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Thermal Expansion in a Glass and Aluminum Window: Part 1

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Thermal Expansion in a Glass and Aluminum Window: Part 1

Cracked Window

Figure 1: Cracked Glass
By Andrew Chan

[FreeImages.com Content License]

Thermal expansion of dissimilar materials is a challenge in designing equipment and structures that are subject to large temperature changes.  This is often called “CTE mismatch.” CTE is the acronym for coefficient of thermal expansion.  In this blog, we give the fundamentals of thermal expansion calculations used in thermo-mechanical analysis.

These calculations are simple but useful, and easy enough to perform by hand or with a spread sheet. Further, we perform a finite element analysis of a glass and aluminum window and frame to show where the stress is excessive.  This points to the obvious need for a gasket to perform as a thermal interface material, in order to lessen the thermal stress in the glass and avoid fracture.  The need for a gasket is obvious because nearly all of us have seen aluminum window frames with a gasket between the aluminum and the glass.  Nonetheless, it is useful to understand why and how to make that determination.

Single phase materials change in size as their temperature changes, expanding as their temperature increases.  Water expands as it cools and turns to ice, but that is an example of a change in phase, liquid to solid.  Technically, an object’s temperature is just a measure of the energy within it, so increased temperature means increased energy of its atoms.  This leads to greater vibrational magnitude and a shift to larger mean space between atoms.  We commonly see this behavior with gases, since they tend to have the largest change in volume.  It’s common knowledge that warm air is less dense than cold air, which can be calculated using the ideal gas law (Equation 1).

Eqn-ideal-gas(1)

The same thing happens to liquids and solids, though to a smaller degree since their molecules are much more closely packed.  For solids, the potential energy well between atoms is not symmetric, which leads to the average distance between increasing when their energy increases.  The extent that a material expands is governed by that material’s coefficient of thermal expansion (CTE).

An Introduction to Coefficient of Thermal Expansion for Solids

As one might expect, when gases and liquids expand they fill the space they are given.  Solids, however, have a defined shape and tend to keep that shape as they expand and contract.

We can consider the expansion of any solid object in one, two, or three dimensions, and the CTE can be calculated accordingly, using equations 2, 3 and 4.  These equations all use the same format: the parameter’s rate of change per temperature change, divided by the original value of the parameter.  These values is usually determined experimentally, but can then be used to predict behavior later for different sizes or different temperatures.

 

Eqn-CTE(2)

(3)

(4)

Where:
α = CTE
L = Length
A = Area
V = Volume

Engineering for Thermal Expansion

BridgeExpansionJointFigure 2: Thermal expansion joint in a bridge
Matt H. Wade. [CC BY-SA 3.0], via Wikimedia Commons

Thermal expansion must be accounted for in most civil and mechanical engineering.  You’ve probably seen evidence of this on bridges, which use tooth-shaped expansion joints to allow the asphalt to expand and contract without cracking or buckling (Figure 2).  Expansion joints are one way of dealing with a single material that expands or contracts, but a more interesting scenario arises when multiple materials with differing CTEs are used in conjunction.  If one material expands significantly more over a given temperature change, what happens where the materials meet?  If the materials are not bonded, then a gap could open up between them.  Or, if they’re bonded or a higher-CTE material is bounded by a lower-CTE material, then each material will experience internal stress and might fail.  Or, in a third scenario, materials might be constrained from expanding outwards and would expand against each other.

This is a common consideration in semiconductor manufacturing, one of our specialties, where glass, ceramic, crystalline or metal layers, all of which have highly different CTEs are bonded together in thin sheets.  In a situation like this, the common practice is to fill the space between each layer with a thermal interface material of intermediate CTE.  For a more easily-visible application, one only has to look out their window…or, more precisely, at their window.  Glass windows with metal or plastic frames need rubber gaskets between the pane and frame, since the CTE for glass is lower than the frame material.  The frame will expand or contract faster than the glass will, so without a material that can accommodate the changing shapes the glass could fracture (as in Figure 1).

CTE Mismatch for a Glass Window in an Aluminum Frame

Window-cross-sxnFigure 3: Cross-section of a simplified glass window in an aluminum frame
© Glew Engineering

Let’s use an example of a 1 m x 1 m single-pane of glass window with an aluminum frame.  We’ll assume that the aluminum frame can expand freely.  We’ll also assume that the glass is bonded to the aluminum.  The cross-section is shown in Figure 3.  Actually manufacturing a window this way in a commercially viable manner would be challenging, such as though with brazing, think of a stained glass windows or lamp shade.  However, for now we use this simplified setup to illustrate effect of CTE mismatch.  Glass has a linear CTE of 9 x 10-6 K-1 or (9 ppm).  Aluminum has a linear CTE of 22 x 10-6 K-1.  With a temperature 25°C higher than the temperature at which the window was manufactured,a hot summer day, we can use Eqn. 2 to calculate a linear expansion of 0.23 mm for the glass and 0.55 mm for the aluminum frame.  In essence, the glass wants to expand 0.23 mm in length and height, but the vertical and horizontal members of the aluminum frame are expanding 0.55 mm.  That 0.32 mm difference does not sound like much, but is ruinous considering how brittle glass is.  Glass has a very low ultimate strength of 33 MPa in compression, almost a quarter of aluminum’s 110 Mpa.

We can calculate the stress on the glass using the equations for strain with Eqn (5) and Young’s Modulus Eqn (6).

Eqn-strain(5)

(6)

Where:
ϵ = strain
ΔL = change in length
L = original length
E = Young’s Modulus
σ(ϵ) = normal stress due to elongation

The resulting stress in the horizontal and vertical directions is roughly 22 MPa.  The glass will experience the most stress at the corners, where it is in contact with both vertical and horizontal members of the frame, which are each expanding.  However, one need to perform FEA to get this level of detail in a quantitative manner.  We can calculate the principle stresses using the Equation 7.

Eqn-princ-stress(7)

Where:
σa,b = principle stresses
σx = normal stress in the x direction
σy = normal stress in the y direction
τxy = shear stress

Even considering a simplified case where the shear stressτxy= 0, the maximum principle stress is 44MPa.  This surpasses the ultimate strength of glass, 33 MPa, by 33%.  As such, plate glass isn’t installed flush to the inside of a solid aluminum frame, like we created in the example.  Instead, the aluminum frame extends around the side of the pane without contacting the edge.  Holding the window in place is a silicone  gasket or similar, which can easily compress and expand.  That way, both the aluminum and the glass are free to expand or contract as temperatures dictate, but the gap between is still air tight.  Double- or triple-paned windows get even more complicated, not just because they have more components.  Imagine an inner pane that’s shrinking from the cold of an air-conditioned office, while the outer pane is expanding in the heat of a summer day.

FEA Recreation of the Aluminum-Framed Window

Window-stress

Figure 4: Maximum principle stress on simplified glass window in aluminum frame
© Glew Engineering

We recreated the above example in our FEA simulation suite, and the initial result is shown in Figure 4.  One can see that the corners of the glass pane are indeed experiencing the highest stress, at a value around 40 MPa.  This agrees with our calculations, and exceeds the ultimate strength for glass.  We will delve more into the FEA next week, to look at the effects of contraction in colder temperatures.  We’ll also explore the effectiveness of adding a rubber gasket around the window.  The FEA should show even more conclusively the significant effect that changing temperatures can have on materials.  It’s one more factor that mechanical engineers have to keep in mind, whether designing a rocket enginer or a mundane window frame.

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