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Answer :
To determine the pressure difference across the leak, we need to calculate the velocities at both ends using the volumetric flow rate equation and the cross-sectional area. By using Bernoulli's equation, we can find the pressure difference, which will provide insight into the leak.
The given scenario involves a horizontal pipeline carrying benzene, which has developed a leak 43 ft from the high-pressure end. The diameter of the leak is estimated to be 0.1 in. At the time of the leak, the upstream pressure in the pipeline is 50 psig, while the downstream pressure is 40 psig.
To analyze the situation, we can use Bernoulli's equation, which relates fluid pressure, velocity, and elevation. Since the pipe is horizontal, the elevation term can be ignored.
The equation can be written as: P1 + 0.5ρv1^2 = P2 + 0.5ρv2^2, where P1 and P2 are the pressures at the two ends of the leak, ρ is the density of benzene, and v1 and v2 are the velocities of the fluid at the respective ends.
To find the velocity, we can use the equation: v = Q / A, where Q is the volumetric flow rate and A is the cross-sectional area of the pipe.
Since the cross-sectional area is not given, we can assume the pipe to be circular and calculate it using the formula: A = πr^2, where r is the radius of the leak (0.05 in).
Assuming a steady-state flow, we can use the equation: Q = Av, where Q is the volumetric flow rate and v is the velocity of the fluid.
By substituting the values into the equations, we can solve for the velocities at both ends. Then, by rearranging Bernoulli's equation, we can find the difference in pressure between the two ends.
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