A pressure difference of 6.00 * 104 Pa is required to maintain a volume flow rate of 0.800m3/s for a viscous fluid flowing through a section of cylindrical pipe that has radius 0.210 m. What pressure difference is required to maintain the same volume flow rate if the radius of the pipe is decreased to 0.0700 m?

Verified step by step guidance

1. Identify the relevant equation for viscous flow in pipes, which is the Hagen-Poiseuille equation: \(\Delta P = \frac{8 \mu L Q}{\pi r^4}\), where \(\Delta P\) is the pressure difference, \(\mu\) is the dynamic viscosity of the fluid, \(L\) is the length of the pipe, \(Q\) is the volume flow rate, and \(r\) is the radius of the pipe.
2. Note that the dynamic viscosity \(\mu\) and the length of the pipe \(L\) remain constant as they are not mentioned to change in the problem statement.
3. Substitute the initial conditions into the Hagen-Poiseuille equation to find the proportionality constant using the initial radius and pressure difference. This step involves rearranging the equation to solve for \(\frac{8 \mu L}{\pi}\).
4. Using the constant found in the previous step, substitute the new radius of the pipe into the equation to find the new pressure difference required to maintain the same flow rate. This involves plugging the new radius into the rearranged Hagen-Poiseuille equation.
5. Simplify and solve for the new pressure difference \(\Delta P\) required for the decreased radius, ensuring all units are consistent and calculations are checked for accuracy.

Key Concepts

Poiseuille's Law

Poiseuille's Law describes the flow of a viscous fluid through a cylindrical pipe. It states that the volume flow rate (Q) is directly proportional to the fourth power of the radius (r) of the pipe and the pressure difference (ΔP) across the length of the pipe, while being inversely proportional to the viscosity (η) of the fluid and the length (L) of the pipe. This relationship is crucial for understanding how changes in pipe radius affect flow rates and pressure differences.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. It quantifies how thick or sticky a fluid is, influencing how easily it flows through pipes. In the context of fluid dynamics, higher viscosity results in greater resistance to flow, which affects the pressure required to maintain a specific flow rate. Understanding viscosity is essential for analyzing how different fluids behave under varying conditions.

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics that states that the mass flow rate must remain constant from one cross-section of a pipe to another, assuming incompressible flow. This means that if the radius of the pipe decreases, the velocity of the fluid must increase to maintain the same flow rate. This concept is vital for understanding how changes in pipe dimensions impact flow characteristics and pressure requirements.