INDEX entropy

Equation 18.34 is the basis of the use of the entropy index defined in the following section it should be borne in mind, however, that, strictly speaking, equation 18.34 is derived for a mono-sized suspension of fine particles only. 

In an ideal world, solid-liquid separation separates the solids and the liquid in a suspension into a stream of dry solids going one way and a stream of pure liquid going the other way. The entropy of the whole is reduced from a given 

In a real world, we always have to accept some liquid with the separated solids (in the filter cake or in the system underflow ) and some, usually fine, solids with the liquid (in the filtrate of overflow ). In other words, neither the separation of the solids nor that of the liquid is perfect. In evaluating the efficiency of a separation process, therefore, both the separation of the solids and the separation of the liquid must be considered. In most practical applications, the emphasis on either of these two is different and they are kept separate in thickening, for example, the goal is the separation of the liquid and the (complete) separation of the sohds is secondary. In solids recovery or liquid clarification, the completeness of the separation of the solids has priority over the separation of the liquid. 

If we wish to consider both the separation of the solids and that of the liquid with equal emphasis, entropy gives us an ideal tool. The efficiency of the overall separation can thus be evaluated as a fraction or percentage decrease in the entropy of the system, taking the initial entropy of the suspension (from equation 18.34) as being 100%. This definition of separation efficiency has been called the entropy index and has been used widely in the Russian scientific literature .  

The entropy index as defined in equation 18.36 is potentially very useful in the fundamental evaluation of any separation processes, not just in solid-liquid separation. Besides the Russian references , Ogawa et al. derived the same entropy index (but using mass fractions rather than volumetric ones) from information theory and proposed its use for the evaluation of any separation process. 

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The separation of the solids is usually expressed as mass recovery or total efficiency (in filtration this is also known as retention ) as dealt with in depth in chapter 3, whilst the separation of the liquid is usually characterized by the moisture content of the cake or concentration of solids in the underflow. Separation efficiencies of the solids and the liquid are best considered separately because different applications place different emphasis on the two in thickening, for example, the emphasis is on the high efficiency for the hquid (i.e. high sohds content in cakes or underflows), whilst in recovery or clarification, high efficiency for the solids is required. If the emphasis placed on the two efficiencies is equal then they can be combined in one criterion, the entropy index, discussed in Part II, chapter 18. 

In order to be able to optimize the system, we have to know how the flow ratio affects the separation of the solids. In the case of hydrocyclones, the effect of the ratio is two-fold increasing Rf leads to improvements to separation efficiency through the contribution of dead flux and a further improvement is caused by the reduction in the crowding of the underflow orifice. Both of the effects can be described analytically for certain hydrocyclone geometries and the above-mentioned optimization is therefore possible, using the entropy index as a general criterion for the optimization. 

In general, the evaluation of the separation criterion is subject to much the same criticisms as is the entropy index. In this case, however, the problem is even more difficult due to the fact that absolute changes in entropies and internal energies are needed. More work is needed to establish these quantities for particulate suspensions, with the inevitable particle-particle interactions and excess free energies of mixing. Textbooks on thermodynamics are remarkably silent on the subject of thermodynamics of particulate systems and suspensions. 

Sulla, M. B. and Fikhtman, S. A., Application of the entropy index to evaluate the efficiency of thickening equipment , Vodosnabzhenie i Sanitarnaya Tekhnika, 11, 11-13 (1972)... 

Tables 2,3, and 4 outline many of the physical and thermodynamic properties ofpara- and normal hydrogen in the sohd, hquid, and gaseous states, respectively. Extensive tabulations of all the thermodynamic and transport properties hsted in these tables from the triple point to 3000 K and at 0.01—100 MPa (1—14,500 psi) are available (5,39). Additional properties, including accommodation coefficients, thermal diffusivity, virial coefficients, index of refraction, Joule-Thorns on coefficients, Prandti numbers, vapor pressures, infrared absorption, and heat transfer and thermal transpiration parameters are also available (5,40). Thermodynamic properties for hydrogen at 300—20,000 K and 10 Pa to 10.4 MPa (lO " -103 atm) (41) and transport properties at 1,000—30,000 K and 0.1—3.0 MPa (1—30 atm) (42) have been compiled. Enthalpy—entropy tabulations for hydrogen over the range 3—100,000 K and 0.001—101.3 MPa (0.01—1000 atm) have been made (43). Many physical properties for the other isotopes of hydrogen (deuterium and tritium) have also been compiled (44).

Table 1 Spectral analysis for Nino 3.4 index, streamflow, and AF using the Maximum Entropy Method. Only results for the low frequency band are showed ...

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

The Gini index and the cross entropy measure are differentiable which is of advantage for optimization. Moreover, the Gini index and the deviance are more sensitive to changes in the relative frequencies than the misclassification error. Which criterion is better depends on the data set, some authors prefer Gini which favors a split into a small pure region and a large impure one. 

Temperature, Heat capacity. Pressure, Dielectric constant. Density, Boiling point. Viscosity, Concentration, Refractive index. Enthalpy, Entropy, Gibbs free energy. Molar enthalpy. Chemical potential. Molality, Volume, Mass, Specific heat. No. of moles. Free energy per mole. 

Both kinetic and thermodynamic approaches have been used to measure and explain the abrupt change in properties as a polymer changes from a glassy to a leathery state. These involve the coefficient of expansion, the compressibility, the index of refraction, and the specific heat values. In the thermodynamic approach used by Gibbs and DiMarzio, the process is considered to be related to conformational entropy changes with temperature and is related to a second-order transition. There is also an abrupt change from the solid crystalline to the liquid state at the first-order transition or melting point Tm. 

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