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Answer :
Final answer:
The increase in pressure from the smaller pipe to the larger pipe can be calculated by applying the principle of continuity and Bernoulli's equation. The principle of continuity states that the product of the cross-sectional area and the velocity of the fluid remains constant along the flow. Bernoulli's equation relates the pressure and velocity of the fluid. By solving these equations simultaneously, we can find the increase in pressure.
Explanation:
To find the increase in pressure from the smaller pipe to the larger pipe, we can apply the principle of continuity and Bernoulli's equation.
The principle of continuity states that the product of the cross-sectional area and the velocity of the fluid remains constant along the flow. Mathematically, this can be expressed as:
A1V1 = A2V2
Where A1 and A2 are the cross-sectional areas of the smaller and larger pipes respectively, and V1 and V2 are the velocities of the fluid in the smaller and larger pipes respectively.
Now, let's use Bernoulli's equation to relate the pressure and velocity of the fluid. Bernoulli's equation states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
Mathematically, Bernoulli's equation can be expressed as:
P1 + 0.5ρV12 + ρgh1 = P2 + 0.5ρV22 + ρgh2
Where P1 and P2 are the pressures in the smaller and larger pipes respectively, ρ is the density of the fluid, V1 and V2 are the velocities of the fluid in the smaller and larger pipes respectively, g is the acceleration due to gravity, and h1 and h2 are the heights of the fluid in the smaller and larger pipes respectively.
Since the flow is assumed to be incompressible and inviscid, we can neglect the potential energy terms (ρgh) in Bernoulli's equation. Therefore, the equation simplifies to:
P1 + 0.5ρV12 = P2 + 0.5ρV22
Now, we can solve these equations simultaneously to find the increase in pressure (P2 - P1) from the smaller pipe to the larger pipe.
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