Velocity of Water Calculation

[mr.]

Homework Statement

Water is in laminar flow, in a horizontal pipe 2.5 cm in radius. The pressure drop in a 20m section of the pipe is 3000Pa. The viscosity of the water is 1.2 x 10^-3 N *s/m^3. Calculate the velocity of the water along the axis of the pipe.

Homework Equations

The Bernoulli equation

The Attempt at a Solution

I attempted to apply the Bernoulli equation to this problem, but am not sure what to do with the viscosity of water. Every answer I have come up with is not correct.

How does the viscosity fit into this problem (if at all).

 

[Pyrrhus]

Aren't you forgetting about the energy lost to friction added to Bernoulli's equation?

 

[ogb p]

As a scientist, you are correct in applying the Bernoulli equation to this problem. However, the viscosity of the water does play a role in determining the velocity of the water.

The Bernoulli equation states that the total energy of a fluid is constant along a streamline. In this case, we can consider the pressure drop as a change in the potential energy of the water, and the velocity as the kinetic energy of the water. The equation can be written as:

P1 + (1/2)ρv1^2 + ρgh1 = P2 + (1/2)ρv2^2 + ρgh2

Where P is pressure, ρ is density, v is velocity, g is acceleration due to gravity, and h is height.

To solve for the velocity, we can rearrange the equation to isolate v2:

v2 = √((2(P1-P2)/ρ) + (2gh1) - (2gh2))

The viscosity of the water comes into play when we consider the Reynolds number, which is a dimensionless quantity used to determine the type of flow (laminar or turbulent). In laminar flow, the Reynolds number is less than 2300, and in this case, we can use the Hagen-Poiseuille equation to calculate the velocity:

v = (P1-P2)πr^4 / 8ηL

Where r is the radius of the pipe, η is the viscosity, and L is the length of the pipe.

Plugging in the given values, we get:

v = (3000Pa)(π(0.025m)^4) / (8(1.2x10^-3 N*s/m^2)(20m))

v = 0.005 m/s

So, the velocity of the water along the axis of the pipe is 0.005 m/s. We can see that the viscosity played a crucial role in determining the velocity in this case, as it was used in the Hagen-Poiseuille equation to calculate the velocity.