design of hemispherical pressure vessel is most considerable in designing of pressure vessel. in present industry this type of vessels are widely used.

1. HEMISPHERICAL
PRESSURE VESSEL
NAME: JAIMIN AJITKUMAR KEMKAR
EN. NO.: 160123119014
CLASS AND BATCH: 5TH D2
SUBJECT: DESIGN OF MACHINE ELEMENTS (2151907)
SUBJECT GUIDE: PROF. C. V. BHATIA

2. CONTENTS
• PRESSURE VESSEL
• TYPES OF WELD JOINT USED IN PRESSURE VESSELS
• HEMISPHERICAL PRESSURE VESSEL

3. PRESSURE VESSEL
• Pressure vessels are the containers or pipelines used for
storing, receiving or carrying the fluids under the
pressure.
• The fluid being stored may remain as it is, as in case of
storage vessels or may undergo a change of state while
inside the pressure vessels, as in case of steam boilers or
it may combine with other reagents, as in case of
chemical processing vessels.
• Most process equipment unit may be considered to be
pressure vessels with various modifications necessary
to enable the units to perform required functions.
• The material of the pressure vessels may be brittle
such as cast iron or ductile such as plain carbon steel
and alloy steel.

4. TYPES OF WELDED JOINTS USED IN
PRESSURE VESSELS
Types of weld joint
Weld joint efficiency 𝜂
FULLY
RADIOGRAPHED
SPOT
RADIOGRAPHED
NOT
RADIOGRAPHED
1. DOUBLE WELDED BUTT
JOINT WITH FULL
PENETRATION
1.0 0.85 0.7
1. SINGLE WELDED BUTT
JOINT WITH BACKING
STRIP
0.9 0.8 0.65
1. SINGLE WELDED BUTT
JOINT WITHOUT
BACKING STRIP
- - 0.60
1. SINGLE FULL FILLET
LAP JOINT - - 0.55

5. HEMISPHERICAL PRESSURE VESSEL
• The hemispherical heads are strongest of all the formed heads. They are free from
discontinuities and hence used in high pressure vessels.
• However, the amount of forming required to produce the hemispherical shape is
more, resulting in more forming cost.
• The thickness of the hemispherical head is given by,
𝒕 𝒉 =
𝒑𝒊 𝒅𝒊
𝟒𝝈 𝒂𝒍𝒍 𝜼 − 𝟎. 𝟒𝒑𝒊
+ 𝑪

6. • For this head,
𝑺 𝒇 = 𝟑𝒕 𝒉 𝒐𝒓 𝟐𝟎𝒎𝒎 whichever is larger
• The volume of fluid contained within the hemispherical head excluding the
straight flange portion is given by,
𝑽 𝒉 = 𝟎. 𝟐𝟔𝟐𝒅 𝟏
𝟑
(𝒎𝒎 𝟑)

7. EXAMPLE
• The following data refers to the vertical pressure vessel with hemispherical ends,
used to store the gas at a pressure of 1.5 Mpa and temperature of 20 degree Celsius.
The vessel shell as well as the hemispherical ends are made of plain carbon steel.
• 𝑖𝑛𝑛𝑒𝑟 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑠𝑠𝑒𝑙 𝑠ℎ𝑒𝑙𝑙 = 2𝑚
• 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑠𝑠𝑒𝑙 𝑠ℎ𝑒𝑙𝑙 𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑓𝑙𝑎𝑛𝑔𝑒
𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑒𝑛𝑑 = 3𝑚
• 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑣𝑒𝑠𝑠𝑒𝑙 = 1200 𝑘𝑔
𝑚3
• 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑤𝑖𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 = 200 𝑘𝑚
ℎ𝑟
• 𝑡𝑜𝑟𝑞𝑢𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑜𝑓𝑓𝑠𝑒𝑡 𝑝𝑖𝑝𝑖𝑛𝑔 𝑜𝑛 𝑣𝑒𝑠𝑠𝑒𝑙 𝑠ℎ𝑒𝑙𝑙 = 2 𝑘𝑁 ∙ 𝑚
• 𝑢𝑙𝑡𝑖𝑚𝑎𝑡𝑒 𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑠𝑠𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 = 450 𝑁
𝑚𝑚2
• 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑠𝑠𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 = 7800 𝑘𝑔
𝑚3
• Efficiency of welded joints = 85%
• Design the pressure vessel shell and the hemispherical end.

8. 1. Thickness of hemispherical head:
• The thickness of the hemispherical head is given by,
𝑡ℎ =
𝑝𝑖 𝑑𝑖
4𝜎 𝑎𝑙𝑙 𝜂 − 0.4𝑝𝑖
+ 𝐶 =
1.575 × 2000
4 × 150 × 0.85 − 0.4 × 1.575
+ 3
∴ 𝑡ℎ = 9.18 𝑚𝑚 ≅ 10 𝑚𝑚
• Straight flange length is given by,
𝑆𝑓 = 3𝑡ℎ 𝑜𝑟 20 𝑚𝑚 𝑤ℎ𝑖𝑐ℎ𝑒𝑣𝑒𝑟 𝑖𝑠 𝑙𝑎𝑟𝑔𝑒𝑟
= 3 × 10 𝑜𝑟 20 𝑚𝑚
= 30 𝑜𝑟 20 𝑚𝑚
𝑆𝑓 = 30 𝑚𝑚
2. Thickness of vessel shell:
• The thickness of the vessel shell is given by,
𝑡 𝑠 =
𝑝𝑖 𝑑𝑖
4𝜎 𝑎𝑙𝑙 𝜂𝑙 − 𝑝𝑖
+ 𝑐 =
1.575 × 2000
2 × 150 × 0.85 − 1.575
+ 3
∴ 𝑡 𝑠 = 15.43 𝑚𝑚 ≅ 16 𝑚𝑚
𝑡 = 𝑡 𝑠 − 𝑐 = 16 − 3 = 13 𝑚𝑚

9. 3. Stress in circumferential direction (𝝈 𝒕):
𝜎𝑡 =
𝑝𝑖(𝑑𝑖 + 𝑡)
2𝑡
=
1.575 × (2000 + 13)
2 × 13
= 121.94 𝑁
𝑚𝑚2 (𝑡𝑒𝑛𝑠𝑖𝑙𝑒)
4. Stress in longitudinal (or axial) direction (𝝈 𝒕):
a) Stress in longitudinal direction due to internal pressure:
𝜎𝑡1 =
𝑝𝑖 𝑑𝑖
4𝑡
=
1.575 × 2000
4 × 13
= 60.57 𝑁
𝑚𝑚2 (𝑡𝑒𝑛𝑠𝑖𝑙𝑒)
b) Stress in longitudinal direction due to weight of vessel and its contents:
• Weight of the gas in the vessel is,
𝑊𝑐 = 𝑉 × 𝜌 𝑔 × 𝑔 × 𝑁 =
𝜋
4
𝑑𝑖
2
𝑙 𝑠 + 2
𝜋
12
𝑑𝑖
3
× 𝜌 𝑔 × 𝑔
𝑊𝑐 =
𝜋
4
× (2)2× 3 + 2 ×
𝜋
12
× (2)3 × 1200 × 9.81 = 160258.92 𝑁
• Weight of the hemispherical end is,
𝑊ℎ = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 × 𝑔
= 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 × 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 × 𝜌𝑠 × 𝑔

10. = 2𝜋𝑟2 + 2𝜋𝑟 ∙ 𝑆𝑓 ∙ 𝑡ℎ ∙ 𝜌𝑠 ∙ 𝑔 =
𝜋(𝑑𝑖 + 𝑡ℎ)2
2
+ 𝜋(𝑑𝑖 + 𝑡ℎ) ∙ 𝑆𝑓 ∙ 𝑡ℎ ∙ 𝜌𝑠 ∙ 𝑔
=
𝜋(2.0 + 0.01)2
2
+ 𝜋(2.0 + 0.01) ∙ 0.03 ∙ 0.01 ∙ 7800 ∙ 9.81 = 5000.92 𝑁
• Total weight W = weight of the gas in vessel + weight of the lower cover
𝑊 = 𝑊𝑐 + 𝑊ℎ = 160258.92 + 5000.92 = 165259.84 𝑁
• The stress in longitudinal direction due to weight of vessel and its contents is,
𝜎𝑡2 =
𝑊
𝜋 𝑑𝑖 + 𝑡 𝑡
=
165259.84
𝜋(2000 + 13) × 13
= 2.01 𝑁
𝑚𝑚2 (𝑡𝑒𝑛𝑠𝑖𝑙𝑒)
5. Shear stress due to offset piping:
𝜏 =
𝑇
𝐽
𝑟 𝑚𝑎𝑥
=
2𝑇
𝜋(𝑑𝑖 + 𝑡)2 𝑡
=
2 × 2 × 106
𝜋(2000 + 13)2× 13
= 0.024 𝑁
𝑚𝑚2

11. 6. Maximum resultant stress in vessel shell:
𝜎 𝑅 = 𝜎𝑡
2 − 𝜎𝑡 ∙ 𝜎𝑙 + 𝜎𝑙
2 + 3𝜏2
(121.94)2−121.94 × 63.724 + (63.724)2+3(0.024)2
= 105.64 𝑁
𝑚𝑚2
Here,
𝜎 𝑅 < 150 𝑁
𝑚𝑚2
𝜎𝑡 < 150 𝑁
𝑚𝑚2
Hence the design is safe.

12. references
• BOOK:
• Books India
• R. S. khurmi
• WEB:
• Me-mechanical.com