Numerical investigation and experimental verification of the Joule heating effect of polyacrylonitrile-based carbon fiber tows under high vacuum conditions

Abstract

Joule’s first-law dictates that when electric current passes through a conductor heat is generated. The scope of this work is to investigate the Joule heating effect of two types of carbon fiber tows in dry form. The tows are used as primary heating elements on various preform structures that require accurate and uniform temperature control, using a DC power supply. The investigated temperatures, ranged from 300 K up to 650 K. A finite difference scheme was developed, in order to predict and quantify the transient phenomenon and is verified against experimental results. In order to exploit in-depth the phenomenon and achieve very good agreement between numerical and experimental results, temperature dependence of thermal and electrical properties (specific heat capacity, total hemispherical emissivity, and volume resistivity) was examined under controlled vacuum environment, ensuring no fiber oxidation.

Keywords

Joule heating resistive heating carbon fibers electrical conductivity emissivity

Introduction

Applying electric potential on the carbon preform makes it act as an Ohmic resistor. Joule’s first law dictates that when electric current passes through a conductor heat is generated, this is known as Joule heating effect or Ohmic heating or resistive heating. The generated heat and thus the preform temperature is easy to control, being affected only by the applied electric power. Carbon fibers (CFs), despite their excellent mechanical properties, also feature good thermal behavior (low coefficient of thermal expansion (CTE) in the transverse direction and almost zero along the fiber length). The ability of carbon fibers to conduct electric current also can be used in applications such as resin curing for space applications,¹ heating pipes, or de-icing layers on aircrafts. Also CFs can be used for clothing applications similarly to metallic fibers.²

The authors have utilized Joule heating (or resistive heating method) to achieve resin flow acceleration during the liquid composite molding process and also, curing and post-curing cycles. Omitting the need of costly ovens and metallic molds, by heating directly the preforms. The temperature distribution obtained during this process (curing and post-curing stages) is depicted in Figure 1, presenting satisfactory thermal response and uniformity.³ Additionally this method provides an easier way to process composite parts (impregnation and curing) using high performance resins, with curing temperatures around 600 K, such as cyanate-esters. As far as CF composite materials are concerned, limited work is available on using electric current to heat carbon preforms and to cure resin.^4,5 Joseph and Viney⁴ conducted electric current through carbon prepregs and showed that due to the Joule heating effect it was possible to cure the prepregs with no significant variation in the bending properties. The use of embedded resistive heating elements provide significant improvement in cure cycle time and cure uniformity.^6,7 Also, the tensile properties of composite laminates manufactured by two preform types remained unchanged.³

Figure 1.

Cure cycle and temperature distribution on CF Fabric/Epocast52 using vacuum bagging method.³

Innovative electrically heated CFRP molds have also been developed⁸ for the resin infusion manufacturing process, featuring low energy consumption and high mold dimensional stability. The use of carbon fibers as primary heating elements, presents excellent results in terms of geometric adaptability and uniformity in temperature distribution. The type of CF reinforcement widely used in applications is the woven fabrics. The presence of woven tows at different directions gives geometrical adaptability to fabrics, making them ideal for complex geometries.

The resistivity of the fabrics is strongly anisotropic^9,10 depending on the fabric architecture, fiber volume fraction, fiber electrical conductivity and temperature. In Figure 2, a single carbon fiber tow is presented in different magnification levels. The longitudinal electric volume resistivity at room temperature takes values from 0.6 × 10⁻⁵Ωm up to 1.9 × 10⁻⁵Ωm¹¹ and is temperature dependent. CF fabrics can be used as stand-alone heating elements or can be embedded inside a matrix material (ceramic or polymer matrix).

Figure 2.

A carbon fabric in different magnification level: (a) fabric, (b) tow, (c) tow cross section detail.

The scope of the current work is to investigate the carbon fiber tow (CF-tow) Joule heating effect under high vacuum conditions. In order to perform this work, the behavior of CF specific electric resistance, specific heat capacity and total hemispherical emissivity with respect to temperature changes had to be evaluated. Understanding the behavior of these properties resulted in accurate prediction and control of the CF-preform temperature. Numerical predictions and experimental results were in excellent agreement. The accumulated knowledge can be utilized in space and other applications requiring accurate temperature control.

Governing equations and Joule heating effect

The current density inside a conductor is expressed by Εquation (1):

\vec{J} = \bar{\bar{σ}} \vec{E}

(1)

Where

\bar{\bar{σ}}

is a second-order tensor when the material is anisotropic or a scalar quantity if the material is isotropic. The current density expression takes the form of Equation (2). Since the electric current conduction in the present work is approximated as a 1D phenomenon

\vec{J} = σ_{f}^{el} \vec{E}

(2)

For 1D problem in the fiber direction Figure 3, the electric voltage distribution is given by Equation (3).¹²

σ_{f}^{el} \frac{\partial^{2} ϕ}{\partial x^{2}} = 0

(3)

Where φ is the electric potential and the electric field (

\vec{E} = - \nabla ϕ

.) is the gradient of the electric potential.

Figure 3.

Schematic representation of the problem under vacuum conditions.

In a conductive media, charged particles are accelerated by an electric field. When the electric current flows through the electrical conductive media, electric energy is dissipated to heat in the conductive media. The electrical energy per unit of time and volume is equal to the scalar product $\vec{J} \cdot \vec{E}$ ,¹³ (Equation (4)).

{\overset{\cdot}{Q}}_{el} = \vec{J} \cdot \vec{E}

(4)

The governing differential equation for the temperature distribution on a conductor^13,14 is given by Equation (5).

(5)

The third term on the right hand of Equation (5) is the Thomson effect term, describing the presence of electric current inside a material in the absence of electric field, simply due to temperature gradient along the material.

In the present work, the Thomson effect is considered to be negligible since temperature uniformity is achieved. Thus Equation (5) takes the form of (6).

(6)

Heat transfer between the CF tow and the environment is performed via convection and radiation, so Equation (6) is re-written for 1D analysis in the following form:

ρ c_{p} (T) \frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^{2}} + \frac{J^{2}}{σ_{f}^{el} (T)} - \frac{A_{rad}}{V} ɛ (T) σ (T^{4} - T_{\infty}^{4}) - {\overset{\cdot}{Q}}_{conv}

(7)

Assuming thermal conductivity of the carbon fibers to be constant and due to the absence of surrounding air (or any fluid in general since the present study was performed under vacuum conditions), heat is transferred only by radiation (Figure 3).

As a result the governing partial differential equation is expressed by Equation (8).

\begin{matrix} ρ c_{p} (T) \frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^{2}} + \frac{J^{2}}{σ_{f}^{el} (T)} - \frac{A_{rad}}{V} ɛ (T) σ (T^{4} - T_{\infty}^{4}) \Leftrightarrow; \\ ρ c_{p} (T) \frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^{2}} + ρ_{f}^{el} (T) J^{2} - \frac{A_{rad}}{n_{f} \frac{π d_{f}^{2}}{4} L} ɛ (T) σ (T^{4} - T_{\infty}^{4}) \end{matrix}

(8)

Experimental procedure

An apparatus was manufactured approximating a black body cavity and the tows were placed inside (Figure 4). The internal conditions can either be high vacuum or elevated pressure. The selected environment was high vacuum (1.3 mPa or less) according to section 6.5.2 of ASTM C835-95 standard.¹⁵ Under these conditions the CF tow is chemically stable and no oxidation occurs. The tow was connected using two copper electrodes. Voltage is applied on the tow resulting in a uniform temperature increase of the material due to the Joule effect. This apparatus was able to measure the electric power provided to the tow, the radiated power from the CF tow and the temperature distribution as a function of time along the tow length. Based on these data, the volume resistivity and total hemispherical emissivity as a function of the tow temperature can be calculated.

Figure 4.

(a) Manufactured apparatus resembling black body environment, (b) Apparatus schematic as described in the ASTM Standard.¹⁵

Setup and materials

The testing apparatus described in Figure 4 was designed and manufactured according to the rules dictated by the ASTM C835-95 Standard.¹⁵ Two different PAN based CF tows having different number of fibers each, were tested. The first CF tow was the (HTA40J/E-3K-E13/E13 type with ca. 1.3%, sizing based on epoxy resin) consisting of 3000 fibers and the second one was the (T700S-12K-60E/sizing type and amount 60E 0.3%) consisting of 12,000 fibers. Both types of fibers feature the same volume resistivity equal to 1.6 × 10⁻⁵Ωm at room temperature.^16,17 The length of the tested tows was 0.1 m. All fibers in a tow should be in contact with the copper electrodes to ensure a smooth distribution of the current and to avoid errors in measurements. To this end, the tow edges were infused with a small amount of silver paint. Subsequently, using Sn99.5/Cu0.5/Co soldering material, the copper electrodes were locally soldered. Voltage was applied using a stabilized DC power supply able to regulate either voltage or current. Vacuum was applied using a vacuum pump and measured with a vacuum gage. A thermocouple was placed in the middle and at the centre of the CF tow. The measurements were stored in a PC via a USB data receiver. Every test was performed with constant voltage input having different amplitude. All measurements were obtained once steady-state conditions were reached, assuming that no temperature gradient is present through the tow thickness.

After every measurement session the tow was left to cool to room temperature and its resistance was measured again. This step was used as a verification measurement ensuring that no chemical reaction took place during the test, which could alter the tow composition, and that all fibers were still in contact with the copper electrodes. The circuit’s net resistance was measured using a multimeter (KITHLEY 3456) and the value was subtracted from the measurements. Details for the manufactured apparatus can be found in Table 1.

Table 1.

Details of the experimental apparatus

Component	Specifications
Temperature range	300–650K
Stainless steel vacuum chamber	Diameter: 0.3 m/Length: 0.4 m Black coating (ε > 0.95)
Power supply	100 W, 0–25 V, 0–4 A
Multimeter	KITHLAY 3456/Seven decimals accuracy
Thermocouples	J-type (25 µm)
Sample size	HTA40-3K/2.5 mm × 0.27 mm × 0.1 m
Sample size	T700S-12K/6 mm × 0.5 mm × 0.1 m

As mentioned earlier, the C835-95 ASTM standard was used in order to investigate the carbon fiber tow total hemispherical emissivity behavior and also to prevent fiber oxidation due the elevated temperatures. This calorimetric test method covers the determination of total hemispherical emittance of metal surfaces, graphite surfaces and coated metal surfaces in the range of 300–1700 K. According to the standard the size of the thermocouple wire should be the minimum possible, suggesting that wire diameter less than 0.13 mm provides acceptable results.¹⁵ In this work because of the specimen size, the 0.13 mm thermocouple wires induced substantial error. Using K-type thermocouples with wire diameter of 0.13 mm results to recorded temperature of almost 40°C lower than the recorded temperature resulted by the use of J-type thermocouples having wire diameter 25 µm. The last ones were eventually used for all the measurements. This discrepancy of the results provided by the use of K-type thermocouples occurs because the thermocouple size is comparable to the carbon fiber tow size (thickness and width) resulting to a significant thermal gradient at the contact area leading to erroneous measurements.

Electrical insulation of the thermocouples is also crucial, since a small current leak can alter the measurement significantly. Failure of insulation can be detected as spikes in the measured signal and nonrepetitive measurements. Another indication of insulation failure is by the fact that when the current of the CF-tow is reversed, the spikes mentioned previously have the opposite direction. Electrical insulation of the thermocouple tip can be achieved by using low viscosity and high thermal stability liquids that solidify in room temperature or with a slight temperature rise, for example liquid release agents used in composite parts manufacturing. During the tests, the measured signal was smooth, spike-free, and stable in the steady state and repeatability was achieved.

Reliability of volume resistivity tests

The crucial parameter that influences measurement repeatability/reliability is the way the tow fibers are connected to the copper electrodes. Verification of the electrode-tow connections together with the validity check of the assumption that all fibers conduct current, was performed by measuring the resistance of every tow and comparing them against theoretical ones. Assuming that the tow is intact (all fibers are bridging the electrodes) its theoretical resistance at room temperature is given by Equation (9).

R_{(298 K)} = ρ_{(298 K)} \frac{L}{A} = ρ_{(298 K)} \frac{4 L}{n_{f} π d_{f}^{2}}

(9)

In Table 2 the measured mean values and standard deviation of the tow resistance are presented. It can be observed that the relative error is less than 3%, thus supporting the reliability of the adopted methodology.

Table 2.

Carbon fiber tow resistance at room temperature and geometric details

CF Tow	Theoretical value R (Ω) (L = 0.1 m)	Experimental value S.D., R (Ω) (L = 0.1 m)	Relative error δ (%)	Geometrical characteristics of a tow	Geometrical characteristics of a fabric
HTA40J/E-3K-E13	13.500	13.435 ± 0.052	0.5	100 mm × 2.5 mm × 0.27mm	400 tows/m²
T700S-12K-60E	3.465	3.563 ± 0.091	2.83	100 mm × 6 mm × 0.5mm	166 tows/m²

Experimental investigation of electro-physical properties

Material thermal and electrical properties such as volume resistivity, specific heat capacity, thermal conductivity, and total hemispherical emissivity vary with temperature. In the present work, the thermal conductivity coefficient is assumed to be constant and equal to the room temperature value. All other properties were experimentally estimated for the two CF tows.

Volume resistivity as a function of temperature

The value of the volume resistivity with respect to temperature was calculated from the voltage and current measurements at different tow temperatures. As shown in Figure 5 there is a linear correlation between volume resistivity and temperature. Using the linear relation

ρ_{el} (T) = ρ_{el}^{o} [1 + a (T - T_{o})]

(10)

the temperature resistivity coefficient calculated from the experimental results for the two types of CF tow were a_HTA40 = −0.000308 K⁻¹ and a_T700S =−0.000247 K⁻¹. As a comparison the amorphous carbon temperature resistivity coefficient is a = −0.0005 K⁻¹.¹⁸ The negative value of this coefficient is a characteristic of semiconductors. The coefficient of thermal expansion (CTE) of the tows are –0.1 × 10⁻⁶ K⁻¹ and –0.38 × 10⁻⁶ K⁻¹ for HTA40 and T700S, respectively. Considering these low values, the length change (shrinkage) of the tows was neglected.

Figure 5.

Volume resistivity as a function of temperature for the T700S-12K and HTA40-3K carbon fiber tows.

The maximum volume resistivity change in the investigated temperature range was approximately 10%, indicating that this effect cannot be neglected and should be considered during calculations. As already mentioned, the whole test was performed under vacuum atmospheric conditions to prevent fiber oxidation. Moreover, after every heating/cooling cycle the tow resistance remained unchanged indicating that tow integrity was maintained and that all the initial fibers were intact.

Specific heat capacity as a function of temperature

The variation of specific heat capacity versus temperature was measured with DSC tests of the fibers under investigation. The procedure was the one described in the operation manual of the DSC apparatus.¹⁹ At first the aluminum sample pans were run empty and the heat flow was measured. This curve was the reference curve. Then the sample tin was filled with a small amount (approximately 6–10 mg) of carbon fibers, it was sealed with a lid and put inside the chamber. The same as before heating profile was applied and the heat flow was measured. The difference between the two heat flow curves (reference empty pans and fiber filled), along with the calibration coefficient of the apparatus, were used for the derivation of the relationship of specific heat capacity versus temperature.

The results indicate that a linear relation exist between specific heat capacity and temperature, as shown in Figure 6. The experimental results are in good agreement with Pradere et al.²⁰

Figure 6.

Specific heat capacity c_p as a function of temperature for the T700S-12K and HTA40-3K carbon fiber tows.

Total hemispherical emissivity as a function of temperature

The tow reaches steady-state thermal conditions, when its temperature does not change in time, and therefore

{mc}_{p} \frac{\partial T}{\partial t} = 0

(11)

At steady state and since there is no convection in the vacuum, the electric power consumed on the tow is transferred to the environment by radiation. Therefore, the electric power is equal to the radiative heat transfer, obtained by the Stefan–Boltzmann equation.^15,21,22 Using the measured tow temperature, the value of the total hemispherical emissivity of the specimen surface can be calculated. Based on the assumption that the test specimen is a small radiating body surrounded by a large absorbing cavity, the product of the total hemispherical emissivity and radiation surface, εΑ_rad, of the specimen can be calculated, as

\frac{V^{2}}{R (T)} = ɛ (T) A_{rad} σ (T^{4} - T_{\infty}^{4}) \Leftrightarrow; ɛ (T) A_{rad} = \frac{(\frac{V^{2}}{R (T)})}{σ (T^{4} - T_{\infty}^{4})}

(12)

A thermocouple was attached to the inner wall of the vacuum chamber, monitoring any variation of the wall temperature. The observed temperature rise of the inner surface of the measurement chamber was not greater than 2 K throughout all tests. In Figure 7 the temperature dependence of the product εA_rad is presented. The larger values observed for the T700-12K tow are due to the larger emitting surface of this tow in comparison to the HTA-3K tow. If the radiating area A_rad is considered known, then the total hemispherical emissivity can be calculated. The results are presented in Figure 8. It can be observed that for the investigated temperature range there is a decreasing trend of the total hemispherical emissivity versus temperature.

Figure 7.

Total hemispherical emissivity ε and radiation area Α_rad as a function of temperature for the T700S-12K and HTA40-3K tows.

Figure 8.

Total hemispherical emissivity ε as a function of temperature for the T700S-12K and HTA40-3K.

Power consumption

The maximum tow temperature has been measured in relation to the supplied electric power. Temperature versus current and applied voltage was studied for the investigated tows in high vacuum environment and the required electric power was extracted.

The temperature dependence on the applied voltage and electric current postulates a linear correlation and are both presented in Figure 9.

Figure 9.

Steady-state temperature vs applied: (a) current, (b) voltage, for the investigated tows HTA40-3K and T700S-12K.

In Figure 10 the required electric power and the achieved steady-state temperature are presented for the investigated tows normalized for 1 m length and 1 m² area. The T700-12K tow has a larger surface area, as presented in Table 2, so its radiative cooling is greater, requiring more power to reach the same steady-state temperature as the 3K tow. For applications where a large surface area (S) needs to be heated and controlled, an array of parallel tows can be used as parallel resistors (Figure 11(a) and (b)).

Figure 10.

Steady-state temperature as a function of provided electric power: (a) per meter of tow length, (b) per square meter of tow area.

Figure 11.

(a) UD carbon fabric with 10 wt% glass fibers, (b) plain weave carbon fiber fabric, (c) carbon fiber tow detail.

CF tow temperature as function of time and electric field

An explicit numerical scheme with 200 nodes, based on Equations (3) and (8), was developed, combining all the above presented experimentally measured thermal and electrical parameters.

The boundary and initial conditions are expressed by Equation (13).

{\begin{matrix} T (0, t) = T (L, t) = 293 K \begin{matrix} for & t > 0 \end{matrix} \\ T (x, 0) = 293 K \\ φ = φ_{applied} \begin{matrix} \begin{matrix} for & t > 0 \end{matrix} \end{matrix} \end{matrix}}

(13)

By numerically solving the boundary value problem (BVP) of Equations (3), (8), and (13), the tow temperature distribution in time and space (1D) can be predicted. This solution scheme was used for the investigated tows.

The results of temperature versus time at the central point of the fiber tows are presented in Figure 12(a) and (b) for the 12K tow and the 3K tow respectively, and compared against the experimental data. The numerical results correlate pretty well with the experimental values.

Figure 12.

Carbon fiber temperature as a function of time and electric potential T(V,t) at the center of the tow. (a) T700S-12K, (b) HTA40-3K.

In order to investigate the tow heating and cooling rates a simple test with electric voltage pulses was performed. It was observed that a temperature change of the order of 270K in the case of the 3K tow was achieved in only 8 s, while in the case of fibertow of 12K, 30 s were required.

It has to be mentioned that due to the relatively low thermal conductivity of carbon fibers, end conduction effects are negligible, and a uniform temperature distribution was managed across the length of the tow, as shown in Figure 13, in the case of 3K tow, using 5.5 V electric potential for heating at 20 s after the heating initiation.

Figure 13.

Temperature distribution along the fiber CF in the case of 3K tow, at t = 15 s.

Uniform temperature distribution along the carbon fiber tows, makes the CFs preforms suitable for applications requiring accurate temperature distribution during heating. Also the flexibility/adaptability of carbon fabrics make them an ideal choice for complex geometry heating.

Furthermore, heating and cooling cycles using step voltage were performed. In the first part of Figure 14, a 2.5 V voltage was applied on the 12K fiber tow, for a period of 10 s during heating, while the cooling period was also 10 s. Then this heating-cooling pattern was repeated. In the second part of Figure 14, the same voltage was applied but the heating/cooling duration was equal to 15 s. There is a very good correlation between experimental and modeling results.

Figure 14.

Temperature response of the T700S-12K tow to the applied voltage pulses at the tow center. Heating and cooling cycles.

Conclusions

Τhe variation of thermal and electrical properties of the CF tows versus temperature was investigated under controlled high vacuum environment. The investigated temperature range was 300 K up to 650 K. These data were used for the solution of the BVP that correlates the applied voltage/power to the temperature profile developed on the investigated CF tows, based on the Joule effect.

The investigated Joule-heating temperature control methodology can be used in various heating applications such as, surface or pipe heating where temperature uniformity is required. This can be achieved using single tows or fabrics (where many tows are present), either in dry form or embedded inside an epoxy matrix material. Considering the radiation terms during the energy calculations, it is noticed that this effect is very important since the radiation influences the overall response and steady-state temperature for a given applied electric power. Further investigation is required for the calculation of the effect of convection in the presence of air. Presence of air causes fiber oxidation (up to 600 K), thus limiting their use in dry form. In conclusion, using carbon fibers as heating elements for uniform temperature distribution on various surfaces is an efficient, sensitive, agile and economic method. Prediction and control of the Joule heating response of carbon fibers, requires knowledge of the thermal and electrical properties as a function of temperature, since these properties influence the phenomenon in great extent.

Footnotes

Nomenclature

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interests

None declared.

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