## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000400038

**THERMODYNAMIC ANALYSIS OF MULTICOMPONENT DISTILLATION COLUMNS: IDENTIFYING OPTIMAL FEED CONDITIONS**

**M. L.O.Maia and R.J. Zemp **DESQ, FEQ,Unicamp, C.P. 6066,

13083-970, Campinas - SP, Brazil

E-mail: lmaia@desq.feq.unicamp.br

*(Received: October 1, 1999 ; Accepted: April 7, 2000 )*

Abstract -A new methodology for the optimisation of feed conditions as well as the calculation of minimum reflux ratio of distillation columns is presented. The reversible profile approach used for saturated liquid feeds is extended to consider other feed conditions. For flashed feed, the liquid fraction of the feed stream is used to compute the column pinch conditions and the minimum reflux ratio. The modifications required for subcooled liquid and superheated vapor feed are discussed, and a procedure to estimate the minimum reflux for those conditions is proposed. The methodology presented allows the identification of the optimal feed condition, without having to resort to a full stage-by-stage procedure.

Keywords: mininum reflux, near-reversible distillation, exergy analysis

**INTRODUCTION**

The synthesis of cost-effective distillation columns sequences as well as the design of stand-alone distillation columns demand the determination of the minimum energy requirement for the separations. Moreover, different feed thermal conditions should be considered not only to optimize the energy consumption, but also to consider heat integration among the columns and with the background process. A number of methods which estimates the minimum reflux ratio is found in the literature, however, they present varying degrees of precision and demanded effort. In addition, the columns are usually designed considering the feed condition as saturated liquid. Therefore, there is a need for methods that allow the design engineer to quickly compute minimum reflux conditions and to determine the optimum feed condition, without previously designing the column.

In this work, the near-reversible column profile is used to determine the minimum reflux condition of a distillation column, for a wide range of feed condition. Firstly, the near-reversible profile is obtained for feed condition varying from saturated liquid to saturated vapor. Secondly, an additional modification is proposed to consider subcooled liquid and superheated vapor feed.

Results attested that the optimum feed thermal state for a separation under minimum reflux is very close to the optimum feed condition for a column operating under real reflux conditions. Furthermore, the pinch properties found for subcooled liquid feed and superheated vapor feed have shown a close agreement between the proposed method and a more rigorous, but time consuming method.

**MINIMUM REFLUX FOR SATURATED LIQUID FEED**

In 1991, Koehler and co-workers (1991) proposed a method that determines the minimum reflux condition by making use of reversible column profiles. This new procedure is based on the colinearity principle. Consider a ternary system being separated in a column operating under minimum reflux conditions. It has been observed that for such case, the rectifier and stripping section pinch compositions and the feed composition are located on a straight line, on the triangular composition diagram. This colinearity criteria is originally attributed to Underwood (Koehler et al, 1993).

Figure 1 illustrates the principle for an arbitrary ternary system ABC. Under minimum reflux condition the column has two pinches. The pinch compositions can be located on a triangular composition diagram, together with the feed composition. Underwood verified that for systems in which the assumptions of constant molal overflow (CMO) and constant relative volatilities (CRV) are valid, the minimum reflux pinch compositions (*P _{s}* and

*P*) are located on a straight line together with the feed composition (

_{r}*F*).

Koehler and coworkers (1993) extended the previous concept to non-ideal systems considering that the colinearity criteria should be relaxed and substituted by a minimum angle criteria, due to the violation of the CMO and CRV restrictions. In this case, the vector given by the lines connecting the pinch compositions to the feed composition should present the smallest angle. The determination of the minimum reflux conditions is thus reduced to a search for a pair of pinches, related to each other by the column heat balance, which forms a minimum angle with the saturated liquid feed composition. Once the pinch properties are known, the minimum energy requirement can easily be computed.

For multicomponent mixtures, the minimum angle is determined by the dot-product (eq. 1) of two vectors (Kreyszig, 1988). These vectors are the lines connecting each pinch composition to the feed composition.

(1) |

The substitution of the appropriate vectors produces the following expression for the angle between them:

(2) |

According to Koehler's proposal, the angle should be minimum for minimum reflux conditions. The objective is then to determine the pinch compositions that minimize equation (2) or

(3) |

Koehler and coworkers successfully applied this procedure for the determination of the minimum reflux conditions in both ideal and non-ideal systems.

**PROPOSED METHODOLOGY**

One drawback in Koehler’s method is that it is restricted to saturated liquid feed, making the method unsuitable for general analysis of optimal feed condition. Moreover, the need for repeated physical properties and pinch calculations required for the angle minimisation procedure make the whole process very time consuming. Both issues are now discussed further and a solution presented.

Extensive analysis of rigorously simulated minimum reflux columns (using a very large number of stages) has shown that the angle between the feed composition and each pinch composition is not at a minimum when the feed is not a saturated liquid. However, tests with saturated vapor feed showed the angle to be minimum if the vapor phase pinch compositions are used, thus suggesting that there might be a modified feed composition that satisfies the minimum angle criteria.

Furthermore, it was observed that when the feed composition is substituted by the composition of the liquid fraction of the flashed feed, the angle formed is a minimum. Thus, the original procedure proposed by Koehler et al (1993) can be extended to feed conditions other than saturated liquid by flashing the feed and taking the liquid phase composition as the one to be used in the angle procedure, as shown in Figure 2.

During the search for the minimum angle, a large number of pinch sets have to be computed, and each pinch set itself is determined by a root-finding procedure. Analysis and run-time tests of the program generated showed that a large percentage of the execution time is spent calculating physical properties.

The solution proposed to reduce the computational effort is based on the observation that while the positions of both column reversible sections are related to each other by the column energy balance (and therefore of the feed condition), the shape of the reversible section are not. Thus, the reversible column profiles for the rectifying and stripping sections need to be computed only once for a given column, and saved as splines, avoiding unnecessary physical pinch computations.

**The New Procedure**

The new procedure for the calculation of minimum energy consumption of distillation columns is based on the following steps:

1.for a given column specification, compute product flows and composition

2.compute energy balance

3.compute the reversible profile for the rectifying section (a procedure for this can be found in Zemp, 1994)

4.compute the reversible profile for the stripping section

5.compute spline coefficients for both profiles (composition, heat loads, temperatures and flows)

6.start minimum angle search procedure

6.1 chose rectifying pinch

6.2 compute corresponding stripping pinch (based on energy balance of the column)

6.3 compute angle between pinch compositions and composition of liquid fraction of feed

6.4 repeat until a minimum angle is obtained

Although the above procedure has been successfully used for flashed feed and saturated vapor feed, it failed in the determination of the minimum reflux conditions when the feed is a subcooled liquid or superheated vapor. In order to consider these feed conditions, the above method has to be modified.

Extensive analysis has shown that the heavy key composition in the rectifying pinch, obtained when the feed condition ranges from saturated liquid to saturated vapor, can be properly adjusted by a quadratic function of the feed condition. Consequently, this composition can be calculated for subcooled liquid feed or a superheated vapor feed, prior to the determination of the minimum reflux conditions. The linear function as well as the exponential function, both simpler than the quadratic one due to the less adjustable parameters, were tested for extrapolation. However, these functions failed to fit the desired variables for various tests performed. The quadratic fit was chosen due to its relative simplicity and reasonable level of accuracy. However, one should be aware of the danger of extrapolating feed condition well above or below the saturated states.

The aim of this new procedure is to use the adjusted heavy key pinch composition to determine the remaining rectifying pinch properties. The corresponding stripping pinch is determined straightforwardly, as both pinches are related by the energy balance of the column. Therefore, the need for a search of a minimum angle between the feed composition and both pinches compositions no longer exists. The new method is based on the following steps:

1. compute the heavy key composition in the rectifying section for feed conditions ranging from saturated liquid to saturated vapor.

2. adjust the values by a quadratic function.

3. compute the heavy key composition for the desired feed condition using the quadratic function.

4. start the near-reversible profile determination:

3.1 compute the reversible profile for the rectifying and stripping section of the specified column, along with the spline coefficients. (steps 1 to 5 of the preceding procedure)

3.2 for the calculated heavy key composition, compute the other rectifying pinch properties using the spline coefficients.

3.3 compute the corresponding stripping pinch, based on the energy balance of the column.

Both methodologies discussed allow the feed thermal condition to vary from subcooled liquid to superheated vapour. For flashed feed, a modified feed composition is used in the minimum angle criteria for the pinches determination. For feed conditions laying outside the saturated range, the adjusted value of the heavy key composition is used instead.

The practical result of both methods are the boundaries of the two reversible sections in the column, one above the rectifying pinch and other below the stripping pinch. There is also one irreversible section between the pinches. This profile is known as near-reversible profile. The properties of the rectifying pinch allow the computation of the minimum reflux ratio of the column for a given feed condition, and therefore its the energy requirement.

**FEED CONDITION OPTIMIZATION THROUGH EXERGY ANALYSIS**

So far the two procedures for the determination of the minimum reflux conditions of distillation columns have been discussed for a wide range of feed thermal conditions. These can now be used to identify the best feed condition for a given separation. For such a goal, *Exergy Analysis* plays an important role.

Faria (1995) used the exergy loss profile of a real column to correctly identify the optimum feed stage and thermal condition and other changes which improve column efficiency. Exergy analysis can also be applied to distillation columns operating under minimum reflux condition. However, as an exergy loss profile is not available, a global exergy balance (equation 4) must be done and the global exergy loss (D*Ex _{lost}*) is used instead:

(4) |

The great advantage of working with a minimum reflux column is that the design engineer has not to be concerned with the optimum feed stage and the number of stages, when identifying the optimum thermal condition of the feed.

Having identified the near-reversible profile, the exergy balance can be applied to the column in a straightforward manner. The exergy change across the column streams (D*Ex _{streams}*) is a state function, depending only on the conditions of the feed, distillate and bottom product streams. Also the near-reversible column and a column operating under minimum reflux have the same energy balance, meaning that the heat loads of both columns are the same. Therefore, the exergy requirement of the minimum reflux column can be calculated and used as an optimisation parameter.

The actual exergy requirement of a distillation column (D*Ex _{req}*) is dictated by the exergy flowing in and out through the reboiler and condenser respectively, as shown in equation 5 below, where

*Q*is the heat load at minimum reflux:

_{i}
(5) |

For feed condition apart from saturated liquid, the exergy flow in the pre-heater (or pre-cooler) must be included in the equation above. Thus, from equation 4 and 5 the exergy being lost in the column can be computed, and the feed condition leading to its minimum value determined.

**RESULTS AND DISCUSSION**

The proposed procedure was tested for a number of multicomponent systems (ternary and 5 components). The physical properties were computed using Peng-Robinson equation of state. First, the results of the quadratic adjustment of the heavy key composition will be shown. Two systems were used as examples of the effectiveness of the method.

Figure 3 presents the results obtained for the system 1: n-pentane, n-hexane, n-heptane [0.33, 0.34, 033 feed composition, with 98% recovery of n-pentane at the top, pressure = 101.3 kPa]. The adjusted composition of the n-hexane (heavy key) was compared with the composition obtained form a rigorous simulation of a column (high number of stages). As shown in figure 3, a very close agreement was reached.

A system of five components (system 2) was also analysed: propane, i-butane, n-butane, i-pentane, n-pentane [0.05, 0.15, 0.25, 0.20, 0.35 feed composition, with recovery of 98% n-butane at the top, pressure = 800 kPa]. A very close agreement between the adjusted composition and the rigorous simulation was also obtained. The results are presented in Figure 4.

Changes of the light key component specification as well as changes of the systems pressure were also tested. For all simulated cases, the quadratic function has proved to fit satisfactorily the composition of the heavy key component over a reasonable range of feed conditions. For feed conditions far beyond saturation, the deviation may be large, leading to significant error in the predicted minimum reflux values.

Having defined the best function for the heavy key composition fitting, the adjusted composition was used to calculate pinch properties and the exergy balance of the column for subcooled liquid and superheated vapor feeds. Tables 1a and 1b show the exergy loss calculated for those columns, along with the results obtained in rigorous simulations. The deviations found were less than 3%. The exergy balances seem not to be affected by the deviations in the quadratic fit for feed conditions far beyond saturation.

The minimum reflux ratio using the new procedure gives very good results when compared with the commercial process simulator HYSIM, at a much lower CPU time. A wide range of feed conditions was used. The results for systems 1 and 2 are presented in Tables 2 and 3.

For both systems the optimum feed condition was found by an exergy balance of the column. Figures 5a and 5b illustrates the behaviour of exergy loss when thermal feed condition (*q*) vary from superheated vapour (*q*<0) to subcooled liquid (*q*>1). The optimum feed condition of a column operating under minimum reflux condition was found to be very close to the optimum feed condition of a real column. The results of the real columns were obtained using the commercial simulator HYSIM. However, for the rigorous column simulation the optimal feed stage had to be determined for each feed condition through a tedious trial-and-error procedure, a task that is not required for the minimum reflux approach. In addition, the use of splines in the near-reversible column exergy analysis reduces dramatically the computing time.

**CONCLUSIONS**

A new methodology for minimum energy calculation of distillation columns, which operates under a wide range of feed conditions, is proposed in this work. In order to consider feed conditions other than saturated liquid, modifications in the calculation of the minimum reflux column pinches have been proposed. The use of the composition of the liquid fraction of the feed instead of the composition of the feed allowed the extension of the near-reversible column profile approach to flashed feeds. For feed below saturated liquid or above saturated vapor the use of a quadratic fit of the pinch composition as a function of feed condition yielded good estimates of the minimum reflux value.

Comparison with results obtained using rigorous column simulation has shown a very close agreement. Also, the use of splines for interpolating the properties of the reversible column profiles increase dramatically the speed of convergence for the minimum reflux calculation. In addition, the optimum feed condition was found by a straightforward exergy balance of the minimum reflux column. The results obtained have shown very good agreement with those obtained for a real column. This means that the designer engineer can easily determine the feed condition for a given separation without having to design the column in advance. The time required to compute the best feed condition is at least one order of magnitude lower than that for a full rigorous column simulation for one feed condition.

**ACKNOWLEDGEMENTS**

The authors would like to acknowledge CNPq (Brazilian Council for Research and Development) for the financial support.

**NOMENCLATURE**

E | exergy |

Nc | number of components |

Q | heat duty (kJ/h) |

T | temperature |

T_{0 } | ambient temperature |

x | liquid mole fraction |

y | vapour mole fraction |

q | angle between saturated liquid feed and liquid composition of the two pinches of the column |

D | exergy diference defined in equation 4 |

*Subscripts*

i | component defined in equation 3 |

ii | reboiler, condenser and pre-heater defined in equation 5 |

F | feed |

Lost | exergy loss |

P | pinch |

r | rectifying section |

req | exergy requirement |

s | stripping section |

streams | exergy stream |

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