A round pipe of varying diameter carries petroleum from a wellhead to a refinery. At the wellhead the pipe's diameter is 56.3 cm (0.563 m) and the flow speed of the petroleum is 13.5 m/s. At the refinery the petroleum flows at 5.65 m/s. What is the volume flow rate of the petroleum along the pipe and what is the pipe\'s diameter at the refinery?

flow rate = Q = pi r^2 v

= amount per second that flows through the pipe
so
Q = pi (.563)^2 (13.5)

if no oil has been added or subtracted or compressed then Q is the same everywhere
so

pi (D^2/4) (5.65) = Q

round pipe of varying diameter carries petroleum from a wellhead to a refinery. At the wellhead, the pipe's diameter is 57.7 cm

and the flow speed of the petroleum is 11.1 m/s.
At the refinery, the petroleum flows at 5.51 m/s.
What is the volume flow rate of the petroleum along the pipe, and what is the pipe's diameter at the refinery?

Using the same formula as before, we have:

Volume flow rate at wellhead = Q = pi (0.577/2)^2 * 11.1 = 0.1391 m^3/s

Since the volume flow rate is constant along the pipe, we can set up another equation:

pi (D/2)^2 * 5.51 = 0.1391

Solving for D, we get:

D/2 = sqrt(0.1391/(pi*5.51)) = 0.236

So the diameter of the pipe at the refinery is:

D = 0.472 m = 47.2 cm

Therefore, the volume flow rate of the petroleum is 0.1391 m^3/s and the pipe's diameter at the refinery is 47.2 cm.

To find the volume flow rate of the petroleum along the pipe, we can use the formula:

Q = A1 * V1 = A2 * V2

where:
Q is the volume flow rate,
A1 and A2 are the cross-sectional areas of the pipe at the wellhead and refinery respectively,
V1 and V2 are the flow speeds at the wellhead and refinery respectively.

First, let's calculate the cross-sectional area of the pipe at the wellhead. The formula for the area of a circle is:

A = π * r^2

where:
A is the area,
r is the radius of the circle.

Since we have the diameter at the wellhead, we can find the radius by dividing it by 2:

r1 = 0.563 m / 2 = 0.2815 m

Now, we can calculate the area at the wellhead:

A1 = π * r1^2 = π * (0.2815 m)^2

Next, we can use the formula for volume flow rate to calculate the volume flow rate at the wellhead:

Q = A1 * V1 = (π * (0.2815 m)^2) * 13.5 m/s

Next, let's find the diameter of the pipe at the refinery. Again, we can calculate the radius by dividing the diameter at the refinery by 2:

r2 = d2 / 2 = ? / 2

We need to find the unknown diameter at the refinery (d2). To do this, we can rearrange the equation for volume flow rate:

A2 = Q / V2

Now, we can calculate the area at the refinery:

A2 = Q / V2 = ? / 5.65 m/s

Finally, we can use the formula for the area of a circle to find the diameter at the refinery:

d2 = 2 * r2 = 2 * sqrt(A2 / π)

Now that we have explained how to find the volume flow rate and the diameter at the refinery, you can plug in the known values and calculate the answers.