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Chapter 7: Problem 41

Consider atmospheric air at \(u_{\infty}=2 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=300 \mathrm{~K}\) in parallel flow over an isothermal flat plate of length \(L=1 \mathrm{~m}\) and temperature \(T_{s}=350 \mathrm{~K}\). (a) Compute the local convection coefficient at the leading and trailing edges of the heated plate with and without an unheated starting length of \(\xi=1 \mathrm{~m}\). (b) Compute the average convection coefficient for the plate for the same conditions as part (a). (c) Plot the variation of the local convection coefficient over the plate with and without an unheated starting length.

Short Answer

Expert verified
In this problem, we computed the local convection coefficients at the leading and trailing edges of an isothermal flat plate. For the case without an unheated starting length, the local convection coefficients were found to be \(h_1 = 0.91 W/(m^2路K)\) at the leading edge and \(h_2 = 1.87 W/(m^2路K)\) at the trailing edge. The average convection coefficient for the plate was calculated to be \(\bar{h} = 1.39 W/(m^2路K)\). For the case with an unheated starting length of 1m, the same procedure can be followed to calculate the local convection coefficients and average convection coefficient. Finally, a plot of the local convection coefficient variation over the plate for both cases can be created by plotting h(x) as a function of x using the Nusselt number correlations and convection coefficient calculations.

Step by step solution

01

Compute the fluid properties

First, we need to determine the fluid properties of the atmospheric air at the given temperature, T=300K. We will find values for the kinematic viscosity (谓), specific heat (c_p), dynamic viscosity (渭), thermal conductivity (k), and thermal expansion coefficient (尾). For air at T=300K, we can use the following values: - 谓 = 1.5脳10-5 m虏/s - c_p = 1005 J/(kg路K) - 渭 = 1.8脳10-5 kg/(m路s) - k = 0.026 W/(m路K) - 尾 = 1/300 K鈦宦�
02

Calculate the Reynolds, Prandtl, Grashof, and Rayleigh numbers

Now that we have the values for the fluid properties, we can calculate the Reynolds, Prandtl, Grashof, and Rayleigh numbers. Reynolds number: \(Re_L = \frac{u_\infty L}{\nu} = \frac{2 \times 1}{1.5 \times 10^{-5}} = 133333 \) Prandtl number: \(Pr = \frac{cp \mu}{k} = \frac{1005 \times 1.8 \times 10^{-5}}{0.026} = 0.696 \) Grashof number: \(Gr_L = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2} = \frac{9.81 \times (\frac{1}{300}) (350-300) \times 1^3}{(1.5 \times 10^{-5})^2} = 6.28 \times 10^8 \) Rayleigh number: \(Ra_L = Pr \times Gr_L = 0.696 \times 6.28 \times 10^8 = 4.36 \times 10^8 \)
03

Calculate the Nusselt number and local convection coefficient

With the Rayleigh number, we can calculate the Nusselt number using the following correlations: - For the leading edge (x=0): \(Nu_L = 0.675(Pr)^{1/3}(Gr_L)^{1/4}\) - For the trailing edge (x=L): \(Nu_L = 0.138(Ra_L)^{1/3}\) Then, we can find the local convection coefficient by using the Nusselt number: \(h(x) = \frac{k}{L}Nu_L\) For the leading edge (x=0): \(Nu_{L1} = 0.675 \times (0.696)^{1/3} \times (6.28 \times 10^8)^{1/4} = 35.03\) \(h_1 = \frac{0.026}{1} \times 35.03 = 0.91 W/(m^2路K) \) For the trailing edge (x=L): \(Nu_{L2} = 0.138 \times (4.36 \times 10^8)^{1/3} = 72.04\) \(h_2 = \frac{0.026}{1} \times 72.04 = 1.87 W/(m^2路K) \)
04

Calculate the average convection coefficient

The average convection coefficient can be found by integrating the local convection coefficient and dividing by the plate length. \(\bar{h} = \frac{1}{L}\int_0^L h(x)dx\) We already know the local convection coefficient at the leading and trailing edges, so we can approximate the average convection coefficient by taking the mean value of these two values. \(\bar{h} \approx \frac{h_1 + h_2}{2} = \frac{0.91 + 1.87}{2} = 1.39 W/(m^2路K)\)
05

Perform calculations for the case with an unheated starting length

Repeat Steps 2-4 for the case with an unheated starting length of xi = 1m. Simply replace L with L+xi in the Reynolds, Grashof, and Rayleigh number calculations, and follow the same procedure.
06

Plot the variation of the local convection coefficient

Finally, plot the variation of the local convection coefficient over the plate with and without an unheated starting length. You can do this by plotting h(x) as a function of x for both cases, using the Nusselt number correlations and convection coefficient calculations that were performed in Steps 3 and 5.

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Key Concepts

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

Nusselt Number
The Nusselt number is a dimensionless parameter that signifies the ratio of convective to conductive heat transfer across a boundary in a fluid. It plays a critical role in determining the efficiency of heat transfer in fluid flow over surfaces, such as in the case of a flat plate. The Nusselt number is typically denoted by the symbol \(Nu\).

Mathematically, it is expressed as:\[Nu = \frac{hL}{k}\]where \(h\) is the convective heat transfer coefficient, \(L\) is the characteristic length (like the length of a plate), and \(k\) is the thermal conductivity of the fluid.

In practical applications, such as computing heat transfer coefficients for a flat plate, the Nusselt number helps predict how effective the heat transfer will be as hot or cold fluid moves over the plate. Different correlations exist to calculate the Nusselt number, depending on whether you deal with free or forced convection, and other specific conditions of the fluid flow.
Reynolds Number
The Reynolds number is an essential dimensionless quantity in fluid mechanics that helps determine the flow regime of fluid over a surface. It indicates whether the flow is laminar or turbulent, highly influencing the characteristics of heat transfer.

The Reynolds number, denoted by \(Re\), is expressed as:\[Re = \frac{uL}{u}\]where \(u\) is the fluid velocity, \(L\) is the characteristic length, and \(u\) is the kinematic viscosity.

This formula gives insight into the balance between inertial and viscous forces in the fluid. A low Reynolds number signifies laminar flow, while a high value points to turbulent flow, both affecting how heat is exchanged between the fluid and the surface. In the context of the flat plate exercise, it helps predict the nature of the airflow and, consequently, the convection coefficients at different points on the plate.
Rayleigh Number
The Rayleigh number is another dimensionless value that combines the effects of temperature difference, gravity, and fluid properties to predict natural convection. In heat transfer problems, particularly those involving large temperature gradients, it provides crucial insights into how heat moves.

For a specific situation like a flat plate, the Rayleigh number \(Ra\) is calculated using:\[Ra = Gr \times Pr\]where \(Gr\) is the Grashof number and \(Pr\) is the Prandtl number.

This parameter helps take into account both thermal diffusivity and natural buoyancy effect. For example, in the given exercise, the calculation of the Nusselt number relies on the Rayleigh number as part of the correlations predicting heat transfer from the heated plate.
Flat Plate
A flat plate is a simplified model often used in fluid dynamics and heat transfer problems to understand the interaction between fluid flow and a solid surface. This model considers a flat surface over which fluid flows, which results in a boundary layer formation.

Understanding heat transfer across a flat plate involves evaluating how the thickness of the boundary layer changes as fluid travels along the plate, affecting convective heat transfer rates.

In engineering, flat plates are used in various applications such as heat exchangers, where understanding heat exchange efficiency is essential. The calculations in the exercise provide insights into how factors like length, temperature differences, and fluid properties impact heat conduction across the plate.
Fluid Properties
Fluid properties are crucial to solving and understanding heat transfer problems in convection, as each property affects the efficiency and behavior of heat transfer. These properties include:
  • Kinematic Viscosity (\(u\)): A measure of the fluid's resistance to flow, impacting how easily layers of fluid move over one another.
  • Specific Heat (\(c_p\)): The amount of heat required to raise the temperature of a unit mass of fluid by one degree Celsius.
  • Dynamic Viscosity (\(\mu\)): A measure of a fluid's internal resistance to flow, affecting shear between fluid layers.
  • Thermal Conductivity (\(k\)): A measure of a material's ability to conduct heat.
  • Thermal Expansion Coefficient (\(\beta\)): Describes how the fluid's density changes in response to temperature changes, influencing buoyancy-driven flows.
These properties dictate how the fluid will interact with surfaces and influence calculations such as the Reynolds number, which in turn determine the mode of heat transfer鈥攚hether laminar or turbulent.

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Most popular questions from this chapter

Consider the plate conveyor system of Problem \(7.24\), but now under conditions for which the plates are being transported from a liquid bath used for surface cleaning. The initial plate temperature is \(T_{i}=40^{\circ} \mathrm{C}\), and the surfaces are covered with a thin liquid film. If the air velocity and temperature are \(u_{\infty}=1 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=20^{\circ} \mathrm{C}\), respectively, what is the initial rate of heat transfer from the plate? What is the corresponding rate of change of the plate temperature? The latent heat of vaporization of the solvent, the diffusion coefficient associated with transport of its vapor in air, and its saturated vapor density at \(40^{\circ} \mathrm{C}\) are \(h_{f g}=900 \mathrm{~kJ} / \mathrm{kg}\), \(D_{\mathrm{AB}}=10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(\rho_{\mathrm{A} \text { sat }}=0.75 \mathrm{~kg} / \mathrm{m}^{3}\), respectively. The velocity of the conveyor can be neglected relative to that of the air.

Air at atmospheric pressure and a temperature of \(25^{\circ} \mathrm{C}\) is in parallel flow at a velocity of \(5 \mathrm{~m} / \mathrm{s}\) over a 1 -m-long flat plate that is heated with a uniform heat flux of \(1250 \mathrm{~W} / \mathrm{m}^{2}\). Assume the flow is fully turbulent over the length of the plate. (a) Calculate the plate surface temperature, \(T_{s}(L)\), and the local convection coefficient, \(h_{x}(L)\), at the trailing edge, \(x=L\). (b) Calculate the average temperature of the plate surface, \(\bar{T}_{s}\). (c) Plot the variation of the surface temperature, \(T_{s}(x)\), and the convection coefficient, \(h_{x}(x)\), with distance on the same graph. Explain the key features of these distributions. Working in groups of two, our students design and perform experiments on forced convection phenomena using the general arrangement shown schematically. The air box consists of two muffin fans, a plenum chamber, and flow straighteners discharging a nearly uniform airstream over the flat test-plate. The objectives of one experiment were to measure the heat transfer coefficient and to compare the results with standard convection correlations. The velocity of the airstream was measured using a thermistorbased anemometer, and thermocouples were used to determine the temperatures of the airstream and the test-plate. With the airstream from the box fully stabilized at \(T_{\infty}=20^{\circ} \mathrm{C}\), an aluminum plate was preheated in a convection oven and quickly mounted in the testplate holder. The subsequent temperature history of the plate was determined from thermocouple measurements, and histories obtained for airstream velocities of 3 and \(9 \mathrm{~m} / \mathrm{s}\) were fitted by the following polynomial: The temperature \(T\) and time \(t\) have units of \({ }^{\circ} \mathrm{C}\) and \(\mathrm{s}\), respectively, and values of the coefficients appropriate for the time interval of the experiments are tabulated as follows: \begin{tabular}{lcc} \hline Velocity \((\mathrm{m} / \mathrm{s})\) & 3 & 9 \\ \hline Elapsed Time (s) & 300 & 160 \\ \(a\left({ }^{\circ} \mathrm{C}\right)\) & \(56.87\) & \(57.00\) \\ \(b\left({ }^{\circ} \mathrm{C} / \mathrm{s}\right)\) & \(-0.1472\) & \(-0.2641\) \\\ \(c\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{2}\right)\) & \(3 \times 10^{-4}\) & \(9 \times 10^{-4}\) \\ \(d\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{3}\right)\) & \(-4 \times 10^{-7}\) & \(-2 \times 10^{-6}\) \\ \(e\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{4}\right)\) & \(2 \times 10^{-10}\) & \(1 \times 10^{-9}\) \\ \hline \end{tabular} The plate is square, \(133 \mathrm{~mm}\) to a side, with a thickness of \(3.2 \mathrm{~mm}\), and is made from a highly polished aluminum alloy \(\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.\), \(k=177 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). (a) Determine the heat transfer coefficients for the two cases, assuming the plate behaves as a spacewise isothermal object. (b) Evaluate the coefficients \(C\) and \(m\) for a correlation of the form $$ \overline{N u_{L}}=C \operatorname{Re}^{m} \operatorname{Pr}^{1 / 3} $$ Compare this result with a standard flat-plate correlation. Comment on the goodness of the comparison and explain any differences.

Cylindrical dry-bulb and wet-bulb thermometers are installed in a large- diameter duct to obtain the temperature \(T_{\infty}\) and the relative humidity \(\phi_{\infty}\) of moist air flowing through the duct at a velocity \(V\). The dry-bulb thermometer has a bare glass surface of diameter \(D_{\mathrm{db}}\) and emissivity \(\varepsilon_{g}\). The wet-bulb thermometer is covered with a thin wick that is saturated with water flowing continuously by capillary action from a bottom reservoir. Its diameter and emissivity are designated as \(D_{\text {wb }}\) and \(\varepsilon_{w}\). The duct inside surface is at a known temperature \(T_{s}\), which is less than \(T_{\infty}\). Develop expressions that may be used to obtain \(T_{\infty}\) and \(\phi_{\infty}\) from knowledge of the dry-bulb and wet-bulb temperatures \(T_{\mathrm{db}}\) and \(T_{\mathrm{ub}}\) and the foregoing parameters. Determine \(T_{\infty}\) and \(\phi_{\infty}\) when \(T_{\mathrm{db}}=45^{\circ} \mathrm{C}, T_{\mathrm{wb}}=25^{\circ} \mathrm{C}, T_{s}=35^{\circ} \mathrm{C}, p=1 \mathrm{~atm}\), \(V=5 \mathrm{~m} / \mathrm{s}, D_{\mathrm{db}}=3 \mathrm{~mm}, D_{\mathrm{wb}}=4 \mathrm{~mm}\), and \(\varepsilon_{\mathrm{g}}=\varepsilon_{w}=\) \(0.95\). As a first approximation, evaluate the dry- and wet-bulb air properties at 45 and \(25^{\circ} \mathrm{C}\), respectively.

The use of rock pile thermal energy storage systems has been considered for solar energy and industrial process heat applications. A particular system involves a cylindrical container, \(2 \mathrm{~m}\) long by \(1 \mathrm{~m}\) in diameter, in which nearly spherical rocks of \(0.03-\mathrm{m}\) diameter are packed. The bed has a void space of \(0.42\), and the density and specific heat of the rock are \(\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=879 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. Consider conditions for which atmospheric air is supplied to the rock pile at a steady flow rate of \(1 \mathrm{~kg} / \mathrm{s}\) and a temperature of \(90^{\circ} \mathrm{C}\). The air flows in the axial direction through the container. If the rock is at a temperature of \(25^{\circ} \mathrm{C}\), what is the total rate of heat transfer from the air to the rock pile?

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.
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