I've come across an ideal gas process which can be represented by a positively sloped line in the $pV$ diagram. I found something about this on KhanAcademy, but it's simply a worked example on how to compute the work done by such a process. My question is what kind of a physical process does this represent? It's clearly not isochoric, isobaric, isothermal or adiabatic and it doesn't seem to appear in any basic heat engines.
Here's an image of the process (from KhanAcademy):
It does not have a name, and I am not sure how practical is to do it, it is just an expansion in which the pressure increases linearly with the volume. In order to do that you need to put the gas in contact with a variable thermal source whose temperature will augment quadratically with the volume: $T=(a+bV)V/nR$
Any path on the $PV$ graph (or on any other thermodynamic plane) represents a quasistatic process that can, in principle, be conducted experimentally. How easily, that's another question. Some paths represent simple processes, for example isothermal, isobaric, isentropic etc. Others require elaborate setups.
For the linear path on the $PV$ plane we might construct a temperature controller that adjusts the temperature to make the path a straight line. For an ideal gas, the controller would set the temperature according to the current volume as follows:
$$ T = \frac{V \left(P_0 V-P_0 V_1-P_1 V+P_1 V_0\right)}{n R \left(V_0-V_1\right)} $$
If you plug this temperature in the ideal gas law it will produce a linear path between $(P_0,V_0)$ and $(P_1,V_1)$. That's the same quadratic formula of @Wolphram jonny, evaluated for the specific end points of the path.
