RTD Modeling of a Non-Ideal Coiled-Tube Reactor through Experimental Investigation for Pulse Input Using Methylene Blue Dye

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 3, Pages: 1220-1231

J. Environ. Treat. Tech.

ISSN: 2309-1185

Journal web link: http://www.jett.dormaj.com

https://doi.org/10.47277/JETT/8(3)1231

RTD Modeling of a Non-Ideal Coiled-Tube

Reactor through Experimental Investigation for

Pulse Input Using Methylene Blue Dye

Avdesh Singh Pundir *, Kailash Singh , Varshika Singhal , Mayank Pandey , Garvit Gupta ,

Moh. Rohaan , Sunil Rajoriya and Girish Tyagi

Department of Chemical Engineering, MIET, Meerut, U.P-250005

Department of Chemical Engineering, MNIT, Jaipur, Rajasthan-302017

Received: 01/07/2020

Accepted: 07/08/2020

Published: 20/09/2020

Abstract

This paper focuses on the grasp of a deep understanding of flow behavior in a coiled tube reactor through Residence Time

Distribution (RTD) studies. The reactors, in general, are classified ideally: mixed and plug-flow patterns. Unfortunately, in the real

world, it has been observed that they show very different behavior from that expected. Thus, the characterization of the nonideal coiled

tube reactor is needed to carry out. The calculations were carried out in the Matlab for distribution of residence time of the coiled tube

reactor that is used in the Chemical Reaction Engineering Laboratory at MIET College. Pulse input tests were used significantly to

analyzed the flow behavior using methylene blue (MB) tracer. A significant disparity in RTD curves in the presence of the secondary

flow was examined and data were recorded. Finally, a suitable mathematical model was selected from the Tank in Series (TIS) and Axial

Dispersion Models (ADMs) based on residual error and was used to validate these outcomes. The deconvoluted of the signal was used

to get Cin for the verification of the pulse input behavior. The results were compared with the experimental data that concluded the

modeling of the reactor is in good agreement.

Keywords: Pulse input, Methylene blue, RTD, Coiled tube reactor, Non-ideal

Introduction¹

modified with time, therefore, producing a frequency-response

layout for the system. Each approach can be modified into the

other two as per our convenience. However, practically, it is a

good deal to handle a less complicated approach to monitoring

a tracer response that approximates a step-change or a pulse

enter due to measurement complexity related to sinusoidal

variations, besides, to consume much greater time and want for

a unique measuring device (3).

The applications of continuous processing in the chemical

industry have been increasing day by day. There are many

causes why a chemical product manufacturer may choose to run

chemical reactions in continuous mode instead of batch. The

main causes for choosing continuous instead of batch reaction

system are rapid kinetics, exothermic reaction, hazardous

chemicals, high temperatures, and pressures. In general, a

reactor used in chemical industries has non-ideal behavior.

Therefore, it’s important to know the real behavior of the

reactor through modeling and simulation in practice. To

simulate a real reactor, it’s necessary to estimate RTD. RTD

describes how much time a particle spends inside the reactor

For an ideal reactor, the flow and mixing conditions are

needed to be exactly known, which allows us to develop the

theoretically mathematical model equation (1). However, in

reality, for built reactors, these complex characteristics much

deviate to some extent from ideal behavior due to various

possible reasons such as a consequence of a short-circuiting,

absence of turbulence, macroscopic internal currents, and

stagnant zones (2). In an actual system, the velocity distribution

profile leads to

a residence time distribution (RTD).

Experimental determination of RTD is carried out via

measuring the outlet response of an inert tracer which injected

into the flow stream to be study and so-called stimulus-

response technique. In general, tracers are normally a soluble

dye, acid, or color compound that can be effortlessly detected

the concentration at the exit of a reactor with the assistance of

an online or manually measuring device. The applied

techniques are all primarily based on the measurement of some

properties at the output flow of a reactor based on regarded

modifications to the input flow which may additionally consist

of the following strategies for making use of the tracer to a

system (a) a step enter method, in which the regarded input

concentration is modified from one regular stage to some other

regular level;(b) a pulse enter method, in which a pretty small

recognized quantity of tracer is injected into the feed circulate

in the negligible feasible time; and (c) a sinusoidal enter

method, in which the frequency of the sinusoidal variant is

(

4). For a better understanding of flow reactors, two different

ideal reactor models i.e. Continuous Stirred Tank Reactor

CSTR) and Plug Flow Reactor (PFR) are used at the industry

(

level. An Ideal CSTR assumes perfect mixing, while an ideal

PFR assumes no mixing. No real reactor has consisted

accurately the characteristics of either of the two ideal reactors,

in general. Many researchers have shown that a real reactor can

Corresponding author: (a and b) Avdesh Singh Pundir and Sunil Rajoriya, Department of Chemical Engineering, MIET, Meerut,

U.P-250005. Emails: avdesh.pundir@miet.ac.in and sunil.rajoriya@miet.ac.in (c) Kailash Singh, Department of Chemical Engineering,

MNIT, Jaipur, Rajasthan-302017. E-mail: ksingh.mnit@gmail.com

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2020, Volume 8, Issue 3, Pages: 1220-1231

be modeled as reactor data tank-in-series (TIS) and axial

dispersion model (ADM) which are widely used (5). In many

practical situations, the fluid in the reactor is neither well mixed

nor plug flow. Therefore, there is a need for time to model a

real reactor and it can be possible in several ways. The RTD

describes us how long the various reacting fluid elements have

been retained inside the reactor, however, it does not convey

any information regarding the exchange of matter among the

fluid elements. The mixing of different reacting species is one

of the main factors responsible for controlling the behavior of

chemical reactors. For the different order reactions system,

more especially for the first-order system, the knowledge of the

time with which each molecule remains in the reactor is

required to predict the conversion of a reactant. Therefore, once

the RTD is determined, it is quite easy to predict the reaction

conversion that will be achieved in a real reactor with the

known value of the specific reaction rate for the first-order

reaction. For reactions other than the first-order only

knowledge of RTD is not adequate to predict conversion. In

such cases, in addition to RTD, the degree of mixing of

molecules must be known (6). In this study, no reaction mixture

has been considered while an investigation of flow behavior has

been carried out using a methylene blue dye as a tracer for pulse

input.

rate of water can be adjusted by operating the provided needle

valve and measured with the help of a rotameter. The

compressed air is used for the circulation of water. The reactor

is a helical coil tube type made up of stainless-steel pipe. The

methylene blue dye solution enters at the lower end coming out

of the top of the coil from where samples are collected for

analysis of outcomes. To investigate the flow behavior by using

the RTD characteristics, a simple arrangement is made

available to inject MB as a tracer into the lower end of the

reactor, using a syringe, manually. Pressure regulator &

pressure gauge are fitted in the compressed airline for ease of

varying the water flow rate. The coiled tubular flow reactor

consists of 60.96 cm length of the tube and 12.33 mm inner

diameter provided with one inlet/outlet at one end opposite to

each other and inclined at 90˚ with the axis of the tube. The

diameter of the pipe before the inlet point of the reactor is 10

mm and through which a pulse input is given. The volume of

the reactor is 2.911x10 m . The coiled tube reactor is kept in

an isothermal rectangular tank fitted with a temperature sensor.

The temperature of the stirred water bath tank is kept constant

with the help of a PID controller and uniform by stirring action.

Effect of shape of the coiled tube reactor

The coiled1tube reactor consists of two main flows: a

The main aim of this study is to develop a reliable model

for a continuous reactor, especially for a coiled-tube reactor

which is mostly used in many chemical and allied industries. A

model is to develop for the flow of Newtonian fluid (because

mostly organic solvents may have similar properties to water)

in a coiled-tube reactor. This model is used to define the design

rules for the reactor. The model describes the RTD as a function

of the rheological behavior of the system and operational

parameters for fixed tube dimensions. The first emphasis lies

on determining the suitable model for a coiled tube reactor from

experimental data and validation of the assumption by

primary flow occurring along the axial direction of the fluid

motion and a secondary flow acting perpendicularly to this due

to centrifugal force. By the secondary flows, drag effects in the

proximity of wall surfaces come in the picture (7). The

secondary flow in helical pipes was firstly investigated by Dean

(8). He used the dimensionless group to characterize the flow

behavior inside the coiled-tube reactor, normally called Dean

number, De:

푅ꢀ

휌푢푑푡 푑푡

퐷푒 = _√

ꢁ_ꢂ_푐

(1)

휆

휇

determining the dispersion coefficient,

Axial

During

experimentation, it is normally assumed that the input tracer

concentration has perfect pulse shape, and then a suitable

analytical expression is normally used without verifying the

pulse input shape. That is why a different way for the

calculation is presented, not only taking the moments but the

whole distribution into account. The geometry configuration of

the coiled tube reactor introduces a secondary motion which

ultimately affects the dispersion. To accomplish this, the

solutions of the three models are fitted to the measured RTD

data. At first, a one-parametric tank in a series model is used

due to its simple nature without taking into account complex

phenomena such as bypass or channeling. Second dispersion

models have been applied to solve the dispersion convection

equations and fitted to the measured RTD data by using the

least-squares method. At last, by using the deconvolution

technique, a validation of the pulse input shape has been carried

out along with an optimization scheme for an overdetermined

system. All calculations were carried out in Matlab. This study

has been carried out and presented in such a manner so that it

can be more accessible to many postgraduate and research

students who are working presently in this field. The prime

objective of the present study is to determine a suitable model

that predicts the output in a known manner from data gathered

in an undergraduate laboratory through a lot of experiments.

This study adds “off the shelves” method to the chemical

engineering discipline.

where ꢃ is the ratio of coil-to-tube diameter. ꢄ and ꢅ are the

density and dynamic viscosity of the fluid and ꢆ is the mean

axial velocity, respectively. For a better understanding of the

physical meaning of Dean number, 퐷푒, it can be re-written in

terms of the forces which come in the picture are inertia,

centrifugal acceleration, and viscous forces.

ꢇ(ꢈꢀ푛ꢉ푟푖ꢊ푢푔푎푙 ꢊ표푟ꢈꢀ) ×(푖푛ꢀ푟ꢉ푖푎 ꢊ표푟ꢈꢀ)

퐷푒 훼 푓(

)

(2)

(푣푖푠ꢈ표푢푠 ꢊ표푟ꢈꢀ)

Under the condition of low values of Dean numbers (퐷푒 ≪

), viscous forces are dominant and the secondary flow is

approximately absent (9). Conversely, with the increment in

퐷푒, centrifugal and inertial forces overcome against drag which

ultimately leading to the formation of secondary flow.

Therefore, the Dean number is more informative than

Reynold's number to describe the flow behavior in a

curved/helically coiled-tube reactor. There are many analytical

solutions available for the equation which describes the flow

for incompressible laminar flow and lower values of curvatures

radius (high ꢃ). However, the nondimensional solution

provided by analytical depends heavily on 퐷푒. The action of

centrifugal force due to the curvature of the reactor tube

develops two opposite-rotating vortices, which supports

secondary mixing. The centrifugal force pushes hard on the

fluid elements which are moving in and around the center of

the tube, and from the fluid mechanics' theory, it is clear that

the axial velocity is maximum at the center point of the tube.

The central region's fast-moving fluid is forced to move

outward and it is constantly replaced by the fluid near the wall,

so there is some sort of inter-transfer movement of inner and

Experimental setup

The experimental setup consists of two feed tanks and one

tank is used through which water is fed to the reactor. The flow

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2020, Volume 8, Issue 3, Pages: 1220-1231

∞

outer fluid elements. Due to that reason, the position of

maximum axial velocity is off-centered and moves toward the

outer wall.

( )

function: ∫ 퐸 ꢋ ꢌꢋ = ꢍ. Under the normalization condition,

the shaded area under the residence time density function is

unity. It means that the fraction of fluid elements remains into

the reactor for time limits between 0 and ∞ is 1. There are two

important type reactors which normally encounter in chemical

industries which are the plug flow and the mixed flow

(continuously stirred tank reactor, CSTR) reactors. In the case

of an ideal plug flow reactor, the RTD is a delta function that is

centered at the point of reactor space-time 휏 and under ideal

condition, it is assumed that there is no distribution of residence

times, every fluid element lives for the same amount of time 휏

inside the reactor.

Results and discussion

There are many approaches available to model a real non-

ideal reactor but two models have been significantly in use.

These two models have one unknown parameter. One

parameter model is also known as the Tank-In-Series (TIS)

model and Axial Dispersion Model (ADM). There are two most

common injection methods available: the first one is being

called pulse input (mathematically, called “Dirac delta

function” in case of ideal input) and the second one is called

step input. In this study, the pulse input injection method has

been used. Figure1 shows the actual experimental setup used in

this study.

퐸_푃_퐹_푅ꢋ = 훿(ꢋ − 휏)

( )

(3)

The spent period called macroscopic residence time

normally depends on the reactor geometry and the operating

conditions. For a tubular reactor with constant cross-section:

.1 Measurement of the RTD

The main aim of this study is to introduce the reader to the

analysis of the coiled tube chemical reactor. In it, reactions with

complex kinetics can be carried out. It is normally work in

unusual condition, such as unsteady-state operation. The

푉

휏 =

(4)

휗

where ꢎ and 퐿 are the volume and length of the reactor,

respectively, ꢏ is the volumetric flow rate and ꢆ is the averaged

fluid velocity. To calculate the space-time for any kind of

reactor, the knowledge of the reactor volume and the fluid flow

rate is a must.

The major flow pattern in RTD theory other than a plug

flow reactor is a CSTR. Each element of fluid at the inlet is

instantaneously and perfectly mixed with the solution already

present in the reactor. It can be easily shown that the RTD for

a CSTR is the exponential function (12):

mathematical analysis is

necessary tool for the

characterization of flow and kinetic models. A software tool is

also necessary to solve a complex problem in the design of a

coiled tube reactor (10). In the last four decades, many

researchers have published literature but out of them, the

detailed review has been published by Nauman (11) along with

the introduction of the development of the RTD theory. He

adopted both the experimental as well as the modeling

techniques and many applied applications. The RTD is defined

as the product of function 퐸(ꢋ) and t such that as 퐸(ꢋ)ꢌꢋ. E(t)

represents the fraction of fluid elements whose residence time

in the reactor lies in the time range dꢋ. It is called the residence

time density function. Alternatively, the fraction of fluid

ꢑ푡

퐸_퐶_푆_푇_푅(ꢋ) = _ꢐ

퐸

ꢒ

(5)

1 2

elements residing in the vessel between time limits ꢋ and ꢋ is

The RTD of a real reactor can be approached to the

behavior of these ideal reactors and the extent of the deviation

must be investigated by using well established RTD studies.

There are many experiments possible which permit the RTD of

a real reactor to be derived.

ꢉ2

given by ∫_ꢉ₁퐸(ꢋ) ꢌꢋ . From the other point of view, the fraction

of fluid element with age lessor than ꢋ can be represented by

ꢉ

∫

퐸(ꢋ) ꢌꢋ. As the nature of being a probability distribution

function, 퐸(ꢋ) function can be presented as the normalization

Stirrer

Coiled Tube Reactor

Injector

Dye Solution

Outlet

Heater

Water Inlet

Cout

Stirred Tank Heater

Cin

Product Tank

Time

Figure 1: Schematic representation of the RTD experimental setup with a pulse input

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In this study, only the simplest ones are introduced which

doesn't demand any special skill or much chemical and

sophisticated equipment for injection. The experimental

measurement of the RTD is carried out by injecting a tracer,

which is methylene blue (MB) in this study, into the coiled-tube

reactor at a fixed time. This MB is an inert chemical. The MB

concentration at the outlets of the reactor is measured by the

colorimeter with the progress of time. The necessary condition

for the selection of a trace is that it must be inert, soluble in the

reacting mixture, and easily detectable. Besides, it should not

have any interaction with the walls of the reactor so that it can

be used to reflect, as best as possible, the real behavior of the

reacting materials which are flowing through the reactor. Table

predefined time and then measuring the tracer

concentration, C, (= Cout) in the effluent stream with the

progress of time. In a pulse experiment, however, the uniform

distribution of the MB dye is not practically possible which

emphasizes the investigation of deconvolution study. The

measured concentration of the MB at the outlet plotted against

the time which is called C-curve and from this C-curve, E-curve

was produced. The E-curve gives the “fraction” of the volume

element exiting the system at a particular time. The E-curve can

also be produced as a normalized distribution which means that

the total area under the curve is unity. From the above-

discussed theory, the E-curve can be determined:

퐶(ꢉ)

represents the values of the process parameters used and

퐸(ꢋ) = _∫_ꢘ

ꢗ

(6)

퐶푑ꢉ

evaluated in this study.

Once the values of 퐸(ꢋ) are known, the mean residence time

Table 1: Various parameters used in this study

∞

can be calculated as ꢋ

= ∫₀퐸(ꢋ)ꢌꢋ This is called the first

moment of the distribution. The mean residence time ꢋ

is equal

to the space-time only under the ideal flow behavior condition.

It is also useful to calculate the variance of the distribution,

which is the second moment about the mean:

퐷_푚

ꢄ

6.47E-10 (m /sec)

21.83 (sec)

430.26

∞

ꢙ_ꢉ²= ∫ (ꢋ − ꢋ

)²퐸(ꢋ)ꢌꢋ.

(7)

950 (kg/m )

N_R_e

Dꢓ

퐿

0.6096 (m)

49.44

117.62

The calibration experiments were carried out on the UV-

VIS detection system and presented in figure 2. The fitted

calibration equation was used to estimate the concentration of

the MB dye at the reactor outlet. Figure 3 depicts the outlet

concentration measured with the progress of time. The

evaluated cross ponding E and F curves from figure 3 have been

shown by figures 4 and 5 along with its dimensionless forms.

These figures show the smooth curve as expected.

퐿/ꢌ_ꢉ

ꢔ푒_ꢕ

ꢎ

13.38

10.01

D_A_x_i_a_l

0.0017 (m /sec)

600 (ml)

22.08(sec)

0.2025

0.0279 (m/sec)

87.42

ꢋ

퐵ꢖ

d_t

0.01233 (m)

.2 C, E and F curves

The RTD is determined experimentally by injecting a small

amount of MB dye, called a tracer, into the coiled tube reactor

Figure 2 Calibration and residual error curves for concentration measurement

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Figure 3: Concentration at the outlet of the coiled tube reactor

(a)

(b)

Figure 4: E curves of the coiled tube reactor with time (a) and without time dimension (b)

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2020, Volume 8, Issue 3, Pages: 1220-1231

(a)

(b)

Figure 5: F curves of the coiled tube reactor with time (a) and without time dimension (b)

̅^ꢟ

ꢉ

)퐸(ꢋ)ꢌꢋ =

∞

ꢙ = ∫ (ꢋ − ꢋ

(9)

푛

.3 Tank in series model (TIS)

In this approach, we study the RTD of a coiled tube reactor

It is clear from figure 7 and 8, as we increase the number

of tanks, the error gets reduce but after n = 6, the value of error

gets an increase. Therefore, the best-fitted value for the TIS

model is n = 6. But from figure 7, it is also clear that this value

has a large error which is not acceptable for the real system.

to determine the number of CSTRs in series that will represent

approximately the same RTD behavior as of the coiled tube

reactor. We started the investigation with two CSTRs in series

expression for the expression of the RTD, and then generalize

it for “n” reactors connected in series. In this way, it is obtained

an equation that allows us to calculate the number of tanks that

best correlates the RTD data of the coiled tube reactor(13).

.4 Axial dispersion model (ADM)

In this study, the simplest model was to describe flow

_ꢉꢚꢑꢛ

푛ꢜ1!ꢉ_ꢝ^ꢑ^ꢚ

inside the coiled reactor which consists of both the convection

and the diffusion terms which are most important of the axial

dispersion model as reported by many researchers (15–17). The

model is represented by the following equation:

ꢉ

ꢓꢞp(− )

ꢉꢝ

^퐸퐶푆푇푅푠

(8)

Here t = t/n, where t in the numerator represents the total

volume of all reactors divided by the volumetric flow rate.

Figure 6 shows the actual approach of the TIS model. Figure 7

shows the RTD mapping for different CSTR numbers in series.

As n increases, the behavior is closer to plug flow behavior.

The number of reactors in series can be calculated from the

dimensionless variance (14):

ꢟ

휕퐶

휕ꢉ

휕 퐶

휕푥^ꢟ

휕퐶

휕푥

퐷_퐴_푥_푖_푎_푙

− ꢆ

(10)

where ꢠ, the concentration of a tracer, is a function of time and

the axial coordinate of the system x, 퐷Axial is the axial

dispersion coefficient and ꢆ is a constant mean axial velocity,

which does not vary with x. The model described a one-

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dimensional dispersion process of a plug-flow reactor. Here, it

is assumed that the axial dispersion coefficient is independent

of the axial position as well as the tracer concentration. It

describes the rate of axial dispersion inside the reactor. To

make the model more approachable from the numerical

solution point of view, we have converted the model in a non-

dimensional form (18).

model which is generally known as the Peclet number, ꢔ푒퐿 =

ꢆ퐿/퐷Aꢣial. However, the literature shows different definitions of

this parameter under different conditions which differ in the

way of representing the characteristic length and the diffusion

coefficient (molecular or axial diffusion coefficient). The

analytical solutions available are different when subject to

different boundary conditions (19–21). The changing

dispersion parameter due to the reactor geometry and operating

conditions can be estimated by comparing the RTD

experimental data with the analytical solution available.

ꢟ

휕퐶

휕휃

휕 퐶

휕퐶

− _휕_푧

(11)

푃ꢀꢡ 휕푧^ꢟ

ꢉ

푢ꢕ

with ꢢ = _ꢐ= _ꢕand z = x/퐿, where 퐿 shows the length of the

reactor. It is clear that this model is a single parameter-based

Coiled Tube Reactor

A Small Segment of a Coiled Tube Reactor

Cout

Cin

Series of Stirred Tank Heater

Figure 6: Schematic diagram for the tank in series model (TIS)

Figure 7: RTD mapping of the tank in series (TIS) model with the experimental data

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Figure 8: RTD mapping error of the tank in series model with the experimental data

Once we knew that unknown parameter then one can define

the extent of hydrodynamic dispersion inside the reactor using

that parameter value. To do so it is more convenient to work

stage which consists both of diffusion as well as convective

mechanism. For the coiled tube reactor, there is no straight

forward relation available for the applicability of the model.

The residual error of the fitting may be used for validation

purposes. It is reported that for Dean number, De < 0.5,

applicability validation relation for straight tube can be used

퐿

with dimensionless groups. The inverse of ꢔ푒 is the vessel

dispersion number, N . This is a measure of the spread of tracer

in the whole reactor and is so defined:

(

23). But in this study, Dean number, De, has a value of

ꢕ

ꢂꢤꢥꢝꢦꢧ

푢ꢕ

⁄

ꢉꢨꢩꢚꢪꢫ푐푡ꢝꢩꢚ

^≈ꢉꢬꢝꢭꢭꢮꢯꢝꢩꢚ

117.61. However, for crosschecking purposes, we have applied

that relation too. The analytical solutions provide by equations

푢

푁 =

ꢕ

= _ꢕ

ꢟ

(12)

푃ꢀꢡ

⁄

ꢂꢤꢥꢝꢦꢧ

(

13 & 14) depend on the vessel dispersion number N which is

The experimental RTD values were compared with the

obtained analytical solution of the axial dispersion model (22),

not known before fitting of the experimental data. Therefore,

both solutions need to be applied to find a suitable model. To

check the applicability of the axial dispersion model, the

following relation has been used.

which holds in this study (퐵ꢖ = 0.20, 퐿/ꢌ

ꢋ

= 49.44) . Levenspiel

(17) suggested the analytical solution for small deviations from

plug-flow (푁

conditions as (21)

퐿

> 0.01) under open-open ends boundary

ꢉ

휏_퐴_ꢂ_ꢳ= _푑_푡

(17)

^ꢟ/ꢂꢲ

ꢴꢵꢫ

(ꢕꢜ푢푡)^ꢟ

) for large dispersion

4ꢂꢤꢥꢝꢦꢧꢉ

퐸 푀ꢖꢌ푒ꢰ ꢍ =

ꢓꢞp(−

^v^a^lⁱ^d^c^oⁿ^dⁱ^tⁱ^oⁿ^:^ꢶ^<^ꢃ^<^ꢱ⁸^ꢶ^ꢷ^ꢸ^ꢌ^푁푅ꢀ < ꢍꢶꢶꢶ and the mean

residence time of the vessel, ꢋ = ꢆ/퐿.Our study parameters

ꢇ4휋ꢂꢤꢥꢝꢦꢧꢉ

number, N

> 0.01

(13)

come under this valid range. The geometry of the coiled tube

reactor is responsible for the development of two major

mechanisms: the first one is secondary flow and the second one

is interchanging velocity between adjacent fluid layers. The

development of these mechanisms can be verified by Dean

number, De. Dean's number expresses the magnitude of the

developed secondary flow. If De < 1.5, it can be assumed that

the secondary flow is absent while 1.5 < De < 3, fully

developed secondary flow can be assumed. Therefore, the

secondary flow comes in the picture in this study. For

푢³

(ꢕꢜ푢푡)^ꢟ

퐸 푀ꢖꢌ푒ꢰ ꢱ = ꢁ₄

ꢓꢞp(−

) for small

ꢕ

휋ꢂꢤꢥꢝꢦꢧꢕ

4ꢂꢤꢥꢝꢦꢧ ⁄

푢

dispersion number, N

< 0.01

(14)

(15)

푢푑푡

^B^o^d^eⁿ^s^t^eⁱⁿⁿ^u^m^b^e^rⁱ^s^,^퐵^ꢖ⁼ꢂꢲ

where 퐿 is the length of the reactor, ꢆ the cross-section average

velocity and 퐷Aꢣial the axial dispersion coefficient, which need

be calculated from the Taylor expression for dispersion (5):

methylene blue dye in water, molecular diffusivity goes around

.74±1.32×10-6 cm /sec. In general, the liquid diffusivity has

푢 푑푡^ꢟ

ꢟ

exponent in the range of -6. In equations 13 and 14, the only

unknown parameter is the axial diffusion. The length of the

coiled tube reactor was measured and the mean internal

퐷_퐴_푥_푖_푎_푙= 퐷_푚⁺1

(16)

92 ꢂꢲ

if Bo < 10, diffusion dominates over convection, the second

term on the right-hand side of the above equation can be

omitted. But for Bo > 100, the first term can be neglected. This

criterion seems not to fit well in our study. It seems from the

experimental evidence that this study comes in an intermediate

velocity was calculated as ꢆ = ꢏ ̇/ ꢹ with ꢏ̇ the volumetric flow

rate and ꢹ the cross-section area of the coiled tube reactor. The

unknown axial dispersion coefficient was estimated using the

least-squares minimization technique.

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2020, Volume 8, Issue 3, Pages: 1220-1231

Figure 9: Fitting of Model 1 and Model 2 to the experimental data.

Figure 10: Models ’error with respect to the experimental data

Figure 9 depicts the output of the Model1 and Model 2

with time. It is clear that Model 1 is more reliable and seems

good to fit in comparison to Model 2. The same conclusion

can also be drawn from figure 10.

If the input and impulse responses of a system are Cin[o] and

E[n], respectively, then the output Cout[o] of the system is given

by convolution operation (24).

∞

ꢠ_표_푢_ꢉ[ꢍ × ꢖ] = ꢠ [ꢍ × ꢻ] ∗ 퐸[ꢻ × ꢖ] = ∑ ꢠ_푖_푛[ꢍ ×

푖푛

ꢜ∞

.5 Convolution and deconvolution

ꢻ]퐸[ꢻ × ꢖ]

(18)

The convolution integral is an important property to

investigate RTD data. From our previous experience about

carrying out the RTD experiments, the major challenge is

generally related to the injection of the tracer. In addition, the

injection point of MB dye may be located at a fixed distance

from the inlet boundary of the vessel, the shape of the MB dye

stimulus produced by the dye injector change as it moves

towards the vessel entrance. As a result, the RTD curve cannot

always be derived directly from the experimental measurement

The convolution implemented between the discrete-time

signals is usually called a convolution sum. If Cin[m] is m

points sequence and E[n] is an n points sequence then Cout[o]

will be (m+n-1) points sequence. Deconvolution operation is

performed by the reversed operation of the convolution (25). It

is used to separate the mixed signals. This is an important part

of this study. To the best of authors ‘knowledge, no such study

over the coiled tube reactor has been reported so far. For

illustration purposes, a simple representation has been shown

in figure 11. In this study, Cout is the only measurable. Different

samples were taken at the outlet point of the coiled tube reactor

and the corresponding concentrations, Cout, were measured

using a colorimeter. Using Cout, EExp was calculated both in the

time domain and in the dimensionless form.

of the transient concentration in the effluent stream, ꢠꢖꢆꢋ

However, if one knows ꢠꢺꢸ, by detecting it as a function of time

at the inlet, the convolution integral method allows the

extracting of the RTD of the coiled tube reactor. This

methodology is called deconvolution, 퐸 being one of the factors

of the convolution product. The convolution operation is used

to get the output response of a linear and time-invariant system.

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2020, Volume 8, Issue 3, Pages: 1220-1231

Convolution

Cin

Cout

Deconvolution

Time

Figure 11: Convolution and deconvolution of the concentration signals

Figure 12: Optimization function values with respect to the number of iterations

Figure 13: Deconvolution signal Cin from E and Cout signals

The concentration of tracer is in mg/l at the outlet of a

calculated RTD signals. In this study as shown in figure 11, the

reactor distribution is EExp and the signal output is Cout, and we

need to deconvolute both signals to get Cin (26):

continuous system. The deconvolution theorem has been

applied to a de-combination of the RTD and Cout of the coiled

tube reactor. In general, to calculate the signal exit of a reactor

is known along with calculated RTD when the input signal is

unknown. The major changeling in such a study is associated

with the sampling of the output signals and the corresponding

∞

ꢠ_표_푢_ꢉ= ∑ 퐸

⊗ ꢠ_푖_푛

(19)

ꢼ푋푃

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2020, Volume 8, Issue 3, Pages: 1220-1231

It is noted that the deconvolution of the two signals is not

possible if each signal has a different sampling increment

of figures, competing interests, and compliance with policies

on research ethics. Authors adhere to publication requirements

that submitted work is original and has not been published

elsewhere in any language.

interval. For deconvolution purpose, EExp = [E1,

has n = 17 values and Cout = [Cout1,

3 ,…..

]

out2,

out3 ,…..

out22] has o

22 values.

Symbols used

^ꢠ표푢ꢉ1

^ꢠ표푢ꢉ2

ꢠ_푖_푛₁

ꢠ_푖_푛₂

Cout (=

^퐸1 ^ꢶ^…^ꢶ

tracer outlet, inlet concentrations

[mg L^–¹]

ꢿ

ꢾ

ꣂ

ꣁ

ꢿ

ꢾ

ꣂ

ꣁ

C), Cin

퐸 퐸 … ꢶ

= ꣃ

꣄

(20)

[m]

[m s ]

[–]

[s]

diameter of the reactor

axial dispersion coefficient

residence time distribution (RTD)

dimensionless RTD

ꢶ

ꢶ … 퐸₁₇_표_×_푚

ꢠ

–2

ꢽ

표푢ꢉ표ꣀ_표_×₁

ꢽ

푖푛5ꣀ_푚_×₁

D_A_x_i_a_l

퐸_휃

For simplification point of view, it can be written in the

following matrix form

E_E_x_p

experimental RTD

cumulative distribution function

(CDF)

ꢠ

= 퐸

⊗ ꢠ

푖푛푚×1

(21)

표푢ꢉ표×1

ꢼ푋푃표×푚

ꣅ_휃

[–]

[m]

[–]

dimensionless CDF

length of the reactor

Reynolds number

where EExp being the “matrix convolution,” with values of E in

the form of columns. The computation was done using a simple

Matlab script. In Matlab, the sizing of the sample plays an

important role. Here, ‘o’ represents the number of rows of the

time of measurement

[–]

–

Cout vector,.i.e. o = m+n-1, where ‘n’ and ‘m’ represent the

[m s ]

[–]

average tracer speed

Bodenstein number

Dean number

number of columns of EExp and Cin vectors, respectively. The

calculations of the time vector associated with Cin were

evaluated using cross ponding time vector of Cout and EExp. As

we can see, we have m = 5 unknowns and o = 22 equations, so

the system is overdetermined and it has multiple solutions. For

calculating the Cin curve, it is necessary to do the optimization

by minimizing an objective function (O.F.) in the form of the

sum of squared residuals between the Cout curve which is

known and the product [EExp⋅Cin]:

Peclet number

length of Cout column vector

length of Cin column vector

length of EExp column vector

vessel dispersion number

molecular diffusivity

mean residence time

[–]

–2

[m s ]

–1

ꢋ

[ s ]

⁻^퐸^ꢼ^푥^푝^표^×^푚^⊗^ꢠ푖푛푚×1)²

(22)

푂ꣅ = ∑( ꢠ

표푢ꢉ표×1

Greek letters

ꢢ

ꢄ

ꢃ

휏

[–]

dimensionless time vector

[kg m ] density

[–]

[s]

Figure 12 represents the objective function values with

respect to the number of iterations. The function value falls

sharply after 20 iterations and continuously decreases with the

number of iterations. The system calculation gets converged

after 620 iterations and function obtains a minimum value of

length to coil diameter ratio

residence time of coiled tube reactor

Competing interests

The authors declare that no conflict of interest would

prejudice the impartiality of this scientific work.

.000732. Figure 13 shows the evaluated deconvolution signals

from E and Cout signals.

Authors’ contribution

All authors of this study have a complete contribution to

data collection, data analyses, and manuscript writing.

Conclusions

The study of the distribution of the residence time in a

reactor helps to characterize the mixing phenomena and

internal transport in the reactor. The comparison between the

theoretically computed and experimentally determined RTD

was guided us to develop a realistic mathematical model of the

reactor understudy. Different mathematical models were tested

to determine the performance of the coiled tube reactor. TIS

model was tested first and n = 6 tanks found to be reasonable

but it had shown the large deviation from the experimental data.

Therefore, axial dispersion models were fitted. Two models

were tested which are valid only for open-open boundary

conditions. Model1 was more realistic to the experimental data

and had fewer errors. In all the methods of testing and

validation used, the results agree well. To validate the perfect

pulse assumption, deconvolution of outlet concentration and

exit age distribution signals was carried out. The evaluated inlet

concentration seems to fit well with assumed assumptions.

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