Short Answer:

When a fluid flows through a sudden expansion in a pipe, its velocity decreases and pressure increases due to the change in cross-sectional area. According to the energy principle, part of the kinetic energy is lost due to turbulence and eddies formed at the expanded section. This loss of energy is known as head loss due to sudden expansion and is proportional to the square of the velocity difference before and after expansion.

The energy principle (Bernoulli’s equation) helps relate the pressure, velocity, and elevation between the two sections of flow. The head loss represents the energy converted into heat and turbulence because of the abrupt change in flow area. This concept is important in fluid mechanics to design smooth pipe transitions and reduce energy losses in fluid systems.

Detailed Explanation :

Flow through Sudden Expansions using Energy Principle

When fluid moves through a pipe and suddenly encounters an increased cross-sectional area, the velocity of the fluid decreases while pressure increases. This phenomenon is called flow through sudden expansion. The change in energy distribution and pressure can be explained by the energy principle, also known as Bernoulli’s principle.

Bernoulli’s equation states that for steady and incompressible flow, the sum of pressure energy, kinetic energy, and potential energy per unit weight of the fluid remains constant, neglecting losses. Mathematically,

where,
= pressure, velocity, and elevation at section 1 (before expansion),
= pressure, velocity, and elevation at section 2 (after expansion),
and  = head loss due to sudden expansion.

Derivation of Head Loss

Consider two sections of a horizontal pipe — section 1 (smaller diameter) and section 2 (larger diameter). Since the pipe is horizontal, the elevation term is the same ().
By applying Bernoulli’s equation:

Now, applying the momentum principle between the two sections:

where  and  are the cross-sectional areas at section 1 and section 2 respectively, and  is the discharge.

Simplifying and combining the above equations gives:

This is the head loss due to sudden expansion. It shows that the loss of head depends on the square of the difference in velocities before and after expansion.

Explanation of Energy Loss

When the flow expands suddenly, the fluid particles cannot follow the abrupt change in pipe wall. This causes flow separation and the formation of eddies or vortices near the wall. These turbulent regions result in loss of mechanical energy in the form of heat and sound.

The velocity distribution after the expansion becomes irregular initially but gradually becomes uniform further downstream. The loss in kinetic energy due to turbulence appears as pressure loss or head loss in the flow system.

The energy lost due to sudden expansion can also be calculated using the relation:

This means the energy lost is directly proportional to the difference in velocity squares before and after expansion.

Practical Applications

1. Pipeline Design: In designing pipelines, gradual expansions are preferred over sudden expansions to minimize head loss.
2. Hydraulic Machines: Understanding this principle helps in the design of turbines and pumps where sudden area changes occur.
3. Energy Efficiency: Engineers try to avoid sudden expansions to reduce unnecessary energy loss and maintain efficient flow.
4. Water Distribution Systems: Proper expansion design ensures steady pressure distribution and minimizes cavitation risk.

Conclusion

Flow through sudden expansion involves a decrease in velocity and a rise in pressure, but with an energy loss due to turbulence and eddy formation. The energy principle (Bernoulli’s equation) helps calculate this loss, which is given by . In engineering applications, it is crucial to reduce such energy losses by using smooth transitions and gradual expansions. Understanding this principle ensures efficient and safe fluid system designs.