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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.

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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.

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Most popular questions from this chapter

An object of irregular shape has a characteristic length of \(L=1 \mathrm{~m}\) and is maintained at a uniform surface temperature of \(T_{s}=400 \mathrm{~K}\). When placed in atmospheric air at a temperature of \(T_{x}=300 \mathrm{~K}\) and moving with a velocity of \(V=100 \mathrm{~m} / \mathrm{s}\), the average heat flux from the surface to the air is \(20,000 \mathrm{~W} / \mathrm{m}^{2}\). If a second object of the same shape, but with a characteristic length of \(L=5 \mathrm{~m}\), is maintained at a surface temperature of \(T_{s}=400 \mathrm{~K}\) and is placed in atmospheric air at \(T_{\infty}=300 \mathrm{~K}\), what will the value of the average convection coefficient be if the air velocity is \(V=20 \mathrm{~m} / \mathrm{s}\) ?

A streamlined strut supporting a bearing housing is exposed to a hot airflow from an engine exhaust. It is necessary to run experiments to determine the average convection heat transfer coefficient \(\bar{h}\) from the air to the strut in order to be able to cool the strut to the desired surface temperature \(T_{x}\). It is decided to run mass transfer experiments on an object of the same shape and to obtain the desired heat transfer results by using the heat and mass transfer analogy. The mass transfer experiments were conducted using a half-size model strut constructed from naphthalene exposed to an airstream at \(27^{\circ} \mathrm{C}\). Mass transfer measurements yielded these results: \begin{tabular}{rr} \hline \multicolumn{1}{c}{\(\boldsymbol{\boldsymbol { e } _ { \boldsymbol { L } }}\)} & \(\overline{\boldsymbol{S h}}_{\boldsymbol{L}}\) \\ \hline 60,000 & 282 \\ 120,000 & 491 \\ 144,000 & 568 \\ 288,000 & 989 \\ \hline \end{tabular} (a) Using the mass transfer experimental results, determine the coefficients \(C\) and \(m\) for a correlation of the form \(\overline{S h}_{L}=C R e_{L}^{m} S c^{1 / 3}\). (b) Determine the average convection heat transfer coefficient \(\bar{h}\) for the full-sized strut, \(L_{H}=60 \mathrm{~mm}\), when exposed to a free stream airflow with \(V=60 \mathrm{~m} / \mathrm{s}\), \(T_{\infty}=184^{\circ} \mathrm{C}\), and \(p_{\infty}=1 \mathrm{~atm}\) when \(T_{s}=70^{\circ} \mathrm{C}\). (c) The surface area of the strut can be expressed as \(A_{s}=2.2 L_{H} \cdot l\), where \(l\) is the length normal to the page. For the conditions of part (b), what is the change in the rate of heat transfer to the strut if the characteristic length \(L_{H}\) is doubled?

Forced air at \(T_{\infty}=25^{\circ} \mathrm{C}\) and \(V=10 \mathrm{~m} / \mathrm{s}\) is used to cool electronic elements on a circuit board. One such element is a chip, \(4 \mathrm{~mm} \times 4 \mathrm{~mm}\), located \(120 \mathrm{~mm}\) from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form Estimate the surface temperature of the chip if it is dissipating \(30 \mathrm{~mW}\).

Consider conditions for which a fluid with a free stream velocity of \(V=1 \mathrm{~m} / \mathrm{s}\) flows over an evaporating or subliming surface with a characteristic length of \(L=1 \mathrm{~m}\), providing an average mass transfer convection coefficient of \(\bar{h}_{\mathrm{m}}=10^{-2} \mathrm{~m} / \mathrm{s}\). Calculate the dimensionless parameters \(\overline{S h}_{L}, R e_{L}, S c\), and \(j_{m}\) for the following combinations: airflow over water, airflow over naphthalene, and warm glycerol over ice. Assume a fluid temperature of \(300 \mathrm{~K}\) and a pressure of \(1 \mathrm{~atm}\).

Species A is evaporating from a flat surface into species B. Assume that the concentration profile for species A in the concentration boundary layer is of the form \(C_{\mathrm{A}}(y)=D y^{2}+E y+F\), where \(D, E\), and \(F\) are constants at any \(x\)-location and \(y\) is measured along a normal from the surface. Develop an expression for the mass transfer convection coefficient \(h_{w}\) in terms of these constants, the concentration of \(A\) in the free stream \(C_{\mathrm{A}, \infty}\) and the mass diffusivity \(D_{\mathrm{AB}}\). Write an expression for the molar flux of mass transfer by convection for species \(A\).
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