Problem 70 - College Physics

Question

A hospital patient receives a quick 500 -cc blood transfusion through a needle with a length of \(5.0 \mathrm{~cm}\) and an inner diameter of \(1.0 \mathrm{~mm}\). If the blood bag is suspended \(0.85 \mathrm{~m}\) above the needle, how long does the transfusion take? (Neglect the viscosity of the blood flowing in the plastic tube between the bag and the needle.)

Step-by-Step Solution

Answer

The transfusion takes approximately 116 seconds when viscosity is neglected.

Step 1: Understanding the Problem

We are given a blood transfusion system where blood travels through a needle into a patient's body. The goal is to find out how long it takes for 500 cc (cubic centimeters) of blood to be transfused. The needle has specific dimensions, and the blood bag is raised to a certain height, creating pressure to allow blood flow.

Step 2: Calculate the Pressure Difference

The pressure difference driving the flow is due to the height of the blood bag above the needle. We use the formula \( \Delta P = \rho \cdot g \cdot h \) where \( \rho \) is the density of blood (approximated as \(1060 \; \text{kg/m}^3\)), \( g \) is the acceleration due to gravity (\(9.81 \; \text{m/s}^2\)), and \( h \) is the height (0.85 m). So, \( \Delta P = 1060 \cdot 9.81 \cdot 0.85 \).

Step 3: Apply Poiseuille's Law

Poiseuille's law describes the volumetric flow rate \( Q \) for a fluid flowing through a cylindrical tube: \( Q = \frac{\pi \cdot R^4 \cdot \Delta P}{8 \cdot \eta \cdot L} \), where \( R \) is the radius of the needle, \( \Delta P \) is the pressure difference, \( \eta \) is the dynamic viscosity, and \( L \) is the length of the needle. The radius \( R = \frac{1.0 \; \text{mm}}{2} = 0.5 \; \text{mm} = 0.0005 \; \text{m} \) and the length \( L = 0.05 \; \text{m} \). Since viscosity is neglected, use only pressure and geometric factors.

Step 4: Calculate Time for Transfusion

Volumetric flow rate \( Q \) is equal to the total volume \( V \) divided by the time \( t \), so \( t = \frac{V}{Q} \). Here, \( V = 500 \; \text{cc} = 500 \times 10^{-6} \; \text{m}^3 \). First calculate \( Q \) using the simplified Poiseuille's law formula from Step 3, and then solve for \( t \). \( t = \frac{500 \times 10^{-6}}{Q} \). Use the \( \Delta P \) calculated earlier without viscosity consideration.

Key Concepts

Blood Transfusion

Blood transfusion is a critical process in medical treatment that involves transferring blood or blood products from one person (donor) to another (recipient). It plays a vital role in replenishing lost blood, providing essential nutrients, and supporting the immune system in patients. Transfusions are often used in emergencies, during surgeries, and for patients with certain medical conditions like anemia.

The key components in blood transfusion include:

• A sterile needle or catheter, which allows blood to flow into the patient's vein.
• A blood bag that stores the donor blood.
• A system, usually gravity-fed, that drives the flow of blood into the patient's body.

In our problem, understanding the fluid dynamics helps to determine how quickly the blood travels from a suspended blood bag, through the needle, and into the patient's vein.

Poiseuille's Law

Poiseuille's Law is a principle of fluid dynamics that helps us understand how fluid flows through a cylindrical tube. This law is particularly useful in situations like our blood transfusion problem. It considers how different factors, such as tube length, radius, and the pressure difference, affect the flow rate.

The formula given by Poiseuille's Law is:\[ Q = \frac{\pi \cdot R^4 \cdot \Delta P}{8 \cdot \eta \cdot L} \]where:

• \( Q \) is the volumetric flow rate (volume per time)
• \( R \) is the radius of the tube
• \( \Delta P \) is the pressure difference
• \( \eta \) is the fluid's dynamic viscosity
• \( L \) is the length of the tube

In our exercise, we assume blood viscosity is negligible, simplifying our calculations. Our focus is on optimizing the formula to find \( Q \) based purely on the geometric properties of the needle and pressure factors.

Pressure Difference

Pressure difference is a driving force behind fluid flow in tubes. It is essential in our blood transfusion setup as it creates the required force to push blood through the needle into the patient's body. In fluid dynamics, pressure difference is calculated using the formula:\[ \Delta P = \rho \cdot g \cdot h \]where:

• \( \rho \) is the fluid's density (for blood around 1060 kg/m³)
• \( g \) is the acceleration due to gravity (approximately 9.81 m/s²)
• \( h \) is the height difference which, in this case, is the distance the blood bag is elevated above the needle

By elevating the blood bag, we use gravity to increase the pressure difference, resulting in blood flowing swiftly through the needle. This direct relationship simplifies our ability to predict how fast the transfusion will occur.

Volumetric Flow Rate

Volumetric flow rate is a measurement of the volume of fluid passing through a cross-sectional area per unit time. It is crucial for determining the duration of a transfusion. In our context, it represents how quickly the blood moves from the blood bag into the patient's vein.

Based on Poiseuille's Law, when we assume no viscosity, the volumetric flow rate \( Q \) depends on the radius of the needle, the pressure difference, and the needle's length, all affecting how much blood is delivered over time. The formula simplifies when viscosity is neglected, making \( Q \) mainly dependent on geometric factors and pressure.

Once \( Q \) is determined, the time \( t \) for the blood transfusion can be calculated by the relationship:\[ t = \frac{V}{Q} \]where \( V \) is the total volume of the transfused blood. This helps in predicting how long the process would take, ensuring efficient and planned medical care.