Pressure Measurements

1. Pressure Measurements

2. Pascal’s Law
• “ The intensity of pressure at any point in a liquid at rest, is the same in all
directions”
Proof: Let us consider a very small wedge shaped element LMN of a liquid.
Let, 𝑝𝑥 = 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑
𝑝𝑦 = 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑
𝑝𝑧
= 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑑 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑒𝑙𝑒𝑚
α = Angle of the element of the liquid
𝑃𝑥 = 𝑇𝑜𝑡𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑠𝑖𝑑𝑒 𝐿𝑁 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑
𝑃𝑦 = 𝑇𝑜𝑡𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑠𝑖𝑑𝑒 𝑀𝑁 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑
𝑃𝑧 = 𝑇𝑜𝑡𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑠𝑖𝑑𝑒 𝐿𝑀 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑
Now, 𝑃𝑥 = 𝑝𝑥 x LN ……(i)
𝑃𝑦 = 𝑝𝑦 x MN………….(ii)
𝑃𝑧 = 𝑝𝑧 x LM……………(iii)

3. • As the element of the liquid is at rest, therefore
the sum of horizontal and vertical components
of the liquid pressures must be equal to zero
• Resolving the forces horizontally:
• 𝑃𝑥 = 𝑃𝑧 sin α
• 𝑝𝑥 x LN = 𝑝𝑧 x LM sin α
• 𝑝𝑥 x LN = 𝑝𝑧 x LM x
𝐿𝑁
𝐿𝑀
• 𝑝𝑥 = 𝑝𝑧 ….. (iV)

4. • Resolving the forces vertically:
𝑃𝑦 − 𝑊 = 𝑃𝑧 cos α (W= weight of the liquid element)
Since the element is very small, neglecting its weight we have
• 𝑃𝑦 = 𝑃𝑧 cos α
• 𝑝𝑦. MN = 𝑝𝑧 LM cos α
• 𝑝𝑦. MN = 𝑝𝑧 LM
MN
LM
• 𝑝𝑦 = 𝑝𝑧 ……..(v)
• From (iv) & (v) we get
• 𝑝𝑥 = 𝑝𝑦 = 𝑝𝑧

5. Pressure Head of a Liquid
• A liquid is subjected to pressure due to its own
weight, this pressure increases as the depth of
the liquid increases.
• Consider a vessel containing liquid, the liquid
will exert pressure on all sides and bottom of
the vessel.
Now let cylinder be
made to stand in the liquid

6. • Let h = height of liquid in the cylinder,
• A = area of the cylinder base,
• W = specific weight of the liquid,
• P = intensity of pressure
Now Total pressure on the base of the cylinder =
weight if liquid in the cylinder
i.e., p A = w x A x h
𝑝 =
𝑤𝐴ℎ
𝐴
= 𝑤ℎ
𝑝 = 𝑤ℎ (ℎ =
𝑝
𝑤
)
The intensity of pressure in a liquid due to its depth
will vary directly with depth

7. Measurement of Pressure
• 1. Manometers
These are defined as the devices used for
measuring the pressure at a point in a fluid by
balancing the column of fluid by the same or
another column of liquid. These are classified as
a) Simple manometers
i. Piezometer
ii. U-tube manometer
iii. Single column manometer
b) Differential manometers

8. • Mechanical gauges:
• These are the devices in which the pressure is
measured by balancing the fluid column by
spring or dead weight.
• These gauges are used for measuring high
pressure and high precision is not required.
• Bourdon tube pressure gauge
• Bellow pressure gauge
• Diaphragm pressure gauge
• Dead-weight pressure gauge

9. Simple manometers
• 1. Piezometer: A piezometer is the simplest form of
manometer which can be used for measuring moderate
pressures of liquids.
• It consists of a glass tube inserted in the wall of a vessel or
of a pipe, containing liquid whose pressure is to be
measured. The tube extends vertically upward to such a
height that liquid can freely rise in it without overflowing.

10. • U- tube manometer:
• A U-tube manometer consists of a glass tube
bent in U-shape, one end of which is
connected to a point at which pressure is to
be measured and other end remains open to
the atmosphere.