# Thread: How thrust relates to velocity (Warning: long technical discussion)

1. ## How thrust relates to velocity (Warning: long technical discussion)

I got to thinking, thrust is merely the force that accelerates our watercraft. So then how is it that you can change the nozzle diameter from an 83mm to an 81mm and get more top speed, but less acceleration, or go the other way and get more acceleration and less top speed? I mean force is force, but is there somehow a different “quality” of force depending on the nozzle size and somehow it knows the difference between low speed and top speed? So, I set out to find the answers. For those of you that have read some of my posts, you know that the math and the numbers have to work for me or I am just not happy.

So, I started looking at the thrust formulas to see if I could get my head wrapped around it. And, I did. Here’s what I’ve figured out:

Thrust of jet type device be it air or water is governed by the equation

F = M * (Vj - Vc)
F = Force (thrust)
M = Mass Flow of water
Vj = the velocity of the water jet exiting the pump nozzle and
Vc = the velocity of the craft.

By putting in a smaller nozzle, you should have a faster water jet Vj, but a smaller mass flow M. There has to be an optimal combination to get the most thrust, but why does it then change where the thrust gets applied (low speed acceleration vs top speed)?

It has to do with the term Vc. With the ski sitting still, Vc = 0 so the thrust is F = M x Vj. But, as the ski begins to move Vc starts to go up and the effective thrust goes down. As the speed of the ski increases, it becomes more important that you have a higher speed jet exiting than the total volume (or mass flow) of water. As the craft goes faster and faster, you start to catch up the exit velocity of the water jet and the effective thrust starts approaching 0. You can then put in a smaller nozzle and the water jet speeds up so that you now can go faster before your effective thrust diminishes. But, because the nozzle increases the restriction in the pump, your mass flow rate goes down, so your maximum thrust number is lower.

This all makes sense: Larger nozzle ring = more mass flow M, but slower water exit velocity Vj. As long and the craft is stopped or moving slow, this produces maximum thrust. Smaller nozzle ring = faster water exit velocity Vj, but with less mass flow. As the ski speeds up, this helps to offset the term (Vj-Vc) so that you can go faster before Vj-Vc get too close to 0 but does not produce as much thrust when stopped or going a slow speeds.

To illustrate this, I put together a spreadsheet utilizing the thrust formula. I estimated the decrease in mass flow M for each subsequently smaller nozzle size follows:

83mm, Mass flow M = 100%
82mm, M = 95% of 83mm
81mm = 90% of 82mm
80mm = 85% of 81mm
79mm = 80% of 80mm

I estimated the increase in Vj with each subsequent nozzle change as follows:
83mm, Velocity of the water jet, Vj = 100%
82mm, Vj = 110% of 83
81mm = 110% of 82
80mm = 110% of 81
79mm = 110% of 80

I did it this way so as to make sure that the values would approximate the graph curves I saw in my Fluid Dynamics book.

I then figured flow rate by using the formula for water horsepower:

Pwhp = SG Qgal h / 3960 where
Q = volume flow rate (gpm)
SG = specific gravity

Doing some quick algebra the equation becomes:

Qgal = Pwhp * 3960 / (SG * h )

So, if we say that the actual efficiency of the pump and motor is a little over 50% in transferring the power to the water, we could estimate that 110 hp actually gets transferred to the water.

Estimating 105 psi max pressuring in the pump converts to approximately 242 feet of head.

Nozzle diameter: 83 mm
Engine Horsepower: 207
Pwhp, Pump horsepower: 110
SG for water is: 1.0
h = Pump pressure: 105psi = 242 ft head

Qgal = 110 x 3960/ (1 x 242) = 1792 gpm. And, the flow velocity for 1792 gpm through 83mm = 179 ft/s

The graphical results were interesting. It shows that as the ski speed increases, the available thrust decreases. This can be somewhat made up for by going to a smaller nozzle until a point, but will decrease the thrust while stopped or moving slow. Here is the spread sheet and accompanying graph. I bolded the values for maximum thrust for each speed range so you can see which nozzle would give the maximum thrust. Please keep in mind that I made some assumtions and estimations, and I am note saying that these valules are extremely accurate, but they do show the relationship of thrust vs. speed for various nozzle sizes. I hope some of you find this information interesting and useful.

2. hmmmmm...so if the angle of the dangle is in direct proportion to the heat of the meat the lust for the bust remains constant..........right

3. So reading this over it comes down to optimizing the intake to the size of the propeller/pitch and nozzle ring.
the larger nozzle ring allows more water to flow but the diameter of the immediate engery or force being pushed against the water medium is less per square cm.
Now the smaller nozzle is homing the pump output into a smaller more concentrated flow creating more energy or force into the same water medium per sq cm. But the smaller nozzle over time will ultimately move less water and thus lessing the max speed..

it makes sense to me

4. Are these all just static numbers on how much water will flow through the intake,pump and venturi for a given nozzle size? What about the prop and the difference in pitch's, that is going to have an effect on total thrust output. Just because a prop is pitched to allow a higher rpm doesnt mean it is producing more thrust. Just wondering

5. Originally Posted by suck my wake
Are these all just static numbers on how much water will flow through the intake,pump and venturi for a given nozzle size? What about the prop and the difference in pitch's, that is going to have an effect on total thrust output. Just because a prop is pitched to allow a higher rpm doesnt mean it is producing more thrust. Just wondering
Yes these numbers are approximations based on a the various nozzles sizes and prop matched to a stock motor.

A higher pitched prop would produce more thrust if properly matched to a higher output engine. You pitch down to bring the rpm up so that you can make max power, which in turn makes the maximum amount of thrust.

6. That is a good point you make.

7. WOT, I hav no idea WTH you are talking about there, but I applaud you for the effort.

8. ummm WHAT?
I sure hope there will be a Cliff-Note posted when you boys are done...
JP

9. Originally Posted by YoYamma

WOT, I hav no idea WTH you are talking about there, but I applaud you for the effort.
Ok, he could have said, " the football goes further&faster throwing it out of the back of a truck when the truck is not moving." Now go 20mph and throw the ball....now 50mph.....at 80 that ball ain't goin' nowhere. On the serious side, that does break it down, and is the very same reason acceleration goes down as speed increases. Add to that the exponential increase in water resistance at speed (not linear) and u can start to see why it takes so much to get a little more speed.

10. maybe u guys can call NASA and work with them on this??????????? im just gonna stick to my LIL HILLBILLY WORLD amd not DELVE TO DEEP! my mind cant handle all this big verbage with the linear and exponential stuff......... call me when its time for dinner though!!!!!!!!!!!!!