Question

Water, in a 100-mm-diameter jet with speed of 30 m/s to the right, is deflected by a cone that moves to the left at 14 m/s. Determine (a) the thickness of the jet sheet at a radius of 230 mm. and (b) the external horizontal force needed to m

Answer 1

Step-by-step explanation:

The velocity at the inlet and exit of the control volume are same
`V_i=V_e=V`

Calculate the inlet and exit velocity of water jet

`V=V_j+V_e\\\\V=30+14\\\\V=44m/s`

The conservation of mass equation of steady flow

`\sum ^e_i\bar V. \bar A=0\\\\(-V_iA_i+V_eA_e)=0`

`A_i\ \texttt {is the inlet area of the jet}`

`A_e\ \texttt {is the exit area of the jet}`

since inlet and exit velocity of water jet are equal so the inlet and exit cross section area of the jet is equal

The expression for thickness of the jet

`A_i=A_e\\\\(\pi)/(4) D_j^2=2\pi Rt\\\\t=(D^2_j)/(8R)`

R is the radius

t is the thickness of the jet

D_j is the diameter of the inlet jet

`t=((100*10^(-3))^2)/(8(230*10^(-3)) \\\\=5.434mm`

(b)

`R-x=\rho(AV_r)[-(V_i)+(V_c)\cos 60^o]\\\\=\rho(V_j+V_c)A[-(V_i+V_c)+(V_i+V_c)\cos 60^o]\\\\=\rho(V_j+V_c)((\pi)/(4)D_j^2 )[V_i+V_c](\cos60^o-1)]`

`1000kg/m^3=\rho\\\\44m/s=(V_j+V+c)\\\\100*10^(-3)m=D_j`

`R_x=[1000*(44)(\pi)/(4) (10*10^(-3))^2[(44)(\cos60^o-1)]]\\\\=-7603N`

The negative sign indicate that the direction of the force will be in opposite direction of our assumption

Therefore, the horizontal force is -7603N