Question

Water is circulating through a closed system of pipes in a two-floor apartment. On the first floor, the water has a gauge pressure of 4.5 × 105 Pa and a speed of 2.3 m/s. However, on the second floor, which is 4.0 m higher, the speed of the water is 4.4 m/s. The speeds are different because the pipe diameters are different. What is the gauge pressure of the water on the second floor?

Answer 1

the gauge pressure of the water on the second floor is Pg2 = 403765 Pa = 4.03 * 10⁵ Pa

Step-by-step explanation:

According to Bernoulli's equation ( conservation of mechanical energy, therefore we neglect the friction inside the pipes)

P1 + ρ*g*h1 + 1/2*ρ*v1² = P2 + ρ*g*h2 + 1/2*ρ*v2²

where ρ= density of water , h=height , P= absolute pressure, g= gravity ( 9.8 m/s²)

since the gauge pressure Pg is

Pg = P - Pa (atmospheric pressure)

Pg1 +Pa + ρ*g*h1 + 1/2*ρ*v1² = Pg2 + Pa + ρ*g*h2 + 1/2*ρ*v2²

Pg1 + ρ*g*h1 + 1/2*ρ*v1² = Pg2 + ρ*g*h2 + 1/2*ρ*v2²

since our reference point to measure heights can be chosen, we choose h1=0 and h2= 4m. Thus

Pg1 + 1/2*ρ*v1² = Pg2 + ρ*g*h2 + 1/2*ρ*v2²

therefore

Pg2 = Pg1 + 1/2*ρ*v1² -ρ*g*h2 - 1/2*ρ*v2² = Pg1 -ρ*g*h2 -1/2*ρ*(v2²-v1²)

replacing values and assuming ρ=1000 kg/m³

Pg2 = Pg1 -ρ*g*h2 -1/2*ρ*(v2²-v1²) = 4.5 * 10⁵ Pa - 1000kg/m³*9.8 m/s²*4m -1/2* 1000 kg/m³ * [(4.4 m/s)² - (2.3 m/s)² ] = 403765 Pa = 4.03 * 10⁵ Pa

Pg2 =4.03 * 10⁵ Pa