You need to siphon water from a clogged sink. The sink has an area of 0.36 and is filled to a height of 4.0 . Your siphon tube rises 45 above the bottom of the sink and then descends 85 to a pail as shown in the figure. The siphon tube has a diameter of 2.4
A)Assuming that the water level in the sink has almost zero velocity, estimate the water velocity when it enters the pail.
B)Estimate how long it will take to empty the sink
sorry, the units didn't show up. it's .36 m^2, 4.0cm, 45cm, 85cm, 2.4cm
To estimate the water velocity when it enters the pail, we can use the principle of conservation of energy. We need to consider the potential energy and kinetic energy of the water.
A) First, let's calculate the potential energy of the water at the surface of the sink. The potential energy is given by the formula:
Potential Energy = mass * gravitational acceleration * height
Given that the area of the sink is 0.36 m², and the height of the water is 4.0 m, we can calculate the mass of the water using the formula:
Mass = density * volume
The density of water is approximately 1000 kg/m³. The volume of water in the sink is given by:
Volume = area * height
Now, let's calculate the potential energy using the formula mentioned earlier.
Potential Energy = mass * gravitational acceleration * height
Next, in order to calculate the kinetic energy of the water at the bottom of the siphon tube, we use the formula:
Kinetic Energy = 0.5 * mass * velocity²
Since the potential energy is converted to kinetic energy, we can equate the two.
Potential Energy = Kinetic Energy
mass * gravitational acceleration * height = 0.5 * mass * velocity²
We can rearrange the equation and solve for velocity:
velocity = √(2 * gravitational acceleration * height)
Substituting the values of gravitational acceleration (approximately 9.8 m/s²) and height (45 m), we can calculate the velocity.
B) To estimate the time it will take to empty the sink, we need to calculate the flow rate of water.
The flow rate can be calculated using the equation:
Flow Rate = cross-sectional area * velocity
Since the cross-sectional area is given by the formula:
Cross-sectional Area = (π * diameter²)/4
Substituting the values of diameter (2.4 m), and the velocity calculated in part A, we can calculate the flow rate.
Finally, to estimate the time it will take to empty the sink, we divide the volume of water in the sink by the flow rate.
Volume = area * height (which we calculated earlier)
Finally, we divide the calculated volume by the flow rate to get the estimated time to empty the sink.
These calculations can be done using units consistent with the given values to obtain accurate results.
water .04 m deep.
volume = .04*.36 = .0144 m^3
bottom of sink to pail height difference is .85-.45 = .4 m
We will have to assume no friction in the pipe or entrance or exit losses unless you know those . Thefore use bernoulli
p + rho g * change in h + (1/2) rho v ^2 = constant
p is air pressure, v at start = 0 so
rho (9.81) (.4) = .5 rho v^2
v^2 = 7.848
v = 2.8 m/s
then
radius = .024/2 = .012 m
pi r^2 = .000452 m^2
so
volume flow rate = 2.8*.000452 = .001267 m^3/s
volume of water = .0144 so
time = .0144/.001267 = 11.36 seconds