Water at 40 C (density = 992.3 kg/m3 v = 0.664 10⁶ m²/s) flows at a velocity of 2 m/s through a pipe with a diameter of 20 mm and a friction factor of f = 0.004. The water enters the pipe at point A (Figure Q2) into a bath shower mixer unit out of which, the water can exit through the spout via a gate valve (Figure Q2a) or through the showerhead (Figure Q2b).

a. If the water exits through the spout (Figure Q2a) via a fully opened gate valve at B, with a diameter of 12 mm, determine the water pressure at point A. The minor loss coefficient is K_spout = 0.6 for the spout at B. Account for the minor loss in the elbow, the tee, and the gate valve. The gate valve at E is closed.

b. If the water emerges through the showerhead (Figure Q2b, point C) that consists of 100 identical holes, each with a diameter of 1.5 mm. Determine the pressure of the water at A if the minor loss coefficient is K_showerhead = 0.45 for the showerhead. The gate valve at B is closed. Account for the minor loss in the three elbows, the fully opened gate valve at E, and the tee.

M₁ = 1.110m

M₂ = 2.110m

M₃ = 0.905m

Table of Loss Coeffcients

Loss Coeffcients	KL
Gate Valve - Fully Open	0.19
90 Bend	0.9
45 Bend	0.4
Tee for Valve along Branch	1.8

Figure Q2. Bath shower mixer unit with water entering the system at point A and exiting through (a) spout located at point B and (b) shower head located at point C