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.