TECHNOLOGY New, simpler equations calculate pressure-compensated temperatures

Jerald Linsley
Continental Controls Inc.
Houston

• Equations [50162 bytes]

In addition, the author has developed new, simpler equations for calculating PCTs. These equations are suitable for advanced computer control systems and have been implemented in such systems.

The motivation for using PCTs is to substitute a continuously measured process variable for an infrequently measured one in a control scheme. For example, because process temperature can imply composition, temperature measured by a thermocouple can be substituted for composition measured by on-line chromatographic analyses or, worse, laboratory analyses.

For many process or control engineers, pressure compensation of temperature is a "black box" portion of the instrumentation. Lack of understanding can lead to misuse of PCTs. The author has seen operators apply to gas streams pressure compensations intended for condensates streams, and vice versa.

The new equations simplify PCT calculations, and an industrial example illustrates why PCTs are better control variables than raw, uncompensated temperatures.

The use of PCTs

In the operation of distillation columns, engineers have long used temperature as an indication of composition at various points in the column. In modern parlance, temperature is reffered to as an inferential or surrogate variable for composition. The thermodynamic justification for this is that the streams or systems of interest are in phase (vapor/liquid) equilibrium; therefore, the Gibbs Phase Rule applies (Equation 1, Equations and Nomenclature).¹

For homogeneous systems with only one liquid phase and the vapor phase, f = 2, the number of degrees of freedom equals the number of components (F = C). For a system of C components, the composition can be fixed by specifying the number of composition variables (mole fractions, weight percents, etc.) at C-1.

The one remaining composition variable can be fixed by difference because the sum of the concentrations, in mole fractions, is fixed at 1.0, or because the sum of the number of moles of each component must equal the total number of moles (fixed as a result of system definition). This leaves one additional variable to be fixed.

If the temperature is fixed, the pressure can be determined. If the pressure is fixed, the temperature can be determined. The determined variable is determinant because the system is in thermodynamic (vapor/liquid) equilibrium.

Temperature can be used directly as an indication of composition in columns that operate at constant pressure. A constant temperature of a stream or tray implies a constant, fixed composition. Many distillation columns, however, do not operate at constant pressure.

Column pressure is a variable that is often manipulated to control the performance of the column. When the pressure varies, even for a stream or tray of constant composition, the temperature must vary.

By means of a vapor-pressure equation (saturated vapor pressure defined as a function of temperature and composition), a temperature at a given pressure can be "corrected" to some constant, base pressure. This is what is meant by the term "pressure compensation."

It should be pointed out that this so-called correction is made with respect to a base composition in addition to a base pressure.

The Gibbs phase rule shows that, for this base composition or any other fixed composition, the vapor-pressure relationship will form a locus of points in the P-T plane, and that, for a given composition and a given base pressure, the compensated temperature (the temperature corrected to this base pressure) will be constant.

Thus, use of this PCT as an inferred-composition variable eliminates the effects of pressure on temperature; hence, deviations of the PCT from a desired or predicted value will result from changes in composition only. This is the reason for using PCTs.

Vapor pressure is, in general, a nonlinear function of temperature and composition (Equation 2). For a fixed composition, vapor pressure is generally a monotonically increasing function of temperature.

Most vapor-pressure equations are empirical functions adapted from an integration of the Clausius-Clapeyron equation.² These equations usually are written for a single, pure chemical component for which empirical constants have been determined.

It would be possible to use these component vapor pressures in conjunction with some mixture rule to arrive at a mixture vapor-pressure equation. Some candidate mixture rules are the ideal gas law, Dalton's law, Amagat's law, or critical properties of mixtures used in conjunction with the law of corresponding states.^{3 4}

Instead, the author has applied a vapor-pressure equation for a pure component directly to a mixture, and determined empirical coefficients as if the mixture were some pseudo-pure component. This will be illustrated in more detail. For the purposes of this article, the concentration dependence of vapor pressure will be dropped, and the vapor pressure equation will be written as shown in Equation 3.

Some authors use a technique of linearization of this vapor pressure equation.^{5 6} However, the form of some vapor pressure equations is such that they can be used directly and no linearization is required. The inverse of Equation 3 gives a "vapor temperature" equation (Equation 4). This form will prove useful.

The Antoine equation has been found to be an adequate vapor pressure equation for use in PCT calculations (Equation 5).⁷ Equation 5 can be rewritten in the form of Equation 3, to produce Equation 6. The vapor-temperature form analogous to Equation 4 is Equation 7.

The common logarithm (base 10) also could be used in Equation 7. In fact, a logarithm to any base could be used.

To calculate the PCT, a few definitions are needed. PB is the base pressure (in absolute units) and PC is the current, raw, operating pressure (same units). TC is the current, raw, operating temperature. This is the temperature to which a correction will be applied to arrive at the PCT.

TB is the pressure-corrected temperature (the current, raw operating temperature corrected to the base pressure, PB). In addition, TCX is defined as the calculated current temperature at the current pressure and base composition at which the Antoine coefficients for the mixture were determined. TCX is calculated from the Antoine equation.

TBX is defined as the calculated temperature at the base pressure and base composition. TBX could be called a "base" temperature if such a designation were meaningful or useful. TBX is also calculated from the Antoine equation.

With these definitions, TB, the PCT, can be written as Equation 8, with DTx defined as shown in Equation 9. Substituting Equation 7 into Equation 9 produces Equation 10. This is the correction term that is applied to the raw temperature, TC, to give the PCT, TB.

As stated previously, each mixture is considered a pseudo-pure component. Ideally, one would have laboratory or operating data for a number of vapor pressures at a number of temperature points, and laboratory or on-line analytical data verifying that the composition of the mixture at all these points is constant. Then a statistical regression package could be used to determine the Antoine coefficients, A, B, and C, which best fit these data. This is seldom practical.

The three constants of the Antoine equation can be determined from a minimum of three points. If three vapor pressures, P1, P2, P3, are available at three temperatures, T1, T2, T3, equations developed elsewhere can be used to calculate the three constants (Equations 11-13).⁸

It is seldom practical to obtain three such measurements along with analyses showing constant composition at the three points. Such data can be obtained, however, by utilizing steady-state process simulators. At least one of the three points should be supported by lab analysis.

For the purposes of Equations 11-13, the three points-P1, P2, and P3 at, respectively, T1, T2, and T3-can be in any order. A good practice is to let them be in ascending order. Then the first point would represent a minimum while the third would represent a maximum in the range of interest. The second, intermediate point is used as the base pressure to which the raw temperature is corrected to produce the PCT.

The pressure and temperature ranges for determining the Antoine coefficients should approximate the column operating range. The proper choice of range enables the PCT to serve its control function.

De-ethanizer example

A commercial de-ethanizer has a bottoms stream with the composition and thermal state given in Table 1 [10901 bytes]. This stream is at 410 psig pressure. The operating range for this stream was determined to be 400-440 psia.

Three pressures (400, 420, and 440 psia), somewhat displaced from the sample conditions shown in Table 1, were chosen for determination of the Antoine coefficients. The second pressure, 420 psia, is the base pressure.

The resulting bubble point temperatures, as determined from a steady-state simulation, are:

• Point No. 1 @ 400.0 psia-191.7° F.

• Point No. 2 @ 420.0 psia-193.4° F.

• Point No. 3 @ 440.0 psia-195.1° F.

The Antoine coefficients resulting from using these data in Equations 11-13 are:

• Coefficient A-4.590958

• Coefficient B-1,017.306

• Coefficient C-323.5772.

A look at variations in the composition of the bottoms stream from this de-ethanizer will illustrate the utility of the PCT concept.

One could look just at variations in the concentration of ethane, propane, or any other component, but it is common practice, when analyzing streams such as this, to look at the C2/C3 ratio. To do this, the concentration of C4+ components should be held at 34% and the C2/C3 total at 66%; the C2/C3 ratio should be varied from 0.050 to 0.150. This is shown in Table 2 [14290 bytes].

Also shown in Table 2 are the bubble point temperatures (raw, uncompensated temperatures) that correspond to various, randomly chosen raw operating pressures. These latter data are repeated in Table 3 [10096 bytes], along with the PCTs calculated by Equations 8, 9, and 10 for these various streams at various C2/C3 ratios.

The data in Table 3 are depicted graphically in Fig. 1 [11849 bytes]. One can see a general trend in the raw, uncompensated temperatures as related to the composition. But the raw pressures were chosen randomly, and this zigzagging curve could have just as easily fallen below the PCT line where it now rises above it. Nevertheless, the zigzags probably still would fall near the PCT curve. In addition, the trend shown in Fig. 1 would likely be present for most other randomly chosen operating pressures within the operating region.

This general trend can partially explain the tendency of some operators or process engineers to always work with the raw, uncompensated temperatures because they are somehow more "real" than the calculated compensated values. It has been shown, however, that the relationship between the PCT and the composition variable is much more uniform.

PCT is a better variable than the raw, uncompensated temperature for composition control in a distillation column.

References

1. Walas, Stanley M., Phase Equilibria in Chemical Engineering, Butterworth Publishers, Stoneham, Mass., 1985, p. 255.

2. Reid, Robert C., Prausnitz, John M., and Poling, Bruce E., The Properties of Gases and Liquids, 4th ed., McGraw-Hill Book Co., New York, N.Y., 1987, p. 206.

3. Balzhiser, Richard E., Samuels, Michael R., and Eliassen, John D., Chemical Engineering Thermodynamics, Prentice-Hall Inc., Englewood Cliffs, N.J., 1972, p. 390.

4. Model, M., and Reic, R.C., Thermodynamics and its Applications, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974, P. 532, et seq.

5. Shinskey, F.G., Distillation Control, McGraw-Hill Book Co., New York, N.Y., 1977, p. 254.

6. Buckley, Page S., Luyben, William L., and Shunta, Joseph P., Design of Distillation Column Control Systems, Instrument Society of America, Research Triangle Park, N.C., 1985, p. 237.

7. Antoine, C., Vapor Pressure: A New Relationship Between Pressure and Temperature. Comptes Rendus, 107, pp. 681-96, 836-37, 1888.

8. Linsley, Jerald, "The Estimation of Antoine Vapor Pressure Coeffcients for Input to Process Simulators," Summer Computer Simulation Conference, Wade, S.D., ed., Boston, July 23-25, 1984, pp. 522-24.

The Author