Math Problem Statement
Blood plasma (at 37.0°C) is to be supplied to a patient at the rate of 2.80 × 10−6 m3/s. If the tube connecting the plasma to the patient’s vein has a radius of 2.00 mm and a length of 54.0 cm, what is the pressure difference between the plasma and the patient’s vein? Viscosity of blood plasma is 1.30 × 10−3 Pa·s.
Solution
To solve for the pressure difference required to supply blood plasma through a tube at a certain flow rate, we can use Poiseuille’s Law for laminar flow, which is:
where:
- = pressure difference (what we want to find),
- = viscosity of the fluid (given as ),
- = volumetric flow rate (given as ),
- = length of the tube (given as 54.0 cm, or 0.54 m),
- = radius of the tube (given as 2.00 mm, or 0.002 m).
Step 1: Substitute the values into Poiseuille’s equation
Step 2: Calculate each term
1. Calculate
2. Calculate the numerator
3. Substitute and solve for
Answer
The pressure difference required to supply blood plasma at the given rate is approximately:
Would you like more details on any of these steps?
- How would changes in the viscosity affect the pressure difference?
- How would altering the radius impact the flow rate or pressure difference?
- What assumptions are made when using Poiseuille's Law for blood plasma flow?
- How would temperature variations affect blood plasma viscosity?
- Could this calculation apply to other fluids besides blood plasma?
Tip: Always ensure units are consistent in physics equations; small discrepancies can lead to significant errors in results.
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Math Problem Analysis
Mathematical Concepts
Fluid Dynamics
Laminar Flow
Pressure Difference
Formulas
Poiseuille's Law: ΔP = (8ηQL) / (πr^4)
Theorems
Poiseuille's Law
Suitable Grade Level
Grades 11-12
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