Modeling Real Flow in a Continuous Stirred Liquid–Liquid System

Article pubs.acs.org/IECR

Modeling Real Flow in a Continuous Stirred Liquid−Liquid System Anabela G. Nogueira,†,‡ Dulce C. M. Silva,‡ and Cristina M. S. G. Baptista*,† †

CIEPQPF-Department of Chemical Engineering, University of Coimbra, P-3030-790 Coimbra, Portugal CUF-Químicos Industriais, Quinta da Indústria, Beduído, 3860-680 Estarreja, Portugal

‡

S Supporting Information *

ABSTRACT: The behavior of flow in an industrial adiabatic reactor was studied for assessing nonuniform regions or bypassing. Mathematical models for a continuous stirred tank reactor were developed using adjustable parameters to be obtained from residence time distribution data. Owing to reaction the flow rates of the two liquid phases change. These models combine ideal vessels in series: a plug flow followed by one or two stirred tanks. The interchange of fluid between two stirred reactors was regarded, as well as a stagnant region in the middle reactor. The experimental study was carried out by injecting a tracer in the organic phase of the liquid−liquid system and quantifying it in the outlet stream. This procedure was followed for different flow rates and RTD data used to assess the best-fit combined model for the flow in this reactor which was a plug flow followed by two stirred tanks with interchange of fluid between them.

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being fed. This allows to calculate the mean residence time, tr̅ , using eq 1:16

INTRODUCTION Over 60 years elapsed since Danckwerts1 published his first work on residence time distribution (RTD) of fluid elements in a process vessel. It is still used for assessing deviations from ideal flow in chemical systems or to estimate process parameters. The major advantage of this technique is its usefulness for diagnosing stagnant zones and bypassing of the flow through a wide range of vessels, which are used in several processes such as crystallization,2 wastewater treatment,3−7 and chemical reactors.8 Deviations from ideal flow are often responsible for not achieving design targets. Nevertheless, once information on the real flow pattern is available, it may be used to adjust operating conditions and improve performance. Estimation of kinetic parameters can be achieved based on RTD data5,9 as well as mass transfer in liquid−liquid systems.10 Residence time analysis may allow evaluating transition conditions from mixed to plug flow in the riser of a circulating fluidized bed,11 or provide information to correlate circulation mass flows with measurable operational data in a chemical looping combustion process.12 Residence time and its distribution has been used to assess continuous flow in homogeneous and heterogeneous systems11−13 and in microreactors.14,15 While RTD data alone are insufficient to describe all possible flow combinations, once combined with the development of mathematical models it provides very useful information. These mathematical models take into account the different time periods the fluid elements have spent in the reactor and the mixing or exchange between these elements. In liquid−liquid reaction systems mixing is crucial for achieving a good dispersion, which determines both the mass transfer between phases and the extent of the reaction processes that take place in the reactor. In RTD studies, F(t) is referred to as cumulative age distribution and, by definition, it is the fraction of fluid elements in the outlet stream which spent in the system less than time t. When using a step input one wants to determine the fraction of the effluent flow that entered after the instant the tracer started © XXXX American Chemical Society

tr̅ =

∫0

∞

[1 − F(t )] dt

(1)

Comparison of the measured F(t) to the cumulative distribution for a perfectly mixed continuous stirred tank reactor (CSTR) will confirm ideal flow, or diagnose bypassing, or encounter dead zones or stagnant regions, but other nonuniformities will not be detected.17 Although RTD information is relevant, it is not sufficient to evaluate reactor behavior which also depends on how early or how late fluid mixing takes place.18 A practically meaningful mathematical model for the reactor is needed to improve performance, and this will include adjustable parameters correlated to flow conditions, which will allow a fit of experimental data. One-parameter models such as the dispersion model or the tanks-in-series model are mostly used to describe tubular reactors. Himmelblau and Bischoff17 state that one-adjustable parameter models do not provide a good representation for stirred tank reactors. These models can include only bypassing of fluid or dead zones.18 Two-parameter models that combine ideal reactors (plug flow and/or CSTR), in series or in parallel, allow to model the real reactor and to describe both bypassing and dead zones, with or without interchange. The adjustable parameters are the size of the reactors and the flow rates between them.19 Depending on RTD data, different combinations of several ideal reactors may need to be included, increasing the number of adjustable parameters. A compromise between accuracy, physical meaning, and mathematical complexity is to be reached for the sake of tractability. The experimental determination of RTD is achieved by injecting a known amount of a nonreactive tracer into the Received: September 1, 2015 Revised: December 10, 2015 Accepted: December 15, 2015

A

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research reactor inlet stream. In a stimulus-response approach20 a particular stimulus, usually a pulse or a step input of tracer is injected,9,21 and its concentration in the outlet stream is registered. Achieving a measurable tracer concentration when the vessel volume is large, as is the case in an industrial plant, requires injecting a large amount of tracer, which makes it difficult to achieve a pulse input. A step input is more feasible to run accurately, even if the test takes longer and in spite of difficulties to maintain a constant tracer concentration. The modeling of the flow in the industrial reactor will make use of RTD data. In a liquid−liquid reaction the flow rates of the two phases, here designated as organic and aqueous phases, may depend on reaction extent. Therefore, the density is not constant and the total volumetric flow rates, in and out of the reactor, are different and calculated based on a mass balance of the reactor. A tracer response analysis was used assuming that the tracer and the fluid that entered the continuous stirred reactor at the same time have the same RTD.22 In this study, included in ref 23, a preliminary analysis considered a stirred tank with ideal flow, also evaluating the presence of stagnant zones and/or bypass. A more complex description of the system combining ideal plug flow and continuous stirred reactors was then assessed. Interchange between two CSTRs and a stagnant region were alternatively included, and the resulting mathematical models were tested to establish the flow model of the industrial nitration reactor.

operating conditions were available. Relevant input for the study on residence time distribution were inlet flow rates of the organic and aqueous phases, QO,in and QAq,in. Later in this work the precision of the two flow meters was taken into account to set the bounds imposed on these parameters in mathematical model fitting. The tracer was pumped into the organic phase inlet stream and the average flow rate of tracer (QMCH,inj·CMCH,inj, in Figure 1) was the same in every experiment. Therefore, inlet tracer

Figure 1. Illustration of the streams nomenclature in the tracer injection tests.

concentration, CMCH,in, in the organic phase inlet stream, QO,in in Figure 1, decreased with increasing production. A material balance allowed the calculation of the flow rate of the organic phase leaving the reactor, QO,out.

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PRELIMINARY ASSESSMENT OF RESULTS Experimental operating conditions used in the measurement of residence time distribution are compiled in Figure 2 and Table

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METHODOLOGY Carrying out experimental measurement of distribution functions in industrial equipment and not compromising the catalyst9 or the specifications of final product requires a careful selection of the tracer. It should not undergo reaction or build up in the plant. In a two-phase system it is also important to check the density of the tracer and its solubility in the two phases. In the first stage of assessment of real flow in the nitrator reported in this work, the flow of a dispersed organic phase was considered. Methyl cyclohexane (MCH), the organic tracer, was confirmed not to nitrate under industrial process conditions. Its properties24,25 are in the range of the dispersed phase properties reported by Schiefferle et al.26 and Zaldivar.27 Location of tracer inlet, near the bottom, and sampling point at nitrator top outlet were governed by plant layout. Despite efforts to minimize piping length, it was a few meters long due to plant layout. An up-step injection of tracer was used. Following the procedure suggested by Teefy28 the injection lasted 4 times the nominal space time. On average, 51 samples of the outlet stream were collected during each experiment, and the time gap between sampling increased as time elapsed. The liquid−liquid system leaving the nitrator does not allow online measurement of tracer concentration. The samples collected were allowed to separate phases, which was a fast process.29 The organic phase was prepared and analyzed30 in a gas chromatograph (Agilent, series 6890, using a HP-1 methyl siloxane column 30 m × 0.32 mm, DN = 4 μm). The RTD experiments in the nitration plant, six in total (Supporting Information, Table S1), were carried out during normal operation. To gather information at different flow rates, which lead to different productions, this study lasted for approximately one year, as plant production depends on market needs. Daily operation of this industrial nitration reactor is controlled according to production targets. Records of

Figure 2. Mean residence time, theoretical and experimental, in the RTD experiments in an industrial reactor, at different productions (normalized data).

S1. For the sake of confidentiality, data in this work were normalized. The mean residence time, tr̅ , was calculated with eq 1. Tracer concentration data in the outlet stream were used to calculate the experimental mean residence time, tr̅ ,exp, according to eq 2, where CMCH,max is the steady state tracer concentration in the organic phase of the outlet stream.18 tr,exp ̅ =

1 CMCH,max

∫0

C MCH,max

t dC (2)

To assess the quality of the data, the experimental tracer outlet concentrations, CMCH,out, and mass balance tracer concentration, Cmb, were plotted. Figure 3 shows this information for experiment 4. In Figure 3 it is possible to observe that, despite some scatter, CMCH,out reaches steady state, confirming that the tracer injection provided enough data. Tracer concentration in the outlet stream, CMCH,out, both predicted and experimental values, were lower than CMCH,in in B

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 5. Model-0Plug flow reactor followed by a CSTR: model parameters.

Figure 3. Tracer dimensionless concentration in the outlet stream experiment 4.

Figure 6. Model-1Plug flow reactor followed by two CSTRs: model parameters.

Table S1, and this is explained by the increase in organic flow rate due to reaction. On the other hand, experimental data never reach the mass balance concentration profile. As the solubility of MCH in the aqueous phase is negligible, this difference may be partially due to evaporation of tracer during sampling. Another striking feature in Figure 3 is the initial time lag. The piping length in the tracer inlet route referred before, undoubtedly contributed to these results. This can also be confirmed in Figure 2 where the experimental mean residence time, tr̅ ,exp, is higher than the mean residence time, tr̅ . This result will influence the combination of reactors in the mathematical models to be developed in the next section. The first approach used to model reactor flow was to explore one- and two-adjustable parameter models21 to fit the RTD data and evaluate deviations from ideal flow. Two oneparameter models: one accounting for stagnant regions also termed as dead zones, the other for bypassing (BP) or channeling, did not fit tracer data (Figure 4). A two adjustable

vessels (Figures 7 and 8). In the next section the fitting of experimental data to these models will be sequentially tested.

Figure 7. Model-2Plug flow reactor followed by two CSTRs with interchange between reactors.

Figure 8. Model-3Plug flow reactor followed by two CSTRs. The first CSTR has a stagnant region with fluid interchange.

The plug flow reactor was the first vessel in the assemblage of reactors in series. This vessel, where reaction does not take place, was introduced in order to account for the initial time lag in tracer concentration (Figures 1, 3, 4) attributed to the length of piping before the tracer enters the reactor. It is worth mentioning again that, due to chemical reaction, the volumetric flow rates of the two liquid phases change during the process. This was addressed in the CSTR mathematical models. In steady state the tracer outlet concentration is

Figure 4. Tracer experimental concentration in the effluent stream vs expected concentration calculated using one-parameter models (BPbypassing) and two-parameter model (BP+DZ- bypassing and dead zones)experiment 1, normalized data.

parameter model, considering dead zone (DZ) and bypassing, was also not able to replicate the tracer concentration data, regardless the flow rate tested. This is illustrated in Figure 4 with data from the first experiment. The failure of these models was predictable as explained by Himmelblau and Bischoff,17 who point out that only a combined model will provide a suitable model for stirred tank reactors.

CMCH,out(t = ∞) = C MCH,max = Q O,inCMCH,in /Q O,out

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(3)

COMBINED MODELS To seek for a suitable model for the flow in the continuous stirred nitrator, combined models were developed.23 These models considered first plug flow and a perfectly mixed reactor (Figure 5), then a second mixed region was added (Figure 6), later including exchange of fluid elements between stirred

From this equation it is possible to know reaction extent, as the outlet organic phase flow rate, QO,out, depends on conversion. Mass balances to the tracer, MCH, in each subsystem, lead to equations involving ODE. The application of Laplace transformation simplifies the mathematical problem, as the ODE will be reduced to algebraic equations.5 C

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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MODEL-0: PLUG FLOW REACTOR FOLLOWED BY A CSTR The conservation equation for the tracer concentration in an element volume of the plug flow reactor, where reaction does not take place, leads to eq 4 u

∂CMCH(t , z) ∂CMCH(t , z) + =0 ∂z ∂t

αVεO

dt

(4)

C̅ MCH,1(s) =

i. c. :

t = 0, CMCH(0, z) = 0

f . c. :

z = 0, CMCH(t , 0) = CMCH,in

(1 − α)VεO

dt

(12)

CMCH,out(t ) ⎧ ⎫ QQ O,inC̅ MCH,in(s) exp( −θs) ⎪ ⎪ ⎬ = 3−1⎨ ⎪ ⎪ ⎩ [(αVεOs + Q )((1 − α)VεOs + Q O,out)] ⎭

* Q O,inC̅MCH,in (s )

(14)

Model-2: Plug Flow Reactor Followed by Two CSTRs with Interchange between Reactors. The model of a real reactor needs to involve some realism, as Fogler21 pointed out. When two agitated tank reactors are used to model the nitrator it seems realistic to include flow interchange between the two tanks that represent two regions in the reactor, although this approach introduces another model parameter, QR, as illustrated in Figure 7. The mass balance to the tracer in the plug flow reactor is not altered and CMCH,in * is calculated with eq 5. Between the two CSTRs a second stream, with a QR volumetric flow rate, is now included in Model-2 (Figure 7). A mole balance on the inert tracer in the first CSTR gives

(7)

(8)

where C̅ MCH.in(s) is the tracer concentration in the Laplace domain. When using an up-step tracer injection, eq 9 together with eq 8, enables the quantification of this tracer concentration.

CMCH,in s

dt

Introducing eq 11 in eq 13 and solving the mathematical model for CMCH,out, the tracer concentration in the exit stream during the experiment, gives

Introducing eq 5 in eq 7 and solving the mathematical model for CMCH,out, the tracer concentration in the exit stream during the experiment, gives

C̅ MCH,in(s) =

dCMCH,out(t )

(13)

where εO is the volume fraction of organic phase. Laplace transformation was used to obtain the equation for tracer concentration at the outlet of this reactor in the s domain, eq 7:

⎧ ⎫ ⎪ Q O,inC̅ MCH,in(s) exp( − θs) ⎪ ⎬ CMCH,out(t ) = 3−1⎨ ⎪ ⎪ VεOs + Q O,out ⎭ ⎩

(11)

(1 − α)VεOsC̅ MCH,out = QC̅ MCH,1(s) − Q O,outC̅ MCH,out(s)

* = Q O,inCMCH,in − Q O,outCMCH,out(t )

VεOs + Q O,out

αVεOs + Q

or, in Laplace domain

(5)

(6)

C̅ MCH,out(s) =

* Q O,inC̅ MCH,in (s )

= QCMCH,1(t ) − Q O,outCMCH,out(t )

A mass balance to the tracer in the continuous stirred reactor (Figure 5) is illustrated by eq 6 dCMCH,out(t )

(10)

The equation of conservation of tracer for the second CSTR, with volume (1 − α)V, is therefore

the solution of eq 4 applying the Laplace transform method gives the tracer concentration at the outlet of the plug flow reactor in the Laplace domain, where θ is the space time in this reactor, or the time lag due to piping. * C̅ MCH,in = C̅MCH,in exp( −θs)

* = Q O,inCMCH,in − QCMCH,1(t )

where α is the fraction of nitrator volume corresponding to this CSTR and Q is the volumetric flow rate between the two CSTRs (Figure 6). Laplace transformation was used to obtain the equation for tracer concentration at the outlet of this reactor in s domain, eq 11:

with

VεO

dCMCH,1(t )

αVεO

(9)

dCMCH,1(t ) dt

* = Q O,inCMCH,in + Q RCMCH,out − QCMCH,1(t )

Model-1: Plug Flow Reactor Followed by Two CSTRs. According to Fogler,21 the configuration of both inlet and outlet in the vessel, recommends adding stirred tanks to describe the nitrator. Moreover, the shape of tracer concentration profiles in the effluent stream suggested a second order system. Therefore, in Model-1 the industrial nitrator was represented by a plug flow followed by two perfectly mixed reactors, Figure 6. The sum of the volumes of the two CSTRs equals the volume of the nitrator. This new combined model, Model-1, adds a CSTR to Model-0 assuming the same outlet concentration in the plug flow vessel. Therefore, the tracer concentration at the outlet of the tubular reactor is given by eq 5. A mass balance to the tracer in the first continuous stirred reactor (Figure 6) is illustrated by eq 10

(15)

and, in the Laplace domain, CMCH,1 is C̅MCH,1(s) =

* Q O,inC̅MCH,in (s) + Q RC̅ MCH,out(s) αVεOs + Q

(16)

The same specifications are valid for the second CSTR, and the balance to the tracer in this vessel is (1 − α)VεO

dCMCH,out(t ) dt

= QCMCH,1(t ) − Q O,outCMCH,out(t ) − Q RCMCH,out(t ) (17) D

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research In the Laplace domain the tracer outlet concentration, C̅ MCH,out, is C̅ MCH,out(s) =

Laplace transformation was used to write the algebraic equation for tracer concentration in Zone 1, C̅ MCH.z1(s), in eq 21

QC̅ MCH,1(s) (1 − α)VεOs + Q O,out + Q R

C̅MCH,z1(s) =

(18)

Solving eq 18 together with eq 16, and including the tracer concentration leaving the plug flow reactor, C̅ *MCH,in, leads to eq 19 for CMCH,out during the tracer injection. For an up-step tracer injection eq 9 is to be considered.

αz2VεO

⎧ ⎫ QQ O,inC̅ MCH,in(s)exp(−θs) ⎪ ⎪ ⎬ 3−1⎨ ⎪ ⎪ ⎩ [(αVεOs + Q )((1 − α)VεOs + Q O,out + Q R ) − QQ R ] ⎭

dt

C̅ MCH,z2(s) =

= Q z1CMCH,z1(t ) − Q z2CMCH,z2(t )

Q z1C̅ MCH,z1(s) αz2VεOs + Q z2

(23)

Combining eq 21 and eq 23, in Laplace domain, gives the tracer concentration in Zone 1 as C̅MCH,z1(s) =

* (αz2VεOs + Q z2)Q O,inC̅MCH,in (s ) (αz2VεOs + Q z2)(αz1VεOs + Q z1 + Q ) − Q z1Q z2 (24)

For the second CSTR with volume (1 − α1 − α2)V, and in series with Zone 1, the mass balance to the tracer is (1 − αz1 − αz2)VεO

dCMCH,out(t ) dt

= QCMCH,z1(t ) − Q O,outCMCH,out(t )

* = Q O,inCMCH,in + Q z2CMCH,z2(t )

− Q z1CMCH,z1(t ) − QCMCH,z1(t )

dCMCH,z2(t )

and, following the same mathematical approach, the equation for tracer concentration in Zone 2 is

Model-3: Plug Flow Reactor Followed by Two CSTRsStagnant Region with Fluid Interchange in the First CSTR. The occurrence of stagnant zones in the nitrator under study should not be excluded, given the volume of the industrial reactors. As α1, the volume fraction of the first stirred reactor in Figure 8 is expected to be high, a stagnant zone with fluid interchange was considered in the new Model-3 developed in this subsection. The first CSTR in Models 1 and 2 was here divided into two vessels, or two zones, with Zone 2 describing the stagnant region. Therefore, Figure 8 now shows four reactors where the sum of the volumes of the three stirred reactors is equal to the V volume of the nitrator. Besides one more parameter to account for the volume of the stagnant zone, αz2, this model introduces two new interchange streams, Qz1 and Qz2, and the tracer concentration in the stagnant zone, CMCH,z2. In this new model the concentration of tracer leaving the plug flow reactor is also given by eq 5. The mole balance on the tracer for Zone 1, with volume αz1V (Figure 8), has to consider two inlet streams and two streams out dt

(21)

(22)

(19)

αz1VεO

αz1VεOs + Q z1 + Q

Another mass balance on the tracer is needed for the stagnant region, Zone 2 in Figure 8

C MCH,out(t ) =

dCMCH,z1(t )

* Q O,inC̅MCH,in (s) + Q z2C̅MCH,z2(s)

(25)

Solving eq 25 together with the other equations for this Model3 one obtains eq 26 for the tracer concentration in the effluent stream

(20)

⎧ ⎫ (αz2VεOs + Q z2)QQ O,inC̅ MCH,in(s) exp( −θs) ⎪ ⎪ ⎬ CMCH,out(t ) = 3−1⎨ ⎪ ⎪ ⎩ [(1 − αz1 − αz2)VεOs + Q O,out][(αz2VεOs + Q z2)(αz1VεOs + Q z1 + Q ) − Q z1Q z2] ⎭

As a step injection of tracer was adopted, C̅ MCH,in is given by eq 9.

(26)

summarized in Table S3. The fitting of experimental data to each combined model was carried out using GAMS 23.5 together with CONOPT library.31 Bounds of ±5% on the measured values were imposed. Model-0. The model fit results were compiled in Table S2 and Table S3. Comparison of the organic phase flow rate, QO,in, and tracer inlet concentration, CMCH,in, as calculated (Table S3), with the operating conditions in Table S1 shows that these variables are underestimated by the model. On the other hand, QO,out, is overestimated, with the exception of experiment 2. The space time in the plug flow reactor, Figure 5, is expected to decrease with increasing production. Nevertheless, Table S2 shows that it reaches a steady value after experiment 3. A further assessment of Model-0 may rely on the coefficients of determination analysis, which were lower than 0.900 (R2 < 0.900), apart from one experiment. Having in mind that

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VALIDATION OF COMBINED MODELS FOR THE LIQUID−LIQUID NITRATOR The six tracer injection experiments carried out in the industrial plant, and summarized in Table S1, provided data that will be used to validate the combined Model-0 to Model-3 to describe flow in the industrial nitrator. In this chemical process it is not possible to measure every operating condition online. Although some conditions were available (Table S1), when modeling the nitrator the organic phase flow rates (Q O,in and Q O,out ) and tracer inlet concentration (CMCH,in), were added to the other model parameters to be estimated. This strategy was intended to overcome analytical and measurement errors. The values of these operating conditions calculated by each model were E

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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scatter in experiment 3 may explain the very different value for α, 0.50, a finding supported by the lowest R2 in Tables 1 and 2, also registered in Table S2. The most striking feature in Table 2 is that in experiments 2 and 6 the interchange flow is estimated to be null and this leads to Model-1. The higher coefficients of determination in Table 2 confirm that this third combined model, with interchange between the two CSTRs, enables a better fitting of the data. This is particularly true in the first section of the concentration profile where the plug flow reactor depicts the time lag due to the length of tracer piping before the reactor inlet, as illustrated in Figure 9. The improved performance of Model-2 over Model-0 and Model-1 can be explained by stirrer design leading to two mixed regions with an exchange of fluid.

experiments were performed in an industrial plant, this is a good result. Model-1. Combined Model-1 was solved with an additional restriction ensuring that flow rate Q is comprised between QO,in and QO,1. Table 1 summarizes the adjustable parameters Table 1. Model-1: Adjustable Parameters Predicted by the Model θ α Q R2 Radj2

expt 1

expt 2

expt 3

expt 4

expt 5

expt 6

1.36 0.88 0.101 0.987 0.986

0.81 0.94 0.135 0.915 0.911

0.47 0.51 0.158 0.825 0.818

0.60 0.95 0.159 0.965 0.964

0.73 0.93 0.191 0.852 0.846

0.91 0.96 0.200 0.891 0.886

obtained. When the predicted organic phase exit flow rates are compared with the experimental values in Table S1, the model estimates are higher but, on the other hand, QO,in and CMCH,in are lower. Regarding model parameters, the first CSTR is the largest vessel, and its volume is greater than 88% of the industrial nitrator, not taking into account experiment 3 results. The plug flow reactor parameter, the space time θ, was expected to decrease with increasing production, and this pattern is not registered. Moreover, in experiment 1 the predicted space time is longer than the time lag in tracer concentration data. Coefficients of determination, R2, depend not only on the fitting ability of the model but also on the scatter of experimental data. Although not illustrated in this manuscript, in experiments 3 and 5 tracer outlet concentration data, CMCH,out, exhibit more scatter and this is in agreement with a R2 lower than in the other experiments. Model-2. The results in Table 1 can be considered good, as experimental data are being used, but they led to search for a better fit to the RTD data. In Model-2, in Figure 7, model complexity is increased by one adjustable parameter, QR, the fraction of total volumetric flow exchanged between the two CSTRs, to be estimated together with the other model parameters: θ, α, and Q and the operating conditions. Because of the increase in flow rate, it is important to include the following restriction in the model: Q ≥ QO,in + QR. The solution of the model by using GAMS 23.5 together with CONOPT library led to the results in Table 2.

Figure 9. Fitting of Model-2 to normalized experimental tracer concentration for different flow rates. In the horizontal axis tnorm = t/ tr̅ ,exp (, Model-2).

Despite good fittings already achieved, the large volume fraction allocated to the first CSTR suggested that a refinement of its modeling might improve results. This might be achieved by Model-3. Model-3. The number of adjustable parameters in Model-3 (Figure 8) is six, and this includes the fraction of total volume allocated to the two zones in the first CSTR (αz1 and αz2) and volumetric flow rates in and out of Zone 2 (Qz1 and Qz2). This new combined model led to eq 26 for MCH concentration in the effluent stream, CMCH,out. The same computational tools were used and the adjustable parameters values are collected in Table 3. The estimation of operating parameters by Model-3, Table 3, led to the same values as in Table 2, except for experiment 6 with a higher QO,1. Reaction may take place in every CSTR, and in this particular liquid−liquid system flow rate increases with reaction extent. Therefore, Zone 2 inlet flow rate, Qz1, should be less than the outlet flow rate, Qz2, and this is confirmed in

Table 2. Model-2: Adjustable Parameters Predicted by the Model θ α Q QR × 102 R2 Radj2

expt 1

expt 2

expt 3

expt 4

expt 5

expt 6

1.27 0.88 0.101 0.88 0.989 0.988

0.80 0.95 0.135 0.00 0.915 0.910

0.44 0.50 0.159 0.50 0.825 0.814

0.58 0.99 0.159 1.5 0.973 0.971

0.58 0.99 0.191 1.38 0.860 0.851

0.93 0.99 0.200 0.00 0.894 0.887

Table 3. Model-3: Adjustable Parameters Predicted by the Model θ αz1 αz2 Q Qz1 Qz2 R2 R2adj

Assessment of the differences between Model-1 and Model-2 parameters will be first focused on the prediction of operating parameters: QO,in, QO,1, and CMCH,in to conclude that the differences in Tables 1 and 2 may be considered insignificant. The size of the first CSTR, which in Model-1 was already the largest vessel, is confirmed to increase with production. Model2 estimated this vessel to be greater than 95% of total industrial reactor volume, reaching 99% in three experiments. The data F

expt 1

expt 2

expt 3

expt 4

expt 5

expt 6

1.38 0.98 0.01 0.101 0.071 0.080 0.989 0.988

0.80 0.93 0.01 0.135 0.148 0.148 0.915 0.906

0.47 0.51 0.003 0.159 0.134 0.145 0.825 0.806

0.60 0.97 0.03 0.159 0.185 0.193 0.973 0.970

0.58 0.94 0.05 0.191 0.203 0.210 0.860 0.844

0.93 0.81 0.18 0.200 0.209 0.209 0.894 0.882

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Table 3, except for experiments 2 and 6. Moreover, as expected, Qz1, increases with production (experiments 1−6). In Table 3, the volume of the first CSTR (αz1 + αz2) in experiment 1 is closer to the range of the other experiments, excepting the third one, where Model-3 confirms the smaller volume fraction already estimated by Model-1 and Model-2. Another trend in Table 3 is the low fraction of total volume allocated to the second CSTR (1 − αz1 − αz2) suggesting this reactor could be removed from the model. A combined model, with a plug flow reactor and a CSTR with stagnant zone, was tested, but did not provide a better fitting of the data. Model Selection. The following Figure 10 was built for a comprehensive analysis of the different models based on their

Figure 12. Determination coefficients for the models assessed. Experiment 1: ●, R2; , Radj2. Experiment 4: ▲, R2; ×, Radj2.

Radj2 values for the four models in experiments 1 and 4. Model2 and Model-3 show improved determination coefficients over Model-0 and Model-1. Nevertheless it is the decrease in Radj2 coefficients that confirms Model-3 does not add value to the fitting of the RTD data. The criteria used when evaluating the combined models developed were focused on the physical meaning of the parameters and their value. It is worth mentioning that interchange flow between the two CSTRs was not included in Model-3 and the small QR flow predicted by Model-2 may be envisaged as a validation of this assumption. The differences were not always significant, especially between Model-2 and Model-3. Nevertheless, Model-2 has the advantage of being simpler, involving fewer parameters and with a higher Radj2, therefore point out to its adoption for describing the hydrodynamics of the industrial reactor.

Figure 10. Relative deviation in residence time prediction by the combined models.

ability to estimate the delay in systems response due to length of piping. In every model this corresponds to the space time, θ. This parameter should be able to explain the difference between mean residence time, tr̅ , and (tr̅ ,exp − θ) as reaction in the plug flow vessel was not considered. The relative deviations (%), ((tr̅ ,exp − θ) − tr̅ )/tr̅ × 100, in Figure 10, depend on the experiment. Apart from Model-0, for each experiment the differences between model predictions are small. However, an overall analysis shows that Model-2 provides better estimates. To highlight the small differences between the four models, Figure 11 illustrates the fitting to the F-curve at the beginning of experiment 4. It can be seen that Model-2 and Model-3 fit the data better than the other models.

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CONCLUSIONS In an industrial plant, meeting production targets and product specifications are the main goals in daily operation. This is better achieved with increased information on the chemical process, the plant, and equipment operation. Mathematical models are good tools to reach this objective, particularly when they are based on data and operating conditions of the plant. In this study the modeling of flow in the industrial continuous reactor was based on residence time distribution experiments. An up-step injection of tracer proved to be feasible on the plant scale and provided reliable data for model fitting. Flow models with an increased complexity were developed to assess tracer concentration data fitting. As expected, one- and two-parameter models failed to fit the data and were not able to describe the flow in this continuous reactor involving a liquid− liquid system. Four combined models, with adjustable parameters correlated to flow conditions, were tested. Each model included vessels in series: plug flow and one or two stirred reactors. Model complexity and number of adjustable parameters increased as deviations from ideal flow were considered, such as interchange and stagnant regions. Differences between the fittings of the combined models to residence time data were small. A good compromise between accuracy, physical meaning, and mathematical complexity was found to be reached by Model-2, which takes into account fluid interchange between the two continuous stirred tanks. This new flow model for the nitrator will enable discarding ideal flow assumption used up to now when modeling this industrial process.

Figure 11. Fitting of the four flow models to the F-curve−experiment 4.

The assessment of the fitting for these four combined models was not only based on the determination coefficient, but also on the adjusted determination coefficient, Radj2. In fact, this coefficient is usually calculated to evaluate the influence of the increase in the number of parameters in the model, as its value only changes if the new parameter added to the model has impact on the model explanation.32 Figure 12 shows the R2 and G

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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ASSOCIATED CONTENT

S Supporting Information *

REFERENCES

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03236. Table S1compiles the operating conditions used in the RTD experiments. Table S2 includes Model-0 parameter and the determination coefficients for this model. Table S3 summarizes the operating conditions in the industrial nitrator estimated by the different combined models, Model-0 to Model-3 (PDF)

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: cristina@eq uc.pt. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from Fundaçaõ para a Ciência e Tecnologia (FCT) for Ph.D. Grant SFRH/BDE/33907/2009 is gratefully acknowledged.

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NOMENCLATURE Cexp = molar concentration of tracer, experimental Ci,j = molar concentration of compound i in the stream j Cmb = molar concentration of tracer, mass balance CMCH,j = molar concentration of MCH in the stream j C̅ MCH,j = molar concentration of MCH in the stream j in Laplace domain CMCH,max = outlet molar concentration of MCH, in steady state, for an up-step injection ODE = ordinary differential equations Q = volumetric flow rate between two CSTR Qf,j = volumetric flow rate of phase f in the stream j QMCH,j = volumetric flow rate of MCH in the stream j QR = fraction of total volumetric flow exchanged between the two CSTR R2 = determination coefficient Radj2 = adjusted determination coefficient t = time tr̅ = mean residence time tr̅ ,exp = experimental mean residence time u = average linear velocity V = reactor volume z = axial distance

Subscripts

Aq = aqueous phase in = inlet inj = injection O = organic phase out = outlet t = tank z1 = Zone 1 z2 = Zone 2 Greek symbols

α = fraction of the total reactor volume V εO = volumetric fraction of organic phase in the system θ = space time in the plug flow reactor H

DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.iecr.5b03236 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX