Select thermodynamic models for process simulation
A Practical Guide to a Three Steps Methodology

Example 2.8: Energy balance in a column feed

An adiabatic distillation column is used to separate a mixture of n-butane (1) and n-heptane (2). The liquid feed is introduced directly on the third stage as shown on figure 1.

image Figure 1 Energy balance in a column feed

The column operates at an isobaric pressure of 2.26 atm. Some additional pieces of information concerning the characteristics of the feed and the surrounding stages are also given as follows:

Table 1: Data of the example
Position Feed Stage 2 Stage 4
Temperature (°C) 47.5
Vapour molar flow (mol/s) 0 43.5
Liquid molar flow (mol/s) 100 0.46 118.2
Vapour molar composition y1 0.5 0.565 0.990
Liquid molar composition x1 0.969

For the simplicity of the model, some basic expressions have been selected for enthalpy calculation and for distribution coefficient predictions: the expressions are image (with T in °C, and h in cal/mol) for both phases and image (with T in K). These equations apply to each component and the values of the coefficients are found below:

Table 2: Parameters for enthalpies and equilibrium constants
Vapour enthalpy Liquid enthalpy Equilibrium constant
Component image image image image image image
n-Butane 23.3 5470 34 0 -2530.4 8.5426
n-Heptane 39.7 9128 54 0 -4124.6 10.412

Analysis:

The properties to be evaluated are the temperature of the various equilibria on the 3rd stage around the feed. Pressure is fixed and part of the compositions is given.

image

The fluids are regular light hydrocarbons. All properties are well known. Particular expressions are recommended and no binary interaction is considered (ideal mixture, i.e. image).

All fluids will be found in both phases as vapour or liquid due to the under-critical pressure zone.

Solution:

See complete results in file (xls):

Some help on nomenclature and tips to use this file can be found here.

Plate number 2 is defined in both phases. So the distribution coefficients of each component can be calculated (it is a binary mixture!).

image

and

image

Normally, knowledge of T on the 2nd stage will give both image and image, so there are two equations for one unknown. Both equations lead to:

image

and

image
.

The average is 316.08.

For the 4th stage, the bubble calculation has to be undertaken. The iterative procedure must start with an initial value (perhaps the temperature of stage number 2, or any other chosen by the user). If the temperature of stage 2 is used, equation image yields

image

The temperature must be lowered to reduce the sum. After a few iterations, a value of 298.66 K is found for plate number 4.

Material and energy balances around the 3rd stage must be made.

image

This implies that enthalpies of each flow must be calculated. For example, the enthalpy of the vapour leaving stage 2, it is expressed by:

image

Using this technique, the material and enthalpy content of each of the streams entering the 3rd plate can be calculated and summed as shown in table 3.

Table 3: Material and enthalpy content of the stream entering plate 3
Component image image image image
n-Butane (mol/s) 50 42.15 109.93 202.08
n-Heptane (mol/s) 50 1.35 8.27 59.62
Total nC4+nC7 100 43.5 118.2 261.7
h (cal/mol) 2090 6605 903 2303
H (cal/s) 209000 287336 106740 603077

Introducing the equilibrium condition in the above material balances (2 first equations), they can be treated as mass balance leading to the Rachford Rice equation. For a binary mixture, the vapour fraction can be calculated directly:

image

We are now left with two equations (the enthalpy balance and the above Rachford Rice equation) with two unknowns (θ and T). The complete algorithm becomes very simple:

A complete iteration is shown in table 4.

Table 4: Results of one iteration, using an assumed temperature of 40C, yielding a vapour fraction θ=0.4366 and
a total enthalpy of 989.444 kcal/s
Temperature (°C) = 40 image0.4366
Component image image image image image image image image
n-Butane (mol/s) 202.08 0.772 1.587 0.615 0.976 90.62 111.46 202.08
n-Heptane (mol/s) 59.62 0.228 0.063 0.385 0.024 56.83 2.79 59.62
Total nC4+nC7 261.7 1 1 1 147.45 114.25 261.7
h (cal/mol) 1668 6507 3781
H (cal/s) 603077 246002 743442 989444

As observed in the results above, calculated at a temperature of 40 °C, the vapour fraction of the flash is 0.4366 and the total enthalpy of image is 989444, greater than the value of 603077 for image. The temperature must therefore be less than 40 °C. A few additional iterations lead to a final value of 34.29 °C with the corresponding compositions and vapour fraction as seen below:

Table 5: Result of the enthalpy balance
Temperature (°C) = 34.29 0.1904
Component image image image image image image image image
n-Butane (mol/s) 202.08 0.772 1.367 0.722 0.986 152.93 49.15 202.08
n-Heptane (mol/s) 59.62 0.228 0.050 0.278 0.014 58.94 0.69 59.62
Total nC4+nC7 261.7 1 1 1 211.87 49.84 261.7
h (cal/mol) 1357 6327 2303
H (cal/s) 603077 287442 315209 603077