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A 0.20-inch hole develops in a pipeline containing toluene. The pressure in the pipeline at the point of the leak is 100 psig. Determine the leakage rate. The specific gravity of toluene is 0.866.

Answer :

Final answer:

To determine the leakage rate of the toluene from the 0.20 inch hole in the pipeline, we can use Bernoulli's equation and the principle of continuity. The leakage rate is calculated by multiplying the area of the hole by the velocity at the point of the leak.

Explanation:

To determine the leakage rate of the toluene from the 0.20 inch hole in the pipeline, we can use Bernoulli's equation and the principle of continuity. Bernoulli's equation states that the total pressure at any point inside a fluid is equal to the sum of the static pressure, the dynamic pressure, and the potential energy per unit volume of the fluid. The principle of continuity states that, for an incompressible fluid, the mass flow rate is constant along a streamline.

Using these principles, we can calculate the leakage rate as follows:

  • First, we need to convert the 0.20 inch hole diameter to meters. 1 inch is equal to 0.0254 meters, so the hole diameter is equal to 0.20 * 0.0254 = 0.00508 meters.
  • Next, we can calculate the velocity of the toluene at the point of the leak using the principle of continuity. Since the mass flow rate is constant, we can assume that the velocity of the toluene at the point of the leak is equal to the velocity of the toluene in the pipeline. We can calculate the velocity using the equation A1v1 = A2v2, where A1 is the area of the hole and v1 is the velocity of the toluene at the point of the leak. Rearranging the equation, we get v1 = (A2v2) / A1. Plugging in the values, we get v1 = (π * (0.00508/2)² * 100) / (π * (0.0254/2)²), which simplifies to v1 = 1000 m/s.
  • Finally, we can calculate the leakage rate by multiplying the area of the hole by the velocity at the point of the leak. The area of the hole is equal to pi * (0.00508/2)², and the leakage rate is equal to (π * (0.00508/2)²) * 1000.

Therefore, the leakage rate of toluene from the 0.20 inch hole in the pipeline is (pi * (0.00508/2)²) * 1000 meters cubed per second.

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