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An analysis of Axial Flow Turbines using both the |Euler and Aerofoi;l Theories

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AXIAL FLOW TURBINES

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13108/img_axt1.jpg

As with a Pump consider an element at radius r.

13108/img_axt2.jpg

$v = v_1 = \frac{2\pi RN}{60}$

(1)

$Q = \pi \left(R_0^2 - R_{Boss}^2 \right)V_f$

(2)

$V_1 = V_{f1} = V_f$

(3)

The Euler Theory

Work done by the water on the Vanes per lb. $= \frac{V_w\,v}{g}$ ( It is assumed that there is no whirl at exit)

Applying Bernoulli's equation across the Vanes and assuming no losses,

$\frac{p}{w} + \frac{V^2}{2g} = \frac{p_1}{w} + \frac{V_1^2}{2g} + \frac{V_w\,v}{g}$

(4)

$\frac{p - p_1}{w} = \frac{V_w\,v}{g} + \frac{V_1^2 - V^2}{2g}$

(5)

$But\;since\;V^2 = V_w^2 + V_f^2 = V_w^2 + V_1^2$

(6)

$\frac{p - p_1}{w} = \frac{V_w\,v}{g} - \frac{V_w^2}{2g}$

(7)

$Hydraulic\;\eta = \frac{V_w\,v}{gH}$

(8)

The Aerofoil Theory used for Turbines.

Combining the Velocity Triangles

13108/img_axt3.jpg

The following diagram is a sketch through one of the Vanes.

13108/img_axt4.jpg

Consider the Vanes as Aerofoils in a fluid stream of Velocity $V_{r\,ave.}$ in a direction $theta _{ave.}$

For an elemental thickness dR at a radius R

$Lift\;\;\L = C_L\times \frac{1}{2}\;\rho\; V_{r\,ave.}^2\times Chord\times dR\;\;\;lbs.$

(9)

$Drag\;\;\D = C_d\times \frac{1}{2}\;\rho\; V_{r\,ave.}^2\times Chord\times dR\;\;\;lbs.$

(10)

The vector sum of L and D is the resultant force P on the Vane element.

The components of P in the tangential Direction is given by:-

$P_t = L\;Sin\,\theta _{ave} - D\;Cos\,\theta _{ave.}\;\;\;lbs.$

(11)

And the Axial component is :-

$P_a = L\;Cos\,\theta _{ave} + D\;Sin\,\theta _{ave.}\;\;\;lbs.$

(12)

For n Vanes

The Total Tangential Force = n Pt

The Torque T = n Pt X R

The Horse Power output

$= \frac{2\pi NT}{550\times 60}$

(13)

Total Axial Force = n Pa = The Thrust on the Bearings