Question 4.10: Water at 20^◦ C is forced from a 2-cm-diameter syringe throu...

(a) In order to determine the ideal velocity of the forced jet as it leaves the tip of the needle 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. Assuming the datum is at point 1 yields the following:

z_{1}: = 0 m                 z_{2}: = 3 cm                 F: = 0.75 N                D_{1}: = 2 cm                D_{2}: = 0.1 cm
A_{1}: = \frac{\pi . D^{2}_{1} }{4} = 3.142 \times 10^{-4} m^{2}                      A_{2}: = \frac{\pi . D^{2}_{2} }{4} = 7.854 \times 10^{-7} m^{2}
p_{1}: = \frac{F}{A_{1}} = 2.387 \times 10^{3} \frac{N}{m^{2}}                   p_{2}: =0 \frac{N}{m^{2}}
\rho : = 998 \frac{kg}{m^{3}}                g: = 9.81 \frac{m}{sec^{2}}                \gamma : = \rho .g = 9.79 \times 10^{3} \frac{kg}{m^{2}s^{2}}

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

Given

\frac{p_{1}}{\gamma } + z_{1} + \frac{V^{2}_{1} }{2.g} = \frac{p_{2}}{\gamma } + z_{2} + \frac{V^{2}_{2} }{2.g}
v_{1}. A_{1} = v_{2}. A_{2}
\left ( \begin{matrix} v_{1} \\ v_{2} \end{matrix} \right ) : = Find (v_{1},v_{2})
v_{1} = 5.121 \times 10^{-3} \frac{m}{s}                 v_{2} = 2.048 \frac{m}{s}

Application of the Bernoulli equation for the forced jet between points 1 and 2 illustrates a conversion of pressure energy to kinetic energy.
(b) In order to determine the ideal height of the trajectory at point 3, the Bernoulli equation is applied between points 2 and 3 as follows:

p_{3}: = 0 \frac{N}{m^{2}}                             v_{3}: = 0 \frac{m}{sec}
Guess value:                z_{3}: = 10 cm

Given

Application of the Bernoulli equation for the forced jet between points 2 and 3 illustrates a conversion of kinetic energy to potential energy.