ANSWER TO QUESTION 3:
(a) Thermal equilibrium exists between two systems when there is no net flow of heat between them. In our analogy we are therefore looking for situations where there is no flow of water. This is the case when the tap is turned off and the level of water in the bath has relaxed to its equilibrium state which is at the same height as water in the lake, i.e. 10 cm above the bottom of the bath.
(b) Thermal steady state exists between two systems when the temperatures are constant (but different) and the heat flows are constant. This is represented in our bath analogy when the tap is on and the level of water in the bath is steady. This level is higher than the level of water in the lake. Note also that the rate of flow of water through the tap is the same as that through the plug hole.
(c) Thermal transients occur when temperatures and heat flows are changing with time. This is the situation when neither thermal equilibrium or thermal steady state apply. In our example it is when the level of water in the bath and the water flow through the plug hole are varying.
End of Example.