter

effective length of conical shell x 944.7 mm

effective diameter of conical shell x 1156.8 mm

PED-ex.6 22

DE = 2392 mm

)(*100**45.9;0.1

1

12111cs

D

l

DBBMinB E

E

E

Parameter B1 is specified from equation

56.41)10.15(*100

2392*

9.710

2392*45.912

B

B1 = 1.0

2392

)115(*100*

2392

)115(*100*

9.710

2392*

0.1*4.2

10*206*10*8.20236

Ep

MPap E 574.1

Allowed external overpressure in elastic state (region) is

like for the cylinder

(x for 45° it was 0.622 MPa)

PED-ex.6 23

Then is the maximal allowed external overpressure for the preliminary specified cone wall thickness s1 = 15 mm

MPaMPap 100.0880.0

574.1

061.11

061.1

2

Such conical shell is for the wall thickness s1 = 15 mm again too

overdesigned. Therefore we will repeat these calculations for a

newly estimated cone wall thickness.

Note: When calculating the cylindrical part of the vessel to the external overpressure the calculated wall thickness was sR = 4.8 mm, it means that the realized wall thickness (with allowances for corrosion etc.) was about 6 mm.

x 0.526 MPa (for 45°)

8.8 x overdesigned preliminary estimation

5.3 x overdesigned preliminary estimation

PED-ex.6 24

b) New estimation of the preliminary designed wall cone thickness together with all allowances

s1 = sR1 + c = 5 + 1 6 mm Calculating diameter of a smooth conical shell without torus in connection with the cylinder (used in next calculations) DC = D – 1.4 * a1* sina1

where is

)(*cos

*7.0 1

1

1 csD

a a

mma 2.103)16(*70cos

1486*7.01

(x for 45° it was s1 = 5.0 mm)

(x for 45° it was 64.1 mm)

PED-ex.6 25

DC = 1486 – 1.4 * 103.2 * sin70 = 1350 mm

Checking of the preliminary specified wall thickness for the external overpressure Allowed external overpressure

2

1

E

P

P

p

p

pp

Allowed external overpressure in plastic state (region)

EEE

E

U

E

D

cs

D

cs

l

D

Bn

Ep

)(*100*

)(*100**

*

*10*8,201

2

1

1

6

where effective dimensions of a conical shell are:

mmdD

lE 9.71070sin*2

1501486

sin*2 1

1

a

21;

EEEDDMaxD

Allowed external overpressure in elastic state (region)

MPa

csD

csp

K

DP 354.0

)16(70cos

1350

)16(*140*2

)(cos

)(**2

1

1

1

a

x for 70° and preliminary estimation s1 = 15 mm it was 0.556 MPa

(x 1422.5 mm)

PED-ex.6 26

mmn

dDD

U

E 239270cos*4.2

1501486

cos* 1

11

a

1

1

11

1

2 **)(*31.0cos

aa

tgcs

dDdD

DDE

mmtgDE 2303370*16

1501486*)1501486(*31,0

70cos

14862

DE = 2392 mm

)(*100**45.9;0.1

1

12111cs

D

l

DBBMinB E

E

E

5.69)16(*100

2392*

9.710

2392*45.912

B

B1 = 1.0

(x 1156.8 mm)

PED-ex.6 27

2392

)16(*100*

2392

)16(*100*

9.710

2392*

0.1*4.2

10*206*10*8.20236

Ep

MPap E 120.0

Then is the maximal allowed external overpressure for this specified cone wall thickness s1 = 6 mm

MPaMPap 100.0114.0

120.0

354.01

354.0

2

Realized wall thickness s1 = 8 mm of the conical cover is sufficient for this higher angle too.

(x for 45° and s1 = 5.0 mm it was 0.148 MPa)

(x for 70° and s1 = 15.0 mm it was 0.880 MPa)

(x for 45° and s1 = 5.0 mm it was 0.154 MPa)

PED-ex.6 28

Summary of results half apex angle internal overpressure

100 kPa external overpressure

100 kPa

calculated wall thickness (mm)

α = 0 ° (cylinder)

0.54 4.8

α = 45 ° (cone)

1.0 3.5

α = 70 ° (cone)

2.0 5.0

Note: • From the viewpoint of internal overpressure is preferable cylindrical vessel. • From the viewpoint of external overpressure (stability = collapse) is preferable

sharper cone

PED-ex.6 29

Design of conical bottom for a = 70 ° according European standard ES 13445-3

These calculations are performed according the ES 13445-3, part

8.6.3. that was several times amended. The last change was in

2010.

These calculations are valid for α ≤ 75 °.

Because the standard uses quite different symbols I show the

sketch of the conical cover again with these new symbols.

PED-ex.6 30

Internal radius of cylindrical shell Rmax = Di / 2 = 1486 / 2 = 743 mm Analyzed wall thickness of the conical shell without allowances ea = s1 – c = 6 – 1 = 5 mm (it corresponds to the calculated thickness)

(we consider the wall thicknes specified according the ČSN, that is checked according the EN)

L a

tube f 159x4,5

s2

s1 (ea)

f Di

f De

Rmax

Rn

L/2 L/2

(as in the case of ES for cylinder is not the analyzed thickness calculated, but is estimated )

Characteristic dimensions

PED-ex.6 31

Mean radius of the conical shell Rn = (Rmax + RTR) / 2 = (743 + 150 / 2) / 2 = 409 mm Height of the conical shell L = 240 mm

Check of the shell against a stability loss between two reinforcements (it is cylindrical parts = shell and neck).

According part. 8.4.3 (eq. 8.4.3-1) is the allowed elastic limit σe = RP0.2/t / 1.25 = 210 / 1.25 = 168 MPa where RP0.2/t = Remin = 210 MPa is the yield point for working temperature. (x according ČSN is σD = 140 MPa)

L = H; Rmax = Di/2; RTR = d1i/2

(safety coefficient 1.25)

(the cone is without any reinforcement L = H)

d1

D

lEα1H Rn

PED-ex.6 32

Then is a pressure PY, at what a mean tangential stress reaches the yield point in the center of shell between reinforcements

PY = (ea * σe * cosα) / Rmax

PY = (5 * 168 * cos70) / 743 = 0.387 MPa

Now we can specify a theoretic pressure for an elastic loss of stability of the shell wall

Pm = (E * ea * ε * cos3α) / Rn

where the value ε is specified from fig. 8.5-3 (see the next page) for

L / (2 * Rn * cosα) = 240 / (2 * 409 *cos70) = 0.858 instead (L/2R) for cylinder in the ČSN (dimensionless length) and

(2 * Rn * cosα) / ea = (2 * 409 * cos 70) / 5 = 55.95 instead (s/R) for cylinder in the ČSN (dimensionless wall thickness) .

s = p*R/σD*cosα1 p = s*σD*cosα1/R

x 0.354 MPa

PED-ex.6 33

0,858 56

0,004 ko

nst

*p/E

L/D

s/D

ČSN 690010

x according ČSN

2R/ea = reciprocal value of dimensionless thickness of the wall

PED-ex.6 34

It follows from the diagram

ε ≈ 0,004

Then we can specify the value Pm = theoretical pressure at the elastic stability loss of conical shell

Pm = (206000 * 5 * 0,004 * cos370) / 409 = 0.403 MPa

Now we specify a ratio Pm / PY = 0.403 / 0.387 = 1.04.

(teor. pressure at elastic stability loss / teor. pressure when is reached allowed stress)

For curve 1 from following diagram 8.5-5 a ratio Pr / Py ≈ 0,53 is

found and from it the low limit of the calculating external

overpressure for the stability loss.

Pr = 0.53 * PY = 0.53 * 0.387 = 0.205 MPa

E (MPa) s (mm) ε (-) Rn (mm)

(similar way like was for the cylinder)

PED-ex.6 35

y = -0,0569x3 + 0,077x2 + + 0,4769x + 0,0005 for Pm/Py = 0-1,25

0,53

1,04

cylinder, cone

sphere

cylindrical and conical shells

spherical shells and vaulted covers

Fig. 8.5-5 – Values Pr/Py as function of Pm/Py

PED-ex.6 36

Maximal calculating overpressure for the realized wall thickness

must conform to the condition (8.6.3-5)

P ≤ Pr / S

where S = 1.5 is safety coefficient according part 8.4.4

P ≤ 0,205 / 1.5 = 0.137 MPa

Real maximal theoretic external overpressure is 0.100 MPa. From

it follows that according the ES is the conical shell O.K.

Comparison of results for a1 = 70°

According new ČSN s = 6 mm 0,114 MPa

According old ČSN s = 5.4 mm 0,100 MPa

s = 6 mm 0,111 MPa

According ES s = 6 mm 0,137 MPa

max. external overpressure calculated wall thickness