What is the absolute pressure required by the water pump to reach the 13th floor?

Question

A water pump is located on the first floor of a 13 floor building. The water pump releases water at a pressure of 35lb/in^2. If the distance between each floor is 10ft, what is the absolute pressure required by the water pump to reach the 13th floor?

Answer 1

Step 1) Collect given data

P₂ = 35 psi

h = 13 stories

Each story is 10 ft.

Fluid is water.

ρ = 62.4 lb/m³

Step 2) Use Bernoulli’s Theorem

(P + 0.5ρv²+ρgh)₁ = (P + 0.5ρv²+ρgh)₂

(P + 0.5ρv²+ρgh)₁ = (P + 0.5ρv²+ρgh)₂

P₁ = (P + ρgh)₂

Step 3) Evaluate

P₁ = (35 + 62.4*10*13/144)₂ = 91 psig

Step 4) Calculate P_abs

P_abs = P_atm + P_psig = 14.7 + 91 = 105.8 psia

Comments:

When using imperial units, too often students will include the acceleration due to gravity. In imperial units, the density term already has the gravity term included. Hence the imperial density term is lb/ft³ or simply weight per cubic foot. This should not be confused with the SI density term of kg/m³ or mass per cubic meter.

The other mistake students make is not converting between pounds per square inch and pounds per square foot.

Answer 2

Hey Derek, here are the steps for this calculation:

1. Determine the height difference: Calculate the total height the water needs to be pumped. Each floor is 10 feet high, and there are 13 floors. Therefore, the total height is: 10 ft/floor × 13 floors=130 ft
2. Convert height to pressure: Use the relationship between height and pressure in a fluid. The pressure increase due to a height difference hhh in feet can be calculated using the equation: P=ρgh. where ρ is the density of water (62.4 lb/ft³), g is the acceleration due to gravity (32.2 ft/s²), and h is the height in feet. The pressure in lb/ft² can be converted to lb/in² by dividing by 144 (since there are 144 in² in 1 ft²).
3. Calculate pressure increase: The pressure increase ΔP in lb/ft² is: ΔP = 62.4 lb/ft3 × 32.2 ft/s2 × 130 ft = 261,504 lb/ft2 (≈ 1,815.33 lb/in²)
4. Total required pressure: The water pump initially releases water at 35 lb/in². Add the pressure increase to the initial pump pressure to find the absolute pressure required at the pump: Ptotal=35 lb/in² + 1,815.33 lb/in² = 1850.33 lb/in²

Thus, the absolute pressure required by the water pump to ensure water reaches the 13th floor is approximately 1850.33 lb/in².

Best,

Andrew