Question

A blood transfusion is being set up in an emergency room for an accident victim. Blood has a density of 1060 kg/m³ and a viscosity of 4.0 × 10^-3 Pa·s. The needle being used has a length of 3.0 cm and an inner radius of 0.25 mm. The doctor wishes to use a volume flow rate through the needle of 3.9 × 10^-8 m³/s. What is the distance h above the victim's arm where the level of the blood in the transfusion bottle should be located? As an approximation, assume that the level of the blood in the transfusion bottle and the point where the needle enters the vein in the arm have the same pressure of one atmosphere. (In reality, the pressure in the vein is slightly above atmospheric pressure.)

Answer 1

We know that the Poiseulle flow, the relation between pressure differential ?p,the flow rate Q, the radius of the circular cross section R, length of the needle L and viscosity ? is given as

Q = ? ?p R? / ( 8 ? L)

the pressure differential thus is given as

?p = 8 ? L Q / (? R?).

But in hydrostatic case, we know that

?p = ? g ?h

Therefore , we have


?h = 8 ? L Q / (? g ? R?).

substituting given values, we get  

?h =[ 8* 4.0*10^-3 kg/(m s) * (0.030 m) * (3.9*10^-8 m³/s) ]/[ ( 1060 kg/m³ * 9.81 m/s²*3.1415*(25*10^-5 m)⁴ ]
= 2.93 x 10^-1 m = 0.293 m = 29.3 cm