Derivation of Xxxxxxx Channel Capacity Sample Clauses

Derivation of Xxxxxxx Channel Capacity. Theorem from Xxxxxx’x Law‌ Xxxxxx’x law states that the maximum efficiency max by which useful work can be extracted from a heat engine is ultimately determined by the temperature difference between the engine (assumed to take in energy at a high temperature, THI ) and the external environment (at a lower temperature, TLO ), such that hmax = 1- TLO THI . Perfect efficiency can thus only be achieved either for infinitely high temperature, or the environment being at absolute zero – both unphysical concepts. Figure 33(a) indicates the classic textbook schematic diagram of the engine (indicated by a circle), which can be considered as lying between an (infinite) heat reservoir at temperature THI , and a cooler reservoir at temperature TLO . The useful work extracted from the engine is indicated by the arrow pointing to the right. During operation, the heat engine allows a quantity of energy U to flow out of the high temperature reservoir, such that a maximum fraction of the energy W is output as useful work, and the balance U W flows into the lower reservoir, with the maximum efficiency therefore given by max  W U . From an entropy point of view, this is equivalent to the flow of a quantity of entropy DH = U THI (i.e. the change in energy in the upper reservoir divided by its temperature) from the hot reservoir to the cold reservoir, where the change in entropy is also given by DH = (U -W) TLO . The ‘useful’ output work (which, by definition, is in itself of 100% useful value) has no entropy associated with it, DHW = 0 , i.e. the ‘useful’ output work can be characterised by an infinitely high temperature. Figure 33: (a) Schematic diagram of a classical thermodynamic heat engine; (b) Equivalent representation showing the entropy states of the reservoirs and the entropy flows. Conceptually, the heat engine can alternatively be considered as a consisting of an engine placed between a low entropy reservoir (equivalent to the hot reservoir), and a higher entropy reservoir below it (i.e. the cooler reservoir), as indicated in Figure 33(b). In this case, we consider the flow of entropy from the low entropy reservoir to the high entropy reservoir. No entropy flows out via the ‘useful’ output work, since as mentioned above, it has zero entropy associated with it. It may appear paradoxical that the upper hot reservoir should be characterised by low entropy, since a hot material (e.g. a gaseous form) is generally in a higher state of disorder compared with its colde...