Experiment 7 Energy Loss in a Hydraulic Jump

Transcription

1 Experiment 7 Energ Loss in a Hdraulic Jump n Purpose: The purpose of this experiment is to examine the transition from supercritical (rapid) flow to subcritical (slow) flow in an open channel and to analze the resulting energ changes. n Apparatus: A 0 m long tilting recirculation flume with rectangular cross section will be used to create a hdraulic jump in the laborator. Flow rate will be controlled b a variable speed pump. Supercritical flow will be produced b releasing the water at the upstream end of the flume over a weir and adjusting the slope of the flume. Velocit heads at different depths and locations will be measured with pitot tubes. Figure 7- Recirculation flume.

2 Figure 7- Pitot tube. n Procedure:. Adjust the slope of the flume to approximatel.5%.. Fill up the flume with water up to 8 inch at the downstream end. 3. Turn on the pump and adjust speed of the pump to approximatel 0.0 Hz. 4. Adjust the slope of the flume carefull until a stable hdraulic jump occurs approximatel at the middle portion of the channel. 5. Measure depth of flow ( and ) at relativel stable reaches before and after the jump. v 6. Measure velocit head ( ) with a pitot tube at 0%, 40%, 60% and 80% of the g flow depth measured from the water surface. For each of these measurements, measure the water surface elevation in the flume and height of water inside the pitot tube. You should have a total of 8 velocit head measurements for upstream and downstream section. 7. Increase pump speed to approximatel.0 Hz and.5 Hz and repeat steps 4 through 6 for each case. n Questions: (interpretation of results). From the measured velocit heads, calculate velocit at four depths for both upstream and downstream sections. Using these velocities, plot the velocit distribution profiles. You should have a total of 6 profiles for three different flow rates.. From each velocit profile, find the average velocit graphicall. 3. Using the measured depths of flow ( and ) calculate the Froude numbers Fr (Equation 7-) at each flow rate. You should have three Froude numbers. 4. Using Froude numbers found in question 3 and equation 7-3, find the average velocities of flow. Compare these velocities with those found in question 5. Calculate the flow rates using average velocit found in question 4 and area of cross-section of flow determined from measured depth of flow. 6. Calculate the specific energ (E and E ) for three different flow rates (Equation 7-) 7. Calculate the energ dissipation E ( = E E ) and the relative loss E ( = * 00 ) for each flow rate. Was the hdraulic jump an efficient energ E dissipater? 8. Calculate specific force F = γ for each flow rate.

3 9. Draw Fr Vs.. Notes:. In open channel flow, the free surface coincides with the hdraulic grade line provided that the pressure distribution is hdrostatic. The pressure distribution will be hdrostatic if the accelerations and curvature of the streamlines in a vertical plane are negligible and if the slope of the bed is small (< 0%). Under these conditions the expression for the total energ ma be written as, H = p/γ + z + v /(g) = + z + v /(g) (7-) in which = the vertical distance from the bed to the water surface (depth of flow) and z is defined as the height of the bed above the datum. If we consider the channel bed as the datum (z = 0), the Eq. 7- reduces to, H = + v /(g) = E (7-) where E is termed the specific energ. For this experiment, since z = 0, the total energ and the specific energ are the same. Furthermore, for this experiment with a rectangular channel cross-section with constant width b, (area of cross-section), A = b (7-3), (flow rate or discharge), Q = v A = v b (7-4), (discharge per unit width), q = Q/b = v b /b = v (7-5). Therefore, E = + q /(g ) (7-6). The relationship expressed b Eq. 7-6 is cubic in terms of and there must be three solutions for a given set of values for E and q. However, onl two of the solutions are real. Thus, there are two possible depths of flow for a given specific energ level, E, and discharge, q. The two depths are referred to as alternate depths. This provides for two regimes of flow, either slow and deep, or fast and shallow, referred to as subcritical flow and supercritical flow, respectivel. If there is a transition from one regime to the other for a given E and q, then the flow must go through an intermediate condition known as critical flow. This critical condition describes the state of flow at which the specific energ is a minimum for a given flow rate per unit width, q. Conversel, at this critical condition for a given E, the flow rate, q, must reach a maximum value. The depth at which critical discharge occurs is called the critical depth, c, and the velocit is the critical velocit, v c. From Eq. 7-6 assuming critical conditions, v c = g c (7-7) q 3 = g c (7-8) and, c = /3 E (7-9). From Eq. 7-7, at critical flow conditions, F r = v c /g c = (7-0)

4 in which F r = the Froude number. At the critical flow condition, F r =. When F r < the flow is subcritical and when F r > the flow is supercritical. The hdraulic jump allows a reasonabl abrupt transition from supercritical to subcritical flow. It can be accompanied b considerable energ loss and turbulence. The energ loss varies with the Froude number of the jump and is generall an unknown. The energ loss can be found b computing the flow energ upstream of the jump and the reduced flow energ downstream of the jump. The energ loss is the difference between the two energ levels. Conservation of momentum can be used to derive the equations for the upstream and downstream depths, which are needed to calculate the energies and the energ loss. The relationships between the upstream and downstream depths are given b, = [ + 8F r ] (7-) = [ + 8F r ] (7-) where the subscripts and appl to conditions upstream and downstream of the jump respectivel. and are termed the conjugate or sequent depths. The upstream Froude number is given b, v Fr = (7-3) g and the downstream Froude number is given b, v Fr = (7-4) g (Note that the conjugate depths are not the same as the alternate depths discussed above since there is a loss of energ from the upstream side to the downstream side of the hdraulic jump.)

5 Figure 7-3 Hdraulic Jump n Data: Width of the channel = 0.75 cm Water temperature= Slope of channel = Case No. 3 Pump Speed (Hz) (cm) (cm) Velocit Head (cm) Pitot tube (upstream) Pitot tube (downstream)