Distillation control structure selection

Generally, variables that are needed to be controlled for a binary distillation column are composition of the distillate 𝜒_𝐷, composition of the bottom product 𝜒_𝐵, liquid level in the reflux drum, liquid level in the base drum and pressure in the column. Usually, the manipulated variables are reflux flow, L, reboiler vapour flow, V, distillate flow, D, bottom product flow, B, and condenser duty. Column pressure is usually controlled by the condenser duty and various distillation column control configurations refer to the

pairing of other controlled and manipulated variables. Some typical distillation column control schemes include LV, DV, and LB control configurations.

In the LV control configuration, the top product composition is regulated by adjusting the reflux flow L and the bottom product composition is controlled by adjusting the reboiler’s energy which is equivalent to reboiler vapour flow V. Distillate rate is used to control the condenser level and bottom product rate B is used to control the reboiler level. Similarly for the DV control configuration, L is used to control the condensor level and D is used to control the top composition while the reboiler level is controlled by B and the bottom product composition is controlled by V.

4.3.1 RGA analysis

A multi-input multi-output (MIMO) system usually has interactions among the control loops. For better control of a process, control loop interactions should be minimised as a high degree of loop interaction makes the control difficult. Relative gain array (RGA) proposed by Bristol (Bristol, 1966) is a tool that can be used to quantify control loop interactions. Relative gain is the ratio of the steady state gain when the loops are open to the steady state gain with all other loops closed.

The relative gain between the ith controlled variable and the jth manipulated variable is represented mathematically as 𝛼𝑖𝑗 = (∆𝑦𝑖 ∆𝑢𝑗)_{𝑎𝑙𝑙 𝑙𝑜𝑜𝑝𝑠 𝑜𝑝𝑒𝑛} (∆𝑦𝑖 ∆𝑢𝑗)_{𝑎𝑙𝑙 𝑙𝑜𝑜𝑝𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑒𝑥𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝑢𝑗 𝑙𝑜𝑜𝑝} 4.11 = _{𝑐𝑙𝑜𝑠𝑒𝑑 𝑙𝑜𝑜𝑝 𝑔𝑎𝑖𝑛}𝑜𝑝𝑒𝑛 𝑙𝑜𝑜𝑝 𝑔𝑎𝑖𝑛

RGA is then obtained when the relative gains for all the pairing combinations in a multi-loop control system are calculated and put in an array.

𝑅𝐺𝐴 = [ 𝛼₁₁ 𝛼₁₂ … 𝛼_1𝑛 𝛼₂₁ 𝛼₂₂ … 𝛼_2𝑛 ⋮ ⋮ ⋱ ⋮ 𝛼𝑛1 𝛼𝑛2 … 𝛼𝑛𝑛 ] 4.12

A relative gain of 1 on the diagonal of RGA indicates that there are no control loop interactions. The strategy is then to match the controlled and manipulated variables when 𝛼𝑖𝑗 is nearest to 1 and to avoid the pairings with close to zero or negative relative

4.3.2 Thermodynamic analysis

Exergy is from a combination of the 1st and 2nd laws of thermodynamics. It is a key aspect of providing better understanding of the process and quantifying sources of inefficiency and distinguishing quality of energy used (Jin et al., 1997, Rosen and Dincer, 1997, Doldersum, 1998). Exergy analysis is a measure of the quality of energy and is the maximum work produced or the minimum required depending on whether the system produces or requires work in bringing the system through reversible process with the environment. It is a tool for determining how efficient a process is (Dhole and Linnhoff, 1993, Demirel, 2004).

Exergy represents the part of energy, which can be converted into maximum useful work. It is used to establish criteria for the performance of engineering devices (Asada and Boelman, 2004). Unlike energy, exergy is not conserved and gets depleted due to irreversibilities in the processes (Sengupta et al., 2007). The greater the extent of irreversibilities is, the greater the entropy production is. Therefore, entropy can be used as a quantitative measure of irreversibilities associated with a process. Minimization of irreversibility in processes implies increase in energy efficiency of such process. Exergy analysis of processes gives insights into the overall energy usage evaluation of the process, potentials for efficient energy usage of such processes can then be identified and measures for improving energy usage of the processes can be suggested.

The total exergy of a stream is calculated as

𝐸𝑥𝑡𝑜𝑡𝑎𝑙 = 𝐸𝑥𝑝ℎ𝑦+ 𝐸𝑥𝑐ℎ𝑒𝑚+ 𝐸𝑥𝑚𝑖𝑥𝑖𝑛𝑔 4.13

𝐸𝑥𝑝ℎ𝑦 = 𝐻 − 𝐻0 − 𝑇0(𝑆 − 𝑆0) 4.14

𝐸𝑥_𝑝ℎ𝑦 = ∆𝐻 − 𝑇₀∆𝑆 4.15

∆𝐸𝑥_{𝑐ℎ𝑒𝑚=}∑ 𝑛_𝑖𝑏_𝑐ℎ𝑖+ 𝑅𝑇₀∑ 𝑛_𝑖𝑙𝑛𝛾_𝑖 4.16

In the above equations, 𝑏𝑐ℎ𝑖 is the chemical exergy for component i, 𝛾𝑖 is the activity

coefficient of component i, 𝐻 is the total enthalpy, 𝑆 is the total entropy, 𝑇₀ is the reference temperature, 𝐻₀ and 𝑆₀ are enthalpy and entropy respectively measured at reference conditions.

For a heat source such as the reboiler, if Q_z is a heat source at an absolute temperature,

z

T , and if T0 is the ambient temperature, then the work equivalent of heat is given by

𝑊𝑍 = ∫ (1 −_𝑇𝑇0

𝑍) 𝜕𝑄𝑍

𝑓𝑖𝑛𝑎𝑙

𝑖𝑛𝑡𝑖𝑎𝑙 4.17

where 𝜕𝑄𝑍 is an incremental heat transfer at absolute temperature 𝑇𝑍 and the integral is

from initial state to final state.

If the temperature of the heat source is constant, the work equivalent of heat is given by (Dincer and Rosen, 2012)





This is the absolute theoretical maximum work recoverable. Equation 4.18 is used in calculating the exergy of the reboiler and the condenser.

Exergy efficiency of a system is calculated as 𝜑 =∑ 𝐸𝑥𝑜𝑢𝑡

∑ 𝐸𝑥𝑖𝑛 4.19

While the exergy loss of a system is given as

𝐼 = ∑ 𝐸𝑥𝑖𝑛− ∑ 𝐸𝑥𝑜𝑢𝑡 4.20

It takes a good engineering judgement to determine the streams that are qualified as in and those that are qualified as out.

For a binary distillation system the total exergy in and total exergy out are given as

𝑇𝑜𝑡𝑎𝑙 𝐸𝑥𝑖𝑛=𝐸𝑥𝑓𝑒𝑒𝑑+ 𝐸𝑥𝑅𝑒𝑏𝑜𝑖𝑙𝑒𝑟 4.21

𝑇𝑜𝑡𝑎𝑙 𝐸𝑥_𝑜𝑢𝑡 = 𝐸𝑥_{𝐷𝑖𝑠𝑡𝑖𝑙𝑙𝑎𝑡𝑒} + 𝐸𝑥_{𝐵𝑜𝑡𝑡𝑜𝑚𝑠} 4.22

In the above equations, Exfeed, ExReboiler, ExReflux, ExBoilup, ExDistillate, and ExBottom are,

respectively, the exergy in the feed stream, reboiler, reflux stream, boil up stream, distillate product stream, and bottom product stream.

4.3.3 Relative exergy array

Relative exergy gain is defined as “the ratio of the gain change in the steady state exergy of the controlled stream with respect to that of the manipulated stream when all loops are open to the gain change in the steady state exergy of the controlled stream with respect to that of the manipulated stream when all other loops are closed and in perfect control” (Montelongo-Luna et al., 2011). This is given in equation 4.23.

𝛽𝑖𝑗 = (∆𝐸𝑥(𝑦𝑖) ∆𝐸𝑥(𝑢𝑗))_{𝑎𝑙𝑙 𝑙𝑜𝑜𝑝𝑠 𝑜𝑝𝑒𝑛} (∆𝐸𝑥(𝑦𝑖) ∆𝐸𝑥(𝑢𝑗))_{𝑎𝑙𝑙 𝑙𝑜𝑜𝑝𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑒𝑥𝑐𝑒𝑝𝑡 𝑡ℎ𝑒 𝑢𝑗 𝑙𝑜𝑜𝑝} 4.23

REA is based on the RGA concept by replacing relative gain with relative exergy gain. The exergy gain ratio is usually calculated after a step input change in the manipulated variable. It gives the amount of exergy change in the controlled variable resulting from the exergy change in the manipulated variable and hence provides information on the thermodynamic efficiency of the pairing. This permits a good insight to the energy efficiency of a process right from the design stage and allows for the choice of optimum combination of loops.

Putting all the relative exergy gains in an array gives the relative exergy array:

𝑅𝐸𝐴 = [ 𝛽₁₁ 𝛽₁₂ … 𝛽_1𝑛 𝛽₂₁ 𝛽₂₂ … 𝛽_2𝑛 ⋮ ⋮ ⋱ ⋮ 𝛽_𝑛1 𝛽_𝑛2 … 𝛽_𝑛𝑛 ] 4.24

REA indicates the exergy efficiency effects of pairing each of the manipulated variables to each of the controlled variables. It is defined analogous to the relative gain array. If the value of a relative exergy gain on the diagonal of REA is equal to 1, then it indicates the thermodynamic efficiency of the control loop under consideration is not affected by the other control loops (Montelongo-Luna et al., 2011, Munir et al., 2013c, Munir et al., 2012b). This control loop pairing will be good in terms of thermodynamic efficiency. The value of a relative exergy gain greater than 1 implies that the exergy change from the open loop is much more pronounced. In this case, interaction from the variables in the process will decrease the process exergy change. The value of a relative exergy gain less than 1 indicates the exergy change due to open loop is less and hence an increase in exergy changes when the loops are closed. If the sign is negative, closing the control loop will improve the thermodynamic efficiency of the process but if on the other hand the sign is positive, this shows that the thermodynamic efficiency of the process will be decreased by the control loop. In control structure selection, a control loop paring with relative exergy gain close to one is preferred.