Theoretical Principles of Fractional Distillation - Part 2 #Download no.14

1.5   K VALUES

 The fact that components in a non-ideal mixture do influence the volatility of the other components results in deviations from Raoult – Dalton’s law which are of the same order of magnitude as the ones which can be corrected for by the introduction of fugacities.  As the theoretical treatment of this subject is very complicated, a solution of the problem is found by introducing the concept of “k values”.

 For a number of pure hydrocarbons and a few other compounds the xi/yi ratios for mixtures have been determined experimentally over a wide range of temperatures and pressures, and the values thus obtained, the so-called k values, have been plotted versus P and T in "k value charts" (Figures 1.3 through 1.15).

It should be mentioned that some of the k charts are based on experimental data and some of the graphs are calculated from fugacities.

Using k values, equation (4) becomes:

Applications

Using the simple relationship:

It is possible to calculate boiling points, dew points, vapour pressures and dew pressures of mixtures.

1.6 BOILING POINT CALCULATION

 The boiling point of a mixture (at a given pressure) is defined as the temperature at which, when the temperature of the liquid mixture is raised, the first infinitesimal amount of vapour escapes as a bubble.  This temperature is also known as the "bubble point".  The vapour will contain a certain amount of each component, depending on the volatility of that component, and it is clear that if the concentration in the vapour phase (Yi) is expressed in mole fraction, sum Y_i must equal 1.

Starting with a liquid of known composition the problem has to be solved by trial and error, since at a given pressure the temperature has to be found, at which the corresponding k_i values, when multiplied by x_i, result in Y_i values adding up to 1.

The following example illustrates the procedure:

System pressure:  150 psia

By graphical interpolation we find a boiling point of 169°F under 150 psia.  With this temperature a check calculation should be made and, if necessary, the procedure should be repeated within narrower limits until the correct value is found.  In those cases, where no k values of the components are available, the vapour pressure will have to be used.

Vapour pressures should not, however, be used in combination with k values.

Vapour Pressure Calculation for Hydrocarbon Mixtures

 The calculation of the vapour pressure of mixtures at a given temperature is very simply made if Raoult-Dalton's law is valid.  The mole fraction of each component is multiplied by the vapour pressure of the component at the desired temperature and the values obtained are added together.  The sum represents the vapour pressure of the mixture.

A more exact method would be to use k values and to determine the boiling pressure for the given temperature in essentially the same fashion as has been shown in the preceding section for the determination of the bubble point at a given pressure.

1.7 DEW POINT CALCULATION

 The dew point of a mixture is the temperature at which, when the temperature of the vapour is lowered, the first infinitesimal droplet of condensate is formed.  This temperature is of practical importance for establishing the top temperature of a fractionating column.  In the case of a dew point calculation the vapour composition is known and by dividing the Y_i values by the corresponding ki values one arrives at x_i values which should add up to 1.  At trial and error calculation similar to that carried out for the bubble point leads to the dew point.  The same remarks regarding the use of k values and vapour pressures as made under "Boiling Point Calculation" also apply to the dew point calculation.

 Calculation of Dew Pressure

 The dew pressure at a given temperature is the pressure at which condensation starts to take place and is, therefore, an important value in connection with partial condensation for making reflux.  According to Raoult-Dalton's law:

For the calculation of p (the dew pressure) the molal concentration of each component is divided by its vapour pressure and the values obtained are added together.  The reciprocal of the sum is the dew pressure.

Also in this case a more exact solution can be obtained by using k values and determining dew pressures at the given temperature by trial and error.

1.8 FLASH CALCULATION

 So far we have only considered vaporization and condensation of infinitesimal amounts of material at the boiling and dew points.  In many cases it is desired to know the amount and composition of the vapour in equilibrium with liquid at temperature and pressure conditions in between the bubble point and the dew point.  For instance, the degree of vaporization of a crude oil or some other feedstock after leaving the furnace is an important item in design calculations. Often it is desired to know the amount vaporized at a series of temperatures and pressures for a given material.  When such a series of values is plotted against temperature with pressure as a parameter, the resulting graph is called a "flash curve".  A flash calculation is based on a combination of a material balance and vapour liquid equilibrium relations.

Assume a feed of n components and take F moles of total feed as a basis.

Let    V   =   total moles of feed-vapour

L   =   total moles of feed-liquid

 So V + L = F

x_i   =   mole fraction of component i in the liquid phase

Y_i   =   mole fraction of component i in the vapour phase

Z_i   =   mole fraction of component i in the total feed

k_i   =   k value for component i at flash conditions.

The following material balance for a component holds true:

By keeping the pressure constant and changing the temperature, a series of values of V (all in mole fractions) at different temperatures can be obtained and a flash curve for the pressure under consideration can be plotted.

As an illustration of the use of the flash equation one point of a flash curve (at 50 psia) of a crude oil will be calculated.  In order to be able to carry out this calculation the composition of the crude should be known.  Above C5, narrow boiling cuts (10-20°F boiling range) are considered to be "components".  For obtaining the necessary data a precision distillation of the crude has to be carried out in the laboratory and the volume percentage distilled is plotted against the top temperature. This type of distillation is carried out in an apparatus allowing a high degree of fractionation and should, therefore, not be confused with the conventional ASTM distillation.  The yield vs. temperature curve obtained by the precision distillation is known as a "true boiling point curve" (TBP curve) and is the basis of our flash calculations.  As the precision distillation proceeds, the gravities of the distillates are determined and plotted in a similar manner.

Figure 1.2 shows the TBP curve of a crude oil.  This curve is already plotted on a weight basis, but usually the volumetric mid points of the fractions (the sum of the volumes of all lighter fractions plus half the volume of the fraction in question) are plotted versus the cutting temperature of each fraction and the weight percentages are calculated from the corresponding volume yields as read from the TBP curve and the gravities of the fractions.  The mid-boiling temperature of each cut is its reference boiling point when reading the vapour pressure from the Shell Data Book Chart.  The amount of vapour flashed off from given crude at 400°F at a pressure of 50 psia is now calculated.  The various data are tabulated in Table 1.3.  For the sake of convenience somewhat wider fractions (30-50°F) were used in this example.

Tables and Charts

Theoretical Principles of Fractional Distillation - Chapter 1 - Part 1