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Use Back-of-the-Envelope Calculations to Save Time and Understand Your Problems
When I was in engineering school, one of my professors told the class that an engineer is only as good as their approximations. Over 20 years of working on packaging designs, I’ve found that holds true for many situations we come across in designing bottles for different applications. While computer simulations and finite-element techniques have become more widespread and faster, there is still a use for simpler approaches that can get us to a close answer using only some basic property data, some scrap paper, a pencil and a calculator.
Even basic tools like these can save us time (and money) on running simulations for designs that are unlikely to work, or on carrying out actual trial-and-error approaches in the lab. They can also help us to better understand how the different factors interact, and arrive at a first approximation for a design that we later take into more complex simulations. In this post, we’ll look at a situation which lends itself to some back-of-the-envelope calculation, and also think about when it makes sense to bring more sophisticated analytical power to bear.
Two common performance specifications for plastic bottles are burst pressure resistance and percent growth under pressure. The question we often have is what wall thickness is needed to meet those specifications; knowing the wall thickness allows for an estimation of the package weight and therefore its cost. The goal for this kind of calculation is not high precision, but to come close enough to get to the next step, whether that’s going into a finite element model, or just to do some cost calculations based on the approximate weight of the bottle.
The first step in estimating the thickness is understanding the physical situation, and knowing how the factors involved relate mathematically. A diagram and the appropriate equation for stress in the hoop direction (a rearrangement of Barlow’s formula), are shown in Figure 1. The derivation of this equation comes from the problem of stress in a thin-walled cylinder, which was originally solved for structures such as steam boilers, but that is also applicable for a pressurized plastic bottle. The assumption of the wall thickness being much smaller than the diameter (D/t>20) is a good one, since for a 2-L bottle the result is generally over 300; this means that we can ignore the different stresses from inside to outside the bottle sidewall.
Given a bottle of diameter D that has to withstand a pressure P, we want to know the bottle thickness t that we need to design into the bottle. To work that out, we also need the maximum force required to pull apart the material we’re using to make the bottle, in this case PET. Figure 2 contains the stress-strain curves for PET that has been highly oriented in the hoop direction, and well oriented in the axial direction as well. This is data taken from tensile tests in the axial (red trace) and hoop (blue trace) directions.
For a burst-test specification, we need the stress at break in the hoop direction, which is about 24,000 PSI. A typical situation to work out might be for the 2-liter bottle mentioned above, which is about four inches in diameter, and might have a 140-PSI burst specification. Now, we’re ready to plug in our numbers and get an answer. In this case, we get:
If our 2-liter bottle has a surface area below the support ledge of 950 cm2 and a finish weight of 3.8 grams, we can work out an approximate preform weight:
950 cm2 · 0.30 mm · (1 cm/10 mm) · 1.335 g/cm3 + 3.8 g = 41.8 g (2)
The weight at 42 grams doesn’t factor in small amounts of extra material in the base and in the neck, but it’s not a bad estimate for a few minute’s work. It also shows why the bottle wall thickness should drive the package weight in most cases; it’s what drives the physical performance. The approach is also, of course, nothing new; the original patent by DuPont for a PET bottle shows the same formula and similar calculations; see page six on the patent.
While the approach above is good for evaluating the simple case of right cylinders under pressure, we know that the real world is more complex. For example, the petaloid base that is now common on pressurized bottles is best understood using finite-element techniques. That’s because the base has changing thickness, material properties, and geometry as we look at different points of the foot.
That said, for simple cases, or for first approximations that we want to take into the next level of analysis, the quick calculation still has its place. Running a finite element model usually takes a few days at least to set up and run, and actual prototypes often take several weeks to produce and test; either option usually costs several thousand dollars to accomplish. Doing some quick calculations can either give us more confidence in committing to those steps, or save us from trying something that is doomed to failure from the start. In the next entry in this series, we will look at the amount that we can expect the bottle to expand under pressure.