Energy-State Models

Energy-State Models - Energy-State Models Eugene M. Cliff...

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Unformatted text preview: Energy-State Models Eugene M. Cliff April 23, 1998 1 Introduction The purpose of these notes is to supplement the text material related to energy mangement in atmospheric flight. Energy models provide an alternative to classical climb calculations and are particularly useful for vehicles capable of supersonic flight. Classical climb focuses on altitude change; that is, in changing the potential energy of the vehicle. Indeed, the analysis includes a force equilibrium require- ment which implies unaccelerated flight. Since the true airspeed ( V ) is constant the kinetic energy is likewise, constant. 2 Correcting for Acceleration It has long been understood that the classical climb analysis has an embedded inconsistency. Recall that we are led to choose a speed V to maximize the rate-of-climb, at a given altitude h . This analysis is repeated at a sequence of altitudes and the resulting family of best speeds defines a function V opt ( h ). As the aircraft climbs and the altitude changes, the choice of best speed will vary according to this function. It’s clear then that the resulting speed is generally changing with time [ V ( t ) = V opt ( h ( t )) ]. Since our choice of speed was based on maximizing the unaccelerated rate-of-climb there is an inconsistency. This was well appreciated in the days before WWII and various ‘corrections’ were suggested. It is useful to consider this issue. We begin with the equation describing the velocity change from Newton’s Laws: m ˙ V = T- D- W sin γ. Using the V ( t ) function implied above we are led to compute the time-derivative via the chain-rule. m d V d h ˙ h = T- D- W sin γ. (1) The result (1) is re-arranged to yield T- D W = (1 /g ) d V d h ˙ h + sin γ, 1 and using the kinematic relation ˙ h = V sin γ we have T- D W = ( 1 + ( V/g ) d V d h ) sin γ. (2) Equation (2) is solved for sin γ and used in the kinematic climb-rate expression to produce ˙ h a ≡ ( T- D ) V W 1 + ( V/g ) d V d h- 1 . (3) Result (3) is the rate of climb expression including the effects of acceleration. Note that the first term on the right is our old friend P s , the specific excess power. In the earlier unaccelerated climb analysis we had ˙ h u = P s so that we might also write (3) as ˙ h a = ˙ h u 1 + ( V/g ) d V d h . (4) This makes it clear that the term in the denominator can be interpreted as a correction applied to the original calculation. 2.1 Climb at Constant EAS To illustrate these ideas let’s suppose that we perform a climb at constant equivalent airspeed (EAS). Since V = V e / p σ ( h ), it is clear that the airplane will be changing its true airspeed as it climbs. In particular we have d V d h =- V e σ- 3 / 2 σ / 2 , so that the correction factor becomes 1 + ( V/g ) d V d h = 1- ( V 2 e 2 g ) σ σ 2 ....
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Energy-State Models - Energy-State Models Eugene M. Cliff...

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