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Cee4476b Lecture 04 Annotated (1)

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  CEE 4476b Environmental Hydraulics Design Instructor: Dr. Andrew Binns Email: abinns2@uwo.ca Office: CMLP 1302 Phone: (519) 661-2111 ext. 88633 ! #$ &'()#* +,-#$    #$%& '()*++,,,-.'/&01$230$-%4-5/+)/$61+7289,%%9:;<-'.&= Week 2, Lecture 4 Monday, January 14, 2013 CEE 4476b: Environmental hydraulics design Today   s lecture ã   Discharge diagram ã   Contractions and expansions with head loss ã   Critical depth in non-rectangular channels ã   Examples Discharge diagram ã   In addition to transitions in channel bottom elevation, transitions in channel width b  can also be analyzed by the specific energy (E) concept Transition from channel width b 1  to channel width b 2 Note:  For rectangular channels, it is no longer true that the flow rate per unit width ( q  = Q / b ) remains constant as we saw in the last lecture Discharge diagram ã   With negligible energy loss, the energy equation simply states: But this requires that the flow regime move from one specific energy curve to another that is appropriate for the new value of q    Discharge diagram ã   Shown graphically: Discharge diagram ã   Shown graphically: Discharge diagram ã    Alternative approach:  –   Quantity that remains constant is not q , but E   in a width contraction ã   Neglecting energy losses ã    Assuming a horizontal channel bottom Discharge diagram ã   Discharge function for a given specific energy E  1 : q 2 =  2 gy 2 (  E  1    y )   q =  y  2 g (  E  1  y ) () 1/2  Discharge diagram ã   There is clearly a unique functional relationship between discharge per unit width  q  and depth y   for a constant value of specific energy q =  y  2 g (  E  1  y ) () 1/2   Discharge diagram ã   Unique relationship between q  and y   for constant specific energy E Discharge diagram ã   Decrease in y   from 1 to 2 corresponds to an increase in q  in the discharge diagram   Discharge diagram ã   From the above equation, we can determine the maximum value of the discharge function by setting dq / dy   = 0 ã   Which is the same relationship we determined for critical depth in a rectangular channel q =  y  2 g (  E  1  y ) () 1/2    y =  (2/3)  E  1  Discharge diagram ã   Critical depth y  c   is not only the depth of minimum specific energy for constant q  –   But also the depth of maximum discharge for a given specific energy ã   Note: The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of E because y   = E   when q  = 0 in the discharge diagram  y =  (2/3)  E  1 Discharge diagram ã   Note:  The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of E because y   = E   when q  = 0 in the discharge diagram Discharge diagram ã   Note:  The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of E because y   = E   when q  = 0 in the discharge diagram Choking in contractions  ã   Choking can be caused in a contraction by decreasing the width to a value such that the available E   no longer is sufficient to pass the flow through the contraction without an increase in upstream depth

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