Concept explainers
The rotating parts of a hydroelectric power plant having power capacity W have a rotational synchronous speed it. The weight of the rotating parts (the hydroturbine and its electric generator) is supported in a thrust bearing having amulet form between D and d diameters as sketched the thrust hearing is operated with a very thin oil film of thickness e and dynamic viscosity. It is armed that the oil is a Newtonian fluid and the velocity is approximated as linear in the hearing. Calculate the ratio of lost power in the thrust heating to the produced power in the hydraulic power plant. Use
The ratio of lost power in the thrust bearing.
Answer to Problem 127P
The ratio of lost power in the thrust bearing is
Explanation of Solution
Given information:
The larger diameter of the bearing is
Write the expression for the area in differential form.
Here, the area of the plant is
Write the expression for the radius of the smaller bearing.
Here, the radius of the smaller bearing is
Write the expression for the radius of the larger bearing.
Here, the radius of the larger bearing is
Write the expression for the volume of the plant in differential form.
Here, the volume of the plant is
Write the expression for the angular velocity.
Here, the speed of the plant is
Write the expression for the shear stress due to viscosity.
Here, the shear stress due to viscosity is
Write the expression for the force due to shear stress.
Here, the force due to shear stress is
Write the expression for the power loss due to viscosity of oil.
Here, the power loss is
Write the expression for the net power produced.
Here, the net power produced is
Write the expression for the ratio of power loss.
Here, the ratio of power loss is
Calculation:
Substitute
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Substitute
Substitute
Substitute
Substitute
Integrate Equation (XIII) under the lower limit
Substitute
Integrate Equation (XIV) under the lower limit
Substitute
Substitute
Integrate Equation (XVI) under the lower limit
Substitute
Substitute
Conclusion:
The ratio of lost power in the thrust bearing is
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Chapter 2 Solutions
Fluid Mechanics: Fundamentals and Applications
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