Tait equation

An empirical representation of PVT data of polymer systans is given by the Tait equation. It can be viewed as an isothermal compressibility model. The general form of the Tait equation can be written as follows  

The Tait equation was first developed over 120 years ago [10] in order to fit compressibility data of freshwater and seawater. Modifications of the Tait equation are used for fitting liquid density data worldwide. The constant C in Equation (2.27) is taken as 0.0894 and is a universal constant. The zero pressure isotherms are given by 

The Tail equation is a four-parameter representation of the P-V-T behavior of polymers. The four parameters are the zero pressure isotherm, Vq. Tait parameter, B, thermal expansion coefficient, a, and activation energy, In some industrial systems polynomial expressions are used for the zero-pressure isotherm and Tait parameter as follows  

The zero-isotherm and Tait parameter for some polymer systems were provided for 56 polymers by Rodgers [9]. The values for some systems are given in Table 2.1. For a few polymers, the temperature, T, is expressed as °C. 

Polymer Zero-Pressure Isotherm Vg Tait Parameter B 

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Tait equation Takadiastase Takagi reagent Takahax process Takanelite Takasweet Takatherm... 

The radius and velocity of the bubble waU are given by R and U respectively. The values for H, the enthalpy at the bubble waU, and C, the local sound speed, maybe expressed as foUows, using the Tait equation of state for the Hquid. 

The Tait equation for the pressure—volume behavior of Hquids (1) correlates data accurately, and is expressed mathematically as... 

It was recently revealed by Hayward (134) that two generations of workers were misled into using what they came to call "the Tait equation". They did not realize that this well known equation was not Tait s original equation. The Tait equation was first misquoted by Tammann (135). The misquoted form of the Tait equation (which we shall refer to as the Tammann equation) is... 

The pressure dependence of (3P/3v) can be tested by fitting [(3P/3v)P - (3P/3v) ]/P vs. pressure. It is clearly shown in Figure 14 that the values of [(3P/3v)p - (3P/3v) ]/P for pure water and 35 °/oo salinity seawater determined from sound speed data (112,123),increase almost linearly with pressure. This indicates that P or even higher order terms are needed to represent (3P/3v) over the pressure range of 0 to 1000 bars. In other words, the Tammann equation and the original Tait equation do not represent the PVT properties for pure or saline water within the accuracy of the data. 

Hayward (134) extensively investigated different forms of equations of state and suggested the reciprocal form of the Tait equation... 

Macdonald (144) analyzed several equations of state which had a variety of mathematical forms including the Tammann equation and the secant bulk modulus equation chosen by Hayward. (In his statistical analysis, Macdonald used the PVT data of Kell and Whalley (26) which has been shown to be in error (29) Thus, the conclusions of Macdonald may be questionable.) He disagreed with Hayward and selected the Murnaghan equation to be superior to either the Tammann equation or the linear secant modulus equation chosen by Hayward. If, however, the Tammann equation and the Murnaghan equation were both expanded to second order in pressure, then Macdonald found that the results obtained from both equations would agree. As shown earlier, the expansion of the Tammann equation to second order is equivalent to the bulk modulus form of the original Tait equation. 

The key assumption to be made in interpreting the pressure data in terms of solvational change is that neither the ligands in the first coordination sphere of the complex nor the solvating solvent molecules are significantly compressible relative to bulk solvent (volume Vs), the compression of which is described by the modified Tait equation... 

A widely accepted form of modeling the density or specific volume is the Tait equation. It is often used to represent the pvT-behavior of polymers and it is represented as,... 

Tait equation constants for various materials based upon fitting data [13]... 

Figure 2.14 pvT diagram from the Tait equation for PC (Table 2.3). 

In a numerical solution, we can include temperature dependent density and thermal conductivity. The temperature dependent density can be modeled interpolating throughout a pvT diagram. The temperature dependence of the thermal conductivity is not always available, as is the case for many properties used in modeling. Chapter 2 presents the Tait equation, which can be used to model the pvT behavior of a polymer. 

The Tait equation predicts that the isothermal compressibility of a liquid approaches zero as the pressure becomes infinitely large. Is this reasonable in terms of the van der Waals model of real fluids ... 

TABLE 4.14 Constants of the Tait-equation for polymer melts... 

Simha et al. (1973) showed that the Tait relation is also valid for polymers in the glassy state. In this case the value of bx is about the same as for polymer melts, but b2 is smaller ( 2 3 x 10 3). Even nowadays frequently use is made of the Tait equation. 

This equation was verified by application of the p-D-T-data of the melts of 23 polymers of very different structure, adapting the reducing parameters Ba/ vQ and T0 to the closest fit with the experiments. The obtained values are shown in the left part of Table 4.15. The average deviation between calculated and experimental v(P, T) data is the same as obtained with the Tait relation. The advantage of Hartmann s equation is that it contains only three constants, whereas the Tait equation involves 4. 

Taber abraser, 841 Tacticity, 147, 167 Tait equation, 101 Telechelic polymers, 8 Telescopic flow, 528 Temperature... 

This method is to be used to calculate the specific volume of a pure polymer liquid at a given temperature and pressure. This procedure uses the empirical Tait equation along with a polynomial expression for the zero pressure isobar. The method requires only the equation constants for the polymer. 

The application of this procedure is restricted to polymers for which the Tait equation constants have been determined from experimental data. No methods are available to predict these constants. The sensitivity of the equation constants to different molecular weight species of the same polymer is not known. 

This method uses the form of the Tait equation which gives the best representation of P-V-T data for most of the polymers. Other forms of the Tait equation which use a different expression for V(0,T) or a different value for the parameter C are available. For a few specific polymers, they might provide a better representation of the P-V-T behavior. 

This example shows how to use the Tait equation to calculate the specific volume of a polymer included in the data bank. 

Highlight (polycarbonate) for the polymer and press ENTER. Note the pressure and temperature limits for the Tait equation. In this case, the pressure limits are 0.0 and 1.77E + 08 pascals and the temperature limits are 430 and 610 K. 

Over an extended pressure range we can represent V by the modified Kirkwood-Tait equation, V Vo/(l + where B and C are positive constants, B in the range of... 

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