Numerical modelling of raw materials atomization and vaporization in a heat carrier gas flow in technical carbon production based on the Euler approach

. Previously, the authors of this article proposed a 3D Euler-Eulerian model of raw materials spraying and evaporation in the heat-carrier gas flow in the carbon production reactor. The raw materials' droplets fragmentation and evaporation are considered as the carrier and dispersed phase interaction. The continuous phase is a heat carrier gas whose motion is described by the Navier-Stokes equations. Dispersed phase (droplets of raw material and products of its evaporation) motion is described using a non-uniform convective-diffusion equation for the particle concentration. The model was tested with water as a raw material. In this work, hydrocarbons are specified as raw materials. It is proposed next to solve the inverse problem with the target functional providing zero contact of the unevaporated raw materials with the reactor wall. Then the number of solid coke particles in the final product will be minimal. An analysis is made of the unevaporated raw material area sensitivity to the inlet velocity of the heat-carrier gas flow. This parameter has the most impact on the distribution area of the unevaporated raw materials and is assumed to be varied in the optimization problem.


Introduction
The furnace process of technical carbon production is a rigid form of hydrocarbons homogeneous pyrolysis in a heat carrier gas flow (decomposition of hydrocarbon raw materials in a high-temperature heat carrier flow) [1].The heat carrier gas is a mixture of natural gas combustion products with a temperature of up to 1900 °C and composition: N2 ~ 73%, H2O ~ 15%, CO2 ~ 7%, and O2 ~ 5%.The fuel complete combustion products enter the zone where they are mixed intensively with injected raw materials, accompanied by rapid raw material drops fragmentation and evaporation.
Figure 1 shows the area near the injection site of raw materials into the reactor of technical carbon production.The raw materials injection takes place using six nozzles evenly distributed in the circumferential direction of the section perpendicular to the reactor axis.The degree of atomization of raw materials depends on the technical carbon structure and the probability of grit formation [2].An important task is to assess the effect of the distribution of the concentration of hydrocarbons in the reactor reaction zone on the structure of carbon black.There are several empirical rules for regulating this indicator in the process of carbon black production, but there is no definitive understanding of the fundamental laws of aggregation yet.
It is necessary to completely evaporate the raw materials before the start of pyrolysis since technical carbon has been forming during the decomposition of hydrocarbons in a gaseous state in obtaining technical сarbon.If raw materials liquid droplets decompose at high temperatures, coke particles polluting the product are formed.One of the ways of coke formation is the deposition of unevaporated raw materials on the reactor's walls.To obtain optimal conditions for E3S Web of Conferences 459, 04019 (2023) https://doi.org/10.1051/e3sconf/202345904019XXXIX Siberian Thermophysical Seminar spraying and evaporation of raw materials, a solution is proposed based on the developed model of the inverse problem: determining the input parameters of the production process that ensure the maximum quality of carbon.As the target functional, zero contact of the unevaporated raw materials with the reactor wall is considered since such contact leads to the formation of coke, which pollutes carbon.The input parameters that have the greatest influence on the region of unevaporated raw materials distribution are the rate of supply of the coolant gas to the reactor, the size, and the speed of the raw material droplets at the nozzle outlet.
The experimental analysis of the technology has its limits associated with the absence or limited capabilities of structural or functional materials and with the insufficiency of the technological level achieved.In this regard, the importance of computational and theoretical analysis of such problems increases.
There are many articles dedicated to the simulation of liquid jet decay in a gas flow.Research [3] develops an effective technique for the numerical simulation of the liquid jet decay in a compressed gas flow based on the Reynolds equations and the volume of fluid method (VOF).The technique describes the primary drops formation and their transition to the Eulerian dispersed phase, where they undergo secondary fragmentation and evaporation.In [4], the authors present an Eulerian multicomponent model of liquid atomization and evaporation based on the WAVE model of jet decay.Special attention is paid to the thermal processes occurring inside moving droplets modeling.A successful simulation of the complex dynamic dense liquid jets' behavior is carried out.In [5], the model of large vortices (LES) was used to describe the spraying of diesel fuel.There was, in particular, studied the assignment of turbulent boundary conditions.The comparison of the results obtained experimentally and numerically according to the proposed model showed good accuracy.In [6], the accuracy of the new Eulerian model of two-phase turbulent flows, which contains the transport equation for the averaged area of the boundary between a liquid and a gas, is estimated.The quality of the liquid spray obtained using this model is also evaluated.In [7], the authors used CFD code ANSYS-CFX 12.0 to analyze the stationary process of the decay and spraying of a liquid fuel jet at various time scales.The Lagrangian approach to tracking the trajectory of liquid particles is applied.In [8], a review of the studied jet decay process is given.Various jet characteristics affecting its decay and can be used to form the maximum interfacial surface due to the fragmentation of the liquid jet into drops are discussed.These characteristics are jet velocity, pressure drop, nozzle geometry, which determines the shape of the jet initial section, the temperature of both phases, their densities, viscosities, surface tensions, and Weber number.In [9], the drops shedding from the liquid surface by an oncoming airflow is numerically studied.The dependence of the spray drops average Sauter diameter on the liquid layer thickness is constructed.In [10], the evolution of fuel aerosol in the combustion chamber of an automobile engine is numerically studied.The CFD analysis, carried out in the study, is turned out to be the only chance to get a complete picture of the fuel atomization process and makes it possible to design the optimal chamber geometry.In [11], a numerical investigation of the kerosene dispersion into the air by the centrifugal single-component atomizer was carried out.The calculation results are compared to the experimental data determining the average Sauter diameter of the spray droplets.It has been established that the ligament constant of the mathematical model is its main parameter, which affects the average Sauter diameter.In [12], a hybrid procedure for the numerical simulation of two-phase flows in a combustion chamber is proposed.It is a two-stage combination of the Euler and Lagrangian approaches.Euler's method gives a preliminary two-phase flow field.The Lagrangian method traces the particles' movement with a detailed description of the secondary droplets' decay.A 30% computational acceleration was achieved compared to the modeling based on the Lagrangian method alone.
In the presented work, the study of the features of the carbon production process in the reactor is continued based on the model created earlier by the authors [13].In this case, real hydrocarbons are set as raw materials.The calculation involves the volume fractions of raw materials, their evaporation product, and gas.The numerical model makes it possible to determine the dispersed composition of the sprayed raw material and the degree of its evaporation.The object of study is a cylindrical chamber of variable diameter, schematically shown in Figure 1.Natural gas combustion products are injected into the reactor's chamber (N2 = 73%, H20 = 15%, CO2 = 7%, O2 = 5%).At a predetermined distance from the inlet, there are six nozzles evenly along the circumference of the cross-section, through which liquid hydrocarbon raw materials enter the chamber.To formulate and solve the problem of the raw materials spraying and evaporation conditions optimization, which ensures the minimization of the unevaporated raw materials with the reactor wall contact, an analysis of the model output data sensitivity to the rate of supply of the heat carrier gas to the reactor was carried out.The results of this analysis are presented.

Mathematical model
The raw materials droplets fragmentation and evaporation in the heat carrier gas flow are modeled within the Euler approach describing the interaction of continuous (carrier) and dispersed phase.The continuous phase is a mixture of natural gas combustion products and evaporated raw materials.The dispersed phase describes the liquid raw materials.In continuous phase, the Navier-Stokes and turbulence equations, which describe the motion, and the equation for the energy and mass transfer of the components are used.For the dispersed phase, the mass, momentum, and energy transfer equations are taken.The transfer of the dispersed phase is described using the convectivediffusion equation for the concentration of particles, as well as the model of droplet fragmentation.

Continuous phase
The equations of continuity and momentum balance in a continuous medium have the form: Here  с ,  с ,   ,  are the volume fraction, the density, the velocity, and the pressure of the continuous phase,    is the source term,   and   are the dispersed phase particle concentration and mass,  ̂ is the effective viscous stress tensor, which depends on the continuous phase dynamic coefficient of molecular and turbulent viscosities.
Turbulence is described using the KEFV model [14].The dispersed phase influences the carrier phase through the terms on the right-hand side of the Navier-Stokes equations describing the motion of the carrier phase (coolant gas).The equation for the energy balance of a continuous phase, written with respect to the total enthalpy  с , has the form Here   is the effective enthalpy diffusion flux,  , and   ℎ are source terms.The mass fraction of the evaporated raw material is determined from: are the molecular and turbulent Schmidt numbers and   is the continuous phase diffusion coefficient of the i-th substance.The evaporated raw materials is added to the heat carrier gas, forming the carrier phase.

Approaches to modeling merging/fragmentation and evaporation/condensation of particles (droplets)
The fragmentation (merging) of drops obeys the WAVE model [15].This model is well-suited for describing high-velocity injections in combustion chambers.According to the model, the decay of a liquid particle occurs due to the development of a perturbation on its surface, associated with the Kelvin-Helmholtz instability.The model makes it possible to calculate the radius of a liquid particle, which is present in the righthand sides of the momentum balance and mass transfer equations for the dispersed phase.Evaporation/condensation processes on the surface of particles are described by the particle mass transfer equation written for the     variable: The values of particle concentration    ,   +1 on two adjacent layers in time are assumed to be known during the equation integration.The source member    is figured out as Here ̇ is the specific rate of the particle phase mass change, ̇ , is the specific rate of i-th substance of the particle phase mass change,   is the number of the dispersed phase substances,  is the particle diameter.

Particle mass, momentum, and energy balance equations
The particles transport is described by the inhomogeneous convective-diffusion equation for the particles concentration: , =  , where   is the dispersed phase (particles) velocity,  , and  , are the dispersed and continuous phase kinematic coefficients of the turbulent viscosity and  , -turbulent Schmidt number.The velocity of the dispersed phase (particles) in the main flow   is determined by the particle momentum transfer equation, which is a non-uniform convectivediffusion equation for the conservative variable       : The force   should be understood as the total force acting on the particle.It is obtained from the addition of repulsive, lift and drag forces.The values of the particle concentration    ,   +1 and their masses    ,   +1 on two adjacent layers in time are assumed to be known during the equation integration.
The dispersed phase energy transfer is described by an inhomogeneous convective-diffusion equation for the conservative variable ℎ      : where ℎ  is the thermodynamic enthalpy of the particle phase.

Problem adaptation to FlowVision SP
To speed up the calculations, the simulation is performed in the 60° sector of the reactor with one nozzle under the assumption that the patterns of interaction of the raw materials with the heat carrier gas and the evaporation of the raw materials in the remaining five segments of the reactor are the same as in the considered one.Symmetry conditions are specified on the side faces of the sector, i.e., there is no normal to the boundary velocity component on them; there is no sticking on this boundary, that is, the flow does not slow down anything; the tangential velocity component at the boundary is equal to the velocity in the boundary cell.A condition with zero pressure is defined (in increments relative to the reference value) at the exit from the computational domain.A condition with a turbulent boundary layer is set on the reactor walls, which is characterized by a logarithmic law of change in the tangential velocity component.The thermal insulation condition is set for the temperature on the walls.The scheme of raw materials and gas injection in the reactor's camera is shown in Figure 2.
To describe the physical properties of raw materials, tabular dependences of viscosity, density and specific heat on temperature are used.For example, at 20 °C, the kinematic viscosity is 24.1 mm 2 /s, the density is 1050 kg/m 3 , and the specific heat capacity is 1693 J/kg•°C.Other coefficients were taken as constants: the thermal conductivity coefficient is 0.13 W/m•K, the molar mass is 183.12 g/mol, the average boiling point is 280 °C, and the heat of vaporization is 270 kJ/kg.The given dependences of the saturated vapor pressure and the surface tension coefficient on temperature were also taken into the calculation.
The conditions for its velocity and temperature (1860 °C) are set at the inlet boundary of the heat carrier gas.The heat carrier gas composition is given through the values of the mass fractions of its components:   2 = 0,73;   2  = 0.15;   2 =0.07;   2 =0.05.In the nozzle spray cylinder, the distribution of its volume fraction is set equal to one.The mode of spraying raw materials from the nozzle is characterized by a particle diameter (1.4 mm), particle velocity (27.07 m/s) and particle temperature (30 °С).

Results
The results of three calculations were obtained with different heat carrier gas flow rates: 350 m/s (variant 1), 453 m/s (variant 2), and 500 m/s (variant 3).The flow patterns depicted from the results of the raw materials spraying modeling in the camera are shown in the plane of symmetry of the reactor sector.
In Figure 3 shows the distribution of the volume fraction of raw materials.At a distance of several nozzle diameters, the volume fraction of the raw material drops by an order of magnitude.The drop jet reaches the chamber axis at the lowest gas velocity.In this case, the interaction of jets, which is not described within the framework of the chosen reactor's sector, from different nozzles is possible.As the velocity of the heat carrier gas increases, a greater deviation of the jet along its flow and a decrease in the area of unevaporated raw materials can be observed.At a maximum speed of 500 m/s, part  of the raw material is in contact with the chamber wall adjacent to the nozzle.
In Figure 4 shows the distribution of the raw materials' droplet diameter.The droplet diameter at the inlet is 1.4 mm.More intensive fragmentation of the raw materials droplets in the reactor volume can be seen with an increase in the flow rate of the heat carrier gas.For a gas flow rate of 350 m/s, drops with a diameter of the order of the initial one reach the axis and propagate along the flow in the second and third sections of the chamber.The droplet diameter drops by an order of magnitude at high speeds when several nozzle diameters are reached.To estimate the change in the parameter, the average values of the particle diameter in the control sections of the reactor were calculated.These sections are located as shown in Figure 5.The values of the raw materials droplet diameter in the indicated sections are presented for three calculation variants in Table 1.
In Figure 6 shows the distribution of the evaporated raw materials mass fraction, defined as       .At a maximum gas flow rate of 500 m/s, due to more intensive fragmentation of the raw material droplets, its complete evaporation is already observed in the second cylindrical section of the chamber.The area of maximum raw material vapor concentration is rapidly blown away by the main gas flow.At the lowest flow rate of 350 m/s, the evaporation area of the raw material is noticeably larger.Complete raw materials evaporation occurs in the last cylindrical section of the chamber.There is a more intense mixing of the heat  carrier gas and raw material vapor toward the chamber outlet.
In Figure 7 shows the gas temperature distribution.An increase in the gas velocity leads to a significant change in the temperature fields: the region of low temperatures shifts upward from the axis of the reactor chamber, and the temperature in the third section of the camera increases on average.The average gas temperature was calculated for the given distributions at the outlet of the camera.The average temperature was 1459 °C in the first variant, 1550 °C in the second and 1590 °C in the third.

Conclusion
Based on the 3D mathematical model proposed in [13], atomization and evaporation of liquid hydrocarbon injected through a nozzle in a carbon production reactor were simulated at three different flow rates of the heat carrier gas.To obtain optimal conditions for spraying and evaporation of raw materials, it is proposed to formulate and solve an inverse problem based on the developed model: determining the input parameters of the production process that ensure the maximum quality of carbon.As the target functional, zero contact of the unevaporated raw materials with the reactor wall is considered since such contact leads to the formation of coke, which pollutes carbon.The input parameters that affect the region of distribution of unevaporated raw materials are the rate of supply of the heat carrier gas to the reactor, the size and velocity of the raw materials droplets at the outlet of the nozzle.An analysis of the non-evaporated raw materials region sensitivity to the rate of the heat-carrier gas supply was carried out.Based on the simulation results, complete evaporation of the raw material was obtained in all considered cases.The droplet flow reaches the chamber axis at the lowest gas velocity (350 m/s).As the velocity of the heat carrier gas increases, a greater deviation of the raw materials jet along its flow and a reduction of the unevaporated raw materials area are observed.At a maximum speed of 500 m/s, part of the raw material is in contact with the adjacent wall of the chamber.With an increase in the flow rate of the heat carrier gas, more intensive raw materials droplets fragmentation is observed -the average droplet diameter value at the outlet of the first cylindrical section of the chamber varies from 105 µm to 52 µm.The average gas outlet temperature is 1500 -1600 °C at 1860 °C at the inlet.
(      )  +  • (        ) = − • (   , ) +  , where   and  , are the continuous phase mass fraction and the effective diffusion flux of the i-th substance,  ,  is the continuous phase i-th substance mass source due to dispersed phase mass change (if the dispersed phase consists of one substance, then  ,

Fig. 3 .
Fig. 3. Distribution of the raw materials volume fraction.

Table 1 .
Average raw materials drop diameter change in the control sections.Gas temperature distribution.