/*! 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 7 Parallel flow of atmospheric air... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 7

Parallel flow of atmospheric air over a flat plate of length \(L=3 \mathrm{~m}\) is disrupted by an array of stationary rods placed in the flow path over the plate. Laboratory measurements of the local convection coefficient at the surface of the plate are made for a prescribed value of \(V\) and \(T_{x}>T_{x}\). The results are correlated by an expression of the form \(h_{x}=0.7+13.6 x-3.4 x^{2}\), where \(h_{x}\) has units of \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(x\) is in meters. Evaluate the average convection coefficient \(\bar{h}_{L}\) for the entire plate and the ratio \(\bar{h}_{L} / h_{L}\) at the trailing edge.

Short Answer

Expert verified
The average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\).

Step by step solution

01

Calculate average convection coefficient \(\bar{h}_L\)

First, we need to find the average convection coefficient \(\bar{h}_L\) for the entire plate using the given equation for the local convection coefficient \(h_x = 0.7 + 13.6x - 3.4x^2\). To do this, we integrate \(h_x\) over the length of the plate (\(0\) to \(L\), where \(L = 3\,\text{m}\)) and divide by the length of the plate: \[\bar{h}_L = \frac{1}{L} \int_{0}^{L} h_x \,\mathrm{d}x\] Substitute the given equation for \(h_x\): \[\bar{h}_L = \frac{1}{3} \int_{0}^{3} (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x\]
02

Integrate the given expression of \(h_x\)

Now integrate the expression of \(h_x\) with respect to \(x\): \[\int (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x = 0.7x + \frac{13.6}{2}x^2 - \frac{3.4}{3}x^3 + C\] Since the integration is definite, we don't need to include the constant \(C\). Now we can evaluate the integral at the limits: \begin{align*} \int_{0}^{3} (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x &= \left[ 0.7x + \frac{13.6}{2}x^2 - \frac{3.4}{3}x^3 \right]_{0}^{3} \\ &= \left(0.7(3) + \frac{13.6}{2}(3)^2 - \frac{3.4}{3}(3)^3\right) - \left(0.7(0) + \frac{13.6}{2}(0)^2 - \frac{3.4}{3}(0)^3\right) \\ &= 12.6 \end{align*}
03

Calculate the average convection coefficient \(\bar{h}_L\)

Now that we have the integral value, we can calculate the average convection coefficient \(\bar{h}_L\): \[\bar{h}_L = \frac{1}{3}(12.6) = 4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\] The average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\).
04

Calculate \(h_L\) at the trailing edge

To find the local convection coefficient at the trailing edge, evaluate \(h_x\) at \(x = L = 3\,\text{m}\): \[h_L = 0.7 + 13.6(3) - 3.4(3)^2 = 7.6\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\]
05

Calculate the ratio \(\bar{h}_L / h_L\) at the trailing edge

Now, we can calculate the ratio \(\bar{h}_L / h_L\) at the trailing edge: \[\frac{\bar{h}_L}{h_L} = \frac{4.2}{7.6} = 0.5526\] The ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\). In summary, the average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\).

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.

Heat Transfer
Heat transfer is a fundamental concept in engineering and physics, describing the movement of thermal energy from one object to another.
This process is essential for understanding how objects warm up or cool down.
In heat transfer, thermal energy can move through three main methods: conduction, convection, and radiation. - **Conduction** occurs when heat moves through a solid material from a high temperature area to a low temperature region.
This process relies on the direct contact of molecules. - **Convection** involves the transfer of heat through fluids, which can be liquids or gases.
When a fluid flows over a surface, it can carry away or supply heat, depending on its temperature relative to the surface.
The convection coefficient is a measure of how effectively this transfer takes place. - **Radiation** involves the transfer of heat through electromagnetic waves without requiring a medium. Understanding these processes helps in calculating the convection coefficient, which is central to the problem.
The convection coefficient describes how well a fluid can absorb or disperse heat while flowing over a surface like a flat plate.
Parallel Flow
In fluid mechanics, parallel flow refers to when a fluid moves over a surface with streamline patterns remaining relatively uniform and consistent.
This type of flow is common in heat transfer problems, especially when analyzing air or liquid moving over flat surfaces.
Parallel flow ensures that all points along the flow path experience relatively similar conditions. With parallel flow, evaluating the heat transfer becomes more straightforward.
One can model the heat transfer using known equations that depend on factors like fluid velocity, surface characteristics, and temperature differences.
In exercises like the one provided, laboratory measurements of convection coefficients help illustrate how parallel flow affects heat distribution across the surface. Parallel flow in particular allows us to easily integrate heat transfer equations over a specified surface length.
This enables the calculation of properties like the average convection coefficient, which offers insight into the overall heat transfer effects throughout the surface area.
Understanding parallel flow is essential for predicting how efficient heat transfer will be during different scenarios.
Flat Plate
The flat plate is a simple yet widely used model in heat transfer analysis.
Researchers and engineers employ flat plates to understand fundamental concepts of thermal energy distribution and transfer. - A flat plate provides a consistent surface over which controlled experiments can take place.
This includes analyzing air or fluid flow and measuring temperature changes as heat moves through the material. - The flat plate assumption simplifies many mathematical models. This simplification is vital because it allows for basic, accurate predictions about how convection and other forms of heat transfer occur. In the exercise, the flat plate is crucial for the analysis of the convection coefficient.
Using the formula provided, the convection coefficient changes along the length of the plate as defined by the length parameter, illustrating how local conditions can vary. Understanding the role of a flat plate makes it easier to grasp more complex heat transfer scenarios.
It provides an essential foundational model for assessing the effectiveness and dynamics of heat transfer mechanisms in engineering applications.

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

An expression for the actual water vapor partial pressure in terms of wet-bulb and dry-bulb temperatures, referred to as the Carrier equation, is given as $$ p_{v}=p_{g w}-\frac{\left(p-p_{g w}\right)\left(T_{d b}-T_{\mathrm{wb}}\right)}{1810-T_{\mathrm{wb}}} $$ where \(p_{v}, p_{g w}\) and \(p\) are the actual partial pressure, the saturation pressure at the wet-bulb temperature, and the total pressure (all in bars), while \(T_{\mathrm{db}}\) and \(T_{\mathrm{wb}}\) are the dry- and wet-bulb temperatures in kelvins. Consider air at \(1 \mathrm{~atm}\) and \(37.8^{\circ} \mathrm{C}\) flowing over a wet-bulb thermometer that indicates \(21.1^{\circ} \mathrm{C}\). (a) Using Carrier's equation, calculate the partial pressure of the water vapor in the free stream. What is the relative humidity? (b) Refer to a psychrometric chart and obtain the relative humidity directly for the conditions indicated. Compare the result with part (a). (c) Use Equation \(6.65\) to determine the relative humidity. Compare the result to parts (a) and (b).

As a means of preventing ice formation on the wings of a small, private aircraft, it is proposed that electric resistance heating elements be installed within the wings. To determine representative power requirements, consider nominal flight conditions for which the plane moves at \(100 \mathrm{~m} / \mathrm{s}\) in air that is at a temperature of \(-23^{\circ} \mathrm{C}\). If the characteristic length of the airfoil is \(L=2 \mathrm{~m}\) and wind tunnel measurements indicate an average friction coefficient of \(\bar{C}_{f}=0.0025\) for the nominal conditions, what is the average heat flux needed to maintain a surface temperature of \(T_{s}=5^{\circ} \mathrm{C}\) ?

An industrial process involves evaporation of a thin water film from a contoured surface by heating it from below and forcing air across it. Laboratory measurements for this surface have provided the following heat transfer correlation: $$ \overline{N u_{L}}=0.43 R e_{L}^{0.58} P r^{0.4} $$ The air flowing over the surface has a temperature of \(290 \mathrm{~K}\), a velocity of \(10 \mathrm{~m} / \mathrm{s}\), and is completely dry \(\left(\phi_{\infty}=0\right)\). The surface has a length of \(1 \mathrm{~m}\) and a surface area of \(1 \mathrm{~m}^{2}\). Just enough energy is supplied to maintain its steady-state temperature at \(310 \mathrm{~K}\). (a) Determine the heat transfer coefficient and the rate at which the surface loses heat by convection. (b) Determine the mass transfer coefficient and the evaporation rate \((\mathrm{kg} / \mathrm{h})\) of the water on the surface. (c) Determine the rate at which heat must be supplied to the surface for these conditions.

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 \(0 \leq x \leq L\).

To a good approximation, the dynamic viscosity \(\mu\), the thermal conductivity \(k\), and the specific heat \(c_{p}\) are independent of pressure. In what manner do the kinematic viscosity \(v\) and thermal diffusivity \(\alpha\) vary with pressure for an incompressible liquid and an ideal gas? Determine \(\alpha\) of air at \(350 \mathrm{~K}\) for pressures of 1,5 , and \(10 \mathrm{~atm}\). Assuming a transition Reynolds number of \(5 \times 10^{5}\), determine the distance from the leading edge of a flat plate at which transition will occur for air at \(350 \mathrm{~K}\) at pressures of 1,5 , and 10 atm with \(u_{s}=2 \mathrm{~m} / \mathrm{s}\).
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Solid State Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Modern Physics

Read Explanation

Engineering 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.