Fundamental Definition
Volume flow rate is defined by the limit:
I.e., the flow of volume of fluid V through a surface per unit time t.
Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar quantity. The change in volume is the amount that flows after crossing the boundary for some time duration, not simply the initial amount of volume at the boundary minus the final amount at the boundary, since the change in volume flowing through the area would be zero for steady flow (which isn't the case, since an amount of volume has passed the boundary in the time duration).
Read more about this topic: Volumetric Flow Rate
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