1 HW #3: Cnservatin f Linear Mmentum, Cnservatin f Energy, Cnservatin f Angular Mmentum and Turbmachines, Bernulli s Equatin, Dimensinal Analysis, and Pipe Flws Prblem 1. Cnservatins f Mass and Linear Mmentum fr A Cntrl Vlume as bserved frm A Statinary Frame f Reference [Adapted frm Munsn et al., 2002, Prblem 5.41, p. 283.] The hydraulic dredge is used t dredge sand frm a river bttm, and the sand/water mixture is discharged as a free-jet as shwn belw. The discharge has a crss sectinal area A. The jet discharge speed is V, which is riented at an angle with respect t the hrizntal. Assume that the bell-muth suctin has relative large crss sectinal area such that the sand/water mixture speed at the suctin can be neglected. The specific gravity f the sand/water mixture is SG and water density is. 1.1. Estimate the thrust needed frm the prpeller t hld the bat statinary. 1.2. Under this statinary perating cnditin, is the buyancy frce n the bat equal t the weight W f the bat (including all the lads n the bat)? If nt, hw des the buyancy frce change frm that when the hydraulic dredge is nt perated? A V Prblem 2. Bernulli s Equatin and Cnservatin f Linear Mmentum A nzzle is attached t a vertical pipe and discharges water int the atmsphere as shwn in the figure belw. The discharge rate is 0.1 m 3 /s. The nzzle has a weight f 200 N, and the vlume f water in the nzzle is 0.012 m 3. 2.1. Find the gage pressure at the nzzle inlet. 2.2. Find the frce vectr F F iˆ F ˆj that is required t supprt the nzzle at the flange supprt. x y Assume that the water in the nzzle is accelerated at a rate such that the frictinal effect can be neglected. Nte: In the past, we have slved the cnservatin f linear mmentum with pressure, hence pressure frce, being given. With the knwledge f the Bernulli s equatin under the assumptin, amng thers, f invisicid flw, we can use the Bernulli s equatin t slve fr the unknwn pressure first, then use the pressure infrmatin t find the pressure frce fr the C-Mm equatin.
2 0.25 m Prblem 3. Cnservatin f Energy [Fx et al., 2010, Prblem 4.202, p. 160.] Air enters a cmpressr at 96 kpa, 27 C with negligible speed and is discharged at 480 kpa, 260 C with a speed f 152 m/s. If the pwer input is 2.38 MW and the flw rate is 9 kg/s, determine the rate f heat transfer. Prblem 4. Idealized Machines: Is it a pump r a turbine? 4.1. Idealized Axial-Flw Turbmachines Fr the given blade angles, 1. qualitatively sketch the blade shape, 2. qualitatively sketch the inlet and exit velcity diagrams, 3. use apprpriate gverning equatins t shw whether the design is suitable fr a pump r a turbine. 4. Als, state necessary assumptins yu make in rder t arrive at these results. flw z U 1 a. 20, 2 70 1 b. 70, 2 20 flw flw U U
3 4.2. Idealized Radial-Flw Turbmachines [Adapted frm Munsn et al., 2002, Prblem 12.3, p. 816.] The rtr shwn belw has straight, thugh backwardly inclined, blades and cnstant passage width frm inlet t exit. It rtates with an angular velcity f 2,000 rpm. Assume that the fluid enters the impeller with the abslute flw velcity purely in the radial directin and the relative flw velcity is tangent t the blades acrss the entire rtr. Use 1) apprpriate velcity diagram and 2) gverning equatins t shw whether the device is a pump r a turbine. Relevant velcity cmpnents and angles shuld be calculated in rder t shw clearly that this device is a pump r a turbine. Prblem 5. Radial-Flw Machine and The Effect f The Exit Blade Angle A centrifugal radial water pump has the dimensins shwn in the figure belw. The vlume rate f flw is 0.25 ft 3 /s, and the abslute inlet velcity is directed radially utward. The angular velcity f the impeller is 960 rpm. The exit velcity as seen frm a crdinate system attached t the impeller can be assumed t be tangent t the vane at its trailing edge. Calculate the (ideal) pwer required t drive the pump.
4 Prblem 6. Radial-Flw Machine [Adapted frm Fx et al. 2010, Prblem 10.4, pp. 537-538.] Dimensins f a centrifugal pump impeller are Parameter Inlet Outlet Radius, r (mm) 175 500 Blade width, b (mm) 50 30 Blade angle, (deg) 65 70 The pump handles water and is driven at 750 rpm. At the current perating cnditin, the vlume flwrate is 0.75 m 3 /s. Assume that the relative flw enters and leaves the blade at the blade angles 1 and 2, respectively (shckless entry/exit cnditin). 1. Draw the inlet and exit velcity diagrams. 2. Find inlet and exit flw angles, 1 and 2. 3. Find the theretical shaft trque. 4. Find the theretical shaft pwer. 5. Find the theretical hydraulic pwer. 6. Find the theretical hydraulic head. 7. If instead the pump handles a fluid with density 1,200 kg/m 3 at the same kinematical cnditin (i.e., the same velcity diagrams), find the theretical hydraulic pwer and head. Cmpare these t thse f water. Prblem 7. Dimensinal Analysis fr Turbmachines [Adapted frm Fx et al., 2004, Prblem 7.61, p. 308.] The pwer, P, required t drive a fan is assumed t depend n fluid density,, vlume flwrate, Q, impeller diameter, D, and angular speed,. If a fan with D 1 = 200 mm delivers Q 1 = 0.4 m 3 /s f air at 1 = 2400 RPM, 1. what vlume flwrate culd be expected fr a gemetrically similar fan with D 2 = 400 mm at 2 = 1850 RPM, and 2. what ratis f the required driving pwer (P 2 / P 1 ) and trque (T 2 /T 1 ) culd be expected? In the case f the trque rati, use at least tw methds in calculating the rati and cmpare the results f the tw methds. 3. If viscus effect is nt negligible, what additinal variable shuld be included and what the resulting dimensinless grup culd be expected?
5 Prblem 8. Dimensinal Analysis and Similarity A mdel test is perfrmed t determine the flight characteristics f a Frisbee (a plastic disk fr tssing between players). Dependent parameters are drag frce, F D, and lift frce, F L. The independent parameters are speed f the Frisbee, V, the size f the Frisbee (diameter f the Frisbee), D, the prperties f air; density,, and dynamic viscsity,, the Frisbee angular speed,, and the Frisbee rughness height, h. 1. Determine the suitable dimensinless parameters and express the functinal dependence amng them. 2. The test (using air) n a ¼ - scale mdel Frisbee is t be gemetrically, kinematically, and dynamically similar t the prttype. The prttype values are V p = 6.5 m/s and What values f V m and m shuld be used? p = 100 rpm. Prblem 9. Energy Cnsideratin in Pipe Flw Water frm a reservir is t be pumped and discharged thrugh a nzzle as shwn belw. The elevatin difference measured frm the free surface f the water in the reservir t the nzzle exit is 70 m. Fr the piping system, the straight pipe diameter is 60 mm, the ttal straight pipe length is 30 m, and the relative rughness f the pipe is e/d = 0.0035. The nzzle has an exit diameter f 40 mm. The vlume flwrate f 0.04 m 3 /s is desired. Neglect all minr lsses. Determine the hydraulic pwer required f the pump. If the pump efficiency is 80%, determine the shaft pwer required t drive the pump. [Adapted frm Munsn et al., 2002, Prblem 8.89, p. 528.] Nte: Take water density t be 1,000 kg/m 3 and kinematic viscsity t be The frictin factr can be apprximated by the frmula 8.0 7 10 m 2 /s. 1 6.9 e / D 1.8lg10 f Re 3.7 Haaland, 1983) 1.11. (Frm Çengel and Cimbala, 2006, p. 341; after S. E. 70 m
6 Prblem 10. Energy Cnsideratin in Pipe Flw [Adapted frm Çengel and Cimbala, 2006, Prblem 8.120, p. 396.] Water at 15 C is t be discharged frm an pen reservir at a rate f 18 L/s using tw hrizntal cast irn pipes cnnected in series and a pump between them. The first pipe is 20 m lng and has a 6-cm diameter, while the secnd pipe is 35 m lng and has a 4-cm diameter. The water level in the reservir is 30 m abve the centerline f the pipe. The pipe entrance is sharp-edged, and lsses assciated with the cnnectin f the pump are negligible. 1. Determine the required hydraulic head (in m f water) and hydraulic pwer (in kw) f the pump. 2. If the efficiency f the pump is 75%, determine the driving mechanical pwer at the pump shaft. 3. Rughly, hw many passenger-car engines des this required shaft pwer crrespnd t? [Nte that the flwrate is rughly enugh t fill nine 2-litre Pepsi bttles, r ~ 20-litre water bttle fr an ffice water cler, in a secnd.] 4. Find the static pressure rise acrss the pump, i.e., p pump exit p pump inlet. 5. Find the static pressures at the pump inlet and exit. 6. On the ther hand, if the pump is remved and the tw pipes are cnnected directly t each ther (with sudden cntractin between them), determine the resulting flwrate. What is the percentage f this flwrate t that when the pump is installed? What is the driving frce fr this latter case? Nte Yu can use the Mdy s chart, r the frictin factr can be apprximated by the frmula 1 6.9 e / D 1.8lg10 f Re 3.7 Haaland, 1983) Open reservir. Water surface is subjected t atmspheric pressure. 1.11. (Frm Çengel and Cimbala, 2006, p. 341; after S. E. Water tank 30 m 18 L/s 20 m Pump 35 m D = 6 cm D = 4 cm