Relationship between flow rate and pressure drop

We can consider flow of an incompressible fluid and apply mechanical energy balance at plane 1 and 2 and neglecting frictional losses. We get

(1)

Equation of continuity for incompressible fluid, gives

(2)

Here  and  is average velocity at plane 1 and 2, and  and  are diameter at plane 1 and 2.
By 1 and 2 we get,

(3)

Note that velocity  according to eq. 3 corresponds to maximum velocity when effect of friction on flow is ignored. Pressure difference corresponding to  in eq.3 is the one which one would read at plane 2 in venturi and in orifice at vena- contracta.  It must be noted that the equation 3 is not specific to any flow measuring device; it is applicable to orifice, venturi-meter, nozzle or any other. The equation relates velocity of the fluid to the pressure difference and diameter ratio.

Now the cross section area at vena contracta is not known and hence d₂ at vena contracta is not known. Vena-contracta is created due to the abrupt contraction as the fluid passes through an orifice. The cross sectional area at vena-contracta would depend, among other factors, on shape of the orifice (circular, rectangular or square, etc.) and fluid dynamics. It can be determined experimentally. However, diameter  of the orifice is known. We introduce coefficient of discharge  and replace  in eq. 3 by do we get

(4)

.
Here  and  

Note-  in eq. 4 is  in case of venturi. Equations 3 and 4 can be applied to venturi, and nozzles as well once we know the value of . The discharge coefficient value is specific to the flow measuring device.