fine sand moves through pipes just like fluid. a sandblasting machine moves fine sand through a pipe of inner diameter 2.8 cm at 75L/minutes. the sand comes out from two identical jets at speed 45 m/s. calculated the diameter of the holes through which the jets come out.(answer is 4.21mm)

Question

fine sand moves through pipes just like fluid. a sandblasting machine moves fine sand through a pipe of inner diameter 2.8 cm at 75L/minutes. the sand comes out from two identical jets at speed 45 m/s. calculated the diameter of the holes through which the jets come out.(answer is 4.21mm)

Answer 1

Use continuity equation, namely the volume of sand coming out per second equals the diameter * speed of sand.

Volume = 75 L/min = 1.25 L/s

Total area of jets (d=diam in dm)
=2π(d²)/4

Speed of sand = 45 m/s = 450 dm/s

Equate volumes of sand
450*2π(d²)/4=1.25
d=sqrt(1.25*4/(450*2π))
=0.04205 dm
=4.205 mm

Answer 2

To calculate the diameter of the holes through which the jets come out, we can use the principle of conservation of mass.

First, let's convert the flow rate from liters per minute to cubic meters per second, which is the standard unit for flow rate calculations:

Flow rate = 75 L/minute = (75/1000) m^3/minute = (75/1000)/60 m^3/second

Next, we can calculate the cross-sectional area of the pipe using its inner diameter:

Pipe area = π * (diameter/2)^2 = π * (2.8/2)^2 cm^2

Since we want the cross-sectional area in square meters, we need to convert it:

Pipe area = (π * (2.8/2)^2) * (1 cm/100)^2 m^2

Now, we can use the principle of conservation of mass to relate the flow rate, the jet speeds, and the hole diameters:

Flow rate = (Area of one jet) * (Speed of one jet)

Since we have two identical jets, we can divide the flow rate by 2 and find the area of each jet. We can rearrange the equation to solve for the diameter of the holes:

Area of one jet = Flow rate / (2 * Speed of one jet)

To calculate the diameter, we need to convert the area of one jet from square meters to square millimeters:

Jet area in mm^2 = (Flow rate / (2 * Speed of one jet)) * (1000 mm/1 m)^2

Finally, we can calculate the diameter of the holes using the formula for the area of a circle:

Diameter of the holes = 2 * √(Jet area in mm^2 / π)

Substituting the given values into the equations, we have:

Pipe area = (π * (2.8/2)^2) * (1 cm/100)^2 m^2

Flow rate = (75/1000)/60 m^3/second

Jet area in mm^2 = (Flow rate / (2 * 45)) * (1000 mm/1 m)^2

Diameter of the holes = 2 * √(Jet area in mm^2 / π)

Calculating these values will give us the final answer of 4.21 mm for the diameter of the holes.