Applied Thermal Engineering 130 (2018) 1012–1021

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Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Performance variation of a transient dynamic fluid-structure interaction

system in different life stages and methods for maintaining the
performance
Chaobin Hu, Xiaobing Zhang ⇑
School of Energy and Power Engineering, Nanjing University of Science and Technology, China

h i g h l i g h t s

A model is built to control the combustion proceeded in an expanding chamber.

Performance of the gun decreases with increasing firing cycles.

The gun performance is changed by the influences of wear on the combustion.

Two kinds of methods are provided to maintain the performance of the gun system.

a r t i c l e i n f o a b s t r a c t

Article history: Performance variation in different life stages of dynamic systems subjected to severe fluid-structure
Received 13 December 2016 interactions is unavoidable. A weak coupling model governing the combustion of propellants in a cham-
Revised 13 November 2017 ber with moving boundaries was presented. The model is capable of considering not only the interactions
Accepted 15 November 2017
between the projectile and the barrel but also the volume enlargement caused by wear and erosion. The
Available online 15 November 2017
weak coupling of the model calculation was realized using an interface in the commercial ABAQUS(R)
software. The weak coupling model was validated through comparing with a classical model and exper-
Keywords:
iments. The diameter increments of a gun bore were measured to construct models used in the simula-
Thermodynamics
Dynamic fluid-structure interaction
tions. Based on the validated model, the performance variations of the gun in different life stages were
Wear investigated. Furthermore, pressure and temperature distributions of the core flows along the barrel were
Weak coupling provided. Mechanism of the performance degradation was elucidated. Finally, two kinds of approaches
Finite element method were provided to maintain the performance of the internal combustion system. Considering the sensitiv-
ity of the combustion to the two factors, a method of adjusting the charge mass is advisable in
engineering.
Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction the gun bore and the bottom of the propelled projectile. In the
meantime, the projectile moves forward and impacts the barrel
Wear is a kind of unavoidable phenomenon in mechanical sys- fiercely. As a result, the inner wall of the barrel undergoes mechan-
tems, especially for the ones undergoing extreme conditions such ical, thermal and chemical actions simultaneously. Therefore,
as periodic mechanical interactions [1], high pressure and high every firing cycle is accompanied by a slight wear and erosion of
temperature internal fluids [2]. Internal combustion engines are the barrel which is made of steel. When the firing cycles is great
examples of this kind of mechanical systems. The performance of enough, wear and erosion of the barrel will be accumulated to
a mechanical system changes with increasing powering cycles. obviously influence the motion of the projectile and the combus-
Gun is a kind of single stroke combustion engine, which can propel tion of the propellants. Furthermore, the thermal loads applied
projectiles utilizing energy from propellants [3,4]. While the on the boundary of the combustion chamber are changed. There-
engine is working, huge amounts of high pressure and high tem- fore, it is necessary to establish coupling models to study the influ-
perature gaseous products start to interact with the inner wall of ence of wear and erosion on the combustion of the energetic
materials, which can provide accurate loading conditions for the
heat transfer in the chamber and material ablation caused by the
⇑ Corresponding author.
thermal shocks.
E-mail address: zhangxb680504@163.com (X. Zhang).

https://doi.org/10.1016/j.applthermaleng.2017.11.075
1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.
 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021 1013

The theories and mechanisms of wear and erosion were where r_ is the burning rate of the propellant, u1 the burning rate
explained by Johnston [5], and Sopok [6] et al. in detail. They coefficient, n the burning rate exponent, p the spatial average pres-
thought that the combination of thermal effects, chemical effects sure in the chamber, Z the relative burnt thickness of the propellant
and mechanical effects are leading causes of wear and erosion in and Zk the initial thickness of the propellant. Both u1 and n are
gun bores. And the thermal effects affect a gun life the most, obtained through experiments.
whereas the mechanical influence is the least. In order to protect The chemical energy release of the propellants is determined by
the inner surface of barrels and improve the life time, Underwood, Eq. (2).
Vicenzi, Guilemany, and de Rosset et al. [7–11] studied wear and 8
< vZð1 þ kZ þ lZ Þ ðZ < 1Þ
2
erosion in coated gun bores. In those researches, a general law of >
wear distribution in gun bores was summarized. With the increas- w ¼ vS Z ð1 þ kS Z Þ
Z Z
ð1 6 Z 6 Z k Þ ð2Þ
>
: K K
ing of firing cycles, wear and erosion are increasingly serious along 1 ðZ P Z k Þ
the full length of the gun bore. The forcing cone near the breech
suffers the worst conditions, wear and erosion at this part is much where v, k, l, vS and kS are shape-dependent constants of the propel-
more serious than the other parts of the barrel. Meanwhile, many lant, w the relative burnt mass of the propellant. And the propellants
methods were proposed to protect gun bores from material abla- used in the present work are seven perforated propellant grains, of
tion or slow down the speed of the ablation. For example, de Rosset which the cross section is shown in Fig. 2. The web thickness of
et al. [12] put forward a device that can relieve muzzle erosion. the propellant is 2e1, and dp0 is the inner diameter of the holes.
However, few of them studied how the internal combustion is Based on Newton’s second law, the movement of the projectile
influenced by wear and how to maintain the performance. along the barrel is controlled by Eq. (3).
The nature of an interior ballistic process can be summarized as
mv_ ¼ S  pd  F R ð3Þ
combustion of the propellants in a chamber with moving bound-
aries. Bottom of projectiles play the role of the moving boundaries where m is the projectile mass, v the projectile velocity, v_ the accel-
[13,14]. Not only is the combustion space in the chamber eration, S the effective cross-sectional area of the barrel, pd the pres-
increased, but also the clearance between a projectile and the bar- sure following the projectile, FR the resistance acting on the
rel. As a result, resistance acting on the projectile is influenced, projectile.
which affects the movement of the boundaries. Therefore, the pro- Energy conservation is applicable for isolated systems. We took
pellants combustion in the chamber is not consistent in different the propellant gases as the subject investigated to establish the
life stages of the gun. However, mechanical interactions between energy conservation equation. Based on the first law of thermody-
a projectile and the gun bore are simplified in traditional internal namics, the governing equation of the energy conversion is derived
combustion models. The interaction between a projectile and its and shown in Eq. (4). In this research, the volume increase caused
barrel was studied by Chen, Lisov, Alexander and South et al. by wear of the barrel is considered.
[15–18] with experimental and numerical methods. And List Z
dðxÞ  f d ðxÞ
l
et al. [19] studied the mechanism of friction and wear at high slid- Spðlw þ lÞ þ pp dx
ing speeds. Almost all the researchers concentrated their works on 0 2
the response of gun bores under loads of interior ballistics. The pre- Z l
1
sent work focused on the performance variation of a gun bore over ¼ f xw  ðk  1Þ S  pd ðxÞ  dx  ðk  1Þ  K  mv 2 ð4Þ
0 2
firing cycles and provided methods to maintain the performance.
where lw ¼ l0 ½1  D=qp  Dða  1=qp Þw, l0 is the equivalent length
2. Mathematical model of the chamber under the effective cross-sectional area of the barrel,
l0 ¼ V 0 =S, V0 the initial volume of the chamber, D the charge den-
Function parts of a gun are illustrated in Fig. 1. The system sity, qp the mass density of the propellant, a the equivalent covol-
mainly consists of a barrel, a projectile and propellant grains. The ume of the propellant gases, l the projectile travel, dðxÞ the initial
rotating band is embedded on the projectile. The outer diameter inner diameter of the gun bore, f d ðxÞ the inner diameter increment
of the band dB is larger than the inner diameter of the barrel d0. caused by the wear, f the propellant impetus, x the charge mass, k
Propellant combustion obeys exponential burning rate law in the equivalent adiabatic exponent of the propellant gases, and K is a
Eq. (1). coefficient that represents energy flows that are proportional with

the translational kinematic energy of the projectile,
u1 pn ðZ 6 Z k Þ
r_ ¼ ð1Þ K ¼ x=m=3 þ m=Mð1 þ x=mÞ, M is the effective mass of the recoil
0 ðZ P Z k Þ system.

Copper crusher gauge Combustion chamber

Forcing cone
Rotating band

Gun bore

Propellants and
Igniter Projectile
dB

d0

propellant gases

Fig. 1. Illustration of the main function parts in a gun.

1014 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021

element method [23] etc. Among those methods, FEM is the most
maturely developed and widely used method. Therefore, the
dp0 resistance FR was calculated using FEM based on the commercial
ABAQUS(R) software. A subroutine interface, VUAMP, was
employed to realize the weak coupling simulations. VUAMP is a
user subroutine interface in ABAQUS, which is used to provide
2e1 variable amplitudes. In the present work, parameters such as
the projectile travel, velocity and acceleration are passed into
the user subroutine, which is programmed in FORTRAN, to
obtain the pressure acting on the projectile bottom. Then the
obtained pressure is returned to the FEM calculation through
the interface. A computational procedure is shown in Fig. 3. In
the figure, lg is the travel of projectile when it reaches the muzzle
of the barrel.
The system is divided into two sub-systems. One sub-system
controls the mechanical interactions between the projectile and
the barrel. The governing equations for the system is given in Eq.
(8). The aim of solving the equation is to obtain the position, the
Fig. 2. Cross section of a seven perforated propellant grain. velocity and the acceleration of the moving boundary, which are
traditionally obtained by solving the third and fourth equations
in Eq. (7) in the traditional model.

3. Realization of the model simulation €ðtÞ þ C uðtÞ

Mu _ þ KuðtÞ ¼ Q ðtÞ ð8Þ

Auxiliary equations are provided to close the equation system.

where M, C and Q are the mass matrix, the damping matrix and the
The travel of the projectile is associated with its velocity, which
nodal load matrix, respectively; u; u_ and u
€ are the nodal displace-
is shown in Eq. (5).
ment, velocity and acceleration vector, respectively.
The other equations in Eq. (7) belong to the sub-system that
dl
v¼ ð5Þ controls the propellants combustion in the chamber with
dt
moving boundaries, which is solved with a FORTRAN program.
The active force used in Eq. (3) is the shot-base pressure. How- The two sub-systems provide parameters for each other through
ever, a lumped parameter method was used in the theory of clas- the interface in real time. The models and the related parame-
sical interior ballistics and the pressure used in Eq. (4) is a ters of the two sub-systems are compared with each other in
spatial average pressure. Based on Lagrange hypothesis and the Table 1.
law of mass conservation, the relationship between the shot-base Obviously, there are mainly two aspects of differences between
pressure and the spatial average pressure was deduced and shown the classical model and the modified model. The two aspects are
in Eq. (6). given as following.

x_
pd ¼ p  v ð6Þ
3S
Initial
Then the system of simultaneous equations is shown in Eq. (7).
parameters
8 8
>
>
>
>
>
> vZð1 þ kZ þ lZ2 Þ ðZ < 1Þ dt, dv/dt, v, l
>
> <
>
>
>
> w ¼ vS ZZ ð1 þ kS ZZ Þ ð1 6 Z 6 Z k Þ
>
> >
> K K
>
> >
:
>
> 1 ðZ P Z k Þ
>
>
>
> ( u1 n
>
> Interior ballistic Mechanical
>
> p ðZ 6 Z k Þ
> dZ ¼ e1
< dt calculation interaction calculation
0 ðZ P Z k Þ
>
> (VUAMP) (ABAQUS)
>
> v ¼ dl
>
>
>
> dt
>
>
>
> dv
> dt ¼ S  pd  F R
> m
>
> p, l
>
> Rl Rl
>
>
> Spðlw þ lÞ þ pd 0 pðxÞf2 d ðxÞ dx ¼ f xw  ðk  1Þ 0 S  pd  dx  ðk  1Þ  K  12 mv 2 pd
>
>
>
: x dv
pd ¼ p  3S dt

ð7Þ No
l >lg?
In Eq. (7), the resistance acting on the projectile mainly stems
from the interaction between the projectile and the barrel. In
Fig. 1, the projectile and the gun bore are assembled with interfer- Yes
ence fit. Guided by the forcing cone, the rotating band is gradually
engraved into the barrel. The calculation of the force belongs to the
category of solid mechanics. With the rapid development of com- Stop
puter aided engineering, many methods have been developed to
solve those kinds of problems. The methods include finite element
method (FEM) [20,21], discrete element method [22] and boundary Fig. 3. Procedure of the simulation.
 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021 1015

Table 1
The two sub-systems.

Sub-system The mechanical interactions The internal combustion of the propellant

8 8
The main governing equations € ðtÞ þ C uðtÞ
Mu _ þ KuðtÞ ¼ Q ðtÞ > < vZð1 þ kZ þ lZ 2 Þ ðZ < 1Þ
>
>
>
> w ¼ vS ZZ ð1 þ kS ZZ Þ ð1 6 Z 6 Z k Þ
>
> :
>
> 1
K K
ðZ P Z k Þ
<
u1 n
p ðZ 6 Z k Þ
>
>
dZ
¼ e1
>
>
dt 0 ðZ P Z k Þ
>
> Rl Rl
>
>
> Spðlw þ lÞ þ pd 0 pðxÞf2 d ðxÞ dx ¼ f xw  ðk  1Þ 0 S  pd  dx  ðk  1Þ  K  12 mv 2
: x dv
pd ¼ p  3S dt

Variables provided for the u; u_ and u

€ of a node at the projectile pd (Pressure applied on the projectile bottom.)
other sub-system bottom; dt
The main parameters in the Material parameters for the calculated Charge parameters, such as the mass, the charge density and the projectile mass
sub-system mechanical parts Performance parameters of the propellant, such as the propellant impetus, shape-related
Geometry parameters to determine constants and the burning rate coefficients.
the structures.

(a) In the modified model, the mechanical interactions are 4. Results and discussions
directly calculated using the FEM in ABAQUS to provide
the states of the moving boundary. However, the interac- The modified model presented above was validated firstly. Then
tions are simplified in the classical model and the motion a case was given. For the investigated gun, the geometrical change
of the projectile is obtained by utilizing a coefficient of caused by wear of the barrel was considered in the FEM models.
energy losses. And the FEM models were constructed based on measured data.
(b) In the modified model, the volume enlargement caused by
wear of the barrel is considered in the energy equation. 4.1. Verification and validation

Unifying of the time steps for the two sub-systems should be A medium caliber gun was studied in the present work. Mate-
ensured during the simulations. The internal combustion of the rial parameters of the main parts are provided in Table 2. Consid-
propellants is calculated using a fourth order Runge-Kutta method ering the difference in Young’s modulus, we regarded the barrel
[24]. Meanwhile, a global convergence of the calculation should be and the body of the projectile as rigid bodies. The assumption,
guaranteed, especially for the computational mechanics. A stable which was also used in literature [15], remarkably reduces the
time step of a FEM calculation is determined by mesh size. The computational cost. A J-C model [25] was used to describe the
mesh size was determined using a simple trial and error method. mechanics of the rotating band. Material damage of the rotating
As the projectile travel and velocity are two critical parameters band was considered.
in the coupling process of the modified model, we chose the two In addition to the initial pressure, the initial conditions for the
parameters obtained from the FEM calculation in ABAQUS as the simulations of the two models were the same. Different from the
indicators. The velocity variations over the projectile travel are modified model, the mechanical interactions in the classical model
provided in Fig. 4. In the calculations, the pressure applied on the was simplified using a coefficient that roughly summarize the
projectile bottom is linearly increased to 200 MPa from zero in a energy dissipations [26,27]. The initial pressure in the classical
duration of seven milliseconds, which is close to the loading rate model is used to counteract the influence caused by the simplifica-
in a firing process. Obviously, the results are close to each other tion. Every gun has a unique value of the parameter and it reaches
when the mesh size of the rotating band is smaller than 0.7 mm. 30 MPa for the studied gun. Basing on experiments about ignition
Taking into consideration of the computational cost, we select a uniformity, we assigned a 10 MPa initial pressure to the modified
mesh size of 0.5 mm as the proper mesh size to discrete the rotat- model. Key variables were compared with the classical model to
ing band in the study. verify the simulation. The results are shown in Fig. 5. The velocity
difference is obtained by subtracting the velocity of the modified
model from that of the classical model. The results from different
models agree well with each other.
In addition, the results were compared with experiments in
Table 3. It should be noted that the maximum pressure was mea-
sured using copper crusher gauges located at the breech of the
chamber, which is shown in Fig. 1. The calculated maximum pres-
sure was extrapolated based on the spatial average pressure. In
addition, the muzzle velocity was measured by a Doppler radar
located at the front side of the barrel. Obviously, the comparison
indicates that the modified model is capable of solving such kinds
of transient internal combustion problems.
The model validation was carried out on a new gun. Therefore,
further validation over a worn gun should be added.

4.2. Results

The research focuses on finding the mechanism of the perfor-

mance variation for a gun in different life stages and providing
Fig. 4. The projectile velocity vs. the projectile travel. methods to maintain the performance. Prior to putting forward
1016 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021

Table 2
Material parameters.

Parts Materials Young’s modulus/GPa Poisson’s ratio Density/kg  m3

Projectile/barrel Steel 208 0.3 7800
Rotating band Copper 124 0.34 8920

(a) Pressure vs. travel. (b) Velocity vs. travel.

Fig. 5. The mean pressure and projectile velocity versus the projectile travel.

Table 3 motion of the projectile and the chamber volume are influenced
Results at special moments. by the inner diameter increments. In Fig. 6, the relationship
Maximum pressure/MPa Muzzle velocity/m  s1
between the stages and the corresponding total firing cycles of
the gun is given in Table 4. Obviously, Stage A represents the
CIBM 342.7 1004.7
new barrel and the increment of the diameter is identically zero
This paper 341.9 1002.8
Experiments 336 1000 along the axis.
Based on the data, formulas of the inner radius increments of
the gun in different life stages and the FEM models of the gun were
constructed. Followed the calculation procedure in Fig. 3, perfor-
the simulations, the inner diameter of the barrel in different life mances of the gun in different life stages were simulated.
stages were measured using the technology of electronic measure- The mean pressure and the projectile velocity are displayed in
ment. Results shown in Fig. 6 are the diameter increments. The Figs. 7 and 8, respectively. Figures (a) in Figs. 7 and 8 are the
increment is defined as the measured diameter subtracts the initial variables varying over time. The independent variable in the
diameter at the same position. The original point in Fig. 6 figures (b) is the projectile travel. As shown in the figures, both
represents the muzzle of the gun. Evidently, material ablation near the pressure and the projectile velocity decrease with increasing
the forcing cone is much more severe than the other positions. firing cycles.
Obviously, the increment caused by wear and erosion increases The states of the propellants in combustion are represented by
the interference between the projectile and the barrel. Both the the relative burnt mass w and the relative burnt thickness Z. The
results are shown in Fig. 9. Obviously, the overall burning rate
decreases with increasing firing cycles.
Pressure and temperature distributions of the core flow in the
barrel are pivotal to determine the thickness of the barrel, esti-
mate the firing safety and predict the residual life. Wear and
erosion of guns mainly result from thermal effects [6]. Heat is
transferred to a barrel mainly through heat convection [28].
The temperature of the core flow is pivotal for the calculation
of the heat transfer. The pressure and temperature distributions
are displayed in Figs. 10 and 11, respectively. The maximum
loads locate at the chamber of the barrel. The area in dark red
deceases with increasing firing cycles, which indicates degrada-
tion of the loads.
The temperature variations of the core flow near the forcing
cone are provided in Fig. 12. The results were obtained through
subtracting temperature near the forcing cone from that in the first
cycle of the gun. The main differences locate at the moment when
the pressure reaches the maximum and after the burnt thickness
reaches e1. The maximum differences locate in regions A and B.
Fig. 6. Measured diameter increments of the gun in different life stages. The sign of the two regions are opposite.
 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021 1017

Table 4
The life stages of the gun.

Stage of the barrel A B C D

The total firing cycles 0 150 450 600

(a) Pressure vs. time. (b) Pressure vs. travel.

Fig. 7. The mean pressure in the chamber.

(a) Velocity vs. time. (b) Velocity vs. travel.

Fig. 8. The projectile velocity.

(a) Variables vs. time. (b) Variables vs. travel.

Fig. 9. States of the propellants in combustion.
1018 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021

(a) In Stage A. (b) In Stage B.

(c) In Stage C. (d) In Stage D.

Fig. 10. Pressure distributions.

4.3. Discussions The distances from points I, II and III to the origin represent the
length increments of the chamber. In Fig. 14, the maximum resis-
Results at special moments are compared with experiments in tance decreases with increasing firing cycles. And the position at
Table 5. Good agreements between the results for the worn gun which the maximum resistance locates is moved forward.
were achieved, which further validated that the modified model The performance of the gun is determined by the internal com-
is capable of predicting the performance of interior ballistics of a bustion of the propellant. Consequently, investigation was carried
gun in different life stages. As shown in Table 5, both the maximum out on factors that affect the combustion to elucidate the phenom-
pressure and the projectile muzzle velocity decrease with increas- ena. The propellants are enclosed in the chamber, the volume of
ing firing cycles. which is growing with the increasing of the projectile travel. There-
Similar to the classical model, we define a coefficient of fore, the boundaries of the chamber can be divided into two parts,
energy losses. As shown in Eq. (9), the coefficient is a ratio one belongs to the inner walls of the barrel and the other one
between the work output of propellant gases and the transla- belongs to the projectile movement.
tional kinematic energy of the projectile, which is a constant in On one hand, wear and erosion eliminate materials of the
the classical model. And the coefficients were calculated and barrel exposed to the high temperature core flow. Ideal gas state
shown in Fig. 13. equation was not suitable for high pressure and high tempera-
Rl ture gases. Noble-Abel equation [29] was applied for the propel-
S  pd ðxÞdx
uðlÞ ¼ K þ 0
ð9Þ lant gases. Indicated by the state equation, pressure in the
mv 2 =2 chamber is decreased when the chamber volume is enlarged.
In the initial stage of the internal combustion, work done by the According to Eq. (1), the pressure decrease retards energy release
propellant gases is mainly used to overcome resistances caused by rate of the propellants, which further decreases the pressure. As
the mechanical interactions. Therefore, the coefficients in this a result, wear and erosion slow down the burning rate of the
stage are relatively larger than that in the other stages. After the propellant.
projectile is engraved into the barrel, the coefficient gradually On the other hand, materials ablation enlarges the clearance
decreases with increasing projectile travel. between the projectile and the barrel. Consequently, resistance
This phenomenon reflected the variation tendency of the resis- acting on the projectile is decreased. A projectile in the gun can
tance acting on the projectile. The resistance is very large in the ini- be propelled forward much more easily, which leads to an easy
tial stage of an interior ballistic process, especially in the engraving increase of the chamber volume. Therefore, the propellant combus-
process of the projectile. Based on Eq. (3) and Eq. (6), the resistance tion is suppressed. Based on Newton’s second law, the projectile
was calculated and displayed in Fig. 14. velocity is declined.
 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021 1019

(a) In Stage A. (b) In Stage B.

(c) In Stage C. (d) In Stage D.

Fig. 11. Temperature distributions of the core flow.

In summary, both the two factors decrease the pressure in

the chamber. However, total travel of the projectile is a constant.
Work output of the system can be obtained by integrating the
shot-base pressure over the projectile travel. The decrease of
the work output reduces the amount of energy that flows to
the kinematic energy of the projectile, which leads to the perfor-
mance degradation.
The mean pressure and the projectile velocity in the first fir-
ing cycle of the gun were selected as references. Differences of
the mean pressure and projectile velocity along the barrel in
different life stages are shown in Figs. 15 and 16, respectively.
Consistent with the elucidations, both the pressure differences
and the velocity differences are increasing during the initial
stage of the combustion. Afterwards, both the differences tend
to be reduced.
Performance degradation of mechanical systems is unavoidable.
Fig. 12. Temperature differences near the forcing cone. Several methods were provided to maintain the performance. Prior
to the maximum pressure, the consistence of the muzzle velocity

Table 5
Results at special moments of the gun in different life stages.

Firing cycles Maximum pressure Muzzle velocity Relative travel when propellant burnt out
This paper Experiment This paper Experiment
MPa m  s1
0 341.9 336 1002.8 1000 0.8107
150 329.0 325 997.3 995 0.8376
450 321.7 316 983.1 980 0.8473
600 311.6 304 977.3 972 0.8612
1020 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021

(a) Coefficients vs. time. (b) Coefficients vs. travel.

Fig. 13. Coefficients of energy losses.

Fig. 14. Reverse calculated resistance. Fig. 16. Velocity differences.

Table 6
Adjusting the mass of the charge.

0 150 450 600

Dx/g 0 9 16 20
pm/MPa 313.7 310.2 308.0 305.7
v0/(m/s) 1002.8 1002.2 1003.1 1001.9

Table 7
Adjusting the web-thickness of the propellants.

0 150 450 600

De1/mm 0 0.01 0.02 0.025
pm/MPa 313.7 314.9 313.0 312.7
v0/(m/s) 1002.8 1000.9 1002.5 1002.3

Fig. 15. Pressure differences. is particularly sensitive to the variation of the web thickness.
Therefore, increasing the charge mass is an easier way to maintain
the performance other than controlling the law of energy release in
was guaranteed. Two kinds of methods were provided to achieve engineering.
the consistency. One is to increase the total energy, which can be
realized by adding the charge mass. The other way is to change 5. Conclusions
the energy release law, which can be realized by adjusting the
web-thickness of the propellant. The corresponding results are A modified model, which is suitable for the fluid-structure
shown in Tables 6 and 7, respectively. Obviously, the performance interaction system in different life stages, was established basing
 C. Hu, X. Zhang / Applied Thermal Engineering 130 (2018) 1012–1021 1021

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