Question 4.9: Water flows in a rectangular channel of width 2 m at a veloc...

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

V_{1}:= 1 \frac{m}{sec}                  y_{1}: = 1.75 m                 z_{1}: = 0 m                  z_{2}: = 0.25 m
b: = 2 m                    A_{1}: = b. y_{1} = 3.4 m^{2}                 g: = 9.81 \frac{m}{sec^{2}}

Guess value:                v_{2}: = 1.2 \frac{m}{sec}                  y_{2}: = 1.5 m

Given

y_{1}+ z_{1} + \frac{V^{2}_{1} }{2. g} = y_{2}+ z_{2} + \frac{V^{2}_{2} }{2. g}
V_{1}. b . y_{1} = V_{2}. b . y_{2}
\left ( \begin{matrix} V_{2} \\ y_{2} \end{matrix} \right ) : = Find (V_{2}, y_{2})
V_{2} = 1.183 \frac{m}{s}                      y_{2} = 1.48 m

(b) In order to determine the flowrate in the channel, the continuity equation is applied at either points 1 or 2 as follows:

Application of the Bernoulli equation for the gradual upward step illustrates a conversion of pressure energy to potential energy and kinetic energy.
(c) The EGL and HGL are illustrated in Figure EP 4.9.