Numerical investigation on cooling performance of filter in a pyrotechnic gas generator

Cheng Cheng,Xiaobing Zhang,Chong Wang,Lu Wang

School of Energy and Power Engineering,Nanjing University of Science and Technology,Nanjing,210094,China

Keywords:Pyrotechnic gas generator Cooling performance Filter Metal fiber

ABSTRACT As a key part of the pyrotechnic gas generator,the filter not only removes the particulate matter but also cools the hot gas to a safe level.This paper aims to improve the understanding of the basic heat and flow phenomenon in the gas generator.The pyrotechnic gas generator is modelling by a simplified filter structure with fiber arrays.A finite-volume model of the heat and fluid flow is proposed to simulate the detailed multi-dimensional flow and energy conversion behaviors.Several verification results are in good agreement with data in different references.Simulation results demonstrate that the filter can not only absorb heat from the gas but also cause the high intensity enhancement of the heat transfer.The performance difference between inline and staggered arrays is also discussed.The findings of the study put a further prediction tool for the understanding and design of the filter system with fibers.

1.Introduction

Pyrotechnic gas generator is a device for generating gas by solid propellant grains.It has been widely used in various industrial,military and aerospace platforms due to many advantages,such as small volume,low weight,and high inflation rate.The major application of the pyrotechnic gas generator is in the airbags of automobiles to reduce the severity of a crash[1].Since Hetrick[2],received a patent called safety cushion assembly for automotive vehicles in 1953,a wide range of airbags has been developed for protecting head,neck,face or thorax from injury during a collision.

In a conventional airbag system,there are three main parts:the air bag,the gas generator(inflator),and the crash sensor[1,3].The propellant in the inflator is ignited and produces hot gas and particles,which is decided by the sensor.This mixture flow is forced through the filter in the inflator,which cools the hot gas and removes solid particles.Next the gas inflates the airbag to make it burst from the storage site.Finally,the gas quickly dissipates through tiny holes so the driver can move.As an important component of the airbag system,the studies on the gas generator were also widely carried out with the airbag development during the last decades.For instance,new propellants were as well developed to improve combustion and gas generating performance[4-6].The structures of gas generators were designed or optimized to enhance the integrative performance[7-10].

The combustion and flow in the gas generator have been also studied both theoretically and experimentally.Numerical simulation is a useful tool to get detailed information on the transient combustion process without disturbing the flow field.Therefore numerical simulations are playing a more and more prominent part in the study of gas generators.In 1988,an average-temperature method was proposed to compute the gas mass rate from inflators by Wang et al.[11]and the detailed internal flow and heat transfer processes of the inflator were not considered.Materna[12]proposed an analytical model to predict the thermodynamic processes including the heat transfer,filtration,combustion,fluid flow in the pyrotechnic inflator.Butler et al.[3,13]established mathematical models to simulate the transient,thermochemical events associated with ignition and combustion of both the single-stage and two-stage airbag inflators.The thermal simulation of a pyrotechnic solid propellant gas generator was also presented by Alkam et al.[14].The gas-phase kinetics model and equilibrium analysis were considered in the thermochemical processes[15,16].The theoretical simulation of combustion and inflation processes of single-stage and two-stage airbag inflators were put forward by Hsieh et al.[17].The modelling of an airbag inflator based on combustion of methane-Oxygen mixture was proposed by Labib et al.[18].The analytical MATLAB/Simulink model of pyrotechnical gas generators for airbags was described by Alcala et al.[19].Numerical tests of the one-dimensional inflator model were conducted to analyze the characteristics of an airbag inflator by Seo et al.[6].Recently Numerical models of automotive airbag inflators in conjunction with the LS-DYNA chemistry solver were developed by Im et al.[20].

As a key part of modern airbag inflator,the filter not only removes the particulate matter but also cools the hot gas to a safe level so that the hot gas does not damage the occupants of vehicles.Due to the complex structure of the filter,the current research study on the filter is carried out mainly in the way of experiments.As can be seen from above researches,there are two kinds of theories for the numerical simulation of the filter.One needs to consider the filter as a porous medium[17,21].The pressure drop and heat transfer through the filter are calculated by the classical porous medium model.The other is to take into account the filter as a whole,and a simplified empirical formula for the mass flow rates through the filter is used to calculate the related pressure drop and heat transfer[13,14,20,22].Therefore,the detailed processes of the heat transfer and flow in the filter are not considered fully,and the heat and flow mechanisms have not been investigated clearly.

The focus of this work is to develop and utilize a twodimensional finite-volume model of the heat and fluid flow in a simplified filter with the goal of improving the understanding of the basic heat and flow phenomenon involved.The combustion and gas generating model will be established to describe the pressure and temperature development at the entrance of the filter.A standard discharge tank will be connected with the filter exit to examine the pressure drop and heat transfer in the filter.The findings of the study will be applied to improve the design and performance of the actual filter in gas generators.

2.Mathematical models

In this section,the schematic diagram of the physical model of an airbag system is described in Fig.1.The simplified airbag system has only three main parts:combustion chamber with igniter,constant-volume discharge tank,and filter.Gas generation propellants in the combustion chamber are ignited firstly and the hot gases are released.The pressure and the temperature increase rapidly in the constant-volume combustion chamber.The flows with condensed-phase particles together go through the surrounding filter.The condensed-phase particles are collected by the metal filter,which also acts as a high surface area heat sink to cool the hot combustion gas.The metal foil ruptures between the filter and discharge tank when the pressure reaches a predetermined burst pressure.Finally,the cooling gas discharges into the tank to take the place of the inflation process in the airbag.In order to reduce the computational cost,the burning process is considered in the combustion chamber by lumped parameter method,and the flow and heat transfer processes are considered in the filter and discharge tank by a two-dimensional axisymmetric finite-volume model.

Fig.1.Schematics diagram of a simplified airbag system.

2.1.Theoretical models in combustion chamber

As described above,the physical process in the combustion chamber is described by lumped parameter method,which can be found in our previous work[23,24].Some fundamental assumptions are made in deriving the governing equations for the combustion process in the chamber.The ignition process is neglected and the propellants are ignited simultaneously at the initial time.The condensed-phase product species are not considered in this paper.

All the propellant grain surfaces are burning under average pressure,and follow the laws of the linear burning[25].Thus,the mass generation rate of cylindrical pellets can be obtained by

where N is the total number of propellants,Aris the instantaneous surface area of a pellet,p is the average pressure in the chamber,and b,n are constants for a given propellant material by the experiment.

The gas density in the combustion chamber can be given by following equation

whereωis the propellant mass,V0is the chamber volume.

Also the gas temperature in the combustion chamber can be obtained Nobel-Abel equation of state

where R is the specific gas constant,αis the co-volume parameter,and T is the gas temperature.

Then the pressure can be calculated by

where pingis the pressure after ignition,f is the impetus of propellant.

2.2.Theoretical models in discharge tank and filter

A two-dimensional axisymmetric finite-volume model is established to describe the flow and heat transfer processes and determine the velocity,pressure,and temperature fields in the discharge tank and filter.The governing equations for mass,momentum and energy of the fluid and solid regions are listed as following.

The continuity equation is

The momentum equation is

The energy equation is

whereτij=μ［（?u+?uT）-2/3?·uI］is the stress tensor,u is the velocity vector,I is the unit tensor,Eis the total energy,keffis the effective thermal conductivity.

The turbulence model is employed due to high speed flows and the realizable turbulence model is used.

where GYSσ,εare the turbulence parameters.

The energy equation of the solid region is

whereρSis the density of the solid region,cpis the specific heat and ksis conductivity of solid region.

The solid region receives heat from the high temperature combustion gases mainly in form of convection.The convection equation is

where q is the rate of convection heat transfer,TSis the temperiatue of the solid region,Tgis the hot gas temperature,and h is convective heat transfer coefficient.

In convection,Nusselt number is a common non-dimensional heat transfer coefficient and represents the enhancement of heat transfer through a fluid as a result of convection relative to conduction across the same fluid layer.

whereδis the characteristic length and k is the thermal conductivity of the fluid.

3.Numerical methods

3.1.Numerical model

The simplified axisymmetric simulation model of inflator with detailed filter structures is shown in Fig.2.The axis in Fig.2 is the rotational center axis of the chamber.The fibers in the filter are set as fiber arrays.The ignitor and core combustion chamber are not shown in the simplified model and their physical processes will be calculated by the mathematical model in section 2.1.All geometry and mesh generation are performed by ANSYS ICEM.Quadrilateral grids in Fig.2 are obtained to improve the computational accuracy and reduce the computational cost.Because the different grid scales of each inflator part,the detailed grid cannot be shown clearly and the girds near the metal fiber is given in Fig.2(c).Grid independence will be discussed in the next section.In this study,approximately 900000 elements are used,and 120 nodes are assigned along the circumference of each fiber.A zonal grid technique is applied to save the running time.

Fig.2.Simplified axisymmetric model and grids.

3.2.Solution methods

A two-dimensional axisymmetric finite volume heat transfer and fluid flow package ANSYS Fluent is utilized to solve the governing equations and determine the pressure,velocity in the fluid region and the temperature in both fluid and solid regions.A density-based solver with explicit time discretization is employed to this heat and compressible flow problem.The advection upstream splitting method is used to obtain the second-order discretization.A realizable turbulence model is applied with standard wall functions.The Nobel-Abel equation of state model is compiled into Fluent by user-defined function(UDF).The parallel computational technique is also supplied with 32 processors.

All fluid and solid regions are initially held at ambient pressure and temperature.The combustion process in the chamber is calculated by the author written code with C programming language.

3.3.Initial and boundary conditions

As shown in Fig.2,the axisymmetric boundary condition is applied at the axis.The pressure and temperature at the inlet can be calculated by the theoretical models in the combustion chamber.Hence the pressure inlet boundary condition is implemented at the inlet,which is coupled into Fluent by UDF at each time step.The outlet boundary is set as no slip wall boundary,but it will be changed as interior boundary when the pressure reaches a predetermined burst pressure.A shadow boundary is created automatically by Fluent and the coupled thermal boundary is applied when considering the heat transfer between fluid and fiber solid region.All other walls are adiabatic by applying the wall boundary and no slip condition.

The initial conditions of the gas generator structure are shown in Table 1.The propellant is NaN3.The material of chamber and filter is carbon steel.

Table 1Input variables for gas generator.

4.Results and discussion

Firstly,numerical validation is undertaken on the case of flow past four-cylinder arrays.Numerical results in the discharge tank of the inflator system are also compared with those in published papers.Then the detailed computational results of a typical gas generator are presented in this section.Finally,the effects of different arrays are investigated.

4.1.Numerical validations

4.1.1.Validation result for four-cylinder arrays

Grid independence studies on the case of flow past four-cylinder arrays are investigated to discuss the effect of the grid size.The four-cylinder arrays are the four circular cylinders placed in a 2×2 array,which the longitudinal and transverse distances between the cylinders are set as double of the cylinder diameter.The description in detail can be found in Refs.[26,27].The nodes of a cylinder varied to be 40,90,120,140 and the corresponding grid size of the computational domain are 4221,18698,39345,50845 respectively.Grid convergence results of average Nusselt number around the first and second cylinders at Re=100 are given in Table 2.The relative error is decreased with the increase of grid sizes.When the nodes of a cylinder are 120,the relative error is only 0.83%and 1.13%for the first and second cylinder respectively.Although further increasing the grid size can reduce the error a bit,the computational time increases significantly.By balancing the computational time and numerical accuracy,one-cylinder node of size 120 is sufficient to ensure the numerical accuracy and the grid independent results.

Table 2Grid convergence results of for four-cylinder arrays at Re=100.

Table 2Grid convergence results of for four-cylinder arrays at Re=100.

Nodes of a cylinder First cylinder Second cylinder Nu Error(%)40 5.4007 2.7895 90 5.3276 1.35 2.6366 5.84 120 5.2835 0.83 2.6067 1.13 140 5.2783 0.10 2.6064 0.01 Nu Error(%)

Based on the grid independence study,validation results are compared with the results in references to test the reliability of numerical methods.Fig.3 shows comparisons of local Nusselt number distributions around the second cylinders at Re=200 in different references.The local Nusselt number is very close to the calculation results by other published papers[26,27].

Table 3Comparisons of simulation results.

4.1.2.Validation results with published papers

The pressure and temperature histories of the gas in the combustion chamber are shown in Fig.4.The maximum value of the gas pressure in the combustion chamber can reach about 4.4 MPa.The maximum value of the gas temperature in the combustion chamber can reach more than 1200 K.If it flows into the airbag directly,the airbag will be damaged.Hence,the filter will play an important role in the gas generator.The pressure and temperature shown in Fig.4 are also the boundary conditions for the simulation of flow behavior in the filter and discharge tank.Comparisons of simulation results in different papers and our present results are given in Table 3.The presented pressure and temperature in the tank show good agreements with results in references.

Fig.3.Local Nusselt number distribution for four cylinder arrays at Re=200.

Fig.4.History of the pressure and temperature in the combustion chamber.

4.2.Results with or without heat to fibers

4.2.1.Pressure distributions

The pressure contours at different times in the filter are shown in Fig.5.The detailed pressure development during the whole cooling and inflation process is revealed clearly.The pressure at the filter entrance increases firstly,due to the gas flowing from the combustion chamber.The pressure at the exit of the filter is lower than other parts of the filter,due to the gas flowing into the discharge tank.Before about t=10.5 ms the pressure in the filter always increases.With the burnt up of propellant and the gas discharge into the tank,the pressure in the filter starts to decrease until reaching the pressure balance.

Fig.5.Pressure contour at different times in the filter.

Histories of mean pressure in the filter and discharge tank are shown in Fig.6 and Fig.7 respectively.The development of the average pressure in the filter is very close to the pressure in the combustion chamber,which is shown in Fig.4.The pressure in the discharge tank rises continuously but there is an obvious drop of the rising rate at about t=35 ms.This is because the pressure in the discharge tank and filter comes closer to balance.The different pressure developments in the filter and discharge tank without the heat transfer to fibers are also given in Figs.6 and 7.When the heat transfers to fibers are considered,the pressures in the filter and discharge tank both decrease.

Fig.6.History of mean pressure in the filter.

Fig.7.History of mean pressure in the discharge tank.

4.2.2.Temperature distribution

The gas temperature contours at different times in the filter are shown in Fig.8.The temperature increases continuously in the filter.Overall,the pressure near the filter entrance is bigger than that near the filter exit,and the temperatures at the corners of the filter are also bigger.When about t=10.5 ms,the temperature reaches 1100 K in the most region of the filter.Some lower temperature region can be found in the gaps of fibers.This is because of the fluid flow behavior in the gaps and will be found in the velocity distribution.

The temperature contours of the filter at different times are shown in Fig.9.With the heat transfer from the high temperature gas in the filter,the temperatures of fibers in the filter increase with time.The temperatures of fibers near the entrance,corners and walls are higher than that in other regions.This is very close to the temperature distribution in the filter.

Fig.10 and Fig.11 present histories of filter and fiber increases with time.And the gas temperature reduces from 1245 K at the inlet to 852 K at the outlet.The average fiber temperature increases with time due to the heat transfer from the filter.The temperatures of filter and fiber near the inlet are both bigger than that in other regions.The different temperature developments in the filter without the heat transfer to fibers are also given in Fig.12.The temperature in the filter decrease due to the heat transfer to fibers.temperature at different zones.The development of the average temperature in the filter is also very similar to the temperature in the combustion chamber as shown in Fig.4.But the gas temperature near the outlet keeps increasing with time.This should be discussed with the velocity changing,which will be described furtherly in the next section.When the gas flows through the filter from inlet to outlet,the gas velocity decreases generally.And the pressure difference between filter and tank decreases by the gas flowing into the tank.Then the gas velocity decreases,which can be seen in Fig.14(c)and the static temperature near the outlet

Fig.8.Gas temperature contours at different times in the filter.

Fig.9.Temperature contours of fibers at different times.

The different temperature developments in the discharge tank with or without the heat transfer to fibers are also given in Fig.13.The gas temperature in the tank increases with time due to the gas velocity reduction gradually and hot gas discharging into the tank.We can expect that the gas temperature in the tank will finally increase to a balance value with the gas temperature at the inlet when the heat transfer to fiber and the heat loss is not considered.When the heat transfer to fibers is considered,the temperature in the discharge tank decreases from 650 K to 542 K due to the heat transfer from the gas in the filter to the fibers.However,there is not enough time for the fully heat transfer in the case of this paper.

Fig.10.History of filter temperature at different positions.

Fig.11.History of fiber temperature at different positions.

Fig.12.History of mean temperature in the filter.

Fig.13.History of mean temperature in the discharge tank.

4.2.3.Velocity distribution

Contour of gas velocity in the filter is shown in Fig.14.The overall distributions of gas flow pattern at different times are very similar.The velocity in the filter increases in the early stage and decreases in the late stage,because the pressure difference between the combustion chamber and discharge tank is gradually reducing.The streamline and local vector of gas in the filter at t=31.7 ms are given in Fig.15.It is obviously shown that the gas flows from the entrance to the exit in the filter.The gas velocity is constantly accelerating and slowing between gaps of fibers.The pressure and temperature in the filter change with the velocity variation.When the gas moves through the filter,the gap between the fibers increases the velocity and decreases the temperature due to the gas expansion.In addition,the increased velocity results in a significant increment in the Nusselt number and heat transfer.The fibers placed in arrays can not only absorb heat from the gas but also cause the high intensity enhancement of the heat transfer due to increasement in velocity.This is the main reason why the multifiber structure is used as the coolant in the gas generator.

4.3.Effects of different arrays structures

A staggered-array structure of the metal fibers in the filter is also simulated in Fig.16.The transverse and longitudinal pitches in the staggered arrangement are the same as those in the inline arrangement.The total number of fibers in the staggered arrangement is less than the number of fibers in the inline arrangement.By comparison of the streamlines in Figs.15 and 16,the streamline in the staggered arrangement is more complex than that in the inline arrangement.The diffusion of the streamlines is more intense in the staggered arrangement as compared to the inline arrangement.The flow field is disturbed by the staggered fibers.The gas from the inlet to the outlet should go through more gaps and accelerating behavior is more obvious than that in the inline arrangement.It will improve the performance of the pressure drop and heat transfer in the filter.

The different pressure and temperature developments in the discharge tank for inline and staggered arrays are shown in Fig.17.As the figures show,using staggered arrangement decreases the pressure and temperature in the discharge tank even with fewer numbers of fibers.It means that the pressure drop and the overall heat transfer from a staggered arrangement are higher than that from an inline arrangement.The higher heat transfer of a staggered arrangement is the advantage for the airbag safety,but the higher pressure drop of a staggered arrangement is the disadvantage for the airbag deployment.Hence,the balance between pressure drop and heat transfer should be considered when the inflator is developed.

Fig.14.Contour of gas velocity in the filter.

Fig.15.Streamline and local vector of gas for the inline arrangement in the filter at t=31.7 ms.

Fig.16.Streamline and local vector of gas for the staggered arrangement in the filter at t=31.9 ms.

Fig.17.History of pressure and temperature in the discharge tank for different arrays.

5.Conclusion

Numerical simulations based on the finite-volume model are carried out for studying the transient heat and flow process in the gas generator.The physical phenomena of the transient flow and heat transfer in the filter are described in detail.The main conclusions are summarized as follows:

(1)Validation results of the flow past four-cylinder arrays are in good agreement with the published results.Comparisons of our predicted results and other results in references demonstrate our present the reliability of the simplified filter model.

(2)The gap between the fibers increases the velocity and decreases the temperature due to the gas expansion.The increased velocity results in a significant increment in the Nusselt number and heat transfer.The pressure and temperature in the discharge tank decrease when the heat transfer to fibers considered.

(3)The heat transfer from a staggered arrangement is preferable to that from an inline arrangement.The fibers placed in arrays can not only absorb heat from the gas but also cause the high intensity enhancement of the heat transfer due to increasement in velocity.However,the higher pressure drop should be taken into account as the disadvantage for the airbag deployment with staggered arrangement.

Furthermore,more influential factors,such as different metal materials,different alignments of fibers should be investigated and an optimization design needs to be provided a suggested design scheme for the cooling performance improvement.This is the future direction for this work.

Acknowledgements

This work is supported by the National Natural Science Foundation of China(Grant No.11972194),theFundamental Research Funds for the Central Universities,No.30918011323,China Postdoctoral Science Foundation funded project(Grant No.2015M581797).