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Water Jet Reaction Force

  1. Oct 21, 2015 #1
    Hi all,

    my first post on here and just wanted to check something I'm working on for a project.

    The project is to use a pump to create a jet of water to manouvre small boat.

    I have found a formula to work out the jet reaction force in a fire fighting text book which gives:

    R = 0.157 * P * d^2 where R = Reaction force in Newtons
    P = Pressure in bar
    d = nozzle diameter in mm

    To avoid buying and testing different pumps and nozzles empirically to find the best reaction force I wanted to link a given pumps flow and pressure to find nozzle diameter and then using this and the pressure and the above formula to get the reaction force.

    I have been using

    L = 2/3 * d^2 * sqrt P where L = flow l/min
    d = nozzle diameter in mm
    P = pressure in bar

    re-arranged to give

    d = sqrt (L/ (2/3 * sqrt P))

    this gives d in mm then putting this and the same pressure back into the above reaction formula to get reaction in Newtons.

    as an example:

    a pump giving P = 1.52 bar
    Q = 450 l/min

    d = sqrt (450/ (2/3 * sqrt 1.52))
    = 23.4 mm


    R = 0.157 * P * d^2
    = 0.157 *1.52 *23.4^2
    =130 N


    Are the formulas i'm using valid? If so could someone show me how to get to them from first principles or just tell me they're ok :) If not then why?

    Other losses: I have thought about losses due to pipe work friction but if the nozzle is underwater what sort of losses could I expect due to reduced flow because of higher pressure at/just after outlet

    Thanks in advance for any help.
  2. jcsd
  3. Oct 21, 2015 #2

    Andrew Mason

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    Science Advisor
    Homework Helper

    Welcome to PF acebaraka!

    You could try using: F = dp/dt = d(mv)/dt = v(dm/dt) + m(dv/dt). If the speed of the water jet is constant (ie. dv/dt=0), the force is given by f = v(dm/dt) where v is the speed of the water exiting the nozzle and dm/dt is the mass flow rate out of the nozzle. The mass flow rate is the volume flow rate x mass/unit volume of water (1 kg/l).

    You can determine the speed of the water using Bernoulli's principle: ##\Delta \frac{1}{2}\rho v^2 = -\Delta P## (the change in kinetic energy per unit volume is equal and opposite to the change in pressure).

  4. Oct 21, 2015 #3
    Thanks for your reply, do I have this right ?

    delta P would be:

    system pressure at or just before the nozzle - (for the sake of ease at the moment) atmospheric pressure

    which would be 1.52bar but in SI

    so SqRt (P / (0.5*density)) would give v

    v = SqRt (152000 / (0.5*1000))
    =17.43 m/s


    F= v * (dm/dt) for 450l/min (dm/dt) = 450/60 = 7.5 kg/s
    = 17.43 * 7.5
    = 130N

    which rather wonderfully comes out to the same as what I had with the other equation, to good to be true? :)
  5. Oct 21, 2015 #4

    Andrew Mason

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    Science Advisor
    Homework Helper

    Not at all. I expect that the formulas you had were derived the same way.

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