Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 67

Chapter 14, Problem 67

20°C water flows at 1.5 m/s through a 10-m-long, 1.0-mm-diameter horizontal tube and then exits into the air. What is the gauge pressure in kPa at the point where the water enters the tube?

Verified step by step guidance

Determine the type of flow (laminar or turbulent) by calculating the Reynolds number using the formula:

R e = \frac{ρ}{v}

, where

ρ

is the density of water,

v

is the velocity,

D

is the diameter of the tube, and

μ

is the dynamic viscosity of water at 20°C.

If the flow is laminar (Re < 2000), calculate the pressure drop using the Hagen-Poiseuille equation:

Δ P = \frac{8}{μ} π r^4

, where

L

is the length of the tube,

Q

is the volumetric flow rate, and

r

is the radius of the tube.

To find the volumetric flow rate

Q

, use the formula:

Q = v A

, where

A

is the cross-sectional area of the tube, calculated as

A = π r^2

Add the pressure drop to the atmospheric pressure to find the gauge pressure at the entrance of the tube. Gauge pressure is defined as the pressure relative to atmospheric pressure, so:

P

_gauge = P_entrance - P_atm.

Ensure all units are consistent (e.g., convert diameter to meters, pressure to Pascals, etc.) and express the final gauge pressure in kilopascals (kPa) by dividing the result in Pascals by 1000.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how pressure changes in a fluid as it moves through a tube, especially when considering factors like velocity and elevation.

Continuity Equation

The Continuity Equation is based on the conservation of mass, stating that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a tube to another. This means that if the diameter of the tube changes, the velocity of the fluid must adjust accordingly, which is essential for calculating flow rates and pressures in different sections of the tube.

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 context, it is important to differentiate gauge pressure from absolute pressure, as the question specifically asks for the gauge pressure at the entry point of the tube, which will help determine the fluid dynamics involved.

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