Chapter 3.2, Problem 1P

To determine

Determine the fluid flow velocity in the duct.

Answer to Problem 1P

The fluid flow velocity in the duct is $u_1=\sqrt{\frac{2\left(p_2-p_1\right)}{\rho }}$.

Explanation of Solution

The Bernoulli’s Equation can be used in many places not only in the pipe flow; the following are circumstances where the Bernoulli’s Equation shall be used in tanks as well as in open channels.

Circumference influenced by Bernoulli’s Equation.

• Pitot tube.
• Pitot static tube.
• Venturimeter and orificemeter.
• Flow over notches and weirs.

Pitot tube:

Pitot tube can be used to determine the velocity of fluid flow by connect with U-tube water gauge or with differential pressure gauge.

Sketch the part of Pitot tube of streamlines flow through the blunt body at uniform velocity as in Figure (1).

Apply the Bernoulli’s Equation at duct fluid flow velocity of $u_1$ and the pressure $p_1$ to the velocity $u_2$ at stagnation point $2$ and the pressure $p_2$.

  $\frac{p_1}{\rho g}+\frac{u_1^2}{2g}+z_1=\frac{p_2}{\rho g}+\frac{u_2^2}{2g}+z_2$        (I)

Here, the density fluid is $\rho$, the acceleration due to gravity is $g$ and datum head with respect to center of tube at point 1 and point 2 is $z$.

Conclusion:

From the Figure (1), the datum head at point 1 and 2 will be same $\left(z_1=z_2=z\right)$ and the velocity at stagnation point 2 equals to zero.

Substitute $z$ for $z_1$, $z$ for $z_2$ and $0$ for $u_2$ in Equation (I).

  $\begin{array}{l}\frac{p_1}{\rho g}+\frac{u_1^2}{2g}+z=\frac{p_2}{\rho g}+\frac{0}{2g}+z\\ \frac{p_1}{\rho g}+\frac{u_1^2}{2g}=\frac{p_2}{\rho g}\\ \frac{u_1^2}{2}=\frac{p_2}{\rho }-\frac{p_1}{\rho }\\ u_1=\sqrt{\frac{2\left(p_2-p_1\right)}{\rho }}\end{array}$

Hence, the fluid flow velocity in the duct is $u_1=\sqrt{\frac{2\left(p_2-p_1\right)}{\rho }}$.