This equation is a simplified version of the one derived from the physical construction of a venturi
As you can see, the constant of proportionality (k) shown in the simpler equation is nothing
more than a condensation of the first half of the longer equation: k represents the geometry of the venturi tube. If we define k by the mouth and throat areas (A1, A2) of any particular venturi tube, we must be very careful to express the pressures and densities in compatible units of measurement. For example, with k strictly defined by flow element geometry (tube areas measured in square feet), the calculated flow rate (Q) must be in units of cubic feet per second, the pressure values P1 and P2 must be in units of pounds per square foot, and mass density must be in units of slugs per cubic foot. We cannot arbitrarily choose different units of measurement for these variables, because the units must “agree” with one another. If we wish to use more convenient units of measurement such as inches of water column for pressure and specific gravity (unitless) for density, the original (longer) equation simply will not work.
However, if we happen to know the differential pressure produced by any particular flow element
tube with any particular fluid density at a specified flow rate (real-life conditions), we may calculate a value for k in the short equation that makes all those measurements “agree” with one another. In other words, we may use the constant of proportionality (k) as a unit-of-measurement correction factor as well as a definition of element geometry. This is a useful property of all proportionalities:
simply insert values (expressed in any unit of measurement) determined by physical experiment and solve for the proportionality constant’s value to satisfy the expression as an equation. If we do this, the value we arrive at for k will automatically compensate for whatever units of measurement we arbitrarily choose for pressure and density.