High-Temperature Properties and Applications of Polymeric Materials

Chapter 7

Advanced Modeling of the Generation and Movement of Gases Within a Decomposing Polymer Composite

Downloaded by STANFORD UNIV GREEN LIBR on August 19, 2012 | http://pubs.acs.org Publication Date: October 13, 1995 | doi: 10.1021/bk-1995-0603.ch007

Hugh L . N . McManus and David S. Tai Technology Laboratory for Advanced Composites, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, M A 02139 The release of absorbed water has been shown to be a critical factor in the failure of composite material ablators. A reaction rate equation to model the phase change of water to steam in composite materials is derived from the theories of molecular diffusion and equilibrium moisture concentration. The model is dependent on internal pressure, the microstructure of the voids and channels in the composite materials and the diffusion properties of the material. Hence, it is more fundamental and accurate than the empirical equations currently in use. The model and its implementation into the thermostructural analysis code C H A R are described. Results of parametric studies on the variation of several parameters are presented. When composite nozzle liner materials (geometry shown in Figure 1) are exposed to the high temperature environment inside a rocket nozzle, different reaction zones develop as shown in Figure 2. First, the material on the heated side will begin to decompose and form a pyrolysis zone. When the pyrolysis is complete, it leaves a layer of char behind. As more heat conducts into the material, the pyrolysis zone advances deeper into the virgin material. Ahead of the pyrolysis zone, absorbed moisture is released. A moisture evaporation zone will also develop and advance ahead of the pyrolysis zone in lower temperature material. Gases, which are produced by pyrolysis decomposition and moisture evaporation, flow through the material to the surface. These gases can cause large internal pressures. The thickness of the composite insulation is designed so that the char layer will not reach the back side of the material before the rocket engine is shut down. However, several anomalous events can occur during the flight which can cause the insulation to fail prematurely. One of the severe anomalies is known as ply-lift. Plylift refers to the across-ply failure of the matrix material, and it has been observed in the exit cone liners of post-fired rocket engines. The ply-lift failure mode usually occurs in composites with low ply angles in the region just underneath the pyrolysis zone, as shown in Figure 3. Ply lift failure is discussed from an experimental point of view by Stokes (7). The ply-lift failure has been attributed to the following mechanisms. When the material is heated rapidly, gases are generated and trapped. These gases cause a large increase in internal pressure which forces the plies apart. Since ply-lift usually occurs 0097-6156/95/0603-0114$12.00/0 © 1995 American Chemical Society

In High-Temperature Properties and Applications of Polymeric Materials; Tant, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995.


7. McMANUS & TAI

Generation and Movement of Gases

Figure 1. The geometry of ablative insulation.

Figure 2. Classification of the reaction zones.


115

HIGH-TEMPERATURE PROPERTIES AND APPLICATIONS OF POLYMERS


116

in carbon-phenolic composite materials at temperatures below 400 °C and pyrolysis reactions usually do not begin below 400 °C, it is suspected that the high pressure is mainly caused by the release of absorbed water (7). When the matrix material in the composite decomposes to char, the material's permeability can increase by as much as seven orders of magnitude. Thus, the gases generated inside the pyrolysis zone can escape easily, while steam released in the evaporation zone has more difficulty escaping since it has to pass through the relatively impermeable virgin material between the pyrolysis and moisture evaporation zones. Large internal pressures are built up by gases trapped between these zones. Although not yet pyrolyzing, the material in this region is hot enough to have lost much of its room temperature strength, and it is in this narrow region where ply-lift failure usually occurs. Better modeling of the moisture evaporation process will thus result in more accurate prediction of the internal pressure and ply-lift failure. In this study, a new moisture release model, based on fundamental physical models of moisture diffusion and equilibrium moisture concentration, is derived. This model is coupled into an existing thermo-chemical-structural analysis program to provide a new and more accurate tool for predicting the behavior of ablative composite materials. Background. The techniques used in analyzing decomposing polymer composites are reviewed briefly in the introductory chapter to this book. Here, a specific review of the modeling of decomposition and moisture release in polymer matrix composites, as used in the overall modeling of thermo-mechanical analysis of decomposing structures, will be given. In general, a chemical reaction rate can be modeled by an n-th order Arrhenius rate equation. Many authors have reported on rate equations for the decomposition of phenolic materials typically used in ablative applications, of which the most relevant is that of Henderson et al. (2). More current models for the chemical pyrolysis and gas generation of carbon-phenolic materials include the 4-step Arrhenius model used by Sullivan (5), and Kuhlman (4), originally developed by Loomis, et al. (5). The Arrhenius reaction is not pressure dependent. Therefore, it can predict the generation of gases even if the pressure at the point of generation is higher than the saturation pressure of the assumed gaseous substance. This is physically unreasonable. To model a temperature and pressure dependent moisture evaporation rate equation, McManus (6) proposed using a 2-step reaction including a simple straight-line model for moisture release, in which the reaction rate is constant. He later proposed a hybrid model with straight-line moisture release and Arrhenius pyrolysis, and also an Arrhenius equation with the activation energy as a function of pressure (7). Arrhenius rate reactions coupled with models for re-condensing of pressurized water have also been proposed. However, all these methods are empirical. There is no guarantee that they give an accurate moisture evaporation rate and they provide no insight into rate determining mechanisms. Approach. The new reaction rate model will consider the physical processes of molecular diffusion of moisture to the surface of a pore channel, and the release of moisture (steam) into the channel. For a given initial moisture content, diffusivity constants, and pore channel geometry, the release rate of moisture to steam as a function of temperature and time will be predicted. The new reaction rate equation will be incorporated into the existing C H A R code. Coupling the new reaction rate equation to the thermal, mass continuity and stress equations in C H A R will allow


7. McMANUS & TAI

117


determination of pressure, stress and failure (if any) to be predicted as functions of time and position throughout the ablative structure.


Theory. The new reaction rate equation for moisture evaporation will be derived from a microscopic point of view. First, it is assumed moisture is uniformly distributed within the material. Then the moisture must diffuse to a nearby pore channel, driven by a difference in concentration. The pore channel is either a long crack along the fiber-matrix interface or a closed crenular channel inside a fiber. The geometry considered is shown in Figure 4. A coordinate r is defined perpendicular to the fiber direction. Radial symmetry is assumed. The fiber direction is related to the laminate coordinate system by angles Θ and Φ (Figure 1). The moisture will evaporate at the pore channel's surface and become steam. How much moisture evaporates at the surface is determined by the equilibrium moisture concentration, which depends on the temperature and vapor pressure inside the pore channel. Since moisture evaporation on the surface at high temperature is very fast, it is assumed the surface will achieve equilibrium instantly. Then steam will escape by flowing through the pore channel, driven by a pressure difference. A schematic of the moisture release process is shown in Figure 5. Moisture Diffusion. The moisture diffusion is governed by Fick's diffusion law (8). V-(dVc) = — dt

(1)

where d is the moisture diffusivity, c is the moisture concentration (the mass of moisture inside the material divided by the mass of dry material), and V is the del operator. The diffusivity d is given by d = d cxp(-^ ) 0

(2)

:

where d is a pre-exponential factor, E is an activation energy, R is the universal gas constant, and Τ is the absolute temperature. The moisture control volume is defined such that all moisture inside the control volume will go to one pore channel. In general, the boundary of a moisture control volume is an irregular polygon. This shape is approximated by a circle with radius r . The initial moisture content is assumed to be uniform inside the material, smearing any difference in moisture absorption between the fiber and the matrix. Similarly, the fiber and matrix materials around the pore channel are assumed to have a homogeneous effective diffusivity constant. If it is assumed that r is less than Δζ, where Az is the node spacing for numerical calculations, then it can also be assumed the temperature inside any one control volume is constant so that moisture diffusivity is constant everywhere in the control volume. With the same assumption, the derivative of c in the fiber direction can be neglected, and the moisture release rate can be derived independent of Θ and Φ . These assumptions reduce the problem from three dimensions to one, and equation 1 is reduced to 0

w

g

a

a



118


Figure 3. Geometry of ply lift failure.

Figure 4. Geometry of the moisture release model.


7. McMANUS & TAI

119


A n initial condition c = c and boundary conditions 0

c=0

atr = r

dc _ — =0 dr

at r = r

p

a


are assumed, i.e. moisture is zero at the surface of the central pore channel and moisture cannot cross the boundary to other control volumes. Using separation of variables, let ^ -

= R(r)6(t)

(4)

Then

R

r R

άθ

where R' indicates the derivative of R with respect to r and θ indicates the time derivative of Θ. For the R(r) part, 2

r / r + r/?' + A V / ? = 0

(6)

This ordinary differential equation (ODE) has the form of Bessel differential function of order zero (9), and the general solution is R(r) = AJ (Xr) + BY (Xr) 0

(7)

0

where A and Β are constants, J is a Bessel function of order 0 of the first kind and Y is a Bessel function of order 0 of the second kind. The boundary condition at r = r requires R(r ) = 0, and the boundary condition at r = r requires R'(r ) = 0. These boundary conditions can only be satisfied if the following characteristic equation is satisfied. Details may be found in (10). 0

0

P

P

a

Jo(Kr WnOP

a

r (V , W V . ) = ο 0

(8)

To simplify this characteristic equation, let

where a = rjr

(10)

p

Then J (z )Y (az ) - Y^zJJ^azJ 0

n

x

H

=0


(11)

120


The roots z are solved from equation 11 numerically for any given a , and λ is calculated by equation 9. Then equation 7 becomes H

Λ

^ ) = XCj (r)

(12)

B

n=l

where y (r) = Y a r )J a r) Downloaded by STANFORD UNIV GREEN LIBR on August 19, 2012 | http://pubs.acs.org Publication Date: October 13, 1995 | doi: 10.1021/bk-1995-0603.ch007

n

o

n

p

o

- Μ^ )Υ (

n

ρ

λ κ)

0

(13)

Λ

R(r) is represented by the sum of the magnitude C times the mode shapes y (r). n

n

For the 9(t) part, 2

θ = -λ άθ

(14)

η

Substituting equation 2 into equation 14 15

0 = -A*4,exp(-|^)0

which is an ODE with solution e(t) = e(0)exp(-dXg(t))

(16)

*(*)= fexp(

(17)

where

K J

^—)ds

H V

* Jo R T(s) Since the initial conditions is c(0) = c , and R(r) is assumed to be 1 for r < r < r at time zero, θ(0) = 1 and g

0

p

0(i) = e x p ( - ^ ( O )

a

(18)

Applying R(r)=l for r < r < r to equation 12, p

a

1=ÎCj (r)

r Ρ /Τ) ), c cannot be greater than since the material could not physically absorb more moisture than the maximum amount. In that case, the supersaturated steam will probably condense to water inside the pore channels. This possible phenomena is neglected here so that a single phase flow of gas can be used. 0

v

Μ

Moisture Evaporation Rate. The solution for moisture diffusion and concentration derived in the previous section assumed a boundary condition of zero concentration at


7. McMANUS & TAI


123

the surface. The solution for realistic boundary conditions can be found by a convolution integral (70). The degree of conversion is given by c

w

w = / ( o - £ c . ( i - « ) J à

(32) D

= f(f) - J 7 ( w ) É n *xp(-Jfo[*W " S(u)])du 0

where equilibrium degree of conversion f(t) = c / c is given by Downloaded by STANFORD UNIV GREEN LIBR on August 19, 2012 | http://pubs.acs.org Publication Date: October 13, 1995 | doi: 10.1021/bk-1995-0603.ch007

0

/ ( ί )

=

P P

£=- Α)»

V

High-Temperature Properties and Applications of Polymeric Materials

Recommend Documents