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ME 363 - Fluid Mechanics Homework #7 solutions Due Wednesday, March 12, 2008 Spring Semester 2008 1] Pelton wheel turbines are commonly used in hydroelectric power plants to generate electric power. In these turbines, a high-speed jet at a velocity of Vj impinges on buckets, forcing the wheel to rotate. The buckets reverse the direction of the jet, and the jet leaves the bucket making an angle with the direction of the jet, as shown in Fig. 1. Use the following relationship to obtain the power produced by a Pelton wheel: P = Tshaft, where Tshaft is the total torque produced by the wheel and is an angular velocity of the wheel. Show that the power P = rQ(Vj - r)(1 – cos ), where is the density and Q is the volume flow rate of the fluid. Obtain the numerical value for = 1000 kg m-3, r = 2 m, Q = 10 m3s-1, = 5 s-1, = 160°, and Vj = 50 m/s. 2] A liquid of density ρ flows through a 90° bend as and issues vertically from a uniformly porous section of length L. Neglecting weight, find an expression for the support torque M required at point O. 3] A power plant on a river, as in Fig. 3, must deliver 55 MW of waste heat to the river. The river conditions upstream are Qi = 2.5 m3/s and Ti = 18°C. The river is 45 m wide and 2.7 m deep. If heat losses to the atmosphere and ground are negligible, estimate the downstream river conditions (Q0, T0). 4] A simple turbomachine is constructed from a disk with two internal ducts which exit tangentially through square holes, as in the figure. Water at 20C enters the disk at the center, as shown. The disk must drive, at 250 rev/min, a small device whose retarding torque is 1.5 Nm. What is the proper mass flow of water, in kg/s? 5] Multnomah Falls in the Columbia River Gorge has a sheer drop of 543 ft. Estimate the water temperature change in degrees F caused by this drop. 1 2 4 5 (no figure) 3 Problem 1 A Pelton wheel is considered for power generation in a hydroelectric power plant. A relation is to be obtained for power generation, and its numerical value is to be obtained. Mshaft r Vj - r Shaft Nozzle Vj - r Vj r Assumptions 1 The flow is uniform and cyclically steady. 2 The water is discharged to the atmosphere, and thus the gage pressure at the nozzle outlet is zero. 3 Friction and losses due to air drag of rotating components are neglected. 4 The nozzle diameter is small compared to the moment arm, and thus we use average values of radius and velocity at the outlet. Properties We take the density of water to be 1000 kg/m3 = 1 kg/L. Analysis The tangential velocity of buckets corresponding to an angular velocity of 2n is Vbucket r . Then the relative velocity of the jet (relative to the bucket) becomes Vr V j V bucket V j r We take the imaginary disk that contains the Pelton wheel and rotates with the wheel as the control volume. The inlet velocity of the fluid into this control volume is Vr, and the component of outlet velocity normal to the moment arm is Vrcos . The angular momentum equation can be expressed as M rm V rm V where all moments in the counterclockwise direction are positive, and all in out in the clockwise direction are negative. Then the angular momentum equation about the axis of rotation becomes Vr cos rm Vr M shaft rm or M shaft rm V r (1 cos ) rm (V j r )(1 cos ) V , the shaft power output of a Pelton turbine becomes Noting that W shaft 2nM shaft M shaft and m W shaft Vr (V j r )(1 cos ) which is the desired relation. For given values, the shaft power output is determined to be 1 MW W shaft (1000 kg/m 3 )(10 m 3 /s)( 2 m) (15.71 rad/s)(50 - 2 15.71 m/s)(1 - cos160) 6 11.3 MW 10 N m/s where 2n 2 (150 rev/min) 1 min 15 .71 rad/s 60 s Discussion The actual power will be somewhat lower than this due to air drag and friction. Note that this is the shaft power; the electrical power generated by the generator connected to the shaft is be lower due to generator inefficiencies. Problem 2 Problem 3 Problem 4 Problem 5