Section 6.5: Bernoulli's Principle

Bernoulli’s Principle states that for an incompressible, non-viscous fluid, the total mechanical energy along a streamline remains constant:

Bernoulli Equation

\[ P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} \] where:
\( P \) = fluid pressure (Pa)
\( \rho \) = fluid density (kg/m³)
\( v \) = fluid velocity (m/s)
\( g \) = acceleration due to gravity (m/s²)
\( h \) = height above reference (m)

Example 1: Horizontal Flow

A fluid flows through a horizontal pipe where the velocity increases from 2 m/s to 5 m/s. If the pressure at the first section is 150 kPa, find the pressure at the second section. Assume density = 1000 kg/m³.

Using Bernoulli’s equation:
\( P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \)
\( P_2 = P_1 + \frac{1}{2}\rho(v_1^2 - v_2^2) \)
\( P_2 = 150{,}000 + 0.5 \times 1000 \times (4 - 25) = 150{,}000 - 10{,}500 = 139{,}500 \, \text{Pa} \)
The pressure at the second section is 139.5 kPa.

Example 2: Vertical Flow

Water flows from a tank and exits a hole 2 m below the water surface. Find the exit speed.

Use Bernoulli between water surface and hole:
\( P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \)
Assuming \( v_1 \approx 0 \), \( P_1 = P_2 = \) atmospheric, \( h_1 - h_2 = 2 \):
\( \frac{1}{2} \rho v_2^2 = \rho g \Delta h \Rightarrow v_2 = \sqrt{2 g h} = \sqrt{2 \cdot 9.8 \cdot 2} \approx 6.26 \text{ m/s} \)
The exit speed is approximately 6.26 m/s.

Practice Problems

  1. Water flows through a horizontal pipe of varying diameter. Velocity increases from 1 m/s to 4 m/s. Pressure drops from 200 kPa. Find the final pressure. (Density = 1000 kg/m³)
  2. A fluid exits a tank through a hole 3 m below the surface. Calculate exit speed.
  3. Air flows over the top of a roof at 10 m/s, bottom at 5 m/s. Find pressure difference. (Density = 1.2 kg/m³)
  4. A horizontal water pipe narrows from 0.2 m² to 0.05 m². Velocity at wide section is 2 m/s. Find velocity at narrow section and pressure difference if pressure at wide section is 150 kPa.
  5. A river’s surface water flows faster than deeper layers. Explain using Bernoulli’s principle.
← Previous Next →