A thin rectangular plate having a width w and a height h is located so that it is normal to a moving stream of fluid. Assume the drag, Fp, that the fluid exerts on the plate is a function of w and h, the fluid viscosity and density, µ and p respectively and the velocity uo of the fluid approaching the plate. Determine the dimensionless drag and the dimensionless numbers that influence its magnitude for this flow.
A thin rectangular plate having a width w and a height h is located so that it is normal to a moving stream of fluid. Assume the drag, Fp, that the fluid exerts on the plate is a function of w and h, the fluid viscosity and density, µ and p respectively and the velocity uo of the fluid approaching the plate. Determine the dimensionless drag and the dimensionless numbers that influence its magnitude for this flow.
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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![**Problem Description:**
A thin rectangular plate with a width \( w \) and a height \( h \) is positioned normal to a moving fluid stream. The drag force \( F_D \), which the fluid exerts on the plate, depends on \( w \), \( h \), the fluid's viscosity \( \mu \), density \( \rho \), and the velocity \( u_\infty \) of the fluid approaching the plate.
**Objective:**
Determine the dimensionless drag and the dimensionless numbers that influence its magnitude for this flow.
**Explanation:**
To analyze this problem, we apply dimensional analysis. Common dimensionless numbers for fluid flow problems include the Reynolds number and the drag coefficient:
1. **Reynolds Number (\( \text{Re} \)):** This number characterizes the flow regime, indicating whether it is laminar or turbulent. It is defined as:
\[
\text{Re} = \frac{\rho \cdot u_\infty \cdot L}{\mu}
\]
where \( L \) is a characteristic length (often taken as \( h \) or \( w \) for a plate), \( \rho \) is the fluid density, \( u_\infty \) is the fluid velocity, and \( \mu \) is the fluid viscosity.
2. **Drag Coefficient (\( C_D \)):** This is a dimensionless quantity that represents drag per unit area relative to the fluid's dynamic pressure:
\[
C_D = \frac{F_D}{0.5 \cdot \rho \cdot u_\infty^2 \cdot A}
\]
where \( A \) is the reference area (often \( w \times h \) for a rectangular plate).
These dimensionless numbers will help in understanding how the drag force \( F_D \) varies with different fluid conditions and plate dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58e30080-5af3-43c1-bb19-7e2f27cfd9f6%2F3b35bd06-0f60-4759-935c-d9742fe7af53%2F7bbxvtf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
A thin rectangular plate with a width \( w \) and a height \( h \) is positioned normal to a moving fluid stream. The drag force \( F_D \), which the fluid exerts on the plate, depends on \( w \), \( h \), the fluid's viscosity \( \mu \), density \( \rho \), and the velocity \( u_\infty \) of the fluid approaching the plate.
**Objective:**
Determine the dimensionless drag and the dimensionless numbers that influence its magnitude for this flow.
**Explanation:**
To analyze this problem, we apply dimensional analysis. Common dimensionless numbers for fluid flow problems include the Reynolds number and the drag coefficient:
1. **Reynolds Number (\( \text{Re} \)):** This number characterizes the flow regime, indicating whether it is laminar or turbulent. It is defined as:
\[
\text{Re} = \frac{\rho \cdot u_\infty \cdot L}{\mu}
\]
where \( L \) is a characteristic length (often taken as \( h \) or \( w \) for a plate), \( \rho \) is the fluid density, \( u_\infty \) is the fluid velocity, and \( \mu \) is the fluid viscosity.
2. **Drag Coefficient (\( C_D \)):** This is a dimensionless quantity that represents drag per unit area relative to the fluid's dynamic pressure:
\[
C_D = \frac{F_D}{0.5 \cdot \rho \cdot u_\infty^2 \cdot A}
\]
where \( A \) is the reference area (often \( w \times h \) for a rectangular plate).
These dimensionless numbers will help in understanding how the drag force \( F_D \) varies with different fluid conditions and plate dimensions.
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