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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 $310K$.

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  • $\begingroup$ Depends on the mechanism of heat transfer. For convection, if you know the heat transfer coefficient then you can find the heat flux between the systems, and from the heat capacities you can find the resultant changes in temperature. Combine the two to get a differential equation... $\endgroup$
    – lemon
    Apr 8, 2016 at 10:45
  • $\begingroup$ To bound the answer for the maximum possible amount of time required, you could neglect convection within the mercury and treat the problem as transient heat conduction between two solids. That being said, even with transient heat conduction problems, it helps to have some experience to get a quick answer without having to resort to numerical methods or complicated analytical approaches. $\endgroup$ Apr 8, 2016 at 11:05

2 Answers 2

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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:

Temperature evolution

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.

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In a nutshell: it takes an infinite time to reach perfect equilibrium.

For realistic measurements though, the time until it is 'near enough' depends on material constants and what you consider 'near enough'.

For measuring human body temperature, an exactness of .05 K is probably good enough, and the thermometers are made for good heat transfer, so it would be probably less than ten seconds.

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