HYDROLOGY - TUTORIAL 2 TRAPEZOIDAL CHANNELS

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1 HYDROLOGY - TUTORIAL TRAPEZOIDAL CHANNELS In this ttorial yo will Derive eqations assoiated with flow in a trapezoidal hannel. Derive eqations for optimal dimensions. Solve slope of ed sing Chezy and manning formlae. Solve qestions from past papers. This ttorial is a ontination of ttorial 1 whih shold e stdied first. D.J.DUNN 1

2 TRAPEZOIDAL SECTION This topi ors reglarly in the Engineering Conil Exam. The trapezoidal setion is widely sed in anals to aommodate the shape of oats and rede the erosion of the sides. BEST DIMENSION Figre 1 The hannel dimensions that give the maximm flow rate for a fixed ross setional area is the one with the least amont of frition. This means that it mst have the minimm wetted srfae area and hene the minimm wetted perimeter P. If this vale is then sed in any formlae for the flow rate, we will have the maximm disharge possile. Using the notation shown on the diagram we proeed as follows. Area A (B + ) h from whih B (A/ h ) (A/ h ) h /tanθ Wetted Perimeter P B + h /sinθ A h h A 1 Sstitte for B P + + h h tanθ sinθ h sinθ tanθ For a given ross setional area the minimm vale of P ors when dp/dh 0 dp A Eqate to zero and A h and sstitte for A dh h sinθ tanθ sinθ tanθ h 1 h 1 B + h h B + h tanθ sinθ tanθ tanθ sinθ tanθ B h or B hk where K sinθ tanθ sinθ tanθ It an e shown that when this is the ase, the ottom and sides are oth tangents to a irle of radis h. When θ 90 o K 1 and when θ 45 o K and in fat K is almost a linear fntion sh that K θ/90 WORKED EXAMPLE No.1 Callate the dimensions of a trapezoidal hannel with sides at 45 o if it mst arry.5 m /s of water with minimm frition given that C 50 in the Chezy formla and the ed has a gradient of 1 in 1000 The Chezy formla is o C (R h S) ½ or A C (R h S) ½ B h 0.88h h /tan 45 o h sin45 tan45 A (B + ) h (0.88 h + h ) h 1.88 h 1 P B + h 0.88h +.88h.656h sin45 R h A/P 0.5 h h x 50(0.5 h /1000) 1/ h 4 (0.5 h /1000) h 5 h m B 0.88 h m D.J.DUNN

3 SELF ASSESSMENT EXERCISE No.1 1. Callate the dimensions of a trapezoidal hannel with sides at 60 o to the horizontal if it mst arry 4 m /s of water with minimm frition given that C 55 in the Chezy formla and the ed has a gradient of 1 in 100. (h 1.4 m B m). Callate the dimensions of a trapezoidal hannel with sides at 0 o to the horizontal if it mst arry m /s of water with minimm frition given that C 49 in the Chezy formla and the ed has a gradient of 1 in 000. (h 1.05 m B m) CRITICAL DEPTH It reqires a lot of Algera to get to the ritial vales. Start as efore h s h + o /g Rearrange to make the sjet o { g( hs h )} A o A o A (B + )h (B + ) h o Sstitte for o ( hs h )( B + ) h g We annot differentiate this expression ease is a fntion of h so we make a sstittion first. g g ( h h ) B + h s h tanθ h /tan θ h ( hs h ) Bh + tanθ Now we need to mltiply ot Bh h hs Bhhs h Bh + B hhs + + B h + + θ tanθ ( ) h h B h + tan s h 4 Now differentiate with respet to h to find the maximm flow rate for a given speifi energy head. 4 d 4hhs 6Bhhs 5h 8Bh B hhs + + B h gdh For maximm Flow rate eqate d/dh to zero. 4 4hhs 6Bhhs 5h 8Bh 0 B hhs + + B h We an simplify y sstitting ak h /tan θ 0 B h h + 4 h h + 6Bh h B h 5 h 8Bh s s s B hs + 4 hhs + 6Bhs h B hs + 4 hhs + 6Bhs h B B 0 h ( ) ( 8B) 0 ( B ) ( ) ( B B) Rearrange to get the ritial depth h h h s Chs ( B B) ( B B) ( B + 4)( B + ) ( B + 4) C ( B B) ( B + 5) (B + ) ( B + 5) D.J.DUNN

4 h ( B + 4) hs ( B + 5) or h s ( B + 5) h ( B + 4) 4hs If B 0 we have a Vee setion h h as efore. 5 hs If 0 we have a retanglar setion h h as efore. There are ompter programs for making the allations sh as the one at To find the ritial veloity flow rate sstitte h o ( B + 5) ( B + 4) g h h gh 1 gh gh ( B + 5) ( B + 4) B + ( B + 4) 1 If B 0 we have a Vee setion gh gh ( B + 5) s h into o { g( hs h )} ( B + 4) ( B + 5) ( B + 4) ( B + 5) ( B + 4) ( B + 4) as efore. If 0 we have a retanglar setion we have { gh } as efore. To find the ritial flow rate sstitte se A (B + )h gh B + ( B + 4) If B 0 we have a Vee setion h / If 0 we have a retanglar setion we have (B + )h A (B + )h / g B + ( B + 4) g as efore in a slightly different form Bh / g as efore. Smmary for trapezoidal setion The ritial depth is h ( B + 4) hs ( B + 5) The ritial veloity is gh B + ( B + 4) The ritial flow is (B + )h / g B + ( B + 4) The major prolem exists that solving with these formlae reqires a vale for and this depends on the answer itself. D.J.DUNN 4

5 WORKED EXAMPLE No. A anal has a trapezoidal setion with a ase 5 m wide and sides inlined at 50 o to the horizontal. It is reqired to have a depth of m, what wold the flow rate e if the speifi energy head is a minimm? Callate the depth, flow rate and mean veloity for this ondition. What is the Frode Nmer? For minimm speifi energy, the flow and depth mst e ritial so h m. /tan50 o B 5 / / (B + )h g (6.678) g 5.89 m /s A (B + )h x 1.56 m /A.96 m Fr / (gh ) 0.89 B + ( B + 4) WORKED EXAMPLE No. A hannel has a trapezoidal setion with a ase 0.5 m wide and sides inlined at 45 o to the horizontal. It mst arry 0. m /s of water at the ritial depth. Callate the depth and mean veloity. There is no simple way to solve this prolem ease of the omplexity of the formla. (B + )h / g B + ( B + 4) Evalate and plot for varios vales of h and we get the following graphs. where h /tanθ From the graph we see that when 0. m /s, h 0.75 m A ( )(0.75) 0.1 m /A 1.41 m/s Figre SELF ASSESSMENT EXERCISE No. 1. A hannel has a trapezoidal setion with a ase m wide and sides inlined at 60 o to the horizontal. It mst arry 0.4 m /s of water with the minimm speifi energy head. Callate the depth and mean veloity for this ondition. (0.157 m and 1. m/s). A anal has a trapezoidal setion with a ase 4 m wide and sides inlined at 40 o to the horizontal. It is reqired to have a depth of 1.5 m, what wold the flow rate e if the speifi energy head is a minimm? Callate the flow rate and mean veloity for this ondition. (9.1 m /s and.5 m/s) D.J.DUNN 5

6 WORKED EXAMPLE No. 4 An open hannel has a trapezoidal ross setion with sides inlined at 45 o to the vertial. The hannel mst arry 1 m /s with a veloity of m/s with minimm frition. Determine the smallest slope of the ed for these onditions and the orresponding depth and dimensions of the hannel. The onstant n in the Manning formla is Show that this is a s ritial flow. Figre 1 m /s o m/s A / 7 m For minimm frition the optimal vale of B is B h 0.884h h sin45 tan45 1 B h sinθ 1 tanθ A (B + )h 7 (0.884h + h )h h h (7/1. 884) m B 1.61 m m P B + /sin x 1.957/sin A 7 R h 7/ m (Note that for 45 o R h 0.5 h ) / 1/ / 1/ R S S Manning formla n 0.01 S The speifi energy head is h s /g.416 m ( B + 4) x x The ritial depth is h hs ( B + 5) x x Sine the atal depth is larger the flow is s ritial m D.J.DUNN 6

7 SELF ASSESSMENT EXERCISE No. These are exam standard qestions. 1. An open hannel has a trapezoidal setion with sides inlined at 45 o to the vertial. The hannel mst arry 0 m /s of water with a mean veloity of.5 m/s. Determine the smallest slope of the ed possile and the orresponding depth and dimensions of the hannel. The onstant n in the Manning formla is Show that this is a s ritial flow. (R / S 1/ )/n (Answer S , h.1, B.1 m and.1 m.). A hannel has a trapezoidal setion 5 m wide at the ottom. The sides slope at 1 metre p for eah horizontal. The ed has a slope of 1/600 and n in the manning formla is Callate the flow rates orresponding to mean veloities of 0. and 0.6 m/s. (Ans m /s and 4.81 m /s) D.J.DUNN 7