Experimental Simulation of a Two-Dimensional Attainable Region and

Article pubs.acs.org/IECR

Experimental Simulation of a Two-Dimensional Attainable Region and Its Application in the Optimization of Production Rate and Process Time of an Adiabatic Batch Reactor Nana Asiedu,* Diane Hildebrandt, and David Glasser Centre of Materials and Process Synthesis, School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, Gauteng, South Africa ABSTRACT: Attainable region analysis is a graphical optimization technique which enables design engineers to synthesize optimal reactor networks to achieve a given objective. This is accomplished if a given system of reactions with known reaction kinetic models and feedstock is available. This paper shows experimentally how the geometry of a two-dimensional attainable region can be generated by reaction and mixing as the only fundamental processes occurring in the reactor of a hydrolysis process without taking the reaction kinetic model of the process into consideration. The experiment was conducted using an adiabatic batch reactor fitted with a thermistor for temperature measurement. The results were used to develop a novel technique to run the adiabatic batch reactor by applying the attainable region analysis. From the analysis the production rate of the product and the process time of the adiabatic batch reactor were optimized by graphical techniques, and it is shown that the maximum production rate increased by 72% as compared with the standard method of operating the batch reactor and the corresponding minimized process time was reduced to 0.36 h compared with 1.26 h of the standard batch operation. It is also shown that the limit of the optimization process generated the optimal process configuration required for a continuous operation to achieve the highest production rate in a minimum time possible.

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INTRODUCTION Batch reactors play a very important role in the chemical process industries. Batch reactors are receiving great attention worldwide following the advent of generic drugs and high volume demand for specialty chemicals and polymers.1 These process are characterized not only by small scale flexible production and high added value products but also by reaction systems that may be quite complex. For this reason there is the need for the design engineer to develop new techniques to improve the production rates in order for producers to meet demands. Studies with adiabatic batch reactors have been reported by several authors. Most of them have studied the influence of different initial reagent concentrations and started their reaction at 373 K in order to obtain the kinetic model. However, in this paper the kinetics of the reaction of the hydrolysis of acetic anhydride was studied using variation of temperature with time in an adiabatic batch reactor. Previous works on studies of the kinetics of acetic anhydride hydrolysis include, Glasser et al.,2 King and Glasser,3 and Shatynski et al.4 However, in this paper the emphasis is not necessarily on the kinetics of the hydrolysis of acetic anhydride but the profiles of the temperature- time curves has enable the authors to use the results to develop a new technique of operating the batch reactor to maximize its production rate and minimize the process time by using the attainable region (AR) theory, and also to propose a reactor system for the continuous operations by considering reaction and mixing as the only permitted fundamental processes occurring in the reactor. The initial work of AR focused on reaction systems and it was a brainchild of Horn.5 The AR concept was defined by Horn as the set of all possible outputs from all physically realizable reactors. Glasser and Hildebrandt6 approached the idea from a geometrical point. They considered the reactor as a system where the only processes © XXXX American Chemical Society

occurring are reaction and mixing. They introduced geometrical interpretations of these processes and derived a set of necessary conditions for the boundary of the AR. They also showed that when the AR is determined, optimization process becomes relatively simple provided the objective function is an algebraic function only of the systems variable. Once the AR is defined, a path between the feed point and any point in the AR can be found. This path could be a combination of reaction and mixing. This combination of different processes could in turn be interpreted as process structure (equipment). Hence by finding the AR, the reactor structure at points in the AR can be determined. There has been a fair amount work devoted to AR over the past decade. Most of the early works in AR research7 were based on geometric interpretation of the fundamental processes such as reaction and mixing and heat exchange. Subsequent work in AR research has relied on properties and results from the geometric interpretations which include publications by Hildebrandt and Glasser,8 Glasser, Hildebrandt, and Crow,9 Glasser and Hildebrandt,10 Fienberg and Hildebrandt,11 and Nicol.12 The AR technique also has been applied to many synthesis problems such as isothermal reactor network synthesis,13 nonisothermal reactor synthesis,14,15 optimal control problems,16 classical variation and dynamic problems,17 feasible products in separation problems,18 reaction, separation, and recycle problems,19 distillation with and without reaction,20,21 and batch distillation energy optimization.22 Received: March 20, 2014 Revised: May 21, 2014 Accepted: July 23, 2014

A

dx.doi.org/10.1021/ie501194c | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

Figure 1. Composition−time profile of a batch reactor.

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ATTAINABLE REGION (AR) THEORY

THEORETICAL DEVELOPMENTS For an adiabatic batch reactor the system model is made up of a set differential equations resulting from mass and energy balances referred only to the reaction mixture, because there is no heat transfer. The stoichiometry of the system understudy can be represented as

The AR as stated above is a technique used to synthesis processes and predicts optimal operating conditions and policies. Hildebrandt’s23 extension of the idea led to the following necessary conditions. The following needs to be performed in order to complete an AR analysis: • Choose the fundamental processes • Choose the state variables • Define process variables • Define the geometry of the process units • Construct the region • Interpret the boundary as process flow sheet • Determine the optimum of your objective Some necessary conditions for AR derived from the work of Hildebrandt23 can be summarized as follows: • The AR includes all feed(s) to the system • The AR is convex • No process vector points out of the AR boundary • No rate vector(s) in the complement of the AR when extended backward intersects the AR The procedure entails finding all possible achievable output from the process feed for a given system, using all combinations of the fundamental processes. The region is tested against the necessary conditions and recursively updated so that violated condition(s) is eliminated. The algorithm ceases when all conditions are satisfied, such that all process vectors along the boundary are either tangent, zero, or point inward. When the tangency criteria are satisfied, the complement of AR (ARC) is searched for the rate vectors pointing back into the ARc. If no rate vector exists, then the resulting ARc is considered to satisfy the necessary conditions.

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(CH3CO)2 O + H 2O ↔ 2CH3COOH ΔHrxn(298 K) = −52.16 kJ/mol

(1)

where the acetic anhydride is the limiting reactant and the water is in excess. For a constant-volume batch reactor: −rA = −

dC 1 dNA =− A V dt dt

(2)

The energy balance can be established as follows: Heat generated by the reaction = Heat absorbed by reactor content + Heat transferred through the reactor walls ( −ΔHrxn)( −rA )V dt + Q dt = mCP dT + UA(ΔT )dt

(3)

For an adiabatic system, no heat is transferred through the reactor walls. The heat emitted by the stirrer was assumed negligible, thus eq 3 becomes ( −ΔHrxn)( −rA )V dt = mCP dT + UA(ΔT )dt

(4)

By rearranging and integrating eq 4 becomes (T − T0) +

UA mCP

∫0

t

(T − T0)dt =

( −ΔHrxn) ·Δε mCP

(5)

From eq 5 it can also be shown that the energy balance equation for an adiabatic batch reduces to a linear form given by eq 6 T = T0 + ΔTadx

AIMS

(6)

where ΔTad is the adiabatic temperature rise. Theoretically ΔTad is by definition obtained when x = 1 with respect to the reagent of interest and the value can be computed beforehand. Eq 6 allows one to find conversion and or extent of reaction at any instant using only one measure of temperature. Then, concentrations of reagents and products can be found, as shown in eq 7.

The aims of this paper are to use experimental results generated from temperature−time measurements to simulate 2-D attainable region without considering the kinetic model and to show how the production rate and processing time of adiabatic batch and continuous reactors can be optimized. B


Industrial & Engineering Chemistry Research T = T0 + ΔTad·(1 − C)

Article

(7)

The LHS eq 5 was used as a correction term to adjust experimental data to suit adiabatic conditions. With eq 5, if it assumed that the heat of reaction is independent of temperature after correcting the experimental data to suit adiabatic conditions, the heat of reaction (ΔHrxn) of the process can then be determined using eq 8 ( −ΔHrxn) ·Δε = ΔTad mCP

(8)

where Δε is the extent of the reaction.

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BATCH REACTOR PROFILE AND PRODUCTION RATE DEFINITION The batch reactor can be defined mathematically by eq 9: dC = −rA(C), C(t = 0) = C0 (9) dt The state-space curve describing a batch reactor in the AR state-space is a trajectory. The fundamentals of differential algebra define the batch reactor trajectory as being directional. The unique nature of the rate−vector at any point means that the trajectories of batch reactor progresses in such a way that they do not cross each other. There exists one unique batch reactor trajectory for any given initial feed point. Figure 1 depicts a typical batch reactor trajectory in concentration−time space. The straight line AB has significant property. Let the slope of AB be (m) given by eq 10: t − t2 (m ) = 1 C1 − C2 (10)

Figure 2. Block diagram of experimental setup.

engineering supports data management. The data acquisition system called Clarity has the following part numbers: C50 Clarity Chromatography SW, single instrument, 3 × 55 Clarity Add-on instrument SW, 194 (INT9 quad channel A/D converter card). The hardware is INT9 PCI A/D 24 bit converters. The properties are as follows: Input signal range 100 mV−10 V, Acquisition frequency 10−100 Hz, Internal A/D converter (INT9-1 to 4 channel PCI A/D converter). All physically available analog inputs and outputs as well as virtual channel are all automatically monitored and the process values are stored. The process values are transmitted in such a way that the computer screen displays profiles of voltage−time curves. Data acquisition software was used to convert the compressed data form of the history file on the hard disk into text file format. The text files are converted to an Excel spreadsheet, and the data are then transported into MATLAB 2010a for analysis. The thermistor used in the experiments was a negative temperature coefficient with unknown thermistor constants. The calibrations involve the determination of the thermistor constants and establishment of the relationship between the thermistor’s resistance and temperature. Thermistor resistance (RTH) and temperature (T) in Kelvin was modeled using the empirical equation given developed by Considine25

Let the production rate of a component, acetic acid be P, given by the equation P = V·

C1 − C2 t1 − t 2

(11)

Combining eqs 10 and 11 gives eq 12 as the production rate in terms of reactor volume (V) and the slope of the line AB is given by P=

V ( − m)

⎛B ⎞ RTH = exp⎜ + C ⎟ ⎝T ⎠

(12)

Thus, from eq 12 for a constant volume batch reactor the production rate is inversely proportional slope AB as shown in eq 12. From line AB any concentration (C*) can be obtained by mixing with initial feed material (C0). The relationship between C* and C0 is given by the lever-arm rule as shown in eq 13 C* = αC0 + (1 − α)C1 ,

0≤α≤1

(14)

or B +C (15) T where the parameters (B) and (C) are the thermistor constants and were obtained from experimental calibration using warm water. The constants were determined by fitting the “best” leastsquares straight line plot of ln(RTH) against 1/T, giving the thermistor equation as 1773.8 ln(RTH) = − + 18.178 (16) T ln(RTH) =

(13)

where α is the mixing ratio, and line AB is a mixing line.

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EXPERIMENTAL DESCRIPTIONS The experimental setup as shown in Figure 2 was used in all the reactions studied. The adiabatic batch reactor used in the experiments is an 18/8 stainless steel thermos-flask of total volume of 500 mL equipped with a removable magnetic stirrer. The flask is provided with a negative temperature coefficient thermistor connected online with a data-logging system. The signal from the sensor (thermistor) is fed to a measuring and a control unit amplifier and a power interface. The acquisition units are connected to a data processor. A process control

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THE REACTOR (THERMOS-FLASK) CALIBRATION The reaction vessel used was an ordinary dewer thermos-flask with a removable screw cap lid. The flask has a total volume of 500 mL. The calibration involves the determination of the heat transfer coefficient of the flask and fitting the experimental data to the model described in eq 17: C



Article

Figure 3. Thermos-flask cooling curve at T0 = 361 K.

⎛ UA ⎞ T = TS + (T0 − TS) exp⎜ − t⎟ ⎝ mCP ⎠

(17)

In this experiment 400.00 g of distilled water at 361 K (T0) was injected into the reaction vessel and left the system temperature to fall over a period of time until the temperature−time profile reaches its asymptotic state or the steady-state temperature (TS). Figure 3 shows the temperature−time profile of the cooling process. From eq 17 the values of T0 and TS were obtained from Figure 3. Rearrangement of eq 17 gives ⎛ T − TS ⎞ ⎛ UA ⎞ t⎟ ln⎜ ⎟ = ln(β) = ⎜ − ⎝ mCP ⎠ ⎝ T0 − TS ⎠

(18)

Since T and t values are known least-squares regression analysis was performed, and a straight plot of ln(β) against time is shown in Figure 4. The slope of the straight line of Figure 4 is given by 0.0013 s−1 which corresponds to the value of the heat transfer coefficient of the flask.

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Figure 4. Regression line of heat transfer coefficient of reactor.

ESTIMATION OF UA/mCP TERM IN EQ 5 Experimental data profile slopes gradually downward after it has gone through its maximum temperature. This is as a result of cooling since the system is not perfectly adiabatic. Considering any close two data points on the downward (cooling) slope on the experimental curve, X(T1, t1) and Y(T2, t2), the slope of the line XY is given by eq 19 dT UA Slope = = (Tav − Tamb) dt mCp

and Tamb is the ambient temperature. Thus, given the slope, Tav and Tamb

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(20)

EXPERIMENTAL PROCEDURES AND RESULTS Analytical reagent grade acetic anhydride and distilled water were used in all experiments. In the first set of experiments 1 mol of acetic anhydride was injected into the reaction vessel followed by 10 mol of distilled water. These quantities (volumes) were used so that at least 60% of the length of the sensor (thermistor) would be submerged in the resulting initial reacting mixture. In the course of the reaction, the stirrer speed was set at 1000 rev/

(19)

where

Tav =

UA was determined for any particular run mCp

T1 + T2 2 D



Article

Figure 5. a. Adiabatic temperature−time profile at initial temperature at 305 K. b Conversion−time profile.

min, the resulting voltage −time profiles were captured, and the corresponding temperature−time curve was determined using the thermistor equation derived above. Runs were carried out adiabatically at the following initial temperatures: 305, 310, and 328 K. In the second experiment the same procedure was used, but this time the molar ratio of acetic anhydride to distilled water was 1:1 with an initial temperature of 294 K.

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the profiles of conversion−time of experiments (A), (B), (C), and (D), respectively. It is seen that the profiles depict the typical adiabatic batch reactor profiles as is expected in the solution of eq 9. Corrections were made to adjust the experimental data to make the system adiabatic as explained using eq 5. A summary of the characteristics of the reaction profiles are shown in Tables 1 and 2. The figures are as stated by Love,24 these trajectories are like Plug-Flow Reactor trajectories, and they are unique and directional. Thus, there exists a unique batch reactor trajectory for any given initial feed point. It can be seen that the profiles show concavities, but the concavities diminish with increasing initial temperature of the feed material. This is due to a high

RESULTS AND DISCUSSIONS

Figures 5a, 6a, 7a, and 8a show experimentally recorded temperature versus time curves obtained for experiments (A), (B), (C), and (D), respectively. Figures 5b, 6b, 7b, and 8b show E



Article

Figure 6. a. Adiabatic temperature−time profile initial temperature at 310 K. b. Conversion−time profile.

dimensional attainable region and use it to optimize process time and production rate using an adiabatic batch reactor. Experimental profiles of experiment (D) appear better since they show pronounced concavities and will be used for the graphical optimization.

initial reaction rate at the start of the process. The evolution of concavities thus gives room for the application of the attainable region concept for optimization of operating parameters of reactors.

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ATTAINABLE REGION (AR) AND THE OPTIMIZATION TECHNIQUE Optimization of batch and continuous reactors operations are considered among other parameters including process time and production rates. In batch and continuous reactors operations minimizing process time and maximizing production rate simultaneously has always been a challenge for process engineers. By the attainable region theory, optimization of these parameters (process time and production rate) is made possible by graphical means by using real experimental results. Discussions of optimization using the AR have always been theoretical in the past. Hence this paper is attempting to generate a two-

ILLUSTRATING THE ATTAINABLE REGION OPTIMIZATION TECHNIQUE USING THE EXPERIMENTAL RESULTS OF PROCESS (D) Optimization of batch reactor operations is considered among other parameters including process times and production rates. In batch reactor operations minimizing process time and maximizing production rate simultaneously has always been a challenge since the current methods and techniques have not been able to handle parameters concurrently. By the attainable region theory, optimization of these parameters (process time and production rate) is made possible by graphical techniques. F



Article

Figure 7. a. Adiabatic temperature−time profile at initial temperature at 328 K. b. Conversion−time profile.

Considering a feed concentration of 1 at time t = 0, we can see from Figure 10 the batch trajectory on the concentration−time profile. This trajectory is concave at some part of the trajectory. This concavity can be eliminated by mixing with a feed material (A) with coordinate A(1,0) and concentration B with coordinate B(0.53, 4545), where the straight line AB is a tangent or a mixing line to the curve as shown in Figure 10. The reaction vector in this case is a function of concentration and temperature. The slope of the line AB in Figure 10 is (−9670 s/M). From eq 11 or 12, for a unit volume of the reaction mixture, the production rate of the product will be 0.000100 mol/s. We can see that by a simple downward translation of the curve the reaction vector will vary along the line AB. It can also be seen that there is a part of the line over which the reaction vector points outward. This implied that we have not yet found the lower boundary of the attainable region. By allowing a bypass and mixing from any point along the trajectory with another point

along the trajectory we obtain the attainable region for a single batch reactor trajectory, and the boundary is given by ADBC. The line AB fills the concavity of the trajectory. It can be seen that we can operate another batch reactor such that the boundary of the attainable region is lowered as much as possible. The batch trajectory has a fixed shape, thus the trajectory from feed point (A) is the same as the initial trajectory ADBC. Since temperature is proportional to concentration in an adiabatic system, the temperature at point (X) is the same as point (D) (see Figure 11). Hence shifting the initial trajectory we still fulfill the adiabatic relationship that exists between temperature and concentration. Thus, when the initial trajectory (AXBC) is translated downward it thus extends the region. The trajectory of the second batch is given by DTKC. The second batch operating from point (D) has a coordinate of (0.85, 1750), thus the second batch operation will have to have an initial mixture of concentration 0.85 M consisting of feed material (A) G



Article

Figure 8. a. Adiabatic temperature−time profile at initial temperature at 294 K. b. Conversion−time profile.

production rate will be given by 0.000225 mol/s which is much higher than the previous production rate by a factor of 2.25. The percentage change in the production rate is about 56%. The third reactor operates along the curve PSNC as shown in Figure 12, and the boundary of the attainable region for the three-stage attainable region system would be given by the line AS. The third reactor will start operating from point P(0.80,1000), thus the required mixture to start the third reactor should have a concentration of 0.8 M consisting of the feed material (A) and point T (0.55, 2000); hence from eq 13 the mixing policy required to start the third reactor is α = 0.56. It can be seen that the change in the boundary of the attainable between two and three stages is much smaller than between one and two stages. Each additional stage would give a much limited region until in the limit we reach the lowest part of the boundary of the whole attainable region-ASKC. The slope of the line at point S (0.53,1300) and the feed point (A) represents the minimum slope achievable (−2766 s/M) which also corresponds to the maximum production rate achievable (0.00036 mol/s) in the shortest possible process time of about 1300 s. The production

Table 1. Summary of the Characteristics of the Figures of Processes A, B, and C process

T0 (K)

Tmax (K)

ΔTad (K)

Xmax

ΔHrxn (kJ/mol)

A B C

305 310 328

351 356 366

52.68 46.57 38.03

0.95 0.83 0.68

−53.32 −52.94 −52.77

Table 2. Summary of the Characteristics of the Figure of Process D process

T0(K)

Tmax (K)

ΔTad (K)

Xmax

Cf (M)

ΔHrxn(kJ/mol)

D

294

385

91.68

0.47

0.53

−51.69

and material (B), and by applying the lever-arm rule (mixing rule) the second batch will have mixing policy of α = 0.67. We can then mix any point that can be reached by the system with any other point on line AB to obtain a two-reactor attainable region with boundary given by ATKC. At point T, the coordinate is given by T(0.55,2000), thus the slope AT is given by (−4444 s/ M); hence from eq 11 or 12 with a unit reactor volume, the H



Article

Figure 9. Concentration−time profile.

Figure 10. Stage 1: standard batch reactor profile of the process: 1-reactor AR.

the mixing line from the feed is a tangent to a CFSTR locus and represents the conditions in the system in which the rate of production is maximized. The point (S) is structurally shown in Figure 13. From point (S) a PFR is operated giving the trajectory (SKC) (see Figure 14). This then results in a region defined by the ASKC which satisfies all the necessary conditions of the attainable region, and it is represented in terms of process structure as shown in Figure 14. The concavity of the initial trajectory is eliminated completely at the limit of the operation by the mixing line AS. This is represented structurally as a CFSTR with a bypass as shown in Figure 15. The attainable region is in fact the region above the boundary as any process time larger than the minimum is possible.

rate is much higher at this stage than the second stage by 38%. The boundary ASKC of this attainable region satisfies all the necessary conditions, and it is therefore the attainable region that could be reached by considering the fundamental processes of reaction and mixing only. It can be seen that the overall change in the production rate has increased considerably to 72%. This is not likely to be achieved by the conventional trial-and-error methods within the minimum process time as shown in the optimization method described above.

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INTERPRETING THE BOUNDARY STRUCTURE IN TERMS OF CONTINUOUS PROCESSES It can also be seen that in the final boundary structure defined by the profile ASKC we can represent the optimal process configuration for a continuous process which in most cases is used in real industrial processes. The point (S) is the point where I



Article

Figure 11. Stage 2: generation of 2-reactor AR.

Figure 12. Stage 3: generation of 3-reactor AR.

Figure 14. CFSTR operating at point (S) followed by a PFR.

Figure 13. CFSTR operating at point (S).

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CONCLUSIONS We have by means of experiments simulated the attainable region in concentration−time space for an adiabatic batch reactor without considering the kinetic model of the hydrolysis process. We have also used the geometric ideas to develop a novel

The optimal reactor structure which will enable one to obtain all points on the surface of the boundary of the attainable region as shown in Figure 16. J



Article

Figure 15. CFSTR with a bypass operating between point (A) and point (S).

Figure 16. Optimal reactor structure.

technique to operate the batch reactor and also to minimize the reactors process time for any desired production rate. This is an indication of the practical usefulness of the attainable region theory in solving two-dimensional problems. Thus, this technique can be applied to any reaction scheme with unknown kinetics as opposed to the rather inconclusive methods of most classical reactor texts. The attainable region theory has revealed that upper and lower limits exist for the optimization of the batch operations. It is important to realize that the optimum continuous structure for the process was obtained directly from the boundary of the experimentally simulated attainable region, rather than relying on the ingenuity of the process synthesis approach. The traditional optimization techniques could be used to find the optimum changeover point for a CFSRT followed by a PFR but could not obtain the optimum structure as shown in Figure 16 above and also proves the results could not be improved by another reactor structure. We also see the usefulness of the tangency conditions in the attainable region theory in the optimization process. The real power of the technique is that adiabatic batch reactor operations can be manipulated to achieve any desire maximum production rate with minimum process time. It should be noted that the work carried out only addresses a simple irreversible exothermic process, and it is the desire of the authors to explore other more complicated processes which will lead to higher dimensional problems.

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C1, C2 = concentrations at points 1 and 2 (mol/L) CAo = concentration of feed material (mol/L) CA* = desired concentration (mol/L) CSFTR = continuous flow stirred tank reactor CP = constant heat capacity of reaction mixture (kJ/mol·K) ΔHrxn = heat of reaction (kJ/mol) ms = slope m = mass of reaction mixture (kg) P = production rate (mol/s) PFR = plug-flow reactor t, t1, t2 = time (s) T = reactor temperature (K) Tf = final reactor temperature (K) T0 = basis temperature (K) Tav = average temperature (K) Tamb = ambient temperature (K) Tmax = maximum temperature (K) ΔTad = adiabatic temperature change (K) U = heat transfer coefficient (J/m2 s·K) V = reactor volume (m3) X = conversion Δε = extent of reaction (mol) α = scalar quantity (mixing policy)

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS N.A. would like to thank the National Research Fund (NRF) of South Africa, Directors of Centre of Material Processing and Synthesis (COMPS), University of Witwatersrand, Johannesburg, South Africa for financial support of this work.

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NOMENCLATURE AR = Attainable Region ARC = Candidate Attainable Region C = concentration (mol/L) Cf = final concentration (mol/L) C0 = initial concentration (mol/L) K



Article

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Experimental Simulation of a Two-Dimensional Attainable Region and

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