Thermal design basics: materials and calculations

This article is a basic look at thermal materials that are available today, and general design considerations for thermal design. This article covers thermal conductivity, thermal interface materials, and metallic materials for passive heat spreading. We will also look at some high conductivity materials that are commercially available.

Basic conduction equation

The equation defined by Fourier for conduction within is a system is as follows:

Q = -k (dT/dx)

Q = is the heat flow per unit area
K = thermal conductivity (W/m-K),
dT/dx = temperature change per change in length
T = temperature
x = length.

Additionally, Q = q/A, where q = Joules / second (Watts) and A = cross-sectional area perpendicular to the direction of heat flow. For simple one dimensional design problems, calculations in network resistance models or first past trade studies, this equation can be simplified to:

q = k*A*ΔT
x

Using this equation some basic design estimations can be made on the range of thermal conductivity that will be required. Selecting a material or determining the size or shape of a heat spreader, heat sink or heat exchanger can be quickly estimated with simple hand calculations.

Typically, the material selected for a heat spreader or heat exchanger is an Aluminium alloy. When a higher-conductivity material is needed a Copper alloy is most often selected. How do we select which alloys to use? There are many options to choose from and we must study the tradeoffs between the different material options, the complexity of our design, the cost of our design and the manufacturing techniques that will be used to create the parts and assemblies. These are all important, and the material properties must also be considered to ensure that the material will have a high enough thermal conductivity, and/or thermal capacitance. Additional physical properties that may be in consideration are yield strength and flexural modulus as well as the coefficient of thermal expansion, density and the specific heat of the material.

See Table 1 for a list of common materials, and their properties, that can be considered for design studies.

In many design applications, rather than having a dedicated heat exchanger or heat spreader, the structure (chassis or housing), is often used as the method of heat removal.

One example of this is an electronic circuit board, where the passive heat removal technique is a landing or mounting surface (Fig.1). Good thermal contact is assured in this area and the heat loads are conducted from that interface to the external surfaces of the chassis. The heat removal mechanism is natural convection to the ambient air. This leaves us to consider a wider range of materials for solving our thermal management requirements.

Fig. 1: Example of passive heat dissipation

CTE (coefficient of thermal expansion)

The coefficient of thermal expansion (CTE) is an additional material property that can often be overlooked in the design of mechanisms, instruments or other devices. The expansion and contraction of materials can directly affect function and performance. If the worst case tolerance stack-up makes your fit very marginal at room temperature, what happens to the clearance of parts at the hot or cold extreme? An example is a running clearance fit with dissimilar materials. These may bind when the product is then taken to the extremes of its operating temperature range, and/or with additional size variation of your parts when manufacturing tolerances are considered.

Thermal Resistance

Contact resistance or thermal interface resistance is the ability of the heat to flow between two surfaces in direct contact. The interface of the two parts does not have a connection at the molecular level so this affects the ability of electrons of the two different items to easily interact and transfer energy across the boundary. If the surface contact is viewed under a microscope it can be readily seen that the interfaces do not make a smooth contact. The micro scale topology is a rough series of ridges and valleys with interstitial gaps (Fig.2).

Fig.2: Surface topology between thermal interfaces

A fine surface finish can reduce thermal resistance, but such surface finishes come at a price. They are typically precision-machined or lapped, and if you are using cast or MIM (Metal Injection Molded) parts, a post process machining operating would be required for very fine surface finishes. These extra processes will increase the part cost, perhaps significantly. A tradeoff that can be made is to use commercially available thermal interface materials. These are available from an array of vendors. Table 2 shows some options that are available.

Thermal Interface Materials

It should be noted that thermal resistance is not listed in Table 2 because it varies with contact pressure and material interface. The designer should carefully select the thickness and design the compression range or gap that can be obtained with manufacturing tolerances. For thermal greases, epoxies and compounds, the assembly process or the design itself should control the bond line thickness to a defined value in order to properly design the thermal resistance required for the design to meet the demands of the project.

The thermal compound and gap pad options available on the market are numerous. The industry is very competitive and the design engineer will find many options to choose from with some simple searches. The thermal conductivity of these materials varies from about 1.0 W/m-K to levels as high as 17 W/m-K.

Thinner interface materials can help reduce thermal contact resistance by filling the gaps between contact points, and the fact that they are thinner than gap pads mean the temperature rise through the material will be lower than for the thicker materials. Another option beyond gap gad materials are thermal compounds or foams that can be dispensed into place. These materials are used to fill the gap between heat generating components, or a surface with varying heights, such as a circuit board, where heat must be removed from multiple surfaces into the heat sink or heat spreader.

Metallic Thermal Interfaces

For more challenging applications, metallic thermal interface materials can be used. Graphite materials are readily available on the market today. Indium alloy metallic foils are also available but are slightly more expensive. The advantage of metallic foils is not only the higher thermal conductivity, but their softness can also create a very low thermal resistance at the boundary between materials. This is true for the graphite products as well as the Indium and Silver based metallic foils.

There will be a price difference between the polymer based thermal interface materials and the metallic foil based materials. The metallic materials will come at a higher cost. The design trade-off must be made based on project requirements, budget allowances and the temperature rise across a thermal interface that can be tolerated.

Thermal Grease

Thermal grease can also be considered at an interface where it can be tolerated within the system design. This is a well-known method and many resources are available to help find the right solution. This can be a low cost design solution that can help significantly reduce thermal resistance at an interface. However, it can come at a cost in either performance (as eventual dry out of the interface may occur), internal contamination of other components, or additional complexity in terms of ease of assembly or repair, such as automotive or military electronics and equipment where depot repair is part of the design consideration.

High thermal conductivity epoxies and pastes

For higher heat flux densities or designs with more complex geometries, high thermal conductivity epoxies and pastes can be used. Some silver loaded epoxies are as high as 60 W/m K and higher limits are being continuously pushed by innovators. Epoxies are often used to aid in the assembly of bolting together housings, mounting heat exchangers to housings, or in more complex precision assemblies such as laser crystal mounting or die attach packaging of semi-conductor components.

A wide range of epoxies are available commercially. In the near future diamond loaded thermal interface materials and compounds are rapidly being developed, and are pushing capabilities higher and higher and as will be discussed later. Graphene sheets are currently the holy grail of today’s latest TIM research.

Transient Considerations

For transient thermal considerations attention should be given to the specific heat value of a material. This is the ability of a material to hold energy per unit mass. This can be an important part of the design when heat loading conditions are intermittent or cyclical. Proper selection of materials can help optimize a system’s transient thermal performance. If you consider the specific heat multiplied by density, the resulting value can add significant insight to your materials’ ability to hold heat or its volumetric heat capacity. The units obtained from multiplying specific heat times density are [Joules/(Kelvin*m³] or [Joules/°C*m³].

It then follows that for each cubic meter of an object of specified material, it would take “X” Joules to raise the temperature of the object 1°C. For most of today’s electronics, mechatronics, and robotics applications, you will most likely be working with volumes of material much smaller than 1 cubic meter, so it can be seen that transiently the local temperature rise could easily increase several degrees with just a few Watts [Joules/second] of waste heat.

Bulk Heat Transfer

If the amount of waste heat from a system is known the bulk temperature can be calculated. The bulk temperature heat transfer equation is:

Where,
Q = amount of heat transferred to system, Watts [Joules/second]
Cp = specific heat of material (J/kg·K)
Dt = time energy is applied (seconds)
ρ = density of material (kg/m³)
V = Volume of material (m³)
DT = Temperature rise (K)

Thermal Diffusivity equation

Where,
k = thermal conductivity, W/(m·K)
ρ = density (kg/m³)
Cp = specific heat capacity, J/(kg·K)

Alloy	Conductivity W/m-K	Specific Heat J/kg K	Density, kg/m3	Heat Capacity J/(m3-K)	Diffusivity m2/second
Aluminum 1100 H14	220	904	2716	2.45E+06	8.96186E-05
Aluminum 2024-T3	121	875	2774	2.43E+06	4.98544E-05
Aluminum 6061-T6	167	896	2704	2.42E+06	6.89176E-05
Aluminum 6063-0	218	900	2704	2.43E+06	8.95645E-05
Aluminum 6063-T6	200	900	2704	2.43E+06	8.21692E-05
Magnesium AZ91	72.7	1047	1803	1.89E+06	3.85124E-05
Magnesium AZ31	96	1000	1775	1.78E+06	5.40776E-05
Copper UNS C10200, H04 Temper	388	391	8959	3.50E+06	0.000110759
Brass Free Cutting; 15% Cold Draw; C36000	115	380	8516	3.24E+06	3.55388E-05
Naval Brass; UNS C46400, O61; (tubing)	116	380	8432	3.20E+06	3.62016E-05
Commercial Bronze (90-10) H04, Flat Products	189	376	8821	3.32E+06	5.69867E-05
AISI 1080 Steel, as rolled	47.7	490	7878	3.86E+06	1.23575E-05
Stainless Steel. 302, Annealed, Bar	16.2	500	7878	3.94E+06	4.11295E-06
Stainless Steel 17-4 H900	17	460	7767	3.57E+06	4.75838E-06
Titanium Ti-6Al-4V (Grade 5), Annealed Bar	6.7	523	4438	2.32E+06	2.88655E-06
Carpenter Invar 36® Alloy, Cold Drawn Bars	10.1	515	8072	4.16E+06	2.42967E-06
Al-SiC – Metal Matrix Composite	170	750	2607	1.96E+06	8.69454E-05

Table 3: Thermal diffusivity of materials

The thermal conductivity of your material is now important transiently as it directly correlates to the thermal gradient that occurs within the system during the transient thermal load. Transiently the value obtained when the Specific Heat is multiplied by density and then by the thermal conductivity of the material is the thermal diffusivity. In some designs transient temperature rise may be low enough that conductivity or heat spreading is less important and a material can be used with lower thermal conductivity with a higher density and specific heat value such as steel or stainless steel.

This leads to another type of material that can be considered for circuit board level component cooling: phase change thermal interface materials:

Phase change thermal interface materials

The majority of these materials are designed such that during a phase change they are retained within the local volume of material that is present at room temperature. Often many design engineers, including myself, may be hesitant to use this type of material because of the concern that as the material experiences phase change, particles of the material could potentially escape to the surrounding components. Most vendors are aware of this issue and their products are designed such that this is a non-issue. The use of this type of material in the same area or volume of precision optical systems is not recommended without careful design consideration.

Other materials such as paraffin wax have been used as phase change reservoirs or cavities that store heat for transient designs. A design augmentation for removal of steady state heat loads with consideration for additional transient loads is coupling metallic cooling paths with embedded phase change materials or waxes. Designs can integrate a conduction path such as metal fins or metallic foam with embedded phase change material. This design allows for removal of steady state loads with additional ability to handle higher cyclical loads.

High conductivity materials

Beyond Copper we start to find more exotic and expensive materials. These materials are typically a carbon based material. The caveat is that these high conductivity materials come with thermal conductivities that are anisotropic, meaning, the conductivity varies with direction. These materials tend to have high conductivity in X-Y and a much lower value through the thickness of the material (Z). A chart (Table 4) is provided to show the effective thermal conductivities of commercially available materials.

 

Description	K – W/(m-K)
Pyrolitic Graphite Aluminum Composites	250 to 380
Meso Pitch Carbon Fiber	500 to 600
Pyrolytic Graphite	1200 to 1500
CVD Diamond	~2500
Heat Pipes & 2D Plates	>10,000

Table 4: High conductivity materials

These materials can be used effectively as heat spreaders to reduce local thermal gradients at the interface of a heat generating component and then mated to a material with a higher “Z” direction conductivity.

Some more advanced materials have become commercially available such as Graphite- Aluminum composites, which have low density and high conductivity. Even these next generation materials are applicable in special design scenarios. A special situation must exist where an item’s weight requirement outweighs cost. The item can be used in such a manner its lower yield strength and elastic modulus do not affect the structural performance of the system or device.

With a quick search of the web, many research institutions and universities are pushing forward with research into Graphene (5000 W/m-K) as a potential thermal interface material, and carbon nanotubes (3500 W/m-K). There are firms popping up that have made some very small scale versions of these technologies. The real trick to getting this into the design will be convincing the program manager that the technology will be worth the cost and then convincing the reliability engineer that using this in the design will not produce any long term reliability problems, corrosion or contamination issues.

Thermal designers have one additional option when very high conductivities are needed, and that is the heat pipe where expected effective thermal conductivity starts at about 6,000 W/m-K but can be much higher than 10,000 W/m-K. We leave that for future discussions.

The intent here was not to provide a list of vendors for these materials, but rather provide a basic review of readily available materials for thermal management and design. We hope this article has provided many readers with a review or basic introduction of thermal management design considerations and materials available to select from in your next design.

References

1. matweb.com
2. Mills, AF. Heat Transfer, R.D. Irwin Inc. Boston, MA, 1992.
3. Donachie, MJ. & Donachie, SJ. Super Alloys, A technical Guide, 2^nd Ed., ASM International, Materials Park, Ohio, 2002.