Question

show that the efficiency of a free jet striking normally on a series of flat plates mounted on the periphery of a wheel can never exceed 50%?​

Answer 1

Given:

A free jet striking normally on a series of flat plates mounted on the periphery of a wheel.

To Find:

Show that the efficiency of jet can never exceed 50%.

Solution:

Let we take:

$\mathbf{\textrm{Density of fluid }\ = \rho \ \dfrac{kg}{m^{3}}}$

$\mathbf{\textrm{Velocity of fluid }\ = v \ \dfrac{m}{s}}$

$\mathbf{\textrm{Speed of wheel }\ = u \ \dfrac{m}{s}}$

$\mathbf{\textrm{Exit area of nozzle }\ = a \ (m^{2})}$

Mass of fluid strike with blade is :

$\mathbf{\textrm{Mass of fluid }\ = m =\rho \times a\times V \ \ \ \ \ \dfrac{kg}{s}}$

$\mathbf{\textrm{kinetic energy of fluid }\ = K.E =\dfrac{1}{2}\rho \times a\times V \times V^{2}\ \ \ \ \ (J)}$      ...1)

Relative velocity of jet strike with blade with respect to blade = V- u

Work done by jet on blade:

$\mathbf{\textrm{Work done }\ = W =\rho \times a\times V(V-u)\times u\ \ \ \ \ \ (J)}$        ...2)

$\mathbf{\textrm{Efficiency }\ = \eta =\dfrac{Work\ done\ by\ fluid}{Kinetic \ energy\ of\ fluid}}$

$\mathbf{\textrm{Efficiency }\ = \eta =\dfrac{\rho \times a\times V(V-u)\times u}{\dfrac{1}{2}\rho \times a\times V \times V^{2}}}$

On simplify:

$\mathbf{\textrm{Efficiency }\ = \eta =\dfrac{2\times (V-u)\times u}{V^{2}}}$         ...3)

From here, we see that efficiency of jet depend upon speed of wheel for a particular value of speed of jet:

On differentiating efficiency with respect to blade speed , we get:

$\mathbf{u = \dfrac{V}{2}}$                       ...4)

This is required condition of blade speed for maximum efficiency of jet:

Putting value of u from equation 4  to equation 3):

$\mathbf{\textrm{Maximum Efficiency }\ = \eta _{max}=\dfrac{2\times (V-\frac{V}{2})\times \frac{V}{2}}{V^{2}}}$

On solving , we get:

$\mathbf{\textrm{Maximum Efficiency }\ = \eta _{max}=0.5 = 50}$%

Hence prove that the efficiency of a free jet striking normally on a series of flat plates mounted on the periphery of a wheel can never exceed 50%.