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Chapter 7: Problem 8

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.

Short Answer

Expert verified
The required heat generation in the module is \(\dot{q} = 480.96 \mathrm{~W/m^2}\) and the maximum temperature in the heat-generating module is \(T_{max} = 275.25^\circ \mathrm{C}\).

Step by step solution

01

Determine the Reynolds number at 700 mm from the leading edge

First, we need to determine the Reynolds number at \(x = 700 \mathrm{~mm}\) from the leading edge. The Reynolds number is given by: \[Re_x = \frac{Ux}{\nu}\] where \(U = 30 \mathrm{~m/s}\) is the flow velocity, \(x = 0.7 \mathrm{~m}\) is the distance from the leading edge, and \(\nu\) is the kinematic viscosity of the air. Assuming the air viscosity to be \(\nu = 1.5 \times 10^{-5} \mathrm{~m^2/s}\), we have: \[Re_x = \frac{30 \times 0.7}{1.5 \times 10^{-5}} = 1.4 \times 10^6\]
02

Calculate the Nusselt number using a correlation for laminar flow over a flat plate

Since the Reynolds number is larger than \(5\times10^5\), the flow is turbulent. We can use the correlation for turbulent flow over a flat plate to find the Nusselt number: \[\frac{Nu_x}{Re_x^{1/5}} = 0.037Pr^{1/3}\] where \(Nu_x = \frac{h x}{k_f}\) is the Nusselt number, \(h\) is the convective heat transfer coefficient, and \(k_f\) is the thermal conductivity of the fluid. For air, we have \(k_f = 0.026 \mathrm{~W/m \cdot K}\) and \(Pr = 0.7\). Thus, the Nusselt number is: \[Nu_x = 0.037(1.4\times10^6)^{1/5}Pr^{1/3}=103.6\]
03

Determine the convective heat transfer coefficient

Now we can find the convective heat transfer coefficient \(h\): \[h = \frac{Nu_x k_f}{x}\] \[h = \frac{103.6 \times 0.026}{0.7} = 3.84 \mathrm{~W/m^2 \cdot K}\]
04

Calculate the required power generation in the module

We can now find the required power generation in the module using the following equation: \[\dot{q} = h(T_s - T_\infty)\] where \(T_s = 150^\circ \mathrm{C}\) is the surface temperature, and \(T_\infty = 25^\circ \mathrm{C}\) is the atmospheric temperature. Thus, we have: \[\dot{q} = 3.84 (150 -25) = 480.96 \mathrm{~W/m^2}\]
05

Find the maximum temperature in the heat-generating module

We can find the maximum temperature in the module by adding the temperature rise produced by the heat generation to the surface temperature \(T_s\). First, we need to find the temperature rise using the equation: \[\Delta T = \frac{\dot{q}}{h} = \frac{480.96}{3.84} = 125.25^\circ \mathrm{C}\] Then, the maximum temperature in the module is: \[T_{max} = T_s + \Delta T = 150 + 125.25 = 275.25^\circ \mathrm{C}\] Finally, we have found that the required heat generation in the module is \(\dot{q} = 480.96 \mathrm{~W/m^2}\) and the maximum temperature in the heat-generating module is \(T_{max} = 275.25^\circ \mathrm{C}\).

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Key Concepts

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

Reynolds Number
Understanding the Reynolds Number is key when analyzing fluid flow over surfaces. It is a dimensionless number that helps predict flow patterns. The Reynolds Number, denoted by \(Re \), is determined using the formula: \[ Re_x = \frac{Ux}{u} \] where \( U \) is the velocity of the fluid, \( x \) is the characteristic length (in this case, the distance from the leading edge), and \( u \) is the kinematic viscosity of the fluid. The Reynolds Number indicates whether the flow is laminar or turbulent. For flow over a flat plate, a \(Re \) less than \(5 \times 10^5\) suggests laminar flow, while a \(Re \) greater than \(5 \times 10^5\) indicates turbulent flow. Turbulent flow is characterized by chaotic and eddying motion, which enhances mixing and thus affects heat transfer rates.
  • Laminar Flow: Streamlined and orderly.
  • Turbulent Flow: Disordered and mixed.
Nusselt Number
The Nusselt Number \(Nu\), another dimensionless number, provides insight into the convective heat transfer occurring at the surface. It is defined as: \[ Nu_x = \frac{hx}{k_f} \] where \( h \) is the convective heat transfer coefficient and \( k_f \) is the thermal conductivity of the fluid. The Nusselt Number represents the ratio of convective to conductive heat transfer across a boundary. The larger the \(Nu \), the more significant the convective heat transfer. In this context, a higher Nusselt Number typically results when the flow is turbulent.
The relationship between the Reynolds Number and the Nusselt Number is often given through empirical correlations depending on whether the flow is laminar or turbulent. For turbulent flow over a flat plate, the correlation used is: \[ \frac{Nu_x}{Re_x^{1/5}} = 0.037 Pr^{1/3} \] where \(Pr \) is the Prandtl Number, a measure of fluid flow properties that links viscosity and thermal diffusivity. This correlation helps determine \(Nu_x\), essential for calculating the heat transfer coefficient.
Convective Heat Transfer Coefficient
The Convective Heat Transfer Coefficient \(h\) is a fundamental part of analyzing heat transfer between a surface and a fluid. It is found from the Nusselt Number with the expression: \[ h = \frac{Nu_x k_f}{x} \] The coefficient \(h\) represents the heat transfer capability due to convection, measured in \( \text{W/m}^2 \cdot \text{K} \). It is indicative of the rate at which heat dissipates from the surface to a fluid in motion. Higher values of \(h\) suggest more efficient heat dissipation. In our context, achieving a higher \(h\) is crucial for preventing overheating of surfaces, especially when dealing with heat-generating components.
Understanding \(h\) is essential for designing cooling processes, improving thermal performance, and ensuring safe operating temperatures in technological applications.
Thermophysical Properties
Thermophysical Properties of materials are crucial in determining how heat is transferred in systems. They include:
  • Thermal conductivity \(k\): Measures a material's ability to conduct heat. High \(k\) indicates good thermal conducting ability.
  • Specific heat \(c_p\): The amount of heat per unit mass required to raise the temperature by one degree Celsius. Important for understanding heat capacity.
  • Density \(\rho\): The mass per unit volume, impacting how a material absorbs and retains heat.
These properties define how components in thermal systems interact, affecting heat diffusion and overall thermal performance. Understanding these helps in selecting materials for specific thermal requirements, ensuring stability and efficiency in heat-related applications.
Turbulent Flow
Turbulent Flow occurs when a fluid moves in a chaotic, irregular manner, often described as having swirling vortices. This type of flow significantly enhances heat transfer compared to laminar flow. The transition to turbulence typically occurs at higher Reynolds Numbers, as observed in our exercise.
Turbulent flow leads to:
  • Enhanced mixing of fluid layers.
  • Increased heat and mass transfer rates.
  • Greater energy dissipation.
The turbulence's chaotic nature promotes mixing, making it easier for heat to move from the surface to the moving fluid, resulting in a higher convective heat transfer coefficient. As such, understanding and predicting turbulent behavior is vital for engineering processes, as it directly influences the design of heat exchangers and cooling systems.

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Most popular questions from this chapter

A circular pipe of 25 -mm outside diameter is placed in an airstream at \(25^{\circ} \mathrm{C}\) and 1 -atm pressure. The air moves in cross flow over the pipe at \(15 \mathrm{~m} / \mathrm{s}\), while the outer surface of the pipe is maintained at \(100^{\circ} \mathrm{C}\). What is the drag force exerted on the pipe per unit length? What is the rate of heat transfer from the pipe per unit length?

An uninsulated steam pipe is used to transport hightemperature steam from one building to another. The pipe is of \(0.5-\mathrm{m}\) diameter, has a surface temperature of \(150^{\circ} \mathrm{C}\), and is exposed to ambient air at \(-10^{\circ} \mathrm{C}\). The air moves in cross flow over the pipe with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). (a) What is the heat loss per unit length of pipe? (b) Consider the effect of insulating the pipe with a rigid urethane foam \((k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Evaluate and plot the heat loss as a function of the thickness \(\delta\) of the insulation layer for \(0 \leq \delta \leq 50 \mathrm{~mm}\).

A square ( \(10 \mathrm{~mm} \times 10 \mathrm{~mm}\) ) silicon chip is insulated on one side and cooled on the opposite side by atmospheric air in parallel flow at \(u_{\infty}=20 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=\) \(24^{\circ} \mathrm{C}\). When in use, electrical power dissipation within the chip maintains a uniform heat flux at the cooled surface. If the chip temperature may not exceed \(80^{\circ} \mathrm{C}\) at any point on its surface, what is the maximum allowable power? What is the maximum allowable power if the chip is flush mounted in a substrate that provides for an unheated starting length of \(20 \mathrm{~mm}\) ?

Copper spheres of \(20-\mathrm{mm}\) diameter are quenched by being dropped into a tank of water that is maintained at \(280 \mathrm{~K}\). The spheres may be assumed to reach the terminal velocity on impact and to drop freely through the water. Estimate the terminal velocity by equating the drag and gravitational forces acting on the sphere. What is the approximate height of the water tank needed to cool the spheres from an initial temperature of \(360 \mathrm{~K}\) to a center temperature of \(320 \mathrm{~K}\) ?

Consider the following fluids, each with a velocity of \(V=5 \mathrm{~m} / \mathrm{s}\) and a temperature of \(T_{\infty}=20^{\circ} \mathrm{C}\), in cross flow over a 10-mm-diameter cylinder maintained at \(50^{\circ} \mathrm{C}\) : atmospheric air, saturated water, and engine oil. (a) Calculate the rate of heat transfer per unit length, \(q^{\prime}\), using the Churchill-Bernstein correlation. (b) Generate a plot of \(q^{\prime}\) as a function of fluid velocity for \(0.5 \leq V \leq 10 \mathrm{~m} / \mathrm{s}\).
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