# Underwater Jet Dispersal Rates

A long lance 38mm ID mounted vertically into a pool the last 300mm of the lance is bent 45 deg, so the end of the lance is 305mm above the pool floor.

It is supplied by a submersible pump that is 1m below the pool surface and provides 3.5m/s and 4.25 l/s

I'm trying to figure out how far the water jet will travel in the water, before losing all its momentum

i.e. How far would the water disperse the larger mass of the pool.

The lances function is to disturb and plume a silt like layer of material that is approx 80% dead plant algii

Pond 95 by 35 m by 5m deep

What would the result of doubling the flow to say 9 l/s?

Would fitting a nozzle or several small nozzles to the lance head improve the dispersal rate?

Any other optioneering thoughts would be most welcome? Would an educator be a better method for example?

Also if you could explain your workings as much as possible.

Â© BrainMass Inc. brainmass.com December 24, 2021, 11:39 pm ad1c9bdddfhttps://brainmass.com/physics/hydrodynamics/underwater-jet-dispersal-rates-589551

## SOLUTION This solution is **FREE** courtesy of BrainMass!

Here (below) are the governing equations, from which you will be able to re-arrange the variables (by simple algebra) and solve for items that you are interested in changing or evaluating.

Please carefully consider that your plume of silt/algii is the primary body that is "reacting" the momentum of the volume of fluid that exits the lance. Therefore, by "F=m*a", this is a transient problem. In other words, the increment of time over which these two different fluid volumes interact is an important component for your understanding.

So, you will want to treat your plume as stationary, and your lance fluid volume as being in flux.

And, you will need to take some educated guesses about the effective mass properties of your plume at the depth/elevation of the outlet of the lance.

Conservation of Momentum:

In general-

sum of all external forces acting on the control volume of fluid, which is exiting your lance,

F_ext = m * a

where

m = mass of the fluid volume

a = acceleration of the fluid volume

or

F_ext = m * (d/dt * v)

where

v = velocity of fluid volume

m = (rho)*(dVol)

rho = fluid density

rho = (gamma)/(g_n)

gamma = specific weight [N/m^3]

g_n = acceleration due to gravity [9.81 m/s^2]

dVol = differential volume of fluid <-- a constant value for your regime, with incompressible (ie. non-gaseous) fluid

V = dVol

so, for your incompressible fluid volume:

F_ext = (v_out - v_in)*Q*rho

where

v_out = exit velocity, at submerged end of lance

v_in = velocity of fluid at entrance to lance (exit of pump)

Q = volume flow rate [m^3/s]

- and -

by conservation of mass for your fluid, from the inlet to the outlet of the lance,

Q = A_in * v_in = A_out * v_out

^--- so, if you know the cross section area (A) of the inlet and outlet, you can know the respective datum velocity [m/s]

- now-

to determine how the input energy of the pump influences the inherent momentum of your fluid volume, recall the terms of Bernoulli's equation

[ z_in + (P_in/gamma) + (v_in)^2/g_n ] = [ z_out + (P_out/gamma) + (v_out)^2/g_n ] + h_L + h_m

where

z = the local datum height, or elevation [m]

P = pressure [Pa]

v = velocity = Q/A

h_L = "head loss" due to pipe friction

h_m = "head loss" due to the pump <--- can be a negative quantity!

...in your case, "h_L" may be negligible compared to the other variables

however, here is the Darcy-Weisbach equation:

h_L = f * (l/d) * [(v_in)^2 - (v_out)^2]/g_n

where

f = pipe roughness factor

l = linear length of pipe [m]

d = diameter of pipe [m]

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Sep 04, 2014 10:47 am

From You

Hello Peter

I attempted to reply to you last evening (for me), about 10-1/2 hours ago.

However, the BrainMass site wasn't loading into any web browser.

I hope that my reply, now, is still timely for your use.

(Where, in Europe?, are you located?)

There are a couple of ways you could estimate (low/high) dispersal distances of the lance fluid jetting off into the plume. Neither of the methods is exact. They are only going to be bounding approximations.

You must try to quantify each of the following fluid characteristics, for both of your fluids:

rho = density

nu = kinematic viscosity

alpha = thermal diffusivity coefficient

beta = coefficient of thermal expansion

In addition, you should try to estimate temperature at the outlet of the lance (T_o), as well as the nominal temperature of the large, static body all around (T_f).

With these, you can attempt to solve for the "characteristic length" term that serves as the anchor in the "Rayleigh number".

An additional method to try is to solve for the "characteristic length" term of the "Weber number". However, this is only based on surface tension (sigma) for the flowing fluid, ie. exiting the lance.

Finally, you could treat the larger, static plume body of fluid as a temporary (dt ~1 sec?) conduit for the fluid exiting the lance. Based on viscosity and density of this plume body, compared to the lance fluid, you can attempt to estimate an effective friction factor in the "h_L" term, due to this submerged fluid interaction. In this case, you're re-arranging the Bernoulli equation and solving for the (l/d) term of "h_L". And, in this case, the "v_in" is the outlet lance velocity and "v_out" is zero ... because you want to know (l/d) when this lance fluid stops flowing.

I guess I've given you three approaches to take.

And, I think (if it were me), I would actually start with the last approach based on Bernoulli's equation because that is essentially and energy-balance approach, from some initial to some final state.

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