This article provides a summary of all the important equations relating to Pressure Measurement in an industrial environment.

1. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 1 of 6
Industrial Instrumentation
Pressure Measurement
Sl. No.
1.
Pressure
𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 (𝑷) =
𝑭𝒐𝒓𝒄𝒆
𝑨𝒓𝒆𝒂
=
𝑭
𝑨
Pressure for an ideal gas is expressed as
𝒑 =
𝟏
𝟑
𝒎𝒏𝒗𝒓𝒎𝒔
𝟐
𝑛 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝑚 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠; 𝑣𝑟𝑚𝑠 = 𝑟𝑚𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒
Pressure Sensitivity (S)
𝑺 =
𝑶𝒖𝒕𝒑𝒖𝒕
𝑰𝒏𝒑𝒖𝒕
=
∆𝒉
∆𝒑
=
𝒉
𝒈𝒉𝝆𝒎
=
𝟏
𝒈𝝆𝒎
2.
Pascal – SI Unit for Pressure
𝟏 𝑷𝒂 = 𝟏 𝑵𝒆𝒘𝒕𝒐𝒏 𝒇𝒐𝒓𝒄𝒆 𝒂𝒑𝒑𝒍𝒊𝒆𝒅 𝒑𝒆𝒓 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒆𝒕𝒆𝒓 =
𝟏 𝑵
𝟏 𝒎𝟐
𝑷𝒂 = 𝑵 𝒎𝟐
⁄
3.
Absolute Pressure
𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝑮𝒂𝒖𝒈𝒆 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 + 𝑨𝒕𝒎𝒐𝒔𝒑𝒉𝒆𝒓𝒊𝒄 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆
Relationship between
Absolute 𝒑𝒂, Gauge 𝒑𝒈, Atmospheric 𝒑𝒔and Vacuum Pressure 𝒑𝒗
𝒑𝒈 = 𝒑𝒂 − 𝒑𝒔
𝒑𝒗 = 𝒑𝒔 − 𝒑𝒂
𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 = 𝑻𝒐𝒕𝒂𝒍 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 − 𝑺𝒕𝒂𝒕𝒊𝒄 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆
4.
Units
𝟏 𝒂𝒕𝒎 = 𝟏𝟒. 𝟔𝟗𝟔 𝒑𝒔𝒊 = 𝟏𝟎𝟏. 𝟑𝟐𝟓 𝒌𝑷𝒂
𝟏 𝒎𝒊𝒍𝒍𝒊𝒃𝒂𝒓 = 𝟏𝟎𝟎 𝒅𝒚𝒏𝒆 𝒄𝒎𝟐
⁄ = 𝟏𝟒. 𝟓 ∗ 𝟏𝟎−𝟑
𝒑𝒔𝒊
𝟏 𝒎𝒊𝒄𝒓𝒐𝒏 = 𝟏𝟎−𝟔
𝒎 𝒐𝒓𝟏𝟎−𝟑
𝒎𝒎 𝑯𝒈 = 𝟏𝟗. 𝟑𝟒 ∗ 𝟏𝟎−𝟔
𝒑𝒔𝒊
𝟏 𝑩𝒂𝒓 = 𝟏𝟎𝟓
𝑷𝒂
𝟏 𝒂𝒕𝒎 = 𝟕𝟔 𝒄𝒎 𝒐𝒇 𝑯𝒈 (𝑴𝒆𝒓𝒄𝒖𝒓𝒚) = 𝟏𝟒. 𝟕 𝒑𝒔𝒊 = 𝟏𝟎𝟏. 𝟑 𝒌𝑷𝒂;
𝟏𝒄𝒎 𝒐𝒇 𝑯𝒈 = 𝟏𝟑𝟑𝟑. 𝟐𝟐 𝑷𝒂; 𝟏 𝑷𝒂 = 𝟎. 𝟎𝟎𝟎𝟕𝟓 𝒄𝒎 𝒐𝒇 𝑯𝒈

2. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 6
5.
Manometers:
Pressure Balance equation for U – tube manometer
𝑷𝟏 + 𝒈𝒉𝝆𝒇 = 𝑷𝟐 + 𝒈𝒉𝝆𝒎
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝑷𝒓𝒆𝒔𝒔𝒖𝒓𝒆 (∆𝑷) = 𝑷𝟏 − 𝑷𝟐 = 𝒈𝒉(𝝆𝒎 − 𝝆𝒇) = 𝒈𝒉𝝆𝒎
𝜌𝑚 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑟𝑐𝑢𝑟𝑦;
𝜌𝑓 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑓𝑙𝑢𝑖𝑑
General Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
U-Tube Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
Differential U-Tube Manometer
∆𝑷 = (𝑷𝟏 − 𝑷𝟐) = (𝝆𝒎 − 𝝆𝒇)(𝒉𝟏 − 𝒉𝟐)𝒈 = (𝝆𝒎 − 𝝆𝒇)𝒉𝒈
Inverted U-Tube Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝟏𝒈𝑯𝟏 + 𝝆𝟐𝒈𝑯𝟐 − 𝝆𝒎𝒈(𝑯𝟏 − 𝑯𝟐)
Inclined Tube Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈(𝒙 + 𝒚) = 𝝆𝒈𝒅 (
𝑨𝟐
𝑨𝟏
+ 𝒔𝒊𝒏𝜽)
[𝑦 = 𝑑𝑠𝑖𝑛𝜃, 𝐴1𝑥 = 𝐴2𝑑], 𝑑 = 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑢𝑏𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑑𝑟𝑢𝑚 𝑑𝑎𝑡𝑢𝑚 𝑝𝑜𝑖𝑛𝑡.
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉′(𝟏 + 𝑨𝟐 𝑨𝟏
⁄ ) = 𝝆𝒈𝑹𝒔𝒊𝒏𝜽(𝟏 + 𝑨𝟐 𝑨𝟏
⁄ )
ℎ′
= 𝑅𝑠𝑖𝑛𝜃, 𝑅 = 𝑆𝑐𝑎𝑙𝑒 𝑟𝑒𝑎𝑑𝑖𝑛𝑔, 𝜃 = 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛
Micro Manometer
∆𝑷 = 𝑷𝟐 − 𝑷𝟏 = 𝝆𝒈(𝒉𝟐 − 𝒉𝟏) = 𝝆𝒈(𝒍𝒔𝒊𝒏𝜽 − 𝒉𝟏)
Well-Type Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 = 𝝆𝒈𝒉 (𝟏 +
𝑨𝟐
𝑨𝟏
) = 𝝆𝒈𝒉
[𝑖𝑓 𝐴2 𝐴1
⁄ 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑡𝑜𝑜 𝑠𝑚𝑎𝑙𝑙]
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑤𝑒𝑙𝑙 (𝑑) = ℎ − ℎ′
𝑑𝐴1 = ℎ𝐴2

3. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 3 of 6
𝒉 = 𝒉′
(𝟏 +
𝑨𝟐
𝑨𝟏
)
𝐴2 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒; 𝐴1 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑙𝑙; ℎ = 𝑚𝑒𝑟𝑐𝑢𝑟𝑦 𝑐𝑜𝑙𝑢𝑚𝑛 ℎ𝑒𝑖𝑔ℎ𝑡
Ring-Balance Manometer
∆𝑷 = 𝑷𝟏 − 𝑷𝟐 =
𝑮 ∗ 𝑺
𝑨 ∗ 𝒓
𝐬𝐢𝐧 𝜶
𝛼 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑎𝑛𝑔𝑙𝑒, 𝑟 = 𝑡𝑢𝑏𝑒 𝑟𝑎𝑑𝑖𝑢𝑠, 𝐴 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛,
𝐺 = 𝑏𝑎𝑙𝑎𝑛𝑐𝑖𝑛𝑔 𝑤𝑒𝑖𝑔ℎ𝑡, 𝑆 = 𝑡𝑢𝑏𝑖𝑛𝑔 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒.
Barometer
𝑷𝒂𝒕𝒎 = 𝝆𝑯𝒈𝒈𝒉
𝑷𝑺 = 𝑷𝑺𝟎 (𝟏 −
𝑩𝒁
𝑻𝟎
)
𝟓.𝟐𝟔
𝑃𝑆 = 𝑙𝑜𝑐𝑎𝑙 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒,
𝑃𝑆0 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑠𝑒𝑎 𝑙𝑒𝑣𝑒𝑙, 760 𝑚𝑚 𝑜𝑓 𝐻𝑔,
𝑍 = 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝑖𝑛 𝑚𝑒𝑡𝑟𝑒, 𝐵 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 0.0065 𝐾 𝑚
⁄ ,
𝑇0 = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 = 273.16 𝐾
6.
Manometer Transfer Function
𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 𝑒𝑥𝑒𝑟𝑡𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 = 𝜌𝑎𝑔(𝐻 − ℎ) = 2𝜌𝑎𝑔𝑥
𝐹𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑑𝑢𝑒 𝑡𝑜 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = 𝜌𝑉
𝑑2
𝑥
𝑑𝑡2
𝐹𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑑𝑢𝑒 𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 = 𝐵
𝑑𝑥
𝑑𝑡
𝑉 = 𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑙𝑢𝑖𝑑; 𝑎 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑢𝑏𝑒;
𝐵 = 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦; 𝑥 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑟𝑛𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑; 𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑
By balancing the forces, we get
𝑇𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑝
= 𝑇𝑜𝑡𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑙𝑖𝑞𝑢𝑖𝑑 + 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑖𝑛𝑒𝑟𝑡𝑖𝑎
+ 𝐹𝑜𝑟𝑐𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛
𝑝𝑎 = 𝜌𝑉
𝑑2
𝑥
𝑑𝑡2
+ 𝐵
𝑑𝑥
𝑑𝑡
+ 2𝜌𝑔𝑎𝑥
∴ 𝒑 =
𝝆𝑽
𝒂
𝒅𝟐
𝒙
𝒅𝒕𝟐
+
𝑩
𝒂
𝒅𝒙
𝒅𝒕
+ 𝟐𝝆𝒈𝒙
Taking Laplace Transfer on both sides, we get
𝑝(𝑠) = [
𝜌𝑉
𝑎
𝑠2
+
𝐵
𝑎
𝑠 + 2𝜌𝑔] 𝑋(𝑠)

4. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 4 of 6
𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = 𝐺(𝑠) =
𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡
𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡
=
𝑋(𝑠)
𝑃(𝑠)
𝐺(𝑠) = 2𝜌𝑎𝑔.
𝑋(𝑠)
𝑝(𝑠)
𝑮(𝒔) =
𝟐𝒂𝒈 𝑽
⁄
𝒔𝟐 +
𝑩
𝝆𝑽
𝒔 +
𝟐𝒂𝒈
𝑽
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝜔𝑛 = √2𝑎𝑔 𝑉
⁄ ; 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜𝑛 𝜉 =
𝐵
2𝜌√2𝑎𝑔𝑉
𝑪(𝒔) =
𝝎𝒏
𝟐
𝒔𝟐 + 𝟐𝝃𝝎𝒏𝒔 + 𝝎𝒏
𝟐
For Elastic Pressure Transducer
𝑷 = 𝑷𝒇𝒊𝒏𝒂𝒍 + (𝑷𝒊𝒏𝒊𝒕𝒊𝒂𝒍 − 𝑷𝒇𝒊𝒏𝒂𝒍)𝒆−𝒕 𝝉
⁄
𝑃 = 𝑎𝑛𝑦𝑡𝑖𝑚𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑃𝑓𝑖𝑛𝑎𝑙 = 𝐹𝑖𝑛𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑃𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒,
𝜏 = 𝑡𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑡 = 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒 𝑖𝑛𝑠𝑡𝑎𝑛𝑡,
7.
Vacuum and Low Pressure
McLeod Gauge
𝑉
𝑑𝑝2
𝑑𝑡
= 𝐾(𝑃1 − 𝑃2)
𝑉 = 𝑏𝑢𝑙𝑏 𝑣𝑜𝑙𝑢𝑚𝑒, 𝐾 = 𝑓𝑙𝑜𝑤 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒,
𝑑𝑝2
𝑑𝑡
= 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
𝑷 =
𝒉𝟐
𝑨𝒕
𝑽 − 𝒉. 𝑨𝒕
𝑷 =
𝒉𝟐
𝑨𝒕
𝑽
ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑓𝑙𝑢𝑖𝑑 𝑐𝑜𝑙𝑢𝑚𝑛; 𝐴𝑡 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛; 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑢𝑙𝑏
Ionization Gauge
𝑆 =
𝑂𝑢𝑡𝑝𝑢𝑡
𝐼𝑛𝑝𝑢𝑡
=
𝑰𝑷
𝑃𝑰𝑮
𝑷 =
𝟏
𝑺
∗
𝑰𝑷
𝑰𝑮
𝑆 𝑖𝑠 𝐺𝑎𝑢𝑔𝑒 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦, 𝐼𝑃 𝑖𝑠 𝑃𝑙𝑎𝑡𝑒 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 & 𝐼𝐺 𝑖𝑠 𝐺𝑟𝑖𝑑 𝐶𝑢𝑟𝑟𝑒𝑛𝑡
Knudsen Gauge
𝑷 =
𝒌𝑭
√𝑻𝑷 𝑻𝒈
⁄ − 𝟏
= 𝟒𝑭
𝑻𝒈
𝑻𝑷 − 𝑻𝑽

5. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 5 of 6
𝐹 = 𝑁𝑒𝑡 𝑓𝑜𝑟𝑐𝑒; 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 4,
𝑇𝑔 = 𝑇𝑒𝑚𝑝. 𝑜𝑓 𝑔𝑎𝑠; 𝑇𝑉 = 𝑣𝑎𝑛𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝑃 = 𝑝𝑙𝑎𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
8.
High Pressure
Bridgman Gauge
𝑹 = 𝑹𝟏(𝟏 + 𝒃∆𝑷)
𝑅 = 𝑤𝑖𝑟𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑅1 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 1 𝑎𝑡𝑚,
𝑏 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 2.5 ∗ 10−11
𝑃𝑎−1
, ∆𝑃 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
9.
Elastic Pressure Transducer
Bourdon Tube:
∆𝒂 = 𝟎. 𝟎𝟓
𝒂𝑷
𝑬
(
𝒓
𝒕
)
𝟎.𝟐
(
𝒙
𝒚
)
𝟎.𝟑𝟑
(
𝒙
𝒕
)
𝟑
∆𝑎 = 𝑡𝑖𝑝 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑎 = 𝑡𝑖𝑝 𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡, 𝐸 = 𝑌𝑜𝑢𝑛𝑔′
𝑠𝑀𝑜𝑑𝑢𝑙𝑢𝑠,
𝑟 = 𝑑𝑖𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠, 𝑥 = 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑦 = 𝑤𝑖𝑑𝑡ℎ, 𝑡 = 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠,
Bellow:
𝒅 =
𝟎. 𝟒𝟓𝟑𝑷. 𝒓. 𝒏. 𝑫𝟐
√𝟏 − 𝝑𝟐
𝑬𝒕𝟑
𝑑 = 𝑏𝑒𝑙𝑙𝑜𝑤 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑃 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑟 = 𝑟𝑎𝑖𝑠𝑢 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑐𝑜𝑟𝑟𝑢𝑔𝑎𝑡𝑖𝑜𝑛,
𝑛 = 𝑛𝑜. 𝑜𝑓 𝑠𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑐𝑜𝑟𝑟𝑢𝑔𝑎𝑡𝑖𝑜𝑛, 𝑡 = 𝑤𝑎𝑙𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠,
𝐷 = 𝑚𝑒𝑎𝑛 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟, 𝐸 = 𝑌𝑜𝑢𝑛𝑔′
𝑠𝑚𝑜𝑑𝑢𝑙𝑢𝑠, 𝜗 = 𝑃𝑜𝑖𝑠𝑜𝑛′
𝑠 𝑟𝑎𝑡𝑖𝑜,
Diaphragm:
𝒅𝒎 =
𝟑𝑷
𝟏𝟔𝑬𝒕𝟑
𝑹𝟒(𝟏 − 𝝑𝟐)
𝒅𝒓 =
𝟑𝑷(𝟏 − 𝝑𝟐)
𝟏𝟔𝑬𝒕𝟑
(𝑹𝟐
− 𝒓𝟐)𝟐
𝒇𝒏 =
𝟐. 𝟓𝒕
𝝅𝑹𝟐
[
𝑬
𝟑𝝆
(𝟏 − 𝝑𝟐)]
𝟏 𝟐
⁄
𝑑𝑚 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑑𝑟 = 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑟𝑎𝑑𝑖𝑢𝑠,
𝐸 = 𝑌𝑜𝑢𝑛𝑔′
𝑠 𝑚𝑜𝑑𝑢𝑙𝑢𝑠, 𝑃 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒, 𝑡 = 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, 𝜗 = 𝑃𝑜𝑖𝑠𝑠𝑜𝑛′
𝑠 𝑟𝑎𝑡𝑖𝑜,
𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠, 𝑟 = 𝑟𝑎𝑑𝑖𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛, 𝑓𝑛 = 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦, 𝜌 = 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,
10.
Electrical Pressure Transducers
Strain Gauge
Strain Gauge Resistance (R)
𝑹 = 𝝆
𝒍
𝑨

6. PRESSURE MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 6 of 6
𝑅 = 𝑆𝑡𝑟𝑎𝑖𝑛 𝐺𝑎𝑢𝑔𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝜌 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒,
𝑙 = 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ, 𝐴 = 𝑐𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒.
Strain (𝜺)
𝜺 =
𝟏
𝑲
∆𝑹
𝑹
𝜀 = 𝑆𝑡𝑟𝑎𝑖𝑛, 𝐾 = 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,
∆𝑅 𝑅
⁄ = 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑡𝑟𝑎𝑖𝑛 𝑔𝑎𝑢𝑔𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
Hooke’s Law for Stress (𝝈)
𝝈 = 𝜺. 𝑬 =
𝟏
𝑲
∆𝑹
𝑹
. 𝑬
𝜎 = 𝑆𝑡𝑟𝑒𝑠𝑠 (𝑁 𝑚2
⁄ ), 𝜀 = 𝑠𝑡𝑟𝑎𝑖𝑛(𝑚 𝑚
⁄ ), 𝐸 = 𝑌𝑜𝑢𝑛𝑔′
𝑠𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 𝑚𝑜𝑑𝑢𝑙𝑢𝑠
Capacitive Pressure Gauge
𝑪 =
∈𝟎∈𝒓 𝑨
𝒅
C = capacitance in Farad; A = area of each plate (𝑚2), 𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑙𝑎𝑡𝑒𝑠 (𝑚),
𝜖0 = 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 8.854 ∗ 10−12
𝐹 𝑚2
⁄ ,
𝜖𝑟 = 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 1 𝑓𝑜𝑟 𝑎𝑖𝑟 𝑜𝑟 𝑓𝑟𝑒𝑒 𝑠𝑝𝑎𝑐𝑒