The Calculations For The Total Work Done By The Heart

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The diagram shown below in Figure 1 describes the circulatory system of the heart, where the vena cavae brings $O_{2}$-depleated blood to the right atrium, and then the blood passes through a valve and flows to the right ventricle, then through the pulmonary arteries to the lungs, where $O_{2}$-depleated blood becomes $O_{2}$ rich after the exchange of $CO_{2}$ and $O_{2}$. Following this the pulmonary vein delivers the $O_{2}$-rich blood to the left atrium, and then it flows into the left ventricle. Heart muscles contract and pump blood out through the aortic valve into the aorta. Blood flowing through the aorta is circulated through the arteriovenous system of the body. Figure 1 Schematic description of circulatory system. By considering the right and left sides of the heart as systems 1 and 2 , respectively, calculate the total work done by the heart. Hint; use the first law of thermodynamics. All assumptions, sources of data used e.g. density of blood etc. must be provided. The output of your assignment should include an excel spreadsheet of all calculations and a graph of total work done by the heart vs beats per minute. \begin{tabular}{|l|c|l|c|} \hline System 1 & & System 2 \\ \hline [Right side of the heart] & Numbers & [Left side of the heart] & Numbers \\ \hline Density of blood $=$ & $0.000994kg/ml$ & Density of blood $=$ & $0.000994kg/ml$ \\ \hline Pressure & $0.483419psi$ & Pressure & $0.193368psi$ \\ \hline Force & $0.005kg/mm2$ & Force & $0.007kg/mm2$ \\ \hline volume & $5L$ & volume & $5L$ \\ \hline Heat of the heart & $38.8_{∘}C$ & Heat of the heart & $38.8_{∘}C$ \\ \hline Heat of the blood & $37_{∘}C$ & Heat of the blood & $37_{∘}C$ \\ \hline \end{tabular}The Calculations For The Total Work Done By The Heart

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Step 1/5

The calculations for the total work done by the heart:

To calculate the total work done by the heart, we can use the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system:

ΔU = Q – W

where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.

Explanation:

Step 2/5

We can assume that the heart is a closed system, so there is no heat transfer between the heart and its surroundings. Therefore, the total work done by the heart is equal to the change in internal energy.

To calculate the change in internal energy, we can use the equation:

ΔU = ΔH – ΔPV

where ΔH is the change in enthalpy, ΔP is the change in pressure, and ΔV is the change in volume.

Explanation:

Step 3/5

For system 1 (right side of the heart):

ΔH = (5 L) x (0.000994 kg/ml) x (4.18 kJ/kg°C) x (38.8°C – 37°C) = 0.087 kJ

ΔP = 0.483419 psi – 0.193368 psi = 0.290051 psi = 0.002 kPa

ΔV = 5 L = 0.005 m^3

ΔU = ΔH – ΔPV = 0.087 kJ – (0.002 kPa x 0.005 m^3) = 0.077 kJ

Explanation: The Calculations For The Total Work Done By The Heart

In system 1 (right side of the heart), the change in enthalpy (ΔH) is calculated by multiplying the mass of blood by the specific heat capacity of blood and the difference in temperature between the blood and the heart. The change in pressure (ΔP) is obtained by subtracting the pressure in the right atrium from the pressure in the right ventricle. The change in volume (ΔV) is given as 5 L, which is the volume of blood flowing through the system.

Using the equation ΔU = ΔH – ΔPV, the change in internal energy (ΔU) for system 1 is calculated as 0.077 kJ. This represents the total work done by the right side of the heart.

Step 4/5

For system 2 (left side of the heart):

ΔH = (5 L) x (0.000994 kg/ml) x (4.18 kJ/kg°C) x (38.8°C – 37°C) = 0.087 kJ

ΔP = 0.193368 psi – 0 psi = 0.193368 psi = 0.001 kPa

ΔV = 5 L = 0.005 m^3

ΔU = ΔH – ΔPV = 0.087 kJ – (0.001 kPa x 0.005 m^3) = 0.082 kJ

Explanation:

In system 2 (left side of the heart), the calculations are similar to those for system 1. The change in enthalpy (ΔH) is calculated in the same way, and the change in pressure (ΔP) is obtained by subtracting the pressure in the left atrium from the pressure in the left ventricle. The change in volume (ΔV) is again given as 5 L.

Using the same equation ΔU = ΔH – ΔPV, the change in internal energy (ΔU) for system 2 is calculated as 0.082 kJ. This represents the total work done by the left side of the heart.

Step 5/5

The equation to plot the graph of total work done by the heart vs beats per minute:

The total work done by the heart is a function of the heart rate, which is measured in beats per minute (bpm). Therefore, we can plot the total work done by the heart as a function of heart rate.

Assuming a linear relationship between total work done and heart rate, the equation for the graph would be:

Total work done = m x Heart rate + b

where m is the slope of the line and b is the y-intercept. To find the values of m and b, we would need to perform a linear regression analysis using data obtained from experiments or simulations.

Alternatively, we could plot the total work done by the heart for a range of heart rates, assuming that the other parameters (such as volume and pressure) remain constant. The resulting graph would show how the total work done by the heart varies with heart rate, and could be used to identify the optimal heart rate for minimizing the work done by the heart while still maintaining sufficient blood flow to the body.

Final answer

Finally, to obtain the total work done by the heart, the change in internal energy for system 1 and system 2 are added together, resulting in a total work done of 0.159 kJ.

Therefore, the total work done by the heart is:

W = ΔU system 1 + ΔU system 2 = 0.077 kJ + 0.082 kJ = 0.159 kJ

The Calculations For The Total Work Done By The Heart