gokarn ship propulsion part 2

8/16/2019 Gokarn ship propulsion part 2

1/9

Propeller Tbeory

29

;

ropeller heory

the shape of the propeller. The blade element theories, on the other hand,

expla in the effec t of propeller geometry on it s performance but give the er-

roneous result that the ideal efficiency of a propeller is 100 percent. The

divergence between the two groups of theories is expla ined by the ci rcula-

t ion theory (vortex theory) of propellers initial ly formulated by Prandtl and

Betz (1927) and then developed by a number of others to a stage where it

is not only in agreement with experimental results but may also be used for

the practical design of propellers.

CHAPTER 3

3.2

Axial Momentum Theory

3.1 Introduction

In the axial momentum theory, the propeller is regarded as an actuator

disc which imparts a sudden increase in pressure to the fluid passing through

i t. The mechanism by which this pressure increase i s obtained is ignored.

Further , it is assumed that the result ing acceleration of the fluid and hence

the thrust generated by the propeller are uniformly distributed over the disc,

the flow is frictionless, there is no rotation of the fluid, and there is an un-

limited inflow of fluid to the propeUer. The acceleration of the fluid involves

a contraction of the fluid column passing through the propeller disc and,

since this cannot take place suddenly, the acceleration takes place over some

dis tance forward and some dis tance aft of the propeller disc. The pressure

in the f luid decreases gradually as it approaches the d isc, it is suddenly in-

creased at the disc, and it then gradually decreases as the flu id leaves the

disc. Consider a propeller (actuator disc) of area Ao advancing into undis-

turbed fluid with a velocity VA. A uniform velocity equal and opposite to VA

is imposed on this whole system, so that there is no change in the hydrody-

namic forces but one considers a stationary disc in a uniform flow of velocity

VA Let the pressures and velocit ies in the fluid column passing through the

propeller disc be Po and VA far ahead, Pi and VA+ Vi j ust ahead of the disc,

P~ and VA+vdust behind the disc, and P2 and VA V2 far behind the disc, as

shown in Fig. 3.1. From considerations of continuity, the velocity just ahead

and just behind the d isc must be equal, and since there is no rotation of the

fluid, the pressure far behind the propel ler must be equal to the pressure far

ahead, i .e. P2 = Po.

The mass off luid flowing through the propel ler disc per uni t t ime is given

by:

A study ofthe theory of propel le rs is important not only for understanding

the fundamenta ls of propelle r action but al so because the theory provides

results that are useful in the design of propellers. Thus, for example, pro-

peller theory shows that even in ideal conditions there is an upper limit to

the efficiency ofa propelle r, and that this efficiency decreases as the thrust

loading on the propel le r increases. The theory also shows that a propel ler

is most efficient if all its radial sections work at the same efficiency. Fi-

nal ly, propelle r theory can be used to determine the deta iled geometry of a

propeller for optimum performance in given operating conditions.

Although the screw propeller was used for ship propulsion from the be-

ginning of the 19th Century, the first propelle r theories began to be devel-

oped only some fif ty years late r. These early theories followed two schools

of thought. In the momentum theor ies as developed by Rankine, Green-

hill and R.E. Froude for example, the origin of the propeller thrust is ex-

plained entirely by the change in the momentum of the fluid due to the

propeller. The blade element theories, associated with Weissbach, Redten-

bacher, W. Froude, Drzewiecki and others, rest on observed facts rather

than on mathematica l principles , and explain the act ion of the propeller in

terms of the hydrodynamic forces experienced by the radia l sections (blade

elements) ofwhich the propelle r blades are composed. The momentum the-

ories are based on correct fundamental principles but give no indica tion of

28


2/9


3/9

32

Basic Ship Propulsion

Propeller Theory

If CTL reduces to zero, i.e. T = 0, the ideal efficiency 'T/ibecomes equal to 1.

If, on the other hand, VA tends to zero, 'T/ialso tends to zero, although the

propeller still produces thrust. The relation between thrust and delivered

power a t zero speed of advance is of interest s ince this condit ion represents

the practical situations of a tug applying a static pull at a bollard or of a

ship at a dock trial. For an actuatotdisc propeller, the delivered power is

given by:

where a

=

Vl/VA is the axial inflow factor, and vland V2 are the axial induced

velocities at the propeller and far behind it. The efficiency 'T/iis called the

ideal efficiency because the only energy loss considered is the kinetic energy

lost in the f luid column behind the propeller , i .e. in the propeller s lipstream,

and the other losses such as those due to viscosity, the rotation of the fluid

and the creat ion ofeddies are neglec ted.

The thrust loading coefficient of a propelle r is def ined as:

which gives:

T

CTL

=-

pAo VA2

Substituting the value of

T

from Eqn. (3.2) and noting that Vl = a¥,.t,

tI2=

2a A,

and

a

=

(1/ T/i)

- 1, one obtains:

(3.11)

2

'T/i

= 1 + VI +

Cn

(3.12)

This is an important result, for it shows that the maximum efficiency of a

propeller even under ideal condi tions is limited to a value less than 1, and

that this eff ic iency decreases as the thrust loading increases . I t therefore fol-

lows tha t for a given thrus t, the larger the propeller the greater i ts efficiency,

other things being equal.

VI

= 0.9425 ms-I

a

=

0.2356

V2 = 1.8850ms-1

1

1/i

=

_

=

0.8093

+a

PD = T VA = 30.0 X4.0

1/i 0.8093

T

30.0 x 1000

CTL

= - = -

-

= 1.1645

= 148.27kW

TVA 1

./

PD

= - =

2 T VA(1+ V 1+ Cn)

'T/i

33

(3.13)

Example 1

As

VA

tends to zero, 1+ VI +

CTL

tends to

VCTL,

sothat inthe Ihnit:

A propeller of 2.0m diameter produces a thrust of 30.0kN when advancing at a

speed of4.0m per seein sea water. Determine the power delivered to the propeller,

the.velocities in the slipstream at the propeller disc and at a section far astern, the

thrust loading coefficientand the ideal efficiency.

Ii

D

=

2.0m Ao

=

~ D2

=

3.1416m2

4

T

=

30.0kN . p=1025kgm-3

VA = 4.0ms-1

that is,

T

=

pAo (VA

+

vd

2 VI

so that

1025 x 3.1416 (4.0 + VI) 2 VI

=

30.0x 1000

]

~

1 2 2 T

PD = T A

VCn

= [4 T VA p Ao VA

2

[

T3

]

~

=

2pAo

TrT

PDY~

= )2,

VA

=

0 (3.14)


4/9

,,\

34


Propeller Theory

35

This rel at ion between t hrust and del ivered power at zero vel oci ty of ad-

vance for a propeller in ideal conditions thus has a value of.;2. In actual

pract ice , the value of thi s relat ion i s considerably less.

FAR ASTERN

ACTUATOR DISC

AREA Ao

ANGULAR VELOCITY w

FAR AHEAD

D

= 3.0m

Ao

= ~

D2

=

7.0686m2-

4

p = 1025kgm-3

PD

= 700kW

--

/,,- ./~~/~---~~==~

7-

,- ,- ,. /

1

, ~- ---~-

.

,, ,,,, ,,,, ,,,, ,,-,- ----------

- --,-.~/ -

,

..L . - - -

,. ,. ,. , , , -- -- ------

/ , J/ _./ ,,,, ,,,, -,,,,,,;,,,, ,,- :--:---::----------

- - - -:::.-::::::::::::===

Example 2

A propeller

of 3.0m

diameter absorbs 700kW in the s ta ti c cond it ion in sea water .

What i s i ts thrus t?

FLUID VELOCITIES

VA v2 AXIAL

VA+v,

T3

=

2pAoP b

= 2x 1025

x 7.0686

x 700 x 1000 2kgm-3m2 Nms-1 2

= 7100.39x 1012

N3

W2

ANGULAR -

w,

VA

0

Figure

3.2:

Action of a Propeller in the Impulse Theory

T

= 192.20kN

The thrus t developed by the e lement i s determined from the change in the

axi al momentum of t he f lui d per unit time:

Momentum h ory Including Rotation

dT

=

dm [ VA

+

V2

-

VA]= PdAo VA+ VI .V2

(3.16)

In t his t heory, al so somet imes call ed the impulse theory, t he propell er i s

regarded asimparting both axial and angular acceleration to the fluid flowing

through the prope ller disc. Consider a propel ler of disc area

Ao

advancing

i nto undist urbed water wi th an axi al vel ocit y VA whil e revolvi ng wit h an

angular veloc ity UI.Impose a uni form veloc ity equa l and opposi te to VA on

the whole sys tem so that the prope ller i s revolving with an angular veloc ity

c.J at a fi xed posi tion . Let the axi al and angu lar vel oci ti es of the fl uid then

be VA+ vI and Ul lat t he propell er d isc and VA+ V2 and Ul2far downstream,

as shown in Fig. 3.2. The mass of fluid flowing per unit time through an

annular element between the radii r and r +

dr

is given by:

The torque ofthe e lement i s s imilar ly obtained from the change in angular

momentum per uni t t ime:

dQ

= dm

r2 (Ul2

-

0)

=

pdAo VA+

VI) Ul2r2

(3.17)

The work done by the element thrust is equal to the increase in the axial

ki net ic energy of the fl uid flowing t hrough the annular el ement. Per unit

time, th is i sg iven by:

dT VA

+ VI)

~ dm [ VA + V2?

-

VA2]

dm

=

p d Ao VA + VI

(3.15)

that is,

where dAo i s the area of the annular e lement.

p dAo VA+ V2 v2 VA+ VI)

=

~ p d Ao VA+ VI) v2 (2 VA + V2Y


5/9

36

.


Propeller Theory

37

1

VI

= 2V2

(3.18)

where a =

UJI/UJand

a = VI/~ are the rotational and axial inflow factors,

VI and V2 are the axial induced velocities at the propeller and far down-

stream, UJIand UJ2being the corresponding angular induced velocities. It

may be seen by comparing this expression for efficiency,Eqn. (3.20), with

the expression obtained in the axial momentum theory, Eqn. (3.10), that the

effect of slipstream rotation is to reduce the efficiencyby the factor (1- a ).

By making the substitutions:

.;\so that:

This is the same result as obtained in the axial momentum theory,

Eqn. (3.5). The work done per unit time by the element torque is simi-

larly equal to the increase in the rotational kinetic energy of the fluid per

unit t ime, i .e.:

dQUJl

=

dmr2 [UJ~- 0]

= ~ pdAo ~ + VI)UJ22W2

= dQUJ2

dAo = 27rr dr,

UJl = a UJ

VI = a~,

UJ2 =

2a

UJ

V2

= 2a~

in Eqns. (3.16) and (3.17), one obtains:

dT = 47rprdr VA2a 1 + a)

so that,

dQ

= 47r

pr3 dr VAUJ

a (1+ a)

(3.21)

(3.22)

1

UJl

= 2

UJ2

(3.19)

The efficiencyof the annular element is then given by:

Thus, half the angular velocity of the fluid is acquired before it reaches

the propeller and half after the fluid leaves the propeller.

The total power expended by the element must be equal to the increase

in the total kinetic energy (axial and rotational) per unit t ime, or the work

done by the element thrust and torque on the fluid passing through the

element per unit time:

dTVA 47rprdrVA2 a l+a)VA a VA2

1/=-= =--

dQw 47rpr3drVAUJa I+a)UJ a UJ2r2

(3.23)

Comparing this with Eqn. (3.20), one then obtains:

a

VA2 I-a

1/-----

- a UJ22 - 1+ a

or,

dQUJ = dT (~+ VI)+ dQUJl

a I- a )

UJ2r2

=

a (1 + a) tA,2 (3.24)

that is ,

This gives the relation between the axial and rotational induced velocities

in a propeller when friction is neglected.

dT VA+ Vl) = dQ

(UJ - UJl)

Example 3

and the efficiency of the element is then:

dTVA

(UJ- UJI)VA

1

- ~ 1- a

1/--- - UJ_-

- dQUJ - (~+ VI)UJ- 1+ ~ - 1+ a

(3.20)

A propeller ofdiameter 4.0m has an rpm of 180when advancing into sea water at

a speed of6.0m per sec. The element ofthe propeller at 0.7

R

produces a thrust of

200kN per m. Determine the torque, the axial and rotational inflowfactors, and

the efficiency of the element.


6/9

38

Basic

Sbip Propulsion

,I

{'Ii

;l

Propeller Tbeory

39

I

..'

D

= 4.0m

n

= 180rpm = 3.0s-1

VA

= 6.0ms-1

r

= 0.7

R

=

0.7 x 2 ;0

=

104m

~=

200kN m-1

w

= 211

n =

611 adians per see

4

Blade Element Theory

411 1025x 1.4x 6.02

a(l

+

a)

=

200 x 1000

The blade element theory, in contrast to the momentum theory, is concerned

with how the propeller generates its thrust imd how this thrust depends upon

the shape of the propeller blades. A propeller blade isregarded as being com-

posed of a series of blade elements, each of which produces a hydrodynamic

force due to its motion through the fluid. The axial component ofthis hydro-

dynamic force is the element thrust while the moment about the propeller

axis of the tangential component is the element torque. The integration of

the element thrust and torque over the radius for all the blades gives the

total thrust and torque of the propeller.

L

dT = 411 prVA2a(1+a)

dr

so that,

which gives,

a

= 0.2470

a

(1-

a )

w2r2 =

a

(1+

a) VA2

D

t hat is,

a (l -

a )(671 )2

1.42 = 0.2470(1+ 0.2470)x 6.02

1

s

1

J c f

or,

at

= 0.01619

dQ = 411 r3 VAwa (1 + a)

dr

= 471'x 1025X1.43x 6.0 x 611 0.01619x 1.2470

= 80.696kNmm-1

Figure

3.3:

Lift and Drag of

a

Wing.

1)

=

1-

a

- 1 - 0.01619

1+

a-I

+ 0.2470 = 0.7889

dT

-VA

= .. k - 200 x 6.0

~

w

- 80.696X 611 =

0.7889

Consider a wing of chord (width) c and span (length)

s

at' an an-

gle of attack Q: t o an incident flow of velocity

V

in a fluid of den-

sity

p,

as shown in Fig. 3.3. The wing develops a hydrodynamic force

whose components normal and parallel to

V

are the lift

L

and the drag

D.

One defines non-dimensional l if t and drag coefficients as follows:


7/9

40

Basic Ship. Propulsion

L

CL

=

~pAV2

D

CD

=

~pAV2

where

A

=

s

c is the area of the wing plan form. These coefficients depend

upon the shape of the wing section, the aspect ratio

sj

c and the angle of

attack, and are often determined experimentally in a wind tunnel. These

experimental values may then be used in the blade element theory, which

may thus be said to rest on observed fact.

(3.25)

dL

VA

(0) WITHOUTINDUCEDVELOCITIES

dL

..LdQ

rZ

Figure 3.4: Bl ad e El em ent Velo ci ti es a nd F or ces.

Now consider a propeller with

Z

blades, diameter

D

and pitch ratio

PjD

advancing into undisturbed water with a veloci ty VA while turning at a rev-

olution rate n. The blade element between the radii rand r + dr when

expanded will have an incident flow whose axial and tangential velocity.

components are VA and 21rn r respectively , giving a resultant veloci ty VR

I

I

i

Propeller Theory

at an angle ofattack a, as shown in Fig.3.4(a). The blade element will then

produce a lift

dL

and a drag

dD,

where:

dL

=

CL ~pcdrVA

dD

=

CD ~pcdrVA

(3.26)

If the thrus~ and torque produced by the elements between r and r +

dr

for all the

Z

blades are

dT

and

dQ,

then from Fig.3.4(a):

i dT = dL

cos {3.-

dD

s in {3

= dL

cos {3 (1 - ~~ tan {3)

1 .

(

dD

Z dQ = dL

sm{3

- dD

cos{3=

dL

cos{3 tan {3+

dL

(3.27)

where

~

tan{3

= -

21r

n

r

Putting tan, =

dDjdL,

and writing

dL

and

dD

in termsof

CL

and

CD,

one obtains:

dT

=

Z

C

L . ~ pc dr VAcos{3(1 -

tan {3tan,)

dQ

=

rZCL ~pcdrVA

cos{3(tan{3+ tan,)

(3.28)

The efficiencyof the blade element is then:

dT~

~ 1- tan{3 tan, tan{3

1}- - -

-

dQ

21r

n

- 21r

n

r tan {3+ tan, - tan ({3+ ,)

It will be shown later that for a propeller to have the maximum efficiency

in given conditions, all i ts blade elements must have the same efficiency.

Eqn. 3.29 thus also gives the efficiencyof the most efficient propeller for

the specified operating conditions.

If the propeller works in ideal conditions, there is no drag and hence

tan, = 0, resultingin the bladeelementefficiencyandhencethe efficiency

3.29


8/9

42


j

Propeller

Theory

43

of the most effic::ientpropeller being

TJ

= 1. This is at variance with the

., results of the momentum theory which indicates that if a propeller produces

a thrust greater than zero, its efficiencyeven in ideal conditions must be less

than 1.

Example 4

The primary reason for this discrepancy lies in the neglect of the induced

velocities, Le. the inflowfactors a,a . If the induced velocities are taken into

account, as shown in Fig.3.4(b), one obtains:

A fourbladed propeller of3.0m diameter and 1.0 constant pitch ratio has a speed of

advance of4.0m per sec when running at 120rpm. The blade section at

0.7R

has a

chord of0.5m, a no-lift angle of2 degrees, a lift-drag ratio of30 and a lift coefficient

that increases at the rate of6.0per radian for small angles ofattack. Determine the

thrust, torque and efficiencyofthe blade element at 0.7

R

(a) neglecting the induced

velocities and (b) given that the axial and rotational inflow factors are 0.2000and

0.0225 respectively.

dT = Z CL ~pcdr VJ coslh (1

-

tan/h tan-y)

dQ = rZCL.~pcdrVJcos{3r (tan{3r+tan-y)

(3.30)

.~

Z=4

D = 3.0m

p

15 = 1.0

VA

=

4.0ms-l

and:

n

= 120 rpm =

2.0 S-1

dTVA

TJ

=

dQ27rn

VA 1- tan{3r tan-y - tan{3

27rnr tan{3r+tan-y - tan({3r+-y)

r

x =

Ii

= 0.7

c = 0.5m

0 0 = 2°

CL

= 30

CD

- tan{3 tan{h - 1 - a tan{3r

- tan{3r tan {3r+ -y) - 1 + a tan {3r+-y)

(3.31)

a CL = 6.0 per radian

aO

p

= 1025kgm-3

smce,

VA

tan{3 = 27rnr

and

VA(1 + a) 1+ a

tan{3r = ( ) = tan{3---,

7rn r 1 - a 1 - a

(a) Neglecting induced velocities:

P /D

1.0

tan


9/9

. .

44


Propeller Theory

dQ

dr = rZCL~pcVJcos/3 tan/3+tan Y)

one obtains:

45

Substituting the numerical values calculated:

-1

dT =

113.640kNm

dr

-1

dQ = 48.991kNmm

dr

dT

=

185.333kNm-1

dr

dQ

=

66.148kNmm-1

dr 1 -

a

tan/31 1 - 0.0225 0.3722

0 7

/=- = x-= .383

1+

a tan /31

+ Y) 1+ 0.2000 0.4104

tan/3

=

0.8918

1/ =

3.5

irculation heory

VJ = [ 1 + a) VA]2+ [ 1- a ) 271 nr]2

=

[ 1 + 0.2000) 4.0]2 + [ 1 - 0.0225) 271 x 2.0 X 1.05] 2

The circulation theory or vortex theory provides a more satisfactory

explanat ion of the hydro dynamics of propeller act ion t han the mo ment um

and blade element theories. The lift produced by each propeller blade is

explained i n terms of the circulat ion around it in a manner anal ogous to t he

l if t p ro du ced b y a n a ir cr af t wi ng , a s d es cr ib ed i n t he f ol lo wi ng .

b)

a =

0.2000

a =

0.0225

iven:

= 23.0400+ 166.3535

= 189.3935 m2 s-2

/3

- VA 1+ a) - 4.0 1+ 0.2000) -

0 3722

tan I - - - .

271 nr l- a )

271x 2.0 x 1.05 1- 0.022 5)

/31 = 2 0. 41 31 0

v r =k

a

=

cp-

/31 = 24.4526- 20.4131

= 4.03950

0)

VORT X

LOW

aCL

2 + 4.0395

CL = _

ao

+

a)

=

6.0 180

= 0.6325

a -

71

A-

~-v

~

~

V-v~

-_-r

.-

b) UNIFORM

FLOW

c) VORT X

IN

UNIFORM FLOW

Figure 3.5: Flow of an Ideal Fluid around a Circular Cylinder.

C on si de r a f lo w i n w hi ch t he f lu id p ar ti cl es mo ve i n c ir cu lar p at hs s uch t hat

t he v el oc it y i s i nv er sel y p ro po rt io nal t o t he r ad iu s o f t he c ir cl e, F ig .3 .5 a ).

S uch a f lo w i s c al led a v or te x f lo w, an d t he ax is a bo ut w hi ch t he f lu id p ar ti cl es

move in a three dimensional flow is called a vortex line. In an ideal fluid, a

v or tex l in e c an no t e nd ab ru pt ly i ns id e t he f lu id b ut m ust e it her f or m a cl ose d

curve or end on the boundary of the fluid Helrnholz theorem). A circular

c yl in der p lace d i n a u ni fo rm f lo w o f a n i dea l f lu id , F ig . 3 .5 b ), wi ll e xp er ien ce

Substituting these values in:

dT

-

= ZCL

~

pcVJ cos/31 1- t an/31 t an Y)

dr

dQ

=

r Z CL ~pc

VJ

cos/31 tan/31

+ t an Y)

dr

-.

gokarn ship propulsion part 2

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