Boo Ho Seo, Young Jun Cho, Jae Roun Youn, Kwansoo Chung, Tae Jin Kang
School of Materials Science and Engineering, Seoul National University, Seoul 151-742, Korea
Jong Kyoo Park
Composite Laboratory, Agency for Defense Development, Yuseong, P.O. Box 35-5, Daejeon 305-600, Korea
A study on the thermal conductivities of spun yarn-type carbon/phenolic (spun C/P) composites using a thermalelectrical analogy method is presented. This method is based on the similarity between the partial differential equation that governs the thermal potential and electrical potential distribution. The unit cell of a laminate composite is divided into several conduction elements. By constructing an equivalent thermal resistance network in series, and in parallel based on analogy, we were able to predict the thermal conductivity of the composite. The prediction values obtained from the model are compared with known thermal conductivities on a carbon/ epoxy composite with an eight-harness satin (8HS) texture. It is shown that the model provides a good estimate of the thermal conductivity of the spun yarn fabric-reinforced composite. With the use of this model, the thermal conductivity of the spun C/P composites with 8HS was validated experimentally. Good agreement was found between the present approach and the experimental results.
POLYM. COMPOS., 26:791–798, 2005. © 2005
Society of Plastics Engineers
INTRODUCTION
The magnitude of thermal conductivity in composites is, on average, much lower than that of metals, and their response is anisotropic. Hence, it is generally much more difficult to dissipate heat in a fiber-reinforced composite compared to metals, and in some applications this may be an important consideration [1]. The physical and thermal properties of fiber-reinforced composites may be anisotropic depending on the fiber orientation. The determination of the effective properties of composite materials is of great importance in the effective design and application of composite materials [2].
Correspondence to: Y.J. Cho; e-mail: jaussi@empal.com
Contract grant sponsor: Ministry of Science and Technology of Korea/
National Research Laboratory, Seoul National University.
DOI 10.1002/pc.20142
Published online in Wiley InterScience (www.interscience.wiley.
com).
© 2005 Society of Plastics Engineers
Carbon/phenolic (C/P) composites are one of the most important materials utilized for ablative thermal protection.
Carbon fibers have been widely used as reinforcement in composites used for thermal protection because of their dimensional stability at high temperatures, low density, nonflammability, and outstanding physical properties. It is well known that rayon-based carbon fiber-reinforced phenolic composites exhibit low thermal conductivity along with good interlaminar shear strength and cross-ply tensile properties. However, because of difficulties in source availability and manufacturing processes, the use of rayon-based carbon fibers has been limited.
With the development of polyacrylonitrile (PAN)-based carbon fibers in the early 1960s, the rayon-based carbon fibers have gradually been replaced by PAN-based carbon fibers in most applications. Commercial PAN-based carbon fibers, in general, show good mechanical properties and high electrical and thermal conductivities. Mechanical properties and thermal conductivities of carbon fibers should be compromised to some extent for the ablative applications that low thermal conductivity is primarily more important than mechanical properties of the carbon fiber. For solid rocket motor ablative applications, one of the key requirements is that the carbon fiber should have low thermal conductivity to minimize the thickness of the pyrolyzed carbon layer and the temperature rise at the back-face of the composite [3]. Therefore, PAN-based carbon fibers with low thermal conductivity similar to that of rayon-based carbon fibers are in great demand in terms of both cost and performance.
With a carbon fiber-reinforced polymer matrix composite, one approach to decrease the thermal conductivity of the composite is primarily to decrease the thermal conductivity of carbon fiber rather than that of polymer matrix because the conductivity of carbon fibers is much greater than that of polymer matrix and it is also effectively managed through a variety of heat-treatment processes. A few attempts have been made to obtain carbon fibers with less thermal conductivity.
One possible way to produce a low thermal conductivity carbon fiber is to carbonize the stabilized PAN fiber at a
POLYMER COMPOSITES—2005

FIG. 1.
Analogy between Ohm’s law and Fourier’s law.
lower temperature compared to the standard processing temperature [4]. Partially carbonized fibers, which are known as heat-treated low-temperature carbon (LTC) fibers, have attracted much interest because of their low thermal conductivity, as well as their unique physical, chemical, and semiconductive properties at the intermediate range between the stabilized PAN fiber and the carbon fiber. It is well known that the thermal conductivity and tensile strength of carbon fibers increase with increasing heattreatment temperature (HTT). Therefore, it is important to develop PAN-based carbon fiber composites that have similar thermal conductivity compared to rayon and also have acceptable mechanical properties. Recently, Park and Kang
[3] reported that the use of LTC fibers heat-treated at about
1100°C to reinforce the phenolic composite improves the ablative and insulation properties of the composite.
Another possible route to obtain low thermal conductivity of the composite is to introduce the spun yarn type PAN-based carbon fiber as a reinforcement instead of a continuous filament yarn type carbon fiber. It has been reported that the phenolic composites reinforced with spun yarn type PANbased carbon fabric reduce the thermal conductivity along the fiber direction about 30% in comparison with conventional continuous filament type PAN-based carbon fiber composites
[5]. Park et al. [6] reported the interfacial, thermal, and ablative properties of spun and filament yarn type carbon fiber composites. The results showed that the longitudinal thermal conductivity of the spun yarn type carbon fiber composite is about
7% lower than that of the filament yarn type carbon fiber composite.
In this study we aimed to predict the thermal conductivity of a spun yarn type carbon fiber composite using a thermal-electrical analogy method. Series-parallel thermal resistance network models of eight-harness satin (8HS) fabric composites in both in-plane and transverse directions were established. The model was validated by a comparison with previously reported works and experimental results.
Good agreement was found between the present approach and the experimental results.
Approaches to the Model
Thermal-Electrical Analogy.
Of the numerous theoretical, empirical, and numerical methods available to predict the effective thermal conductivity of composites, the thermal-electrical analogy has been shown to be the most simple and effective. Dasgupta and Agarwal [7] presented a new three-dimensional series-parallel thermal resistance network to predict the orthotropic thermal conductivity of plain weave fabric composites. Using a three- dimensional thermal resistance network, Ning and Chou [8 –10] developed simple geometric models for woven fabric composites and obtained a closed-form solution for composite thermal conductivities. However, they dealt with simple geometry woven fabric composites, such as rectangular yarn cross section. The real yarn cross section can be presented by elliptic, lenticular, or “horse-track” shapes. The thermalelectrical analogy is based on the fact that Fourier’s law, which governs the heat flow, is mathematically and phe-
792 POLYMER COMPOSITES—2005
FIG. 2.
Representative structure of the 8HS fabric lamina.

FIG. 3.
Repeated unit cell of the 8HS fabric lamina.
nomenologically equivalent to Ohm’s law, which governs the current flow. The analogy between Ohm’s law and
Fourier’s law is presented in Fig. 1.
Based on this analogy, the effective thermal conductivity of a composite material can be evaluated by determining the equivalent electrical resistance [9]. A thermal resistance of material, R i
, is defined as:
R i
⫽
L i k i
A i
, (1) where L i
, k i
, A i are the material length, the thermal conductivity of conduction element in the direction of heat flow, and the material cross-sectional area perpendicular to the heat flow, respectively. Thus the thermal conductivity of material, k , is calculated by determining the equivalent thermal resistance of material, and its unit is W/mK. This
FIG. 4.
Horse-track yarn cross-section contour.
FIG. 5.
Thermal resistance network of the unit cell in in-plane direction.
POLYMER COMPOSITES—2005 793

FIG. 6.
Partitions and thermal resistance network of elements. a–f: Thermal resistance networks of elements
1– 6, respectively.
study examines the effective thermal conductivity of a spun woven carbon fabric composite using this analogy method.
In this approach, some assumptions are made regarding the modeling of effective thermal conductivities of woven fabric composite as follows: 1) thermal contact resistance between the fiber and the matrix is negligible; 2) heat flow is along the x- and z-axes in the in-plane and transverse directions, respectively (Fig. 2); 3) the thermal conductivity of the unit cell is the same as that of laminate; and 4) yarn fiber bundle can be regarded as a unidirectional composite.
In the following sections we focus on the model for the thermal conductivity of lamina, mainly in the in-plane direction. For the transverse direction, a similar procedure can be utilized.
In-Plane Thermal Resistance Network.
Satin weave is characterized by long floats of warp and weft yarns covering the face of the fabric. The representative structure of the 8HS fabric lamina, composed of eight warps and eight wefts, is shown in Fig. 2. In this analogy method, the number of conduction elements and the way the elements are connected are considered, but the sequence of conduction elements in the direction of heat flow is not considered. Hence, the simplified structure
(one-eighth of the whole lamina; Fig. 3) can be regarded as a repeated unit cell. The yarn cross-section is assumed to be a horse-track contour as shown in Fig. 4, and the reinforcing fiber materials and parameters of the warp and weft yarns are identical.
794 POLYMER COMPOSITES—2005

FIG. 6.
( Continued )
The parameters g , h z
, h m
, and a ⫹ h z denote the space between the two adjacent yarns, the height of the yarn, the thickness of the matrix, and the yarn width, respectively.
The mean fiber inclination angle is defined as follows:
៮ ␪ ⫽
再
␪
៮
៮
␪
1
, for the inclined parts of a yarn
2
, for the rest of a yarn
, (2) where
␪ ៮ i
( i
⫽
1, 2) is calculated by the methods presented in Refs. 7 and 10.
The unit cell can be divided into six subelements (elements
1–6). The unit cell contains eight “1” elements, two “2” elements, six “3” elements, six “4” elements, two “5” elements, and eight
“6” elements (Fig. 3). These elements are connected in series and parallel in the in-plane heat flow direction, as shown in Fig. 5.
Thus the equivalent thermal resistance of the unit cell in the in-plane direction, R in-plane
, can be computed as:
1
R
In-plane
⫽
1
8 R
关
1
兴
⫹
2 R
关
2
兴
⫹
6 R
关
3
兴
⫹
1
6 R
关
4
兴
⫹
2 R
关
5
兴
⫹
8 R
关
6
兴
,
(3) where R
[1]
⬃
R
[6] represent the thermal resistances of elements 1– 6, respectively. The relationship between the thermal resistance of the unit cell and the in-plane effective thermal conductivity, k
In eff
, of the unit cell is:
R
In-plane
⫽ k In
L unit eff
A unit
, (4)
POLYMER COMPOSITES—2005 795

FIG. 6.
( Continued ) where L unit is the length and A unit is the area of the unit cell perpendicular to the direction of heat flow. By substituting Eq.
3 and the dimensions of the unit cell into Eq. 4 , the in-plane effective thermal conductivity of composite can be obtained.
Each element partitioned into several sections and their thermal resistance are shown in Fig. 6a–f. The thermal resistance of the corresponding section can be calculated from Eq. 1 .
To take into account the contribution of the local fiber orientation to the global thermal conductivities of the composite, the impregnated yarn is considered. The thermal conductivities of the impregnated yarn with the fiber inclined at an angle
␪ i in the conduction element can be expressed as the following transformation equation [11]: k
␪ i
⫽ k ya cos 2
៮ ␪ i
⫹ k yt sin 2 ␪ ៮ i
共 i
⫽
1, 2
兲 共 in the direction of in-plane
兲
, (5) k
␪ i
⫽ k ya sin 2
៮ ␪ i
⫹ k yt cos 2 ␪ ៮ i
共 i
⫽
1, 2
兲 共 in the direction of transverse
兲
.
(6)
The thermal conductivities along and transverse to the impregnated yarns, k ya and k yt
, have to be determined to evaluate the effective thermal conductivities in both the in-plane and transverse directions.
EXPERIMENTAL
Materials and Composites
Stabilized staple PAN fibers (Pyron®; Zoltek Co., USA) were used as precursor fibers to prepare spun carbon yarns to fabricate the spun yarn type C/P composites. The physical properties of staple stabilized PAN fibers are summarized in
796 POLYMER COMPOSITES—2005

Table 1. The stabilized staple PAN fibers were converted into the stabilized PAN spun yarns using a semiworsted spinning process. The stabilized PAN spun yarns were woven into a fabric form with an 8HS structure. The woven spun yarn type fabrics were proprietarily heat-treated up to
1100°C at a heating rate of 1°C/min with a purging N
2 gas in a batch-type carbonization furnace, and then successfully converted into spun yarn type carbon fabrics.
Resol-type phenolic resin was used as a matrix for the composites. The resin content of spun C/P prepregs was about
36% (w/w). The spun C/P composites were fabricated at
150°C for 2 hr, with an identical curing cycle using a hydroclave. The debulking process at 105°C for 1 hr was carried out to remove possible entrapped air and consequently remaining voids in the resulting composite. A pressure of about
1000 p.s.i. was applied. The resin content of spun C/P composites after fabrication was about 32% (w/w).
Thermal Measurement
The thermal conductivities of spun C/P specimen were measured by employing a comparative steady-state method against a reference sample using a tailor-made apparatus for thermal conductivity measurements based on ASTM
E1225-87. At steady-state conditions, the thermal conductivity was derived by comparing the temperature gradients between the reference sample and the target one. Figure 7 shows the schematic illustration of an apparatus used for thermal conductivity measurements. Stainless steel STS 304 was used as a reference sample. The thermal conductivity of target sample, k t
, was calculated using the following equation derived from Fourier’s law of heat conduction: k t
⫽ k r
⌬
T r
⌬
T t
䡠
␦ t
␦ r
䡠 d 2 r d 2 t
K
关
W/mK
兴
, (7) where k r is the thermal conductivity of the reference sample of diameter d r
,
⌬
T is the temperature difference in the target of sample, and
␦ is the distance between two junctions of the thermocouples (where the subscript letter t designates the target sample, and the subscript letter r designates the reference sample). K is the correction factor that compensates for the effect of the negligible conditions (usually the value of K is within the limits of 1.0 –1.02). Cylindrical specimens (20 mm long, 12 mm in diameter) were used for the measurements. The
TABLE 1.
Physical properties of staple stabilized PAN fiber.
Physical property
Density (g/cm
3
)
Tensile modulus (GPa)
Tensile strength (MPa)
Maximum strain (%)
Fiber length (mm)
Carbon content (%)
Fiber diameter (
␮ m)
Value
240
22
1.37
7.4
101
62
13
FIG. 7.
Schematic illustration of the apparatus used to measure thermal conductivity.
measurements were carried out along the directions parallel and perpendicular to the laminar plane of a composite at ambient temperature in air (Fig. 7).
RESULTS AND DISCUSSION
To validate the model presented, the effective thermal conductivities of 8HS fabric composite in the in-plane and transverse directions were calculated. The 8HS graphite fabric/epoxy composite was used for comparison with a previous work [10], which examined thermal conductivity only in the transverse direction. We also adopted the dimension of the unit cell and material properties from Ref.
10. The fiber volume fraction in the yarn,
␯ f was 0.63, and the fiber volume fraction of the laminate was 0.35. The thermal conductivity of the graphite was 8.40 W/mK along the fibers, 0.84 W/mK in the transverse direction to the fiber axis, and 0.19 W/mK in the epoxy resin [7]. Thus, the thermal conductivity of the yarn was 5.36 W/mK along the fibers, and 0.46 W/mK transverse to the fiber.
Figure 8 shows a comparison of the effective transverse thermal conductivities of graphite/epoxy composite between the present prediction and the previous study. As the fiber volume fraction increased, the transverse thermal conductivity of the composite also increased. Reasonable agreement was found between the present study and the previous work [10].
Figure 9 illustrates the effect of the fiber volume fraction on the predicted in-plane thermal conductivity of the 8HS fabric composite. The effective in-plane conductivity of the composite is governed mainly by the longitudinal yarn because the yarn’s transverse conductivity is low in comparison to its longitudinal conductivity. Graphite yarn has a much higher conductivity (by 10-fold) along the fibers
POLYMER COMPOSITES—2005 797

FIG. 8.
Comparison of the predicted effective transverse thermal conductivities of graphite/epoxy composite with previous results.
FIG. 10.
Comparison of the thermal conductivities of a spun C/P composite between present predictions and experimental results.
compared to transverse to the fiber. Therefore, as the fiber volume fraction increases, the in-plane thermal conductivity also increases. To our knowledge, no previous work on the in-plane thermal conductivity of 8HS fabric composites has been reported. The values of k ya and k yt were 6.38 and 0.71
W/mK, respectively. The mean fiber inclination angles,
៮ ␪
1 and
៮ ␪
2
, as computed using the method described in Refs. 7 and 10, were 15.12 and 7.12, respectively.
A comparison of the in-plane and transverse thermal conductivities of the spun C/P composite between the present predictions and experimental results is shown in Fig. 10. Good agreement was found in both the in-plane and transverse directions for the thermal conductivities of the spun C/P composite. The predicted values were somewhat higher than the experimental results. We believe this overestimation reflects the facts that heat flow was assumed in only one direction for both the in-plane and transverse directions, separately, and there was no heat loss between each element.
Another possible reason for the predicted results being higher than the test results is the assumption of zero contact resistance between the fiber and matrix.
CONCLUSIONS
This paper presents a thermal conductivity model of a spun
C/P composite using the thermal resistance method. This method uses the analogy between the heat diffusion and the electrical charge. The structure of the 8HS spun carbon fabric composite is divided into several conduction elements. By constructing the equivalent thermal resistance network in series and in parallel, in a model based on analogy, the thermal conductivities of composite were predicted. The predicted values obtained from the model were compared with the previously reported thermal conductivities of a carbon/epoxy composite with 8HS texture. It was found that the thermalelectrical analogy method provided a good estimate of the thermal conductivity in the composite with 8HS texture. Using this model, the thermal conductivities of spun C/P composites with 8HS texture were estimated and then validated experimentally. Reasonable agreement was found between the present approach and the experimental results.
FIG. 9.
Prediction of the effective in-plane thermal conductivities of a graphite/epoxy composite.
REFERENCES
1. G. Kalaprasad, P. Pradeep, G. Mathew, C. Pavithran, and S.
Thomas, Compos. Sci. Technol.
, 60 , 2967 (2000).
2. I.H. Tavman and H. Akinci, Int. Commun. Heat Mass Transfer , 27 , 253 (2000).
3. J.K. Park and T.J. Kang, Carbon , 40 , 2125 (2002).
4. H.A. Katzman, P.M. Adams, T.D. Le, and C.S. Hemminger,
Carbon , 32 , 379 (1994).
5. R.L. Noland, U.S. Patent 5,298,313 (1994).
6. J.K. Park, D.H. Cho, and T.J. Kang, Carbon , 42 , 795 (2004).
7. A. Dasgupta and R.K. Agarwal, J. Compos. Mater.
, 26 , 2736 (1992).
8. Q.G. Ning and T.W. Chou, J. Compos. Mater.
, 29 , 2280 (1995).
9. Q.G. Ning and T.W. Chou, Compos. Sci. Technol.
, 55 , 41 (1995).
10. Q.G. Ning and T.W. Chou, Composites Part A , 29A , 315 (1998).
11. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids ,
Clarendon Press, Oxford (1959).
798 POLYMER COMPOSITES—2005