Examiner: Dr. Mohamed Elsharnoby Time: 180 min. Attempt all the following questions Solve the following five questions, and assume any missing data

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1 Benha University Cllege f Engineering at Banha Department f Mechanical Eng. First Year Mechanical Subject : Fluid Mechanics M111 Date:4/5/016 Questins Fr Final Crrective Examinatin Examiner: Dr. Mhamed Elsharnby Time: 180 min. Attempt all the fllwing questins Slve the fllwing five questins, and assume any missing data 1-a) Define the system ; write dwn the mass, mmentum and energy cnservatin equatins and define each term in these equatins. b) Liquid enters a circular pipe f radius with a linear velcity prfile as a functin f the radius with maximum velcity f U max. After magical mixing, the velcity became unifrm. (i) Write the equatin which describes the velcity at the entrance. (ii) What is the magical averaged velcity at the exit? Assume n slip cnditin. (iii) Calculate the mmentum flux crrectin factr fr this flw. -a) Fr the nzzle shwn in Figure 1, flw rate f 0.01 [kg/sec]. The entrance pressure is 3[Bar] and the entrance velcity is 5 [m/sec]. The exit is unifrm but unknwn. The exit pressure is 1[Bar]. The entrance area is [m ] and the exit area is [cm ]. What is the exit velcity? What is the frce acting the nzzle? Assume that the density is cnstant ρ = 1000[kg/m 3 ] and the vlume in the nzzle is [m 3 ]. b) Given is steady isthermal flw f water at 0 C thrugh the device in Fig.. Heat-transfer, gravity, and temperature effects are negligible. Knwn data are D1 9 cm, Q1 0 m 3 /h, p1 150 kpa, D 7 cm, Q 100 m 3 /h, p 5 kpa, D3 4 cm, and p3 65 kpa. Cmpute the rate f shaft wrk dne fr this device and its directin. 3-a) The insulated tank in Fig. 3 is t be filled frm a high-pressure air supply. Initial cnditins in the tank are T 0 C and p 00 kpa. When the valve is pened, the initial mass flw rate int the tank is kg/s. Assuming an ideal gas, estimate the initial rate f temperature rise f the air in the tank. b) Define the fllwing: i) bundary layer, ii) fully develped flw, iii) entrance length L e (write expressins fr L e in laminar and turbulent flws) 4-a) Incmpressible steady flw in the inlet between parallel plates in Fig.4 is unifrm. u = U = 8 cm/sec, while dwnstream the flw develps int the parablic laminar prfile u = az (z -z), where a is cnstant. If z = 4 cm and the fluid is SAE 30 il whse viscsity µ = 0.9 kg/m.s, and mass density ρ = 891 kg/m 3 i) u max in cm/sec. ii) The skin frictin cefficient C F. iii) What is the axial pressure gradient (dp/dx)? 4-b) Cmpute the displacement thickness, the mmentum thickness, and the shape factr assuming the velcity prfile f turbulent bundary layer is given by: 1

2 4-c) Tw pipes cnnect tw reservirs (A and B) which have a height difference f 10m. Pipe 1 has diameter 50mm and length 100m. Pipe has diameter 100mm and length 100m. Bth have entry lss k L = 0.5 and exit lss k L =1.0 and Darcy f f Calculate: rate f flw fr each pipe 5-a) The pwer P generated by a certain windmill design depends upn its diameter D, the air density ρ, the wind velcity V, the rtatin rate Ω, and the number f blades n. (i) Write this relatinship in dimensinless frm. A mdel windmill, f diameter 50 cm, develps.7 kw at sea level when V 40 m/s and when rtating at 4800 rev/min. (ii) What pwer will be develped by a gemetrically and dynamically similar prttype, f diameter 5 m, in winds f 1 m/s at 000 m standard altitude? (iii) What is the apprpriate rtatin rate f the prttype? 5-b) Sketch curves represent the perfrmance and perating pints f tw pumps perating singly and cmbined in parallel and in series Figure 1 Figure Figure 3 Figure 4 Figure 5 GOOD LUCK

3 Benha University Elabrated by: Dr. Mhamed Elsharnby Cllege f Engineering at Banha Department f Mechanical Eng. Subject: Fluid Mechanics Mdel Answer f the Final Exam Date: 4/5/016 اجابة امتحان الدكتور محمد عبد اللطيف الشرنوبي ميكانيكا الموائع م 111 السنة األولى ميكانيكا 1-a ecall frm therm class, that a system is defined as a vlume f mass f fixed identity. Cnservatin f mass states that the mass f a system is cnstant. This can be written as the fllwing equatin: Cnservatin f linear mmentum which is a restatement f Newtn's Secnd Law. Newtn s Secnd Law In equatin frm this is written as: Where mv = the linear mmentum f the system. Cnservatin f Energy equatin: Fr this, use the First Law f Thermdynamics in rate frm t btain the fllwing Where E = the ttal energy f the system. In the abve equatin is the rate f change f system energy. 3

4 is the rate f heat added t the system is the rate f wrk dne by the system. Because wrk is dne by the system, the negative sign is in the equatin fr the first law f thermdynamics. Nw, these cnservatin laws must always hld fr a system. Cnservatin f Angular Mmentum We will have time t study this 1-b The velcity prfile is linear with radius Figure 1. Additinally, later a discussin n relatinship between velcity at interface t slid als referred as the (n) slip cnditin will be prvided. This assumptin is gd fr mst cases with very few exceptins. It will be assumed that the velcity at the interface is zer. Thus, the bundary cnditin is Figure 1 U(r = ) = 0 and U(r = 0) = Umax Therefre the velcity prfile is (i) Where is radius and r is the wrking radius (fr the integratin). The magical averaged velcity is btained using the equatin. Fr which r Umax 1 rdr Uave 0 The integratin f the equatin gives Umax Umax Uave Uave (ii) 3 3 Calculating the mmentum flux crrectin factr 1 U 1 r da 3 1 rdr A U ave 0 9 r r (1 ) rdr 0 4

5 r r r 18 ( ) 0 ( ) 1.5 (iii) a) The chsen cntrl vlume is shwn in Figure Figure. First, the velcity has t be fund. This situatin is a steady state fr cnstant density. Then U 1A1 UA and after rearrangement, the exit velcity is A U U1 5 5m / sec A The mmentum equatin is applicable but shuld be transfrmed int the z directin which is The cntrl vlume des nt crss any slid bdy (r surface) there is n external frces. Hence, All the frces that act n the nzzle are cmbined as The secnd term r the bdy frce which acts thrugh the center f the nzzle is Ntice that in the results the gravity is nt bld since nly the magnitude is used. The part f the pressure which act n the nzzle in the z directin is 5

6 The last term in the equatin is Which results in Cmbining all transfrm equatin int F z ( ) F z N -b) Fr cntinuity, Q3 Q1 Q 10 m3/hr. Establish the velcities at each prt (figure 3) Figure 3 With gravity and heat transfer and internal energy neglected, the energy equatin becmes Slve fr the shaft wrk: W s 998( ) W Ans. (negative dentes wrk dne n the fluid) 3-a) Fr a CV surrunding the tank, Figure 4, with unsteady flw, the energy equatin is 6

7 Figure 4 where and T are the instantaneus cnditins inside the tank. The CV mass flw gives Cmbine these tw t eliminate (d /dt) and use the given data fr air: 3-b i) This regin, where there is a velcity prfile in the flw due t the shear stress at the wall, we call the bundary layer. Bundary layer is the regin near a slid where the fluid mtin is affected by the slid bundary. ii) Once the bundary layer has reached the centre f the pipe the flw is said t be fully develped. (Nte that at this pint the whle f the fluid is nw affected by the bundary frictin.) At a finite distance frm the entrance, the bundary layers merge and the inviscid cre disappears. The flw is then entirely viscus, and the axial velcity adjusts slightly further until at x = L e it n lnger changes with x and is said t be fully develped, v = v(r) nly. iii) The length f pipe befre fully develped flw is achieved is different fr the tw types f flw. The length is knwn as the entry length. The entrance length L e is estimated fr laminar flw t be : L e /D = 0.06 e D fr laminar L e /D = 4.4 e 1/6 D fr turbulent flw Where L e is the entrance length; and e D is the eynlds number based n Diameter 4-a) figure 5 7

8 The flw rate per unit width f the area z Q az( z z0 z0 z0 z) dz a( ) a 3 6 Figure 5 Q Uz cm 3 / sec a 3 a 3 6 z U max at the middle where z cm umax 3 (4 ) =1 cm/sec (i) du du The shear stress at the wall i.e z=0 az0 =0.9x3x4=3.48 \n/m dz dz The skin frictin cefficient 3.48 CF 1.05 (ii) 1 U (0.08) dp dx dp z 87Pa / m dx z (iii) 4-b) 8

9 5-a) Apply Bernulli t each pipe separately. Fr pipe 1: Figure 6 p A and p B are atmspheric, and as the reservir surface mve s slwly u A and u B are negligible, s And flw rate is given by: 9

10 Fr pipe : Again p A and p B are atmspheric, and as the reservir surface mve s slwly u A and u B are negligible, s And flw rate is given by: 11

11 Perfrmance and perating pints f tw pumps perating singly and cmbined in parallel Perfrmance and perating pints f tw pumps perating singly and cmbined in series 11