Fluid Mechanics for International Engineers HW #4: Conservation of Linear Momentum and Conservation of Energy

Transcription

1 Fluid Mechanics for International Engineers 1 Problem 1 RTT and Time Rate of Change of Linear Momentum and The Corresponding Eternal Force Notation: Here a material volume (MV) is referred to a closed sstem (Ss) in Fo et al s notation For each of the fluid stream in a control volume shown below, state whether there is a time rate of change of linear momentum of the material volume (ie, closed sstem) that instantaneousl coincides with the control volume for both components of the linear momentum, ie, P MV, (ie, P ss, ) and P MV, (ie, P ss, ) And if so, also state whether MV, / dt ( ie, ss, / dt ) and MV, / dt ( ie, ss, / dt ) is, and whether there is a net/resultant force acting on the CV and, if so, in which direction [Note that the fact that there is no net/resultant force acting on the CV does not mean that there is no force acting on the CV It simpl means that the net/resultant force of the force sstem that acts on the CV is zero] Otherwise stated, assume that the CV is stationar and non-deforming, the observer is in the inertial frame of reference, velocit and all properties are uniform over each cross section, no flow across an part of the CS that there is no arrow indicating the direction of the flow, gravit is in the z direction Recall that the RTT for the linear momentum read MV, Ss, CV, v v = + u( ρ V f / s da), P CV, ( t) = dt dt dt u( ρ dv ) CS MV Ss, CV, v = + v ( ρ V f / s da), P CV, ( t) = dt dt dt v( ρ dv ), v CS NOTE: Although the current problem does not ask ou to reason/derive/prove our answer, ou should be able to do that since our final eam will probabl do CV CV V 2 = V1 velocit fields (a) MV, / dt = 0? (es/no) F on CV? (es/no) MV, / dt = 0? (es/no) Using case (a) as a baseline case [no change in speed and direction of the fluid steam from inlet to eit], the following problems illustrate various effects as stated with the problem number V 2 > V1 velocit fields (but the flow is not incompressible) (b) Increase in speed / No change in direction (but not incompressible) MV, / dt = 0? (es/no) F on CV? (es/no) MV, / dt = 0? (es/no)

2 2 V 2 < V1 (c) Decrease in speed / No change in direction [Diffuser: opposite to (c)] MV, / dt = 0? (es/no) F on CV? (es/no) velocit fields MV, / dt = 0? (es/no) velocit fields V 2 = (d) No change in speed / Change in direction MV, / dt = 0? (es/no) F on CV? (es/no) MV, / dt = 0? (es/no) V 2 = velocit fields (e) No change in speed / Change in direction [Reverse direction from (e)] MV, / dt = 0? (es/no) F on CV? (es/no) MV, / dt = 0? (es/no)

3 3 Problem 2 A cascade is a series of airfoil sections usuall arranged as shown below It is used, eg, as guide vanes to deflect a fluid stream or in aial turbomachines A stationar cascade for an aial turbomachine is shown below Assume that the flow is incompressible with fluid densit ρ, the velocit field is stead and uniform over each cross section, the fluid stream enters the cascade with the magnitude of the velocit and the direction such that the velocit vector is tangent to the blade camber, which makes an angle β 1 as shown, the fluid stream leaves the cascade with the direction of the velocit such that the velocit vector is tangent to the blade camber, which makes an angle β 2 as shown, the static pressures at inlet and eit are given as p 1 and p 2, respectivel, the spacing between airfoils is s, the span width of the cascade in the z direction is w, the gravitational force is in the z direction, and the frictional force is negligible β 1 p 1 s β 2 p 2 Image from Visualized Flow, The Japan Societ of Mechanical Engineers, Pergamon Press, 1988, p101) Choose an appropriate control volume to find the magnitude of the fluid velocit at the eit V 2, and v the net force vector F = F iˆ + F ˆj on an one airfoil Make sure that ou choose the appropriate CV such that ou can reall see this net force vector

4 Problem 3 Conservations of Mass and Linear Momentum for A Control Volume as observed from A Stationar Frame of Reference [Adapted from Munson et al, 2002, Problem 541, p 283] The hdraulic dredge is used to dredge sand from a river bottom, and the sand/water miture is discharged as a free-jet as shown below The discharge has a cross sectional area A The jet discharge speed is V, which is oriented at an angle θ with respect to the horizontal Assume that the bell-mouth suction has relative large cross sectional area such that the sand/water miture speed at the suction can be neglected The specific gravit of the sand/water miture is SG and water densit is ρ a Estimate the thrust needed from the propeller to hold the boat stationar b Under this stationar operating condition, is the buoanc force on the boat equal to the weight W of the boat (including all the loads on the boat)? If not, how does the buoanc force change from that when the hdraulic dredge is not operated? 4 A θ V

5 Problem 4 Conservation of Mass, Conservation of Linear Momentum ( and ), and Conservation of Energ A stead, laminar fountain of a fluid (densit ρ ) is issued from a nozzle with an eit area A 1 The speed and angle of the fluid at the nozzle eit are and θ 1, respectivel 41 Use the conservation of -momentum (and conservation of mass) to prove that the component of the velocit V v at an cross section 2 obes the projectile relation, ie, V2 = V1 42 Use the conservation of energ (not et the Bernoulli s equation) to find V2 ( ) 43 Use the conservation of mass (and mass flu equation) to find A 2 ( ) 44 Use the conservation of -momentum to find the weight of the fluid in the arc from 1 to 2, W 2 ( ) 45 Find the maimum height H that the arc of the fountain can reach 46 Find the locus () of the arc Point 2 is an arbitrar point on the arc Assume that the velocit is uniform over the cross sectional area whose normal vector is aligned with the local velocit vector 5 A 2 V v 2 V v 1 H Laminar Fontain A 1 a θ 1 b Laminar flow nozzles send coherent streams of water in the form of graceful arcs for a fountain at McCormick Place in Chicago From Said Shakerin (2001), Engineering Art, ASME s Mechanical Engineering Magazine, Water flows from a group of Laminar Flow Nozzles manufactured b Roman Fountains, 8600 Paseo Alameda NE The specialit nozzles, which were being tested Wednesda, take in water, line it up and slow it down, causing water to flow at a consistent rate The Albuquerque compan, specializing in decorative fountains, is a consultant on the World Trade Center memorial project in New York Cit (Michael J Gallegos/Tribune) From Case Phillips, Jul , Oasis of success, The Albuquerque Tribune,

6 Problem 5 RTT and Conservation of Linear Momentum for A Control Volume as observed from A Stationar, Inertial Frame of Reference [Adapted from Fo et al, 2010, Problem 484, p 149] A stream of water flows steadil through a curved nozzle assembl that discharges the stream to the atmosphere The nozzle mass is 45 kg and its internal volume is 0002 m 3 Assume that all frictional forces can be neglected 51 Consider the volume inside the nozzle (ecluding the nozzle) as the control volume (CV1) and also consider the coincident material volume (MV1) r r 511 Find the time rate of change of linear momentum of the coincident material volume MV ss dt dt Note that this is a vector quantit 512 If it is not zero, state clearl which force is responsible for this change Draw a free-bod-diagram (FBD) to illustrate our answer 513 Determine the net force vector due to water pressure on the nozzle s inner wall 52 Now, consider the curved nozzle itself and the water inside it as the control volume (CV2) and also consider the coincident material volume (MV2) r r 521 Find the time rate of change of linear momentum of the coincident material volume MV ss dt dt 522 Is it equal to that of MV1 in 21a?; and if so, wh? 523 Determine the reaction force vector eerted on the nozzle at its flange 524 Is the force in 22c equal to that in 21c? Eplain 6 = 2 m/s p 1 = 125 kpa gage D = 75 cm 1 g D 2 = 25 cm θ = 30 o V 2 Problem 6 Conservation of Energ [Fo et al, 2010, Problem 4202, p 160] Air enters a compressor at 96 kpa, 27 o C with negligible speed and is discharged at 480 kpa, 260 o C with a speed of 152 m/s If the power input is 238 MW and the flow rate is 9 kg/s, determine the rate of heat transfer

7 Problem 7 Conservation of Energ for A Stead, Incompressible, Fluid Stream with No Applied Heat Transfer 71 Water is flowing between two identical clindrical tanks (A B) as shown below At this configuration, the elevation difference between the two free surfaces is H, and the resulting mass flow rate is m& Find the mechanical power loss as the stream of fluid flows from 1 to A H B 2 d D m& D 72 On the other hand, if we are to pump the water from 2 to 1 (B A) at the same mass flowrate m& as in 71, find the required hdraulic power of the pump