• Thermodynamics is defined as the study of equilibrium states of a system which has been subjected to some energy transformation. More specifically, thermodynamics is concerned with transformations of heat into mechanical work and of mechanical work into heat.

• A system is a specific sample of matter. In the atmosphere a parcel of air is a system. A system is called open when it exchanges matter and energy with its surroundings (Figure 1.1). In the atmosphere all systems are more or less open. A closed system is a system that does not exchange matter with its surroundings. In this case, the system is always composed of the same pointmasses (a point-mass refers to a very small object, for example a molecule). Obviously, the mathematical treatment of closed systems is not as involved as the one for open systems, which are extremely hard to handle. Because of that, in atmospheric thermodynamics, we assume that most systems are closed. This assumption is justified when the interactions associated with open systems can be neglected. This is approximately true in the following cases. (a) The system is large enough to ignore mixing with its surroundings at the boundaries. For example, a large cumulonimbus cloud may be considered as a closed system but a small cumulus may not. (b) The system is part of a larger homogeneous system. In this case mixing does not significantly change its composition. A system is called isolated when it exchanges neither matter nor energy with its surroundings.

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The first law of thermodynamics arose from the conservation of energy principle. The first law, even though it implies that we cannot create or destroy energy, places no limits on how energy can be transformed from one form to another. Thus, on the basis of the first law, heat can be transformed into work, work into heat, work can be done at the expense of internal energy, and so on. However, if no other laws existed the first law would allow certain phenomena to happen that never happen in reality. For example, consider a heavy body falling on the ground. We will observe that during the impact the body will warm. The opposite phenomenon according to which a body at rest on the ground begins to rise by itself while it is cooling is impossible. Similarly, no engine has yet been built which, for instance, would receive heat from the sea, transform it to work, and then set a ship in motion. Both the above examples are not in disagreement with the first law since the work would be done at the expense of the internal energy of the soil or the sea. The impossibility of these phenomena is due to the second law of thermodynamics, often hailed as the supreme law of nature. We will start our discussion of this law with the following example.

The Carnot cycle

The Carnot cycle is a thermal engine. A thermal engine is one that receives from some source an amount of heat, part of which it transforms into work.

Our atmosphere is basically a two-component system. One component is dry air and the other is water existing in vapor and possibly one of the condensed phases (liquid water or ice). According to Dalton's law, in a mixture of ideal gases each gas can be assumed to behave as if the other gases were absent. As such, in a mixture of dry air, water vapor, and a condensed phase, the “water” system (water vapor + condensed phase) can be treated as being independent of the dry air. In this case, the concepts developed in the previous chapter (for the one-component heterogeneous system “water”) are valid for the two-component heterogeneous system “dry air + water”. We will call the system consisting of dry air and water vapor “moist air” and it can be unsaturated or saturated with water vapor. Since liquid water is absent, moist air is a two-component system with one phase present. As such, according to equation (6.1) we need three state variables to specify the system's state. Usually these variables are taken to be pressure, temperature, and a new variable (to be defined soon) called mixing ratio. If the condensed phase is present and in equilibrium with the vapor phase, then two variables are needed (typically temperature and pressure).

For clarity in our notation we will use the subscripts d, w, v to indicate dry air, liquid water, and water vapor, respectively. The only exception will be for the vapor pressure, which we will denote simply as e. Variables with no subscripts will correspond to a mixture of dry air and water vapor.

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word “hydrodynamic” is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition.

For both types of problem, we can view the mathematical problem as one of determining the consequence of adding an initial perturbation in the velocity, pressure, temperature, or solute concentration fields to a basic unperturbed state. If the perturbation grows in time, the original unperturbed state is said to be unstable.

In the preceding chapters, we focused mainly on fluid dynamics problems, with only an occasional problem involving heat or mass transfer. In this chapter, we change our focus to problems of heat (or single-solute mass) transfer. Specifically, we address the problem of heat (or mass) transfer from a finite body to a surrounding fluid that is moving relative to the body. In this chapter, we concentrate on problems in which the fluid motion is viscous in nature, and thus is “known” (or can be calculated) from creeping-flow theory. Later, after we have considered flows at nonzero Reynolds number, we will also consider heat (or mass) transfer for this situation.

In all of the fluid mechanics problems that we have considered until now, the nonlinear inertia terms in the equations of motion were either identically zero or small compared with the viscous terms. We begin this chapter by considering the corresponding heat (or mass) transfer problem, in which the fluid motion is “slow” in a sense to be described shortly, so that convection effects are weak and the transport process is dominated by conduction. When convection terms in the thermal energy equation can be neglected altogether, the resulting pure conduction problem is mathematically and physically analogous to the creeping flows that we have been studying in the preceding two chapters. The transport of heat is purely “diffusive” in this limit, i.e., conduction, just as the transport of momentum (or vorticity) in a creeping flow is also “diffusive.”

Although the application of the “thin-film” approximation to analyze lubrication problems is one of its most important successes, there is an even larger body of problems in which the thin-film approximation can still be applied but in which the upper surface (or in some cases both surfaces of the thin film) is an interface. Examples include such diverse applications as gravity currents in geological phenomena, such as the gravitationally driven spread of molten lava; the dynamics of foams and or emulsions for which the thin films between bubbles (or drops) play a critical role in the dynamics; the dynamics of thin films in coating operations, and a variety of other materials processing applications; and thin films in biological systems, such as the coatings of the lung. Not only are the areas of application very diverse, but such films can and do display an astonishing array of complex phenomena, in spite of the limitations inherent in the thin-film assumptions. In part this is a consequence of the wide variety of physical effects that can play a role, including the capillary and Marangoni phenomena associated with surface tension, the possibility of a significant role for nonhydrodynamic effects such as van der Waals forces across the thin film and the possibility of transport processes such as evaporation/condensation.

In this chapter, we derive the governing equations for this class of thin films and show how they can be modified to account for the presence or absence of the various physical phenomena that were mentioned above.

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2–20), (2–88) with the last term set equal to zero, and (2–93).

The first thing to note is that this set of equations is highly nonlinear. This can clearly be seen in the term u · grad u in (2–88). However, because the material properties such as ρ, Cp, and k are all functions of the temperature θ, and the latter is a function of the velocity u through the convected derivative on the left-hand side of (2–93), it can be seen that almost every term of (2–88) and (2–93) involves a product of at least two unknowns either explicitly or implicitly. In contrast, all of the classical analytic methods of solving partial differential equations (PDEs) (for example, eigenfunction expansions by means of separation of variables, or Laplace and Fourier transforms) require that the equation(s) be linear. This is because they rely on the construction of general solutions as sums of simpler, fundamental solutions of the DEs.

In Chap. 9 we considered strong-convection effects in heat (or mass) transfer problems at low Reynolds numbers. The most important findings were the existence of a thermal boundary layer for open-streamline flows at high Peclet numbers and the fundamental distinction between open- and closed-streamline flows for heat or mass transfer processes at high Peclet numbers. An important conclusion in each of these cases is that conduction (or diffusion) plays a critical role in the transport process, even though Pe → ∞. In open-streamline flows, this occurs because the temperature field develops increasingly large gradients near the body surface as Pe → ∞. For closed-streamline flows, on the other hand, the temperature gradients are O(1) – except possibly during some initial transient period – and conduction is important because it has an indefinite time to act.

In this chapter we continue the development of these ideas by considering their application to the approximate solution of fluid mechanics problems in the asymptotic limit Re → ∞, with a particular emphasis on problems in which boundary layers play a key role. Before embarking on this program, however, it is useful to highlight the expected goals and limitations of the analysis in which we formally require Re → ∞ but still assume that the flow remains laminar. In practice, of course, most flows will become unstable at a large, but finite, value of Reynolds number and eventually undergo a transition to turbulence, and this is the flow we will see in the lab.

This book represents a major revision of my book Laminar Flow and Convective Transport Processes that was published in 1992 by Butterworth-Heinemann. As was the case with the previous book, it is about fluid mechanics and the convective transport of heat (or any passive scalar quantity) for simple Newtonian, incompressible fluids, treated from the point of view of classical continuum mechanics. It is intended for a graduate-level course that introduces students to fundamental aspects of fluid mechanics and convective transport processes (mainly heat transfer and some single solute mass transfer) in a context that is relevant to applications that are likely to arise in research or industrial applications. In view of the current emphasis on small-scale systems, biological problems, and materials, rather than large-scale classical industrial problems, the book is focused more on viscous phenomena, thin films, interfacial phenomena, and related topics than was true 14 years ago, though there is still significant coverage of high-Reynolds-number and high-Peclet-number boundary layers in the second half of the book. It also incorporates an entirely new chapter on linear stability theory for many of the problems of greatest interest to chemical engineers.

The material in this book is the basis of an introductory (two-term) graduate course on transport phenomena. It starts with a derivation of all of the necessary governing equations and boundary conditions in a context that is intended to focus on the underlying fundamental principles and the connections between this topic and other topics in continuum physics and thermodynamics.