Technological Institute of Aeronautics São José dos Campos - São Paulo - Brazil Requirement 3: Second Segment Climb

 Background

Takeoff Climb Gradient

JAR 25.121 Climb: One-engine-inoperative.
(b) Take-off; landing gear retracted. In the take-off configuration existing at the point of the flight path at which the landing gear is fully retracted, and in the configuration used in JAR 25.111 but without ground effect, the steady gradient of climb may not be less than
      2.4 percent for two-engined airplanes,
      2.7 percent for three-engined airplanes, and
      3.0 percent for four-engined airplanes,
at V2 and with -
(1) The critical engine inoperative, the remaining engines at the take-off power or thrust available at the time the landing gear is fully retracted, determined under JAR 25.111, unless there is a more critical power operating condition existing later along the flight path but before the point where the airplane reaches a height of 400 ft above the take-off surface(...); and
(2) The weight equal to the weight existing when the airplane's landing gear is fully retracted, determined under JAR 25.111.

In figure 1 the second segment while the takeoff is presented according to JAR.

Next: equations for calculation

The second segment is based on the situation with one engine of the aircraft inoperative.
Therefore, the thrust of the remaining engine(s) has to be enough to climb as required. From the equilibrium condition can be got two equations:

Since these equilibrium conditions have to be achieved with one engine inoperative, the required thrust at takeoff with all engines operating has to be higher by a factor N/(N-1). N is the number of engines. The thrust to weight ratio following from the requirement is:

The climb gradient in the equation above is small. Therefore it can be simplified:

 Data

Lift to Drag Ratio

• Oswald's airplane efficiency factor e is 0.7,
• wing aspect ratio A is provided as input
• drag coefficient at zero lift c_{D, 0} can be calculated by using Roskam's Class I approach
• slat drag coefficient c_{D, slat} is neglected and set to zero,
• In a simple tay, theflap drag coefficient c_{D, flap}:
  for c_L = 1.3 : flap deflection 15° => c_{D, flap} = 0.01
  for c_L = 1.5 : flap deflection 25° => c_{D, flap} = 0.02
  for c_L = 1.7 : flap deflection 35° => c_{D, flap} = 0.03 .
• Windmilling drag can be estimated according to a formula elaborated by Torenbeek

• Drag caused by rudder deflection

In order to counterbalance the adverse moment caused by a failed engine, the airplane rudder must be actuated. This causes drag that can be easily accounted by considering the rudder as a plain flap.

Roskam provides a methodology for drag estimation of several flap types.

For plain flaps, the induced and interference components can be discarded. The profile drag is estimated according to the following expression:

The delta term on the right side is obtained from a graph provided by Roskam and, in general, can be assumed to be 0.02. The area relation is the vertical tail-to-wing area ratio and can be assumed to be 0.140 for most jet transport airplanes. Considering an averaged 35⁰ for the quarter-chord sweep angle of the vertical tailplane, a value of 0.0023 or 23 counts is obtained for the drag caused by rudder deflection.

Statistics of aspect ratio   Equation characters - explanation page

 Typical figures for thrust-to-weight ratio

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