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Very Simple Kaplan Turbine Design

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Very Simple Kaplan Turbine Design
Grant Ingram 30th January 2007

Nomenclature b blade height g gravitational acceleration H head k loss coeﬁcient m mass ﬂow rate ˙ P power output Q volumetric ﬂow rate r radial direction R radius U blade speed V absolute velocity, subscripts denote stations and components W relative velocity, subscripts denote stations and components x axial direction α absolute ﬂow angle β relative ﬂow angle ω rotational speed θ tangential direction η efﬁciency 1, 2, 3, 4 stations through the machine

1

School of Engineering, Durham University

1

2

3 draft tube

4

inlet stator rotor

r x

Figure 1: General Arrangement of Kaplan

1 Introduction
This short note indicates how a preliminary design of an axail ﬂow Kaplan turbine can be carried out - see Figure 1 for a cross section of the device. Note that this analysis is approximate and is useful for a ﬁrst appoximation only. In order to carry out a preliminary blade analysis consider a mean radius through the machine. In order to draw or manufacture the blades you will need to know the inlet and exit angles of the stator (α1 and α2 ) and the rotor (β1 and β2 ). This is shown in Figure 2 In this analysis the effect of blade shape or number is not considered - to a ﬁrst approximation you can ignore them - simply pick a reasonable shape and a reasonable number of blades. The approach is to set the ﬂow rate through the machine and then calculate the power output. Once this has been determined the losses and therefore the overall head required to drive the machine are determined - if this exceeds the site head you will need to iterate your design. Parameters you need to set or guess:

• α1 Stator inlet angle, set up to line up with incoming ﬂow. • α2 Stator exit angle. A value needs to be selected and then checked following the results of this calculation (determines power output)

• β2 Rotor inlet angle, set to line up with relative inlet ﬂow at station 2. • β3 Rotor exit angle. A value needs to be selected and then checked following results of this calculation (determines power output)

• Q Flow rate, guess a value and then use this calculation to see if it is reasonable

• R2m , R3m Mean radii at station 2 and 3, pick at start of analysis • b2 , b3 Blade heights, pick at start of analysis

2

School of Engineering, Durham University

2

rotation

2 3

1

stator x rotor

Figure 2: Angles that need to determined for design

• ω Rotation speed of machine
All other quantities are derived from these selections.

2 Power Output
The Euler turbomachinery equation will give us the power output:

P = mω(R3mV3θ − R2mV2θ ) ˙
From velocity triangle at station 2:

(1)

V2θ = V2x tan α2 where: (2)

V2x =
From velocity triangle at station 3:

Q 2πR2m b2

(3)

V3θ = ωR3m +V3x tan β3 where: (4)

V3x =

Q 2πR3m b3

(5)

Remember that the angles are positive in the direction of rotation! To set β2 to line up with the exit ﬂow from the stator use the following expression:

β2 = tan−1

V2θ − ωR2m V2x

(6)

3

School of Engineering, Durham University

1

2 rotation V2 V3
3

3

V1

2 2

U W3

3

U W2

x

Figure 3: Velocity Triangles for the Axial Flow Kaplan

3 Efﬁciency and Required Head
Losses are modelled as ∆H = k different categories.
(velocity)2 2g

and can be classiﬁed into a number of

l • Pipe Loss The L2 course will explain how to calculate this, k = 4 f D
2 • Guide Vane Loss ∆Hgv = k 2g where k = 0.05 typically. 3 • Runner Loss ∆Hr = k 2g where k = 0.06 typically.

V2

W2

• Draft Tube Loss ∆Hdt = k

where k = 0.15 typically. To work out V4 select your draft tube radius and then use continutiy. i.e. ρA3V3x = ρA4V4x where A3 = 2πR3m b3 and A4 = πR2 4
V2

V32 −V42 2g

2 • Bend Losses ∆Hb = k 2g where k = 0.05 typically.

The actual head required is given by:

∆Hactual = ∆Hideal + ∑ ∆Hlosses
Typically ∆Hlosses = ∆Hgv + ∆Hr + ∆Hdt and:

(7)

∆Hideal =

P mg ˙

(8)

Finally the machine efﬁciency is given by:

η=

P mg∆Hactual ˙

(9)

4

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