If we have two bodies a and b, with temperatures $T_a$ and $T_b$ respectively, with $T_a > T_b$ and we also know that body b is surrounded by body a completely. How do we calculate the relaxation time for the bodies?
A simple example illustrating my question would be a mercury thermometer at room temperature $T_b = 298 K$ placed into a human mouth with temperature $T_a = 310 K$, we would want to calculate how long it would take for the thermometer to give an accurate reading such that the final temperature of the thermometer would be $310K$.
Best Answer
In the case of the mercury thermometer we can model its temperature rise based on the following assumptions and parameters.
$T(t)$ is thermometer temperature, $T_m$ is mouth temperature. $C$ is total heat capacity of the part of the thermometer placed in mouth. $h$ is the Heat Transfer Coefficient between mouth and thermometer, $A$ is the total surface area of the part of the thermometer placed in mouth. $q$ is Heat Energy, $t$ represents time.
We assume heat transfer to be convective, the temperature of the part of the thermometer placed in mouth to be homogeneous (no spatial temperature gradients) and the mouth temperature constant.
In accordance with Newton's cooling law (here applied to heating) we can write the heat flux from mouth to thermometer as:
$$\frac{dq}{dt}=hA[T_m-T(t)]$$
The heat flux causes a temperature rise of the thermometer:
$$dq=CdT(t)$$
So that the rate of temperature rise is given by:
$$\frac{dT(t)}{dt}=\frac{hA}{C}[T_m-T(t)]$$
When we integrate this simple Differential Equation between $t=0,T(0)=T_i$ and $t, T(t)$ ($T_i$ is the initial temperature of the thermometer), we get:
$$t=\frac{C}{hA}\ln \Big[\frac{T_m-T_i}{T_m-T(t)}\Big]$$
So this tells us how long it takes to get from $T_i$ to a specified temperature $T(t)$ but note that for $T(t) \to T_m$, then $t \to +\infty$: in theory it takes an infinite amount of time for the thermometer to fully reach $T_m$! This problem can be 'avoided' by setting the final thermometer temperature to for instance $99$ percent of $T_m$, so $T(t)=0.99T_m$. That would allow some comparative calculations for various values of the parameters involved.
Schematically the temperature evolution of the thermometer is as follows:
The value of $C$ can be estimated from the mass of the thermometer, in particular the mass of glass and mercury and their specific heat capacities.
For the Heat Transfer Coefficient $h$ various engineering websites will provide estimated values.