WO2019241800A1 - Distillation methods - Google Patents

Abstract

Provided herein are dynamic operating paradigms for distillation processes (e.g., binary distillation columns) that have the potential to generate a desired product slate (defined in terms of bottoms and distillate purities and flow rates) at a lower energy consumption than an equivalent process that is operated at a single steady state.

Description

DISTILLATION METHODS

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claim benefit of U.S. Provisional Application No. 62/685,382, filed June 15, 2018, which is hereby incorporated herein by reference in its entirety.

BACKGROUND

The production of chemicals and petrochemicals follows a simple principle: raw materials undergo chemical and physical transformations, resulting in value-added products. Practice, however, often deviates significantly from this simple paradigm: raw materials are rarely pure and reactions typically neither selective nor carried to completion. As a consequence, the outlet stream of a chemical reaction unit must undergo one or more separation steps: products are separated, sold or undergo further processing, unused reactants are recovered and sent back to the reaction section, while impurities and undesired byproducts are removed and discarded.

Distillation has been and remains the workhorse separation process in industrial settings, constituting the approach of choice for separating liquid mixtures of two or more components. The design and operation of distillation technology is considered to be well understood; nevertheless, even the best conventional distillation towers can absorb more than half of a plant 's operating cost. In total, distillation processes have been estimated to account for 40% of the energy consumption in the chemical industry, and as much as 3% of total U.S. energy consumption.

As a consequence, there is an intense interest in improving the design of chemical plants (and their respective separation sections) in order to lower operating cost and increase energy efficiency.

SUMMARY

Approaches to improving the design of chemical plants broadly fall into three categories. First, conventional wisdom dictates exploiting the economy of scale, whereby larger plants have lower per-unit production cost. Second, new technologies that take different processing routes may provide lower operating cost. For example, in the case of separations, membrane-based technologies appear to offer significant promise, but have not yet gained significant ground against distillation.

A third route towards reducing capital and operating expenditure is process intensification (PI), typically defined as a chemical engineering development that leads to a substantially smaller, cleaner and more energy-efficient technology. Over the past few decades, this rather general definition has been viewed and interpreted— by both academics and industry practitioners— in ter s of outlining a set of innovative design principles that bring the chemical and physical phenomena occurring in a chemical process into closer physical proximity. Intensified process systems thus feature multiple phenomena (which would often be occurring in dedicated unit operations) taking place in one physical device to achieve advantages such as energy and capital cost savings. In the separations realm, these ideas have materialized in innovative systems such as reactive distillation and dividing-wall distillation columns (DWC).

To date, process intensification progress has generally relied on innovative changes to the design of a process. In general, intensified processes transcend the conventional unit operations framework and rely on devices that minimize transfer or transport resistance, overcome equilibrium limitations and thereby reduce equipment size and cost. However, such devices are usually assumed to operate at steady-state. Much less emphasis has been placed on intensification via operational changes, which rely on exploiting process dynamics to achieve capital and/or operating cost savings and increases in efficiency.

Motivated by the desire to improve the efficiency of separatory processes, provided herein are improved separatory methods which are an outgrowth of dynamic process intensification. Dynamic process intensification can be defined as "any change to the dynamics, operation strategy and/or control of a conventional or intensified system, that leads to a substantially more efficient processing path. " These methods can be used to improve the efficiency distillation processes that feature nonlinear behavior.

Provided herein is a dynamic operating paradigm for distillation processes (e.g., binary distillation columns) that has the potential to generate a desired product slate (defined in terms of bottoms and distillate purities and flow rates) at a lower energy consumption than an equivalent process that is operated at a single steady state.

The methods can involve producing a relatively energy-intensive (e.g., having a relatively high reboiler duty Q_B*) product P*, with target purity y_d*, as a mixture of two auxiliary products Pi and P 2 , having purities y _,i (higher than the target purity) and, respectively, purity y , 2 (lower than the target purity). The two auxiliary products can be produced by the same column at two other different operating points and are obtained by alternating periodically between the two corresponding operating points. The auxiliary products are then blended in a (sufficiently large) tank, such that, on average, over time, the blended product reaches the desired specification. Possible off-specification products made during the dynamic transition can also be added to and mixed in the tank. The reboiler duties corresponding to the auxiliary products can be defined as QB,I and QB,2 respectively. In some embodiments, both QB,I and QB,2 are lower than QB* and, as a consequence, the mixture of the two auxiliary products, chosen in the appropriate proportion, can have the same average purity as target purity yd*, but lower average energy consumption than QB*.

DESCRIPTION OF DRAWINGS

Figure 1 illustrates that the amount of energy required to increase distillate purity by one unit is considerably higher than the energy saved when purity is decreased by one unit. The value of the corresponding“gain” can vary considerably. Data represent a 20 equilibrium stage column separating an equimolar methanol-water mixture fed to stage 10 (reflux ratio = 1, distillate rate = 50 kmol/h, feed flow rate = 100 kmol/h, and condenser pressure = 1.1 bar)

Figures 2A and 2B illustrate the input (Figure 2A) and output (Figure 2B steady- state multiplicity.

Figure 3 illustrates the steady-state output multiplicity for a binary distillation column separating methanol and isopropanol.

Figure 4 illustrates the steady-state output multiplicity for a binary distillation column separating propanol and acetic acid.

Figure 5 is a conceptual representation of dynamic intensification based on output multiplicity: the desired product CV* (with production cost MV* can be made as a mixture of auxiliary products CVi and CV2, both having lower production cost (MVi and MV2 respectively, though in this case MV_I=MV2).

Figure 6 illustrates steady-state output multiplicity defined based on a static input- output relationship between the manipulated variable MV (input) and the controlled variable CV (output). The states of a process may be different between the steady-state points Pi = (MVi, CVi) and P2 = (MV2, CV2) (note that in this figure MVi = MV2).

Figures 7A-7C illustrate multiplicity exhibited by the three-stage propanol-acetic acid column. Figure 7A show the mole fraction of propanol in distillate as a function of reboiler duty. Figure 7B shows vapor boilup rate as a function of reboiler duty. Figure 7C shows the mole fraction of propanol in distillate as a function of vapor boilup.

Figure 8 shows the configuration of propanol-acetic acid column.

Figure 9 shows the dependence of distillate flow rate on product purity for the propanol-acetic acid column.

Figures 10A-10B illustrate the effect of column pressure on the distillate flow rate (Figure 10 A) and reboiler duty (Figure 10B). Figures 11A-11D show a simulation of switching between operating at Pi and P2. The setpoint trajectory for column pressure follows a step change, while the vapor boilup rate is“ramped up” and“ramped down.” The vapor boilup setpoint signal is designed such that the constant portion has a duration of 50 min (0.83 h) at each operating point. Figure 11 A shows the condenser pressure and ramp change in vapor flow rate. Figure 11B shows the distillate and bottoms flow rates. Figure 11C shows the purity of the propanol distillate. Figure 11D shows the total energy consumption.

Figures 12A-12D show a simulation of three cycles of the dynamic intensification solution. Figure 12A shows the condenser pressure and vapor flow. Figure 12B shows the distillate and bottoms flow rates. Figure 12C shows the purity of the propanol distillate. Figure 12D shows the total energy consumption.

Figures 13A-13D show a simulation of the dynamic intensification solution. Figure 13A shows the condenser pressure and vapor flow. Figure 13B shows the distillate and bottoms flow rates. Figure 13C shows the purity of propanol distillate. Figure 13D shows the total energy consumption.

Figure 14 illustrates a schematic of the column configuration.

Figures 15A-15B illustrate the nonlinear steady-state characteristics of

methanol-propanol binary system. Figure 15A is a plot showing distillate purity versus reboiler duty under fixed boilup ratio = 6. Figure 15B is a plot showing distillate purity versus reboiler duty under fixed reflux rate = 5 kmol/min. Data are obtained from an eight equilibrium-stage column separating an equimolar saturated liquid mixture (feed flow rate = 1 kmol/min).

Figure 16 is a Conceptual representation of energy savings via operational changes. Target CV* can be achieved by switching between CVi and CV2 based on their corresponding MVi and MV2.

Figure 17 illustrates the nonlinear relationship between distillate flow rate and product purity for methanol-propanol binary system follows the same trend as purity versus reboiler duty.

Figures 18A-18B illustrate the effect of pressure on the distillate flow rate (Figure 18A) and reboiler heat duty (Figure 18B) under fixed reflux rate = 5 kmol/min.

Figures 19A-19B illustrate the effect of reflux rate on the distillate flow rate at high purity (Figure 19A) and reboiler heat duty (Figure 19B) at operating pressure of 1 bar.

Figures 20A-20C illustrate switching between the desired product and auxiliary products in Table 4. Figure 20A is a plot showing the mole fraction of methanol in distillate versus reboiler duty. Figure 20B is a plot showing the vapor boilup rate versus reboiler duty. Figure 20C is a plot showing the mole fraction of methanol in distillate versus vapor boilup rate.

Figures 21 A-21D show a simulation of switching between operating at Pi and P2. Figure 21A shows the operating pressure, reflux rate, and vapor flow rate. Figure 21B shows the distillate and bottoms flow rates. Figure 21C shows the purity of the methanol distillate. Figure 21D shows the total energy consumption.

Figures 22A-22D show a simulation of dynamic switching between Pi and P2 operating for 2.91 h at each point. Figure 22A shows the operating pressure, reflux rate, and vapor flow rate. Figure 22B shows the distillate and bottoms flow rates. Figure 22C shows the purity of the methanol distillate. Figure 22D shows the total energy consumption.

Figure 23 illustrates that the column reaches a cyclic steady state after six cycles.

Figure 24 is a nonlinear non-monotonic representation of target and two auxiliary products in terms of distillate purity and reboiler duty.

Figure 25 shows the schematic configuration of methanol-propanol binary column and operation conditions for reference steady state.

Figures 26A shows the effect of reflux rate on reboiler duty at operating pressure of 1 bar and feed pressure 1.03 bar.

Figure 26B shows the effect of operating pressure on reboiler duty under fixed reflux rate 300 kmol/h.

Figure 27 shows the effect of saturated liquid feed pressure on reboiler duty under operating pressure 1 bar and reflux rate 300 kmol/h.

Figure 28 shows the Aspen Plus flowsheet.

DETAILED DESCRIPTION

Provided herein are improved distillation methods for separating a mixture of volatile components (e.g., binary mixtures comprising a more volatile component and a less volatile component) that employ dynamic process intensification (DPI). Generally, these methods can involve (i) defining a target product (in terms of purity and production rate),

(ii) identifying two auxiliary products (e.g., with lower reboiler duties than the target product), and (iii) a periodic operation pattern that comprises switching production between the auxiliary products, such that the resulting blend has, on average, the same properties as the target product but features lower energy use.

This concept has been illustrated below using binary distillation as one of the most prevalent operations in the chemical industry. Specifically, it was demonstrated that the nonlinear features of distillation columns (notably, output multiplicity) can be exploited to generate a periodic operating pattern that lowers energy consumption without

compromising product quality.

Accordingly, provided herein are methods for obtaining a target product having a target purity from a mixture via a distillative process performed in a distillation apparatus (e.g., a distillation column). These methods can comprise (i) operating the distillation apparatus under a first set of conditions to obtain a first auxiliary product from the mixture, wherein the first auxiliary product has a purity higher than the target purity; (ii) operating the distillation apparatus under a second set of conditions to obtain a second auxiliary product from the mixture, wherein the second auxiliary product has a purity lower than the target purity; and (iii) combining the first auxiliary product and the second auxiliary product to afford the target product having the target purity.

In some cases, the distillation apparatus can comprise a distillation column. In some embodiments, the first auxiliary product and the second auxiliary product can comprise distillate products from the distillation column. In other embodiments, the first auxiliary product and the second auxiliary product comprise bottoms products from the distillation column.

Steps (i) and (ii) can be performed in any order, and repeated any number of times. For example, step (i) can be performed prior to step (ii), step (ii) is performed prior to step (i), or a combination thereof. In certain embodiments, the method can comprise periodically switching operation of the distillation apparatus between the first set of conditions and the second set of conditions. Periodically switching operation of the distillation apparatus between the first set of conditions and the second set of conditions can comprise cycling between the first set of conditions and the second set of conditions at least five times, at least 25 times, at least 50 times, or at least 100 times.

The first auxiliary product and the second auxiliary can be combined at a split ratio (a) which satisfies the following:

a(ydi) + (1 - ) (yd₂) = yd

where y_d represents the target purity, y _i represents the purity of the first auxiliary product, and y 2 represents the purity of the second auxiliary product.

The first auxiliary product and the second auxiliary can be combined at a split ratio (a) which satisfies the following:

a(PFRi) + (1 - a) (PFR₂) = PFR_d where PFR_d represents the target product flow rate when the distillation apparatus is operated

at a steady-state to obtain the target product from the mixture, PFRi represents the first auxiliary product flow rate when the distillation apparatus is operated under the first set of conditions, and PFR represents the second auxiliary product flow rate when the distillation apparatus is operated under the second set of conditions.

Step (i) and step (ii) can be performed for relative lengths of time which satisfy the following:

a(Timci ) + (1 - a) (Time2) = Total Operation Time where a is the split ratio, Timei comprises a total time during which the distillation apparatus is operated under the first set of conditions, Time2 comprises a total time during which the distillation apparatus is operated under the second set of conditions, and Total Operation Time comprises a total time during which the distillation apparatus is operated under the first set of conditions and the second set of conditions (e.g., the total operating time less the time required to cycle between the first set of conditions and the second set of conditions).

Switching between the first set of conditions and the second set of conditions comprises altering column pressure (P), feed pressure (P_F), boilup ratio (B ratio), reflux rate (Reflux), distillate flow rate, bottoms flow rate, or a combination thereof in the distillation apparatus. In some embodiments, switching between the first set of conditions and the second set of conditions can comprise altering the column pressure (P), and wherein altering the column pressure (P) comprises adjusting coolant flow rate. In some embodiments, switching between the first set of conditions and the second set of conditions can comprise altering the feed pressure (PF), and wherein altering the feed pressure (PF) comprises adjusting a feed valve controlling the feed pressure. In some embodiments, switching between the first set of conditions and the second set of conditions can comprise altering the boilup ratio (B ratio), and wherein altering the boilup ratio (B ratio) comprises adjusting a steam flow rate to a reboiler. In some embodiments, switching between the first set of conditions and the second set of conditions can comprise altering the reflux rate (Reflux), and wherein altering the reflux rate (Reflux) comprises adjusting a reflux valve controlling the reflux flow rate.

The total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions can be lower that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture. If desired, the first auxiliary product and the second auxiliary product can be selected (e.g., optimized) to achieve a desired energy savings relative to the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

In some embodiments, the total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions can be at least 0.5% less

(e.g., at least 1% less, at least 1.5% less, at least 2% less, at least 2.5% less, at least 3% less, at least 4% less, at least 5% less, at least 6% less, at least 7% less, at least 8% less, at least 9% less, or more) than that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture. In some embodiments, the total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions can be from 1% to 10% less (e.g., from 1% to 5% less) than that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

Step (iii) can comprise collecting the first auxiliary product and the second auxiliary product in a mixing tank, so as to accumulate a combination the first auxiliary product and the second auxiliary product (i.e., the target product).

Optionally, methods can further comprise obtaining a second target product from the distillation apparatus. For example, methods are provided for obtaining a first target product having a first target purity and a second target product having a second target purity from a mixture via a distillative process performed in a distillation apparatus (e.g., a distillation column). These methods can comprise (i) operating the distillation apparatus under a first set of conditions to obtain a first auxiliary product and a third auxiliary product from the mixture; (ii) operating the distillation apparatus under a second set of conditions to obtain a second auxiliary product and a fourth auxiliary product from the mixture; (iii) combining the first auxiliary product and the second auxiliary product to afford the first target product having the firt target purity; and (iv) combining the third auxiliary product and the fourth auxiliary product to afford the second target product having the second target purity.

The first auxiliary product can have a purity higher than the first target purity and the second auxiliary product can have a purity lower than the first target purity. The third auxiliary product can have a purity higher than the second target purity and the fourth auxiliary product can have a purity lower than the second target purity. In some cases, the distillation apparatus can comprise a distillation column. In some embodiments, the first auxiliary product and the second auxiliary product can comprise distillate products from the distillation column, and the third auxiliary product and the fourth auxiliary product can comprise bottoms products from the distillation column.

While the methods described above are described in the context of a single distillation apparatus operating at two states, it will be understood that a similar energy savings can likewise be achieved by forming the first auxiliary product and the second auxiliary product using two separate distillation apparatus. Accordingly, also provided are methods for obtaining a target product having a target purity from a mixture via a distillative process that comprise (i) operating a first distillation apparatus (e.g., a first distillation column) under a first set of conditions to obtain a first auxiliary product from the mixture, wherein the first auxiliary product has a purity higher than the target purity; (ii) operating a second distillation apparatus (e.g., a second distillation column) under a second set of conditions to obtain a second auxiliary product from the mixture, wherein the second auxiliary product has a purity lower than the target purity; and (iii) combining the first auxiliary product and the second auxiliary product to afford the product having the target purity.

The total energy utilized to operate the first distillation apparatus under the first set of conditions and the second distillation apparatus under the second set of conditions is lower that the energy required to operate a single distillation apparatus at a steady-state to obtain the target product from the mixture.

By way of non-limiting illustration, examples of certain embodiments of the present disclosure are given below.

EXAMPLES

Example 1: Dynamic Process Intensification of Binary Distillation Based on Output Multiplicity

Process intensification focuses largely on process and equipment design. Much less emphasis has been placed on operational changes to achieve cost savings and increased efficiency. This example introduces the concept of dynamic intensification, defined as changes to the dynamics, operating strategy and/or control of a process that lead to a substantially more efficient processing path. This idea is illustrated in the context of binary distillation. Output multiplicity properties are exploited to establish a new periodic operating mode based on switching between two auxiliary products, which, on a time- average basis, is more energy efficient than steady-state operation. An extensive case study is presented concerning the distillation of a propanol-acetic acid mixture, confirming the theoretical developments. In contrast to previous research (e.g., on cyclic distillation), the present concept has significant advantages as it relies on existing hardware and exploiting system nonlinearity, rather than using specialized equipment operated in a discontinuous fashion.

Dynamic Intensification of Distillation Operations

The dynamic behavior of chemical processes can be highly complex; the presence of nonlinearity and steady-state multiplicity has been acknowledged. Further complications (such as unstable zero dynamics/inverse response) can arise from the fact that multiple competing phenomena, often with opposite effect on the process states and outputs, can take place simultaneously. In this section, two features and operating strategies that are of interest in the rest of this example are reviewed: periodic operation of and steady-state multiplicity in distillation processes.

Nonlinear features of distillation columns. The nonlinear behavior of distillation columns (particularly those operating at high purity) has been recognized, largely in the context of the control difficulties that it creates. This nonlinearity is reflected, amongst others, in the gain between system inputs and outputs. Consider, for example, the relationship between reboiler heat duty and the purity of the distillate. As shown in Figure 1, at high purities, the amount of energy required to increase distillate purity by one unit is considerably higher than the energy saved when purity is decreased by one unit. As a consequence, the value of the corresponding“gain,” defined as the ratio of the change in the controlled variable CV (the purity y_d) to the imposed change in the manipulated variable MV (the reboiler duty Q),

ACV

K = Dga

(Equation 1)

AQ AMV

can vary considerably depending on the operating point.

Equation 1 represents a relationship that is nonlinear but monotonic (here, the gain K is continuously decreasing as the duty and purity increase). Further research has revealed that it is possible that such input-output relations be nonmonotonic (i.e., the gain K may change sign), leading to multiplicities (Figure 2). An input multiplicity denotes the case where multiple sets of MV values lead to the same value of the CVs. Conversely, in the case of an output multiplicity, multiple solutions for the dependent variables (CVs) are possible for the same set of independent variables (MVs).

Multiplicity has been demonstrated both computationally and experimentally for distillation columns. For example, Figure 3 shows the steady-state behavior of a binary distillation column separating methanol and isopropanol. As shown in Figure 3, multiple values of, for example distillate purity and molar boilup rate (and, hence, reboiler duty), are possible for the same value of the volumetric reflux. Likewise, a binary column processing a propanol-acetic acid mixture reveals, among others, an output multiplicity in the distillate purity with the reboiler being the MV (Figure 4). Multiple values of the conversion of a key impurity in a reactive distillation column are also possible for a given value of the reboiler duty.

Dynamic intensification based on output multiplicity: Concept. The presence of such multiplicities— in particular, the presence of an output multiplicity— offers an intriguing opportunity, which we will exploit in this example. Referring now to Figure 5, assume that it is desired that a column manufacture a product at the steady-state operating point P^*, which is characterized in terms of a value CV^* of the controlled (output variable), and the corresponding value MV^* of the input, P* = (MV^*, CV^*). Assume now that:

The output variable (CV) of interest is intensive and represents a characteristic (e.g., concentration), which can be used to uniquely define a product.

· The input variable considered (MV) represents a direct or indirect measure of the production cost. Without loss of generality, let us assume that the production cost is directly proportional to the value of the MV, i.e., that a product corresponding to a high value of the MV is more expensive to produce than a product corresponding to a low MV value.

Under these assumptions, we can postulate the following dynamic intensification concept: the product of interest, CV*, can be manufactured by mixing two auxiliary products, CVi and CV₂, with input values (costs) MVi and MV₂, respectively, i.e., Pi = (MVi, CVi) and P₂ = (MV₂, CV₂). The auxiliary products are made by periodically switching the

operation of the column between the two corresponding operating points. The two products are stored in the same (sufficiently large) tank, which will, via mixing and time averaging, contain the desired product CV*. Any off-specification product made during the transition can either be discarded or“blended away” by adding it to the tank. Both auxiliary products have a lower“cost” than MV*. As a consequence, the time-average cost of the product CV* obtained via the periodic operation strategy described above will be lower than MV*.

In the context above, (output) steady-state multiplicity is defined based on an input- output relationship between the manipulated variable MV (input) and the controlled variable CV (output). While two (or multiple) steady-state points described by input-output pairs of the type (MVi, CVi), (MVi, CV₂), etc. exist, the remaining states of the process may be quite different (and, in fact, uniquely defined) at each such point (Figure 6). This can present both an opportunity and a challenge.

On the one hand, switching between operating points IT = (MVi, CVi), GT = (MV₂, CV2) can be effected by controlling other states/outputs than CV, potentially also using other manipulated inputs than MV, thereby avoiding the control difficulties associated with a nonlinear input-output relationship of the type illustrated in Figure 2.

On the other hand, the fact that other process states can vary between operating points (MVi, CVi) and (MV₂, CV₂) can be deleterious to the dynamic intensification concept described above. Consider for example the situation where the output CV represents the purity of the distillate or bottoms products, x, and MV is the reboiler duty Q (see, e.g., Figure 4). The desired product purity x^* and the operating point (Q^*, x^*) correspond to, e.g., a specific product flow rate. The objective is to make a product of purity x^* using less energy than Q^*. While the reboiler duty at the auxiliary operating points (Qi, xi) and (Q₂, x₂) is lower than Q^*, it is possible that the product flow rate of the column when making the auxiliary products may be lower than the production rate at (Q^*, x^*). In this case, the energy consumption per unit product may increase when implementing the proposed strategy, thereby eliminating any potential economic benefit. Such situations should be dealt with by taking advantage of other available manipulated variables, a point that will be elaborated on in the case study presented later in the example.

The proposed dynamic intensification strategy also calls for determining a split coefficient (a), which dictates the relative amounts of the auxiliary products to be produced. As a very basic level, it can be computed as:

CV^* = aCV_j + (1 — a)CV₂ (Equation 2)

However, as we will show later, other process variables may need to be taken into account.

Placing dynamic intensification in the historical context of cyclic distillation.

Periodic/cyclic distillation has been investigated since the l960s. Early cyclic distillation operating strategies involves, for example, discrete switching between two separate regimes: a vapor flow period and a liquid flow period. This“spliting” of the column operation was made possible by the use of special trays with no downcomers, which can block the flow of liquid from a stage to the stage below. This controlled cycling required fast switching (in the order of seconds) to completely separate between continuous vapor flow and sequential liquid flow, thereby reducing unnecessary mixing between the two phases and between the liquid materials in adjacent stages. Further studies claimed higher efficiency and up to twice more throughput (compared to a conventional column of the same size) based on tests from sieve and screen plate columns as well as packed columns. In-depth studies (both computational and experimental) continued to be carried out through the l980s to probe the effect of periodic column operation on separation/stage efficiency.

After an apparent publication hiatus of about two decades, research on cyclic distillation has regained momentum, focusing on improving the hardware that allows for separate movement of the liquid and vapor phases. Current tray designs use valves and sluice chambers to alternate between the two flow regimes, again with fast switching in the order of seconds. Pilot studies carried out on an ethanol-water system reported a 30% improvement in energy efficiency and 2.6 times fewer stages than an equivalent conventional column.

It must be reemphasized that this cyclic distillation concept relies on discrete switching between different flow regimes. In other words, significant discontinuities appear in the states of the column, with the associated modeling and control difficulties. It is also noteworthy that cyclic distillation requires special, customized trays and other hardware, and can potentially carry higher capital costs than conventional columns, particularly in the case of retrofits.

In this context, the main features of the dynamic intensification strategy proposed in the present work are as follows:

First, the proposed strategy can be implemented in a conventional distillation column, requiring no hardware changes (as opposed to a complete hardware overhaul needed for cyclic distillation).

Second, the proposed strategy exploits the nonlinearity of the system and involves alternating the operation of the column among different steady states that have the same flow regime. In contrast to cyclic distillation, the states of the system are continuous, which can be advantageous from modeling, simulation and control points of view.

Case Study: Dynamic Intensification of Propanol-Acetic Acid Distillation

In this case study, the separation via distillation of a propanol-acetic acid mixture is analyzed. Others have investigated the behavior of a two-stage separator with an equimolar propanol-acetic acid feed, and demonstrated that the distillate purity (defined in terms of propanol concentration), y_d, exhibits the behavior shown in Figure 4 when the reflux rate is fixed and the reboiler duty Q_B varies. Flere, y and Q_B take the roles of CV and, respectively, MV, as defined earlier in the example.

For the sake of completeness, this finding was reexamined in the initial stages of the current work, using a three-stage distillation column model, operating at 1 bar condenser pressure and a fixed 0.01 bar pressure drop per tray, with an equimolar saturated liquid feed at 1 bar with 1 kmol/min flow rate. A 9.5 kmol/min reflux rate was used. The system was simulated in Aspen Plus^®, and the van Laar equation was used to perform the phase equilibrium calculations. A sensitivity study of y_d to the changes in QB was carried out. The results shown in Figures 7A-7C.

Subsequently, a different binary distillation column with seven equilibrium stages was designed, along with a control system as shown in Figure 8. This column was intended to purify propanol in distillate, focusing for illustration purposes on generating the most energy intensive product CV^*, i.e., a distillate whose purity corresponds to the highest reboiler duty as illustrated in Figure 7. The seven equilibrium stages include a total condenser and the reboiler. The pressure drop across each tray is 0.01 bar. Sufficiently large mixing tanks are used to average any oscillations in product purity. Under the fixed reflux rate, the column is designed to reach propanol purity y ^* = 96.64 in the distillate.

According to the dynamic intensification principle stated earlier, product CV^* can be made with lower energy consumption by switching between two operating modes, whereby the distillate stream has higher and, respectively, lower purity. The finished product is obtained as a blend of the two auxiliary products, whose time-average concentration meets the purity specification CV^*.

Also according to our earlier observations, this strategy can be fraught with difficulties: exploiting the above input-output relationship that does have an impact on the process states. Flere, the distillate flow rate drops monotonically as purity increases (Figure 9) and, as a consequence, it is not possible to maintain both the purities of the two auxiliary products and the corresponding production rates, such that the overall time-average production rate and product purity meet or exceed the purity and production rate of the product CV^*.

This challenge can be overcome by choosing a second manipulated variable, MVo, which, together with the original MV, is used to redefine the operating points where the two auxiliary products are produced. Figures 10A-10B show that the operating pressure of the column is a suitable candidate for this role: as pressure increases, for a given distillate composition, the distillate flow rate increases, while the reboiler duty decreases slightly.

Thus, the operating points for producing the auxiliary products can be dehned qualitatively as“low vapor flow rate (low duty QB), high pressure” (Th) and“high vapor flow rate (high duty QB), low pressure” (Th) where IT_, i £ { 1,2} are now represented by the triple (MV;, MVi, and CV;). Table 1. Operating Points for the Auxiliary Products, the Desired Product CV^*, and the

Time-Average Values for the Mixed Product.

Table 1 lists the steady-state data corresponding to the target product and the auxiliary products, as well as a comparison of the weighted average properties of a mix of II_I and IT with the properties of IT In order to obtain the target distillate purity, the split ratio a is calculated based on Equation 3, where y* refers to the desired purity, Di, y u refer to the distillate flow rate and purity for the high-pressure operating point Pi, and D₂, y_d2 denote the same variables for the low-pressure operating point II₂. The operating strategy calls for weighting the high-pressure operating point (IT) with a = 0.922 (i.e., spending 92.2% of the operating time at this point), while the low-pressure operating point IT should be weighted by 1 - a = 0.078. Thus, the weighted average operation results in 1.63% energy savings and a 1.47% production increase in the target product.

Dynamic Implementation of the Proposed Periodic Operation Strategy

In this section, dynamic simulation is used to verify the steady-state results presented above. The aforementioned results assume that this continuous process can remain at each operating point for a well-defined fraction of the operating time; the steady- state analysis is inherently agnostic to the transient/dynamic properties of the process. Thus, the objective of this dynamic simulation study is to confirm that the column is able to:

Rapidly switch between the two steady-state operating points IT and IT_;

Operate stably at each of these points; and Generate the auxiliary products with the same characteristics (notably, purity and flow rate) as those specified in the steady-state analysis.

To this end, a flow-driven dynamic simulation was developed using Aspen Dynamics V8.8. In this case study, the feed temperature was fixed at l09°C to guarantee that the feed stream is in the liquid phase for any feed pressure (which varies between 1.13 bar and 1.63 bar depending on the operating point). To reflect the production of the two auxiliary products, the simulation scenario consists of two different periods (high and low pressures). During the high-pressure period, the vapor flow rate is set to 1669.3 kmol/h, the feed pressure is set to 1.63 bar, and the condenser pressure to 1.60 bar with a (fixed) reflux rate of 1330.2 kmol/h. During the low-pressure period, the vapor flow rate is switched to

1829.8 kmol/h, the feed pressure drops to 1.13 bar, and the condenser pressure is set to 1.10 bar with the same reflux rate of 1330.2 kmol/h.

In order to impose these changes, as shown in Figure 8, two dedicated control loops are used. First, the condenser pressure is controlled by adjusting the condenser duty/cooling water (CW) flow rate, while the vapor boilup rate is controlled by adjusting the steam flow rate to the reboiler (and effectively the heat duty Q_B). Note that the relation between the vapor boilup rate and distillate purity is monotonic (Figure 7C), and therefore the state of the system can be uniquely defined by setting the condenser duty and the vapor boilup rate. The two control loops were tuned by trial and error to provide fast set-point tracking. Finally, the aforementioned sequences of set-point changes are imposed by generating the appropriate signals using a Visual Basic for Applications (VBA) script in Microsoft Excel. The latter communicates with the Aspen Dynamics simulator via ActiveX.

An initial set of simulations showed that using step changes in the setpoint signals can lead to aggressive control moves and changes in the manipulated variables. In particular, it was noted that the bottoms or distillate flow rates could be reduced to very small values during transitions. In order to avoid this, the setpoint changes for the vapor boilup rate were imposed in the form of ramp signals, with a decrease rate of -5.74 kmol/h per time step (with the time step being fixed at 0.01 h) and an increase rate of 10.7 kmol/h per time step (Figures 11A-1 ID).

Subsequently, the dynamic simulation was used to study the influence of the transitions between auxiliary products on the performance of the process and to determine the actual length of each operating period, which on average results in the desired product quality CV^*. Empirically, it was determined that when the duration of the low-pressure operating period exceeds 0.83 h (and, correspondingly, more than 9.88 h are spent in the high-pressure operating period), the effect of the transition becomes negligible, and the desired average performance is reached. The results are presented in Figures 12A-12D and summarized in Table 2, and they confirm the outcome of the steady-state analysis in terms of reduction in the total energy use.

Table 2. Dynamic simulation results for three cycles including a 0.83 h low-pressure period and a 9.88 h high-pressure period (time-average shown).

Further Performance Improvements from Dynamic Intensification

Based on Figure 9, operating point Pi can empirically be labeled as having“high purity, low production rate.” Continuing this line of reasoning, operating point Th can be viewed as“low purity, high production rate,” and the operating strategy described above then constitutes a trade-off between making a high purity, low volume product and a low purity, high volume product, both having lower or just slightly higher specific energy requirements than MV^*. Moreover, the dynamic operating sequence determined using steady-state data involves operating predominantly at Pi, which suggests that switching to Th represents a“correction” to the high-purity operation mode, meant to adjust product concentration and increase the production rate.

It is important to note, however, that this correction need not be static; that is, dynamic intensification also accounts for the contribution of the transition between Pi and Tb. Equivalently, the process need not spend as much time operating at P2 as forecast by the steady-state analysis (which, as was noted earlier, is not privy to any dynamic effects) to achieve the same outcome, i.e., to generate a product with the desired time-average purity and production rate.

The duration of the time period corresponding to P2 in the setpoint sequence, as well as the ratio between the duration of the operating periods corresponding to Pi and P2, can be determined by solving the following optimization problem:

(Equation 4)

where the objective is to obtain, after time averaging, a product with the same purity and flow rate as P^*, but with lower specific energy consumption. Here, tm and tm are, respectively, the time spans for which the setpoints are set to the values corresponding to Pi and Eh, and T is the total time for which the operation is run. Given the small number of decision variables, a line search was performed to solve this problem. The dynamic intensification solution that was obtained involved maintaining the controller setpoints for 0.46 h at values corresponding to Th and 8.26 h at Pi. This solution is not unique given the nonconvex nature of the problem. The resulting dynamic intensification setpoint sequence was simulated for three production cycles (Figures 13A-13D). The time-average results are presented in Table 3, revealing a similarly significant reduction (1.66%) in total energy consumption compared to operating point P^*. A 1.53% increase in production rate compared to operating point P^* was also observed. Figures 13A-13D reveal that— in true dynamic intensification fashion— the system does not reach the steady state when the setpoints are switched to values corresponding to Th (note the evolution of the production rates).

Table 3. Results of dynamic intensification simulation.

Conclusions In this example, dynamic intensification was proposed as a new process intensification concept. Dynamic intensification relies on exploiting the nonlinear behavior of the process (where both static and dynamic nonlinearities are considered) to achieve improvements in performance (defined in terms of economic, environmental, and/or safety metrics).

This concept was illustrated using binary distillation as one of the most prevalent operations in the chemical industry. Specifically, it was demonstrated that the nonlinear features of distillation columns (notably, output multiplicity) can be exploited to generate a periodic operating pattern that lowers energy consumption without compromising product quality. An extensive case study on the separation of a propanol-acetic acid mixture was presented, confirming these results.

These concepts represent a significant contribution in the context of efforts that centered on periodic distillation thus far. Compared to, e.g., cyclic distillation, the present work employs existing hardware and exploits the system nonlinearity, rather than using specialized distillation equipment operated in a discontinuous fashion.

In a broader context, it is worth bearing in mind that other process units (and possibly entire plants) exhibit similarly relevant nonlinear features, and that dynamic intensification potentially holds significant promise for improving process operations without requiring major capital expenditure. To this end, future efforts can be directed at (1) documenting the relevant behaviors, (2) identifying appropriate dynamic

intensification/operating strategies, and (3) improving dynamic optimization capabilities to fully characterize the benefits that dynamic intensification can provide. These endeavors should also seek to alleviate any potential down- sides of periodic operation; on the process equipment side, these include potentially higher equipment wear and tear, and degradation of efficiency due to operating at off-design points. From the deployment perspective, efforts should be geared toward operator training and fostering acceptance of a transient operating strategy by plant personnel, who are typically trained and accustomed to running their facilities at or around a steady state. Example 2. Dynamic Process Intensification of Binary Distillation via Periodic Operation.

This example applies the concept of dynamic intensification (defined as changes to the dynamics, operation strategy, and/or control of a process that lead to a substantially more efficient processing path) to binary distillation columns. The resulting strategy includes manufacturing a target product as a blend of two auxiliary products, both having lower energy demands than a reference value, which corresponds to producing the target product(s) in a column operating at steady state. A discussion of the appropriate control structures and switching strategies between the two auxiliary products is provided. An extensive case study concerning the separation of a methanol- 1 -propanol mixture was carried out, demonstrating that energy savings in the order of 1.4% are possible with no disruption in product quality or production rate.

Background

Distillation remains the workhorse separation technology for liquid mixtures in the chemical industry. Distillation relies on the difference in boiling points between the components of a mixture to achieve separation. In other words, the mixture must be (at least partially) vaporized to be separated. Providing the required heat input (as dictated by the latent heat of vaporization) makes distillation highly energy intensive, with distillation operations accounting for an estimated 40% of the energy consumption of chemical plants. The need to improve the economics of distillation processes has spurred significant research efforts. In this example, we highlight the role of process intensification as a design philosophy that emphasizes“doing more with less”. In the realm of distillation, intensification led to the development and implementation of new configurations combining the functionality of two or multiple columns in a single device. The most prominent representative of this concept is dividing wall columns. In a different vein, cyclic distillation relies on separating the vapor and liquid traffic in the column such that mixing inefficiencies are reduced. However, cyclic distillation requires specialized hardware (column trays), which can come at considerable cost, particularly for retrofits. Motivated by this, we have introduced the concept of dynamic process intensification, which relies on operational changes (particularly, periodic operation), typically applied to existing hardware and equipment, with the goal of improving energy efficiency.

Focusing specifically on distillation columns, in Example 1 we demonstrated that the static nonlinear characteristics of columns can be exploited to lower their average energy consumption ln this case, dynamic process intensification involves (i) defining a target product (in terms of purity and production rate), (ii) identifying two auxiliary products with lower reboiler duties than the target product, and (iii) a periodic operation pattern that comprises switching production between the auxiliary products, such that the resulting blend has, on average, the same properties at the target product but features lower energy use. In this context, the dynamic intensification strategy outlined in Example 1 exploited the nonlinearity associated with the thermodynamic properties of a specific class of binary mixtures, where the more volatile (lower boiling point) component has a higher latent heat of vaporization than the less volatile component. This class of mixtures exhibits a favorable “negative gain” between distillate purity and reboiler duty (namely, in a specific operating range, duty may decrease as purity increases). We continue to seek and exploit this property in this Example, which extends the dynamic intensification strategy to the distillation of a broader set of binary mixtures. We demonstrate that an appropriate choice of the column degrees of freedom and specifications can lead to similarly desirable nonlinear properties. We also show that the dynamic intensification strategy can be applied to obtain both the distillate and bottoms products at the same flow rate and purity but with lower energy use, compared to a reference distillation column operating at steady state. An extensive case study concerning the separation of a methanol- 1 -propanol mixture is presented.

Motivating Example

In this Example, we will consider the separation of a binary methanol-propanol mixture via distillation. This process has been investigated considering an eight-stage column with an equimolar saturated liquid feed consisting of methanol and 1 -propanol.

The column operates at 1 bar condenser pressure with fixed 0.01 bar pressure drop per tray and a feed stream at 1.03 bar with 1 kmol/min flow rate. The eight equilibrium stages include a total condenser and a reboiler with feed introduced on the fourth stage from the top. Figure 14 shows the column configuration and the corresponding control loops (which are discussed later in this Example). The system was simulated at steady state in AspenPlus¹⁰ using the Wilson activity coefficient model for the liquid phase and assuming that the vapor phase has an ideal gas behavior to perform the phase equilibrium

calculations. In particular, a set of sensitivity studies of the distillate purity, y_d, to the reboiler duty, Q_B, was carried out. Our investigations sought to delineate the effect of the product purity on reboiler duty under two operating regimes: fixed reflux rate and fixed boilup ratio.

The results of the simulation studies (Figures 15A-15B) reveal an interesting feature, namely, that the choice of operating specifications has a strong impact on the static characteristics of the system. In effect, depending on the choice of specification, the (nonlinear) dependence between the aforementioned variables of interest can follow opposite directions. In order to quantify the difference between the static effects of the specifications, we define the system gain as follows:

Ay_d ACV

k = (Equation 5)

AQB AMV

which represents the ratio between changes in system input/ manipulated variable Q_B (the “MV”) and corresponding changes in the output/controlled variable y_d (the“CV”). The gain defined above provides useful insights on the energy consumption of the column and can quantify the energy consumption effect of specifying the design distillate purity y . To this end, an intuitive metric is to compare the change in Q_B for an increase and a decrease of equal magnitudes in y_d. Figure 15 A shows that under a fixed boilup ratio and changing reflux rate the gain is monotonic and positive, and its value is approaching zero as purity increases. In this operating case, AQ_B is higher for a unit increase in y_d compared to a unit decrease. By contrast, Figure 15B shows a completely opposite but more economically interesting relationship. The gain is monotonic but negative. The magnitude of the gain also approaches zero as purity increases. Under a fixed reflux rate (boilup ratio or vapor flow rate is allowed to change), a unit increase in y_d can reduce reboiler duty (and save energy) by a larger amount than the duty increase when decreasing y_d by one unit.

Dynamic Intensification Concept

The arguments above and the data in Figure 15B suggest a new dynamic operating paradigm for binary distillation columns that has the potential to generate a desired product slate (defined in terms of bottoms and distillate purities and flow rates) at a lower energy consumption than an equivalent column that is operated at a single steady state.

We rely on Figure 16, which provides a generic representation of the relationship revealed by Figure 15B, to explain this concept. Let us assume that the desired purity of the top product is CV*. Based on the corresponding value CV* of the controlled (output variable) and using the monotonicity of the relationship between the manipulated and controlled variables, the corresponding value MV* of the input can be computed. The desired product can thus be uniquely defined in terms of the pair P* = (MV*,CV*).

Observe that a product with purity CV* can either be manufactured by operating the column at steady state at (MV*,CV*) or by mixing two auxiliary products, Pi =

(MVi CVi) and Tb = (MV2,CV2). The two auxiliary products are produced by the same column at two other different operating points and are obtained by alternating periodically between the two corresponding operating points. The auxiliary products are then blended in a (sufficiently large) tank, such that, on average, over time, the blended product reaches the desired specification CV*. Possible off-specification products made during the dynamic transition can also be added to and mixed in the tank.

In Example 1, we postulated that ideal auxiliary product candidates should both have a lower“cost” (defined in terms of the manipulated variable) than MV* to guarantee a time-average lower energy cost of the product with purity CV*. The relationship presented in Figure 15B indicates that such ideal candidates do not exist for this binary mixture. Nevertheless, the figure suggests that energy savings could still be achieved by producing a product with purity CV* as a blend of products with purities CVi and CV₂, given that the energy consumption for the high purity product is lower than the energy consumption of the low purity product, i.e., CVi > CV₂ while MVi < MV₂. The corresponding mixture can be defined based on the split coefficient a, which dictates the proportion (i.e., the relative amount of time) of operating points Pi = (MVi,CVi) and P₂ = (MV₂,CV₂) required to obtain the desired CV*:

(Equation 6) The implementation of the proposed dynamic intensification approach can be described via the following procedural steps:

Step Sl : Establish the operating specihcations of the column in terms of a hxed reflux rate for a given product slate (product purities and flow rates).

Step S2: Dehne the two auxiliary products, such that the corresponding“quality variables” (product purities) are higher and, respectively, lower than that of the target product(s).

Step S3: Compute the split ratio a.

Step S4: Implement a control system capable of transitioning effectively between the operating points corresponding to the two auxiliary products. Dehne a switching scheme that imposes the split ratio a between the products, such that, on average, the purity of the blended products meets the product slate specihcations.

Below, we present several general remarks and potential challenges to the above concepts; these are further elaborated in the case study presented later in this Example.

First, the premise of the proposed dynamic intensihcation approach is

counterintuitive given the common wisdom that increasing purity requires a higher energy consumption (also as indicated by the results shown in Figure 15B). However, the results in Figure 15B can be understood by considering the converse perspective, that energy can be saved by sending less vapor toward the top of the column. Second, Figure 17 shows that the distillate flow rate drops monotonically as purity increases under constant reflux operation. This drop is particularly noticeable as the purity of the distillate increases. Thus, the proposed strategy can experience difficulties in maintaining separation performance. Changes in the vapor flow rate, while reducing energy consumption, may result in a higher purity distillate that is produced at a low flow rate, and the loss of production rate could result in a product that in fact requires a higher per unit energy expenditure than the product obtained during steady-state operation. In turn, this suggests that one or more additional degrees of freedom (and control loops) should be employed in Step S2 (and later in Step S4) to ensure that the desired production rate is met while also meeting product quality constraints.

Third, the proposed approach may present control difficulties, as it entails operating at high purities for at least part of the time.

Fourth, a major benefit of this strategy is that it relies on exploiting the nonlinear features of existing equipment designs, rather than on specialized equipment as is the case with cyclic distillation (discussed above). As a consequence, the capital expenditure required to implement it is low and involves the purchasing and installation of the blending tanks required to "average out" the product quality.

Case Study: Dynamic Intensification of Methanol-Propanol Distillation

Below, we rely on the motivating example presented earlier to illustrate the dynamic process intensification concepts introduced above.

Steady-State Considerations. In order to carry out Steps Sl and S2, we must choose a feasible location of y*_d (CV*) in Figure 15B and the corresponding two auxiliary products. Figure 15B was generated based on a fixed 5.0 kmol/min reflux rate, and the column is designed to reach a methanol purity y*_d = 87.36%. We maintain this value as the product purity target for periodic operation.

In Example 1, we showed that increasing the product purity (naturally) led to a drop of product flow rate (in that case, distillate). A second manipulated variable (pressure) was utilized to compensate for this effect, and it was shown that the nature of the binary mixture was such that higher pressure increased flow with lower energy consumption. In this Example, utilizing pressure as a secondary manipulated variable MV' may assist with increasing distillate flow rate but may compromise energy savings (Figures 18A-18B). As a consequence, we utilize reflux flow rate as a third manipulated variable, MV", which can be used to maintain the desired product flow rates if applied simultaneously with pressure changes due to its direct influence on both distillate flow rate and purity. Thus, the auxiliary operating points can be defined in term of the corresponding reflux, vapor flow rates, and operating pressure. Consider the dependence of the distillate flow rate on the reflux rate shown in Figures 19A-19B. Focusing particularly on the high purity operating regime (which was the operating region of concern above), we note that, for a given distillate composition, the distillate flow rate can slightly increase for a slight increase in reflux rate. Interestingly, the distillate flow rate may also slightly increase for a slight drop in reflux. The behavior of the reboiler duty is as expected, where duty increases as the reflux rate increases.

Next, the auxiliary products are dehned. In general, we seek a combination of an auxiliary Pi, whose high purity compensates for the drop in production rate, and an auxiliary Tb whose concentration should be lower than but close to that of P*, such that its contribution to energy savings is high without having a signihcant negative impact on the quality of the blended product. Table 4. Operating Points for Desired Product CV*, Auxiliary Products, and

Corresponding Weighted Average Values for the Blended Product. The values for Pi and Tb are presented in ter s of deviations from P^*.

The choice of auxiliary products is shown in Table 4, where the steady-state operating parameters are listed along with a comparison between weighted average properties of IT and IT (which would reflect the blended product obtained from periodic operation) and those of P*. The split coefficient is calculated based on Equation 7 and calls for weighting the high purity operating point (IT) with a = 0.037 (i.e., spending 3.7 % of the operating time at this point), and the low purity operating point IT by 1 - a = 0.963, a result consistent with the arguments presented in the previous paragraph. The three operating points are represented graphically in Figures 20A-20C. Based on steady-state arguments, dynamic intensification via periodic operation can potentially result in 1.44% energy savings with minimal impact on distillate and bottoms product qualities, compared to producing a product of purity and flow rate corresponding to P* at steady state.

«£>iydi+ ( l-a)D₂y_d2

Ύa = (Equation 7)

<xD₁ + (l-a)D₂

Dynamics and Control Considerations. A multiloop linear control system was implemented with the purpose of imposing transitions between the operating regimes corresponding to the two auxiliary products. The control loop pairings were as follows: the vapor boilup rate was adjusted by the steam flow rate to the reboiler (and effectively the heat duty Q_B), while the reflux rate was adjusted by the reflux valve. The column pressure was controlled by manipulating the coolant flow rate to the condenser, while the levels of the sump and distillate drum were stabilized by manipulating the bottoms and distillate flow rates, respectively.

It is important to emphasize that this control configuration serves the sole purpose of studying the feasibility and effect of imposing the transitions between the auxiliary operating points. It is not intended to track a product purity set point or to reject disturbances. An additional, supervisory control system would be required to track product purity in closed loop. The design of this system must account for the fact that the relationship between QB and distillate purity y_d becomes nonlinear and nonmonotonic when the auxiliary operating points are defined in terms of both the reflux flow rate and the vapor boilup/reboiler duty. This point is well illustrated by comparing Figure 15B and Figure 20A, which show the multiplicity between input QB and output y when the other manipulated variables are considered. Note also that the mission of this supervisory control system is facilitated by the blending tanks installed for the distillate and bottoms products (Figure 14), which act as buffers that attenuate the impact of any disturbances that affect composition. Dynamic Simulation Results. A dynamic simulation study was carried out to verify the steady-state results presented in Table 4. The steady state results rely on the key assumption that the process can be continuously run at the desired operating points for a well-defined amount of time and does not account for the transient effects that are inherent to periodic operation. Thus, the purpose of the dynamic simulation study is two-fold:

• Confirm the ability of the control system to reach and maintain the steady-state operating points corresponding to the two auxiliary products and ascertain that the steady states of the dynamical system correspond (notably, in terms of purity and flow rate) to those specified in the steady-state analysis.

· Demonstrate the feasibility of rapidly switching between the two steady-state operating points.

As emphasized above, we are not interested in tracking product purity per se or in studying the disturbance rejection capabilities of the control system. A study of these characteristics (along with the design of the aforementioned supervisory control system) is planned as future work.

The dynamic simulation study was carried out as a flow-driven dynamic simulation using Aspen Dynamics V8.8. A Visual Basic for Applications (VBA) script in Microsoft Excel, communicating with Aspen Dynamics via ActiveX, was used to impose the set point changes required to switch between the auxiliary operating points Pi and Th. The set points were imposed as square-wave signals; for producing the product with high purity CVi, the vapor flow rate was set to 294.371 kmol/h, with a reflux rate of 306 kmol/h. During the low purity period (P₂), the vapor flow rate was switched to 290.990 kmol/h with reflux rate set to 294 kmol/h.

The actual length of each operating period was calculated based on the dynamic performance of the system evaluated with regard to operating at the high purity state. Operation at the high purity state was assumed to be complete when the steady state (defined as the distillate purity being within 0.03% of the value for CVi from Table 4) was reached. The time constant of the column under consideration (calculated as the ratio between liquid holdup in the sump and condenser drum, and the feed flow rate) is about 0.17 h. The dynamic simulation results indicated that the above (admittedly restrictive) steady-state condition was reached in a time interval spanning about 16 time constants.

For the purpose of simulating periodic operation, the simulation was extended for one more time constant (total 17 time constants) at Pi, after which the relevant set points were switched to the values corresponding to Th for the amount of time indicated by the split fraction a. Thus, each cycle of the dynamic simulation consists of maintaining set points at values corresponding to P2 for 76.51 h then switching to Pi for 2.91 h as shown in Figures 21A-21D. Time-average data for three consecutive cycles are shown in Table 5, confirming the fact that periodic operation can achieve the energy savings and product purity targets predicted by the steady-state analysis.

Table 5. Dynamic Simulation Results for Three Cycles (time average shown).

Impact of Transition Times and Choice of Switching Frequency. In order to define the impact of transitions on the proposed periodic operation strategy, we define the transition time as the amount of time required by the process variables to be within 0.05% of the target value for Pi or P2 once a set point change (from, respectively P2 or IT) has been initiated. Transition times were extracted from the previous simulation data: due to the nonlinearity of the system, the system requires 2.52 h to transition from P2 to IT, while the transition from IT to IT requires 1.69 h.

The transitions do not have the same impact on the time-average performance of the system. Simulation of a single cycle (with a 2.91 h switching time) is shown in Figures 22A-22D. It suggests that the longer transition from P2 to P 1 impacts performance negatively. Purity is always below P 1 target, and the distillate rate is predominantly below its P 1 steady-state value. On the other hand, the shorter transition from P 1 to P 2 has qualitatively the opposite effect. Quantitatively, however, the impact of this latter transition is not sufficient to offset the performance penalty incurred by transitioning from P 2 to P 1.

These findings suggest that transition times should be accounted for when designing the periodic operation policy. The switching frequency must consider transitions in addition to following the split ratio a prescribed by the steady-state analysis, and requires a dynamic optimization calculation which is planned as future work.

We examined the effect of the choice of switching frequency empirically via simulation. Considering the 2.52 h (about 14.8 time constants) transition time from Th to Pi as the fastest possible switching time (referred to as the base case ), we carried out simulations with gradually decreasing switching frequencies (94%, corresponding to the example in the previous subsection, 50%, and finally 10% of this value), which were imposed by correspondingly increasing the switching times. Table 6 lists the deviation in performance and cost with respect to P* for each case, time-averaged for three cycles. The results indicate decreasing the switching frequency leads the system to more closely mimic the results of the steady-state analysis; equivalently, the impact of transitions on overall time-average system performance is attenuated if the transitions are less frequent.

Table 6. Time- Average (over three cycles) Percent Deviation of Key Process Variables from P^* as a Function of Switching Frequency.

Finally, we note that the initial conditions of the system also impact its time average performance if the number of cycles/switches is small. This is the case with all periodic operations, and the effect largely disappears once the system goes through a sufficient number of cycles and reaches a periodic steady-state. We verified that the distillation column considered here does indeed reach what can be construed as a periodic steady state based on six switching cycles, as illustrated by the phase plane plot shown in Figure 23.

Conclusions

We presented a dynamic intensification strategy for binary distillation columns. The underlying principle consists of manufacturing the target product as a blend of two auxiliary products, both having lower energy demand than the value corresponding to producing the target product in a column operating at steady state. We demonstrated that this is a viable strategy for a binary column provided that an appropriate choice of the column degrees of freedom and specifications is made. For the near-ideal mixtures considered, this amounted to specifying the reflux rate, boilup/vapor flow rate and operating pressure. An extensive case study concerning the separation of a methanol- 1 -propanol mixture was carried out, demonstrating that energy savings in the order of 1.4% are possible with no disruption in product quality or production rate. This is an important result given that (i) it is obtained with only operational changes (i.e., no dedicated column hardware is required) and (ii) it can potentially be replicated to a large fleet of operating columns.

The proposed framework is still in the conceptual stage but provides motivation and impetus for future work aimed toward its practical implementation. Future efforts will include (i) (dynamic) optimization studies for determining the most appropriate auxiliary products and switching strategies, (ii) devising supervisory control algorithms that ensure product purity tracking in the face of disturbances, and (iii) developing screening tools for determining mixtures whose distillation is amenable to dynamic intensification.

Example 3. Maximizing Energy Savings Attainable by Dynamic Intensification of Binary Distillation.

Dynamic intensification of distillation columns has shown significant promise in achieving energy savings with minimal investment in new equipment. Conceptually, it entails making a desired product as a blend of two auxiliary products (one with higher purity, the other with lower purity, but both having lower energy consumption). Practically, dynamic intensification means periodically switching between two operating states corresponding to the aforementioned products. Past work has relied on ad-hoc choices of auxiliary products. In this paper, we introduce an optimization framework for selecting auxiliary products for dynamic intensification. An extensive case study concerning the separation of an equimolar methanol/propanol mixture is then presented. We show that optimizing the choice of auxiliary products can lead to significant energy savings (more than 3.6% compared to a column operated at steady state) derived from dynamic intensification.

Background The chemical industry turns raw materials into value-added products via physical and/or chemical transformations. In most circumstances, the feedstock of chemical plants contains (traces of) impurities. Moreover, most chemical reactors are not designed for complete conversion, and many chemical reactions produce a (set of) desired product(s) and (several) undesired byproduct(s). As a consequence, chemical plants comprise reaction and separation units, interconnected via material and energy recycle streams. Separation units account for a significant portion of the capital and operating cost of a chemical plant, with distillation being the dominant approach for separating liquid mixtures. In the United States alone, it was estimated that there are about 40,000 distillation columns in operation, which handle 90-95% of total separation needs. Distillation columns are flexible and robust in dealing with fluctuations in feed quality and product constraints. However, distillation is a thermal process that exploits the difference in volatility between the components of the mixture. This requires that the mixture to be separated be brought to a boiling state, which, in turn, entails a significant energy input. The theoretical energy use of distillation columns is driven by the nature of the mixture, and increases as the throughput of the system increases. The energy demand (typically described in terms of the amount of steam supplied to the reboiler) of columns used in practice is further increased by inefficiencies related to heat loss, heat transfer, etc, with the overall thermal efficiency of a distillation tower being as low as 10%.

Significant research and engineering efforts have been directed at lowering the energy consumption of distillation columns. In the design realm, we mention thermal integration concepts (whereby a heat source— typically a condenser— within a column or a distillation train is used to meet heat demand in a sink - typically a reboiler) and intensification (whereby the function of two or more distillation columns is combined in a single shell, compartmented by a septum/wall).

In the operations area, work has focused on imposing cyclic operating patterns that segregate liquid and vapor traffic in the column, with the purpose of minimizing energy inefficiencies associated with (re)mixing. The advent of these ideas can be traced back to the l960s, and research continues to the present. The implementation of cyclic distillation concepts can lead to significant energy savings (compared to a conventional column of the same capacity) but entails major capital cost. Separating the movement of the vapor and liquid phases requires special trays and control strategies that can manage frequent (every minute or faster) flow redirection. The capital expenditure is significant for new projects, and can be prohibitive for retrofits. In Examples 1 and 2, we have introduced dynamic process intensification (DPI) as a novel operational approach for lowering energy use in distillation columns. DPI exploits the nonlinearities inherently present in the static behavior of distillation columns to create periodic, dynamic operating patterns that produce the same results (in terms of time- averaged flow rates and purities of the products) as an equivalent conventional column, but with lower energy consumption. Importantly, DPI relies on existing distillation hardware and can therefore be deployed on a significant number of columns already in operation.

Our previous work in DPI focused on demonstrating the concept empirically; in this paper, we propose a rigorous optimization framework for defining the maximum energy savings attainable by DPI. This Example is organized as follows. In the next section, we introduce the DPI framework and the underlying physics. Next, we define the optimization framework for computing the maximum achievable energy savings. A case study, focused on the separation of an equimolar methanol/propanol mixture is presented, demonstrating considerable energy savings compared to the empirical case. Finally, we draw conclusions and propose potential directions for future work.

Dynamic Intensification of Binary Distillation: Concept

We begin by defining dynamic process intensification in the general case, as any “changes to the dynamics, operation strategy, and/or control of a process that lead to a substantially more efficient processing path.” This general statement was translated to the intensification of binary distillation columns by exploiting their intrinsically nonlinear behavior. Specifically, work has revealed an economically interesting output multiplicity. This consists of a nonlinear steady-state characteristic whereby the same reboiler duty Q_B (manipulated input/cost) could lead to two different values of the distillate purity y_d (controlled variable/output), as shown in Figure 24.

This observation served as the basis for formulating the DPI paradigm for binary distillation columns in terms of producing an energy-intensive (i.e., having high reboiler duty Q_B*) product P*, with target purity y *, as a mixture of two auxiliary products Pi and P 2 , having purities y_d,i (higher than the target purity) and, respectively, purity y_d,2 (lower than the target purity). The reboiler duties corresponding to the auxiliary products are, respectively, QB.I and QB,₂. Importantly, both QB.I and QB,₂ are lower than QB* and, as a consequence, the mixture of the two auxiliary products, chosen in the appropriate proportion, can have the same average purity as target purity y_d*, but lower average energy consumption than QB*.

Of critical importance here is the split coefficient a, defined such that: a Product Flow Rate Pi + (1 - a) Product Flow Rate P2 = Product Flow Rate P* a Product Purity Il_l + (1 - a) Product Purity P2 = Product Purity P*

(1)

That is, a is weighting the product qualities of the auxiliary products, such that the weighted average of the respective values is equal to the desired/target value for the desired product P^*. From a practical perspective, the implementation of this concept entails operating a single distillation column in a transient, periodic fashion, switching between making products Pi and fh with a frequency dictated by a. The desired product P* is obtained based on a time-averaged mixture of the auxiliary products. Naturally, in practice, this requires the installation of holding tanks for the distillate and bottoms products of the distillation column, where the high purity and low purity auxiliary products are mixed and “time averaging” occurs.

The DPI concept described above is not applicable immediately to binary distillation columns due to an additional complication. The conditions abovecannot be simultaneously met by altering only the boilup rate. First, considering, e.g., the case of the distillate product, it is intuitive that, as the distillate purity increases, its flow rate will drop. In turn - based on the column overall and component material balances - the bottoms flow rate will increase, and the purity of the bottoms product (in terms of the heavy component) will drop to accommodate the decrease in the amount of light component that leaves the column as distillate.

In the examples above, we have resolved this problem by defining the auxiliary product in terms of broader operating states, characterized as a function of the values of multiple manipulated variables. In addition to boilup rate, the other include, e.g., reflux rate, column pressure. In this manner, we demonstrated that, with the appropriate choice of the split coefficient and auxiliary operating states, a periodically operated distillation column can meet, on average over time, all the product specifications (purity, flow rate) of a conventional, steady-state column, but with lower energy consumption (defined in terms of reboiler duty or the sum of reboiler and condenser duties).

Optimizing Auxiliary Products for Dynamic Intensification

Our previous work relied on ad-hoc choices of the auxiliary products and related operating states. These were largely based on empirical exploration, via steady-state simulation, of the static nonlinear response of a column to changes in manipulated variables frequently used in practice. In this section, we aim to set this exploration on a rigorous basis. Below, we describe an optimization problem formulation that captures the search for the optimal auxiliary operating states and split coefficient for dynamic intensification.

The inputs of the problem are as follows: we assume that the parameters of the binary feed mixture and target properties of the distillate and bottoms products (flow rate, composition) are known, and that the desired separation is feasible with a finite number of theoretical stages. Further, we assume that a steady-state model of the column, reflecting the material and energy balances, as well as relevant constitutive relations (e.g., phase equilibria) of the column operating at the respective state, is available. Finally, we state that the choice of n manipulated variables (MV;, i=l,...,n) to be used in imposing periodic operation is fixed a priori (that is, the optimization procedure will not select which column inputs to manipulate; rather, it will set their values within known upper and lower bounds). Implicitly, we assume that a control scheme can be designed, to impose the periodic transitions between the two auxiliary operating states.

Under these circumstances, the goal of solving the optimization problem is to identify the values of the split coefficient and the values MVy and MV;, 2 of the manipulated variables at each of the two operating states, such that the weighted average energy consumption of the column OCQB.I + (1 - OC)QB,₂ is lower than that of the aforementioned steady-state design. The constraints of the problem include, i) ensuring that flow rate and quality constraints (1) are met for distillate and bottoms, ii) that the manipulated variables are within their bounds, and iii) that the material and energy balance equations are satisfied at both auxiliary operating states.

Thus, the problem statement is as follows:

mm «QB,I + (1 - «)QB,2

a, MVi,i, MVi,₂

s.t.

quality and flow constraints (1) (2)

MVi,rnin £ MVi£ MVi i„,max

F (MVu) = 0

F (MVi,₂) = 0 where F(MVi) = 0 reflects the constraint that the material and energy balance equations of the column be satisfied. We note here that the problem (2) is nonlinear and non-convex, and multiple local minima are to be expected. One of these minima is in fact the original design of the column, where MVy = MV;, 2 = MV;^*. In order to steer the optimization solver away from this trivial solution, additional constraints should be included; these can be of the form

MVj njn, /— M Vi A MVi,max,l

MVi,min,2 £ MV i,2 £ MVi,max,2 (3)

MVi rnax ,1— MVi,min,2

Case Study: Optimal Dynamic Intensification of Methanol-Propanol Binary Distillation Column.

In this section, we build on the extensive case study of the dynamic intensification of a methanol-propanol binary distillation column and identify the optimal (rather than ad- hoc) auxiliary operating states. Figure 25 shows the design and control configurations of column and lists operating conditions under reference steady state for the target distillate and bottoms products. The column has total 8 stages, including a total condenser and a reboiler, and was modeled in AspenPlus. An equimolar mixture of methanol-propanol enters column at stage 4 and is saturated liquid under 1.03 bar. The reference steady state is operated under 1.00 bar pressure, 300 kmol/h reflux rate and 11.51 boilup ratio. The target distillate purity is y_d* = 87.37% with distillate flow rate 34.26 kmol/h. Two blending tanks are used to reflect the needs of dynamic intensification. These tanks were not explicitly modeled and are assumed to be sufficiently large to filter fluctuations in product flow rates and compositions. Six control loops are implemented:

boilup ratio/rate is adjusted using steam flow rate to the reboiler;

the reflux rate is adjusted using the reflux valve;

column pressure is controlled using coolant flow rate;

the feed flow rate is controlled using the feed valve; and

condensate drum and sump levels are stabilized using distillate and bottoms flow rates, respectively.

Four manipulated variables were used to define the auxiliary operating states used for dynamic intensification: reflux rate, boilup rate, column pressure and feed stream pressure. Figure 26A shows the effect of varying reflux rate on reboiler duty under fixed pressure. As expected, lower reflux rates require less energy to reach same purity than base case, at the cost of a drop in distillate flow rate. Figure 26B presents the influence of column pressure on reboiler duty under fixed reflux rate. Somewhat counter-intuitively, higher pressures can save energy while maintaining same purity. The reason is that increasing pressure diminishes the amount of material vaporized.

Figure 27 shows that pressure of saturated liquid feed has a similar effect on reducing reboiler duty as column pressure. Under same purity, higher feed pressure is favorable to reduce reboiler energy consumption. The reason comes from shift of vapor- liquid equilibrium. As pressure goes up, temperature of feed simply raises and therefore less steam is required to reach same bottom temperature with minimal impact on separation.

In our previous Examples, we empirically selected two auxiliary operating states, reaching 1.4% energy savings for periodic operation/dynamic intensification compared to the steady-state column. Below, we implement the optimization-based strategy outlined above to maximize these savings.

The optimization problem includes nine bounded decision variables: one defining the overall operation (a); four for each auxiliary product/operating point, specifically column pressure (P), feed pressure (P_F), boilup ratio (B ratio) and reflux rate (Reflux).

Min O:QB,I + (1 - «)QB,2

Subject to:

aDistillate_{. i} + (1 - a)Distillate,2 = Distillate*

ayd,i + (1 - ) yd, 2 = yd*

P min.1 — Pi — P max.1

Pl niin. l — PFI — Pl max.1

Bratiominu £ Bratioi < Bratio_max,i (4)

RefluXmin.l £ RefluXl < RcfluXmax. l

Pmin,2— P2— Pmax,2

PFmin,2 A PF2 A PFmax,2

Bratiomin,2 £ Bratio2 £ Bratio_max,2

RefluXmin,2 £ RefluX2 £ RefluXmax,2

CCmin— ^— C( _max

Inequality constraints reflect upper and lower bounds for eight decision variables are subjected to change according to different target point and auxiliary operating points. The upper and lower bounds for the split coefficient, a, are, respectively, 0.01 and 0.99 to guarantee at least 1 % of contribution from one of the auxiliary operating points. The problem was solved for two product purities, y * = 83.37% (Table 7 lists bounds used for this case), and y_d* = 92.97% (Table 9). We note that quality and flow rate constraints were only set on the distillate product; since this is a binary column, the desired flow rate and composition of the bottoms stream are achieved implicitly by virtue of closing the material balance.

Table 7. Upper and lower bounds for decision variables, y_d* = 83.37%.

Table 8. Optimal operating conditions, y * = 83.37%.

Table 9. Upper and lower bounds for decision variables, y_d* = 92.97%.

The problem was implemented and solved in AspenPlus V8.8. The flowsheet (Figure 28) uses two column units (represented as RadFrac blocks) to represent the two auxiliary operating states (which correspond to a low purity and, respectively, high purity product). The splitter blocks are used to reflect the effect of the split coefficient, while a mixer block mimics the mixing tank where the final blended product is collected. The optimization problem was solved using the DMO solver. The objective convergence tolerance was set to le-6 and residual convergence tolerance to le-5. To facilitate the numerical solution, the equality constraints on distillate flow rate and purity in (4) were reformulated as inequalities, with tolerance 5e-4%. Given a feasible initial solution, the problem could be solved in a matter of seconds on an Intel Core i7 computer with 32GB RAM running Windows 10.

Table 7 summarizes the results of the optimization calculations for y_d* = 83.37%. The results show a potential 3.63% energy savings with no impact on product quality. The values of the manipulated variables follow the trends expected based on the discussion above (Figures 26A-26B and Figure 27). Both operating and feed pressure approached to their upper limits and reflux rate reached its lower limit.

The second run of the optimization problem focused on dynamic intensification at a higher purity point, y * = 92.97%. Table 10 lists the corresponding optimized auxiliary operating points, based on bounds of manipulated variables set in Table 9. Optimization results show a similar energy saving for the higher purity target, with closely matched average stream qualities.

Table 10. Optimal operating conditions, y_d* = 92.97%.

A couple of remarks are in order. First, the energy savings are quite significant, given that they largely entail changes in operating strategy with minimal hardware modifications. Second, we note that the implementation of results similar to the ones presented above in the form of a transient, periodic operation strategy was successfully demonstrated via dynamic simulation in our previous Examples.

Conclusions

In this Example, we present recent developments in maximizing the economic and energy saving benefits derived from dynamic intensification of distillation column operations. Conceptually, this entails making a desired product as a blend of two auxiliary products (one with higher purity, the other with lower purity, but both having lower energy consumption). Practically, dynamic intensification means periodically switching between two operating states corresponding to the aforementioned products. Past work has relied on ad-hoc choices of auxiliary products. Here, we formulated identifying said products as an optimization problem. An extensive case study concerning the separation of an equimolar methanol/propanol mixture demonstrated that optimizing the choice of auxiliary products can lead to significant energy savings (more than 3.6% compared to a column operated at steady state) with minimal hardware additions.

The methods of the appended claims are not limited in scope by the specific methods described herein, which are intended as illustrations of a few aspects of the claims. Any methods that are functionally equivalent are intended to fall within the scope of the claims. Various modifications of the methods in addition to those shown and described herein are intended to fall within the scope of the appended claims. Further, while only certain representative method steps disclosed herein are specifically described, other combinations of the method steps also are intended to fall within the scope of the appended claims, even if not specifically recited. Thus, a combination of steps, elements, components, or constituents may be explicitly mentioned herein or less, however, other combinations of steps, elements, components, and constituents are included, even though not explicitly stated.

The term“comprising” and variations thereof as used herein is used synonymously with the term“including” and variations thereof and are open, non-limiting terms. Although the terms“comprising” and“including” have been used herein to describe various embodiments, the terms“consisting essentially of’ and“consisting of’ can be used in place of“comprising” and“including” to provide for more specific embodiments of the invention and are also disclosed. Other than where noted, all numbers expressing geometries, dimensions, and so forth used in the specification and claims are to be understood at the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claims, to be construed in light of the number of significant digits and ordinary rounding approaches.

Unless defined otherwise, all technical and scientific terms used herein have the same meanings as commonly understood by one of skill in the art to which the disclosed invention belongs. Publications cited herein and the materials for which they are cited are specifically incorporated by reference.

Claims

WHAT IS CLAIMED IS:

1. A method for obtaining a target product having a target purity from a mixture via a distillative process performed in a distillation apparatus, the method comprising:

(i) operating the distillation apparatus under a first set of conditions to obtain a first auxiliary product from the mixture, wherein the first auxiliary product has a purity higher than the target purity;

(ii) operating the distillation apparatus under a second set of conditions to obtain a second auxiliary product from the mixture, wherein the second auxiliary product has a purity lower than the target purity; and

(iii) combining the first auxiliary product and the second auxiliary product to afford the target product having the target purity.

2. The method of claim 1 , wherein the total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions is lower that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

3. The method of claim 2, wherein the total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions is at least 0.5% less than that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

4. The method of claim 3, wherein the total energy utilized to operate the distillation apparatus under the first set of conditions and the second set of conditions is from 1% to 10% less (e.g., from 1% to 5% less) than that the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

5. The method of any of claims 1-4, wherein the method comprises periodically switching operation of the distillation apparatus between the first set of conditions and the second set of conditions.

6. The method of claim 5, wherein periodically switching operation of the distillation apparatus between the first set of conditions and the second set of conditions comprises cycling between the first set of conditions and the second set of conditions at least five times, at least 25 times, at least 50 times, or at least 100 times.

7. The method of any of claims 1-6, wherein step (i) is performed prior to step (ii), wherein step (ii) is performed prior to step (i), or a combination thereof.

8. The method of any of claims 1-7, wherein step (iii) comprises collecting the first auxiliary product and the second auxiliary product in a mixing tank.

9. The method of any of claims 1-8, the first auxiliary product and the second auxiliary are combined at a split ratio (a) which satisfies the following:

a(ydi) + (1 - a) (y_d2) = y_d

where y_d represents the target purity, y_di represents the purity of the first auxiliary product, and y_d2 represents the purity of the second auxiliary product.

10. The method of any of claims 1-9, the first auxiliary product and the second auxiliary are combined at a split ratio (a) which satisfies the following:

a(PFRi) + (1 - a) (PFR₂) = PFR_d

where PFR_d represents the target product flow rate when the distillation apparatus is operated at a steady-state to obtain the target product from the mixture, PFRi represents the first auxiliary product flow rate when the distillation apparatus is operated under the first set of conditions, and PFR₂ represents the second auxiliary product flow rate when the distillation apparatus is operated under the second set of conditions.

11. The method of any of claims 9-10, wherein step (i) and step (ii) are performed for relative lengths of time which satisfy the following:

«(Time i ) + (1 - a) (Time₂) = Total Operation Time where a is the split ratio, Timei comprises a total time during which the distillation apparatus is operated under the first set of conditions, Time₂ comprises a total time during which the distillation apparatus is operated under the second set of conditions, and Total Operation Time comprises a total time during which the distillation apparatus is operated under the first set of conditions and the second set of conditions.

12. The method of any of claims 1-11, further comprising selecting the first auxiliary product and the second auxiliary product to achieve a desired energy savings relative to the energy required to operate the distillation apparatus at a steady-state to obtain the target product from the mixture.

13. The method of any of claims 1-12, wherein switching between the first set of conditions and the second set of conditions comprises altering column pressure (P), feed pressure (P_F), boilup ratio (B ratio), reflux rate (Reflux), distillate flow rate, bottoms flow rate, or a combination thereof in the distillation apparatus.

14. The method of claim 13, wherein switching between the first set of conditions and the second set of conditions comprises altering the column pressure (P), and wherein altering the column pressure (P) comprises adjusting coolant flow rate.

15. The method of any of claims 13-14, wherein switching between the first set of conditions and the second set of conditions comprises altering the feed pressure (PF), and wherein altering the feed pressure (PF) comprises adjusting a feed valve controlling the feed pressure.

16. The method of any of claims 13-15, wherein switching between the first set of conditions and the second set of conditions comprises altering the boilup ratio (B ratio), and wherein altering the boilup ratio (B ratio) comprises adjusting a steam flow rate to a reboiler.

17. The method of any of claims 13-16, wherein switching between the first set of conditions and the second set of conditions comprises altering the reflux rate (Reflux), and wherein altering the reflux rate (Reflux) comprises adjusting a reflux valve controlling the reflux flow rate.

18. The method of any of claims 1-17, wherein the distillation apparatus comprises a distillation column.

19. The method of claim 18, wherein the first auxiliary product and the second auxiliary product comprise distillate products from the distillation column.

20. The method of claim 18, wherein the first auxiliary product and the second auxiliary product comprise bottoms products from the distillation column.

21. The method of any of claims 1-20, further comprising obtaining a second target product from the distillation apparatus.

22. A method for obtaining a target product having a target purity from a mixture via a distillative process, the method comprising:

(i) operating a first distillation apparatus under a first set of conditions to obtain a first auxiliary product from the mixture, wherein the first auxiliary product has a purity higher than the target purity;

(ii) operating a second distillation apparatus under a second set of conditions to obtain a second auxiliary product from the mixture, wherein the second auxiliary product has a purity lower than the target purity; and

(iii) combining the first auxiliary product and the second auxiliary product to afford the target product having the target purity.

23. The method of claim 22, wherein the first distillation apparatus comprises a first distillation column and the second distillation apparatus comprises a second distillation column.

24. The method of any of claims 22-23, wherein the total energy utilized to operate the first distillation apparatus under the first set of conditions and the second distillation apparatus under the second set of conditions is lower that the energy required to operate a single distillation apparatus at a steady-state to obtain the target product from the mixture.