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# Unit 4[1]

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### Unit 4[1]

1. 1. Mechanical Engineering Department, KNUST, MARCH 2012 FLUID DYNAMICS (ME 252)Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Department of Mechanical Engineering INSTIT UTE OF DISTANCE LEARNING, BSc. MECHANICAL ENGINEERING, BRIDGING C.K.K. SEKYERE
2. 2. UNIT : FLOW IN OPEN CHANNELS OUTLINE2 Flow in channels  Classification of Flows in Channels  Discharge through Open Channels by Chezy’s Formula  Empirical Formulae for the value of Chezy’s Constant  Most Economical Sections of Channels Energy of liquid in open channels  Non Uniform Flow through Open Channels  Specific Energy and Specific Energy Curve  Hydraulic Jump or Standing Wave  Gradual Varied Flow C.K.K. SEKYERE; MARCH 2012
3. 3. FLOW IN OPEN CHANNELS3 CLASSIFICATION OF FLOW IN CHANNELS The flow in open channel is classified into the following types: 1. Steady flow and unsteady flow, 2. Uniform flow and non-uniform flow, 3. Laminar flow and turbulent flow, and 4. Sub-critical, critical and super critical flow Steady Flow and Unsteady Flow: If the flow characteristics such as depth of flow, velocity of flow, rate of flow at any point in open channel flow do not change with respect to time, the flow is said to be steady flow. Mathematically, steady flow is expressed as …………………………………………….(4.1) Where V = velocity, Q = rate of flow and y = depth of flow Unit 3
4. 4. If at any point in open channel flow, the velocity of flow, depth of flow or rate of flow changes with respect to time, the flow is said to be unsteady flow. Mathematically, unsteady-flow means4 …………………………….(4.2) Uniform Flow and Non-uniform Flow If for a given length of the channel, the velocity of flow, depth of flow, slope of the channel and cross-section remain constant, the flow is said to be uniform Unit 13
5. 5. On the other hand, if for a given length of the channel, the velocity of flow, depth of flow, etc. do not remain constant, the flow is said to be non-uniform flow5Mathematically, uniform and non-uniform flow are written as: …………………….(4.3) and …………………………………..(4.4) Non-uniform flow in open channels is also called varied flow, which is classified as: •Rapidly varied flow (R.V.F), and •Gradually varied flow (G.V.F). Unit 4
6. 6. Rapidly Varied Flow (R.V.F): Rapidly varied flow is defined as that flow in which depth of flow6 changes abruptly over a small length of the channel  As shown in Fig. 4.1 when there is any obstruction in the path of flow of water, the level of water rises above the obstruction and then falls and again rises over a short length of the channel Fig. 4.1 Unit 4
7. 7. Gradually Varied Flow (G.V.F)if the depth of flow in a channel changes gradually over a long lengthof the channel, the flow is said to be gradually varied flow and is7denoted by G.V.F Laminar Flow and Turbulent FlowThe flow in open channel is said to be laminar if the Reynoldsnumber (Re) is less than 500 Reynolds number in case of open channel is defined as: ………………………………..….(4.5) R = Hydraulic radius or Hydraulic mean depth Cross  sec tional area of flow normal to the direction of flow R Wetted perimeter Unit 4
8. 8. If the Reynolds number is more than 2000, the flow is said to beturbulent in open channel flow If Re lies between 500 and 2000, the flow is considered to be in8transition state Sub-critical, Critical and Super Critical FlowThe flow in open channel is said to be sub-critical if the Froudenumber (Fe) is less than 1.0Sub-critical flow is also called tranquil or streaming flow The flow is called critical if Fe = 1.0if Fe > 1.0 the flow is called super critical or shooting or rapid ortorrential. ………………………………………..(4.6)D = Hydraulic depth of channel and is equal to the ratio of wetted Unit 4area to the top width of channel = A/T................................(4.7)
9. 9. Where T = Top width of channel9 DISCHARGE THROUGH OPEN CHANNEL BY CHEZY’S FORMULA Consider flow of water in a channel as shown in Fig. 4.2. As the flow is uniform, it means the velocity, depth of flow and area of flow will be constant for a given length of the channel Unit 4
10. 10. Consider sections 1-1 and 2-2. Let10 L = Length of channel, A = Area of flow of water, i = Slope of bed, V = Mean velocity of flow of water, P = Wetted perimeter of the cross-section, f = Frictional resistance per unit velocity per unit area The weight of water between sections 1-1 and 2-2 W = Specific weight of water x volume of water = w x A x L........................(4.8) Component of W along direction of flow =W x sin i = wAL sin i ......(4.9) Unit 4
11. 11. ……..(4.10) The value of n is found experimentally to be equal to 2 and the surface area =P x L11 ……………………(4.11) The forces acting on the water between sections 1-1 and 2-2 are1. Component of weight of water along the direction of flow,2. Frictional resistance against flow of water,3. Pressure force at section 1-1,4. Pressure force at section 2-2,As the depths of water at the sections 1-1 and 2-2 are the same, thepressure forces on these two sections are the same and acting inopposite directionsHence they cancel each other Unit 4
12. 12. In case of uniform flow, the velocity of flow is constant for the given length of the channel12 Hence there is no acceleration acting on the water  Hence the resultant force acting in the direction of flow must be zero Resolving all forces in the direction of flow, we get ………………………(4.12) ………………………(4.13) ………………………(4.14) Unit 4
13. 13. ……………………………………….(4.15)13 (4.14), V  C m sin i …………………………………….(4.16) ……………………(4.16) ……………………(4.17) Unit 4
14. 14. EMPIRICAL FORMALAE FOR THE VALUE OF CHEZY’S CONSTANT14 Equation (4.16) is known as Chezy’s formula after the name of a French Engineer, Antoine Chezy who developed this formula in 1975 C is known as Chezy’s constant; C is not dimensionless The dimension of C is Hence the value of C depends upon the system of units The following are the empirical formulae, after the name of their inventors, used to determine the value of C: Unit 4
15. 15. 1.Bazin formula (in MKS units): ………………………..(4.18)15 where K = Bazin’s constant and depends upon the roughness of the surface of channel, whose values are given in Table 4.1 m = Hydraulic mean depth or hydraulic radius. 2. Ganguillet-Kutter Forumula. The value of C is given by (in MKS unit) as ……………………………………(4.19) where N = Roughness co-efficient which is known as Kutter’s constant, whose value for different surfaces are given in Table 4.2. i = Slope of the bed m = Hydraulic mean depth Unit 4
16. 16. 16 Unit 4
17. 17. Manning’s Formula The value of C according to this formula is given as17 ……………………………………………………….(4.20) N = Manning’s constant which is having same value as Kutter’s constant for the normal range of slope and hydraulic mean depth. The values of N are given in Table 4.2. Unit 4
18. 18. MOST ECONOMICAL SECTION OF CHANNELS A section of a channel is said to be most economical when the cost18 of construction of the channel is minimum  But the cost of construction of a channel depends upon the excavation and the lining To keep the cost down or minimum, the wetted perimeter, for a given discharge, should be minimum This condition is utilized for determining the dimensions of economical sections of different forms of channels Most Economical Rectangular Channel Fig. 4.3 Rectangular channel Unit 4
19. 19. Let b = width of channel, d = depth of flow, area of flow, A,= bxd……………………….(4.21)19 ……………………………….(4.22) From equ. (4.21), equ. (4.22) become …………………………………………..(4.23) …………………………..(4.24) Unit 4
20. 20. ……………………………(4.25)20 ….(4.26) …………(4.27) ……………………………..(4.28) From equations (4.25) and (4.28), it is clear that rectangular channel will be most economical when: 1. either b = 2d 2. Or m=d/2 Unit 4
21. 21. Most Economical Trapezoidal Channel21 Fig. 4.4 Trapezoidal channel Fig. 4.4The side slope is given as 1 vertical to n horizontal ……………(4.29) Unit 4
22. 22. ……………………………………..………….…(4.30)22 ……………………………………..………….…(4.31) ……………………………………..………….…(4.32) …………………(4.33) ………………………………(4.34) Substituting the value of b from (4.32) into (4.34) Unit 4
23. 23. …………………(4.35)23 …………………………………………(4.36) ……………………………………………….(4.37) Substituting the value of A from equ. (4.31) ...............(4.38) Unit 4
24. 24. But from fig. 4.1 Equ. 4.11 is the required condition for a trapezoidal section to be24 most economical, which can be expressed as half of the top width must be equal to one of the sloping sides of the channel Hydraulic mean depth ………………..(4.39) From eq. (4.31) …….(4.39) …………………(4.40) Unit 4
25. 25. Hence for a trapezoidal section to be most economical hydraulic mean depth must be equal to half the depth of flow25 Best side slope for most economical trapezoidal section …………………(4.41) From eq.(4.41) …………………(4.42) …………………(4.43) Wetted perimeter of channelSubstituting the value of b from (4.42) Unit 4
26. 26. ………………………..(4.44)26 For the most economical trapezoidal section, the depth of flow, d and area A are constant Then n is the only variable. Best side slope will be when section is most economical or in other words, P is minimum Hence differentiating equation (4.44) with respect to n, …………………………..(4.45) Unit 4
27. 27. 27 Squaring both sides …………….(4.45)If the sloping sides make an angle θ, with the horizontal, then we have …………….(4.46) Unit 4
28. 28. Hence best side slope is at 60ᵒ to the horizontal or the value of n for best side slope is given by equation (4.45)28 For the most economical trapezoidal section, we have Half of top width = length of one sloping side ………………………………(4.47) Substituting the value of n from equation (4.45), we have Unit 4
29. 29. ……………………………………..(4.48)29 ……………………….(4.49) Unit 4
30. 30. Flow through Circular Channel The flow of a liquid through a circular pipe, when the level of30 liquid in the pipe is below the top of the pipe is classified as an open channel flow The rate of flow through circular channel is determined from the depth of flow and angle subtended by the liquid surface at the centre of the circular channel Fig. 4.5 Circular Channel Unit 4
31. 31. ……………………………..(4.50)31 ……………………………….(4.51) Unit 4
32. 32. 32 Most Economical Circular Section  Incase of circular channels, the area of flow cannot be maintained constant With the change of depth of flow in a circular channel of any radius, the wetted area and wetted perimeter changes Thus in case of circular channels, for most economical section, two separate conditions are obtained They are: 1. Condition for maximum velocity, and 2. Condition for maximum discharge Unit 4
33. 33. Condition for Maximum Velocity for Circular Section33 The velocity of flow according to Chezy’s formula Unit 4
34. 34. The velocity of flow through a circular channel will be maximum when the hydraulic mean depth m or A/P is maximum for a given34 value of C and i. In case of circular pipe, the variable is θ only ……………………………….(4.52)where A and P both are functions of θ The value of wetted area, A is given by equation (4.51) as ……………………………(4.52) The value of wetted perimeter, P is given by equation (4.50) as ……………………………(4.53) Unit 4
35. 35. Differentiating equation (4.52) with respect to θ, we have35 …………………(4.54) From equ. (4.52) into (4.54) Unit 4
36. 36. 36 Unit 4
37. 37. 37 ately The depth of flow for maximum velocity from fig. 4.5 is Unit 4
38. 38. 38 …………………………(4.55) Unit 4
39. 39. Thus for maximum velocity of flow, the depth of water in the circular channel should be equal to 0.81 times the diameter of the channel.39 Hydraulic mean depth for maximum velocity is Unit 4
40. 40. 40 …………………..(4.56) Thus for maximum velocity, the hydraulic mean depth is equal to 0.3 times the diameter of circular channel Unit 4
41. 41. 41 Unit 4
42. 42. …………………………………..(4.57)42 But from equ. (4.53) (4.58) Unit 4
43. 43. 43 Unit 4
44. 44. Depth of flow for maximum discharge (see fig. 4.5)44 …………………………….(4.59) Thus for maximum discharge through a circular channel the depth of flow is equal to 0.95 times its diameter Unit 4
45. 45. PROBLEM 1 Find the velocity of flow and rate of flow of water through a45 rectangular channel of 6 m wide and 3 m deep, when it is running full. The channel is having bed slope as 1 in 2000. Take Chezy’s constant C = 55. SOLUTION Unit 4
46. 46. 46 Unit 4
47. 47. 47 PROBLEM 2Find the slope of the bed of a rectangular channel of width 5 m whendepth of water is 2 m and rate of flow is given as 20 m3/s. TakeChezy’s constant, C=50. Unit 4
48. 48. 48 Unit 4
49. 49. ENERGY OF LIQUID IN OPEN CHANNELS49 Non Uniform (Varied ) Flow through Open Channels A flow is said to be uniform if the velocity of flow, depth of flow, slope of the bed of the channel and area of cross section remain constant for a given length of the channel if velocity of flow, depth of flow area of cross-section and slope of the bed of channel do not remain constant for a given length of pipe, the flow is said to be non-uniform Non-uniform flow is further divided into Rapidly Varied Flow (R.V.F) and, gradually varied flow (G.V.F) depending upon the depth of flow over the length of the channel If the depth of flow changes abruptly over a small length of channel, the flow is said to be rapidly varied flow if the depth of flow in a channel changes gradually over a long length of channel, the flow is said to be gradually varied flow Unit 4
50. 50. EQUATION OF GRADUALLY VARIED FLOW50 Before deriving an equation for gradually varied flow, the following assumptions are made: •the bed slope of the channel is small •the flow is steady and hence discharge Q is constant •accelerative effect is negligible and hence hydrostatic pressure distribution prevails over channel cross-section •the kinetic energy correction factor, α is unity •the roughness co-efficient is constant for the length of the channel and it does not depend on the depth of flow •the formulae, such as Chezy’s formula, Manning’s formula, which are applicable, to the uniform flow are also applicable to the gradually varied flow for determining the slope of energy line •the channel is prismatic Unit 4
51. 51. 51 Fig. 4.6 Specific energy Unit 4
52. 52. The total energy of a flowing liquid per unit weight is given by,52 …………………………………..(4.58) Let L = Length of channel If the channel bottom is taken as the datum as shown in Fig.4.6, then the total energy per unit weight of the liquid will be, ………………………………………….(4.59) The energy given by equation (4.58) is known as specific energy. Unit 4
53. 53. Hence specific energy of a given liquid is defined as energy per unit weight of the liquid with respect to the bottom of the channel.53 Specific Energy Curve It is defined as the curve which shows the variation of specific energy with depth of flow. It is obtained as follows: From equation (4.59), the specific energy of a flowing liquid, ………………………………….(4.60) Unit 4
54. 54. Consider a rectangular channel in which a steady but non-uniform flowis taking place54 Let Q=discharge through the tunnel b=width of the channel h=depth of flow, and q=discharge per unit width Q Q Then q    cons tan t ……………………………(4.61) width b Velocity of the fluid, V  Disch arg e  Q  q ………..(4.62) area bh h ……………………………….(4.63) Unit 4
55. 55. Specific Energy and Specific Energy Curve55 Fig. 4.7 Specific energy ……..(4.64) Unit 4
56. 56. equation (4.63) gives the variation of specific energy (E) with the depth of flow(h)56 hence for a given discharge Q, for different values of depth of flow, the corresponding values of E may be obtained then a graph between specific energy (along x- axis) and the depth of flow, h (along y-axis) may be plotted the specific energy curve may be obtained by first drawing a curve for potential energy (i.e. Ep=h) which will be a straight line passing through the origin, making an angle of 45° with the X-axis as shown in fig 4.7 drawing another curve for kinetic energy (i.e. Ek = q2/(2gh2)) which will be a parabola as shown in fig 4.7 by combining these two curves, we can obtain the specific energy curve in Fig 4.8, curve ACB denotes the specific energy curve Unit 4
57. 57. Critical Depth (hc) the critical depth, denoted by hc, is defined as that depth of flow of57 water at which the specific energy is minimum in Fig 4.7, the curve ACB is a specific energy and point C corresponds to the minimum specific energy the depth of flow of water at C is known as the critical depth The mathematical expression for critical depth is obtained by differentiating the specific energy equation (4.63) with respect to depth of flow and equating the same to zero ……………………………………………………….(4.65 Unit 4
58. 58. 58 ………………………………(4.66)Hence, the critical depth is ………………………………(4.67) Critical Velocity (Vc) the velocity of flow at the critical depth is known as the critical velocity (Vc)  The mathematical expression for critical velocity is obtained from the equation (4.67) as ………………………………(4.68) Unit 4
59. 59. Taking the cube of both sides, we get ..(4.69)59 ….......(4.70)Substituting (4.70) into (4.69) ……………….(4.71) ………………………………….(4.72) Unit 4
60. 60. Minimum Specific Energy in terms of Critical Depth60 Specific energy equation is given by equation (4.72) When specific energy is minimum depth of flow is critical depth and hence above equation becomes ………………………………..(4.73)But from the equation (4.67) Substituting the value of into equ. (4.73) ………………………..(4.74) Unit 4
61. 61. Hydraulic Jump or Standing Wave61 Fig. 4.8 Hydraulic jump Unit 4
62. 62. consider the flow of water over a dam as shown in Fig. 4.8  the height of water at the section 1-1 is small as we move towards62 downstream, the height or depth of water increases rapidly over a short length of the channel this is because at the section 1-1, the flow is a shooting flow as the depth of water at section 1-1 is less than the critical depth shooting flow is an unstable type of flow and does not continue on the downstream side then this shooting flow will convert itself into a streaming or tranquil flow and hence depth of water will increase this sudden increase of depth of water is called a hydraulic jump or a standing wave thus, hydraulic jump is defined as: the rise of water level, which takes place due to the transformation of the unstable shooting flow (Super-critical) to the stable streaming flow (sub- critical flow) Unit 4
63. 63. Expression for depth of hydraulic jump63 Assumptions: the flow is uniform and the pressure distribution is due to hydrostatic before and after the jump losses due to friction on the surface of the bed of channel are small and hence neglected the slope of the bed of the channel is small, so that the component of the weight of the fluid in the direction of flow is negligibly small Consider two sections 1-1 and 2-2 before and after a hydraulic pump as shown in fig. 4.9 Unit 4
64. 64. 64 Fig. 4.9 Hydraulic jump Unit 4
65. 65. 65Consider unit width of the channelThe forces acting on the mass of water between section 1-1 and 2-2are: Unit 4
66. 66. (iii) Frictional force on the floor of the channel, which is assumed to be66 negligible ……………………..(4.75) Unit 4
67. 67. 67 Unit 4
68. 68. 68 ……………………………………(4.76) But from the momentum principle, the net force acting on a mass of fluid must be equal to the rate of change of momentum in the same direction Rate of change of momentum in the direction of the force = mass flow rate of water x change ofvelocity in the direction of force Unit 4
69. 69. …(4.77) Therefore, according to the Newton’s 2nd law, equ. (4.76) = equ.69 (4.77) …………….(4.78) But from the equation (4.75), Dividing by ρ Dividing by (d2-d1) …………………………(4.79) Unit 4
70. 70. 70 ………………………………..(4.80) Equation (4.80) is a quadratic equation in d2 and hence its solution is Unit 4
71. 71. 71 ………………………………(4.81) …………………………(4.82) ………………………….(4.83) Unit 4
72. 72. Expression for loss of Energy due to Hydraulic Jump when hydraulic jump takes place, a loss of energy due to eddies72 formation and turbulence occurs this loss of energy is equal to the difference of specific energies at sections 1-1 and 2-2 Unit 4
73. 73. …(4.84)73But from (4.84) (4.85) Substituting (4.85) into (4.84) Unit 4
74. 74. ……………………….(4.86)74 Expression for depth of Hydraulic Jump in terms of upstream Froude Number …………………………(4.87) Unit 4
75. 75. (4.83)75 …………………………(4.88) But from the equation (4.87), Unit 4
76. 76. Substituting this value in the equation (4.88) we get76 (4.89)LENGTH OF HYDRAULIC JUMP is defined as the length between twosections where one section is taken before the hydraulic jump and thesecond section is taken immediately after the jump for a rectangular channel from experiments, it has been found equalto 5 to 7 times the height of the hydraulic jump. Unit 4