Problem 71 - Physics

Question

A 1.3-m length of horizontal pipe has a radius of \(6.4 \times 10^{-3} \mathrm{~m}\). Water within the pipe flows with a volume flow rate of \(9.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}\) out of the right end of the pipe and into the air. What is the pressure in the flowing water at the left end of the pipe if the water behaves as (a) an ideal fluid and (b) a viscous fluid \(\left(\eta=1.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) ?

Step-by-Step Solution

Answer

Ideal fluid pressure: \(3.52 \times 10^6 \; Pa\). Viscous fluid pressure: \(111265 \; Pa\).

Step 1: Calculate the Area of the Pipe

The cross-sectional area of the pipe is needed for further calculations. The radius of the pipe is \( r = 6.4 \times 10^{-3} \; m \). The area \( A \) can be calculated using the formula for the area of a circle: \[ A = \pi r^2 \].Substitute the given radius into the formula:\[ A = \pi (6.4 \times 10^{-3})^2 \; m^2 \]Calculate the area:\[ A \approx 1.29 \times 10^{-4} \; m^2 \]

Step 2: Calculate the Flow Speed

To find the flow speed of the water, use the volume flow rate formula \( Q = Av \), where \( Q \) is the given volume flow rate \( 9.0 \times 10^{-3} \; m^3/s \), and \( v \) is the flow speed:\[ v = \frac{Q}{A} = \frac{9.0 \times 10^{-3}}{1.29 \times 10^{-4}} \; m/s \]Calculate the flow speed:\[ v \approx 69.77 \; m/s \]

Step 3a: Apply Bernoulli’s Equation for Ideal Fluid

When considering water as an ideal fluid, we use Bernoulli’s equation:\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]Since the pipe is horizontal, \( h_1 = h_2 \), and with an open outlet, \( P_2 = 0 \) is atmospheric pressure. The density of water \( \rho \) is approximately \( 1000 \; kg/m^3 \) and \( v_1 = v_2 \) at both ends.\[ P_1 = \frac{1}{2} \rho v^2 + P_0 \]Substitute the known values for \( \rho \), \( v \), and atmospheric pressure \( P_0 \approx 101325 \; Pa \):\[ P_1 = \frac{1}{2} (1000) (69.77)^2 + 101325 \; Pa \]Calculate \( P_1 \):\[ P_1 \approx 3.52 \times 10^6 \; Pa \]

Step 3b: Calculate Pressure using Viscous Fluid Approach (Hagen-Poiseuille Law)

For a viscous fluid, use the Hagen-Poiseuille law, which states:\[ \Delta P = \frac{8 \eta L Q}{\pi r^4} \]Where \( \eta = 1.00 \times 10^{-3} \; Pa \cdot s \), \( L = 1.3 \; m \), \( Q = 9.0 \times 10^{-3} \; m^3/s \), and \( r = 6.4 \times 10^{-3} \; m \). Compute the pressure difference:\[ \Delta P = \frac{8 \cdot 1.00 \times 10^{-3} \cdot 1.3 \cdot 9.0 \times 10^{-3}}{\pi (6.4 \times 10^{-3})^4} \]Calculate \( \Delta P \):\[ \Delta P \approx 9940 \; Pa \]The pressure at the left end considering this pressure drop is:\[ P_1 = P_2 + \Delta P = 101325 + 9940 \approx 111265 \; Pa \]

Key Concepts

Ideal Fluid

Simply put, an ideal fluid is a theoretical concept used in fluid dynamics. It signifies a fluid that is incompressible and has no viscosity. This means it can flow without any internal resistance. Understanding ideal fluids helps in solving problems more easily because it eliminates complexities like friction and energy loss that you would find in real-world fluids.

When dealing with an ideal fluid, like water flowing smoothly through a pipe, we use Bernoulli’s Equation to describe the flow behavior. Bernoulli’s equation is a preferred method because it considers the conservation of energy among an ideal fluid system. It combines kinetic energy, potential energy, and energy due to pressure to determine how they balance as the fluid flows.

• In pressure calculations, since an ideal fluid experiences no loss, the flow speed and height remain constant.
• The equation simplifies issues to involve only the fluid speed and pressure.

Understanding ideal fluids and their simple behavior is necessary for grasping more complex situations later.

Viscous Fluid

Now, in contrast to ideal fluids, viscous fluids have actual thickness and offer resistance to flow. Viscosity is essentially the friction within the fluid, and it affects how easily the fluid can move. It is an intrinsic property of real-world fluids like oil, honey, and even water at a certain scale.

This resistance impacts fluid dynamics significantly because it influences the rate of flow and the pressure required to maintain fluid motion. It is especially relevant when you’re dealing with narrow pipes or slow-moving streams.

• The concept of viscous fluid introduces the need to consider additional factors like shear stress and strain when analyzing fluid behavior.
• Understanding viscosity helps explain why different fluids behave differently under the same conditions.

Hence, considering viscosity becomes important in realistic applications and engineering solutions.

Bernoulli's Equation

Bernoulli's equation is an essential principle in fluid dynamics that describes the conservation of energy in flowing fluids. It states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or potential energy of the fluid.

It is particularly useful for analyzing fluid flow within pipes. For example, in the exercise, Bernoulli’s principle helps determine the pressure at the left end of the pipe when the fluid is considered ideal.

• Components of Bernoulli's equation include pressure energy, kinetic energy, and potential energy due to height.
• It works under the assumption that the flow is steady and the fluid is non-viscous and incompressible.

This fundamental equation provides insight into how fluids behave under different constraints and is extensively applied in aerodynamics, hydraulics, and various engineering systems.

Hagen-Poiseuille Law

The Hagen-Poiseuille law is crucial for understanding how viscous fluids flow through narrow constrictions or pipes. It provides a formula for calculating the pressure drop due to viscosity.

In essence, this law establishes a relationship between the pressure difference across the pipe, the fluid’s viscosity, the length of the pipe, the flow rate, and the radius of the pipe. It is mostly applicable in laminar flow conditions where the fluid flows in parallel layers without disruption.

• The law emphasizes how increasing the viscosity or length of the pipe increases resistance and thus pressure drop.
• It also illustrates how changes in the radius affect the flow, with smaller radii causing higher pressure drops, requiring more pressure to maintain the same flow rate.

The Hagen-Poiseuille equation offers precise predictions in engineering and medical fields, particularly in designing pipelines and understanding blood flow in capillaries.