# Thermal time constant in heat transfer: Definition, Unit, Formula [with Pdf]

Contents

## What is Thermal time constant?

For the body undergoing cooling or heating, the thermal time constant is the time to reach the temperature gradient equals 63.21 % of the initial temperature gradient.

The thermal time constant is denoted by the symbol \tau_{th}.

E.g.:- If the object with initial temperature Ti is cooled by the surrounding medium at temperature T_{\infty} then at the time equals to the thermal time constant the relation between temperature is given by,

\Delta T=0.6321\times\Delta T_{INITIAL}

(T-T_{\infty})=0.6321(T_{i}-T_{\infty})

The SI unit of the thermal time constant is second.

## Thermal time constant equation:

The thermal time constant is given by the following equation,

\tau_{th}=\frac{\rho VC}{hA_{S}}

Where,
ρ = Density of the body
V = Volume of the body
C = Specific heat of an object
h = Convective heat transfer coefficient
As = Surface area of the body

Therefore thermal time constant depends on the above factors.

By using a thermal time constant, the lumped system equation is also rewrite as

\frac{T-T_{\infty}}{T_{i}-T_{\infty}}=e^{-\frac{t}{\tau}}

## Significance of thermal time constant:

1. It indicates the response of the system or object to the change in the surrounding temperature.
2. The system with a lower time constant takes less time to achieve temperature change.
3. The system with a higher value of time constant takes much time to achieve temperature change.

## Relation between temperature gradient at t=\tau_{th}

To check the relation between temperature gradient at t=\tau_{th}, put the time interval equal to \tau_{th} in the equation of lumped system analysis.

\frac{T-T_{\infty}}{T_{i}-T_{\infty}}=e^{-\frac{t}{\tau}}

\frac{T-T_{\infty}}{T_{i}-T_{\infty}}=e^{-1}

\therefore (T-T_{\infty})=0.367(T_{i}-T_{\infty})

\therefore \Delta T_{AFTER}=0.367\Delta T_{INITIAL}

\therefore \Delta T_{AFTER}=36.7\%\Delta T_{INITIAL}

From the above equation, it is clear that after the interval of t=\tau_{th} the temperature difference between the system and surroundings is equal to 36.7% of the initial temperature difference.

Therefore at t=\tau_{th}, the system achieves a temperature gradient equal to 63.2% of the initial temperature gradient. Pratik is a Graduated Mechanical engineer. He enjoys sharing the engineering knowledge learned by him with people.