The document provides comprehensive formulas and principles related to pressure, flow, temperature, pH, and viscosity measurement in industrial instrumentation. It details various methods for measuring pressure (including manometers and gauges), flow (using orifice, venturimeter, pitot tube, and flow meters), temperature (through RTDs, thermocouples, and pyrometers), and viscosity (with viscometers). Additionally, it includes mathematical relationships and equations essential for engineers and technicians in instrumentation and control systems.

1. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 1 of 13
Industrial Instrumentation
Mathematical Expressions & Relationships
Pressure Measurement
1. Pressure for an ideal gas is expressed as
𝒑 =
𝟏
𝟑
𝒎𝒏𝒗𝒓𝒎𝒔
𝟐
𝑛 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝑚 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠; 𝑣𝑟𝑚𝑠 = 𝑟𝑚𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒
2. Relationship between Absolute 𝒑𝒂, Gauge 𝒑𝒈, Atmospheric 𝒑𝒔and Vacuum Pressure
𝒑𝒗
𝒑𝒈 = 𝒑𝒂 − 𝒑𝒔
𝒑𝒗 = 𝒑𝒔 − 𝒑𝒂
3. Pressure Balance equation for U – tube manometer
𝒑𝟏 + 𝒈𝒉𝝆𝒇 = 𝒑𝟐 + 𝒈𝒉𝝆𝒎
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (∆𝑝) = 𝑝1 − 𝑝2 = 𝑔ℎ(𝜌𝑚 − 𝜌𝑓) = 𝑔ℎ𝜌𝑚
𝜌𝑚 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑛𝑜𝑚𝑒𝑡𝑒𝑟 𝑚𝑒𝑟𝑐𝑢𝑟𝑦;
𝜌𝑓 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑖𝑛𝑔 𝑓𝑙𝑢𝑖𝑑
4. Pressure Sensitivity (S)
𝑺 =
𝑶𝒖𝒕𝒑𝒖𝒕
𝑰𝒏𝒑𝒖𝒕
=
∆𝒉
∆𝒑
=
𝒉
𝒈𝒉𝝆𝒎
=
𝟏
𝒈𝝆𝒎
5. Well Type Manometer – Expression for the mercury height
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑤𝑒𝑙𝑙 (𝑑) = ℎ − ℎ′
𝑑𝐴 = ℎ𝑎
𝒉 = 𝒉′
(𝟏 +
𝒂
𝑨
)
𝑎 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑏𝑒; 𝐴 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑒𝑙𝑙; ℎ = 𝑚𝑒𝑟𝑐𝑢𝑟𝑦 𝑐𝑜𝑙𝑢𝑚𝑛 ℎ𝑒𝑖𝑔ℎ𝑡
6. Pressure Difference in Well Type Manometer
∆𝒑 = 𝒈𝒉′
(𝟏 +
𝒂
𝑨
) 𝝆𝒎
7. Pressure Difference in Inclined Tube Manometer
𝑆𝑖𝑛𝑐𝑒 ∆𝑝 = 𝑝1 − 𝑝2 = 𝑔ℎ′
(1 +
𝑎
𝐴
) 𝜌𝑚
𝑠𝑖𝑛𝜃 =
ℎ′
𝑅
; 𝑅 = 𝑆𝑐𝑎𝑙𝑒 𝑅𝑒𝑎𝑑𝑖𝑛𝑔;
∆𝒑 = 𝒈𝝆𝒎𝑹𝒔𝒊𝒏𝜽 (𝟏 +
𝒂
𝑨
)

2. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 13
8. Pressure Difference I Ring Balance Manometer
∆𝒑 = 𝒑𝟏 − 𝒑𝟐 =
𝟐𝒘𝒈𝑹𝒔𝒊𝒏𝜽
𝑨𝒅
𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑟𝑖𝑛𝑔; 𝑤 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑤𝑒𝑖𝑔ℎ𝑡;
𝑅 = 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑟𝑖𝑛𝑔; 𝜃 = 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡𝑖𝑙𝑡 𝑓𝑟𝑜𝑚 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑥𝑖𝑠
9. Transfer Function of Manometer
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)
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

3. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 3 of 13
𝑪(𝒔) =
𝝎𝒏
𝟐
𝒔𝟐 + 𝟐𝝃𝝎𝒏𝒔 + 𝝎𝒏
𝟐
10. Atmospheric Pressure – Barometer
𝒑𝒔 = 𝝆𝒎𝒈𝒉
11. High Pressure – Bridgman Gauge
𝑹 = 𝑹𝟏(𝟏 + 𝒃∆𝒑)
𝑏 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
12. Low Pressure – Mc Leod Gauge
𝒑 =
𝒉𝟐
𝑨𝒕
𝑽
ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑚𝑒𝑟𝑐𝑢𝑟𝑦 𝑐𝑜𝑙𝑢𝑚𝑛; 𝐴𝑡 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛; 𝑉 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑢𝑙𝑏
13. Low Pressure – Kundsen Gauge
𝒑 =
𝒌𝑭
√𝑻 𝑻𝒈
⁄ − 𝟏
𝐹 = 𝑁𝑒𝑡 𝑓𝑜𝑟𝑐𝑒; 𝑇 = 𝑡𝑒𝑚𝑝. 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒𝑠; 𝑇𝑔 = 𝑇𝑒𝑚𝑝. 𝑜𝑓 𝑔𝑎𝑠; 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
14. Low Pressure – Ionization Gauge
𝑆 =
𝑂𝑢𝑡𝑝𝑢𝑡
𝐼𝑛𝑝𝑢𝑡
=
𝐼𝑖
𝑝𝐼𝑒
𝒑 =
𝑰𝒊
𝑺𝑰𝒆
𝐼𝑖 = 𝑖𝑜𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡, 𝑔𝑎𝑢𝑔𝑒 𝑜𝑢𝑡𝑝𝑢𝑡; 𝐼𝑒 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡

4. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 4 of 13
Flow Measurement
1. Reynolds Number
𝑹𝒆 =
𝒗𝒅𝝆
𝝁
𝑣 = 𝑓𝑙𝑜𝑤 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦; 𝑑 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑝𝑖𝑝𝑒; 𝜌 = 𝑓𝑙𝑢𝑖𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝜇 = 𝑓𝑙𝑢𝑖𝑑 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
2. Venturimeter
𝑝1
𝑤1
+ 𝑍1 +
𝑣1
2
2𝑔
=
𝑝2
𝑤2
+ 𝑍2 +
𝑣2
2
2𝑔
𝑝1 & 𝑝2 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑣1 & 𝑣2 = 𝑎𝑣. 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑤1 & 𝑤2 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝐴1 & 𝐴2 = 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑍1 & 𝑍2 = 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝜌1 & 𝜌2 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑟𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
Considering the venturimeter being held horizontal and fluid at inlet & throat of same density
𝑍1 = 𝑍2; 𝜌1 = 𝜌2; 𝑚 = 𝜌1𝐴1𝑣1 = 𝜌2𝐴2𝑣2
𝑣2
2
− 𝑣1
2
2𝑔
=
𝑝1 − 𝑝2
𝑤
By equation of continuity
𝐴1𝑣1 = 𝐴2𝑣2
𝑣1 = (
𝐴2
𝐴1
) 𝑣2
𝑣2 =
1
√1 − (
𝐴2
𝐴1
)
2
√
2𝑔
𝑤
(𝑝1 − 𝑝2)
𝑣2 = 𝑀√
2𝑔
𝑤
(𝑝1 − 𝑝2); 𝑀 =
1
√1 − (
𝐴2
𝐴1
)
2
Considering few losses, 𝑣2 is multiplied with a factor 𝐶𝑣 called the coefficient of velocity.

5. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 5 of 13
𝑣2(𝑎𝑐𝑡𝑢𝑎𝑙) = 𝐶𝑣𝑀√
2𝑔
𝑤
(𝑝1 − 𝑝2)
Discharge (volume flow rate)
𝑄 = 𝐴2𝑣2 = 𝐶𝑣𝐴2𝑀√
2𝑔
𝑤
(𝑝1 − 𝑝2)
Considering contraction factor 𝐶𝑐
𝑄𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐶𝑐𝐶𝑣𝐴2𝑀√
2𝑔
𝑤
(𝑝1 − 𝑝2)
𝑸𝒂𝒄𝒕𝒖𝒂𝒍 = 𝑪𝒅𝑨𝟐𝑴𝑬√
𝟐𝒈
𝒘
(𝒑𝟏 − 𝒑𝟐)
𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝐶𝑑 = 𝐶𝑐𝐶𝑣; 𝐸 = 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 𝑓𝑜𝑟 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
3. Orifice
Vena-contracta is a point where the liquid jet issued from the orifice has the smallest
diameter. It is located at as distance 𝐷1 2
⁄ from the orifice plate approximately.
Actual velocity at vena=contracta is
𝑣2(𝑎𝑐𝑡𝑢𝑎𝑙) = 𝐶𝑣𝑀√
2𝑔
𝑤
(𝑝1 − 𝑝2) =
𝐶𝑣
√1 − (
𝐴2
𝐴1
)
2
√
2𝑔
𝑤
(𝑝1 − 𝑝2)
The jet of liquid coming out of the orifice plate contracts to a minimum area 𝐴0 at the vena-
contracta.
Area of the vena-contracta is 𝐴0 = 𝐶𝑐𝐴0
∴ 𝑣2(𝑎𝑐𝑡𝑢𝑎𝑙) =
𝐶𝑣
√1 − (
𝐶𝑐𝐴0
𝐴1
)
2
√
2𝑔
𝑤
(𝑝1 − 𝑝2)
𝐷𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 𝑄𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐴2𝑣2 = 𝐶𝑐𝑣2(𝑎𝑐𝑡𝑢𝑎𝑙)
∴ 𝑄𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐶𝑣𝐶𝑐
𝐴0
√1 − (
𝐶𝑐𝐴0
𝐴1
)
2
√
2𝑔
𝑤
(𝑝1 − 𝑝2)
Taking into account the effect of temperature (E)

6. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 6 of 13
𝑸𝒂𝒄𝒕𝒖𝒂𝒍 = 𝑪𝒅𝑨𝟎𝑴𝑬√
𝟐𝒈
𝒘
(𝒑𝟏 − 𝒑𝟐)
Let 𝐾 = 𝐶𝑑𝑀
∴ 𝑸𝒂𝒄𝒕𝒖𝒂𝒍 = 𝑲𝑬𝑨𝟎√
𝟐𝒈
𝒘
(𝒑𝟏 − 𝒑𝟐)
4. Flow Nozzle
The discharge through a flow nozzle is
𝑸𝒂𝒄𝒕𝒖𝒂𝒍 = 𝑲𝑬𝑨𝟎√
𝟐𝒈
𝒘
(𝒑𝟏 − 𝒑𝟐)
𝐾 =
𝐶𝑑
√1 − (
𝐴2
𝐴1
)
2
=
𝐶𝑑
√1 − (
𝐷2
𝐷1
)
2
5. Pitot Tube
Using Bernoulli’s theorem, we have
𝑝
𝑤
=
𝑣2
2𝑔
+
𝑝0
𝑤
𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑖𝑛𝑙𝑒𝑡; 𝑝0 = 𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑣) = √
2𝑔
𝑤
(𝑝 − 𝑝0) = √
2𝑔
𝜌
(𝑝 − 𝑝0)
𝒗𝒎𝒆𝒂𝒏 = 𝑪𝒗√
𝟐𝒈
𝒘
(𝒑 − 𝒑𝟎) = 𝑪𝒗√
𝟐𝒈
𝝆
(𝒑 − 𝒑𝟎)
𝐶𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟
6. Weir
Discharge through rectangular notch
𝑸 =
𝟐
𝟑
𝑪𝒅𝑳√𝟐𝒈𝑯𝟏.𝟓
𝐶𝑑 = 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒; 𝐻 = 𝑙𝑖𝑞𝑢𝑖𝑑 ℎ𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑛𝑜𝑡𝑐ℎ; 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑛𝑜𝑡𝑐ℎ;
Discharge through v-notch

7. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 7 of 13
𝑸 =
𝟖
𝟏𝟓
𝑪𝒅√𝟐𝒈𝑯𝟐.𝟓
𝐭𝐚𝐧
𝜽
𝟐
𝜃 = 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑣 − 𝑛𝑜𝑡𝑐ℎ
Discharge through trapezoidal notch (summation of rectangular and v-notch)
𝑸 =
𝟐
𝟑
𝑪𝒅𝑳√𝟐𝒈𝑯𝟐.𝟓
+
𝟖
𝟏𝟓
𝑪𝒅√𝟐𝒈𝑯𝟐.𝟓
𝐭𝐚𝐧
𝜽
𝟐
7. Flumes
Actual discharge through a flume
𝑸𝒂𝒄𝒕𝒖𝒂𝒍 =
𝐶𝐴2
√1 + (
𝐴2
𝐴1
)
2
+ √2𝑔ℎ
𝐶 = 𝑣𝑒𝑛𝑡𝑢𝑟𝑖 𝑓𝑙𝑢𝑚𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (0.95 𝑡𝑜 1); ℎ = ℎ1 − ℎ2
 The maximum value of flow in a venture flume occurs when ℎ2 = (
2
3
) ℎ
 Maximum discharge through venture flume is given as 𝑄𝑚𝑎𝑥 = 1.7𝑏2𝐻1.5
𝑏2 = 𝑖𝑛𝑙𝑒𝑡 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑣𝑒𝑛𝑡𝑢𝑟𝑖 𝑓𝑙𝑢𝑚𝑒
 Accuracy of flumes are higher that of weirs.
8. Rotameter
By Bernoulli’s theorem and assuming the rotameter to be perfectly vertically aligned, the
energy equation is written as
𝑝2
𝑤
+
𝑣𝑚2
2
2𝑔
=
𝑝1
𝑤
+
𝑣𝑚1
2
2𝑔
𝑜𝑟 𝑣𝑚2
2
− 𝑣𝑚1
2
=
2𝑔
𝑤
(𝑝1 − 𝑝2)
𝑝 = 𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒; 𝑣𝑚 = 𝑚𝑒𝑎𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦; 𝑤 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡
For static equilibrium of the float at any position
𝐴𝑓 (𝑝1 +
𝑣𝑚1
2
2𝑔
𝑤) + 𝑣𝑓𝑤 = 𝐴𝑓𝑝2 + 𝑉𝑓𝑤𝑓
𝑉𝑓 & 𝑤𝑓 𝑎𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑎𝑛𝑑 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑜𝑎𝑡 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
By the continuity equation, we have
𝑄 = 𝑉
𝑚𝐴1 = 𝐶𝑐𝑣𝑚2
𝐴2
𝐴1 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑖𝑛𝑙𝑒𝑡 𝑜𝑓 𝑡𝑎𝑝𝑒𝑟𝑒𝑑 𝑡𝑢𝑏𝑒; 𝐴2 = 𝑎𝑟𝑒𝑎 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑓𝑙𝑜𝑎𝑡 & 𝑡𝑢𝑏𝑒;
𝐶𝑐 = 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛

8. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 8 of 13
Thus we have,
𝑸 = 𝑪𝒄𝑨𝟐√
𝟐𝒈𝑽𝒇
𝑨𝒇
(
𝒘𝒇
𝒘
− 𝟏) = 𝑪𝒄𝑨𝟐√
𝟐𝒈𝑽𝒇
𝑨𝒇
(
𝝆𝒇
𝝆
− 𝟏)
𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑; 𝜌𝑓 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑜𝑎𝑡
𝑸 ∝ 𝒙; 𝑥 = (𝑓𝑙𝑜𝑎𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡)
9. Electromagnetic Flowmeter
𝑬 = 𝑩𝒍𝒗
𝐸 = 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑; 𝐵 = 𝑓𝑙𝑢𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑦; 𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟;
𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟
10. Turbine Flowmeter
𝑬 = −
𝒅𝝋
𝒅𝒕
𝐸 = 𝐴𝐶 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑖𝑐𝑘 𝑢𝑝 𝑐𝑜𝑖𝑙; 𝜑 = 𝑟𝑜𝑡𝑎𝑡𝑖𝑛𝑔 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑓𝑖𝑒𝑙𝑑;
11. Vortex Flowmeter – Vortex Shedding Meter
𝒇 =
𝑵𝒔𝒕𝒗
𝑫
𝑓 = 𝑣𝑜𝑟𝑡𝑒𝑥 𝑠ℎ𝑒𝑑𝑑𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦;
𝐷 = 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑏𝑙𝑢𝑓𝑓 𝑏𝑜𝑑𝑦; 𝑁𝑠𝑡 = 𝑆𝑡𝑟𝑜𝑢ℎ𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟;
12. Ultrasonic Flowmeter
I. Crystal placed inside the tube
∆𝑡1 =
𝑑
𝐶 + 𝑣
; ∆𝑡2 =
𝑑
𝐶 − 𝑣
∆𝒕 = ∆𝒕𝟐 − ∆𝒕𝟏 =
𝟐𝒅𝒗
𝑪𝟐 − 𝒗𝟐
∆𝒕 =
𝟐𝒅𝒗
𝑪𝟐
(𝒂𝒔𝒔𝒖𝒎𝒊𝒏𝒈 𝑪 ≫ 𝒗)
𝐶 = 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑠𝑜𝑢𝑛𝑑 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚; 𝑣 = 𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑;
𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑇 & 𝑅
It is linearly proportional to flow velocity (v).
When sinusoidal signal frequency of 𝑓 Hz travels along the fluid flow, it has a phase
shift of
∆𝜑1 =
2𝜋𝑓𝑑
𝐶 + 𝑣
𝑟𝑎𝑑

9. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 9 of 13
When sinusoidal signal frequency of 𝑓 Hz travels against the fluid flow, it has a phase
shift of
∆𝜑2 =
2𝜋𝑓𝑑
𝐶 − 𝑣
𝑟𝑎𝑑
Velocity of fluid can be measured by either measuring the transient time or the phase
shift.
II. Crystals (T & R) placed outside the tube
∆𝑡 =
2𝑑 cos 𝜃
𝐶2
𝑣
𝒗 =
∆𝒕𝑪𝟐
𝟐𝒅 𝐜𝐨𝐬 𝜽
𝜃 = 𝑖𝑛𝑐𝑙𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑇 & 𝑅
III. US method using feedback
Pulse repetition frequency in forward loop
1
∆𝑡1
= 𝑓1
Pulse repetition frequency in backward loop
1
∆𝑡2
= 𝑓2
∆𝑡1 =
𝑑
𝐶 + 𝑣 cos 𝜃
; ∆𝑓1 =
𝐶 + 𝑣 cos 𝜃
𝑑
∆𝑡2 =
𝑑
𝐶 − 𝑣 cos 𝜃
; ∆𝑓2 =
𝐶 − 𝑣 cos 𝜃
𝑑
∆𝒇 = 𝒇𝟏 − 𝒇𝟐 =
𝟐𝒗 𝐜𝐨𝐬 𝜽
𝒅
IV. US Doppler Flowmeter
∆𝒇 = 𝒇𝒕 − 𝒇𝒓 =
𝟐 𝒇𝒕𝐜𝐨𝐬 𝜽𝒗
𝑪
13. Laser Doppler Anemometer
𝒇 =
𝟐𝒗 𝐬𝐢𝐧 𝜽 𝟐
⁄
𝝀
𝜆 = 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑙𝑎𝑠𝑒𝑟 𝑏𝑒𝑎𝑚

10. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 10 of 13
Temperature Measurement
1. RTD
𝑹𝑻 = 𝑹𝟎[𝟏 + 𝜶∆𝑻]
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 𝑺 =
𝒅𝑹
𝒅(∆𝑻)
=
𝒅
𝒅(∆𝑻)
𝑹𝟎[𝟏 + 𝜶∆𝑻] = 𝜶𝑹𝟎
2. Thermistor
𝑹𝑻 = 𝑹𝟎𝒆𝒙𝒑 [𝜷 (
𝟏
𝑻
−
𝟏
𝑻𝟎
)]
𝑅0 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇0 𝐾
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 𝑺 =
𝒅𝑹𝑻
𝒅𝑻
= 𝑹𝟎𝒆𝒙𝒑 [𝜷 (
𝟏
𝑻
−
𝟏
𝑻𝟎
) −
𝜷
𝑻𝟐
]
3. Thermocouple
𝑬 = 𝒂(∆𝜽) + 𝒃(∆𝜽)𝟐
𝐸 = 𝑒𝑚𝑓 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑; ∆𝜃 = 𝑡𝑒𝑚𝑝. 𝑑𝑖𝑓𝑓. ; 𝑎, 𝑏 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
𝑬 ≅ 𝒂∆𝜽 (𝒊𝒇 𝒂 ≫ 𝒃)
4. Series Connected Thermocouple (Thermopile)
𝑬 = 𝑬𝟏 + 𝑬𝟐 … + 𝑬𝒏 = 𝒏𝑬𝟏 (𝒊𝒇 𝑬𝟏 = 𝑬𝟐 … = 𝑬𝒏
5. High Temperature Measurement (Radiation Pyrometer)
𝒒𝒃 = 𝝈𝑻𝟒
𝑾 𝒎𝟐
⁄
𝑇 = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝. ; 𝜎 = 𝑆𝑡𝑒𝑓𝑎𝑛 𝐵𝑂𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 57.2 ∗ 10−9
𝐸𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 𝜀 =
𝑞
𝑞0
; 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 1
𝑞 = ℎ𝑒𝑎𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 𝑏𝑦 𝑔𝑟𝑎𝑦 𝑏𝑜𝑑𝑦; 𝑞0 = ℎ𝑒𝑎𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 𝑏𝑦 𝑏𝑙𝑎𝑐𝑘 𝑏𝑜𝑑𝑦;
𝒒 = 𝜺𝝈𝑻𝟒

11. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 11 of 13
Level Measurement:
1. Inferential – Capacitive Method:
𝑪 =
𝟐𝝅𝜺𝒉
𝐥𝐨𝐠𝒆 (
𝒅𝟐
𝒅𝟏
)
𝑭
𝜀 = 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑖𝑛𝑠𝑢𝑙𝑎𝑡𝑜𝑟; ℎ = ℎ𝑒𝑖𝑔ℎ𝑡;
𝑑1 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑚𝑒𝑡𝑎𝑙 𝑟𝑜𝑑; 𝑑2 = 𝑒𝑡𝑒𝑟𝑛𝑎𝑙 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑖𝑛𝑠𝑢𝑙𝑎𝑡𝑜𝑟;
Capacitive Voltage Divider Method:
𝑬𝟎 =
𝑪𝟏
𝑪𝟏 + 𝑪𝟐
𝑬𝑨
pH Measurement
1. pH Value:
𝒑𝑯 = − 𝐥𝐨𝐠𝟏𝟎[𝑯+]
2. Nernst’s Equation:
𝑬 = 𝑬𝟎 +
𝑹𝑻
𝒏𝑭
𝐥𝐧(𝒂𝑪)
𝐸 = 𝑒𝑚𝑓 𝑜𝑓 ℎ𝑎𝑙𝑓 𝑐𝑒𝑙𝑙; 𝐸0 = 𝑒𝑚𝑓 𝑜𝑓 ℎ𝑎𝑙𝑓 𝑐𝑒𝑙𝑙 𝑢𝑛𝑑𝑒𝑟 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛;
𝑅 = 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (8.314 𝐽 °𝐶
⁄ ; 𝑇 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑇𝑒𝑚𝑝. ;
𝑛 = 𝑛𝑜. 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑟𝑒𝑑; 𝐹 = 𝐹𝑎𝑟𝑎𝑑𝑎𝑦 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 96493 𝐶𝑚𝑜𝑙−1
;
𝑎 = 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (0 < 𝑎 < 1); 𝑐 = 𝑚𝑜𝑙𝑎𝑟 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑜𝑛𝑠;
3. Calibration for pH:
𝒑𝑯 =
𝑬𝟎 − 𝑬
𝟐𝟑𝟎𝟑 𝑹𝑻 𝑭
⁄
4. Reference Electrode Voltage:
∆𝒗𝟐 = −𝟐. 𝟑
𝑹𝑻
𝝉
𝐥𝐨𝐠
𝑪𝑯
𝑪𝑹
𝑅 = 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝐺𝑎𝑠 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 8314 𝐽 𝑘𝑔. 𝑚𝑜𝑙 − 𝐾
⁄ ; 𝑇 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝. ;
𝐽 = 𝐹𝑎𝑟𝑎𝑑𝑎𝑦′
𝑠𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 0.647 ∗ 107
𝐶 𝑘𝑔. 𝑚𝑜𝑙
⁄
𝐶𝐻 = 𝐻𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑖𝑜𝑛 𝑐𝑜𝑛𝑐. ; 𝐶𝑅 = 𝑐𝑜𝑛𝑐. 𝑜𝑓 𝑔𝑙𝑎𝑠𝑠 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒;

12. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 12 of 13
Viscosity Measurement:
1. Dynamic Viscosity:
𝝁 =
𝑺𝒉𝒆𝒂𝒓 𝒔𝒕𝒓𝒆𝒔𝒔
𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕
=
𝝉
(𝒅𝒗 𝒅𝒚
⁄ )
=
𝑭 𝑨
⁄
𝒅𝒗 𝒅𝒚
⁄
𝜏 = 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑒𝑠𝑠; 𝐹 = 𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒; 𝐴 = 𝑎𝑟𝑒𝑎 𝑜𝑛 𝑤ℎ𝑖𝑐ℎ 𝑓𝑜𝑟𝑐𝑒 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑒𝑑;
2. Kinematic Viscosity:
𝒗 =
𝑫𝒚𝒏𝒂𝒎𝒊𝒄 𝑽𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚
𝑫𝒆𝒏𝒔𝒊𝒕𝒚
=
𝝁
𝝆
3. Unit of Viscosity:
𝑆𝐼 𝑈𝑛𝑖𝑡 = 𝑁 𝑠𝑒𝑐 𝑚2
⁄
1 𝑃𝑜𝑖𝑠𝑒 = 𝑔 𝑐𝑚 − 𝑠
⁄
1 𝑆𝐼 𝑈𝑛𝑖𝑡 𝑜𝑓 𝐾𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 = 104
𝑠𝑡𝑟𝑜𝑘𝑒 (𝑆𝑡)
4. Falling Sphere Viscometer – Calculating Terminal Velocity:
𝑲𝒊𝒏𝒆𝒎𝒂𝒕𝒊𝒄 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝒗 =
𝒌𝒅𝟐
𝒈(𝝆𝒔 − 𝝆𝒍)
𝝆𝒍(𝟏𝟖 − 𝑽𝑻)
𝒎𝟐
𝒔
⁄
𝑘 = 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟; 𝑑 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒;
𝑉𝑇 = 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦; 𝜌𝑠 & 𝜌𝑙 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑖𝑒𝑠 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒 𝑎𝑛𝑑 𝑙𝑖𝑞𝑢𝑖𝑑 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
5. Falling Piston Viscometer:
𝑣
𝑦
=
𝑑𝑣
𝑑𝑦
; 𝜏 = 𝐹 𝐴
⁄
𝑫𝒚𝒏𝒂𝒎𝒊𝒄 𝑽𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝝁 =
𝑭 𝑨
⁄
𝒅𝒗 𝒅𝒚
⁄
=
𝑭𝒚
𝒗𝑨
𝑵 − 𝒔 𝒎𝟐
⁄
𝐹 = 𝑚𝑔; 𝐴 = 𝜋𝐷𝐿; 𝐷 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑝𝑖𝑠𝑡𝑜𝑛; 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑝𝑖𝑠𝑡𝑜𝑛;
𝑚 = 𝑝𝑖𝑠𝑡𝑜𝑛 𝑚𝑎𝑠𝑠; 𝑉𝑇 = 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦;
∴ 𝝁 =
𝒎𝒈𝒚
𝝅𝑫𝑳𝑽𝑻
𝑵 − 𝒔 𝒎𝟐
⁄
6. Capillary Tube Viscometer:
𝑫𝒚𝒏𝒂𝒎𝒊𝒄 𝑽𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚 𝝁 =
𝝅𝝆𝒈𝒉𝒇𝒅𝟒
𝟏𝟐𝟖𝑸𝑳
𝑵 − 𝒔 𝒎𝟐
⁄
𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑; ℎ𝑓 = 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 ℎ𝑒𝑎𝑑 𝑙𝑜𝑠𝑠 𝑜𝑓 𝑓𝑙𝑜𝑤;
𝑑 = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑢𝑏𝑒; 𝑄 = 𝑙𝑖𝑞𝑢𝑖𝑑 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒; 𝐿 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑡𝑢𝑏𝑒;
For Reynolds No. less than 1000 (laminar flow), the volumetric flow rate is given as
𝑸 =
𝝅𝒓𝟒(𝒑𝟏 − 𝒑𝟐)
𝟖𝝁𝑳
𝒎𝟑
𝒔
⁄

13. Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 13 of 13
Relative Humidity Measurement:
1. Aluminium Oxide Hygrometer:
𝒁 = √𝑹𝟐 + (
𝟏
𝝎𝑪
)
𝟐
2. Sling Psychrometer:
𝒑𝒗 = 𝒑𝒈𝒘 −
(𝒑 − 𝒑𝒈𝒘)(𝑻𝑫𝑩 − 𝑻𝑾𝑩)
𝟏𝟓𝟑𝟕. 𝟖 − 𝑻𝑾𝑩
𝑝𝑣 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑝𝑜𝑢𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒; 𝑝 = 𝑡𝑜𝑡𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑖𝑥𝑡𝑢𝑟𝑒;
𝑝𝑔𝑤 = 𝑠𝑎𝑡. 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑟𝑟. 𝑡𝑜 𝑤𝑒𝑡 𝑏𝑢𝑙𝑏 𝑡𝑒𝑚𝑝. ;
𝑇𝑊𝐵 = 𝑤𝑒𝑡 𝑏𝑢𝑙𝑏 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒; 𝑇𝐷𝐵 = 𝑑𝑟𝑦 𝑏𝑢𝑙𝑏 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒;
Finding the relative humidity:
For an ideal gas,
𝒎𝒗
𝒎𝒔𝒂𝒕.
=
𝒑𝒗𝑽 𝑹𝒗𝑻
⁄
𝒑𝒈𝑽 𝑹𝒗𝑻
⁄
=
𝒑𝒗
𝒑𝒈
𝑝𝑣 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 𝑣𝑎𝑝𝑜𝑢𝑟;
𝑝𝑔 = 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 𝑣𝑎𝑝𝑜𝑢𝑟 𝑎𝑡 𝑡ℎ𝑒 𝑡𝑒𝑚𝑝. 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑖𝑥𝑡𝑢𝑟𝑒