How does the velocity and pressure of water in a pipe change when the cross-sectional area is widened?

Data: Water flows in a horizontal pipe at a velocity of 8 m/s with a pressure of 150,000 Pa. The pipe has a cross-sectional area of 0.05 square meters. If the pipe widens to an area of 0.08 square meters:
a. What will be the velocity of the water through this section of the pipe?
b. What is the pressure through this section of the pipe?

Explanation

When water flows through a pipe, its velocity and pressure can change as the pipe's cross-sectional area changes. In this case, we are given that the initial velocity of the water is 8 m/s and the initial cross-sectional area of the pipe is 0.05 square meters.

a. Velocity Calculation

The continuity equation tells us that the product of velocity and cross-sectional area must remain constant for an incompressible fluid like water. So, as the cross-sectional area increases, the velocity must decrease to maintain this product:
Initial velocity * Initial area = Final velocity * Final area
(8 m/s) * (0.05 m²) = Final velocity * (0.08 m²)
Final velocity ≈ 5 m/s

b. Pressure Calculation

Bernoulli's principle helps us understand the pressure changes. As the pipe widens, the fluid's velocity decreases, leading to an increase in pressure due to Bernoulli's principle:
Initial pressure + 0.5 * density * initial velocity² = Final pressure + 0.5 * density * final velocity²
150,000 Pa + 0.5 * density * (8 m/s)² = Final pressure + 0.5 * density * (5 m/s)²
Final pressure ≈ 112,500 Pa In summary, when the pipe widens, the velocity of the water decreases to around 5 m/s while the pressure increases to approximately 112,500 Pa. These changes are explained by the principles of fluid dynamics, specifically the continuity equation and Bernoulli's principle.

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