Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 68

Chapter 14, Problem 68

The 30-cm-long left coronary artery is 4.6 mm in diameter. Blood pressure drops by 3.0 mm of mercury over this distance. What are the (a) average blood speed and (b) volume flow rate in L/min through this artery?

Verified step by step guidance

Convert all given quantities into SI units for consistency. The length of the artery is 30 cm = 0.30 m, the diameter is 4.6 mm = 0.0046 m, and the pressure drop is 3.0 mmHg. Convert the pressure drop to Pascals using the relation: 1 mmHg = 133.322 Pa.

Calculate the radius of the artery from the diameter: \( r = \frac{d}{2} \). Substitute the diameter \( d = 0.0046 \; \text{m} \) to find \( r \).

Use Poiseuille's law to relate the volume flow rate \( Q \) to the pressure drop \( \Delta P \), the radius \( r \), the length \( L \), and the viscosity \( \eta \): \( Q = \frac{\pi r^4 \Delta P}{8 \eta L} \). Substitute the known values, including the viscosity of blood (assume \( \eta = 3.5 \times 10^{-3} \; \text{Pa·s} \)).

Once the volume flow rate \( Q \) is determined, calculate the average blood speed \( v \) using the relationship \( v = \frac{Q}{A} \), where \( A \) is the cross-sectional area of the artery. The area is given by \( A = \pi r^2 \).

Convert the volume flow rate \( Q \) from \( \text{m}^3/\text{s} \) to \( \text{L/min} \) by multiplying by \( 60,000 \) (since 1 m³ = 1000 L and there are 60 seconds in a minute).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Blood Flow Rate

Blood flow rate is the volume of blood that passes through a given cross-sectional area of a blood vessel per unit time, typically measured in liters per minute (L/min). It can be calculated using the formula Q = A × v, where Q is the flow rate, A is the cross-sectional area of the vessel, and v is the average blood speed. Understanding this concept is crucial for determining how efficiently blood circulates through the arteries.

Average Blood Speed

Average blood speed refers to the velocity at which blood travels through a blood vessel. It can be derived from the flow rate and the cross-sectional area of the vessel using the equation v = Q / A. This concept is important for understanding how quickly blood reaches different parts of the body and how changes in vessel diameter can affect circulation.

Pressure Drop

Pressure drop in a blood vessel is the decrease in blood pressure as blood flows through it, which can be influenced by factors such as vessel length, diameter, and resistance. In this context, a pressure drop of 3.0 mm of mercury over a 30-cm length of the artery indicates the resistance the blood encounters. This concept is essential for understanding the relationship between blood flow, vessel characteristics, and overall cardiovascular health.

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