Is time constant the same for linear system for all time?

Question

I have heard that if you take a measurement $T_1$ and wait, then take another measurement $T_2$ and then find $Delta T = T_2 - T_1$. Then 63% of $Delta T$ + $T_1$ will represent the measurement when we can find the time constant.

So I took a PT100 sensor and measure temperature raising from 16C to 98C.

The blue line is the perfect transfer function model:
$$G(s) = frac{1}{900s + 1}$$
And the orange line is the temperature measurement.
Question:
Does the orange line have the same time constant if I pick out two measurement and find the time when $Delta T$ is 63%?

Thomas Fritsch · Accepted Answer

Yes, the time-constant stays the same for all time.
And there is a $63%$-rule valid for all the time.
But as @DKNguyen already said in the first comment, you got it slightly wrong.
The correct rule, valid for all $t$, is
$$frac{T(t+tau)-T(t)}{T(infty)-T(t)}=0.63$$

To understand where this rule comes from, we need to go back
to the differential equation
$$frac{dT(t)}{dt}=frac{1}{tau}(T(infty)-T(t))$$
where $T(t)$ is the temperature measured at time $t$,
$T(infty)$ is the temperature measured after waiting an infinite time,
and $tau$ is the time constant.
This, with the starting condition $T(0)$, has the solution
$$T(t)=T(infty)+(T(0)-T(infty))e^{-t/tau}$$
From this solution, with a little bit of algebra, you can derive
$$frac{T(t+tau)-T(t)}{T(infty)-T(t)}=1-e^{-1}$$
which is just the $63%$-rule given above.

rob · Answer

If the rate of temperature change is proportional to the difference between the current temperature and the eventual temperature,
$$
frac d{dt} T = k( T_text{eq} - T),
tag1
$$
then the temperature will exponentially approach the equilibrium value,
$$
T(t) = T_text{eq} + (T_text{start}-T_text{eq}) e^{-kt}
tag2
$$
with time constant $tau=1/k$.
You're correct that this time constant should be the same along the entire curve.
This model is usually a good first approximation for temperature evolution for a system in a constant-temperature heat bath, because generally the heat flow across a boundary is proportional to the temperature difference, and generally the temperature of a system is proportional to the heat content. However this second assumption breaks down in materials undergoing a phase change, where energy added to the system goes into changing its state rather than changing its temperature.
Here in the twenty-first century we can do better fitting than just choosing two magic points along the curve.  Think of deviations from the exponential approach (2) as giving you information about the accuracy of the model (1). Your data are very pretty, and follow the exponential curve very nicely on average, but have some non-exponential structure that may or may not be physically interesting --- you haven't said enough about your setup for us to be able to tell.

Is time constant the same for linear system for all time?

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