Question

A round pipe of varying diameter carries petroleum from a wellhead to a refinery. At the wellhead the pipe\'s diameter is 52.9 cm and the flow speed of the petroleum is 13.7 m/s. At the refinery the petroleum flows at 5.75 m/s. What is the volume flow rate of the petroleum along the pipe and what is the pipe\'s diameter at the refinery?

Answer 1

(a) At the wellhead, diameter of the pipe, d1 = 52.9 cm = 0.529 m

at this location, flow speed, v1 = 13.7 m/s

Now, expression for the volume flow rate at a point is given by -

V_dot = v*A [A = cross-sectional area of the pipe]

Now,

A = pi*d^2/4

A1 = pi*(0.529 m)^2 / 4 = 0.22 m^2

So,

V_dot = 13.7 m/s * 0.22 m^2 = 3.01 m^3/s

Since, volume flow rate is the same throughout the pipe, hence volume flow rate at refinery = 3.01 m^3/s (Answer)

(b) Now, as we have discussed above, volume flow rate is the same throughout,

So,

V_dot1 = V_dot2

=> v1*A1 = v2*A2

=> v1*pi*d1^2/4 = v2*pi*d2^2/4

=> v1*d1^2 = v2*d2^2

=> d2 = d1*sqrt(v1/v2)

= 52.9 cm * sqrt(13.7 m/s / 5.75 m/s) = 81.7 cm

Therefore, pipe's diameter at the refinery = d2 = 81.7 cm (Answer)