A garden hose attached with a nozzle is used to fill a 20-gal bucket. The inner diameter of the hose is 1 in and it reduces to 0.4 in at the nozzle exit. If the average velocity in the hose is 6 ft/s, determine (a) the volume and mass flow rates of water through the hose, (b) how long it will take to fill the bucket with water, and (c) the average velocity of water at the nozzle exit

In the given case,

(a) the volume and mass flow rates of water through the hose  = 0.0327 ft³/s and 2.04 lb/s

(b) how long it will take to fill the bucket with water = 81.96 seconds

(c) the average velocity of water at the nozzle exit - 37.77 ft/s

Given:

D (diameter) = 1 inch.

1 inch = 0.0833 ft   ...1

V = 20 gallons

 1 gallon = 0.134 ft³

20 gallons = 20 x 0.134 = 2.68 ft³ ....2

D2 = 0.4inches = 0.0332 .... 3

A) The formula for the volume flow rate is;  

V’ = Av  

Where A is the area and v is the velocity.  

Area =  

(A) = (from 1)

= 0.00545 ft²  

So,

V’ = 0.00545 x 6

= 0.0327 ft³/s  

The mass flow rate:  

m’ = ρv  

Where ρ = density.  

Density of water in lb/ft³ = 62.4 lbs/ft³  

Thus mass flow rate (m’) = 62.4 x 0.00327

= 2.04 lb/s  

B) The time it will take to fill the bucket is:

t = V/V’  

So, t = (from 2)

= 81.96 seconds  

C) The velocity at the outlet is gotten:  

v2 =  

v2 =    ( from 3)

= 37.77 ft/s

Thus, In the given case,

(a) the volume and mass flow rates of water through the hose  = 0.0327 ft³/s and 2.04 lb/s

(b) how long it will take to fill the bucket with water = 81.96 seconds

(c) the average velocity of water at the nozzle exit - 37.77 ft/s

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A) Volume flow rate = 0.0327 ft³/s; mass flow rate = 2.04 lb/s

B) 81.96 seconds

C) 37.77 ft/s

Explanation:

A) The formula for the volume flow rate is;

V' = Av

Where A is the area and v is the velocity.

Area = πD²/4

From the question, D(diameter) = 1 inch. So let's convert it to ft since the velocity is in ft.

Thus, 1 inch = 0.0833 ft

Thus Area(A) = π(0.0833)²/4 = 0.00545 ft²

So, V' = 0.00545 x 6 = 0.0327 ft³/s

The mass flow rate is calculated as;

m' = ρv

Where, ρ = density.

Density of water in lb/ft³ = 62.4 lbs/ft³

Thus mass flow rate (m') = 62.4 x 0.00327 = 2.04 lb/s

B) The time it will take to fill the bucket is gotten from the formula;

t = V/V'

From the question, V = 20 gallons

Converting this to ft³, we have;

Since, 1 gallon = 0.134 ft³

20 gallons = 20 x 0.134 = 2.68 ft³

So, t = 2.68/0.0327 = 81.96 seconds

C) The velocity at the outlet is gotten from the formula;

v2 = v1((D1)²/(D2)²)

Since the diameter reduces to 0.4 inches at the exit, D2 = 0.4inches = 0.0332

Thus; v2 = 6(0.0833²/0.0332²) = 37.77 ft/s