/*! 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 37 The boundary layer associated wi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 37

The boundary layer associated with parallel flow over an isothermal plate may be tripped at any \(x\)-location by using a fine wire that is stretched across the width of the plate. Determine the value of the critical Reynolds number \(R e_{x, c, o p}\) that is associated with the optimal location of the trip wire from the leading edge that will result in maximum heat transfer from the warm plate to the cool fluid.

Short Answer

Expert verified
The critical Reynolds number, \(Re_{x,c,op}\), associated with the optimal location of the trip wire from the leading edge of an isothermal plate for maximum heat transfer is achieved when the flow transitions from laminar to turbulent. This transition Reynolds number is around 500,000. The optimal distance for the trip wire, \(x_{op}\), can be found using the expression: \[x_{op} = \frac{500,000 \mu}{\rho u}\] where \(\rho\) is the fluid density, \(u\) is the fluid velocity, and \(\mu\) is the fluid dynamic viscosity.

Step by step solution

01

Define the Reynolds number

The Reynolds number is a dimensionless quantity that expresses the ratio of inertial forces to viscous forces and is commonly used in fluid mechanics. It is defined as: \[Re_x = \frac{\rho u x}{\mu}\] Where: - \(Re_x\) is the local Reynolds number (based on the distance x from the leading edge of the plate), - \(\rho\) is the fluid density, - \(u\) is the fluid velocity, - \(x\) is the distance from the leading edge of the plate, and - \(\mu\) is the fluid dynamic viscosity.
02

Find the transition Reynolds number

The transition Reynolds number, denoted by \(Re_{x,c}\), is the value of the Reynolds number at which the flow transitions from laminar to turbulent. In our context, this occurs at the point where the trip wire is placed. For flow over a flat isothermal plate, the transition Reynolds number is around 500,000.
03

Determine the optimal trip wire location

Since our goal is to find the optimal location of the trip wire from the leading edge of the plate that will result in maximum heat transfer, we can use the transition Reynolds number to do so: \[Re_{x,c,op} = \frac{\rho u x_{op}}{\mu}\] Where \(x_{op}\) is the optimal distance we're seeking. To find \(x_{op}\), we can rearrange the equation to: \[x_{op} = \frac{Re_{x,c,op} \mu}{\rho u}\] Now plugging in the transition Reynolds number \(Re_{x,c} = 500,000\): \[x_{op} = \frac{500,000 \mu}{\rho u}\] This shows that the optimal location of the trip wire from the leading edge that will result in the maximum heat transfer is given by the distance \(x_{op}\).
04

Conclusion

To determine the value of the critical Reynolds number associated with the optimal location of the trip wire from the leading edge of an isothermal plate, we must find the transition Reynolds number, which is around 500,000. From there, the optimal distance for the trip wire can be found using the expression: \[x_{op} = \frac{500,000 \mu}{\rho u}\] This gives us the optimal location for maximum heat transfer between the warm plate and the cool fluid.

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.

Boundary Layer
In fluid mechanics, the boundary layer is a thin region of fluid flow near a solid surface, like an isothermal plate. This zone is where the fluid velocity changes from zero at the surface to the free stream velocity away from it. Understanding the boundary layer is crucial as it affects properties such as friction drag and heat transfer.

When the fluid first contacts the plate, a laminar boundary layer forms where layers of fluid slide smoothly over one another. This continues until a trip wire or another disturbance causes the transition to a turbulent boundary layer, enhancing mixing and heat transfer.

Key characteristics of the boundary layer include:
  • *Thickness*: the distance from the wall to where the fluid velocity reaches nearly free-stream velocities.
  • *Velocity Profile*: varies from zero at the surface to the free-stream velocity.
  • *Thermal Effects*: important for phenomena like heat transfer between the plate and the fluid.
Reynolds Number
The Reynolds number (Re) is a dimensionless quantity used to predict flow patterns in fluid dynamics. It is defined as the ratio of inertial forces to viscous forces, giving insight into whether a flow is laminar or turbulent. The formula is given by:\[Re_x = \frac{\rho u x}{\mu}\]where:
  • \(\rho\) is the fluid density,
  • \(u\) is the fluid velocity,
  • \(x\) is the characteristic length, such as the distance from the leading edge of a plate,
  • \(\mu\) is the dynamic viscosity.

Typically, a low Reynolds number indicates laminar flow, while a high value suggests turbulent flow. For flow over a flat isothermal plate, transition from laminar to turbulent often occurs around a Reynolds number of 500,000.
Heat Transfer
Heat transfer in the context of fluid flow over an isothermal plate involves the movement of thermal energy from the plate to the surrounding fluid. The effectiveness of heat transfer can significantly change depending on whether the boundary layer is laminar or turbulent.

Laminar flow tends to have lower heat transfer rates due to less mixing within the fluid. Transitioning to a turbulent flow increases mixing, improving heat transfer. This results in more efficient energy transfer from the warm plate to the cooler fluid. Here are some important aspects of heat transfer:
  • *Conduction*: Transfer of heat through the boundary layer, where temperature gradients drive heat from hot to cold regions.
  • *Convection*: Often dominates in flowing fluids, where heat is carried away by the motion of the fluid itself.
  • *Enhancement Strategies*: Incorporating devices like trip wires can artificially transition the flow to turbulent, boosting heat transfer.
Laminar to Turbulent Transition
The transition from laminar to turbulent flow is a key concept in fluid mechanics. Laminar flow is characterized by smooth, orderly fluid motion, while turbulent flow is chaotic, with eddies and vortices.

This transition can be triggered by disturbances like a trip wire placed on a plate. Such triggers enhance mixing and increase momentum exchange within the fluid, significantly impacting heat transfer and fluid dynamics.

Key points about this transition include:
  • *Critical Reynolds Number*: This is the threshold at which flow becomes turbulent. For a flat plate, it's around 500,000.
  • *Implications for Design*: Understanding this transition is crucial for optimizing designs to enhance heat transfer or reduce drag.
  • *Control Strategies*: Engineers often manipulate flow by introducing disturbances at strategic locations to provoke a beneficial transition from laminar to turbulent flow.

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

Motile bacteria are equipped with flagella that are rotated by tiny, biological electrochemical engines which, in turn, propel the bacteria through a host liquid. Consider a nominally spherical Escherichia coli bacterium that is of diameter \(D=2 \mu \mathrm{m}\). The bacterium is in a water-based solution at \(37^{\circ} \mathrm{C}\) containing a nutrient which is characterized by a binary diffusion coefficient of \(D_{\mathrm{AB}}=0.7 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\) and a food energy value of \(\mathcal{N}=16,000 \mathrm{~kJ} / \mathrm{kg}\). There is a nutrient density difference between the fluid and the shell of the bacterium of \(\Delta \rho_{\mathrm{A}}=860 \times 10^{-12} \mathrm{~kg} / \mathrm{m}^{3}\). Assuming a propulsion efficiency of \(\eta=0.5\), determine the maximum speed of the E. coli. Report your answer in body diameters per second.

Consider a flat plate subject to parallel flow (top and bottom) characterized by \(u_{\infty}=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=20^{\circ} \mathrm{C}\). (a) Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force associated with an \(L=2\)-m-long, \(w=2-\mathrm{m}\) wide flat plate for airflow and surface temperatures of \(T_{s}=50^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\). (b) Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force associated with an \(L=0.1\)-m-long, \(w=0.1\)-m-wide flat plate for water flow and surface temperatures of \(T_{s}=50^{\circ} \mathrm{C}\) and \(80^{\circ} \mathrm{C}\).

A stream of atmospheric air is used to dry a series of biological samples on plates that are each of length \(L_{i}=0.25 \mathrm{~m}\) in the direction of the airflow. The air is dry and at a temperature equal to that of the plates \(\left(T_{\infty}=T_{s}=50^{\circ} \mathrm{C}\right)\). The air speed is \(u_{\infty}=9.1 \mathrm{~m} / \mathrm{s}\). (a) Sketch the variation of the local convection mass transfer coefficient \(h_{m x}\) with distance \(x\) from the leading edge. Indicate the specific nature of the \(x\) dependence. (b) Which of the plates will dry the fastest? Calculate the drying rate per meter of width for this plate \((\mathrm{kg} / \mathrm{s}+\mathrm{m})\). (c) At what rate would heat have to be supplied to the fastest drying plate to maintain it at \(T_{s}=50^{\circ} \mathrm{C}\) during the drying process?

Consider atmospheric air at \(25^{\circ} \mathrm{C}\) and a velocity of \(25 \mathrm{~m} / \mathrm{s}\) flowing over both surfaces of a 1 - \(\mathrm{m}\)-long flat plate that is maintained at \(125^{\circ} \mathrm{C}\). Determine the rate of heat transfer per unit width from the plate for values of the critical Reynolds number corresponding to \(10^{5}\), \(5 \times 10^{5}\), and \(10^{6}\).

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness \(\delta\) and width \(W\) cooled as it transits the distance \(L\) between two rollers at a velocity \(V\). In this problem, we consider cooling of plain carbon steel by an airstream moving at a velocity \(u_{\infty}\) in cross flow over the top and bottom surfaces of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the sheet or is stationary and through which the sheet passes, and assuming a uniform sheet temperature in the direction of airflow, derive a differential equation that governs the temperature distribution, \(T(x)\), along the sheet. Consider the effects of radiation, as well as convection, and express your result in terms of the velocity, thickness, and properties of the sheet \(\left(V, \delta, \rho, c_{p}, \varepsilon\right)\), the average convection coefficient \(\bar{h}_{W}\) associated with the cross flow, and the environmental temperatures \(\left(T_{\infty}, T_{\text {sur }}\right)\). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For \(\delta=3 \mathrm{~mm}, V=\) \(0.10 \mathrm{~m} / \mathrm{s}, L=10 \mathrm{~m}, W=1 \mathrm{~m}, u_{\infty}=20 \mathrm{~m} / \mathrm{s}, T_{\infty}=\) \(20^{\circ} \mathrm{C}\), and a sheet temperature of \(T_{i}=500^{\circ} \mathrm{C}\) at the onset of cooling, what is the outlet temperature \(T_{o}\) ? Assume a negligible effect of the sheet velocity on boundary layer development in the direction of airflow. The density and specific heat of the steel are \(\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=620 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while properties of the air may be taken to be \(k=0.044\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, \nu=4.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \operatorname{Pr}=0.68\). (c) Accounting for the effects of radiation, with \(\varepsilon=\) \(0.70\) and \(T_{\text {sur }}=20^{\circ} \mathrm{C}\), numerically integrate the differential equation derived in part (a) to determine the temperature of the sheet at \(L=10 \mathrm{~m}\). Explore the effect of \(V\) on the temperature distribution along the sheet.
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Particle Model of Matter

Read Explanation

Translational Dynamics

Read Explanation

Radiation

Read Explanation

Work Energy and Power

Read Explanation

Electrostatics

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.