91Ó°ÊÓ

Reynolds number: The Reynolds number is very important in such fields as fluid flow and aerodynamics. In the case of a fluid flowing through a pipe, the Reynolds number \(R\) is given by $$ R=\frac{v d D}{\mu} $$ Here \(v\) is the velocity of the fluid in meters per second, \(d\) is the diameter of the pipe in meters, \(D\) is the density of the fluid in kilograms per cubic meter, and \(\mu\) is the viscosity of the fluid measured in newton- seconds per square meter. Generally, when the Reynolds number is above 2000 , the flow becomes turbulent, and rapid mixing occurs. \({ }^{10}\) When the Reynolds number is less than 2000 , the flow is streamline. Consider a fluid flowing through a pipe of diameter \(0.05\) meter at a velocity of \(0.2\) meter per second. a. If the fluid in the pipe is toluene, its viscosity is \(0.00059\) newton- seconds per square meter, and its density is 867 kilograms per cubic meter. Is the flow turbulent or streamline? b. If the toluene is replaced by glycerol, then the viscosity is \(1.49\) newton-seconds per square meter, and the density is \(1216.3\) kilograms per cubic meter. Is the glycerol flow turbulent or streamline?

Step by step solution

01

Understand the Formula

The formula for the Reynolds number is \( R = \frac{v d D}{\mu} \), where \( v \) is the fluid velocity in m/s, \( d \) is the pipe diameter in m, \( D \) is the fluid density in kg/m³, and \( \mu \) is the fluid's viscosity in Ns/m².

02

Gather Data for Toluene Flow

For toluene, we have the diameter \( d = 0.05 \) m, velocity \( v = 0.2 \) m/s, viscosity \( \mu = 0.00059 \) Ns/m², and density \( D = 867 \) kg/m³.

03

Calculate Reynolds Number for Toluene

Substitute the toluene values into the formula: \( R = \frac{(0.2)(0.05)(867)}{0.00059} \). Calculate the result.

04

Determine Flow Type for Toluene

Calculate \( R = \frac{(0.2)(0.05)(867)}{0.00059} \approx 14711.9 \). Since \( R > 2000 \), the flow is turbulent.

05

Gather Data for Glycerol Flow

For glycerol, the values are diameter \( d = 0.05 \) m, velocity \( v = 0.2 \) m/s, viscosity \( \mu = 1.49 \) Ns/m², and density \( D = 1216.3 \) kg/m³.

06

Calculate Reynolds Number for Glycerol

Substitute the glycerol values into the formula: \( R = \frac{(0.2)(0.05)(1216.3)}{1.49} \). Calculate the result.

07

Determine Flow Type for Glycerol

Calculate \( R = \frac{(0.2)(0.05)(1216.3)}{1.49} \approx 8.17 \). Since \( R < 2000 \), the flow is streamline.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Dynamics

Fluid dynamics is a branch of physics concerned with the movement of liquids and gases. It helps us understand how fluids behave while flowing through various environments. This field has significant applications in engineering, meteorology, oceanography, and even medicine.
In fluid dynamics, we examine factors such as velocity, pressure, and flow rate to predict and control the behavior of fluids. By applying these concepts, engineers can design efficient systems like pipelines, water treatment facilities, and even aircraft. A key tool in fluid dynamics is the Reynolds number, which helps us determine whether a fluid flow will be turbulent or streamline.

Turbulent Flow

Turbulent flow refers to a chaotic and irregular motion of fluid particles. This type of flow is typically characterized by swirls and eddies that enhance mixing. Turbulent flow occurs when the Reynolds number exceeds 2000.

• This chaotic movement results in greater resistance and energy loss due to friction between the fluid and the pipe walls.
• In turbulent flow, momentum and properties like temperature or solute concentration rapidly mix across the flow cross-section.

Understanding this concept is vital for the design of processes such as chemical reactors and heat exchangers. Turbulence can be desirable when thorough mixing is required, but avoided in cases where friction loss needs to be minimized.

Streamline Flow

Streamline flow, also known as laminar flow, occurs when fluid moves in parallel layers with no disruption between them. This type of flow is observed when the Reynolds number is below 2000. The flow paths, known as streamlines, run parallel, leading to smooth and orderly movement.

• The fluid particles move in a consistent direction, preventing mixing between the layers.
• Streamline flow is energy efficient as it minimizes friction losses due to its orderly nature.

Laminar flow is preferred in applications like lubricant flow in machinery where reduced energy loss and wear are crucial. Conversely, its minimal mixing may require additional considerations in processes that depend on thorough mixing.

Viscosity

Viscosity is a measure of a fluid's resistance to flow. It represents the internal friction arising from the movement of fluid molecules against each other. High viscosity fluids, like honey, flow slower because the particles are more resistant to motion. Conversely, low viscosity fluids, like water, flow more easily.

• Viscosity is affected by temperature, generally decreasing as temperature increases, enabling better flow.
• It plays a significant role in determining the Reynolds number, aiding in identifying whether a flow is turbulent or streamline.

Understanding viscosity is important in applications that require specific flow rates, such as in lubrication systems and fluid transport pipelines.

Fluid Density

Fluid density is the mass of fluid per unit volume. It is a vital factor in fluid dynamics as it influences fluid characteristics such as buoyancy and flow behavior. In the Reynolds number equation, density combined with velocity and diameter helps compute the value that classifies flow patterns.

• Density is often expressed in kilograms per cubic meter (kg/m³).
• The density of a fluid can change with pressure and temperature, influencing the flow dynamics.

In practical scenarios, understanding fluid density aids in applications like aircraft design, where it impacts lift and drag forces. In pipelines, it determines the overall mass flow rate, crucial for efficient transport of substances.