BizJetJock,
With regard to your quote - "a quick perusal of the performance section of the flight manual of any Citation (and many other light jets) gives the result that the second segment climb gradient (Flaps 15, gear up) is often significantly better than the (clean) final segment gradient for the same conditions.", I would be staggered if this were not the case.
(1) Second Segment Climb is conducted at TAKEOFF Thrust, which provides for the maximum possible excess thrust.
(2) Second Segment Climb is conducted at a speed (V2) that is fairly close to VMD for the Flap setting.
(3) The Clean Final Segment is conducted at MAXIMUM CONTINUOUS Thrust, which, in terms of excess thrust, is significantly less than Takeoff Thrust.
(4) I am not familiar with the Citation, but it is quite typical that, for this class of aircraft, Final Takeoff Speed is that for the Maximum Rate of Climb (OEI En-Route climb speed), i.e. significantly above VMD, where climb gradient is NOT a consideration.
(5) Flight Manuals give Net performance data. and, for a 2 engined aircraft, 2nd segment climb is degraded by 0.8%, whereas Final Segment (En-Route) climb is degraded by 1.1%. This will help in making the 2nd segment look better.
For any aircraft, best angle of climb occurs where, for the configuration, the maximum excess thrust is available.
For the Jet aircraft, this is slightly below VMD (but for all practical purposes at VMD). OEI VMD is less than All engines VMD due to Rudder Drag, and engine windmilling Drag, in a speed range where the slight reduction in Net thrust is only slightly lower than static Net thrust. It's not until about M0.3 that thrust decline from static becomes noticeable.
For the Propeller aircraft, thrust decline with increasing forward speed is MUCH more marked (Power = Force X Velocity), and maximum excess thrust will occur somewhat below the 'pure' VMD for the configuration FOR MOST PROPELLERS. I find it almost inconceivable that, for a Constant Speed Propeller, it's efficiency would be greater at a lower speed. (Propellers will vary).
I do not know of, and cannot conceive of, any aircraft which has lower drag with even minimal flap extension - my earlier remark of "If one such aircraft does exist, why not leave the flaps out permanently?" was made very very tongue in cheek, if one does exist I'd be interested to know of it.
Let's take some hypothetical figures, Required Obstacle Clearance Gradient is 5%, Flaps 10 Gradient is 6%, Flaps Up Gradient (Still at Takeoff Power/Thrust) is 8% - Fairly typical figures. Now, if Takeoff Flap setting is 10 degrees, you MUST maintain Flaps 10 and Flaps 10 Vx until obstacles are cleared. What if you decided to avail yourself of the improved gradient available in the clean configuration? You can only achieve this by reducing climb angle allowing the aircraft to accelerate to the Flaps UP Vx, say, we fly for 1000 M at Flaps 10 and a 6% climb gradient, reduce to 3% for another 1000 M to reach the Flaps UP Vx, and then achieve an improved 8%. The Net gradient for the first 2000 M because of the acceleration is 4.5%, LESS THAN the required obstacle gradient, and Splat!!! - You've made the front pages of the papers ( and caused considerable speculation on Pprune as to the 'probable cause'). If some Flight Manual / Operations Manual states "Improved obstacle clearance is available by maintaining Takeoff Flap Vx after Takeoff, they're absolutely correct, for all of the reasons stated above. They're not telling you that you will achieve an improved gradient, you will NOT, they're telling you that obstacle clearance is improved.
All of my current Performance Engineering work is for Jet aircraft, but in bygone years did a huge amount on Turbo-Prop aircraft. In EVERY case, the Flapless takeoff for these aircraft offered the best All engines and OEI climb gradient performance after Takeoff. Of course Prop / Turbo-Prop usually means short runways, so Flapped Takeoffs become necessary because of Field performance, with 2nd segment performance suffering accordingly, BUT, if a Flapped Takeoff was necessary, the Flapped Vx (or V2) had to be flown until clear of obstacles - RIGIDLY!
Dammit Hawk37, you were right, I did get drawn in.
Fly Safe,
Old Smokey