This article provides a comprehensive summary of equations to measure temperature and level in an industrial environment.

1. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 1 of 7
Industrial Instrumentation
Temperature Measurement
Sl. No.
1
Temperature Unit Conversion
Kelvin to Celsius: °𝑪 = [𝑲 − 𝟐𝟕𝟑. 𝟏𝟓]
Kelvin to Fahrenheit: °𝑭 = [(𝑲 − 𝟐𝟕𝟑. 𝟏𝟓) ∗
𝟗
𝟓
+ 𝟑𝟐]
Kelvin to Rankine: °𝑹 = [𝑲 ∗
𝟗
𝟓
]
Kelvin to Réaumur: °𝑹𝒆 = [(𝑲 − 𝟐𝟕𝟑. 𝟏𝟓) ∗
𝟒
𝟓
]
Celsius to Kelvin: 𝑲 = [°𝑪 + 𝟐𝟕𝟑. 𝟏𝟓]
Celsius to Fahrenheit: °𝑭 = [(℃ ∗
𝟗
𝟓
) + 𝟑𝟐]
Celsius to Rankine: °𝑹 = [(℃ ∗
𝟗
𝟓
) + 𝟒𝟗𝟏. 𝟔𝟕]
Celsius to Réaumur: °𝑹𝒆 = [(°𝑪 ∗
𝟒
𝟓
)]
Fahrenheit to Kelvin: 𝑲 = [(°𝑭 − 𝟑𝟐) ∗
𝟓
𝟗
+ 𝟐𝟕𝟑. 𝟏𝟓]
Fahrenheit to Celsius: : °𝑪 = [(℉ − 𝟑𝟐) ∗
𝟓
𝟗
]
Fahrenheit to Rankine: °𝑹 = [℉ + 𝟒𝟓𝟗. 𝟔𝟕]
Fahrenheit to Réaumur: °𝑹𝒆 = [(℉ − 𝟑𝟐) ∗
𝟒
𝟗
]
Rankine to Kelvin: 𝑲 = [°𝑹 ∗
𝟓
𝟗
]
Rankine to Celsius: °𝑪 = [(°𝑹 − 𝟒𝟗𝟏. 𝟔𝟕) ∗
𝟓
𝟗
]
Rankine to Fahrenheit: °𝑭 = [°𝑹 − 𝟒𝟓𝟗. 𝟔𝟕]
Rankine to Réaumur: °𝑹𝒆 = [(°𝑹 − 𝟒𝟗𝟏. 𝟔𝟕) ∗
𝟒
𝟗
]
Réaumur to Kelvin: 𝑲 = [(°𝑹𝒆 ∗
𝟓
𝟒
) + 𝟐𝟕𝟑. 𝟏𝟓]
Réaumur to Celsius: °𝑪 = [(°𝑹𝒆 ∗
𝟓
𝟒
)]
Réaumur to Fahrenheit: °𝑭 = [(°𝑹𝒆 ∗
𝟗
𝟒
) + 𝟑𝟐]
Réaumur to Rankine: °𝑹 = [(°𝑹𝒆 ∗
𝟒
𝟗
) + 𝟒𝟗𝟏. 𝟔𝟕]
2
Thermoelectric Laws:
Seebeck Effect:
𝑱 = 𝝈(−𝜵𝑽 + 𝑬𝒆𝒎𝒇)
𝐸𝑒𝑚𝑓 = 𝐸𝑀𝐹 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑, 𝐽 = 𝑙𝑜𝑐𝑎𝑙 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,
𝑉 = 𝑙𝑜𝑐𝑎𝑙 𝑣𝑜𝑙𝑡𝑎𝑔𝑒, 𝜎 = 𝑙𝑜𝑐𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦.

2. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 7
𝑬𝒆𝒎𝒇 = −𝑺𝜵𝑻
𝐸𝑒𝑚𝑓 = 𝐸𝑀𝐹 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑, 𝛻𝑇 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡, 𝑆 = 𝑆𝑒𝑒𝑏𝑒𝑐𝑘 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
Peltier Effect:
𝑸 = (𝜫𝑨 − 𝜫𝑩)𝑰
𝑄 = 𝑃𝑒𝑙𝑡𝑖𝑒𝑟 𝐻𝑒𝑎𝑡, 𝐼 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑓𝑙𝑜𝑤𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐴 𝑡𝑜 𝐵,
𝛱𝐴 = 𝑃𝑒𝑙𝑡𝑖𝑒𝑟 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟 𝐴,
𝛱𝐵 = 𝑃𝑒𝑙𝑡𝑖𝑒𝑟 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟 𝐵,
Thomson Effect:
𝑸 = 𝑲𝑱𝜵𝑻
𝑲 = 𝑻
𝒅𝑺
𝒅𝑻
𝑄 = 𝐻𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒,
𝐾 = 𝑇ℎ𝑜𝑚𝑠𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡, 𝐽 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,
𝑆 = 𝑆𝑒𝑒𝑏𝑒𝑐𝑘 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝛻𝑇 = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡.
Thomson Relationship:
𝑲 =
𝒅𝜫
𝒅𝑻
− 𝑺
𝜫 = 𝑻𝑺
𝐾 = 𝑇ℎ𝑜𝑚𝑠𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝛱 = 𝑃𝑒𝑙𝑡𝑖𝑒𝑟 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝑆 = 𝑆𝑒𝑒𝑏𝑒𝑐𝑘 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝑇 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒.
Thermoelectric Effect:
𝑱 = 𝝈(−𝜵𝑽 − 𝑺𝜵𝑻)
𝐽 = 𝑙𝑜𝑐𝑎𝑙 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦, 𝜎 = 𝑙𝑜𝑐𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦, 𝑆 = 𝑆𝑒𝑒𝑏𝑒𝑐𝑘 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
Equation for Accumulated Energy 𝒆̇ (temperature and charge changing with time)
𝒆̇ = 𝛁. (𝒌𝛁𝑻) + 𝛁. (𝑽 + 𝜫)𝑱 + 𝒒̇ 𝒆𝒙𝒕
𝑒̇ = 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝐸𝑛𝑒𝑟𝑔𝑦,
𝐽 = 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦, 𝛱 = 𝑃𝑒𝑙𝑡𝑖𝑒𝑟 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝑘 = 𝑡ℎ𝑒𝑟𝑚𝑛𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦,
𝑞̇𝑒𝑥𝑡 = 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑠𝑜𝑢𝑟𝑐𝑒 ℎ𝑒𝑎𝑡 𝑎𝑑𝑑𝑒𝑑,
𝑉 = 𝑙𝑜𝑐𝑎𝑙 𝑣𝑜𝑙𝑡𝑎𝑔𝑒.
3
Thermometer:
𝑻 = 𝑻𝒇 − (𝑻𝒇 − 𝑻𝒊)𝒆−𝒕 𝝉
⁄
𝑇 = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡, 𝑡 = 𝑡𝑖𝑚𝑒, 𝜏 = 𝑇𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑇𝑓 = 𝐹𝑖𝑛𝑎𝑙 𝑇𝑒𝑚𝑒𝑝𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝑖 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒,

3. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 3 of 7
Thermometer Transfer Function:
𝑻𝟎(𝒕) = 𝑻𝒊(𝒕) ∗
𝟏
𝝉𝒔 + 𝟏
=
𝑻𝒊(𝒕)
𝝉𝒋𝝎 + 𝟏
𝑴 = |
𝑻𝟎(𝒔)
𝑻𝒊(𝒔)
| = |
𝑻𝟎(𝒋𝝎)
𝑻𝒊(𝒋𝝎)
| =
𝟏
√(𝝉𝝎)𝟐 + 𝟏
𝑻𝒎𝒂𝒙 = 𝑴. ∆𝑻 + 𝑻𝒎𝒆𝒂𝒏
𝑷𝒉𝒂𝒔𝒆 𝑺𝒉𝒊𝒇𝒕 𝜽 = 𝐭𝐚𝐧−𝟏
𝝎∆𝑻
𝑻𝒊𝒎𝒆𝒍𝒂𝒈 =
𝟏
𝟑𝟔𝟎 ∗ 𝒇
∗ 𝜽
Volume-Temperature Function:
𝑽(𝑻) = 𝑨. 𝒉 = 𝑽(𝟏 + 𝜷𝑻)
𝐴 = 𝑏𝑢𝑙𝑏 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎, ℎ = 𝑏𝑢𝑙𝑏 ℎ𝑒𝑖𝑔ℎ𝑡, 𝑉 = 𝑏𝑢𝑙𝑏 𝑣𝑜𝑙𝑢𝑚𝑒,
𝛽 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
4
Bimetallic Strip:
Timoshenko Formula for Radius of Curvature of Bimetallic Strip (𝑹)
𝑹 =
𝒕 [𝟑. (𝟏 + 𝒎)𝟐
+ (𝟏 + 𝒎. 𝒏) (𝒎𝟐
+
𝟏
𝒎. 𝒏
)]
𝟔(𝜶𝟐 − 𝜶𝟏)(𝑻𝒉 − 𝑻𝑪)(𝟏 + 𝒎)𝟐
𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒, 𝑡 = 𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑖𝑝 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠,
𝑡1 = 𝑆𝑡𝑟𝑖𝑝 1 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠, 𝑡2 = 𝑠𝑡𝑟𝑖𝑝 2 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠,
𝑚 = 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 =
𝑡1
𝑡2
,
𝑛 = 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑌𝑜𝑢𝑛𝑔′
𝑠𝑀𝑜𝑑𝑢𝑙𝑢𝑠 =
𝐸1
𝐸2
,
𝐸1 = 𝑌𝑜𝑢𝑛𝑔′
𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑠𝑡𝑟𝑖𝑝 1,
𝐸2 = 𝑌𝑜𝑢𝑛𝑔′
𝑠𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑠𝑡𝑟𝑖𝑝 2,
𝛼1 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑟𝑖𝑝 1,
𝛼2 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑟𝑖𝑝 2,
𝑇ℎ = 𝐻𝑜𝑡 𝑡𝑒𝑚𝑒𝑝𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑎𝑡𝑒,
𝑇𝐶 = 𝐶𝑜𝑙𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑎𝑡𝑒.
Temperature Measurement – Bimetallic Strip:
𝑻𝒇 = 𝑻𝒊 +
𝟐𝒕
𝟑 ∗ 𝑹 ∗ (𝜶𝑨 − 𝜶𝑩)
𝑇𝑓 = 𝐹𝑖𝑛𝑎𝑙 𝑇𝑒𝑚𝑒𝑝𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝑖 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒,
𝑅 = 𝑆𝑡𝑟𝑖𝑝 𝑟𝑎𝑑𝑖𝑢𝑠, 𝑡 = 𝑠𝑡𝑟𝑖𝑝 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠,
𝛼𝐴 = 𝐸𝑥𝑝𝑎𝑛𝑠𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐴,

4. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 4 of 7
𝛼𝐵 = 𝐸𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐵,
𝑹 =
𝒕
𝟐𝜶(𝑻𝒇 − 𝑻𝒊)
𝛼 =
𝛼𝐴 + 𝛼𝐵
2
Angular Deflection of Strip (𝜽):
𝜃 = 𝑆𝑡𝑟𝑖𝑝 𝐿𝑒𝑛𝑔𝑡ℎ 𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 𝑅𝑎𝑑𝑖𝑢𝑠
⁄
Vertical Strip Displacement (𝒚):
𝒚 = 𝑹(𝟏 − 𝒄𝒐𝒔𝜽)
5
RTD:
𝑹𝑻 = 𝑹𝟎[𝟏 + 𝜶∆𝑻]
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 𝑺 =
𝒅𝑹
𝒅(∆𝑻)
=
𝒅
𝒅(∆𝑻)
𝑹𝟎[𝟏 + 𝜶∆𝑻] = 𝜶𝑹𝟎
𝜶 =
𝑹𝟏𝟎𝟎 − 𝑹𝟎
𝟏𝟎𝟎𝑹𝟎
𝛼 = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡, 𝑅100 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 100 °𝐶, 𝑅0 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 0 °𝐶
Callendar-Van Dusen Equation:
𝑹𝑻 = 𝑹𝟎[𝟏 + 𝑨𝑻 + 𝑩𝑻𝟐
+ 𝑪𝑻𝟑(𝑻 − 𝟏𝟎𝟎)] (−𝟐𝟎𝟎 °𝑪 < 𝑻 < 𝟎°𝑪)
𝑹𝑻 = 𝑹𝟎[𝟏 + 𝑨𝑻 + 𝑩𝑻𝟐] (𝟎 °𝑪 < 𝑻 < 𝟖𝟓𝟎°𝑪)
𝐴 = 3.9083 ∗ 10−3
°𝐶−1
;
𝐵 = −5.775 ∗ 10−7
°𝐶−2
,
𝐶 = −4.183 ∗ 10−12
°𝐶−4
𝑅0 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 0 °𝐶, 𝑅𝑇 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 𝑇 °𝐶.
6
Thermistor
𝑹𝑻 = 𝑹𝟎𝒆𝒙𝒑 [𝜷 (
𝟏
𝑻
−
𝟏
𝑻𝟎
)]
𝑅𝑇 = 𝑇ℎ𝑒𝑟𝑚𝑖𝑠𝑡𝑜𝑟 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑅0 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇0 𝐾
𝑹𝑻 = 𝑹𝑹𝒆𝒇𝒆
𝜷(
𝟏
𝑻
−
𝟏
𝑻𝑹𝒆𝒇
)
𝑅𝑇 = 𝑇ℎ𝑒𝑟𝑚𝑖𝑠𝑡𝑜𝑟 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝐾,
𝑅𝑅𝑒𝑓 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 0 𝐾,
𝛽 = 𝐵 − 𝑣𝑎𝑙𝑢𝑒 𝑠𝑝𝑒𝑐𝑓𝑖𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒𝑟𝑚𝑖𝑠𝑡𝑜𝑟,

5. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 5 of 7
𝑇𝑅𝑒𝑓 = 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 (0 𝐾), 𝑇 = 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒.
𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 𝑺 =
𝒅𝑹𝑻
𝒅𝑻
= 𝑹𝟎𝒆𝒙𝒑 [𝜷 (
𝟏
𝑻
−
𝟏
𝑻𝟎
) −
𝜷
𝑻𝟐
]
Steinhart-Hart Equation:
𝟏
𝑻
= 𝑨 + 𝑩 ∗ 𝒍𝒏(𝑹) + 𝑪(𝒍𝒏(𝑹))𝟑
𝐴, 𝐵, 𝐶 = 𝑆𝑡𝑒𝑖𝑛ℎ𝑎𝑟𝑡 − 𝐻𝑎𝑟𝑡 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠, 𝑅 = 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 𝑇 0°𝐶,
𝐴 = 0.001284, 𝐵 = 2.364 ∗ 10−4
, 𝐶 = 9.304 ∗ 10−8
7
Thermocouple
𝑬 = 𝒂(∆𝜽) + 𝒃(∆𝜽)𝟐
𝐸 = 𝑒𝑚𝑓 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑; ∆𝜃 = 𝑡𝑒𝑚𝑝. 𝑑𝑖𝑓𝑓. ; 𝑎, 𝑏 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
𝑬 ≅ 𝒂∆𝜽 (𝒊𝒇 𝒂 ≫ 𝒃)
𝑬𝒆𝒎𝒇 = −𝑺𝜵𝑻
𝐸𝑒𝑚𝑓 = 𝐸𝑀𝐹 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑, 𝛻𝑇 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡, 𝑆 = 𝑆𝑒𝑒𝑏𝑒𝑐𝑘 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
8
Series Connected Thermocouple (Thermopile)
𝑬 = 𝑬𝟏 + 𝑬𝟐 … + 𝑬𝒏 = 𝒏𝑬𝟏 (𝒊𝒇 𝑬𝟏 = 𝑬𝟐 … = 𝑬𝒏)
𝐸 = 𝑒𝑚𝑓 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑; 𝑛 = 𝑛𝑜. 𝑜𝑓 𝑠𝑒𝑟𝑖𝑒𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑐𝑖𝑟𝑐𝑢𝑖𝑡𝑠.
9
Langmuir Probe:
𝒄𝒔 = √𝑲𝑩(𝒁𝑻𝒆 + 𝜸𝒊𝑻𝒊) 𝒎𝒊
⁄
𝑐𝑠 = 𝑖𝑜𝑛 𝑠𝑝𝑒𝑒𝑑, 𝐾𝐵 = 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 1.38 ∗ 10−23
𝐽 𝐾
⁄
𝑍 = 𝑐ℎ𝑎𝑟𝑔𝑒 𝑠𝑡𝑎𝑡𝑒 𝑜𝑓 𝑖𝑜𝑛𝑠, 𝑚𝑖 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑖𝑜𝑛𝑠,
𝑇𝑒 = 𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝑖 = 𝐼𝑜𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒,
𝛾𝑖 = 𝐼𝑜𝑛𝑠′
𝐴𝑑𝑖𝑎𝑏𝑒𝑡𝑖𝑐 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡.
10
High Temperature Measurement (Radiation Pyrometer)
Stefan-Boltzmann Law: Absolute Power (P) radiated into surface area (A)
𝑷 = 𝑨𝜺𝝈𝑻𝟒
= 𝑨𝜺𝝈(𝑻𝑹 − 𝑻𝑪)𝟒
𝒋∗
= 𝒒𝒃 = 𝝈𝑻𝟒
= 𝑨𝜺𝝈(𝑻𝑹 − 𝑻𝑪)𝟒
𝑾 𝒎𝟐
⁄
𝒋∗
= 𝑞𝑏 = 𝐼𝑟𝑟𝑎𝑑𝑖𝑎𝑛𝑐𝑒,
𝑇 = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒; 𝑇𝑅 = 𝑅𝑎𝑑𝑖𝑎𝑡𝑜𝑟 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒, 𝑇𝐶 = 𝑆𝑢𝑟𝑟𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒,
𝜎 = 𝑆𝑡𝑒𝑓𝑎𝑛 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 56.703 ∗ 10−9
𝑊 𝑚2
𝐾4
⁄
𝐸𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 𝜀 =
𝑞
𝑞0
; 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 1

6. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 6 of 7
𝑞 = ℎ𝑒𝑎𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 𝑏𝑦 𝑔𝑟𝑎𝑦 𝑏𝑜𝑑𝑦; 𝑞0 = ℎ𝑒𝑎𝑡 𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 𝑏𝑦 𝑏𝑙𝑎𝑐𝑘 𝑏𝑜𝑑𝑦;
𝒒 = 𝜺𝝈𝑻𝟒
= 𝜺𝝈(𝑻𝑹 − 𝑻𝑪)𝟒
Thermometer Heat Equation Output:
𝑽(𝑻) = 𝒆𝑲𝑻𝑵; 𝑵 =
𝟏𝟒𝟑𝟖𝟖
𝑰 ∗ 𝑻
𝑉(𝑇) = 𝑇ℎ𝑒𝑟𝑚𝑜𝑚𝑒𝑡𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡, 𝑒 = 𝐸𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦,
𝐾 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑁 = 𝑁 𝑓𝑎𝑐𝑡𝑜𝑟,
𝐼 = 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ.
11
Solid State Temperature Sensor:
𝑽𝑩𝑬 =
𝐾𝐵𝑻
𝒒
𝒍𝒏
𝑰𝒄
𝑰𝒆𝒔
𝑉𝐵𝐸 = 𝐵𝑎𝑠𝑒 𝐸𝑚𝑖𝑡𝑡𝑒𝑟 𝑉𝑜𝑙𝑡𝑎𝑔𝑒, 𝐼𝑐 = 𝐶𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 𝐶𝑢𝑟𝑟𝑒𝑛𝑡,
𝐾𝐵 = 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 1.38 ∗ 10−23
𝐽 𝐾
⁄ ,
𝑞 = 𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑐ℎ𝑎𝑟𝑔𝑒 = 1.6 ∗ 10−19
𝐶,
𝑇 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 (𝐾).
12
Ultrasonic Thermometer:
𝒗𝟐
=
𝜸𝑹𝑻
𝑴𝒘
𝑣 = 𝑠𝑜𝑢𝑛𝑑 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦, 𝛾 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑟𝑎𝑡𝑖𝑜,
𝑅 = 𝐺𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑝𝑒𝑟 𝑚𝑜𝑙𝑒, 𝑇 = 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒,
𝑀𝑤 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑤𝑒𝑖𝑔ℎ𝑡.
Level Measurement
1
Inferential Capacitive Method:
𝑪 =
𝟐𝝅𝜺𝒉
𝐥𝐨𝐠𝒆 (
𝒅𝟐
𝒅𝟏
)
𝑭𝒂𝒓𝒂𝒅
𝐶 = 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑎𝑛𝑐𝑒 (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
Capacitive Tank: Capacitance between point A & B.

7. TEMPERATURE & LEVEL MEASUREMENT – SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 7 of 7
𝑪 = 𝑪𝟏 + 𝑪𝟐 =
𝟐𝝅
𝒍𝒏[(𝒓 + 𝒂) 𝒓
⁄ ]
𝜺𝟎(𝑳 − 𝒙) +
𝟐𝝅
𝒍𝒏[(𝒓 + 𝒂) 𝒓
⁄ ]
𝜺𝟎𝜺𝒓𝒙
∴ 𝑪 = 𝑪𝟏 + 𝑪𝟐 =
𝟐𝝅
𝒍𝒏[(𝒓 + 𝒂) 𝒓
⁄ ]
[𝜺𝟎𝑳 + 𝜺𝟎𝒙(𝜺𝒓 − 𝟏)]
∴ 𝒙 = 𝑪
𝒍𝒏[𝟏 + (𝒂 𝒓
⁄ )]
𝟐𝝅𝜺𝟎(𝜺𝒓 − 𝟏)
−
𝑳
𝜺𝒓 − 𝟏
= 𝑲𝟏𝑪 + 𝑲𝟐
𝑥 = 𝑙𝑖𝑞𝑢𝑖𝑑 𝑙𝑒𝑣𝑒𝑙 𝑖𝑛 𝑡𝑎𝑛𝑘, 𝐿 = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑎𝑛𝑘,
𝜀0 = 𝑎𝑖𝑟 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 = 8.84 ∗ 10−12
𝐹 𝑚
⁄ ,
𝜀𝑟 = 𝑑𝑖𝑒𝑒𝑙𝑐𝑡𝑟𝑖𝑐 𝑚𝑒𝑑𝑖𝑢𝑚 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦,
𝑟 = 𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒 𝑟𝑎𝑑𝑖𝑢𝑠,
𝑎 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑎𝑛𝑘 𝑤𝑎𝑙𝑙 𝑎𝑛𝑑 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑖𝑣𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒,
𝐾1 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝐾2 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡,
2
Differential Pressure Level Indicator:
𝒍 =
𝑷𝟏 − 𝑷𝟐
𝝆𝒈
𝑙 = 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑙𝑒𝑣𝑒𝑙, 𝑃1 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 1, 𝑃2 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 2,
𝜌 = 𝑙𝑖𝑞𝑢𝑖𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦, 𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,
3
Gamma Ray Level Indicator:
𝑰 = 𝑰𝟎𝒆𝒙𝒑(−𝝁𝝆𝒅)
𝐼 = 𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑎𝑓𝑡𝑒𝑟 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙,
𝐼0 = 𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑟𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦,
𝜇 = 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑚𝑎𝑠𝑠 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡,
𝜌 = 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦,
𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙,