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Heat Transfer

Lumped transient cooling

Temperature of a small body cooling in a fluid, with uniform body temperature.

Ref: HTR-04
Rev: v1
Field: Heat Transfer
Result: T [°C]

\frac{T (t) - T_{\infty}}{T_{i} - T_{\infty}} = e^{- \frac{t}{τ}}, τ = \frac{ρ c_{p} V}{h A_{s}}

Uniform initial temperature, constant properties, h and ambient temperature. Check Bi = h(V/A_s)/k_s < 0.1 as a rule of thumb for small internal gradients. Here V/A_s is entered as the characteristic length L_c.

A small hot object cools quickly at first and then more slowly, approaching the temperature of its surroundings.

Calculator

Solve for

TTemperature at time tComputed

°C

T_iInitial temperature

°C

501,000

T_∞Ambient temperature

°C

−40100

ρDensity

kg/m³

50020,000

c_pSpecific heat

J/(kg·K)

1005,000

L_cVolume/area ratio

0.1100

hHeat transfer coefficient

W/(m²·K)

52,000

tTime

13,000

Time constant τ400 s
Fraction of the temperature gap remaining74.08 %

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Temperature at time t (°C) current value 154.6 °C

T_iInitial temperatureKey driver

43.5747

tTime

25.1200

L_cVolume/area ratio

25197

hHeat transfer coefficient

25195

ρDensity

26.4180

c_pSpecific heat

64195

T_∞Ambient temperature

138174

After one time constant the temperature gap falls to 37%; after three it is about 5%.
A large h or a small V/A_s makes the body respond quickly.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (°C)	Local step	Change in result
T_i Initial temperature	50 to 1,000 °C	43.52	747.3	703.8	+10%	+14.8 °C
t Time	1 to 3,000 s	25.1	199.6	174.5	+10%	−3.83 °C
L_c Volume/area ratio	0.1 to 100 mm	25	197.4	172.4	+10%	+3.58 °C
h Heat transfer coefficient	5 to 2,000 W/(m²·K)	25	194.8	169.8	+10%	−3.83 °C
ρ Density	500 to 20,000 kg/m³	26.44	180.2	153.8	+10%	+3.58 °C
c_p Specific heat	100 to 5,000 J/(kg·K)	64.05	194.8	130.8	+10%	+3.58 °C
T_∞ Ambient temperature	−40 to 100 °C	137.8	174.1	36.29	+10%	+0.648 °C

Sweep of Time

Time (s)	Temperature at time t (°C)
1	199.6
375.9	93.38
750.8	51.79
1,126	35.49
1,501	29.11
1,875	26.61
2,250	25.63
2,625	25.25
3,000	25.1

Sweep one variable

Vary

Temperature at time t (°C)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Temperature at time t

Result · °C

What the equation gives you. Everything else feeds it.

T_i

Initial temperature

°C · typical 50 to 1,000

Starting temperature of the body. Raises the temperature.

T_∞

Ambient temperature

°C · typical −40 to 100

Temperature of the surrounding fluid. Raises the final temperature.

Density

kg/m³ · typical 500 to 20,000

Density of the body. Raises the time constant.

c_p

Specific heat

J/(kg·K) · typical 100 to 5,000

Specific heat of the body. Raises the time constant.

L_c

Volume/area ratio

mm · typical 0.1 to 100

V/A_s (for a sphere, diameter/6). Raises the time constant.

Heat transfer coefficient

W/(m²·K) · typical 5 to 2,000

Surface convection coefficient. Shortens the time constant.

Time

s · typical 1 to 3,000

Time since the start. More time brings the body closer to ambient.

Worked example with the values above

τ = ρc_pL_c/h = 8,000 × 500 × 0.005 / 50 = 400 s.
Decay: e^−t/τ = e^−0.3 = 0.7408.
T = 25 + (200 − 25) × 0.7408 = 154.6 °C.

Assumptions and limits

Biot number below about 0.1.
Constant properties and h.

Common mistakes

Using the lumped model for thick or low-conductivity bodies.
Using the total volume without dividing by surface area.

Statistics and uncertainty

The time constant depends on h to within ±20–30%, so the cooling time is only as good as that estimate.

Where it is used

Sensor and thermocouple response.
Cooling of small components and shells.

Maths you need

Exponential decay: e^(−t/τ)
Convection: h

Related formulas

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