Question

A horizontal pipe of diameter d1​ has a sudden expansion to a diameter d2​. At what ratio d1​/d2​ would the differential pressure on either side of the expansion be maximum? What is the corresponding loss of head and differential pressure head?

Answer 1

The question centers on the maximum differential pressure created by a sudden expansion from diameter d1 to d2 in a fluid system, which can be described by Bernoulli's and continuity equations, a fundamental concept in fluid dynamics within Physics.

Step-by-step explanation:

The question relates to fluid dynamics in Physics, particularly concerning the behavior of a fluid as it flows through pipes of different diameters, which is described by the Bernoulli's equation. When considering a sudden expansion from diameter d1 to d2, the maximum differential pressure occurs when the velocity of the fluid before the expansion is at its greatest. According to the principle of conservation of mass (continuity equation) and Bernoulli's principle, this condition is achieved when the ratio of d1 to d2 is such that the velocity decreases substantially while the cross-sectional area increases, leading to a maximum pressure difference across the expansion.

To find the exact ratio of d1 to d2 that maximises this effect, we can set up the appropriate equations using continuity and Bernoulli's principles. Similarly, the head loss can be calculated by considering the conversion of pressure energy to kinetic energy and then back to pressure energy with some loss due to the sudden expansion, commonly referred to as the loss due to enlargement or divergent loss.

The Bernoulli equation gives a way to relate these parameters:

• The change in kinetic energy is reflected in the change of speed between two points in the flow.
• The potential energy per unit volume of the fluid due to height changes is given by ρgh, where ρ is the density, g is the acceleration due to gravity, and h is the height.
• The pressure energy per unit volume is given by the pressure of the fluid (P).

Combining these gives:

• At the smaller diameter (ρgh1 + P1 + 0.5ρv1² = constant)
• At the larger diameter (ρgh2 + P2 + 0.5ρv2² = constant)

The differential pressure head can then be calculated using the difference in the pressure energy terms from Bernoulli's equation for the two points.