A horizontal pipe of diameter 0.845 m has a
smooth constriction to a section of diameter
0.507 m. The density of oil flowing in the pipe
is 821 kg/m3 .
If the pressure in the pipe is 8350 N/m2
and in the constricted section is 6262.5 N/m2 ,
what is the rate at which oil is flowing?
To find the rate at which oil is flowing through the pipe, we can apply Bernoulli's equation, which relates the pressure, velocity, and height of a fluid flowing through a pipe.
Bernoulli's equation states:
P1 + 1/2 * ρ * v1^2 + ρ * g * h1 = P2 + 1/2 * ρ * v2^2 + ρ * g * h2
In this case, P1 is the pressure in the pipe, P2 is the pressure in the constricted section, ρ is the density of the oil, v1 and v2 are the velocities of the oil in the pipe and constricted section respectively, and h1 and h2 are the heights of the oil in the pipe and constricted section respectively (which we can assume to be the same since the pipe is horizontal).
Since the pipe and the constricted section are connected, the oil is incompressible, so the density (ρ) is the same in both sections.
The given information is:
Diameter of the pipe (D1) = 0.845 m
Diameter of the constricted section (D2) = 0.507 m
Pressure in the pipe (P1) = 8350 N/m^2
Pressure in the constricted section (P2) = 6262.5 N/m^2
Density of the oil (ρ) = 821 kg/m^3
To find the velocities (v1 and v2), we can use the equation for the volume flow rate (Q) through a pipe:
Q = A1 * v1 = A2 * v2
Where A1 and A2 are the cross-sectional areas of the pipe and constricted section respectively.
To find the cross-sectional areas, we can use the formula for the area of a circle:
A = π * (d/2)^2
Where d is the diameter.
Calculating the cross-sectional areas:
A1 = π * (0.845/2)^2
A2 = π * (0.507/2)^2
Now, we can rearrange the volume flow rate equation to solve for v1 and v2:
v1 = (A2/A1) * v2
Now, substitute the given values into the equation, and solve for the velocity (v2) in the constricted section:
v2 = (A1/A2) * v1
Next, we can substitute the values of v2 and v1 into Bernoulli's equation and solve for the rate at which oil is flowing (Q):
Q = A1 * v1 = A2 * ((2 * (P2 - P1)) / ρ + v1^2)
By substituting the known values of A1, A2, P1, P2, and ρ into the equation, we can calculate the rate at which oil is flowing.