Work/heat distribution

The exact computation of P W) in this simple one-dipole model is already a very arduous task that, to my knowledge, has not yet been exactly solved. We can, however, consider a limiting case and try to elucidate the properties of the work (heat) distribution. Here we consider the limit of large ramping speed r, where the dipole executes just one transition from the down to the up orientation. A few of these paths are depicted in Fig. 13b. This is also called a first-order Markov process because it only includes transitions that occur in just one direction (from down to up). In this reduced and oversimplified description, a path is fully specified by the value of the field H at which the dipole reverses orientation. The work along one of these paths is given by... 

Methods for numerical analyses such as tlris can be obtained from commercial software, and the advent of the computer has considerably eased the work required to obtain numerical values for heat distribution and profiles in a short time, or even continuously if a monitor supplies the boundary values of heat content or temperature during an operation. 

Another interesting limit is the quasistatic limit r 0. Based on the numerical solution of the saddle point equations (160)-(162), it was suggested in Ref. 117 that T q) converged to a constant value over a finite range of work values. Figure 15a shows the results obtained for the heat distributions, whereas the path temperature is shown in Fig. 15b. A more detailed analysis [134] has shown that a plateau is never fully reached for a finite interval of heat values when r 0. The presence of a plateau has been interpreted as the occurrence of a first-order phase transition in the path entropy s q) [134]. 

Fricke and Rogers work also has implications for global heat distributions in the past. The Cretaceous latitudinal gradient in theropod isotope composition ( 0.2%o per degree latitude) is shallower than observed for meteoric water on Earth today, except in settings such as the North Atlantic, where ocean currents transport an unusual amount of heat pole-ward (Fig. 12). If Fricke and Rogers results are typical for the Cretaceous, then bioapatite isotope compositions can be used to identify increased or reduced pole-ward heat transport in the past and possibly latitudinal temperature gradients within the continents. 

Composition of processes The work or the heat distribution of the process formed by two successive processes following each other is the sum of works or heat distributions of both processes (vectorial sum in the last case). 

Process with w, 2 iscalled/tomogeneoMiifaprocess withQ w,Q (2 foreacha > 0 exists in the universe (e.g., in uniform systems, i.e those without space gradient of properties, changing the mass a-times we change the work and heat distribution in... 

In this section we continue with the method of Silhavy and Kratochvfl [13, 59, 60, 98, 99] here we need only work and heat in the entire process (heat distribution is not necessary and therefore also an approximation by finite numbers of empirical temperatures is redundant) moreover we consider here often cyclic processes. 

Note that the Carnot cycle has the heat distribution Q, composed from only two nonzero members (of different signs) Qa, Qb, cf. (1.1) other members of heat distribution are zero (in adiabatic parts of the Carnot cycle or by definition— most temperatures are not present at all). Also note that from (A.9) follow > q for our case i) , > therefore by (1.17) > 0 and by (1.4) u < 0, i.e., in the Carnot cycle the work w is produced from net absorbed heat q Second law (1.18) is valid. 

Various transient or steady-state forms of Eqn (15.50) are known as integral fluctuation theorems. Fluctuation theorems express universal properties of the probability distribution p Qi) for functionals Q[x(t)], like work, heat or entropy change, evaluated along the fluctuating trajectories taken from ensembles with well-specified... 

The heat pipe vapour chamber is a completely hollow flat plate in which evaporation, and condensation of the working fluid in it, can take place. Again it can be used to isothermalise a surface on which a variable heat distribution is occurring. The variable heat input/output could be a catalysed reaction surface. 

---