Hydrodynamic force on bend-nozzle assembly

Given: pipe $d_1 = 0.10$ m, nozzle exit $d_2 = 0.05$ m, $Q = 2400$ L/min $= 0.04$ m³/s, volume of assembly $= 18.2$ L $= 0.0182$ m³, head loss in bend $0.5V^2/2g$ and in nozzle $2V^2/2g$ ($V$ = velocity in the pipe).

Assumed from the figure: water flows upward in the vertical pipe, passes a bend and leaves through a horizontal nozzle into the atmosphere, 5 m above the inlet section.

   exit ====>  (nozzle, d = 5 cm)
   |
   |  5 m
   |
   ^  inlet 1 (pipe d = 10 cm, flow up)

Velocities

$$\begin{aligned} A_1 &= 7.854\times 10^{-3}\ \text{m}^2,\quad A_2 = 1.9635\times 10^{-3}\ \text{m}^2\\ V_1 &= 5.093\ \text{m/s},\quad V_2 = 20.372\ \text{m/s} \end{aligned}$$

Inlet pressure (Bernoulli with losses)

$$\frac{p_1}{\gamma} + \frac{V_1^2}{2g} = 0 + \frac{V_2^2}{2g} + 5 + (0.5 + 2)\frac{V_1^2}{2g}$$ $$\begin{aligned} \frac{p_1}{\gamma} &= 21.15 + 5 + 2.5(1.322) - 1.322 = 28.13\ \text{m}\\ p_1 &= 276\,009\ \text{Pa} \approx 276\ \text{kPa (gauge)} \end{aligned}$$

(using $V_2^2/2g = 21.15$ m, $V_1^2/2g = 1.322$ m).

Momentum equation

• $p_1A_1 = 276\,009\times 7.854\times 10^{-3} = 2168\ \text{N}$ (up, on the inlet face)
• $W = 9810\times 0.0182 = 178.5\ \text{N}$ (down)
• $\rho Q = 40$ kg/s. Inlet velocity $= (0, +V_1)$; exit velocity $= (+V_2, 0)$.

Forces on the fluid ($R$ = force of the assembly on the fluid):

$$\begin{aligned} x:\ \ R_x &= \rho Q\,V_2 = 40\times 20.372 = 815\ \text{N}\\ y:\ \ p_1A_1 - W + R_y &= \rho Q\,(0 - V_1)\\ 2168 - 178.5 + R_y &= -203.7\\ R_y &= -2193\ \text{N} \end{aligned}$$

The hydrodynamic force exerted by the water on the assembly is the opposite of $R$:

$$F_x = -815\ \text{N},\qquad F_y = +2193\ \text{N}$$ $$F = \sqrt{815^2 + 2193^2} = 2339\ \text{N},\qquad \theta = 110.4^\circ\ \text{from the +x direction (the direction of the exit jet)}$$

Answer: the hydrodynamic force on the system $\approx 2.34$ kN, directed upward and backwards (against the jet), at $69.6^\circ$ above the horizontal, with components 0.815 kN opposite to the jet and 2.19 kN upward.