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

step 1 This problem will calculate the radius of a needle and inside of this needle is a solution that

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 2.5 $\mathrm{cm}$ 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 Law

This principle describes the volumetric flow rate of a Newtonian fluid in a long, cylindrical pipe under laminar flow conditions. It relates the flow rate to the pressure difference across the tube, the tube’s length, the viscosity of the fluid, and especially the fourth power of the pipe’s radius. This sensitivity to the radius is critical in determining the correct dimensions for devices such as needles in medical applications.

Pressure Difference as the Driving Force

The flow of fluid through a tube is driven by the pressure difference between the two ends of the tube. In many practical problems, this difference can be created by factors such as elevation differences (gravitational head) or applied mechanical pressures. Recognizing and quantifying this pressure difference is essential for calculating the necessary dimensions to achieve a desired flow rate.

Viscosity of the Fluid

Viscosity is a measure of a fluid’s resistance to deformation and flow. In fluid dynamics, especially in the study of laminar flow, viscosity plays a central role in determining the ease with which a fluid can move through a confined space. Accurate knowledge of a fluid's viscosity is necessary to apply formulas like the Hagen-Poiseuille equation correctly.

Laminar Flow

Laminar flow refers to a smooth, orderly flow regime where layers of fluid move parallel to each other without mixing turbulently. This condition is assumed in many analytical applications, including the Hagen-Poiseuille calculation, because it simplifies the relationship between the physical parameters of the system and the flow rate. It is especially relevant in small diameter tubes or needles where flow tends to remain laminar.

Volumetric Flow Rate

The volumetric flow rate is defined as the volume of fluid passing through a cross-section of a tube per unit time. It is an essential concept in both engineering and biomedical applications, as many processes require the delivery or removal of fluids at a specified rate. Understanding this concept helps in designing systems that achieve precise fluid delivery under controlled conditions.

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