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I'm writing a weight and balance calculator for a charter operation.

Every aircraft I've ever flight planned for has all the tanks on the same moment arm, so I don't know how you calculate the W&B for an aircraft that has swept wings. Do they have one tank on each wing whose moment changes as it drains, or do they have multiple tanks? And if they have multiple tanks, so you have to calculate for 'drain the forward tanks first' and 'drain the rearward tanks first' and plot both of those on envelope?

Paul Tomblin
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In supplement to @Dave's answer and as he said depending on the airplane, the complexity of calculating the c.g. can vary greatly. The manufacturer of every large aircraft I've programmed weight & balance for has had tables for each tank. In Boeing's case for the 747-400, each table line is a tank volume in gallons and the tank c.g. at that volume in inches. For example, and using JSON notation since the OP is JavaScript fluent:

{"vol":3200,"ba":1453.7}, {"vol":3250,"ba":1454.8}, {"vol":3300,"ba":1456.0}, {"vol":3350,"ba":1457.2}, {"vol":3400,"ba":1458.4}, {"vol":3450,"ba":1459.7},

"vol" is the fuel volume in the tank in gallons

"ba" is the balance arm at that volume in inches

Since fuel is loaded by weight, one has to use the fuel density to convert to volume.

Typically, at least for large aircraft, you have to satisfy the c.g. envelope for:

  1. zero fuel weight
  2. ramp weight after fueling
  3. takeoff weight
  4. landing weight

Also there has to be provision for carrying ballast fuel if that is necessary.

Terry
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You will need to consult the POH for the airframe in question and see what it says in regards to fuel burn from tanks at different moment arms. Ultimately you will need to program in the variable factor of fuel burn. It sounds like the program will have more inputs than just the weight of the passengers, fuel, and cargo. The end user will also need to input their planed fuel burn schedule and the order of the tanks they intended to burn off of. Once you have all this info you can compute inflight CG at different times. What you are going to end up with is a piecewise system that describes your WB at various points in the flight.

One of the contributors here @Terry built a WB app for some 747 variants that he used when he flew. You can find it here, it might be worth reaching out to him if he does not add an answer here.

Dave
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  • +1 for mentioning @Terry's site. (Not that either of our @ references will notify him...) – FreeMan Dec 19 '18 at 18:40
  • @FreeMan We should suggest that @Tagging work in posts and comments, it would be a nice feature (although I can see it getting out of hand in some cases) and opening up more "discussion" style answers. – Dave Dec 19 '18 at 18:57
  • @dave there are already questions about that on MSE and I take it it is by design that @replies are working only in comments. There is not even a simple way for this to work as is since usernames are not unique - there are at least 16 Dave's on Aviation alone. – Ghanima Dec 19 '18 at 21:32
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Most twin engine jets have one very long tank in each wing for each engine and often a center tank that feeds both. The fuel shifts forward as fuel is consumed from the wings (dihedral plus sweep), and aft as fuel is consumed from the center tank, if installed (always before the wings, so if there is a center tank the C of G moves aft until the tank is dry and forward as the wings are consumed). The manufacturer's weights engineering group will determine the overall fuel load's center of gravity moment for different fuel volumes and plot the results on a chart as a curve.

When you do the load sheets on a swept wing airplane the fuel moment/mass curve will be integrated into the chart data so that an accurate fuel moment/mass value is applied to the C of G calculation for takeoff. Also, some form of the fuel curve will be overlaid onto the C of G limits chart to confirm that the C of G shift will stay in limits as fuel is burned during the flight. This is especially important for jets with center fuel tanks, because you may be able to load it with a marginally aft G of G, such that the aircraft is technically in C of G limits at takeoff with a full fuel load, but as the center tank is burned off the C of G could move past the aft limit temporarily until some of the wing fuel is burned off. The result is the C of G with center tank at zero will determine the aft limit for freight/pax.

John K
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  • That's what I was wondering about. With all the tanks on the same arm, I put 4 points on the CG envelope chart - full fuel, take off weight, landing weight and zero fuel weight, and a straight(ish) line between them. I was wondering if I'd have to have more like a diamond with full fuel, empty this tank first, empty that tank first, empty the center tank first, etc. – Paul Tomblin Dec 19 '18 at 20:26
  • You don't need 4 points, you just need to know the overall moment for the combined fuel tanks with whatever quantity is on board, which will be with the CG data for the airplane. The only real difference is that the moment changes as fuel is burned and that has to be taken into account. – John K Dec 20 '18 at 00:43
  • Sorry to come back this after 2 years, but the guy I was writing the W&B calculator for didn't buy the Challenger after all until recently, and so I'm revisiting this. And while he hasn't sent the the relevant pages of the Challenger W&B Handbook, I found this graph online, and it shows a curve. https://code7700.com/images/cl-605_weight_and_balance_chart_example_2.jpg. I assume the three straight segments are center line tanks and the curve is wing tanks? What determines the formula for that curve? – Paul Tomblin Sep 18 '21 at 19:31
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    The chart indicates there probably an aux tank forward that gets burned first, the first zig aft, then aux tanks aft (saddle tanks back around the baggage area and I think they can also come with a tail mounted tank), the second zig sharply forward, then the center tank, the third zig aft, then the wings, the curve. The curve is a plot of the depletion of the fuel volume's C of G as its depleted, and the shape of the curve will be a function of the slope and contours of the tank bottom and fore/aft spars. You'd have to be able to model the fuel tank space accurately to work it out. – John K Sep 18 '21 at 20:21