Bernoulli's Equation — Question 9

Question 9

Jet impact needs momentum after Bernoulli supplies the speed

A large open reservoir feeds a small nozzle $h=1.8\,\mathrm{m}$ below its free surface. The nozzle produces a horizontal water jet of area $a=2.0\,\mathrm{cm^2}$ that strikes a stationary plate normally. The water then leaves sideways with zero velocity component in the incoming direction. Neglect nozzle losses, air drag and height change between nozzle and plate. Use $\rho=1000\,\mathrm{kg/m^3}$ and $g=10\,\mathrm{m/s^2}$.

1. Find the jet speed, mass flow and force exerted on the plate.

2. Compare that force with $\tfrac12\rho v^2a$. Why is multiplying stagnation pressure by the incoming jet area not the correct force calculation here?

Question 9 - Solution

Jet impact needs momentum after Bernoulli supplies the speed

(a) Derivation and calculation

$$\boxed{v=\sqrt{2gh}=\sqrt{2(10)(1.80)}=6.00\,\mathrm{m/s}}.$$$$\boxed{\dot m=\rho av=(1000)(2.0\times10^{-4})(6.00)=1.20\,\mathrm{kg/s}}.$$$$F_{\mathrm{plate\ on\ water,x}}=\dot m(0-v)\Rightarrow\boxed{F_{\mathrm{water\ on\ plate}}=\dot mv=7.20\,\mathrm N}.$$

(b) Result and interpretation

$$\tfrac12\rho v^2a=\tfrac12(1000)(36)(.000200)=3.60\,\mathrm N=F/2.$$$$F=\int_{\mathrm{wetted\ plate}}(p-p_{\mathrm{atm}})\,dA=\rho av^2.$$Stagnation pressure is a local maximum, not a uniform pressure over only the incoming jet area. The actual distribution and spreading region must satisfy momentum balance.

Physical check

Bernoulli determines the approach speed; the change in momentum determines the plate force.