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A blood transfusion is being set up in an emergency room for an accident victim. Blood has a density of \(1060 \mathrm{kg} / \mathrm{m}^{3}\) and a viscosity of \(4.0 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s} .\) The needle being used has a length of \(3.0 \mathrm{cm}\) and an inner radius of \(0.25 \mathrm{mm} .\) The doctor wishes to use a volume flow rate through the needle of \(4.5 \times 10^{-8} \mathrm{m}^{3} / \mathrm{s} .\) What is the distance \(h\) above the victim's arm where the level of the blood in the transfusion bottle should be located? As an approximation, assume that the level of the blood in the transfusion bottle and the point where the needle enters the vein in the arm have the same pressure of one atmosphere. (In reality, the pressure in the vein is slightly above atmospheric pressure.)

Step by step solution

01

Identify Required Formula

To determine the height, we use the Hagen-Poiseuille equation, which relates volume flow rate \( Q \), viscosity \( \eta \), radius \( r \), length \( L \), and pressure difference \( \Delta P \): \[ Q = \frac{{\pi r^4 \Delta P}}{{8 \eta L}} \] In this case, \( \Delta P = \rho gh \), where \( \rho \) is the density of blood and \( h \) is the height, as the pressure at both ends is equal to atmospheric pressure.

02

Rearrange for Height

Rearrange the formula to solve for height \( h \): \[ \Delta P = 8 \eta L \frac{Q}{\pi r^4} \] Substitute \( \Delta P = \rho gh \) to find: \[ h = \frac{8 \eta L Q}{\pi r^4 \rho g} \]

03

Substitute Known Values

Insert the given values into the equation:- viscosity \( \eta = 4.0 \times 10^{-3} \, \text{Pa} \cdot \text{s} \)- length \( L = 0.03 \, \text{m} \)- volume flow rate \( Q = 4.5 \times 10^{-8} \, \text{m}^3/\text{s} \)- radius \( r = 0.00025 \, \text{m} \) (converted to meters)- density \( \rho = 1060 \, \text{kg/m}^3 \)- gravitational acceleration \( g = 9.81 \, \text{m/s}^2 \)The formula becomes:\[ h = \frac{8 \times 4.0 \times 10^{-3} \times 0.03 \times 4.5 \times 10^{-8}}{\pi \times (0.00025)^4 \times 1060 \times 9.81} \]

04

Calculate the Result

Compute the above expression: First calculate \( r^4 \), then the entire denominator and numerator, and finally, the height \( h \):\[ r^4 = (0.00025)^4 = 3.90625 \times 10^{-14} \, \text{m}^4 \]Compute the numerator:\[ 8 \times 4.0 \times 10^{-3} \times 0.03 \times 4.5 \times 10^{-8} = 4.32 \times 10^{-11} \, \text{Pa} \cdot \text{m}^3 \]Compute the denominator:\[ \pi \times 3.90625 \times 10^{-14} \times 1060 \times 9.81 = 1.27827 \times 10^{-10} \]Finally, calculate \( h \):\[ h = \frac{4.32 \times 10^{-11}}{1.27827 \times 10^{-10}} = 0.3379 \, \text{m} \approx 0.34 \, \text{m} \]

05

Conclusion

The transfusion bottle should be placed approximately 0.34 meters above the victim's arm to achieve the desired flow rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hagen-Poiseuille equation

Fluid dynamics in biological systems often involve the motion of liquids through narrow tubes or vessels. Here, the Hagen-Poiseuille equation is vital. This mathematical expression helps us describe how fluids flow through a pipe. The equation states that the volume flow rate (\(Q\)) through a cylindrical pipe is proportionally affected by the fourth power of the radius (\(r^4\)), the pressure difference across the pipe (\(\Delta P\)), and the pipe's length (\(L\)), and is inversely affected by the viscosity of the fluid (\(\eta\)). The formula is given by:\[Q = \frac{{\pi r^4 \Delta P}}{{8 \eta L}}.\]This equation is incredibly useful to analyze systems like blood flow in the circulatory system, where the vessels behave similarly to cylindrical pipes.

Viscosity

Viscosity is a physical property of fluids that describes their resistance to flow. People's everyday perception of viscosity can be related to how thick or sticky a fluid feels. For instance, honey has a much higher viscosity compared to water.
In biological contexts, such as blood flow, viscosity plays a critical role in determining how easily or how hard the blood moves through veins and arteries.
The higher the viscosity, the more resistance the fluid encounters, leading to slower movement. Blood viscosity is measurable and often presented in units known as Pascal-seconds (Pa·s). In the case of blood transfusions, understanding the viscosity is essential to ensure the fluid moves at a desired and regulated rate, ensuring patient safety and treatment efficacy.

Volume Flow Rate

The volume flow rate is a measurement of the amount of fluid passing through a specific section of pipe or vessel in a unit of time. It is usually denoted as \(Q\), with units of cubic meters per second (m³/s).
Volume flow rate plays a crucial role in medical applications, such as administering blood transfusions. Here, knowing the volume flow rate helps ensure that fluids are delivered to the patient at the correct speed, preventing complications.
In this exercise, with the given volume flow rate of \(4.5 \times 10^{-8} \, \text{m}^3/\text{s},\) medical professionals can adjust the height of the transfusion bottle to control this flow, employing the Hagen-Poiseuille equation as a guide.

Density of Fluids

Density is another essential property of fluids, indicating mass per unit volume, typically in kilograms per cubic meter (kg/m³). Understanding the density of a fluid is important as it affects how the fluid behaves under various forces, such as gravity.
In biological scenarios, blood has a particular density that affects how it circulates throughout the body. For example, knowing that the density of blood is \(1060 \, \text{kg/m}^3\) helps in calculations involving pressure differences, as used in the Heights-Poiseuille equation.
Accurate density values are important in calculating the necessary height for fluid delivery systems, like an IV drip, to ensure the fluid transfers at the correct pressure and speed to the patient.