BIO In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 kg/m^3) located at height h above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 Pa, what is the minimum value of h that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity of the liquid.

Verified step by step guidance

1. Identify the relevant pressures involved in the problem. The gauge pressure in the vein is given as 5980 Pa, and the atmospheric pressure acts on the top of the fluid in the reservoir.
2. Understand that the fluid will flow into the vein if the hydrostatic pressure at the needle, due to the fluid column of height h, exceeds the gauge pressure in the vein.
3. Use the hydrostatic pressure formula to calculate the pressure at the bottom of the fluid column: \( P = \rho g h \), where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately 9.81 m/s^2), and \( h \) is the height of the fluid column.
4. Set up the equation to find the minimum height h by equating the hydrostatic pressure to the gauge pressure in the vein. The equation will be \( \rho g h = 5980 \, \text{Pa} \).
5. Solve for h by rearranging the equation: \( h = \frac{5980}{\rho g} \). Substitute the values of \( \rho \) and \( g \) to find the minimum height.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is given by the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this context, it helps determine how high the fluid reservoir must be to overcome the pressure in the vein.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure, meaning it measures the pressure of a fluid above the ambient atmospheric pressure. In this scenario, the gauge pressure of 5980 Pa indicates the pressure inside the vein that must be overcome by the hydrostatic pressure from the fluid reservoir to allow fluid to flow into the vein.

Fluid Dynamics

Fluid dynamics is the study of the behavior of fluids in motion and the forces acting on them. In this problem, it is essential to understand how the pressure difference between the fluid in the reservoir and the pressure in the vein influences the flow of fluid. The principles of fluid dynamics help explain how the height of the fluid reservoir affects the ability of the fluid to enter the vein.