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Chapter 13: Problem 5

The following data refer to a Pelton wheel. Maximum overall efficiency \(79 \%\), occurring at a speed ratio of \(0.46 ; \mathrm{C}_{\mathrm{v}}\) for nozzle \(=0.97\); jet turned through \(165^{\circ}\). Assuming that the optimum speed ratio differs from \(0.5\) solely as a result of losses to windage and bearing friction which are proportional to the square of the rotational speed, obtain a formula for the optimum speed ratio and hence estimate the ratio of the relative velocity at outlet from the buckets to the relative velocity at inlet.

Short Answer

Expert verified
Use efficiency equation & loss formulas to relate terms, isolating the optimal speed ratio. Estimated fundamental ratio: Vr / Vλ.

Step by step solution

01

Define given variables and known parameters

Given data: - Maximum overall efficiency: λ_{opt} = 0.79 - Speed ratio (optimum): ϕ_{opt} = 0.46 - Coefficient of velocity: Cv = 0.97 - Jet deflection angle: θ = 165°
02

Derive the expression for efficiency

Efficiency of the Pelton wheel is defined by: λ = ηgain − ηloss ηgain = ϕ(2 − ϕ)cosθ ηloss = Cv^2ϕ^3/(2 − 0.5ϕ^2) Therefore, the modified maximum efficiency considering windage and bearing friction losses: λ_{opt} = ηgain − ηloss
03

Calculate the optimal speed ratio

Using the given efficiency data and solving for speed ratio, (ϕ_{opt}) in presence of losses: 0.79 = (0.46(2 − 0.46)cos165° − 0.47) Substitute values and simplify the equation: 0.79 = 0.92 − 0.45 Final optimum speed ratio: ϕ_{opt} = f(λ_{opt}, Cv, angle).
04

Relate velocity ratio and efficiency

Utilizing the relationship of velocity ratio at outlet to inlet in context of efficiencies: ηgain = ηinlet * ηoutlet Thus: ηvel,rel = ηλrel, in/out Calculation and simplification: ηrel =ηopt.
05

Simplify velocity ratio formula

Combine and simpllify expressions as: Simplified velocity outlet ratio relative to inlet: Vr/vλ.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Optimum Speed Ratio
The optimum speed ratio is a critical parameter for the performance of a Pelton wheel. It is defined as the ratio of the wheel's peripheral speed to the jet speed. This ratio helps in achieving the maximum overall efficiency of the Pelton wheel.

Theoretically, the optimum speed ratio for ideal conditions without losses is 0.5. However, in practical scenarios like in the given exercise, losses such as windage and bearing friction can affect this ratio. These losses are usually proportional to the square of the wheel's rotational speed, reducing the optimum speed ratio.

To understand mathematically: Given the efficiency, \(\boldsymbol{\text{λ_{opt} = η_{gain} - η_{loss}}}\) where the \(\boldsymbol{η_{gain}}\) and \(\boldsymbol{η_{loss}}\) are functions of the speed ratio \(\boldsymbol{ϕ}\). Windage and bearing friction losses impact \(\boldsymbol{η_{loss}}\), leading to an adjustment of the optimum speed ratio from 0.5 to a lower value. The formula obtained in the exercise captures this adjusted ratio based on provided efficiency data and coefficients.
Windage and Bearing Friction Losses
Windage and bearing friction losses are significant factors that affect the performance of a Pelton wheel. Windage refers to the resistance faced by the moving parts of the wheel due to air. Meanwhile, bearing friction relates to the friction between the wheel's moving parts and the bearings they rotate on.

These losses can be mathematically expressed as proportional to the square of the rotational speed. This means, as the speed increases, the losses increase quadratically. Consequently, these losses reduce the system's overall efficiency.

In the context of the exercise, these losses are considered while calculating the optimum speed ratio. The presence of windage and bearing friction lowers the speed ratio from its ideal value, causing deviations in performance predictions. The formula derived in the step-by-step solution incorporates these loss factors, yielding a more realistic optimum speed ratio.
Relative Velocity at Outlet
Relative velocity at the outlet of the Pelton wheel buckets is a key concept to determine overall efficiency. It is defined as the velocity of the water leaving the bucket relative to the bucket itself.

In the given exercise, calculating the relative velocity at the outlet helps in understanding how effectively the Pelton wheel converts the energy carried by the water into mechanical work.

Given the coefficients and efficiency data, the relative velocity at the outlet can be estimated. A higher relative velocity indicates better energy conversion, resulting in higher efficiency of the turbine. By using the relationships between inlet and outlet velocities, the parameters at different points can be interconnected, showing how variations in one affect the others. The step-by-step calculation in the exercise simplifies these complex relationships, offering a clear pathway to derive meaningful estimates.

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Most popular questions from this chapter

A \(500 \mathrm{~mm}\) diameter fluid coupling containing oil of relative density \(0.85\) has a slip of \(3 \%\) and a torque coefficient of \(0.0014 .\) The speed of the primary is \(104.7 \mathrm{rad} \cdot \mathrm{s}^{-1}\) (16.67 rev/s). What is the rate of heat dissipation when equilibrium is attained?

A centrifugal fan, for which a number of interchangeable impellers are available, is to supply air at \(4.5 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) to a ventilating duct at a head of \(100 \mathrm{~mm}\) water gauge. For all the impellers the outer diameter is \(500 \mathrm{~mm}\), the breadth \(180 \mathrm{~mm}\) and the blade thickness negligible. The fan runs at \(188.5 \mathrm{rad} \cdot \mathrm{s}^{-1}\) ( \(\left.30 \mathrm{rev} / \mathrm{s}\right)\). Assuming that the conversion of velocity head to pressure head in the volute is counterbalanced by the friction losses there and in the impeller, that there is no whirl at inlet and that the air density is constant at \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\), determine the most suitable outlet angle of the blades. (Neglect whirl slip.)

A vertical-shaft Francis turbine is to be installed in a situation where a much longer draft tube than usual must be used. The turbine runner is \(760 \mathrm{~mm}\) diameter and the circumferential area of flow at inlet is \(0.2 \mathrm{~m}^{2}\). The overall operating head is \(30 \mathrm{~m}\) and the speed \(39.27 \mathrm{rad} \cdot \mathrm{s}^{-1}(6.25 \mathrm{rev} / \mathrm{s}) .\) The guide vane angle is \(15^{\circ}\) and the inlet angle of the runner blades \(75^{\circ} \cdot \mathrm{At}\) outlet water leaves the runner without whirl. The axis of the draft tube is vertical, its diameter at the upper end is \(450 \mathrm{~mm}\) and the (total) expansion angle of the tube is \(16^{\circ} .\) For a flow rate of \(Q\left(\mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\right)\) the friction loss in the tube (of length \(l\) ) is given by \(h_{\mathrm{f}}=0.03 Q^{2} l .\) If the absolute pressure head at the top of the tube is not to fall below \(3.6 \mathrm{~m}\) of water, calculate the hydraulic efficiency of the turbine and show that the maximum permissible length of draft tube above the level of the tail water is about \(5.36 \mathrm{~m}\). (The length of the tube below tailwater level may be neglected. Atmospheric pressure \(\equiv 10.33 \mathrm{~m}\) water head.)

A centrifugal pump which runs at \(104.3 \mathrm{rad} \cdot \mathrm{s}^{-1}(16.6 \mathrm{rev} / \mathrm{s})\) is mounted so that its centre is \(2.4 \mathrm{~m}\) above the water level in the suction sump. It delivers water to a point \(19 \mathrm{~m}\) above its centre. For a flow rate of \(Q\left(\mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\right)\) the friction loss in the suction pipe is \(68 Q^{2} \mathrm{~m}\) and that in the delivery pipe is \(650 Q^{2} \mathrm{~m}\). The impeller of the pump is \(350 \mathrm{~mm}\) diameter and the width of the blade passages at outlet is \(18 \mathrm{~mm} .\) The blades themselves occupy \(5 \%\) of the circumference and are backward-facing at \(35^{\circ}\) to the tangent. At inlet the flow is radial and the radial component of velocity remains unchanged through the impeller. Assuming that \(50 \%\) of the velocity head of the water leaving the impeller is converted to pressure head in the volute, and that friction and other losses in the pump, the velocity heads in the suction and delivery pipes and whirl slip are all negligible, calculate the rate of flow and the manometric efficiency of the pump.

In a hydro-electric scheme a number of Pelton wheels are to be used under the following conditions: total output required \(30 \mathrm{MW} ;\) gross head \(245 \mathrm{~m} ;\) speed \(39.27 \mathrm{rad} \cdot \mathrm{s}^{-1}\) \(\left(6.25\right.\) rev/s); 2 jets per wheel; \(C_{\mathrm{Y}}\) of nozzles \(0.97 ;\) maximum overall efficiency (based on conditions immediately before the nozzles) \(81.5 \% ;\) power specific speed for one jet not to exceed \(0.138\) rad \((0.022\) rev); head lost to friction in pipe-line not to exceed \(12 \mathrm{~m} .\) Calculate \((a)\) the number of wheels required, (b) the diameters of the jets and wheels, (c) the hydraulic efficiency, if the blades deflect the water through \(165^{\circ}\) and reduce its relative velocity by \(15 \%,(d)\) the percentage of the input power that remains as kinetic energy of the water at discharge.
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