Hi all,
This is a remake of an earlier thread, incorporating some important new information and feedback from the community. If you read the previous post, some of this information is repeated from there, but do stick around: there is some really interesting new stuff, especially in Section 3. I hope you find it useful. Again, the focus is on tubular box baffles, not karo step or anything like that.
1. Basic Shape
Most baffles are designed and cut with a shape like this in mind:
badbaffle.png
That is, that they are rectangles capped with semicircles or arbitrary circular arcs, the exact shape of which left up to the designer. This shape ends up being pretty close to reality, depending on your goals, but it is not quite right. I'll explain the shape that is a closer fit in a moment, but in order to do so I must explain to you a very specific subset of baffle shapes, and that is the shapes that come about when the baffles are fully filled with down. Or, if you prefer, you can skip right to 1c.
1a. A Fully Filled Baffle
I know what you are thinking: But wait! I always fill my baffles completely. Sometimes I even overfill! While it may be true that you filled the shape you designed completely, the baffle itself still has plenty of room left. Think about what happens when you add overfill to your baffle. To make things especially clear, imagine overfilling it by a lot, just keep piling the down in there. Would it stay the same shape? Of course not. What happens is that the baffle gets rounder: it bulges up at the center to get a taller, and pulls in the edges a little closer to compensate.
To be more specific, what happens is that the baffle expands until it encloses the largest volume possible for the amount of fabric provided. Once it reaches this shape, the baffle cannot move anymore, so the down starts compressing. That means that, in a very important sense, the shape the baffle reaches before the down starts compressing is the shape of a fully filled baffle — and any other shape is an underfilled version of this shape. This isn't necessarily a bad thing per se. There are pros and cons to underfilling the baffles like this, and to a degree is already acknowledged when talking about baffle design (it's why we don't make baffles too wide). But acknowledging this is useful when designing baffles, as we'll see below.
So what is this mysterious fully-filled baffle shape? It is what I like to call a truncated circle.
1b. The Truncated Circle
I'm sure we all remember from grade school that the shape that encloses the most area for a given perimeter is a circle. Baffles are a little more complicated, as they have a pair of straight walls on either end (provided the baffles are equally filled). Fortunately, getting from a circle to our baffle shape is pretty easy: just cut off the ends. If anyone is curious about the proof of this, ask in the comments below, but for the sake of expedience I'll just use the graphic below to explain what I mean:
bafflewh.png
Exactly how much of the end you cut off is up to you, and depends on a number of factors addressed in later sections. There are two benefits to this shape, that go hand in hand: the shape is completely dimensionally stable, meaning that your quilt is more likely to come out the size you designed; and the down is stable, meaning it won't shift at all unless you really try, and even then it will return to normal over time. Unfortunately, this shape comes with a serious disadvantage: weight. Nobody really wants a quilt with baffles thinner than they are deep. A quilt like this will typically use 3x more baffle wall material than a normal quilt! Nobody really wants that. So what is one to do? Well, there are two solutions here, the normal solution and the clever solution. But we'll get more into that in Section 2. Before we do, we need to talk about the broader category of baffle shapes:
1c. Truncated Ellipses (Normal Baffles)
Look, we don't need down shifting to be zero. We just want it to be low enough that performance isn't impacted. So, we want an underfilled baffle, just not too underfilled. So if that doesn't look like a rectangle with elliptical arcs on top, what does that look like? Easy, it looks like a truncated ellipse:
ellipsebaffle.PNG
Like the truncated circle, you just cut the ends off an ellipse — again, how much you cut off is up to you. Now, in the real world, it will never be precisely this shape; things like gravity will distort it since the underfilled down is not able to provide as much resistance. But it is as close as you can get. To be honest, this shape is pretty close to the "capped rectangle" shape people tend to use — usually it's within a few percent. But that few percent is part of the reason that quilts tend to come out with slightly different dimensions than the maker intended. Using this truncated ellipse should help with that (though we are all human, so it'll never be exact). Another minor benefit is that it uses slightly less fabric than the capped rectangle, and so will cut down on weight.
Now, one thing there isn't much point in doing with these baffles is "overfilling" them. Since they are not completely filled to begin with, all overfill accomplishes is changing the shape so that it is more circular. Now, there is nothing wrong with changing the shape like that, and it will cut down on down shifting, as you intend. But if you are going to do that, use a different shape to begin with! Keep the perimeter of your baffle shape the same, but make it slightly taller and slightly less wide overall. That way it comes out the dimensions you expected, instead of the overfill throwing off your measurements.
2. Shapes and Performance
Before I get into designing baffles for quilts, I need to clear up some points on how heat works, and how baffle shape affects performance. There are some pretty major misconceptions here, and it is important to clear them up before continuing.
2a. Major Misconception
People tend to use the amount of of down in a baffle of a given width as a proxy for how warm the baffle is. For example, take two baffles: one is a circular baffle, in a sewn-through construction. Since this baffle is a perfect circle, the width and (center-of-baffle) height are equal. For our sake, lets say that they are both equal to two inches. Now, compare that to a perfectly rectangular baffle. We'll keep the width of our rectangle the same as the circle, so 2 inches. But we'll make the height 2π inches, so that both shapes hold the same amount of down. This also means that both shapes have the same average height. Now, since the edges of the circle are so thin, those will be cold spots on the quilt. But on the other hand, the center of the circle is taller than the rectangle, so it will be warmer in the middle. Sure, the circle baffle would be less comfortable, but both would allow the same overall amount of heat to escape, right? Wrong. Heat is lazy, and will escape though the easiest route. That means that the edges of the circle allow more heat through, and makes the rectangular baffle 19% warmer overall! Before I get too much into the maths, there are two key points I want to cover.
- The more rectangular the shape, the warmer the baffle is relative to a curved baffle containing an equal amount of down. The trade-off is that rectangular baffles experience the most down shifting.
-The relationship between warmth and how rectangular or curved the baffle is is not linear, but inverse. Moving from a sewn-through baffle to really short baffle walls — even as small as 1/8 of the center-of-baffle height — has a huge impact on performance. But as that baffle wall height gets closer to the center of baffle height, there are rapidly diminishing returns.
2b. The Maths
Things get pretty dense here. If you believe me and have no interest in how it works out, feel free to skip to the next section. Otherwise, stick around for some fun! This is also a complete copy/paste from my previous thread, so skip ahead if you already read that one as well.
For those that remain: you are such a nerd! Anyway, some background on how thermal performance is measured. As consumers of quilts, usually all a manufacturer tells us is the "temperature rating", which is somewhat arbitrary. Where I most commonly encounter actual units expressing thermal resistance is home insulation, for which R-values are typically provided. This is also typical in sleeping pads. Since they are what I am most familiar with, I will use R-values throughout this post. Here is the US, R-values are expressed in (h*°F*ft2)/BTU, and are often provided per inch of material depth. The numbers themselves don't have much intuitive meaning, but they are easy to compare. Something that is R8 blocks twice as much heat as something that is R4, and if you put two R4 rectangles on top of each other you get an R8 rectangle. Using the National Institute of Standards and Technology Heat Transmission Properties of Insulating and Building Materials Database, I was able to determine that duck down has an R-value of approximately 3.93 per inch of down (note: this seems a little low to me; if anyone has better numbers I'm happy to take them).
To find the mean R-value for some assemblage of material, one needs to take the mean of the R-values weighted by there areas. But not the geometric mean, which is the kind of mean that we are used to (and would make the previously mentioned rectangular and sewn-through baffles equivalent). Since R-values represent a rate, we need to take the harmonic mean weighted by the areas instead, which is the inverse of the sum of the inverse R-values.
While the harmonic mean works great for discrete shapes, our baffles are curved. That isn't a problem though, all we need to do is take sum up the R-values from arbitrarily small slivers from under the curve to find an approximate area... wait a second, that's the definition of the antiderivative! That's right baby, it's calculus time, and we're breaking out the integral! I bet this isn't where you were expecting this to go. Don't run away yet though: I know not everyone is as excited about math as I am, but the good news is that we get a pretty simple equation at the end.
Since we are dealing with circles, the maths are actually pretty easy. The equation for a semicircle in the plane centered at 0,0 is:
CodeCogsEqn.gif
For our baffles, we can determine the R-value by multiplying that equation by our R-value per inch value (3.93), doubling it (as we have a full circle instead of a semicircle), inverting it, taking the antiderivative over the length of our baffle, dividing by the length of the baffle, and then inverting again. Using our w and h values, we get the integral:
CodeCogsEqn(1).gif
Thankfully, that works out to be equivalent to the much simpler
CodeCogsEqn(3).gif
Now we can use this equation we can use to determine the performance of our baffles. Plugging in w=h=2 for our sewn through baffle, we get that the R value is almost exactly 5, compared to the rectangle which would have an R value of 3.93*π/2, or about 6.2. That's where we get the 19% difference in performance! Of course reality would never work out quite so precisely, as even the fabric has enough of an R-value to cut that percentage to 15% or so. Even considering that, that is enough of a difference that a blanket with smaller box baffles can match the performance using less fabric and down, and thus being lighter than the stitch-through baffle (still more time consuming though).
3. Designing Baffles
Now that we covered the basic shape of baffles, I can talk about how to design them for use in your quilt. I'll start with normal baffle design, before getting into my clever solution I'm sure you are all waiting for.
3a. Normal Baffles
We talked before about wanting to keep down shifting to reasonable levels, if not exactly zero. That means we don't want our shape to be too much wider than it is tall, because the wider it is, the more shifting there will be. Of course, you don't want it to be too narrow either, because then your quilt will be too heavy. This is common knowledge, and we can rely on common knowledge too for a range of height to width ratios that provide acceptable levels of down shifting that minimize weight. In general, people seem to like baffles about 2x wider than they are tall, sometimes as much 2.5x or 3x. Now, this is the height to width ratio of the final baffle — since your baffle comes from cutting the ends off an ellipse, the ellipse itself must be wider. Exactly how much wider is a delicate balance: if you make it too wide, then the baffle becomes more rectangular and has better performance, but also is unstable and allows more down shifting; conversely, narrower ellipses have worse performance.
I'd like to say there is a right answer here, but there is not. We can measure the amount of down shifting by the percentage underfilled the shape is relative to the truncated circle with the same perimeter (there is a little trigonometry, but its not that hard). And we can measure how warm the baffle is. But exactly how warm it needs to be, how much down shifting is acceptable, and how light the quilt must be will vary by person. Most people should just follow the rules of thumb that people generally follow for baffle wall heights and baffle widths and whatnot, just with a slightly different shape. Now, if you use those rules of thumb you can fix some variables, and then you can do some very mathematical optimization if you want (see Part II), but you need to make those somewhat arbitrary choices first. This is not the case in my clever solution though.
And again, as a quick reminder, there is not much point in overfill when using these shapes.
3b. Clever Baffles
Alright, you stuck with the post so far, so here is the really game changing stuff. A quick reminder though: this is still just theory, and I hope to build a demonstration quilt using the baffle outlined here later this month.
Let us go back to our truncated circle shape. The problem with that shape is that the baffle walls weigh too much. But what if there was a way to reduce the baffle wall weight? A way that you cannot use on underfilled shapes? Lucky for us, there is. Let us look at the functions of a baffle wall. In general, it serves two functions:
-To hold the shell material in place, so that you... have baffles.
-To prevent the down from shifting between the baffles.
Here is the key insight: with a truncated circle shape, down does not want to shift between the baffles. That entire second function is moot. But why is that, and how does that help us?
First, the why. With a typical, underfilled baffle, down shifting is a fact of life. Since the shape isn't completely filled, the down offers no resistance. Imagine if you have two baffles, side by side, both about 80% full relative to a truncated circle. Also imagine that the baffle wall material, whatever it is, allows down to pass through it easily. Now, if you grab the baffle from the side and give it a shake, it is easy enough to make it so that one baffle that is 100% full, while the other is 60% full. With time and effort, you can shift the down back so that they are both approximately equal again, though without a scale you'll never get it exactly right. Even if the spread becomes 75% in one and 85% in the other, that is a pretty major difference in both shape and performance, and entirely likely to happen when in use. This tells us it is important that elliptical baffles need baffle walls that not allow down to pass through.
On the other hand, if both baffles are at 100%, they want to stay that way. It is possible to force one baffle to become overfilled at the expense of the other, but that would put the overfilled baffle under pressure. The down would want to shift back so that it is even, and some random jostling of the quilt will allow that to happen. To increase this tendency, you can add some down so both baffles are overfilled. This fact allows the use of down-permeable materials for truncated circle baffles. This is a big deal, as it allows us to save a significant amount of weight.
One thing we can do is take our regular baffle material, like noseeum netting, and cut some pretty big holes in it. How big, and what kind of holes? The exact limits are anyone's guess, but I think it will be very easy to make them 4x lighter, or even more. Imagine taking the baffle wall material, and punching a bunch of circles in it, about as tall as the baffle wall itself is, like so:
perfbafflewall.PNG
This already reduces the material weight by about a factor of 4, and that is significant. Remember when I said that these truncated circle shapes use about 3x as much baffle wall material as traditional baffles? This implies that using a truncated circle baffle with this kind of baffle wall would result in a lighter quilt than the regular baffle equivalents, while being more stable and allowing less down shifting! That is really cool! It will take some experimenting to see exactly how much we can cut from the baffle wall while still holding the shell firmly in place, but this is a really promising area for research.
Another neat fact about this baffle: unlike the elliptical baffle, which has a few "rule of thumb" aspects, this baffle is easy to optimize! The only factor you need to choose is the center-of-baffle height, and the rest follows from your material choices! For more information, see Part II
Of course, theory is one thing. Reality is another. Like I said, I'll be constructing a quilt using this theory later this month. I'll let you all know how it goes.
4. Conclusion
Thanks for reading my thoughts on baffle shape. I hope you all view it as as significant improvement on my first attempt. Again, see Part II for more on baffle optimization. And before you ask about differential baffles, that will be the subject of Part III, which will be posted in a week or so. Here is a quick teaser on shape:
diffbaffle.PNG
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