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# Thermal conductivity - A116

Thermal conductivity
Foundational knowledge article
Document Type Article
Document Identifier 116
Themes
Relevant Class

Material

Tags
Prerequisites

## 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 [1] 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:

## 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 [2]
Ceramics Low 2 - 50 W/m·K thermal insulators [2]
Polymers Low Order of 0.3 W/m·K

(< 1 W/m·K)

thermal insulators [2]

#### 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 [3]
Glass Fibre (S2) 1.45 0.838 [4]
Matrix Materials Polyester 0.2-0.6 0.1-0.3 [5]
Epoxy 0.2-0.6 0.1-0.3 [5]
Tooling Materials Steel 15-52 8.7-30 [2]
Aluminum 247 143 [2]
Invar 10 5.8 [2]

For some materials, the thermal conductivity varies significantly depending on the heat flow direction (anisotropic). For example, carbon reinforcement fibre properties are anisotropic [6], 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 [3], a subset of them is provided below.

#### Selection of anisotropic carbon reinforcement fibre thermal conductivity values (Sourced by Slesinger [3])

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 [6]
CN80 320 11 185 6.36 [7]
T300 100 11 57.8 6.36 [7]
T650 14 5 8.09 2.89 [8]
T700 100 11 57.8 6.36 [7]
YS80 320 11 185 6.36 [7]

## 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.

The following steady-state measurement methods for thermal conductivity are recommended by the Composites Materials Handbook - 17 (CMH-17) [1]:

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 [9])

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 [10].

## Modelling

The following thermal conductivity modelling discussion is adopted from the literature review previously done by Slesinger [3].

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 [11]:

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

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 [12] 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 [13].

Recent works have used finite-element models of unit-cells consisting of several fibres in partial contact, surrounded by the matrix [7]. 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

Introduction to Composites Articles
Foundational Knowledge Articles
Foundational Knowledge Method Documents
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
Practice Documents
Case Studies
Perspectives Articles

## References

1. [Ref] Composite Materials Handbook 17 - Polymer Matrix Composites; Guidelines for Characterization of Structural Materials. 1. SAE International on behalf of CMH-17, a division of Wichita State University. 2012. ISBN 978-0-7680-7811-4.CS1 maint: date and year (link)
2. [Ref] Callister, William D. (2003). Materials Science and Engineering: An Introduction. John Wiley & Sons, Inc. ISBN 0-471-13576-3.CS1 maint: uses authors parameter (link) CS1 maint: date and year (link)
3. [Ref] Slesinger, Nathan Avery (2010). Thermal Modeling Validation Techniques for Thermoset Polymer Matrix Composites (Thesis). doi:10.14288/1.0071063.CS1 maint: uses authors parameter (link)
4. [Ref] MatWeb LLC. "MatWeb: Online Materials Information Resource". Retrieved 9 September 2020.CS1 maint: uses authors parameter (link)
5. [Ref] Ashby, M.F. (2011). Materials Selection in Mechanical Design. Elsevier. doi:10.1016/c2009-0-25539-5. ISBN 9781856176637.CS1 maint: uses authors parameter (link) CS1 maint: date and year (link)
6. [Ref] Johnston, Andrew (1997). An integrated model of the development of process-induced deformation in autoclave processing of composite structures (Thesis). doi:10.14288/1.0088805.CS1 maint: uses authors parameter (link)
7. [Ref] Schuster, J et al. (2009). "Measuring and modeling the thermal conductivities of three-dimensionally woven fabric composites". 45 (2). doi:10.1007/s11029-009-9072-y. ISSN 1573-8922. Cite journal requires |journal= (help)CS1 maint: extra punctuation (link) CS1 maint: uses authors parameter (link)
8. [Ref] Cytec Industries Inc. (2003), Thornel T650/35 product data sheetCS1 maint: uses authors parameter (link) CS1 maint: date and year (link)
9. [Ref] Kaufman, J Gilbert, ed. (2018), ASM Handbook: Aluminum Science and Technology, 2A, ASM International (published 30 November 2018), doi:10.31399/asm.hb.v02a.9781627082075, ISBN 978-1-62708-207-5CS1 maint: date and year (link)
10. [Ref] Struzziero, G et al. (2019). "Measurement of thermal conductivity of epoxy resins during cure". 136 (5). John Wiley & Sons, Ltd. doi:10.1002/app.47015. ISSN 0021-8995. Cite journal requires |journal= (help)CS1 maint: extra punctuation (link) CS1 maint: uses authors parameter (link)
11. [Ref] Peters, S. T., ed. (1998). Handbook of Composites. Springer US. doi:10.1007/978-1-4615-6389-1. ISBN 978-0-412-54020-2.CS1 maint: date and year (link)
12. [Ref] Springer, George S; Tsai, Stephen W (1967). "Thermal Conductivities of Unidirectional Materials". 1 (2). SAGE Publications Ltd STM. doi:10.1177/002199836700100206. ISSN 0021-9983. Cite journal requires |journal= (help)CS1 maint: uses authors parameter (link)
13. [Ref] Zhang, Jing et al.. "Effect of cure cycle on temperature/degree of cure field and hardness for epoxy resin". 10 (1). De Gruyter. doi:10.1515/epoly.2010.10.1.41. Cite journal requires |journal= (help)CS1 maint: extra punctuation (link) CS1 maint: uses authors parameter (link)

## Welcome

Welcome to the CKN Knowledge in Practice Centre (KPC). The KPC is a resource for learning and applying scientific knowledge to the practice of composites manufacturing. As you navigate around the KPC, refer back to the information on this right-hand pane as a resource for understanding the intricacies of composites processing and why the KPC is laid out in the way that it is. The following video explains the KPC approach:

## Understanding Composites Processing

The Knowledge in Practice Centre (KPC) is centered around a structured method of thinking about composite material manufacturing. From the top down, the heirarchy consists of:

The way that the material, shape, tooling & consumables and equipment (abbreviated as MSTE) interact with each other during a process step is critical to the outcome of the manufacturing step, and ultimately critical to the quality of the finished part. The interactions between MSTE during a process step can be numerous and complex, but the Knowledge in Practice Centre aims to make you aware of these interactions, understand how one parameter affects another, and understand how to analyze the problem using a systems based approach. Using this approach, the factory can then be developed with a complete understanding and control of all interactions.

## Interrelationship of Function, Shape, Material & Process

Design for manufacturing is critical to ensuring the producibility of a part. Trouble arises when it is considered too late or not at all in the design process. Conversely, process design (controlling the interactions between shape, material, tooling & consumables and equipment to achieve a desired outcome) must always consider the shape and material of the part. Ashby has developed and popularized the approach linking design (function) to the choice of material and shape, which influence the process selected and vice versa, as shown below:

Within the Knowledge in Practice Centre the same methodology is applied but the process is more fully defined by also explicitly calling out the equipment and tooling & consumables. Note that in common usage, a process which consists of many steps can be arbitrarily defined by just one step, e.g. "spray-up". Though convenient, this can be misleading.

## Workflows

The KPC's Practice and Case Study volumes consist of three types of workflows:

• Development - Analyzing the interactions between MSTE in the process steps to make decisions on processing parameters and understanding how the process steps and factory cells fit within the factory.
• Troubleshooting - Guiding you to possible causes of processing issues affecting either cost, rate or quality and directing you to the most appropriate development workflow to improve the process
• Optimization - An expansion on the development workflows where a larger number of options are considered to achieve the best mixture of cost, rate & quality for your application.