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# Design of a Kaplan Turbine

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The design of the blade does not just depend on the stress analysis; several other factors play significant roles as well. The leading edge is thicker than the trailing edge for a streamlined Flow. Furthermore, the blade should be as thin as possible to improve cavitation Characteristics; it is thicker near the flange becoming thinner and thinner towards the tip. In Addition, the blade has to be distorted on the basis of the tangential velocity.
Blade design is the most complex thing in Kaplan turbine. It consists of six steps. 1. Velocity triangle is evaluated at the leading and Trailing edge of the blade. 2. Angle of distortion of the chord lengths (β∞). 3. Lift Coefficients. 4. Chord length to Spacing (L/t) ratio. 5. Drag Coefficient. 6. Profile. 2.1. Velocity Triangle:
U
U
As shown in the figure different types of velocities occur as the fluid flows from the blades of this turbine. Thorough understanding of the velocity triangle (fig 1. 1) is necessary for a good design.
Figure 1.1
Figure 1.1 β∞ β∞
Wu
Wu
Cu
Cu
Wm
Wm
Cm
Cm

Tangential Flow velocity ……………… (1.2)
Relative Tangential Velocity……………. (1.3)
Relative Axial Velocity…………………… (1.4)
Where,

U = blade Tangential velocity [m/s]
Wm= Axial Component relative velocity [m/s].
Cm = Axial Component flow Velocity [m/s].
Cu= Tangential Component flow velocity [m/s].
Wu= Tangential Component relative velocity [m/s]. ω = Rotational Speed [rad/s]. β∞ = Angle of Distortion [o].

When a cylindrical cut is set at the runner (fig 1.2) and the cut is developed into a drawing pane, Grating like that shown in fig occurs. Velocity triangle 1 occurs directly before the grating and the velocity triangle 2 occurs directly after the grating. The relative velocity components w1m and w2m are equal. The medial relative velocity (Wu∞) can be determined via the average of w1 and w2 and its direction is specified due to the angle (β∞). Value (t) represents the Spacing and L denotes the length chord.

Figure 1.2
Figure 1.2 L L
Trailing edge
Trailing edge
(Wu1+Wu2)/2

(Wu1+Wu2)/2

Wm1=Wm2
Wm1=Wm2
W2
W2
W1
W1
t t U
U
W2
W2
C2
C2
U
U
W1

W1

C1
C1

1.1. 1.2. 1.3.1. 2.2. Angle of Distortion (β∞):

To define the distortion of the blade, the velocity triangles at tip and hub radiuses of the blade are determined. The angle (β∞) of each radius gives conclusions on the distortion of the blade.

Top view of the Runner
Top view of the Runner

Table 1. Results PARAMETER | TIP | HUB | UNITS | d | | | [m] | u | | | [m/s] | Cu1 | | | [m/s] | Cu2 | | | [m/s] | Wu1 | | | [m/s] | Wu2 | | | [m/s] | Wu∞ | | | [m/s] | Wm | | | [m/s] | W1 | | | [m/s] | W2 | | | [m/s] | W∞ | | | [m/s] | β∞ | | | [°] | (180-β∞) | | | [°] |

A table like given above should be created to ease the calculations and the design procedure. Following formulas should be used.

u = π* n * d ………………………………………………………………………………… (1.5)

Cu=Hn * g…………………………………………………………………………………….. (1.6)

Wu=Cu – U……………………………………………………………………………………. (1.7)

Wm=Q/A∞………………………………………………………………………………..…. (1.8)

W= (Wu2 +Wm2).5………………………………………………………………………….. (1.9)

β∞=cos-1(Wu∞/W∞)…………………………………………………………………… (1.10)

2.3. Lifting coefficient (ζa):
Lifting coefficient can be found out by using the following formula

………..…… (1.11)
Where,
W2=relative velocity after the grating [m/s]
W∞= medial relative velocity [m/s]
Patm =atmospheric pressure [pa]
Pmin= minimal water pressure [pa]
Ηs*= efficiency of the energy change [-] =.88-.91
C3=velocity after the runner [m/s]
C4 =outlet velocity [m/s]
K = profile characteristic number [-] =2.6-3

2.4.1. Chord length to Spacing Ratio (L/t):

Following formula is used for calculating the L/t ratio
…………………… (1.12)
Where,
g =acceleration of gravity [m/s2] ηh= hydraulic efficiency [-]
Cm=meridian velocity [m/s] λ = angle of slip [o] u =tangential velocity [m/s]
(180-β∞)= inflow angle [o]

In the above equation the angle of slip λ has to be assumed; the range for the assumption is as follows λ = 2.5°÷3°, Using this assumption, an approximate value of the ratio L/t (tip) can be established.

Note: There is also another criterion for finding (l/t) ratios at hub and the tip*. Guidelines in "Hydraulic Machines and Installations: Volume 2: Water Turbines" by Joachim Raabe are given on p.131, depending on the specific speed. For lower specific speed axial turbines, he recommends: l/t = 1.8 at the hub, 1.0 at the tip; for high specific speed: 1.3 at the hub; 0.7 at the tip. 2.4. Lift coefficient (ζA):

Figure 1.4 showing variation of ζa/ζ Awith t/l
Figure 1.4 showing variation of ζa/ζ Awith t/l

After determining the lifting coefficient (ζa)(at tip) and the ratio(t/L)(at tip). The value of lift coefficient (ζA) (at tip) is determined via this chart.
For example for t/l= 0.9 the value of ζa/ζ A comes out to be 0.5.

2.5. GOE432
GOE432
Figure 1.5 showing variation of ζA with ζw
Figure 1.5 showing variation of ζA with ζw
Drag Coefficient (ζ W):

To use the above chart first, it has to be decided which of the profiles should be chosen, each of the curves represents one of the profiles which is listed beside the chart. Following this, the drag coefficient (at tip) of this profile can be determined by using the graph. An example of GOE 432 is shown.

2.6. Profile selection:

With the following equation, the angle of slip can be calculated:

λ=tan-1(ζw/ζA)………. (1.13)

It has to be checked whether the assumed angle of slip and the calculated angle of slip are similar or not. If the difference is too great, the procedure of the calculations is to be repeated. Steps must be repeated until the angles of slip do not change anymore; however, it is necessary to always choose the same profile, when the angle λ is fixed. After this the ratio l/t and the profile are determined. This step concludes the design of the blade process.
References:

* H.C Radha Krishnay, Hydraulic design of Hydraulic Machinery, Avebury, 1992. * S.L Dixon, Fluid Mechanics and Thermodynamics of Turbo machinery, 1998. * Timor Flashpohler, Final, Thesis University of Tampere Finland, 2007. * Grant Ingram,Basic Concepts in Turbomachinery,Ventus Publishings ApS,2009. * Adam Harvey, Micro-Hydro Design Manual, Immediate Technology Publications, 1999. * Vishnu Parsad, Ruchi Khare, Abhas Chnicholikar, Hydraulic performance of elbow type draft tube for different geometric configuration using CFD, Department of Civil Engineering M.A. National Institute of Technology. IGHEM 2010, Oct2123, 2010, AHEC, IIT Roorkee, India. * R.K Turton, Principles of Turbo Machinery, Chapmann and Hall, 1995.

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