Volume-translation parameter

Peneloux et al. [35] have introduced a clever method of improving the saturated liquid molar volume predictions of a cubic equation of state, by translating the calculated volumes without efffecting the prediction of phase equilibrium. The volume-translation parameter is chosen to give the correct saturated liquid volume at some temperature, usually at a reduced temperature Tr = T/Tc = 0.7, which is near the normal boiling point. It is possible to improve the liquid density predictions further by making the translation parameter temperature dependent. 

For the application to mixtures, the volume translation parameter c can be calculated by a simple linear mixing rule ... 

The required pure component properties Tc, Pc, and co together with the parameters for the Twu a-function are given in Appendices A and K. The volume translation parameter c should be calculated using Eq. (2.179). 

Beyond applications reported in the literature, process simulation is ubiquitous throughout the chemical industry and academia. Every time users select the Soave, or SRK, thermodynamics model, they apply the Soave equation. One motivation for this selection is the long experience with the model and the compilation of correction factors, when the basic model is deficient. For example, binary interaction parameters (ky s) have been compiled for a large number of binary mixtures to improve VLE correlation. It is also possible to compensate for inaccuracies in density through volume translations. ... 

An improvement can be achieved with the volume translation concept introduced by Peneloux et al [55]. The idea is that the specific volume calculated by the equation of state is corrected by addition of a constant parameter c. The volume translation has no effect on the vapor-liquid equilibrium calculation, as both the liquid and the vapor volume are simultaneously translated by a constant value. The procedure has also little effect on the calculated vapor volumes, as c is in the order of magnitude of a liquid volume far away from the critical temperature. 

Calculate the enthalpy of reaction for the ammonia synthesis at 450 C and a pressure of 600 atm using the value of the standard enthalpy of reaction at 450 °C in the ideal gas state calculated in Example 12.1. For the calculation of the residual enthalpies, the group contribution equation of state volume-translated Peng-Robinson (VTPR) should be applied. The required VTPR parameters are given in Appendix K. 

Where and are critical volume translation factor and critical compressibility factor respectively. Equation (11) has two parameters yand that must be determined for each component. These parameters were optimized by minimizing the following objective function ... 

Where N is the number of experimental data points for each component. By using equation (12), the two parameters were determined for all pure components. Now, the volume translation factor, c(T), can be determined therefore, the improved liquid density value was calculated as follows ... 

Table 1. Optimized parameters for temperature-dependent volume translation model to PR EOS...

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

In order to understand monolithic supports and the effects of polymerization parameters, a brief description of the general construction of a monolith in terms of microstructure, backbone and relevant abbreviations is given in Fig. 8.1 [63, 64]. As can be deduced therefrom, monoliths consist of interconnected microstrac-ture-forming microglobules, which are characterized by a certain diameter dp) and microporosity (gp). In addition, the monolith is characterized by an inter-microglobule void volume sfj, which is mainly responsible for the backpressure at a certain flow rate. The sum of gp and g directly translates into the total porosity gf. 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

Mathematically, this is a triple scalar product and can be used to calculate the volume of any cell, with only a knowledge of the lattice translation vectors. If the lattice parameters and interaxial angles are known, the following expression for V can be derived from the vector expression ... 

Model Equations to Describe Component Balances. The design of PVD reacting systems requires a set of model equations describing the component balances for the reacting species and an overall mass balance within the control volume of the surface reaction zone. Constitutive equations that describe the rate processes can then be used to obtain solutions to the model equations. Material-specific parameters may be estimated or obtained from the literature, collateral experiments, or numerical fits to experimental data. In any event, design-oriented solutions to the model equations can be obtained without recourse to equipment-specific fitting parameters. Thus translation of scale from laboratory apparatus to production-scale equipment is possible. 

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