Question

Water flows through a horizontal pipe. The diameter of the pipe at point b is larger than the diameter of the pipe at point a. Where is the water pressure the greatest?.

Answer 1

Water pressure is greatest at point B where the diameter is larger

Step-by-step explanation:

This is due to Bernoulli's Equation:
`P_(A) +(pv _(A)^(2) )/(2) +pgh_(A)=P_(B) +(pv _(B)^(2) )/(2) +pgh_(B)`

and Continuity Equation:
`p_(A) v_(A)A_(A)=p_(B) v_(B)A_(B)`

where...

P = Pressure of Fluid at the Center of the Pipe

ρ = Density of Fluid

v = Velocity of Fluid

g = Gravitational Constant

h = Height of Fluid at the Center of the Pipe

A = Area of Pipe Cross Section

This is the same as saying the following:

Pressure Energy (
`P`) + Kinetic Energy (
`\frac{pv _{}^(2) }{2}`) + Potential Energy (
`pgh_{}`) = Constant

The height of flow at the center of the pipe is the same, so we know that Potential Energy cancels out on both sides of the equation (Δ
`pgh_{}` = 0)

`P_(A) +(pv _(A)^(2) )/(2) =P_(B) +(pv _(B)^(2) )/(2)`

Now that we've simplified Bernoulli's Equation, we need to determine which Pressure is greater using Continuity Equation.

`p_(A) v_(A)A_(A)=p_(B) v_(B)A_(B)`

Density is the same, so we can cancel this out on both sides of the equation (Δρ = 0)

`v_(A)A_(A)=v_(B)A_(B)`

From the problem statement, we know that
`A_(A) < A_(B)`

Since
`A_(A) < A_(B)`, we know that
`v_(A) > v_(B)` due to the Continuity Equation.

Answer: Jumping back to Bernoulli's Equation, we know that
`P_(A) < P_(B)`