Problem 23 - College Physics

Question

A collapsible plastic bag (Fig. P9.23) contains a glucose solution. If the average gauge pressure in the vein is \(1.33 \times 10^{3} \mathrm{~Pa}\), what must be the minimum height \(h\) of the bag in order to infuse glucose into the vein? Assume the specific gravity of the solution is \(1.02\).

Step-by-Step Solution

Answer

The minimum height \( h \) of the glucose bag must be approximately \(0.13 m\) or \(13 cm\) to infuse glucose into the vein.

Step 1 - Compute Pressure of Glucose Solution

To start with this problem, we have to first compute the pressure exerted by the glucose solution in the bag at depth \(h\). According to Pascal’s principle, the pressure at a point in a fluid is determined by the weight of the fluid above it. In mathematical terms, it's given by \[ P = \rho gh \] where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity and \( h \) is the height or depth of the fluid column above the point. The density \(\rho\) can be calculated as the specific gravity times the density of water, which is \( 1.02 \times 10^{3} \, \mathrm{kg/m^3} \)

Step 2 - Solve Pressure Equation

Given the pressure of the vein \(1.33 \times 10^{3} Pa\), we equalize this to the weight of the glucose solution to find the minimum \( h \). So we solve the following equation for h: \( h= \frac{P_{\mathrm{vein}}}{\rho g} \). We substitute the known values: \(1.33 \times 10^{3} \, \mathrm{Pa}\) for \(P_{\mathrm{vein}}\), \(1.02 \times 10^{3} \, kg/m^3\) for \(rho\) and \(9.81 \, m/s^2\) for \(g\).

Step 3 - Compute Minimum Height

After solving the equation, we find the minimum height required to infuse glucose into the vein. This step is mostly arithmetical and requires careful calculation to get an accurate result.

Key Concepts

Pressure Calculation

Calculating pressure in fluid mechanics is essential, especially when dealing with a fluid column. Pressure (P) in a fluid is the force exerted by the fluid per unit area on the walls of its container. In general, it is defined as P = \frac{F}{A}, where F is the force applied and A is the area over which the force is distributed.
For fluids specifically, we use the formula P = \rho gh to calculate pressure due to a fluid column, where:

• \( \rho \) is the density of the fluid
• \( g \) is the acceleration due to gravity
• \( h \) is the height of the fluid column

The weight of the fluid column is responsible for exerting pressure beneath it. Understanding this relationship is crucial for determining how fluids will behave under different conditions.

Specific Gravity

Specific gravity is a dimensionless number that is used to compare the density of a substance to the density of a reference substance, generally water at 4 °C. The formula to find specific gravity is:\[\text{Specific Gravity} = \frac{\text{Density of substance}}{\text{Density of reference}}\]In the context of fluid mechanics, specific gravity is handy because it lets us understand how a liquid is "heavier" or "lighter" than water. For example, a specific gravity of 1.02 signifies that the solution is 1.02 times heavier than water.
To find the actual density of a glucose solution with a specific gravity of 1.02, multiply by the density of water (1000 kg/m³). This gives you a solution density of 1020 kg/m³. Specific gravity helps in calculating other properties of fluids, such as pressure, necessary for understanding fluid dynamics in various systems.

Pascal’s Principle

Pascal's Principle, also known as the principle of transmission of fluid-pressure, states that changes in pressure applied to a contained fluid are transmitted undiminished to every point of the fluid and to the walls of its container. Essentially, this means when you apply pressure at one point in a fluid, that pressure is maintained throughout the fluid.
This principle is vital in explaining how fluid systems work, for instance in hydraulic lifts or in our case, intravenous infusion. If the bag containing the glucose solution is elevated to a sufficient height, the pressure exerted by the fluid overcomes the pressure in the vein allowing infusion. In simple terms, the pressure differential due to gravity ensures the flow from higher to lower pressure areas, ensuring nutrients reach the bloodstream effectively.

Fluid Column Height

The concept of fluid column height is crucial in understanding how fluids create pressure in contained areas. For the infusion of glucose into a vein, the height of the glucose solution bag actually changes the pressure exerted by the fluid.
In the exercise, we calculate the minimum height using the relationship \[ h = \frac{P_{\text{vein}}}{\rho g} \]where \( P_{\text{vein}} \)is the pressure of the vein, \( \rho \)is the density of the fluid, and \( g \)is the gravitational acceleration (9.81 m/s²). The height \( h \)is critical to ensure the fluid overcomes the venous pressure and enters the bloodstream. Elevating the bag to this height ensures gravitational potential energy is converted to pressure, facilitating flow due to not only the density of the fluid but also Pascal’s principle that makes sure pressure is sustained throughout the fluid.