Final answer:
To calculate the kilograms of water poured onto a fire by all three hoses in one hour, one must use the continuity equation, compute the cross-sectional areas of the pipes and hoses, find the velocity of the water in the hoses, and finally multiply the mass flow rate by the number of seconds in one hour.
Step-by-step explanation:
To determine the quantity of water delivered onto a fire by the three hoses, we can use the principle of conservation of mass, also known as the continuity equation, which in this case states that the mass flow rate through the fire hydrant pipe must equal the sum of the mass flow rates through each hose. The mass flow rate can be calculated using the formula ρ × A × v, where ρ is the density of the water (approximately 1000 kg/m3 for fresh water), A is the cross-sectional area of the pipe or hose, and v is the velocity of the water.
Calculating Cross-Sectional Areas
First, we'll calculate the cross-sectional area of the hydrant pipe (A_pipe) and one hose (A_hose):
- A_pipe = π × (radius of hydrant pipe)^2 = π × (0.095m)^2
- A_hose = π × (radius of one hose)^2 = π × (0.023m)^2
As we have three hoses, the total cross-sectional area for all three hoses (A_hoses_total) is three times the area of one hose.
Calculating Mass Flow Rates
Next, we calculate the mass flow rate for the hydrant pipe (m_dot_pipe) and then for all three hoses combined (m_dot_hoses):
- m_dot_pipe = ρ × A_pipe × velocity of water in pipe
- m_dot_hoses = 3 × (ρ × A_hose × velocity of water in hose)
Since the quantity of water must be conserved, m_dot_pipe = m_dot_hoses. We have the velocity of water in the pipe, but we need to find the velocity in the hoses (v_hoses) using the continuity equation:
v_hoses = (A_pipe / A_hoses_total) × velocity of water in pipe
We then use this velocity to calculate the mass flow rate for the hoses and find the total mass of water poured onto the fire over one hour by multiplying this mass flow rate by the number of seconds in an hour (3600).
The final step is to put all these calculations together to find the total kilograms of water used in one hour by all three hoses.