Describe the construction and working of venturimeter and obtain an equation for the volume of liquid flowing per second though a wider entry of the tube

Explanation

Venturimeter : This device is used to measure the rate of flow ( or say flow speed) of the incompressible fluid flowing through a pipe. It works on the principle of Bernoulli.s theorem. It consists of two wider tubes A and A. (with cross sectional area A) connected by a narrow tube B ( with cross sectional area a). A manometer in the fonn of U-tube is also attached between the wide and narrow tubes. The manometer contains a liquid of density `. rho_m . `
Let `P_1` be the pressure of the fluid at the wider region of the tube A. Let us assume that the fluid of density .`rho` . flows .from the pipe with speed `v_1` and into the narrow region, its .speed increases to `v_2` . According to the Bernoulli.s equation, this increase in speed is accompanied by a decrease in the fluid pressure `P_2` at the narrow region of the tube B. Therefore, the pressure difference between the tubes A and B is noted by measuring the height difference `(Delta P = P_1 + P_2)` between the surfaces of the manometer liquid. From the equation of continuity, we can say that `Av_1 = av_2` which means that
` v_2 = A/a v_1`
Using bernoulli.s equation,
` P_1 + rho (v_1^2)/(2) = P_2 + rho (v_2^2)/(2) = P_2 + rho 1/2 (A/a v_1)62`
from the above equation , the pressure difference
` Delta P = P_1 - P_2 = rho (v_1^2)/(2) (A^2 - a^2)/(a^2)`
Thus , the speed of flow of liquid at the wide end of the tube A
` v_1^^2 = (2 (Delta P)a^2)/(rho(A^2 - a^2) ) rArr v_1 = sqrt((2(Delta P)a^2)/(rho (A^2 - a^2) )`
The volume of the liquid flowing out per second is `V = A v_1 = A sqrt( (2 (Delta P )a^2)/(p (A^2 -a^2) ) ) = a A sqrt( (2(Delta P))/(p(A^2 - a^2) ))`