Like most engineering, picking a bellcrank size is a compromise. But to make an attempt to itemize some requirements, choices and consequences:
1. It has to be strong enough to carry worst case flight loads from the lines plus a factor of safety. Since flight loads are approximated by the aircraft speed and weight, it is easy to calculate these loads. It means that the crank has to survive this load; that the support bearing has to survive and not bind up; that the bolt or support rod (axle) has to survive this, too. And the airframe has to be strong enough locally to react and distribute the load. To bound this, the rules have added a safety factor that comes in the form of a pull test. Note that the above is valid whether you are talking about a 2”, 3” or 4” or even larger bellcrank.
2. The larger bellcrank (assuming equal strength of the different sizes) can be used to change the control ratios, allowing you to use a larger handle spacing and keep the ratio closer to 1:1.
3. A second important reason is that the actual control surface deflection loads on the components are reduced. This is most easily described as having more leverage. There is an upper limit that can occur with simple control surfaces, high speeds where even with all of the centrifugal flight loads applied thru one line, you still may not be able to overcome the aero loads. Unlikely you’ll experience this, but imagine the situation would be more likely with a light airplane with huge control surfaces. The aero forces are what they are, but if you have more leverage in your linkage, the loads in the components are reduced. If all you use on a larger bellcrank is the wider line spacing and not a larger pushrod radius, then what you have gained is less torque on your wrist for a given deflection. If you also use a larger radius to the pushrod, and you match this with your flap horn and also on the elevator, then the pushrods see less load. Less load, less friction, less wear.
4. So why not put huge bellcranks into every plane? First, they would be hard to fit in most airplanes. You have to cut away a lot more structure to get them in, and provide clearance for the increased line translation. So ridiculously big is Ridiculous. (Porterism.) If you merely have a large bellcrank, you are going to want large handle spacing to match. Just the opposite problem has plagued 1/2A promoters forever. A guy badmouths a known good 1/2A stunter as just too squirrely to fly. So you volunteer to check it out. Turns out he is flying an 11 oz. airplane with a 2” bellcrank and using a standard E-Z Just handle (huge spacing) and to top it off it has kinks in the cable. But everyone knows 1/2As are squirrelly. Not really. Just some guys that mismatch their control systems.
5.The larger bellcrank (and perhaps larger, beefier attachment hardware) is going to weigh more. Why save grams of balsa weight and then put in a larger bellcrank than you need? On the other hand, if you custom make a bellcrank, you can make it to match the weight of your plane, not some generic PAMPA Sled that weighs 65+ oz. to meet the strength requirement listed in item 1 above. Buying a super-duty crank just so you can get 4” spacing isn’t efficient.
6. Putting a huge bellcrank into a wing with short semi-span can cause a significant bend angle in the leadouts if you try to place them close together. On the other hand, placing the guides far apart may not work well in your wing if the tip is small. Widely spaced leadout exits have detractors that describe yaw disturbances introduced as control loads are introduced. Problems would be exaggerated if using solid leadouts.
7. If you use larger throws on the bellcrank and horns, then the running clearances between the pushrods and these components won’t cause as much control surface slop as if you had a small bellcrank radius and horn radii.
From the limited information given for your particular application, it would seem you should be looking at either a 3” or 4” bellcrank. If the 4” fits without cutting important structure I’d do that. It sounds like we are talking about an R/C 3-D conversion plane, so no need to cut thru an I-Beam spar or thru wing-mounts for landing gear.
This is a generalized discussion, but might help you decide.
Dave