The stylized pipeline:
Is in continuous flow from wellhead to refinery.
i rate of flow being inputted
v volume contained within pipeline defined by the pipeline's dimensions, length, diameter, etc.
o rate of flow being outputted
Some characteristics:
i - dv/dt = o
The dimensions of an ordinary pipeline are fixed; the volume contained within the pipeline is constant: Therefore dv/dt = 0.
Such a pipeline we will say is in dynamic stasis: inputs equal outputs.
i = o
But, and this does require just a slight bit of abstract thinking, imagine a pipeline with expanding dimensions, where v is increasing.
For such a pipeline i will always be greater than o because dv/dt is greater than zero.
i > o
We will term this the condition of dynamic growth.
Case 1: dynamic growth that is steady-state
The volume contained within the pipeline is increasing proportionately to the increase to i, i.e. when v has doubled i has doubled:
di/dt is proportional to dv/dt is proportional to do/dt
Case 2: dv/dt is accelerating in respect to di/dt
Then do/dt will be decelerating in respect to di/dt. Plotted on the same graph against time, the curve o will be falling in respect to i. The curves i and o are diverging.
Later we will develop the concept that either the condition of dynamic stasis or growth that is steady-state is necessary to Say's law. That the more likely condition through time is parametric shift will be taken to be the invalidation of that law.
We will apply the concept to the conventional model of circular flow:
and contrast it to its more realistic alternative derived from the A + B theorem:
Are you with me so far?