Water at 45$^{\circ} \mathrm{C}$ enters a shower head through a circular tube with 15.8 mm inside diameter. The water leaves in 24 streams, each of 1.05 mm diameter. The volume flow rate is 5.67 L/min. Estimate the minimum water pressure needed at the inlet to the shower head. Evaluate the force needed to hold the shower head onto the end of the circular tube. Indicate whether this is a compression or a tension force.

Solutions

VerifiedGiven data:

$T_w=45\mathrm{^{\circ}C}$

$D_1=15.8\mathrm{mm}\cdot\frac{1\mathrm{m}}{1000\mathrm{mm}}=0.0158\mathrm{m}$

$n=24$

$D_s=1.05\mathrm{mm}\cdot\frac{1\mathrm{m}}{1000\mathrm{mm}}=0.00105\mathrm{m}$

$Q=5.67\mathrm{\frac{L}{min}}\cdot\frac{\frac{0.001\mathrm{m^3}}{1\mathrm{L}}}{\frac{60\mathrm{s}}{1\mathrm{min}}}=0.0000945\mathrm{\frac{m^3}{s}}$

Water flows through the circular tube, which leaves in 24 streams, each of 1.05 mm diameter. Water temperature, diameter and the volume flow rate are given in the text of the problem. It is necessary to determine the minimum water pressure needed at the inlet to the tube, as well as the force needed to hold the shower head onto the end of the circular tube.

For this problem, the **energy equation** will be used to determine the water pressure $P_1$ at the inlet. The energy equation is given by:

$\dfrac{P_1}{\rho}+\alpha_1\dfrac{V_1^2}{2}+gz_1 = \dfrac{P_2}{\rho}+\alpha_2\dfrac{V_2^2}{2}+gz_2 +h_{l_T}$