## 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:

- It indicates the response of the system or object to the change in the surrounding temperature.
- The system with a lower time constant takes less time to achieve temperature change.
- 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.