THE MAXIMIZATION OF THE PROPULSION FORCE FOR AN AIRCRAFT
OR SHIP PROPELLER
Mircea Dimitrie CAZACU, Univ.POLITEHNICA of Bucharest, cazacu@hydrop.pub.ro
Abstract: Starting from the classical theory and practice of the airfoil, one presents the
mathematical relations to obtain the greatest axial propulsion force, for a given mechanical
driven power. One put into the evidence the best aerodynamic or hydrodynamic profiles and
also its optimum incidence angle for different radii of the propeller blades.
In the present we try to apply these results for the propulsion of the small environmental
friendly ships, using the solar energy source by means of photo-voltaic panels and which are
foreseen to sail in our biosphere reservation Danube Delta and for ecological tourism.
1. The importance of the proposed problem
The problem of propulsion force increasing for the same consumed mechanical power at
the shaft, is very important not only concerning the operation radius enlargement of an aircraft
or ship, but also by the fossil fuel savings and environmental protection [1]. It is of a greater
importance for the ecological boats using the solar energy [2].
2. The determination of the best peripheral relative angle
Taking into account the expressions of the lift and drag forces, exerted on the profiled
blade, laid at the incidence angle i with respect to the relative angle , corresponding to the
relative velocity W from the velocity triangle (fig.1)


Fy  cy i  W 2 b l R  and Fx  cx i  W 2 b l R  ,
(1)
2
2
we can calculate the axial component of these forces, representing the propulsion force Fa
β
βP = β + i
W = V / sinβ
U = R ω = V / tg β
Fy cos β
Fx sin β
Fx – the drag component
the lift component - Fy
i
V
Fig. 1. Velocity triangle and the components of hydrodynamic resultant


cos 
1 
Fa  Fy cos  Fx sin   V 2bl ( R ) cy (i ) 2  cx (i )
2
sin 
sin  

and also the expression of the shaft driving mechanical power
2Pm
cos 
cos2 
Pm  U ( Fy sin   Fx cos ) or pm 
 cy (i) 2  cx (i) 3 .
V 3bl
sin 
sin 
(2)
(3)
By cancelling the partial differential of the axial force expression (2), with respect to
the relative angle 
(4)
Fa   0  cy (i)sin (1  cos2 )  cx (i)cos (2  cos2 ) ,
one obtains the condition to maximise, or minimise respectively, the propulsion axial force,
denoting by x = sin2 β and the profile fineness f (i) = cy(i) / cx(i) as function of the incidence
angle i, given by the following algebraic relation
(5)
( f 2  1) x2  (4 f 2  1) x  4 f 2  0 ,
having two real solutions and putting into the evidence the relative best and worst respectively
angle β as function of the fineness of the aerodynamic or hydrodynamic profiles (fig.2), for
the positive value under root expression, necessary to assure the non-imaginary solutions
x
4 f 2 1 1 8 f 2
,
2f 2 2
for 1  8 f 2  0 
f (i) 
cy
cx
 0.3536...
(6)
Perypherial relative angle
Beta (degrees)
which condition eliminate a lot of profiles to curved and prefers these, that have the lift force
near by zero for a certain incidence angle i .
100
80
60
+ Radical
- Radical
40
20
0
0
0.1
0.2
0.3
0.4
Profile fineness f = Cy/Cx
Fig.2 The peripheral relative angle  in function of the profile fineness
Once more, introducing the values i of different profiles and β in the axial propulsion
force expression (2), the maximal its value will indicate the best profile to use.
3. The determination of the optimum profile setting angle for other radii
For the other radii, because the peripheral relative angle is already determined by the
relation V = U tg β, applied at the outskirts, we obtain the optimal angular velocity

Uj
V

,
Rp tg p Rj
(7)
which being the same for all the blades, determines the peripheral velocity at any other radius
Rj, at which the relative angle will be done by the relation
tg  j 
Rp
V

tg p
Rj Rj
  j  arc tg
Rp
Rj
tg p ,
(8)
the power maximization following to be obtained only by the election of the optimum
incidence angle in case of different considered profile, as we shall see below
we have determined the blade profile angle p =  + i cancelling the expression of the axial
force with respect to the incidence angle of the profile [3], obtaining the relation

cos j

1 
Fj  V 2bl ( Rj ) (cy0  icy1 ) 2  (cx0  icx1  i 2cx2 )

2
sin  j
sin  j 

(9)
the blade spread being b = R = constant and the blade depth as function of radius Rj having
no importance, we can cancel the axial propulsion force with respect to the incidence angle to
obtain the optimal incidence for each relative radius
cy1
Rj
Fa
1
0
 cx1  2icx2  iopt 
(cy1tg p
 cx1 ) ,
(10)
i
tg  j
2cx2
Rp
considering the variation approximately linear of the lift coefficient of the profile (for
example of the symmetric profile Gö 445 [4]) as function of the incidence angle
(11)
Cy  i  Cy0  Cy1i  0.002 i
and the parabolic approximately variation of the drag coefficient of the profile as function of
the incidence angle
Cx (i ) Cx0  Cx1i  Cx2i 2  0.005  0.004,5 i  0.000,5 i 2 .
(12)
In this manner we can establish the airfoil profile, which realises the best propulsion
axial force, calculated in the juxtaposed table, as also the value of the relative mechanical
driving power.
Table 1
Profile i
Gö 445
0
1
2
3
4
5
Cy(i)
Cx(i)
0
0.002
0.004
0.006
0.008
0.01
0.005
0.01
0.015
0.02
0.025
0.03
f = Cy/Cx
0
0.2
0.266667
0.3
0.32
0.333333
0.35355
Beta(grade)
p(-) mec
0
23.69
32.79
38.14
41.95
45.02
54.64
Infinit
0.1409
0.0783
0.065
0.0597
0.0566
0
Fa (-)
infinit
0.0134
0.0127
0.0135
0.0146
0.0155
0
For the smaller relative radius r = Rj/Rp < 1, where we have already the relative angle
β imposed, to maximise the axial force Fa one calculated in the table 2 the values of the
optimal incidence angle iopt given in the relation (10)
Table 2
r = Rj/Rp
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Betaj(degrees)
32.79816
30.11185
27.27135
24.27816
21.13725
17.85761
14.45267
10.94037
7.343012
3.686629
0
i opt (degrees)
-0.000000803
-8.352E-07
-8.674E-07
-8.996E-07
-9.318E-07
-0.000000964
-9.962E-07
-1.0284E-06
-1.0606E-06
-1.0928E-06
-0.000001125
The obtained results are presented in the figure number 3, with the important remark,
that the increasing of optimum incidence angle for all the radii is practically equal with zero.
4. Conclusions
Relative angle Beta
(degrees)
One have seen that the real solution, having a maximisation physical meaning,
corresponds to the negative sign before the radical and the symmetrical profiles Gö 445 with a
more drag coefficient Cx with respect to the lift coefficient Cy are preferable, the optimum
relative angle being about β = 33 degrees.
40
30
Relative angle
Beta
20
10
0
0
0.5
1
1.5
Relative radius r = Rj/Rp
Optimum incidence angle i
(degrees)
Fig. 3. Relative angle β and profile incidence angle i as function of different relative radii r
0
-0.0000002 0
0.5
1
1.5
-0.0000004
-0.0000006
Opt incid angle i
-0.0000008
-0.000001
-0.0000012
Relative radius r = Rj/Rp
Fig. 4. Optimal incidence angle i as function of relative radius
5. References
[1] M.D.Cazacu. Tehnologii pentru o dezvoltare durabilă. Acad.Oamenilor de Ştiinţă din
România. Congresul ”Dezvoltarea în pragul mileniului al III-lea”, Secţia “Dezvoltarea durabilă”,
27-29 sept. 1998, Bucureşti. Editura EUROPA NOVA, 1999, 533 – 539.
[2] M.D.Cazacu. Microagregat hidroelectric pentru asigurarea autonomiei energetice a
balizelor luminoase sau a unor bărci fluviale. Rev.Transfer de Tehnologii, Bucureşti, 2005.
[3] M.D.Cazacu, S.Nicolaie. Micro-hydroturbine for run-of-riwer power station. A 2Conferinţă “Dorin Pavel ” a Hidroenergeticienilor din România, 24 – 25 mai 2002, Univ. Politehnica,
Bucureşti, Vol.II, 443 – 448.
[4] Hütte – Manualul inginerului. Vol. I, Editura AGIR, Bucureşti, 1947, p. 511.