Laminar free convection heat transfer from vertical cylinders and wires

Abstract

This thesis presents investigations on laminar free?convection heat transfer to a quiescent fluid from vertical cylinders and wires with the following boundary conditions:

Isothermal surface temperature, Non?isothermal surface temperature variation, Uniform heat?flux condition.

A new method transforms the conventional system of partial differential equations governing the problem into ordinary differential equations by incorporating the boundary conditions. The development of a suitable iterative scheme overcomes the difficulties encountered in the numerical solution of these ordinary differential equations. This transformation process and the iterative scheme help to solve the problem for different boundary conditions and provide a clearer understanding of the effect of transverse curvature on heat transfer and fluid flow. Chapter 1 summarizes important, relevant investigations in the literature and indicates the scope of the present study, as well as the range of parameters examined. Chapter 2 states that the inadequacy of conventional analytical methods necessitates the development of a new approach, which begins the solution at the inner boundary represented by a parameter. Such a procedure transforms the set of partial differential equations into ordinary differential equations. Coupled nonlinear two?point, two?search boundary?value problems-as commonly encountered in free?convection studies-present numerical difficulties. The two unknown quantities required to initiate integration from the inner boundary are chosen to satisfy the outer boundary conditions, necessitating iterative refinement. A systematic iteration scheme, developed through numerical experimentation, is presented in Chapter 3. Chapter 4 investigates in detail the problem of vertical isothermal cylinders and wires. The transformations introduced in Chapter 2 and the iterative method of Chapter 3 are used to solve the problem for parametric values of the variable ? and the Prandtl number. The range of ? is from 10 to 1 in steps of 0.1 and 5, covering the complete class of cylinders. Prandtl numbers of 0.01, 0.1, 0.733, and 5 represent liquid metals, gases, and water. Velocity and temperature distributions obtained for these parameters reveal the usual features of free?convection flow. A study of wall heat transfer variation with ? shows that the parameter classifies cylinders into three categories: short cylinders (flat plates), long cylinders, and wires. In agreement with this classification, the heat?transfer and fluid?flow results compare well with many experimental and analytical findings reported in the literature. Additional experiments using water in both cylinder and wire regimes show close agreement (within about 5%) with analytical predictions. The method described in Chapters 1 and 2 is also applied, in Chapter 5, to two types of non?isothermal surface?temperature conditions for vertical cylinders:

Power?law surface?temperature variation, Exponential surface?temperature variation.

Chapter 6 deals with the uniform?heat?flux case. The conclusions drawn from Chapters 5 and 6 are:

The values (Rax?)=0.05(Ra_x^{*}) = 0.05(Rax??)=0.05 and Rax?=10,000Ra_x^{*} = 10,000Rax??=10,000 effectively demarcate wires from long cylinders, and long cylinders from short ones (flat plates), in all cases except exponential surface?temperature variation. In that case, the values 0.01 and 5 (for air) successfully classify the family of cylinders. Heat?transfer parameters are greater in all cases compared with the isothermal case.

Chapter 7 compares results for the different boundary conditions across all cylinder categories and presents the general conclusions. The most important conclusion is that higher heat?transfer rates can be obtained by decreasing the cylinder diameter, with other conditions remaining the same. Since the isothermal case generally yields the lowest heat?transfer rate among all boundary conditions studied, designs based on this case will be conservative.