I’m currently doing undergrad research on using a gravel (basalt 2-5mm grain size) bed as an heat storage. In order to heat up the gravel inside, hot (~100°C) and dry (RF<4%) air with a pressure of 10bar is inserted from above (->see T0). I implemented thermometers, measuring the temperature at different positions. The thermometer T1 is slightly above the gravel, while T2 and T3 are inside the gravel. Obviously, thermometers close to the top (close to the incoming air) heat up first. The temperature increase looks like a logistic function for T0 and T1. T2 and T3 behave a little different, though. At a certain temperature (~50°C) the logistic increase saturates and a linear increase follows. This linear increase takes a lot of time and finally saturates close to the equilibrium temperature. My first guess was, that the saturation of the first curve is due to water condensating inside the gravel bed. Now, this seems rather unlikely, since the incoming air is very dry, and no moisture was found when looking into the gravel bed. Since my advisors are not very experienced with gravel beds, they could not come up with a solution as well. Still, I do believe, I’m not the first one with such an observation, and therefor wanted to ask, whether anyone of you knows what could be happening here. I appreciate every theory, you can come up with (and can be tested).
You would expect a logistic function if you blew air at a constant temperature through the gravel. And this is what you get at $0$ and $1$.
But as the gravel heats, it cools the air. So the air at $2$ is cooler at first. As the gravel above it heats, the air arriving at $2$ gets closer to the expected temperature.
$3$ is the same, but it takes longer.
To address the comment, you might see what you can derive from the $1$D Convection–diffusion equation, given experimental parameters like air temperature, pressure, and velocity; and heat capacity and thermal conductivity of the gravel.
I can't model it off the top of my head. But I would guess you would find this leads to a couple different time scale parameters in the solution. Maybe some thing like the velocity with which the zone of cool gravel retreats, and the width of the intermediate zone where gravel is warming. Maybe something else. If one time scale was short and the other long, you might see a quick rise followed by a gradual rise.
Intuitively, I would expect temperatures near $1$ to follow a logistic curve with a rather short time constant. Farther down, I would expect air temperature to be cool, and slowly rise as all the gravel upstream slowly warms.
Perhaps the air doesn't give up all its heat as it flows through the gravel, but still is cooler than $100$ C as it passes $2$ or $3$. Gravel at those positions quickly warm to the local ambient temperature, and follow it as the local ambient temperature slowly rises.
But I can't say whether you could reasonably get a curve with two time scales at a given point, instead of a single time scale at each point, where the time scale is slower farther downstream.