A patient is to be given a blood transfusion. The blood is to flow through a tube from a raised bottle to a needle inserted in the vein . The inside diameter of the 4.8 cm long needle is 0.48 mm, and the required flow rate is 4.9 cm/3 of blood per minute.
How high should the bottle be placed above the needle? Use pblood = 1.05x10^3 and nblood = 4.0x10E-3 Pa*s. Assume the blood pressure is 18 torr above atmospheric pressure.
I don't understand what nblood is supposed to represent. The units are pressure, but what you need for this problem is a viscosity.
The main flow constriction, where the pressure drop occurs, is the needle. There is a law
(Broken Link Removed)
that relates laminar viscous fluid flow in narrow passages to pressure drop and viscosity. That is what you seem to need for this problem.
To find out how high the bottle should be placed above the needle, we need to understand the factors that affect the flow rate of the blood through the tube.
First, we can use Poiseuille's Law, which describes the flow of a viscous fluid through a cylindrical tube, to calculate the flow rate.
Poiseuille's Law states that the flow rate (Q) is directly proportional to the pressure difference (ΔP) and the fourth power of the radius (r) of the tube, and inversely proportional to the length (L) and viscosity (η) of the fluid. Mathematically, it is expressed as:
Q = (π * ΔP * r^4)/(8 * η * L)
In this case, we are given the flow rate (Q) as 4.9 cm^3 of blood per minute, the length of the needle (L) as 4.8 cm, the inside diameter (2r) as 0.48 mm (which is 0.048 cm), and the viscosity (η) and pressure (ΔP) of the blood.
Before we substitute the values into the formula, we need to convert the flow rate from cm^3/minute to m^3/second and the pressure from torr to Pa.
1 cm^3 = 1x10^(-6) m^3 (conversion factor)
1 minute = 60 seconds (conversion factor)
1 torr = 133.32 Pa (conversion factor)
Therefore,
Q = (4.9 cm^3/min) * (1x10^(-6) m^3/cm^3) * (1/60 min/second)
= 8.17x10^(-8) m^3/second
ΔP = 18 torr * (133.32 Pa/torr)
= 2399.76 Pa
Now we can substitute the values into Poiseuille's Law:
8.17x10^(-8) = (π * 2399.76 * (0.0244)^4)/(8 * (4.0x10^(-3)) * 0.048)
Simplifying the equation will give you the value for the height of the bottle above the needle.