Development of a Mathematical Model for Kinetics of Obtaining Isocyanate via a Non-Phosgene Method for Example Benzylisocyanate

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 4, Pages: 1491-1497

J. Environ. Treat. Tech.

ISSN: 2309-1185

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

https://doi.org/10.47277/JETT/1497

Development of a Mathematical Model for Kinetics

of Obtaining Isocyanate via a Non-Phosgene

Method for Example Benzylisocyanate

Dashkin R. R. , Shishanov M. V.

¹Head of the MUCTR Engineering center, D. Mendeleev University of Chemical Technology of Russia, Moscow, Russia, 20 Geroyev Panfilovtsev str.,

Moscow

Leading engineer of the MUCTR Engineering center, D. Mendeleev University of Chemical Technology of Russia, Moscow, Russia, 20 Geroyev

Panfilovtsev str., Moscow

Received: 02/07/2020

Accepted: 08/10/2020

Published: 20/12/2020

Abstract

The paper considers theprocess of developing a mathematicalmodel for kineticsof obtainingisocyanates via a non-phosgene method

non-catalytic thermolysis of carbamates. A mathematical model is presented, kinetic parameters such as activation energy and a pre-

—

exponential multiplier are ascertained.

Keywords: Isocyanate production, non-phosgene technology, mathematical model, thermolysis of carbamates, kinetics of

decomposition

equilibriumof the decomposition reaction will shifttowards the

Introduction

formation of the target isocyanate, ensuringa satisfactory yield

of the product. For today, the process of obtaining isocyanates

by decomposing carbamates remains poorly understood, in

particular, the kinetic data of the reaction are almost absent. In

the conditions of growing demand for isocyanate raw materials

both in Russia and on the world market, as well as the lack of

industrially applicable phosphorless technology for producing

isocyanates, the study of the method for obtaining isocyanates

by thermal decomposition is an extremely relevant area of

research. Due to the need to studythe kineticsof the carbamates

decomposition, a mathematical model for the kinetics of the

obtaining benzylisocyanate reaction was constructed with the

determination of kinetic parameters – the pre-exponential

Isocyanates are one of the most relevant products of the

chemical industry, as they are primary products for the

polyurethanes production, which are used in construction, in

the production of automotive parts, insulation materials,

paintwork products, adhesives, paints, fillers for upholstered

furniture (1). The main method for obtaining polyurethanes is

the interaction of isocyanates by nucleophilic addition of

polyols (2). In addition, isocyanates are valuable intermediates

in organic synthesis of pesticides and other biologically active

substances (3). Despite the wide range of applications of

isocyanates in the world practice, the main method of

production is still the technology using phosgene (4-7), which

makes production unecological, and modern standards and

requirements for the protection of human health and the

environment leads to an urgent need to develop an

environmentally friendly method for obtaining isocyanates.

It is known that esters of N-substituted carbamic acid can

be precursors in the synthesis of isocyanates. The method of

obtaining isocyanates (8,9), which consists in splitting alcohol

from carbamate when heated or with various catalysts (Fig. 1),

can be called one of the most promising.

multiplier and the activation energy (k

0 A

and E ) from

experimental data. The mathematical model will allow to scale

the process and work out critical parametersat the design stage

for a semi-industrial plant with non-isothermal conditions in

the reactor.

.1 Experimental part

The mathematical model was constructed using

experimental data obtained using a plant for thermal

decomposition of carbamates. To develop the kinetic model,

we conducted a series of experiments using an experimental

setup, the flow diagram of which is shown in figure 2. The

Decomposed carbamate was pre-mixed with an inert gas

(

argon) and the mixture was heated to a predetermined

Figure 1: The general scheme of the carbamates decomposition

reaction

temperature in unit 1. Further, the mixture was delivered in

gaseous form to the unit 2 (reactor), where the process of

carbamate decomposition took place and the formation of a

products mixture with target isocyanate and alcohol. Then the

reaction products were fed to the sorption unit 3, where the

sorption solution was located. In this unit, the product of the

carbamate decomposition reaction (isocyanate) reacted with N-

methylbenzylamine to form urea, and the inert carrier gas

partially together with alcohol went into the waste gases.

The complexity of implementing this method in industry is

related to the reversibility of the reaction. However, there are

ways to shift the balance towards product formation by

reducing the concentration of one of the products. So in

practice, a promising solution to this problem can be the

rectification of a products mixture, as a result of which the

alcohol will be removed from the resulting mixture. Thus, the

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Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 4, Pages: 1491-1497

Carbamate

Urea

Reagent mixing and

heating (200 C)

Isocyanate binding by

Reactor

Unit 2

N-methylbenzylamine

Inert gas and

Inert gas (Ar)

alcohol

Unit 3

Unit 1

Figure 2: The block diagram of the experimental plant of carbamate thermolysis

The inert gas in the carbonate decomposition was used to

reduce the partial pressure in order to shift the equilibrium of

the reaction towards the products. In addition, this method was

able to reduce the vapor pressure of carbamate, which reduces

its boiling point. This made it possible to conduct experiments

in temperatureranges which were convenient and implemented

in practice. The transformation of a substance into a gas phase

in the heating unit is necessary to separate the processes of

evaporation and decomposition in space, which allowed us to

more accurately study the features of the decomposition

reaction. The reaction was performed in a flow-through tube-

type reactor. A sketch of the reactor used is shown in figure 3,

indicating the dimensions and the designation of the inlet and

outlet holes. The choice of this type of reactor is determined by

the type of chemical reaction and the ability to conduct the

process continuously.

If the above conditions are met, the effect of reverse mixing

can be ignored, since the temperature and concentration of the

reaction mixture components do not change along the cross-

section of the reactor. The hydrodynamic mode of the

experimentalreactor was determinedby the abruptintroduction

of the initial carbamate into it without external heating as a

tracer (i.e., by the step perturbation method). Based on the

measurement of its concentration at the reactor outlet, the

system response curves were obtained (Fig. 5).

The reactor was heated using an electric heating element

located on the outer surface of the metal tube of the reactor. In

this case, the temperature profile inside the reactor cannot be

set constant along the length of the entire reactor, because there

are zones of heating the gas phase entering the reactor and

zones of cooling the reactor to the external environment at its

ends. In addition, the temperatureprofile along the length of the

reactor depends on the maximum temperature in the middle of

the reactor and velocity of the carrier gas (Fig. 4).

.2 A mathematical model for the kinetics of carbamates

thermal decomposition

To solve the problem of modeling the decomposition of

carbamate, the necessary calculations of the material and heat

balances were performed. The following conditions were used

for the permanent section tube reactor:

1. The operating mode must ensure uniformity of

parameters across the reactor cross-section, this is provided

either by the turbulent gas flow mode (high flow rates) or by

using distribution nozzles inside the reactor;

Figure 3: Sketch of an experimental reactor.

2. The ratio of the reactor length to the cross-section

diameter l/D must be more than ten units (in our case, the ratio

is 18).

700

650

600

550

500

450

400

350

300

250

200

T_m_a_x=600

T_m_a_x=500

T_m_a_x=400

l, cm

Figure 4: Profiles of temperature along the length of reactor at different temperatures and the carrier gas velocity.

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Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 4, Pages: 1491-1497

0.01

0.008

0.006

0.004

0.002

III

Time,sec

Figure 5: F-curves of response for the ideal displacement reactor (I) and the investigated reactor at different speeds of the carrier gas supply (II-IV)

Figure 6: Diagram of flow modes in the reactor (a – an ideal displacement reactor, b – a laminar flow mode, c – a turbulent flow mode)

The logarithmic nature of the obtained graph allows us to

conclude that the most suitablemodel for describingthe reactor

used is the ideal displacement reactor (Fig. 6). Therefore, this

model was used for further calculations for the tube reactor

used.

ꢁ

where ꢀ is the concentration of the j-th product, mol/L; τ is

astronomical time of the process, sec; w is linear flow rate,

m/sec; υ is stoichiometric coefficient of the substance in the

reaction; r is invariant speed of the i-th stage of the process,

mol/sec; and D is diffusion coefficient of the substance, m /sec.

In the framework of the model of an idealdisplacementreactor,

it can be assumed that there is no transfer of the substance by

diffusion and reverse mixing (푑푖푣(퐷 푔푟푎푑 ꢀ ) = 0). Thus,

.3 Material balance for a perfect displacement reactor

Schematically the decomposition reaction of the

ꢁ

investigated carbamate can be represented as follows:

the change in the concentration of the substance c occurs only

in the coordinate l, and the material balance equation will have

the form:

휕푐푗

휕(푐푗∙ꢆ)

휕푙

푝

ꢂꢃ1

= ∑ 푣_ꢂ_ꢁ∙ 푟_ꢂ

1.2

휕휏

In the course of experiments, the steady temperature

profile, the carrier gas flow rate and the ratio of reagents are

maintained, and the data obtained after the process enters

stationary mode are taken into account when processing the

The general equation of a non-stationary mass transferwith

sources (10):

휕푐푗

휕휏

푝

ꢂꢃ1

푑푖푣(ꢀ ∙ 푤) = ∑^푣ꢂꢁ ∙ 푟 + 푑푖푣ꢄ퐷 푔푟푎푑 ꢀ ꢅ

1.1

^r^e^s^u^l^t^s^,^s^o^w^e^c^aⁿ^a^s^s^u^m^e휕휏 = 0, then:

ꢁ

ꢂ

ꢁ

ꢇ(푐푗∙ꢆ)

ꢇ푙

푝

ꢂꢃ1

∑

푣_ꢂ_ꢁ∙ 푟_ꢂ

1.3

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2020, Volume 8, Issue 4, Pages: 1491-1497

푚

ꢋ

ꢇ

푚ꢜ1

퐸

^푃푡표푡ꢐꢑ

Assuming that c is the concentration of component A

푋 = 푆 ∙ (ꢍ )

∙ 푘 ∙푒푥ꢘ ꢊ−

ꢗ

ꢋ ∙ ꢊ

∙

푁ꢈ

ꢇ푙

퐴

퐴ꢙ

푅푇ꢄ푙ꢅ

푁 ∙푅∙푇ꢄ푙ꢅ

ꢎ

퐶_퐴= _푉

1.4

ꢄꢚ − 푋 ꢅ^푚

2.6

퐴

where где N is the mole flow of component A, mol/sec and V

is volume flow, L/sec. Hence, the material balance for a simple

reaction with varying volume:

Thus, we obtained a differential dependence of the degree

of transformationof substanceA along the lengthof the reactor,

taking into account the molar flow of the inert gas, the molar

flow of substance A at the entrance to the reactor and the

temperature profile.

ꢇꢄꢆ∙ꢉꢈꢅ

−푣 ∙ 푟

1.5

퐴

ꢇ푙

.4 Thermal balance for a perfect displacement reactor

Due to the fact that the temperatureprofile along the length

The equation for the rate of chemical reaction for substance

of the reactor is ascertained and it does not depend on the flow

of a chemical reaction inside the reactor, since the reaction is

performed at high dilution, there is no need to compile a heat

balance. To mathematically describe the temperature profile

along the length of the reactor at different carrier gas flows, a

polynomial function of the temperature of the reactor length

and the volume flow of the inert gas is constructed (2.7). Then

the resulting polynomial is substituted in (2.6).

푚

푟 = 푣 ∙ 푟 = 푘 ∙ ꢄ퐶 ꢅ

퐴

1.6

1.7

퐴

where m is reaction order and substitute in (1.3).

ꢇ

푁ꢈ

ꢊ푤 ∙ ꢋ = −푘 ∙ ꢄ퐶 ꢅ^푚

퐴

ꢇ푙

푉

where = 푆 is a cross sectional area.

ꢆ

ꢟ = ꢟꢄꢠ, ꢍꢅ

2.7

ꢌ

ꢇ푁ꢈ

−푘 ∙ ꢄ퐶_퐴ꢅ^푚

1.8

ꢇ푙

1.5 Determination of kinetic parameters based on

experimental data

Under process conditions, the volume flow depends on

temperature and can be expressed using the ideal gas law with

a correction compressibility coefficient, since ꢍ ≫ ꢍ (the

In equation (2.6), ꢡ (an activation energy), 푘_ꢗ(a pre-

exponential multiplier) and ꢢ (the reaction order) remain

unknown. These parameters are unique for each carbamate. In

this paper, the kinetics of N-benzyl-O-methylcarbamate

decomposition at different temperature profiles is considered.

Due to the fact that the decomposition reaction of carbamate is

reversible, to determine the concentration of the formed

benzylisocyanate at the exit from the reactor, it is bound by

interaction with a secondary amine (methylbenzylamine),

which leads to the urea formation.

퐼

퐴

푁ꢎ

ratio

in experimental conditions from 29 to 1760). This

푁ꢈ

allowed us to get the function of the volume flow from the

temperature.

푁푡표푡ꢐꢑ∙푅∙푇

푍∙푃푡표푡ꢐꢑ

푁ꢎ∙푅∙푇

ꢏ =

2.1

푍∙푃푡표푡ꢐꢑ

where Z is a compressibility factor, Z is equal of 1 for

incompressible gas; ꢍ is the mole flow rate of the carrier gas,

퐼

mol/sec; ꢒ is the gas constant; and ꢓ_ꢔ_ꢕ_ꢔ_ꢖ_푙is the pressure in the

reactor, Pa. The concentration of component A in terms of

molar flows and temperature at a given temperature profile

along the length of the reactor, dependingon the velocity of the

carrier gas:

Figure 7: Reaction of the urea formation in the sorption unit

푁ꢈ

푁ꢈ∙푃푡표푡ꢐꢑ

푁ꢎ ∙푅∙푇ꢄ푙ꢅ

A sample was taken from the resulting solution, and the

concentration of urea and unreacted carbamate was determined

using the HPLC method. Based on the results of the analysis,

the degree of carbamate transformation was calculated for a

seriesof experimentswith different volume flow of an inertgas

and at different temperature profiles. Time, when the mixture

was in the reactor, was derived from the ideal gas law equation,

representing the internal volume of the reactor as the

composition of the cross-section area by the length ꢄ푑ꢏ =

푆푑퐿ꢅ:

퐶_퐴= _푉

2.2

The dependence of the reaction rate constant on the

temperature according to the Arrhenius equation:

퐸

푘 = 푘 ∙푒푥ꢘ ꢊ−

ꢋ

2.3

ꢗ

푅푇ꢄ푙ꢅ

The molar flow of a substance in terms of the degree of

transformation:

ꢌꢇ푙

ꢇ휏

푁ꢎ푅푇ꢄ푙ꢅ

푃푡표푡ꢐꢑ

ꢍ = ꢍ ∙ ꢄꢚ − 푋_퐴ꢅ

2.4

2.8

퐴

퐴ꢙ

Substituting the obtained expressions (2.2), (2.3) and (2.4)

in equation (1.7), we get:

After separating the variables and integrating both parts of

the equation, we get:

ꢌ

ꢇ

퐸

ꢣ

ꢇ푙

푁ꢎ 푅

휏

∙

ꢊꢍ_퐴_ꢙ∙ ꢄꢚ − 푋 ꢅꢋ = − ꢊ푘 ∙푒푥ꢘ ꢊ−

ꢋ ꢋ ∙

퐴

ꢗ

ꢇ푙

푅푇ꢄ푙ꢅ

∫

= _ꢌ_푃

∫ 푑ꢤ

2.9

ꢗ

푇ꢄ푙ꢅ

ꢗ

푚

푡표푡ꢐꢑ

ꢊ푁ꢈ_ꢙ∙ꢄ1ꢜꢝꢈꢅꢋ∙푃푡표푡ꢐꢑ

푁ꢎ∙푅∙푇ꢄ푙ꢅ

ꢛ

ꢞ

2.5

Then, the residence time of the mixture in the reactor is ꢤ,

which will depend on the temperaturefunction along the length

of the reactor.

Open the brackets and make the equation clear:

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Table 1: Dependence of the transformation degree on the temperature mode of the reactor

Maximum temperature in the

The molar flow of the carrier

gas, N (mol/sec)

Time when the mixture was in the

Degree of

transformation, X

№

reactor, Тreactor (˚C)

reactor, τ (sec)

,880

,948

,540

,703

,322

,577

,164

,577

,078

,363

,045

,272

,036

,043

,054

,072

,109

,145

,218

,273

,364

,546

,729

,367

,188

250

300

350

400

450

500

550

600

0,08

0,15

0,25

0,05

0,4

0,664

0,547

0,691

0,878

0,589

0,897

0,791

0,923

0,732

0,964

0,611

0,987

0,755

0,797

0,962

0,786

0,901

0,923

0,948

0,965

0,979

0,985

0,991

0,998

0,994

0,05

0,75

0,05

1,50

0,05

2,50

0,05

3,00

2,5

2,00

1,5

1,00

0,75

0,50

0,40

0,30

0,20

0,15

0,08

0,05

ꢌ푃푡표푡ꢐꢑ

푁ꢎ푅

ꢣ

ꢇ푙

푚ꢜ1

퐸∙1ꢗ3

ꢤ =

∫

3.0

푋_퐴_ꢧ_ꢨ_ꢩ= 푋_퐴_ꢧ+ ꢊ푆 ∙ (ꢍ_А_ꢙ)

∙ 푘 ∙ 푒푥ꢘ ꢊ−

ꢋ ∙

푅∙ꢄ푇ꢄ1ꢗꢗ∙푙 ꢅꢪ27ꢫꢅ

ꢎ

ꢗ 푇ꢄ푙ꢅ

ꢗ

푚

ꢋ

푃푡표푡ꢐꢑ

푚

ꢊ

∙ ꢄꢚ − 푋 ꢅ ꢋ ∙ ℎ

3.1

퐴

As a result, the degree of transformation data was obtained.

푁ꢎ∙푅∙ꢄ푇ꢄ1ꢗꢗ∙푙ꢧꢅꢪ27ꢫꢅ

This conversion rate corresponds to the conversion rate at the

end of the reactor:

푖 = 0. . ꢬ − ꢚ

ꢣ

푋_푓_ꢂ_푛_._ꢥ_ꢦ_푝_.= 푋_ꢄ_푙_푓_ꢂ_푛_._ꢅ

where ℎ = _푛and 푋 = 0. Assuming that the order of the

ꢗ

carbamate decomposition reaction is ꢢ=1, it is possible to find

the remaining kinetic parameters by determining the minimum

deviation of the function from the experimental data. Data for

other orders of reaction ꢢ are not included due to the

inadequacy of the data obtained. For this purpose, the

minimization function (3.3) was established, which allows to

To determine the kinetic parameters, it is necessary to

^r^e^p^r^e^s^eⁿ^t^t^h^e^d^e^g^r^e^e^o^f^t^r^aⁿ^s^f^o^r^m^a^tⁱ^oⁿ^o^f^푋А as a finite-

difference approximation by means of Taylor series expansion.

To solve this problem, it is necessary to represent the length of

the reactor as a set of n segments of finite length and to

determine the degree of transformation 푋 on each segment. In

퐴ꢧ

A 0

find the values of E and k using the gradient descent method

this case, as a result of numerical integration of a nonlinear

differential equation with the parameter:

based on experimentaldata (the reactor temperatureprofile, the

carrier gas flow, and the degree of transformation). A

difference scheme is an explicit Euler scheme.

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ꢄꢡ, 푘ꢗꢅ

The error of the obtaineddata was estimatedusingthe mean

square error (MSE) (3.5) and was 0.009.

푋퐴_ꢙ= 0

ꢍ = ꢚ000

‖

퐿

ℎ =

푚

ꢍ

ꢚ

푀푆ꢡ =

^ꢼ^ꢄ^푋ꢥꢦ푝 ⁻^푋푚ꢕꢇꢅ²

ꢭꢮ푟 ꢯ = 0 … ꢍꢥꢦ푝 − ꢚ

‖

ꢢ

ꢭꢮ푟 ꢯ = 0 … ꢍ − ꢚ

ꢂꢃ1

ꢠꢂ = 푖 ∙ ℎ

‖

ꢡ ∙ ꢚ0^ꢫ

푚ꢜ1

₌₁_.₈₄_·₁₀5

a 0

Thus, the values of E = 54.22 kJ/mol and k

were obtained. The reaction rate constant was calculated from

the Arrhenius equation.

푆 ∙ (ꢍ퐴 )

∙ 푘ꢗ ∙ exp ꢛ−

ꢓꢔꢕꢔꢖ푙

ꢞ ∙

ꢙ

ꢲ

ꢱ

ꢒ ∙ ꢄꢟꢄꢚ00 ∙ ꢠ퐼ꢅ + ꢳꢴꢵꢅ

ꢺ

ꢹ

ꢭ푋퐴_ꢧ_ꢨ_ꢩ= 푋퐴 +

ꢧ

푚

_∙_ꢄ_ꢚ₋_푋_퐴_ꢅ푚

∙

ꢶ

ꢷ

ꢍ퐼 ∙ ꢒ ∙ ꢄꢟꢄꢚ00 ∙ ꢠꢂꢅ + ꢳꢴꢵꢅ

ꢰ

ꢸ

ꢻ

ꢁ = 푋퐴_푗

2 Conclusion

ꢼ

^ꢊ^ꢻ⁻^푋^퐴^ꢽꢾꢿ

ꢋ

ꢄꢵ.ꢳꢅ

During the work, kinetic data for the thermal

decomposition of N-benzyl-O-methylcarbamate were obtained.

Good convergence of the results indicates that the system of

differential equations describes the process of carbamate

decomposition quite accurately. Based on the results a

mathematical model was developed. It allows to carry out

engineering calculations of reactors for obtaining isocyanates

by non-phosgene method. For the reaction of thermal

decomposition of N-benzyl-O-methylcarbamate, the values of

As a minimization criterion in this function, the quadratic

loss function of two vectors was used: the vector of the

transformation degree for experimental data and the vector of

calculated data. As a result, the following values are obtained:

Experiment #

푋_푚_ꢕ_ꢇ

0,651

0,588

0,703

0,863

0,693

0,863

0,692

0,899

0,641

0,963

0,638

0,951

0,731

0,787

0,988

0,92

0,977

0,925

0,933

0,966

0,988

0,992

0,995

0,999

푋ꢥꢦ푝

0,664

0,547

0,691

0,878

0,589

0,897

0,791

0,923

0,732

0,964

0,611

0,987

0,755

0,797

0,962

0,786

0,901

0,923

0,948

0,965

0,979

0,985

0,991

0,998

0,994

the activation energy and the pre-exponential multiplier were

obtained: Eакт = 54.22 kJ/mol and k

= 1.84·10 .

Acknowledgment

Scientific work with the support of the ministry of science

and higher education of the russian federation - federal target

program no. 075-15-2019-1856 of 12/03/19 (unique project

identifier rfmefi60719x0315).

Ethical issue

Authors are aware of, and comply with, best practice in

publication ethics specifically with regard to authorship

(

avoidance of guest authorship), dual submission, manipulation

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.

Competing interests

The authors declare that there is no conflict of interest that

would prejudice the impartiality of this scientific work.

Authors’ contribution

All authors of this study have a complete contribution for

data collection, data analyses and manuscript writing.

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Synthesis of aliphatic, carbocyclic, and fatty-aromatic isocyanates

Experiment #

A.V. Lebedev et al. Russ. J. Gen. Chem. 2006;76(3):. 469-477.

Figure 8: Graphical comparison of the experimentally obtained values

a) of transformation degrees and calculated using the constructed

model (b)

Nagy L, Nagy T, Kuki Á, Purgel M, Zsuga M, Kéki S. Kinetics of

Uncatalyzed Reactions of 2, 4′‐and 4, 4′‐Diphenylmethane‐

Diisocyanate with Primary and Secondary Alcohols. International

Journal of Chemical Kinetics. 2017 Sep;49(9):643-55.

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