DESIGN STEPS OF FRANCIS TURBINE

Main components of the francis turbine are sprial case,stay vanes,guide vanes,turbine runner and
draft tube.

The dimensions of these parts are dependent Practically all dimensions, features and characteristics of the desired turbine can be expressed as a function of the specific speed. Hence, specific speed is logical key to design of turbine and choice of turbine.

Ns=(N√P)/H^(5/4)  where,

N=the normal working speed(rpm)
P=power output of the turbine(kw)
H=effective head(m)

Discharge

H=effective head(ƞ_m)
O=overall efficiency(ƞ_h×ƞ_m )

where,
H=head of turbine
uniformly along the circumference to keep the fluid velocity constant in magnitude along its
path towards the stay vane/guide vane.

Draft tube transform velocity head to static head due to its increasing area and will reduce
effect of cavitation.

Tube Theory

P_2/w=P_a/w-H_s-((V_2^2-V_3^2)/2g-h_f )


, P_2/w is less than atm pressure.

Efficiency of draft tube
ƞ_d=(((V_2^2-V_3^2)/2gh_f ))/((V_2^2)/2g)

Where V2,P2 is velocity, pressure at section 2 V3 is velocity at section 3 Pa is atm pressure h_f is loss of energy between sections 2-2 and 3-3 mainly on the design discharge,head and the speed of the rotor of the generators.

Runner Design
The runner of a francis turbine is required is to be designed to develop a known power P when running at a known speed N rpm under a known head H.

The design of the runner involves the determination of its size and the vane angles.

Specific speed
It is determination of water flow rate transported through the turbine.

Q=P/(ƞ_O wH)

where,P=power output of the turbine(kw)
w=ρg(weight of fluid)

Diameter of Runner

At inlet(External)

D_1=√(Q/(〖(K〗_f √2gH)K_t πn))

At outlet(Internal)

D_2=D_1/2

where,
Q=discharge(m^3/s),
K_f=Flow ratio,K_f=V_f1/√2gH
n=B/D(ratio of width to diameter)
K_t=Vane thickness coefficient(<1)


Velocities

Flow Velocity
Flow velocity is the vector field that is used to describe fluid motion in a mathematical manner. The
entire length of the flow velocity is referred to as the flow speed.

At inlet
Vf1=Q/(K_t1 πD_1 B_1 )

At outlet
V_f1/V_f2 =(K_t2 πD_2 B_2)/(K_t1 πD_1 B_1)

Usually,it is presumed
V_f1=V_f2,K_t1=K_t2,B_2=2B_1

where,B_1=Width of runner vane at inlet(m)

B_2=Width of runner vane at outlet(m)

V_f1=V_f2=Velocity of flow(m/s)
V_w1=velocity of whirl(m/s)

Rim velocity(tangential velocity Tangential velocity is the linear speed of something
moving along a circular path(velocity along the rim).

At inlet
U1=(πD_1 N)/60

At outlet
U2=U1/2
where,

D_1=External diameter of runner(m)

N=the normal working speed(rpm)

Whirl velocity
Whirl velocity is the number of times in a second that a turbine can rotate, moving at a speed(velocity
due to rotation).

Vw1=(ƞ_h gH)/U1

where,
ƞ_h=(V_w1 U_1)/gH(hydraulic efficiency)
U_1=Tangential velocity at inlet(m/s)

Power
Wicket gates around the outside of the turbine's rotating runner control the rate of water flow
through the turbine for different power production rates.

P=ƞ_o wQH
Q=discharge(m^3/s)
ƞ_O=overall efficiency(ƞ_h×ƞ)
w=ρg(weight of fluid)


Number of Vanes
Number of Runner vanes
In order to avoid any pulsations,the runner vanes are Often made an odd number.
n=K√D
K=3.7 for slow runner
K=3 for normal runner
K=2.2 for fast runner

D=Runner diameter

Number of Guide vanes
n′=K′√D
K‘=2.5 for α_1 between 10° & 20°
K‘=3 for α_1 between 20° & 30°
K‘=3.5 for α_1 between 30° & 40°

The number of vanes varies from 16 to 24
Vane angle

Quide vane angle(α)
Tanα=V_f1/V_w2

Runner vane angle(θ) from inlet velocity triangle

Tanθ=V_f1/(V_w1-U1)
Runner vane angle(ø) from outlet velocity triangle

Tanø=V_f2/U2
where,U_1=Tangential velocity at inlet(m/s)
U_2=Tangential velocity at outlet(m/s)
V_f1=V_f2=Velocity of flow(m/s)
V_w1=velocity of whirl(m/s)

Design of Spiral Casing
The cross-sectional area of this casing decreases
At any angle q, the radius of casing is



Discharge per unit: Q

Draft tube





Draft tube transform velocity head to static head due to its increasing area and will reduce
effect of cavitation.

Tube Theory
P_2/w=P_a/w-H_s-((V_2^2-V_3^2)/2g-h_f ) , P_2/w is less than atm pressure.

Efficiency of draft tube

ƞ_d=(((V_2^2-V_3^2)/2gh_f))/((V_2^2)/2g)
Where V2,P2 is velocity, pressure at section 2
V3 is velocity at section
Pa is atm pressure
h_f is loss of energy between sections 2-2 and 3-3

Efficiencies