Answer:
F_y = 151319.01N = 15.132 KN
Explanation:
From the linear momentum equation theory, since flow is steady, the y components would be;
-V1•ρ1•V1•A1 + V2•ρ2•V2•A2 = P1•A1 - P2•A2 - F_y
We are given;
Length; L = 5ft = 1.52.
Initial diameter;d1 = 12in = 0.3m
Exit diameter; d2 = 24 in = 0.6m
Volume flow rate of water; Q2 = 10 ft³/s = 0.28 m³/s
Initial pressure;p1 = 30 psi = 206843 pa
Thus,
initial Area;A1 = π•d1²/4 = π•0.3²/4 = 0.07 m²
Exit area;A2 = π•d2²/4 = π•0.6²/4 = 0.28m²
Now, we know that volume flow rate of water is given by; Q = A•V
Thus,
At exit, Q2 = A2•V2
So, 0.28 = 0.28•V2
So,V2 = 1 m/s
When flow is incompressible, we often say that ;
Initial mass flow rate = exit mass flow rate.
Thus,
ρ1 = ρ2 = 1000 kg/m³
Density of water is 1000 kg/m³
And A1•V1 = A2•V2
So, V1 = A2•V2/A1
So, V1 = 0.28 x 1/0.07
V1 = 4 m/s
So, from initial equation of y components;
-V1•ρ1•V1•A1 + V2•ρ2•V2•A2 = P1•A1 - P2•A2 - F_y
Where F_y is vertical force of enlargement pressure and P2 = 0
Thus, making F_y the subject;
F_y = P1•A1 + V1•ρ1•V1•A1 - V2•ρ2•V2•A2
Plugging in the relevant values to get;
F_y = (206843 x 0.07) + (1² x 1000 x 0.07) - (4² x 1000 x 0.28)
F_y = 151319.01N = 15.132 KN
