1 A garden hose is used to fill a bucket of water as shown i

1. A garden hose is used to fill a bucket of water, as shown in the figure. The mass flow rate of water leaving the hose and entering the bucket is constant at: m =3 Ibm/s . The initial height of water in the bucket Is: h0, = 4 in. Answer the following: (1) By applying the conservation of mass equation, find the rate at which the water level rises in the bucket: dh/dt dr (2) How long will it take for the water to fill the bucket to the top? (3) Assuming the water to be incompressible (p, = 62.4 Ibm/ft^3 ), calculate the average speed at which the water leaves the hose at its exit.

Solution

1)

m = density*volume

= rho*Area*heigh

= rho*A*h

bottom area of bucket = pi*d^2/4

= pi*10^2/4

= 78.54 inch^2

= 78.58 (1/12 ft)^2 (since 1ft = 12 inch)

= 0.5454 ft^2

rate of increases of dm/dt = 3 lbm/s

d(rho*A*h)/dt = 3

rho*A*dh/dt = 3

dh/dt = 3/(rho*A)

= 3/(62.4*0.5454)

= 0.088 ft/s

= 1.058 inch/s

2) time taken to fill the bucket, t = remaining height/(dh/dt)

= (12 - 4)/1.058

= 7.56 s

3) cross sectional area of pipe, A = pi*d^2/4

= pi*1.2^2/4

= 1.13 inch^2

= 1.13 (1/12 ft)^2

= 0.00785 ft^2

mass flow rate, dm/dt = rho*A*v (here v is speed of water)

==> v = (dm/dt)/(rho*A)

= 3/(62.4*0.00785)

= 6.12 ft/s