Answer:
The mass flow rate of water from the tank can be found using the Bernoulli's equation which states that the total energy of a fluid remains constant along a streamline. Applying Bernoulli's equation between the surface of the water in the tank and the orifice at the bottom, neglecting the height difference, we get:
P/ρ + gh + 1/2 * V^2 = P₂/ρ + 1/2 * V₂^2
where P is the pressure at the surface of the water in the tank, ρ is the density of water, g is the acceleration due to gravity, h is the height of the water surface above the orifice, V is the velocity of water at the surface of the water in the tank, P₂ is the pressure at the orifice, and V₂ is the velocity of water at the orifice. Since the orifice is at the bottom of the tank, h is zero, and V₂ is the velocity of water discharging from the orifice, which can be calculated using the continuity equation:
A₁V = A₂V₂
where A₁ is the cross-sectional area of the tank, and V is the velocity of water at the surface of the water in the tank.
Combining these equations and solving for the mass flow rate, we get:
m_dot = A₂ * sqrt(2 * ρ * (P - P₂))
Now, if a pipe of area A₂ and length H/2 is attached to the orifice, the velocity of water at the end of the pipe will be different than the velocity of water discharging from the orifice. We can use the Bernoulli's equation again between the orifice and the end of the pipe to calculate the velocity of water at the end of the pipe:
P₂/ρ + 1/2 * V₂^2 = P₃/ρ + gh + 1/2 * V₃^2
where P₃ is the pressure at the end of the pipe, and V₃ is the velocity of water at the end of the pipe. Again, neglecting the height difference, we get:
P₂/ρ + 1/2 * V₂^2 = P₃/ρ + 1/2 * V₃^2
Since the pipe is attached to the orifice, the pressure at the end of the pipe is atmospheric pressure Po, and the velocity of water at the end of the pipe can be calculated using the continuity equation:
A₂V₂ = A₃V₃
where A₃ is the cross-sectional area of the pipe.
Combining these equations and solving for the increase in mass flow rate, we get:
Δm_dot = A₃ * sqrt(2 * ρ * (P₂ - Po))
To find the pressure at point C for the two cases, we need to apply the Bernoulli's equation between the surface of the water in the tank and point C. Neglecting the height difference again, we get:
P/ρ + 1/2 * V^2 = Pc/ρ + 1/2 * Vc^2
where Pc is the pressure at point C, and Vc is the velocity of water at point C. For the first case, where the pipe is not attached to the orifice, we can assume that the velocity of water at point C is negligible, i.e., Vc = 0. Solving for Pc, we get:
Pc = P - 1/2 * ρ * V^2
For the second case, where the pipe is attached to the orifice, we can assume that the velocity of water at point C