Parcel Temperature Calculator

Calculate parcel temperature over time from initial and ambient temperatures, time constant, or lumped heat transfer properties.

Time constant (τ)

Lumped (m·c)

Enter any 4 values to calculate the missing variable (uses T(t)=Tₐ+(Tᵢ−Tₐ)e^{−t/τ}).

Initial Temperature

Ambient Temperature

Time Constant (τ)

Time

Final Temperature

Related Calculators

Parcel Temperature Formula

The following formulas are commonly used to estimate the temperature of a parcel over time using a lumped-capacitance (Newton’s law of cooling/heating) approximation.

T₍final) & = T₍ambient) + (Tᵢₙᵢₜᵢₐₗ - T₍ambient)) e⁽ - (hA) / (m cₚ) t)\ & = T₍ambient) + (Tᵢₙᵢₜᵢₐₗ - T₍ambient)) e⁽ - t /), = (m cₚ) / (hA)

Variables:

T_final is the final temperature of the parcel (°C, K, or °F)
T_initial is the initial temperature of the parcel (°C, K, or °F)
T_ambient is the ambient temperature (°C, K, or °F)
h is the convective heat transfer coefficient (W/m²·K or Btu/(h·ft²·°F))
A is the surface area of the parcel (m², cm², or ft²)
m is the parcel mass (kg, g, or lb)
c_p (c) is the specific heat capacity (J/(kg·K) or Btu/(lb·°F))
t is the elapsed time (seconds, minutes, or hours; use consistent units in the exponent)
τ is the time constant τ = m c_p / (hA) (same time unit as t)

To calculate the final temperature of a parcel, you can either (1) use the full lumped-capacitance model with h, A, m, and c_p, or (2) use the equivalent time-constant form if you already know τ (for example, from measurement). These models assume the parcel is approximately at a uniform internal temperature (lumped approximation) and that h and ambient conditions are roughly constant over the time interval.

What is Parcel Temperature?

Parcel temperature refers to the temperature of a package or container, which can change over time due to heat transfer with its surroundings. This is particularly important in the context of shipping temperature-sensitive goods, where maintaining a specific temperature range is crucial. In a simple lumped-capacitance model, the rate at which a parcel’s temperature approaches ambient depends on both the external heat transfer (h and surface area A) and the parcel’s thermal mass (m·c_p).

How to Calculate Parcel Temperature?

The following steps outline how to calculate the final temperature of a parcel.

Determine the initial temperature of the parcel (T_initial).
Determine the ambient temperature (T_ambient).
If using the time-constant method, determine the time constant τ (for example, from measurement). If using the lumped method, determine h, surface area A, mass m, and specific heat capacity c_p.
Determine the elapsed time t (use seconds, minutes, or hours, but keep units consistent with τ or within the exponent).
Calculate the final temperature (T_final) using the formula provided (or use the calculator above).

Example Problem :

Use the following variables as an example problem to test your knowledge.

Initial temperature of the parcel (T_initial) = 5°C

Ambient temperature (T_ambient) = 25°C

Heat transfer coefficient (h) = 5 W/m²K

Surface area of the parcel (A) = 0.50 m²

Mass (m) = 5 kg

Specific heat capacity (c_p) = 900 J/kg·K

Time (t) = 3 hours

(For these values, the time constant is τ = m c_p /(hA) = 1800 s = 0.5 hr, giving T_final ≈ 24.95°C after 3 hours.)