Modeling and simulation of solvent behavior and temperature distribution within long stick propellants with large web thickness undergoing drying

Enfa Fu, Qianling Liu, Yu Luan, Yao Zhu, Weidong He, Zhenggang Xiao

Key Laboratory of Special Energy Materials, Ministry of Education, School of Chemistry and Chemical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, People's Republic of China

Keywords:Stick propellants Drying Large web thickness 3D numerical modeling Heat transfer Solvent behavior

ABSTRACT Drying is a complicated physical process which involves simultaneous heat and mass transfer in the removal of solvents inside propellants.Inappropriate drying techniques may result in the formation of a hard skin layer near the surface to block the free access of most solvent through for long stick propellants with large web thickness,which lead to lower drying efficiency and worse drying quality.This study aims to gain a comprehensive understanding of drying process and clarify the mechanism of the blocked layer near the propellant surface.A new three-dimensional coupled heat and mass transfer(3D-CHMT)model was successfully developed under transient conditions.The drying experiment results show that the 3DCHMT model could be applied to describe the drying process well since the relative error of the content of solvent between simulation and experiment values is only 5.5%.The solvent behavior simulation demonstrates that the mass transfer process can be divided into super-fast (SF) and subsequent minorfast(MF)stages,and the SF stage is vital to the prevention of the blocked layer against the free access for solvent molecules inside propellant grains.The effective solvent diffusion coefficient (Deff) of the propellant surface initially increases from 3.4 × 10-6 to 5.3 × 10-6 m2/s as the temperature increases, and then decreases to 4.1 × 10-8 m2/s at 60-100 min.The value of Deff of surface between 0-1.4 mm has a unique trend of change compared with other regions, and it is much lower than that of the internal at 100 min under simulation conditions.Meanwhile, the temperature of the propellant surface increases rapidly at the SF stage (0-100 min) and then very slowly thereafter.Both the evolution of Deff and temperature distribution demonstrate that the blocked layer near the propellant surface has been formed in the time period of approximately 0-100 min and its thickness is about 1.4 mm.To mitigate the formation of blocked layer and improve its drying quality of finial propellant products effectively, it should be initially dried at lower drying temperature(30-40 °C)in 0-100 min and then dried at higher drying temperature (50-60 °C) to reduce drying time for later drying process in double base gun propellants.The present results can provide theoretical guidance for drying process and optimization of drying parameters for long stick propellants with large web thickness.

1.Introduction

Long stick propellants are widely used in artillery due to many advantages, such as simplified charge configuration and improved interior ballistic performance [1-3].The regular packing of long stick propellants may have higher loading densities than that of granular propellants since their packing is random.For long stick propellant with natural flow channels,the proper venting by partial cutting perpendicular to the axis of the stick can provide a rapid escape of burning gases and effectively reduce the pressure-wave during the burning process, thus improve the safety of propelling charge in service [1,3,4].

Nomenclature Ci Concentration/(mol∙m3)Cp Specific heat/(J∙kg-1∙K-1)Di Solvent diffusion coefficient/(m2∙s-1)R Drying rate/(% per h)hm Mass transfer coefficient/(m∙s-1)ht Heat transfer coefficient/(W∙m-2∙K-1)k Thermal conductivity/(W∙m-1∙K-1)K Evaporation rate constant/(s-1)mi Weight/g Me Equilibrium solvent content/%M0 Initial solvent content/%T Temperature/°C t Time/s v Velocity/(m∙s-1)Volume/m3 dt Time interval/s x, y, z Coordinates SF Super-fast MF Minor-fast 3D-CHMT Three-dimensional coupled heat and mass transfer mt The instantaneous mass at drying moment t of propellant sample Ct The specific heat of stick propellant grains at drying moment t▽▽ Divergence Ni The internal solvent diffusion rate Ri The internal solvent evaporation rate μ The velocity vector P The mean relatively percentage deviation modulus V

Drying is an indispensable production process of long stick propellants [5-7].It not only requires a mass of energy to remove excess solvent, but also causes changes in the physicochemical properties of the final product [8,9].During the drying process of propellant grains, inappropriate drying techniques may lead to poorer performance of propellants and deformation of propellant matrix shape[10,11].The drying should be conducted at a suitable temperature and time range.Urbanski [12] proposed that higher drying temperature at the beginning may cause the propellant grains to swell or crack, but no specific solution was given.And longer drying time may cause the unnecessary consumption of energy and lower drying efficiency.Especially, for long stick propellants with large web thickness[13],it is easy to form a blocked layer near the surface of propellant grains which can immensely prevent excess solvent inside the interior grain from transferring to the outside.The above problems can easily arose from improper determination of drying parameters most of which are selected empirically for long stick propellants with large web thickness during present drying techniques.

So far,some experiment methods have been utilized to optimize important drying parameters of propellants [14-16], such as solvent content, drying temperature, air humidity and so on.Chen et al.[17]demonstrated that the content of residual solvents has a significant influence on the mechanical property of TEGDN propellants by using the auto-mass device.Cheng et al.[18] investigated the effect of the drying temperature on the safety performance of modified double-based propellants.The results show that the fast drying of propellant grain can be realized at 80°C by convective heat transfer.The above experimental methods need large amounts of experiments with high costs and longer time.Since the limitation of existing measurement techniques and hostile experimental conditions, mass and heat transfer of the drying process of propellant matrix are difficult to be directly observed in experiments generally, especially the measurement of solvent concentration variations and gradient [19].Meanwhile, few research has been conducted to investigate the form mechanism and prevention method of blocked layer near the propellant theoretically.Consequently,the accurate drying parameters of long stick propellants with large web thickness could not be determined easily.

Thus,it is imperative to develop a suitable and scientific method to clarify the mechanism of block layer near the propellant surface and determine appropriate drying process parameters of long stick propellants with large web thickness theoretically.Although the numerical method has been well used in the determination of drying parameters and quality control of agriculture products such as vegetables and fruit [9,20], few studies on modeling and simulation of the drying process of long stick propellants with large web thickness have been carried out yet.It should be noted that stick propellants have some unique thermophysical properties different from above drying materials [21].So it is necessary to build a new mathematical model for long stick propellants with large web thickness.

In this study, a new three-dimensional coupled heat and mass transfer (3D-CHMT) model with changing solvent diffusivity was developed for long stick propellants with large web thickness during the drying process.The drying experiment was conducted to validate the established model of long stick propellants with large web thickness.And the solvent behavior and temperature distribution within stick propellant grains were both investigated.The formation mechanism of the blocked layer near the propellant surface was further analyzed, and a facile method to inhibit the formation of blocked layer during drying process was established for long stick propellants with large web thickness.

2.Materials and experimental investigations

2.1.Preparation of long stick propellants with large web thickness

The nitrocellulose-nitroglycerin paste was purchased from Sichuan Nitrocellulose Co.Ltd.The propellant samples were prepared by the semi-solvent method[22].First,200 g dried paste was added in a kneader.The mixed solvent of acetone and ethanol(about 70 mL)with the mass ratio of 1:1 was sprayed on the paste.Subsequently,a dough was obtained after gelatinizing the mixture for 2.5 h,and it was added into a mold with a 19-perforation die.A propellant strand with a web thickness of 2.3 mm and a perforation diameter of 0.5 mm was extruded from the mold.Finally, the propellant sample was dried at room temperature for 24 h.The propellant sample was usually cut with a feasible length,i.e less than 80 mm, for the closed bomb test and other experiments.In addition, the longer the length of the propellant sample is, the more computing resources and calculation time the simulation process needs.Therefore,in this study,the long stick propellant with large web thickness was cut with a length of 60 mm for further investigation.

2.2.Drying experiment

The drying experiment was conducted to determine the change of the content of solvent inside the stick propellant grains.Four drying temperatures (30°C, 40°C, 50°C and 60°C) were selected during the drying experiment.

The instantaneous mass of the propellant sample mtwas measured by on-line auto mass measurement experiment setup developed by our group [17], and the experiment setup is illustrated in Fig.1.The mass of propellant sample used in the drying experiment is 10 g.

The instantaneous content of solvent Ctinside the stick propellant grains is calculated by Eq.(1)

where m0and mtis the initial mass and the instantaneous mass at drying moment t of propellant sample, Ctand C0are the instantaneous content of solvent at drying moment t and initial content of solvent of the propellant sample, respectively.

3.Numerical model

A new 3D-CHMT model was established to describe the drying process of long stick propellants with large web thickness.

3.1.Assumptions

Some assumptions used in the modeling of long stick propellants with large web thickness during the drying process are as follows:

(1) Stick propellants are homogenous and isotropic materials.The shrinkage effect of propellant grains is neglected as the solvent content decreases.

(2) The solvent within the stick propellant grains is transported by liquid flow-through,and the evaporation only takes place at the surface of the propellant grains.

(3) The drying air in the interior perforation of the propellant grains is initially uniform before drying.

(4) There is no heat generation within the stick propellant grains.Operating conditions of surface and interior perforation of the stick propellant grains are assumed alike.

(5) The effect of radiation is neglected during drying process.The heat transfer between the drying air and propellant grains are the coupling of conduction, convection, and radiation.However, the radiation effect is extremely small and neglected in this study.

Fig.1.Schematic diagram of automatic-acquisition drying experiment setup.

3.2.Heat and mass transfer equations

During the drying process,the drying air blows over propellant grains, which makes temperature of propellant surface increases due to the convective heat transfer.Meanwhile, the solvent inside stick propellant grains evaporates on account of increase of ambient temperature.Solvent molecule moves to propellant surface on the effect of convection and diffusion,then is carried away by drying air, which results in the gradual reduce of solvent content,as shown in Fig.2.The governing equation of 3D-CHMT model is established, and it includes heat and mass equations between drying air and stick propellant grains.

During the drying process, the solvent molecule diffusion conforms to the second law of Fick diffusion[14],which is an unsteady process changing with time.The equations for mass transfer are shown in Eq.(2) and Eq.(3)

where Ciis the instantaneous concentration of solvent diffusion,Riis the internal solvent evaporation rate,which represented the rate of solvent evaporation inside the propellant grains,Niis the internal solvent diffusion rate, Diis the solvent inter-diffusion coefficient,which is the physical quantity of solvent molecular diffusion ability that can accurately reflect the diffusion behavior of solvent molecule in the propellant matrix [23].

The heat transfer in stick propellant grains is evaluated by Eq.(4)using Fourier's Law, and similar heat equations are provided by Bhargava et al.[24-26].

where ρ is density of stick propellant grains, λ is thermal conductivity of stick propellant grains, Cpis the specific heat of stick propellant grains,Q is the amount of heat lost due to the evaporation.In this study, the solvent used in stick propellant grains is the mixture of acetone and ethanol.

As heat is transferred from drying air to stick propellant grains,solvent molecule becomes excited by the addition of heat which increases the potential and kinetic of solvent molecule.The continuous addition of more heat causes solvent molecule to diffuse outside the propellant grains and evaporate as vapor.The solvent molecule carries heat away as they leave from stick propellant grains, thus a certain amount of heat is lost.The heat transfer for evaporation of solvent Q from propellant grains are evaluated using Eq.(5)

Fig.2.Schematic of the drying process of stick propellant grains.

where Wdis the total weight of stick propellant grains, Qstis the latent heat of the vaporization of solvent, Vtis the total volume of stick propellant grains, dCt/dt is the time differential of solvent.

3.3.Initial and boundary conditions

The initial and boundary conditions for heat transfer process are defined as: homogeneous temperature at the beginning of the drying process, convection on the surface of the propellant material.Similarly, the initial and boundary conditions for the mass transfer process are defined as: homogeneous solvent content at the beginning of drying process,convective condition on surface of propellant materials.A no-slip boundary condition is applied on stick propellant surface, and same boundary conditions have been used in previous research[27,28].

Initial conditions

where T0is the initial temperature, K.C0is the initial content of solvent inside the stick propellant grains, %.

3.4.Numerical solution

To analyze the drying process of long stick propellants with large web thickness clearly, the area between the end face A and face B was defined as the calculation region of this model,as shown in Fig.3.The face B is a hypothetical face, which is located in the middle of the propellant grains.The length from the end face A to face B is 30 mm.A divided grid of the 3D-CHMT model of the propellant grains is shown in Fig.4.The mesh,composed of 340000 quadrilateral elements,was created by the software on the physical field control condition.The time step of simulation process is 10000, and the time interval is 1 s.

Some material parameters are determined experimentally to solve equations in this study, such as heat capacity of stick propellants at constant pressure (Cp), thermal conductivity (k) and effective solvent diffusivity (Deff) of stick propellants, as shown in Table 1.And the density (ρ) and latent heat of vaporization (λ) of stick propellants are instead taken from previous study [17,20].

Fig.3.3D diagram of long stick propellant with large web thickness.

Fig.4.Divided grid of 3D model of the stick propellant grains.

This model was solved according to the flow chart illustrated in Fig.5.Heat Transfer in Solids and Transport of Diluted Species in COMSOL Multiphysics v4.5(COMSOL Inc.,Palo Alto,USA)were used to handle the heat and mass transfer of the drying process.Then,the time-dependent Backward Differentiation Formula solver was used with the Multifrontal Massively Parallel Sparse Direct Solver to perform the simulation with a time step.

4.Results and discussion

4.1.Experimental validation of established 3D-CHMT model

To validate the established 3D-CMHT model, the average content of solvent of the stick propellant grains is estimated from numerical results and the predicated content of solvent is compared with the experimental values.A detailed comparison of the experimental results with predicted solvent content of the propellant sample at various drying temperatures is shown in Fig.6.The solvent loss process of the propellant sample consists of two parts:the curve of Period I(0-48 h)is very steep,and the curve of Period II (48-168 h) is gently inclined.In Period I, about 75%solvent mass of the propellant sample is removed.The solvent gradient within the propellant grains is relatively high at the beginning of drying, and it is decreased as the drying time increases.In Period II,the solvent loss reaches a stable stage,which is approximately below 2%.After 168 h, the content of residual solvent in the stick propellant grains has almost no obvious change.

It is observed that the rate of solvent loss increases with drying temperature since the solvent diffusivity of the stick propellant grains depends on the temperature [17].The drying time to reach steady content of solvent is decreased as drying temperature T increases because this rise in drying temperature increases the solvent diffusion coefficient,that increases the diffusion rate of solvent from interior to propellant surface.Scilicet,the loss rate of solvent in the stick propellant grains is higher when the drying temperature T is higher, leading to shorter drying time.It shows that a higher temperature of drying air takes less drying time to reach a same residual content of solvent.For example,the time to reach 2%residual content of solvent is decreased by approximately 25%from 120 to 96 h as drying temperature increases from 30 to 40°C.The increase in T from 30 to 60°C reduces the drying time to reach the same residual content of solvent about 150% from 120 h to 48 h.

It is observed in Fig.6 that the simulated results agree well with the experimental values.In this study,the capacity of the 3D-CHMTmodel to represent the experimental data is realized in the terms of the mean relatively percentage deviation modulus P[29,30]and it is calculated by Eq.(8)

Table 1Physical parameters of stick propellants used in the simulation.

Fig.5.Flow chart of the calculation strategy in the simulation.

where N is the total number of compared samples,and eiand piare the experimental and predicated solvent content of samples at the same condition, respectively.

The P-value reflects the accuracy and reliability of the mathematics model.Based on Eq.(9), the P between the predicated and experimental results is only 5.5%in this study.The results indicate that the simulation predicts the general trend of the experimental solvent content curve well, which confirms that the established model can be described the drying process of long stick propellants with large web thickness.It also shows that the established mathematics model of long stick propellants with large web thickness is reasonable, and the solvent and temperature distribution could be achieved based on that model.

Fig.6.Comparison between experimental and theoretical content of solvent at various drying temperatures.

4.2.Solvent behavior

The solvent distribution in the long stick propellants with large web thickness during the drying process at drying temperature of 50°C is shown in Fig.7,with red shades indicating higher content of solvent and purple shades indicating the lower content of solvent.The stick propellant grains are with 20.0% initial content of solvent(mixed solvent of acetone and ethanol).At the initial drying stage, the free solvent molecules on the surface of the propellant sample are dried first and the propellant surface reaches lower content of solvent after a few minutes,resulting in solvent gradient inside the stick propellant grains.The solvent gradient is the rate at which solvent changes, or increases and decreases, between one region and another, and it is a major driving force for solvent transport.In turn, the gradient of solvent content inside the stick propellant grains continues to drive the migration of solvent from the interior to the surface.When the drying process proceeds further,the solvent in the interior of the propellant sample diffuses to the surface and is eliminated to surrounding drying air from the surface further.It is mentioned that unbound solvent is initially removed, and bound solvent takes time to transfer, but these mechanisms happen continuously.The content of solvent inside the propellant grains is gently decreased over time.Finally, the propellant surface achieves steady content of solvent after 100 min,and the content gradient of solvent in the surface is almost lower than 0.5%, indicating that the drying reaches to a steady stage.

Fig.7.Surface of predicated solvent profile of stick propellant grains at drying temperature of 50 °C.

To better investigate the solvent behavior of stick propellant grains, the slices are chosen at the height of 7.5,15 and 22.5 mm along the axial direction in the propellant grains,as shown in Fig.8.Fig.8 represents the content of solvent in the interior of the stick propellant grains.There is a significant solvent gradient in the 19 perforations of the stick propellant grains at 0-60 min,and they are all similar with that of the surface in the propellant samples.Due to the existence of perforations in the propellant samples,the solvent can be fast transferred by the evaporation on the perforation surface.But the heat contact area of the perforations with drying air is relatively smaller compared with the whole propellant grains.The results indicate that the perforation in the stick propellants has a slight positive effect on the process of solvent transfer.

Meanwhile, an obvious difference appears at 0-200 min.Specifically,the content of solvent within the propellant grains has an obvious gradient in three different slices at 0-100 min while it has a more uniform distribution after 100 min, showing the early drying process is divided into two periods: super-fast (SF) stage(0-100 min)and subsequent minor-fast(MF)stage(100-200 min).The blocked layer near the propellant surface formed by over-fast drying prevents the residual solvent from transferring to the outside,and the over-fast drying usually takes place in the SF stage[17].Hence, the solvent behavior in the SF stage should be restrained to inhibit the formation of the blocked layer near the propellant surface during the drying process.The aim of this work is used to analyze the solvent behavior inside the stick propellant grains,which can guide the optimization of drying process of long stick propellants with large web thickness.

Fig.8.Various internal slices of predicated solvent profile of the stick propellant grains at drying temperature of 50 °C.

Fig.9.Diagram of profile in the stick propellant grains.

To further investigate the formation mechanism of the blocked layer near the propellant surface(the profile of long stick propellant see Fig.9), the effective solvent (mixed solvent of acetone and ethanol)diffusion coefficient Deffof the propellant grains at drying temperature of 50°C and 30°C was calculated by the simulation results based on Eq.(3),as shown in Fig.10 and Fig.11.The Deffis the rate at which a diffusing substance is transported between opposite faces of a unit cube of a system when there is unit content difference between them, and it is related to the temperature and physical structure of propellant sample[31,32].At the beginning of drying process, the initial value of Deffof the propellant sample is about 3.4×10-6m2/s.During the initial drying process(0-10 min),the value of Deffis rapidly increased over time and its increasing rate of the surface is more than that of the interior.In this period,the solvent is fast transferred from the propellant grains into the drying air.At the following drying stage.At 60-100 min,the value of Deffwithin propellant grains all decreases over time and its decreasing rate in the surface is more than that of the interior.At the ending drying stage (after 100 min), the value of Deffof the surface decreases to only 8.1 × 10-8m2/s but the interior is 2.7 × 10-7m2/s, indicating that the propellant surface is severely dried and has formed a hard skin layer to block the free access of most solvent through.Compared Fig.11 with Fig.10, the similar phenomena do not appear after 100 min and little blocked layer near the propellant surface is observed for propellants dried under 30°C, showing initial lower drying temperature can effective inhabit the formation of blocked layer for long stick propellants with large web thickness.As a result,the solvent is being held at the internal and only few solvent molecules could diffuse out to the surface due to the formation of the blocked layer near the propellant surface.The above results show that the general effective solvent diffusion coefficient of the propellant grains is enormously decreased after the middle drying stage (60-100 min).That is the main reason why the residual solvent within long stick propellants with large web thickness could not be completely removed during the later drying process.And this further indicates that the blocked layer near the propellant surface plays an important role in the mass transfer process and it prevents more residual solvent from transferring to the outside of the propellant grains.

Fig.10.Effective solvent diffusion coefficient contour in section of stick propellant grains at drying temperature of 50 °C.

In this study, the solvent diffusion behavior inside the stick propellant grains conforms to the Vrentas-Duda model[33],and it is largely affected by temperature and physical structure of propellant grains.The blocked layer near the propellant surface could limit the movement of polymer chains(the main component of the stick propellant)and result in the reduce of its free volume,leading to the difficult diffusion of solvent molecules in the propellant grains.So the value-Deffof the propellant surface is decreased as the blocked layer is formed.

Fig.11.Effective solvent diffusion coefficient contour in section of stick propellant grains at drying temperature of 30 °C.

Fig.12.Effective solvent diffusion coefficient distribution on the centerline of face B at drying temperature of 50 °C and 100 min.

Fig.12 shows the effective solvent diffusion coefficient (Deff)distribution on the centerline of face B at drying temperature of 50°C and 100 min, and all data points of Deffare chosen at the equidistance(0.5 mm)along the centerline.It is obviously observed in Fig.12 that the value-Deffat the internal has vibrational distribution along the centerline between the perforations.Combined with solvent transfer results, the solvent content at the surface is much lower than that of the internal at 100 min, and the solvent content is first increased and then decreased along the centerline from the perforation to adjoining perforation.The nitrocellulose chains, the main component of long stick propellants, have lots of hydroxyl groups and may be linked by hydrogen bonds between the chains at the lower solvent content,resulting in the significant reduction of free volume and ultimately decreasing the solvent diffusion coefficient of the propellant surface and inner perforation surface.So the value-Deffat the propellant has similar change rule along the centerline between the perforations at 100 min compared with the gradient distribution of solvent content, indicating that the solvent content has a slight positive effect on the value-Deffof the propellant grains at 100 min.As shown in Fig.10,the effective solvent diffusion coefficient of propellant surface between 0-1.4 mm has unique rate of change compared with other region,and it is much less than that of the internal, further indicating the blocked layer near the propellant surface has been formed and its thickness is about 1.4 mm.

Fig.13.Effective solvent diffusion coefficient distribution on the centerline of face B at drying temperature of 30 °C and 100 min.

Fig.13 shows the effective solvent diffusion coefficient distribution on the centerline of face B at drying temperature of 30°C and 100 min.As can be seen from Fig.13,the value-Deffat the stick propellant grains has vibrational distribution along the centerline between the perforations.Compared Fig.13 with Fig.12,the unique trend of distribution of value-Deffat the propellant surface between 0-1.4 mm disappears as the drying temperature decreases from 50 to 30°C.The results show that the formation of blocked layer has been effective mitigated at drying temperature of 30°C in 0-100 min.These results could be applied to improve the drying quality of final propellant products and prevent the formation of the blocked layer near the propellant surface.For example,the long stick propellants with large web thickness are initial dried at lower drying temperature (30-40°C) at 0-100 min and then dried at higher drying temperature (50-60°C) to reduce drying time for later drying process.For the case of this experiment,it can effective inhabit the formation of blocked layer and improve drying quality for long stick propellant with large web thickness based on this method.

Based on this method (using different drying temperature combination)in this study,the stick propellant grains are first dried at 30°C in 0-100 min and then dried at 60°C after 100 min,and the evolution of solvent content within the propellant grains is shown in Fig.14.Compared with that propellant samples dried at 60°C,the drying time to achieve same solvent content(2%)is reduced by 15% and the residual content of solvent for finally propellant product is reduced by 20% under different drying temperature combination condition, showing the drying quality and drying efficiency are obviously improved under this condition for long stick propellants with large web thickness.

Fig.14.Comparison between content of solvent for propellants dried at 60 °C and different drying temperature combination.

4.3.Temperature distribution

The simulation of temperature distribution is conducted to investigate the heat transfer in the stick propellant grains during the drying process.Fig.15 displays the evolution of average temperature of stick propellant grains under simulation condition.As shown in Fig.15,the increase of temperature at 60°C is faster than that at lower drying temperatures due to the larger temperature difference between stick propellant grains and drying air.As drying temperature increases,heat transfer via convection from drying air to propellant surface and subsequently by the means of conduction to the center of stick propellant grains occurs and causes the increase of temperature of the propellant sample.The slope of simulated temperature curves is very steep and the slope of the curve at 60°C is higher than that at other temperatures, showing that there is a rapid thermal energy transfer process.Moreover,the rate of heat transfer is reduced over time,which is reflected by the slope going to zero.

Fig.15.Simulated data showing the average temperature of stick propellant grains at various drying temperatures.

The temperature contour inside the propellant grains is estimated for all drying temperatures(30,40,50,and 60°C),and they have similar trend of change.In this paper, the temperature contours at drying temperature of 30°C and 50°C for further investigation are respectively displayed in Fig.16 and Fig.17.

Fig.16 clearly shows that the surface of the propellant grains gets heated initially in 10 min because of convection heat transfer between the propellant and the drying air.The heat is conducted into the interior region of stick propellant grains as the interior one of stick propellant grains has a low temperature region compared with the surface region.At the early drying stage, the heat consumed by evaporation exceeds that transferred from propellant surface under lower drying temperature condition and evaporation rate in the internal is larger, resulting in a slightly dropping of internal temperatures [34].When drying process proceeds further,heat transferred from propellant surface is greater than that consumed by evaporation, and internal temperatures start to increase.The propellant grains finally approach the drying temperature and the surface temperature difference is no longer noticeable.The temperature gradient between the surface and interior of the propellant grains are 1.2°C at 10 min and 1.4°C at 30 min.But the temperature gradient decreases to 1.1,1.0 and 0.1°C at 60,100 and 200 min, respectively.It means that the propellant samples will reach their steady state when the drying time is increased till 200 min.

Fig.17 shows the temperature profiles at the drying temperature of 50°C and at different drying times such as 0,10,30,60,100,and 200 min.The nature of Fig.17 is the same as explained in Fig.16.The temperature gradient between the surface and the center of propellant grains is 3.8,2.5,2.0,1.6 and 0.1°C at drying times of 10,30,60, 100 and 200 min, respectively.It shows that the propellant temperature increases when drying time increases, and it reaches its steady state when the drying time is 200 min approximately.

Fig.18 shows the temperature distribution on the centerline of face B at different time points and 50°C.When drying process proceeds further, the temperature difference ΔT (at the same time difference)is decreased over time,and the ΔT at the first 100 min is obviously more than that of later drying stage,showing the transferring heat is decreased over time.The results indicate that the heat transfer between the propellant grains and drying air is not uniform in the time dimension during the drying process.

Fig.16.Three internal slices of predicated temperature profile of the stick propellant grains at drying temperature of 30 °C.

Fig.17.Three internal slices of predicated temperature profile of the stick propellant grains at drying temperature of 50 °C.

Fig.18.Temperature distribution on the centerline of face B at different time points and 50 °C.

Fig.19.Average temperature of long stick propellant grains and effective solvent diffusion coefficient of the propellant surface at drying temperature of 30 °C and 50 °C.

Fig.19 shows the average temperature of long stick propellant grains and effective solvent diffusion coefficient of the propellant surface at drying temperature of 30°C and 50°C, respectively.According to Vrentas-Duda model, the higher the temperature is,the faster solvent molecules diffuse, and the larger the effective solvent diffusion coefficient is.Combined with heat transfer and effective solvent diffusion coefficient results, after 100 min, the temperature of the long stick propellant grains is gently increased until the steady state is reached,the value of Deffof the propellant surface continues to increase over time at drying temperature of 30°C but decrease at drying temperature of 50°C, as shown in Fig.19.Both these results demonstrate the blocked layer near the surface has been formed in the time period of approximately 0-100 min under initial higher drying temperature and it results in the reduction of the effective solvent diffusion coefficient of propellant surface.

For further elucidate its mechanism theoretically,it is deserved to evaluate the solvent behavior,Deffand temperature distribution inside propellant grains under different drying temperature combination.However,the numerical simulation of the combination of different temperature drying process is very complicated and is difficult to converge based on present numerical methods in this study.Moreover, comparison with the actual temperature, all values of the stick propellant grains by established model are predictive due to the limitation of present measure technology.Further efforts need focus on these issues to analyze experimentally the heat transfer effect of different drying conditions on the drying process of long stick propellants with large web thickness.

5.Conclusions

This study contributes valuable knowledge for the clarification of the formation mechanism of blocked layer near the propellant surface and drying behavior of long stick propellants with large web thickness.In this paper, a new 3D-CHMT model was successfully developed and it can be used to investigate the solvent behavior and temperature distribution within the stick propellant grains.

The solvent behavior shows that the predicted change of the content of solvent during the drying process is decreased over time,and the solvent transfer can be divided into SF and MF stages,and the SF stage can be restrained to inhibit the formation of the blocked layer near the propellant surface.The value of Deffof the propellant surface first increases from 3.4 × 10-6m2/s to 5.3 × 10-6m2/s with the increase of temperature, and then decreases to 4.1 × 10-8m2/s at 60-100 min.The value of Deffof the propellant surface between 0-1.4 mm has a unique trend of change compared with the other regions,and it is much lower than that of the internal at 100 min.Meanwhile,the temperature of propellant surface increases rapidly at the SF stage(0-100 min)and then very slowly thereafter.Both the evolution of Deff and temperature distribution results demonstrate that the blocked layer near the propellant surface has been formed in the time period of approximately 0-100 min and the thickness of the formed blocked layer is about 1.4 mm.To avoid the formation of blocked layer and improve drying quality of finial propellant products effectively, it would be better to adjust the drying parameters.For example, for long stick propellants with large web thickness, it should be initially dried at lower drying temperature (30-40°C) in 0-100 min and then dried at higher drying temperature(50-60°C)to reduce drying time for later drying process.The results can provide theoretical guidance for drying process and optimization of drying parameters for long stick propellants with large web thickness.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Grant No.22075146).