Cross-Reference to Related Applications

This application is a continuation-in-part of PCT/CA2021/051804 filed December 14, 2021 , which claims priority to European Patent Application 20215408.4, filed December 18, 2020, the entire contents of all of which are herein incorporated by reference.

Field

This application relates to the production of unsaturated isoolefin copolymers, for example butyl rubbers.

Background

Polymers containing repeating units derived from isoolefins are industrially prepared by carbocationic polymerization processes. Of particular importance is butyl rubber which is a copolymer of isobutylene and a smaller amount of a multiolefin such as isoprene.

The AICI3/H2O initiating system for butyl rubber suffers from variability in the activity of the catalyst. This is attributed to differences in the ratio of active species to inactive species, as aluminum trichloride (AICI3) is known to form aggregates with itself and with water, which generate inactive species that do not initiate polymerization. Variability in the number of active species changes the number of initiating sites in the polymerization reactor, and if increased suddenly without reducing the catalyst addition to the reactor, can result in low molecular weight product, localized temperature increases and fouling of the reactor. Similar issues with a reactor going cold and the reaction stopping can also occur if the number of active species decreases. Reducing the variability of the catalyst activity for the butyl rubber process can increase capacity by reducing fouling and other issues with the initiator system.

There remains a need for reducing variability of catalyst activity in a polymerization process to improve efficiency of a carbocationic polymerization process, especially in processes to produce unsaturated isoolefin copolymers such as butyl rubber. Summary

A process for producing an unsaturated isoolefin copolymer, the process comprising: sonicating a solution of an initiator system in an organic solvent, the initiator system comprising a Lewis acid catalyst, a proton source and a tertiary ether, to produce a sonicated initiator solution, the sonicating performed at a sonication energy input in J/mL that is based on volume of the initiator solution; and then, contacting the sonicated initiator solution with a reaction mixture of at least one isoolefin monomer and at least one copolymerizable unsaturated monomer in an organic diluent to produce the unsaturated isoolefin copolymer; wherein the combination of sonication energy input and molar ratio of tertiary ether to Lewis acid catalyst satisfies Equation 1 :

4.01 < (0.587342) + (64.19198 x Molar Ratio Tertiary Ether) + (0.010807 x Sonication Input Energy) + (0.094975 x Molar Ratio Tertiary Etherx Sonication Input Energy) - (489.22317 x Molar Ratio Tertiary Ether ² ) - (0.000042 x Sonication Input Energy ² )

In some embodiments, the sonication input energy is in a range of from 99 J/mL to 357 J/mL, preferably 229 J/mL.

In some embodiments, the molar ratio is in a range of from 0.05 to 0.125, preferably 0.09.

In some embodiments, the lower limit of sonication input energy “y” for a given molar ratio of tertiary ether “x” is defined by Equation 2: y= 0.0119048 (-0.0000000738315 <(-1.34228656456948 x 10 ²⁵ (x ² ) + 2.35495086848173 x 10 ²⁴ (x) - 8.40594272406205 x 10 ²² ) + 94975 (x) + 10807).

In some embodiments, the upper limit of sonication input energy “y” for a given molar ratio of tertiary ether “x” is defined by Equation 3: y = 0.0119048 (0.0000000738315 <(-1.34228656456948 x 10 ²⁵ (x ² ) + 2.35495086848173 x 10 ²⁴ (x) - 8.40594272406205 x 10 ²² ) + 94975 (x) + 10807).

Both the addition of certain tertiary ethers and sonication of the initiator solution improves catalyst activity, thereby improving conversion of the monomers during production of the unsaturated isoolefin copolymer. Variability of the catalyst activity is reduced with both methods of modification, thereby increasing overall polymerization reactor capacity, reducing reactor fouling and reducing other issues with the initiator system. The use of both a tertiary ether and sonication to improve the catalyst activity is more than additive, demonstrating a larger increase in yield in comparison to either modification alone.

A major benefit of increasing the catalyst activity is to shorten the overall reactor length (residence time) required to achieve a target monomer conversion value (e.g., 82-85 mol%), meaning either that increased flow rates can be achieved through the existing continuous reactors or improved process control can be achieved by ensuring that a consistently high level of monomer conversion near the target value is achieved. In practice, reactors are operated at the highest possible flow rate that can be pushed through the reactor to achieve the target monomer conversion, so having a more active initiator system ensures that the target conversion value is reached and substantially all reactants in the feed mixture have reacted.

Further features will be described or will become apparent in the course of the following detailed description. It should be understood that each feature described herein may be utilized in any combination with any one or more of the other described features, and that each feature does not necessarily rely on the presence of another feature except where evident to one of skill in the art.

Brief Description of the Drawings

For clearer understanding, preferred embodiments will now be described in detail by way of example, with reference to the accompanying drawings, in which:

Fig. 1 is a graph of different catalyst conditions vs. yield of polymer (butyl rubber) in grams, showing the effect of sonication energy input, MTBE and both sonication and MTBE on an initiator system in the copolymerization of isobutene with isoprene to produce butyl rubber.

Fig. 2 shows the statistically analyzed results of a Design of Experiments, showing the delta yield of butyl rubber using a sonicated initiating system with no MTBE, maximizing at +1.32 g when the sonication dosing is 129 J/mL.

Fig. 3 shows the statistically analyzed results of a design of experiments, showing the delta yield of butyl rubber using an MTBE containing initiating system with no sonication of the initiator, maximizing at +2.69 g when the MTBE loading is 0.062 molar ratio to AICI3. Fig. 4 shows the statistically analyzed results of a design of experiments, showing the delta yield of butyl rubber using an MTBE containing and sonicated initiating system, maximizing at +4.63 g when the MTBE loading is 0.09 molar ratio to AlCh and the sonication loading is 229 J/mL.

Fig. 5 shows the statistically analyzed results of a design of experiments, providing a contour plot of the three dimensional region where synergistic combinations of sonication input energy and MTBE molar ratio occur.

Fig. 6 is a simplified version of Fig. 5, showing a two dimensional area bounded by an ellipse where synergistic combinations of sonication input energy and MTBE molar ratio occur.

Fig. 7 is similar to Fig. 2, where delta yield is expressed in terms of a percentage.

Fig. 8 is similar to Fig. 3, where delta yield is expressed in terms of a percentage.

Fig. 9 is similar to Fig. 4, where delta yield is expressed in terms of a percentage.

Fig. 10 is similar to Fig. 5, where delta yield is expressed in terms of a percentage.

Fig. 11 is similar to Fig. 6, where delta yield is expressed in terms of a percentage.

Fig. 12 shows an ellipsoidal surface where a synergistic relationship between sonication input energy and MTBE molar ratio occurs, expressed in terms of delta yield (g).

Fig. 13 is similar to Fig. 12, where delta yield is expressed in terms of a percentage

Detailed Description

Production of the unsaturated isoolefin copolymer involves polymerizing at least one isoolefin monomer and at least one copolymerizable unsaturated monomer in an organic diluent in the presence of an initiator system (a Lewis acid catalyst, a tertiary ether and a proton source) capable of initiating the polymerization process. Polymerization occurs in a polymerization reactor. Suitable polymerization reactors include, for example, flow-through polymerization reactors, plug flow reactor, moving belt or drum reactors, and the like. The process may be a continuous or batch process. In a preferred embodiment, the process is a continuous polymerization process. The process may comprise slurry or solution polymerization of the monomers. In a preferred embodiment, the process is a slurry polymerization process. The unsaturated isoolefin copolymer comprises repeating units derived from at least one isoolefin monomer and repeating units derived from at least one copolymerizable unsaturated monomer, and optionally repeating units derived from one or more further copolymerizable monomers. The unsaturated isoolefin copolymer preferably comprises an unsaturated isoolefin copolymer.

Suitable isoolefin monomers include hydrocarbon monomers having 4 to 16 carbon atoms. In one embodiment, the isoolefin monomers have from 4 to 7 carbon atoms. Examples of suitable isoolefins include isobutene (isobutylene), 2-methyl-1 -butene, 3-methyl-1-butene, 2-methyl-2- butene, 4-methyl-1-pentene, 4-methyl-1 -pentene and mixtures thereof. A preferred isoolefin monomer is isobutene (isobutylene).

Suitable copolymerizable unsaturated monomers include multiolefins, p-methyl styrene, P-pinene or mixtures thereof. Multiolefin monomers include hydrocarbon monomers having 4 to 14 carbon atoms. In some embodiments, the multiolefin monomers are conjugated dienes. Examples of suitable conjugated diene monomers include isoprene, butadiene, 2- methylbutadiene, 2,4-dimethylbutadiene, piperylene, 3-methyl-1 ,3-pentadiene, 2,4-hexadiene, 2- neopentylbutadiene, 2-methyl-1 ,5-hexadiene, 2,5-dimethyl-2,4-hexadiene, 2-methyl-1 ,4- pentadiene, 4-butyl-1 ,3-pentadiene, 2,3-dimethyl-1 ,3-pentadiene, 2,3-dibutyl-1 ,3-pentadiene, 2- ethyl-1 ,3-pentadiene, 2-ethyl-1 ,3-butadiene, 2-methyl-1 ,6-heptadiene, cyclopentadiene, methylcyclopentadiene, cyclohexadiene, 1-vinyl-cyclohexadiene and mixtures thereof. A preferred copolymerizable unsaturated monomer is isoprene.

The unsaturated isoolefin copolymer may optionally include one or more additional copolymerizable monomers. Suitable additional copolymerizable monomers include, for example, styrenic monomers, such as alkyl-substituted vinyl aromatic co-monomers, including but not limited to a C1-C4 alkyl substituted styrene. Specific examples of additional copolymerizable monomers include, for example, a-methyl styrene, p-methyl styrene, chlorostyrene, cyclopentadiene and methylcyclopentadiene. Indene and other styrene derivatives may also be used. In one embodiment, the halogenatable isoolefin copolymer may comprise random copolymers of isobutene, isoprene and p-methyl styrene.

The unsaturated isoolefin copolymer is formed by copolymerization of a monomer mixture. Preferably, the monomer mixture comprises about 80-99.9 mol% of at least one isoolefin monomer and about 0.1-20 mol% of at least one copolymerizable unsaturated monomer, based on the monomers in the monomer mixture. More preferably, the monomer mixture comprises about 90-99.9 mol% of at least one isoolefin monomer and about 0.1-10 mol% of at least one copolymerizable unsaturated monomer. In one embodiment, the monomer mixture comprises about 92.5-97.5 mol% of at least one isoolefin monomer and about 2.5-7.5 mol% of at least one copolymerizable unsaturated monomer. In another embodiment, the monomer mixture comprises about 97.4-95 mol% of at least one isoolefin monomer and about 2.6-5 mol% of at least one copolymerizable unsaturated monomer.

If the monomer mixture comprises the additional copolymerizable monomer with the isoolefins and/or copolymerizable unsaturated monomers, the additional copolymerizable monomer preferably replaces a portion of the copolymerizable unsaturated monomer. When a multiolefin monomer is used, the monomer mixture may also comprise from 0.01 % to 1 % by weight of at least one multiolefin cross-linking agent, and when the multiolefin cross-linking agent is present, the amount of multiolefin monomer is reduced correspondingly.

Suitable organic diluents may include, for example, alkanes, chloroalkanes, cycloalkanes, aromatics, hydrofluorocarbons (HFC) or any mixture thereof. Chloroalkanes may include, for example methyl chloride, dichloromethane or any mixture thereof. Methyl chloride is particularly preferred. Alkanes and cycloalkanes may include, for example, isopentane, cyclopentane, 2,2- dimethylbutane, 2,3-dimethylbutane, 2-methylpentane, 3-methylpentane, n-hexane, methylcyclopentane, 2,2-dimethylpentane or any mixture thereof. Alkanes and cycloalkanes are preferably C6 solvents, which include n-hexane or hexane isomers, such as 2-methyl pentane or 3-methyl pentane, or mixtures of n-hexane and such isomers as well as cyclohexane. The monomers are generally polymerized cationically in the diluent at temperatures in a range of from -120°C to +20°C, preferably -100°C to -50°C, more preferably -95°C to -65°C. The temperature is preferably about -80°C or colder.

The initiator system comprises a Lewis acid catalyst, a tertiary ether and a proton source. The catalyst preferably comprises aluminum trichloride (AlCh). Alkyl aluminum halide catalysts are also useful for catalyzing the polymerization reaction. Examples of alkyl aluminum halide catalysts include methyl aluminum dibromide, methyl aluminum dichloride, ethyl aluminum dibromide, ethyl aluminum dichloride, butyl aluminum dibromide, butyl aluminum dichloride, dimethyl aluminum bromide, dimethyl aluminum chloride, diethyl aluminum bromide, diethyl aluminum chloride, dibutyl aluminum bromide, dibutyl aluminum chloride, methyl aluminum sesquibromide, methyl aluminum sesquichloride, ethyl aluminum sesquibromide, ethyl aluminum sesquichloride and any mixture thereof. Preferred of alkyl aluminum halide catalysts are diethyl aluminum chloride (Et2AICI or DEAC), ethyl aluminum sesquichloride (Eti sAIC s or EASC), ethyl aluminum dichloride (EtAICh or EADC), diethyl aluminum bromide (Et2AIBr or DEAB), ethyl aluminum sesquibromide (Eti.sAIBn.s or EASB) and ethyl aluminum dibromide (EtAIBr2 or EADB) and any mixture thereof. A particularly preferred alkyl aluminum halide catalyst comprises ethyl aluminum sesquichloride, preferably generated by mixing equimolar amounts of diethyl aluminum chloride and ethyl aluminum dichloride, preferably in a diluent. The diluent is preferably the same one used to perform the copolymerization reaction.

The tertiary ether preferably comprising ethers bearing at least one tertiary alkyl or tertiary arylalkyl group, preferably only one tertiary alkyl or tertiary arylalkylaryl group at the ether oxygen. Preferred tertiary alkyl groups include tert.-butyl, and tert.-amyl. Preferred tertiary arylalkylaryl groups include cumyl. Examples of tertiary alkyl ethers include methyl tert.-butyl ether (MTBE), ethyl tert.-butyl ether (ETBE), methyl tert.-amyl ether (MTAE) and phenyl tert.-butyl ether (PTBE) or mixtures thereof, whereby methyl tert.-butyl ether (MTBE), ethyl tert.-butyl ether (ETBE), methyl tert.-amyl ether (MTAE) or mixtures thereof are preferred and whereby methyl tert.-butyl ether (MTBE) is even more preferred. A preferred molar ratio of tertiary ether to aluminum employed is generally from 0.05 to 0.125, more preferably from 0.07 to 0.10, yet more preferably about 0.09.

The proton source includes any compound that will produce a proton when added to the catalyst or a composition containing the catalyst. Protons are generated from the reaction of the catalyst with proton sources to produce the proton and a corresponding by-product. Proton sources include, for example, water (H2O), alcohols, phenols, thiols, carboxylic acids, and the like or any mixture thereof. Water, alcohol, phenol or any mixture thereof is preferred. The most preferred proton source is water. A preferred ratio of catalyst to proton source is from 5:1 to 100:1 by weight, or from 5:1 to 50:1 by weight. The initiator system is preferably present in the reaction mixture in an amount providing 0.0007-0.02 wt% of the catalyst, more preferably 0.001-0.008 wt% of the catalyst, based on total weight of the reaction mixture.

The initiator system is dissolved in an organic solvent to produce an initiator solution, which is then contacted with the reaction mixture to initiate polymerization of the monomers. The organic solvent may comprise any of the organic diluents described above. Preferably, the organic solvent comprises a polar organic solvent. Methyl chloride is particularly preferred. The catalyst is preferably present in the initiator solution at a concentration of 0.01 wt% to 0.6 wt%, based on total weight of the initiator solution, more preferably 0.025 wt% to 0.6 wt%, 0.05 wt% to 0.5 wt% or 0.075 wt% to 0.4 wt%. The initiator system is preferably soluble in the reaction mixture.

The initiator system including the Lewis acid catalyst, tertiary ether and the proton source is preferably present in the reaction mixture in an amount of 0.002 to 5.0 wt.-%, preferably of 0.01 to 0.2 wt.-%, based on the weight of the monomers employed.

In another embodiment, in particular where aluminum trichloride is employed the wt.-ratio of monomers employed to aluminum, in particular aluminum trichloride is within a range of 500 to 20,000, preferably 800 to 10,000.

To improve conversion of monomers to polymer thereby increasing efficiency of the polymerization reaction, the initiator solution is sonicated prior to contacting the initiator solution with the reaction mixture. Sonication of the initiator solution improves catalyst activity thereby improving conversion of the at least one isoolefin monomer, the at least one copolymerizable unsaturated monomer or both during production of the unsaturated isoolefin copolymer. In particular, improved conversions are achieved when energy input from sonication is 98 J/mL or greater, based on volume of the initiator solution. Preferably, the energy input from sonication is in a range of 99 J/mL to 357 J/mL, 100 J/mL to 350 J/mL, 150 J/mL to 300 J/mL, 200 J/mL to 250 J/mL, or about 229 J/mL. Sonication is performed for a sufficient amount of time to improve catalyst activity. Preferably, the initiator solution is sonicated for 0.5 minutes or more, or 1 minute or more, or 0.5-30 minutes, or 1-30 minutes, or 1-20 minutes, or 1-10 minutes, or 0.5-10 minutes, or 0.5-20 minutes.

Sonication has been found to have no deleterious effects on the initiator system, and no negative impact on the molecular weight of the unsaturated isoolefin copolymer at various contents of the at least one copolymerizable unsaturated monomer. Sonication further permits dissolving the catalyst in the organic solvent at higher concentrations than is possible using standard stirring techniques. Sonication further permits dissolving the catalyst in the organic solvent at lower temperatures (e.g. -80°C or colder) than is possible using standard stirring techniques.

Sonication applies sound energy to agitate particles. Because ultrasonic frequencies (>20 kHz) are usually used, sonication is also known as ultrasonication or ultra-sonication. Sonicators are generally well known and any suitably powerful sonicator may be used to sonicate the initiator solution. The power of the sonicator and the amplitude of sound waves generated by the sonicator can be suitably selected to provide energy input in the ranges described above and a sonication time that is suitably short while obtaining the desired monomer conversion. If lower amplitude is desires, a longer sonication time may be used, while sonication time may be reduced by using higher amplitudes of the sound waves.

After the polymerization is complete, the unsaturated isoolefin copolymer may be recovered from the reaction mixture by known methods. For example, the organic diluent, organic solvent and residual monomers may be separated from the unsaturated isoolefin copolymer by flash separation using a heated organic solvent or steam. The unsaturated isoolefin copolymer may then be dried and processed into cements, crumbs, bales or the like for further use, storage or shipping.

EXAMPLES:

Initiator Solution Preparation

0.6 g of AICI3 (99.99% purity) was added to 200 mL of liquid methyl chloride (MeCI) at - 30°C in a 250 mL Erlenmeyer flask, all inside an M Braun™ glovebox filled with nitrogen and equipped with liquid nitrogen cooled pentane baths. The mixture was stirred at approximately 300 rpm using an overhead stirrer for 45 minutes. The solution was then cooled to -95°C and transferred to a 250 mL round bottom with a 45/50 joint. The solution contained a small amount of water as a proton source, the water being present as an impurity in the MeCI in an amount of about 15-50 ppmv.

To prepare tertiary ether containing initiator solution, the liquid ether was added to the initiator solution at -95°C using a micropipette to give the desired molar ratio of ether to aluminum.

To prepare sonicated initiator solutions, the initiator solution was prepared as described above and sonicated using a horn sonicator (QSonica™, 500 Watts, 20 KHz) for a desired period of time to give a desired dosing in J/mL and at a desired amplitude level (50% of full horn movement) to produce the sonicated initiator solution.

Polymerization Reaction

Initiator solutions were then used to prepare butyl rubber (isobutene-co-isoprene) by adding the initiator solution to a mixture of isobutene and isoprene in methyl chloride as follows. Methyl chloride (MeCI) and isobutene (IB) at -96°C and isoprene (IP) at room temperature were added to a reactor that was cooled to -96°C. The reaction mixture was then cooled to about -91°C with stirring at 800 rpm. Then, a desired volume of the initiator solution was added in a manner to provide good initiation without a high temperature increase of the reaction mixture.

During polymerization, the reaction was monitored using an immersion Raman spectrometer to measure conversion of isobutene.

The polymerization reaction was then quenched after 5 minutes by adding to the reaction mixture 1 mL of a solution of 1 wt% NaOH in ethanol. The reaction was terminated if the temperature of the reaction mixture increased by more than 20°C before the end of 5 minutes. The reactor was then removed from the glovebox and 1 mL of dilute antioxidant solution (1 wt% Irganox™ 1076 in hexanes) was added, along with further hexanes to dilute the reaction mixture. The methyl chloride was allowed to evaporate overnight to form a butyl rubber cement in hexanes. The butyl rubber was then coagulated from the hexane cement using ethanol and dried overnight at 60°C under vacuum.

Example 1: Effect of both MTBE and Sonication on total polymer yield

Polymerization reactions to produce butyl rubber were conducted as described above using a series of initiator solutions using 1 mL of initiator solution. Four initiator solutions were used in the series of polymerizations: control, sonication, MTBE, MTBE+sonication. Table 1 shows the results.

Table 1 Fig. 1 shows the delta yield, i.e. the increase in polymer yield as compared with the control, of each polymerization reaction. While the delta yield for the sonication only reaction was 2.1 g (65%), and for the MTBE only reaction was 3 g (94%). The expected additive effect of combining sonication and MTBE is 5.1g (159%); however, the experimentally determined delta yield for sonication + MTBE was 7.7 g (240%). This demonstrates that while an additive effect could be expected, the impact of the combination of sonication and MTBE on the initiating system produced a synergistic effect.

Example 2: Design of Experiments to determine synergistic range

A design of experiments (DoE) was generated using Design-Expert 11 (Stat-Ease, Minneapolis, MN, USA) to determine the range for the synergistic effect of both MTBE and sonication of the initiator solution on polymer yield. A central composite design was generated with sonication range of 0 to 410 J/mL and an MTBE molar ratio to aluminum between 0 and 0.104. Polymerization reactions to produce butyl rubber were conducted as described above using different initiator solution modifications, based on the changes listed in Table 2. Delta yield was used as the primary response.

Table 2

The control with no sonication and no MTBE produced a yield of 0.588 g. Fig. 2 shows that the delta yield for the sonicated-only scenario had a maximum increase in polymer yield of +1.32 g (224%) at 129 J/mL sonication dosing. Fig. 3 shows the delta yield for the MTBE-only scenario had a maximum increase in polymer yield of +2.69 g (457%) at a molar ratio of 0.062. Fig. 7 is similar to Fig. 2 and Fig. 8 is similar to Fig. 3, both depicting delta yield in percentage terms.

Therefore, if the delta yield increase is greater than 1.32 + 2.69 = 4.01 g (671%), the maximum additive delta yield of both improvements, the effect of MTBE and sonication is synergistic. Fig. 4 shows the delta yield, optimized for both sonication dosing and MTBE molar ratio, reached a maximum of 4.63 g (787%) at an MTBE ratio of 0.09 and a sonication dosing of 229 J/mL. The results are summarized in Table 3. Fig. 9 is similar to Fig. 4 and depicts delta yield in percentage terms.

Table 3

The region where the combination of MTBE molar ratio and sonication dosing where the delta yield would be synergistic was calculated using Design Expert 11. Fig. 5 is a contour plot showing a top view of a three dimensional ellipsoidal region shown in Fig. 12 that defines the surface where a synergistic relationship between sonication input energy and MTBE molar ratio occurs. Figs. 10 and 13 correspond to Figs. 5 and 12 and depict delta yield in percentage terms. Synergistic effects are expected from 99 to 357 J/mL for sonication dosing and from 0.05 to 0.125 molar ratio of MTBE to aluminum trichloride. Experimental results falling within the region depicted in Fig. 5 confirmed a delta yield of greater than 4.01 g, indicating a synergistic effect. The equation defining the region where synergistic effects are expected to occur, i.e. combinations of sonication input energy and MTBE molar ratio (molar ratio tertiary ether) that produce delta yield results on or above the ellipsoidal region, is:

Equation 1

4.01 < (0.587342) + (64.19198 x Molar Ratio Tertiary Ether) + (0.010807 x Sonication Input Energy) + (0.094975 x Molar Ratio Tertiary Etherx Sonication Input Energy) - (489.22317 x Molar Ratio Tertiary Ether ² ) - (0.000042 x Sonication Input Energy ² ) Thus, any combination of sonication input energy and MTBE molar ratio satisfying Equation 1 is a combination that produces a synergistic effect on delta yield.

Since the region where a synergistic effect occurs is a function of only two variables, the three dimensional contour plot of Fig. 5 can be simplified to a two dimensional plot that maps the limits of combinations of sonication input energy and MTBE molar ratio where a synergistic effect occurs. This two dimensional plot is shown in Fig. 6 in terms of delta yield in grams; Fig. 11 is similar and depicts delta yield in percentage terms. Taking values from Fig. 6, Table 4 was developed using increments of MTBE molar ratio (along the X axis) with corresponding upper and lower limits of sonication input energy (along the Y axis) that produce a synergistic effect on delta yield.

Table 4 The values from Table 4 can be used to develop equations for the lower limit and upper limit of sonication input energy, denoted as , in terms of MTBE molar ratio (molar ratio of tertiary ether), denoted as “x”. These are presented below as Equation 2 and Equation 3, respectively. Equation 2: Lower Limit of Sonication Input Energy (J/mL) y = 0.0119048 (-0.0000000738315 (-1.34228656456948 x 10 ²⁵ (x ² ) + 2.35495086848173 x 10 ²⁴ (x) - 8.40594272406205 x 10 ²² ) + 94975 (x) + 10807)

Equation 3: Upper Limit of Sonication Input Energy (J/mL) y = 0.0119048 (0.0000000738315 (-1.34228656456948 x 10 ²⁵ (x ² ) + 2.35495086848173 x 10 ²⁴ (x) - 8.40594272406205 x 10 ²² ) + 94975 (x) + 10807)

Thus, for any given molar ratio of tertiary ether, “x”, the lower and upper limits of sonication input energy “y” that will produce a synergistic effect on delta yield are defined by Equation 2 and Equation 3, respectively.