Alexander's choice of angle 45 was easy- Because it was the most optimal one.
This is because any less and the water would fall out before reaching the maximum bottom of the wheel, thus losing out on some of the gravitational potential energy.
And any more and some of the water will be leftover even when the bucket reaches the lowest point, making the apparatus waste energy by again bringing the water up.
"That's all there is to the milling wheel- just take some wood, make them carved on the inside to make them like buckets, attach each of them at 45 degrees with one other, and then hold the entire thing together with thick wooden panels," Alexander finished explaining the most important structure of a waterwheel.
"*Nod*," Diaogosis signaled the understanding with a gentle nod.
"Then there is the transmission system, which is the thing that will transfer the rotation of the wheel into work," Alexander then moved on to the second topic as he indicated with his fingers to flip the page.
"The transmission will be pretty simple, just a log which goes through the center of the milling wheel, which will be attached to the structure using large, thick wooden spokes."
Alexander then twirled his fingers as a visual aid and continued, "So as the wheel spins, the cylindrical structure spins with it. And anything could be attached to the other end to make that spin with it too."
Diaogosis very attentively listened to these, taking mental notes as Alexander said, who then finished his first point by suggesting, "These spokes will go through both ends of the log, and attach itself to two points on opposing sides of the wheel. They will be the ones making the log spin with the wheel and so needs to be made of very good lumber.
"We will also need to use thick lumber then," Diaogosis subconsciously added, as he thought not only about the spokes but also about the challenges of drilling such large and deep holes through wood without cracking it.
"Use thick ropes to tightly bind the log together," Alexander suggested, understanding the artisan was worried that drilling such large holes might compromise the structural integrity of the wood.
"And once iron becomes more available, we will have the transmission fully out of steel," Alexander promised.
If anyone would have said this Diaogosis would have either politely smiled and nodded, while calling the man a lunatic in his mind, or had just outright said so.
But after what Alexander had displayed he was capable in such one short month, Diaogosis knew if Alexander claimed so. it was certainly possible and so eagerly waited to see how Alexander would do it.
So he smiled lightly, and said, "Then I look forward to it, my lord."
"Mmm, hopefully, I will be able to show it soon," Alexander replied with the same smile.
"Now, about the undershot waterwheel," Alexander then moved to the next diagram, as he bought out another piece of paper, "It will be like this and placed on rivers, and other horizontally moving bodies of water."
Alexander then handed it over to Diaogosis so that the man could better read it and said, "Here is how to build that."
As the stonemason laid his eyes on a waterwheel placed on the banks of a river, the classic picture one imagines when thinking of a waterwheel, he found the blueprints to be much more complicated.
There was much more text and even some very complex math.
Well complicated for him, as evidenced by the scrunched-up eyebrows and the confused gaze.
Alexander knew the maths of this time was not advanced enough for most people to be able to read this and so decided to help out.
"First, you will need to measure something called the 'head'. This is basically the distance the water would have fallen if it was a spring and not a river," He began.
"To find that, the primary thing that you will need to know is the speed of the flowing water that will be hitting the wheel."
"This can be calculated by setting two strings of known length near the banks of the water. Let's say it's 100m. Then corks or chips of wood can be thrown on the water and then measure the time taken to travel between the strings. Afterward, simply dividing the distance by the time should give you the speed of the current." Alexander spelled out the experimental procedure.
And then quickly added, "Oh, you will need to do this several times at several points along the stream. and take the average velocity. And remember to do it meters. Not in feet or centimeters." He emphasized.
Diaogosis only silently listened and nodded, understanding most of it, though not why he would need to repeat the procedure several times, and take the average.
This was normal as things like compensating for experimental errors were concepts literally thousands of years of his time.
But for now, he kept his mouth shut and just listened.
Alexander was still speaking, "Once that is done, you will need to take that number and multiply it with itself. And then divide that number by 19.62. That will convert your velocity into the so-called 'head'."
Alexander did not explain where these seemingly arbitrary numbers and procedures came from.
And Diaogosis did not bother to ask.
And even if he did, Alexander would have just instructed him to follow the procedure.
Because the derivation of the formulae used here- v2/2g (velocity squared divided by twice of gravity) would take even a college student some ten-twenty minutes to understand, much less a man with the math skills of a middle schooler.
Another point of consideration was taking g = 9.81m/s2.
Alexander had not measured the gravity of this planet, but he felt that gravity here seemed identical to earth and decided to plug in that value.
After giving Diaogosis some time to digest the 'heavy' maths, Alexander started again, and then spent almost the full next hour explaining in detail the rest of the mathematical procedures in its entirety.
"So, the optimum diameter of the milling wheel will be between three to six (3-6) times the head." Alexander drew attention to that fact, saying, "Any smaller and we will not be using all the energy of water. And any bigger, we will not be getting any additional benefit."
'Just multiply the head with six,' Diaogosis simplified the procedure to himself, as he loudly said, "Understood my lord."
Whatever Alexander said, Diaogosis took as gospel, as, for all he cared, the goddess Gaia might have whispered them to the pasha.
"Okay, next is the spacing between the paddles," Alexander then moved to the next topic, giving the answer, "That will have to be equal to the head."
pαпdα Йᴏνê1,сòМ In the overshot watermill, the buckets were joined together, with no gap, but here they needed to be spaced.
Alexander then reminded, "Oh, remember, here we will have to use paddles and not buckets."
Diaogosis needed no explanation why as the reason for this was obvious, the procedure of making the bottom heavier would be the very antithesis of how a water wheel worked.
It would only make the waterwheel heavier and obstruct movement.
"And lastly the waterwheel needs to be submerged into the river equal to the head length."
With all these instructions, Alexander finally finished the topic that was a waterwheel, making Diaogosis feel his head heavy with all the math.
"I...I will strive," The stonemason felt this simple structure will not be as easy as it looked to build.
"Mmm, that will be the basics. And I will leave the rest to you," Alexander decided to let Diaogosis figure out the rest of the details on his own.
Done with the waterwheel, Alexander then moved on to the thing that would actually do the work- the structure that would be used to crush the clinkers.
To show this, he brought out a third blueprint, and said, "This will answer your question of how to crush the clinkers."
Diaogosis looked down at the new paper and found the new drawing to be much simpler than what he had expected.
It was simply the picture of a huge drum whose diameter read 3m, about one story, with a length of 10m.
And the only thing that was slightly different was that the walls of the drum seemed to have small internal horizontal protrusions.
Below the label read 'Ball Milling machine'.
"The grinding procedure is simple," Alexander reassured, "We will put the clinkers inside and then add many small iron balls with it."
"So, when the drum rotates, these iron balls will rise up riding on those small platforms and then fall from above once it reaches a certain height, smashing against the clinkers and breaking them apart."
Alexander explained.
"Oh! That's ingenious," Diaogosis praised with repeated nods.
"....." Alexander only lightly smiled and gave one last bit of advice, "Make sure to tilt the drum at a slight angle. It will help in the milling."
"I will be sure to remember it, my lord," Diaogosis nodded.
"Okay, that's all there is to the construction for now," Alexander said, though the ominous 'for now' did not escape Diaogosis.
'There's more? What could there be even more? The stonemason dreaded and hoped he was reading too into this.
But Alexander pretended to not see the look of slight horror on the man and simply instructed, "You are to first build a small 5m diameter waterwheel as a prototype. And then I will give you something additional to build." Alexander mysteriously added.
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