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There are many situations where analytic determination of the heat transfer coefficient h is difficult to obtain. As was shown in previous chapters, even after making many simplifying assumptions, analytical determination of h is generally not a simple mathematical problem. When complicating factors such as geometry, variable properties, turbulent flow, boiling, condensation, etc. are involved, the heat transfer coefficient is usually determined experimentally. This does not mean that each time there is a need for h for which there is no analytic solution we must conduct an experiment. Instead, we utilize the experimental results of other researchers. Experimental results are usually correlated and presented as dimensionless equations which are convenient to use. Such equations are known as correlation equations. They are extensively used in the solution of heat transfer problems and therefore deserve special attention.

Research on fluid flow and heat transfer in microchannels was partly driven by miniaturization of microelectronic devices. The need for efficient cooling methods for high heat flux components focused attention on the cooling features of microchannels. Microchannels are used in a variety of engineering and scientific applications. The inkjet printer is a classic example. Extensive use is found in medical applications and in micoelecro- mechanical systems (MEMS) such as micro heat exchangers, mixers, pumps, turbines, sensors and actuators.

This textbook presents the classical topics of conduction heat transfer and extends the coverage to include chapters on perturbation methods, heat transfer in living tissue, and microscale conduction. This makes the book unique among the many published textbook on conduction heat transfer. Other noteworthy features of the book are: The material is organized to provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer. Mathematical techniques are presented in a clear and simplified fashion to be used as instruments in obtaining solutions. The simplicity of one-dimensional conduction is used to drill students in the role of boundary conditions and to explore a variety of physical conditions that are of practical interest. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Students are trained to follow a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification. Solutions to all examples and end-of-chapter problems follow an orderly problems solving approach. (orig.)

There are many applications in which a material undergoes phase change such as in melting, freezing, casting, ablation, cryosurgery and soldering. Conduction with phase change is characterized by a moving interface separating two phases. Such problems are usually referred to as moving boundary or free boundary problems. They are inherently transient during interface motion. The motion and location of the interface are unknown a priori and thus must be determined as part of the solution. Since material properties change following phase transformation and since a discontinuity in temperature gradient exists at the interface, it follows that each phase must be assigned its own temperature function. Furthermore, changes in density give rise to motion of the liquid phase. If the effect of motion is significant, the heat equation of the liquid phase must include a convective term. However, in most problems this effect can be neglected.

Turbulent flow is a complicated physical phenomenon, and a daunting subject for students of engineering. Turbulent flow is disordered, with random and unsteady velocity fluctuations. This description by itself suggests that exact predictions cannot be determined, and that analysis of turbulent flows will not be easy. However intimidating turbulence may be, its importance is clear: turbulent flows are found in many industrial and natural processes, and affect quantities of great practical importance to engineers, like the local velocity distribution, drag force, and heat transfer. Moreover, a better understanding of this flow allows engineers to make design decisions that could either reduce the effects of turbulence or enhance them, in order to improve the performance of their devices.

The mathematical complexity of convection heat transfer is traced to the non-linearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields. The boundary layer concept, first introduced by Prandtl [1] in 1904, provides major simplifications. This concept is based on the notion that under special conditions certain terms in the governing equations are much smaller than others and therefore can be neglected without significantly affecting the accuracy of the solution. This raises two questions

There are various situations where it is desirable to obtain approximate analytic solutions. An obvious case is when an exact solution is not available or can not be easily obtained. Approximate solutions are also obtained when the form of the exact solution is not convenient to use. Examples include solutions that are too complex, implicit or require numerical integration. The integral method is used extensively in fluid flow, heat transfer and mass transfer. Because of the mathematical simplifications associated with this method, it can deal with such complicating factors as turbulent flow, temperature dependent properties and non-linearity.

Two-dimensional steady state conduction is governed by a second order partial differential equation. A solution must satisfy the differential equation and four boundary conditions. The method of separation of variables [1] will be used to construct solutions. The mathematical tools needed to apply this method will be outlined first. Examples will be presented to illustrate the application of this method to the solution of various problems.

We discussed laminar flow in channels in Chapter 6, and many of the features of turbulent flow are similar. We will begin this subject with the criteria for fully developed velocity and temperature profiles, and will focus most of our attention on analyzing fully developed flows. As in Chapter 6, our analysis is limited to general boundary conditions: (i) uniform surface temperature, and (ii) uniform surface heat flux.

In a moving fluid the three fundamental laws, conservation of mass, momentum, and energy, must be satisfied at every point in the domain. Thus the first step is to formulate (cast) the three laws in a form that satisfies this condition. This is accomplished by applying each law to a differential (infinitesimal) element. Following this approach, each law is described by a partial differential equation. Differential formulation of the three laws will be presented using rectangular coordinates. The corresponding forms in cylindrical and spherical coordinates will be stated without details.