Water flows at 0.58 m/s through a pipe 11 cm in diameter when the pressure is 450,000 Pa. Determine the water velocity & the pressure in the pipe if it shrinks down to 5 cm in diameter. You can assume that the pipe is horizontal & ρwater = 1000 kg/m3.
Law of mass continuity first:
area*velocity*density=mass flow
PI(11/2)^2*.58m/s*1000kg/m^3= PI*C kg/sec you calculate C.
then on the smaller pipe
C*PI=PI*(5/2)^2*Vs*1000kg/m^3 solve for Vs the outflow.
Now pressure, Write out Bernoullis equation, and solve for P2 on the outlet.
To determine the water velocity and the pressure in the pipe when it shrinks down to 5 cm in diameter, we can use the principle of mass conservation and Bernoulli's equation.
1. First, let's calculate the cross-sectional area of the pipe at each diameter:
- Cross-sectional area of a pipe is given by the formula: A = πr^2, where r is the radius of the pipe.
For the 11 cm diameter pipe:
- Radius (r) = diameter / 2 = 11 cm / 2 = 5.5 cm = 0.055 m
- Cross-sectional area (A) = π(0.055 m)^2
For the 5 cm diameter pipe:
- Radius (r) = diameter / 2 = 5 cm / 2 = 2.5 cm = 0.025 m
- Cross-sectional area (A) = π(0.025 m)^2
2. Now, let's calculate the water velocity in the 11 cm diameter pipe using the given flow rate:
- Flow rate (Q) = water velocity (v) × cross-sectional area (A)
The flow rate is not directly given, but we are told that water flows at 0.58 m/s through the pipe. Since the flow is presumed to be steady, we can calculate the flow rate as:
- Flow rate (Q) = π(0.055 m)^2 × 0.58 m/s
3. Now that we have the flow rate in the 11 cm diameter pipe, let's find the water velocity in the 5 cm diameter pipe.
We can assume that the mass flow rate remains constant, as the water cannot be created or lost along the pipe:
- Mass flow rate = ρ × Q
The density of water (ρ) is given as 1000 kg/m^3.
4. Now let's find the pressure in the 11 cm diameter pipe using Bernoulli's equation:
- Bernoulli's equation: P₁ + 1/2ρv₁² + ρgh₁ = P₂ + 1/2ρv₂² + ρgh₂
Since the pipe is horizontal, the vertical heights (h₁ and h₂) can be neglected, and the equation becomes:
- P₁ + 1/2ρv₁² = P₂ + 1/2ρv₂²
Since the pressure (P₂) in the 5 cm diameter pipe is not known, we'll leave it as an unknown, and solve for the velocity (v₂).
Now, let's summarize the steps to find the water velocity and the pressure when the pipe shrinks down to a 5 cm diameter:
1. Calculate the cross-sectional areas of the 11 cm and 5 cm diameter pipes.
2. Use the flow rate formula to find the water velocity in the 11 cm diameter pipe.
3. Calculate the mass flow rate using the density of water and the calculated flow rate.
4. Apply Bernoulli's equation to find the water velocity in the 5 cm diameter pipe.
5. Solve the equation to find the pressure in the 5 cm diameter pipe, using the known pressure in the 11 cm diameter pipe and the calculated velocities.
By following these steps, you can determine the water velocity and the pressure in the pipe when it shrinks down to a 5 cm diameter.