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.
