UNIT 3
Euler’s Fundamental Equation
Consider a rotor rotating at an angular speed of ω rad/s
Consider points one and two on the rotor at radii r1 and r2. As in that the fluid enters at point 1 and leaves at point 2.
Let, m = mass flow rate of fluid in kg per second
V1 = absolute velocity of fluid at point 1 in meters per second
V2 = absolute velocity of fluid at point 2 in meters per second
V1 = Vf1 + VR1 + Vw1
V2 = Vf2 + VR2 + Vw2
The energy transferred between the fluid and machine is only due to change in momentum caused by tangential component of velocity.
Therefore, the rate of change of momentum between points 1 and 2
= m Vw1 - m Vw2
The momentum at point 1 = m Vw1 r1
The momentum at point 2 = m Vw2 r2
This rate of change is momentum represent torque produced on the rotor.
Torque, T = rate of change of angular momentum
= m Vw1 r1 - m Vw2 r2
= m (Vw1 r1 - Vw2 r2 ) -------------(1)
In case of power producing machines like turbine, the torque is produced on the rotor due to change of angular momentum, While, in case of power absorbing devices like pumps and compressors, the torque given to the rotor is responsible for causing the change in tangential velocity of fluid.
Consider the machine as turbine. The torque produced by the fluid on rotor is used in producing useful work or developing power. Thus,
Rate of energy transfer, (or Power P).
E = Torque, T x Angular velocity of rotor, ω
i.e. E = T ω = m (Vw1 r1 – Vw2 r2 ) ω -------------(2)
But, peripheral velocity of fluid, (u = ωr) at points 1 and 2 can be written as, u1 = ω r1 and u2 = ω r2.
E = mf (Vw1 u1 – Vw2 u2) -------------(3)
Equation (3) represents the general energy equation for transfer of energy between the fluid and machine.
Energy transfer per unit mass i.e. Work Done or W
E or W = (Vw1 u1 – Vw2 u2) -------------(4)
If H is the head on the machine, then energy transfer can be written as,
E = mf g H -------------(5)
Equating Equations (3) and (5), we get,
H = (1/g) (Vw1 u1 – Vw2 u2) -------------(6)
Equations (2) to (6) are forms of Euler's Equation which are applicable to all turbomachines.
Note:
1. If Vw1 u1 > Vw2 u2, then machine is called turbine
2. If Vw2 u2 > Vw1 u1, then machine is called a pump or compressor
3. If Vw2 is in opposite direction of Vw1, the Equation (2) can be modified as:
E = mf ( Vw1 u1 + Vw2 u2)
Degree of reaction, R of a runner is defined as the ratio of pressure energy change inside the runner to the total energy change inside the runner. Accordingly,
R = (change in pressure energy inside the runner, Hp) / (change in total energy inside the runner, Ht )
But, total energy change inside runner is equal to work done per unit weight of water given by
Ht = (1/g) (Vw1 u1 ± Vw2 u2) -------------(i)
From inlet velocity diagram shown in above diagram, we deduce,
Vw1 = AB + BD = u1 + √{Vr12 - Vf12} = u1 + √{Vr12 - (V12 - Vw12)}
Vw1 - u1 = √{Vr12 - (V12 - Vw12)}
Squaring both the sides,
(Vw1 - u1)2 = Vr12 - (V12 - Vw12)
Expanding the brackets and rearranging the terms,
Vw1 u1 = ½ {u1 + V12 - Vr12} -------------(ii)
Similarly, from the outlet velocity diagram,
Vw2 = EH – EF = √{EG2 - GH2} - EF = √{Vr22 - Vf22} - u2
Vw2 + u2 = √{Vr22 - (V22 - Vw22) }
Squaring both sides, expanding and rearranging,
Vw2 u2 = ½ {Vr22 - u1 - V12} -------------(iii)
Substituting (ii) and (iii) in (i),
Change in total energy in runner,
H = (1/2g) {V12 - V22 + u12 - u22 - Vr22 - Vr12}
{V12 - V22} / 2g = Decrease in KE of water per unit weight.
{u12 - u22} / 2g = Decrease in energy due to centrifugal action per unit weight of water which is converted into pressure energy. Therefore, this energy is available to turbine shaft.
{Vr22 - Vr12} / 2g = Change in static pressure energy per unit weight.
Hp = Ht - (1/2g) {V22 - V12}
R = 1 - [{V22 - V12} / 2g Ht]
References: -
Text Books:
1. G. T. Mase, R. E. Smelser and G. E. Mase, Continuum Mechanics for Engineers, Third
Edition, CRC Press,2004.
2. Y. C. Fung, Foundations of Solid Mechanics, Prentice Hall International, 1965.
3. Lawrence. E. Malvern, Introduction to Mechanics of a Continuous Medium, Prentice Hall
International, 1969.
4. Hydrantic Machine by Jagdish Lal
5. Hydraulics & Hydraulic Machines by Vasandari
6. Hydrantic Machine by RD Purohit