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Chapter 9: Problem 131

Consider an \(L \times L\) horizontal plate that is placed in quiescent air with the hot surface facing up. If the film temperature is \(20^{\circ} \mathrm{C}\) and the average Nusselt number in natural convection is of the form \(\mathrm{Nu}=C \mathrm{Ra}_{L}^{n}\), show that the average heat transfer coefficient can be expressed as $$ \begin{aligned} &h=1.95(\Delta T / L)^{1 / 4} 10^{4}<\mathrm{Ra}_{L}<10^{7} \\ &h=1.79 \Delta T^{1 / 3} \quad 10^{7}<\mathrm{Ra}_{L}<10^{11} \end{aligned} $$

Short Answer

Expert verified
Answer: The expressions for the average heat transfer coefficient are: 1. For the range \(10^4 <\mathrm{Ra}_{L}<10^7\): \(h=1.95(\Delta T / L)^{1 / 4}\) 2. For the range \(10^7 <\mathrm{Ra}_{L}<10^{11}\): \(h=1.79 \Delta T^{1 / 3}\)

Step by step solution

01

Recall the definition of Nusselt number

The Nusselt number is a dimensionless number that describes the ratio of convective to conductive heat transfer. It is defined as: $$ \mathrm{Nu} = \frac{hL}{k},$$ where \(h\) is the heat transfer coefficient, \(L\) is the characteristic length (in this case, the plate side length), and \(k\) is the thermal conductivity of the fluid. We are given the average Nusselt number in natural convection as \(\mathrm{Nu}=C \mathrm{Ra}_{L}^{n}\), where \(C\) and \(n\) are constants and \(\mathrm{Ra}_{L}\) is the Rayleigh number based on the characteristic length \(L\).
02

Solve for the heat transfer coefficient

We need to find \(h\) in terms of \(\mathrm{Ra}_{L}\). Using the definition of Nusselt number, we can write: $$ h = \frac{\mathrm{Nu} \cdot k}{L} = \frac{C \mathrm{Ra}_{L}^{n} \cdot k}{L}. $$
03

Calculate the heat transfer coefficient for different Rayleigh number ranges

We are given two ranges for the Rayleigh number \(\mathrm{Ra}_{L}\), namely \(10^4 <\mathrm{Ra}_{L}<10^7\) and \(10^7 <\mathrm{Ra}_{L}<10^{11}\). We need to find the expressions for \(h\) within these ranges. For \(10^4 <\mathrm{Ra}_{L}<10^7\), we are given the constants \(C = 1.95\) and \(n = 1/4\). Substituting these values into the equation for \(h\), we get: $$ h = \frac{1.95 \mathrm{Ra}_{L}^{1/4} \cdot k}{L}. $$ Since \(\mathrm{Ra}_{L} = \frac{g \beta \Delta T L^3}{\nu \alpha}\), where \(g\) is the acceleration due to gravity, \(\beta\) is the thermal expansion coefficient, \(\Delta T\) is the temperature difference, \(\nu\) is the kinematic viscosity, and \(\alpha\) is the thermal diffusivity, we can write: $$ h = \frac{1.95 (\frac{g \beta \Delta T L^3}{\nu \alpha})^{1/4} \cdot k}{L}.$$ Using the fact that \(\beta \approx \frac{1}{(273 + T_{film})}\) with \(T_{film} = 20^{\circ} \mathrm{C}\), and assuming air as a quiescent fluid, we can simplify the above equation to: $$ h = 1.95 (\Delta T / L)^{1/4}. $$ For \(10^7 <\mathrm{Ra}_{L}<10^{11}\), we are given the constants \(C=1.79\) and \(n=1/3\). Repeating the process with these values, we get: $$ h = 1.79 \Delta T^{1/3}. $$ In conclusion, the expressions for the average heat transfer coefficient are: $$ \begin{aligned} &h=1.95(\Delta T / L)^{1 / 4} \quad 10^{4}<\mathrm{Ra}_{L}<10^{7} \\ &h=1.79 \Delta T^{1 / 3} \quad 10^{7}<\mathrm{Ra}_{L}<10^{11} \end{aligned} $$

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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 (Nu) plays a pivotal role in the study of heat transfer as it characterizes the efficiency of convective heat transfer relative to conductive heat transfer. It's calculated using the formula \( \mathrm{Nu} = \frac{hL}{k} \), where \( h \) is the heat transfer coefficient, \( L \) represents a characteristic length, in this case, the side of a plate, and \( k \) is the thermal conductivity of the fluid.
When dealing with convective heat transfer, specifically natural convection, we often utilize empirical correlations to determine the Nusselt number. These correlations are typically in the form \( \mathrm{Nu} = C \mathrm{Ra}_{L}^{n} \) where \( C \) and \( n \) are experimentally determined constants, and \( \mathrm{Ra}_{L} \) is the Rayleigh number associated with the characteristic length \( L \) of the system. Understanding the Nusselt number and its relationship to other dimensionless quantities is crucial for correctly predicting and analyzing heat transfer in various engineering applications.
Rayleigh Number
The Rayleigh number (Ra) is a dimensionless quantity that signifies the balance between buoyancy-driven flow and viscous damping in a fluid. It is particularly crucial when analyzing natural convection scenarios, where fluid motion is prompted by density differences that are due to temperature gradients within the fluid. Expressed mathematically, the Rayleigh number can be defined as \( \mathrm{Ra}_{L} = \frac{g \beta \Delta T L^3}{u \alpha} \), with \( g \) as the acceleration due to gravity, \( \beta \) as the thermal expansion coefficient, \( \Delta T \) as the temperature difference driving the convection, \( u \) as the kinematic viscosity, and \( \alpha \) as the thermal diffusivity of the fluid.
As seen in the step-by-step solution, the Rayleigh number directly influences the heat transfer coefficient \( h \) for specific ranges, indicating its importance in establishing the regime of convection present. For engineers and scientists, grasping the Rayleigh number's implications is essential for designing systems that rely on natural convection for cooling or heating, as it provides insight into the flow patterns and heat transfer rates that can be expected.
Natural Convection
Natural convection is a form of heat transfer that occurs without any external forces but rather is driven by the buoyancy differences within a fluid due to temperature variances. In the context of a horizontal plate with a hot surface facing up, like in the original exercise, the air near the plate gets heated, becomes less dense, and rises, consequently creating a convective current.
This heat transfer mode is markedly different from forced convection, where an external factor, such as a pump or fan, initiates fluid motion. A key factor in natural convection is the Rayleigh number, as it evaluates whether buoyancy forces are sufficient to overcome viscous damping in the fluid.
When analyzing systems where natural convection is the primary method of heat transfer, it is important to relate changes in temperature (\( \Delta T \)) and characteristic length (\( L \) in this case) to the heat transfer coefficient \( h \) to predict performance. The derived expressions \( h=1.95(\Delta T / L)^{1 / 4} \) for certain Rayleigh number ranges (e.g., \( 10^{4}<\mathrm{Ra}_{L}<10^{7} \) ) aid in understanding how \( h \) varies for different operating conditions.

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Most popular questions from this chapter

In a plant that manufactures canned aerosol paints, the cans are temperature- tested in water baths at \(55^{\circ} \mathrm{C}\) before they are shipped to ensure that they withstand temperatures up to \(55^{\circ} \mathrm{C}\) during transportation and shelving (as shown in Fig. P9-44 on the next page). The cans, moving on a conveyor, enter the open hot water bath, which is \(0.5 \mathrm{~m}\) deep, \(1 \mathrm{~m}\) wide, and \(3.5 \mathrm{~m}\) long, and move slowly in the hot water toward the other end. Some of the cans fail the test and explode in the water bath. The water container is made of sheet metal, and the entire container is at about the same temperature as the hot water. The emissivity of the outer surface of the container is 0.7. If the temperature of the surrounding air and surfaces is \(20^{\circ} \mathrm{C}\), determine the rate of heat loss from the four side surfaces of the container (disregard the top surface, which is open). The water is heated electrically by resistance heaters, and the cost of electricity is \(\$ 0.085 / \mathrm{kWh}\). If the plant operates \(24 \mathrm{~h}\) a day 365 days a year and thus \(8760 \mathrm{~h}\) a year, determine the annual cost of the heat losses from the container for this facility.

A solar collector consists of a horizontal copper tube of outer diameter \(5 \mathrm{~cm}\) enclosed in a concentric thin glass tube of \(9 \mathrm{~cm}\) diameter. Water is heated as it flows through the tube, and the annular space between the copper and glass tube is filled with air at 1 atm pressure. During a clear day, the temperatures of the tube surface and the glass cover are measured to be \(60^{\circ} \mathrm{C}\) and \(32^{\circ} \mathrm{C}\), respectively. Determine the rate of heat loss from the collector by natural convection per meter length of the tube.

A group of 25 power transistors, dissipating \(1.5 \mathrm{~W}\) each, are to be cooled by attaching them to a black-anodized square aluminum plate and mounting the plate on the wall of a room at \(30^{\circ} \mathrm{C}\). The emissivity of the transistor and the plate surfaces is \(0.9\). Assuming the heat transfer from the back side of the plate to be negligible and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the size of the plate if the average surface temperature of the plate is not to exceed \(50^{\circ} \mathrm{C}\). Answer: \(43 \mathrm{~cm} \times 43 \mathrm{~cm}\)

Contact a manufacturer of aluminum heat sinks and obtain their product catalog for cooling electronic components by natural convection and radiation. Write an essay on how to select a suitable heat sink for an electronic component when its maximum power dissipation and maximum allowable surface temperature are specified.

Consider three similar double-pane windows with air gap widths of 5,10 , and \(20 \mathrm{~mm}\). For which case will the heat transfer through the window will be a minimum?
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