Question

Estimate the speed of the water free surface and the time required to fill with water a cone-shaped container 1.5 m high and 1.5m across at the top if the filling rate is 75.7 L/min.

Answer 1

Answer:

speed of water is 0.0007138m/s

Explanation:

From the law of conservation of mass

Rate of mass accumulation inside vessel = mass flow in - mass flow out

so, dm/dt = mass flow in - mass flow out

taking p as density

$d \frac{dQ}{dt} = pq_i_n$

where,

q(in) is the volume flow rate coming in

Q = is the volume of liquid inside tank at any time

But,

dQ = Adh

where ,

A = area of liquid surface at time t

h = height from bottom at time t

A = πr²

r is the radius of liquid surface

$r = (1.5/2) \div 1.5$ $h = \frac{h}{2}$

Hence,

$\pi( \frac{h}{2} )^2\frac{dh}{dt} =q_i_n$

$\frac{dh}{dt} = \frac{q_i_n}{\pi (\frac{h}{2})^2 } =\frac{4q_i_n}{\pi h^2}$

so, the speed of water surface at height h

$v = \frac{dh}{dt} =\frac{4q_i_n}{\pi h^2}$

where,

$q_i_n$ is 75.7 L/min = 0.0757m³/min

h = 1.5m

so,

$v = \frac{4 \times 0.0757}{\pi \times 1.5^2} \\\\v = 0.04283m/min$

v = 0.04283 /60

v = 0.0007138m/s

Hence, speed of water is 0.0007138m/s