ENSC-283Assignment #6 Assignment date: Monday Feb. 23, 2009Due date: Monday Mar. 2, 2009Problem: (Flow under a sluice gate: hydrostatic pressure force)Water in an open channel is held in by a sluice gate. Compare thehorizontal force of the water on the gate (a) when the gate isclosed and (b) when it is open (assuming steady flow, as shown).Assume the flow at sections 1 and 2 is incompressible anduniform, and that (because the streamlines are straight there) thepressure distributions are hydrostatic.1 2V1=1 m/sD1 = 3 mD2 = 0.429 mV2=7 m/swaterSolution:The forces acting on the control volume include: - Force of gravity, W- Friction force, Ff- Components of Rx and Ry of reaction force from gate- Hydrostatic pressure distribution on vertical surface Page1M.Bahrami ENSC 283 Assignment # 6- Pressure distribution along the bottom surface (not shown)Assumptions:- Friction force Ff is negligible. - No body force in x direction.- Steady flow.- Incompressible flow.- Uniform flow at each section.- Hydrostatic pressure distribution at sections 1 and 2. Then,FR 1+FR 2+Rx=(−ρV12w D1)+(ρ V22w D2)The surface forces acting on the CV are due to the pressuredistributions and the unknown force, Rx. From our assumptions,we can integrate the gage pressure distribution on each side tocompute the hydrostatic forces FR 1 and FR 2,FR 1=∫0D1p1(y)dA=w∫0D1ρgydy=ρgwD122Page2M.Bahrami ENSC 283 Assignment # 61 2V1=1 m/sD1 = 3 mD2 = 0.429 mV2=7 m/swaterCVp1( y)p2( y)RxRyWFfFR 2=∫02p2( y )dA=w∫0D2ρgydy=ρgwD222Hence, Rx=ρw(V22D2−V12D1)−ρgw2(D12−D22)The horizontal force per unit width of the gate is Rxw=ρ(V22D2−V12D1)−ρg2(D12−D22)Part (a):When gate is closed, D2=0 and V2=0. From mass conservationV1=0Rxw=−ρg D122Rxw=−999[kgm3]× 9.81[ms2]×(3[m])22=−44.1 kNPart (b):V1=1[m/s], and V2=7 [m/ s], hence,Rxw=999[kgm3]×[(7[ms])2×(0.429[m])−(1[ms])2×(3[m])]−12× 999[kgm3]× 9.81[ms2]×[(3[m])2−(0.429[m])2]=−25.2 kNNote: As can be seen, the force on the open gate is significantlyless than the force on the closed gate as the water acceleratesout under the gate. Page3M.Bahrami ENSC 283 Assignment #