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Thermal Integration of a Distillation Column Through Side-Exchangers
THERMAL INTEGRATION OF A DISTILLATION COLUMN THROUGH SIDE-EXCHANGERS S. Bandyopadhyay Energy Systems Engineering and Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India.
Abstract: Thermal integration of a distillation column with the background process through side exchangers (side reboilers/condensers) is important for designing an energy-efficient distillation column. Thermal integration of a distillation column through side exchangers also improves the exergetic efficiency of the column. Invariant rectifying and stripping (IRS) curves are employed for thermodynamic analysis of a distillation column. Different configurations for side-exchanger integration are evaluated based on the exergetic efficiency of the distillation process and they are also compared with feed preheating options. A novel optimal configuration curve is proposed in this paper that identifies feed composition and its thermal condition for which it is beneficial to have a side reboiler in a distillation column. Keywords: thermodynamic analysis; distillation column; side reboiler; feed preheating; IRS curves; optimal configuration curve.
INTRODUCTION
Correspondence to: Professor S. Bandyopadhyay, Energy Systems Engineering and Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India. E-mail: santanu@ me.iitb.ac.in
DOI: 10.1205/Cherd06108R1 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, January 2007 # 2007 Institution of Chemical Engineers
Distillation is one of the common unit operations used in chemical industries. Operating cost of a chemical process plant mainly depends on the operating cost associated with the separation processes. Fifty to 80% of the operating cost in a chemical process industry is contributed by separation processes (Soave and Feliu, 2002). Total energy requirement of chemical process industry in the United States is over 5 quad year21 and distillation process alone consumes about 2.4 quad year21 (Ognisty, 1995). This amounts to about 3% of the total energy consumed in the United States. Though distillation is energy intensive, it is not energy efficient. Therefore, it is necessary to identify different energy conservation opportunities through thermodynamic analysis of a distillation process. Thermodynamic analysis of a distillation column is important for synthesizing and developing energy-efficient distillation processes. It allows the thermodynamic efficiency of a process to be quantified, regions with poor energy efficiency to be identified and the thermodynamic targets to be defined. Thermodynamic analysis of a distillation column aims at possible reduction in exergy loss, or equivalently, reduction in entropy generation. There are three major sources of entropy generation in a distillation column— heat transfer with finite temperature driving force, mixing of non-equilibrium vapour and
liquid, and pressure drop across the column. Additionally, there may be entropy generation due to heat loss to the ambient from the column surface. There are two different ways of analysing distillation column thermodynamically, viz., exergy analysis and approaches based on the temperature – enthalpy (T – H) curves. The T –H diagram for a distillation column is a useful representation to quantitatively address the energy-saving potential for possible stand-alone modifications such as reflux reduction, feed conditioning and scope for side reboiler/condenser (Naka et al., 1980; Terranova and Westerberg, 1989; Dhole and Linnhoff, 1993; Ognisty, 1995; Hall et al., 1995; Trivedi et al., 1996). The T– H curve of a binary distillation column is generated by solving the coupled mass and enthalpy balances for the reversible separation scheme (Benedict, 1947; Fonyo´, 1974; King, 1980; Fitzmorris and Mah, 1980; Ho and Keller, 1987). Reversible operation corresponds to minimum net work consumption and involves minimization of the driving forces for heat and mass transfer within individual stages (or, in other words, decrease of the gap between the operating and equilibrium curves on the x –y diagram) as discussed in detail by King (1980). In contrast to binary distillation, the sharpness of separation is generally limited in multicomponent reversible distillation (Fonyo´, 1974) and it is impossible to devise a reversible separation scheme for many practical multicomponent 155
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separations (Franklin and Wilkinson, 1982). The limitation in sharpness of reversible multicomponent distillation can be circumvented, for the purposes of generation of the T–H curve, by using the pseudo-binary concept of light and heavy key model (Fonyo´, 1974; Dhole and Linnhoff, 1993). Dhole and Linnhoff (1993) described a procedure for generating a T–H curve (which they called the column grand composite curve) from a converged simulation of a distillation column. The calculation procedure involves determination of the net enthalpy deficit at each stage by generating envelopes from either the condenser end (top-down approach) or the reboiler end (bottom-up approach). However, the values calculated by the two approaches differ for stages with feeds because they do not consider the enthalpy balances at the feed stages. A feed stage correction that rigorously considers the mass and enthalpy balances at feed stages has been recently proposed by Bandyopadhyay et al. (1998) to resolve the discrepancy. Bandyopadhyay et al. (1999) introduced a novel pair of T–H curves, known as the invariant rectifying-stripping (IRS) curves for a distillation column. The IRS curves are invariant to the column configuration (i.e., feed location in the column and number of stages) and depend only on sharpness of separation as well as operating pressure of the column. They are useful for setting quantitative targets such as minimum energy requirement, appropriate feed location, proper feed preheating, scope for side-condensers/ reboilers,as well as thermo-economic optimization of a distillation column. Exergy analysis of a distillation column often provides useful understanding and insight for energy efficient design of distillation processes. Bandyopadhyay (2002) established that the exergy loss in a distillation column may be measured directly by representing the IRS curves on Carnot factor-enthalpy (1 versus H) diagram. Different schemes such as proper feed location (Bandyopadhyay et al., 1999; Lee et al., 2001), feed preheating (Liebert, 1993; Soave and Feliu, 2002; Bandyopadhyay et al., 2003; Deshmukh et al., 2005), side-exchangers (Peterson and Wells, 1977; Aguirre et al., 1997; Agrawal and Herron, 1998; Sharma et al., 1999) and so on are suggested to reduce the energy consumption and to improve energetic efficiency in distillation. By exchanging heat with the bottom product or with any other available low-grade heat sources, thermal condition of the feed may be altered to reduce the reboiler duty. In the case of feed preheating, a portion of the thermal energy given to the feed reduces the reboiler load and the rest increases the condenser load. There exists an efficiency (Peterson and Wells, 1977; Liebert, 1993) associated with feed preconditioning. Bandyopadhyay et al. (2003) developed a procedure to estimate feed preconditioning efficiency through IRS curves. Splitting the feed into two streams and preheating only one fraction of the feed stream, it is possible to improve feed preheat efficiency up to 100% (Wankat and Kessler, 1993; Fidkowski and Agrawal, 1995; Soave and Feliu, 2002; Deshmukh et al., 2005). Thermal integration of a distillation column with the background process through side exchangers (side reboilers/ condensers) is important for energy-efficient design of a distillation column. In refinery distillation columns, it is typically achieved through pump-around reflux (Sharma et al., 1999; Joshi et al., 2001). Thermal integration of a distillation column through side exchangers also improves the exergetic efficiency of the column. For sub-ambient distillation,
significant reduction in energy consumption may be achieved by incorporating side condensers and side reboilers (Kleinberg, 1987; Lynd and Grethlein, 1986; Aguirre et al., 1997). Agrawal and Herron (1998) have studied thermodynamically efficient configurations for adding or removing heat from an intermediate location of an ideal binary distillation column that separates an ideal binary mixture into pure components. However, their equations and methodology are limited to ideal binary mixture with constant relative volatility and are not even applicable for systems where the relative volatility varies moderately from top to bottom of the column. There are different ways an equivalent side exchanger may be integrated with the distillation column. This paper demonstrates different ways to integrate a distillation column thermally with its background process and a comparison with feed preheating is also reported. Feed condition for which thermodynamic efficiency of a distillation column with a side reboiler is more than that of a distillation column with feed preheating, are also identified in this paper. The described procedure is applicable to ideal as well as non-ideal mixtures.
INVARIANT RECTIFYING-STRIPPING (IRS) CURVES Generation procedure and physical significance of the IRS curves are briefly discussed below based on a derivation for a multiple feed distillation column at the minimum thermodynamic condition (MTC). MTC is defined as reversible operation for a distillation column with no entropy generation (King, 1980; Bandyopadhyay et al., 1998). It should be noted that the operating curve coincides with the equilibrium curve at MTC.
Generation of IRS Curves Bandyopadhyay et al. (2004) proposed a method to decompose a complex distillation problem into several separation problems, each consisting of a simple distillation column. Each side-product from a complex distillation may be considered as a feed to the column with a negative flowrate (Kister, 1985; Bandyopadhyay et al., 1998). Therefore, a complex column with multiple feeds and side-products is equivalent to a complex column with only multiple feeds where some of the feeds have negative flowrates. A complex column with n feeds may be decomposed into n simple columns keeping the purity and enthalpy of the end products of the ith column same as that of the original multiple-feed column (Figure 1) as proposed by Bandyopadhyay et al. (2004). For the ith decomposed column, the distillate flowrate (Di), bottom flow-rate (Bi), and enthalpy difference (Di) may be determined through the overall mass, component, and energy balances. Fi ¼ Di þ Bi
(1)
Fi zFi ¼ Di xD þ Bi xB Di ; Qri Qci ¼ Di hD þ Bi hB Fi hFi
(2) (3)
It may be noted that for a side-product, Fj and the corresponding Dj and Bj are negative. In the decomposition process, the overall mass, component, and enthalpy balances are conserved (as D ¼ S Di, B ¼ S Bi and D ; Qr2 Qc ¼ D hD þ B hB2 S Fi hFi ¼ S Di where the summation, S, goes from i ¼ 1 to i ¼ n).
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Figure 1. Decomposition of complex distillation column with n feeds into n single-feed distillation columns.
Let HRi be the minimum condensing load (enthalpy surplus) required to cause separation from x to xD in ith column. Plot of enthalpy surplus as a function of the equilibrium temperature (or equivalently Carnot factor) is termed the invariant rectifying (IR) curve. The overall mass, component, and energy balances for the rectifying section of the ith column may be combined to obtain the following expression for HRi. HRi ¼ Di ½hV (xD x)=(y x) hL (xD y)=(y x) hD
(4)
Similarly, consider the stripping section of a decomposed distillation column with HSi denoting the minimum reboiling load required to cause separation from x to xB. This enthalpy deficit is then plotted as a function of the equilibrium temperature (or equivalently Carnot factor) to give the invariant stripping (IS) curve. HSi ¼ Bi ½hV (x xB )=(y x) hL (y xB )=(y x) þ hB
(5)
For the purpose of generating IR and IS curves, the enthalpies and compositions as a function of temperature as well as for the feeds and products are required. For binary systems, thermodynamic models for enthalpy and vapour–liquid equilibrium may be readily used. In general, the right-hand-sides of equations (4) and (5) may be conveniently calculated for each stage of a distillation column from the output of a converged simulation. From equations (4) and (5), it may be noted that (HRi/Di) and (HSi/Bi) are constant for every feed. Therefore, it is not necessary to perform n simulations for the n decomposed columns in Figure 1 to generate the corresponding IRS curves. In fact, from a single converged simulation of the complex column, IRS curves may be obtained through equations (4) and (5) for each of the n decomposed columns.
The invariance of the IRS curves for binary systems may be proven from the fact that a binary system has only two degrees of freedom as per Gibb’s phase rule. Therefore, these curves become deterministic on specifying the operating pressure and separation required. Thus, HRi and HSi are functions of temperature only and are invariant to the total number of stages and the location of the feed in the column. The invariant property of the IRS curves does not hold rigorously for multicomponent systems. However, IRS curves for multicomponent systems show near-invariance to the total number of stages and the feed locations (Bandyopadhyay et al., 1999).
Order and Location of Feeds Combining component, mass and energy balances around the feed stage, the following relation between the IR curve and the IS curve can be obtained for every decomposed column. HSi ¼ HRi þ Di for the feed stage of the ith column
(6)
Given the fact that enthalpies are relative (i.e., the difference in enthalpy is important rather than the absolute enthalpy), the IR curve and/or the IS curve may be translated horizontally. The following convention may be adopted to translate the IRS curves in accordance with equation (6). Depending on the sign of Di [as defined in equation (3)], the translations may be conveniently classified into two cases: (a) if Di 0, then only the IR curve is translated to the right by Di and (b) if Di , 0, then only the IS curve is translated to the right by jDij. Note that, neither the IR curve nor the IS curve is translated for D ¼ 0. Mathematically, these translations can be conveniently represented as HRTi ¼ HRi þ Di =2 þ jDi =2j HSTi ¼ HSi Di=2 þ jDi =2j
(7) (8)
where HRTi and HSTi are the enthalpy coordinates for the translated IR curve and translated IS curve, respectively. Note that,
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although the IR and IS curves are independent of feed condition, the translated curves depend on the thermal condition of the feed. Equations (6)–(8) may be combined to obtain HSTi ¼ HRTi for the feed stage of the ith column
(9)
Physically, equation (9) signifies the intersection of the q-line for a feed with the equilibrium curve on the x – y diagram. The intersection of ith q-line with the equilibrium curve is independent of the intersections for the remaining feeds. The appropriate location for ith feed may be determined in terms of temperature (TFi) by finding the intersection of the respective translated IRS curves. The appropriate order for the feeds may be determined by simply arranging the target temperatures [e.g., feed F2 is at a higher temperature than feed F1 in Figure 2(a)].
Composite IRS Curves and Energy Targets The portion of the translated IR curve below TFi and the portion of the translated IS curve above TFi may be defined as the active portions of the translated IRS curves for the ith decomposed column and consequently the ith feed. Composite IRS curves are generated for the complex column by
simply adding the enthalpy coordinates of the active translated IRS curves corresponding to all the feeds. The composite IRS curves are shown in Figure 2 for a typical two-feed column based on the translated IRS curves of the two decomposed columns. The composite IRS curves consist of HRT1 þ HRT2, HST1 þ HRT2, and HST1 þ HST2 [Figure 2(a)]. Thus, for section i of an n-feed column HA ¼
n X j¼1
HAj ¼
i X
HSTj þ
j¼1
n X
HRTj
i ¼ 0,1, . . . ,n
(10)
j¼iþ1
In essence, an n-feed complex column may be decomposed into n single-feed columns for the generation of n sets of IRS curves. Then, their enthalpy coordinates may be added to generate (n þ 1) composite IRS curves for the complex column. The active portions of the composite IRS curves (HA) may be circumscribed by a right-angled trapezium. The widths of the parallel sides of the trapezium at the top and bottom define the minimum energy targets for the reboiler and condenser, respectively [see Figure 2(a)]. Figure 2(a) illustrates the case where one of the intersection points of the composite IRS curves determines the pinch. This is observed to be the case whenever the IRS curves of the individual decomposed columns are monotonic in nature. Note that the composite IRS curves for the complex column are, however, not monotonic.
Exergy Loss Target Bandyopadhyay (2002) has shown that the total exergy loss in an adiabatic distillation column may be measured by the area between the active portions of the IRS curves and the circumscribed right-angled trapezium on Carnot factorenthalpy (1 versus H) diagram [Figure 2(b)]. ð dExT ¼ dExR þ dExS ¼ 1S HS 1R HR þ 1dHR ð 1dHS (11) Based on the minimum work (Wmin) for separation and the exergy loss (dExT) in the column, thermodynamic efficiency of the column may be defined as follows.
h ¼ Wmin =(Wmin þ dExT )
(12)
SIDE-EXCHANGER TARGETS
Figure 2. Composite IRS curves for a two-feed column: (a) targeting feed location and minimum energy requirement; (b) targeting exergy loss in a distillation column.
Side exchangers provide increased opportunities for heat integration and reduction in utility costs. As the IRS curves are fundamentally based on the minimum thermodynamic condition, they define the maximum heat load that can be placed on side exchangers at specified temperature levels. Figure 3(a) shows corners truncated out of the right-angled trapezium, employed for targeting minimum energy requirement. The upper corner depicts the maximum scope to supply a portion (Qsr,max) of the required heat through a side reboiler at a temperature (Tsr) below that of the main reboiler. Similarly a lower corner [not shown in Figure 3(a)] shows the maximum potential to remove a portion (Qsc,max) of the excess heat through a side condenser at a temperature (Tsc) above that of the main condenser. The significance of
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the pinch, in the context of side exchangers for distillation, may be stated as follows: no (side-)reboiling below the pinch temperature and no (side-)condensing above the pinch temperature. This implies that for near-ideal system with monotonic IRS curves, side-reboiler and side-condenser may be put in the stripping and the rectifying section of the column, respectively. However, for non-ideal systems with non-monotonic IRS curves, it may be possible to place side-reboiler in the rectifying section or side-condenser in the stripping section of the column. This is consistent with the observations of Naka et al. (1980) and Agrawal and Fidkowski (1996). In actual practice, it is difficult to provide heat in equilibrium to a theoretical stage. In practice, it is easier to draw internal liquid or vapour stream from a distillation column, heat it with external heating medium and place it back to the distillation column. Different possible arrangements are discussed below with their exergetic efficiencies through an ideal binary example.
Base Case
Figure 3. Active IRS curves for different way of incorporating side reboiler: (a) side reboiler targets; (b) withdrawing liquid and returning at the same location; (c) withdrawing 10 kmol h21 of liquid and returning at different location; and (d) withdrawing 36 kmol h21 of liquid and returning at different location.
As an illustrative example, some simplifying assumptions are made that are consistent with the original problem description. The assumptions are no pressure drop losses, ideal vapour phase and ideal liquid solution. Further, it is assumed that the latent heats of vaporization (li) are equal for both the components and independent of temperature (over the operating temperature range of the distillation column), and the vapour pressure of a component is given by the Clausius –Clapeyron equation. These assumptions imply that the mixture to be separated is an ideal binary mixture of constant relative volatility with constant molar overflow. In this example, TA ¼ 353 K and TB ¼ 373 K where TA and TB are the boiling points of the low-boiling component A and the high-boiling component B, respectively. The latent heat of vaporization l is calculated using the Clausius –Clapeyron equation from l ¼ TA TB R ln a/(TB 2 TA) knowing the relative volatility a between components A and B. Assuming a ¼ 2.05, l is found to be 39.291 kJ mol21. However, it should be noted that the methodology described here is not restricted to cases with these assumptions. The described methodology is applicable to ideal as well as non-ideal mixtures. A feed of 50 kmol h21, containing 60% lighter component (A), is fed to the distillation column as saturated liquid. 98% of the lighter component is recovered at the top of the column whereas 97% of the heavier component is recovered at the bottom. From the overall mass and material balances, the product flow rates (in kmol h21) are obtained as D ¼ 30 and B ¼ 20. The IRS curves of the system are shown in Figure 3(a). Appropriate location for the feeds may be targeted as TF ¼ 359.2 K (corresponds to z ¼ 0.6) and the minimum energy targets (in terms of condenser and reboiler loads) are determined from the widths of the circumscribing rectangle (D ¼ 0, in this example). The minimum condenser and reboiler duties are equal (2.897 106 kJ h21), which corresponds to the minimum reflux ratio of 1.458 and minimum reboil ratio of 3.687. Exergetic efficiency of the base column is found to be 56.12%. If the column operates with 20% more reflux, the exergetic efficiency decreases to 46.78%. This is considerably higher than the commonly cited thermodynamic efficiency of 10%. Therefore, design
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of better tray internals for lower pressure drop and the energy efficient design of the reboiler as well as condenser are to be addressed to improve the overall efficiency of the distillation column. Especially, sub-cooled reflux should be avoided to improve the thermodynamic efficiency. It has already been observed that the energy efficiency of a crude distillation unit can be improved upon by avoiding sub-cooled reflux to the main fractionation column (Joshi et al., 2001). As the primary focus is on thermodynamic efficiency and also for brevity, we assume a tall column with minimum reflux. In case an ideal side reboiler is incorporated in the base column, thermodynamic efficiency of the column can be improved significantly. The problem of finding location and duty of a side reboiler can be mathematically translated as cutting a rectangle with maximum area from the IRS curves represented on a Carnot factor-enthalpy (1 versus H) diagram [as explained in Figure 3(a)]. Maximum exergetic efficiency can be achieved when a side reboiler is placed at 363.81 K with a maximum possible duty of 1.414 106 kJ h21 (49% of the main reboiler duty). Exergetic efficiency of the system improves to 70.92% (an improvement of 26.37% over the base column).
Drawing Liquid from the Column and Returning at the Same Point after Heating A portion of the vapour or liquid may be withdrawn from a distillation column and fed back to the column at the same point after heating it up with external utility. Drawing vapour from the column is difficult in practice. Moreover, lower value of heat transfer coefficient of the vapour stream will call for larger heat exchanger. It may also be noted that introduction of superheated vapour will have its own operational problems such as pressure drop, high velocity, product degradation due to liquid carry over, and so on. For all practical purpose, it is better to withdraw liquid from the column. This is also the practice as in the cases of pump-around or pump-back reflux (Sharma et al., 1999; Joshi et al., 2001). For comparison purpose, the heater duty kept constant at 1.414 106 kJ h21. As the liquid is withdrawn at 363.81 K (corresponds to z ¼ 0.359) and put back at the same location, the exergetic efficiency of the column improves to 70.7%. One interesting observation made during the study that the efficiency of the column is independent of the withdrawal flow rate. This implies that the flow rate of the withdrawal may be varied to adjust the appropriate approach temperature of the external heat exchanger. This is equivalent to having a side reboiler at the withdrawal stage. Figure 3(b) shows the active portions of the IRS curves demonstrating this. Small difference in efficiency is due to numerical integration procedure applied to calculate total exergy loss in the column. It may be noted that the withdrawal rate should be such that the resultant feed, after heating, must be saturated or two phase. Otherwise the mass transfer efficiency of the actual stages will decrease significantly as it will be doing mainly heat transfer activities.
curve. This is a new configuration and not analysed before. Withdrawal of 10 kmol h21 from the column at 363.81 K (corresponds to z ¼ 0.359) and heating it with external utility (exchanger duty is kept equal to 1.414 106 kJ h21) changes the thermal condition to q ¼ 22.6. Based on the q-line of this, it may be put back into the column at 370.47 K. Figure 3(c) shows the active portions of the IRS curves demonstrating this. It changes the internal traffic significantly and the resultant efficiency of the column comes out to be only 63.1%, significantly lower than the previous case. This is primarily due to retrograde phenomenon of internal separation. Superheated vapour (as q is negative) has to be separated inside the column such that the required separation may be obtained at the withdrawal stage. Furthermore, there are associated superheated vapour introduction problems, as highlighted before. It may be noted that the efficiency of the column changes with liquid draw rate. Changing the withdrawal rate from 10 kmol h21 to 36 kmol h21, the resultant feed becomes saturated vapour and the thermodynamic efficiency increases up to 67.11% (when located at 367.11 K). Figure 3(d) shows the active portions of the IRS curves demonstrating this. If the withdrawal rate is further increased, the efficiency approaches the limiting value of 70.7% and become equivalent to the previous case.
Comparison with Feed Preheating Options The same amount of heat may be utilized to preheat the feed. By preheating the feed by 1.414 106 kJ h21, the thermal energy of the feed can be changed and consequently the q-value of the feed changes from 1 to 0.28. The location of the feed, in temperature scale, changes to 361.5 K. However, the change in feed quality does not change the reboiler duty alone, it also increase the condenser duty. The minimum condenser duty changes to 3.411 106 kJ h21, corresponds to the minimum reflux ratio of 1.894. The minimum reboiler duty changes to 1.997 106 kJ h21, corresponds to the minimum reboil ratio of 2.541. The preheat efficiency, defined as the portion of the heat given to feed that reduces the reboiler duty, is 63.65%. The thermodynamic efficiency of the column changes to 64.6%. Procedures developed by Bandyopadhyay et al. (2003) may be utilized to calculate the feed preheat efficiency [Figure 4(a)]. As proposed by Deshmukh et al. (2005), feed preheat efficiency can be improved to 100% using the concept of feed splitting. Now, 23.314 kmol h21 of feed may be left unchanged and fed to the column at 359.15 K. Remaining portion of the feed, i.e., 26.686 kmol h21 can be preheated by the external utility heater and fed to the column at 363.73 K. This arrangement ensures 100% feed preheat efficiency [Figure 4(b)] as the condenser duty remain same and the reboiler duty reduced by the same amount. In this case, a maximum thermodynamic efficiency of 77.44% can be achieved.
EFFECT OF SYSTEMS PARAMETERS Drawing Liquid from the Column and Returning at Different Location A portion of the liquid withdrawn from the column may be fed back to the column at another point based on the intersection of the q-line of the heated liquid with the equilibrium
Impact of different system parameters such as relative volatility, feed composition and thermal condition of the feed on thermodynamic performance of a distillation column is studied in details. Typically, feed to a distillation column originates from another distillation column or other unit operations.
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Base Case
Figure 4. Feed preheating: (a) entire feed is preheated and (b) 100% feed preheat efficiency by feed splitting.
Knowing the effect of composition and thermal condition of the feed on the thermodynamic performance of a distillation column, thermodynamic efficiency of the distillation column, as well as that of the overall process, may be improved upon by selecting appropriate thermal integration techniques and/or by adjusting the feed at its appropriate condition from the preceding unit operation. Effect of system parameters on thermodynamic efficiency of a distillation column is discussed in this section. For simplicity an ideal binary system has been considered. Assumptions listed in the previous section are considered. However, recoveries of components are assumed to be 99% in this section. For comparison, thermodynamic efficiencies of the base case distillation column, distillation column with optimum side-reboiler duty, column with entire feed preheating and feed preheating with feed splitting are considered. For all these cases, duties of the external heat exchangers are kept equal to that of the optimum side reboiler. Similar results may be obtained for distillation column with side-condenser. However, for brevity, case of side condenser has not been considered. Effects of relative volatility, feed composition, and thermal condition of the feed on thermodynamic efficiencies of a distillation column are shown in Figures 5–7. These results are discussed below.
For a very low relative volatility (a ! 1), minimum duties for the reboiler and the condenser increase sharply, whereas the efficiency approaches a limiting value. This limiting value depends on the feed composition but is independent of the thermal condition of the feed. In such circumstances, approach temperatures for the reboiler and condenser play a dominant role in defining the efficiency of the column. Thermodynamic efficiency of a distillation column processing saturated liquid feed with composition z is qualitatively comparable, but not exactly equal numerically, with the thermodynamic efficiency of a column processing saturated vapour feed with composition 1 2 z. For saturated liquid feed with more light component (z 0.5) and saturated vapour feed with more heavy component (z 0.5), the efficiency of the base column reduces monotonically with relative volatility (Figure 5). Whereas for saturated liquid feed with more heavy component (z , 0.5) and for saturated vapour feed with more light component (z . 0.5), the efficiency goes through a maximum. As concentration of the lighter component decreases in the saturated liquid feed or concentration of the heavier component decreases in the saturated vapour feed, the optimum shifts towards higher relative volatility. These observations lead to the fact that for a given composition and thermal condition, the relative volatility may be changed by changing the operating pressure to increase the efficiency of distillation. Change in operating pressure is especially important for sub-ambient distillation processes. For saturated liquid feed with more light component (or saturated vapour feed with more heavy component), increase in operating pressure may be encouraged for cryogenic distillation. Higher operating pressure results in lower entropy generation due to lower pressure drop and reduction in refrigeration load. It should be noted that it may be possible to manipulate relative volatility to a small extent by changing the operating pressure of a distillation column, large changes in relative volatility are normally difficult to achieve. It may be worth emphasizing that in most gas separation processes the variations of distillation efficiency with operating pressure is of utmost importance. Depending on the thermal condition of the feed, the operating pressure of distillation may be adjusted. However, costbenefit analysis should be performed before implementing such modifications. Figures 6 and 7 suggest that for a given relative volatility and feed composition, there exists an optimal thermal condition of the feed that maximizes the thermodynamic efficiency of the base column. For a given feed composition, thermal condition of the feed may be adjusted to achieve the maximum thermodynamic efficiency of a distillation column. Approaches based on pinch analysis may be utilised for this purpose.
Column with Side Reboiler Incorporation of a side reboiler always increases the thermodynamic efficiency of a distillation column. This can easily be appreciated from the IRS curves, as described in the earlier section. Similar to the base case, for saturated liquid feed with more light component and saturated vapour feed with more heavy component, the efficiency of a
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Figure 5. Effect of relative volatility on thermodynamic efficiency: (a) z ¼ 0.5 and q ¼ 1, (b) z ¼ 0.25 and q ¼ 1, (c) z ¼ 0.75 and q ¼ 1, (d) z ¼ 0.5 and q ¼ 0, (e) z ¼ 0.25 and q ¼ 0, (f) z ¼ 0.75 and q ¼ 0.
distillation column with a side reboiler reduces monotonically with relative volatility (Figure 5). By incorporating a side reboiler, significant improvement in the thermodynamic efficiency of a distillation column may be obtained when the feed contains more light component (Figure 6). For a saturated vapour feed or a feed with predominant vapour fraction (i.e., low q), it is easier to incorporate side reboiler and feed preheating is thermodynamically not a right option as
introduction of a superheated feed is thermodynamically inefficient as well as has its own operational problems.
Column with Feed Preheating It may be noted that instead of preheating the entire feed, preheating a portion of the feed is always beneficial from thermodynamic efficiency point of view. This is primarily because
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Figure 6. Effect of feed composition on thermodynamic efficiency: (a) a ¼ 1.5 and q ¼ 1, (b) a ¼ 2.5 and q ¼ 1, (c) a ¼ 1.5 and q ¼ 0, (d) a ¼ 2.5 and q ¼ 0.
an extra amount of separation is obtained through feed splitting and 100% feed preheat efficiency is achieved. Feed preheating is beneficial for predominantly liquid feed with more light component (Figures 6 and 7). It may be observed that for feed with more heavy component, it is not beneficial to preheat the feed.
preheating configuration. Whereas thermodynamic efficiency of a distillation column, processing a saturated vapour feed with 50% light component, can be enhanced significantly by incorporating a side reboiler. Such a curve gives a process designer a guideline for choosing appropriate configuration of a distillation column for thermal integration.
Optimal Configuration Curve
CONCLUSION
Figure 5(a) suggests that for a saturated liquid feed with 50% light component, it is always beneficial to have feed preheating with feed splitting. On the other hand, Figure 5(d) suggests that thermodynamic efficiency of a distillation column is always more with a side reboiler. Similar conclusions may be drawn form Figures 6 and 7. Depending on the feed composition and thermal condition of the feed, either side exchanger or feed preheating by feed splitting enhances the thermodynamic efficiency of a distillation column to the maximum extent. Figure 8 represents the optimal configuration curve in terms of the maximum value of q for different feed composition up to which it is beneficial to incorporate side reboiler in a distillation column. Beyond this it is always beneficial to incorporate feed preheater with feed splitting. For example, for a saturate liquid feed with 50% light component, it is beneficial to have feed
Exergy analysis of a distillation column often provides useful understanding and insight for energy efficient design of distillation processes (Atkinson, 1987; Ratkje et al., 1995; Taprap and Ishida, 1996; Agrawal and Herron, 1997; Demirel, 2004). This is particularly true for low-temperature processes, such as gas separation process. Exergy analysis is also useful for thermal integration of a distillation column with other unit operations. In addition to determining total exergy loss in a distillation column, it is desirable to look at its exergetic efficiency. The exergetic efficiency of a distillation column or any other unit operation can provide useful information about its potential for improvement. It is possible to have a large total exergy loss for an operation that still has fairly high exergetic efficiency. For example, even though there is a great incentive to improve further the performance of an efficient heat exchanger, handling a large mass flow, the
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Figure 7. Effect of thermal condition of feed on thermodynamic efficiency: (a) a ¼ 1.5 and z ¼ 0.25, (b) a ¼ 1.5 and z ¼ 0.5, (c) a ¼ 1.5 and z ¼ 0.75, (d) a ¼ 2.5 and z ¼ 0.25, (e) a ¼ 2.5 and z ¼ 0.5, (f) a ¼ 2.5 and z ¼ 0.75.
potential for improvement is low and would be likely to require significant effort and investment. On the other hand, an operation with low exergetic efficiency and also a low exergy loss will not be worth the effort to further improve its performance. Instances where exergetic efficiencies are not very high and where exergy losses are significant provide good potential for improvement. Thermodynamic efficiency of a distillation improves as it is thermally integrated with background process. There are two important ways a distillation column can be energy integrated: through side exchangers and through feed preheating. Depending upon the thermal condition of the feed, feed composition and relative volatility of the system, appropriate
configuration may be selected. An optimal configuration curve, proposed in this paper, may play an important role in helping a process designer to choose an appropriate column configuration for thermal integration. It should be noted that thermodynamic efficiency of a distillation column has been considered in this article as the sole criterion for optimizing the thermal condition of the feed. Applicability and quality of the design of a distillation column cannot possibly be judged solely on the basis of thermodynamic efficiency of the distillation column. Factors such as inefficiencies associated with converting a feed to its optimal conditions, capital and operating cost of the system, operational complexity and flexibility and so on are also important
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165
side condenser side reboiler stripping (translated) vapour
REFERENCES
Figure 8. Optimal configuration curves for different relative volatility. It is represented in terms of the maximum value of q for different feed composition up to which it is beneficial to incorporate side reboiler in a distillation column.
for designing a distillation column. However, thermodynamic analysis of the distillation column does provide useful understanding and insights to the designer. It may also be noted that the exergy analysis performed in this paper, is restricted in analysing a distillation column only. For practical implementation, thermodynamic analysis of the entire process and thermo-economic analysis of the entire system should be performed.
NOMENCLATURE B D F h H L Q q T V W x y z
bottom product molar flow, kmol h21 distillate molar flow, kmol h21 feed molar flow, kmol h21 specific enthalpy, kJ kmol21 enthalpy, kJ h21 liquid molar flow, kmol h21 heat duty, kJ h21 thermal condition of the feed temperature, K vapour molar flow, kmol h21 work, kJ h21 mole fraction in liquid mole fraction in vapour mole fraction in feed
Greek symbols D enthalpy difference, kJ h21 dEx exergy, kJ h21 h thermodynamic efficiency a relative volatility 1 Carnot factor Subscripts and superscripts B bottom product c condenser D distillate F feed L liquid max maximum min minimum r reboiler R rectifying RT rectifying (translated) S stripping
Agrawal, R. and Fidkowski, Z.T., 1996, On the use of intermediate reboilers in the rectifying section and condensers in the stripping section of a distillation column, Ind Eng Chem Res, 35(8): 2801–2807. Agrawal, R. and Herron, D.M., 1997, Optimal thermodynamic feed conditions for distillation of ideal binary mixtures, AIChE J, 43(11): 2984– 2996. Agrawal, R. and Herron, D.M., 1998, Intermediate reboiler and condenser arrangement for binary distillation columns, AIChE J, 44(6): 1316–1323. Aguirre, P., Espinosa, J., Tarifa, E. and Scenna, N., 1997, Optimal thermodynamic approximation to reversible distillation by means of interheaters and intercoolers, Ind Eng Chem Res, 36(11): 4882–4893. Atkinson, T.D., 1987, Second law analysis of cryogenic processes using a three term model, Gas Sep Purif, 1(2): 84–89. Bandyopadhyay, S., 2002, Effect of feed on optimal thermodynamic performance of a distillation column, Chem Eng J, 88(1–3): 175– 186. Bandyopadhyay, S., Malik, R.K. and Shenoy, U.V., 1998, Temperature-enthalpy curve for energy targeting of distillation columns, Comp Chem Eng, 22(12): 1733–1744. Bandyopadhyay S., Malik, R.K. and Shenoy, U.V., 1999, Invariant rectifying-stripping curves for targeting minimum energy and feed location in distillation, Comp Chem Eng, 23(8): 1109–1124. Bandyopadhyay, S., Malik, R.K. and Shenoy, U.V., 2003, Feed preconditioning targets for distillation through invariant rectifyingstripping curves, Ind Eng Chem Res, 42(26): 6851– 6861. Bandyopadhyay, S., Mishra, M. and Shenoy, U.V., 2004, Energybased targets for multiple-feed distillation columns, AIChE J, 50(8): 1837– 1853. Benedict, M., 1947, Multistage separation processes, Trans AIChE, 43(2): 41– 60. Demirel, Y., 2004, Thermodynamic analysis of separation systems, Separation Sci Tech, 39(16): 3897–3942. Deshmukh, B.F., Malik, R.K. and Bandyopadhyay, S., 2005, Efficient feed preheat targeting for distillation by feed splitting, Proceedings of European Symposium on Computer Aided Process Engineering—15 (ESCAPE-15), Barcelona, 29 May– 1 June, 751– 756. Dhole, V.R. and Linnhoff, B., 1993, Distillation column targets, Comp Chem Eng, 17(5/6): 549–560. Fidkowski, Z.T. and Agrawal, R., 1995, Utilization of waste heat stream in distillation, Ind Eng Chem Res, 34(4): 1287– 1293. Fitzmorris, R.E. and Mah, R.S.H., 1980, Improving distillation column design using thermodynamic availability analysis, AIChE J, 26(2): 265– 273. Fonyo´, Z., 1974, Thermodynamic analysis of rectification. I. Reversible model of rectification, Int Chem Eng, 14(1): 18– 27. Franklin, N.L. and Wilkinson, M.B., 1982, Reversibility in the separation of multicomponent mixtures, Trans IChemE, 60(5): 276– 282. Hall, S.G., Ognisty, T.P. and Northup, A.H., 1995, Use process integration to improve FCC/VRU design. Part 1, Hydro Process, March: 63– 74. Ho, F.G. and Keller, G.E., 1987, Process integration, In Recent Developments in Chemical Process and Plant Design Liu, Y.A., McGee, H.A. and Epperly, W.G. (Eds). (Wiley, New York, USA). Joshi, M.K., Roy, D.R. and Bandyopadhyay, S., 2001, Design of energy efficient crude distillation unit, Presented at 4th International Petroleum Conference (PETROTECH), New Delhi, 9– 12 January. King, C.J., 1980, Separation Processes (McGraw-Hill, New York, USA). Kister, H.Z., 1985, Graphically find theoretical trays and minimum reflux for complex binary distillation, Chem Eng, 92(2): 97–104. Kleinberg, W.T., 1987, Dual air pressure cycle to produce low purity oxygen, US Patent No. 4,702,757. Lee, J.-C., Yeo, Y.-K., Song, K.H. and Kim, I.-W., 2001, Operating strategies and optimum feed tray locations of the fractionation unit of btx plants for energy conservation, Korean J Chem Eng, 18(4): 428 –431.
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Liebert, T., 1993, Distillation feed preheat-is it energy efficient? Hydro Process, October: 37–42. Lynd, L.R. and Grethlein, L.R., 1986, Distillation with intermediate heat pumps and optimal sidestream return, AIChE J, 32(8): 1347–1359. Naka, Y., Terashita, M., Hayashiguchi, S. and Takamatsu, T., 1980, An intermediate heating and cooling method for a distillation column, J Chem Eng Japan, 13(2): 123– 129. Ognisty, T.P., 1995, Analyze distillation columns with thermodynamics, Chem Eng Prog, February: 40–46. Peterson, W.C. and Wells, T.A., 1977, Energy-saving schemes in distillation, Chem Eng, 84(20): 78– 90. Ratkje, S.K., Sauar, E., Hansen, E.M., Lien, K.M. and Hafskjold, B., 1995, Analysis of entropy production rates for design of distillation columns, Ind Eng Chem Res, 34(9): 3001–3007. Sharma, R., Jindal, A., Mandawala, D. and Jana, S.K., 1999, Design/ retrofit targets of pump-around refluxes for better energy integration of a crude distillation column, Ind Eng Chem Res, 38(6): 2411– 2417. Soave, G. and Feliu, J.A., 2002, Saving energy in distillation by feed splitting, Appl Thermal Eng, 22(8): 889 –896.
Taprap, R. and Ishida, M., 1996, Graphic exergy analysis of processes in distillation column by energy-utilization diagram, AIChE J, 42(6): 1633–1640. Terranova, B.E. and Westerberg, A.W., 1989, Temperature-heat diagrams for complex columns. 1. Intercooled/interheated distillation columns, Ind Eng Chem Res, 28(9): 1374–1379. Trivedi, K.K., Pang, K.H., O’Young, D.L., Klavers, H.W. and Linnhoff, B., 1996, Optimize a licensor’s design using pinch technology, Hydro Process, May: 113–126. Wankat, P. and Kessler, D.P., 1993, Two-feed distillation. Same-composition feeds with different enthalpies, Ind Eng Chem Res, 32(12): 3061–3067.
ACKNOWLEDGEMENTS The author is thankful to Industrial Research and Consultancy Centre, IIT Bombay, for providing financial support for this work. The manuscript was received 16 July 2006 and accepted for publication after revision 24 October 2006.
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Abstract: Thermal integration of a distillation column with the background process through side exchangers (side reboilers/condensers) is important for designing an energy-efficient distillation column. Thermal integration of a distillation column through side exchangers also improves the exergetic efficiency of the column. Invariant rectifying and stripping (IRS) curves are employed for thermodynamic analysis of a distillation column. Different configurations for side-exchanger integration are evaluated based on the exergetic efficiency of the distillation process and they are also compared with feed preheating options. A novel optimal configuration curve is proposed in this paper that identifies feed composition and its thermal condition for which it is beneficial to have a side reboiler in a distillation column. Keywords: thermodynamic analysis; distillation column; side reboiler; feed preheating; IRS curves; optimal configuration curve.
INTRODUCTION
Correspondence to: Professor S. Bandyopadhyay, Energy Systems Engineering and Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India. E-mail: santanu@ me.iitb.ac.in
DOI: 10.1205/Cherd06108R1 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, January 2007 # 2007 Institution of Chemical Engineers
Distillation is one of the common unit operations used in chemical industries. Operating cost of a chemical process plant mainly depends on the operating cost associated with the separation processes. Fifty to 80% of the operating cost in a chemical process industry is contributed by separation processes (Soave and Feliu, 2002). Total energy requirement of chemical process industry in the United States is over 5 quad year21 and distillation process alone consumes about 2.4 quad year21 (Ognisty, 1995). This amounts to about 3% of the total energy consumed in the United States. Though distillation is energy intensive, it is not energy efficient. Therefore, it is necessary to identify different energy conservation opportunities through thermodynamic analysis of a distillation process. Thermodynamic analysis of a distillation column is important for synthesizing and developing energy-efficient distillation processes. It allows the thermodynamic efficiency of a process to be quantified, regions with poor energy efficiency to be identified and the thermodynamic targets to be defined. Thermodynamic analysis of a distillation column aims at possible reduction in exergy loss, or equivalently, reduction in entropy generation. There are three major sources of entropy generation in a distillation column— heat transfer with finite temperature driving force, mixing of non-equilibrium vapour and
liquid, and pressure drop across the column. Additionally, there may be entropy generation due to heat loss to the ambient from the column surface. There are two different ways of analysing distillation column thermodynamically, viz., exergy analysis and approaches based on the temperature – enthalpy (T – H) curves. The T –H diagram for a distillation column is a useful representation to quantitatively address the energy-saving potential for possible stand-alone modifications such as reflux reduction, feed conditioning and scope for side reboiler/condenser (Naka et al., 1980; Terranova and Westerberg, 1989; Dhole and Linnhoff, 1993; Ognisty, 1995; Hall et al., 1995; Trivedi et al., 1996). The T– H curve of a binary distillation column is generated by solving the coupled mass and enthalpy balances for the reversible separation scheme (Benedict, 1947; Fonyo´, 1974; King, 1980; Fitzmorris and Mah, 1980; Ho and Keller, 1987). Reversible operation corresponds to minimum net work consumption and involves minimization of the driving forces for heat and mass transfer within individual stages (or, in other words, decrease of the gap between the operating and equilibrium curves on the x –y diagram) as discussed in detail by King (1980). In contrast to binary distillation, the sharpness of separation is generally limited in multicomponent reversible distillation (Fonyo´, 1974) and it is impossible to devise a reversible separation scheme for many practical multicomponent 155
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separations (Franklin and Wilkinson, 1982). The limitation in sharpness of reversible multicomponent distillation can be circumvented, for the purposes of generation of the T–H curve, by using the pseudo-binary concept of light and heavy key model (Fonyo´, 1974; Dhole and Linnhoff, 1993). Dhole and Linnhoff (1993) described a procedure for generating a T–H curve (which they called the column grand composite curve) from a converged simulation of a distillation column. The calculation procedure involves determination of the net enthalpy deficit at each stage by generating envelopes from either the condenser end (top-down approach) or the reboiler end (bottom-up approach). However, the values calculated by the two approaches differ for stages with feeds because they do not consider the enthalpy balances at the feed stages. A feed stage correction that rigorously considers the mass and enthalpy balances at feed stages has been recently proposed by Bandyopadhyay et al. (1998) to resolve the discrepancy. Bandyopadhyay et al. (1999) introduced a novel pair of T–H curves, known as the invariant rectifying-stripping (IRS) curves for a distillation column. The IRS curves are invariant to the column configuration (i.e., feed location in the column and number of stages) and depend only on sharpness of separation as well as operating pressure of the column. They are useful for setting quantitative targets such as minimum energy requirement, appropriate feed location, proper feed preheating, scope for side-condensers/ reboilers,as well as thermo-economic optimization of a distillation column. Exergy analysis of a distillation column often provides useful understanding and insight for energy efficient design of distillation processes. Bandyopadhyay (2002) established that the exergy loss in a distillation column may be measured directly by representing the IRS curves on Carnot factor-enthalpy (1 versus H) diagram. Different schemes such as proper feed location (Bandyopadhyay et al., 1999; Lee et al., 2001), feed preheating (Liebert, 1993; Soave and Feliu, 2002; Bandyopadhyay et al., 2003; Deshmukh et al., 2005), side-exchangers (Peterson and Wells, 1977; Aguirre et al., 1997; Agrawal and Herron, 1998; Sharma et al., 1999) and so on are suggested to reduce the energy consumption and to improve energetic efficiency in distillation. By exchanging heat with the bottom product or with any other available low-grade heat sources, thermal condition of the feed may be altered to reduce the reboiler duty. In the case of feed preheating, a portion of the thermal energy given to the feed reduces the reboiler load and the rest increases the condenser load. There exists an efficiency (Peterson and Wells, 1977; Liebert, 1993) associated with feed preconditioning. Bandyopadhyay et al. (2003) developed a procedure to estimate feed preconditioning efficiency through IRS curves. Splitting the feed into two streams and preheating only one fraction of the feed stream, it is possible to improve feed preheat efficiency up to 100% (Wankat and Kessler, 1993; Fidkowski and Agrawal, 1995; Soave and Feliu, 2002; Deshmukh et al., 2005). Thermal integration of a distillation column with the background process through side exchangers (side reboilers/ condensers) is important for energy-efficient design of a distillation column. In refinery distillation columns, it is typically achieved through pump-around reflux (Sharma et al., 1999; Joshi et al., 2001). Thermal integration of a distillation column through side exchangers also improves the exergetic efficiency of the column. For sub-ambient distillation,
significant reduction in energy consumption may be achieved by incorporating side condensers and side reboilers (Kleinberg, 1987; Lynd and Grethlein, 1986; Aguirre et al., 1997). Agrawal and Herron (1998) have studied thermodynamically efficient configurations for adding or removing heat from an intermediate location of an ideal binary distillation column that separates an ideal binary mixture into pure components. However, their equations and methodology are limited to ideal binary mixture with constant relative volatility and are not even applicable for systems where the relative volatility varies moderately from top to bottom of the column. There are different ways an equivalent side exchanger may be integrated with the distillation column. This paper demonstrates different ways to integrate a distillation column thermally with its background process and a comparison with feed preheating is also reported. Feed condition for which thermodynamic efficiency of a distillation column with a side reboiler is more than that of a distillation column with feed preheating, are also identified in this paper. The described procedure is applicable to ideal as well as non-ideal mixtures.
INVARIANT RECTIFYING-STRIPPING (IRS) CURVES Generation procedure and physical significance of the IRS curves are briefly discussed below based on a derivation for a multiple feed distillation column at the minimum thermodynamic condition (MTC). MTC is defined as reversible operation for a distillation column with no entropy generation (King, 1980; Bandyopadhyay et al., 1998). It should be noted that the operating curve coincides with the equilibrium curve at MTC.
Generation of IRS Curves Bandyopadhyay et al. (2004) proposed a method to decompose a complex distillation problem into several separation problems, each consisting of a simple distillation column. Each side-product from a complex distillation may be considered as a feed to the column with a negative flowrate (Kister, 1985; Bandyopadhyay et al., 1998). Therefore, a complex column with multiple feeds and side-products is equivalent to a complex column with only multiple feeds where some of the feeds have negative flowrates. A complex column with n feeds may be decomposed into n simple columns keeping the purity and enthalpy of the end products of the ith column same as that of the original multiple-feed column (Figure 1) as proposed by Bandyopadhyay et al. (2004). For the ith decomposed column, the distillate flowrate (Di), bottom flow-rate (Bi), and enthalpy difference (Di) may be determined through the overall mass, component, and energy balances. Fi ¼ Di þ Bi
(1)
Fi zFi ¼ Di xD þ Bi xB Di ; Qri Qci ¼ Di hD þ Bi hB Fi hFi
(2) (3)
It may be noted that for a side-product, Fj and the corresponding Dj and Bj are negative. In the decomposition process, the overall mass, component, and enthalpy balances are conserved (as D ¼ S Di, B ¼ S Bi and D ; Qr2 Qc ¼ D hD þ B hB2 S Fi hFi ¼ S Di where the summation, S, goes from i ¼ 1 to i ¼ n).
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Figure 1. Decomposition of complex distillation column with n feeds into n single-feed distillation columns.
Let HRi be the minimum condensing load (enthalpy surplus) required to cause separation from x to xD in ith column. Plot of enthalpy surplus as a function of the equilibrium temperature (or equivalently Carnot factor) is termed the invariant rectifying (IR) curve. The overall mass, component, and energy balances for the rectifying section of the ith column may be combined to obtain the following expression for HRi. HRi ¼ Di ½hV (xD x)=(y x) hL (xD y)=(y x) hD
(4)
Similarly, consider the stripping section of a decomposed distillation column with HSi denoting the minimum reboiling load required to cause separation from x to xB. This enthalpy deficit is then plotted as a function of the equilibrium temperature (or equivalently Carnot factor) to give the invariant stripping (IS) curve. HSi ¼ Bi ½hV (x xB )=(y x) hL (y xB )=(y x) þ hB
(5)
For the purpose of generating IR and IS curves, the enthalpies and compositions as a function of temperature as well as for the feeds and products are required. For binary systems, thermodynamic models for enthalpy and vapour–liquid equilibrium may be readily used. In general, the right-hand-sides of equations (4) and (5) may be conveniently calculated for each stage of a distillation column from the output of a converged simulation. From equations (4) and (5), it may be noted that (HRi/Di) and (HSi/Bi) are constant for every feed. Therefore, it is not necessary to perform n simulations for the n decomposed columns in Figure 1 to generate the corresponding IRS curves. In fact, from a single converged simulation of the complex column, IRS curves may be obtained through equations (4) and (5) for each of the n decomposed columns.
The invariance of the IRS curves for binary systems may be proven from the fact that a binary system has only two degrees of freedom as per Gibb’s phase rule. Therefore, these curves become deterministic on specifying the operating pressure and separation required. Thus, HRi and HSi are functions of temperature only and are invariant to the total number of stages and the location of the feed in the column. The invariant property of the IRS curves does not hold rigorously for multicomponent systems. However, IRS curves for multicomponent systems show near-invariance to the total number of stages and the feed locations (Bandyopadhyay et al., 1999).
Order and Location of Feeds Combining component, mass and energy balances around the feed stage, the following relation between the IR curve and the IS curve can be obtained for every decomposed column. HSi ¼ HRi þ Di for the feed stage of the ith column
(6)
Given the fact that enthalpies are relative (i.e., the difference in enthalpy is important rather than the absolute enthalpy), the IR curve and/or the IS curve may be translated horizontally. The following convention may be adopted to translate the IRS curves in accordance with equation (6). Depending on the sign of Di [as defined in equation (3)], the translations may be conveniently classified into two cases: (a) if Di 0, then only the IR curve is translated to the right by Di and (b) if Di , 0, then only the IS curve is translated to the right by jDij. Note that, neither the IR curve nor the IS curve is translated for D ¼ 0. Mathematically, these translations can be conveniently represented as HRTi ¼ HRi þ Di =2 þ jDi =2j HSTi ¼ HSi Di=2 þ jDi =2j
(7) (8)
where HRTi and HSTi are the enthalpy coordinates for the translated IR curve and translated IS curve, respectively. Note that,
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although the IR and IS curves are independent of feed condition, the translated curves depend on the thermal condition of the feed. Equations (6)–(8) may be combined to obtain HSTi ¼ HRTi for the feed stage of the ith column
(9)
Physically, equation (9) signifies the intersection of the q-line for a feed with the equilibrium curve on the x – y diagram. The intersection of ith q-line with the equilibrium curve is independent of the intersections for the remaining feeds. The appropriate location for ith feed may be determined in terms of temperature (TFi) by finding the intersection of the respective translated IRS curves. The appropriate order for the feeds may be determined by simply arranging the target temperatures [e.g., feed F2 is at a higher temperature than feed F1 in Figure 2(a)].
Composite IRS Curves and Energy Targets The portion of the translated IR curve below TFi and the portion of the translated IS curve above TFi may be defined as the active portions of the translated IRS curves for the ith decomposed column and consequently the ith feed. Composite IRS curves are generated for the complex column by
simply adding the enthalpy coordinates of the active translated IRS curves corresponding to all the feeds. The composite IRS curves are shown in Figure 2 for a typical two-feed column based on the translated IRS curves of the two decomposed columns. The composite IRS curves consist of HRT1 þ HRT2, HST1 þ HRT2, and HST1 þ HST2 [Figure 2(a)]. Thus, for section i of an n-feed column HA ¼
n X j¼1
HAj ¼
i X
HSTj þ
j¼1
n X
HRTj
i ¼ 0,1, . . . ,n
(10)
j¼iþ1
In essence, an n-feed complex column may be decomposed into n single-feed columns for the generation of n sets of IRS curves. Then, their enthalpy coordinates may be added to generate (n þ 1) composite IRS curves for the complex column. The active portions of the composite IRS curves (HA) may be circumscribed by a right-angled trapezium. The widths of the parallel sides of the trapezium at the top and bottom define the minimum energy targets for the reboiler and condenser, respectively [see Figure 2(a)]. Figure 2(a) illustrates the case where one of the intersection points of the composite IRS curves determines the pinch. This is observed to be the case whenever the IRS curves of the individual decomposed columns are monotonic in nature. Note that the composite IRS curves for the complex column are, however, not monotonic.
Exergy Loss Target Bandyopadhyay (2002) has shown that the total exergy loss in an adiabatic distillation column may be measured by the area between the active portions of the IRS curves and the circumscribed right-angled trapezium on Carnot factorenthalpy (1 versus H) diagram [Figure 2(b)]. ð dExT ¼ dExR þ dExS ¼ 1S HS 1R HR þ 1dHR ð 1dHS (11) Based on the minimum work (Wmin) for separation and the exergy loss (dExT) in the column, thermodynamic efficiency of the column may be defined as follows.
h ¼ Wmin =(Wmin þ dExT )
(12)
SIDE-EXCHANGER TARGETS
Figure 2. Composite IRS curves for a two-feed column: (a) targeting feed location and minimum energy requirement; (b) targeting exergy loss in a distillation column.
Side exchangers provide increased opportunities for heat integration and reduction in utility costs. As the IRS curves are fundamentally based on the minimum thermodynamic condition, they define the maximum heat load that can be placed on side exchangers at specified temperature levels. Figure 3(a) shows corners truncated out of the right-angled trapezium, employed for targeting minimum energy requirement. The upper corner depicts the maximum scope to supply a portion (Qsr,max) of the required heat through a side reboiler at a temperature (Tsr) below that of the main reboiler. Similarly a lower corner [not shown in Figure 3(a)] shows the maximum potential to remove a portion (Qsc,max) of the excess heat through a side condenser at a temperature (Tsc) above that of the main condenser. The significance of
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the pinch, in the context of side exchangers for distillation, may be stated as follows: no (side-)reboiling below the pinch temperature and no (side-)condensing above the pinch temperature. This implies that for near-ideal system with monotonic IRS curves, side-reboiler and side-condenser may be put in the stripping and the rectifying section of the column, respectively. However, for non-ideal systems with non-monotonic IRS curves, it may be possible to place side-reboiler in the rectifying section or side-condenser in the stripping section of the column. This is consistent with the observations of Naka et al. (1980) and Agrawal and Fidkowski (1996). In actual practice, it is difficult to provide heat in equilibrium to a theoretical stage. In practice, it is easier to draw internal liquid or vapour stream from a distillation column, heat it with external heating medium and place it back to the distillation column. Different possible arrangements are discussed below with their exergetic efficiencies through an ideal binary example.
Base Case
Figure 3. Active IRS curves for different way of incorporating side reboiler: (a) side reboiler targets; (b) withdrawing liquid and returning at the same location; (c) withdrawing 10 kmol h21 of liquid and returning at different location; and (d) withdrawing 36 kmol h21 of liquid and returning at different location.
As an illustrative example, some simplifying assumptions are made that are consistent with the original problem description. The assumptions are no pressure drop losses, ideal vapour phase and ideal liquid solution. Further, it is assumed that the latent heats of vaporization (li) are equal for both the components and independent of temperature (over the operating temperature range of the distillation column), and the vapour pressure of a component is given by the Clausius –Clapeyron equation. These assumptions imply that the mixture to be separated is an ideal binary mixture of constant relative volatility with constant molar overflow. In this example, TA ¼ 353 K and TB ¼ 373 K where TA and TB are the boiling points of the low-boiling component A and the high-boiling component B, respectively. The latent heat of vaporization l is calculated using the Clausius –Clapeyron equation from l ¼ TA TB R ln a/(TB 2 TA) knowing the relative volatility a between components A and B. Assuming a ¼ 2.05, l is found to be 39.291 kJ mol21. However, it should be noted that the methodology described here is not restricted to cases with these assumptions. The described methodology is applicable to ideal as well as non-ideal mixtures. A feed of 50 kmol h21, containing 60% lighter component (A), is fed to the distillation column as saturated liquid. 98% of the lighter component is recovered at the top of the column whereas 97% of the heavier component is recovered at the bottom. From the overall mass and material balances, the product flow rates (in kmol h21) are obtained as D ¼ 30 and B ¼ 20. The IRS curves of the system are shown in Figure 3(a). Appropriate location for the feeds may be targeted as TF ¼ 359.2 K (corresponds to z ¼ 0.6) and the minimum energy targets (in terms of condenser and reboiler loads) are determined from the widths of the circumscribing rectangle (D ¼ 0, in this example). The minimum condenser and reboiler duties are equal (2.897 106 kJ h21), which corresponds to the minimum reflux ratio of 1.458 and minimum reboil ratio of 3.687. Exergetic efficiency of the base column is found to be 56.12%. If the column operates with 20% more reflux, the exergetic efficiency decreases to 46.78%. This is considerably higher than the commonly cited thermodynamic efficiency of 10%. Therefore, design
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of better tray internals for lower pressure drop and the energy efficient design of the reboiler as well as condenser are to be addressed to improve the overall efficiency of the distillation column. Especially, sub-cooled reflux should be avoided to improve the thermodynamic efficiency. It has already been observed that the energy efficiency of a crude distillation unit can be improved upon by avoiding sub-cooled reflux to the main fractionation column (Joshi et al., 2001). As the primary focus is on thermodynamic efficiency and also for brevity, we assume a tall column with minimum reflux. In case an ideal side reboiler is incorporated in the base column, thermodynamic efficiency of the column can be improved significantly. The problem of finding location and duty of a side reboiler can be mathematically translated as cutting a rectangle with maximum area from the IRS curves represented on a Carnot factor-enthalpy (1 versus H) diagram [as explained in Figure 3(a)]. Maximum exergetic efficiency can be achieved when a side reboiler is placed at 363.81 K with a maximum possible duty of 1.414 106 kJ h21 (49% of the main reboiler duty). Exergetic efficiency of the system improves to 70.92% (an improvement of 26.37% over the base column).
Drawing Liquid from the Column and Returning at the Same Point after Heating A portion of the vapour or liquid may be withdrawn from a distillation column and fed back to the column at the same point after heating it up with external utility. Drawing vapour from the column is difficult in practice. Moreover, lower value of heat transfer coefficient of the vapour stream will call for larger heat exchanger. It may also be noted that introduction of superheated vapour will have its own operational problems such as pressure drop, high velocity, product degradation due to liquid carry over, and so on. For all practical purpose, it is better to withdraw liquid from the column. This is also the practice as in the cases of pump-around or pump-back reflux (Sharma et al., 1999; Joshi et al., 2001). For comparison purpose, the heater duty kept constant at 1.414 106 kJ h21. As the liquid is withdrawn at 363.81 K (corresponds to z ¼ 0.359) and put back at the same location, the exergetic efficiency of the column improves to 70.7%. One interesting observation made during the study that the efficiency of the column is independent of the withdrawal flow rate. This implies that the flow rate of the withdrawal may be varied to adjust the appropriate approach temperature of the external heat exchanger. This is equivalent to having a side reboiler at the withdrawal stage. Figure 3(b) shows the active portions of the IRS curves demonstrating this. Small difference in efficiency is due to numerical integration procedure applied to calculate total exergy loss in the column. It may be noted that the withdrawal rate should be such that the resultant feed, after heating, must be saturated or two phase. Otherwise the mass transfer efficiency of the actual stages will decrease significantly as it will be doing mainly heat transfer activities.
curve. This is a new configuration and not analysed before. Withdrawal of 10 kmol h21 from the column at 363.81 K (corresponds to z ¼ 0.359) and heating it with external utility (exchanger duty is kept equal to 1.414 106 kJ h21) changes the thermal condition to q ¼ 22.6. Based on the q-line of this, it may be put back into the column at 370.47 K. Figure 3(c) shows the active portions of the IRS curves demonstrating this. It changes the internal traffic significantly and the resultant efficiency of the column comes out to be only 63.1%, significantly lower than the previous case. This is primarily due to retrograde phenomenon of internal separation. Superheated vapour (as q is negative) has to be separated inside the column such that the required separation may be obtained at the withdrawal stage. Furthermore, there are associated superheated vapour introduction problems, as highlighted before. It may be noted that the efficiency of the column changes with liquid draw rate. Changing the withdrawal rate from 10 kmol h21 to 36 kmol h21, the resultant feed becomes saturated vapour and the thermodynamic efficiency increases up to 67.11% (when located at 367.11 K). Figure 3(d) shows the active portions of the IRS curves demonstrating this. If the withdrawal rate is further increased, the efficiency approaches the limiting value of 70.7% and become equivalent to the previous case.
Comparison with Feed Preheating Options The same amount of heat may be utilized to preheat the feed. By preheating the feed by 1.414 106 kJ h21, the thermal energy of the feed can be changed and consequently the q-value of the feed changes from 1 to 0.28. The location of the feed, in temperature scale, changes to 361.5 K. However, the change in feed quality does not change the reboiler duty alone, it also increase the condenser duty. The minimum condenser duty changes to 3.411 106 kJ h21, corresponds to the minimum reflux ratio of 1.894. The minimum reboiler duty changes to 1.997 106 kJ h21, corresponds to the minimum reboil ratio of 2.541. The preheat efficiency, defined as the portion of the heat given to feed that reduces the reboiler duty, is 63.65%. The thermodynamic efficiency of the column changes to 64.6%. Procedures developed by Bandyopadhyay et al. (2003) may be utilized to calculate the feed preheat efficiency [Figure 4(a)]. As proposed by Deshmukh et al. (2005), feed preheat efficiency can be improved to 100% using the concept of feed splitting. Now, 23.314 kmol h21 of feed may be left unchanged and fed to the column at 359.15 K. Remaining portion of the feed, i.e., 26.686 kmol h21 can be preheated by the external utility heater and fed to the column at 363.73 K. This arrangement ensures 100% feed preheat efficiency [Figure 4(b)] as the condenser duty remain same and the reboiler duty reduced by the same amount. In this case, a maximum thermodynamic efficiency of 77.44% can be achieved.
EFFECT OF SYSTEMS PARAMETERS Drawing Liquid from the Column and Returning at Different Location A portion of the liquid withdrawn from the column may be fed back to the column at another point based on the intersection of the q-line of the heated liquid with the equilibrium
Impact of different system parameters such as relative volatility, feed composition and thermal condition of the feed on thermodynamic performance of a distillation column is studied in details. Typically, feed to a distillation column originates from another distillation column or other unit operations.
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Figure 4. Feed preheating: (a) entire feed is preheated and (b) 100% feed preheat efficiency by feed splitting.
Knowing the effect of composition and thermal condition of the feed on the thermodynamic performance of a distillation column, thermodynamic efficiency of the distillation column, as well as that of the overall process, may be improved upon by selecting appropriate thermal integration techniques and/or by adjusting the feed at its appropriate condition from the preceding unit operation. Effect of system parameters on thermodynamic efficiency of a distillation column is discussed in this section. For simplicity an ideal binary system has been considered. Assumptions listed in the previous section are considered. However, recoveries of components are assumed to be 99% in this section. For comparison, thermodynamic efficiencies of the base case distillation column, distillation column with optimum side-reboiler duty, column with entire feed preheating and feed preheating with feed splitting are considered. For all these cases, duties of the external heat exchangers are kept equal to that of the optimum side reboiler. Similar results may be obtained for distillation column with side-condenser. However, for brevity, case of side condenser has not been considered. Effects of relative volatility, feed composition, and thermal condition of the feed on thermodynamic efficiencies of a distillation column are shown in Figures 5–7. These results are discussed below.
For a very low relative volatility (a ! 1), minimum duties for the reboiler and the condenser increase sharply, whereas the efficiency approaches a limiting value. This limiting value depends on the feed composition but is independent of the thermal condition of the feed. In such circumstances, approach temperatures for the reboiler and condenser play a dominant role in defining the efficiency of the column. Thermodynamic efficiency of a distillation column processing saturated liquid feed with composition z is qualitatively comparable, but not exactly equal numerically, with the thermodynamic efficiency of a column processing saturated vapour feed with composition 1 2 z. For saturated liquid feed with more light component (z 0.5) and saturated vapour feed with more heavy component (z 0.5), the efficiency of the base column reduces monotonically with relative volatility (Figure 5). Whereas for saturated liquid feed with more heavy component (z , 0.5) and for saturated vapour feed with more light component (z . 0.5), the efficiency goes through a maximum. As concentration of the lighter component decreases in the saturated liquid feed or concentration of the heavier component decreases in the saturated vapour feed, the optimum shifts towards higher relative volatility. These observations lead to the fact that for a given composition and thermal condition, the relative volatility may be changed by changing the operating pressure to increase the efficiency of distillation. Change in operating pressure is especially important for sub-ambient distillation processes. For saturated liquid feed with more light component (or saturated vapour feed with more heavy component), increase in operating pressure may be encouraged for cryogenic distillation. Higher operating pressure results in lower entropy generation due to lower pressure drop and reduction in refrigeration load. It should be noted that it may be possible to manipulate relative volatility to a small extent by changing the operating pressure of a distillation column, large changes in relative volatility are normally difficult to achieve. It may be worth emphasizing that in most gas separation processes the variations of distillation efficiency with operating pressure is of utmost importance. Depending on the thermal condition of the feed, the operating pressure of distillation may be adjusted. However, costbenefit analysis should be performed before implementing such modifications. Figures 6 and 7 suggest that for a given relative volatility and feed composition, there exists an optimal thermal condition of the feed that maximizes the thermodynamic efficiency of the base column. For a given feed composition, thermal condition of the feed may be adjusted to achieve the maximum thermodynamic efficiency of a distillation column. Approaches based on pinch analysis may be utilised for this purpose.
Column with Side Reboiler Incorporation of a side reboiler always increases the thermodynamic efficiency of a distillation column. This can easily be appreciated from the IRS curves, as described in the earlier section. Similar to the base case, for saturated liquid feed with more light component and saturated vapour feed with more heavy component, the efficiency of a
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Figure 5. Effect of relative volatility on thermodynamic efficiency: (a) z ¼ 0.5 and q ¼ 1, (b) z ¼ 0.25 and q ¼ 1, (c) z ¼ 0.75 and q ¼ 1, (d) z ¼ 0.5 and q ¼ 0, (e) z ¼ 0.25 and q ¼ 0, (f) z ¼ 0.75 and q ¼ 0.
distillation column with a side reboiler reduces monotonically with relative volatility (Figure 5). By incorporating a side reboiler, significant improvement in the thermodynamic efficiency of a distillation column may be obtained when the feed contains more light component (Figure 6). For a saturated vapour feed or a feed with predominant vapour fraction (i.e., low q), it is easier to incorporate side reboiler and feed preheating is thermodynamically not a right option as
introduction of a superheated feed is thermodynamically inefficient as well as has its own operational problems.
Column with Feed Preheating It may be noted that instead of preheating the entire feed, preheating a portion of the feed is always beneficial from thermodynamic efficiency point of view. This is primarily because
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Figure 6. Effect of feed composition on thermodynamic efficiency: (a) a ¼ 1.5 and q ¼ 1, (b) a ¼ 2.5 and q ¼ 1, (c) a ¼ 1.5 and q ¼ 0, (d) a ¼ 2.5 and q ¼ 0.
an extra amount of separation is obtained through feed splitting and 100% feed preheat efficiency is achieved. Feed preheating is beneficial for predominantly liquid feed with more light component (Figures 6 and 7). It may be observed that for feed with more heavy component, it is not beneficial to preheat the feed.
preheating configuration. Whereas thermodynamic efficiency of a distillation column, processing a saturated vapour feed with 50% light component, can be enhanced significantly by incorporating a side reboiler. Such a curve gives a process designer a guideline for choosing appropriate configuration of a distillation column for thermal integration.
Optimal Configuration Curve
CONCLUSION
Figure 5(a) suggests that for a saturated liquid feed with 50% light component, it is always beneficial to have feed preheating with feed splitting. On the other hand, Figure 5(d) suggests that thermodynamic efficiency of a distillation column is always more with a side reboiler. Similar conclusions may be drawn form Figures 6 and 7. Depending on the feed composition and thermal condition of the feed, either side exchanger or feed preheating by feed splitting enhances the thermodynamic efficiency of a distillation column to the maximum extent. Figure 8 represents the optimal configuration curve in terms of the maximum value of q for different feed composition up to which it is beneficial to incorporate side reboiler in a distillation column. Beyond this it is always beneficial to incorporate feed preheater with feed splitting. For example, for a saturate liquid feed with 50% light component, it is beneficial to have feed
Exergy analysis of a distillation column often provides useful understanding and insight for energy efficient design of distillation processes (Atkinson, 1987; Ratkje et al., 1995; Taprap and Ishida, 1996; Agrawal and Herron, 1997; Demirel, 2004). This is particularly true for low-temperature processes, such as gas separation process. Exergy analysis is also useful for thermal integration of a distillation column with other unit operations. In addition to determining total exergy loss in a distillation column, it is desirable to look at its exergetic efficiency. The exergetic efficiency of a distillation column or any other unit operation can provide useful information about its potential for improvement. It is possible to have a large total exergy loss for an operation that still has fairly high exergetic efficiency. For example, even though there is a great incentive to improve further the performance of an efficient heat exchanger, handling a large mass flow, the
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Figure 7. Effect of thermal condition of feed on thermodynamic efficiency: (a) a ¼ 1.5 and z ¼ 0.25, (b) a ¼ 1.5 and z ¼ 0.5, (c) a ¼ 1.5 and z ¼ 0.75, (d) a ¼ 2.5 and z ¼ 0.25, (e) a ¼ 2.5 and z ¼ 0.5, (f) a ¼ 2.5 and z ¼ 0.75.
potential for improvement is low and would be likely to require significant effort and investment. On the other hand, an operation with low exergetic efficiency and also a low exergy loss will not be worth the effort to further improve its performance. Instances where exergetic efficiencies are not very high and where exergy losses are significant provide good potential for improvement. Thermodynamic efficiency of a distillation improves as it is thermally integrated with background process. There are two important ways a distillation column can be energy integrated: through side exchangers and through feed preheating. Depending upon the thermal condition of the feed, feed composition and relative volatility of the system, appropriate
configuration may be selected. An optimal configuration curve, proposed in this paper, may play an important role in helping a process designer to choose an appropriate column configuration for thermal integration. It should be noted that thermodynamic efficiency of a distillation column has been considered in this article as the sole criterion for optimizing the thermal condition of the feed. Applicability and quality of the design of a distillation column cannot possibly be judged solely on the basis of thermodynamic efficiency of the distillation column. Factors such as inefficiencies associated with converting a feed to its optimal conditions, capital and operating cost of the system, operational complexity and flexibility and so on are also important
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side condenser side reboiler stripping (translated) vapour
REFERENCES
Figure 8. Optimal configuration curves for different relative volatility. It is represented in terms of the maximum value of q for different feed composition up to which it is beneficial to incorporate side reboiler in a distillation column.
for designing a distillation column. However, thermodynamic analysis of the distillation column does provide useful understanding and insights to the designer. It may also be noted that the exergy analysis performed in this paper, is restricted in analysing a distillation column only. For practical implementation, thermodynamic analysis of the entire process and thermo-economic analysis of the entire system should be performed.
NOMENCLATURE B D F h H L Q q T V W x y z
bottom product molar flow, kmol h21 distillate molar flow, kmol h21 feed molar flow, kmol h21 specific enthalpy, kJ kmol21 enthalpy, kJ h21 liquid molar flow, kmol h21 heat duty, kJ h21 thermal condition of the feed temperature, K vapour molar flow, kmol h21 work, kJ h21 mole fraction in liquid mole fraction in vapour mole fraction in feed
Greek symbols D enthalpy difference, kJ h21 dEx exergy, kJ h21 h thermodynamic efficiency a relative volatility 1 Carnot factor Subscripts and superscripts B bottom product c condenser D distillate F feed L liquid max maximum min minimum r reboiler R rectifying RT rectifying (translated) S stripping
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ACKNOWLEDGEMENTS The author is thankful to Industrial Research and Consultancy Centre, IIT Bombay, for providing financial support for this work. The manuscript was received 16 July 2006 and accepted for publication after revision 24 October 2006.
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