Question #233460

The tank of volume 0.25 m^3 and height of 1m has water flowing at 0.005 m^3/min the outlet flow rate is governed by the linear relation Fout-0.1h, where h is the height of the water in the tank in m and Fout is the outlet flow rate in m^3/min. The inlet flow rate. changes suddenly from its nominal value of 0.05 m^3 min to 0.15 m^3/min and remains there. Find the time in minutes at which tank overflows.

Expert's answer

Given that:

The tank of volume 0.25 m^3

Height = 1m

Velocity= 0.005 m^3/min

Linear ratio: Fout-0.1h

Where h is height in meters

Fout is the outlet flow rate in m^3/min

Norminal Change Value: 0.05 m^3 min to 0.15 m^3/min

We can use Bernoulli's theorem:

v=√2gh

Let dh represents decrease in water level during infinitesimally small time interval dt when the water level stands at height h. Therefore,

the rate of decrease in volume of water will be −Adhdt.

Again the rate of flow of water at this moment through the orifice is given by v×a=a√2gh.

These two rates must be same by the *principle of cotinuity*.

Hence a√2gh=−Adhdt

⇒dt=−Aa√2g⋅h−12dh

If the tank is filled to the brim then height of water level will be H and time required to overflow the tank T can be obtained by integrating the above relation.

T=∫T0dt=−Aa√2g⋅∫0Hh−12dh

T=−Aa√2g[h1212]0H

T=(Aa⋅√2g)H12

The time taken for the tank to be full, therefore, is given by:

T'=(Aa⋅√2g)(H2)12

So T'T=1√2

T'=T√2

Substituted with the figures provided, we get time taken for the tank to overflow is **3.2** minutes (192 seconds)

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