Question

A patient is given a blood transfusion through a hypodermic needle
inserted in a vein. The inner diameter of the 3.0-cm-long needle is 0.65 mm, and
the venous pressure of the patient is 20 mm Hg. What should the elevation
difference be between the needle and the bottle containing the blood so that the
transfusion proceeds at the rate of 18 mL/min? Assume that the plastic tube car-
rying the blood from the bottle to the needle introduces negligible resistance to

Answer 1

The vein pressure in Pascals:

$$p_1=20\cdot133.3=2666\text{ Pa}.$$

The density of blood is

$$\rho=1050\text{ kg/m}^3.$$

The viscosity of blood:

$$\eta=0.004\text{ Pa}\cdot\text{s}.$$

Convert flow rate to normal units:

$$Q=18 \text{ cm}^3/\text{min is }3\cdot10^{-7}\text{ m}^3/\text{ s}.$$

Write Poiseuille's equation and express the pressure caused by the liquid in the tube:

$$Q=\frac{(p_2-p_1)\pi r^4}{8\eta L},\\\space\\ p_2=p_1+\frac{8QL\eta}{\pi r^4}.$$

The height depends on pressure according to the following equation:

$$h=\frac{p_2}{\rho g},\\\space\\ h=\frac{1}{\rho g}\bigg(p_1+\frac{8QL\eta}{\pi r^4}\bigg),\\\space\\ h=\frac{1}{1050\cdot9.8}\bigg(2666+\frac{8\cdot3\cdot10^{-7}\cdot0.03\cdot0.004}{3.14\cdot (0.00065/2)^4}\bigg)=\\\space\\ =1.06\text{ m}.$$