We can use Bernoulli's equation to solve this problem, which states that the total pressure at any point in a fluid flow system is the sum of the static pressure and the dynamic pressure. The equation can be written as:
P1 + (1/2)ρv1^2 = P2 + (1/2)ρv2^2
where P is the pressure, ρ is the density, and v is the velocity of the fluid at two different points in the flow.
We can assume that the fluid is incompressible, so the mass flow rate (m_dot) is constant throughout the pipe. The mass flow rate is given by:
m_dot = ρA1v1 = ρA2v2
where A is the cross-sectional area of the pipe at two different points in the flow.
We can use the above equations to solve for the rate at which oil is flowing:
From Bernoulli's equation:
P1 + (1/2)ρv1^2 = P2 + (1/2)ρv2^2
Substituting the given values:
7370 N/m2 + (1/2)821 kg/m3v1^2 = 5527.5 N/m2 + (1/2)821 kg/m3v2^2
From the continuity equation:
A1v1 = A2v2
Substituting the given values:
(π/4)(0.842 m)^2v1 = (π/4)(0.5052 m)^2v2
Simplifying, we get:
v2 = (0.842/0.5052)^2v1 = 2.628v1
Substituting v2 into Bernoulli's equation and simplifying, we get:
v1 = 6.08 m/s
Substituting v1 into the continuity equation and simplifying, we get:
m_dot = ρA1v1 = 177.4 kg/s
Therefore, the rate at which oil is flowing is 177.4 kg/s.