Thermal conductivity - A116

Thermal conductivity
Foundational knowledge article
Document Type	Article
Document Identifier	116
Themes	Thermal management
Relevant Class	Material
Tags	Conductive heating Material parameters Thermal properties
Prerequisites	Heat transfer Conduction Thermal diffusivity

Introduction

Thermal conductivity, \(k\), is defined as the material property measuring a material or medium’s ability to transport heat energy. Materials with a high thermal conductivity are highly conductive materials, and are considered to transport heat internally at a high rate. While insulators are defined as materials with a low thermal conductivity value, and transport heat slowly.

Scope

This page defines thermal conductivity, explains its significance in composites processing, and provides some typical values. This page also discusses the effect of process parameters and material microstructure. Measurement methods are briefly discussed. Links to ASTM measurement techniques are provided, but the techniques are not discussed heavily as this is covered in CMH-17 and in the provided ASTM links.

Significance

A material’s thermal conductivity is central to its thermal response; whether this is for the composite part during its operational use, or for the part and its manufacturing tool during the composite manufacturing process. In the context of manufacturing, examining the thermal conductivities of different materials can be one of the considerations in the selection of the tooling material, tool design, and the employed composite part thermal curing cycle.

Prerequisites

Recommended documents to review before, or in parallel with this document:

• Heat transfer
• Conduction
• Thermal diffusivity

Level I

Level II

Level III

Definition

Thermal conductivity is defined as the material property measuring a material or medium’s ability to transport heat energy. Materials with a high thermal conductivity are highly conductive materials, and are considered to transport heat internally at a high rate. While insulators are defined as materials with a low thermal conductivity value, and transport heat slowly.

It is defined as a physical constant \(k\) from Fourier's Law. In the 1-D heat flow scenario, Fourier's Law can be defined as:

\(q=-k\frac{dT}{dx}\)

Where,

\(q\) = heat flux [J/m²·s]

\(k\) = thermal conductivity [W/m·K]

\(\frac{dT}{dx}\) = temperature gradient [K/m]

Empirically, Fourier's Law describes heat flow to be proportional to the temperature gradient and a physical constant of the material, its thermal conductivity. This relationship is valid under steady-state temperature conditions, when both the temperature gradient and the temperature profile across the material are constant and not changing with time. In the 1-D heat flow case illustrated below, the steady-state condition is defined when the temperature profile across the material span L between constant surface temperatures T₀ and T₁ (temperature gradient) is linear and constant with time.

1-D temperature profile across a material span L - under steady-state conditions.

Thermal conditions when the heat transfer involves a changing temperature profile, referred to as unsteady or transient heat flow, involves the related thermal material property of thermal diffusivity.

Units

The general units of thermal conductivity \(k\) are provided as:

\(k=\frac{Power\, unit}{Length\, unit\cdot Temperature\, unit}\)

The following are common International System of Units (SI) and US Customary Units found in the literature for thermal conductivity:

	SI Units	US Customary Units
Base units	W/m·K	BTU/h·ft·°F
Other common forms	W/cm·K	BTU·in/h·ft2·°F
	W/m·°C

Typical Property Values

Range of thermal conductivity values for gases, liquids, and solids

Thermal conductivity of materials of different phases.

Thermal conductivity values range from high for pure metals (solids) to low for gases such as air. The large differences in thermal conductivity across the different material states is an important consideration in composite processing. For example, a vacuum bag leak in a composite vacuum bag setup can undesirably allow an insulating gas layer to enter the layup stack, and in extreme cases results in insufficient laminate curing temperatures being achieved.

Approximate thermal conductivity ranges of materials of the different solid material classes

Material Class	Thermal Conductivity (\(k\))	Approximate Range		[ref.]
Metals	High	20 - 400 W/m·K	good thermal conductors
Ceramics	Low	2 - 50 W/m·K	thermal insulators
Polymers	Low	Order of 0.3 W/m·K (< 1 W/m·K)	thermal insulators

Examples of typical values seen in composite material constituents and materials involved in the manufacturing processing (e.g. tooling)

Composite Use	Material	Typical Value (SI units) W/m·K	Typical Value (US Customary Units) BTU/h·ft·°F	[ref.]
Reinforcement Fibres	Carbon Fibre	2.5-300	1.4-173
	Glass Fibre (S2)	1.45	0.838
Matrix Materials	Polyester	0.2-0.6	0.1-0.3
	Epoxy	0.2-0.6	0.1-0.3
Tooling Materials	Steel	15-52	8.7-30
	Aluminum	247	143
	Invar	10	5.8

For some materials, the thermal conductivity varies significantly depending on the heat flow direction (anisotropic). For example, carbon reinforcement fibre properties are anisotropic , and can vary in orders of magnitude between the axial (\(k_{11}\)) and the transverse (\(k_{22}\)) directions. However, published values do not consistently report the transverse properties. A selection of anisotropic values of carbon fibres was sourced from the literature by Slesinger , a subset of them is provided below.

Selection of anisotropic carbon reinforcement fibre thermal conductivity values (Sourced by Slesinger )

Carbon Fibre	Axial \(k_{11}\) (W/m·K)	Transverse \(k_{22}\) (W/m·K)	Axial \(k_{11}\) (BTU/h·ft·°F)	Transverse \(k_{22}\) (BTU/h·ft·°F)	[ref.]
AS4	7.7	2.4	4.45	1.39
CN80	320	11	185	6.36
T300	100	11	57.8	6.36
T650	14	5	8.09	2.89
T700	100	11	57.8	6.36
YS80	320	11	185	6.36

Measurement

Thermal conductivity can be measured by either:

• Transient thermal condition test methods
• Steady-state thermal condition test methods

Transient Methods

Transient test methods are more precisely measuring thermal diffusivity, where the thermal conductivity value can be derived and extracted. Transient measurement methods will not be discussed further on this page. Please see the thermal diffusivity page for transient measurement methods.

Steady-State Methods

The following steady-state measurement methods for thermal conductivity are recommended by the Composites Materials Handbook - 17 (CMH-17) :

ASTM C177: (Preferred Method) Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus

ASTM E1225: Standard Test Method for Thermal Conductivity of Solids Using the Guarded-Comparative-Longitudinal Heat Flow Technique

ASTM C518: Standard Test Method for Steady-State Thermal Transmission Properties by Means of the Heat Flow Meter Apparatus

The fundamental principle to all of the listed methods is to induce a controlled steady-state one dimensional constant heat flow across the material being measured with heat flowing in the direction of interest. Simplified, this is achieved by placing the specimen material in good contact between two surfaces of constant but differing temperatures (a hot plate, and a cold plate) forming a constant temperature gradient across the specimen.

Illustration of thermal gradient test setup principle used to measure thermal conductivity. Thermal gradient is induced on specimen by placing between a hot and cold plate at constant temperatures.

By measuring the temperature at the specimen surface(s) and the temperature of the hot and cold plates once steady-state is obtained, the thermal conductivity of the specimen is determined. For further details of the listed methods, please consult directly the ASTM standards.

Material Microstructure Dependence

A material's ability to transport thermal energy is directly dependent on its atomic microstructure and the associated energy transport mechanisms. As a result, thermal conductivity varies greatly between materials of the different phase classes and different solid material classes.

Gas

The high molecular disorder and spacing of gases (compared to liquids and solids), results in the lowest thermal conductivities of the material phases as molecules struggle to collide and transfer vibration energy.

Liquid

The molecular disorder present in liquids also inhibits efficient energy transfer by thermal motion compared to highly ordered close packed solid phase materials.

Solids

For solids, the closed packed and ordered structure provides for efficient heat energy transfer through lattice vibration (wave motion known as phonon transport). This energy transfer is highest for highly ordered crystalline solids. Further, the thermal conductivity of metals is also enhanced by the presence of mobile electrons which provide an additional mechanism for conductive energy transfer.

In solid materials, the mechanisms for thermal energy transport occurs in two ways:

1. Vibration of lattice waves (phonons) (\(k_l\))
2. Free electrons (\(k_e\))

Where the total conductivity energy contributions:

\(k=k_l+k_e\)

The contribution of each of the mechanisms is dependent on the material's chemical and atomic microstructure.

Metals	Ceramics	Polymers
High thermal conductivity (Thermal Conductors)	Low thermal conductivity (Thermal Insulators)	Low thermal conductivity (Thermal Insulators)
large number of free electrons highly pure metals, free electron most efficient and dominant over phonon transport alloying reduces free electron transport efficiency	lacking large number of free electrons phonon transport dominated phonons not as effective as free electrons for heat energy transport	energy transfer by vibration and rotation of polymer chain molecules highly crystalline and ordered polymers have greater conductivity

Process and Environment Dependence

Temperature

Thermal conductivity is influenced by temperature. For gas phases, thermal conductivity increases with increasing temperature, except at high pressures, where the opposite is true. Liquid thermal conductivity is also a stronger and nearly linear function of temperature. The thermal conductivity in a solid is a function of temperature, with most solids showing a maximum thermal conductivity at low temperatures, which may actually be several orders of magnitude higher than at room temperature. The trend of thermal conductivity change with temperature for solids is very much material dependent. Sudden material structure changes can be occurring, e.g. material state changes such as solid-solid phase transformations can occur at specific temperatures.

Thermal conductivity values should be looked up in handbooks or measured for the material at the temperature of interest.

Example: Pure aluminum (99.99+ %) thermal conductivity changes with temperature (values from: ASM Handbook Vol.2A )

Temperature	Thermal Conductivity (\(k\)) SI Units (W/m·K)	Thermal Conductivity (\(k\)) US Customary Units (BTU/h·ft·°F)
-223 °C	1350	781
-50 °C	235	136
0 °C	236	136
25 °C	237	137
50 °C	239	138
100 °C	240	139
127 °C	240	139
200 °C	237	137
227 °C	236	136

Pressure

Thermal conductivity is influenced by pressure. Increasing pressure will slightly increase the thermal conductivity of a gas. Although pressure can affect liquid thermal conductivity, the effect is negligible low to moderate pressures. Very high pressures of hundreds to thousands of atmospheres are usually required to produce a significant change in liquid thermal conductivity. Therefore, pressure dependence is typically ignored. For solids, pressure affects thermal conductivity causing a linear increase with increasing pressure.

Material State

Changes in material state can lead to changes in thermal conductivity. Two examples are described below.

Solid-solid phase transformations

Sudden changes in the thermal conductivity values for some metals at distinct temperatures is attributed to solid-solid phase transformations. In these cases, the solid crystal structure phase change can transition to either promote improved energy transfer or decrease the energy transfer efficiency of the solid.

Degree of cure (DOC)

During cure, thermoset resins like epoxies undergo polymerization and cross-linking. The thermal conductivity of curing epoxies has been shown to increase linearly with increasing degree of cure development. The influence is attributed to improved phonon transport within the resin due to the tightening of the structure and increased interaction between molecular groups with cross-linking .

Modelling

The following thermal conductivity modelling discussion is adopted from the literature review previously done by Slesinger .

The thermal conductivity of composite laminates can be predicted from the thermal conductivity of the reinforcement fibre and resin constituents. Because of the anisotropic thermal properties of the reinforcement fibres, the laminate has different in-plane and through-thickness conductivities.

In-plane laminate thermal conductivity

For in-plane conductivity \(k_{11C}\), the rule-of-mixtures is used :

\(k_{11C}=V_fk_{11f}+(1-V_f)k_r\)

Where,

\(V_f=\) volume fraction fibre

\(k_{11f}=\) fibre (longitudinal) thermal conductivity [W/m·K]

\(k_r=\) resin thermal conductivity [W/m·K]

As fibre longitudinal thermal conductivities are higher than the resin, increasing \(V_f\) increases the composite laminate conductivity.

Through-thickness laminate thermal conductivity

For through-thickness, the Springer-Tsai relationship used for calculating the laminate thermal conductivity \(k_{22C}\). The presented form is for cylindrical fibres and square packing array.

\(\frac{k_{22C}}{k_r}=(1-2\sqrt{V_f/\pi})+\frac{1}{B}\Biggl(\pi-\frac{4}{\sqrt{1-(B^2V_f/\pi)}}\tan^{-1}\frac{\sqrt{1-(B^2V_f/\pi)}}{1+\sqrt{B^2V_f/\pi}}\Biggr)\)

\(B=2\Biggl(\frac{k_r}{k_f}-1\Biggr)\)

Where,

\(V_f=\) volume fraction fibre

\(k_f=\) fibre thermal conductivity [W/m·K]

\(k_r=\) resin (matrix) thermal conductivity [W/m·K]

The Springer-Tsai relationship assumes isotropic conductivity in the fibres and a packing arrangement with isolated fibres in a square grid of resin. From a micro-mechanical point of view, the assumption of no fibre contact in the Springer-Tsai relationship is incorrect. The absent fibre-fibre contact raises the through-thickness conductivity, and higher \(V_f\) increases the frequency of fibre-fibre contact .

Recent works have used finite-element models of unit-cells consisting of several fibres in partial contact, surrounded by the matrix . The unit-cell result can be more accurate than assuming no fibre contact in a closed-form solution, but it is a time consuming approach. A new unit-cell must be analyzed for every change in fibre architecture, and verification that the composite micro-structure matches the unit-cell geometry is difficult.

Related pages

Page type	Links
Introduction to Composites Articles
Foundational Knowledge Articles	Basic Definitions of Material Properties - A211 Composite properties - A214 Conduction - A118 Glass transition temperature (Tg) - A210 Heat of reaction - A114 Material properties - A150 Polymer properties - A212 Reinforcement properties - A213 Specific heat capacity - A117 Thermal conductivity - A116 Thermal diffusivity - A143 Viscosity (resin) - A203
Foundational Knowledge Method Documents	How to measure gel time - M101 How to measure reinforcement content (and corresponding matrix content) - M109
Foundational Knowledge Worked Examples
Systems Knowledge Articles
Systems Knowledge Method Documents
Systems Knowledge Worked Examples
Systems Catalogue Articles
Systems Catalogue Objects – Material
Systems Catalogue Objects – Shape
Systems Catalogue Objects – Tooling and consumables
Systems Catalogue Objects – Equipment	Rheometer - A357
Practice Documents
Case Studies
Perspectives Articles	Heat Transfer in Composites Processing - A321

References

About	Help

Engineered materials (designed to have specific properties) made from two or more constituent materials with different physical or chemical properties. The constituents remain separate and distinct on a macroscopic level within the finished structure.

A quantitative measure of how a material will respond to transient thermal conditions. It is defined as the ratio of thermal conductivity to the volumetric heat capacity of the material (density times the specific heat capacity).

Materials with large thermal diffusivity have a high thermal conductivity relative to their capacity to store energy (heat capacity) and will rapidly distribute the thermal energy throughout its volume and rapidly change temperature to reach the new equilibrium temperature.

Materials with small thermal diffusivity have a low thermal conductivity relative to their capacity to store energy (heat capacity) and will have a sluggish temperature response to a change in thermal conditions (large temperature lag). This is because it takes more energy per unit volume to change the temperature, and the low thermal conductivity means it takes longer to distribute the thermal energy throughout its volume.

Periodic 3-D, repeating array of molecules.

Thermosets are a class of polymer that undergo polymerization and crosslinking during curing with the aid of a hardening agent and heating or promoter. Initially they behave like a viscous fluid. During curing, they change from viscous fluid to rubbery gel (viscoelastic material) and finally glassy solid.

If heated after curing, initially they become soft and rubbery at high temperatures. If further heated, they do not melt but decompose (burn)

Comes in two parts: part A (resin) and B (hardener). When mixed, curing reaction starts and is not reversible.

Examples include epoxy or polyester.

Degree of cure (DOC) is an indication of how far the chemical curing reaction (crosslinking process) has advanced in a thermoset resin.

DOC is defined with a number between 0 and 1 (or 0% and 100%) where 100% is a fully cured resin. It does not have to fully reach 100% for the resin to become solid or the part to be used. In some aerospace applications, resins are only cured to about 90%. Higher the degree of cure, higher the mechanical properties.

For polymer matrix composites (PMCs), resin refers to the matrix; the continuous material phase that binds the reinforcement together, maintains shape, and transfers load. Resins are divided into two main groups: thermosets and thermoplastics.

The use of multiphysics models to predict the outcome of real-world scenarios. May be analytical (closed form, "hand calculations") or computational (implementation on computers is required due to the large number of calculations involved. e.g. finite element method, finite difference method)

Most often in composite materials engineering, modelling refers to either:

• Process modelling (predicting manufacturing outcomes, or the inverse problem of predicting manufacturing parameters to achieve a desired outcome), or
• Performance modelling (predicting the stiffness/strength/suitability of a structure, or the inverse problem of the material properties and dimensional requirements to achieve a desired stiffness/strength/suitability of a structure)

(same as "Simulation")

Volume fraction of either matrix or fibres with respect to total composite volume (matrix + fibre).

The continuous material phase that binds the reinforcement together, maintains shape, transfers load, protects the reinforcement from environment and damage, and provides the composite support in compression.

Desirable characteristics:

• Moisture/chemical resistance
• Low density
• Processability

Fibre architecture refers to how the fibre is assembled.

Most common architectures:

• Unidirectional
• Chopped strand mat (CSM) and chopped fibre (chopper gun)
• Continuous filament mat (CFM)
• Woven fabric
• Non crimp fabric (NCF)
• Braided
• Prepreg

With regards to modelling & simulation, verification confirms the implementation of a computational model to ensure that it represents the mathematical model and equations used to describe it (requirements are met).