Question

A hypodermic needle is 4.0 cm long and has an inner diameter of 0.25 mm. Assume the syringe contents have the same properties as water and is at 20 C. a. What pressure is needed to deliver a steady flow of 0.1 cm3 per second? b. If the inner diameter of the needle were increased by 20%, what pressure would be needed to deliver the same flow?

Answer 1

Given,

Inner diameter, D = 0.25 mm = 2.5*10^-4 m

Cross sectional area, A = (D/2)²

                                      = 3.14*(2.5*10^-4/2)² = 3.14*1.25*1.25*10^-8

                                      = 4.91*10^-8 m²

Steady flow, Q = 0.1 cm³/s = 1*10^-7 m³/s

Let the velocity be v

Now,

=> Q = A*v

=> 10^-7 = 4.91*10^-8 *v

=> v = 10^-7/(4.91*10^-8)

          = 2.01 m/s

Now,

Density of water, = 1000 kg/m³

We know,

Pressure, P = 1/2**v² = 0.5*1000*2.01*2.01

                   = 2020.01 Pa

(b)

Now,

Inner diameter is increased by 20%

So,

New inner diameter, D = 0.25 mm + (20/100)*0.25 = 0.25+0.05

                                      = 0.30 mm

                                      = 3*10^-4 m

Cross sectional area, A = (D/2)²

                                      = 3.14*(3*10^-4/2)² = 3.14*1.5*1.5*10^-8

                                      = 7.1*10^-8 m²

Steady flow, Q = 0.1 cm³/s = 1*10^-7 m³/s

Let the velocity be v

Now,

=> Q = A*v

=> 10^-7 = 7.1*10^-8 *v

=> v = 10^-7/(7.1*10^-8)

          = 1.41 m/s

Now,

Density of water, = 1000 kg/m³

We know,

Pressure, P = 1/2**v² = 0.5*1000*1.41*1.41

                   = 994.1 Pa