Lumped Capacitance Calculator
With Bi < 0.1 the body cools or heats exponentially with time constant ρcLc/h: a 1 mm thermocouple bead reaches 199 °C in 7.31678 s.
7.31678012972498 s
Full answer, then cut to fewer decimals (not rounded):
- 7.31678012972498 s
- 7.3167801297249 s
- 7.316780129724 s
- 7.31678012972 s
- 7.3167801297 s
- 7.316780129 s
- 7.31678012 s
- 7.3167801 s
- 7.316780 s
- 7.31678 s
- 7.3167 s
- 7.316 s
- 7.31 s
- 7.3 s
- Biot number Bi = hLc/k
- 0.00333333
- Lumped model valid
- yes (Bi < 0.1)
- Characteristic length Lc = V/As
- 0.166667 mm
- Time constant τ = ρcLc/h
- 1.41667 s
- Temperature ratio θ/θᵢ
- 0.00571429
Step-by-step solution
- (T − T∞)/(Tᵢ − T∞) = e^(−t/τ)
- τ = ρcLc/h = 8500 × 400 × 0.000166667 ÷ 400 = 1.41667 s
- t = −τ ln[(T − T∞)/(Tᵢ − T∞)] = −1.41667 × ln(0.00571429) = 7.31678 s
Lumped Capacitance Formula
(T − T∞)/(Tᵢ − T∞) = e^(−t/τ), τ = ρcLc/h
Lc = V/As (D/6 sphere, D/4 cylinder), valid when Bi = hLc/k < 0.1
Worked Example
A 1 mm thermocouple bead (ρ = 8500, c = 400, k = 20) at 25 °C plunged into gas at 200 °C with h = 400 W/m²·K, reaching 199 °C:
- (T − T∞)/(Tᵢ − T∞) = e^(−t/τ)
- τ = ρcLc/h = 8500 × 400 × 0.000166667 ÷ 400 = 1.41667 s
- t = −τ ln[(T − T∞)/(Tᵢ − T∞)] = −1.41667 × ln(0.00571429) = 7.31678 s
Lumped model valid: yes (Bi < 0.1) · Characteristic length Lc = V/As: 0.166667 mm · Time constant τ = ρcLc/h: 1.41667 s · Temperature ratio θ/θᵢ: 0.00571429
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Frequently asked questions
What does a Biot number below 0.1 mean?
Internal conduction resistance is small compared with surface convection, so the body stays at nearly one temperature and the lumped model holds.
How long does it take to reach 63.2 % of the change?
One time constant τ; after 5τ more than 99 % of the change has occurred.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Lumped Capacitance Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/lumped-capacitance-calculator/
IEEE
Scientific Converters, "Lumped Capacitance Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/lumped-capacitance-calculator/