First of all, what is the correct equation to determine the thrust produced by a water rocket?
$$
F_T = \dot{m} V_e \tag{1}
$$
$$
F_T = \dot{m} V_e + (p_e – p_0)A_e \tag{2}
$$
Using the first equation it should be possible to increase the thrust by using a convergent nozzle because the water's exhaust velocity $V_e$ increases while the mass flow rate $ṁ$ out of the rocket is constant.
The second equation additionally incorporates the cross sectional area of the exhaust as well as the difference between the exhaust and ambient pressure.
Given the mass flow rate out of the rocket is unchanged, can a nozzle change the thrust of a water rocket?
Best Answer
prerequisite
Thrust produced by a nozzle can be given by
$$F_T = \dot{m} V_e + (p_e - p_0)A_e$$
Thrust component in a nozzle can be split into two component that is pressure thrust ($(p_e - p_0)A_e$) and momentum thrust ($\dot{m} V_e$). In most of the nozzles we try to achieve exit pressure equal to ambient pressure, this phenomena is called fully expanded nozzle. But some time, this may not be possible in converging-diverging nozzle, if that is not operating in designed mach number so pressure thrust will be added or subtracted from momentum thrust. Momentum thrust is higher than pressure thrust for a given pressure, so we always try to achieve thrust by momentum thrust. Pressure thrust is because of design problem or limitations.
Kind of thrust in Converging nozzle and Converging -Diverging nozzle: Converging nozzle always try to achieve ambient pressure at exit after reaching equilibrium state. Time to reach this equilibrium state is negligible for most of the calculations. Because pressure waves travel at the speed of sound, so time taken by this pressure waves to tell the exit pressure information to the pressure chamber is very less because length of the nozzle is very small compared to speed of sound. In C-D nozzle exit pressure not necessarily equal to atmospheric pressure because characteristic waves do not tell that information to upstream of flow due to hyperbolic nature of the flow, its completely depends on area ratio after reaching sonic velocity.
Answer:
Most of the water rockets up-to my knowledge used converging nozzle, since converging nozzle expands to ambient pressure (explained earlier) there is no pressure thrust . You can use any equation 1 or 2, both of them will give same result for converging nozzle. In-fact equation 1 is sub-case of equation 2!