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Chapter 6: Problem 18

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

Short Answer

Expert verified
The air velocity is found to be \(V_c = 9.375 \mathrm{~m/s}\). The expressions for the average convection coefficient in the laminar and turbulent regions are, respectively, \(\bar{h}_{lam}(x) = \frac{2(8.845)}{\sqrt{x}} \mathrm{W/m^2 \cdot K}\) and \(\bar{h}_{turb}(x) = \frac{5(49.75)}{4(x - 0.5)^{0.8}}(x^{0.8} - 0.5^{0.8}) \mathrm{W/m^2 \cdot K}\). To plot the local and average convection coefficients as a function of distance from the leading edge, use a graph plotting software with the given expressions, considering the distance range \(0\leq x\leq L\), where \( L = 1\mathrm{~m}\).

Step by step solution

01

Calculate the air velocity

At the critical point, we can use the equation: \(Re_{x, c} = \frac{\rho V_c x_c}{\mu}\) given that \(Re_{x, c} = 5 \times 10^5\), \(x_c = 0.5 \mathrm{~m}\) and we evaluate the thermophysical properties of air at \(350 \mathrm{~K}\), we have: \(V_c = \frac{Re_{x, c} \mu}{\rho x_c}\) Using the properties of air at 350 K: \(\rho = 1.177 \mathrm{~kg/m^3}\), and \(\mu = 2.21 \times 10^{-5} \mathrm{Pa \cdot s}\), the air velocity can be calculated as: \(V_c = \frac{5 \times 10^5 \times 2.21 \times 10^{-5} \mathrm{Pa \cdot s}}{1.177 \mathrm{~kg/m^3} \times 0.5 \mathrm{~m}}\) Upon calculating, we get: \(V_c = 9.375 \mathrm{~m/s}\) #b) Develop an expression for the average convection coefficient in the laminar region#
02

Average convection coefficient in the laminar region

In the laminar region, we can integrate the local convection coefficient equation to find the average convection coefficient, as follows: \(\bar{h}_{lam}(x) = \frac{1}{x} \int_{0}^{x}C_{lam}x'^{-0.5}dx'\) After integrating, we get the expression for the average convection coefficient in the laminar region as: \(\bar{h}_{lam}(x) = \frac{2C_{lam}}{\sqrt{x}}\) We are given \(C_{lam} = 8.845 \mathrm{W/m^{3/2} \cdot K}\), so the equation becomes: \(\bar{h}_{lam}(x) = \frac{2(8.845)}{\sqrt{x}} \mathrm{W/m^2 \cdot K}\) #c) Develop an expression for the average convection coefficient in the turbulent region#
03

Average convection coefficient in the turbulent region

In the turbulent region, we can integrate the local convection coefficient equation to find the average convection coefficient, as follows: \(\bar{h}_{turb}(x) = \frac{1}{x - x_c} \int_{x_c}^{x}C_{turb}x'^{-0.2}dx'\) After integrating, we get the expression for the average convection coefficient in the turbulent region as: \(\bar{h}_{turb}(x) = \frac{5C_{turb}}{4(x - x_c)^{0.8}}(x^{0.8} - x_c^{0.8})\) We are given \(C_{turb} = 49.75 \mathrm{W/m^{1.8} \cdot K}\) and \(x_c = 0.5 \mathrm{m}\), so the equation becomes: \(\bar{h}_{turb}(x) = \frac{5(49.75)}{4(x - 0.5)^{0.8}}(x^{0.8} - 0.5^{0.8}) \mathrm{W/m^2 \cdot K}\) #d) Plot the local and average convection coefficients as a function of distance from the leading edge#
04

Plot the local and average convection coefficients as a function of distance

We have the following expressions: 1. \(h_{lam}(x) = C_{lam} x^{-0.5}\) 2. \(h_{turb}(x) = C_{turb} x^{-0.2}\) 3. \(\bar{h}_{lam}(x) = \frac{2C_{lam}}{\sqrt{x}}\) 4. \(\bar{h}_{turb}(x) = \frac{5C_{turb}}{4(x - x_c)^{0.8}}(x^{0.8} - x_c^{0.8})\) Use any graph plotting software or calculator to plot these expressions in the same graph considering the distance range \(0\leq x\leq L\), where \( L = 1\mathrm{~m}\). This will give you the graphical representation of local and average convection coefficients as a function of distance from the leading edge.

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Key Concepts

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

Reynolds Number
The concept of Reynolds Number is fundamental in fluid dynamics and helps in predicting flow patterns in different fluid flow situations. Reynolds Number, often denoted by \( Re \), is calculated using the formula:
  • \( Re = \frac{\rho VD}{\mu} \)
where:- \( \rho \) is the fluid density- \( V \) is the velocity of the fluid- \( D \) is the characteristic length (like diameter or length of the plate)- \( \mu \) is the dynamic viscosity of the fluid.
Reynolds Number helps us determine whether the flow will be laminar or turbulent. When \( Re \) is below about 2000 for pipe flow (and around 5 x 10^5 for flow over a flat plate), the flow is generally laminar. Above this, the flow tends to become turbulent. It's a dimensionless quantity that provides a powerful way to relate all these factors involved in the flow. Understanding the Reynolds Number is critical in engineering to predict how fluids will behave across surfaces and obstacles.
Laminar Flow
Laminar Flow refers to a type of fluid motion characterized by smooth, parallel layers of fluid that flow in the same direction, often with little to no disruption between them. This type of flow occurs when the Reynolds Number is below a critical threshold, typically \( Re < 2000 \) for flow in pipes.
Several notable characteristics define laminar flow:
  • The flow is steady and layers do not mix.
  • Velocity profile is usually parabolic or linear, depending on the system.
  • Energy loss in the system is minimal when compared with turbulent flow.
In the context of a flat plate, the transition from laminar to turbulent flow happens at a critical Reynolds Number, which affects the convection coefficients and hence the thermal performance. Laminar flow on a flat plate begins at the leading edge and transitions to turbulent as the Reynolds Number increases past the threshold, changing the nature of heat transfer across the plate.
Turbulent Flow
Turbulent Flow is a complex flow regime characterized by chaotic and irregular fluid motion, where layers of fluid mix thoroughly. This type of flow typically occurs when the Reynolds Number is higher than 4000. Here's what to know about turbulent flow:
  • Flow is erratic, with rapid fluctuations in velocity and pressure.
  • Mixes layers and increases the rate of momentum, heat, and mass transfer.
  • Typically leads to higher friction losses compared to laminar flow.
When air moves quickly over a flat plate, it can transition from laminar to turbulent flow past a certain distance, as depicted by a critical Reynolds Number. In turbulent flow regime, local convection coefficients are influenced significantly, often increasing the heat transfer to a higher level due to the improved mixing efficiency. Understanding turbulent flow allows engineers to resolve many practical fluid dynamics challenges in aerodynamics and hydrodynamics, optimizing designs for performance and safety.

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Most popular questions from this chapter

Experiments to determine the local convection heat transfer coefficient for uniform flow normal to a heated circular disk have yielded a radial Nusselt number distribution of the form $$ N u_{D}=\frac{h(r) D}{k}=N u_{o}\left[1+a\left(\frac{r}{r_{o}}\right)^{n}\right] $$ where both \(n\) and \(a\) are positive. The Nusselt number at the stagnation point is correlated in terms of the Reynolds \(\left(R e_{D}=V D / v\right)\) and Prandtl numbers $$ N u_{o}=\frac{h(r=0) D}{k}=0.814 \operatorname{Re}_{D}^{1 / 2} \mathrm{Pr}^{0.36} $$ Obtain an expression for the average Nusselt number, \(\overline{N u}_{D}=\bar{h} D / k\), corresponding to heat transfer from an isothermal disk. Typically, boundary layer development from a stagnation point yields a decaying convection coefficient with increasing distance from the stagnation point. Provide a plausible explanation for why the opposite trend is observed for the disk.

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 \(u_{s}=2 \mathrm{~m} / \mathrm{s}\).

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 the evaporation of water from a liquid film that forms on a contoured surface. Dry air is passed over the surface, and from laboratory measurements the convection heat transfer correlation is of the form $$ \overline{N_{L}}=0.43 \operatorname{Re}_{L}^{0.58} P r r^{\Omega .4} $$ (a) For an air temperature and velocity of \(27^{\circ} \mathrm{C}\) and \(10 \mathrm{~m} / \mathrm{s}\), respectively, what is the rate of evaporation from a surface of \(1-\mathrm{m}^{2}\) area and characteristic length \(L=1 \mathrm{~m}\) ? Approximate the density of saturated vapor as \(\rho_{A, \text { sat }}=0.0077 \mathrm{~kg} / \mathrm{m}^{3}\). (b) What is the steady-state temperature of the liquid film?

The naphthalene sublimation technique involves the use of a mass transfer experiment coupled with an analysis based on the heat and mass transfer analogy to obtain local or average convection heat transfer coefficients for complex surface geometries. A coating of naphthalene, which is a volatile solid at room temperature, is applied to the surface and is then subjected to airflow in a wind tunnel. Alternatively, solid objects may be cast from liquid naphthalene. Over a designated time interval, \(\Delta t\), there is a discernible loss of naphthalene due to sublimation, and by measuring the surface recession at locations of interest or the mass loss of the sample, local or average mass transfer coefficients may be determined. Consider a rectangular rod of naphthalene exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{\mathrm{s}}=300 \mathrm{~K}\), as in Problem 6.10, except now \(c=10 \mathrm{~mm}\) and \(d=30 \mathrm{~mm}\). Determine the change in mass of the \(L=500\)-mm-long rod over a time period of \(\Delta t=30 \mathrm{~min}\). Naphthalene has a molecular weight of \(M_{\mathrm{A}}=128.16 \mathrm{~kg} / \mathrm{kmol}\), and its solid-vapor saturation pressure at \(27^{\circ} \mathrm{C}\) and \(1 \mathrm{ltm}\) is \(p_{\text {A, } a t}=1.33 \times 10^{-4}\) bar.
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