NASA/TM--1999-209444

Porous Media and Mixture Models for

Hygrothermal Behavior of

Phenolic Composites

Roy M. Sullivan

Glenn Research Center, Cleveland, Ohio

Eric H. Stokes

Southern Research Institute, Birmingham, Alabama

Prepared for the

International Mechanical Engineering Congress and Exposition

sponsored by the American Society of Mechanical Engineers

Nashville, Tennessee, November 14-19, 1999

National Aeronautics and

Space Administration

Glenn Research Center

October 1999

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POROUS MEDIA AND MIXTURE MODELS FOR HYGROTHERMAL BEHAVIOR OF

PHENOLIC COMPOSITES

Roy M. Sullivan

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, OH 44135

Eric H. Stokes

Southern Research Institute

Birmingham, AL 35211

ABSTRACT

Theoretical models are proposed to describe the interaction

of water with phenolic polymer. The theoretical models involve

the study of the flow of a viscous fluid through a porous media

and the thermodynamic theory of mixtures. From the theory a

set of mathematical relations are developed to simulate the

effect of water on the thermostructural response of phenolic

composites. The expressions are applied to simulate the

measured effect of water in a series of experiments conducted

on carbon phenolic composites.

NOMENCLATURE

A

Ai

E

M RU

P

MW

Np

N iRu

N W

P

PR>I

T

VW

g

Specific Helmholtz Free Energy

Partial Helmholtz Free Energy of ith Constituent

Polymer Network Stiffness

Molecular Weight of the Polymer Repeating Unit

Molecular Weight of Water

Number of Moles of Polymer Molecules per Unit

Volume

Number of Moles of Polymer Repeating Unit per Unit

Volume

Number of Moles of Water Molecules per Unit Volume

Solid Mixture Pressure

Partial Pressure of i'h Constituent

Total Pressure on the Material

Temperature

Water Vapor Velocity Vector

Specific Gibbs Free Energy

gi

k

P

Pw

S

S i

V

"¢i

0:'

Ett

14"

P

P

P_

1)i

Partial Gibbs Free Energy, of i 'h Constituent

Permeability

Pore Vapor Pressure

Vapor Pressure of Water

Specific Entropy

Partial Entropy of im Constituent

Specific Volume

Partial Volume of i th Constituent

Thermal Expansion Coefficient for Polymer Network

Material Strain Normal to the Fabric Plane

Material Strain Normal to the Fabric Plane due to

Water

Porosity

Vapor Viscosity

Density

Partial Density of i" Constituent

Mixture Volume Fraction of i_hConstituent

Subscripts

n

P

W

material direction normal to the fabric plane

polymer

water

Superscripts

p Pore Volume

s Solid Mixture

NASA/TM--1999-209444 1
INTRODUCTION

Phenolic composites are used as thermal insulation in a

variety of high temperature applications. Phenolic composites

are quite often fabricated as a laminate structure using a carbon

or ceramic woven fabric reinforcement. Although these

materials char and ablate at ultra high temperatures, their

structural integrity must be maintained at moderately high

temperatures in order to fulfill their protective function.

The thermostructural response of the phenolic composite

material is significantly influenced by the presence of water in

the material. The effect of water on the thermal expansion

behavior and the thermal stresses generated in carbon fabric

reinforced phenolic composites has been measured by Stokes

(1987) and Hubbert (1989). This effect is most pronounced in

the direction normal to the fabric plane, the direction which is

most significantly influenced by the physical events occurring

in the polymer.

In this paper, we propose theoretical models, which

describe the interaction of water with the phenolic composite

material. These theoretical models involve the study of porous

media and the flow of a viscous fluid through a porous media.

We also employ mixtures theory and thermodynamic principles

of condensed mixtures to develop mathematical relations for

simulating the effect of water on the thermostructural response.

The mathematical relations are applied to simulate the measured

effect of water on the thermal expansion and thermal stress

response of carbon fabric reinforced phenolic composites. In

the end, the mathematical relations prove to be quite accurate in

sinmlating the effect of water as measured by Stokes (1987) and

Hubbcrt (1989).

MATHEMATICAL MODELS

Basic Assumptions: A Review of the Paradigm

In order to develop the mathematical theory to describe the

effect of water on the thermostructural behavior, let us consider

the scenario which is depicted by Fig. I. The water molecules

(represented-by circles), given their polar nature, occupy

charged sites on the polymer chains resulting from local

induced or permanent dipole formations on the phenolic

network. It's also possible that clusters of water molecules are

associated with local concentrations of charged sites on the

thermoset resin. As the polymer network is heated past the glass

transition temperature, it will undergo significant elongation.

When the water molecules have absorbed enough energy to

overcome the bond energy, they will be liberated from the

polymer chains and essentially evaporate into the free volumes

of the polymer (Fig. I). The effusion of the water molecules

from the material occurs through the flow of water molecules

through the free volume of the polymer. The water molecules

that remain attached to the polymer network have a significant

impact on the thermal expansion behavior of the polymer

network and on the thermal stresses that are generated in the

material.

J

Solid Mixture

Volume, l-d)

/,,,

I

I, I

tI

• t, - O O uO0 _"

Fig.1. Sketch of the Polymer Containing Water

We propose that these physical processes are sufficiently

represented by a porous media model where water is evaporated

from a solid skeleton into a pore network and that, once

evaporated, the water flows in the form of water vapor through

the pore network. The polymer free volume is represented by

the pore volume with volume fraction _. The polymer and

attached water molecules are represented by the solid skeleton

and occupy the volume I-¢. We assume that the evaporation

of water molecules from their sites on the solid skeleton is

thermodynamically equivalent to the evaporation of water

molecules from a condensed mixture, a mixture where only the

water constituent is volatile. The tbrcc responsible for the water

vapor flow though the pore network is the pore pressure

gradient.

This thermodynamic system may be represented by the

sketch shown in Fig. 2. Each cell in the sketch represents an

open thermodynamic system consisting of a condensed mixture

of polymer and water in equilibrium with its vapor. Each

mixture is itself a homogeneous mixture, although the

concentration may vary from cell to cell. Water may enter or

exit each cell through the pore network in the tbrm of water

vapor. Within each local thermodynamic system (cell),

chemical equilibrium between the water molecules in the

mixture and the water molecules in the vapor state is always

maintained. Water may either evaporate from or condense into

the mixture depending upon the local concentration of water,

the temperature and the vapor pressure.

NASA/TMI1999-209444
Fig. 2. Sketch of the Equivalent Thermodynamic System

As water will exist at both the solid sites and in the pore

volumes in the form of water vapor, the volume average

expression for the density of water in the composite material is

p,,, =(_- e)p;_,+ Cp,(', (I)

where p l' is the density of the water in the pores, p_, is the

density of the water in the solid mixture and @ is the volumetric

porosity.

Thermodynamics of Mixtures: Derivation of the

Mixture Pressure Expression

The thermodynamic relations for incremental changes in

the specific Helmholtz tree energy and specific Gibbs tree

energy lot a homogeneous body are given by

dA = -sdT - Pdv (2)

dg = -sdT + vdP (3)

where A, g, s, v, T and P are the specific Helmholtz free

energy, the specific Gibbs free energy, the specific entropy, the

specific volume, the temperature and the pressure, respectively.

For a homogeneous mixture of two constituents, the usual

form of the mixt'ures relations tot the extensive variables are

pA = PlAl + P2A2 (4)

Pg = Plgl + P292 (5)

ps = PlSl + p2s2 (6)

where p is the density of the mixture and where p] and P2

are the partial densities of constituents I and 2, respectively. In

equations (4) through (6), A1 , gl and s I are the partial

Helmholtz flee energy, the partial Gibbs free energy and the

partial entropy of constituent I, respectively. The quantities A2 ,

g2 and s-, are the partial Helmholtz free energy, the partial

Gibbs free energy and the partial entropy of constituent 2,

respectively. The differential form of (5) can be written as

pdg = Pldgl + P2dg2 (7)

provided the density and partial densities do not vary

considerably over the differential increment.

The same relations as given in equations (2) and (3) can be

used to relate the partial thermodynamic quantities for each

constituent in the mixture. That is the partial Helmholtz free

energies and the partial Gibbs free energies l_r constituents I

and 2 are given by

dA I = -s IdT - PtdVl (8)

dA 2 =-sodT - P2dv2 (9)

dg I =-sjdT + vldP I (10)

dg 2 = -s,,,dT + v_dP_ (11)

where v I and v_, are partial volumes and PI and P2 are the

partial pressures.

Rewriting (3) and multiplying through by the density of the

mixture p, we get

pvdP = pdg + psdT . (12)

Substituting (6) and (7)into (12), we obtain

pvdP= pldg I + P2dg2 + PlsldT + p2s2dT. (13)

Finally, upon substituting (10) and (11) into (13), we obtain

pvdP = plvldP! + p_v_dP, . (14)

Since the density of the mixture is the inverse of the specific

volume, pv = I. Equation (14) therefore becomes

dP = plvldPl + p2v2dP2. (15)

Equation (15) expresses the total pressure acting on the mixture

in terms of the partial pressures of the two constituents.

NASA/TM--1999-209444 3
Notethatthepartialdensities/9i are the mass of the ith

constituent per unit total volume, whereas the partial volumes

vi are the volumes occupied by each unit mass of the ith

constituent as measured within the occupied volumes. As such,

the products PlVl and P2v2 represent volume fractions of

constituent I and 2, respectively and (15) may be rewritten as

dP = VldP l + w, dP. (16) _

where Pi are the volume fractions.

In the application of the mixtures theory to the mixture of

phenolic polymer and water, the subscripts I and 2 in equations

(15) and (16) are replaced with w and p to obtain

dP = p;++.v,.dC,. +ppvedP p (17)

and

dP = v., dP. + v t' dPl' (18)

In (17), the superscript s has been added to P_v to denote the

density of water in the condensed (solid) mixture.

Application of Mixtures Theory to the RTG

Experiments

The Restrained Thermal Growth (RTG) test was designed

to measure the thermal stress generated in the material while

subjected to high temperatures and simultaneously restrained

from expansion in one of the principal material directions. In

the tests conducted by Stokes (1987) and Hubbert (1989), the

stress (or pressure) in the direction normal to the fabric plane

which is required to maintain zero strain in that direction was

recorded as a tunction of temperature.

I. We note lhal (16) has precisely the form of the volume average stress

expression used in porous media theory to express Ihe total stress of a porous

body _ in terms of the solid skeleton stress _' and the pore volume stress

OP. namely _=(1-¢)_+_ p where _ is the volumetric porosity. It

should be noted, however, that equation (16) is not of Ihe same form as

Dalton's lay, of partial pressures which defines the total pressure of gaseous

mixtures as the sum of the partial pressures, P = Pt + 1'_ + ..... F34uation (16)

would have the same form as Dahon's law if each gas constituent

simuhaneously occupies the same volume, namely the total volume, since in

thatcase v l=u 2 =v_=...=l.

In order to apply the mixtures theory to simulate the effect

of water in the RTG experiment, it is necessary to make certain

assumptions regarding the thermodynamic state of the water in

the material. After all, equations (17) and (18) do not specify

the variation of the partial pressure with temperature, but

merely relate the partial pressures to the total pressure.

Knowing how the thermodynamic state of the water in the

polymer/water mixture varies with temperature will necessarily

specit_, the variation of its partial pressure with temperature.

To this end, we first assume that as the mixture is heated

through a continuous series of equilibrium states, the partial

Gibbs fl'ee energy of the water does not vary, dg. = 0, Under

this condition, equation (10) can be rewritten as

swdT = vwdP .. (19) -_

after replacing the subscript 1 with w to represent the water

constituent. Substituting (19), equation (17) may be written

dP = P;_I.swdT + ppvpdPp. (20)

The total pressure is obtained by integration of (20) as

P = P" + f P_'I, ,%mdT+ f p l, Vl, dPp (21)

where P" is the total pressure at some reference temperature.

The last term in equation (21 ) represents the response of a

RTG specimen devoid of water since it represents the pressure

response when p_, =0. The second term on the right hand

side in (21 ) represents the additional stress due to the presence

of water. This term, of course, accounts for the variation in

moisture content through the value of p_'_, .

We also make the assumption that s w in (21 ) is the entropy

of condensed water, which follows from our paradigm

represented by Fig. 2 that the mixture in each cell is condensed.

The variation of sw with temperature follows the variation of

the entropy of condensed water with temperature.

2. Equation (19) implies that the increment in the complimentary heal absorbed

by the water is equal to the increment in the complimentary work perlbrmed by

the water.

NASA/TM--1999-209444
InFig.3,therestrainingstresspredictedbyequation(21) is

plottedasafunctionoftemperaturealongwithmeasuredRTG

resultsfromHubbert(1989).Themeasuredresultsarefrom

testsconductedonFM5055carbonphenolicspecimensheated

at a constantrateof 5.55°C/sec(10°F/sec). Themeasured

resultsareshownfor specimenswith0%,4%and8%initial

moistureandareplottedas dashedlines.Theresponses

predictedbyequation(21)for4%and8%moisturearcplotted

assolidlines.Thiscomparisonhasbeenshownpreviouslyin

Sullivan(1996).

8o ..... -; ...........

i ,,,° .°

-- _o_ , -- ..... ,,

50 _ -- 4%_,Molslure ,

201°t _" ,.,. "a" _."

o//'

0 1O0 200 300 400 500

Temperature (C)

Fig. 3. Comparison of Measured and Predicted RTG Results

Our assumption that the thermodynamic state of water

molecules in the polymer network may be approximated by the

thermodynamic state of condensed water results in close

agreement between the predicted and measured RTG results.

This assumption seems justified in light of the fact that, in the

RTG experiment, the mechanical constraint and the presence of

the polymer molecules restrict the movement of the water

molecules, limit their available volume and maintain a low

entropy state which is approximately equal to that of condensed

water. Furthermore. we may argue that, under the constraints

imposed in the RTG experiment, there is no significant free

molar volume available lor the evaporation of water molecules.

Definition of Volume Fractions

The thermodynamic theory of polymer solutions utilizes a

liquid lattice model to establish relations for the entropy of

mixing in polymer solutions (Flory, 1953). In the lattice model,

every site in the liquid lattice is assumed to be occupied by

either a solvent molecule or a segment of the polymer chain.

Using the lattice model, the volume fractions for the water and

polymer in the mixture may be written as

N _,_,

v_, - (22a)

N _t + xN p

and

xN p

Vp - N w + xN I'

(22b)

respectively, where N,, is the number of moles of water per

unit volume and NI, is the number of moles of polymer

molecules per unit volume. The quantity x is defined as the

ratio of molar volumes of polymer and water. Its presence in the

volume traction relations is necessary in order to account for

the large differences in molar volumes between the two

constituents.

If we assign x to be the average number of repeating units

in the polymer chains _, then the number of moles of the

MRU RU repeating units per unit volume "'1' is given by Np = xNp.

The water volume fraction is therefore given by

N w

'o., - (23)

RU

N w +Np

The volume fraction now has the form of a mole fraction, since

we have accounted for the relative molecular size of the two

constituents.

The number of moles of water and polymer repeating unit

per unit volume in the mixture are related to the partial densities

• MRU RU by N, - P_I/M. and ..1, = pl,/MI, , respectively, where

RU

M w is the molecular weight of water and Mp is the

molecular weight of the repeating unit. The water volume

traction may therefore be written in terms of the partial

densities of water and polymer as

P;'I

v. - __ (24)

p_,, + Mp I,

-- RU

where M = M, / M 1, . Alternatively, we may write

S -- lJ 'd --

Pw 1- V_, MPI'" (25)

3. This definition is not the same as the previous definition of .I-. Hov, e,,rer, the

two definitions are consistent if we assume that each site in the liquid lattice is

occupied by either a water molecule or a polymer repeating unit. i.e. that the

repeating unit is the fundamental mixing unit of the polymer.

NASA/TM--1999-209444 5
Porous Media Model for Evaporation and Flow

For an ideal gas flowing through a porous solid body, the

local mass conservation equation is written as

O + 9 v- ap w

•(p.,Vw)=-(l-O) a--7- (26)

where _,. is the average velocity vector tbr the water and

3()at denotes the time derivative. In equation (26), the first

term represents the time rate of change of pore water density

per unit bulk volume. The second term represents the rate of

change in pore water density per unit bulk volume due to flow

through the pores. The last term represents the rate at which

water is supplied to the pores by evaporation from the solid.

In order to determine the rate at which water evaporates

from the solid, we consider a thermodynamic system consisting

of a condensed mixture containing water in chemical

equilibrium with its vapor. As this mixture is heated through a

continuous series of equilibrium states, chemical equilibrium

between the mixture and its vapor leads to the relation

dvw=vwldP( P dpw}p. (27)

where p.. is the vapor pressure of pure water and p represents

the pressure of thc vapor. Equation (27) results front

differentiation of Raoult's law. p = O..lh, 4.

Using (27). the time derivative of the water volume fraction

may bc written as

av., 3t - v,, {1 ap 1 ap_t, aT} . (28) p Ot p. aT at

Using the chain rule of differentiation, we have

ap;'i, op;_,,ao,,,

at _V., Ot

(29)

Differentiating (25) with respect to o. and using (28) and (24),

equation (29) becomes

3t M---p p _t Pw aT Ot " (30)5

Substituting (30), equation (26) becomes

¢ +_, •(p_,Vv- n,) =

-(]-¢)

(31)

Jo_St'(_'"_Mpp){ l ap ' _Pw aT 1

Mpp p at Pw aT at

Substituting the ideal gas relation for the pore volume density.

p_{',= p/RT, into equation (31) leads finally to

{ 1

(1 -O)P_] tP;], + Mpp )

Cp,{',+ _ .

Mp p ap

p at

--1- V (p..v.) 1_

[¢p,¢ (J-¢)m_,.(p;_,.

=I-7-+ Mp p

(32)

}

+ MOp I 3p. 3T

p. aT Ot

The water vapor velocity vector vw is calculated using

Darcy's law. which is given by v., =-(k//.t)V. p where k is

the permeability and /.t is the viscosity of the vapor.

Numerical Simulation of Water Effusion in the

Thermoqravimetric Experiments

Thermogravimetric tests involved heating a sample of

material in an atmosphere of inert gas and simultaneously

recording the sample's weight as a function of temperature. The

data generated is the weight fraction remaining versus

temperature. Tests were conducted on cylindrical samples

which were 0.635 cm (0.25 inches) in diameter and 1.27 cm

4. The classic expression of Raouh's law uses a mole fraction rather than a

volume fraction. The use of the volume fraction instead of the mole fraction

comes from the fact that. in polymer solutions, the volume traction is a more

accurate representation of the activity (Flory. 1953).

5. The use of the volume fraction in Raouh's taw results in a slightly more

complicated relation for _)p,',/_)t than that given in Sullivan and Stokes 11997)

where the water mass concentration was used to relate the pore vapor pressure

to the vapor pressure of pure water.

NASA/TM--1999-209444 6
(0.50inches)long.Thesampleswereheatedradiantlywiththe

electricalresistanceheater,the exactheatingratebeing

controlledmanuallyor with an electronictemperature

controller. Eachrunconsistedofa20°C/minuterampto100°C,

lollowedbya 10-minuteholdat 100°C,andfinallya linear

ramprateof 3°C/minute. A detaileddescriptionof thetest

procedureandthermogravimetric facilityaregivenin Stokes

(1993,1994a& 1994b).

The weightlossdataas a functionof thermocouple

temperaturewasfittedwithacubicsp[ine.Derivativeweight

losscurveswereproducedtbreachrunbytakingthederivative

of thesplinecurvewithrespectto temperature. A derivative

weightlosscurveconsistsof a seriesof peaks,eachpeak

representingaweightlossevent.

Thermogravimetricestswereconductedon specimens

with()c_moistureand49_moisture. Thederivativeweightloss

curveslk)rall the0_ moisturespecimenswereaveragedto

obtainan average0% derivativeweightlosscurve.The

derivativeweightlosscurvesforallthe4CAmoisturespecimens

wereaveragedtoobtainanaverage4_ derivativeweighttoss

curve.Thenetderivativeweightlosscurvewasobtainedby

subtractingtheaverage()cAderivativeweightlosscurvefrom

theaverage4_ derivativeweightlosscurve.Assuch,thenet

derivativeweightlosscurveconsistsof onepeak.which

represents theweightlossduetotheeffusionof moisturefrom

the4_,moisturespecimen.

Equation(32)isusedtomodeltheeffusionof waterfrom

carbonphenolicspecimensduringthe thermogravimetric

experiments. We assumethat the flow of waterin the

cylindricalspecimensis purelyradial.Theone-dimensional,

axisymmetricform of equation(32) is cast into a matrix

equation using the finite element method and solved

numerically at discrete time increments. The details of the finite

element approach for solving (32) and the numerical solution

scheme are described in Sullivan and Stokes (1997).

The solution to equation I"32) yields the pore vapor

pressures at the finite element nodes at each time step. The

average pore vapor pressure in each finite element is obtained

by averaging the element nodal values. The water volume

fraction is updated lot each finite element using (27). The

partial density of water in the solid mixture P_si,is updated for

each finite element using equation (25). The density of water

vapor in the pores p_! is calculated from the pore vapor

pressure and the temperature using the ideal gas law. The time

step is then incremented and the new temperature is calculated.

The vapor pressure of pure water Pw and the temperature

derivative dt?w/dT are updated for the new temperature using

a tabular input. The values for these quantities as a function of

temperature were obtained from Keenan, et al., (1969). The

solution to (32) is pertormed again and the process is repeated

until the end of the simulation.

The results of the thermogravimetric simulation are shown

in Figs. 4 through 6. These numerical results were obtained by

using the numerical values for porosity, permeability and

molecular weight of the repeating unit listed in Table 1.

Table 1. List of Properties Used for Thermogravimetric

Simulation

Property

O (%)

Value

11.5

Reference

Stokes (1992a)

k (sq. cm.) 2.69x10 -l_, Stokes (1992b)

Mp R_'(g/mole) 107 Theoretical

The pore vapor pressure as a function of spatial position in

the specimen at various temperatures is plotted in Fig. 4. Since

the specimens are cylindrical, the pore vapor pressure is plotted

versus the radial position using a non-dimensional radius ratio.

A radius ratio value of zero represents the centerline of the

cylindrical specimen and a radius ratio value of unity represents

the outer radius of the specimen. At temperatures below 150°C,

the pore vapor pressure is uniform throughout the specimen.

Above 200°C, the vapor pressure no longer remains uniform

due to water vapor flow through the pore network. From room

temperature to 300°C, the pore vapor pressure increases with

increasing temperature. Above 300°C, the vapor pressure

decreases with increasing temperature due to the depletion of

moisture in the specimens. Less water in the solid mixture

results in lower pore vapor pressures.

5 _ ................

300C

4 1 ....

_ 35 ..........

31_ .....

E. 2s _ - -

1 . C ....

0

0 2 O 4 0 6 08 1 1 2

Radius Ratio (-)

Fig. 4. Predicted Pore Vapor Pressure Versus Radial

Position at Various Temperatures for the

Thermogravimetric Experiment

NASA/TM--1999-209444 7
Thewatervolumefractionandthewaterdensityin the

solidskeletonwerevolumeaveragedateachtimestep.The

volumeaveragedensityof waterinthesolidskeletonandthe

volumeaveragewatervolumefractionareplottedasafunction

oftemperatureinFig.5.Theseresultsindicatethatasignificant

amountof evaporationfrom the solidmixturebeginsat

approximately200°Candthatall of thewateris evaporated

above400°C.

.... !02

,_ o o3

01

"6 oo2t _ .... 1

! k too,

100 200 300 400 500 6130 700

Temperature (C)

o

3=

Fig. 5. Predicted Average Density of Water in the Solid

Mixture and Average Water Volume Fraction Versus

Temperature in Thermogravimetric Experiment

The mathematical theory for evaporation and flow and the

numerical solution scheme are verified in Fig. 6, where the

derivative weight loss obtained from the numerical solution is

compared to the measured, net derivative weight loss reported

in Sullivan and Stokes (1997). The numerical solution results

are plotted as a solid line and the measured results are plotted as

hollow circles. The derivativc weight loss is obtained from the

numerical solution by taking the temperature derivative of

equation (1) using the wdume average values tbr p;_) and pP.

Notice the close correlation between the measured net

derivative weight loss and that predicted by the numerical

simulation.

Mechanics of Free Expansion

The thermal expansion o1' FM5055 carbon phenolic was

measured by Stokes (1987). The thermal expansion in the

direction normal to the fabric plane is shown in Fig. 7 for

specimens containing 0G, 4% and 8c_ moisture and heated at

5.55°C/sec (10°F/see). Cylindrical specimens 0.635 cm (0.25

inch) in diameter and 2.54 cm (I.0 inch) long were used for the

thermal expansion measurements. Notice the significant effect

of water content on the thermal expansi_m behavior.

o 03 .................................

0 025

_ 002

=

£

.t

_ 001

_ 0005

0

0

0 ̧005 ""

Temperature (C)

Fig. 6. Comparison Between Measured and Predicted

Derivative Weight Loss

The thermal expansion response normal to the fabric plane

may be separated into two parts: the inherent thermal expansion

behavior, due solely to the thermal excitation of the polymer

network, and that which is due to the presence of water in the

polymer network. This is expressed mathematically as

£n = _' (33) °_ndT + £,

where the first term on the right hand side represents the

inherent thermal expansion behavior of the polymer network

and the second term represents the expansion strain due to the

presence of water in the network. The thermal expansion

response of the 0_ moisture specimen shown in Fig. 7 is the

inherent thermal expansion behavior of the polymer network.

1.1' The strain due to water e n is dependent upon the amount of

water in the material and may be obtained from the thermal

expansion measurements as the difference between the 0%

specimen response and the response of the 4_ and 8%

specimens.

1oo .................. 8% Moisture

4% Moisture _

•_ '°!- / # / .... "7.\ ...... :

L IOO.M ..... \.,\ ....

= i x ," ..J \\"-.

-2o[ .....................

4° t ..................................................................................................

Temperature (C)

Fig. 7. Measured Thermal Expansion Versus Temperature

for Various Moisture Contents

NASArI'M-- 1999-209444
n'

In order to establish an expression for e,, , we note that the

differential increment in the total pressure of the material dPt,;t

may be given by the volume average expression as

dPt,,t = ( I - O )dP + Odp (34)

where again P is the pressure of the solid mixture and p is the

pore vapor pressure. We also note that, during the tree

expansion tests, the increment in the total pressure normal to the

fabric plane is zero. Setting equation (34) to zero and neglecting

the pore vapor pressure term leads to dP = 0 _. Using (18), we

may obtain

dPp - vw dPn, , (35) 7

1 - V+,,

since u w + Vl, = 1. Thus, although the total stress increment is

zero, the partial pressure in the polymer network is not. This

14'

polymer network partial pressure is the source of e,+ .

Assuming that the polymer network is linear elastic and

restricting our attention to one-dimensional expansion, the

polymer network strain normal to the fabric plane e,+ is given

by e_[ = Pp/E,+ where E,+ is the polymer network stiffness

normal to the fabric plane 8. The strain can thus be written in

terms of the water partial pressure and the volume fraction as

I [" Ow dP. C','=--j-- _,.

E n ( I - V_ )

Once again, we must assume the thermodynamic state of

water in the mixture in order to estimate the variation of its

partial pressure with temperature Pw = Pn" (T). Again, we

6. The relative insignificance of the pore vapor pressure is illustrated by

comparing the results shown in Figs. 3 and 4. The pore vapor pressure values

shown in Fig. 4 are less than 5 MPa. whereas the calculated partial pressures of

water in the mixture, as evidenced by Fig. 3. are significantly higher.

7. The negative sign results since the water partial pressure is compressive and

the polymer partial pressure is tensile.

8. The normal stiffness k/,, is also dependent upon the amount of water in the

polymer network, since water will effect the degree of cross-linking in the

polymer network and will have an effect on the glass transition temperature in

glassy polymers. The latter effect is evident in Fig. 7.

9. In [36), we have dropped the negative sign since it has already been

acknowledged that P,, is compressive.

assume that as the water/polymer mixture is heated, the partial

Gibbs free energy of water in the mixture remains constant.

This leads again to the relation given in (19). Substituting (19)

into (36) leads to

e,_] = I_] e o n Sn dT (37)

E,_ J (1 - o w ) v+v

where s n, and v n, are assumed to be the entropy and partial

volume of condensed water, respectively.

Numerical Simulation of the Effect of Water in the

Thermal Expansion Measurements

We will now use the previously discussed theories to model

the effect of water on the thermal expansion response of carbon

phenolic. We use the evaporation and flow model to calculate

the amount of water in the thermal expansion specimens as a

function of time and spatial location. Again. we assume that the

effusion of water from the specimen occurs through a pore

network, that the pore vapor pressure gradients drive the

effusion of water from the specimen and that the water vapor

flow in the cylindrical specimens is purely radial.

The pore vapor pressures, the density of water in the solid

mixture and the water volume fractions are Calculated at each

time step in the same manner as described in the previous

section. The normal tensile strain el| is calculated by equation

(37) at each time step using a volume average value for the

water volume fraction. The numerical values for s w and vw as

a |'unction of temperature was obtained from Keenan, et al.

(1969). The tensile stiffness E, was obtained from

measurements made by Stokes (1987. 1990a. 1990b). The

variation of E, with temperature for both 4% and 8c_ initial

moisture contents is plotted in Fig. 8. The numerical values

listed in Table 1 for the porosity, permeability and molecular

weight of the repeating unit are once again used for the

simulation.

In Fig. 9, the numerical results t.or e,_v obtained by (37) are

plotted as a function of temperature and compared to the

measured responses. The measured response for e][ for the 4e_

and 8% moisture specimens was obtained by subtracting the

thermal expansion response for the 0c/c moisture specimens

from the thermal expansion response |'or the 4c_ and 8%

moisture, respectively. The numerical results are plotted as solid

lines and the measured responses are plotted as dashed lines

with hollow circles.

NASA/TM--1999-209444 9
12O00 _4% Intllal _lslure _ _ _

100_

0

50 100 _SG 200 250 300 3517

Temperature (C)

Fig. 8. Tensile Stiffness Versus Temperature for 4% and 8%

Moisture

At temperatures below 250°C. the strain response obtained

by the numerical solution and equation (37) compares well with

the measured response of the 8c_ moisture specimens. The

predicted results for the 4_ specimen overestimates the

measured response at all temperatures. For both moisture

contents, the peak in the measured strain response occurs at

lower temperatures and lower strain values than that predicted

by the theory and numerical solution. The reason for this

discrepancy is that the permeability was held constant in the

numerical solution, whereas, in reality, the permeability is a

function of strain state. The effect of strain on permeability has

been measured by Stokes (1992c). As the material strains, the

permeability will increase. This will cause the moisture flow

rates to increase. The end result is the exhaustion of moisture

from the specimens at temperatures lower than that predicted by

the present solution.

g

150

125

75

50

25

0

Temperature (C)

300 350 400

Fig. 9. Comparison Between Measured and Predicted

Normal Strains Versus Temperature

DISCUSSION

Theoretical models have been proposed which describe the

interaction of water with phenolic polymer. These models are

proposed in order to derive a set of equations which simulate

the hygrothermal behavior of reinforced phenolic composites.

The evaporation and flow equation was successful in

simulating the effusion behavior of water from FM5055 carbon

phenolic specimens during thermogravimetric experiments.

This equation was derived from porous media principles as well

as from the thermodynamic principles of condensed mixtures. It

was developed based on the assumptions that the liberation of

water molecules from the polymer is thermodynamically

equivalent to the evaporation of water from a condensed

mixture and that the bound water on the polymer network is in

chemical equilibrium with the free water in the frec molar

volumes. Furthermore, it was assumed that the water transport

occurs through the free molar volumes.

In treating the polymer network and the water molecules

which are attached to the polymer network as a condensed,

binary mixture, two equations were derived from the

thermodynamics of mixtures principles. One equation simulates

the thermal stresses generated by the water. The other simulates

the stress-induced thermal strains due to water. The accuracy of

these two equations was examined by applying them to simulate

the effect of water in the RTG and thermal expansion

experiments. In both cases, the response predicted by the

theoretical models was in close agreement with the measured

response.

REFERENCES

Hubbert, T. E., 1989, "Mechanical Properties of FM5055

Baseline Carbon Phenolic," SRI-MME-89-1149-6071-23.

Flory, P. J., 1953, Principles of Polwner Chemisto', Cornell

University Press, New York.

Keenan, J. H., Keyes, E G., Hill, P. G. and Moore, J. G., 1969.

Steam Tables: Thermodynamic Properties qf Water hTcluding

Vapol, Liquid and Solid Phases, John Wiley and Sons, New

York.

Stokes, E. H., 1987, "The Effect ot" Moisture on the Mechanical

and Thermal Response of FM 5055 Carbon Phenolic

Composite," SRI-EAS-87-1244-6245-6.

Stokes, E. H., 1988, "The Physical, Mechanical and Thermal

Properties of the FM 5055 404 Aft Inlet Ring, Thermal Tests,

Gas Permeability and Thermal Expansion, Part 1." SRI-EAS88-201-6032-1, March, 252 p.

NAS A/TM-- 1999 -,09444 10
Stokes,E.H., 1990a,"AcrossPlyMechanicalndThermal

Characteristicsof FM 5055(AvtexPre-Shutdown) as a

Functionof theMaterial'sMoistureContent"SRI-MME-90-

1109-6526-5.

Stokes,E. H., 1990b,"MoistureEffectson AcrossPly

Mechanical andThermalCharacteristics of FM5055Carbon

PhenolicUtilizingRapidIsothermal Heating,"In:Proceedings

JANNAF Propulsiolz Meeting, Anaheim, CA, CPIA Pub. 550,

pp. 505-512.

Stokes, E. H., 1992a, "Anomalous Swelling Behavior of FM

5055 Carbon Phenolic Composite," In: Proceedings 33rd

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics

and Materials Conference, pp. 2932-2938, AIAA Paper 92-

2428, April. see also: AIAA Journal, Vol. 31, No. 3, March,

1993, pp. 584-589.

Stokes, E. H., 1992b, "Permeability of Carbonized Rayon

Based Polymer Composites," In: Proceedings ASME

Symposium on Computational Mechanics of Porous Solid

Matelqals and Their Thermal Decomposition, Scottsdale, AZ,

AMD-Vol. 136, April, pp. 145-156.

Stokes, E. H., 1992c, "Gas Permeability of RSRM Carbon

Phenolic Composites as a Function of Across Ply Tensile Stress,

Strain, and Temperature," In: PIvceedings JANNAF-RNTS

Meeting, Sunnyvale, CA, edited by M. J. Paul, CPIA Pub. 592,

December, pp. 457-466.

Stokes, E. H., 1993, "Kinetics of Pyrolysis Mass Loss From

Phenolic Resin Based Composites," In: Proeeedings JANNAFRNTS Meeting, Huntsville. AL, edited by M. J. Paul, CPIA Pub.

605, October, pp. 235-246.

Stokes, E. H., 1994a, "Mass Loss from Phenolic Resin Based

Composites Subjected to High Heating Rates," In: Proceedings

JANNAF-RNTS Meeting, Seattle, WA. edited by M. J. Paul.

CPIA Pub. 618, November 15-17, pp. 367-379.

Stokes, E.H., 1994b, "Kinetics of Pyrolysis Mass Loss from

MX 4926 Standard Density NARC Based Carbon Phenolic

Composite," SRI-MME-94-043-6672.18A.

Sullivan, R. M., 1996, "The Effect of Water on Thermal

Stresses in Polymer Composites," ASME Journal of Applied

Mechanics, Vol. 63, pp. 173- t79.

Sullivan, Roy M. and Stokes, Eric H., 1997, "'A Model lbr the

Effusion of Water in Carbon Phenolic Composites." Mechanics

_!fMaterials, Vol. 26, pp. 197-207.

NASA/TM--1999-209444 11
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

October 1999 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Porous Media and Mixture Models for Hygrothermal Behavior of

Phenolic Composites

6. AUTHOR(S)

Roy M. Sullivan and Eric H. Stokes

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44t35-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington. DC 20546-0001

WU-242-23-53-O0

8. PERFORMING ORGANIZATION

REPORT NUMBER

E- I 1930

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-- 1999-209444

11. SUPPLEMENTARY NOTES

Prepared for the International Mechanical Engineering Congress and Exposition sponsored by the American Society of

Mechanical Engineers. Nashville, Tennessee, November 14-19, 1999. Roy M. Sullivan, NASA Glenn Research Center,

and Eric H. Stokes, Southern Research Institute, Birmingham, Alabama 35211. Responsible person,

Roy M. Sullivan. organization code 5920, (216)433-3249.

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This publication is available from the NASA Center for AeroSpace Information. (301) 621-(t390.

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13. ABSTRACT (Maximum 200 words)

Theoretical models are proposed to describe the interaction of water with phenolic polymer. The theoretical models

involve the study of the tlow of a viscous fluid through a porous media and the thermodynamic theory of mixtures. From

the theory a set of mathematical relations are developed to simulate the effect of water on the thermostructural response of

phenolic composites. The expressions are applied to simulate the measured effect of water in a series of experiments

conducted on carbon phenolic composites.

14. SUBJECT TERMS

Carbon phenolic composites: Thermodynamics of mixtures; Porous media;

Thermogravimetric analysis; Moisture effusion

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