A large storage tank is filled with fuel oil (density = 900 kg/m^3) to a height

Question

A large storage tank is filled with fuel oil (density = 900 kg/m^3) to a height of H = 100 m above the ground. A horizontal outlet pipe 10 cm in diameter is attached to the tank at a height h = 5 m above the ground (see drawing).

A) How much time would be required to to fill a tanker truck with a capacity of 30,000 L from this pipe? Assume that only gravity is used to pump the oil out of the tank, and that the volume of the tank is great enough that the surface level of oil does not change significantly.

B) A small catchment pond is located a horizontal distance x = 30 m from the base of the storage tank (underneath wehere the pipe joins the tank), up a slight incline so the edge of the pond is y = 1.5 m above the base of the storage tank (see drawing). If the pipe were to accidentally break off of the tank, would the resulting stream of outflowing oil reach the pond. Back up your answer with numbers- no credit just for guessing the answer!

Explanation / Answer

a) This is an exercise in Torricelli's result. It states that the exit velocity of fluid from a large tank like that with a small hole punched in the side is

(v = sqrt{2gD})

where D is the depth. Since the tank is 100m tall, and the pipe is 5m off the ground, the depth is 95m:

(v = sqrt{2*9.81m/s^2*95m} = 43.17 m/s)

Now, the flow rate of water in a tube is defined as

(f = Av = pi r^2v = pi(0.1m)^2(43.17m/s) = 1.36 m^3/s)

Now, 1m^3 = 1000L, so f = 1360L/s. How long will it take to fill a tanker to 30,000L?

(t = V/f = (30,000L)/(1360L/s) = 22.0s)

b) The exit velocity of the oil from the tank is completely horizontal (initially). How long would it take to fall from y_0 = 5m to y = 1.5m?

(y = y_0 + v_{0y}t -(1/2)gt^2 = y_0 - (1/2)gt^2)

--> (2(y - y_0) = -gt^2)

--> (t^2 = -2(y-y_0)/g)

--> (t = sqrt{-2(y-y_0)/g} = sqrt{2(1.5m-5m)/(9.81m/s^2)} = 0.845s)

Now, how far will it get?

(D = v_{0x}t = (43.17m/s)(0.845s) = 36.47 m)

Since the pond is only 30m away, yes, the oil will fall into the pond.

Hope this helps.

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