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2015 | Book

Thermal Circuits Read chapter Read first chapter

Heat Transfer Modeling

An Inductive Approach

Author: George Sidebotham

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This innovative text emphasizes a "less-is-more" approach to modeling complicated systems such as heat transfer by treating them first as "1-node lumped models" that yield simple closed-form solutions. The author develops numerical techniques for students to obtain more detail, but also trains them to use the techniques only when simpler approaches fail. Covering all essential methods offered in traditional texts, but with a different order, Professor Sidebotham stresses inductive thinking and problem solving as well as a constructive understanding of modern, computer-based practice. Readers learn to develop their own code in the context of the material, rather than just how to use packaged software, offering a deeper, intrinsic grasp behind models of heat transfer. Developed from over twenty-five years of lecture notes to teach students of mechanical and chemical engineering at The Cooper Union for the Advancement of Science and Art, the book is ideal for students and practitioners across engineering disciplines seeking a solid understanding of heat transfer.

This book also:

· Adopts a novel inductive pedagogy where commonly understood examples are introduced early and theory is developed to explain and predict readily recognized phenomena

· Introduces new techniques as needed to address specific problems, in contrast to traditional texts’ use of a deductive approach, where abstract general principles lead to specific examples

· Elucidates readers’ understanding of the "heat transfer takes time" idea—transient analysis applications are introduced first and steady-state methods are shown to be a limiting case of those applications

· Focuses on basic numerical methods rather than analytical methods of solving partial differential equations, largely obsolete in light of modern computer power

· Maximizes readers’ insights to heat transfer modeling by framing theory as an engineering design tool, not as a pure science, as has been done in traditional textbooks

· Integrates practical use of spreadsheets for calculations and provides many tips for their use throughout the text examples

Table of Contents

Frontmatter

Modes of Heat Transfer

Frontmatter
1. Thermal Circuits
Abstract
This introductory chapter develops the concept of a thermal circuit and analytical and numerical methods used to analyze simple circuits. Subsequent chapters will apply and build on them using heat transfer examples from common experience. Key elements, namely, resistors, capacitors, batteries, and current sources, are introduced. Resistive/capacitive (RC) network diagrams are used as modeling tools to visualize the flow of heat, an abstract massless quantity. Hence, the term “thermal circuit.” The focus of this book is to develop heat transfer modeling as a design tool, rather than as a pure science. The principles apply to other areas in engineering (and life) practice. Specific tools are developed in this textbook that pertain to heat transfer phenomena, most of which can be captured in thermal circuits.
George Sidebotham
2. Lumped Capacity Systems and Overall Heat Transfer Coefficients
Abstract
This chapter introduces two practical heat transfer concepts; lumped capacity and overall heat transfer coefficients. The term “lumped capacity system” means that a finite region of space is considered to be at some average temperature, even though there could be hot or cold spots in it. In numerical analysis, subdivisions of a physical object can be made, treating each subdivision as a lumped capacity element. The “overall heat transfer coefficient, U” is a parameter that captures the basic idea of heat transfer, namely, energy transfer that is driven by a temperature difference. Experimental results for a cooling mug of coffee are presented and integrated with a theoretical model that combines both concepts. The focus of this chapter is on the thermal capacity of systems, which depends on the material properties of density and specific heat. The next chapter will focus on the individual thermal resistances of systems, where thermal conductivity and emissivity are key material properties. The remaining chapters will build analysis tools for a widening range of applications.
George Sidebotham
3. Heat Transfer Modes: Conduction, Convection, and Radiation
Abstract
Heat transfer is energy transfer driven by a temperature difference. A first step in estimating the rate of heat transfer between, say, a hot mug of coffee and its environment is to identify the modes of heat transfer and the thermal channels of heat flow between two locations. There are different pathways, or thermal channels, that heat can take from hot coffee to ambient; directly from the coffee to the air above the free surface, through the side wall of the mug, or through the bottom of the mug and through the floor on which the mug rests. Within each of these pathways, heat flow encounters discontinuities in material properties, from coffee, to mug, to air. The mathematical expression of the different underlying physics associated with these discontinuities gives rise to three modes of heat transfer: conduction, convection, and radiation. In contrast, there are two mechanisms of heat transfer (conduction and radiation) that refer to the physical phenomena that give rise to the energy transfer. The mode of heat transfer by convection is based on a conduction mechanism. A thermal channel is defined as a “pathway” for heat to flow from a hot fluid or surface to a cold one, expressible in terms of a series combination of heat transfer modes. In this chapter, the practical application of a single-paned window is used to develop the engineering tools (and the thinking process behind them) used in heat transfer analysis. This example involves a single thermal channel. The basic method is applied more generally, and to the coffee/mug problem with its multiple channels, in subsequent chapters.
George Sidebotham

Transient Conduction

Frontmatter
4. 1-Node Transient Models
Abstract
A basic philosophy of this book is to develop a modeling approach that considers “the big picture” first, and then more detail only if needed. Since heat transfer rates are driven by temperature differences, the first step in analyzing heat transfer problems is usually to solve for the temperature field, which depends on spatial position and time. Numerical analyses involve the discretization of space (breaking the physical region into discrete subelements) and time (taking finite steps forward in time). This chapter considers “1-node” models, where the region of space in question is lumped into a single node. Time dependence often results in easily solved ordinary differential equations, or ones that can be easily solved numerically. By treating a defined system as a single object with average properties, that is, as a single node, a first approximation can often be made that is either sufficient to make design decisions, or directs attention to intelligently develop a more detailed approach. Furthermore, key functional relationships and associated insights are obtained with relatively little mathematical complexity.
George Sidebotham
5. Few-Node Transient Models
Abstract
Try this experiment. Fill a ceramic mug with boiling water and immediately grab its side. Close your eyes and concentrate on what it feels like. You’ve just experienced the mug heating phase, the topic of this chapter. In order to observe the separate behavior of the coffee and the mug, the capacitance of the coffee and the mug must be treated separately, creating a 2-node model. The goal of this chapter is to develop a general scheme for doing so. The term “Few-Node Model” is loosely defined as one in which space is discretized into clearly identifiable regions that can be given clear names (as opposed to numbers). In the next chapter, called “multi-node,” space will be discretized into finer elements in a way that there is no clear sense of how many nodes there should be beforehand.
George Sidebotham
6. Multi-Node Transients
Abstract
This chapter builds on the few-node numerical approach, detailing the side wall of the mug of coffee problem. Conceptually, multi-node problems involve a simple extension of the principles developed for the few-node model. However, in programming a multi-node simulation, some or all of the recursion formulas can be written in terms of an input numerical parameter, say “n” and the governing equations are derived in terms of “n”. These problems cannot be readily handled that way with a spreadsheet solution, where a commitment to a specific number of nodes must be made at the outset (set up the number of columns). Nevertheless, a multi-node model with predetermined number of nodes can be developed in a spreadsheet, as presented in this chapter to observe more detailed thermal response during the mug heating phase.
George Sidebotham

Steady-State Conduction

Frontmatter
7. Heat Transfer Fins (and Handles)
Abstract
This chapter introduces a lumped capacity approach to steady-state conduction problems using the important practical application of heat transfer fins (and handles). The approach is applied to the coffee/mug problem, this time with half a mug, and the heat transfer into the rim and out to ambient is detailed. Extended surfaces are widely used as either heat transfer fins (devices designed to increase the overall rate of heat transfer between a solid and a fluid) or handles (something you can grab without burning your hand). For fin applications, since the rate of heat transfer by convection (and possibly radiation in parallel) is proportional to the exposed surface area, the basic idea is to increase that exposed surface area. However, because the temperature in the fin decreases with the distance from the wall (x), there is a limit to how far the fin can extend and still be effective. A “perfect” fin is one in which the temperature of the fin is the same everywhere as its base. In a real fin, the average temperature of the fin exposed to the fluid will be less than that, and therefore the fin transfers heat at a lower rate. If the fin is long enough, the temperature at the end of the fin approaches the ambient temperature. That’s a handle you can grab. The approach taken is to develop a finite element approach. First, the fin is considered as a single lumped element, a 1-node model. This approach yields a simple closed-form solution that is surprisingly versatile and captures all the key functional dependencies of the exact solution. Next, a multi-element approach (“few node”) is developed considering the fin broken into three equal elements. This method is easily extended to two or three dimensions. Finally, a detailed methodology of the 1-node model is applied to the case of the rim of a mug of coffee.
George Sidebotham
8. Steady-State Conduction
Abstract
After the initial mug heating phase when hot coffee is poured into a ceramic mug, the coffee and mug gradually cool together. During this time, the rate of heat transfer from the coffee to the inner wall of the mug is approximately equal to the rate of heat transfer from the outer wall to ambient. That near balance defines a quasi-steady state situation for the mug itself, and is why the lumping of its capacity with the coffee is reasonable. For problems that are steady-state in nature (or quasi-steady state), a transient model can be developed and then run to equilibrium to obtain the steady-state solution. However, it is possible to solve directly for the steady-state solution, which involves somewhat different mathematics to execute. Methods for doing so were initiated in the previous chapter on fins, and elaborated on in this chapter. Furthermore, the mathematics involved is similar to that required to execute a step forward using an implicit transient solution. The method is first developed and applied to the mug of coffee, and then to a classic 2D problem. Extension to 3D is straightforward.
George Sidebotham

Open Systems: Convection and Heat Exchangers

Frontmatter
9. Nusselt Number Correlations
Abstract
It can be argued that the exchange of energy between a solid and a fluid is important in most, if not virtually all, practical situations for which heat transfer plays a role. The goal of this chapter is to determine the convective heat transfer coefficients between the coffee and the mug and between the mug and ambient that have been used in the various thermal models. The general approach to using Nusselt number correlations is developed, and applied to the mug inner and outer walls. Rough order of magnitude “Rules of Thumb” are thereby tested for this case study. The next chapter explores more fundamentally the physics of the thermal boundary layer. Nusselt number correlations can be obtained from a variety of printed and online sources, and only a few relevant examples are introduced here, as needed.
George Sidebotham
10. Convection Fundamentals
Abstract
Convection coefficients are used as a boundary condition in transient and steady-state conduction problems. An appreciation for the underlying physics of Newton’s “Law” of Cooling requires a closer look at the fluid in the immediate vicinity of a solid object at a different temperature. In this chapter, a 1-node analysis of the flow across a thin flat plate is developed that reveals the underlying physics of convection. The concept of a boundary layer is central to understanding convection coefficients. A boundary layer is a layer of fluid immediately adjacent to a solid that is generally thin compared to a characteristic dimension of the solid object. Outside the boundary layer, the fluid has the properties of the free stream (temperature and flow velocity). At the solid surface, the fluid properties are those of the solid. In between, there are large spatial gradients in velocity and temperature that occur across the boundary layer. In heat transfer, there are two boundary layers that develop simultaneously; a momentum boundary layer and a thermal boundary layer. The retarding effects of viscosity occur within the momentum boundary layer. In fluid mechanics, the forces exerted by a fluid on an object are of primary interest and these are determined by the momentum boundary layer. In heat transfer, heat exchange between the solid object and the free stream occurs across a thermal boundary layer. However, the thermal boundary layer is influenced by the momentum boundary layer, so an understanding of both is needed. Models are developed for both forced convection (with an imposed fluid velocity) and natural convection (where the fluid flow is driven by gravity).
George Sidebotham
11. Internal Flows Models
Abstract
It is common for engineering devices to be designed with pipes or ducts in which fluids (gas or liquid) flow. The fluid enters at an inlet temperature and the walls of the duct are at a different temperature. This chapter addresses internal flows in which a single ducted stream is involved. The next chapter considers heat exchangers, where one stream transfers heat to another. When mass flows into a control volume, it carries the energy associated with it. This mechanism of energy transfer is termed advection and will be modeled as a fluid temperature-dependent current source. There are also momentum considerations, as the flows are driven by pressure (or gravity) and opposed by friction, so that pumps or compressors are generally needed to move the fluid. Several 1-node models are developed (thermally long, thermally short, average temperature, and well mixed) and contrasted. The choice of a model in practice depends on whether temperatures or flow rates are considered to be input parameters.
George Sidebotham
12. Heat Exchanger Fundamentals
Abstract
A heat exchanger can be defined as an engineering device whose primary function is to transfer heat from one fluid to another. The previous chapter focused on the flow of a single stream that exchanges heat with its environment. This chapter considers the exchange of heat between two streams at different temperatures separated by a solid barrier. The aim is to develop a simplified modeling approach that can be used as a preliminary design tool. A closed-form 2-node model is developed that captures the key relationships between fluid types, their flow rates and temperatures, and the heat transfer surface between them. A more detailed model of a specific design configuration (a tube in a tube, or double-pipe) is developed as a case study for a more detailed investigation. There are many different configurations in engineering practice, and there are numerous resources available to obtain more comprehensive treatments of heat exchangers. In the end, practitioners and designers must learn how to interpret manufacturer specifications, which can be presented in various forms. A good fundamental understanding of the key principles involved facilitates that practice.
George Sidebotham
13. Evaporation and Mass Transfer Fundamentals
Abstract
This chapter introduces fundamental mass transfer concepts through the modeling of the evaporation process that takes place at the free surface of a mug of coffee. An evaporative heat transfer coefficient, hevap, is defined that can be included in any of the transient numerical simulations from earlier chapters (i.e., lumped 1-node, few node, multi-node). Evaporation is driven by concentration gradients, not a temperature difference, and defining an evaporation coefficient this way allows direct comparison of the relative effects of evaporation, convection, and radiation on the cooling of the free surface. As will become apparent, this approach makes sense only for cases when the liquid surface temperature differs significantly from ambient. The case where the coffee reaches thermal equilibrium with the environment and the slower evaporation continues is treated differently, and it will be seen that the coffee attains a temperature below ambient, a form of evaporative cooling.
George Sidebotham
Metadata
Title
Heat Transfer Modeling
Author
George Sidebotham
Copyright Year
2015
Electronic ISBN
978-3-319-14514-3
Print ISBN
978-3-319-14513-6
DOI
https://doi.org/10.1007/978-3-319-14514-3

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