Prof. Daniel J. Bodony AE433, Fall 2015 HW #1 Due 11 September 2015 in class Problem 1 Consider the steady flow of a water jet impinging on a stationary vane as shown below. Determine the force required to hold the vane in place. Assume the jet follows the surface of the vane in a smooth manner and has the same shape and area at the exit. Problem 2 Consider the same vane as in Problem 2 now in uniform motion along the x-direction with speed V v which is less than V j . Find the force necessary to keep the vane moving at a constant velocity, that is, to prevent it from accelerating.
Transcript
Prof. Daniel J. BodonyAE433, Fall 2015
HW #1Due 11 September 2015 in class
Problem 1 Consider the steady flow of a water jet impinging on a stationary vane as shown below. Determine theforce required to hold the vane in place. Assume the jet follows the surface of the vane in a smooth manner and hasthe same shape and area at the exit.
Problem 2 Consider the same vane as in Problem 2 now in uniform motion along the x-direction with speed Vv
which is less than V j. Find the force necessary to keep the vane moving at a constant velocity, that is, to prevent itfrom accelerating.
Problem 3 In the analysis of turbo-machinery we will often have to use the principle of conservation of angularmomentum. Recall the basic statement of conservation of angular momentum, which follows directly from Newton’ssecond law, that the net applied torques M are equal to the time rate-of-change of H, the system angular momentum,in which the velocities are written in an inertial frame of reference:
M =ddt
∫Vmat
H dV.
H is the system angular momemtum per unit volume and may be written
H = r × ρu
where ρ is the fluid density and u is the fluid velocity. Let Fs be the sum of the applied surface forces on a controlvolumeV with bounding surface S. If r is the radius vector from a suitable origin, then show that the conservation ofangular momentum using a stationary control volume is
ddt
∫V
r × ρu dV +∮S
r × ρu[u · dS] = r × Fs + Mshaft +
∫V
ρr × g dV
where g is the body force vector associated with gravity and Mshaft is any applied shaft torque.
Problem 4 In the above problem we saw that for a fixed control volume V the statement of angular momentumconservation for a fluidic system was
ddt
∫V
r × ρu dV +∮S
r × ρu[u · dS] = r × Fs + Mshaft +
∫V
ρr × g dV
where g is the body force vector associated with gravity and Mshaft is any applied shaft torque, and H is the systemangular momemtum per unit volume and was written H = r × ρu.
Extend your derivation to now an accelerating, rigidly rotating control volume Vrot. Moreover, write all fluidvelocities from the reference from ofVrot. Referring to the figure below, let a particle P have position vector X(t) withrespect to the fixed frame OXYZ and r(t) with respect to the moving frame oxyz. The origin of the moving frame islocated at position R(t) with respect to the fixed frame. The angular velocity of the moving frame is ω(t). Supposethe rotating CV rotates with angular velocity ω(t) and has a centroid that follows the trajectory R(t). Let u denote thevelocity vector relative to an inertial reference frame and let v denote the velocity vector relative to the control volumeVrot.
Use this information to show that the conservation of angular momentum statement implies
Problem 5 In the above problems we derived expressions for the conservation of angular momentum, first in aninertial frame and second in a rigidly accelerating and rotating non-inertial frame. Use those expressions to solve thefollowing problem.
A small lawn sprinkler is shown in the sketch below. At an inlet gage pressure of 20 kPa, the total volume flowrate of water through the sprinkler is 7.5 liters per minute and it rotates at 30 rpm. The diameter of each jet is 4 mm.Calculated the jet speed relative to each sprinkler nozzle. Evaluate the friction torque at the sprinkler pivot.
For the same problem, use the conservation of angular momentum in the
(a) inertial frame; and,
(b) non-inertial frame
and show that both methods yield exactly the same answer.
Problem 6 A centrifugal pump (see below) pumps water at a rate of 1.0 ft3/s. The water enters the impeller inan axial direction. The diameter of the impeller is 10 inches and the vanes are 1.0 in. high and radial at the outsidediameter. Determine the power input to the rotor which turns at 1000 rpm.
Problem 7 A Pelton wheel is a form of impulse turbine well adapted to situations of high pressure and low flowrate. Consider the Pelton wheel and single-jet arrangement shown, in which a jet stream strikes the bucket tangentiallyand is turned through an angle θ. Obtain an expression for the torque exerted by the water stream on the wheel andthe corresponding power output. Show that the maximum power occurs when the bucket speed, U = rω, is half the jetspeed. Hint: use a rotating CV and neglect the mass of the water on the wheel.
