Bulk Density and Spherical Packing (need to increase weight of fixed-volume cylinder)

gitano

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I have looked extensively for a solution to the below defined problem, and have found none. Let me save "you" time: I am NOT seeking the answer to "how many balls can I fit in a cylinder". I am well aware of the "64 to 74%" rules of thumb. Neither of those 'solutions' solve my problem.

The problem stated as clearly as I can is:
1) I have a well-defined, uniform, unvarying volume. (I need to increase the WEIGHT of a cylinder.)
2) I need to add as much weight to that cylinder as I can using spheres of pure lead. (RhoPb = 11.34g/cc)
3) It is obvious that the smaller the sphere I use, the higher the bulk density will be. However, there are practical reasons (handling problems) for not using spheres smaller than a given size.
4) The 'handling problem' does not have a "threshold". Meaning that regardless of the 'starting' sphere diameter, as sphere diameter gets smaller, the handling problems increase.
5) The 'issue' is that I want to maximize the weight of the cylinder, but I don't want to have to deal with the handling problems associated with "small" spheres. In other words there is a point of diminishing returns, where whatever gain is realized in increased weight is offset by handling issues.

I want to be able to calculate BULK density in a FIXED SIZE cylinder as a function of sphere DIAMETER when sphere DENSITY is constant. The cylinder diameter and height are >> than the largest sphere diameter.

Let me restate the problem in "simple" language:
How does sphere diameter affect bulk density in a given volume?

Thanks!

Paul
 
I have looked extensively for a solution to the below defined problem, and have found none. Let me save "you" time: I am NOT seeking the answer to "how many balls can I fit in a cylinder". I am well aware of the "64 to 74%" rules of thumb. Neither of those 'solutions' solve my problem.

The problem stated as clearly as I can is:
1) I have a well-defined, uniform, unvarying volume. (I need to increase the WEIGHT of a cylinder.)
2) I need to add as much weight to that cylinder as I can using spheres of pure lead. (RhoPb = 11.34g/cc)
3) It is obvious that the smaller the sphere I use, the higher the bulk density will be. However, there are practical reasons (handling problems) for not using spheres smaller than a given size.
4) The 'handling problem' does not have a "threshold". Meaning that regardless of the 'starting' sphere diameter, as sphere diameter gets smaller, the handling problems increase.
5) The 'issue' is that I want to maximize the weight of the cylinder, but I don't want to have to deal with the handling problems associated with "small" spheres. In other words there is a point of diminishing returns, where whatever gain is realized in increased weight is offset by handling issues.

I want to be able to calculate BULK density in a FIXED SIZE cylinder as a function of sphere DIAMETER when sphere DENSITY is constant. The cylinder diameter and height are >> than the largest sphere diameter.

Let me restate the problem in "simple" language:
How does sphere diameter affect bulk density in a given volume?

Thanks!

Paul
Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/for
 
I did read the "Read before Posting" thread BEFORE I posted. Let's have that here for reference. I acknowledge that your response seems to be your standard response to all posts here. As I read through this website's other posts, looking for an answer to my problem, I found your standard response unhelpful (at best) to most of the other people asking for help. I find it just unhelpful to me. However, in the name of 'completeness', let's examine the contents of the thread you demand everyone "read first" (even if they have).

Please take the time to read the following before you make your first post. It will help you to get your math questions answered promptly and in the most helpful manner.

Don't post a list of homework problems. Please show us any work you have done towards answering the problem, and try to explain what specifically is giving you trouble. Also, please try to limit the number of questions you ask to a reasonable number per day. There's no exact limit, but try to be polite and not abuse the service.
This is not a "homework problem".

Try to use halfway-decent English. No, this isn't Englishhelp.com, but you'll get more help if you spell correctly. Actually, I don't care if you spell a few words wrong, but the IM speak will probably cause some people to skip over "ur" question. If we can't read it, we can't help you :smile:
If you think my explanation was "less than halfway-decent" English, please feel free to correct were appropriate, or ask for clarification of my poor English.

Be nice. Everyone here is a volunteer, so treat them with respect and we will (probably) be nice in return. If we aren't... then it's probably just been a long day. If you have any conflicts, please contact me personally rather than starting a flamewar on the forum.
I thought I was being "nice". I usually respond in the spirit in which I am addressed.

Don't spam. Okay, so the spammers aren't going to read this anyway, but it's probably good to have it in writing.
The OP was not spam by any reasonable person's definition.

Post to an appropriate category. Questions about derivatives (calculus) do not belong in "Beginning Algebra". Questions about finding the area of a circle (geometry) do not belong in "Other Math". Please show appropriate care and consideration.
I chose what I thought was the most appropriate category. If you think it should be in another category, please feel free to 'move' it there.

Preview or edit your posts for clarity. When composing your message, you may include formatting that doesn't "take" when the message is finally posted. For instance, the forum script strips away spaces at the beginnings of lines. Kindly "edit" your post to correct errors if they occur, and "preview" your posts to minimize errors. (For formatting advice, try "Formatting Math as Text".)
I didn't provide any formulae or any other "math text".

Post the complete text of the exercise. This would include the full statement of the exercise and its instructions, so the tutors will know what you are working on. If there is a graphic or table or some other non-textual information necessary to the exercise, include a detailed description.
There is no "exercise". I wasn't aware that this was a "help high school students do their math homework" site. If so, please let me know, and I'll take my question elsewhere.

Show all of your work. If you've shown no work at all, the tutors may assume that you're needing help getting started, and may suggest only how to do the first step. Even if you're asking only about the very end of the solution process, still include all of your intermediate steps. Errors may have occurred earlier than you'd realized; correcting part (b) may clear up your confusion on part (d).
There is no "work". I felt,, (apparently incorrectly), that I explained that I had looked for a solution to this problem without success. The "work" one would do is take the "Edison approach" and fill the described cylinder repeatedly with varying sizes of spheres, record the bulk density for each, and decide what one 'liked'. Taking that approach to engineering eliminates the need for both engineering and engineering math. I thought I might try to avoid the brutish trial and error approach.

Have patience. There is no paid staff waiting on-hand to give instant replies. Many of the volunteer tutors have "real" jobs, and they all have to sleep from time to time. The people "viewing" your posts may be fellow students. Please don't be offended if there are "views" but no replies. It may take hours, even days, for a tutor, qualified in your topic's area, to respond.
I have not "demanded" that anyone respond AT ALL, let alone "quickly".

Don't post links to image-hosting sites that add extra images to their pages. We have no control over these extra images, and some of them violate forum policy. (This is a family-friendly site.) Pick an image-hosting site that does not add extra images, or reference the hosted image using IMG tags instead of posting an URL. You may also upload images to the FreeMathHelp server. (For questions, see the FAQ.)
I did not post any links.

Don't create duplicate threads or posts. There's no reason to do that.
I have posted ONE post.

Well that's all I can come up with right now. I know there must be other useful things to have in this post, so go ahead and reply or PM me if you would like to add something.
I have nothing to add.

Thanks for visiting, and please come back!

Ted

Please advise which of the above components of the "Read before Posting" thread that I violated with my OP.

For clarity's sake, I'll add the following information. I don't normally do this for several reasons, including that when I provide additional information for clarity's sake I invariably get whines about being too "verbose". (Although the whiners usually don't have sufficient command of their native tongue to use the term "verbose".) However, after 20 years "on the net", I realize that "you're damned if you do, and your damned if you don't". So, having been 'damned' for NOT doing, I'll "do" with expectations of the 'next level' of 'damning'.

1) I'm not a student looking for help with my homework.
2) I'm 67 years old, a retired research scientist (Ph.D.) with a 'little' problem that I wanted a little assistance with. I really don't need a lecture on how to ask for help, and as you can see, don't really appreciate being patronized. (Even more so when the admonition was inappropriate/not factually based.)
3) If you can't/won't help, you might consider the absolute necessity of responding.
4) If you - or ANYONE - needs more information about the problem, please ask. I'm only too happy to provide further clarification.
5) If I have "read" this place wrong, and it IS just a place to help high-school students do their homework, please accept my sincere apology. (NO sarcasm there. Seriously.) I won't waste 'your' time further.

Paul
 
The problem stated as clearly as I can is:
1) I have a well-defined, uniform, unvarying volume. (I need to increase the WEIGHT of a cylinder.)
2) I need to add as much weight to that cylinder as I can using spheres of pure lead. (RhoPb = 11.34g/cc)
3) It is obvious that the smaller the sphere I use, the higher the bulk density will be. However, there are practical reasons (handling problems) for not using spheres smaller than a given size.
4) The 'handling problem' does not have a "threshold". Meaning that regardless of the 'starting' sphere diameter, as sphere diameter gets smaller, the handling problems increase.
5) The 'issue' is that I want to maximize the weight of the cylinder, but I don't want to have to deal with the handling problems associated with "small" spheres. In other words there is a point of diminishing returns, where whatever gain is realized in increased weight is offset by handling issues.

I want to be able to calculate BULK density in a FIXED SIZE cylinder as a function of sphere DIAMETER when sphere DENSITY is constant. The cylinder diameter and height are >> than the largest sphere diameter.

Let me restate the problem in "simple" language:
How does sphere diameter affect bulk density in a given volume?

I have looked extensively for a solution to the below defined problem, and have found none. Let me save "you" time: I am NOT seeking the answer to "how many balls can I fit in a cylinder". I am well aware of the "64 to 74%" rules of thumb. Neither of those 'solutions' solve my problem.
The reason for those "rules of thumb" is that the topic of "sphere packing" is surprising complex and "bigly" unsolved. Perhaps somebody doing current research in the area might be willing (for a fee) to design a computer program or something for you, but I kind of doubt that there is a nice fixed formula that a volunteers here has been hiding from academia. :shock:

Meanwhile, there may be some volunteers here who would be willing to explore the topic with you. If you're open to that, then please reply with your current efforts, so we can "see" what you're attempting. Thank you! ;)
 
I did read the "Read before Posting" thread BEFORE I posted. Let's have that here for reference. I acknowledge that your response seems to be your standard response to all posts here. As I read through this website's other posts, looking for an answer to my problem, I found your standard response unhelpful (at best) to most of the other people asking for help. I find it just unhelpful to me. However, in the name of 'completeness', let's examine the contents of the thread you demand everyone "read first" (even if they have).



Please advise which of the above components of the "Read before Posting" thread that I violated with my OP.

For clarity's sake, I'll add the following information. I don't normally do this for several reasons, including that when I provide additional information for clarity's sake I invariably get whines about being too "verbose". (Although the whiners usually don't have sufficient command of their native tongue to use the term "verbose".) However, after 20 years "on the net", I realize that "you're damned if you do, and your damned if you don't". So, having been 'damned' for NOT doing, I'll "do" with expectations of the 'next level' of 'damning'.

1) I'm not a student looking for help with my homework.
2) I'm 67 years old, a retired research scientist (Ph.D.) with a 'little' problem that I wanted a little assistance with. I really don't need a lecture on how to ask for help, and as you can see, don't really appreciate being patronized. (Even more so when the admonition was inappropriate/not factually based.)
3) If you can't/won't help, you might consider the absolute necessity of responding.
4) If you - or ANYONE - needs more information about the problem, please ask. I'm only too happy to provide further clarification.
5) If I have "read" this place wrong, and it IS just a place to help high-school students do their homework, please accept my sincere apology. (NO sarcasm there. Seriously.) I won't waste 'your' time further.

Paul

Please read:

https://arxiv.org/pdf/1203.3373.pdf

You'll find "bounds" of solution.

If that is not sufficient, my advise would be go to local university/college and consult with a professor of mathematics. If s/he cannot solve your problem, s/he might be able to guide you to the correct consultant.
 
Thank you for taking the time to respond.

I have read the above paper by Mughal, and his subsequent one including simulations. I am 'dissatisfied'. (That's not criticism, just emotional response to not getting the solution to a problem.) I am 'uncomfortable' with "independent of sphere diameter" when coupled with lower and upper boundaries of 64 and 74 percent.

I can "perform the experiment", and will do so to 1) convince myself of the 'universality' of the 64-74% rule of thumb, AND 2) to test that rule when the ratio of the sphere diameter and cylinder diameter is greater than an order of magnitude. As a quick addendum:

Assuming a cylinder of diameter of 1 (dc=1) and length 10 (Lc=10), and spheres of diameter 1 (ds=1), I will be able to fit 10 spheres in the cylinder.
The total volume of the spheres is 10 * 4/3 * pi * (ds/2)^3 or 5.236. The volume of the cylinder is 10 * pi * (ds/2)^2 or 7.85. The ratio of the volume of the spheres to the volume of the cylinder is 5.236:7.85 or 66.6%. This is within the range provided by the "64 to 74%" rule of thumb. Continue to reduce the size of the spheres, and you approach the 74% figure of "ideal packing". However, continue to reduce the diameter of the spheres to "microscopic", and I question whether the difference in bulk density is only going to be 10%. (64% for "large" spheres, to 74% for "dust".)

For completion's sake, I'll report back here with the result of the trial and error exercise.

Paul
 
Thank you for taking the time to respond.

I have read the above paper by Mughal, and his subsequent one including simulations. I am 'dissatisfied'. (That's not criticism, just emotional response to not getting the solution to a problem.) I am 'uncomfortable' with "independent of sphere diameter" when coupled with lower and upper boundaries of 64 and 74 percent.

I can "perform the experiment", and will do so to 1) convince myself of the 'universality' of the 64-74% rule of thumb, AND 2) to test that rule when the ratio of the sphere diameter and cylinder diameter is greater than an order of magnitude. As a quick addendum:

Assuming a cylinder of diameter of 1 (dc=1) and length 10 (Lc=10), and spheres of diameter 1 (ds=1), I will be able to fit 10 spheres in the cylinder.
The total volume of the spheres is 10 * 4/3 * pi * (ds/2)^3 or 5.236. The volume of the cylinder is 10 * pi * (ds/2)^2 or 7.85. The ratio of the volume of the spheres to the volume of the cylinder is 5.236:7.85 or 66.6%. This is within the range provided by the "64 to 74%" rule of thumb. Continue to reduce the size of the spheres, and you approach the 74% figure of "ideal packing". However, continue to reduce the diameter of the spheres to "microscopic", and I question whether the difference in bulk density is only going to be 10%. (64% for "large" spheres, to 74% for "dust".)

For completion's sake, I'll report back here with the result of the trial and error exercise.

Paul
A BIG assumption in that 64-74% rule is that the size of the spheres (diameters) is constant. If you uses spheres with a distribution of sizes in the same experiment, the packing density will go up. There were some research in this domain.
 
Thank you Subhotosh Khan.

As I said above, I'll just 'do the work' and post it here.

Bulk%20Density%20Exercise_zpsscw1kcw9.jpg


On the left are the constants: Measured cylinder volume and measured pellet density: 8 ccs and 11 g/cc respectively.
Just for completeness, column "C" is the ratio of the diameters of the different pellet sizes to the diameter of the cylinder.
The numbers in column "K" (the difference between the actual cumulative weight of the pellets to the calculated cumulative weight of the pellets), verify the density of the pellets - 11.00 g/cc.
The data in the rest of the columns should be self-explanatory. The last column labeled "Ratio of absolute Density to Bulk Density" was what I was after.
The r2 value on the graph suggests that a linear model explains 87% of the variability in the data. 89% for a Q&D little exercise isn't bad. I made no effort to have the column of pellets "settle" in the cylinder, so this is "random packing". While I'm fairly confident that the relationship between sphere diameter and bulk density isn't linear, I think most of the remaining variation IN THIS DATA SET is due to 'operator error'. You'll note that the actual results in column "M" don't get to the 64-74% rule of thumb, but are consistent with comments in the publications regarding the results of "random packing" of spheres in cylinders. However, where the publications stopped - probably due to editorial heavy-handedness for "rigor" - was an empirical model of what that relationship was. THAT is precisely why I posted the OP.

I have my 'answer'. I had to do the trial and error 'thing', which I had hoped to avoid, but got yet another example of "If you want something done, just do it yourself". I could have saved three days and all of the internet 'pleasantries'.

My thanks to those that took their time to respond.

Paul
 
Thank you Subhotosh Khan.

As I said above, I'll just 'do the work' and post it here.

Bulk%20Density%20Exercise_zpsscw1kcw9.jpg


On the left are the constants: Measured cylinder volume and measured pellet density: 8 ccs and 11 g/cc respectively.
Just for completeness, column "C" is the ratio of the diameters of the different pellet sizes to the diameter of the cylinder.
The numbers in column "K" (the difference between the actual cumulative weight of the pellets to the calculated cumulative weight of the pellets), verify the density of the pellets - 11.00 g/cc.
The data in the rest of the columns should be self-explanatory. The last column labeled "Ratio of absolute Density to Bulk Density" was what I was after.
The r2 value on the graph suggests that a linear model explains 87% of the variability in the data. 89% for a Q&D little exercise isn't bad. I made no effort to have the column of pellets "settle" in the cylinder, so this is "random packing". While I'm fairly confident that the relationship between sphere diameter and bulk density isn't linear, I think most of the remaining variation IN THIS DATA SET is due to 'operator error'. You'll note that the actual results in column "M" don't get to the 64-74% rule of thumb, but are consistent with comments in the publications regarding the results of "random packing" of spheres in cylinders. However, where the publications stopped - probably due to editorial heavy-handedness for "rigor" - was an empirical model of what that relationship was. THAT is precisely why I posted the OP.

I have my 'answer'. I had to do the trial and error 'thing', which I had hoped to avoid, but got yet another example of "If you want something done, just do it yourself". I could have saved three days and all of the internet 'pleasantries'.

My thanks to those that took their time to respond.

Paul
That old adage is specially true when you are looking for free work from volunteers. And that is why we encourage our students to do things themselves.

To improve the "quality" of your inference from the experiments, I suggest that you include a "sieve analysis" of your pellets to estimate the distribution of pellet size distribution.

Another control variable - shaking the tube after adding the pellets to the cylinder - could be added.

Over all very nice experiment. I have not come across any rigorous theoretical model yet.

This analysis is also very important for "hydro-fracturing".
 
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Assuming you are actually addressing your 'regular clientele', (lurkers and high school students), and not me personally; one of them interested experimentation in this subject, would be well-advised to consider your suggestions before pursuing this matter further. However...

Given that you quoted my post - and used the pronoun "you" after the quote - the more parsimonious assumption is that you are actually advising me about experimental design. You cannot imagine how humorous that is.

Let me try to clarify my interest level in this, one last time.

I have no interest in 'formal' research into this subject. I have only marginal scientific interest in the 'problem' at all. My 'problem' was, a purely practical matter. I needed to add some weight to a specific object to increase inertia. I had available to me lead pellets that had to fit in a specific space - the cylinder. I had a range of pellet sizes the smallest of which were about ~60X smaller than the diameter of the cylinder and a pain in the butt to "handle", (both now, and in the future), and some as large as ~25X smaller than the diameter of the cylinder, that were easier to handle but not as "efficient" in terms of increasing weight. I simply wanted the relationship between bulk density and sphere size. A relationship that anyone appreciating, and able to see, reality, would acknowledge exists. And by the way, in the end, I acknowledge that the difference in "efficiency" between the smallest pellets I have and largest I have, is not worth "worrying about". BUT... Neither is it "64-74%.

There may be no formal equation to describe my bulk density issue. HOWEVER, I would point out that the 64% to 74% RoT is not "formal" either, and everyone here seems only too happy to espouse its 'virtues'. A simple physical experiment, (as opposed to a simulation, that was performed in all of the citations), conducted with only a modicum of rigor, would yield another "rule of thumb" more useful and just as 'valid' as the 64-74% RoT. That level (64-74%) of BULK density is ONLY possible with the addition of considerable effort; as noted by the authors of most of the papers cited herein. I thought, maybe, someone had actually performed that experiment, and reported the results in some obscure place where I couldn't find it but someone here might have seen. I would also point out that while some of the cited authors noted that "random packing result is more likely about 50% "density", "y'all" was 'asserting' that the 64-74% RoT was what I "needed" to use. That was wrong, as I stated in the OP.

If I had known all that this was going to entail, I would NEVER have raised the issue. I could have expended a great deal less effort and received a great deal less 'grief', (here and elsewhere), by not being too lazy to just "do the work myself" to start. I am not too old to learn new lessons. I am appropriately 'schooled' on where I should, and shouldn't, seek answers to simple questions.

Paul
 
The poor guy. He had to do some work, and people did not meet his expectations. My heart bleeds.
 
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