The thought experiment which we are here calling the Optimization Problem is intellectually interesting and is leading to some other findings. We'll take a break and chase one of those findings here.
If you think of the paper machine in the abstract, it is really a piece of pipe. A very fancy and complicated piece of pipe, but nevertheless, a piece of pipe.
How do we choose a diameter for a pipe line? We usually design pipelines to allowable Reynolds Numbers--an expression of the friction and boundary layer effects on the internal walls plus the volume desired to be moved within a certain period of time. This is complicated by pump sizing and head requirements.
Doing some rough calculations, a nominal 10 inch (25 cm) pipe can carry 56% more than a nominal 8 inch (20 cm) pipe, at a given flow velocity. If a designer is being conservative about their Reynolds Number selection and pump sizing, they may choose the larger pipe, "just to be safe."
Today, other issues come into play, especially our overall lifetime environmental and carbon effects. There is a possibility here to trade one environmental effect for another, reaching a minimum if properly designed.
One know from work we did long ago at the Light Green Machine Institute that process equipment represents about 1/3 the total weight of any process, or for every kilogram of process equipment there are 2 kilograms of structural steel, concrete and pilings. We used to not pay much attention to these structural materials--they were relatively cheap.
However, today, it is customary to look at the environmental effects of producing and placing structural materials in the construction of any facility. We think the pulp and paper industry has some catching up to do in this area, but the awareness is just on the horizon.
We say all the above to say this: on a qualitative level (we haven't figure out the math yet, so we can't get into the details), it looks like pipe designing pushing the limits of the Reynolds Number in any given application may yield a lower overall environmental footprint for any given facility.
Some exact cases need to be examined, for such a design stance may possibly up the pump energy requirements, but this still may produce a favorable offset.
We have more work to do here, but we thought we would share this as we have discovered it. If you, dear reader, know of any other work along these lines, we would appreciate hearing about it.