Fig. 1 Thermal expansion curves for a number of different metals. Notice that each metal expands at a different rate than any of the other metals (This chart adapted from the American Welding Society’s AWS Brazing Handbook, 5th Edition, 1997, p. 29)

As described In previous articles I’ve written about the thermal expansion of metals, it’s important to note the obvious fact that all metals will expand (grow) when heated up, and then contract (shrink) when they are cooled. It was also noted that, unfortunately, each different type of metal expands/contracts at a different rate from all of the other types of metal, which is something that can lead to significant brazing problems when someone is trying to braze two different types of metal together (such as tungsten-carbide to stainless steel), because at the high temperatures of brazing (which may often exceed 2000°F/1100°C) the different expansion rates between those metals may cause the joint to literally close up completely, or perhaps to become too large to braze effectively. It is very important for designers of brazed components to understand, and take into account in their designs, the different expansion rates between any two metals being joined (also known as “differential metal expansion” rates, or “DME” rates) so that gap-clearances at brazing temperature will be correct, and thus allow good, effective brazing to be accomplished.

Fig.1 shows some thermal expansion data for a number of common metals, and does so from room-temperature all the way up to brazing temp, i.e., to temperatures in excess of 2000°F (1100°C). This is very important since you can see that the curves shown in the charts in Fig. 1 are NOT truly linear (i.e., straight lines) from room temperature up to whatever brazing temperature may be used, but do indeed have a curvature to them, and some even have significant breaks in them. Such information is VERY important when designing assemblies/components that will be joined by brazing.

Interestingly, most thermal-expansion data published in metals-handbooks shows only the expansion rates for metals between room temperature and the boiling point of water, i.e., from 0°C – 100°C (32°F-212°F). So, although such info might be a nice “starting point” for figuring out what to do when designing components to be brazed, it is actually NOT truly enough information to fully understand what happens to brazed-joint gap clearances when that joint is heated all the way up to the very high temps involved in brazing. Let’s explore further…..

### Coefficient of Thermal Expansion (CTE)

The “coefficient” of thermal expansion is a number assigned to a particular metal or alloy and represents the “average slope” (mathematical term) of the “linear expansion” curves (such as those shown in Fig. 1) for any given metal on that chart over any particular temperature range desired. As mentioned above, many handbooks decided to merely use the narrow temperature range from 0-100°C (32-212°F). For brazing design considerations, that range might not be sufficient, and a much wider range (up to brazing temp) may be better.

The formula/method used to calculate the “slope” of any particular curve rising from left to right on the chart is illustrated in Fig. 2. Using the information shown, let’s look back again at Fig. 1, and try to calculate the slope of the line that is shown on that chart for 6061 aluminum (first curve on the left of chart). It appears to be a fairly straight line. So, as you move from room temperature (the beginning of the line at the bottom left side) up to 800°F (425°C), you see that the line for that 6061 alloy has risen along the left axis from 0.000” up to about 0.0105” (0.27 mm). The slope of that line, according to the formula in Fig. 2, is then found by dividing the 0.0105” rise by the 800°F run-up in temp, the result of that division then being 0.000013125 inches (usually expressed instead as 13.1 x 10-6 inches) and represents the amount of expansion for each degree Fahrenheit (F) that the part has been heated. The slope of that line would then be designated as the “coefficient” used to specify the rate of expansion for that particular aluminum alloy. Such coefficients are specified as a whole number, saving the 10-6 factor until later when calculating the overall change in dimensions of the part that would occur during the brazing process.

Fig. 2 Chart showing how the slope of any given curve (such as the curves shown in Fig. 1) is calculated.

Notice that in most metals handbooks they list the CTE for 6061, as well as for other aluminum alloys, as being 13.1 in/in/F x 10-6, which is identical to the slope of the 6061-alloy line that we just calculated from the chart in Fig. 1.

The Coefficient of Thermal Expansion (CTE) for any given metal specifies a growth-rate for that particular metal and is always denoted by the Greek letter Alpha (“⍺”). Each metal’s CTE is expressed dimensionally as an amount of expansion (in inches, or mm), and applies to EACH inch (or mm) of the length of the part), and for EACH degree (F or C) of heating that occurs from room temperature up to the temperature of brazing. Thus, CTE’s are given as a specific number “X” followed by “in/in/°F x 10-6 ” or by “mm/mm/°C x 10-6 ”. Thus, the CTE for 6061 aluminum would be shown in handbook tables only as “13.1” when using British units of measure, or “23.8” when using metric numbers.

Remember — when a metal is heated, the atoms in that metal’s structure oscillate faster and faster and move further and further apart, causing the metal to grow in size. We tend to assume that this growth will be linear in all dimensions, but that is usually not the case, especially for metals that go through phase-changes (and thus change the way atoms line up with each other at different temperatures). Notice the radical expansion changes that occur with 1018 steel in Fig. 1 (discussed in one of my previous articles that I wrote on the topic of differential metal expansion (DME)).

### How Are CTE’s Actually Measured?

There are a number of methods used in industry to accurately measure, and then chart, the expansion rates of metals. CTE’s can be determined via the use of: (1) a strain gauge; or (2) interferometer (highest precision method), or (3) x-ray diffractometry, among other methods. It is not the purpose of this article to investigate any of these specific testing procedures. Instead, we will accept the charts that are developed/created by such methods, and go on from there. Please understand that, from a practical brazing standpoint, the charts and data developed by these methods, like the one shown in Fig. 1, as well as the CTE data published in metal handbooks and alloy datasheets for each different base-metal, are, in fact, a reasonable starting point to use when calculating how gap-clearances may change as metals are heated. This can be especially helpful when trying to calculate gap-clearances changes when heating up tubular assemblies to brazing temps, especially when the tubes are made of different metals. In fact, let’s look at an example of just such a situation:

**Example (tubular joint): **Let’s suppose you wanted to furnace-braze a tube of Metal-A into a larger diameter tube of Metal-B, using a brazing filler metal (BFM) that was ideal for joining them, and which had, for this example, a recommended brazing temp of 1870°F/1020°C). Let’s assume the outside-diameter (OD) of the Metal-A tube is exactly 2.000”(50.80 mm) at room temp, and the internal diameter (ID) of Metal-B is exactly 2.005” (50.93 mm) at room temp — in real life you’d also have to take the dimensional tolerances (min/max) into account, but we’ll overlook that for now for this example — Thus, at room temperature, the Metal-A tube slides easily into the Metal-B tube with a smooth slip-fit, and has a diametrical clearance of 0.005” (0.127 mm)

For this example, let’s assume that Metal-A has a published CTE of 9.5 (17.1) and Metal-B has a published CTE of 7.5 (13.6), the first number being British units, while the metric numbers are placed in parentheses next to the British units.

Assuming that the temperature in the brazing shop is 70°F (21°C) and that the two tubes have been properly cleaned and prepped for brazing, what will happen to that joint when it is brazed? Can we calculate what the joint-clearance will be at brazing temp? Yes, we can get a good approximation of what the joint clearance will be at brazing temp, and thus make our brazing decisions accordingly. Here’s how:

1. Determine how much the base metals will grow during brazing:

a) Each tube can be considered to be a flat sheet rolled into a tubular shape so that the ends of the sheet butt-up against each other. Thus, when looking at the expansion of the metal, calculate how much each “length” of metal will grow, which can then be used to calculate what its new diameter will be at brazing temp.

Metal-A: It’s a 2.000”(50.8 mm) diameter tube. When straightened out to a flat sheet, the length of Metal-A would be equivalent to its circumference (C), which in turn is equal to its diameter multiplied by “pi” (π = 3.1416). The same applies to Metal-B.

Thus: Metal-A’s OD = 3.1416 x 2.000” = 6.2832” (159.59 mm) Metal-B’s ID = 3.1416 x 2.005” = 6.2989” (159.99 mm)

b) At brazing temp the metals will each grow in accordance with their respective CTE’s as follows: Metal-A has a CTE of 9.5 in/in/°F x 10-6 (17.1 mm/mm/°C x 10-6). Thus Metal-A will grow 9.5 x 10-6 inches (that is, it will grow 9.5 millionths of an inch) for each inch of length of the sheet (6.2832 in.) for EACH degree F that the part has been heated (1870°F brazing temp — minus room temp of 70°F means it has seen a change in temp (i.e., a delta-T, or ∆T) of 1800°F (1870°F – 70°F = 1800°F).

Thus, change (growth) in the length of the metals:

Metal-A growth = 6.2832 in. x 9.5 millionths per in. x 1800 = 0.1074” (2.73 mm) Metal-B growth = 6.2989 in. x 7.5 millionths per in. x 1800 = 0.0850” (2.16 mm)

c) Add this growth to the original “lengths” of each metal, to find their new total length (tube circumference) at brazing temperature.

Thus Metal-A’s new length (circumference) is: 6.2832” + 0.1074” = 6.3906” (162.321 mm) and Metal-B’s new length (circumference) is: 6.2989” + 0.0850” = 6.3839” (162.151 mm)

2. Determine the new diameter for each tube while it is at brazing temp. This is done by dividing each new length (circumference) by “pi” (π = 3.1416) to determine each tube’s diameter at brazing temp:

Thus Metal-A’s new OD is: 6.3906”/3.1416 = 2.0342” (51.668 mm) and Metal-B’s new ID is: 6.3839”/3.1416 = 2.0321” (51.614 mm)

3. Gap-clearance change:

The new diametrical gap-clearance at brazing temp is the new 2.0321” ID of Metal-B, minus the new 2.0342” OD of Metal-A that is inside Metal-B, or -0.0021” (-0.053 mm). Thus, at brazing temp, the gap-clearance in the joint has closed completely to a negative fit (a “press fit” so to speak), and the brazing filler metal (BFM) will probably not be able to penetrate into the joint, resulting in either a non-brazed joint or one with very little BFM in it.

Based on the calculations above, this joint needs to be re-designed to make it more brazeable. Perhaps the designer can open the room-temp gap more so that it will not close off completely at brazing temp.

IMPORTANT NOTE: These calculations, even though I’ve carried some of them out to four decimal places, are still only approximations, since we started with a HUGE assumption about the CTE’s of the metals in question, those CTE’s merely being the “average” slope of a very limited portion of each of those curves in Fig. 1 (and other similar charts for other metals, as shown in some metals handbooks). You can see that if you take different temperature ranges into account in Fig. 1, you can get different values of CTE for each metal.

### Conclusion

I caution designers and brazers to only use published CTE data as a “general guide” for designing assemblies and evaluating their fit-up at brazing temperatures. Since published CTE’s are only approximations for each different metal, don’t depend too much on any calculated results you get from using them, but instead, use them merely as a broad generalization of what might happen to a metal assembly when it’s heated up to brazing temp.

IF – and I repeat — IF you need more accurate information about the expansion of a particular assembly that you want to braze, then actual furnace experimentation may be needed. Here’s what one of my colleagues did:

**Procedure he followed:** My colleague had a 48-inch diameter aero-engine turbine ring that was to be brazed inside another outer ring, and he wanted to have a tight, but non-distorting close connection between them after they were brazed together. So, although he did the calculations from the handbook data, as we did in the previous example, he decided that he would also run some simple furnace experiments to see how the results compared to what he had calculated. So, he centered the turbine ring (the one that he wanted to braze later on) on top of a large, flat, smooth-surfaced low-expanding ceramic plate, which he then set on top of his furnace grate, and heated it up to the brazing temperature that he intended to use.

But, before heating the fixtured ring, he carefully poured some high-temp powder all around the outside of the ring on the ceramic plate, being sure to push the powder uniformly up around the OD of the ring.

**NOTE: You can use a number of different hi-temp powders for such an application, such as hexagonal boron nitride (HBN) powder, aluminum-oxide powder, etc (among others). If metal powders are used, they may react with the base-metals in your assembly, which you don’t want to happen.**

The ring was then heated up to brazing temperature, held there for a few minutes, and then brought back down to room temp.

Examination of the flat ceramic plate showed that the aero-ring had indeed expanded, pushed the powder out to the final diameter the ring had at brazing temp, and then when cooling occurred, the ring shrank back down to its original diameter, leaving an open space between the OD of the ring and the ID of the powder circle (as shown in the rough-sketch at the bottom of Fig. 3 below). The results clearly showed how far the ring had expanded during that test run in the furnace.

By carefully measuring the distance between the ring and the larger powder ring, and taking into account the expansion of the ceramic plate on which the part was sitting, he was then able to reasonably determine the amount of expansion that had occurred, compare it to his calculated figures (such as I described earlier in this article) and then plan his brazing procedure accordingly.

Fig. 3 The use of a non-reactive, hi-temp powder on the outside of a metal part, placed so that the metal, when it expands, will push the powder outwards to the point where the metal stops expanding (at the particular temperature used for furnace brazing).

My colleague then did the same type of testing with the outer ring to which the inner ring was to be brazed.

Thus, the result of all the work he did was this: The final assembly, brazed in accordance with the info he learned from his calculations and his testing, came out very well, with no distortion, and with a uniform, thin well-brazed joint around the periphery of the entire assembly.