Throughout my many years of conducting initial and recurrent King Air training courses, a topic that consistently stirs some consternation is the fuel calculations associated with the Aircraft Weight and Balance form. The problem arises because the fuel tanks in a King Air do not all reside at the same location in the wing. Hence, there is not one location – one “arm,” the distance aft of the Weight and Balance (W & B) datum point – that applies to all of the fuel.
In fact, the W & B section of the POH does not even list a fuel arm! It lists arms for every other cockpit and cabin option – pilots, passengers, baggage, cabinet contents – but no fuel arm is shown. Huh? Then how do we do our W & B calculations?!
Not to worry. We need an arm to be able to calculate the fuel’s moment, using the familiar formula, Moment = Weight x Arm. Instead of giving the Arm, Beech simply provides the associated Moment for each different Weight of fuel on board. If we wish to actually know the Arm – and there is really no need to know that, other than for curiosity’s sake – then we can rearrange the formula to see that Arm = Moment/Weight. Let me give you a couple of examples, using the fuel in a King Air model 200 as an example.
A chart in section six of the POH shows that 200 gallons of Jet A fuel, with a density of 6.7 pounds per gallon, gives a weight of 1,340 pounds and a moment of 2,437. However, there is a little gotcha here: 2,437 is not the moment, but is the moment divided by 100.
Back in the day when none of us carried around a calculator in our smartphone, and we preferred to avoid the laborious task of doing longhand multiplication and division, Beech did the work for us by providing charts that gave the Moments for different Weights and the various locations. The actual numbers were rather large. For example, a 180-pound pilot at an Arm of 129 inches gives a Moment of 23,220. A fully loaded airplane might have a Total Arm in excess of 2,000,000! Therefore, Beech presented all Moments as “Moments divided by 100.” Instead of the 23,220 of our example, it was listed as 232. This simplified the longhand addition of the individual Moments into the Total Moment, yet the inaccuracy caused by this “rounding off” method was so tiny as to be negligible.
Back to the fuel chart. If curiosity compelled me to know the Arm for that 200 gallons of fuel, the formula Arm = Moment/Weight would now be Arm = (2,437 X 100)/1,340 or 181.87 inches. No pilot cares where the Arm is to the nearest hundredth of an inch, so it is normally stated to the nearest whole inch: 182 inches.
For “fun” – satisfying that itch of curiosity – let’s calculate the Arm for the first 10 gallons put into the airplane and for the last 10 gallons added before the tanks are full.
The chart shows, for 10 gallons, a weight of 67 and a Moment/100 of 103. The Arm, calculated by us, is 154 inches. The 10 gallons that brings the chart from 530 to 540 gallons adds the same weight, 67 pounds, but increases the Moment/100 by 138 for an Arm of 206 inches. Wow, that fuel arm changed a lot! It moved aft by 52 inches, nearly a yard and a half!
This shift in Arm is not surprising when one understands the fuel system on the model 200. When the refueler pumps in the first ten gallons into an empty airplane – maybe the fuel had been drained for a re-weighing of the airplane – he uses the filler cap at the wingtip, fueling the main tank before the aux tank. The fuel he adds flows downhill, propelled by gravity, and ends up at the bottom of the nacelle tank – the lowest and most forward position of the main complex of fuel tanks. As fuel continues to be added, it gradually begins to fill the other tanks in the main complex, some of which are forward of the main spar and some of which are located between the main and rear spars in the outboard wing. Only after the mains are topped, would the fueler now go to the aux tank cap between the nacelle and the fuselage to input fuel – a single bladder tank located between the spars of the center section. As you can see, the first fuel added sits more forward than the last fuel added.
In the Weight and Balance section of the POH, where we find the fuel chart we have been using, Beech also provides a “Weight and Balance Loading Form” chart. In fact, they even fill it in with actual numbers for both a forward and an aft loading example using the true Empty Weight figures for this exact Serial Number airplane as it was when it left the factory. How nice!
Line #11 on that form, labeled “Less Fuel to Destination,” has a column for “Weight” and another for “Mom/100.” I will wager that those who decide to use this form for their own W & B calculations make an error on this line more often than not. For example, let’s say they started the flight with 380 gallons – almost full mains – and landed with 100 gallons remaining, for a total consumption of 280 gallons. To find the numbers to enter into line 11, they refer to the fuel chart, go to 280 gallons, and find a weight of 1,876 pounds and a Moment/100 of 3,437. Our curiosity shows that the average Arm for this fuel is 183 inches. Entering 1,876 pounds for the Weight on line one is correct; using 3,437 for the Moment/100 is incorrect.
Why? Because that Moment is showing the increase in Moment that the first 280 gallons, added to an empty airplane, contributed to the airplane. Yet the 280 gallons we consumed during this flight are not those same gallons. To calculate the correct Moment to use, we must find how much the Moment decreased as we went from our starting fuel to our landing fuel. The Moment/100 of the initial 380 gallons is 4,700, and for the remaining 100 gallons is 1,196. The difference – the Moment/100 decrease that our 280 gallons of consumed fuel provided – is 3,504. This is the value that should go in the Mom/100 column on line 11. The average Arm for this fuel is 187 inches. Although the error caused by using a fuel arm that is four inches off is not huge, it definitely makes for a CG error that is noticeable.
Two thoughts about how to fight this tendency to use the wrong Moment figure on line 11 of Beech’s form: First, reread my discussion here and be certain you are using the difference between starting and ending fuel moments to enter on line 11. Second, avoid using this form!
A different and perhaps easier way to calculate the CG for your landing condition is to start from the Zero Fuel Weight condition that you have already calculated and add to it the Weight and the Moment of the remaining fuel on board at landing. The Moment figure can come straight out of the chart in Section six, since this fuel is the same as the chart shows: The fuel that remains sits at the same location as that much fuel does when added to an empty airplane.
“Hey, you out there in readership land! Yes, I am talking to you, Mr. Foreflight, and also to you, Mr. Fltplan.com! I presume you are doing this correctly and not providing slightly erroneous answers to my peeps, right? You are? Great! Thanks.”
If you have a question you’d like Tom to answer, please send it to Editor Kim Blonigen at kblonigen@cox.net.
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