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Cairo University Second Year Students Faculty of Engineering Sheet [ 4 ] 2007-2008 Aerospace Eng. Department. Fluid Mechanics Integral Analysis of Flow (Applications on Linear Momentum Equation) (1) [5.5R*] Water flows through the right angle valve at the rate of 1000 lbm/s as is shown in figure. The pressure just upstream of the valve is 90 psi and the pressure drop across the valve is 50 psi. The inside diameter of the valve inlet and exit pipes are 12 and 24 in. If the flow through the valve occurs in a horizontal plane. Determine the x and y components of the force exerted by the valve on the water. (2) [5.7R*] An axisymmrtric device is used to partially “plug” the end of the round pipe shown in the figure. The air leaves in a radial direction with a speed of 50 ft/s as indicated. Gravity and viscous forces are negligible, Determine the: (a) Flow rate through the pipe. (b) Gage pressure at point (1). (c) Gage pressure at the tip of the plug, point(2). (d) Force, F, needed to hold the plug in place. 1 (3) [5.8R*] A nozzle is attached to an 80 mm inside-diameter flexible hose. The nozzle area is 500 mm2. If the delivery pressure of the water at the nozzle inlet is 700 kPa, could you hold the hose and the nozzle stationary? Explain. (4) [5.31*] A nozzle is attached to a vertical pipe and discharges water into the atmosphere as shown in the figure. When the discharge is 0.1 m 3/s, the gage pressure at the flange is 40 kPa. Determine the vertical component of the anchoring force required to hold the nozzle in place. The nozzle has weight of 200N, and the volume of the water in the nozzle is 0.012 m 3. Is the anchoring force directed upward or downward ? (5) Determine the magnitude and direction of the x and y components of the anchoring force required to hold in place the horizontal 180 elbow and nozzle combination shown in figure. 2 (6) [5.33*] Water flows as two free jets from a tee attached to the pipe shown in the figure. The exit speed is 15 m/s. If viscous effects and gravity are negligible, determine the x and y components of the force that the pipe exerts on the tee. (7)Water flows through a horizontal bend and discharges into the atmosphere as shown in figure. When the pressure gauge reads 70 kPa (gauge), the resultant x- direction anchoring force, FAx, in the horizontal plane required to hold the bend in place is 6400 N in the shown direction. Knowing that the flow is not frictionless: Determine the flow rate through the bend in m3/s. Determine the y-direction anchoring force, FAy , required to hold the bend in place (8) [5.36*] For the conditions of problem (4) of sheet (3), Determine the frictional force exerted by the pipe wall on the air flowing between sections (1) and (2). Assume uniform velocity distribution at each section. (9)[5.42*] Water flows vertically upward in a circular cross-sectional pipe as shown in the figure. At section (1), the velocity profile over the cross- sectional area is uniform. At section (2) the velocity profile is: V = wc [ (R – r) / R )1/7 ] k Where V = local velocity vector, wc = centerline velocity in the axial direction, R = pipe radius, and r = radius from pipe axis. Develop an 3 expression for the fluid pressure drop that occurs between sections (1) and (2). (10) Water discharges into the atmosphere through the device shown in figure. Determine the x component of force at the flange required to hold the device in place. Neglect the effect of gravity and friction and assume water density to be 1.94 slug/ft3 (11) A 10 cm diameter fire hose with a 3 cm diameter nozzle discharges water at a rate of 1.5 m3/min to the atmosphere. Assuming frictionless flow, find the force exerted by the flange bolts to hold the nozzle on the hose. 4 (12) Water flows through a 5 cm diameter pipe which has a 180 vertical bend. The total length of pipe between flanges 1 and 2 is 75 cm. The mass flow rate is 24 kg/s, P1 = 165 kPa (abs), P2 = 134 kPa (abs), and Patm = 101.3 kPa. Neglecting pipe empty weight, determine the total force, which the flanges must withstand for this flow. (13) [5.6R*] a horizontal circular jet of air strikes a stationary flat plate as indicated in the figure. The jet velocity is 40 m/s and the jet diameter is 30 mm. If the air velocity magnitude remains constant as the air flows over the plate surface in the directions shown, determine: (a) The magnitude of FA, the anchoring force required to hold the plate stationary. (b) The fraction of mass flow along the plate surface in each of the two directions shown. (c) The magnitude of FA the anchoring force required to allow the plate to move to the right at constant speed of 10 m/s. (14) [5.9R*] A horizontal air jet having a velocity of 50 m/s and a diameter of 20 mm strikes the inside surface of a hollow hemisphere as indicated in the figure. How large is the horizontal anchoring force needed to hold the hemisphere in place? The magnitude of the velocity of the air remains constant. 5 (15) Air flows into the atmosphere from a pipe and nozzle combination and strikes a vertical plate as shown in figure. The pipe cross-section is 0.01 m2, while the nozzle exit area is 0.003 m2. A horizontal force of 12 N is required to hold the plate in place. Determine the reading on the pressure gauge. Assume the flow to be incompressible and frictionless. (16) [5.58*] The four devices shown in the figure rest on frictionless wheels are restricted to move in the x direction only and initially held stationary. The pressure at the inlets and outlets of each is atmospheric, and the flow is incompressible. The content of each device is not known. When released, which devices will move to the right and which to the left? Explain. (17)[5.65*] A 3 in diameter horizontal jet of water strikes a flat plate as indicated in the figure. Determine the jet velocity, if a 10-Ib horizontal force is required to: (a) Hold the plate stationary. (b) Allow the plate to move at constant speed of 10 m/s to the right. 6 (18) [5.66*] A vane directs a horizontal, circular cross-sectional jet of water symmetrically as indicated in the figure. The jet leaves the nozzle with a velocity of 100 ft/s. determine the x direction component of anchoring force required to: (a) Hold the vane stationary. (b) Confine the speed of the vane to a value of 10 ft/s to the right. The fluid speed magnitude remains constant along the vane surface. (19) [5.35*] Thrust vector control is a new technique that can be used to greatly improve the maneuverability of the military fighter aircraft. It consists of using a set of vanes in the exit of a jet engine to deflect the exhaust gases as shown in the figure. (a) Determine the pitching moment (the moment tending to rotate the nose of the aircraft up) about the aircraft’s mass center (cg) for the condition indicated in the figure. (b) By how much is the thrust (force along the centerline of the aircraft) reduced for the case indicated compared to normal flight when the exhaust is parallel to the centerline. (20) The jet engine on a test stand, shown in the figure, admits air at 20 oC and 1 atm at section 1, where A1 = 0.5 m2 and V1 = 250 m/s. the fuel-to-air ratio is 1:30. The air leaves section 2 at atmospheric pressure and higher temperature, where V2 = 900 m/s and A2 = 0.4 m2 calculate the horizontal test stand reaction Rx needed to hold the engine fixed. 7 (21) [5.41*] The exhaust gas from the rocket shown in the figure (a) leaves the nozzle with a uniform velocity parallel to the x axis. The gas is assumed to be discharged form the nozzle as a free jet. (a) Show that the thrust developed is equal to AV2, where A = D2/4. (b) The exhaust from the rocket nozzle shown in the figure (b) is also uniform, but rather than being directed along the x-axis, it is directed along rays from point O as indicated. Determine the thrust for this rocket. (22) Propeller blades exert thrust on the fluid as it is accelerated through the plane of the propeller. Although the flow filed in the vicinity of the propeller must be unsteady, its action can be replaced by an accelerating “actuator disk” to reduce the problem to an equivalent steady one. The flow field is then approximately as shown with V 1 < V4 and P1 = P4. for this idealized theory, show that V2 = V3 = the average velocity (V1+V4)/2. (23) To propel a light aircraft at an absolute velocity of 240 km/h against a head wind of 48 km/h a thrust of 10300 N is required. Assuming a theoretical efficiency of 75% and a constant air density of 1.2 kg/m3, determine the diameter of ideal propeller required and the power needed to drive it. (24) A boot traveling at 12 m/s in fresh water has a 600mm-diameter propeller which takes 4.25 m3 of water per second between its blades. Assuming that the effects of the propeller hub and the boot hull on flow conditions are negligible, calculate the thrust on the boot, the theoretical efficiency of the propulsion, and the power input to the propeller. 8 (25) A helicopter has a mass of 2500 kg and has a 10-m diameter rotor. What is the average velocity of air through the rotor blades when the helicopter hovers at rest near the ground in a standard air. (26) A 30-cm diameter axial flow fan supplies 2.1 m3/s of air from atmosphere to a 60 cm diameter pipe by means of a diffuser fitted between the fan and pipe. A manometer is connected across the fan. The manometer indicates a pressure rise of 50 mm of water. Neglecting the losses calculate: (a) The gauge pressure at the entry to the 60-cm diameter pipe. (b) The axial force transmitted by the fan to its drive motor. (c) The axial force at the flanges between the diffuser and the 60-cm diameter pipe.air = 1.225 kg/m3. (27) [5.40*] The result of a wind tunnel test to be determine the drag on the body(see the figure) are summarized below. The up stream[section (1)] velocity is uniform at 100 ft/s. the static pressure are given by P 1 = P2 = 14.7 psia. The downstream velocity distribution, which is symmetrical about the centerline. Given by: u = 100 – 30(1- y /3) y ≤3 ft u = 100 y >3 ft where u is the velocity in ft/s and y is the distance on either side of the centerline in feet. Assume that the body shape dose not change in the direction normal to the paper. Calculate the drag force (reaction force in x direction) exerted on the body by the air per unit length normal to the plane of the sketch. 9 (28) A cylinder is placed in a uniform and parallel flow of an incompressible fluid density velocity profile are measured at 2D upstream and at 10 D downstream from the central plane of the cylinder, find the drag per unit length FD on the cylinder, and calculate the drag coefficient defined as: CD = FD /0.5U2D. Hint: take a large control volume about the cylinder as shown and assume that the fluid pressure at the control surface is constant. (29) A fluid of a density flows along a flat plat. At the leading edge of the plate the velocity is uniform and parallel to the velocity plate, but at the trailing edge of the plate the velocity component u is reduced in a thin region called the boundary layer. Within this layer the velocity profile is given by: 10 u/U = 2where yand is the boundary layer thickness. Determine the skin-friction drag force on a unit width of the plate. The variation of the fluid pressure is negligible. (30) The drag force acting on an airfoil can be calculated by determining the change in the momentum of the fluid as it flows past the airfoil. As part of this exercise, the velocity profiles are measured well upstream and well downstream of the airfoil, at surface (1) and (2) of a rectangular control volume, as shown in figure. If the flow is incompressible, two-dimensional, and steady: (a) Determine the total volumetric flow rate across the horizontal surfaces (3) and (4) per unit span. (b) Determine the drag force per unit span, knowing that the pressure is P over the entire surface of the control volume. * Selected problems from the text book: Fundamental of Fluid Mechanics, 3rd Edition, by: Munson, Young and Okiishi, 1998. 11 12