Thermodynamics of Energy Conversion and Transport

This chapter deals with the general class of processes in which solar energy is absorbed and in which the absorbed energy is converted in multistage processes until it is eventually in a storable form. Several such processes are discussed, as well as the difficulties of modeling the processes in the framework of statistical mechanics. The description pays special attention to loss mechanisms, including the temporal aspects. Possibilities and limits for optimization, arising from both absolute limits and from irreducible losses, are also discussed.

Furthermore, the connection between statistical mechanics and information theory is analyzed in the context of solar energy conversion. In this context, the usefulness of the exergy concept, as an efficiency measure inthe analysis of solar energy conversion processes, is discussed. The results for the exergy, of general incoherent photon exergy as well as for a more general kind of radiation field, are reviewed.

The conversion of black-body radiation, in the form of free photons within a certain volume, is used as a benchmark case. Several problems with this widely used model are described.

It is clear that the actual achievable exergetic efficiency is in some cases considerably less than for the ideal benchmark case and also that the difference is not quantifiable with the existing models.

The main aim of any energy conversion model is to establish upper limits for the conversion efficiency. This chapter deals with solar energy conversion into mechanical work. The first section refers to models which allow an easy (handy) computation of the conversion efficiency while the second section deals with models which usually require the use of computers.

A solar cell is a thermodynamic engine working between two heat reservoirs, one at high temperature T ₁ (= the temperature of the Sun = 5762 K) and one at low temperature T ₂ (= the temperature of the Earth = 288 K). Its electric current consists of two parts: the light current, strongly dependent on T ₁, and the dark current, strongly dependent both on T ₂ and on material constants and technology parameters.

A survey is given of various theoretical approaches to estimating solar cell efficiencies. We start (Section 4.2) with a development of the usual solar cell equation which is widely used and assume the so-called shift theorem. It is itself an approximation as is shown again here. A theory of the heterojunction solar cell is then developed (Section 4.4), following a brief survey of properties of efficiencies in general (Section 4.3). In this section we also give an introduction to the problem of estimating the effects of impact ionization. This is done by introducing a probability that a current carrier which has enough energy to impact ionize will actually do so. Following simpler special cases (Section 4.5), a more detailed theory of heterojunction cells with impact ionization is then presented (Section 4.6).

As well known, conversion efficiencies can be increased by connecting two or more cells in series, i.e., proceeding from a heterojunction ortandem cell to several cells, or even many cells. This problem is discussed in Section 4.7. It involves radiation theory, based on some elementary quantum mechanics and statistical mechanics. Thermophotovoltaic conversion (Section 4.8) has the benefit of yielding relatively high conversion efficiencies because the energy loss due to the thermalization of the current carries which occurs in a normal solar cell is here reduced. this is due to the fact that the solar energy is first absorbed by a material that reemits radiation at a lower temperature.

A building that has the capacity to control the thermal flow interactions with the outside, the in and out flows, and in addition is able to control the internal thermal flows between adjacent zones, formally acts as a thermodynamic automaton. The automaton rule minimizes the distance of the temperature of the core (living space) from a predetermined desired temperature. The automaton rule must manage the chaotic variations of the external inputs and also its own capacity to learn whether the external inputs belong to summer or winter, etc., because according to season the automaton strategy changes. If backup injection of heating (or cooling) is necessary, how can this be operated? In fact, the building is “alive,” and the injection of energy automatically perturbs the automaton logic; it may fool the automaton into making it believe that it is summer when instead it is winter (or vice versa). A problem of similar nature is known to pilots of highly automated airplanes, for instance in the process of automatic landing, where the intervention of the human action has caused disasters. It is also known in medical science where the introduction of a drug always carries basically unknown side-effects. We are studying the strategy of the “back up” for a class of thermodynamic automata.

The performance of a Diesel engine is analyzed for a model which includes losses due to mechanical friction and heat losses through the cylinder walls. Using the work output of the Diesel engine as an objective, the optimal piston trajectories for the compression and power stroke are determined simultaneously. Results for a linear approximation of the heat leakage are compared to a more realistic, empirical heat transfer law due to Annand. Optimal operating conditions are found and discussed and significant improvements in the engine’s efficiency relative to conventionally designed engines are obtained.

In Section 8.1 we overview the classical theory of heat conduction formulated in thermodynamic terms. The dissipative character of pure heat conduction is manifested in the heat conductional inequality, the maximum principle, and other related properties (Section 8.2). These properties can be stated in general, far beyond the linear theory. The classical heat equation yields an infinite velocity of propagation. The hyperbolic heat equation has been proposed to overcome this paradox. However, the maximum principle is not valid for a hyperbolic equation; a satisfactory theory involving the maximum principle, as well as finite propagation, is not yet known (Section 8.3). The required basic properties may be used as postulates in searching for a new theory. An attempt of this kind is outlined. For a homogeneous one-dimensional-medium it is demonstrated in Section 8.4 that a theory for stationary heat conduction can be derived. The solution of the initial-boundary value problem for the classical linear heat conduction equation of parabolic type has several special characteristic properties, like contractivity in time, nonoscillatory behavior, exponential convergence, and others. In Section 8.5 we list some such properties. As is typical, the continuous problem cannot be solved analytically. In Section 8.6 some numerical processes are applied. This means that we define the approximate solution at the discrete points of a mesh. Obviously, the basic question is the convergence, that is, when refining the mesh, the numerical solution should be convergent to the solution of the original continuous problem. It is no less important to require the preservation of the discrete analogues of the basic qualitative properties of the continuous solution mentioned the above. In Section 8.7 the exact conditions for the preservation of some qualitative properties are established. There are damped traveling wave solutions of the sourceless parabolic heat equation. In Section 8.8 we list all the “shape-preserving signal forms,” that is, all the signal forms that can propagate inside the body without distortion, after a transient period. The solutions of this “shape-preserving” type have an attractive property: the asymptotic solutions belonging to different initial conditions tend to solutions of that kind.

Energy transfer in collisions of single neutral and ionized particles with surfaces is reviewed. Collision energy domains (thermal to MeV) are only briefly characterized, as the main emphasis is on energy transfer in collisions of slow particles. Basic features of energy transfer in surface collisions of thermal and slightly hyperthermal neutral atoms, molecules, and clusters are described. Results on surface collisions of ions (hyperthermal energy region from 1 eV to about 1 keV) are discussed separately for atomic, simple molecular, polyatomic, and cluster ions. Neutralization processes of ions at surfaces are outlined. The important problem of energy transfer and dissociation in polyatomic ion-surface interactions is treated in more detail.

The aim of this chapter is to investigate some general aspects of classical thermodynamics from a geometrical point of view. Contact and Riemannian geometries are mainly employed, which correspond to the first and second laws of thermodynamics, respectively. These two structures have been denned on the so-called thermodynamic phase space (TPS). For a thermodynamic system having n degrees of freedom, TPS is a (2n + l)-dimensional manifold. Its contact structure may be given by a nondegenerate Pfaff form 9, for instance, by θ = dU - T dS +P dV - μ dN for n = 3. It is important that all variables in θ are treated as independent. In that respect, the developed formalism is similar to the symplectic formalism of classical Hamiltonian mechanics, with TPS and the contact form playing, in a sense, the role analogous to the phase space and symplectic form in mechanics.

A quantitative notion of statistical distinguishability led R. A. Fisher to his idea of statistical distance which has since been developed into Riemannian geometries on the space of statistical ensembles. Parallel to, though independently, of this progress, Riemannian geometries were being proposed on spaces of quantum states and also of thermodynamic states. Riemannian geometries in various fields have found various applications as different as population dynamics and fractional distillation, just to mention the first and the most recent ones. For decades, however, little attention was paid to the common theoretical basis of these geometric methods.

This chapter intends to fill the gap. We present an elementary introduction to the concept and mathematics of statistical distance in order to help understand the emergence of Riemannian geometrical structures. While we put more emphasis on the thermodynamical aspects, the main goal is still the interpretation of different applications on equal footing and using a unified framework.

The thermal efficiency of a distillation column may be improved by permitting heat exchange on every tray rather than only in the reboiler and the condenser. Thermodynamic length optimizations on discrete systems specify the optimal temperature of each tray and, consequently, the amount of heat to be added or withdrawn in order to maintain that temperature.