A water hose fills 10 gallon (1 gallon 3.785 liter) bucket in 50 second. Hose inside diameter is 2 cm, outlet nozzle diameter is 0.8 cm. Find: a- Volumetric flow rate b- Mass flow rate c- Water flow velocity at hose nozzle outlet

To tackle this problem, we need to break it down into a few manageable parts. We’ll calculate the volumetric flow rate, mass flow rate, and the water flow velocity at the hose nozzle outlet step by step. Let’s dive in!

Calculating the Volumetric Flow Rate

The volumetric flow rate (Q) is defined as the volume of fluid that passes through a given surface per unit time. We can calculate it using the formula:

Q = V / t

Where:

• V = volume of water (in liters)
• t = time (in seconds)

From the problem, we know that the hose fills a 10-gallon bucket in 50 seconds. First, we need to convert gallons to liters:

10 gallons = 10 × 3.785 liters = 37.85 liters

Now we can plug the values into the formula:

Q = 37.85 liters / 50 seconds = 0.757 liters/second

Finding the Mass Flow Rate

The mass flow rate (ṁ) can be calculated using the formula:

ṁ = ρ × Q

Where:

• ρ = density of water (approximately 1000 kg/m³)
• Q = volumetric flow rate (in m³/s)

First, we need to convert the volumetric flow rate from liters per second to cubic meters per second:

0.757 liters/second = 0.757 / 1000 m³/second = 0.000757 m³/second

Now, substituting the values into the mass flow rate formula:

ṁ = 1000 kg/m³ × 0.000757 m³/second = 0.757 kg/second

Determining Water Flow Velocity at the Hose Nozzle Outlet

The flow velocity (v) at the nozzle can be calculated using the formula:

v = Q / A

Where:

• A = cross-sectional area of the nozzle (in m²)

First, we need to find the area of the nozzle. The area can be calculated using the formula for the area of a circle:

A = π × (d/2)²

Here, the diameter (d) of the nozzle is 0.8 cm, which we need to convert to meters:

0.8 cm = 0.008 m

Now, calculating the area:

A = π × (0.008/2)² = π × (0.004)² ≈ 5.0265 × 10⁻⁵ m²

Now we can find the flow velocity:

v = 0.000757 m³/second / 5.0265 × 10⁻⁵ m² ≈ 15.04 m/s

Summary of Results

To summarize our findings:

• Volumetric Flow Rate: 0.757 liters/second
• Mass Flow Rate: 0.757 kg/second
• Water Flow Velocity at Hose Nozzle Outlet: 15.04 m/s

This breakdown shows how we can systematically approach problems involving fluid dynamics, using basic principles and formulas to derive meaningful results. If you have any further questions or need clarification on any of these steps, feel free to ask!