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 h should the bottle be placed above the needle? Use Pblood}=1.05 10^-3 kg/m^3, Tblood}=4.0 10^{-3}Pa\s. Assume the blood pressure is 18 torr above atmospheric pressure.

Question

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 h should the bottle be placed above the needle? Use Pblood}=1.05 10^-3 kg/m^3, Tblood}=4.0 10^{-3}Pa\s. Assume the blood pressure is 18 torr above atmospheric pressure.

Answer 1

To determine the height at which the bottle should be placed above the needle, we need to consider the flow rate and the properties of the blood.

The height of the bottle will create a pressure difference, which will drive the flow of blood through the tube. We can use the Bernoulli's equation to calculate the pressure difference:

P1 + ρgh1 + ½ρv1^2 = P2 + ρgh2 + ½ρv2^2

Where:
P1 and P2 are the pressures at points 1 and 2
ρ is the density of the fluid (blood in this case)
g is the acceleration due to gravity
h1 and h2 are the heights at points 1 and 2
v1 and v2 are the velocities at points 1 and 2

We want to find h1, which represents the height of the bottle above the needle. Since the blood is at atmospheric pressure just before it enters the needle, we can simplify the equation to:

Pblood + ρgh = Patm + ½ρv^2

Rearranging the equation gives us:

h = (Patm - Pblood) / (ρg) + (v^2) / (2g)

Now we have all the necessary information to calculate h.

First, let's convert the given blood pressure from torr to pascal:

Patm = 18 torr = 18 * 133.32 Pa = 2399.76 Pa

Next, let's calculate the velocity of blood flow using the given flow rate:

Flow rate = 4.9 cm^3/min = 4.9 * 10^(-6) m^3/s

The cross-sectional area of the needle can be calculated using the diameter:

Diameter = 0.48 mm = 0.48 * 10^(-3) m
Radius = 0.24 * 10^(-3) m
Area = π * (0.24 * 10^(-3))^2 = 1.809 * 10^(-7) m^2

Velocity = Flow rate / Area

Now we can substitute the known values into the equation:

h = (2399.76 - 1.05 * 10^(-3)) / (1.05 * 10^(-3) * 9.8) + [(4.9 * 10^(-6)) / (2 * 9.8 * 1.05 * 10^(-3))]

Calculating the equation will give us the required height h in meters.