HW_Week6 - L EFIGURE P4.4O4 V Volt M4“ sintcoril 4.41...

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Unformatted text preview: L». EFIGURE P4.4O4 V: Volt + M4“ sintcoril 4.41 Water flows over the crest of a dam with speed Vas shown in Fig. P441. Determine the speed if the magnitude of the normal ac- celeration at point (i) is to equal the acceleration of gravity, g. EFIGUFIE P4.41 4.42 Assume that the streamlines for the wingtip vortices from an airplane (see Fig. P419 and Video V4.6) can be approximated by circles of radius r and that thé speed is V = K/r, where K is a constant. Determine the streamline acceleration, as, and the normal acceleration, at", for this flow. 4.43 A fluid flows past a sphere with an upstream velocity of V0 = 40 m/s as shown in Fig. P443. From a more advanced theory it is found that the speed of the fluid along the front part of the sphere is V = gift, sin 0. Determine the streamwise and normal components of acceleration at pointA if the radius of the sphere is a = 0.20 m. IFIGUFIE P4.43 *4.44 For flow past a sphere as discussed in Problem 4.43, plot a graph of the sueamwise acceleration, (2,, the normal acceleration, a,,, and the magnitude of the acceleration as a function of 9 for 0 s 6 s 90° with V0 = 50 ft/s anda = 0.1, 1.0, and 10 it. Repeat for V0 = 5 ft/s. At what point is the acceleration a maximum; 3 minimum? ”34.45 The velocity components for steady flow through the nozzle shown in Fig. P445 are L! = —Vo IN and 1) = Vo[1 + (yr/8)}, EFIGURE 134.45 183 Problems where V0 and t? are constants. Determine the ratio of the magnitude of the acceleration at point (1) to that at point (2). *4.46 A fluid flows past a circular cylinder of radius a with an upstream speed of V0 as shown in Fig. P446. A more advanced the» ory indicates that if viscous effeéts are negligible, the velocity of the fluid along the surface of the cylinder is given by V = 2% sin 6. Determine the streamline and normal components of acceleration on the surface of the cylinder as a function of V0, a, and 6 and plot graphs of a, and an for 0 S 6 S 90° with V0 = 10 m/s and a = 0.01, 0.10, 1.0, and 10.0 m. V .I' V ._°>. _'_/; \ EFIGURE P4.46 4.47 Determine the x and y components of acceleration for the flow given in Problem 4.11. If c > 0, is the particle at point x = x0 > 0 and y = 0 accelerating or decelerating? Explain. Repeat if x0 < 0. <4.4§‘>When flood gates in a channel are opened, water flows h'long the channel downstream of the gates with an increasing speed given by V = 4(1 ~l— 0.1:) ft/s, for 0 S t S 20 s, where I is in seconds. For t > ‘20 s the speed is a constant V = 12 file. Consider a location in the curved channel where the radius of curvature of the streamlines is 50 ft. For t = 10 5 determine (a) the component of acceleration along the streamline, (b) the component of acceleration normal to the streamline, and (c) the net acceleration (magnitude and direction). Repeat for I: 3.0 s. @9 ater flows steadily through the funnel shown in igf”P4.49. Throughout most of the funnel the flow is approxi- .mately radial (along rays from O) with a velocity of V = c/rz, where r is the radial coordinate and c is a constant. If the veloc— ity is 0.4 m/s when 'r x 0.1 m, determine the acceleration at points A and B. FIGURE P4.49 4.50 Water flows though the slit at the bottom. of a two— dimensional water trough as shown in Fig. P450. Throughout most of the trough the flow is approximately radial (along rays from O) with a velocity of V = c/r, where r is the radial coordinate and c is a constant. If the velocity is 0.04 m/s when r m 0.1 m, determine the acceleration at points A and B. 4.62 In the region just downstream of a sluice gate, the water may develop a reverse flow region as is indicated in Fig. P4.62 and Video VlO.9. The velocity profile is assumed to consist of two uniform regions, one with velocity V,l = 10 fps and the other with Vb = 3 fps. Determine the net flowrate of water across the portion of the control surface at section (2) if the channel is 20 ft wide. I. Sluice gate Vb = 3 fits (1f EFIGUHE P4.62 4.63 At time t = 0 the valve on an initially empty (perfect vac- uum, p = 0) tank is opened and air rushes in. Ifthe tank has a vol- ume of V0 and the density of air within the tank increases as p = pm(i — 8“"), where b is a censtant, determine the time rate of change of mass within the tank. "{4.64 From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration: d 13(3) 116 [ix (it —l ream] gateway: were? 11 Discuss how this formula is related to the time derivative of the total amount of a property in a system and to theReynolds transport theorem. 4.65 Water enters the bend of a river with the uniform velocity profile shown in Fig. P465. At the end of the bend there is a re- gion of separation or reverse flow. The fixed control volume ABCD coincides with the system at time t = 0. Make a sketch to indicate (a) the system at time t = 5 s and (b) the fluid that has entered and exited the control volume in that time period. -_;_ Control votume EFIGUFIE 134.65 4.66 A layer of oil flows down a vertical plate as“ shown in 'Fig. P466 with a velocity of V = (VD/hi) (231x — A?” where V0 and h are constants. (a) Show that the fluid sticks to the plate and that the shear stress at the edge of the layer (x = h) is zero. (b) De- termine the flowrate across surface AB. Assume the width of the plate is 1;. (Note: The velocity profile for laminar flow in a pipe has a similar shape. See Video V6.13.) 185 Problems nix) Plate BFIGUFIE P4.66 4.67 ater flows in the branching pipe shown in Fig. P467 with uniform velocity at each inlet and outlet. The fixed control volume indicated coincides with the system at time t = 20 3. Make a sketch to indicate (a) the boundary of the system at time t : 20.1 s, (b) the fluid that left the control volume during that 0.1-5 interval, and (c) the fluid that entered the control volume during that time interval. ,, 7 Control volume EFIGURE P4.67 4.68 TWO plates are pulled in opposite directions with speeds of 1.0 his as shown in Fig. P468. 'Ilre oil between the plates moves with a velocity given by V = 10 yi ft/s, where y is in feet. The fixed control volume ABCD coincides with the system at time t = 0. Make a sketch to indicate (a) the system at time t = 0.2 s and (h) the fluid that has entered and exited the control volume in that time period. Control 1 it/s EFIGUFIE P4.68 4.69 Wateris squirted from a syringe with a speed of V = 5 m/s by pushing in the plunger with a speed of V!J = 0.03 m/s as shown in Fig. P469. The surface of the deforming control volume consists of the sides and end of the cylinder and the end of the plunger. The sys- tem consists of the water in the syringe at r = 0 when the plunger is at section (1) as shown. Make a sketch to indicate the control sur- face and the system when t = 0.5 s. 5.11 Air flows steadily between two cross sections in a long, straight section of 0.1-m inside diameter pipe. The static tempera» ture and pressure at each section are indicated in Fig. Pill. If the average air velocity at section (1) is 205 m/ s, determine the average air velocity at section (2). Section (1) . Section (2) p1 = 77 kPa (abs) p2 = 45 kPa tabs} T1=268K T2=240K V1 = 205 mfs HFIGURE 135.11 5.12 A hydraulic jump (see Video V1010) is in place downstream from a spillway as indicated in Fig. P112. Upstream of the jump, the depth of the stream is 0.6 ft and the average stream velocity is 18 ft/s. Just downstream of the jump, the average stream velocity is 3.4 ft/s. Calculate the depth of the stream, h, just downstream of the jump. 3 FIGURE P5.12 5.13 'An evaporative cooling tower (see Fig. P513) is used to cool water from 110 to 80°F. Water enters the tower at a rate of 250,000 lbm/ hr. Dry air (no water vapor) flows into the tower at a rate of 151,0001bm/hr. If the rate of wet air flow out of the tower is 156,900 lbrn/ hr, determine the rate of water evaporation in lbrn/iu‘ and the rate of cooled water flow in lbrn/ hr. Wet alr rh = 156,900 lbmihr 1‘ Warm water -—)- th = 250.000 tbmlhr _ Dry air w»)— m =151.000lbmihr EFIGUHE P5.13 Cooled water 5.14 At cruise conditions, air flows into a jet engine at a steady rate of 65 lbm/s. Fuel enters the engine at a steady rate of 0.60 lbm/s. The average velocity of the exhaust gases is 1500 ft/ s relative to the engine. If the engine exhaust effective cross-sectional area is 3.5 ftz, estimate the density of the exhaust gases in lbrn/ft3. 247 Problems 5.15 Water at 0.1 111313 and alcohol {SG=0.8) at 0.3 msls are mixed in a y—duct as shown in Fig. 5.15. What is the average density of the mixture of alcohol and water? Water and alcohol mix —l>- Water 0 \ Q = 0.1 male \ Alcohol (so = 0.8} Q a 0.3 mals -> FIGURE P5.15 5.16 Freshwater flows steadily into an open 55-gal drum initially filled With seawater. The freshwater mixes thoroughly with the sea— water and the mixture overflows out of the drum. If the freshwater flowrate is 10 galjmin, estimate the time in seconds required to de« crease the difference between the density of the mixture and the density of freshwater by 50%. Section 5.1.2 Fixed, Nondet‘orming Control Volume— Nonum‘form Velocity Profile 5.17 A water ‘et pump (see Fig. Pit?) involves a jet cross~sectional area of 0.01 111 , and a jet velocity of 30 m/s. The jet is surrounded by entrained water. The total cross—sectional area associated with the jet and entrained streams is 0.075 m2. These two fluid streams leave the pump thoroughly mixed with an average velocity of 6 m/s through a cross—sectional area of 0.075 ml. Determine the pumping rate (i.e., the entrained fluid flowrate) involved in liters/s. EFIGURE P5.‘i7 5.18 ’IWo rivers merge to form a larger river as shown in Fig. P5.18. At a location downstream from the junction (before the two streams completely merge), the nonuniform velocity profile is as shown and the depth is 6 ft. Determine the value of V. E FIGURE £35.18 248 Chapter 5 E Finite Control Volume Analysis 5.19 Various types of attachments can be used with the shop vac 5.23 An incompressible flow velocity field (water) is given as shown in Video V5.2. Two such attachments are shown in Fig. P5.19 1 1 ma nozzle and a brush. The flowrate is 1 ft3/s. (a) Determine the V = __ 9r + _ 36 m/s average velocity through the nozzle entrance, V”. (b) Assume the air I‘ enters the brush attachment in a radial direction all around the brush with a velocity profile that varies linearly from 0 to Vb along the length of the bristles as shown in the figure. Determine the value of Vb. h where r is in meters. (3) Calculate the mass flowrate through the cylindrical surface at r = 1 m from 7. = 0 to z = 1 In as shown in Fig.P5.23a. (b) Show that mass is conserved in the annular control voiurnefromrz lintorz 2mandz= Otoz = 1 masshown Q = 1 “3,5 in Fig. P5231). Z 1m Vn a Fl G U Fl E P5.19 (a) W ‘ EFEGUHEP5.23 5.20 An appropriate turbulent pipe flow velocity profile is R _ m, 5.24 Flow of a viscous fluid over a flat plate surface results in the V g ”a (4) i development of a region of reduced velocity adjacent to the wetted R surface as depicted in Fig. P524. This region of reduced flow is h 2 t l' 1 . : l . z . d‘ , called a boundary layer. At the leading edge of the plate, the veloc— :vfl de €622 nit‘ifielciad r1321: 12:3]er eht erfiffiggims’. Re th 6133353131]: ity profile may be considered uniformly distributed with a value U. erage velocity, 5, to centerii n e velocity, u” for (a) n : 4, (b) n z 6, All along the outer edge of the boundary layer, the fluid velocity = 8, d = 1 .C th d'ff t 1 ‘ fil _ component parallel to the plate surface is also U. If thexdirection (GE—M ( ) n 0 ompare e l eren “3 oc1ty pro es velocity profile at section (2) is @Ps shown in Fig. P521, at the entrance to a 3-ft—wide channel 1 1,? e velocity distribution is uniform with a velocity V. Further down- i = (X) stream the velocity profile is given by u : 4y H 23:2, where u is in U 5 ft/3 andy is in ft- Detel‘miflfi the “11116 0f V- develop an expression for the volume flowrate through the edge of the boundary layer from the leading edge to a location downstream at x where the boundary layer thickness is 8. Section (2) Section {1} \ U Outer edge IFIGURE P5.21 5.22 A water flow situation is described by the velocity field equation v 2 (3x + 2)i + (2y e 4)} — 5zi< ft/s wherex, y, and z are in feet. (a) Determine the mass flowrate through E F | G u n E P5.24 the rectangular area in the plane corresponding to z = 2 feet having comers at (x, y, z) x (0, 0, 2), (5, 0, 2), (5, 5, 2), and (0, 5, 2) as shown Section 5.1.2 Fixed, Nondeforming Control Volume— in Fig P5.22a. (b) Show that mass is conserved in the control volume Unsteady Flow having corners at (x, y, z) = (0,0, 2), (5, O, 2), (5, 5, 2), (0, 5, 2), (0, O, 0), )1: . _ _ _ (5’ 0’ 0), (5, 5, a), and (0, 5, 0), as shown in Fig. pink (2; at standard conditions enters the compressor shown 1n Fig. P525 at a rate of 10 ft3/s. It leaves the tank through a I.24n.-diame— ter pipe with a density of 0.0035 slugs/ft3 and a uniform speed of 700 ft/s. (a) Determine the rate (slugs/s) at which the mass of air in the tank is increasing or decreasing. (b) Determine the average time rate of change of air density within the tank. Compressor Tank volume = 20113 10 rats. (a) (1;) 9.00233 slugsma FIGURE 135.22 - _EFIGURE 135.25 ...
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