Velocity triangles in Fluid Mechanics and Machinery | Unsymmetrical vane | Turbomachinery

This video explains the force exerted by a jet, on asymmetrical curved blade, which is being used in hydroelectric power plants. The velocity triangles, at the inlet and outlet of the blades are drawn. At the inlet, the jet is impinging with a velocity V1.
As the blade is moving simultaneously, let’s take the velocity of the blade or vane, at the inlet as u1
The blade and the jet are moving simultaneously, the relative velocity between the jet and the blade is taken as V R1.
The angle made between the jet velocity and the movement of the blade is taken as alpha.
Likewise, the angle made between the relative velocity and the movement of the blade is taken as theta.
The horizontal component of the V1 will make the blade to swirl or move. Hence the horizontal component is named as Swirl velocity at the inlet, V W1.
The vertical component of the V1 will just flow over the blade. Hence the vertical component is named as flow velocity at the inlet, V F1.
The same thing in the outlet side of the vane is called outlet triangle and all the terms are given with a suffix 2.
We can observe that the jet velocity at the outlet is reduced, as some of the energy is spent to move the blade.
However, we can assume that the blade is smooth, and the relative velocity at the inlet, is equal to the relative velocity at the outlet.
If you assume that the blade is moving linearly, the vane velocity u1 is equal to u2
If we want to quantify the force exerted by the jet on the vane, we must know the mass of water striking the jet per second.
Mass flow of water per second is equal to rho, A, V R1
Therefore, force exerted in the direction of motion of the jet, is equal to, mass flow rate into, initial velocity minus final velocity, along the direction x.
Here the final velocity at the vane outlet, is against the direction of motion of the vane, hence the minus sign is provided there.
We can cancel the u1 and u2 as they are assumed to be the same.
Hence the force exerted by the jet on asymmetric vane is equal to rho, a, v r1 into V, W1 plus V, W2
This equation is valid only, if the vane angle at the outlet, beta is acute
If the angle beta is 90 degrees, the V W2 becomes zero. And the equation becomes as follows.
And, if the angle beta becomes obtuse, the v W2’s direction flips and the V W2 becomes negative. And the equation becomes as follows.
Hence, in general, the force exerted by a jet on an asymmetrical vane, is equal to Rho, A, V R1 into V W1, plus or minus V W2
We know that work done, is equal to force into velocity, hence the work done expression is derived

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