FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 1 - FLUID FLOW THEORY S.A.E. No. 1 1. Describe the principle of operation of the following types of viscometers. a. Redwood Viscometers. b. British Standard 188 glass U tube viscometer. c. British Standard 188 Falling Sphere Viscometer. d. Any form of Rotational Viscometer The solutions are contained in part 1 of the tutorial. S.A.E. No. 2 1. Oil flows in a pipe 80 mm bore diameter with a mean velocity of 0.4 m/s. The density is 890 kg/m3 and the viscosity is 0.075 Ns/m2. Show that the flow is laminar and hence deduce the pressure loss per metre length. ρud 890 x 0.4 x 0.08 Re = = = 379.7 µ 0.075 Since this is less than 2000 flow is laminar so Poiseuille’s equation applies. 32µ2µ 32 x 0.075 x 1 x 0.4 ∆p = = = 150 Pa d2 0.08 2 2. Oil flows in a pipe 100 mm bore diameter with a Reynolds’ Number of 500. The density is 800 kg/m3. Calculate the velocity of a streamline at a radius of 40 mm. The viscosity µ = 0.08 Ns/m2. ρu d R e = 500 = m µ 500µ 500 x 0.08 um = = = 0.5 m/s ρd 800 x 0.1 Since Re is less than 2000 flow is laminar so Poiseuille’s equation applies. 32µ2µ 32 x 0.08 x L x 0.5 ∆p = = = 128L Pa d2 0.12 ∆p R 2 − r 2 128L 0.05 2 - 0.04 2 u= = = 0.36 m/s 4Lµ 4L x 0.08 3. A liquid of dynamic viscosity 5 x 10-3 Ns/m2 flows through a capillary of diameter 3.0 mm under a pressure gradient of 1800 N/m3. Evaluate the volumetric flow rate, the mean velocity, the centre line velocity and the radial position at which the velocity is equal to the mean velocity. ( ) ( ) 32 µ u m ∆P = 1800 = u m = 0.10125 m/s L d2 umax = 2 um = 0.2025 m/s ∆p R 2 − r 2 1800 0.00152 - r 2 = u = 0.10125 = 4Lµ 4 x 0.005 ( ) ( ) r = 0.0010107 m or 1.0107 mm 4. a. Explain the term Stokes flow and terminal velocity. b. Show that a spherical particle with Stokes flow has a terminal velocity given by u = d2g(ρs - ρf)/18µ Go on to show that CD=24/Re c. For spherical particles, a useful empirical formula relating the drag coefficient and the Reynold’s number is 24 6 + + 0.4 CD = Re 1+ Re Given ρf = 1000 kg/m3, µ= 1 cP and ρs= 2630 kg/m3 determine the maximum size of spherical particles that will be lifted upwards by a vertical stream of water moving at 1 m/s. d. If the water velocity is reduced to 0.5 m/s, show that particles with a diameter of less than 5.95 mm will fall downwards. a) For Re<0.2 the flow is called Stokes flow and Stokes showed that R = 3πd µ u hence R = W = volume x density difference x gravity πd 3g (ρ s − ρ f ) = 3πd µ u R=W= 6 ρs = density of the sphere material ρf = density of fluid d = sphere diameter πd 3g(ρ s − ρ f ) d 2 g(ρ s − ρ f ) = u= 18 π d µ 18 µ b) CD = R/(projected area x ρu2/2) C D = πd 3g(ρ s − ρ f ) 4dg(ρ s − ρ f ) = 2 2 (ρ u /2)6 π d /4 3ρ u 2 4 x 9.81x (1630 - 998) d d = 21.389 2 2 3x 998 x u u 24 6 d CD = + + 0.4 = 21.389 2 Re 1+ R e u CD = d 24 6 d 24 6 − − = 0.4 − =x let 21.389 2 − 2 R e 1+ R e R e 1+ R e u u Re = ρud/µ = 998 x 1 x d/0.89 x 10-3 = 1.1213 x 106 d 21.389 Make a table D Re x 0.001 0.003 0.01 1121.3 3363.9 11213 -0.174 -0.045 0.156 0.02 22426 0.387 0.03 33639 0.608 Plot and find that when d = 0.0205 m (20.5 mm) x = 0.4 c) u = 0.5m/s d = 5.95mm Re = ρud/µ = 998 x 0.5 x 0.00595/0.89 x 10-3 = 3336 d C D = 21.389 2 = 0.509 u 24 6 CD = + + 0.4 = 0.509 3336 1 + 3336 Since CD is the same, larger ones will fall. 5. Similar to Q5 1998 A simple fluid coupling consists of two parallel round discs of radius R separated by a a gap h. One disc is connected to the input shaft and rotates at ω1 rad/s. The other disc is connected to the output shaft and rotates at ω2 rad/s. The discs are separated by oil of dynamic viscosity µ and it may be assumed that the velocity gradient is linear at all radii. Show that the Torque at the input shaft is given by πD 4 µ (ω1 − ω 2 ) T= 32h The input shaft rotates at 900 rev/min and transmits 500W of power. Calculate the output speed, torque and power. (747 rev/min, 5.3 Nm and 414 W) Show by application of max/min theory that the output speed is half the input speed when maximum output power is obtained. SOLUTION Assume the velocity varies linearly from u1 to over the gap at any radius. Gap is h = 1.2 mm Τ = µ du/dy = µ (u1 - u2)/h For an elementary ring radius r and width dr shear force is Force = τ dA = τ 2πr dr u − u2 x2π r dr dF = µ 1 h Torque due to this force is Substitute u = ωr Integrate Rearrange and substitute R = D/2 u2 the u1 − u 2 x 2 π r 2 dr h ( ω1 − ω 2 ) dT = rdF = µ x 2 π r 3 dr h dT = rdF = µ R ( ω1 − ω2 ) T=µ x 2 π r 3 dr h T=µ ∫ 0 4 (ω1 − ω2 ) x π D h 32 =µ (ω1 − ω2 ) x 2 π R 4 h 4 (ω1 − ω2 ) x π 0.34 = 0.33(ω1 − ω 2 ) 0.012 32 60P 60 x 500 = = 5.305 Nm N = 900 rev/min P = 500 W Power = 2πNT/60 T = 2π N 2π x 900 The torque input and output must be the same. ω1 = 2πN1 /60 = 94.25 rad/s 5.305 = 0.33(94.251 − ω2 ) hence ω2 = 78.22 rad/s and N2 = 747 rev/min P2 = 2πN2T/60 = ω2T = 78.22 x 5.305 = 414 W (Power out) For maximum power output dp2/dω2 = 0 P2 = ω2T = 0.33 ω1ω 2 − ω 22 dP2 Differentiate = 0.33(ω1 − 2ω 2 ) dω 2 Equate to zero and it follows that for maximum power output ω1 = 2 ω2 And it follows N1 = 2 N2 so N2 = 450 rev/min Put D = 0.3 m, µ = 0.5 N s/m2, h = 0.012 m T = 0.5 ( ) 6. Show that for fully developed laminar flow of a fluid of viscosity µ between horizontal parallel plates a distance h apart, the mean velocity um is related to the pressure gradient dp/dx by um = - (h2/12µ)(dp/dx) A flanged pipe joint of internal diameter di containing viscous fluid of viscosity µ at gauge pressure p. The flange has an outer diameter do and is imperfectly tightened so that there is a narrow gap of thickness h. Obtain an expression for the leakage rate of the fluid through the flange. Note that this is a radial flow problem and B in the notes becomes 2πr and dp/dx becomes -dp/dr. An integration between inner and outer radii will be required to give flow rate Q in terms of pressure drop p. The answer is Q = (2πh3p/12µ)/{ln(do/di)} FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 1 - FLUID FLOW THEORY ASSIGNMENT 3 1. A pipe is 25 km long and 80 mm bore diameter. The mean surface roughness is 0.03 mm. It carries oil of density 825 kg/m3 at a rate of 10 kg/s. The dynamic viscosity is 0.025 N s/m2. Determine the friction coefficient using the Moody Chart and calculate the friction head. (Ans. 3075 m.) Q = m/ρ = 10/825 = 0.01212 m3/s um = Q/A = 0.01212/(π x 0.042) = 2.411 m/s Re = ρud/µ = 825 x 2.4114 x 0.08/0.025 = 6366 k/D = 0.03/80 = 0.000375 From the Moody chart Cf = 0.0083 hf = 4 Cf L u2/2gd = 4 x 0.0083 x 25000 x 2.41142/(2 x 9.91 x 0.08) = 3075 m 2. Water flows in a pipe at 0.015 m3/s. The pipe is 50 mm bore diameter. The pressure drop is 13 420 Pa per metre length. The density is 1000 kg/m3 and the dynamic viscosity is 0.001 N s/m2. Determine i. the wall shear stress (167.75 Pa) ii. the dynamic pressure (29180 Pa). iii. the friction coefficient (0.00575) iv. the mean surface roughness (0.0875 mm) τo = ∆p D/4L = 13420 x 0.05/4 = 167.75 Pa um = Q/A = 0.015/(π x 0.0252) = 7.64 m/s Dynamic Pressure = ρu2/2 = 1000 x 7.642/2 = 29180 Pa Cf = τo/Dyn Press = 167.75/29180 = 0.00575 From the Moody Chart we can deduce that ε = 0.0017 = k/D k = 0.0017 x 50 = 0.085 mm 3. Explain briefly what is meant by fully developed laminar flow. The velocity u at any radius r in fully developed laminar flow through a straight horizontal pipe of internal radius ro is given by u = (1/4µ)(ro2 - r2)dp/dx dp/dx is the pressure gradient in the direction of flow and µ is the dynamic viscosity. Show that the pressure drop over a length L is given by the following formula. ∆p = 32µLum/D2 The wall skin friction coefficient is defined as Cf = 2τo/( ρum2). Show that Cf = 16/Re where Re = ρumD/µ and ρ is the density, um is the mean velocity and τo is the wall shear stress. 3. Oil with viscosity 2 x 10-2 Ns/m2 and density 850 kg/m3 is pumped along a straight horizontal pipe with a flow rate of 5 dm3/s. The static pressure difference between two tapping points 10 m apart is 80 N/m2. Assuming laminar flow determine the following. i. The pipe diameter. ii. The Reynolds number. Comment on the validity of the assumption that the flow is laminar ASSIGNMENT 4 1. Research has shown that tomato ketchup has the following viscous properties at 25oC. Consistency coefficient K = 18.7 Pa sn Power n = 0.27 Shear yield stress = 32 Pa Calculate the apparent viscosity when the rate of shear is 1, 10, 100 and 1000 s-1 and conclude on the effect of the shear rate on the apparent viscosity. µ app = This fluid should obey the Herchel-Bulkeley equation so put γ = 1 and µapp = 50.7 put γ = 10 and µapp = 6.682 put γ = 100 and µapp = 0.968 put γ = 1000 and µapp = 0.153 The apparent viscosity reduces as the shear rate increases. τy γ& + Kγ& n −1 = 32 + 18.7γ& 0.27−1 &γ 2. A Bingham plastic fluid has a viscosity of 0.05 N s/m2 and yield stress of 0.6 N/m2. It flows in a tube 15 mm bore diameter and 3 m long. (i) Evaluate the minimum pressure drop required to produce flow. The actual pressure drop is twice the minimum value. Sketch the velocity profile and calculate the following. (ii) The radius of the solid core. (iii) The velocity of the core. (iv) The volumetric flow rate. du τ = τY + µ The minimum value of τ is τy dy Balancing forces on the plug τy x 2πrL = ∆pπr2 2L ∆p = τ Y and the minimum ∆p- is at r = R r b ∆p = 0.6 ∆p = 2 x 480 = 960 Pa From the force balance ∆p = τ Y r = τY 2x3 = 480 Pa 0.0075 2L r 2L 2x3 = 0.6 = 0.00375 m or 3.75 mm ∆p 960 The profile is follows Poiseuille’s equation ∆p 960 u= R2 − r2 = 0.0075 2 − 0.003752 = 0.0675 m/s 4µ L 4 x 0.05 x 3 ( ) ( ) Flow rate of plug =Au = π(0.003752)x0.0675 = 2.982 x 10-6 m3/s dQ = u (2πr dr) = ∆p(2 π r ) 2 2 R −r 4µ L ( ) ∆p(2 π ) ⎡ R 2 r 2 r 4 ⎤ − ⎥ Q= ⎢ 4µ L ⎣ 2 2 ⎦r ∆p(2 π ) ⎧⎪⎡ R 4 R 4 ⎤ ⎡ R 2 r 2 r 4 ⎤ ⎪⎫ − ⎥⎬ Q= − ⎨⎢ ⎥−⎢ 4 µ L ⎪⎩⎣ 2 4 ⎦ ⎣ 2 4 ⎦ ⎪⎭ ∆p π ⎡ R 4 R 2 r 2 r 4 ⎤ Q= − − ⎥ ⎢ 2µ L ⎣ 4 2 4⎦ 960 π ⎡ 0.00754 0.00752 x 0.003752 0.003754 ⎤ −6 3 − − Q= ⎢ ⎥ = 4.473x10 m /s 2 x0.05 x 3 ⎣ 4 2 4 ⎦ -6 -6 3 Total Q = (4.473 + 2.982) x 10 = 7.46 x 10 m /s ∆p(2 π ) 2 3 Q=∫ rR − r 4 µ L r R ( ) R n ⎛ du ⎞ 3. A non-Newtonian fluid is modelled by the equation τ = K ⎜ ⎟ where n = 0.8 and ⎝ dr ⎠ 0.8 2 K = 0.05 N s /m . It flows through a tube 6 mm bore diameter under the influence of a pressure drop of 6400 N/m2 per metre length. Obtain an expression for the velocity profile and evaluate the following. (i) The centre line velocity. (0.953 m/s) (ii) The mean velocity. (0.5 m/s) FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 2 – APPLICATIONS OF BERNOULLI SELF ASSESSMENT EXERCISE 1 1. A pipe 100 mm bore diameter carries oil of density 900 kg/m3 at a rate of 4 kg/s. The pipe reduces to 60 mm bore diameter and rises 120 m in altitude. The pressure at this point is atmospheric (zero gauge). Assuming no frictional losses, determine: i. The volume/s (4.44 dm3/s) ii. The velocity at each section (0.566 m/s and 1.57 m/s) iii. The pressure at the lower end. (1.06 MPa) Q = m/ρ = 4/900 = 0.00444 m3/s u1 = Q/A1 = 0.00444/(π x 0.052) = 0.456 m/s u2 = Q/A2 = 0.00444/(π x 0.032) = 1.57 m/s 2 2 h1 + z1 +u1 /2g = h2 + z2 +u2 /2g h2 = 0 z1 = 0 h1 + 0 + 0.5662/2g = 0 + 120 + 1.572/2g h1 =120.1 m p = ρgh = 900 x 9.81 x 120.1 = 1060 kPa 2. A pipe 120 mm bore diameter carries water with a head of 3 m. The pipe descends 12 m in altitude and reduces to 80 mm bore diameter. The pressure head at this point is 13 m. The density is 1000 kg/m3. Assuming no losses, determine i. The velocity in the small pipe (7 m/s) ii. The volume flow rate. (35 dm3/s) h1 + z1 +u12/2g = h2 + z2 + u22/2g 3 + 12 +u12/2g = 13 + 0 + u22/2g 2 2 2 = (u2 - u1 ) /2g (u22 - u12 ) = 39.24 u1 A1= Q = u2 A2 u1 = u2 (80/120)2 = 0.444 u2 2 2 2 39.24 = u2 – (0.444 u2) = 0.802 u2 u2 = 6.99 m/s u1 = 3.1 m/s 2 3 3 Q = u2 A2 = 6.99 x π x 0.04 = 0.035 m /s or 35 dm /s 3. A horizontal nozzle reduces from 100 mm bore diameter at inlet to 50 mm at exit. It carries liquid of density 1000 kg/m3 at a rate of 0.05 m3/s. The pressure at the wide end is 500 kPa (gauge). Calculate the pressure at the narrow end neglecting friction. (196 kPa) A1 = πD12/4 = π(0.1)2/4 = 7.854 x 10-3 m2 A2 = πD22/4 = π(0.05)2/4 = 1.9635 x 10-3 m2 u1 = Q/A1 = 0.05/7.854 x 10-3 = 6.366 m/s u2 = Q/A2 = 0.05/1.9635 x 10-3 = 25.46 m/s p1 +ρu12/2 = p2 + ρu22/2 500 x 103 + 100 x (6.366)2/2 = p2 + 1000 x (25.46)2/2 p2 = 196 kPa 4. A pipe carries oil of density 800 kg/m3. At a given point (1) the pipe has a bore area of 0.005 m2 and the oil flows with a mean velocity of 4 m/s with a gauge pressure of 800 kPa. Point (2) is further along the pipe and there the bore area is 0.002 m2 and the level is 50 m above point (1). Calculate the pressure at this point (2). Neglect friction. (374 kPa) 800 x103 + 800x 42/2 + 0 = p2 + 800 102/2+ 800 x 9.81 x 50 p2 = 374 kPa 5. A horizontal nozzle has an inlet velocity u1 and an outlet velocity u2 and discharges into the atmosphere. Show that the velocity at exit is given by the following formulae. u2 ={2∆p/ρ + u12}½ and u2 ={2g∆h + u12}½ p1 +ρu12/2 + ρgz1 = p2 + ρu22/2+ ρgz2 z1= z2 2 2 p1 +ρu1 /2 = p2 + ρu2 /2 p1 - p2 = (ρ/2)( u22 - u12) 2(p1 - p2)/ ρ = ( u22 - u12) 2 u2 = √(2∆p/ρ + u1 ) Substitute p = ρgh and u2 =√{2g∆h + u12}½ SELF ASSESSMENT EXERCISE 2 1. A pipe carries oil at a mean velocity of 6 m/s. The pipe is 5 km long and 1.5 m diameter. The surface roughness is 0.8 mm. The density is 890 kg/m3 and the dynamic viscosity is 0.014 N s/m2. Determine the friction coefficient from the Moody chart and go on to calculate the friction head hf. L = 5000 m d = 1.5 m k = 0.08 mm ε = k/D 0.8/1500 = 533 x 10-6 From the Moody Chart Cf = 0.0045 2 ρ = 890 kg/m3 µ = 0.014 Ns/m u = 6 m/s Re = ρuD/µ = 890 x 6 x 1.5/0.014 = 572 x 103 hf = 4 Cf Lu2/(2 g d) = 110 m 2. The diagram shows a tank draining into another lower tank through a pipe. Note the velocity and pressure is both zero on the surface on a large tank. Calculate the flow rate using the data given on the diagram. (Ans. 7.16 dm3/s) h1 + z1 + u12/2g = h2 + z2 + u22/2g 0 + z 1 + 0 = 0 + 0 + 0 + hL hL = 20 20 = 4 Cf Lu2/(2 g d) + minor losses 20 = {4 x 0.007 x 50u2/(2 x 9.81 x 0.05)} + 0.5 u2/(2 x 9.81) + u2/(2 x 9.81) = 29.5 u2/(2 x 9.81) u = 20(2 x 9.81)/ 29.5 = 3.65 m/s A = 0.00196 m2 Q = Au =0.00196 x 3.65 = 0.00716 m3/s or 7.16 dm3/s 3. Water flows through the sudden pipe expansion shown below at a flow rate of 3 dm3/s. Upstream of the expansion the pipe diameter is 25 mm and downstream the diameter is 40 mm. There are pressure tappings at section (1), about half a diameter upstream, and at section (2), about 5 diameters downstream. At section (1) the gauge pressure is 0.3 bar. Evaluate the following. (i) The gauge pressure at section (2) (0.387 bar) (ii) The total force exerted by the fluid on the expansion. (-23 N) u1 = Q/A1 = 0.003/(π x 0.01252) = 6.11 m/s u2 = Q/A2 = 0.003/(π x 0.022) = 2.387 m/s hL (sudden expansion) = (u12- u22)/2g = 0.7067 m u12/2g + h1 = u22/2g + h2 + hL h1 - h2 = 2.3872/2g – 6.112/2 + 0.7067 = -0.9065 p1 - p2 = ρg(h1 - h2) = 997 x 9.81 x (-0.9065) = -8866 kPa p1 = 0.3 bar p2 = 0.3886 bar p1 A1- + ρ Q u1 = p2 A2- + ρ Q u2 + F 0.3 x105x 0.491 x10-3 + 997 x 0.003 x 6.11 = 0.38866 x105 x 1.257 x10-3 + 997 x 0.003 x 2.387 + F F = -23 N If smooth hL = 0 h1 - h2 = -1.613 and p2 = 0.45778 bar 4. A tank of water empties by gravity through a siphon into a lower tank. The difference in levels is 6 m and the highest point of the siphon is 2 m above the top surface level. The length of pipe from the inlet to the highest point is 3 m. The pipe has a bore of 30 mm and length 11 m. The friction coefficient for the pipe is 0.006.The inlet loss coefficient K is 0.6. Calculate the volume flow rate and the pressure at the highest point in the pipe. Total length = 11 m Cf = 0.006 Bernoulli between (1) and (3) h1 + u12/2g + z1 = h3 + u32/2g + z3 + hL 0 + 6 + 0 = 0 + 0 + 0 + hL hL= 6 hL= Inlet + Exit + pipe 6 = 0.6 u2/2g + u2/2g + (4 x 0.006 x 11/0.03) u2/2g 6 = 0.6 u2/2g + u2/2g + 8.8 u2/2g = 10.4 u2/2g u = 3.364 m/s Q = A u = (π x 0.032/4) x 3.364 Q = 0.002378 m3/s Bernoulli between (1) and (2) h1 + u12/2g + z1 = h2 + u22/2g + z2 + hL 0 + 0 + 0 = h2 + 2 + u2/2g + hL hL= Inlet + pipe = 0.6 u2/2g + (3/11) x 8.8 u2/2g hL= 0.6 x 3.3642/2g + (3/11) x 8.8x 3.364/2g hL= 1.73 m 0 = h2 + 2 + 3.3642/2g + 1.73 h2 = -4.31 m 5. (Q5 1989) A domestic water supply consists of a large tank with a loss free-inlet to a 10 mm diameter pipe of length 20 m, that contains 9 right angles bends. The pipe discharges to atmosphere 8.0 m below the free surface level of the water in the tank. Evaluate the flow rate of water assuming that there is a loss of 0.75 velocity heads in each bend and that friction in the pipe is given by the Blasius equation Cf=0.079(Re)-0.25 The dynamic viscosity is 0.89 x 10-3 and the density is 997 kg/m3. (0.118 dm3/s). 6. A pump A whose characteristics are given in table 1, is used to pump water from an open tank through 40 m of 70 mm diameter pipe of friction factor Cf=0.005 to another open tank in which the surface level of the water is 5.0 m above that in the supply tank. Determine the flow rate when the pump is operated at 1450 rev/min. (7.8 dm3/s) It is desired to increase the flow rate and 3 possibilities are under investigation. (i) To install a second identical pump in series with pump A. (ii) To install a second identical pump in parallel with pump A. (iii) To increase the speed of the pump by 10%. Predict the flow rate that would occur in each of these situations. Head-Flow Characteristics of pump A when operating at 1450 rev/min Head/m 9.75 8.83 7.73 6.90 5.50 3.83 Flow Rate/(l/s) 4.73 6.22 7.57 8.36 9.55 10.75 7. A steel pipe of 0.075 m inside diameter and length 120 m is connected to a large reservoir. Water is discharged to atmosphere through a gate valve at the free end, which is 6 m below the surface level in the reservoir. There are four right angle bends in the pipe line. Find the rate of discharge when the valve is fully open. (ans. 8.3 dm3/s).The kinematic viscosity of the water may be taken to be 1.14 x 10-6 m2/s. Use a value of the friction factor Cf taken from table 2 which gives Cf as a function of the Reynolds number Re and allow for other losses as follows. at entry to the pipe 0.5 velocity heads. at each right angle bend 0.9 velocity heads. for a fully open gate valve 0.2 velocity heads. Re x 105 0.987 1.184 1.382 Cf 0.00448 0.00432 0.00419 8. (i) Sketch diagrams showing the relationship between Reynolds number, Re, and friction factor, Cf , for the head lost when oil flows through pipes of varying degrees of roughness. Discuss the importance of the information given in the diagrams when specifying the pipework for a particular system. (ii) The connection between the supply tank and the suction side of a pump consists of 0.4 m of horizontal pipe , a gate valve one elbow of equivalent pipe length 0.7 m and a vertical pipe down to the tank. If the diameter of the pipes is 25 mm and the flow rate is 30 l/min, estimate the maximum distance at which the supply tank may be placed below the pump inlet in order that the pressure there is no less than 0.8 bar absolute. (Ans. 1.78 m) The fluid has kinematic viscosity 40x10-6 m2/s and density 870 kg/m3. Assume (a) for laminar flow Cf =16/(Re) and for turbulent flow Cf = 0.08/(Re)0.25. (b) head loss due to friction is 4Cf V2L/2gD and due to fittings is KV2/2g. where K=0.72 for an elbow and K=0.25 for a gate valve. What would be a suitable diameter for the delivery pipe ? FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 2 – APPLICATIONS OF BERNOULLI SELF ASSESSMENT EXERCISE 3 Take the density of water to be 997 kg/m3 throughout unless otherwise stated. 1. A Venturi meter is 50 mm bore diameter at inlet and 10 mm bore diameter at the throat. Oil of density 900 kg/m3 flows through it and a differential pressure head of 80 mm is produced. Given Cd = 0.92, determine the flow rate in kg/s. 2∆ p Q = C d A1 r = A1/A2 = 25 ∆p = ρg∆h = 900 x 9.81 x 0.08 =706.3 x 103 Pa 2 ρ r −1 ( ) 0.92 x π x 0.052 Q= 4 2 x 706300 = 909.59 x 10 −6 m 3 /s 2 900 25 − 1 ( ) m = ρQ = 0.0815 kg/s 2. A Venturi meter is 60 mm bore diameter at inlet and 20 mm bore diameter at the throat. Water of density 1000 kg/m3 flows through it and a differential pressure head of 150 mm is produced. Given Cd = 0.95, determine the flow rate in dm3/s. Q = Cd A1 Q= 2ρg ∆ h ρ r2 − 1 ( ) 0.95 x π x 0.06 2 4 r=9 2 x 1000 x 9.81 x 0.15 = 515 x 10 −6 m 3 /s or 0.515 dm3/s 2 1000 9 − 1 ( ) 3. Calculate the differential pressure expected from a Venturi meter when the flow rate is 2 dm3/s of water. The area ratio is 4 and Cd is 0.94. The inlet c.s.a. is 900 mm2. 2∆ p ρ r 2 −1 Q = 0.002 = C d A1 ( 0.002 = 0.94 x 900x10 −6 ) r=4 2∆ p 1000 4 2 − 1 ( ) ∆p 7500 2.3641 = 5.589 = ∆p 7500 ∆p = 41916 Pa 4. Calculate the mass flow rate of water through a Venturi meter when the differential pressure is 980 Pa given Cd = 0.93, the area ratio is 5 and the inlet c.s.a. is 1000 mm2. r=5 2∆ p 2 x 980 m = ρ C d A1 = 1000 x 0.93 x 1000 x 10 -6 = 0.2658 kg/s 2 ρ r −1 1000 5 2 − 1 5. Calculate the flow rate of water through an orifice meter with an area ratio of 4 given Cd is 0.62, the pipe area is 900 mm2 and the d.p. is 586 Pa. (ans. 0.156 dm3/3). ( ) ( ) r=4 Q = C d A1 2∆ p 2 x 586 -6 900 x 10 x 0.62 = = 155.9 x 10 −6 m 3 /s 2 2 ρ r −1 1000 4 − 1 ( ) ( ) 6. Water flows at a mass flow rate 0f 0.8 kg/s through a pipe of diameter 30 mm fitted with a 15 mm diameter sharp edged orifice. There are pressure tappings (a) 60 mm upstream of the orifice, (b) 15 mm downstream of the orifice and (c) 150 mm downstream of the orifice, recording pressure pa, pb and pc respectively. Assuming a contraction coefficient 0f 0.68, evaluate (i) the pressure difference (pa - pb) and hence the discharge coefficient. (ii)the pressure difference (pb - pc) and hence the diffuser efficiency. (iii) the net force on the orifice plate. do = 15 mm dj = jet diameter Cc = 0.68 = (Ab/Ao) = (db /15)2 db = 12.37 mm No Friction between (a) and (b) so Cv = 1.0 Cd = Cc Cv = Cc 2∆ p m = ρ A oCd β = 15/30 = 0.5 ρ 1 - Cc 2β 4 ( 0.8 = 997 ) π x 0.0152 2∆ p x 0.68 4 997 1 - 0.68 2 x 0.5 4 ( ) ∆p ∆p = p a − p b = 21581 Pa 484 Note the same answer may be obtained by applying Bernoulli’s equation between (a) and (b) Now apply Bernoulli’s equation between (b) and (c) 6.677 = pb + ρ ub2/2= pc + ρ uc2/2 + loss loss = ρ (ub - uc) 2 /2 m 0.8 ub = = = 6.677 m/s ρA b 997 x π x 0.01237 2 /4 m 0.8 loss = 997 (6.677 – 1.135) 2 /2 = 15311 Pa uc = = = 1.135 m/s 2 ρA c 997 x π x 0.03 /4 pc - pb = (997/2)( 6.6772 - 1.135 2 ) – 15311 = 6271 Pa η = 15.31/21.581 = 71% Energy recovered = 6.27/21.58 = 29% Force = π x 0.032/4 x 15310 = 10.8 N (on the control section) 7. The figure shows an ejector (or jet pump) which extracts 2 x 10-3 m3/s of water from tank A which is situated 2.0 m below the centre-line of the ejector. The diameter of the outer pipe of the ejector is 40 mm and water is supplied from a reservoir to the thin-walled inner pipe which is of diameter 20 mm. The ejector discharges to atmosphere at section C. Evaluate the pressure p at section D, just downstream of the end of pipe B, the velocity in pipe B and the required height of the free water level in the reservoir supplying pipe B. (-21.8 kPa gauge, 12.9 m/s, 6.3 m). It may be assumed that both supply pipes are loss free. AB = π x 0.022/4 = 314.2 x 10-6 m2 AD = AC - AB = 942.48 x 10-6 m2 Apply Bernoulli from A to D AC = π x 0.042/4 = 1256 x 10-6 m2 uD = QD/AD = 0.002 x 10-6/0.94248 x 10-6 = 2.122 m/s hA + u2 u 2A + zA = h D + D + zD 2g 2g u 2D 2.1222 − zD = − − 2 = −2.23 m pD = ρghD = -21.8 kPa 2g 2g Next apply the conservation of momentum between the points where B and D join and the exit at C. This results in the following. hD = − ⎧ 1 ⎧ 1 1 ⎫ 2Q B Q D p B A C 1 ⎫ − + + Q 2D ⎨ − Q 2B ⎨ ⎬− ⎬=0 AC ρ ⎩ AB AC ⎭ ⎩ AD AC ⎭ ⎧ 1 1 ⎫ ⎧ 10 6 10 6 ⎫ − − a=⎨ ⎬=⎨ ⎬ = 2386 ⎩ A B A C ⎭ ⎩ 314.2 1256 ⎭ b= 2Q D 2 x 2x10 −3 = = 3.1847 A C 1.256 x 10 −3 ( ) 6 pBAC 10 6 ⎫ 1 ⎫ - 21800 x 1256 x 10 -6 2 ⎧ 1 −3 2 ⎧ 10 c= + QD ⎨ − + 2 x 10 − ⎬= ⎨ ⎬ 1000 ρ ⎩AD AC ⎭ ⎩ 942.48 1256 ⎭ c = −27.38x10 −3 + 1.06 x 10 -3 = -26.32 x 10 -3 aQ 2B + bQ B + c = 0 QB = − b ± b 2 − 4ac 2a − 3.1847 ± 3.1847 2 + 4 x 2386 x 0.02632 2 x 2386 − 3.1847 ± 16.17 QB = = −0.00272 or 0.00405 m 3 /s 2 x 2386 uB = QB/AB = 12.922 m/s QB = Apply Bernoulli between E and point D z = hB + uB2/2g = 6.282 m 8. Discuss the use of orifice plates and venturi-meters for the measurement of flow rates in pipes. Water flows with a mean velocity of 0.6 m/s in a 50 mm diameter pipe fitted with a sharp edged orifice of diameter 30 mm. Assuming the contraction coefficient is 0.64, find the pressure difference between tappings at the vena contracta and a few diameters upstream of the orifice, and hence evaluate the discharge coefficient. Estimate also the overall pressure loss caused by the orifice plate. It may be assumed that there is no loss of energy upstream of the vena contracta. do = 30 mm dj = jet diameter = db Cc = 0.64 = (Ab/Ao) = (db /30)2 db = 24 mm Q = 0.6 x π x 0.052/4 = 0.001178 m3/s No Friction between (a) and (b) so Cv = 1.0 Cd = Cc Cv = Cc 2∆ p Q = A oCd β = 30/50 = 0.6 ρ 1 - Cc 2β 4 ( 0.001178 = ) π x 0.032 2∆ p x 0.64 4 997 1 - 0.64 2 x 0.6 4 ( ) ∆p ∆p = p a − p b = 3200 Pa 472 Note the same answer may be obtained by applying Bernoulli’s equation between (a) and (b) Now apply Bernoulli’s equation between (b) and (c) 2.06 = pb + ρ ub2/2= pc + ρ uc2/2 + loss Q 0.001178 ub = = = 2.6 m/s Ab π x 0.024 2 /4 q 0.001178 = = 0.6 m/s uc = Ac π x 0.035 2 /4 loss = ρ (ub - uc) 2 /2 loss = 997 (2.6 – 0.6) 2 /2 = 2000 Pa 9. The figure shows an ejector pump BDC designed to lift 2 x 10-3 m3/s of water from an open tank A, 3.0 m below the level of the centre-line of the pump. The pump discharges to atmosphere at C. The diameter of thin-walled inner pipe 12 mm and the internal diameter of the outer pipe of the is 25 mm. Assuming that there is no energy loss in pipe AD and there is no shear stress on the wall of pipe DC, calculate the pressure at point D and the required velocity of the water in pipe BD. Derive all the equations used and state your assumptions. AB = π x 0.0122/4 =113.1 x 10-6 m2 AD = AC - AB = 377.8 x 10-6 m2 Apply Bernoulli from A to D AC = π x 0.0252/4 = 491 x 10-6 m2 uD = QD/AD = 0.002 x 10-6/377.8 x 10-6 = 5.294 m/s u 2D u 2A hA + + zA = h D + + zD 2g 2g u 2D 5.294 2 − zD = − − 3 = −4.429 m pD = ρghD = -43.4 kPa 2g 2g Next apply the conservation of momentum between the points where B and D join and the exit at C. This results in the following. hD = − ⎧ 1 ⎧ 1 1 ⎫ 2Q B Q D p B A C 1 ⎫ Q 2B ⎨ − + + Q 2D ⎨ − ⎬− ⎬=0 AC ρ ⎩ AB AC ⎭ ⎩ AD AC ⎭ ⎧ 1 1 ⎫ ⎧ 10 6 10 6 ⎫ − − a=⎨ ⎬=⎨ ⎬ = 6805 ⎩ A B A C ⎭ ⎩113.1 491⎭ b= - 2Q D 2 x 2x10−3 = = −8.149 AC 491 x 10 −6 ( ) 6 p BA C 1 ⎫ - 43400 x 491 x 10 -6 10 6 ⎫ −3 2 ⎧ 10 2 ⎧ 1 + QD ⎨ − + 2 x 10 − c= ⎬= ⎨ ⎬ ρ 1000 ⎩ AD AC ⎭ ⎩ 377.8 491⎭ c = -18.886 x 10 -3 aQ 2B + bQ B + c = 0 QB = − b ± b 2 − 4ac 2a 8.149 ± 8.149 2 + 4 x 6805 x 0.0188 2 x 6805 Q B = −0.001172 or 0.002369 m 3 /s uB = QB/AB = 20.95 m/s QB = FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 3 – BOUNDARY LAYERS SELF ASSESSMENT EXERCISE 1 1. A smooth thin plate 5 m long and 1 m wide is placed in an air stream moving at 3 m/s with its length parallel with the flow. Calculate the drag force on each side of the plate. The density of the air is 1.2 kg/m3 and the kinematic viscosity is 1.6 x 10-5 m2/s. Rex = u L/ν = 3 x 5/1.6 x 10-5 = 937.5 x 103 CDF = 0.074 Rex -1/5 = 4.729 x 10-3 Dynamic Pressure = ρuo2/2 = 1.2 x 32/2 = 5.4 Pa τw = CDF x dyn press = 0.0255 Pa R = τw x A = 0.0255 x 5 = 0.128 N 2. A pipe bore diameter D and length L has fully developed laminar flow throughout the entire length with a centre line velocity uo. Given that the drag coefficient is given as CDf = 16/Re where Re = ρu o D , show that the drag force on the inside of the pipe is given as R=8πµuoL and µ hence the pressure loss in the pipe due to skin friction is pL = 32µuoL/D2 CDF = 16/Re R = τw x ρuo2/2 = CDF x (ρuo2/2) x A R = (16/Re)(ρuo2/2) A R = (16µ/ρuoD)(ρuo2/2) πDL R = (16 µ uo π L/2) = 8 π µ uo L pL = R/A = 8 π µ uo L /(πD2/4) = 32 µ uo L/D2 SELF ASSESSMENT EXERCISE No. 2 1. Calculate the drag force for a cylindrical chimney 0.9 m diameter and 50 m tall in a wind blowing at 30 m/s given that the drag coefficient is 0.8. The density of the air is 1.2 kg/m3. CD = 0.8 = 2R/(ρu2A) R = 0.8 (ρu2/2)A = 0.8 (1.2 x 302/2)(50 x 0.9) = 19440 N 2 Using the graph (fig.1.12) to find the drag coefficient, determine the drag force per metre length acting on an overhead power line 30 mm diameter when the wind blows at 8 m/s. The density of air may be taken as 1.25 kg/m3 and the kinematic viscosity as 1.5 x 10-5 m2/s. (1.8 N). Re = u d/ν = 8 x 0.03/1.5 x 10-5 = 16 x 103 From the graph CD = 1.5 R = CD (ρuo2/2)A = 1.5 (1.25 x 82/2)(0.03 x 1) = 1.8 N SELF ASSESSMENT EXERCISE No. 3 1. a. Explain the term Stokes flow and terminal velocity. b. Show that the terminal velocity of a spherical particle with Stokes flow is given by the formulae = d2g(ρs - ρf)/18µ. Go on to show that CD=24/Re Stokes flow –for ideal fluid - no separation - Re <0.2 R = Buoyant weight = (πd3/6)g (ρs – ρf) = 3 π d µ ut ut = d2 g (ρs – ρf)/18 µ R = CD(ρ ut2/2)(π d2/4) CD = 26µ/(ρ ut d) = 24/Re 2. Calculate the largest diameter sphere that can be lifted upwards by a vertical flow of water moving at 1 m/s. The sphere is made of glass with a density of 2630 kg/m3. The water has a density of 998 kg/m3 and a dynamic viscosity of 1 cP. CD =(2/ρ u2A)R = {(2 x 4)/ (ρ u2 πd2)} (πd3/6)g (ρs – ρf) = 21.38 d Try Newton Flow first D = 0.44/21.38 = 0.206 m Re =( 998 x 1 x 0.0206)/0.001 = 20530 therefore this is valid. 3. Using the same data for the sphere and water as in Q2, calculate the diameter of the largest sphere that can be lifted upwards by a vertical flow of water moving at 0.5 m/s. (5.95 mm). CD = 85.52 d Try Newton Flow D = 0.44/85.52 = 0.0051 Re =( 998 x 0.5 x 0.0051)/0.001 = 2567 therefore this is valid. 4. Using the graph (fig. 1.12) to obtain the drag coefficient of a sphere, determine the drag on a totally immersed sphere 0.2 m diameter moving at 0.3 m/s in sea water. The density of the water is 1025 kg/m3 and the dynamic viscosity is 1.05 x 10-3 Ns/m2. Re = (ρud/µ) =(1025 x 0.3 x 0.2/1.05 x 10-3) = 58.57 x 103 From the graph CD = 0.45 R = CD (ρu2/2)A = 0.45(1025 x 0.32/2)(π x 0.22/4) = 0.65 N 5. A glass sphere of diameter 1.5 mm and density 2 500 kg/m3 is allowed to fall through water under the action of gravity. The density of the water is 1000 kg/m3 and the dynamic viscosity is 1 cP. Calculate the terminal velocity assuming the drag coefficient is CD = 24 Re -1 (1+ 0.15Re 0.687) CD = F/(Area x Dynamic Pressure) πd 3g(ρ s − ρ f ) πd 3g (ρ s − ρ f ) 4d 3g (ρ s − ρ f ) R= = CD = C D 6 (π d 2 /4)(ρ u 2 /2) 3ρ f u 2 d 2 Arrange the formula into the form CD Re2 as follows. CD = 4d 3g (ρ s − ρ f ) ρ f µ 2 4d 3g(ρ s − ρ f )ρ f 4d 3g(ρ s − ρ f )ρ f µ2 1 = = x x x 3ρ f u 2 d 2 ρf µ 2 3µ 2 ρ f2 u 2 d 2 3µ 2 R 2e 4d 3g(ρ s − ρ f )ρ f and evaluating this we 3µ 2 get 66217 24 1 + 0.15R e 0.687 we may solve by From C D = Re plotting CDRe2 against Re From the graph Re = 320 hence u = Re µ/ρd = 0.215 m/s C D R 2e = [ ] 6. A glass sphere of density 2 690 kg/m3 falls freely through water. Find the terminal velocity for a 4 mm diameter sphere and a 0.4 mm diameter sphere. The drag coefficient is CD = 8F/{πd2ρu2} This coefficient is related to the Reynolds number as shown for low values of Re. Re 15 20 25 30 35 CD 3.14 2.61 2.33 2.04 1.87 The density and viscosity of the water is 997 kg/m3 and 0.89 x 10-3 N s/m2. For Re >1000 CD = 0.44 For a 4 mm sphere we might guess from the question that Re is greater than 1000 and hence CD = 0.44 πd 3g(ρ s − ρ f ) πd 3g(ρ s − ρ f ) 4d 3g (ρ s − ρ f ) R= CD = = C D 6 (π d 2 /4)(ρ u 2 /2) 3ρ f u 2 d 2 4d 3g(ρ s − ρ f ) Putting in values ρ = 997 µ = 0.0089 d = 0.004 CD = 0.44 3ρ f C D d 2 u = 0.45 m/s Check Re = ρud/µ = 2013 so this is valid 8F For the 0.4 mm sphere we might guess from the question that C D = 2 2 πd ρu u= 8F ρµ 2 8Fρ µ2 8Fρ 1 CD = 2 2 x 2 = x 2 2 2 = x 2 πd ρu ρµ πµ ρ u d π µ 2 R e2 C D R e2 = 8Fρ = 3.205x109 F 2 πµ πd 3g(ρ s − ρ f ) (The buoyant weight) 6 For a 0.4 mm sphere F = 556.55 x 10-9 N C D R e2 = 3.205x10 9 F = 1784 Plot graph for the 0.4 mm sphere The 0.4 mm sphere fits the table u = Re µ/ρd = 0.066 m/s R= 7. A glass sphere of diameter 1.5 mm and density 2 500 kg/m3 is allowed to fall through water under the action of gravity. Find the terminal velocity assuming the drag coefficient is CD = 24 Re-1(1+ 0.15Re0.687) πd 3g(ρ s − ρ f ) R= (The buoyant weight) 6 πx 0.0015 3 x 9.81 (2500 − 997 ) R= = 26.056 x 10 -6 N 6 8F 8 x 26x10 −6 29.578 x 103 CD = 2 2 = = πd ρu π (0.0015) 2 x 997 x u 2 u2 CD = [ 29.578 x 10 3 24 = 1 + 0.15R 0.687 e 2 R u e ] 0.687 ⎤ 29.578 x 10 3 24 x 0.00089 ⎡ ⎛ 997 u (0.0015) ⎞ = ⎢1 + 0.15⎜ ⎥ ⎟ 997 u (0.0015) ⎣⎢ u2 ⎝ 0.00089 ⎠ ⎦⎥ [ ] 2.0709 = u 1 + 24.657u 0.687 = u + 24.657u1.687 Solve for u and u = 0.215 m/s (plotting might be the best way) FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 3 – BOUNDARY LAYERS SELF ASSESSMENT EXERCISE 4 1. The BL over a plate is described by u/u1=sin(πy/2δ). Show that the momentum thickness is 0.137δ. 2 2 δ⎡ δ⎡ ⎡ u ⎤⎡ u ⎤ u ⎛u⎞ ⎤ πy ⎫ ⎛ ⎧ πy ⎫ ⎞ ⎤ ⎧ ⎥ ⎢ − ⎜ ⎟ dy = ∫ ⎢sin ⎨ ⎬ − ⎜⎜ sin ⎨ ⎬ ⎟⎟ ⎥dy θ = ∫ ⎢ ⎥ ⎢1 − ⎥ dy = ∫ u1 ⎜⎝ u1 ⎟⎠ ⎥ u u1 ⎦ 2δ 2δ 0 ⎣ 1 ⎦⎣ 0⎢ 0⎢ ⎣ ⎩ ⎭ ⎝ ⎩ ⎭ ⎠ ⎥⎦ ⎦ ⎣ We need the trig identity sin2A = ½ - ½ cos 2A δ ⎡ ⎧ πy ⎫ 1 1 ⎛ ⎧ πy ⎫ ⎞⎤ θ = ∫ ⎢sin ⎨ ⎬ − + ⎜⎜ cos ⎨ ⎬ ⎟⎟⎥dy ⎩ 2δ ⎭ 2 2 ⎝ ⎩ 2δ ⎭ ⎠⎦ 0⎣ δ δ ⎡ 2δ ⎧ πy ⎫ y δ ⎧ πy ⎫⎤ θ = ⎢− cos⎨ ⎬ − + sin ⎨ ⎬⎥ ⎩ 2δ ⎭ 2 2π ⎩ 2δ ⎭⎦ 0 ⎣ π δ ⎤ ⎡ ⎤ ⎡ 2δ θ = ⎢− 0 − + 0⎥ − ⎢− − 0 + 0⎥ = 0.137δ 2 ⎦ ⎣ π ⎦ ⎣ 2. The velocity profile in a laminar boundary layer on a flat plate is to be modelled by the cubic expression u/u1=a0+a1y + a2y2+a3y3 where u is the velocity a distance y from the wall and u1 is the main stream velocity. Explain why a0 and a2 are zero and evaluate the constants a1 and a3 in terms of the boundary layer thickness δ. Define the momentum thickness θ and show that it equals 39δ/280 Hence evaluate the constant A in the expression δ/x = A (Rex)-0.5 where x is the distance from the leading edge of the plate. It may be assumed without proof that the friction factor Cf = 2 dθ/dx At y = 0, u = 0 so it follows that ao = 0 d2u/dy2 = 0 @ y = 0 so a2 = 0. Show for yourself that this is so. The law is reduced to At y = δ, u = u1 so hence u/ul = aly + a3y3 1 =a1δ + 3a3 δ2 al =(1-a3δ3)/δ Now differentiate and at y = δ, du/dy is zero so du/dy = ul (a1 + 3a3y2) 0 =a1+ 3a3 δ2 so a1 = -3a3δ2 Hence by equating a1 = 3/2δ and a3 = -1/2δ3 Now we can write the velocity distribution as u/u1=3y/2δ - (y/δ)3/2 and du/dy = u1 {3/2δ + 3y2/2δ3} If we let y/δ = η u/u1={3η/2 + (η)3/2} The momentum thickness is δ ⎡ u ⎤⎡ u ⎤ θ = ∫ ⎢ ⎥ ⎢1 − ⎥ dy but dy = δdη u1 ⎦ u 0 ⎣ 1 ⎦⎣ ⎡ 3η η 3 ⎤ ⎡ 3η η 3 ⎤ θ = ∫ ⎢ − ⎥ ⎢1 − − ⎥ dη 2 2⎦ 2 2 ⎦⎣ 0⎣ ⎡ 3η 2 η 4 9η3 η 7 3η5 ⎤ Integrating gives: − − − + θ = δ⎢ ⎥ 4 8 12 28 10 ⎦ ⎣ Between the limits η = 0 and η =1 this evaluates to θ = 39δ/280 1 Now must first go back to the basic relationship. du/dy = u1 {3/2δ + 3y2/2δ3} At the wall where y = 0 the shear stress is τo =µ du/dy = µ ul {3/2δ + 3y2/2δ3} = ( µ ul/δ) δ [(3/2δ) + 3y2/2δ3] Putting y/δ= η we get at the wall η = 0 τo = (µ ul /δ) δ [(3/2δ) +3δ2/2δ] τo = (µ ul /δ) [(3/2) +3δ2/2] τo = (µ ul /δ) (3/2)...... ...........(2.1) The friction coefficient Cf is always defined as Cf = 2τo /(ρ u 2 2 )............. (2.2) It has been shown elsewhere that Cf = 2dθ/dx. The student should search out this information from test books. Putting θ = 39δ/280 (from the last example) then C f = 2dθ/dx = (2 x 39/280) dδ/dx ..............(2.3) Equating (2.2) and (2.3) gives τo =( ρu12)(39/280)dδ/dx .............(2.4) Equating (2.1) and (2.4) gives (ρu12)(39/280)dδ/dx = (ρu/δ)(3/2) Hence (3 x 280)/(2 x 39)(µ dx)/ρu) = δdδ Integrating 10.77(µx/ ρul) = δ2 /2 + C Since δ = 0 at x = 0 (the leading edge of the plate) then C=0 Hence δ= {21.54 µx/ρul }½ Dividing both sides by x gives δ/x = 4.64(µ/ρulx)- ½ = 4.64Re- ½ NB Rex =ρulx/µ. and is based on length from the leading edge. 3.(a) The velocity profile in a laminar boundary layer is sometimes expressed in the formula 2 3 4 u y ⎛ y⎞ ⎛ y⎞ ⎛ y⎞ = a 0 + a1 + a 2 ⎜ ⎟ + a 3 ⎜ ⎟ + a 4 ⎜ ⎟ u1 δ ⎝δ⎠ ⎝δ⎠ ⎝δ⎠ where u1 is the velocity outside the boundary layer and δ is the boundary layer thickness. Evaluate the coefficients a0 to a4 for the case when the pressure gradient along the surface is zero. (b) Assuming a velocity profile u/u1=2(y/δ) - (y/δ)2 obtain an expression for the mass and momentum fluxes within the boundary layer and hence determine the displacement and momentum thickness. Part A 2 3 4 u y ⎛ y⎞ ⎛ y⎞ ⎛ y⎞ = a 0 + a1 + a 2 ⎜ ⎟ + a 3 ⎜ ⎟ + a 4 ⎜ ⎟ u1 δ ⎝δ⎠ ⎝δ⎠ ⎝δ⎠ Boundary conditions Where y = 0, u = 0 hence a0 = 0 1 = a 1 + a 2 + a 3 + a 4 ....................(A) Where y = δ, u = u1 Differentiate with respect to y ⎛ y2 ⎞ ⎛ y3 ⎞ 1 du a 1 ⎛ y⎞ = + 2a 2 ⎜ 2 ⎟ + 3a 3 ⎜⎜ 3 ⎟⎟ + 4a 4 ⎜⎜ 4 ⎟⎟ u1 dy δ ⎝δ ⎠ ⎝δ ⎠ ⎝δ ⎠ Where y = δ, du/dy = 0 a ⎛1⎞ ⎛1⎞ ⎛1⎞ 0 = 1 + 2a 2 ⎜ ⎟ + 3a 3 ⎜ ⎟ + 4a 4 ⎜ ⎟ δ ⎝δ⎠ ⎝δ⎠ ⎝δ⎠ 0 = a 1 + 2a 2 + 3a 3 + 4a 4 ..........................(B) Differentiate a second time. ⎛ y2 ⎞ 1 d 2u ⎛ 1 ⎞ ⎛ y⎞ ⎜ ⎟ = + + 2a 6a 12a ⎜ ⎟ ⎜ ⎟ 2 3 4⎜ 4 ⎟ 2 3 u1 dy 2 ⎝δ ⎠ ⎝δ ⎠ ⎝δ ⎠ ⎛ 1 ⎞ Where y = 0 , d2u/dy2 = 0 hence 0 = 2a 2 ⎜ 2 ⎟ Hence a2 = 0 ⎝δ ⎠ (A) becomes 1 = a1 + a 3 + a 4 0 = a 1 + 3a 3 + 4a 4 (B) becomes 1 = 0 − 2a 3 − 3a 4 .................(C) Subtract The second differential becomes ⎛ y2 ⎞ 1 d 2u ⎛ y⎞ ⎜ ⎟ = + 6a 12a ⎜ ⎟ 3 4⎜ 4 ⎟ 3 u1 dy 2 ⎝δ ⎠ ⎝δ ⎠ 2 2 Where y = δ, d u/dy = 0 ⎛ y2 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ y⎞ 0 = 6a 3 ⎜ 3 ⎟ + 12a 4 ⎜⎜ 4 ⎟⎟ 6a 3 ⎜ 2 ⎟ + 12a 4 ⎜ 2 ⎟ = 6a 3 + 12a 4 .................(D) ⎝δ ⎠ ⎝δ ⎠ ⎝δ ⎠ ⎝δ ⎠ Divide through by 3 0 = 2a 3 + 4a 4 .............................(E) Add (C) and (E) a4 = 1 Substitute into (E) 0 = 2a3 + 4 a3 = -2 Substitute into (A) 1 = a1 -2 + 1 a1 = 2 Hence 3 u y ⎛ y⎞ ⎛ y⎞ = 2 − 2⎜ ⎟ + 2⎜ ⎟ u1 δ ⎝δ⎠ ⎝δ⎠ 4 PART B δ ⎡ u⎤ δ = ∫ ⎢1 − ⎥dy u1 ⎦ 0⎣ u = 2η − η 2 u1 ∗ 1 [ ] δ = δ ∫ 1 − 2η + η 2 dη ∗ 0 1 ⎡ η3 ⎤ δ ∗ = δ ⎢η − η 2 + ⎥ 3 ⎦0 ⎣ [ 1 1⎤ δ ⎡ δ ∗ = δ ⎢1 − 1 + ⎥ = 3⎦0 3 ⎣ ][ ] δ δ ⎡ u ⎤⎡ u ⎤ θ = ∫ ⎢ ⎥ ⎢1 − ⎥ dy = ∫ 2η − η 2 1 − 2η + η 2 dy u u1 ⎦ 0 0 ⎣ 1 ⎦⎣ 1 ( ) θ = δ ∫ 2η − 5η 2 + 4η3 − dη 4 dη [ ] 1 θ = δ η 2 − 5η 2 / 3 + η 4 − η5 / 5 0 0 θ = δ[1 − 5 / 3 + 1 − 1 / 5] θ = 2δ/15 4. When a fluid flows over a flat surface and the flow is laminar, the boundary layer profile may be represented by the equation u/u1= 2(η) - (η)2 where η = y/δ y is the height within the layer and δ is the thickness of the layer. u is the velocity within the layer and u1 is the velocity of the main stream. Show that this distribution satisfies the boundary conditions for the layer. Show that the thickness of the layer varies with distance (x) from the leading edge by the equation δ=5.48x(Rex)-0.5 It may be assumed that τo = ρu12 dθ/dx Where y = 0, Where y = δ, Where y = δ, η = y/δ = 0 so the condition is satisfied. u = u1 η=1 u/u1= 2(η) - (η)2 = 1 so the condition is satisfied. 1 du 2 ⎛ y ⎞ 4 du/dy = 0 = + 2⎜ 2 ⎟ = u1 dy δ ⎝δ ⎠ δ u=0 1 d 2u ⎛ 2 ⎞ =⎜ ⎟ Where y = 0, d2u/dy = 0 u1 dy 2 ⎝ δ 2 ⎠ The last two are apparently not satisfactory conditions. 2 du ⎧ 2 2y ⎫ = u1 ⎨ + 2 ⎬ dy ⎩δ δ ⎭ At the wall where y =0 the shear stress is τo=µ du/dy = µu1 {2/δ + 2y/δ2} =(µu1/δ) [2 +2y/δ] Putting y/δ= η we get τo = (µ u1/δ) δ [2+ 2η] Starting with at the wall η=0 τo = (2µu1/δ)………........(1) Putting θ =2δ/15 (last example) then τo= ( ρu12) dθ/dx =( ρu12)(2/15)dδ/dx ....(2) Equating (1) and (2) Hence ( ρu12)(2/15)dδ/dx = (2µu1/δ) 15 (µ/ρu1)dx = δ dδ Integrating 15(µx/ ρu1) = δ2/2 + C Since δ = 0 at x = 0 (the leading edge of the plate) then C=0 Hence δ2 = 30(µx/ ρu1) δ = 5.478(µx /ρu1) = 5.478Rex-½ 5. Define the terms displacement thickness δ * and momentum thickness θ. Find the ratio of these quantities to the boundary layer thickness δ if the velocity profile within the boundary layer is given by u/u1=sin(πy/2δ) Show, by means of a momentum balance, that the variation of the boundary layer thickness δ with -0.5 distance (x) from the leading edge is given by δ = 4.8(Rex) It may be assumed that τo = ρu12 dθ/dx Estimate the boundary layer thickness at the trailing edge of a plane surface of length 0.1 m when air at STP is flowing parallel to it with a free stream velocity u1 of 0.8 m/s. It may be assumed without proof that the friction factor Cf is given by Cf = 2 dθ/dx N.B. standard data µ = 1.71 x 10-5 N s/m2. ρ = 1.29 kg/m3. DISPLACEMENT THICKNESS δ* The flow rate within a boundary layer is less than that for a uniform flow of velocity u1. If we had a uniform flow equal to that in the boundary layer, the surface would have to be displaced a distance δ* in order to produce the reduction. This distance is called the displacement thickness. MOMENTUM THICKNESS θ The momentum in a flow with a BL present is less than that in a uniform flow of the same thickness. The momentum in a uniform layer at velocity u1 and height h is ρhu12. When a BL exists this is reduced by ρu12θ. Where θ is the thickness of the uniform layer that contains the equivalent to the reduction. 2 2 δ⎡ δ δ⎡ ⎡ u ⎤⎡ u ⎤ u ⎛u⎞ ⎤ ⎧ πy ⎫ ⎛ ⎧ πy ⎫ ⎞ ⎤ ⎥ ⎢ ⎜ ⎟ dy = ∫ ⎢sin ⎨ ⎬ − ⎜⎜ sin ⎨ ⎬ ⎟⎟ ⎥dy − θ = ∫ ⎢ ⎥ ⎢1 − ⎥ dy = ∫ u1 ⎜⎝ u1 ⎟⎠ ⎥ u u1 ⎦ 2δ 2δ 0⎢ 0 ⎣ 1 ⎦⎣ 0⎢ ⎣ ⎩ ⎭ ⎝ ⎩ ⎭ ⎠ ⎥⎦ ⎦ ⎣ We need the trig identity sin2A = ½ - ½ cos 2A δ ⎡ ⎧ πy ⎫ 1 1 ⎛ ⎧ πy ⎫ ⎞⎤ θ = ∫ ⎢sin ⎨ ⎬ − + ⎜⎜ cos⎨ ⎬ ⎟⎟⎥dy ⎩ 2δ ⎭ 2 2 ⎝ ⎩ 2δ ⎭ ⎠⎦ 0⎣ δ ⎡ 2δ ⎧ πy ⎫⎤ ⎧ πy ⎫ y δ θ = ⎢− cos⎨ ⎬ − + sin ⎨ ⎬⎥ ⎩ 2δ ⎭ 2 2π ⎩ 2δ ⎭⎦ 0 ⎣ π δ ⎤ ⎡ 2δ ⎤ ⎡ θ = ⎢− 0 − + 0⎥ − ⎢− − 0 + 0⎥ = 0.137δ 2 ⎦ ⎣ π ⎦ ⎣ τo = µ du/dy = µ u1 sin(πy/2δ) = µ u1(π/2δ)cos(πy/2δ) At the wall y = τo = µ u1 (π/2δ) .................(1) C f = 2τo/ρu1 2 ..........(2) Cf = 2 dθ/dx and θ = 0.137δ Cf=2 d(0.137δ)/dx = 0.274δ dδ/dx ..........(3) Equate (2) and (3) 2τo/ρu1 2 = 0.274δ dδ/dx τo = ρ u 1 2(0.137δ) dδ/dx............... (4) Equate (1) and (4) µ u 1 (π/2δ) = ρ u 1 2(0.137δ)dδ/dx 2 2 µπx/(0.274ρ u 1 )= δ /2 C but where x =0, δ = 0 so C = 0 δ/x = {(2π)/0.274)}½ Rex -½ = 4.8 Rex -½ x = 0.1 m u = 0.8 m/s ρ = 1.29 kg/m3 µ =1.7I x 10-5 Rex = (1.29)(0.8)(0.1)/ 1.71 x10-5 = 6035 δ/0.1 = 4.8(6035) -½ δ = 0.006 m Extra ... τ o =µ u1π/2δ δ/x = 4.8 Rex -½ (from fluids tables) C f = 2τo ρ u12 2(µ ul π/2δ) / ρ u 1 2 = µ π x / ( ρ u 1 δ x ) = π x / Rex δ C f = (π / Rex ) (Rex½ /4.8) = 0.65 Rex -½ 6. In a laminar flow of a fluid over a flat plate with zero pressure gradient an approximation to the velocity profile is u/u1=(3/2)(η) - (1/2)(η)3 η = y/δand u is the velocity at a distance y from the plate and u1 is the mainstream velocity. δ is the boundary layer thickness. Discuss whether this profile satisfies appropriate boundary conditions. Show that the local skin-friction coefficient Cf is related to the Reynolds’ number (Rex) based on distance x from the leading edge by the expression Cf =A (Rex)-0.5 and evaluate the constant A. It may be assumed without proof that Cf = 2 dθ/dx and that θ is the integral of (u/u1)(1 - u/u1)dy between the limits 0 and δ This is the same as Q2 whence θ = 39δ/280 Cf = 2 dθ/dx = (78/280)dδ/dx τo = ρu2π/2δ Cf = 2 τo/ρu2 = (ρ u1 π x)/δ ρ u12x) = (π x /δ) Rex½ δ/x = 4.64 Rex½ Cf = π/(4.64 Rex½) x (1/ Rex) = 0.65 Rex½ SELF ASSESSMENT EXERCISE No. 5 1. Under what circumstances is the velocity profile in a pipe adequately represented by the 1/7 th 1/7 power law u/u1=(y/R) where u is the velocity at distance y from the wall, R is the pipe radius and u1 is the centre-line velocity ? The table shows the measured velocity profile in a pipe radius 30 mm. Show that these data satisfy the 1/7 th power law and hence evaluate (i) the centre-line velocity (ii) the mean velocity um (iii) the distance from the wall at which the velocity equals um. 1.0 1.54 2.0 1.70 5.0 1.94 10.0 2.14 15.0 2.26 20.0 2.36 y (mm) u (m/s) Limitations are that the flow must be turbulent, with Re >107 and the velocity gradient must be the same at the junction between laminar sub layer and the turbulent layer. u/u1 = (y/R)1/7 a = radius = 30 mm Evaluate at various values of y y 1 2 u 1540 1700 ul 2503 2503 5 1940 2506 10 2140 2504 15 2260 2495 20 2350 2500 mm mm/s mm/s Since ul is constant the law is true. Take ul = 2502 mm/s R 1/7 ⎛y⎞ Q = 2π ∫ (R − y )⎜ ⎟ ⎝R⎠ 0 R ( ) = 2 π x 2502 ∫ R 6/7 y1/7 − y 8/7 R −1/7 dy 0 ⎡ 49R ⎤ Q = 2π x 2502 ⎢ ⎥ ⎣ 120 ⎦ 2 2π x 2502 ⎡ 49R 2 ⎤ Mean velocity um = Q/A = ⎥ =2403 ⎢ πR 2 ⎣ 120 ⎦ 2043/2502 = (y/30)1/7 y = 7.261 mm. Note this fits with um = (49/60) ul and if this was the starting point the question would be simple. 2. 1/7 (a) Discuss the limitations of the 1/7th power law u/u1=(y/R) for the velocity profile in a circular pipe of radius R, indicating the range of Reynolds numbers for which this law is applicable. (b) Show that the mean velocity is given by 49u1/60. (c) Water flows at a volumetric flow rate of 1.1 x 10-3 m3/s in a tube of diameter 25 mm. Calculate the centre-line velocity and the distance from the wall at which the velocity is equal to the mean velocity. -0.25 (d) Assuming that Cf=0.079(Re) evaluate the wall shear stress and hence estimate the laminar sub-layer thickness. µ = 0.89 x 10-3 N s/m2. ρ = 998 kg/m3. Limitations are that the flow must be turbulent with Re>107 and the velocity gradient must be the same at the junction between the laminar sub layer and the turbulent level. Flow through an elementary cylinder. For a pipe, the B.L. extends to the centre so δ = radius = R. Consider an elementary ring of flow. The velocity through the ring is u. The volume flow rate through the ring is 2πrudr The volume flow rate in the pipe is Q = 2π∫rudr 1/7 Since δ = R then u = u1(y/R) also r = R-y 1/7 -1/7 (R-y)y dy Q = 2π∫ (R-y)udr = 2π∫u1R -1/7 1/7 8/7 Q =2πu1R [Ry -y ] -1/7 8/7 15/7 Q =2πu1R [(7/8)Ry - (7/15)y ] 2 Q =(49/60)πR u1. 2 The mean velocity is defined by um=Q/πR hence um=(49/60)u1 Q = 1.1 x 10-3 m3/s R = 0.025/2 = 0.0125 m umean = 2.241 m/s u = u1 (y/R)1/7 y = 3.0285 m when u = umean u1 = 2.744 m/s At the junction, the gradients are the same. Laminar sub layer τo = µ du/dy Re = ρ u D/µ = 2.241 x 0.025 x 998/0.89 x 10-3 = 62820 Cf = 0.005 = 2τo/ρu2 τo = 0.005 x 998 x 2.2412/(2 x 0.005) = 12.5 N/m2 For the turbulent layer τo = µ du/dy ⎧⎪ ⎛ y ⎞1/7 ⎫⎪ d ⎨u 1 ⎜ ⎟ ⎬ ⎪ ⎝ R ⎠ ⎪⎭ 12.5 = 0.89x10 −3 ⎩ dy y −6/7 = 7 x 14045 x 0.01251/7 = 52.196 x 10 −6 2.744 y = 10.09 x 10-6 m Some text uses the following method. y= 5µ/ρu* where u*= √(τo/ρ) = √(12.7/998) = 0/112 In this case y = 39.81 x 10-6 m (the thickness of the laminar sub-layer) FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 4 – FLOW THROUGH POROUS PASSAGES SELF ASSESSMENT EXERCISE No.1 Q.1 Outline briefly the derivation of the Carman-Kozeny equation. 2 dp 180 µ u (1 − ε ) =− dl d s2 ε 3 dp/dl is the pressure gradient, µ is the fluid viscosity, u is the superficial velocity, ds is the particle diameter and ε is the void fraction. A cartridge filter consists of an annular piece of material of length 150 mm and internal diameter and external diameters 10 mm and 20 mm. Water at 25oC flows radially inwards under the influence of a pressure difference of 0.1 bar. Determine the volumetric flow rate. (21.53 cm3/s) For the filter material take d = 0.05 mm and ε = 0.35. µ=0.89 x 10-3 N s/m2 and ρ= 997 kg/m3. The solution for part 1 is as given in the tutorial. dp 180 µ u (1 − ε ) =− dr d s2 ε 3 The surface area of the annulus is 2πrL L = 0.15 m and d = 0.05 x 10-3 m The velocity is u = Q/2πrL 2 dp 180 µ (1 − ε ) Q =− x 2 3 dr 2πrL ds ε 2 For radial flow we change dl to dr dp 180 x 0.89 x10 -3 (1 − 0.35)2 Q x =− 2 dr 2 π r x 0.15 0.05 x 10 -3 0.353 ( dp = −670 x 10 -6 Q ) dr r ⎛r ⎞ ∆p = −670 x 10 -6 Q ln⎜⎜ 1 ⎟⎟ = −670 x 10 -6 Q ln (2 ) ⎝ r2 ⎠ 5 6 ∆p = -0.1 x 10 = -464.4 x 10 Q Q = 21.53 x 10-6 m3/s Integrate Q.2 (a) Discuss the assumptions leading to the equation of horizontal viscous flow through a packed bed 2 dp 180µu (1 − ε ) =− dL d s2 ε 3 ∆p is the pressure drop across a bed of depth L, void fraction ε and effective particle diameter d. u is the approach velocity and µ is the viscosity of the fluid. (b) Water percolates downwards through a sand filter of thickness 15 mm, consisting of sand grains of effective diameter 0.3 mm and void fraction 0.45. The depth of the effectively stagnant clear water above the filter is 20 mm and the pressure at the base of the filter is atmospheric. Calculate the volumetric flow rate per m2 of filter. (2.2 dm3/s) (Note the density and viscosity of water are given in the instructions on all exams papers) µ=0.89 x 10-3 N s/m2 and ρ= 997 kg/m3 Part (a) is as stated in the tutorial. ∆p = ρ g h = 997 x 9.81 x 0.02 = 195.61 Pa dp 195.61 180 x 0.89 x 10 −3 u (1 − 0.45) =− − dL 0.015 (0.3 x 10 −3 ) 2 0.453 u = 0.0022 m/s Q = u A = 0.022 m3/s per unit area 2 Q3. Oil is extracted from a horizontal oil-bearing stratum of thickness 15 m into a vertical bore hole of radius 0.18 m. Find the rate of extraction of the oil if the pressure in the bore-hole is 250 bar and the pressure 300 m from the bore hole is 350 bar. Take d= 0.05 mm, ε= 0.30 and µ = 5.0 x 10-3 N s/m2. 2 dp 180 µ u (1 − ε ) =− dr d s2 ε 3 dp 180 µ (1 − ε ) Q x =− 2 3 dr 2πrL ds ε The surface area of the annulus is 2πrL The velocity is u = Q/2πrL 2 dp 180 µ (1 − ε ) Q x =− 2 3 dr 2πrL ds ε 2 L = 15 m and d = 0.05 x 10-3 m dp 180 x 5 x 10 -3 (1 − 0.3) Q x =− 2 dr 2 π r x 15 0.05 x 10 -3 0.33 2 ( ) dp = −69.32 x 10 6 Q dr r ⎛r ⎞ ⎛ 300 ⎞ 6 ∆p = −69.32 x 10 6 Q ln⎜⎜ 1 ⎟⎟ = −69.32 x 10 6 Q ln⎜ ⎟ = 514.3 x 10 Q ⎝ 0.18 ⎠ ⎝ r2 ⎠ ∆p = 250 – 350 = -100bar Integrate Q = 100 x 105/514.3 x 106 = 0.01944 m3/s FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 5 – POTENTIAL FLOW SELF ASSESSMENT EXERCISE No.1 1.a. Show that the potential function φ = A(r + B/r)cosθ represents the flow of an ideal fluid around a long cylinder. Evaluate the constants A and B if the cylinder is 40 mm radius and the velocity of the main flow is 3 m/s. b. c. d. Obtain expressions for the tangential and radial velocities and hence the stream function ψ. Evaluate the largest velocity in the directions parallel and perpendicular to the flow direction. (6 m/s for tangential velocity) A small neutrally buoyant particle is released into the stream at r = 100 mm and θ = 150o. Determine the distance at the closest approach to the cylinder. (66.18 mm) Part (a) is as given in the tutorial. Normally this equation is given as φ = (A r + B/r)cosθ but both are the same but the constants represent different values. Part (b) The values of the constants depend upon the quadrant selected to solve the boundary conditions. This is because the sign of the tangential velocity and radial velocity are different in each quadrant. Which ever one is used, the final result is the same. Let us select the quadrant from 90o to 180o. At a large distance from the cylinder and at the 90o position the velocity is from left to right so at this point vT = -u. From equation 4 we have dφ ⎧ B⎫ vT = φ = A ⎨r + ⎬cosθ rdθ ⎩ r⎭ A ⎧ B⎫ ⎧ B⎫ ⎨r + ⎬sinθ = −A ⎨1 + 2 ⎬sinθ r ⎩ r⎭ ⎩ r ⎭ dφ ⎛ B⎞ = A⎜1 − 2 ⎟cosθ vR = dr ⎝ r ⎠ Putting r = infinity and θ = 90o and remembering that +vT is anticlockwise +u is left to right, we have vT = -3 m/s. B/r2 →0 ⎧ B⎫ v T = − A ⎨1 + 2 ⎬sinθ = −3 = − A{1 + 0} Hence A=3 ⎩ r ⎭ At angle 180o with r = 0.04, vR = 0 B ⎞ ⎛ B⎞ ⎛ v R = A⎜1 − 2 ⎟cosθ = 0 = 3⎜1 − Hence B = 0.0016 ⎟(−1) ⎝ r ⎠ ⎝ 0.04 2 ⎠ ⎛ 0.0016 ⎞ ⎧ 0.0016 ⎫ v T = −3⎨1 + v R = 3⎜1 − ⎟cosθ ⎬sinθ 2 r r2 ⎠ ⎝ ⎭ ⎩ ⎛ 0.0016 ⎞ ⎛ 0.0016 ⎞ ⎛ 0.0016 ⎞ dψ = v R r dθ = 3⎜1 ⎟ cos θ dθ ψ = 3⎜ r ⎟ sin θ ⎟r cos θ dθ = 3⎜ r 2 r ⎠ r ⎠ r ⎝ ⎝ ⎝ ⎠ Part (c) The maximum velocity is 2u = 6 m/s (proof is in the tutorial) Part (d) R = 0.1 m θ = 150o 0.0016 ⎞ 0.0016 ⎞ ⎛ 0.0016 ⎞ ⎛ ⎛ ψ = 3⎜ r ⎟ sin θ = 3⎜ 0.1 ⎟ sin 150 = 3⎜ 0.1 ⎟ sin 150 = 0.244 r ⎠ 0.1 ⎠ 0.1 ⎠ ⎝ ⎝ ⎝ The closest approach is at θ = 90o ⎛ 0.0016 ⎞ 0.0813r = r 2 - 0.0016 ψ = 0.244 = 3⎜ r ⎟ sin 90 r ⎠ ⎝ vT = − ( r2 - 0.0813 r – 0.0016 = 0 ) solve the quadratic and r = 0.098 m or 98 mm 2.a. Show that the potential function φ = (Ar + B/r)cosθ gives the flow of an ideal fluid around a cylinder. Determine the constants A and B if the velocity of the main stream is u and the cylinder is radius R. b. Find the pressure distribution around the cylinder expressed in the form (p - p')/(ρu2/2) as a function of angle. c. Sketch the relationship derived above and compare it with the actual pressure profiles that occur up to a Reynolds number of 5 x 105. Part (a) is in the tutorial. dφ Part (b) vT = rdθ 1 ⎧B ⎫ v T = − ⎨ + Ar ⎬sinθ r⎩r ⎭ ⎫ ⎧B v T = −⎨ 2 + A ⎬sinθ ⎭ ⎩r Putting r = infinity and θ = 90o and remembering that +vT is anticlockwise +u is left to right, we have ⎧B ⎫ v T = − u = − ⎨ 2 + A ⎬ sin θ = −{0 + A}x1 ⎩r ⎭ Hence vT = -A = -u so A = u as expected from earlier work. At angle 180o with r = R, the velocity is only radial in directions and is zero because it is arrested. dφ ⎛ B ⎞ = ⎜ − 2 + A ⎟cosθ From equation 3 we have vR = dr ⎝ r ⎠ Putting r = R and vR = 0 and θ = 180 we have ⎛ B ⎞ ⎛ B ⎞ 0 = ⎜ − 2 + A ⎟(−1) = ⎜ 2 − A ⎟ ⎝ R ⎠ ⎝R ⎠ B 0 = 2 − u B = uR 2 Put A = u R Substituting for B = uR2 and A = u we have ⎧ uR 2 ⎫ ⎧B ⎫ + ur ⎬cosθ φ = ⎨ + Ar ⎬cosθ = ⎨ ⎩r ⎭ ⎩ r ⎭ At the surface of the cylinder r = R the velocity potential is φ = {uR + uR }cosθ = 2uRcosθ The tangential velocity on the surface of the cylinder is then ⎫ ⎧ uR 2 dφ ⎫ ⎧B vT = v T = −2usinθ = −⎨ 2 + A ⎬sinθ v T = −⎨ 2 + u ⎬sinθ rdθ ⎭ ⎩r ⎭ ⎩ r This is a maximum at θ = 90o where the streamlines are closest together so the maximum velocity is 2u on the top and bottom of the cylinder. The velocity of the main stream flow is u and the pressure is p'. When it flows over the surface of the cylinder the pressure is p because of the change in velocity. The pressure change is p - p'. The dynamic pressure for a stream is defined as ρu2/2 The pressure distribution is usually shown in the dimensionless form 2(p - p' )/(ρu2) For an infinitely long cylinder placed in a stream of mean velocity u we apply Bernoulli's equation between a point well away from the stream and a point on the surface. At the surface the velocity is entirely tangential so : p' + ρu2/ 2 = p + ρvT2/2 From the work previous this becomes p' + ρu2/2 = p + ρ(2u sinθ)2/2 p - p' = ρu2/2 - (ρ/2)(4u2sin2θ) = (ρu2/2)(1 - 4sin2θ) (p - p')/(ρu2/2) = 1 - 4 sin2θ If this function is plotted against angle we find that the distribution has a maximum value of 1.0 at the front and back, and a minimum value of -3 at the sides. Research shows that the drag coefficient reduces with increased stream velocity and then remains constant when the boundary layer achieves separation. If the mainstream velocity is further increased, turbulent flow sets in around the cylinder and this produces a marked drop in the drag. This is shown below on the graph of CD against Reynolds’s number. The point where the sudden drop occurs is at a critical value of Reynolds’s number of 5 x105. 3.Show that in the region y>0 the potential function φ = a ln[x2 + (y-c)2 ] + a ln [x2 + (y+c)2 ] gives the 2 dimensional flow pattern associated with a source distance c above a solid flat plane at y=0. b. Obtain expressions for the velocity adjacent to the plane at y = 0. Find the pressure distribution along this plane. c. Derive an expression for the stream function φ. The key to this problem is knowing that two identical sources of strength m equal distance above and below the origin produces the pattern required. φA = -(m/2π)ln r2 φB = -(m/2π)ln r1 2 2 Now use pythagoras ½ r2 = {x + (y – c) } r1 = {x2 + (y + c)2} ½ φp = φA + φB { } { } ⎤⎥⎦ } { }] [{ = a [ln{x + (y + c ) }+ ln{x + (y + c ) }] a = -m/2π 1/2 m⎡ ln x 2 + (y + c )2 + ln x 2 + (y + c )2 2π ⎢⎣ m φp = − ln x 2 + (y + c )2 + ln x 2 + (y + c )2 4π φp = − φp 2 2 1/2 2 2 At y = 0 φ = 2a ln(x2 + c2) v = -dφ/dy = 0 at all values of x so it is the same as an impervious plane. u = -dφ/dx = − At very large values of x, 4ax x + c2 2 u = 0 and p = po ρ ⎛ 4ax ⎞ Apply Bernoulli and po = p + ρu /2 = p o + ⎜ 2 2 ⎟ 2⎝ x +c ⎠ 2 2 4. A uniform flow has a sink placed in it at the origin of the Cartesian co-ordinates. The stream function of the uniform flow and sink are ψ1= Uy and ψ2= Bθ Write out the combined stream function in Cartesian co-ordinates. Given U=0.001 m/s and B= -0.04 m3/3 per m thickness, derive the velocity potential. Determine the width of the flux into the sink at a large distance upstream. ψ1 = u y ψ2 = Bθ ψ = u y + Bθ B = Q/2π for a sink B⎞ dψ urcosθ + B ⎛ dφ = ⎜ − ucosθ + ⎟dr − =− r⎠ rdθ r ⎝ φ = (− u r cosθ + B lnr ) { φ = ⎛⎜ − u x + B ln x 2 + y 2 φ = (− u x + B lnr ) ⎝ } 1/2 ⎞ ⎟ ⎠ u = 0.001 B = -0.04 1/2 φ = ⎛⎜ − 0.001 x + 0.04 ln x 2 + y 2 ⎞⎟ φ = − 0.001 x + 0.02 ln x 2 + y 2 ⎠ ⎝ Q=ut t = Q/u Q = 2π x 0.04 u = 0.001 t = (2π x 0.04)/0.001 = 80π metres { } ( { }) SELF ASSESSMENT EXERCISE No.2 1. Define the following terms. Stream function. Velocity potential function. Streamline Stream tube Circulation Vorticity. All these definitions are in the tutorial. 2.A free vortex of with circulation K = 2πvTR is placed in a uniform flow of velocity u. Derive the stream function and velocity potential for the combined flow. The circulation is 7 m2/s and it is placed in a uniform flow of 3 m/s in the x direction. Calculate the pressure difference between a point at x = 0.5 and y = 0.5. The density of the fluid is 1000 kg/m3. (Ans. 6695 Pascal) Free Vortex vT = k/2πr r k ⎛r⎞ k ln⎜ ⎟ dr = ψ=∫ 2π ⎝ a ⎠ 2ππ a Uniform flow ψ = -u y = - u r sin θ Combined Flow k ⎛r⎞ ψ= ln⎜ ⎟ − u r sin θ 2π ⎝ a ⎠ dψ = vT dr = (k/2πr) dr θ θ k k θ rdθ = 2π 2π 0 φ = ∫ v T r dθ = ∫ 0 φ = u r cos θ φ= k θ + u r cos θ 2π vθ = √(vT2 + vR2) k = 7 u = 3 vR = dφ/dr = u cos θ vT =dψ/dr = k/2πr - u sin θ Point A vT = 7/(2 π 0.5) - 3 sin 90o = -0.7718 m/s vR = 0 vθ = √(vT2 + vR2)= 0.7718 m/s Pont B θ = 0o vR = u cos θ = 3 vT = 7/(2 π 0.5) - 3 sin 0o = 2.228 m/s vθ = √(vT2 + vR2)= 3.74 m/s Bernoulli between stream and A p = pA + ρ vθ 2/2 pA – pθ = (ρ/2)(vθB2 - vθA2 ) = (1000/2)(3.742 – 0.77182) = 6695 Pa FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 6 – DIMENSIONAL ANALYSIS SELF ASSESSMENT EXERCISE No. 1 1. It is observed that the velocity 'v' of a liquid leaving a nozzle depends upon the pressure drop 'p' and the density 'ρ'. 1 ⎛ p ⎞2 Show that the relationship between them is of the form v = C⎜⎜ ⎟⎟ ⎝ρ⎠ a b -1 -1 -2 v = C {p ρ } [v] = LT [p] = ML T [ρ] = ML-3 M0L1T-1 = (ML-1T-2)a( ML-3)b (T) (M) -1 = -2a 0=a+b a=½ b=-½ 1 v = C {p 1/2 ρ -1/2 ⎛ p ⎞2 v = C⎜⎜ ⎟⎟ ⎝ρ⎠ } 2. It is observed that the speed of a sound in 'a' in a liquid depends upon the density 'ρ' and the bulk modulus 'K'. 1 ⎛ K ⎞2 Show that the relationship between them is a = C⎜⎜ ⎟⎟ ⎝ρ⎠ a b -1 a = C {K ρ } [a] = LT [K] = ML-1T-2 [ρ] = ML-3 M0L1T-1 = (ML-1T-2)a( ML-3)b (T) (M) -1 = -2a 0=a+b a=½ b=-½ 1 1/2 v = C {K ρ -1/2 ⎛ K ⎞2 v = C⎜⎜ ⎟⎟ ⎝ρ⎠ } 3. It is observed that the frequency of oscillation of a guitar string 'f' depends upon the mass 'm', the length 'l' and tension 'F'. 1 Show that the relationship between them is ⎛ F ⎞2 f = C⎜ ⎟ ⎝ ml ⎠ f = C {Fa mb lc } [f] = T-1 [F] = MLT-2 [m] = M T-1 = (MLT-2)a( M)b( L)c (T) (M) (L) -1 = -2a 0=a+b 0=a+c a=½ b=-½ c=-½ 1 1/2 f = C {F -1/2 -1/2 m l } ⎛ F ⎞2 f = C⎜ ⎟ ⎝ ml ⎠ [l] = L SELF ASSESSMENT EXERCISE No.2 1. The resistance to motion 'R' for a sphere of diameter 'D' moving at constant velocity 'v' on the surface of a liquid is due to the density 'ρ' and the surface waves produced by the acceleration of gravity 'g'. Show that the dimensionless equation linking these quantities is Ne = function(Fr) Fr is the Froude number and is given by Fr = v2 gD R = function (D v ρ g) = C Da vb ρc gd There are 3 dimensions and 5 quantities so there will be 5 –3 = 2 dimensionless numbers. Identify that the one dimensionless group will be formed with R and the other with K. Π1 is the group formed between g and D v ρ Π2 is the group formed between R and D v ρ R = Π1 Da vb ρc g = Π2 Da vb ρc --2 [R] = MLT-2 [g] = L T [D] = L [D] = L -1 -1 [v] = LT [v] = LT -3 -3 [ρ] = ML [ρ] = ML --2 a -1 b -3 c -2 a -1 b -3 c MLT = L (LT ) (ML ) LT = L (LT ) (ML ) --2 a+b-3c c -b 1 -2 a+b-3c c -b LT = L M T ML T =L M T Time -2 = -b b = 2 Time -2 = -b b=2 Mass c=0 Mass c=1 Length 1 = a + b -3c Length 1 = a + b -3c 1=a+2 –0 a = -1 1=a+2 –3 a=2 g = Π2 D1 v2 ρ 0 R = Π1 D2 v2 ρ 1 gD R Π 2 = 2 = Fr −2 Π1 = 2 2 = Ne ρv D v Π1 = φΠ2 Ne = φ(Fr) 2. The Torque 'T' required to rotate a disc in a viscous fluid depends upon the diameter 'D' , the speed of rotation 'N' the density 'ρ' and the dynamic viscosity 'µ'. Show that the dimensionless -5 -2 -1 2 -1 equation linking these quantities is {T D N ρ } = function {ρ N D µ } Use the other method here. Identify d as the unknown power. T = f(D N ρ µ) = C Da Nb ρc µd ML2T-2 = (L)a (T-1)b (ML-3)c (ML-1T-1)d (T) -2 = -b - d b=2–d (M) 1=c+d c=1-d (L) 2 = a - 3c – d = a – 3(1 - d) – d a = 5 – 2d T = C D5-2d N2-d ρ1-d µd = C D5 N2 ρ (D -2 N-1 ρ-1 µ1 )d T D-5 N-2 ρ-1 = f(D -2 N-1 ρ-1 µ1 ) ⎛ D2 N ρ ⎞ T ⎟ = f ⎜⎜ ⎟ µ D 5 N 2ρ ⎝ ⎠ SELF ASSESSMENT EXERCISE No.3 1.The resistance to motion 'R' of a sphere travelling through a fluid which is both viscous and compressible, depends upon the diameter 'D' , the velocity 'v' , the density 'ρ' , the dynamic viscosity 'µ' and the bulk modulus 'K'. Show that the complete relationship between these quantities is : Ne = function{Re } {Ma} -1 -2 -2 -1 0.5 where Ne = R ρ v D Re = ρ v D µ Ma = v/a and a = (k/ρ) This may be solved with Buckingham’s method but the traditional method is given here. R = function (D v ρ µ K) = C Da vb ρc µd Ke First write out the MLT dimensions. 1 -2 [R] = ML T [D] = L -1 [v] = LT -3 [ρ] = ML -1 -1 [µ] = ML T -1 --2 [K] = ML T 1 -2 -3 -1 -1 -1 --2 ML T = La (LT-1)b (ML )c(ML T )d (ML T )e 1 -2 a+b-3c-d-e c+d+e -b-d-2e ML T =L M T Viscosity and Bulk Modulus are the quantities which causes resistance so the unsolved indexes are d and e. TIME -2 = -b – d -2e hence b = 2 - d - 2e is as far as we can resolve b MASS 1=c+d+e hence c = 1 - d - e LENGTH 1= a + b - 3c - d - e 1 = a + (2 –d - e) -3(1-d - e) - d - e 1=a–1–d a=2-d Next put these back into the original formula. R = C D2-d v2-d-2e ρ1-d - e µd Ke R = C D2 v2 ρ1 (D-1 v-1ρ-1µ)d (v-2ρ-1K)e ⎛ µ ⎞ R ⎟⎟ = ⎜⎜ 2 2 ρv D ⎝ ρvD ⎠ Ne = f(Re)(Ma) d ⎛ K ⎞ ⎜ 2⎟ ⎜ ρv ⎟ ⎝ ⎠ e SELF ASSESSMENT EXERCISE No.4 1.(a) The viscous torque produced on a disc rotating in a liquid depends upon the characteristic dimension D, the speed of rotation N, the density ρ and the dynamic viscosity µ. Show that : 2 5 2 {T/(ρN D )} =f(ρND /µ) (b) In order to predict the torque on a disc 0.5 m diameter which rotates in oil at 200 rev/min, a model is made to a scale of 1/5. The model is rotated in water. Calculate the speed of rotation for the model which produces dynamic similarity. 3 2 For the oil the density is 750 kg/m and the dynamic viscosity is 0.2 Ns/m . 3 For water the density is 1000 kg/m and the dynamic viscosity is 0.001 Ns/m2. (c) When the model is tested at 18.75 rev/min the torque was 0.02 Nm. Predict the torque on the full size disc at 200 rev/min. ⎛ D 2 Nρ ⎞ T ⎜ ⎟ = f ⎜ µ ⎟ Part (a) is the same as in SAE 2 whence D 5 N 2 ρ ⎝ ⎠ Part (b) For dynamic similarity ⎛ D 2 Nρ ⎞ ⎛ D 2 Nρ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ µ ⎟ ⎜ µ ⎟ ⎝ ⎠ model ⎝ ⎠ object ⎛ (0.2D )2 N m x 1000 ⎞ ⎛ D 2 200 x 750 ⎞ ⎟ ⎜ ⎟ = ⎜⎜ ⎟ ⎟ ⎜ 0.001 0.2 ⎠ object ⎠ model ⎝ ⎝ Nm = 18.75 rev/min ⎛ T ⎞ ⎛ T ⎞ ⎜ 5 2 ⎟ ⎜ 5 2 ⎟ = ⎜D N ρ⎟ ⎜ ⎟ ⎝ ⎠ model ⎝ D N ρ ⎠ objectl ⎛ Tm D o5 N o2 ρ o ⎞ ⎛ 0.02 x 55 200 2 x 750 ⎞ ⎟== ⎜ ⎟ To = ⎜⎜ 5 2 ⎟ ⎜ 15 x 18.75 2 x 1000 ⎟ D N ρ ⎠ ⎝ m m m ⎠ ⎝ To = 5333 N 2. The resistance to motion of a submarine due to viscous resistance is given by : d ⎛ ρvD ⎞ R ⎟⎟ where D is the characteristic dimension. = f ⎜⎜ 2 2 ρv D ⎝ µ ⎠ The submarine moves at 8 m/s through sea water. In order to predict its resistance, a model is made to a scale of 1/100 and tested in fresh water. Determine the velocity at which the model should be tested. (690.7 m/s) 3 The density of sea water is 1036 kg/m 3 The density of fresh water is 1000 kg/m 2 The viscosity of sea water is 0.0012 N s/m . 2 The viscosity of fresh water is 0.001 N s/m . When run at the calculated speed, the model resistance was 200 N. Predict the resistance of the submarine. (278 N). ⎛ ρvD ⎞ ⎛ ρvD ⎞ ⎛ 1000 x v m x D m ⎞ ⎛ 1036 x 8 x D ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎜ ⎟=⎜ ⎟ 0.001 ⎝ ⎠ ⎝ 100 x 0.0012 ⎠ ⎝ µ ⎠m ⎝ µ ⎠o vm = 690.7 m/s ⎛ R ⎞ ⎛ R ⎞ ⎜ 2 2 ⎟ =⎜ 2 2 ⎟ ⎟ ⎜ ⎜ ρv D ⎟ ⎠ m ⎝ ρv D ⎠ m ⎝ ⎞ ⎛ ⎛ Ro 200 ⎞ ⎟= ⎜ ⎜ 1036 x 8 2 (D /100 )2 ⎟ ⎜⎝ 1000 x 690.7 2 D 2 ⎟⎠ m ⎠ ⎝ Ro = 278 N 3. The resistance of an aeroplane is due to, viscosity and compressibility of the fluid. Show that: ⎛ R ⎞ ⎜ 2 2 ⎟ = f (M a )(R e ) ⎜ ρv D ⎟ ⎝ ⎠ An aeroplane is to fly at an altitude of 30 km at Mach 2.0. A model is to be made to a suitable scale and tested at a suitable velocity at ground level. Determine the velocity of the model that gives dynamic similarity for the Mach number and then using this velocity determine the scale which makes dynamic similarity in the Reynolds number. (680.6 m/s and 1/61.86) The properties of air are sea level 30 km -5 µ= 1.7897 x 10 -5 µ= 1.4745 x 10 a= 340.3 m/s a= 301.7 m/s 3 ρ = 1.225 kg/m 3 ρ = 0.0184 kg/m When built and tested at the correct speed, the resistance of the model was 50 N. Predict the resistance of the aeroplane. For dynamic similarity (Ma)m = (Ma)o and (Re)m = (Re)o ⎛ ρvD ⎞ ⎛ ρvD ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ ⎝ µ ⎠m ⎝ µ ⎠o and (v/a)m = (v/a)o = 2 vm = 340.3 = 680.6 m/s -5 ⎛ D o ⎞ ⎛ ρ m v m µ o ⎞ 1.225 x 680.6 x 1.4745 x 10 ⎟⎟ = = 61.87 ⎜ ⎟ = ⎜⎜ -5 ⎝ D m ⎠ ⎝ ρ o v o µ m ⎠ 0.0184 x 603.4 x 1.7897 x 10 ⎛ R ⎞ ⎛ R ⎞ ⎜ 2 2 ⎟ =⎜ 2 2 ⎟ ⎜ ρv D ⎟ ⎜ ρv D ⎟ ⎠m ⎠o ⎝ ⎝ ⎛ R ρ v 2 D 2 ⎞ 50 x 0.0184 x 602.4 2 R o = ⎜⎜ m o2 o 2 o ⎟⎟ = x 61.87 2 2 ρ v D 1.225 x 680.6 ⎝ m m m ⎠ Ro = 2259 N 4. The force on a body of length 3 m placed in an air stream at 1 bar and moving at 60 m/s is to be found by testing a scale model. The model is 0.3 m long and placed in high pressure air moving at 30 m/s. Assuming the same temperature and viscosity, determine the air pressure which produced dynamic similarity. The force on the model is found to be 500 N. Predict the force on the actual body. ⎛ F ⎞ The relevant equation is ⎜⎜ 2 2 ⎟⎟ = f(R e ) ⎝ ρv D ⎠ For dynamic similarity (Re)m = (Re)o (ρvD)m = (ρvD)o For a gas p = ρRT ⎛ ρvD ⎞ ⎛ ρvD ⎞ ⎜⎜ ⎟⎟ = ⎜⎜ ⎟⎟ µ µ ⎝ ⎠m ⎝ ⎠o µo = µm The temperature T is constant so ρ ∝ p = c p pm x 30 x 0.3 = po x 60 x 3 = 1bar x 60 x 3 ⎛ F ⎞ ⎛ F ⎞ ⎜ 2 2 ⎟ =⎜ 2 2 ⎟ ⎟ ⎜ ⎜ ρv D ⎟ ⎠ m ⎝ ρv D ⎠ o ⎝ pm = 20 bar ⎞ ⎞ ⎛ ⎛ Fo 500 ⎟ ⎟=⎜ ⎜ 2 2 2 2 ⎜ c p x 30 x 0.3 ⎟ ⎜ c p x 60 x 3 ⎟ o m ⎠ ⎝ ⎝ ⎠ Fo = 10 000N 5. Show by dimensional analysis that the velocity profile near the wall of a pipe containing + + turbulent flow is of the form u = f(y ) + 1/2 + 1/2 and y = y(ρτo) /µ u = u(ρ/τo) When water flows through a smooth walled pipe 60 mm bore diameter at 0.8 m/s, the velocity + + profile is u = 2.5ln(y ) + 5.5 Find the velocity 10 mm from the wall. -0.25 The friction coefficient is Cf = 0.079 Re . This is best solved by Buckingham’s Pi method. τo = φ(y u ρ µ) ( ) ( ) Form a dimensionless group with µ and leave out u that means the group y x1 u y1 ρ z1 µ1 has no dimensions ( M 0 L0 T 0 = (L )x1 LT -1 0 = 1 – 2 z1 0 = y1 + z1 + 1 0 = x1 - 3 y1 - z1 - 1 Time Mass Length The group is µ y-1 ρ - ½ τo - ½ or ) (ML ) (ML T ) y1 -3 z1 -1 1 -1 z1- ½ y1 = - ½ x1 = -1 µ y(ρτ o )1/2 Form a dimensionless group with u and leave out µ that means the group y x 2 u y2 ρ z2 u1 has no dimensions ( M 0 L0 T 0 = (L ) 2 LT -1 0 = 2 z2- 1 0 = y2 + z2 0 = x2 - 3 y2 - z2 + 1 x Time Mass Length ) (ML ) (LT ) y2 -3 z2 -1 1 z2- ½ y2 = ½ x2 = 0 1/2 0 The group is u y ρ ½ τo -½ ⎛ρ ⎞ or u⎜⎜ ⎟⎟ ⎝ τo ⎠ 1/2 Hence ⎛ ⎛ρ ⎞ µ u⎜⎜ ⎟⎟ = f ⎜ ⎜ y(ρτ )1/2 ⎝ τo ⎠ o ⎝ ⎞ ⎟ or ⎟ ⎠ + + u = f(y ) Cf = 2τo/(ρu2)= Wall Shear Stress/Dynamic Pressure Cf = 0.079 Re-0.25 0.079 (ρ u D/µ) -0.25 = 5.1879 x 10-3 -3 5.1879 x 10 = 2τo/(997 x 0.82) τo = 1.655 Pa + 1/2 1/ + u(ρ/τo) = 2.5 ln {(y/µ)(ρτo) } + 5.5 u = 2.5 ln y + 5.5 1/2 1/2 u(997/1.6551) = 2.5 ln {(0.01/0.89x10-3)(1.655 x 997) } + 5.5 24.543 u = 2.5 ln 451.42 + 5.5 u = 0.85 m/s FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 7 – FLUID FORCES SELF ASSESSMENT EXERCISE No. 1 1. A pipe bends through an angle of 90o in the vertical plane. At the inlet it has a cross sectional area of 0.003 m2 and a gauge pressure of 500 kPa. At exit it has an area of 0.001 m2 and a gauge pressure of 200 kPa. Calculate the vertical and horizontal forces due to the pressure only. Fh = 500 000 x 0.003 = 1500 N→ Fv = 200 000 x 0.001 = 200 N↓ 2. A pipe bends through an angle of 45o in the vertical plane. At the inlet it has a cross sectional area of 0.002 m2 and a gauge pressure of 800 kPa. At exit it has an area of 0.0008 m2 and a gauge pressure of 300 kPa. Calculate the vertical and horizontal forces due to the pressure only. Fp1 = 800 000 x 0.002 = 1600 N Fpx1 = 1600 N Fpy1 = 0 Fp2 = 300 000 x 0.0008 = 240 N Fpy2 = 240 sin 45o = 169.7 N Fpx2 = 240 cos 45o= 169.7 N Totals Fh =1600 – 169.7 = 1430 N Fv = 0 - 169.7 = -169.7 N 3. Calculate the momentum force acting on a bend of 130o that carries 2 kg/s of water at 16m/s velocity. Determine the vertical and horizontal components. ∆v = 16 sin 130/sin 25 = 29 m/s F = m ∆v = 2 x 29 = 58 N Fv = 58 sin 25 = 24.5 N Fh = 58 cos 25 = 52.57 N 4. Calculate the momentum force on a 180o bend that carries 5 kg/s of water. The pipe is 50 mm bore diameter throughout. The density is 1000 kg/m3. v1 = Q/A = m/ρA = 5/(1000 x π x 0.0252) =2.546 m/s v2 = -2.546 m/s ∆v = 2.546 – (-2.546) = 5.093 m/s F = m ∆v = 5 x 5.093 = 25.25 N 5. A horizontal pipe bend reduces from 300 mm bore diameter at inlet to 150 mm diameter at outlet. The bend is swept through 50o from its initial direction. The flow rate is 0.05 m3/s and the density is 1000 kg/m3. Calculate the momentum force on the bend and resolve it into two perpendicular directions relative to the initial direction. ∆v = (2.82sin50) 2 + (2.825 sin50 − 0.707) 2 = 2.4359m/s 2.825sin50 ⎞ = 62.8o φ = tan - 1 ⎛⎜ ⎟ ⎝ 2.82cos50 − .707 ⎠ F = m ∆v = 50 x 2.43 = 121.5 N Fv = 121.5 sin 6.84 = 108.1 N Fh = 121.5 cos 62.84 = 55.46 N SELF ASSESSMENT EXERCISE No. 2 Assume the density of water is 1000 kg/m3 throughout. 1. A pipe bends through 90o from its initial direction as shown in fig.13. The pipe reduces in diameter such that the velocity at point (2) is 1.5 times the velocity at point (1). The pipe is 200 mm diameter at point (1) and the static pressure is 100 kPa. The volume flow rate is 0.2 m3/s. Assume there is no friction. Calculate the following. a) The static pressure at (2). b) The velocity at (2). c) The horizontal and vertical forces on the bend FH and FV. d) The total resultant force on the bend. u2 = 1.5 u1 D1 = 200 mm p1 = 100 kPa Q = 0.02 m3/s m = 200 kg/s A1 = πD12/4 = 0.0314 m2 u1 = Q/A1 = 6.37 m/s u2 = 1.5 u1 = 9.55 m/s Bernoulli p1 +ρu12/2 = p2 +ρu22/2 Gauge pressures assumed. 100 000 + 1000 x 6.372/2 = p2 + 1000 x 9.552/2 p2 = 74.59 kPa A2 = Q/u2 = 0.0209 m2 Fp1 = p1 A1 = 3140 N → Fp2 = p2 A2 = 1560 N ↓ Fm1 = m∆v (hor) = 200 (0 – 6.37) = -1274 N on water and 1274 N on bend → Fm2 = m∆v (vert) = 200 (9.55 - 0) = 1910 N on water and -1910 N on bend ↓ Total horizontal force on bend = 3140 + 1274 = 4414→ Total vertical force on bend = 1560↓ + 1910 = 3470 N ↓ F = √(44142 +34702) = 561 N φ = tan-1(3470/4414) = 38.1o 2. A nozzle produces a jet of water. The gauge pressure behind the nozzle is 2 MPa. The exit diameter is 100 mm. The coefficient of velocity is 0.97 and there is no contraction of the jet. The approach velocity is negligible. The jet of water is deflected 165o from its initial direction by a stationary vane. Calculate the resultant force on the nozzle and on the vane due to momentum changes only. cv = 0.97 ∆p = 2 MPa ρ = 1000 kg/m3 v1 = cv √(2∆p/ρ) = 0.97√(2 x 2 x 106/1000) = 61.35 m/s m = ρA1 v1 = 1000 x π x 0.12/4 x 61.35 = 481.8 kg/s Force on Nozzle = m ∆v = 481.8 x (61.35 – 0) = 29.56 kN Force on vane = m ∆v ∆v = 61.35√{2(1 – cos165o)} = 121.6 m/s Force on vane = m ∆v = 481.8 x 121.6 = 58.6 kN 3. A stationary vane deflects 5 kg/s of water 50o from its initial direction. The jet velocity is 13 m/s. Draw the vector diagram to scale showing the velocity change. Deduce by either scaling or calculation the change in velocity and go on to calculate the force on the vane in the original direction of the jet. v1 = 13 m/s m = 5 kg/s ∆v = 13 sin 50o/sin 65o = 10.99 m/s Force on vane = m ∆v = 5 x 10.99= 54.9 N Horizontal component is Fcos 65o = 23.2 N 4. A jet of water travelling with a velocity of 25 m/s and flow rate 0.4 kg/s is deflected 150o from its initial direction by a stationary vane. Calculate the force on the vane acting parallel to and perpendicular to the initial direction. v1 = 25 m/s m = 0.4 kg/s ∆v = 25 √{2(1 – cos150o)} = 48.3 m/s F = m ∆v = 0.4 x 48.3 = 19.32 N Fv = 19.32 sin15o = 5 N Fh = 19.32 cos15o = 18.66 N 5. A jet of water discharges from a nozzle 30 mm diameter with a flow rate of 15 dm3/s into the atmosphere. The inlet to the nozzle is 100 mm diameter. There is no friction nor contraction of the jet. Calculate the following. i. the jet velocity. ii. the gauge pressure at inlet. iii. the force on the nozzle. The jet strikes a flat stationary plate normal to it. Determine the force on the plate. Q = 0.015 m3/s ρ = 1000 kg/m3 m = 15 kg/s A1= π x 0.12/4 = 0.00785 m2 v1 = Q/A1= 0.015 ÷ 0.00785 = 1.901 m/s A2 = π x 0.032/4 = 0.0007068 m2 v2 = Q/A2= 0.015 ÷ 0.0007068 = 21.22 m/s Bernoulli p1 + ρv12/2 = p2 + ρv22/2 Gauge pressures assumed. p1 + 1000 x 1.9012/2 = 0 + 1000 x 21.222/2 p1 = 223.2 kPa Force on nozzle = (p1A1- p2A2) + m(v2 – v1) v1 is approximately zero. = (223.2 x 103 x 0.00785 - 0) + 15(21.22 – 0) = 2039 N← Force on Plate = m ∆v ∆v in horizontal direction is 21.22 Force on Plate = 15 x 21.22 = 311.8 N → Some common sense is needed determining the directions. SELF ASSESSMENT EXERCISE No.3 1. A vane moving at 30 m/s has a deflection angle of 90o. The water jet moves at 50 m/s with a flow of 2.5 kg/s. Calculate the diagram power assuming that all the mass strikes the vane. ρ = 100 kg/m3 m = 2.5 kg/s u = 30 m/s v = 50 m/s Diagram Power = mu (v – u) = 2.5x 30 (50 – 30) = 1500 Watts 2. Figure 10 shows a jet of water 40 mm diameter flowing at 45 m/s onto a curved fixed vane. The deflection angle is 150o. There is no friction. Determine the magnitude and direction of the resultant force on the vane. The vane is allowed to move away from the nozzle in the same direction as the jet at a velocity of 18 m/s. Draw the vector diagram for the velocity at exit from the vane and determine the magnitude and direction of the velocity at exit from the vane. STATIONARY VANE ∆v = 45√{2(1-cos150o)} = 86.93 m/s m = ρAv = 1000 x π x 0.042/4 x 45 = 56.54 kg/s F = m ∆v = 4916 N MOVING VANE The relative velocity at exit is ω2 = 27 m/s The absolute velocity v2 = √(13.52 + 5.382) = 14.53 m/s FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 8A –TURBINES SELF ASSESSMENT EXERCISE No. 1 1. The buckets of a Pelton wheel revolve on a mean diameter of 1.5 m at 1500 rev/min. The jet velocity is 1.8 times the bucket velocity. Calculate the water flow rate required to produce a power output of 2MW. The mechanical efficiency is 80% and the blade friction coefficient is 0.97. The deflection angle is 165o. D = 1.5 m N = 1500 rev/min v = 1.8 u η = 80% k = 0.97 θ = 165o Diagram Power = 2MW/0.8 = 2.5 MW u = πND/60 = 117.8 m/s v = 1.8 x 117.8 = 212 m/s DP = m u (v-u) (1- kcosθ) = 2.5 x 106 m x 117.8 (94.24)(1- 0.97cos 165) = 2.5 x 106 m = 2.5 x 106/21503 = 116.26 kg/s 2. Calculate the diagram power for a Pelton Wheel 2m mean diameter revolving at 3000 rev/min with a deflection angle of 170o under the action of two nozzles , each supplying 10 kg/s of water with a velocity twice the bucket velocity. The blade friction coefficient is 0.98. If the coefficient of velocity is 0.97, calculate the pressure behind the nozzles. (Ans 209.8 MPa) D = 2m N = 3000 rev/min θ = 170o v = 2u k = 0.98 cv = 0.97 m = 2 x 10 = 20 kg/s u = πND/60 = 314.16 m/s DP = m u (v-u) (1- kcosθ) = 20 x 314.16 x 314.16(1- 0.98 cos170o) = 3.879 MW v = cv √2∆p/ρ ∆p = (314.16 x 2/0.97)2 x 1000/2 = 209.8 MPa 3. A Pelton Wheel is 1.7 m mean diameter and runs at maximum power. It is supplied from two nozzles. The gauge pressure head behind each nozzle is 180 metres of water. Other data for the wheel is : Coefficient of Discharge Cd= 0.99 Coefficient of velocity Cv= 0.995 Deflection angle = 165o. Blade friction coefficient = 0.98 Mechanical efficiency = 87% Nozzle diameters = 30 mm Calculate the following. i. The jet velocity (59.13 m/s) ii. The mass flow rate (41.586 kg/s) iii The water power ( 73.432 kW) iv. The diagram power ( 70.759 kW) v. The diagram efficiency (96.36%) vi. The overall efficiency (83.8%) vii. The wheel speed in rev/min (332 rev/min) D = 1.7 m ∆H = 180 m cd = 0.99 cv = 0.995 cc = cd/cv = 0.995 ρ = 1000 kg/m3 Power is maximum so v = 2u 2 nozzles v = cv √2g ∆H = 0.995 √(2g x 180) = 59.13 m/s m = cc ρ A v = 0.995 x 1000 (π x 0.032/4) x 59.13 = 41.587 kg/s per nozzle Water Power = mg ∆H = 41.587 x 9.81 x 180 = 73.43 kW per nozzle. u = v/2 = 29.565 m/s Diagram Power = m u (v-u) (1- kcosθ) DP = 41.587 x 29.565 (29.565)(1 – 0.98cos165) = 70.76 kW per nozzle ηd = 70.76/73.43 = 83.8% Mechanical Power = 70.76 x 87% = 61.56 kW per nozzle. ηoa = 61.56/73.43 = 83.8% N = 60u/πD = 29.565 x 60/(π x 1.7) = 332.1 rev/min NP1/2 ρ1/2 (gH) 5/4 Calculate the specific speed of a Pelton Wheel given the following. d = nozzle diameter. D = Wheel diameter. u = optimum blade speed = 0.46 v1 v1= jet speed. η = 88% Cv = coefficient of velocity = 0.98 4. Explain the significance and use of 'specific speed Ns = v j = c v 2gH = 0.98 2gH = 4.34H1/2 u =0.46 vj = πND/60 1/2 0.46 x 4.34H x 60 H1/2 = = 38.128 N= πD πD D Q = Aj vj = (πd2/4) x 4.34 H½ = 3.41 H½ d2 0.46 v j x 60 P = η m g H = η x ρg Q H = 0.88 x 1000 x 9.81 x 3.41 x H½ d2 = 29438 H½ d2 H ( H1/2 28438 H 3/2 d 2 Ns = 38.128 x D ρ1/2 (gH) 5/4 ) 1/2 = 38.128 x 284381/2 H1/2 H 3/4 d d x = 11.9 1/ 2 5/ 4 5/4 D 1000 9.81 DH 5. A turbine is to run at 150 rev/min under a head difference of 22 m and an expected flow rate of 3 85 m /s. 3 A scale model is made and tested with a flow rate of 0.1 m /s and a head difference of 5 m. Determine the scale and speed of the model in order to obtain valid results. When tested at the speed calculated, the power was 4.5 kW. Predict the power and efficiency of the full size turbine. N1 = 150 rev/min Q1 = 85 m3/s Q2 = 0.1 m3/s ∆H1 = 22 m ∆H2 = 55 m For similarity of Head Coefficient we have D 22 5x150 2 5114 ∆H1 ∆H 2 = = 2 = D12 22N 22 N2 N12 D12 N 22 D 22 For similarity of Flow Coefficient we have D 32 0.1x150 0.176 Q1 Q2 = = = 85N 2 N2 D13 N1D13 N 2 D 32 Equate D 2 71.51 0.560 = = 1/2 D1 N2 N2 D2 = 0.0496 D1 D2 5114 71.51 = = D1 N2 N 22 D2 71.51 0.560 =3 = 1/2 D1 N2 N2 N 2/3 2 = 71.51 0.56 N2 = 1443 rev/min Note if we use N1Q1/2 1 H13/4 Power Coefficient = N 2 Q1/2 2 H 3/4 2 we get the same result. 5 P1 P2 = 3 5 ρN1 D1 ρN 32 D 52 P1 = ( P2 ρN13 D15 ρN 32 D 52 ) ⎛ 1 ⎞ 4.5 x 150 x⎜ ⎟ 3 5 P2 N1 D1 0.05 ⎠ ⎝ = = = 16.2MW N 32 D 52 14433 3 Water Power = mg∆H = (85 x 1000) x 9.81 x 22 = 18.3 MW Η = 16.2/18.3 = 88% SELF ASSESSMENT EXERCISE No.2 1. The following data is for a Francis Wheel Radial velocity is constant No whirl at exit. Flow rate=0.4 m3/s D1=0.4 m D2=0.15 m k =0.95 α1=90o Head at inlet = 56 m head at entry to rotor = 26 m Entry is shock less. Calculate i. the inlet velocity v1 (24.26 m/s) ii. the guide vane angle (30.3o) iii. the vane height at inlet and outlet (27.3 mm, 72.9 mm) iv. the diagram power (175.4 MW) v. the hydraulic efficiency (80%) N=1000 rev/min head at exit = 0 m v1 = (2gh)½ = {2 x 9.81 x (56 – 26)} ½ = 24.26 m/s u1 = πND/60 = π x 1000 x 0.4/60 = 20.94 m/s β1 = cos-1 (20.94/24.26) = 30.3o ω1 = vr1 = 12.25 m/s Q = 0.4 = π D t k vr t1 = 0.4/(π x 0.4 x 0.95 x 12.25) =0.0273 m t2 = 0.4/(π x 0.15 x 0.95 x 12.25) =0.0729 m vw2 = 0 vw1 = 20.94 P = mu1 vw1 = 400 x 20.94 x 20.94 = 174.4 kW Water Power = m g H = 400 x 9.81 x 56 = 219.7 kW η = 174.4/219.7 = 80% 2. A radial flow turbine has a rotor 400 mm diameter and runs at 600 rev/min. The vanes are 30 mm high at the outer edge. The vanes are inclined at 42o to the tangent to the inner edge. The flow rate is 0.5 m3/s and leaves the rotor radially. Determine i. the inlet velocity as it leaves the guide vanes. (19.81 m/s) ii. the inlet vane angle. (80.8o) iii. the power developed. (92.5 kW) Radial Flow Turbine Inlet is the outer edge. = πND/60 = π x 600 x 0.4/60 = 12.57 m/s vr1 = Q/πDt = 0.5/(π x 0.4 x 0.03) = 13.26 m/s 13.26/vw1 = tan 42o vw1 = 14.72 m/s v1 = (13.262 + 14.722) ½ = 19.81 m/s u1 13.26/(14.72 – 12.57) = tan α1 α1= 80.8o vw2 = 0 DP = mu = vw1 DP = 500 x 12.57 x 14.72 = 92.5 kW 3. The runner (rotor) of a Francis turbine has a blade configuration as shown. The outer diameter is 0.45 m and the inner diameter is 0.3 m. The vanes are 62.5 mm high at inlet and 100 mm at outlet. The supply head is 18 m and the losses in the guide vanes and runner are equivalent to 0.36 m. The water exhausts from the middle at atmospheric pressure. Entry is shock less and there is no whirl at exit. Neglecting the blade thickness, determine : i. ii. iii. iv. v. The speed of rotation. The flow rate. The output power given a mechanical efficiency of 90%. The overall efficiency. The outlet vane angle. INLET Useful head is 18 – 0.36 = 17.64 m m u1 vw1 = m u2 vw2 u1 vw1 = u2 vw2 (u1 vw1/g) = ∆H = 17.64 sine rule (v1/sin 60) = (u1 /sin 100) v1 = 0.879 u1 (vr1/ v1) = sin 20 v1 = 2.923 vr1 Equate 0.879 u1=2.923 vr1 vr1 = 0.3 u1 vw1 = vr1/tan 20 = 0.824 u1 17.64 = u1 x 0.824 u1 /g u12 = 210 u1 = 14.5 m/s vr1 = 0.3 u1 = 4.35 m/s EXIT u = π N D N = u1 / π D1= u2 / π D2 u2 = u1 D1 / D2 = 14.4 x 300/450 = 9.67 m/s N = u1/ π D1 = 14.5 x 60/(π x 0.45) = 615 rev/min vr = Q/πDh vr1 =4.35 = Q/πD1h1= Q/(π x 0.45 x 0.0625) Q = 0.384 m3/s vr2 = Q/πD2h2 = Q/(π x 0.3 x 0.1) = 10.61 Q = 4.08 m/s 4.08/9.67 = tan β2 β2 = 22.8o P = m g ∆H = 384 x 9.81 x 17.64 = 66.45 kW Output Power = 66.45 x 90% = 59.8 kW Overall efficiency = 59800/(m g ∆H) = 58805/(384 x 9.81 x 18) = 88.2 % FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 8B – CENTRIFUGAL PUMPS SELF ASSESSMENT EXERCISE No. 1 1. A centrifugal pump must produce a head of 15 m with a flow rate of 40 dm3/s and shaft speed of 725 rev/min. The pump must be geometrically similar to either pump A or pump B whose characteristics are shown in the table below. Which of the two designs will give the highest efficiency and what impeller diameter should be used ? Pump A D = 0.25 m N = 1 000 rev/min Q (dm3/s) H (m) η% Ns = 8 48 11 8.1 55 15 7.9 62 Pump B D = 0.55 m N = 900 rev/min Q (dm3/s) H (m) η% 8 42 65 6 55 9 36 66 19 7.3 56 6.1 11 33 58 27 NQ1/2 725 x 0.041/2 = = 19 H 3/4 153/4 PUMP A Q (m3/s) H (m) η% Ns 0.008 8.1 48 18.6 0.011 7.9 55 22.26 Q (m3/s) H (m) η% Ns 0.06 42 55 13.36 0.008 36 65 17.32 0.009 33 66 19.6 Pump B gives the greater efficiency when Ns = 19 Drawing a graph or interpolating we find Q = 0.085 m3/s H = 34.5 m , η = 65.5% when Ns=19 Q1 Q2 0.04 0.085 = = 3 3 3 N1D1 N 2 D 2 725D1 900 x 0.553 or using the head coefficient ∆H1 ∆H 2 0.15 34.5 = = 2 2 2 2 2 2 N1 D1 N 2 D 2 725 D1 19002 2 x0.55 2 D = 0.46 m D = 0.45 m Take the mean D = 0.455 m for the new pump To commence pumping 2gH = πND/60 D= 60 2g x 15 π x 725 Hence D = 0.452 m This seems to give the right answer more simply. 2. Define the Head and flow Coefficients for a pump. Oil is pumped through a pipe 750 m long and 0.15 bore diameter. The outlet is 4 m below the oil level in the supply tank. The pump has an impeller diameter of 508 mm which runs at 600 rev/min. Calculate the flow rate of oil and the power consumed by the pump. It may be assumed -0.25 3 -3 Cf=0.079(Re) . The density of the oil is 950 kg/m and the dynamic viscosity is 5 x 10 N s/m2. The data for a geometrically similar pump is shown below. D = 0.552 m N = 900 rev/min Q (m3/min) H (m) η% 0 34.1 0 1.14 37.2 22 2.27 39.9 41 3.41 40.5 56 4.55 38.1 67 5.68 32.9 72 ∆H = Head Added to the system by the pump. This may be put into Bernoulli’s equation. hA + zA + uA2/2g + ∆H = hB + zB + uB2/2g + hL 0 + 4 + 0 + ∆H = 0 + 0 + uB2/2g + hL 4 + ∆H = uB2/2g + hL uB = Q/A = Q/(π x 0.0752) = 56.588 Q hL = 4 Cf Lu2/2gD Re = ρuD/µ = 950 x (56.588 Q)0.15/0.005 = 1612758Q Cf = 0.079 Re-0.25 = 0.079 (1612758Q)-0.25 = 0.002217 Q-0.25 hL = 4 (0.002217 Q-0.25) x 750 (56.588 Q)2/(2g x 0.15) = 7236 Q1.75 4 + ∆H = (56.588 Q)2/2g + 7236 Q1.75 ∆H = 163.2 Q2 + 7236 Q1.75 - 4 If Q is given in m3/min this becomes ∆H = 0.0453 Q2 + 5.594 Q1.75 - 4 ow create a table for the system. Q (m3/min) ∆H (m) 0 -4 Q1 (m3/min) ∆H1 (m) η% Q2 (m3/min) ∆H1 (m) 0 1.14 34.1 37.2 0 22 0 0.59 12.83 14 1 1.64 4 5 60 90.6 3 N D Q N 2 D 2 ∆H Now compile a table for a similar pump using Q 2 = 2 2 3 1 ∆H 2 = 2 22 2 1 N1D1 N1 D1 N2 = 600 D2 = 508 N1 = 900 D1 = 552 produces Q 2 = 0.52Q 1 ∆H 2 = 0.38∆. 1 Plotting ∆H for the system against Q and η produces a matching point ∆H = 15 m Q = 2 (m3/min) and η = 59% P = mg∆H/η = 2 x (950/60) x 9.81 x 15/0.59 = 7.89 kW 2 15 3 34.7 2.27 39.9 41 1.18 15 3.41 40.5 56 1.8 15.2 4.55 38.1 67 2.4 14.3 5.68 32.9 72 2.9 12.4 6.86 25.9 65 3,6 9.8 6.86 25.9 65 SELF ASSESSMENT EXERCISE No. 2 1. The rotor of a centrifugal pump is 100 mm diameter and runs at 1 450 rev/min. It is 10 mm deep at the outer edge and swept back at 30o. The inlet flow is radial. the vanes take up 10% of the outlet area. 25% of the outlet velocity head is lost in the volute chamber. Estimate the shut off head and developed head when 8 dm3/s is pumped. (5.87 m and 1.89 m) vR2 = Q/A2 = 0.008/(π x 0.1 x 0.01 x 0.9) = 2.829 m/s vw1 = 0 OUTLET u2 = πND2 = π x (450/60) x 0.1 = 7.592 m/s vw2 = 7.592 – 2.829/tan 30o = 2.692 m/s v2 = (2.6922 + 2.8292)½ = 3.905 m/s Kinetic Head = v22/2g = 3.9052/2g = 0.777 m Loss in chamber = 25% x 0.777 = 0.194 m Manometric Head = u2 vw2/g = 7.592 x 2.692/9.81 = 2.08 m Developed Head = 2.08 – 0.194 = 1.89 m ∆h = u2 vw2/g = (u2 – Q/A2tanα2) When there is no flow Q = 0 so ∆h = u2 vw2/g - u2 = (7.592/9.81) x 7.592 = 5.875 m 2. The rotor of a centrifugal pump is 170 mm diameter and runs at 1 450 rev/min. It is 15 mm deep at the outer edge and swept back at 30o. The inlet flow is radial. the vanes take up 10% of the outlet area. 65% of the outlet velocity head is lost in the volute chamber. The pump delivers 15 dm3/s of water. Calculate i. The head produced. (9.23 m) ii. The efficiency. (75.4%) iii. The power consumed. (1.8 kW) u2 = πND2 = π x (1450/60) x 0.17 = 12.906 m/s vR2 = Q/A2 vR2 = 0.015/(π x 0.17 x 0.015 x 0.9) = 2.08 m/s vw1 = 0 OUTLET vw2 = 12.906 – 2.08/tan 30o = 9.3 m/s v2 = (9.32 + 2.082)½ = 9.53 m/s Kinetic Head = v22/2g = 9.532/2g = 4.628 m Head Recovered = 35% x 4.628 = 1.62 m Head Loss = 3 m Manometric Head = u2 vw2/g = 12.906 x 9.3/9.81 = 12.23 m Developed Head 12.23 – 3= 9.23 m η man = 9.23/12.23 = 75.3% DP = m u2 vw2 = 15 x 12.906 x 9.3 = 1.8 kW WP = m g ∆h = 15 x 9.81 x 9.23 = 1.358 kW η = 1.358/1.8 = 75.4% FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 8C – PUMPED PIPED SYSTEMS SELF ASSESSMENT EXERCISE No. 1 1. A pump has a suction pipe and a delivery pipe. The head required to pass water through them varies with flow rate as shown. The pump must deliver 3 m3/s at 2 000 rev/min. Determine the specific speed. The vapour pressure is 0.025 bar and atmospheric pressure is 1.025 bar. Calculate the NPSH and the cavitation parameter. From the graph at 3 m/s hd = 13 m hs = -6 m 5 NPSH = {1.025 x 10 /(9.81 x 1000) – 6 } – 0.025 x 105/(9.81 x 1000) NPSH = 4.448 – 0.2548 = 4.19 m σ = NPSH/hd = 4.19/13 = 0.323 Specific speed Ns = NQ½ / H3/4 = 2000 x 3½/193/4 = 380.6 2. Define the term "Net Positive Suction Head" and explain its significance in pump operation. 1.2 kg/s of acetone is to be pumped from a tank at 1 bar pressure. The acetone is at 40oC and the pump is 1.5 m below the surface. The suction pipe is 25 mm bore diameter. Calculate the NPSH at the pump inlet. Losses in the suction pipe are equal to three velocity heads. The vapour pressure of acetone is 55 kPa. The density is 780 kg/m3. The Net Positive Suction Head is the amount by which the absolute pressure on the suction side is larger than the vapour pressure (saturation pressure) of the liquid. u = m/ρA = 1.2/(780 x π x 0.0252) = 3.134 m/s hA + zA + uA2/2g = hB + zB + uB2/2g + hL hL = 3 uB2/2g 0 + 1.5 + 0 = hB + 0 + 3.1342/2g + 3 uB2/2g hB = 1.5 - 4 uB2/2g = -0.5 m gauge Atmospheric pressure = 1.0 bar ρ = 780 kg/m3 Convert to pressure head h = p/ρg = 13.06 m Absolute head at pump = 13.06 – 0.5 = 12.56 m Vapour pressure head = 55 x 103 /ρg = 7.19 m NPSH = 12.56 – 7.19 = 5.37 m 3. A centrifugal pump delivers fluid from one vessel to another distant vessel. The flow is controlled with a valve. Sketch and justify appropriate positions for the pump and valve when the fluid is a) a liquid and b) a gas. (a) Minimum suction is required to avoid cavitation so put the valve on the pump outlet and this will also keep the pump primed when closed. The pump should be as close to the tank as possible. (b) With gas cavitation is not a problem but for minimal friction the velocity must be kept low. If the gas is kept under pressure by putting the valve at the end of the pipe, it will be more dense and so the velocity will be lower for any given mass flow rate. The pump should be close to the supply tank. SELF ASSESSMENT EXERCISE No. 2 The density of water is 1000 kg/m3 and the bulk modulus is 4 GPa throughout. 1. A pipe 50 m long carries water at 1.5 m/s. Calculate the pressure rise produced when a) the valve is closed uniformly in 3 seconds. b) when it is shut suddenly. (a) (b) ∆p = ρLu/t = 1000x 50 x1.5/3 = 25 kPa ∆p = u(Kρ)0.5 = 1.5 x (4 x 109 x 1000) 0.5 = 3 MPa 2. A pipe 2000 m long carries water at 0.8 m/s. A valve is closed. Calculate the pressure rise when a) it is closed uniformly in 10 seconds. a) b) it is suddenly closed. (a) (b) ∆p = ρLu/t = 1000 x 2000 x 0.8/10 = 160 kPa ∆p = u(Kρ)0.5 = 0.8 x (4 x 109 x 1000) 0.5 = 1.6 MPa SELF ASSESSMENT EXERCISE No. 3 1. Derive the water hammer equation for a long elastic pipe carrying water from a large upstream reservoir with a constant water level to a lower downstream reservoir. Flow is controlled by a valve at the downstream end. Sketch the variation in pressure with time for both ends and the middle of the pipe following sudden closure of the valve. Sketch these variations for when friction is negligible and for when both friction and cavitation occur. Assuming the effective bulk modulus is given by K' = {(D/tE) + 1/K}-1 and that the maximum stress in the pipe is σ, derive a formula for the maximum allowable discharge. Part (a) is given in the tutorial. Part (b) below – no friction on left. When cavitation occurs the minimum pressure is the vapour pressure so the bottom part of the cycle will be at this pressure. Part (c) 2tσ pD For a thin walled cylinder σ = u = Q/A = 4Q/πD2 p= 2t D 1⎫ ⎧ D + ⎬ ⎨ ρ 2tσ 4Q ρ σ π D t ⎩ 2tE K ⎭ ∆p = u = Q= 2 1⎫ 1⎫ D πD ⎧ D 2 ρ ⎧ D + ⎬ + ⎬ ⎨ ⎨ ⎩ 2tE K ⎭ ⎩ 2tE K ⎭ 2a. Explain the purpose and features of a surge tank used to protect hydroelectric installations. b. Derive an expression for the amplitude of oscillation of the water surface in a surge tank of cross sectional area AT connected to a pipe of cross sectional area Ap and length L following a complete stoppage of the flow. The normal mean velocity in the pipe is uo and friction may be ignored. The general solution to the standard second order differential equation d2z c2 2 2 + m z = c is z = Esin(mt) + Fcos(mt) + 2 dt 2 m Part (a) On hydroelectric schemes or large pumped systems, a surge tank is used. This is an elevated reservoir attached as close to the equipment needing protection as possible. When the valve is closed, the large quantity of water in the main system is diverted upwards into the surge tank. The pressure surge is converted into a raised level and hence potential energy. The level drops again as the surge passes and an oscillatory trend sets in with the water level rising and falling. A damping orifice in the pipe to the surge tank will help to dissipate the energy as friction and the oscillation dies away quickly. Part (b) Mean velocity in surge tank u T = dz Q = dt A T Q = AT dz dt Q dz A T Substitute for Q u p = ...........(1) Ap dt A p The diversion of the flow into the surge tank raises the level by z. This produces an increased pressure at the junction point of ∆p = ρgz The pressure force produced F= Ap∆p= Ap∆gz The inertia force required to decelerate the water in the pipe is F = mass x deceleration = - mass x acceleration = - ρApL du/dt Equating forces we have the following. du du L du A p ρgz = - ρA p L gz = - L z=..................(2) dt dt g dt Putting (1) into (2) we get gA p d2z L AT d 2z z==− z ...........................(3) 2 2 LA T g A p dt dt By definition this is simple harmonic motion since the displacement z is directly proportional to the acceleration and opposite in sense. It follows that the frequency of the resulting oscillation is 1 g Ap The periodic time will be T=1/f f= 2π L A T The amplitude and periodic time are referred to as the APO (amplitude and period of oscillation). Mean velocity in the pipe u p = Equation (3) maybe re-written as follows. gA p d2z = − z = −ω 2 z 2 LA T dt 1 d 2z +z=0 ω 2 dt 2 d 2z + ω2 z = 0 2 dt The standard solution to this equation is z = z o sin(ωt ) zo is the amplitude, that is, the amount by which the height in the tank will move up and down from the mean level. The following is a direct way of finding the amplitude. z The mean change in height = o 2 The weight of water entering the surge tank = ρgA T z o The potential energy stored in the tank = ρgA T z o zo z2 = ρgA T o 2 2 u2 u2 = ρLA p The kinetic energy lost = Mass x 2 2 z o2 u2 = ρgA T Equate the energies. ρLA p 2 2 The equation for the motion in full is z = uo zo = u o LA p LA p gA T sin(ω t ) gA T The peak of the surge occurs at T/4 seconds from the disturbance. 3.a. A hydroelectric turbine is supplied with 0.76 m3/s of water from a dam with the level 51 m above the inlet valve. The pipe is 0.5 m bore diameter and 650 m long. Calculate the pressure at inlet to the turbine given that the head loss in the pipe is 8.1 m. Calculate the maximum pressure on the inlet valve if it is closed suddenly. The speed of sound in the pipe is 1200 m/s. b. The pipe is protected by a surge tank positioned close to the inlet valve. Calculate the maximum change in level in the surge tank when the valve is closed suddenly (ignore friction). Calculate the periodic time of the resulting oscillation. A = π x 0.52/4 = 0.1963 m2 u = Q/A = 0.76/0.1963 = 3.87 m/s hA + zA + uA2/2g = hB + zB + uB2/2g + hL 0 + 189 + 0 = hB + 138 + 3.872/2g + 8.4 hB = 41.83 m pB = ρg hB = 0.41 x 106 Pa Sudden closure ∆p = ρ u a’ a’ = 1200 m/s ∆p = 998x 3.87 x1200 = 4.635 x 106 Pa The maximum pressure is 0.41 + 4.635 = 5.045 MPa This will occur at T/4 seconds Part (b) uo = Q/Ap dz/dt = Q/AT ∆p = ρ g z uo = (AT/Ap)dz/dt ..............(1) ∆F = Ap ρ g z This force decelerates the fluid and the mass decelerated is m = ρ Ap L Acceleration is –duo/dt Ap ρ g z = -ρ Ap L duo/dt g z = - L duo/dt z = - (L/g) duo/dt ...........(2) L AT d2z ..................(3) Put (1) into (2) z = − g A p dt 2 ∆F = m a 1/2 ⎧ Ap L ⎫ L AT Simple Harmonic Motion so ω = and the amplitude is u o ⎨ ⎬ g Ap ⎩AT g ⎭ AT = π 42/4 = 12.566 m2 ∆p = 4.635 x 106 Pa ∆h = ∆p/ρg = 473.4 m 2 1/2 ⎧ 0.1963 650 ⎫ x Amplitude = 3.87⎨ ⎬ = 3.987 m ⎩12.566 9.81⎭ 4. A pipe 2 m bore diameter and 420 m long supplies water from a dam to a turbine. The turbine is located 80 m below the dam level. The pipe friction coefficient f is 0.01 (f = 4Cf). Calculate the pressure at inlet to the turbine when 10 m3/s of water is supplied. Calculate the pressure that would result on the inlet valve if it was closed suddenly. The speed of sound in the pipe is 1432 m/s. Calculate the fastest time the valve could be closed normally if the pressure rise must not exceed 0.772 MPa). A = π 22/4 = 3.142 m2 u = Q/A = 10/3.142 = 3.18 m/s Sudden closure ∆p = ρ u a’ = 998 x 3.18 x 1432 = 4.55 MPa Gradual closure ∆p = ρLu/t = 998 x 420 x 3.18/t = 1.333/t MPa hB = 80 – hL Loss in pipe = 4Cf Lu2/2gD = f Lu2/2gD = 0.01 x 420 x 3.182/(2 x 9.81 x 2) = 1.082 m hB = 78.92 m p = ρg hB = 0.772 MPa To avoid cavitation ∆p is about 0.772 MPa T = 1.333/0.772 = 1.72 seconds 5. a) Sketch the main features of a high-head hydro-electric scheme. b) Deduce from Newton's laws the amplitude and period of oscillation (APO) in a cylindrical surge tank after a sudden stoppage of flow to the turbine. Assume there is no friction. c) State the approximate effect of friction on the oscillation. d) An orifice of one half the tunnel diameter is added in the surge pipe near to the junction with the tunnel. What effect does this have on the APO ? All the answers to this question are contained in the tutorial. FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL9 – COMPRESSIBLE FLOW SELF ASSESSMENT EXERCISE No. 1 1. Calculate the specific entropy change when a perfect gas undergoes a reversible isothermal expansion from 500 kPa to 100 kPa. R = 287 J/kg K. T is constant so ∆s = mRln(p1/p2) = 1 x 287 x ln(5/1) = 462 J/kg K 2. Calculate the total entropy change when 2 kg of perfect gas is compressed reversibly and isothermally from 9 dm3 to 1 dm3. R=300 J/kg K. ∆s = mRln(V2/V1) = 1 x 300 x ln(1/9) = 470 J/kg K 3. Calculate the change in entropy when 2.5 kg of perfect gas is heated from 20oC to 100oC at constant volume. Take cv= 780 J/kg K (Answer 470 J/K) ∆s = m cv ln(T2/T1) = 2.5 x 780 x ln(373/293) = -1318 J/kg K 4. Calculate the total entropy change when 5 kg of gas is expanded at constant pressure from 30oC to 200oC. R = 300 J/kg K cv= 800 J/kg K (Answer 2.45 kJ/K) ∆s = m cp ln(T2/T1) cp = R + cv = 1100 J/kg K ∆s = = 5 x 1100 x ln(473/303) = 2450 J/kg K 5. Derive the formula for the specific change in entropy during a polytropic process using a constant volume process from (A) to (2). s2-s1 = (sA-s1) - (sA-s2) s2-s1 = (sA-s1) + (s2-sA) For the constant temperature process (sA-s1) = R ln(p1/pA) For the constant volume process (s2-sA) = (cv/R) ln(T2/TA) Hence p T ∆s = R ln 1 + C p ln 2 + s2-s1 TA= T1 pA TA ⎛T ⎞ ⎛p ⎞ ∆s = s2 - s1 = ∆s = Rln⎜⎜ 1 ⎟⎟ + c v ln⎜⎜ 2 ⎟⎟ ⎝ TA ⎠ ⎝ pA ⎠ ⎛T ⎞ ⎛p ⎞ c Divide through by R ∆s/R = ln ⎜⎜ 1 ⎟⎟ + v ln⎜⎜ 2 ⎟⎟ ⎝ p A ⎠ R ⎝ TA ⎠ Then From the relationship between cp, cv, R and γ we have cp/R = γ /(γ-1) From the gas laws we have pA/TA = p2/T2 pA = p2 TA /T2= p2 T1 /T2 Hence 1+ ⎛ p ⎞⎛ T ⎞ ⎛T ⎞ ⎛p ⎞ ∆s 1 ln⎜⎜ 2 ⎟⎟ = ln⎜⎜ 1 ⎟⎟⎜⎜ 2 ⎟⎟ = ln⎜⎜ 1 ⎟⎟ + R ⎝ p 2 ⎠⎝ T1 ⎠ ⎝ p 2 ⎠ γ − 1 ⎝ T1 ⎠ 1 γ −1 γ γ ⎞ −1 ⎛ p ⎞⎛ T = ln⎜⎜ 1 ⎟⎟⎜⎜ 2 ⎟⎟ ⎝ p 2 ⎠⎝ T1 ⎠ 6. A perfect gas is expanded from 5 bar to 1 bar by the law pV o 200 C. Calculate the change in specific entropy. R = 287 J/kg K γ =1.4. 1−1/n ⎛p ⎞ T2 = T1 ⎜⎜ 2 ⎟⎟ ⎝ p1 ⎠ ( 5) = 473 1 1−1/1.6 γ 1.6 = C. The initial temperature is = 258.7 K 1.4 ⎛ p ⎞⎛ T ⎞ γ−1 ⎛ 258.7 ⎞ 0.4 ∆s = Rln⎜⎜ 1 ⎟⎟⎜⎜ 2 ⎟⎟ = 287 ln (5)⎜ ⎟ = −144 J/K ⎝ 473 ⎠ ⎝ p 2 ⎠⎝ T1 ⎠ γ 7. A perfect gas is expanded reversibly and adiabatically from 5 bar to 1 bar by the law pV = C. The initial temperature is 200oC. Calculate the change in specific entropy using the formula for a polytropic process. R = 287 J/kg K γ =1.4. ( 5) T2 = 473 1 1−1/1.4 = 298.6 K γ 1.4 ⎛ p ⎞⎛ T ⎞ γ−1 ⎛ 298.6 ⎞ 0.4 ∆s = Rln⎜⎜ 1 ⎟⎟⎜⎜ 2 ⎟⎟ = 287 ln (5)⎜ ⎟ =0 ⎝ 473 ⎠ ⎝ p 2 ⎠⎝ T1 ⎠ SELF ASSESSMENT EXERCISE No. 2 Take γ = 1.4 and R = 283 J/kg K in all the following questions. 1. An aeroplane flies at Mach 0.8 in air at 15o C and 100 kPa pressure. Calculate the stagnation pressure and temperature. (Answers 324.9 K and 152.4 kPa) ∆T k −1 1.4 = M2 = 0.8 2 = 0.128 ∆T = 0.128 x 288 = 36.86 K T 2 2 k p 2 ⎛ 2 k − 1 ⎞ k −1 = ⎜M + 1⎟ = 1.1283.5 = 1.5243 p1 ⎝ 2 ⎠ p2 = 100 x 1.5243 = 152.43 kPa 2. Repeat problem 1 if the aeroplane flies at Mach 2. ∆T k −1 1.4 = M2 = 22 = 0.8 T 2 2 T2 = 288 + 230.4 = 518.4 K ∆T = 0. 8 x 288 = 230.4 K k p 2 ⎛ 2 k − 1 ⎞ k −1 = ⎜M + 1⎟ = 1.83.5 = 7.824 p1 ⎝ 2 ⎠ p2 = 100 x 7.824 = 782.4 kPa 3. The pressure on the leading edges of an aircraft is 4.52 kPa more than the surrounding atmosphere. The aeroplane flies at an altitude of 5 000 metres. Calculate the speed of the aeroplane.( Answer 109.186 m/s) From fluids tables, find that a = 320.5 m/s p1 = 54.05 kPa γ = 1.4 k p 2 58.57 k − 1 ⎞ k −1 ⎛ = = 1.0836 = ⎜ M 2 + 1⎟ p1 54.05 2 ⎝ ⎠ 1.4 ( ) 3.5 1.4 − 1 ⎞ 1.4−1 ⎛ = 0.2 M 2 + 1 1.0836 = ⎜ M 2 + 1⎟ 2 ⎝ ⎠ 2 1.0232 = 0.2 M + 1 M = 0.3407 = v/a v = 109.2 m/s 4. An air compressor delivers air with a stagnation temperature 5 K temperature. Determine the velocity of the air. (Answer 100.2 m/s) ∆T v12 (k − 1) = T1 2γRT1 v12 (1.4 − 1) ∆T = =5K 2 x 1.4 x 287 v1 = 100.2 m/s above the ambient SELF ASSESSMENT EXERCISE No. 3 1. A Venturi Meter must pass 300g/s of air. The inlet pressure is 2 bar and the inlet temperature is 120oC. Ignoring the inlet velocity, determine the throat area. Take Cd as 0.97. Take γ =1.4 and R = 287 J/kg K (assume choked flow) m = Cd A 2 ⎡ 2γ ⎤ ⎧ ⎢ γ − 1⎥ p1ρ1 ⎨( ⎩ ⎣ ⎦ 2 rc γ ) 1 rc 1+ γ −( ) γ γ ⎞ −1 ⎛ 2 ⎟⎟ rc = ⎜⎜ ⎝ γ +1⎠ ⎫ ⎬ ⎭ ⎛ 2 ⎞ =⎜ ⎟ ⎝ 2.4 ⎠ 3.5 = 0.528 ρ1 = p1/RT1 = 2 x 105 /(287 x 393) = 1.773 kg/m3 { } 0.3 = 0.97A 2 7 x 2 x 105 x 1.773 (0.528)1.428 − (0.528)1.714 = 0.97A 2 166307 A2 = 758 x10-6 m2 and the diameter = 31.07 mm 2. Repeat problem 1 given that the inlet is 60 mm diameter and the inlet velocity must not be neglected. m = Cd A 2 1 ⎡ 2γ ⎤ ⎧ 2 1+ ⎫ ( ) ( ) − p ρ r r γ⎬ γ c ⎢ γ − 1⎥ 1 1 ⎨ c ⎩ ⎭ ⎣ ⎦ 2 ⎛A ⎞ ⎛p ⎞ 1 − ⎜⎜ 2 ⎟⎟ ⎜⎜ 2 ⎟⎟ ⎝ A1 ⎠ ⎝ p1 ⎠ 2 1 – (A2/A1) x 0.4 = 1738882A22 166307 0 .3 = C d A 2 2/γ 2 ⎛A ⎞ 1 − ⎜⎜ 2 ⎟⎟ (0.4 ) ⎝ A1 ⎠ 2 2 A1 = (π x 0.06 /4)2 = 7.99 x10-6 m2 1 - 50062 A22 = 1738882A22 A22 = 559 x 10-9 A2 = 747.6 x10-6 m2 The diameter is 30.8 mm. Neglecting the inlet velocity made very little difference. 3. A nozzle must pass 0.5 kg/s of steam with inlet conditions of 10 bar and 400oC. Calculate the throat diameter that causes choking at this condition. The density of the steam at inlet is 3.263 kg/m3. Take γ for steam as 1.3 and Cd as 0.98. m = Cd A 2 γ 1 ⎡ 2γ ⎤ ⎧ 2 1+ ⎫ ⎢ γ − 1⎥ p1ρ1 ⎨(rc ) γ − (rc ) γ ⎬ ⎭ ⎩ ⎣ ⎦ 0.5 = 0.98A 2 8.667 x 3.2626 x 10 x 105 ⎛ 2 ⎞ γ −1 ⎛ 2 ⎞ ⎟⎟ = ⎜ rc = ⎜⎜ ⎟ ⎝ 2.3 ⎠ ⎝ γ +1⎠ {(0.5457) 1.538 4.33 = 0.5457 } − (0.5457)1.538 = 0.98A 2 1.4526 x 106 A2 = 423 x 10-6 m2 and the diameter =23.2 mm 4. A Venturi Meter has a throat area of 500 mm2. Steam flows through it, and the inlet pressure is 7 bar and the throat pressure is 5 bar. The inlet temperature is 400oC. Calculate the flow rate. The density of the steam at inlet is 2.274 kg/m3. Take γ= 1.3. R = 462 J/kg K. Cd = 0.97. From the steam tables v1 = 0.4397 m3/ kg so ρ1 = 1/0.4397 = 2.274 kg/m3 1⎫ ⎡ 2γ ⎤ ⎧ 2 m = Cd A 2 ⎢ p1ρ1 ⎨(rc ) γ − (rc )1+ γ ⎬ ⎥ ⎭ ⎩ ⎣ γ − 1⎦ { ⎡ 2 x 1.3 ⎤ 1.538 1.764 7 x10 5 x 2.274 (5/7 ) m = 0.97 x 500 x 10 -6 ⎢ − (5/7 ) ⎥ ⎣ 1 .3 − 1 ⎦ -6 m = 0.379 kg/s m = 485x10 x 783 } 5. A pitot tube is pointed into an air stream which has an ambient pressure of 100 kPa and temperature of 20oC. The pressure rise measured is 23 kPa. Calculate the air velocity. Take γ = 1.4 and R = 287 J/kg K. γ p 2 123 γ − 1 ⎞ γ −1 ⎛ = = 1.23 = ⎜ M 2 + 1⎟ p1 100 2 ⎝ ⎠ 0.0609 = 0.2M2 1.0609 = 0.2M2 + 1 a = γRT1/2 = (1.4 x 287 x 293) 1/2 = 343.1 m/s v = 0.5519 x 343.1 = 189.4 m/s ( ) 1.23 = 0.2M 2 + 1 3.5 M 0.5519 6. A fast moving stream of gas has a temperature of 25oC. A thermometer is placed into it in front of a small barrier to record the stagnation temperature. The stagnation temperature is 28oC. Calculate the velocity of the gas. Take γ= 1.5 and R = 300 J/kg K. (Answer 73.5 m/s) ∆T = 3 K ∆T/T1 = v2/γRT cp = γR/(γ-1) ∆T = 3 = v2/2 cp = v2(γ - 1)/(2 γR) = v2(1.5 - 1)/(2 x 1.5 x 300) v2 = 5400 v = 73.48 m/s FLUID MECHANICS D203 SAE SOLUTIONS TUTORIAL 9 – COMPRESSIBLE FLOW SELF ASSESSMENT EXERCISE No. 4 1. Air discharges from a pipe into the atmosphere through an orifice. The stagnation pressure and temperature immediately upstream of the orifice is 10 bar and 287 K at all times. Determine the diameter of the orifice which regulates the flow rate to 0.03 kg/s. Determine the diameter of the orifice which regulates the flow rate to 0.0675 kg/s. Atmospheric pressure is 1 bar, the flow is isentropic and the air should be treated as a perfect gas. The following formulae are given to you. /( -1) To = T{1 + M2(γ-1)/2 } p1/p2 = (T1/T2) γ γ The relationship between areas for the flow of air through a convergent- divergent nozzle is given by A/A* = (1/M){(M2 + 5)/6}3 where A and A* are cross sectional areas at which the Mach Numbers are M and 1.0 respectively. Determine the ratio of exit to throat areas of the nozzle when the Mach number is 2.44 at exit. Confirm that an exit Mach number of 0.24 also gives the same area ratio. Pressure ratio is 10/1 so it is clearly choked. To Tt = M = 1 at throat 1 + 0.2M 2 287 Tt = = 239 K 1.2 3.5 p o ⎛ 287 ⎞ =⎜ pt = 10/1.893 = 5.28 bar ⎟ = 1.893 p t ⎝ 239 ⎠ ρt = p/RT = 5.28 x 105 /(287 x 239) = 7.7 kg/m3 a = (γ R T)½ = 310 m/s m = 0.03 = ρ A a = 77 A x 310 A = 12.57 x 10-6 m2 m = 0.0675 = ρ A a = 77 A x 310 A = 28.278 x 10-6 m2 3 Diameter = √(4A/π) = 4 mm Diameter = √(4A/π) = 6 mm 3 A 1 ⎛ M2 + 5 ⎞ 1 ⎛ 2.44 2 + 5 ⎞ ⎜ ⎟ = ⎜ ⎟ = 2.49 = ⎟ 2.44 ⎜⎝ 6 A * M ⎜⎝ 6 ⎟⎠ ⎠ With M = 0.24 3 A 1 ⎛ 2.4 2 + 5 ⎞ ⎜ ⎟ = 2.49 = A * 2.4 ⎜⎝ 6 ⎟⎠ Hence this is a correct solution and this is the theoretical result when m = 0.24 at inlet and 1.0 at the throat. 2. Air discharges from a vessel in which the stagnation temperature and pressure are 350 K and 1.3 bar into the atmosphere through a convergent-divergent nozzle. The throat area is 1 x 10-3 m2. The exit area is 1.2 x 10-3 m2. Assuming isentropic flow and no friction and starting with the equations Cp To= CpT + v2/2 pρ-γ = constant a = (γRT)1/2 Determine the mass flow rate through the nozzle , the pressure at the throat and the exit velocity. Tt/T1 = (pt/p1) γ γ hence Tt = 291.7 K pt = 0.528 p1 = 0.686 bar if chocked. at = (γ R Tt)½ = 342.35 m/s ρt = pt/RTt = 0.686 x 105 /(287 x 291.7) = 0.82 kg/m3 m = ρ A a = 0.82 x 1 x 10-3 x 342 = 0.28 kg/s ( -1)/ T2/T0 = {1 + M2(γ -1)/2 } hence To = 350 K T2 = To (p2/pt) ( γ-1)/ γ ( γ-1)/ = 350(1.013/1.3) γ = 325.9 K ρ2 = p2/RT2 = 1.013 x 105 /(287 x 325.9) = 1.083 kg/m3 c2 = m/ (ρ2 A2) = 0.28/(1.083 x 1.2 x 10-3) = 215.4 m/s a = (γRT)½ 3. Show that the velocity of sound in a perfect gas is given by Show that the relationship between stagnation pressure, pressure and Mach number for the /( -1) isentropic flow of a perfect gas is po/p = {1 + (γ-1)M2/2}γ γ It may be assumed that ds = Cp d(lnv) + Cv d(lnp) where v is the specific volume. A convergent - divergent nozzle is to be designed to produce a Mach number of 3 when the absolute pressure is 1 bar. Calculate the required supply pressure and the ratio of the throat and exit areas. The solution for part 1 requires the derivations contained in the tutorial. γ 3.5 γ − 1 ⎞ γ −1 ⎛ ⎛ 0.4 ⎞ + 1⎟ = 1 x ⎜ 3 2 + 1⎟ = 36.73 bar p o = p⎜ M 2 2 2 ⎝ ⎠ ⎝ ⎠ For chocked flow pt = 0.528 po = 19.39 bar ct = 1 Mt = (γ R Tt )½ ce = 3 Me = 3(γ R Te )½ 1 ⎞γ ρe ⎛ pe =⎜ ⎟ ρ t ⎜⎝ p t ⎟⎠ A t ρece = Ae ρtct 1 A t ⎛ pe ⎞ γ =⎜ ⎟ x3x A e ⎜⎝ p t ⎟⎠ 1−1 / γ ⎛ pe ⎞ ⎜⎜ ⎟⎟ ⎝ pt ⎠ 1 ⎞γ A t ⎛ pe =⎜ ⎟ x3x A e ⎜⎝ p t ⎟⎠ 1+ γ ⎛ p ⎞ 2γ ⎛ 1 ⎞ = 3 ⎜⎜ e ⎟⎟ = 3⎜ ⎟ ⎝ 19.39 ⎠ ⎝ pt ⎠ γRTe γRTt m = ρ t A t c t = ρe Ae ce 1− Te ⎛ p e ⎞ =⎜ ⎟ Tt ⎜⎝ p t ⎟⎠ 2.4/2.8 = 0.236 1 γ SELF ASSESSMENT EXERCISE No. 5 1. An air storage vessel contains air at 6.5 bar and 15oC. Air is supplied from the vessel to a machine through a pipe 90 m long and 50 mm diameter. The flow rate is 2.25 m3/min at the pipe inlet. The friction coefficient Cf is 0.005. Neglecting kinetic energy, calculate the pressure at the machine assuming isothermal flow. m = pV/RT = 6.5 x 10 5 x 2.25/(60 x 288 x 287) = 0.295 kg/s (1-p22/p12)= (64 m2RT Cf L)/( π2 D5 p12) 2 64 x 0.2952 x 287 x 288 x 0.005 x 90 ⎛p ⎞ 1− ⎜ 2 ⎟ = = 0.1589 2 ⎝ 6.5 ⎠ π 2 x 0.055 x 6.5 x 105 2 ⎛ p2 ⎞ 1 − 0.1589 = 0.841 = ⎜ ⎟ ⎝ 6.5 ⎠ ( p22 – 35.5 ) p2 = 5.96 bar SELF ASSESSMENT EXERCISE No. 6 1. A natural gas pipeline is 1000 m long and 100 mm bore diameter. It carries 0.7 kg/s of gas at a constant temperature of 0o C. The viscosity is 10.3 x 10-6 N s/m2 and the gas constant R = 519.6 J/kg K. The outlet pressure is 105 kPa. Calculate the inlet pressure. Using the Blazius formula to find f. (Answer 357 kPa.) 4m 4 x 0.7 = = 865 x 10 3 −6 πµD π x 10.3 x 10 x 0.1 C f = 0.079R e−0.25 = 0.00259 Re = 2 ⎛p ⎞ 64 x 0.7 2 x 519.6 x 273 x 0.00269 x 1000 1 − ⎜⎜ 2 ⎟⎟ = 2 π 2 x 0.15 x (p1 ) ⎝ p1 ⎠ 116.7 x 109 + 11.025 x 109 1= p1= 357 kPa (p1 )2 2. A pipeline is 20 km long and 500 mm bore diameter. 3 kg/s of natural gas must be pumped through it at a constant temperature of 20oC. The outlet pressure is 200 kPa. Calculate the inlet pressure using the same gas constants as Q.1. 4m 4x3 = = 741693 πµD π x 10.3 x 10 −6 x 0.5 C f = 0.079R e−0.25 = 0.00269 Re = 2 ⎛ 200 x 10 3 ⎞ 64 x 32 x 519.6 x 293 x 0.00269 x 20000 15.296 x 109 ⎟ = 1 − ⎜⎜ = 2 ⎟ 2 5 p p12 ( ) π x 0.5 x p 1 ⎠ ⎝ 1 p12 = 15.296 x109 + 40 x 109 p1= 235 kPa 3. Air flows at a mass flow rate of 9.0 kg/s isothermally at 300 K through a straight rough duct of constant cross sectional area of 1.5 x 10-3 m2. At end A the pressure is 6.5 bar and at end B it is 8.5 bar. Determine a. the velocities at each end. (Answers 794.8m/s and 607.7 m/s) b. the force on the duct. (Answer 1 380 N) c. the rate of heat transfer through the walls. (Answer 1.18 MJ) d. the entropy change due to heat transfer. (Answer 3.935 KJ/k) e. the total entropy change. (Answer 0.693 kJ/K) It may be assumed that ds = Cp dT/T + R dp/p v2 = mRT/p2A = 9 x 287 x 300/(6.5 x 105 x 1.5 x 10-3) = 794.8 m/s v1 = mRT/p1A = 9 x 287 x 300/(8.5 x 105 x 1.5 x 10-3) = 607.7 m/s p1A1 + m v1 = p2A2 + m v2 + F F = 1.5 x 10-3 (2 x 105) + 9 (607.76 – 794.48) =300 – 1680 = -1380 N The force to accelerate the gas is greater than the pressure force. Φ + P = mcp∆T + (m/2)( v22 – v12 ) ∆T = 0 P=0 2 2 Φ = cp∆T + (m/2)( v2 – v1 ) Φ = 0 + (9/2)( 794.82 – 607.72 ) = 1.18 MJ ∆s = Φ/T = 1180/300 = 3.935 kJ/k Φ = ∫ T ds = T ∆ s ∆s = mRln(p1/p2) = 9 x 287 ln(8.5/6.5) = 693 J/K 4. A gas flows along a pipe of diameter D at a rate of m kg/s. dp 32C f m 2 RT = Show that the pressure gradient is − dL π 2 p D5 Methane gas is passed through a pipe 500 mm diameter and 40 km long at 13 kg/s. The supply pressure is 11 bar. The flow is isothermal at 15oC. Given that the molecular mass is 16 kg/kmol and the friction coefficient Cf is 0.005 determine a. the exit pressure. b. the inlet and exit velocities. c. the rate of heat transfer to the gas. d. the entropy change resulting from the heat transfer. e. the total entropy change calculated from the formula ds = Cp ln(T2/T1) - R ln(p2/p1) The derivation is given in the tutorial. p2 L ⎛ p 22 − p12 ⎞ 32C f m 2 RTL R o 8314.4 32C f m 2 RT ⎜ ⎟ R= ~ = = 520 J/kgK − ∫ pdp = − dL ∫ 2 5 ⎜ 2 ⎟= 16 N π 2D5 π D ⎝ ⎠ o 0 64 x 0.005 x 132 x 520 x 288 x 40000 64C f m 2 RTL 2 5 2⎤ 2 2 ⎡ = − p 2 − 11 x 10 − p 2 − p1 = ⎢⎣ ⎥⎦ π 2D5 π 2 x 0.55 2 2 − ⎡p 22 − 11 x 105 ⎤ = 1.05 x 1012 ⎡ 11 x 105 ⎤ - 1.05 x 1012 = p 22 p2 = 3.99 bar ⎥⎦ ⎥⎦ ⎢⎣ ⎢⎣ v1 = mRT1/p1A1 = 13 x 520 x 288/(11 x 105 x π x 0.252) = 9.014 m/s v2 = mRT2/p2A2 = 13 x 520 x 288/(3.99 x 105 x π x 0.252) = 24.85 m/s Φ + P = mcp∆T + (m/2)( v22 – v12 ) ∆T = 0 P=0 2 2 Φ = cp∆T + (m/2)( v2 – v1 ) Φ = 0 + (13/2)(24.852 – 9.0142 ) = 3.484 kW ∆s = Φ/T = 3484/288 = 12.09 J/k ∆s = -Rln(p2/p1) = -520 ln(3.99/11) = 526 J/K ( ( ) ( ) ( ) ) SELF ASSESSMENT EXERCISE No. 7 1. Write down the equations representing the conservation of mass, energy and momentum across a normal shock wave. Carbon dioxide gas enters a normal shock wave at 300 K and 1.5 bar with a velocity of 450 m/s. Calculate the pressure, temperature and velocity after the shock wave. The molecular mass is 44 kg/kmol and the adiabatic index is 1.3. R = 8314/44 = 188.95 J/kg K a1= √(γRT1) = √(1.3 x 188.95 x 300) = 271.5 m/s M1 = v1/a1 = 450/271.5 = 1.6577 2 M12 + 1.6577 2 + 2/0.3 γ −1 M 22 = = = 0.4126 M2 = 0.643 2 γ M12 2 x 1.3 x 1.657 2 − 1 −1 γ −1 ( p 2 1 + γM12 = p1 1 + γM 22 ( ) ) p 2 1 + 1.3 x 1.6577 2 = = 2.976 p1 1 + 1.3 x 0.6432 ( ) 1.6577 2 M12 1 + (0.3) T2 T2 2 2 = 1.329 = = 2 T1 T1 0.6432 M2 1 + (0.3) 1 + (γ − 1) 2 2 a2= √(γRT2) = √(1.3 x 188.95 x 398.8) = 313 m/s p2 = 1.5 x 105 x 2.976 = 446 kPa 1 + (γ − 1) T2 = 300 x 1.329 = 398.8 K v = a2 M2 = 201 m/s 2. Air discharges from a large container through a convergent - divergent nozzle into another large container at 1 bar. the exit mach number is 2.0. Determine the pressure in the container and at the throat. When the pressure is increased in the outlet container to 6 bar, a normal shock wave occurs in the divergent section of the nozzle. Sketch the variation of pressure, stagnation pressure, stagnation temperature and Mach number through the nozzle. Assume isentropic flow except through the shock. The following equations may be used. Energy Balance from o to e o is the stagnation condition uo = 0 ue= 2√(γRTe) γ/(γ-1)RTo + 0 = γ/(γ-1)RTe + ue2/2 3.5 RTo + 0 = 3.5RTe + 2γRTe γ p o ⎛ To ⎞ γ−1 To ⎛ p o ⎞ =⎜ ⎟ =⎜ ⎟ p e ⎜⎝ Te ⎟⎠ Te ⎜⎝ p e ⎟⎠ ⎛ po ⎞ ⎜⎜ ⎟⎟ ⎝ pe ⎠ γ-1 γ = γ-1 γ ⎛p ⎞ 3.5RTe = ⎜⎜ o ⎟⎟ ⎝ pe ⎠ (3.5 + 2.8) = 1.8 p 3.5 e γ-1 γ = 3.5RTe + 2γγR e po = 1.8(1/0.286) = 7.82 bar = 1 bar γ ⎛ p ⎞ ⎛ 2 ⎞ γ −1 The throat is chocked ⎜⎜ t ⎟⎟ = ⎜⎜ ⎟⎟ = 0.528 ⎝ po ⎠ ⎝ γ −1 ⎠