/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 A fan that can provide air speed... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 19

A fan that can provide air speeds up to \(50 \mathrm{~m} / \mathrm{s}\) is to be used in a low-speed wind tunnel with atmospheric air at \(25^{\circ} \mathrm{C}\). If one wishes to use the wind tunnel to study flatplate boundary layer behavior up to Reynolds numbers of \(R e_{x}=10^{8}\), what is the minimum plate length that should be used? At what distance from the leading edge would transition occur if the critical Reynolds number were \(R e_{x, c}=5 \times 10^{5}\) ?

Short Answer

Expert verified
The minimum plate length required to study the flat plate boundary layer behavior up to a Reynolds number of \(10^8\) is approximately 31.187 meters. The transition from the laminar to turbulent boundary layer occurs at a distance of approximately 1.559 meters from the leading edge of the plate.

Step by step solution

01

Calculate the minimum plate length

To find the minimum plate length, we will use the formula for the Reynolds number: \[Re_x = \frac{\rho u_x L_x}{\mu}\] where \(Re_x\) is the Reynolds number, \(\rho\) is the fluid density, \(u_x\) is the airspeed, \(L_x\) is the plate length. We are given \(u_x = 50 \, \mathrm{m/s}\), \(Re_x = 10^8\), and the air temperature \(T = 25^{\circ}\, \mathrm{C}\). We will use the properties of air at this temperature. At \(T=25^{\circ}\, \mathrm{C}\), we have \(\rho \approx 1.184 \, \mathrm{kg/m^3}\) and \(\mu \approx 1.85 \times 10^{-5} \, \mathrm{Pa\cdot s}\). Plugging the given values into the Reynolds number formula, we can solve for the minimum plate length \(L_x\): \[L_x = \frac{Re_x \cdot \mu}{\rho \cdot u_x}\]
02

Find the minimum plate length

Calculate the minimum plate length with the given values: \[L_x = \frac{10^8 \cdot 1.85 \times 10^{-5} \, \mathrm{Pa\cdot s}}{1.184 \, \mathrm{kg/m^3} \cdot 50 \, \mathrm{m/s}}\] \[L_x \approx 31.187 \, \mathrm{m}\] Thus, the minimum plate length required to study the flat plate boundary layer behavior up to a Reynolds number of \(10^8\) is approximately 31.187 meters.
03

Calculate the distance from the leading edge where the transition occurs

Now we will use the critical Reynolds number, \(Re_{x,c}\) = \(5 \times 10^5\) to find the distance from the leading edge where the transition from laminar to turbulent boundary layer occurs. First, we need to rewrite the plate length formula in terms of distance for the given critical Reynolds number \(Re_{x,c}\): \[x_c = \frac{Re_{x,c}\cdot \mu}{\rho \cdot u_x}\]
04

Find the distance from the leading edge

Calculate the distance from the leading edge with the given values: \[x_c = \frac{5 \times 10^5 \cdot 1.85 \times 10^{-5} \, \mathrm{Pa\cdot s}}{1.184 \, \mathrm{kg/m^3} \cdot 50 \, \mathrm{m/s}}\] \[x_c \approx 1.559 \, \mathrm{m}\] Therefore, the transition from the laminar to turbulent boundary layer occurs at a distance of approximately 1.559 meters from the leading edge of the plate.

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 91影视!

Key Concepts

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

Flat Plate Boundary Layer
In fluid dynamics, a flat plate boundary layer refers to the region where the fluid comes into contact with the surface of a flat plate. This contact slows down the fluid particles, creating a region where the speed gradually changes from the surface to the free stream velocity. Understanding this concept is crucial because it directly affects the aerodynamic properties and friction experienced by an object moving through a fluid.
The boundary layer typically has a laminar flow at the beginning, where layers of fluid slide smoothly past one another, and gradually becomes turbulent, characterized by irregular fluctuations. Analyzing a flat plate boundary layer helps engineers design surfaces to minimize drag and optimize performance.
Key aspects of boundary layers include:
  • Laminar Flow: Smooth and orderly movement of fluid layers with minimal mixing.
  • Boundary Layer Thickness: The distance from the plate where fluid velocity reaches 99% of the free stream velocity.
  • Turbulent Flow: Chaotic and highly mixed motion of fluid layers that increases energy dissipation.
Transition from Laminar to Turbulent
The transition from laminar to turbulent flow is a crucial aspect in the study of boundary layers, affecting drag and heat transfer. This change occurs at a specific point known as the 'transition point,' and it's typically measured as a critical Reynolds number. The critical Reynolds number, denoted as \(Re_{x,c}\), is pivotal in determining where this transition happens along a flat plate.
In the example provided, the critical Reynolds number \(Re_{x,c}\) is \(5 \times 10^5\). To locate the transition point, we use the formula:
\[x_c = \frac{Re_{x,c} \cdot \mu}{\rho \cdot u_x}\]
This formula takes into account the viscosity \(\mu\), density \(\rho\), and velocity \(u_x\) of the fluid. Understanding when and why flow transitions are important for optimizing design to reduce drag and increase efficiency in applications like aircraft and automobiles.
Wind Tunnel
Wind tunnels are essential tools for studying aerodynamic properties and behaviors such as boundary layer characteristics. In a wind tunnel, a controlled and uniform air stream allows researchers to simulate real-world conditions. This environment is perfect for testing flat plate models to understand boundary layer dynamics, especially when dealing with transitions from laminar to turbulent flow.
The benefit of using a wind tunnel lies in its ability to create repeatable conditions, ensuring precise measurements. By controlling factors like airspeed and temperature, experiments can provide detailed insights into how an object will perform. This capability is invaluable for sectors like automotive and aerospace, where understanding airflow and boundary effects can inform designs to improve efficiency and performance.
When studying models like flat plates, adjusting the wind tunnel conditions can help reach desired Reynolds numbers, making it possible to observe and measure exactly where transitional behaviors occur in a real or scaled-down scenario.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For flow over a flat plate of length \(L\), the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge of the plate. What is the ratio of the average Nusselt number for the entire plate \(\left(\overline{N u}_{L}\right)\) to the local Nusselt number at \(x=L\left(N u_{L}\right)\) ?

Experiments have been conducted to determine local heat transfer coefficients for flow perpendicular to a long, isothermal bar of rectangular cross section. The bar is of width \(c\) parallel to the flow, and height \(d\) normal to the flow. For Reynolds numbers in the range \(10^{4} \leq R_{d} \leq 5 \times 10^{4}\), the face-averaged Nusselt numbers are well correlated by an expression of the form The values of \(C\) and \(m\) for the front face, side faces, and back face of the rectangular rod are found to be the following: \begin{tabular}{llll} \hline Face & cld & \(\boldsymbol{C}\) & \(\boldsymbol{m}\) \\ \hline Front & \(0.33 \leq\) cld \(51.33\) & \(0.674\) & \(1 / 2\) \\ Side & \(0.33\) & \(0.153\) & \(2 / 3\) \\ Side & \(1.33\) & \(0.107\) & \(2 / 3\) \\ Back & \(0.33\) & \(0.174\) & \(2 / 3\) \\ Back & \(1.33\) & \(0.153\) & \(2 / 3\) \\ \hline \end{tabular} Determine the value of the average heat transfer coefficient for the entire exposed surface (that is, averaged over all four faces) of a \(c=40\)-mm-wide, \(d=30\)-mm-tall rectangular rod. The rod is exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{x}=300 \mathrm{~K}\). Provide a plausible explanation of the relative values of the face-averaged heat transfer coefficients on the front, side, and back faces.

Photosynthesis, as it occurs in the leaves of a green plant, involves the transport of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) from the atmosphere to the chloroplasts of the leaves. The rate of photosynthesis may be quantified in terms of the rate of \(\mathrm{CO}_{2}\) assimilation by the chloroplasts. This assimilation is strongly influenced by \(\mathrm{CO}_{2}\) transfer through the boundary layer that develops on the leaf surface. Under conditions for which the density of \(\mathrm{CO}_{2}\) is \(6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) in the air and \(5 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) at the leaf surface and the convection mass transfer coefficient is \(10^{-2} \mathrm{~m} / \mathrm{s}\), what is the rate of photosynthesis in terms of kilograms of \(\mathrm{CO}_{2}\) assimilated per unit time and area of leaf surface?

Consider airflow over a flat plate of length \(L=1 \mathrm{~m}\) under conditions for which transition occurs at \(x_{c}=0.5 \mathrm{~m}\) based on the critical Reynolds number, \(R e_{x, c}=5 \times 10^{5}\). (a) Evaluating the thermophysical properties of air at \(350 \mathrm{~K}\), determine the air velocity. (b) In the laminar and turbulent regions, the local convection coefficients are, respectively, \(h_{\text {lam }}(x)=C_{\text {lam }} x^{-05}\) and \(h_{\text {marb }}=C_{\text {marb }} x^{-0.2}\) where, at \(T=350 \mathrm{~K}, C_{\text {lum }}=8.845 \mathrm{~W} / \mathrm{m}^{3 / 2} \cdot \mathrm{K}, C_{\text {tub }}=\) \(49.75 \mathrm{~W} / \mathrm{m}^{1.8} \cdot \mathrm{K}\), and \(x\) has units of \(\mathrm{m}\). Develop an expression for the average convection coefficient, \(\bar{h}_{\mathrm{hm}}(x)\), as a function of distance from the leading edge, \(x\), for the laminar region, \(0 \leq x \leq x_{x}\). (c) Develop an expression for the average convection coefficient, \(\bar{h}_{\text {art }}(x)\), as a function of distance from the leading edge, \(x\), for the turbulent region, \(x_{c} \leq x \leq L\). (d) On the same coordinates, plot the local and average convection coefficients, \(h_{x}\) and \(\bar{h}_{x}\), respectively, as a function of \(x\) for \(0 \leq x \leq L\).

For laminar flow over a flat plate, the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge \((x=0)\) of the plate. What is the ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) ?
See all solutions

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Read Explanation

Measurements

Read Explanation

Particle Model of Matter

Read Explanation

Atoms and Radioactivity

Read Explanation

Magnetism

Read Explanation

Waves Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.