Spatial-temporal semi-empirical dynamic modeling of
thermal gradient CVI processes
D. K. Rollins, Sr.a, b,, D. J. Rollinsc, and A. D. Jones, Jr.d
a
Department of Chemical and Biological Engineering, Iowa State University, Ames, Iowa, USA
b
Department of Statistics, Iowa State University, Ames, Iowa, USA
c
Materials Science and Engineering Department, The Ohio State University, Columbus, Ohio, USA
d
Department of Mathematics, Florida A&M University, Tallahassee, Florida, USA
Abstract
Thermal gradient chemical vapor infiltration (CVI) appears to have much promise as a
process to produce carbon/carbon composites and has several advantages over conventional
isothermal CVI. Using a graphite heater in the center of porous disk preforms, Zhao et al. (2006)
reported excellent densification and no surface preform plugging. The aforementioned work
produced a large amount of spatial and dynamic temperature data but falls short of providing a
model of temperature over space (r) and time (t). Hence, using the data reported by Zhao et al.
(2006), this article develops and presents two dynamic models for deposition temperature as a
continuous function of t and distance from the heater surface, r. One model uses only two
adjustable parameters and the other five and both capture the phenomenological structural
behavior quite well. Hence, the approach presented in this work provides a methodology to
model real processes using a limited amount of data and produces important continuous dynamic
and spatial behavior that can prove valuable in controlling and optimizing thermal gradient CVI
processes.
Keywords: Carbon/carbon composites; Chemical vapor infiltration; modeling

Correspondance to: Professor D. K. Rollins, Sr., Department of Chemical Engineering, Iowa State University,
Ames, IA 50011, USA.
E-mail: drollins@iastate.edu
1
INTRODUCTION
Chemical vapor infiltration is a relatively new process for forming carbon reinforced
carbon composites. This is an important process because carbon/carbon composites have been
proven to have excellent high-temperature mechanical properties and high thermal stabilities (Li
et al., 2005; Zhang and Hüttinger, 2003; Vignoles et al., 2004; Delhaes, 2002; Birakayala and
Edward, 2002). This material exhibits high strength, stiffness, toughness, thermal shock, wearresistance, ablation performance, friction resistance and is also lightweight (Zhang and
Hüttinger, 2003). These composites are already being used where high temperatures and friction
are an issue such as in brake pads for planes and on space crafts (Birakayala and Edward, 2002);
unfortunately though because of high processing times this material is extremely expensive,
limiting its possible usage.
The conventional CVI process consists of infiltration of fiber preforms with a carrier gas
such as helium and a low molecular weight hydrocarbon such as methane or ethane at constant
temperature and pressure. The temperature is usually around 1000 C and the hydrocarbons react
rapidly to form larger molecules that deposit into the carbon matrix at a high rate typically
plugging the entrance of pores and inhibiting infiltration and thus, densification. A promising
alternative to overcome the limitations of the isothermal CVI process is thermal gradient CVI
(Tang et al., 2003; Golecki et al., 1995; Probst et al., 1999; Jiang et al., 2002). As the name
suggest, thermal gradient CVI seeks to control carbon deposition by controlling the temperature
profile in the preform. This is done by creating a higher temperature where deposition is to occur
and maintaining a lower temperature where deposition is not to occur. Filling of the pores is
controlled by temperature in the zones of the preform. Thermal gradient CVI was demonstrated
in Zhao et al. (2006) using an electrical furnace consisting of a heated rod between two
2
electrodes with cylindrical donut shaped preforms slid over the rod. Thus, the preforms were
heated in the radial direction from the center out. Natural gas was introduced into the reactor
from the cold outer surface at a controlled rate and reached the hot deposition zone by diffusion.
The deposition zone is controlled by temperature and a thermocouple inserted into the preforms
that moves in the radial direction outward as the temperature rises and densification is
completed.
This work develops two types of dynamic semi-empirical models of the temperature field
of the preform continuously over time and space for the process in Zhao et al. (2006) using only
the data they reported. Moreover, this article will demonstrate a new model building process for
this application and provide theorists and practitioners with a means to obtain accurate spatial
and dynamic response surface behavior. The models give the temperature (T) as a function of
time (t) and radial distance from the heater surface (r). The difference between the models is the
dynamic form they use. Model 1 uses a third order, critically damped transfer function and
depends only on two adjustable coefficients. Model 2 uses a sigmoidal transfer function and
depends only on five adjustable coefficients. Model 2 fits the data better but Model 1 has fewer
adjustable coefficients. By “adjustable coefficients,” we mean model parameters that are
estimated under least squares estimation.
Our procedure consists of basically three (3) steps. The first step uses temperature versus
time data, at fixed values of r, to obtain the dynamic models for temperature. Next we determine
models for the initial temperature and model parameters as a function of r. In the final step we
incorporate these models for the parameters, which are functions of r, into T(t,r).
We present this work using the following outline: in Section 2 we use the data in Zhao et
al. (2006) and develop the dynamic models. Section 3 describes the model building procedures
3
to obtain initial temperature and the dynamic parameters as a function of r. We present and
compare both our final models with the three dimensional surface plot in Zhao et al. (2006) in
Section 4 and give concluding remarks in Section 5.
DYNAMIC MODELING
2.1 Collecting the modeling data
In this section we develop dynamic fits for eleven (11) fixed values of r. The data we
used come from Figs 5-6 in Zhao et al. (2006) and are given in Fig. 1. The plot on the left gives
temperature versus time (t) data at constant values of r and the other one gives temperature
versus r data at constant values of t. We used the nine (9) curves in the left plot to obtain T0 for
the values of r shown. We estimated these values by mildly extrapolating to t = 0. We added two
values of r (80 and 100 mm) using information from the right plot. Their T0 values were obtained
from the t = 3 hr curve and their dynamic data were obtained from all the curves at their values
of r. Note that the initial temperatures across r represent the temperature profile in the radial
direction at t = 0, where we are assuming that t = 0 is the time that the flow of gas was started,
i.e., deposition began. Based on the surface temperature plot in Zhao et al (shown later in Fig. 7)
we used a maximum temperature of 1000 oC. Next, we use this information to dynamically
model temperature at the eleven values of r for Model 1 first and then Model 2.
4
Fig. 1: The plots taken from Zhao et al. (2006)1 used to provide the modeling data. The plot on
the left gives temperature versus t curves for different values of r and the one on the right give
temperature versus r curves for different values of t. The plot on the left was used to obtain T0
data at each value of r shown except r = 80 mm and r = 100 mm. These two were estimated
using the 3 hr curve from the plot on the right.
2.2 Model 1
From the data in Fig. 1, for each of the eleven values of r, we obtained a “best fit”
dynamic transfer function for Model 1 using a third order, critically damped transfer function,
with a step input. This model form was accepted after evaluating a few low order functions. The
Laplace domain form of this transfer function is given by Eq. 1 below:
1000  Tr 0 
Tˆr s  
sˆr s  13
(1)
where “^” is used to represent “estimate.” In the time domain,
Tr t   Tr t   Tr 0
(2)
where Tr0 is the initial temperature at a distance r from the heater surface and ̂ r is the estimated
time constant for the transfer function at a distance r from the heater surface. By applying the
method of partial fraction expansion and then transforming to the time domain (see Seborg et al.,
2004) and by use of Eq. 2 we get
1
Reprinted from Zhao et al. (2006) with permission from Elsevier
5
t


1000  Tr 0  2 
ˆr
ˆ
ˆ
Tr (t ) 
 1 e
 r 
ˆr
 
t
t

2
 ˆ  ˆr t  ˆr
 e
   r t e
2


 T
 r0

(3)
At each value of r, the estimate ̂ r is obtained using the data corresponding to this value of r and
nonlinear regression (see Bates and Watts, 1988) to minimize
nr

SSE r   Tr ( t rk )  T̂r ( t rk )

2
(4)
k 1
where nr is the number of samples corresponding to the value of r and trk represents the time of
the kth sample at a distance r from the heater surface. SSE is called the “sum of squared errors”
and minimization of SSE is called the “Least Squares Criterion,” or “Least Squares Objective
Function.” Here, our specific goal is to find the value of ̂ r that satisfies the Least Squares
Criterion for each value of r. To achieve this objective we used the “Solver” command in the
Microsoft Excel software package. After completion of this step we have a value of ̂ r as well as
Tr0 for each r. Therefore, our next step is to produced functional relationships that are dependent
on the value of r, that is, ˆr  and T0 r  . We will do this after completing Step 1 for Model 2.
2.3 Model 2
Dynamically, the advantage of Model 2, the sigmoidal model, over Model 1 is that it
allows a steeper approach to the steady state temperature, i.e., the maximum temperature of 1000
o
C. Hence, we modeled this process using a special sigmoidal function we developed to have the
initial and ultimate response behavior of dynamic transfer functions as shown in Eq. 5 below:
 

1
1


C


 2 

  r  t   r 
2
1

e


T
Tr  t   1000  Tr 0 
r0
1 C






where
6
(5)
1
1

C  2

  


 1  e r r 2
(6)
 1  C
Note that, from Eq. 5, Tr 0  0  Tr 0  Tr 0 and Tr     1000  Tr 0  
  Tr 0  1000 o C. A
 1  C
set of adjustable parameters, ˆ r and ˆ r , were estimated using nonlinear regression according to
Eq. 4 for each of the eleven values of r. Next we discuss how we obtained the required
parameter dependencies on r.
 r  , τˆ r , αˆ r  and βˆ r 
DETERMINING T
0
3.1 T0 r 
To determine the temperature continuously over all values of r and t, for both Models 1
and 2, continuous functions of T0 r ,  r ,  r , and   r  must be determined. (Note that we are
using “(r)” to represent a functional relationship that depends on r and the subscript r to represent
the value obtained at one of the eleven values of r using either Eq. 3 for Model 1 or Eq. 5 for
Model 2.) We use the results and data from the previous section to accomplish this goal in this
section, first for T0 r  which is used by both models, then for ˆr  for Model 1, and then for
 r  and   r  for Model 2. A plot of the data for Tr0 against r is given in Fig. 2. As shown, Tr0
monotonically decreases as r increases and reaches a steady state value of 97 C, which must be
the temperature at the outer surface of the preform at t = 0. Also, note that the initial temperature
at r = 0, T0(0), is shown to be 900 C. This boundary condition was not reported in Zhao et al.
(2006) and we estimated it to be 100 C less than the maximum temperature of 1000oC based on
Fig. 4 in Zhao et al. (2006), their three dimensional surface plot. We attempted to fit polynomial
regression to this data but did not use this form since it could not produce a function with
7
monotonically decreasing behavior and an asymptote at 97 C. Our fitted equation meeting these
criteria is


r

T0 r   T0 0  T0 120 e 120 r  T0 120

(7)
where T0(120) = 97 C and ̂ = 3.143 (determined by nonlinear regression). Note that Eq. 7 has
exponentially decaying behavior between r = 0 and r = 120 and the correct limiting behavior
(i.e., T0(0) = 900 C and T0(120) = 97 C). As Fig. 2 shows, this equation, for the initial
temperature as a function of r, fit the data quite well.
800
Data
Fit
o
T0 ( C)
600
400
200
0
0
20
40
60
80
100
120
Distance from heater surface (mm)
Fig. 2: The fit of T0 r  using Eq. 7.
3.2 ̂r  for Model 1
A plot of the values for ̂ r versus r is given in Fig. 3. In this plot, ̂ r appears to
monotonically increase as r increases, and begins to level out at the largest values of r. At r = 0,
we set ̂0 = ˆ0 = 4 hr and the maximum value, ˆmax , equal to 98 hr for an optimum fit. Notice
that there are no estimated values of the time constant for r < 20 mm. However, one would
expect the time constant to continue to decrease monotonically as r approaches zero since the
temperature continues to increase in the radial direction towards the heater. In addition, at values
of r close to zero, since the dynamic response is fairly fast, model performance should be fairly
8
robust to the choice of r. Nonetheless, in practice, if one desire to more accurately determine the
value of the time constant at r = 0, it would be a matter of just collecting the data over time. The
fitted equation for the curve given in this figure is:
ˆ (r) 

r

ˆ max  
ˆ 0  ˆ2 

ˆ


  1  e
2
ˆ

 

r
r
2

 

ˆ r e ˆ  r e ˆ   

ˆ 0

2



(8)
where ˆ0 = 4 hr, ˆmax  98 hr, and ˆ  15 .710 mm as determined by nonlinear regression. As
shown, Eq. 8 fit these values fairly accurately. Note that Eq. 8 has a form similar to Eq. 3.
120
100
(r)
80
Estimated Values
Fitted Curve
60
40
20
0
0
20
40
60
80
100
120
Distance from heater surface (mm)
Fig.3:
The fit of ̂r  for Model 1 using Eq. 8.
3.3  r  and   r  for Model 2
Plots of the values for  r and  r versus r are given in Fig. 4. In these plots, both
 r and  r monotonically decrease as r increases, and begin to level out at the largest values of r.
At r = 0, we set ̂ 0 = ̂0 = 0 hr and ̂ 0 = ̂ 0  = 0.07 hr-1. Similarly, at r = 115, we set ̂115
= ̂115 = -208.972 hr and ̂115 = ̂115 = 0.0217 hr-1 for optimum fits as shown in Fig. 4. As
in the case for ̂ r , there are no estimated values for ˆ r and ˆ r for r < 20 mm. However, based on
physical considerations, we allowed these values to continue to increase monotonically as r
9
decreased. The fitted equations for both  r  and   r  are sigmoidal functions and are given
below.
 
1
1
   C
 2 
    r    
2
 r    115  1  e
1  C









(9)
where ̂ = -31.3947 mm and ̂  = 0.06433 mm-1, and
1
1

C  2 
 
 




1 e
2
(10)
 

1
1
   C 
2 
    r    
2


 r    115   0  1  e


1  C




where ̂ = 173.433 mm and ̂  = 0.0315 mm-1, and


(11)

1
1
C  2 
 
    
1 e
2
(12)
0
r (hr)
-1
r (hr )
0.05
0
-210
0
20
40
60
80
100
120
r(mm)
0
20
40
60
r (mm)
Fig. 4: The fits of  r  and   r  for Model 2 using Eqs. 9 and 11, respectively.
10
80
100
120
 t,r 
PRESENTING T
In this section we present our dynamic and spatial models for the temperature field for
this process, i.e., Tt,r  . The equation for Models 1 and 2 are similar to their dynamic equations
(Eqs. 3 and 5) with the subtle differences that the r subscript is no longer necessary since they
are valid for any value of r and not just those with measured values and that the initial
temperature and the parameters can be determined for any value of r as derived above. Thus, to
reflect these properties, the final equation for Model 1 is written as:
t 
t
t 



1000  Tˆ0 r   2 
t 2  ˆ r   ˆ
ˆ r  
ˆ r 
ˆ
ˆ




Tˆ (t , r ) 

r
1

e


r
t
e

e
 T r 



 0
2
ˆr 2




(13)
where T0 r  is given by Eq. 7 and ˆr  is given by Eq. 8. Similarly, for Model 2,
 

1
1
   C 
 2 
  ( r ) t   ( r ) 
2
  T 0 (r )
T t , r   1000  T 0 ( r )  1  e


1 C





with

1
1

C  2

 ( r )  ( r ) 


1 e
2
(14)
(15)
where  r  and   r  are given by Eqs. 9 and 11, respectively. In Figs. 5 and 6 we give the fits
of these models to data. Figure 5 gives the temperature versus time (t) responses at constant
values of r, in relation to the left plot in Fig. 1 and Fig. 6 gives temperature versus r responses at
constant values of t, in relation to the right plot of Fig. 1. Although both plots show excellent fit
to the data, Model 2 does the best job in handing the “steep” response behavior. This is best seen
by comparing the fits for r = 20 and 30 mm in Fig. 5 and t = 121 and 160 hr in Fig. 6.
11
20 mm
30 mm
35 mm
40 mm
45 mm
50 mm
55 mm
60 mm
80 mm
100 mm
115 mm
20 mm
30 mm
35 mm
40 mm
45 mm
50 mm
55 mm
60 mm
80 mm
100 mm
115 mm
1000
1000
800
800
600
600
400
400
200
200
0
0
0
20
40
60
80 100
Time (hr)
120
140
160
0
20
40
60
80
100
Time (hr)
120
140
160
Fig. 5: Temperature versus time for the eleven values of r for both models corresponding to the
left plot in Fig. 1.
1000
1000
3 hr
121hr
40 hr
160 hr
800
40 hr
160 hr
83 hr
0 hr
83 hr
3 hr
121 hr
0 hr
0 hr
83 hr
0 hr
83 hr
800
600
600
400
400
200
200
3 hr
121 hr
3 hr
121 hr
40 hr
160 hr
40 hr
160 hr
0
0
0
20
40
60
80
100
0
120
20
40
60
80
100
120
Distance from heater surface (mm)
Distance from the heater surface (mm)
Fig. 6: Temperature versus r for six values of t for both models corresponding to the right plot in
Fig. 1.
Eq. 13 (Model 1) depends only on two adjustable parameters, ̂ through Eq. 7 and ̂
through Eq. 8. Similarly, Eq. 14 (Model 2) depends only on five adjustable parameters, ̂
through Eq. 7, ̂ and ̂  through Eq. 9, and ̂ and ̂  through Eq. 11. With dependence on such
a small number of parameters and the excellent fits, the implication is that these model structures
capture the phenomenological behavior rather well. This modeling approach falls under the semiempirical class. This class is developed from a combination of data and model structures with
phenomenological meaning. This is in contrast to empirical modeling that strongly relies on data
and model forms that typically are not physically or biologically interpretable. An advantage of
semi-empirical modeling over theoretical or semi-theoretical modeling is that semi-empirical
12
modeling does not require the rigorous development from first principles but is able to obtain
forms with phenomenological meaning and model parameters that are often physically
interpretable; for example, under Model 1, ̂ r is an estimate of the time constant a distance r
from the surface of the heater. (For examples of work using block-oriented modeling, which is a
class of semi-empirical modeling that addresses non-linear static and dynamic behavior, see
Gomez and Baeyens, 2004; Westwick and Verhaegan, 1996; Shu, 2002; Rollins et al., 2006a;
Rollins et al., 2006b; Rollins and Bhandari, 2004; Bhandari and Rollins, 2003; Rollins et al.,
2003).
In Fig. 7 we show surface plots of temperature over r and t from Zhao et al. (2006) (top
plot) and for Models 1 (left plot) and 2 (right plot). As we noted earlier, we set the maximum
temperature to 1000 oC in agreement with the figure from Zhao et al. (2006) as shown. As this
figures shows, our two surface plots of temperature over r and t compares quite nicely with the
top plot from Zhao et al. (2006). Therefore, in summary, we recommend Models 1 and 2 for
determining temperature at any value of r and t for the process given in Zhao et al. (2006) and
the procedure we have presented in this work for developing models for similar thermal gradient
CVI processes.
13
1000
500
0
150
0
100
40
50
15 0
0
0
100
80
40
50
120
80
0
120
Fig. 7: The surface plots for temperature. The top plot is from Zhao et al. (2006)2, the one on the
left from Model 1 and the one on the right is from Model 2. With the maximum temperature as
900 oC as indicated in Zhao et al. (2006), the agreement is excellent.
CONCLUDING REMARKS
In this article we presented a semi-empirical model for the temperature field of a thermal
gradient CVI process as a function of space and time. In addition to the model itself, a critical
contribution of this work is the illustration of the model building process. This procedure could
be used to build these types of models from limited process data and provide critical knowledge
2
Reprinted from Zhao et al. (2006) with permission from Elsevier
14
on the process behavior important to optimal control and optimization. For any real process a
much larger study could involve different heater temperatures, gas flow rates, and other
boundary and initial conditions. For a fix set of these conditions, as in Zhao et al. (2006), our
proposed method offers a way to obtain an accurate knowledge of the thermal field from a set of
collected data consisting of even fewer values of r as long as the process is as well behaved over
r as it was for the process in Zhao et al. (2006).
Our modeling approach is semi-empirical and has several advantages over empirical
modeling approaches such are artificial neural networks (ANN). One is that our approach is
highly more structured as evidence by the low number of adjustable parameters. To capture the
dynamic behavior using an ANN model would require the use of lag variables that carries at least
one parameter for each lag variable. Hence, the number of parameters can be enormous
depending on the sampling rate and the dynamics. Also adding to the number of variables is the
need for one variable for each r value. Thus empirical models can require an extremely large set
of parameters that can be quite challenging to estimate under nonlinear regression techniques.
Secondly, our proposed approach is a continuous-time one and does not place restrictions on
sample data such as constant and equally spaced values as required by empirically-based lag
methods. Thirdly, since our approach uses highly structured models with a phenomenological
skeletons (i.e., transfer forms), mild extrapolation is less risky using our approach than an
empirical model with highly nonlinear behavior and many adjustable parameters. (Note the
emphasis we placed on our models having the correct limiting behavior.) Lastly, unlike the
empirical model parameters, the model parameters in our approach have phenomenological
interpretation. For example, the time constant in Model 1 is τ(r) and in Model 2 it is β(r).
15
The modeling approach that we have presented in this work comes from the area known
as system or process identification in the systems engineering literature (Nells, 2001; Gomez and
Baeyens, 2004; Westwick and Verhaegan, 1996; Zhu, 2002). Moreover, the work that we have
presented in this article has broader impact to the area of systems engineering and transfer
function modeling than this application and our recent work in this area (Rollins et al., 2006a;
Rollins et al., 2006b; Rollins and Bhandari, 2004; Bhandari and Rollins, 2003; Rollins et al.,
2003). To our knowledge, this is the first application to produce a continuous-spatial-temporal
model of a real process in analytical form (i.e., closed form) under low parameterization. We
were able to do this here because we were able to develop continuous models for the dynamic
model parameter and the initial value of the response variable as a function of the spatial
variable. Consequently, in other similar type of modeling situations we encourage the use of this
approach because the development of analytical continuous-spatial-temporal models offers
unique benefits in process understanding, control and optimization.
ACKNOWLEDGEMENTS
This research was funded by the Future Affordable Multi-utility Vehicle Program DAAD19-03D-0003 under the Department of Defense Appropriations Act administered through Florida
A&M University in Tallahassee, Florida. We are also grateful to Ailing Teh for her assistance in
the revised manuscript.
16
NOMENCLATURE
C
CVI
nr
r
SSE
t
trk
T
T0
Tr
Tr0
The constant given by Eq. 15
Chemical Vapor Infiltration
number of samples corresponding to the value of r
radial distance from the heater surface
Sum of Squares Errors
Time
time of the kth sample at distance r from the heater surface
Temperature
initial temperature
temperature at distance r from the heater surface
initial temperature at distance r from the heater surface
Greek letters
α
adjustable parameter
β
adjustable parameter
γ
adjustable parameter
λ
adjustable parameter
τr
time constant at distance r from the heater surface
τ0
time constant at distance r = 0 from the heater surface
Superscript
Estimate
^
17
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