Question 4.7: Water flows in a rectangular channel of width 3m at a mild s...

(a) In order to determine the ideal velocity at either point 1 or point 2, the Bernoulli equation is applied between points 1 and 2. However, in order to determine the relationship between the ideal velocities at points 1 and 2, the continuity equation is applied between points 1 and 2; thus, we have two equations and two unknowns. Furthermore, in order to determine the ideal flowrate in the pipe, the continuity equation is applied at either point 1 or 2 as follows:

b: = 3 m                 y_{1}: = 2 m                 y_{2}: = 0.95 m                 z_{1}: = 0 m                 z_{2}: = 0 m
g: = 9.81 \frac{m}{sec^{2}}                  A_{1}: = b. y_{1}= 6 m^{2}                 A_{2}: = b. y_{2}= 2.85 m^{2}

Guess value:            v_{2}: = 2 \frac{m}{sec}               v_{2}: = 2 \frac{m}{sec}             Q: = 1 \frac{m^{3}}{sec}

Given

y_{1} + z_{1} + \frac{v^{2}_{1} }{2.g} = y_{2} + z_{2} + \frac{v^{2}_{2} }{2.g}
v_{1}.A_{1} = v_{2}.A_{2}
Q = v_{2}.A_{2}
\left ( \begin{matrix} v_{1} \\ v_{2} \\ Q \end{matrix} \right ) : = Find (v_{1},v_{2},Q)

v_{1}= 2.45 \frac{m}{s}                  v_{2}= 5.158 \frac{m}{s}                  Q= 14.7 \frac{m^{3}}{s}

Application of the Bernoulli equation for the sluice gate illustrates a conversion of pressure energy to kinetic energy.
(b) The EGL and HGL are illustrated in Figure EP 4.7.