Water flows into a cylindrical tank through pipe 1 at a rate of 25 ft/s and leaves through pipe 2 at 10 ft/s and through pipe 3 at 12 ft/s. Pipe 4 is an open air vent. Internal diameters of the pipes are: D1=3 in, D2= 2 in, D3=2.5 in, D4=2 in. Calculate a) dh/dt and b) the average velocity of the air escaping through pipe 4 assuming incompressible flow. (Tank's diameter: 2 ft)
find dV/dt out of the exit pipe. (if you are not in calculus,dV/dt is the flow rate of the water out). That will equal the flow rate of the air coming in, then convert that flow rate to velocity.
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Jiskha wouldn't let me post the url so I added spaces but this site has the answer
The tank shown below is 1/2 of water.then a spherical wrecking ball with radius of 2 ft is dropped in the tank.how far Will the water level rise once the sphere is completely submerged?
To calculate the values of dh/dt (rate of change of the water height) and the average velocity of the air escaping through pipe 4, we can apply the principle of conservation of mass and Bernoulli's equation to analyze the fluid flow in the system.
Let's start by calculating the cross-sectional areas of the pipes:
A1 = π * (D1/2)^2
A2 = π * (D2/2)^2
A3 = π * (D3/2)^2
A4 = π * (D4/2)^2
where π is a constant (approximately equal to 3.14159).
a) Now, let's find the rate of change of the water height (dh/dt):
The rate of change of the water height in the tank is equal to the difference between the inflow and outflow rates. Therefore, we need to calculate the volume flow rates for pipes 1, 2, and 3.
Q1 = A1 * v1
Q2 = A2 * v2
Q3 = A3 * v3
where v1, v2, and v3 represent the velocities of water flow through pipes 1, 2, and 3, respectively.
The volume flow rate for pipe 4 (Q4) is not needed for calculating dh/dt. It is only used for calculating the average velocity of the air escaping through pipe 4.
Now, we can find dh/dt by considering the principle of conservation of mass:
Q1 - (Q2 + Q3) = A * (dh/dt)
where A is the cross-sectional area of the tank.
Substituting the values, we have:
π * (D1/2)^2 * v1 - (π * (D2/2)^2 * v2 + π * (D3/2)^2 * v3) = π * (D/2)^2 * (dh/dt)
Now, substitute the given values:
π * (3/2)^2 * 25 - (π * (2/2)^2 * 10 + π * (2.5/2)^2 * 12) = π * (2/2)^2 * (dh/dt)
Simplifying the equation will give us the value of dh/dt.
b) To find the average velocity of the air escaping through pipe 4, we need to calculate the volume flow rate for pipe 4 (Q4) and then divide it by the cross-sectional area of pipe 4.
Q4 = A4 * v4
Substituting the values:
Q4 = π * (2/2)^2 * v4
To calculate the average velocity, we divide Q4 by the cross-sectional area of pipe 4:
Average velocity = Q4 / A4
Substituting the values, we can find the average velocity of the air escaping through pipe 4.