Chapter 10, Problem 47P

To determine

The relation $v_1=A_2\sqrt{\frac{2\left(P_1-P_2\right)}{\rho \left(A_1{}^2-A_2{}^2\right)}}$ which used to calculate the flow velocity is to be proved.

Explanation of Solution

Introduction:

According to the Bernoulli’s principle, the pressure inside the fluid system inversely deals with the volume of the system. The Pressure is decreased with the increase in the volume of the fluid system and vice-versa. When there is a pressure difference across the body, it causes a push on the body from the region of high pressure to that of low pressure.

The Bernoulli’s equation is written as,

  $P_1+\rho g{h}_1+\frac{1}{2}\rho v_1{}^2=P_2+\rho g{h}_2+\frac{1}{2}\rho v_2{}^2$

To prove:

Due to the conversion mass and incompressibility of fluid, the volumetric flow rate through each section is same. So, the volumetric flow rate becomes equals velocity time cross sectional area, such that

  $Q=\frac{V\pi d^2}{4}$

From the equation of continuity,

  $\begin{array}{l}{A}_1{v}_1=A_2{v}_2\\ v_2=v_1\frac{A_1}{A_2}\end{array}$

Now both sections in the venturi meter have the same average elevation which means that $h_1=h_2$ . So, the Bernoulli’s equation becomes,

  $\begin{array}{l}{P}_1+\rho g{h}_1+\frac{1}{2}\rho v_1{}^2=P_2+\rho g{h}_2+\frac{1}{2}\rho v_2{}^2\\ P_1+\frac{1}{2}\rho v_1{}^2=P_2+\frac{1}{2}\rho v_2{}^2\\ P_2-P_1=\frac{1}{2}\rho \left(v_1{}^2-v_2{}^2\right)\\ \left(v_1{}^2-v_2{}^2\right)=\frac{2\Delta P}{\rho }\end{array}$

Now use the value of velocity in the Bernoulli’s equation,

  $\begin{array}{l}\left(v_1{}^2-{\left(v_1\frac{A_1}{A_2}\right)}^2\right)=\frac{2\Delta P}{\rho }\Rightarrow v_1=\sqrt{\frac{2\left(P_2-P_1\right)}{\rho \left(1-{\left(\frac{A_1}{A_2}\right)}^2\right)}}=A_2\sqrt{\frac{2\left(P_2-P_1\right)}{\rho \left(A_2{}^2-A_1{}^2\right)}}\\ v_1=A_2\sqrt{\frac{2\left(P_2-P_1\right)}{\rho \left(A_2{}^2-A_1{}^2\right)}}\end{array}$

Conclusion:

Hence, the flow velocity expression is given by the relation $v_1=A_2\sqrt{\frac{2\left(P_1-P_2\right)}{\rho \left(A_1{}^2-A_2{}^2\right)}}$ is proved.