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Chapter 5: Problem 30

A hypodermic syringe (see Fig. \(P 5.30\) ) is used to apply a vaccine. If the plunger is moved forward at the steady rate of \(20 \mathrm{mm} / \mathrm{s}\) and if vaccine leaks past the plunger at 0.1 of the volume flowrate out the needle opening, calculate the average velocity of the needle exit flow. The inside diameters of the syringe ard the needle are \(20 \mathrm{mm}\) and \(0.7 \mathrm{mm}\)

Short Answer

Expert verified
The average velocity of the vaccine at the needle exit is approximately 18 m/s.

Step by step solution

01

Calculation of Volume Flowrate

First, calculate the volume flowrate pushed by the plunger using formula for the volume of a cylinder, \(V_p = \pi r_{s}^2 H_p\). Where \(r_s\) is radius of the syringe and \(H_p\) is plunger’s displacement rate. Plug the given values (note the conversion from mm to m for consistency), \(V_p = \pi(0.01m)^2 (0.02m/s) = 6.28 \times 10^{-6} m^3/s\).
02

Calculation of leakage past the plunger

Based on the volume flowrate and provided rate of leakage, compute the leakage volume. Since the leakage is 0.1 of the volume flowrate, the leakage volume \(V_l\) = \(0.1V_p = 0.628 \times 10^{-6} m^3/s\).
03

Calculation of Total Flowrate

Combine the flowrate pushed by the plunger and the leakage volume to determine the total flowrate. Therefore, the total flowrate \(V_t = V_p + V_l = 6.908 \times 10^{-6} m^3/s\).
04

Calculate the average velocity

Use the total volumetric flowrate and the cross-sectional area of the needle to calculate the average needle exit speed. The formula used is \(v = \frac{V}{A_n}\), where \(A_n\) is the cross-sectional area of the needle given by \(\pi r^2\) (with \(r = 0.7mm/2 = 0.00035m\)). Therefore, \(v = \frac{6.908 \times 10^{-6} m^3/s}{\pi (0.00035m)^2} = 18 m/s\).

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