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Chapter 7: Problem 3

Consider steady, parallel flow of atmospheric air over a flat plate. The air has a temperature and free stream velocity of \(300 \mathrm{~K}\) and \(25 \mathrm{~m} / \mathrm{s}\). (a) Evaluate the boundary layer thickness at distances of \(x=1,10\), and \(100 \mathrm{~mm}\) from the leading edge. If a second plate were installed parallel to and at a distance of \(3 \mathrm{~mm}\) from the first plate, what is the distance from the leading edge at which boundary layer merger would occur? (b) Evaluate the surface shear stress and the \(y\)-velocity component at the outer edge of the boundary layer for the single plate at \(x=1,10\), and \(100 \mathrm{~mm}\). (c) Comment on the validity of the boundary layer approximations.

Short Answer

Expert verified
We compute the boundary layer thickness, 未, for laminar flow over a flat plate using the Blasius equation. With given air temperature, free stream velocity, and kinematic viscosity, we find 未 at \(x = 1, 10, 100\) mm. For boundary layer merger distance, we equate the sum of the boundary layer thicknesses between both plates to the distance between the plates, and find the distance from the leading edge. Surface shear stress, 蟿, is computed using air density, free stream velocity, and skin friction coefficient. To find the \( y \) -velocity component at the outer edge of the boundary layer, we use the velocity formula at distance \( y \) from the plate. The boundary layer approximations are reasonable for basic understanding, but real applications may have additional factors influencing the flow.

Step by step solution

01

(a) Boundary layer thickness computation

To solve this, we can use the Blasius equation for the boundary layer thickness (未) at the different distances (x) for a laminar flow over a flat plate: \[ \delta = 5 \sqrt{\frac{\nu x}{U_\infty}} \] where - \( \delta \) = Boundary layer thickness - \( \nu \) = Kinematic viscosity - \(x\) = Distance from the leading edge - \(U_\infty \) = Free stream velocity The air temperature is given as 300 K, and we can find the kinematic viscosity (\( \nu \)) of the air at this temperature in a standard table: \( \nu \approx 1.566 \times 10^{-5} m^2/s \). Given, the free stream velocity (\(U_\infty\)) is 25 m/s. Now, plug in these values to calculate the boundary layer thickness at each distance (\(x\)): - \(x = 1\) mm \[ \delta_1 = 5 \sqrt{\frac{1.566 \times 10^{-5} * 0.001}{25}} \] - \(x = 10\) mm \[ \delta_2 = 5 \sqrt{\frac{1.566 \times 10^{-5} * 0.01}{25}} \] - \(x = 100\) mm \[ \delta_3 = 5 \sqrt{\frac{1.566 \times 10^{-5} * 0.1}{25}} \] Compute these using a calculator, and we have the boundary layer thicknesses at each distance.
02

(a) Boundary layer merger distance

Given that a second plate is installed parallel to and at a distance of 3 mm from the first plate, we can find the distance at which the boundary layer merger occurs when the combined thicknesses of the boundary layers are equal to the distance between the plates. \[ \delta + \delta' = 3 \times 10^{-3} \] Since both plates have the same conditions, the boundary layer thickness will be the same for them, so we can write: \[ 2\delta = 3 \times 10^{-3} \] Solve for \( \delta\): \[ \delta = 1.5 \times 10^{-3} \] Now, substitute the Blasius equation and solve for \(x\) to find the distance from the leading edge: \[ 1.5 \times 10^{-3} = 5 \sqrt{\frac{1.566 \times 10^{-5} * x}{25}} \] Solve this equation to get the value of \(x\) at which the boundary layer merger occurs.
03

(b) Surface shear stress and y-velocity computation

Surface shear stress (蟿) can be calculated using the formula: \[ \tau = \frac{1}{2} \rho U_\infty^2 C_f \] where - \( \rho \) = Air density at 300 K (approx. 1.18 kg/m鲁) - \( C_f\) = Skin friction coefficient For a laminar boundary layer, the skin friction coefficient ( \( C_f \) ) can be determined by: \[ C_f = \frac{1.328}{\sqrt{Re_x}} \] where \( Re_x = \frac{U_\infty x}{\nu}\) is the local Reynolds number. We can compute the surface shear stress at the three given distances: - 蟿1 at x = 1 mm \[ \tau_1 = \frac{1}{2} (1.18) (25^2) \frac{1.328}{\sqrt{\frac{25 * 0.001}{1.566 \times 10^{-5}}}} \] - 蟿2 at x = 10 mm \[ \tau_2 = \frac{1}{2} (1.18) (25^2) \frac{1.328}{\sqrt{\frac{25 * 0.01}{1.566 \times 10^{-5}}}} \] - 蟿3 at x = 100 mm \[ \tau_3 = \frac{1}{2} (1.18) (25^2) \frac{1.328}{\sqrt{\frac{25 * 0.1}{1.566 \times 10^{-5}}}} \] Calculate the values of \( \tau_1, \tau_2\), and \( \tau_3 \) using a calculator. To get the \(y\)-velocity component at the outer edge of the boundary layer (\(U_\infty\)), we can use the formula for the velocity at distance \(y \) from the plate: \[ U(y) = U_\infty (1-e^{\frac{-yU_\infty}{\nu}}) \] Since we want to find the \(y\)-velocity component at the outer edge of the boundary layer, we'll substitute the boundary layer thickness \(\delta\) for \(y\): \[ U(\delta) = U_\infty (1-e^{\frac{-\delta U_\infty}{\nu}}) \] Use the values of \(\delta_{1}, \delta_{2}, \delta_{3} \) obtained earlier to compute the \(y\)-velocity component at the outer edge of the boundary layer for each of the given distances.
04

(c) Validity of boundary layer approximations

The boundary layer approximations are based on certain assumptions such as incompressible flow, laminar flow, and steady-state conditions. In this exercise, the flow over a flat plate can be assumed to be incompressible since the velocities involved are low and temperature changes are negligible. We simplify the flow as being laminar, although in reality, boundary layers can transition to turbulence depending on the Reynolds number. Also, we consider a steady-state scenario for ease of analysis. Thus, the approximations made in this problem are reasonable for getting an understanding of the basic concepts of boundary layer flow, but one should be aware that in real applications, flow might transition to turbulence, and further factors might influence the behavior of the flow.

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Key Concepts

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

Blasius equation
The Blasius equation is an essential mathematical relationship for understanding boundary layer behavior in fluid dynamics. It describes the thickness of the boundary layer as a function of the distance from the leading edge of a flat plate in a laminar flow. Named after the physicist Heinrich Blasius, the equation takes the form:

[ \delta = 5 \sqrt{\frac{u x}{U_{\infty}}} ]
where:\begin{itemize}\item \( \delta \) is the boundary layer thickness\item \( u \) is the kinematic viscosity of the fluid\item \( x \) is the distance from the leading edge of the flat plate\item \( U_{\infty} \) is the free stream velocity, which is the velocity of the fluid far removed from the effects of the boundary layer\end{itemize}
The simplicity of the Blasius equation allows for a relatively straightforward calculation of the boundary layer thickness given the above parameters. However, it applies only to laminar flows with steady, incompressible conditions over flat plates. The equation is crucial in predicting flow patterns, which can then be used to calculate surface shear stress and the velocity components within the boundary layer.
Laminar flow
Laminar flow represents a fluid motion where layers of the fluid slide smoothly over one another with minimal mixing and without eddies or swirls. Such a flow pattern is characterized by its orderly motion and is typically observed at lower velocities and smaller scales, where the inertial forces are relatively weak compared to the viscous forces. In the scenario of air flowing over a flat plate, the laminar flow can be described effectively using the boundary layer concept.

Laminar boundary layers are significant because they offer less resistance to flow, resulting in lower drag forces as compared to turbulent boundary layers. This makes understanding and predicting laminar flow crucial for optimizing the design of various objects in engineering, such as aircraft wings, to reduce energy consumption and enhance efficiency. However, it should be noted that beyond a certain critical Reynolds number, laminar flow may destabilize and transition to turbulence, which can drastically change the behavior of the fluid flow and associated boundary layer thickness.
Surface shear stress
Surface shear stress is a measure of the force of friction per unit area exerted by a fluid as it flows over a surface. It's an important factor in the study of boundary layers as it directly affects the drag experienced by the surface in contact with the fluid. For the calculation of surface shear stress caused by laminar flow over a flat plate, a formula tying the shear stress \( \tau \) to the local Reynolds number and the flow velocity is used:

[ \tau = \frac{1}{2} \rho U_\infty^2 C_f ]
where:\begin{itemize}\item \( \rho \) is the density of the fluid\item \( U_{\infty} \) is the free stream velocity\item \( C_f \) is the skin friction coefficient, which for a laminar flow can be calculated as \( C_f = \frac{1.328}{\sqrt{Re_x}} \), with \( Re_x \) being the local Reynolds number\end{itemize}
Understanding and predicting surface shear stress is vital in applications ranging from industrial processes to the aerodynamics of vehicles. It helps in estimating the overall drag force that affects an object鈥檚 movement through a fluid and it's crucial in designing surfaces that can withstand the forces imposed by fluid flow, such as pipelines, ship hulls, or skyscraper exteriors faced with wind currents.

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Most popular questions from this chapter

A flat plate of width \(1 \mathrm{~m}\) is maintained at a uniform surface temperature of \(T_{s}=150^{\circ} \mathrm{C}\) by using independently controlled, heat-generating rectangular modules of thickness \(a=10 \mathrm{~mm}\) and length \(b=50 \mathrm{~mm}\). Each module is insulated from its neighbors, as well as on its back side. Atmospheric air at \(25^{\circ} \mathrm{C}\) flows over the plate at a velocity of \(30 \mathrm{~m} / \mathrm{s}\). The thermophysical properties of the module are \(k=5.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=320 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\), and \(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\). (a) Find the required power generation, \(\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\), in a module positioned at a distance \(700 \mathrm{~mm}\) from the leading edge. (b) Find the maximum temperature \(T_{\max }\) in the heatgenerating module.

In an extrusion process, copper wire emerges from the extruder at a velocity \(V_{e}\) and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, \(T(x)\), along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire \(\left(V_{e}, D, \rho, c_{p}, \varepsilon\right)\), the convection coefficient associated with the cross flow \((\bar{h})\), and the environmental temperatures \(\left(T_{w}, T_{\text {sur }}\right)\). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For \(V_{c}=0.2 \mathrm{~m} / \mathrm{s}\), \(D=5 \mathrm{~mm}, V=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}\), and an initial wire temperature of \(T_{i}=600^{\circ} \mathrm{C}\), compute the temperature \(T_{o}\) of the wire at \(x=L=5 \mathrm{~m}\). The density and specific heat of the copper are \(\rho=8900 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while properties of the air may be taken to be \(k=0.037 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=3 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(P r=0.69\). (c) Accounting for the effects of radiation, with \(\varepsilon=0.55\) and \(T_{\text {sur }}=25^{\circ} \mathrm{C}\), numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at \(L=5 \mathrm{~m}\). Explore the effects of \(V_{e}\) and \(\varepsilon\) on the temperature distribution along the wire.

An array of electronic chips is mounted within a sealed rectangular enclosure, and cooling is implemented by attaching an aluminum heat \(\operatorname{sink}(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The base of the heat sink has dimensions of \(w_{1}=w_{2}=\) \(100 \mathrm{~mm}\), while the 6 fins are of thickness \(t=10 \mathrm{~mm}\) and pitch \(S=18 \mathrm{~mm}\). The fin length is \(L_{f}=50 \mathrm{~mm}\), and the base of the heat sink has a thickness of \(L_{b}=10 \mathrm{~mm}\). If cooling is implemented by water flow through the heat sink, with \(u_{\infty}=3 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=17^{\circ} \mathrm{C}\), what is the base temperature \(T_{b}\) of the heat sink when power dissipation by the chips is \(P_{\text {elec }}=1800 \mathrm{~W}\) ? The average convection coefficient for surfaces of the fins and the exposed base may be estimated by assuming parallel flow over a flat plate. Properties of the water may be approximated as \(k=0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=995 \mathrm{~kg} / \mathrm{m}^{3}\), \(c_{p}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \nu=7.73 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\), and \(\operatorname{Pr}=5.2\).

Consider the plasma spraying process of Problems \(5.25\) and \(7.82\). For a nozzle exit diameter of \(D=\) \(10 \mathrm{~mm}\) and a substrate radius of \(r=25 \mathrm{~mm}\), estimate the rate of heat transfer by convection \(q_{\text {cany }}\) from the argon plasma to the substrate, if the substrate temperature is maintained at \(300 \mathrm{~K}\). Energy transfer to the substrate is also associated with the release of latent heat \(q_{\text {lat }}\), which occurs during solidification of the impacted molten droplets. If the mass rate of droplet impingement is \(\dot{m}_{p}=0.02 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2}\), estimate the rate of latent heat release.

Fluid velocities can be measured using hot-film sensors, and a common design is one for which the sensing element forms a thin film about the circumference of a quartz rod. The film is typically comprised of a thin \((\sim 100 \mathrm{~nm})\) layer of platinum, whose electrical resistance is proportional to its temperature. Hence, when submerged in a fluid stream, an electric current may be passed through the film to maintain its temperature above that of the fluid. The temperature of the film is controlled by monitoring its electric resistance, and with concurrent measurement of the electric current, the power dissipated in the film may be determined. Proper operation is assured only if the heat generated in the film is transferred to the fluid, rather than conducted from the film into the quartz rod. Thermally, the film should therefore be strongly coupled to the fluid and weakly coupled to the quartz rod. This condition is satisfied if the Biot number is very large, \(B i=\bar{h} D / 2 k \geqslant 1\), where \(\bar{h}\) is the convection coefficient between the fluid and the film and \(k\) is the thermal conductivity of the rod. (a) For the following fluids and velocities, calculate and plot the convection coefficient as a function of velocity: (i) water, \(0.5 \leq V \leq 5 \mathrm{~m} / \mathrm{s}\); (ii) air, \(1 \leq V \leq 20 \mathrm{~m} / \mathrm{s}\). (b) Comment on the suitability of using this hot-film sensor for the foregoing conditions.
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