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Chapter 6: Problem 56

What are the advantages of nondimensionalizing the convection equations?

Short Answer

Expert verified
Short Answer: Nondimensionalizing the convection equations provides several benefits which include simplifying the equations, identifying crucial dimensionless parameters, facilitating scaling analysis, reducing computational effort, and improving the interpretation of results. These advantages lead to a better understanding and management of fluid mechanics and heat transfer problems associated with convection.

Step by step solution

01

Understanding Convection Equations

Convection is the transfer of heat and mass in a fluid due to the flow of the fluid and the substance being carried within it (heat, mass, or momentum). The convection equations describe this transfer behavior mathematically. These equations involve several variables and parameters, such as flow velocity, density, viscosity, thermal conductivity and heat capacity, which makes them quite complex to solve.
02

Concept of Nondimensionalization

Nondimensionalization is a technique used to simplify the presentation and analysis of problems in fluid mechanics and heat transfer. It is done by scaling the variables and parameters in the equations (like length, time, and velocity) with appropriate reference values, so that we arrive at dimensionless quantities. These dimensionless quantities help in identifying the critical parameters governing the problem and help to simplify the analysis of the problem.
03

Advantage 1: Simplification of Equations

Nondimensionalizing the convection equations simplifies the equations by reducing the number of variables and eliminating the need for units. It makes the analysis and interpretation of the results more manageable, allowing easy comparison between different cases.
04

Advantage 2: Identification of Dimensionless Parameters

Dimensionless parameters such as Reynolds number, Prandtl number, and Grashof number play a crucial role in characterizing the fluid flow and heat transfer behavior. By nondimensionalizing the convection equations, these important dimensionless parameters come to light and help determine the dominant terms or effects in the problem.
05

Advantage 3: Scaling Analysis

By nondimensionalizing the convection equations, it becomes easy to compare and contrast the results obtained for different cases or configurations with different scales. This allows for identifying trends or patterns in the data, aiding in the decision-making process for research or engineering applications.
06

Advantage 4: Reduction in Computational Effort

Nondimensionalizing the convection equations leads to a reduction in the number of variables in the problem, which in turn reduces the computational effort required to solve the equations. This is particularly valuable when dealing with complex or large-scale problems, where computational resources are limited.
07

Advantage 5: Better Interpretation of Results

Nondimensionalization helps in the better interpretation of results by offering a clear view of the dominant variables and parameters in the problem. By working with dimensionless quantities, it is easier to understand the underlying physics governing the convection behavior and the relative importance of different effects. In summary, nondimensionalizing the convection equations offers several advantages that include simplifying the equations, identifying important dimensionless parameters, enabling scaling analysis, reducing computational effort, and better interpreting results. These advantages help in better understanding and managing the convection problems in fluid mechanics and heat transfer.

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Most popular questions from this chapter

In an effort to prevent the formation of ice on the surface of a wing, electrical heaters are embedded inside the wing. With a characteristic length of \(2.5 \mathrm{~m}\), the wing has a friction coefficient of \(0.001\). If the wing is moving at a speed of \(200 \mathrm{~m} / \mathrm{s}\) through air at 1 atm and \(-20^{\circ} \mathrm{C}\), determine the heat flux necessary to keep the wing surface above \(0^{\circ} \mathrm{C}\). Evaluate fluid properties at \(-10^{\circ} \mathrm{C}\).

A 6-cm-diameter shaft rotates at \(3000 \mathrm{rpm}\) in a 20 -cm-long bearing with a uniform clearance of \(0.2 \mathrm{~mm}\). At steady operating conditions, both the bearing and the shaft in the vicinity of the oil gap are at \(50^{\circ} \mathrm{C}\), and the viscosity and thermal conductivity of lubricating oil are \(0.05 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) and \(0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). By simplifying and solving the continuity, momentum, and energy equations, determine \((a)\) the maximum temperature of oil, \((b)\) the rates of heat transfer to the bearing and the shaft, and \((c)\) the mechanical power wasted by the viscous dissipation in the oil.

A metal plate \(\left(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2800 \mathrm{~kg} / \mathrm{m}^{3}\right.\), and \(\left.c_{p}=880 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) with a thickness of \(1 \mathrm{~cm}\) is being cooled by air at \(5^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the initial temperature of the plate is \(300^{\circ} \mathrm{C}\), determine the plate temperature gradient at the surface after 2 minutes of cooling. Hint: Use the lumped system analysis to calculate the plate surface temperature. Make sure to verify the application of this method to this problem.

For what types of fluids and flows is the viscous dissipation term in the energy equation likely to be significant?

A flat plate is subject to air flow parallel to its surface. The average friction coefficient over the plate is given as $$ \begin{aligned} &C_{f}=1.33\left(\operatorname{Re}_{L}\right)^{-1 / 2} \text { for } \operatorname{Re}_{L}<5 \times 10^{5} \text { (laminar flow) } \\ &C_{f}=0.074\left(\operatorname{Re}_{L}\right)^{-1 / 5} \text { for } 5 \times 10^{5} \leq \operatorname{Re}_{L} \leq 10^{7} \text { (turbulent flow) } \end{aligned} $$ The plate length parallel to the air flow is \(1 \mathrm{~m}\). Using EES (or other) software, determine the effect of air velocity on the average convection heat transfer coefficient for the plate. By varying the air velocity for \(0
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