Conical Tank Linerisation

iyerm2 · Jul 15, 2011

Hi
I need some help to figure out what the equation is for calculating the volume of a conical tank.
So far i know that the volume equation is:

(pi*r^2*h)/3 = V
Since level changes "r" will also change so
r = tanx * h
therefore
V = pi*(tanx*h)^2*h/3
BUT how do we linearise this equation since Volume and height are not linear in a conical tank

Thanks for help

Tom Jenkins · Jul 15, 2011

iyerm2 said:
BUT how do we linearise this equation since Volume and height are not linear in a conical tank

You can't linearize it, since it is inherrently a non-linear function.

You can, however, easily calculate it assuming you know the angle of the hopper wall. If the angle between the vertical centerline and wall is A then the radius at any given point is h x tan(A) as you indicate.

V = 1/3 x pi x (tan(A) x h)^2 x h
V = 1/3 x pi x tan(A) x h x tan(A) x h x h

So, if by linearize you mean get rid of the square, just multiply tan(A) x h by itself!

Steve Etter · Jul 15, 2011

iyerm2 - While both you and Tom are both correct that Tom's equation is not technically a linear equation (by definition a linear equation plots as a straight line) I have to wonder; is that fact really important to you or are you simply looking to dynamically calculate volume as a function of height in your PLC? If so, you should be able program Tom's equation into just about any PLC with higher level math functionality.

Steve

curlyandshemp · Jul 15, 2011

iyerm2 said:
Hi
I need some help to figure out what the equation is for calculating the volume of a conical tank.
So far i know that the volume equation is:

(pi*r^2*h)/3 = V
Since level changes "r" will also change so
r = tanx * h
therefore
V = pi*(tanx*h)^2*h/3
BUT how do we linearise this equation since Volume and height are not linear in a conical tank

Thanks for help

What i do is use a lookup table with known amounts of water metered into the tank.

Assummimg level sensor is at the bottom of the cone and scaled 0 - XXX" of water column, meter in water until sensor reads
1", record that meter amount,
2", record amount and amount of change from 1"
3", record amount and amount of change from 2"
...etc
until " of w.c. are in linear section of tank, than standard litres/" of w.c. take over.

Then PLC logic to look at current value of level sensor with product use limit check for " and jump to labels for calcs based on where teh level is using the lookup table.

Once level is in linear section of tank, volume is then:

amount of litres recorded at Start of Linear section +
( recorded " of W.C. - Start Of Linear " of W.C.)* Litres per " of W.C.

Peter Nachtwey · Jul 15, 2011

Using a table is nuts when there is a simple equation

The equation can be manually simplified so that it is simply v=c*h^3. This requires only 3 mulitplies at run time. Looking up values in a table and interpolating between two adjacent entries is more more complicated.

The 1/3 PI and ratio between the radius and height can all be combined into one constant c.

curlyandshemp · Jul 15, 2011

Peter Nachtwey said:
The equation can be manually simplified so that it is simply v=c*h^3. This requires only 3 mulitplies at run time. Looking up values in a table and interpolating between two adjacent entries is more more complicated.

The 1/3 PI and ratio between the radius and height can all be combined into one constant c.

True, but is works. I have done this a long time and have had customers complain about the accuracy when using the calculated method. Calcualted method assumes the data for the tank is correct.

There are two worlds , the theoretical and the actual.
I have to deal with customers in the actual

TConnolly · Jul 15, 2011

This thread shows you how to program a conical tank volume computation using constant*height*height*height and how to determine the constant.
http://www.plctalk.net/qanda/showthread.php?t=21418

Tom Jenkins · Jul 15, 2011

curlyandshemp said:
Calcualted method assumes the data for the tank is correct.

So, if the data for the tank is innaccurate how did you get the points for the look up table?

curlyandshemp · Jul 15, 2011

Tom Jenkins said:
So, if the data for the tank is innaccurate how did you get the points for the look up table?

by metering in known amounts of water.

Peter Nachtwey · Jul 15, 2011

That argument is bogus

curlyandshemp said:
True, but is works. I have done this a long time and have had customers complain about the accuracy

The real equation is probably closer to V=k*(h-c)^3
Where c is an offset from the true height.
So you adjust k and c until it is right. It is easy enough to use a table of data to find the k and c that best fits the data. I use optimizing ( minimizing ) algorithms that will minimize the sum of squared error between measured and calculated and basically lets me fit equations to any data. In the above example I must first subtract an offset from the measure height before doing 4 multiplies but this is still faster than doing a table look up and interpolating between two readings.

Interpolating between measured data in a table assumes that each reading is accurate.

jkerekes · Jul 15, 2011

I agree. Just plot the data in an Excel chart. It has a curve fitting function. Find the type of equation that gives the best fit and then use that equation in the PLC. Much easier than lookup tables.

curlyandshemp · Jul 15, 2011

jkerekes said:
I agree. Just plot the data in an Excel chart. It has a curve fitting function. Find the type of equation that gives the best fit and then use that equation in the PLC. Much easier than lookup tables.

How do you go about obtaining tank data when the tank manufacturer refuses to release drawings because they have not been paid by the end user?

Or when there is a confined space permit policy inforced by local authorities preventing a PLC guy crawling inside a tank to take measurements?

These are all real world conditions experienced by a controls integrator on a job site that must get the job done.

Peter Nachtwey · Jul 16, 2011

curlyandshemp said:
How do you go about obtaining tank data when the tank manufacturer refuses to release drawings because they have not been paid by the end user?

That isn't your problem and don't let anybody tell you it is. The mechanical designer ( kludger ) shoud supply you with that data.

jkerekes · Jul 16, 2011

curlyandshemp said:
How do you go about obtaining tank data when the tank manufacturer refuses to release drawings because they have not been paid by the end user?

Or when there is a confined space permit policy inforced by local authorities preventing a PLC guy crawling inside a tank to take measurements?

These are all real world conditions experienced by a controls integrator on a job site that must get the job done.

No, I meant that if you already put known amounts of water in the tank and took level measurements, plot that. then use Excel to curve fit to obtain the best fitting equation.

curlyandshemp · Jul 16, 2011

jkerekes said:
No, I meant that if you already put known amounts of water in the tank and took level measurements, plot that. then use Excel to curve fit to obtain the best fitting equation.

good idea

Conical Tank Linerisation

iyerm2

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Tom Jenkins

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Steve Etter

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curlyandshemp

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Peter Nachtwey

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curlyandshemp

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TConnolly

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Tom Jenkins

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curlyandshemp

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Peter Nachtwey

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jkerekes

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curlyandshemp

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Peter Nachtwey

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jkerekes

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curlyandshemp

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