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Author Topic:   Propeller diameter and fuel economy

cooper1958nc posted 10-09-2013 02:16 PM ET (US)
Propellers produce thrust by accelerating water sternward. The force, thrust, is due to the momentum change of the water. Momentum is mass times speed change. The formula is:
Thrust= Rho * vp * ((1/2)*pi*(diam/2)^2) * (vp-vb)
where Rho - density of water
vp= speed of water discharged from the prop,
assumed to be rpm*pitch (times conversion constants)
vb = boat speed
diam= diameter of prop
However the power (fuel) required to accelerate the water is not mass times speed but mass times speed squared.
Powerprop = Rho * vp * ((1/2)*pi*(diam/2)^2 * (vp-vb)^2
For a boat going 30 mph, swinging a 16d by 15p prop at 2500 prop rpm, the thrust is 5108N (*.225 to give 1149 pounds). The hp delivered to the boat is Thrust * vb = 92 hp. The power required (powerprop) is 169 hp.
Compare this with the same boat, same speed, same RPM, swinging a 14d by 15.6p prop. The thrust is 5226N, virtually the same (because we set it that way), the hp delivered to the boat is the same about 92 hp, but the hp required (powerprop) is now 213, up from 169.
All other things being equal (they are not, however), it pays to move a lot of water slower (big diameter). This may be one basis for the "fuel economy" propeller discussed earlier.
I have omitted unit conversions for clarity. If anyone wants the detail I will be happy to oblige.

jimh posted 10-09-2013 02:59 PM ET (US)
Very beautifully set out explanation, and much appreciated.
jimh posted 10-09-2013 03:04 PM ET (US)
In looking at the math, it seems that the key element is the definition of THRUST. Because of the way that THRUST is defined, it is intrinsic that larger diameter produces more THRUST. Further, it seems that the definition of THRUST is set up so that thrust varies with the diameter raised to the exponent 2.0. As a result, it follows necessarily that evaluation of the relationship at different propeller diameters and solving for power will produce the result shown.
A more critical examination of the relationship between a propeller and the volume of water it can accelerate might show that there were other variables that affected the relationship besides propeller diameter.
Jerry Townsend posted 10-09-2013 03:44 PM ET (US)
Coop* -- The efficiency of the prop including slip and spillage off the tip of the blades alters the results. While this affects the calculated results, and all props are not born equal - the general trend of the comparison is still correct.--- Jerry/Idaho
jimh posted 12-01-2013 08:38 AM ET (US)
The analysis presented in the initial article is known as the Simple Momentum Theory. NASA explains:

quote:
...we know that the amount of thrust depends on the mass flow rate through the propeller and the velocity change through the propulsion system. Let us denote the free stream conditions by the subscript "0", the conditions at the propeller by the subscript "p", and the exit conditions by the subscript "e". The thrust F is equal to the mass flow rate m dot times the difference in velocity V.
F = [m dot * V]e - [m dot * V]0
There is no pressure-area term because the pressure at the exit is equal to the free stream pressure. The mass flow through the propulsion system is a constant, and we can determine the value at the plane of the propeller. Since the propeller rotates, we can define an area A that is swept out by the propeller of blade length L. Through this area, the mass flow rate is density r times velocity Vp, times area.
m dot = r * Vp * A
Substitute this value for the mass flow rate into the thrust equation to get thethrust in terms of the exit velocity, entrance velocity, and velocity through the propeller.
F = r * Vp * A * [Ve - V0]
The interesting part is the sentence that says, "Since the propeller rotates, we can define an area A that is swept out by the propeller of blade length L" This is how the diameter of the propeller comes into the calculation.
Cf.: http://www.grc.nasa.gov/WWW/k-12/airplane/propanl.html

OMCrobert posted 12-02-2013 11:24 AM ET (US)
That backs up the brand new design of the Enertia Eco which uses a much larger diameter vs any other propeller.
jimh posted 12-03-2013 10:56 AM ET (US)
The diameter of propellers for use on an outboard motor is limited by the size of the propeller aperture on the engine. In a prior discussion it was already observed that a number of propellers retain the maximum diameter or near maximum diameter while varying the pitch, and that this approach was not limited to a single new propeller recently introduced, but, rather was an established practice. As a result of the limit on diameter imposed by the outboard engine propeller aperture, it is impossible that any one outboard propeller could be of a diamter that "is much larger diameter vs any other propeller." The difference in diameters is confined to a variation of about 0.125-inch in a diameter of about 16-inches. This is a variation of 0.125/16 = 0.0078 or less than one-percent. It is not reasonable to attribute a variation in diameter of less than one percent as being enabling of a radical improvement in propeller efficiency at creating thrust.
The analysis of the propeller that was presented by cooper is called the Simple Momentum Theory. The key word is probably "simple." In the NASA page I cited, there is the following footnote:

quote:
Note that this thrust is an ideal number that does not account for many losses that occur in practical, high speed propellers like tip losses. The losses must be determined by a more detailed propeller theory, which is beyond the scope of these pages. The complex theory also provides the magnitude of the pressure jump for a given geometry. The simple momentum theory, however, provides a good first cut at the answer and could be used for a preliminary design.
What the "simple" theory establishes is a general trend for propeller efficiency to increase with propeller diameter. However, we can see that this trend is not without limitations by looking at the diameter of propellers powering large ships: there seems to be some limit to diameter, that is, on ship propellers you don't see propeller diameters that are enormously disproportionate to outboard propellers, even though on a ship the propeller aperture could be made to be very large if it meant gaining efficiency by using a larger diameter propeller.

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Author	Topic: Propeller diameter and fuel economy
cooper1958nc	posted 10-09-2013 02:16 PM ET (US) Propellers produce thrust by accelerating water sternward. The force, thrust, is due to the momentum change of the water. Momentum is mass times speed change. The formula is: Thrust= Rho * vp * ((1/2)pi(diam/2)^2) * (vp-vb) where Rho - density of water vp= speed of water discharged from the prop, assumed to be rpmpitch (times conversion constants) vb = boat speed diam= diameter of prop However the power (fuel) required to accelerate the water is not mass times speed but mass times speed squared. Powerprop = Rho vp * ((1/2)pi(diam/2)^2 * (vp-vb)^2 For a boat going 30 mph, swinging a 16d by 15p prop at 2500 prop rpm, the thrust is 5108N (.225 to give 1149 pounds). The hp delivered to the boat is Thrust vb = 92 hp. The power required (powerprop) is 169 hp. Compare this with the same boat, same speed, same RPM, swinging a 14d by 15.6p prop. The thrust is 5226N, virtually the same (because we set it that way), the hp delivered to the boat is the same about 92 hp, but the hp required (powerprop) is now 213, up from 169. All other things being equal (they are not, however), it pays to move a lot of water slower (big diameter). This may be one basis for the "fuel economy" propeller discussed earlier. I have omitted unit conversions for clarity. If anyone wants the detail I will be happy to oblige.
jimh	posted 10-09-2013 02:59 PM ET (US) Very beautifully set out explanation, and much appreciated.
jimh	posted 10-09-2013 03:04 PM ET (US) In looking at the math, it seems that the key element is the definition of THRUST. Because of the way that THRUST is defined, it is intrinsic that larger diameter produces more THRUST. Further, it seems that the definition of THRUST is set up so that thrust varies with the diameter raised to the exponent 2.0. As a result, it follows necessarily that evaluation of the relationship at different propeller diameters and solving for power will produce the result shown. A more critical examination of the relationship between a propeller and the volume of water it can accelerate might show that there were other variables that affected the relationship besides propeller diameter.
Jerry Townsend	posted 10-09-2013 03:44 PM ET (US) Coop* -- The efficiency of the prop including slip and spillage off the tip of the blades alters the results. While this affects the calculated results, and all props are not born equal - the general trend of the comparison is still correct.--- Jerry/Idaho
jimh	posted 12-01-2013 08:38 AM ET (US) The analysis presented in the initial article is known as the Simple Momentum Theory. NASA explains: quote: ...we know that the amount of thrust depends on the mass flow rate through the propeller and the velocity change through the propulsion system. Let us denote the free stream conditions by the subscript "0", the conditions at the propeller by the subscript "p", and the exit conditions by the subscript "e". The thrust F is equal to the mass flow rate m dot times the difference in velocity V. F = [m dot * V]e - [m dot * V]0 There is no pressure-area term because the pressure at the exit is equal to the free stream pressure. The mass flow through the propulsion system is a constant, and we can determine the value at the plane of the propeller. Since the propeller rotates, we can define an area A that is swept out by the propeller of blade length L. Through this area, the mass flow rate is density r times velocity Vp, times area. m dot = r * Vp * A Substitute this value for the mass flow rate into the thrust equation to get thethrust in terms of the exit velocity, entrance velocity, and velocity through the propeller. F = r * Vp * A * [Ve - V0] The interesting part is the sentence that says, "Since the propeller rotates, we can define an area A that is swept out by the propeller of blade length L" This is how the diameter of the propeller comes into the calculation. Cf.: http://www.grc.nasa.gov/WWW/k-12/airplane/propanl.html
OMCrobert	posted 12-02-2013 11:24 AM ET (US) That backs up the brand new design of the Enertia Eco which uses a much larger diameter vs any other propeller.
jimh	posted 12-03-2013 10:56 AM ET (US) The diameter of propellers for use on an outboard motor is limited by the size of the propeller aperture on the engine. In a prior discussion it was already observed that a number of propellers retain the maximum diameter or near maximum diameter while varying the pitch, and that this approach was not limited to a single new propeller recently introduced, but, rather was an established practice. As a result of the limit on diameter imposed by the outboard engine propeller aperture, it is impossible that any one outboard propeller could be of a diamter that "is much larger diameter vs any other propeller." The difference in diameters is confined to a variation of about 0.125-inch in a diameter of about 16-inches. This is a variation of 0.125/16 = 0.0078 or less than one-percent. It is not reasonable to attribute a variation in diameter of less than one percent as being enabling of a radical improvement in propeller efficiency at creating thrust. The analysis of the propeller that was presented by cooper is called the Simple Momentum Theory. The key word is probably "simple." In the NASA page I cited, there is the following footnote: quote: Note that this thrust is an ideal number that does not account for many losses that occur in practical, high speed propellers like tip losses. The losses must be determined by a more detailed propeller theory, which is beyond the scope of these pages. The complex theory also provides the magnitude of the pressure jump for a given geometry. The simple momentum theory, however, provides a good first cut at the answer and could be used for a preliminary design. What the "simple" theory establishes is a general trend for propeller efficiency to increase with propeller diameter. However, we can see that this trend is not without limitations by looking at the diameter of propellers powering large ships: there seems to be some limit to diameter, that is, on ship propellers you don't see propeller diameters that are enormously disproportionate to outboard propellers, even though on a ship the propeller aperture could be made to be very large if it meant gaining efficiency by using a larger diameter propeller.