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Performance Characteristics of Francis Turbine

INTRODUCTION

The Francis turbine is a reaction turbine, in which the fluid changes pressure as it moves through the turbine, giving up its energy. The turbine is located between the high pressure water source and the low pressure water exit. The inlet of turbine is in spiral shape. The guide vanes direct the water tangentially to the turbine wheel which is known as a runner. This radial flow of water acts on the runner's vanes, causing the runner to spin. The guide vanes may be adjustable to allow efficient turbine operation for a range of water flow conditions. At the point of exit, the water leaves with no swirl and very little kinetic or potential energy. The shape of the turbine's exit tube helps to decelerate the flow of water and recover the pressure.

Sectional View of Francis Turbine
Source: (https://www.researchgate.net/figure/Basic-features-of-a-Francis-turbine-6_fig1_326934034)

Main parts:

1. Spiral casing: The runner is completely enclosed in an air-tight spiral casing. The casing and runner are always full of water. The water from the penstocks enters the casing in which area of cross-section of the casing goes on decreasing gradually. Since the casing is spiral shape, the water enters the runner at constant velocity throughout the circumference of the runner.

2. Guide vanes: The guide vanes allow the water to strike the vanes fixed on the runner without any shake at the inlet. Also by a suitable arrangement, the width between two adjacent vanes of a guide’s mechanism can be adjusted, so that the amount of water striking the runner can be varied.

3. Runner/Impeller: It is a circular wheel on which a series of radial curved vanes are fixed and the surface of the vanes is made very smooth. The radial curved vanes are so shaped that the water enters and leaves the runner without shock. The runners are made of cast steel, cast iron or stainless steel. The runner is connected to the shaft.

4. Draft tube: Draft tube is or pipe of gradually increasing area which is used for discharging the water from the turbine to the tail race. The pressure at the exit of the runner is generally less than atmosphere pressure. Hence the water is discharged to tail race through the draft tube.

Let,

R_1 = Radius of wheel at inlet of the vane

R_2 = Radius of wheel at outlet of the vane

$\omega$ = Angular speed of the wheel

Tangential speed of the vane at inlet = $u_1 = \omeaga R_1$

Tangential speed of the vane at outlet = $u_2 = \omeaga R_2$

The velocity triangles at inlet and outlet are drawn as shown in figure. $\alpha$ and $\beta$ are the angles between the absolute velocities of jet and vane at inlet and outlet respectively. $\theta$ and $\psi$ are vane angles at inlet and outlet respectively.

The mass of water striking a series of vanes per second = $\rho a V_1$

where, a is the area of jet or flow and V_1 is the velocity of flow at inlet.

The momentum of water striking a series of vanes per second at inlet is given by the product of mass of water striking per second and the component of velocity of flow at inlet = $\rho a V_1 \times V_{w1}$ , where $V_{w1}$ is the velocity component of flow at inlet along tangential direction. Similarly momentum of water striking a series of vanes per second at outlet is given by $\rho a V_1 \times V_{w1}$ , where $-V_{w1}$ is the velocity component of flow at inlet along tangential direction and the -ve sign is because the velocity component is acting in the opposite direction. Now angular momentum per second at inlet is given by the product of momentum of water at inlet and its radial distance = $\rho a V_1 \times V_{w1}\times R_1$ and angular momentum per second at inlet =− $\rho a V_1 \times V_{w2}\times R_2$

Torque exerted by water on the wheel is given by impulse momentum theorem as the rate of change of angular momentum

$T = \rho aV_1 \times V_{w1} \times R_1 - (-\rho aV_1\times V_{w2}\times R_2)$ $T = \rho a V_1 (V_{w1}R_1+ V_{w2} R_2)$

Work done per second on the wheel = Torque x Angular velocity

= $T.\omega$ = $\rho aV_1 (V_{w1}R_1+ V_{w2}R_2) \times \omega$ = $\rho aV_1(V_{w1}R_1\times\omega+V_{w2}R_2\times\omega)$

As $u_1 = \omega R_1$ and $u_2 = \omega R_2$

We can simplify the above equation as

$WD/s = \rho aV_1 (V_{w1}u_1+ V_{w2}u_2)$

In the above case, always the velocity of whirl at outlet is given by both magnitude and direction as

$V_{w2} = (V_{r2} \cos\phi -u_2 )$

If the discharge is radial at outlet, then $V_{w2} = 0$ and

Hence the equation reduces to,

$WD/s = \rho a u_1 V_1 V_{w1}$ $KE/s = \frac{1}{2} \rho aV_1^3$

Efficiency of the reaction turbine is given by,

$\eta = \frac{workdone/sec}{power developed/sec} = \frac{\rho a (V_{w1}+ V_{w2})}{\frac{1}{2} \rho aV_1^3}$

Note: The value of the velocity of whirl at outlet is to be substituted as $V_{w2}= (V_{r2} \cos \phi -u_2)$ along with its sign.

= $\frac {2(u_1V_{w1}+ u_2V_{w2})}{V_1^2}$ Speed ratio = $\frac {u_1}{\sqrt{2gH}}$

Where, H is the Head on turbine, it varies from 0.6 to 0.9.

Flow ratio= $\frac {V_{f1}}{\sqrt{2gH}}$

Where, $V_{f1}$ is the velocity of flow at inlet, it varies from 0.15 to 0.3.

Discharge flowing through the reaction turbine is given by

$Q = \frac{\rho Q(V_{w1}u_1 \pm V_{w2}u_2 )}{\rho g H Q} = \frac{1}{gh}(V_{w1}u_1 \pm V_{w2}u_2 )$

Where, D_1 and D_2 are the diameter of the runner at inlet and outlet. B_1 and B_2 are the width of the runner at inlet and outlet. $V_{f1}$ and $V_{f2}$ velocity at inlet and outlet. If the thickness (t) of the vane is to be considered, then

The area through which flow takes place is given by

$=(\pi D_1-nt)$

Where, n is the number of vanes mounted on the runner.

Discharge flowing through the reaction turbine is given by

$Q= (\pi D_1-nt) B_1V_{f1} = (\pi D_2-nt) B_2V_{f2}$

The head (H) on the turbine is given by

$= \frac{P_1}{\rho g }+\frac{V_1^2}{2g}$ $u_1 = \frac{\pi DN}{60}$ and $u_2 = \frac{\pi DN}{60}$

Hydraulic efficiency is

$= \frac{\rho Q(V_{w1}u_1 \pm V_{w2}u_2 )}{\rho g H Q} = \frac{1}{gh}(V_{w1}u_1 \pm V_{w2}u_2 )$

If the discharge at the exit is radial then $V_{w2}=0$ ,

Hence hydraulic efficiency is

$\frac{1}{gH}(V_{w1}u_1)$

Comparison of Francis turbine and Kaplan turbine

Francis Turbine	Kaplan Turbine
Normal range of head:30-550m	Normal range of head:1.5-80m
Number of vanes:15-25	Number of vanes:3-8
Specific speed:50-250 60-400	Specific speed:300-1000
Cavitation susceptibility	Greater cavitation susceptibility
Overall efficiency is good	Overall efficiency is better than Francis turbine
Requires large area	Requires less area
Normal speed of runner:90-1000	Normal speed of runner:70-600
Flow of water is radially and mixed flow type	Flow of water is axial
Flow regulation is by guide vanes	Flow regulation is by needle valve fitted into nozzle
Turbines works by medium discharge at average head	Turbines works by high discharge low head

Cavitation:

Cavitation is an effect mainly on the reaction turbine when water pressure drops below the critical pressure (generally vapour pressure) at constant temperature and vapour cavities are formed and grow based on the dynamic pressure reduction. Cavitation in hydraulic machines negatively affects their performance and may cause severe damages. Cavitation commonly occurs in hydraulic turbines, around runner exit and in the draft tube.

Cavitation is a phenomenon occurs when the static pressure of the liquid falls below its vapour pressure, the liquid boils and large number of small bubbles of vapours are formed. These bubbles mainly formed due to low pressure and carried by the stream to higher pressure zones where the vapours condense and the bubbles suddenly collapse, as the vapours are condensed to liquid again. This results in the formation of a cavity and the surrounding liquid rushes to fill it from all direction which collide at the centre of cavity giving rise to a very high local pressure whose magnitude may be as high as 7000 atm. Formation of cavity and high pressure are repeated many thousand times a second. This causes pitting on the metallic surface of runner blades or draft tube. The material then fails by fatigue, added by corrosion.

There are two ways to reduce the cavitation damage. One involves optimizing the hydraulic design of equipment and the other involves developing coatings for the substrates of wetted parts, which can prolong the overhaul interval of hydraulic components.

Cavitation in general is slow process but the effect of cavitation is severe. Damaged caused by cavitation, if summarized are: erosion of material from turbine parts, distortion of blade angle, loss of efficiency due to erosion/distortion (Santa JF et al, 2009) . Prof. D Thoma Suggested a dimensionless number called as Thoma's Cavitation factor σ (sigma), which can be used for determining the region where cavitation takes place in reaction turbines

$\sigma = \frac{H_a - H_v - H_s}{H}---(1)$

Where,

H_a is the atmospheric pressure head in m of water

H_s is the suction pressure at outlet of reaction turbine in m of water or height of turbine runner above the tail water surface,

H_v is vapour pressure head,

is the net head on the turbine in m of water

The value of $\sigma$ depends on N_s (specific speed) of the turbine and for a turbine of given N_s the factor $\sigma$ can be reduced up-to a certain value up to which its efficiency, $\eta_o$ remains constant. A further decrease in value of $\sigma$ can be reduced up-to a certain value up to which its efficiency, $\eta_o$ remains constant. A further decrease in value of $\sigma$ results in a sharp fall in $\eta_o$ . The value of $\sigma$ at this turning point is called critical cavitation factor $\sigma_c$ . The value of $\sigma$ for different turbines may be determined with the help of following empirical relationships:

For Francis Turbine:

$\sigma^c = 0.625 \times \frac{N_s}{380.78}^2 ----(2)$

The values of $\sigma$ from equation (1) is compared with the value of $\sigma_c$ from equation (2) and if value of $\sigma$ is greater than $\sigma_c$ , cavitation will not occur in that turbine.

Francis Turbine Working Principle

Source: (https://www.ishwaranand.com/2020/09/francis-turbine.html)

Performance characteristics of Francis Turbine:

Performance characteristics are of three types:

a. Constant head characteristics:

In order to obtain these curves the tests are performed on the turbine by maintaining a constant head and a constant gate opening and the speed is raised by changing the load on the turbine.

Constant Head Characteristics
Source: (http://mechanics4change.blogspot.com/2017/12/turbine-characteristic-curve.html)

b. Constant speed characteristics:

In order to obtain these curves the tests are performed on the turbine by operating them at constant speed.

Constant Speed Characteristics
Source: (http://mechanics4change.blogspot.com/2017/12/turbine-characteristic-curve.html)

c. Constant efficiency curves:

These curves show the efficiency of the turbine for all conditions of running and hence these are also known as universal characteristic curves.

Constant Speed Characteristics
Source: (https://www.researchgate.net/figure/Efficiency-hill-chart-of-existing-turbine-Q-11opt-073-N-11opt-65-and-opt-92_fig7_245356930)