Reduced temperatures

It is convenient to use the reduced temperature T/T, where is the nematic temperature, instead of temperature T, in the temperature range near Fig. 6 shows the observed values of 1/S versus T/Tj, for PBDG in chloroform, at various concentrations. 

The slope B depends on the concentration, and it increases with concentration. If the cholesteric pitch was measured at constant concentration, the following equation holds  

As shown in the table, these values are clearly constant and independent of concentration. 

In order to study the mechanism of the transition of cholesteric to nematic structure, several physical properties were measured as a function of temperature from room temperature through this transition region. 

This is also supported by the fact that a X-like endothermic peak was observed near the nematic temperature in the differential scanning calorimetry. 

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Equations (2) and (3) are physically meaningful only in the temperature range bounded by the triple-point temperature and the critical temperature. Nevertheless, it is often useful to extrapolate these equations either to lower or, more often, to higher temperatures. In this monograph we have extrapolated the function F [Equation (3)] to a reduced temperature of nearly 2. We do not recommend further extrapolation. For highly supercritical components it is better to use the unsymmetric normalization for activity coefficients as indicated in Chapter 2 and as discussed further in a later section of this chapter. 

The critical temperature of methane is 191°K. At 25°C, therefore, the reduced temperature is 1.56. If the dividing line is taken at T/T = 1.8, methane should be considered condensable at temperatures below (about) 70°C and noncondensable at higher temperatures. However, in process design calculations, it is often inconvenient to switch from one method of normalization to the other. In this monograph, since we consider only equilibria at low or moderate pressures in the region 200-600°K, we elect to consider methane as a noncondensable component. 

At pressures above the highest real data point, the extrapolated data were generated by the correlation of Lyckman et al. (1965), modified slightly to eliminate any discontinuity between the real and generated data. This modification is small, only a few percent, well within the uncertainties of the Lyckman method. The Lyckman correlation was always used within its recommended limits of validity--that is, at reduced temperatures no greater than 1.5 to 2.0. 

Spencer and Danner, 1972). This equation has been further modified by O Connell for reduced temperatures greater than 0.75. The saturated-liquid molar volume is given by the equation... 

Tj. is the reduced temperature, T is the critical temperature, is the critical pressure, and is the modified Rackett parameter as given in the supplemental table for pure-component properties. 

Adiabatic operation. If adiabatic operation leads to an acceptable temperature rise for exothermic reactors or an acceptable fall for endothermic reactors, then this is the option normally chosen. If this is the case, then the feed stream to the reactor requires heating and the efiluent stream requires cooling. The heat integration characteristics are thus a cold stream (the reactor feed) and a hot stream (the reactor efiluent). The heat of reaction appears as elevated temperature of the efiluent stream in the case of exothermic reaction or reduced temperature in the case of endothermic reaction. 

The expression in terms of reduced temperature provides a way to calculate the following properties ... 

In the first type of method, the density at saturation pressure is calculated, then this density is corrected for pressure. The COSTALD and Rackett methods belong to this category. Correction for pressure is done using Thompson s method. These methods are applicable only if the reduced temperature is less than 0.98. 

For reduced temperatures higher than 0.98, a second type of method must be used that is based on an equation of state such as that of Lee and Kesler. 

The density at saturation pressure is expressed as a function of reduced temperature ... 

The density of a liquid depends on the pressure this effect is particularly sensitive for light liquids at reduced temperatures greater than 0.8. For pressures higher than saturation pressure, the density is calculated by the relation published by Thompson et al. in 1979 ... 

When the reduced temperature is iess than 0.85, d(T depends very little on pressure. Table 4.8 gives values for and for enthalpy correction factorsi calculated by the Lee and Kesler method. 

T = reduced temperature of the mixture P = reduced pressure of the mixture... 

The coefficient can be modified to include an experimental vTscdsIly af a reduced temperature between 0.85 and 0.95. This method applies only if the reduced density is less than 2.5 and the reduced temperature is greater than 0.85. Its average accuracy is about 30%. 

Thermal conductivity is expressed in W/(m K) and measures the ease in which heat is transmitted through a thin layer of material. Conductivity of liquids, written as A, decreases in an essentially linear manner between the triple point and the boiling point temperatures. Beyond a reduced temperature of 0.8, the relationship is not at all linear. For estimation of conductivity we will distinguish two cases < )... 

When the reduced temperature is greater than 0.8, the conductivity of pure hydrocarbons can be calculated by the method of Stiel and Thodos (1963) ... 

The effects of pressure are especially sensitive at high temperatures. The analytical expression [4.71] given by the API is limited to reduced temperatures less than 0.8. Its average error is about 5%. 

This relation should be used preferentially when the reduced temperature for the fraction under consideration is greater than 0.8. 

When the reduced temperature is less than 0.8, it is better to estimate the C starting from the Co, of the fraction in the liquid state by the following... 

For petroleum fractions, there is a problem of coherence between the expression for liquid enthalpy and that of an ideal gas. When the reduced temperature is greater than 0.8, the liquid enthalpy is calculated starting with the enthalpy of the ideal gas. On the contrary, when the reduced temperature is less than 0.8, it is preferable to calculate the enthalpy of the ideal gas starting with the enthalpy of the liquid (... 

The conductivity of a pure hydrocarbon in the ideal gas state is expressed as a function of reduced temperature according to the equation of Misic and Thodos (1961) ... 

This method applies only to liquids whose constituents have reduced temperatures less than 1. The average error is about 10% the most important differences are observed in mixtures of components belonging to different chemical families. 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

K/2R. The reduced temperatures that appear on the isothenns in figure A2.5.15 are tlien defined as =... 

Figure A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ...

Figure A2.5.23. Reduced temperature = T/T versus reduced density p. = p/p for Ne, Ar, Kr, Xe, N2, O2, CO, and CH. The frill curve is the cubic equation (A2.5.26). Reproduced from [10], p 257 by pennission of the American Institute of Physics.

Figure A2.5.25. Coexistence-curve diameters as functions of reduced temperature for Ne, N2, C2H4, and SFg. Dashed lines indicate linear fits to the data far from the critical point. Reproduced from [19] Pestak M W, Goldstein R E, Chan M H W, de Bniyn J R, Balzarini D A and Ashcroft N W 1987 Phys. Rev. B 36 599, figure 3. Copyright (1987) by the American Physical Society.

Sample Preservation Without preservation, many solid samples are subject to changes in chemical composition due to the loss of volatile material, biodegradation, and chemical reactivity (particularly redox reactions). Samples stored at reduced temperatures are less prone to biodegradation and the loss of volatile material, but fracturing and phase separations may present problems. The loss of volatile material is minimized by ensuring that the sample completely fills its container without leaving a headspace where gases can collect. Samples collected from materials that have not been exposed to O2 are particularly susceptible to oxidation reactions. For example, the contact of air with anaerobic sediments must be prevented. 

As we did in the case of relaxation, we now compare the behavior predicted by the Voigt model—and, for that matter, the Maxwell model—with the behavior of actual polymer samples in a creep experiment. Figure 3.12 shows plots of such experiments for two polymers. The graph is on log-log coordinates and should therefore be compared with Fig. 3.11b. The polymers are polystyrene of molecular weight 6.0 X 10 at a reduced temperature of 100°C and cis-poly-isoprene of molecular weight 6.2 X 10 at a reduced temperature of -30°C. 

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