View Full Version : Maybe OT A Math and Truck question

Stepside

02-13-2015, 08:53 PM

A friend of mine asked me the following question. If you roll a piece of 8-1/2 x 11 paper into a cylinder

8-1/2inches tall or a cylinder 11 inches is the enclosed volume the same? Doing "Gestimation Math" I came to the answer of "No". That is the correct answer. I will let you play with a calculator and get the percent of difference. So far so good.

Now, today driving from Kirkland to Kelso and back we saw a large number of Tank Trucks. So my question is why were all the trucks hauling petroleum equipped with oval tanks? Those hauling Milk were using cylindrical tanks. The compressed gas ones were understandably a cylinder as well. Those hauling bulk Cement are some goofy shape and some unlabeled ones were a kind of cylinder with a sag in the middle of the bottom.

After I get some answers I will explain why the question.

Thanks in advance

Pete

J Harp

02-13-2015, 09:03 PM

Ex gas passing tanker yanker here. I was told that the oval petroleum tanks allow a slightly lower center of gravity.

Milk tanks are round and without bulkheads or baffles to allow for proper cleaning between loads.

TGTool

02-13-2015, 09:06 PM

Some of that just may be where the company buys its trucks. Around here milk trucks are straight trucks with oval tanks. Semis are all kinds of shapes. Some skinny cylinders are because they're hauling liquids with high specific gravity so max weight takes a smaller volume. Lots of the trailers with the sag are for unloading, often bulk flour in these parts where there's a lot of wheat and it looks to me like many of them are equipped with compressed air to loosen the powders and get them flowing out. Any of those solids can bridge in particular conditions so the stuff in the upper part of the chamber is temporarily stuck.

Ohio Mike

02-13-2015, 10:14 PM

You can get more volume from a oval and you can stay within legal width and height. Fuel only weighs 6-7.5 lbs per gallon depending on type, water is 8.34 lbs, and milk is 8.6 lbs. Milk tanks are also insulated.

kf2qd

02-13-2015, 10:22 PM

Cement trucks are designed like a bag of cement being carried by it's ends. Design is also affected by the cone angle of a pile of the contents. Gravel, stones, flour all form a different angle cone. If you want the truck to self unload the bottom has to form an angle less than the cone angle for that material.

Lee Cordochorea

02-13-2015, 11:35 PM

The paper cylinder problem reminds me of a few years back. I had to find the ratio of height & diameter to minimize the ratio of surface area to volume. Without doing any math, I know for certain the 8.5" tall cylinder will have less surface area per unit volume than the 11" cylinder. :) I would ask which would have more surface area period, but that would be a truck question with a typo. :D

Paul Alciatore

02-14-2015, 12:25 AM

What?

If you form a cylinder with a 8.5" x 11" piece of paper, the cylindrical walls will be the exact same area no matter which way you bend it. That area is constant. But, the 11" high cylinder would have SMALLER ends (A = pi * r^2) than the 8.5" high one because the ends of the 11" tall cylinder will have an 8.5" circumference and a 8.5/pi diameter while the 8.5" tall cylinder will have ends with a 11" circumference and an 11/pi diameter. The larger diameter of the shorter cylinder will require larger ends so it will have the larger surface area.

I just did the math and there was an apparent error so I need to run through it again. Back in a minute.

The paper cylinder problem reminds me of a few years back. I had to find the ratio of height & diameter to minimize the ratio of surface area to volume. Without doing any math, I know for certain the 8.5" tall cylinder will have less surface area per unit volume than the 11" cylinder. :) I would ask which would have more surface area period, but that would be a truck question with a typo. :D

Paul Alciatore

02-14-2015, 12:35 AM

An oval does not have more area that a circle of equal length. It is a well known and rigorously proven mathematical fact that a circle encloses the maximum area for a given length. There IS NO shape that will have a greater area if the length of the outline is the same. Since the tanks are generally straight in their length, this principle applies to their volume too.

The oval shape would tend to equalize the pressure on the skin of the tank so the oval tank would be a bit stronger for a given size and thickness of the walls. Many water tower tanks also have a oval or flattened bottom for this reason. This applies to the bottom side of the tank only so you see the sides of those water tower tanks go straight up above that oval bottom. Apparently no one has applied this idea of straight up sides above the bottom to a truck or rail car tank. Probably because it would be too expensive to fabricate and also more difficult to clean.

You can get more volume from a oval and you can stay within legal width and height. Fuel only weighs 6-7.5 lbs per gallon depending on type, water is 8.34 lbs, and milk is 8.6 lbs. Milk tanks are also insulated.

winchman

02-14-2015, 02:07 AM

First pass with no checking yields:

8.5 high by 11 circumference has a volume of 81.8

11 high by 8.5 circumference has a volume of 63.1

Oil is too expensive to spill and clean up, and nobody cries over spilled milk, so they use a stronger truck for oil. The area of a circle increases by the square of the radius. Bigger diameter trumps for the same square area of wall surface.

Paul Alciatore

02-14-2015, 04:09 AM

Hummm, I get the same results.

First pass with no checking yields:

8.5 high by 11 circumference has a volume of 81.8

11 high by 8.5 circumference has a volume of 63.1

A more general question would be, given a constant surface area, INCLUDING THE ENDS, what proportions of length and diameter provide the largest volume for a tank (cylinder). I found to nine decimal places, a 1:1 ratio of length to diameter provides the largest diameter. I am sure with a little more work, I could prove this completely, but I suspect it is the solution. An outline of my work follows:

K = Total Area of the sides and ends - this is a constant

r = Radius of the ends

Vt = Total volume of the tank

l = Length of the tank

Area of one end (Ae):

Ae = pi*r

Area of both ends (Aee):

Aee = 2*pi*r

Remaining Area (Ar):

Ar = K - Aee

Ar = K - (2*pi*r)

Area of Sides (As):

As = 2*pi*r*l

Since the area of the sides is the remaining area:

As = Ar

2*pi*r*l = K-(2*pi*r^2)

Solving for l:

l = (K-(2*pi*r^2)/(2*pi*r))

l = (K/(2*pi*r))-r

Volume of tank (V):

V = Ae*l

V = pi*r^2*((K/(2*pi*r))-r)

V = (K*r/2)-(pi*r^3)

Inflection points (points where the change in volume changes from + to -):

V = (K*r/2)-(pi*r^3)

Differentiating:

dV = (K/2)-(3*pi*r^2)

Set dV to zero for zero slope:

0 = (K/2)-(3*pi*r^2)

3*pi*r^2 = K/2

r^2 = K/(2*3*pi)

r = (K/(2*3*pi))^-2 (square root)

Setting K = unity (1) (this allows us to multiply the results by whatever the volume actually is)

r = (1/(6*pi))^-2

and finally

r = 0.230329433

For the diameter

d = 0.460658866

And the corresponding length to keep the Area the same (K):

l = (K/(2*pi*r))-r

Again, setting K = 1 or unity:

l = (1/(2*pi*r))-r

and

l = 0.460658866

Thus, for a given surface area, the greatest volume occurs when the length and diameter are equal.

bob ward

02-14-2015, 05:32 AM

That takes me back. Those sorts of area/volume problems were typical high school calculus questions.

A.K. Boomer

02-14-2015, 06:10 AM

First pass with no checking yields:

8.5 high by 11 circumference has a volume of 81.8

11 high by 8.5 circumference has a volume of 63.1

Someone else stated the opposite but I believe this is correct, I can't verify the exact numbers because im incapable of doing so...

what I can do is offer this, take things to extreme,

say a 1" by 25" piece of paper, now fold it both one way and the other, in one way it's just a very skinny 25" long tube, very skinny,,,, as in close to 3/8" skinny,

the other way is a pretty good size 1" tall little mini swimming pool - about the size of a frisbee,

at least in my minds eye there is simply no comparison as to the extreme offset in volume, and Im going with Winchmans calculations as to how this goes, meaning if you use the short side of the paper for your height you will always increase the overall volume..

Stepside

02-14-2015, 09:56 AM

So far the answers are just about what I expected. As far as the paper is concerned, I should have stated that the ends are not part of the issue. But the answer from Paul A is also of great value. As to the Bulk trailers, the "angle of repose" is part of the answer.

If you work out equal volume you will find out that the ellipse will have a greater surface area compared to a circle.

So the next question is "Assuming using the same piece of paper, make equal side polygons and compare the volumes.

So why the question? Because a teacher friend of mine was trying to apply math to the "real world". He calls the truck part of the question an "extension". He would appreciate any suggestions as to other "extensions" related to the paper problem.

Thanks again for all the replies

Pete

Ohio Mike

02-14-2015, 10:41 AM

An oval does not have more area that a circle of equal length. It is a well known and rigorously proven mathematical fact that a circle encloses the maximum area for a given length. There IS NO shape that will have a greater area if the length of the outline is the same. Since the tanks are generally straight in their length, this principle applies to their volume too.

The oval shape would tend to equalize the pressure on the skin of the tank so the oval tank would be a bit stronger for a given size and thickness of the walls. Many water tower tanks also have a oval or flattened bottom for this reason. This applies to the bottom side of the tank only so you see the sides of those water tower tanks go straight up above that oval bottom. Apparently no one has applied this idea of straight up sides above the bottom to a truck or rail car tank. Probably because it would be too expensive to fabricate and also more difficult to clean.

Sorry that definitely didn't come out clear at all. What I was attempting to say is you can get more volume within the prescribed height and width limitations because the tank can be wider than it is tall. If you require the tank profile to be a circle then you can only make it as big a round as the smaller distance. Good point about the oval being stronger. Tank strength, unloading and cleaning requirements are major factors in design too.

boslab

02-14-2015, 11:49 AM

http://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx

Oh what fun that was, forgot how to do it now, aren't tankers oval to lower CG?

Mark

Paul Alciatore

02-14-2015, 02:13 PM

There's probably a number of reasons why they are oval. Don't discount the cool factor: they just plain look good. The round ones look antiquated. Sales appeal!

http://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx

Oh what fun that was, forgot how to do it now, aren't tankers oval to lower CG?

Mark

boslab

02-14-2015, 03:44 PM

Can't argue with that, a big tanker does look cool, and if there was a major advantage I suppose railcars would be oval too.

Mark

Willy

02-14-2015, 04:47 PM

The reason trucks and trailers built for transporting petroleum utilize an oval design has nothing to do with aesthetics and everything to do with functionality. As always form follows function.

Fuel having a relatively low specific gravity compared to most liquids needs a container that maximizes volume in order to take advantage of size and weight laws as they pertain to highway transportation laws. An oval or elliptical shape is best at maximizing the weight/volume capabilities of tankers carrying fuel.

I've driven fuel tankers that were repurposed to transport a liquid chemical with a specific gravity that was almost twice that of fuel. When these tankers were loaded to legal axle load limits the liquid's level in the tank was much lower than they would have been with fuel. In spite of baffled multi-compartment trailers these units where ill handling pigs to put it mildly.

J Harp

02-14-2015, 05:07 PM

My favorite fuel tankers were not only oval, but also double coned. Looking at the front you would see an oval shape. Viewed from from the side the top is straight, but the bottom slopes from both ends in a shallow V to the loading/unloading pipes. This lets the tank drain dry very quickly when unloading. A flat bottom (lenghtwise) tank will almost never drain dry.