Moisture diffusion in multi-layer carbon composite structures is difficult to model using finite difference methods due to the discontinuity in concentrations between adjacent layers of differing materials. Applying a mass conserving approach at these boundaries proved to be effective at accurately predicting moisture uptake for a sample exposed to a fixed temperature and relative humidity. Details of the model developed are presented and compared with actual moisture uptake data gathered over 130 days from a graphite epoxy composite sandwich coupon with a Rohacell foam core

Nurge, Mark A.

Starr, Stanley O.

Youngquist, Robert C.

NASA Technical Reports Server

· , 
Mass conservation in modeling moisture diffusion in multi-layer carbon 
composite structures 
M A Nurge , R C Youngquist, S 0 Starr 
National Aeronautics and Space Administration, Mail Code: KT-D-l, Kennedy Space Center, Florida 32899 
Abstract 
Moisture diffusion in multi-layer carbon composite structures is difficult to model using finite difference methods due 
to the discontinuity in concentrations between adjacent layers of differing materials. Applying a mass conserving 
approach at these boundaries proved to be effective at accurately predicting moisture uptake for a sample exposed to 
a fixed temperature and relative humidity. Details of the model developed are presented and compared with actual 
moisture uptake data gathered over 130 days from a graphite epoxy composite sandwich coupon with a Rohacell® 
foam core. 
Key words: Finite difference modeling, Moisture diffusion, Rohacell®, Gaphite epoxy composites 
1. Introduction 
Rockets, such as the Delta and Atlas, are exposed to 
a varied environment, including high levels of humidity 
and moisture, which can adversely affect various com-
ponents and structures. Of particular interest are rocket 
fairings , often fabricated from graphite epoxy compos-
ite (GEC). These have been shown to absorb signifi-
cant amounts of water while in a high humidity envi-
ronment leading to structural weakening during ascent 
heating[I]. Consequently, methods for determining the 
water content of GEC structures are of interest, and one 
such method is by computation using a diffusion model. 
This approach assumes the GEC is dry at fabrication , 
requires continuous monitoring of the GEC tempera-
ture and humidity environment, and needs well estab-
lished material parameters, but is inexpensive and non-
intrusive. 
This paper discusses a one dimensional finite differ-
ence based diffusion model for a three layer composite 
structure composed of a GEC-foam-GEC sandwich ar-
rangement, similar to that seen in some rocket fairings. 
Others[3] have created similar models, but lack a thor-
ough explanation of the boundary conditions needed to 
enforce water mass conservation and have not compared 
Email addresses: Mark. A. NurgeIDnasa . gov (M A Nurge), 
Robert. C. YoungquistlDnasa. gov (R C Youngquist), 
Stanley . O. StarrlDnasa . gov (S 0 Starr) 
their models against long term experimental data. In 
the present discussion we explain in detail how mass 
conservation over the GEC-foam interface is treated 
and then compare the resultant model against the mois-
ture uptake seen by a GEC-foam-GEC sandwich sample 
over a 130 day period. 
2. Three Layer Diffusion Model 
The fairing material physically consists of five lay-
ers, formed by" bonding a GEC face sheet to each side 
of a Rohacell® foam core using a thin adhesive. Due 
to their thin size with respect to the GEC and foam, the 
glue layers were neglected for modeling the overall dif-
fusion behavior. However, the three layer model created 
can easily be extended to include the thin adhesive layer. 
The model consists of two main computational tech-
niques that will be discussed below. The first uses a 
standard finite difference equation based on Fick's law 
to compute the water concentration for the next step in 
time. This technique cannot be used at the boundary 
between dissimilar materials because the moisture con-
centration is discontinuous across the interface. There-
fore, a second mass conserving technique was created to 
compute the moisture concentration across that bound-
ary. 
Preprint submitted to Composites Part A: applied science and manufacturing January 5, 2009 
https://ntrs.nasa.gov/search.jsp?R=20110011490 2019-08-30T15:41:49+00:00Z
2.1. Discrete representation of Fick 's law 
Fick's law of diffusion in differential form is 
(I) 
where D is the diffusivity and c is the water concen-
tration [2] . After integrating this equation over a vol-
ume and applying the Divergence Theorem, the result-
ing form tells us that the change in water in that volume 
is the net difference between what goes in and out of the 
surface that bounds the volume. The present problem 
can be reduced to a simple one dimensional equation 
by assuming the diffusivity and concentration only vary 
across the thickness of the sample. Integrating Equa-
tion lover a cuboid shaped volume with each face per-
pendicular to a coordinate axis and assuming that the 
concentration varies only along the x-axis yields the fol-
lowing equation: 
l h Be ac Be A -dx = A(D- II -D- 12), o at ax ax (2) 
where A is the area of each of the cuboid faces normal to 
the x-axis, numbered 1 and 2 above, and h is the distance 
between the two faces. 
For sufficiently small time and spacial intervals, the 
following discrete form can be used to perform numeric 
computations: 
where T is the time interval between adjacent time steps, 
nand n+ 1, and h is the spacial interval between adjacent 
spacial nodes, i and i + 1. This equation was used to find 
the time evolution of the concentration c7+ I at each point 
on the grid except at the boundary layers. This method 
is often referred to as the forward in time and centered in 
space (FTCS) approach and is stable if 2DT / h2 < 1 [4] . 
2.2. Boundary conditions and conservation of mass 
The FTCS method works well until a discontinuity in 
moisture concentration is encountered at the border be-
tween two dissimilar materials. Springer [5] approaches 
this in the "W8GAIN" Fortran code by applying two 
boundary conditions at the interface between the two 
layers. 
The first is a reasonable statement that the relative hu-
midity, ¢, is equal across the boundary. This can be used 
to join together the empirical relationships between rel-
ative humidity and concentration for each of the mate-
rials, cIt = Plal¢f and c2\+1 = p2a2¢~D where c1 and 
2 
c2 are the moisture concentrations on either side of the 
boundary in materials 1 and 2, shown in Figure 1. PI 
and P2 are the dry densities of each of the materials and 
a i, bl , a2, b2 are the empirical coefficients of materials 
1 and 2, respectively. Equating ¢ on each side of the 
boundary yields: 
(4) 
However, the second condition mentioned by 
Springer, 
acl Be2 
Dlax = D2ax , (5) 
is not reasonable. This condition states that any mois-
ture that flows in from one face of a small cuboid vol-
ume leaves the opposing face with no accumulation, 
which is not physical since no water remains to accu-
mulate at the boundary. Therefore, another boundary 
condition is needed and can be found by enforcing con-
servation of mass in the finite volume surrounding the 
border between layers (refer to Figure 1). 
Let Cb be the accumulated moisture concentration at 
the boundary, determined by what flows into the left 
hand side of finite volume in material 1 and what flows 
out of it's right hand side in material 2. This is repre-
sented mathematically by Equation 6 below: 
;+1 ; T el~_ N - cl~ c2i - c2~ 
cb = cb + -h(D I 3 - D2 3 ), (6) 4h 4h 
where material 1 has been divided into k cells. The 
nodes for elk and c21 are ±h away from Cb and ~h away 
from el~_ N and c2; , respectively. 
From Figure 1, its easy to see that in order to conserve 
mass, 
(7) . 1 Ah . 1 Ah . 1 Ahc'+ - - c1'+ + - c2'+ b -2 k 2 1 ' 
Simplifying and rearranging terms yields, 
(8) 
Now Equations 4 and 8 provide enough information 
to progress across the boundary. Substitution of Equa-
tion 4 into Equation 8 gives us an equation for C~Hl as a 
function of c1i+l: 
C;'+ I = NL2EcN ~H N + p2a2EcN~HN /Plad2 / bl ) . (9) 
The value of C~H f is determined from Equation 6, leaving 
only cl~HN as unknown. Unfortunately, due to its non-
linearity, this equation is difficult to invert. Some [6] as-
sume the concentration is linearly related to the relative 
humidity, but this leads to error when b l *' b2. However, 
, 1 " 
M ter" 11 a la Interface M t . 12 a ena 
I I I I I I 
I .. h • 1 I I I I 
.. 
h 
• I Ch.2 I 
.. 3/4Ih. I I I 
Ch.1 I I I I 
c1. I I I f-+-
b .. NPL4~h I I 
C21 I I f....-
C22 I I I C23 
I 1 
c 
~x 
Figure 1: Finite volumes of width h are shown near the boundary 
between materials I and 2. The volume around node Cb is made up of 
two different materials, one with concentration C I k and the other c21. 
by allowing cN~HN to vary between 0 andpla, in fine in-
crements, a polynomial fit can be applied to the data to 
get cN~HN as a function of C~H l. A tenth order fit was 
performed for this work, but lower orders may be used 
depending on desired accuracy and the degree to which 
the empirical values for b vary from 1. Once cN ~H N is 
calculated, c2't 1 is easily found from Equation 8. 
Equation 3 works well for the remaining nodes out-
side the boundary. However, it must be modified 
slightly when calculating c1 k- 1 and c22 to account for 
the change in node spacing near the boundary (3/4 h vs. 
h). Once each of concentrations for the next time step 
are computed, the final step is to sum up the water mass 
in each finite volume for a given time. This is then used 
to construct the plot of the total water mass vs. time. 
As a self consistency test, the moisture rate flowing in 
through the outer faces was integrated with respect to 
time and proved to be identical to the total achieved by 
summing up the moisture distribution through the sam-
ple at each step in time. 
3. Experimental Data 
A graphite epoxy composite sandwich with Rohacell 
foam core was analyzed to determine its moisture up-
take properties. Moisture uptake measurements were 
made on a sample of the foam core and a complete 
sandwich coupon. The manufacturer, United Launch 
Alliance (ULA), provided the key information on the 
graphite epoxy face sheet. The samples were vacuum 
3 
dried and then subjected to 37.8°C and 85.0% RH in 
an environmental chamber. Periodic weight measure-
ments were made during the wetting process to deter-
mine change in mass over time . 
3.1. Foam Core Results 
Two samples (Figure 2) of 110 WF Rohacell were 
tested, which according to the manufacturer has an av-
erage density of 0.110 gjcm3 over a large sheet. The 
measured densities of the samples tested are 0.11 g/cm3 
and 0.13 g/cm3, showing some construction variation. 
The mathematical expression for the normalized mois-
ture uptake is given by Springer [5] as 
M(t) - M; = 1-! ~ 1 e-(2)+ I? rr2Dt /h2 (10) 
Mm-M; 7r2 ko (2)+ 1)2 ' 
where, M(t) is the measured mass as a function of 
. elapsed time, t, M; is the initial mass, Mm is the max-
imum mass, h is the thickness of the sample, and D is 
the diffusivity. Mathematica@ [8] was used to find the 
diffusivity by restricting the summation to the first 100 
terms and applying the Fit function to the data sets. The 
resulting plots are shown below 
Figure 2: Two samples of 110 WF Rohacell Foam were vacuum dried 
and subjected to moisture uptake testing in an environmental chamber. 
Equation 10 applies to an infinite sheet of fixed thick-
ness. Since the sample's length and width aren't suf-
ficiently large when compared to the thickness, a cor-
rection needs to be applied to take out the effect of the 
increased uptake due to the open sides. This corrected 
diffusivity, Dc is provided in terms of the length, width, 
and thickness (I ,w, and h) by Springer as: 
ac =aLENH~H~F2. 
I w 
(11) 
N.MrJJtJJrJJJJJJHJJWWWWW~J=J"D_f!__=N 
0.81-----+--+--
~ 
I 
WiM.SNJJJJHJJHJII"JJJJJL~JJJJJJjJJfJJJJj 
~ 10.4 
6 
0.2 
400 600 800 1000 1200 1400 
Sqrt(seconds) 
Figure 3: Normalized moisture uptake data and fitted curves for each 
of the RohaceLl foam samples. The fitted diffusivities are 4.2 x 10- 7 
cm2/sec for the 0.11 g/cm3 sample and 3.2 x 10-7 cm2/sec for the 0.13 
g/cm3 sample. 
The corrected values of diffusivity for our samples are 
2.8 x 10-7 cm2jsec for the 0.11 g/cm3 sample and 2.0 x 
10-7 cm2jsec for the 0.13 g/cm3 sample. 
3.2. GEC-Foam-GEC Sample Results 
Epoxy and aluminum tape were applied to the edge 
of a GEC-foam-GEC sample coupon to minimize mois-
ture uptake directly into the foam core. The sample 
was dried in a vacuum chamber for several weeks and 
then placed in the environmental chamber on a precision 
scale for approximately 130 days at constant tempera-
ture and humidity (37.8°C and 85.0% RH). The mass 
was collected from the scale at regular intervals and 
stored along with a time-stamp on a laptop computer. 
With the exception of the diffusivity of the Rohacell , 
the diffusion properties of the sample's materials have 
been previously determined. ULA provided the needed 
values for input to our model program, which are sum-
marized in Figure 4 below. 
Density a b D Area h (thickness 
(l!/cc)' (cm'/sec) (cm') (cm) 
facesheet 1.61 1.5 1.7 2.00E-09 248 0.142 
core 0.111 II 1.9 2.80E-07 248 1.27 
Figure 4: This contains a summary of the sandwich material parame-
ters . 
The data from the moisture uptake test is plotted 
along with the predicted output in Figure 5. The un-
usual data behavior after the 100 day mark is believed 
to be due problems with the scale, which failed entirely 
after about 130 days. 
4 
... 
o 
1.5 
«i 
~M .R 
0.0 
o 20 
, 
40 60 80 100 120 
Time (days) 
Figure 5: This plot shows the moisture uptake of a fairing coupon in an 
environmental chamber at 37.8°C and 85 .0% RH for about 130 days. 
Since the scale was also exposed to this environment, the anoma-
lous data after the lOOth day may be related to possible condensation 
within the scale electronics, which experienced a hard failure around 
the I 30th day. The 2nd plot shows the output predicted by our diffu-
sion model using the di ffusion parameters for the facesheet and foam 
core. 
4. Conclusion 
The finite difference method coupled with a mass 
conserving approach at the boundaries proved to be ef-
fective at predicting moisture uptake for fixed RH and 
temperature. The output of the multi-layer diffusion 
model is in close agreement with data collected on the 
moisture uptake of a GEC-foam-GEC sample coupon. 
Future work will focus on incorporating variations in 
temperature and relative humidity into the model to bet-
ter represent the environment seen by fairing material 
between fabrication and liftoff. 
References 
[ I ) Epstein G, Ruth S. Honeycomb Sandwich Structures: Vented 
Versus Unvented Designs for Space Systems. The Aerospace 
Corp. Contract Number:F04701-88-C-0089. Report Number: 
A3 I 7672. 
(2) Incropera FP, DeWitt DP, Bergman TL, and Lavine AS. Fun-
damentals of Heat and Mass Transfer. 6th Edition (2007), John 
Wiley and Sons, ISBN:987-0-45728-2, Chapter 14 
(3) Katzman HA, Castaneda RM, Lee S. Moisture diffusion in com-
posite sandwich structures. Composites: Part A 39(2008), p. 
887-892 
(4) Garcia AL. Numerical Methods for Physics. 2nd Edition (2000), 
Prentice-Hall Inc., ISBN:O-13-906744-2, p.198-200 
(5) Springer GS. Environmental Effects on Composite Materi-
als . 1 st Edition (1981), Technomic Publishing Company, Inc., 
ISBN:087762-300-7, Chapter 13 
(6) Augl 1M, Berger AE, Moisture Diffusion Analysis in Multilayer 
Composite Materials by Finite Difference Analysis. Naval Sur-
face Warfare Center, Carderock Division, NSWCCARDIV-TR-
95/013, March 1996. 
(7) RohaceLl product information, http://www.rohacell .com/ 
(8) Mathematica product information, http://www.wolfram.com 
" \ 


Mass Conservation in Modeling Moisture Diffusion in Multi-Layer Carbon Composite Structures

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