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1 Chapter 2 Energy Depth Relationships

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2 2.1 SPECIFIC ENERGY The total energy of a channel flow referred to a datum is given by Eq. (1.39) as If the datum coincides with the channel bed at the section, the resulting expression is known as specific energy and is denoted by. Thus (2.1) When and, (2.2)

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3 The concept of specific energy, introduced by Bakhmeteff, is very useful in defining critical depth and in the analysis of flow problems. It may be noted that while the total energy in a real fluid flow always decreases in the downstream direction, the specific energy is constant for a uniform flow and can either decrease or increase in a varied flow, since the elevation of the bed of the channel relative to the elevation of the total energy line, determines the specific energy. If the frictional resistance of the flow can be neglected, the total energy in non-uniform flow will be constant at all sections while the specific energy for such flows, however,

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4 will be constant only for a horizontal bed channel and in all other cases the specific energy will vary. To simplify the expressions it will be assumed, for use in all further analysis, that the specific energy is given by Eq. (2.2) i.e. and.This is with the knowledge that and can be appended to and terms respectively, without difficulty if warranted.

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5 2.2 CRITICAL DEPTH Constant Discharge Situation Since the specific energy (2.2a) for a channel of known geometry,. Keeping =constant =, the variation of with is represented by a cubic parabola (Fig. 2.1). It is seen that there are two positive roots for the equation of indicating that any particular discharge can be passed in a given channel at two depths and still maintain the same specific energy. In Fig. 2.1 the ordinate represents

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6 the condition for a specific energy of E1. The depths of flow can be either or. These two possible depths having the same specific energy are known as alternate depths.

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7 In Fig.2.1, a line (OS) drawn such that (i.e. at to the abscissa) is the asymptote of the upper limb of the specific-energy curve. It may be noticed that the intercept or represents the velocity head. Of the two alternate depths, one ( )is smaller and has a large velocity head while the other ( ) has a larger depth and consequently a smaller velocity head. For a given, as the specific energy is increased the difference between the two alternate depths increases. On the other hand, if is decreased, the difference ( ) will decrease and at a certain value,

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8 the two depths will merge with each other (point in Fig. 21). No value for can be obtained when,denoting that the flow under the given conditions is not possible in this region. The condition of minimum specific energy is known as the critical- flow condition and the corresponding depth is known as the critical depth. At critical depth, the specific energy is minimum. Thus differentiating Eq.(2.2a) with respect to (keeping constant) and equating to zero, (2.3)

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9 But = top width, i.e. width of the channel at the water surface. Designating the critical-flow conditions by the suffix, (2.4) or (2.4a) If an value other than unity is to be used, Eq. (2.4) will become (2.5)

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10 Equation (2.4) or (2.5) is the basic equation governing the critical-flow conditions in a channel. It may be noted that the critical-flow condition is governed solely by the channel geometry and discharge (and ). Other channel properties such as the bed slope and roughness do not influence the critical flow condition for any given. If the Froude number of the flow is denned as (2.6) it is easy to see that by using in Eq. (2.4), at the critical flow and. we thus get an important result that the critical flow corresponds to the minimum specific energy and at this condition the Froude number of the flow is unity.

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11 For a channel with large longitudinal slope and having a flow with an energy correction factor of, the Froude number will be defined as (2.6a) Referring to Fig. 2.1, considering any specific energy other than, (say ordinate at )the Froude number of the flow corresponding to both the alternate depths will be different from unity as or.

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12 At the lower limb, CR of the specific-energy curve, the depth. As such, and. This region is called the supercritical flow region. In the upper limb,. As such and. This denotes the subcritical flow region. Discharge as a Variable In the above section the critical-flow condition was derived by keeping the discharge constant. The specific-energy diagram can be plotted for different discharges ( =1,2,3...), as in Fig In this figure, and is constant along the respective vs plots.

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13 Consider a section in this plot. It is seen that for the ordinate,. Different curves give different intercepts. The difference between the alternate depths decreases as the value increases.

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14 It is possible to imagine a value of at a point at which the corresponding specific-energy curve would be just tangential to the ordinate. The dotted line in Fig. 2.2 indicating represents the maximum value of discharge that can be passed in the channel while maintaining the specific energy at a constant value ( ). Any specific-energy curve of higher value (i.e. ) will have no intercept with the ordinate and hence there will be no depth at which such a discharge can be passed in the channel with the given specific energy. Since by Eq. (2.2a)

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15 (2.7) The condition for maximum-discharge can be obtained by differentiating Eq. (2.7) with respect to and equating it to zero while keeping =constant. Thus Putting and (2.8) This is same as Eq.(2.4)and hence represent the critical-flow conditions.

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16 EXAMPLE 2.1 A rectangular channel 2.5m wide has a specific energy of 1.5m when carrying a discharge of Calculate the alternate depths and corresponding Froude numbers. Solution From Eq. (2.2a)

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17 Solving this equation by trial and error, the alternate depths and are obtained as and. Froude number At, ;and at, The depth is in the subcritical flow region and the depth is the supercritical flow region.

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18 EXAMPLE 2.2 A flow of is passing at a depth of 1.5 m through a rectangular channel of width 2.5 m. The kinetic energy correction factor α is found to be what is the specific energy of the flow? what is the value of the depth alternate to the existing depth if α =1.0 is assumed for the alternate flow ? Solution

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19 Specific energy For the alternate depth, i.e. By trial and error

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CALCULATION OF THE CRITICAL DEPTH Using Eq. (2.4), expressions for the critical depth in channels of various geometric shapes can be obtained as follows: Rectangular Section For a rectangular section, and (Fig. 2.4). Hence by Eq. (2.4) or (2.9)

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21 Specific energy at critical depth (2.10) Note that Eq. (2.10) is independent of the width of the channel. Also, if = discharge per unit width =, i.e. (2.11) Since, from Eq. (2.6) the Froude number for a rectangular channel will be defined as (2.12)

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22 Triangular Channel For a triangular channel having a side slope of horizontal: 1 vertical (Fig. 2.4), and. By Eq. (2.4a), (2.13) Hence (2.14)

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23 The specific energy at critical i.e. (2.15) It is noted that Eq. (2.15) is independent of the side slope m of the channel. Since the Froude number for a triangular channel is denned by using Eq. (2.6) as (2.16)

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24 Circular Channel Let be the diameter of a circular channel (Fig. 25) and be the angle in radians subtended by the water surface at the centre.

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25 Top width and

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26 Substituting these in Eq. (2.4a) yields (2.17) Since explicit solution for cannot be obtained from Eq. (2.17), a non-dimensional representation of Eq. (2.17) is obtained as (2.18) This function is evaluated and is given in Table 2A.1

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27 of Appendix 2A at the end of this chapter as an aid for the estimation of. Since, the Froude number for a given at any depth will be

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28 Trapezoidal Channel For a trapezoidal channel having a bottom width of and side slopes of horizontal : 1 vertical (Fig.(2.6)) Area and Top width

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29 At the critical flow Here also an explicit expression for the critical depth is not possible. The non-dimensional representation of Eq. (2.19) facilitates the solution of by the aid of tables or graphs. Rewriting the right- hand side of Eq. (2.19) as

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30 where gives (2.20) or (2.20a) Equation 2.20(a) can easily be evaluated for various value of and plotted as vs. It may be noted that if, can be defined as

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31 One such plot, shown in Fig. 2.7, is very helpful in quick estimation of critical depth and other parameters related to the critical-flow condition in trapezoidal channels. Table 2A.2 which gives values of for various values is useful for constructing a plot of vs as in Fig. 2.7 on a lager scale. Since the Froude number at any depth is

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33 for a given discharge. Further the specific energy at critical depth, is a function of ( ) and it can be shown that (Problem 2.7) where

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SECTION FACTOR Z The expression is a function of the depth for a given channel geometry and is known as the section factor. Thus (2.22) At the critical-flow condition, and (2.23) As a corollary of Eq. (2.23), if is the section factor for any depth of flow, then (2.24) where represents the discharge that would make the depth critical and is known as the critical discharge.

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FIRST HYDRAULIC EXPONENT M In many computations involving a wide range of depths in a channel, such as in the GVF computations, It is convenient to express the variation of with in an exponential form. The ( )relationship (2.25) Is found to be very advantageous. In this equation = a coefficient and =an exponent called the first hydraulic exponent. It is found that generally is a slowly-varying function of the aspect ratio for most of the channel shapes. The variation of and for a trapezoidal channel is indicated in Fig. 2.8.

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37 The value of for a given channel can be determined by preparing a plot of vs on a log- log scale. If is constant between two points (, ) and (, )in this plot, the value of is determined as (2.26) In Eq.(2.26), instead of Z, a non-dimensionalised Z value can also be used. For a trapezoidalchannel,Eq.(220a)represents a non- dimensionalised value of Z if the suffix ‘c‘ is removed.

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38 Hence the slope of vs on a log-log plot, such as in Fig. 2.7, can be used to obtain the value of at any value of. It may be noted that for a trapezoidal channel is a unitque function of and will have a value in the range 3.0 to5.0. An estimate of can also be obtained by the relation (2.27)

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39 EXAMPLE 2.3 Obtain the value of for (a) a rectangular channel and (b) a triangular channel. Solution For a rectangular channel, and by Eq. (2.25) By equating the exponent of on both sides, For a triangular channel of side slope horizontal : 1 vertical,, and by By equating the exponent of on both sides,

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COMPUTATIONS The problems concerning critical depth involve the following parameters: geometry of the channel, or or. EXAMPLE 2.4 Calculate the critical depth and the corresponding specific energy for a discharge of in the following channels: (a) Rectangular channel, = 2.0 m (b) Triangular channel, = 0.5 (c) Trapezoidal channel, = 2.0 m, = 1.5 (d) Circular channel, = 2.0 m

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41 Solution (a) Rectangular Channel Since for a rectangular channel, (b) Triangular Channel From Eq. (2.14)

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42 Since for a triangular channel (c) Trapezoidal Channel Using Table 2A.2 the corresponding value of

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43 (d) Circular Channel From Table 2A.1 showing vs, the corresponding value of by suitable linear interpolation is

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44 Also, from Table 2A.1 for Hence

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45 EXAMPLE 2.5 Calculate the bottom width of a channel required to carry a discharge of as a critical flow at a depth of 1.2m, if the channel section is (a) rectangular and (b) trapezoidal with side slope of 1.5 horizontal : 1 vertical. Solution (a) Rectangular Section The solution here is straightforward

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46 (b) Trapezoidal Channel The solution in this case id by trial-and-error By trial-and-error

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47 EXAMLE 2.6 Find the critical depth for a specific energy head of 1.5 m in the following channels: (a) Rectangular channel, = 2.0 m (b) Triangular channel, = 1.5 (c) Trapezoidal channel, = 2.0 m, = 1.0 (d) Circular channel, = 1.5 m Solution (a) Rectangular channel By Eq. (2.10)

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48 (b) Triangular channel By Eq. (2.15) (c) Trapezoidal channel Since by Eq. (2.4a) Solving by trial and error,

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49 (d) Circular channel By non-dimension with respect to the diameter. From Table2A.1, values of and for a chosen are read and a trial and error procedure is adopted to solve for. It is found that

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TRANSITIONS The concepts of specific energy and critical depth are extremely useful in the analysis of problems connected with transitions. To illustrate the various aspects, a few simple transitions in rectangular channels are presented here. The principles are nevertheless equally applicable to channels of any shape and other types of transitions.

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Channel with a Hump (a)Subcritical Flow Consider a horizontal, frictionless rectangular channel of width carrying at a depth. Let the flow be subcritical. At a section 2 (Fig. 2.9) a smooth hump of height is built on the floor. Since there are no energy losses between sections 1 and 2, and construction of a hump causes the specific energy at section 2 to decrease by. Thus the specific energies at sections 1 and 2 are given by (2.28)

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52 Since the flow is subcritical, the water surface will drop due to a decrease in the specific energy. In Fig. 2.10, the water surface which was at at section 1 will come down to point at section 2. The depth will be given by (2.29)

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54 It is easy to see from Fig that as the value of is increased, the depth at section 2, i.e., will decrease. The minimum depth is reached when the point coincides with, the critical-depth point. At this point the hump height will be maximum, say=, =critical depth and. The condition at is given by the relation (2.30) The question naturally arises as to what happens when. The upstream depth has to increase to cause an increase in the specific energy at section 1. If this modified depth is represented by,then

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55 (2.31) Recollecting the various sequences, when the upstream water level remains stationary at while the depth of flow at section 2 decreases with reaching a minimum value of at (Fig. 2.11). With further increase in the value of, i.e. for, will change to while y 2 will continue to remain at. The variation of and with in the subcritical regime can be clearly noticed in Fig

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56 (b) Supercritical Flow If is in the supercritical flow regime, Fig shows that the depth of flow increases due to the reduction of specific energy. In Fig point, corresponds to and point to depth at the section 2. Up to the critical depth, increases to reach at.

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57 the depth over the hump will remain constant and the upstream depth will change. It will decrease to have a higher specific energy. The variation of the depths and with in the supercritical flow is shown in Fig

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58 EXAMPLE 2.7 A rectangular channel has a width of 2.0 m and carries a discharge of 4.80 with a depth of 1.60 m. At a certain section a small, smooth hump with a flat top and of height 0.10 m is proposed to be built. Calculate the likely change in the water surface. Neglect the energy loss. Solution Let the suffixes 1 and 2 refer to the upstream and downstream sections respectively as in Fig. 2.9.

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59, hence the upstream flow is subcritical and the hump will cause a drop in the water-surface elevation. At section 2,

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60 The minimum specific energy at section 2, is less than, the available specific energy at that section. Hence and the upstream depth will remain unchanged. The depth is calculated by solving the specific-energy relation i.e. Solving by trial and error,

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61 EXAMPLE 2.8 In Example 2.7, if the height of the hump is 0.5 m, estimate the water surface elevation on the hump and at a section upstream of the hump. Solution Form Example 2.7 : =0.391, =1.715 m and. Available specific energy at section The minimum specific energy at section 2 is greater than,the available specific energy at that section. Hence, the depth at section 2 will be at the critical depth. Thus = = m. The upstream depth will increase to a depth such that the new specific energy at the upstream section 1 is

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62 Thus

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63 Solving by trial and error and selecting the positive root which gives, = m. The nature of the water surface is shown in Fig. 2.13

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64 EXAMPLE 2.9 A rectangular channel 2.5 m wide carries 6.0 of flow at a depth of 0.5 m. Calculate the height of a flat topped hump required to be placed at a section to cause critical flow. The energy loss due to the obstruction by the hump can be taken as 0.1 times the upstream velocity head. Solution hence the flow is supercritical

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65 Since the critical flow is desired at section 2 By the energy equation between section 1 and 2, Where Hence

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Transition with a Change in Width (a) Subcritical Flow in a Width Constriction Consider a frictionless horizontal channel of width carrying a discharge at a depth as in Fig At section 2 the channel width has been constricted to by a smooth transition. Since there are no losses involved and since the bed elevations at sections 1 and 2 are same, the specific energy at section 1 is equal to the specific energy at section 2.

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68 It is convenient to analysis the flow in terms of the discharge intensity. At section 1, and at section 2,. Since,. In the specific energy diagram (Fig. 2.15) drawn with the discharge intensity as the third parameter, point

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69 on the curve corresponds to depth and specific energy. Since at section 2, and, point will move vertically downward to point on the curve to reach the depth. Thus, in subcritical flow the depth.If is made smaller, then will increase and will decrease. The limit of the contracted width is obviously reached when corresponding to, the discharge intensity, i.e. the maximum discharge intensity for a given specific energy (critical-flow condition) will prevail 1. At this minimum width, =critical depth at Section 2, and (2.33)

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70 For a rectangular channel, at critical flow Since (2.34) and i.e. (2.35) If, the discharge intensity will be larger than the maximum discharge intensity consistent with.The flow will not, therefore, be possible with the given upstream conditions.

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71 The upstream depth will have to increase to so that a new specific energy is formed which will just be sufficient to cause critical flow at section 2. It may be noted that the new critical depth at section 2 for a rectangular channel is

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72 and Since, will be larger than. Further,. Thus even though critical flow prevails for all, the depth at section 2 is not constant as in the hump case but increases as,and hence, rises. The variation of, and with shown schematically in Fig.216.

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73 (b) Supercritical Flow in a Width Constriction If the upstream depth is in the supercritical flow regime, a reduction of the flow width and hence an increase in the discharge intensity cause a rise in depth. In Fig.2.15, point, corresponds to and point to.

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74 Choking In the case of a channel with a hump, and also in the case of a width constriction, it is observed that the upstream water-surface elevation is not affected by the conditions at section 2 till a critical stage is first achieved. EXAMPLE 2.10 A rectangular channel is 3.5 m wide and conveys a discharge of 15.0 at a depth of 2.0 m. It is proposed to reduce the width of the channel at a hydraulic structure. Assuming the transition to be horizontal and the flow to be frictionless determine the water surface elevations upstream and downstream of the constriction when the constricted width is (a) 2.50 m and (b) 2.20 m

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75 Solution Let suffixes l and 2 denote sections upstream and downstream of the transition respectively. Discharge The upstream flow is subcritical and the transition will cause a drop in the water surface.

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76 Let = minimum width at section 2 which does not cause choking. Then Since

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77 (a) When =2.50 m and hence choking conditions prevail. The depth at the section.The upstream depth will increase to. Actual At the upstream section 1: with new upstream depth of such that.

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78 Hence Solving by trial and error and selecting a root that gives subcritical flow, (b) when =2.20 m As choking condition prevail. Depth at section

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79 At upstream section 1 : New upstream depth= and Hence

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General Transition A transition in its general form may have a change of channel shape, provision of a hump or a depression, and contraction or expansion of channel width, in any combination. In addition, there may be various degrees of loss of energy at various components. EXAMPLE 2.11 A discharge of 16.0 flows with a depth of 2.0 m in a rectangular channel 4.0 m wide. At a downstream section the width is reduced to 3.5 m and the channel bed is raised by. Analysis the water-surface elevations in the transitions when (a) =0.20m and (b) =0.35m.

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81 Solution Let the suffixes 1 and 2 refer to the upstream and downstream sections respectively. At the upstream section, The upstream flow is subcritical and the transition will cause a drop in the water surface elevation.

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82 (a) When = 0.2 m Hence the depth and the upstream depth will remain unchanged at.

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83 Solving by trial and error, = m. Hence when = 0.20 m, = 2.00 m and = m (b) When = 0.35, Hence the contraction will be working under choked conditions. The upstream depth must rise to create a higher total head. The depth of flow at section 2 will be critical with. If the new upstream depth is

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84 i.e. By trial-and-error, The upstream depth will therefore rise by m due to the choked condition at the constriction. Hence, when = 0.35 m

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85 PROBLEMS 2.3 A rectangular channel 5.0 m wide carries 20 of discharge at depth of 2.0 m. The width beyond a certain section is to be charged to 3.5 m. If it is desired to keep the water-surface elevation unaffected by this change, what modifications are needed to the bottom elevation? 2.4 Find the alternate depths corresponding to a specific head of 2.0 m and a discharge of 6.0 in (a) trapezoidal channel, = 0.9 m, = 1.0, (b) triangular channel, = 1.5, (c) circular channel, = 2.50m. (Use the trial and error method. For Part (c) use Table 2A.1.)

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A trapezoidal channel has a bottom width of 6.0 m and side slope of 1:1. The depth of flow is 1.5 m at a discharge of 15. Determine the specific energy and alternate depth A triangular channel has an apex angle of 60 and carries a flow with a velocity of 2.0 and depth of 1.25 m. (a) Is the flow subcritical or super-critical? (b) What is the critical depth? (c) what is the specific energy? (d) What is the alternate depth possible for this specific energy?

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Fill the missing data in the following table connected with critical depth computations in rectangular channels: 2.20 Fill the missing data in the following table connected with critical depth computations in triangular channels:

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