What radius needle should be used to inject a volume of 500...

What radius needle should be used to inject a volume of $500 \mathrm{~cm}^{3}$ of a solution into a patient in $30.0 \mathrm{~min}$ ? Assume the length of the needle is $2.50 \mathrm{~cm}$ and the solution is elevated $1.00 \mathrm{~m}$ 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

Flow Rate

Flow rate measures the volume of fluid that moves through a cross-section of a tube per unit time. It is a central concept in fluid dynamics and is directly connected to the dimensions of the needle, the pressure differential, and the viscosity of the fluid, as described by the Hagen–Poiseuille law.

Pressure Differential

The pressure differential, or pressure drop, is the driving force for fluid flow through a tube. In the context of this problem, it can include contributions from gravitational head due to elevation differences as well as any other pressure differences between the fluid reservoir and the exit. Correctly determining this pressure difference is essential for calculating the necessary dimensions of the needle.

Viscosity

Viscosity is a measure of a fluid's internal resistance to deformation or flow. It is a critical parameter in fluid dynamics, as higher viscosity means a greater resistance to flow under a given pressure difference. The Hagen–Poiseuille equation incorporates viscosity to predict how easily a fluid can pass through a tube.

Hagen–Poiseuille Law

This law quantitatively describes the laminar flow of an incompressible, Newtonian fluid through a cylindrical tube. It shows that the volumetric flow rate is proportional to the pressure drop and the fourth power of the tube's radius, while it is inversely proportional to the fluid's viscosity and the tube's length. This relationship is key to calculating the needle radius required for a given flow rate.

Laminar Flow

Laminar flow refers to fluid motion in which layers move smoothly and parallel to one another with little to no mixing. This regime, typical for flows in small tubes like needles, allows for the accurate application of the Hagen–Poiseuille law, as it assumes a steady, non-turbulent, and uniform flow profile.

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