Incompressible oil of density 850 kg/m^3 is pumped through a cylindrical pipe at a rate of 9.5 liters per second. The second section of the pipe has a diameter of 4.0 cm. What are the flow speed in that section? (Unit: m/s)
user
Sure, I'd be happy to help with this calculation.

The flow rate, Q, in liters per second is given, and the density, ρ, of the oil in kg/m^3 is provided. To calculate the flow speed, we'll use the following formula:

Q = πr²v

Where:
- Q = flow rate (L/s)
- π (pi) = 3.14159 (a constant)
- r = radius (m)
- v = velocity (m/s)

But because we're dealing with volume (liters) instead of mass (kg), we'll first need to convert the density from kg/m^3 to kg/L. The conversion factor is:

1 kg/m^3 = (1000 kg)/(1 m³) = (1000 kg)/(1000 L/m³) = 1000 kg/L

So, 850 kg/m^3 becomes 850 kg/L x 1000 L/m³ = 850000 kg/L.

Next, we'll convert the flow rate from liters/second to cubic meters/second, since velocity is a linear speed (m/s). The conversion factor is:

1 L/s = (1000 L)/(1 s) = (1000 m³)/(1 s) = 1000 m³/s

So, 9.5 L/s becomes 9.5 L/s x 1000 L/m³/L = 9500 m³/s.

Now, let's substitute these values into the above formula:

Q = πr²v
Q = 3.14159 * (0.04 m/2)² * v
Q = 1.59154 * v²

Q = 9500 m³/s

Solving for v:

v = Q / (1.59154 * r²) = 9500 m³/s / (1.59154 * (0.04 m)²) = 34.81 m/s

So, the velocity of the oil through the 4.0 cm diameter section is approximately 34.81 m/s. This calculation assumes that the velocity is constant throughout the entire cross-sectional area of the pipe. This is known as the assumption of "uniform flow," and it's often used in fluid mechanics calculations.
assistant
Incompressible oil of density 850 kg/m^3 is pumped through a cylindrical pipe at a rate of 9.5 liters per second. The second section of the pipe has a diameter of 4.0 cm. What are the flow speed in that section? (Unit: m/s)
user
First, let's convert the flow rate from liters per second to cubic meters per second, since theСР flow speed will be in meters per second.

1 liter = 0.001 cubic meters

So,
9.5 liters/second = 9.5 \* 0.001 = 0.0095 cubic meters/second

Next, let's find the cross-sectional area of the second section of the pipe. The formula for the area of a circle is A = πr^2, where r is the radius of the circle. The radius is half the diameter, so:

r = diameter / 2 = 4.0 cm / 2 = 2.0 cm = 0.02 meters (converting cm to meters)

A = π \* (0.02)^2 = 0.00125663706144 pi square meters

Now that we have the flow rate in cubic meters per second and the cross-sectional area in square meters, we can find the flow speed using the formula:

flow speed = flow rate / cross-sectional area

flow speed = 0.0095 cubic meters/second / 0.00125663706144 pi square meters
flow speed ≈ 7.55928098377 m/s

Therefore, the flow speed in the second section of the pipe is approximately 7.56 m/s (rounded to two decimal places).
assistant