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One can make a theoretical estimate of the proper diameter for a penstock in an impulse turbine installation, as in Fig. P11.95. Let L and H be known, and let the turbine performance be

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idealized by Eqs. (11.38) and (11.39). Account for friction loss hf in the penstock, but neglect

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minor losses. Show that (a) the maximum power is generated when hf=H/3, (b) the optimum jet velocity is

, and (c) the best nozzle diameter is Dj =

. Where fis the pipe

friction factor.

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The maximum power can be obtained when

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Where, vane velocity is Jet velocity is a) Hence the maximum power is given by,

Here, the density of fluid is Flow rate is Exit angle is

.

Jet area is, . Where, flow rate is Q Velocity of the jet is Hence, maximum power reduces to,

…… (1)

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Apply steady flow energy equation between the reservoir and the outlet jet.

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Chapter 11, Problem 95P

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…… (2)

Here,

is the head,

is the friction factor,

the pipe velocity and

is the pipe length,

is the pipe diameter,

is

is the jet velocity.

According to continuity equation,

…… (3) Where, the cross section area of pipe is Substitute

Substitute

.

in equation (2),

in equation (2),

The condition for maximum power,

…… (4) But, the friction head loss of pipe is,

Substitute

in equation (4)

Therefore, the maximum power is generated when

is proved.

b) From equation (2), the optimum jet velocity is,

Hence, the optimum jet velocity is

. Therefore, it is proved.

Comment

Step 2 of 2 c) We have,

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Chapter 11, Problem 95P

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…… (5) From equation (3) …… (6) From equations (5) and (6), …… (7)

Substitute

in equation (7)

Therefore, the best nozzle diameter is

.

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