Problem 14 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

During systole, blood is ejected from the left ventricle at a velocity of \(125 \mathrm{~cm} / \mathrm{s}\). The diameter of the aortic valve is \(24 \mathrm{~mm}\), and there is no heat transfer or temperature change within the system. Assume that systole lasts for \(0.25 \mathrm{~s}\), that the height difference is \(5 \mathrm{~cm}\), and that there is no change in area within this distance. Determine the amount of work performed by the heart during systole and the power that the heart generates. The density of blood is \(1050 \mathrm{~kg} / \mathrm{m}^{3}\), and we will assume there are no internal energy changes associated with blood over this short time interval.

Step-by-Step Solution

Answer

The work done by the heart is approximately 0.072 J, and the power generated is about 0.288 W.

Step 1: Calculate the Cross-Sectional Area of the Aortic Valve

First, we find the cross-sectional area of the circular aortic valve. The formula for the area of a circle is \[ A = \pi \left(\frac{d}{2}\right)^2 \]where \( d \) is the diameter of the valve. Given \( d = 24 \; \text{mm} = 0.024 \; \text{m} \), substituting into the formula gives:\[ A = \pi \left(\frac{0.024}{2}\right)^2 = \pi \times 0.012^2 \approx 4.52 \times 10^{-4} \; \text{m}^2 \]

Step 2: Determine the Volume of Blood Ejected

Volume of blood ejected can be determined by multiplying the cross-sectional area by the velocity and the duration of systole. The formula is:\[ V = A \cdot v \cdot t \]Substituting the known values, we get:\[ V = 4.52 \times 10^{-4} \; \text{m}^2 \times 1.25 \; \text{m/s} \times 0.25 \; \text{s} \] \[ V = 1.41 \times 10^{-4} \; \text{m}^3 \]

Step 3: Calculate Work Done Using Potential Energy Change

Work done by the heart can be estimated as the change in potential energy. The change in potential energy \( \Delta PE \) is given by:\[ \Delta PE = \rho \cdot g \cdot h \cdot V \]where \( \rho = 1050 \; \text{kg/m}^3 \), \( g = 9.8 \; \text{m/s}^2 \), \( h = 0.05 \; \text{m} \), and \( V = 1.41 \times 10^{-4} \; \text{m}^3 \).Substituting:\[ \Delta PE = 1050 \times 9.8 \times 0.05 \times 1.41 \times 10^{-4} \approx 0.072 \; \text{J} \]

Step 4: Calculate Power Developed by the Heart

Power developed by the heart is the work done divided by the time over which it is done. Using the relationship:\[ P = \frac{W}{t} \]where \( W = 0.072 \; \text{J} \) and \( t = 0.25 \; \text{s} \), we find:\[ P = \frac{0.072}{0.25} \approx 0.288 \; \text{W} \]

Key Concepts

Hemodynamics

In the study of biofluid mechanics, hemodynamics plays a crucial role as it deals with the flow of blood in the circulatory system. Understanding hemodynamics is essential for analyzing how the heart and blood vessels work together to circulate blood through the body. Several parameters are vital in this area:

• Blood Velocity: This refers to the speed at which blood travels through vessels. In our exercise, the peak velocity of blood from the left ventricle is given as 125 cm/s.
• Cross-sectional Area: It affects how much blood can flow at any given time. For the aortic valve, we've calculated this area using its diameter of 24 mm.
• Time of Systole: Blood is expelled over a brief period known as systole, lasting 0.25 seconds in this scenario.

Studying these factors helps in calculating critical metrics like blood volume and pressure, which are crucial for determining the heart's efficiency.

Cardiovascular System

The cardiovascular system is responsible for the circulation of blood that transports oxygen and nutrients to cells while removing waste products. The heart, arteries, veins, and capillaries make up this intricate system.

• The Heart: It acts as a pump, generating force to keep blood circulating through vessels. In our problem, the left ventricle's contraction (systole) causes blood to flow with a calculated velocity.
• Arteries: These vessels carry oxygenated blood from the heart to tissues. The aorta is the largest artery where the cross-sectional area was calculated with the aortic valve diameter in our exercise.
• Pressure and Volume Relationship: The heart's job is to maintain sufficient pressure to ensure blood reaches all organs efficiently, seen here through calculations determining work and power.

Understanding these components helps us appreciate how synergistically they function to maintain vital bodily functions.

Fluid Dynamics

Fluid dynamics concerns the flow of fluids, specifically focusing on how fluids like blood move through the body. It draws on principles from physics to explain circulation in the cardiovascular system.

• Continuity Equation: This principle states that the mass flow rate should remain constant across different parts of the vessel, reflecting in the calculation of blood ejected volume during systole.
• Equation of Motion: Newton's laws help us derive how blood's momentum aids its circulation, characterized here by calculations for potential energy and work done by the heart.
• Bernoulli's Principle: Uses in vascular flow analysis. As we assume no heat loss or energy change, the heart’s efficiency in shifting energy forms (pressure to kinetic) is evident here.

Fluid dynamics provides essential insights into understanding and optimizing systems for health and disease alike.

Biomechanics

Biomechanics looks at biological systems through mechanics, including forces and pressures in the cardiovascular system.

• Work: Performed by the heart, calculated using potential energy changes. In this exercise, the work done during systole is determined as it overcomes the gravitational pull to move blood upwards a slight height.
• Power: The rate of work, reflecting the heart's ability to sustain blood flow, calculated as 0.288 watts based on the work done over time.
• Material Properties of Blood: Fluid mechanics coalesce with biology, where understanding properties like blood density (1050 kg/m³ in our case) aids in calculating the forces at play.

Biomechanics integrates principles from physics and biology, providing insights into how physical laws operate within living organisms, crucial for fields like biomedical engineering and medical diagnostics.