Polytropic model

Polytropic models have played a very important role in the development of stellar structure theory, but although popular before the computer age, still involve the solution of quite messy differential equations. 

A useful insight into the structure of nearly homogeneous stars or parts thereof can be gained from the study of so-called polytropic stellar models which depend on a combination of the principle of hydrostatic equilibrium with an assumed equation of state of the form... 

Reynolds number, p 46), etc 61-72 (Shock relationships and formulas) 73-98 (Shock wave interactions formulas) 99-102 (The Rayleigh and Fanno lines) Ibid (1958) 159-6l(Thermal theory of initiation) 168-69 (One-dimensional steady-state process) 169-72 (The Chapman-Jouguet condition) 172-76 (The von Neumann spike) 181-84 (Equations of state and covolume) 184-87 (Polytropic law) 188, 210 212 (Curved front theory of Eyring) 191-94 (The Rayleigh transformation in deton) 210-12 (Nozzle thepry of H. Jones) 285-88 (The deton head model) ... 

If we plot Wilson s (1982) numerical model of 10 M star on a plane of homology invariants we see that the structure of the core stays very close to that of polytrope of index N=3 except for the very outer mantle. 

We will need some background before we describe a further application of the polytropic EOS, namely estimation of the magnitude of the spike pressure. The VonNeumann-Zeldovitch-Doering model of detonation can be summarized graphically as shown in Fig 3 (sketch taken from Ref 18) ... 

Following this study, Wilk et al. [230] simulated the composition-time profiles for selected alkenes and oxygenated products that were formed from n-butane and i-butane combustion, and also mixtures of these fuels, in a motored engine. An engine cycle was simulated within a spatially uniform zone of varying volume. The volume history was specified in such a way that the predicted pressure history matched the measured polytropic pressure history in non-reactive conditions. Composition profiles were compared with those measured experimentally. Some of the kinetic features that distinguish the reactivities of the two fuels and their modes of reaction involving alkylperoxy and dialkylperoxy radicals were elucidated in this work. The n-butane oxidation model had also been applied to the... 

Since polytropic means "timriirg marry w s polytropic process suggests a model of some versatility. Witir 6 a constairt, it is defiired as a process for which... 

In fact, an experimental determination of the exponent, m, making use of equation (17.83) (or an equivalent) will introduce a small error due to the implicit inclusion of an imperfect model of compressibility effects. To compensate for this, manufacturers sometimes introduce an additional factor, /, into the equation for polytropic specific work, although in most cases... 

Because of the affinity laws, however, the modeller need work with only the characteristic giving polytropic head vs. volume flow at the design point, which may be stored in the form of a curve of Qo vs. Hpo. Then for any off-design point, he may calculate the current polytropic head, H p, from the ratio of the inlet and outlet pressures and the inlet temperature using equation (17.85). Given the current operating speed, N, he may calculate the polytropic head at the design... 

The general multistage compressor model may have several functions polytropic and isentropic compressor, positive displacement compressor, inter-cooler between stages, isentropic turbine. Single phase, as well as two- or three-phase calculations, is possible. 

Under adiabatic conditions heat flows have to be modelled accordingly with respect to the mass balances. For exothermic reactions usually a stimulation of the chemical reaction is required due to the higher energy level at the beginning of the reaction. In general, the interactions between heat and mass balances are more difficult to model. More complicated cases occur when both conditions are mixed, i.e. under polytropic conditions. Here, solutions for heat and mass balances are hardly available analytically. Instead,... 

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