Modeling Refractory Material Production by Selfpropagating High Temperature Synthesis (SHS)
Conduction Heat Transfer
MEAE 6630
Dianbo Li
ABSTRACT
Self-propagating high temperature synthesis (SHS) can be used to prepare near net refractory shapes which
are difficult to produce by other methods. Once initiated, highly exothermic reactions can become selfsustaining and will propagate through the reactant mixture in the form of a combustion wave, and reactants
turn into products as the combustion wave goes through. Physical processes involved in SHS include
chemical kinetics, macroscopic transport phenomena and phase transformations. This project describes a
simplified mathematical model using numerical method to simulate the process of the SHS and investigate
behaviors of SHS process.
INTRODUCTION
The basic mathematical foundation for self-propagating exothermic reaction was developed more than 35
years ago. Over the past 20 years many materials have been synthesized by this method, including metallic,
ceramic, and composite phases. Credits should be given to formal Soviet Union because vast majority of
literature accounts of the SHS process has been done in CCCP. Formal USSR had commercial production
using SHS process. Recent reports show that Russia had produced more than 300 materials using SHS
process. USSR’s progresses in the field of SHS have attracted international attention as a result of the
special importance given to this process. SHS has many advantages as a process for the synthesis of
materials. The most important of these include the energy savings associated with the use of self-sustaining
reactions, the simplicity of the process, the relative purity of the products, and the possibility of
simultaneous formation and densification of the product. Also, products created by the SHS process are
superior to those created by conventional methods in some cases. The shape-recovery force associated with
SHS alloys is believed to be greater than that observed in alloys produced by conventional methods. The
products of synthesis by combustion can be in powder form. The powder products have been used as filters
or as crucible materials. Also, SHS process can products highly dense bodies which involving the
application of pressure and molten state of the products.
THEORETICAL CONSIDERATION
The SHS process depends on a highly exothermic reaction, which occurs in the form of a traveling or
combustion wave. The process in initialed by providing a heat source, such as tungsten coil or laser, at one
end of a compacted powder sample. The sample end is heated until the temperature reaches a point where
the exothermal reaction begins. This temperature, which initiates the reaction, is defined as the ignition
temperature Tig. Another important temperature parameter is the adiabatic temperature of combustion T ad.
Tad is defined as the temperature to which the product is raised under adiabatic conditions due to the heat
generated by the exothermic reaction. When the exothermic reaction is initiated, a combustion wave forms
at the heated end of the sample and proceeds to propagate along the sample’s axis of symmetry. The SHS
process is self-propagating. So the energy required to continue the exothermic reaction is provided solely
from the combustion wave itself. Therefore, no external heat source is required other than that to initiate the
process. During the process, the rate of propagation of a combustion wave through the reactant mixture is
dictated by thermophysical and heat transfer criteria. In this project one assumption is that convection is
negligible and only radiation occurred at end of material is considered.
PROBLEM & FORMULATION
Consider a compacted powder cylindrical rod of uniform stoichiometric metal-nonmetal composition. As a
class of materials, nitrides possess relatively unique properties which make them highly desirable in a
variety of applications. They are generally hard, refractory, and chemically stable materials. In this project
mixture is titanium powder and nitrogen gas. For the general case of a reaction between porous solid and
gaseous nitrogen, two considerations must be examined. The local availability of nitrogen and the pressure
of the nitrogen gas. Pressure PN must exceed the thermodynamically calculated dissociation pressure of the
nitride phase at the ignition temperature. However, this is not the focus of this paper. We just simply assume
the environment meets these pressure criteria.
The chemical reaction is
Ti + N2 ---- > TiN + N + Heat
Since the heat flux to the plane of a combustion wave is the sum of the heat transferred to it prior to the
arrival of the wave and the heat generated by the chemical reaction associated with the wave itself. The
following expression has been derived for self-propagating combustion reactions:
Qin – Qout + Qgenerated = Qstored
Here Qin
rate of energy entering the control volume
Qout
rate of energy exiting the control volume element
Q generated
energy generation rate within the control volume element
Q stored
rate of energy stored within the control volume element
In order to simplify the problem, following assumptions are made:
a.
Thermal properties are constant and independent of temperature. Although we know these properties
are not really independent of temperature (. i.e. specific heat is a function of temperature.), we set all
thermal properties constant to approach a simplified computer model.
b.
One-dimensional transient heat conduction. If we consider convection and side radiation are
significant, a two-dimensional model has to be developed.
c.
Assuming velocity of combustion wave is constant through the rod. Sometimes the reaction can be so
complex as to lead to the formation of multiple wave propagation. The existence of two-wave
combustion in nitride formation is believed to be the consequence of the complex nature of the reaction
between gaseous nitrogen and the refractory metals.
d.
Any phase change will occur prior to the arrival of the combustion wave, so that temperature profile is
not affected.
e.
the chemical reaction is completed after the passage of the combustion wave along any point on the
rod, reaction granularity is not significant, and the width of the wave (reaction zone) is narrow in
comparison with thermally affected zone.
within the rod, energies entering and exiting the control volume are simply due to heat conduction through
the rod material. Therefore
Q in = q x
Q out = q x+x where q = -K T/x
K is the thermal conductivity
Formula for heat generated by chemical reaction is as follow
Q generated = Hr p Ko (1-) n e(- E /RT )
here Hr is the heat of reaction,  is the density, Ko is a constant and  is the fraction reacted of the reactant
mixture, n is the reaction kinetic order, E is the apparent activation energy of the reaction, and R is the
specific gas constant.
The energy stored within the control volume element represents the time lag that occurs for heat conduction
through the element and is based on the properties of the material:
Q stored =  Cp T/t
therefore, governing equation is as following:
E
K
 2T
T
n
 H r Ko 1    e RT  CP
2
x
t
Also, at each end of the rod, radiation heat transfer is significant and can’t be ignored.
Q radiation =  T2
Where  is radiation constant, then governing equation is
E
K
 2T
T
n
 H r Ko 1    e RT  T 4  CP
2
x
t
Solution
The governing equation shown previously is a non-linear one-dimensional Fourier equation. In this project,
numerical solution techniques will be introduced to solve this equation. This differential equation can be
approximated using an explicit finite difference method.
Ti n 1  rTi n1  (1  2r )Ti n  rTi n1
where r =  t / (x2) < 0.5
Substitute the upper equation into governing heat transfer equation and rearrange, the resulting equation is
as follows:
Ti
E
t   Ti j 1  2Ti j  Ti j 1 
n RT 
K
 Ti 
  H r Ko 1    e 
CP  
x 2


j 1
j
while at the end of the rod, involving radiation heat transfer:
Ti
j 1
E
t   Ti j 1  2Ti j  Ti j 1 
n RT 

 Ti 
K
 H r Ko 1    e    (Ti j ) 4

2

C P  
x


j
The initial and boundary condition used in solving the upper equation is:
T 10 = T start
T0j = T 1jT nj = T jn-1
A simple computer model was developed using the above finite difference scheme to approximate the
temperature distribution at different times using the SHS process. The resulting algorithm was code in
Fortran 77 and run on a Sun workstation. Input parameters are as table I
Table I
Material properties approximated for titanium nitride.
Sample Length
0.01 m
To ( initial temperate)
298 K
R (Gas constant )
8.31 kJ/(kg mol K)
K (thermal conductivity)
50 W/m K
 (density)
5430 kg/ m3
Cp
1373 J/kg K
E (activation energy)
2.0 e5 J/kg mol
n (reaction order)
1
Hr (enthalpy of reaction)
3e5 J/m3
Atomic weight of TiN
61.88
RESULTS AND DISCUSSIONS
In this project, Several points are being investigated to see their impacts on SHS process. When varying one
point, all other parameters are kept constant. Points of interest are:
1.
The effect of QA (igniting heat flux) to induce the SHS process.
2.
The influence of the reaction kinetic order (n) on the solution of the mathematical model.
3.
The effect of the combustion process fraction reacted, i.e. the rate of conversion from reactants to
products.
Below is a general shape plot of the computed temperature distribution. X axis represents different section
of sample in its axial direction. Y-axis is temperature in Calvin. And different color lines represent different
time step.
1) Initial heat flux (QA) & Time on for QA
To determine the effect of QA on the SHS process. The Fortran code was run using various initial heat
fluxes while other parameters are the same. The simulation time used in the code is 0.01 s. From following
plots, we can tell that Ti rod has normal heat conduction transfer as long as the temperature not reach T ignite.
According to plots, T ignite is around 1600 K. Once temperature passes T ignite, chemical reaction happens
rapidly (See Fig 3 & 4). Heat releases from combustion along with initial heat flux, push the temperature up
to 4500 K. Away from the rod, maximum temperature is almost a constant which is close to maximum
adiabatic temperature of combustion. However, T ad at location away from left end does not have obvious
change when QA increases. Fig 3 and Fig 5 show QA = 1E9 W/m2 and QA = 2E9 W/m2 and prove that Tad
keeps constant while QA changes.
Fig 1
Fig 2
d
Fig 3
Fig 5
Fig 4
Enlarge Area of Fig 3
Fig 6 enlarge area of Fig 5
2. Effect of kinetic reaction order to SHS process.
The following plots are created using ENE (reaction order) of 0.5, 1, 2, and 3. Temperature distributions at
different reaction orders vary rapidly. When other parameters are unchanged, SHS process only happens at
narrow zone. Plots indicate that stable combustion wave happens around order of 1. When change of
reaction order is significant, combustion can't occur ( Fig 7)or combustion is unstable (Fig 9 & 10). The
instability of the combustion wave, which leads to the transformation from a steady state to a non steadystate mode, can lead ultimately to the extinction of the combustion process. Change of reaction order may
result in insufficient heat generation and therefore cause combustion to stop.
3
Fig 7
Fig 8
Fig 9
Fig 10
Impact of the rate of conversion from reactants to products.
In the computer model, rate of conversion is expressed as  = ACH  E-ECH/(RT). Therefore, we can vary the
value of ACH ( frequency factor) to change the value of reaction rate. ACH values for plots 11~14 are
9.0E7, 1.0E8, 2.0E8 and 10E8. According to the plots, when the rate of reaction is small, then combustion
is not going to happen. Also, when the rate keeps increasing, T ad is decreasing. When the Tad is drop below
the ignition temperature, SHS process will extinct due to insufficient heat generation. Therefore, there is a
zone of reaction rate which SHS process can happen.
Fig 11
Fig 13
Fig 12
Fig 14
SUMMARY
The mathematical model discussed in this project is based on many assumptions made previously. The
model provides a simple means to approximate the SHS process. To increase the accuracy of the results,
the model can be improved by disregarding some of the assumptions. For example, expanding the model to
two-dimensional to include the surface heat losses; considering the impact of phase changes occurring near
combustion wave and eliminating other assumptions will provide us a higher accuracy model. It is worth
investigating the behavior of SHS process since SHS process has so many advantages in preparing material
with superior properties comparing to conventional methods.