Does a vehicle manufacturer do something to make the fuel gauge compensate though, so it wasn’t just reading the distance/level? It could be pretty confusing with a really irregularly-shaped tank? There must be something that can be done. Think about aircraft wing tanks. They’re irregular shapes, and the wings are often at a bit of an upward angle (on low-wing planes). But those gauges read remaining fuel accurately; it could be fatally catastrophic if they didn’t.
Truck Fuel Tank Dip Stick
The original assumptions about the
markings of a dip stick for a truck’s fuel
tank were remarkably close to a mathemaical
solution, i.e. before everyone said, “wait a
minute”.
Assuming a cylindrical tank laid on
its (long) side, and that the “full” line on
the stick is marked and “empty” is the end
of the stick, I found that the 3/4 mark
should be .2533 the distance of the stick
from the full line. As expected, the 1/2
mark is at half the distance from full and
that the 3/4 mark should be .7467 from full.
Long story short, an angle of 14.677
degrees between radii of a semi-circle
defines dimensions that render half of the
area of the semi-circle. Hint: Think right
triangles
Ok, I’ve been out of school way to long to attempt to do calculus, and Rich doesn’t need a precise answer to 17 decimal places - just a pretty good idea.
So, being the visual person I am, I took a 7" sauce pan from the kitchen, covered it with clear plastic wrap, secured the plastic wrap with rubber bands, turned the sauce pan on its side and filled it halfway with water. The horizontal diameter at the water line, of course, was at 3.5" To fill the pan halfway with water took almost exactly 5 cups of water.
Pour out the 5 cups of water, pour back in 2.5 cups of water. The horizontal chord of the water line is now at 2 inches (give or take a little bit).
Therefore, if “half full” is at 3.5 inches, and “half of half full” - or a quarter of a tank is 2 inches when measured from the bottom, then the ratio of the level of a quarter of tank to a half of a tank is 2/3.5 or .57.
Another way to say it would be the level of a quarter tank is 57% of the level of a half of a tank.
So if the diameter of the tank is 20 inches, and a half of a tank measures 10 inches, a quarter of a tank should read 5.7 inches. 3/4 of a tank should read 5.7 inches from the top of the tank, or 14.3 inches up the stick. Of course, all of this assumes that the measurements are taken on his stick if it is inserted directly from the exact top of the tank straight down into the tank.
I thought I was really smart, until I went back and saw the pringles can and the jelly jar suggestions… at least I was warm!
Hey GearG,
my solution is on apx post 95 and 96 under user name “andrewpogany”. I too am too old to remember calc, trig, geo, etc. (how the hell did I get A’s in all that crap if I can’t remember anything?).
I still think my solution is best and only requires 5th grade math. Can you look at it and tell me what do you think? Everyone seems to want to argue about math and not actually solve the problem in the most simple and accurate and least inconvenient way possible (esp. for a trucker).
As a truck mechanic who has also filled his fair share of these thing, remember:
-
The fill cap is not directly above the top of the tank but slightly off set. Otherwise, you are trying to shove that big nozzle into a hole under the cab. I’ve had to turn these tanks that worked loose and the cap was right under the cab and near impossible to fill.
-
Law requires the tank not be filled to more than 95% of liquid capacity. It’s stamped right on the tank next to the fill cap. This is to allow for heat expansion and to prevent overflow.
Also, yes the tanks are connected by a cross-over pipe. There is also a shut off valve on each tank so you can drain one at a time if one needs replaced. We ran into this on some brand new Freightliners that were built in Mexico. They installed one valve backwards and they literally ran off of one tank from Mexico to Idaho. LOTS of fuel stops.
- Go to your local bulk lubricant supplier. They have measuring sticks all ready marked.
Ditto, Andrew
I got A’s in calc in high school, and even then I had no idea what was really going on.
I absolutely love your solution - it’s simple and it gets you to the information you need. You don’t need to know what a quarter tank is down to 1/100th of an ounce. Your solution is simple, and is actually practical for an over the road truck driver. If I were still driving truck, I would definitely use your suggestion.
Lookks like a graph of what I put forth in post 141 (?)avnapoli 11/09/10 2:56pm
saddle tanks on 18 wheelers
Until recently I have held you guys in such
high esteem. You were the asymptote to the
car maven ideal! But to get stumped on that
saddle tank problem, I’m crushed, started
dipping into my girlfriends Prozac —
If he knows the tank size, lets say 100
gallons and he knows that half a tank lets
say measures at 10 inches, then when he is
at half a tank, (50 gallons) all he has to
do is put 25 gallons in and notch the stick
making the 3/4 tank mark. Finally measuring
that distance from the halfway mark and
duplicating that measurement in the opposite
direction will give him the 1/4 tank
measurment. Ta Daa! And yes, calculus will
do this too but this is so much easier.
Sincerely yours,
William H. James, Clinical Psychologist (and
by the way, I would be happy to see either
one of you guys for a free evaluation.
Law requires the tank not be filled to more than 95% of liquid capacity. It's stamped right on the tank next to the fill cap. This is to allow for heat expansion and to prevent overflow.I've seen this stamped on the tanks of my father's trucks that he hauls grain on the farm with. I figured maybe old Rich ignores this if he's filling up and leaving again right away - before the fuel would get hot enough to expand he would burn enough to leave room. I've seen pilots do that when I worked at the local airport in college. If they were taking off immediately and flying a cross-country flight, they might insist on it being as completely full as possible, even though it might exceed the full mark.
Can I have your number? You offering to work pro-bono? LOL! I’m serious. I need all the help I can get!!!
Attached is an image illustrating the computation of the “Magliozzi Dipstick Constants” which quickly allow the computation of the dip stick depths for ANY size horizontal tank, given the radius R.
Basically solving the equation: [area circlular segment projected by theta] = [1/4 pi r^2]
for theta. The angle that projects a circular segment with the area of 1/4 the circle itself is 132.35 degrees.
This angle theta can then be used to simply compute the “Magliozzi Dipstick Constants” for 1/4 and 3/4 full dipstick lines, as illustrated.
For any tank of radius R - any units:
1/4 Tank Dipstick Line h=0.5961R 3/4 Tank Dipstick Line h=1.404R
So for a tank of 20" the radius is 10"
and therefore 1/4 full mark at 5.96" and the 3/4 mark at 14.04"
I made a YouTube video of my solution (using the integral method) available here:
You win, this is the best solution here.
Mark stick at 6" (1/4 tank), 10" (1/2 tank), 14" (3/4 tank) and 20" (full tank).
Geometry or Calculus will give same equation to solve interatively, but its a lot easier to get to that equation using geometry. Calculus is integrating under curve and subtract out rectangle. Geometry is fraction of circle and subract out right triangle. Work - in glorious technicolor - attached!
Pretty much got the same answer as all you guys using the geometric method with numerical solver:
1/4 = 5.96027"
1/8 = 3.6530"
3/4 = 14.0397"
Here was my writeup:
But you can make it easier on yourself by recognizing that you really only need to calculate HALF the volume of the circle, which simplifies the equations (just don’t forget to multiply the theta by 2 when done!)
http://mattfife.net/wordpress/?p=437
Just a few notes in case it hadn’t been spelled out before:
- It doesn’t matter how long the cylindrical tank is - the mark heights will always be the same
- It doesn’t matter if 2 tanks are connected (or 50 tanks for that matter), the fill lines will still be the same (so long as all tanks are equal volume)
- It doesn’t even matter the radius of the tank if you have theta. Simply plug it back into radius*cos(theta/2), and you can calculate the fill mark on ANY size tank.
Sorry, this probably didn’t come across well. I am certainly not meaning to deprecate engineering. I agree with you completely, this is an engineering problem. My meaning was that people are writing in with “mathematical” solutions which are not mathematical but engineering. Using calculus in the first step does not make it mathematical if the final step is using a computer to calculate it, any more than translating it into Chinese for a first step would mean that someone found a Chinese solution.
By no means do I mean to imply the engineering solution is inferior, in fact this case is an example of a problem where there is no exact mathematical solution, like the other examples you give. Despite my background in math I work as an engineer, and fully appreciate the right solution for a problem. Sorry if my careless and brief post sent you or anyone else the wrong message.
A simple, closed form integration
I thought I?d try to find a solution to the mathematical-cyclinder-fill problem that might be something Tom and Ray could explain on their show. That limits the visuals, the long calculations, and the references to interative solutions to non-closed form expressions. I found a number of simple geometric approximations that use rectangles in various positions inside and outside the circle, but those solutions are still too taxing visually for a strictly verbal description. Here is a relatively easy mathematical solution that is so simple it (almost!) doesn?t require visuals.
Let me first explain the three ?tricks? involved in this solution:
[list]As suggested on-air, we can convert the original problem to a two dimensional one. By using considerations of symmetry, we only need consider a quarter circle rather than a full circle. That quarter circle with radius 1 can be conveniently placed in quadrant I of the usual Cartesian Coordinate system with its center at the origin.[/list]
[list]We have to know something about the trigonometric cosine curve: the cosine curve from 0 to 0.5 radians is almost identical to the curve of a quarter circle.[/list]
[list]For convenience, we are going to imagine that a two dimensional picture of the fuel tank with its ? tank of fuel at the bottom is rotated 90 degrees counter-clockwise. In this position, the fuel is to the right. Note that the area filled by the fuel in this quarter circle picture is the same as the area not filled within the quarter circle. For convenience we will make calculations on the space that is not filled with fuel.[/list]
To calculate the area under the arc of the circle, we need only integrate the cosine function from 0 to X, where X is the horizontal position of a vertical line representing the fuel level. Since the area of a circle is pi times the radius of the circle squared, and since the radius is assumed to be 1, the area of the entire circle is pi times 1 squared, or pi. This area represents a full tank. If we are to measure ? of a tank, the required area is pi over 4. Since our picture takes advantage of symetry, we actually only show half of the pi over 4 area or pi over 8.
We are almost done. To find the area under the circle from 0 to X we integrate the cosine funtion from 0 to X. The integral of the cosine function is the sine function. Since the sine of 0 is 0, the required area is simply sine X. That has to be set equal to pi over 8 as just explained. Taking the inverse sine of X gives X and the inverse sine of pi over 8 gives 0.403564607. Thus X has a value of about 0.4. The fuel level is on the right hand side of our vertical line so the distance from the bottom of the tank to the fuel is 1 minus 0.4 or 0.6. If the tank had radius 10 inches, we would scale that to a level of 10 time 0.6 or 6 inches.
For those who prefer to see the equations and a picture of this setup, I?ve included them in the attached .pdf file. By the way, the error introduced by using the cosine rather than a circle is very small, a difference of about 0.000408146 for the value of X above.
There may well be even more elegant (simple and understandable) solutions to this mathematical problem that requires little visualization. I?d be interested in seeing what others come up with.
You don’t have to drain the tank. Drive until it is below 1/2. Add fuel until the level is at 10". Add another 1/4 of the capacity of the tank. The level shown on the stick is now 3/4. Being on level ground isn’t necessary if the fill hole is in the middle of the tank. I didn’t even try the calculus method; it’s been too long…
Jim
Tank the tank problem
Dipping your stick to measure the fill,
will not prove conclusive, sadly, not much
will.
One must consider, as I’m sure you both know,
infinite variables, including fluid flow.
The tank’s prescious cargo, upon which we
all so depend, sloshes and surges in waves
end to end.
To measure the fuel level while lying there
still, won’t be good data when climbing a
hill.
To instill more confusion, of which there’s
no limit, consider the fuel inlet and how
the fuel gets in it!
So please! advise your caller, to toss his
dowel in the fire, and seek out a truck tech
to fix his fuel gage for hire.
The tech will be happy having work for the
week, and the truck driver with deliveries
when the tech’s paycheck isn’t as bleek.
And each one of us will benefit as well,
when we’re not stuck in traffic behind a
stalled truck on a hill.
JC-n-KC