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Chapter 13: Problem 8

A vertical-shaft Francis turbine, with an overall efficiency of \(90 \%\), runs at \(44.86 \mathrm{rad} \cdot \mathrm{s}^{-1}(7.14 \mathrm{rev} / \mathrm{s})\) with a water discharge of \(15.5 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1} .\) The velocity at the inlet of the spiral casing is \(8.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and the pressure head at this point is \(240 \mathrm{~m}\), the centre-line of the casing inlet being \(3 \mathrm{~m}\) above the tail-water level. The diameter of the runner at inlet is \(2.23 \mathrm{~m}\) and the width at inlet is \(300 \mathrm{~mm}\). The hydraulic efficiency is \(93 \%\). Determine \((a)\) the output power, \((b)\) the power specific speed, ( \(c\) ) the guide vane angle, \((d)\) the runner blade angle at inlet, (e) the percentage of the net head which is kinetic at entry to the runner. Assume that there is no whirl at outlet from the runner and neglect the thickness of the blades.

Short Answer

Expert verified
Overall net efficiency is determinable by velocity params 93%. Specific power speed is with dynamic pressure param}

Step by step solution

01

- Determine the net head

Calculate the net head available to the turbine using the pressure head at the inlet, velocity head, and the height difference. The net head, ////added specific variables with description
02

- Calculate the hydraulic power

Use the hydraulic efficiency and net head to determine the hydraulic power available to the turbine. Replace efficiency with net hydraulic performance
03

- Determine the output power

Use the overall efficiency of the turbine and the hydraulic power calculated in Step 2 to determine the output power. The hydraulic power calculated in Step 2. Adjust the stated efficiency params as per usage.
04

- Calculate power specific speed

Determine the power specific speed using the given rotational speed, water discharge, and output power.
05

- Determine the velocity triangle parameters

Calculate the velocity of water at the inlet using discharge, and then calculate the components of velocity to use for determining angles of the guide vane and runner blade at the inlet.
06

- Determine the guide vane angle

Using the calculated velocity and net parameters of dynamic pressure and relative velocity from the previous steps, calculate the guide vane angle using relationships between tangential and total velocity. Specifically expand on the tangential velocity equation
07

- Determine the runner blade angle at inlet

Using the calculated velocity and net parameters of hydraulic velocity to keep overall efficiency calculation.
08

- Calculate the percentage of the net head

Calculate the net kinetic energy and overall efficiency cross ratio. Determine the percentange of the kinetic energyit has net energy.

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Key Concepts

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

hydraulic efficiency
Hydraulic efficiency is a measure of how well a turbine converts the available hydraulic energy into mechanical energy. For a Francis turbine, it is particularly important as it indicates the effectiveness of the turbine in utilizing the water flow.
Hydraulic efficiency ( \eta_{h}) can be calculated using the equation: \[ \eta_{h} = \frac{P_h}{P_w} \], where \( P_h \) is the hydraulic power and \( P_w \) is the potential power of the water. Given in this exercise, the hydraulic efficiency is \( 93\% \). This efficiency reflects the portion of the total energy that is effectively utilized by the turbine.
  • Hydraulic power \( P_h \) can be calculated using the formula \[ P_h = \rho \cdot g \cdot Q \cdot H_n \] where \( \rho \) is the density of water, \( g \) is the acceleration due to gravity, \( Q \) is the discharge flow rate, and \( H_n \) is the net head.
net head determination
The net head ( \( H_n \) ) represents the absolute energy head available to the turbine for doing work. It takes into account both pressure and velocity heads along with any elevation differences. To compute the net head:
  • Calculate the pressure head ( \( H_p \) ). Given as 240 meters in the exercise.
  • Calculate the velocity head ( \( H_v \) ) using the inlet velocity: \[ H_v = \frac{v^2}{2g} \], where \( v \) is the inlet velocity and \( g \) is the gravity.
  • Height difference ( \( z \) ) is provided as 3 meters.
The net head is obtained by summing these components: \[ H_n = H_p + H_v - z \]. This integration is crucial as it ensures all energy contributions and losses are accounted for.
output power calculation
Output power ( \( P_{out} \) ) is the mechanical power delivered by the turbine to the shaft. To determine the output power, you need to consider the overall efficiency ( \( \eta_{o} \) ) of the turbine, given as 90%. The output power can be calculated from the hydraulic power:
  • First, calculate the hydraulic power ( \( P_h \) ) as mentioned in the hydraulic efficiency section.
  • Then, use the overall efficiency to find the output power: \[ P_{out} = P_h \cdot \eta_{o} \].
This calculation ensures that only the effective energy contributing to mechanical work is considered, providing realistic power output values.
power specific speed
Power specific speed ( \( N_s \) ) is a dimensionless parameter that helps categorize turbines based on their speed and power characteristics. It is defined using the rotational speed, discharge, and output power: \[ N_s = N \cdot \sqrt{Q} \cdot \frac{1}{H_n^{5/4}} \], where \( N \) is the rotational speed, \( Q \) is the discharge, and \( H_n \) is the net head.
This parameter is crucial for comparing turbines of various designs and sizes, determining their suitability for specific applications based on their efficiency and processing power at a given head.
guide vane angle
The guide vane angle ( \( \alpha \) ) controls the direction of the water entering the turbine’s runner. It ensures proper flow direction and optimal energy extraction. To calculate the guide vane angle:
  • Determine the tangential component of the inlet velocity ( \( v_u \) ) and the total velocity ( \( v \) ).

Use trigonometric relationships where \[ \tan( \( \alpha \) ) = \frac{v_u}{v} \].
This angle is critical in ensuring the water flow matches the design parameters and operational efficiency of the turbine.
runner blade angle
Runner blade angle ( \( \beta \) ) is the positioning of the blades at the entry to the runner. It affects the reduction of velocity losses and increases efficiency. Calculation involves:
  • Analyzing the velocity triangle at the inlet, including the absolute velocity of water, the relative velocity to the blade, and the blade speed.

The angle can be found using specific geometric formulations, considering that the runners convert hydraulic energy to mechanical energy effectively thereby ensuring high efficient operations.
kinetic energy percentage
Kinetic energy percentage refers to the portion of the net head ( \( H_n \) ) that is kinetic energy as water enters the runner. This is relevant to understand how much of the water's energy is due to its movement. It can be calculated using:
  • Velocity head ( \( H_v \) ) from earlier and determining the ratio: \[ \frac{H_v}{H_n} \].
This metric is important as it shows the amount of dynamic energy, facilitating adjustments for maximizing turbine efficiency.

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Most popular questions from this chapter

The impeller of a centrifugal fan has an inner radius of \(250 \mathrm{~mm}\) and width of \(187.5 \mathrm{~mm} ;\) the values at exit are \(375 \mathrm{~mm}\) and \(125 \mathrm{~mm}\) respectively. There is no whirl at inlet, and at outlet the blades are backward-facing at \(70^{\circ}\) to the tangent. In the impeller there is a loss by friction of \(0.4\) times the kinetic head corresponding to the relative outlet velocity, and in the volute there is a gain equivalent to \(0.5\) times the kinetic head corresponding to the absolute velocity at exit from the runner. The discharge of air is \(5.7 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) when the rotational speed is \(84.8 \mathrm{rad} \cdot \mathrm{s}^{-1}(13.5 \mathrm{rev} / \mathrm{s}) .\) Neglecting the thickness of the blades and whirl slip, determine the head across the fan and the power required to drive it if the density of the air is sensibly constant at \(1.25 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) throughout and mechanical losses account for \(220 \mathrm{~W}\).

The following duties are to be performed by rotodynamic pumps driven by electric synchronous motors, speed \(100 \pi / n \mathrm{rad} \cdot \mathrm{s}^{-1}(=50 / n \mathrm{rev} / \mathrm{s})\), where \(n\) is an integer: (a) \(14 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) of water against \(1.5 \mathrm{~m}\) head; \((b)\) oil (relative density \(0.80\) ) at \(11.3 \mathrm{~L} \cdot \mathrm{s}^{-1}\) against \(70 \mathrm{kPa}\) pressure; \((c)\) water at \(5.25 \mathrm{~L} \cdot \mathrm{s}^{-1}\) against \(5.5 \mathrm{MPa}\). Designs of pumps are available with specific speeds of \(0.20,0.60,1.20,2.83,4.0\) rad. Which design and speed should be used for each duty?

An inward-flow reaction turbine has an inlet guide vane angle of \(30^{\circ}\) and the inlet edges of the runner blades are at \(120^{\circ}\) to the direction of whirl. The breadth of the runner at inlet is a quarter of the diameter at inlet and there is no velocity of whirl at outlet. The overall head is \(15 \mathrm{~m}\) and the rotational speed \(104.7 \mathrm{rad} \cdot \mathrm{s}^{-1}\) (16.67 rev/s). The hydraulic and overall efficiencies may be assumed to be \(88 \%\) and \(85 \%\) respectively. Calculate the runner diameter at inlet and the power developed. (The thickness of the blades may be neglected.)

A large centrifugal pump is to have a specific speed of \(1.15 \mathrm{rad}\) \((0.183 \mathrm{rev})\) and is to discharge liquid at \(2 \mathrm{~L} \cdot \mathrm{s}^{-1}\) against a total head of \(15 \mathrm{~m}\). The kinematic viscosity of the liquid may vary between 3 and 6 times that of water. Determine the range of speeds and test heads for a one-quarter scale model investigation of the full-size pump, the model using water.

During a laboratory test on a water pump appreciable cavitation began when the pressure plus velocity head at inlet was reduced to \(3.26 \mathrm{~m}\) while the total head change across the pump was \(36.5 \mathrm{~m}\) and the discharge was \(48 \mathrm{~L} \cdot \mathrm{s}^{-1} .\) Barometric pressure was \(750 \mathrm{~mm} \mathrm{Hg}\) and the vapour pressure of water \(1.8 \mathrm{kPa}\). What is the value of \(\sigma_{\mathrm{c}}\) ? If the pump is to give the same total head and discharge in a location where the normal atmospheric pressure is \(622 \mathrm{~mm} \mathrm{Hg}\) and the vapour pressure of water \(830 \mathrm{~Pa}\), by how much must the height of the pump above the supply level be reduced?
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