What diameter needle should be used to inject a volume of 500 cm^3 of a solution into a patient

step 1 This problem covers the concept of the Poison's Law and first we need to find the change in pres

What diameter needle should be used to inject a volume of...

What diameter needle should be used to inject a volume of $500 \mathrm{~cm}^{3}$ of a solution into a patient in $30 \mathrm{~min} ?$ Assume the length of the needle is $2.5 \mathrm{~cm}$ and the solution is elevated $1.0 \mathrm{~m}$ above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.

Key Concepts

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation describes the flow of a viscous fluid through a cylindrical pipe under laminar flow conditions. It connects the volumetric flow rate with the pressure difference across the tube, the fluid’s viscosity, the tube's length, and its radius, making it essential for determining parameters such as the needle diameter in medical injections.

Laminar Flow

Laminar flow refers to the smooth, orderly motion of fluid layers, which is typical in small-diameter tubes or needles. Under laminar conditions, the velocity profile is parabolic, and the flow behavior can be accurately modeled by approaches like the Hagen-Poiseuille equation.

Hydrostatic Pressure

Hydrostatic pressure arises from the weight of a fluid at rest, which creates a pressure difference due to gravity. In this context, the elevation of the fluid above the point of injection generates a pressure head that drives the flow through the needle.

Viscosity

Viscosity is a measure of a fluid’s resistance to deformation or flow. It plays a critical role in determining the rate of flow through narrow passages. For problems involving injection through a needle, using the known viscosity of the fluid (such as water) is essential to calculate the required needle diameter for achieving a specified flow rate.

Volumetric Flow Rate

The volumetric flow rate is the volume of fluid passing through a cross-section of the tube per unit time. In applications like medical injections, knowing the desired flow rate is necessary to link the physical dimensions of the needle with the driving forces and the fluid’s properties.

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