General open water characteristics

6 Propeller performance characteristics

6.1 General open water characteristics

The forces and moments produced by the propeller are expressed in their most fundamental form in terms of a series of non-dimensional characteristics: these are completely general for a specific geometric configur- ation. The non-dimensional terms used to express the general performance characteristics are as follows:

thrust coefficient KT= T ρn²D⁴ torque coefficient KQ= Q

ρn²D⁵

(6.1) advance coefficient J=V_a

nD

cavitation number σ=p₀−e

1 2ρV²

where in the definition of cavitation number,Vis a rep- resentative velocity which can either be based on free stream advance velocity or propeller rotational speed.

Whilst for generalized open water studies the former is more likely to be encountered there are exceptions when this is not the case, notably at the bollard pull con- dition whenVa=0 and henceσ0→ ∞. Consequently, care should be exercised to ascertain the velocity term being employed when using design charts or propeller characteristics for analysis purposes.

To establish the non-dimensional groups involved in the above expressions (equation (6.1)), the principle of dimensional similarity can be applied to geometrically similar propellers. The thrust of a marine propeller when working sufficiently far away from the free surface so as not to cause surface waves may be expected to depend

upon the following parameters:

(a) The diameter (D).

(b) The speed of advance (V_a).

(c) The rotational speed (n).

(d) The density of the fluid (ρ).

(e) The viscosity of the fluid (μ).

(f ) The static pressure of the fluid at the propeller station (p0−e).

Hence the thrust (T) can be assumed to be proportional toρ,D,Va,n,μand (p0−e):

T∝ρ^aD^bV_a^cn^dμ^f(p0−e)^g

Since the above equation must be dimensionally correct it follows that

MLT⁻²=(ML⁻³)^aL^b(LT⁻¹)^c(T⁻¹)^d

×(ML⁻¹T⁻¹)^f(ML⁻¹T⁻²)^g and by equating indices forM,LandTwe have for mass M: 1=a+f +g

for length L: 1= −3a+b+c−f−g for time T: −2= −c−d−f−2g from which it can be shown that

a=1−f −g b=4−c−2f −g d=2−c−f −2g Hence from the above we have

T∝ρ^(1−f−g)D^{(4−c−2f−g)}V_a^cn^{(2−c−f−2g)}μ^f(p0−e)^g from which

T=ρn²D⁴ Va

nD _c

· μ

ρnD² _f

· p0−e

ρn²D² _g These non-dimensional groups are known by the following:

thrust coefficient K_T= T ρn²D⁴ advance coefficient J= Va

nD

Reynolds number R_n=ρnD² μ cavitation number σ0= p0−e

1 2ρn²D²

∴ KT∝ {J,Rn,σ0} that is

K_T=f(J,R_n,σ0) (6.2)

The derivation for propeller torqueK_Qis an analogous problem to that of the thrust coefficient just discussed.

The same dependencies in this case can be considered to apply, and hence the torque (Q) of the propeller can be

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90 Marine propellers and propulsion

considered by writing it as a function of the following terms:

Q=ρ^aD^bV_a^cn^dμ^f(p0−e)^g

and hence by equating indices we arrive at Q=ρn²D⁵

V_a nD

c μ ρnD²

f

· p₀−e

ρn²D² g

which reduces to

KQ=g(J,Rn,σ0) (6.3)

where the torque coefficient KQ= Q

ρn²D⁵

With the form of the analysis chosen the cavita- tion number and Reynold’s number have been non- dimensionalized by the rotational speed. These numbers could equally well be based on advance velocity, so that

σ0=p0−e

1

2ρV² and Rn= ρVD μ

Furthermore, by selecting different groupings of indices in the dimensional analysis it would be possible to arrive at an alternative form for the thrust loading:

T=ρV_a²D²φ(J,R_n,σ0)

Figure 6.1 Open water diagram for Wageningen B5-75 screw series (Courtesy: MARIN)

which gives rise to the alternative form of thrust coefficientC_Tdefined as

C_T= T

1

2ρV_a²(πD²/4)= 8T

πρV_a²D² (6.4) CT=(J,Rn,σ0)

Similarly it can be shown that the power coefficientC_P can also be given by

CP=φ(J,Rn,σ0) (6.5)

In cases where the propeller is sufficiently close to the surface, so as to disturb the free surface or to draw air, other dimensionless groups will become important.

These will principally be the Froude and Weber num- bers, and these can readily be shown to apply by intro- ducing gravity and surface tension into the foregoing dimensional analysis equations for thrust and torque.

A typical open water diagram for a set of fixed pitch propellers working in a non-cavitating environ- ment at forward, or positive, advance coefficient is shown in Figure 6.1. This figure defines, for the particu- lar propeller, the complete set of operating conditions at positive advance and rotational speed, since the pro- peller under steady conditions can only operate along the characteristic line defined by its pitch ratioP/D.

The diagram is general in the sense that, subject to scale effects, it is applicable to any propeller having the same

Propeller performance characteristics 91 geometric form as the one for which the characteristic

curves were derived, but the subject propeller may have a different diameter or scale ratio and can work in any other fluid, subject to certain Reynolds number effects.

When, however, theK_T,K_QversusJ diagram is used for a particular propeller of a given geometric size and working in a particular fluid medium, the diagram, since the density of the fluid and the diameter then become constants, effectively reduces from general definitions ofKT,KQandJto a particular set of relationships defin- ing torque, thrust, revolutions and speed of advance as follows:

Q n², T

n²

versus V_a

n

The alternative form of the thrust and torque coefficient which stems from equations (6.4) and (6.5), and which is based on the advance velocity rather than the rotational speed, is defined as follows:

CT= T

1 2ρA0V_a² CP= pD

1 2ρA0V_a³

(6.6)

From equation (6.6) it can be readily deduced that these thrust loading and power loading coefficients can be expressed in terms of the conventional thrust and torque coefficient as follows:

CT= 8 π

KT

J² and

CP= 8 π

KQ

J³ (6.7)

The open water efficiency of a propeller (ηo) is defined as the ratio of the thrust horsepower to delivered horsepower:

ηo= THP DHP

Now since THP=TV_a

and DHP=2πnQ

whereTis the propeller thrust,Va, the speed of advance, n, the rotational speed of the propeller andQ, the torque.

Consequently, with a little mathematical manipulation we may write

ηo= TV_a 2πnQ that is

ηo= KT

K_Q J

2π (6.8)

TheK_Q,K_TversusJ characteristic curves contain all of the information necessary to define the propeller performance at a particular operating condition. Indeed, the curves can be used for design purposes for a par- ticular basic geometry when the model characteristics are known for a series of pitch ratios. This, however, is a cumbersome process and to overcome these prob- lems Admiral Taylor derived a set of design coefficients termedBpandδ; these coefficients, unlike theKT,KQ

andJ characteristics, are dimensional parameters and so considerable care needs to be exercised in their use.

The termsBpandδare defined as follows:

Bp=(DHP)^1/2N V_a^2.5 δ= ND

Va

(6.9)

where DHP is the delivered horsepower in British or metric units depending on the, diagram used N is the propeller rpm

Vais the speed of advance (knots) Dis the propeller diameter (ft).

From Figure 6.2, which shows a typical propeller design diagram, it can be seen that it essentially com- prises a plotting ofBp, as abscissa, against pitch ratio as ordinate with lines of constantδand open water effi- ciency superimposed. This diagram is the basis of many design procedures for marine propellers, since the term Bpis usually known from the engine and ship character- istics. From the figure a line of optimum propeller open water efficiency can be seen as being the locus of the points on the diagram which have the highest efficiency for a give value ofB_p. Consequently, it is possible with this diagram to select values ofδandP/Dto maximize the open water efficiencyηofor a given powering con- dition as defined by theB_p parameter. Hence a basic propeller geometry can be derived in terms of diame- terD, sinceD=δV_a/N, andP/D. Additionally, this diagram can be used for a variety of other design pur- poses, such as, for example, rpm selection; however, these aspects of the design process will be discussed later in Chapter 22.

It will be seen that theBpversusδdiagram is limited to the representation of forward speeds of advance only, that is, whereVa>0, sinceBp→ ∞whenVa=0. This limitation is of particular importance when consider- ing the design of tugs and other similar craft, which can be expected to spend an important part of their service duty at zero ship speed, termed bollard pull, whilst at the same time developing full power. To over- come this problem, a different sort of design diagram was developed from the fundamentalK_T,K_QversusJ characteristics, so that design and analysis problems at or close to zero speed of advance can be considered.

This diagram is termed theμ−σdiagram, and a typical example of one is shown in Figure 6.3. In this diagram

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92Marinepropellersandpropulsion

Figure 6.2 Original B4-70 B_p−δdiagram (Courtesy: MARIN)

Propeller performance characteristics 93

Figure 6.3 Original B3.65μ−σdiagram (Courtesy: MARIN)

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94 Marine propellers and propulsion the following relationships apply:

μ=n ρD⁵

Q

φ=V_a ρD³

Q σ= TD

2πQ

(6.10)

where Dis the propeller diameter (m) Qis the delivered torque (kgf m) ρis the mass density of water (kg/m⁴s²) Tis the propeller thrust (kgf)

nis the propeller rotational speed (rev/s) Vais the ship speed of advance (m/s).

Diagrams of the type shown in Figure 6.3 are non- dimensional in the same sense as those of the funda- mentalKT, KQcharacteristics and it will be seen that the problem of zero ship speed, that is whenVa=0, has been removed, since the functionφ→0 asVa→0.

Consequently, the line on the diagram defined byφ=0 represents the bollard pull condition for the propeller.

It is important, however, not to confuse propeller thrust with bollard pull, as these terms are quite distinct and mean different things. Propeller thrust and bollard pull are exactly what the terms imply; the former relates to the hydrodynamic thrust produced by the propeller, whereas the latter is the pull the vessel can exert through a towline on some other stationary object. Bollard pull Table 6.1 Common functional relationships (British units) K_Q=9.5013×10⁶

P_D N³D⁵

(salt water)

B_P=23.77 ρKQ

J⁵ J =101.33V_a

ND =101.33 δ μ= 1

√K_Q =3.2442×10⁻⁴

N³D⁵

P_D (salt water) φ= J

√K_Q =Jμ σ=ηo

J =ηoμ φ = K_T

2πK_Q where:

P_Dis the delivered horsepower in Imperial units Qis the delivered torque at propeller in (lbf ft) Tis the propeller thrust (lbf)

Nis the propeller rotational speed in (rpm) nis the propeller rotational speed in (rev/s) Dis the propeller diameter in (ft)

V_ais the propeller speed of advance in (knots) vais the propeller speed of advance in (ft/s)

ρis the mass density of water (1.99 slug/ft³sea water;

1.94 slug/ft³for fresh water).

Table 6.2 Common functional relationships (Metric units)

KQ=2.4669×10⁴ P_D

N³D⁵

(salt water)

BP=23.77 ρK_Q

J⁵ J=30.896V_a

ND =101.33 δ μ= 1

√K_Q =6.3668×10⁻³

N³D⁵

P_D (salt water) φ= J

√K_Q =Jμ σ=ηo

J =ηoμ φ = K_T

2πK_Q where:

P_Dis the delivered horsepower (metric units) Qis the delivered propeller torque (kp m) Tis the propeller thrust (kp)

Nis the propeller rotational speed (rpm) nis the propeller rotational speed (rev/s) Dis the propeller diameter (m)

V_ais the propeller speed of advance (knots) vais the propeller speed of advance (m/s)

ρis the mass density of water (104.48 sea water) (101.94 fresh water)

is always less than the propeller thrust by a complex ratio, which is dependent on the underwater hull form of the vessel, the depth of water, the distance of the vessel from other objects, and so on.

In the design process it is frequently necessary to change between coefficients, and to facilitate this pro- cess. Tables 6.1 and 6.2 are produced in order to show some of the more common relationships between the parameters.

Note the termσin Tables 6.1 and 6.2 and in equation (6.10) should not be confused with cavitation number, which is an entirely different concept. The termσ in the above tables and equation (6.10) relates to theμ−σ diagram, which is a non-cavitating diagram.

6.2 The effect of cavitation on open