2. The Energy Principle in Open Channel Flows

Transcription

1 . The Energy Priniple in Open Channel Flows. Basi Energy Equation In the one-dimensional analysis of steady open-hannel flow, the energy equation in the form of Bernoulli equation is used. Aording to this equation, the total energy at downstream setion defers from the total energy at upstream setion by an amount equal to the loss of energy between the setions. (Fig.). Figure. Energy in gradually varied open-hannel flow Chapter of 5

2 Figure. Energy in gradually varied open-hannel flow H V = Z + d + α g. Speifi energy and ritial depth The total energy of a hannel flow referred to a datum is given by equation below: If the datum oinides with the hannel bed at the setion, the resulting expression is known as speifi energy and is denoted as E. thus V E = dos θ + α g For a hannel of small slope and α=, V E = y + for V = A g E y + = [Eqn.] ga Chapter of 5

3 For a hannel of known geometry, E = f ( y, ), keeping onstant it an be seen that, the speifi energy in a hannel setion is a funtion of the depth of the flow only. The variation E with y is represented by a ubi parabola (Fig.). it is seen that there are two positive roots for the equation of E indiating that any partiular disharge an be passed in a given hannel at two depths and still maintain the same speifi energy E. in the Figure. the ordinate PP represents the ondition for a speifi energy of E. the depth of flow an be either PR=y or PR =y. These two possible depths have the same speifi energy are known as alternate depths. In the Figure., a line OS drawn suh that E=y is the asymptote of the upper limb of the speifi-energy urve. It may be notied that the interept P R or P R represents the veloity head of the two alternate depths, one (PR=y ) is smaller and has a larger veloity head while the other (PR =y ) has a larger depth and onsequently a smaller veloity head. The ondition of minimum speifi energy is known as the ritial-flow ondition and the orresponding depth y is known as ritial depth. Fig.. Definition sketh of speifi energy Thus, at the ritial state the two alternate depths apparently beome one. When the depth of flow is greater than the ritial depth, the veloity of flow is less than the ritial veloity for the given disharge, and, hene, the flow is subritial. When the depth of flow is less than the ritial depth, the flow is superritial. Hene, y is the depth of superritial flow, and y is the depth of subritial flow. At the ritial depth, the speifi energy is minimum. Thus differentiating Eqn. with respet to y (keeping onstant) and equating to zero, = da = 0 but da = dy T top width, width of hannel at the water surfae. de dy ga dy Designating the ritial-flow ondition by the suffix, T ga = Chapter of 5

4 = g A T F If an α value other than unity is used the above equation will be: α = g A T Critial flow ondition is governed by the hannel geometry and disharge (and α). If the Froude number is defined as: V = ga/ T It is easy to see that at the ritial flow y=y F=F =.. Critial depth for a variable disharge In the above setion the rital-flow ondition was derived by keeping the disharge onstant. The speifi energy diagram an be plotted for different disharge = i =onstant (i=,,..). in the figure, < < <.. and is onstant along the respetive E vs y plot. Fig. speifi energy for varying disharge Consider a setion PP in this plot, for the ordinate PP, E=E=onstant. Different urves give different interepts. It is possible to imagine a value of = max at a point C at whih the orresponding speifi energy urve would be just tangent to the ordinate PP. The dotted line indiating = max represents the maximum value disharge that an be passed in the hannel while maintaining the speifi energy at onstant value E. E = y + = A ga g( E y) The ondition for maximum disharge an be obtained by differentiating the above equation with respet to y and equating to zero while keeping E = onstant. Chapter 4 of 5

5 d dy = Putting ga da ga g( E y) = 0 dy g( E y) da = T and = g( E y ) dy A T this is the same as the ritial flow onditions. Hene, the ritial flow ondition also = orresponds to the ondition of maximum disharge in a hannel for a fixed speifi energy. Setion fator Z The expression A A T is a funtion of the depth y for a given hannel geometry aand is known as the setion fator Z. Those: Z = A A T At the ritial flow ondition y=y and Z = A A T = g Chapter 5 of 5

6 Triangular Channel For a triang ular hann el havin g a side slope of m horizo ntal: verti al fig.4 A=my and T=m y. Chapter 6 of 5

7 Chapter 7 of 5

8 Chapter 8 of 5

9 .4 Channel transitions.4. Channel with a Hump /rise in bed level/ A. subritial flow Consider a horizontal, fritionless retangular hannel of width B arrying at a depth y. Let the flow be subritial. At setion a smooth hump of height ΔZ is built on the floor. Sine there is no energy losses between setion and, and onstrution of a hump auses the speifi energy at setion to derease by ΔZ. the speifi energy at setion and are given by: Chapter 9 of 5

10 V = y g E ΔZ E + E = Figure.4 hannel transition with a hump Sine the flow is subritial, the water surfae will drop due to a derease in the speifi energy. In figure.5, the water surfae whih was at P at setion will ome down to point R at setion. The depth y will be given by: V E = y + = y + g gb y Fig.5 Speifi energy diagram It is easy to see from fig.5 that as the value of ΔZ is inreased, the depth at setion, will derease. The minimum depth is reahed when the point R oinides with C, the ritial depth point. At this point the hump height will be maximum, say ΔZ m, y =y ritial depth and E =E. the ondition at ΔZ m is given by- the relation: E ΔZ = E = E = y + m gb y Chapter 0 of 5

11 For the hump height greater than ΔZ m the flow is not possible with the given speifi energy. The upstream depth has to inrease to ause an inrease in speifi energy at setion. if this modified depth is represented by y, then ' ' E = y + with {E ' > E and y >y } gb y At setion the flow will ontinue at the minimum peifi energy level, i.e. at the ritial ondition. At this ondition y =y and ' E ΔZ m = E = E = y + gb y When 0<ΔZ<ΔZ m the upstream water level remains stationary at y while the depth of flow at setion derease with ΔZ a minimum value of y at ΔZ=ΔZ m. With further inrease of ΔZ for ΔZ>ΔZ m, y will hange to y while y will ontinue to remain at y. Figue variation of y and y in subritial flow over a hump. B. superritial flow If y is in the superritial flow regime, fig.5 shows that the depth of flow inrease due to the redution of speifi energy. In figure.5 point P orresponds to y and point R to a depth at setion. Up to the ritial depth, y inreases to reah y at ΔZ=ΔZ m. For ΔZ>ΔZ m, the depth over the hump y =y will remain onstant and the upstream depth y will hange. It will derease to have a higher speifi energy E. The variation of the depths y and y with ΔZ in the super ritial flow is shown below. Figure y and y in subritial flow over the hump Variation of Chapter of 5

12 .4. Transition with hange in width a. Subritial flow in a width onstrution onsider a frition less horizontal hannel of width B arring a disharge at a depth y as in figure.6. at setion the hannel width has been onstrited to B by a smooth transition. Sine there are no losses involved and sine the bed elevation at setion and are the same, the speifi energy at setion and are the same. V V E = y + = y + and E = y + = y + g gb y g gb y Fig..6 transition with width onstrution It is onvenient to analyses the flow in terms of the disharge intensity q = B.at setion, q = B and at setion, q = B. Sine B < B, q >q. The speifi energy diagram fig.7 drawn with disharge intensity as the third parameter, point P on the urve q orresponds to a depth y and speifi energy E. Figure.7 speifi energy diagram. Chapter of 5

13 Sine at setion, E=E and q=q, point P will move vertially downward to point R on the urve q to reah the depth y. Those in subritial flow the depth y < y. If B is made smaller, then q will inrease and y will derease. The limit of the ontrated width B =B m is obviously reahed when orresponding to E, the disharge intensity q =q m, i.e. the maximum disharge intensity for a given speifi energy (ritial flow ondition) will prevail. At this minimum width, y = ritial depth at setion, y m and E = Em = ym + gb y m m For a retangular hannel, at ritial flow Sine E =E m y = ym = Em = E and And = g A T y = E y = or B m = Bm gy i.e B m = 7 8gE m If B <B m, the disharge intensity q will be larger than q m the maximum disharge intensity onsistant with E. the flow will not, therefore, be possible with the given upstream ondition. The ' ' upstream depth will have to inrease to y so that a new speifi energy E = y + is ' gb y formed whih will just be suffiient to ause ritial flow at setion. The new ritial depth at setion for a retangular hannel is: q = = B g g y V E = y + =. 5 g y and Sine B <B m, y will be larger than y m. further E =E =.5y. thus even though ritial flow prevails for all B <B m, the depth at setion is not onstant as in the hump ase but inrease as y and hene E rises. The variation of y, y and E with B / B is shown shematially in figure.8. Chapter of 5

14 Figure.8 Variation of y and y in subritial flow in a width onstrition. B. superritial flow in width onstrition If the upstream depth is superritial flow regime, a redution of the flow width and hene inrease in disharge intensity ause a rise in depth y. in figure.7, point P orresponds to y and point R to y. as the width B is dereased, R moves up till it beomes ritial at B =B m. any further redution in B auses the upstream depth to derease to y so that E rises to E. At setion, ritial depth y orresponds to the new speifi energy E will prevail. The variation of y, y and E with B /B in superritial regime is shown below. Exerise:. in a retangular hannel F and F are the Froude numbers orresponding to the alternate depths at aertain disharge. Show that: F + F = F + F. show that in a triangular hannel the Froude number orresponding to alternate depth are given by : F = F 5 ( 4 + F ) ( 4 + F ) 5. If y and y are alternate depths in a retangular hannel show that y y y + y y + y = y ( ) and hene the speifi energy E = y + y ( y + y ) 4. Prove that the alternate depths in an exponential hannel (A=k y a ) are given by a a ay y ( y y ) a+ E = y a a and = + ( y y ) y a 5. What is the ritial depth orresponding to a disharge of 5m /s in a) trapezoidal hannel of B=0.8 and.5: slope b) a irular hannel of D=.5m 6. A irular ulvert.m diameter is flowing half full and flow is in ritial state. Estimate the disharge and the speifi energy. 7. A retangular hannel is 4.0m wide and arries a disharge of 0m /s at a depth of.0 m. at a ertain setion it is proposed to build a hump. Calulate the water surfae elevations at Chapter 4 of 5

15 upstream of the hump and over the hump if the hump height is a) 0.m and b) 0.m (assume no loss of energy at the hump) 8. A retangular hannel is.5m wide and onveys a disharge of.75m /s at a depth of 0.9m. A onstrition of width is proposed at a setion in this anal. Calulate the water surfae elevations in the ontrated setion as well as in an upstream.5m wide setion when the width of the proposed ontration is a).0m b).5m (neglet energy losses in the transition). 9. Water flows at a veloity of m/s and depth of.0 m in an open hannel of retangular ross setion and bed width of.0m. at a ertain setion the width is redued to.8m and bed is raised by 0.65m. Will the upstream depth be affeted and if so, to what extent? 0. Water flows in a retangular hannel.0m wide at a veloity of.5 m/s and a depth of.8m. If at a setion there is a smooth upward smooth step of 0.m, what width is needed at that setion to enable the ritial flow to our on the hump without any hange in the u/s depth? Chapter 5 of 5