Climb and Descent Performance
Climb is possible when the available engine thrust is more than the aircraft drag; the excess thrust (i. e., thrust minus aircraft drag, (T – D)) is converted into the potential energy of height gain. The total energy of an aircraft is the sum of its PE and KE, expressed as follows:
E = mgh + (mgV2/2g) = mg(h + V2/2g)
The term for the rate of change of specific energy is specific excess power (SEP):
SEP = dh/dt + V/g(dV/dt) = V(T – D)/(mg) = dhe/dt (13.7)
Equation 13.7 shows that he > h by the term V2/2g in Equation 13.6. In other words, an aircraft can continue to climb by converting KE to PE until the speed is decreased to the point where the aricraft is unable to sustain the climb.
An enroute climb is performed in an accelerated climb. The equation for an accelerated climb is derived as follows (Figure 13.10). For simplicity, the subscript to to represent aircraft velocity is omitted. From Figure 13.10, the force equilibrium gives:
(T – D) = mg sin y + (m)dV/dt This gives the gradient:
sin y = [T – D – (W/g)dV/dt]/W = [(T – D)/W] – [(1/g) x dV/dt] (13.8)
dV/dt = (dV/dh) x (dh/dt)
Then, rate of climb:
R/Cacd = dh/dt = Vsin y = V(T – D)/ W – (V/g) x (dV/dh) x (dh/dt) (13.9) By transposing and collecting dh/dt:
V[(T – D)/ W]
1 + (V/g)(dV/dh)
Combining Equations 13.7 and 13.9, the rate of climb is written as: dhe /dt = V(T – D)/ W – (V/g)(dV/dt) + V/g(dV/dt) = V(T – D)/ W (13.11)
The rate of climb is a point performance and is valid at any altitude. The term V (dV) is dimensionless. It penalizes the unaccelerated rate (i. e., the numerator in Equation 13.10) of climb depending on how fast an aircraft is accelerating during the climb. Part of the propulsive energy is consumed for speed gain rather than altitude gain. Military aircraft make an accelerated climb in the operational arena when the g (ddh) term reduces the rate of climb depending on how fast the aircraft is accelerating. Conversely, civil aircraft has no demand for a high-accelerated climb; rather, it makes an enroute climb to cruise altitude at a quasi-steady-state climb by holding
Table 13.7. Y (dY) value (dimensionless quantity)
Below tropopause Above tropopause
At constant EAS 0.566 m2 0.7 m2
At constant Mach number -0.133 m2 0 (Mach held constant) the climb speed at a constant EAS or Mach number. A constant-EAS climb causes the TAS to increase with altitude gain. A constant speed indication eases a pilot’s workload. During a quasi-steady-state climb at a constant EAS, the contribution by the V (dY) term is minimal. The magnitude of the acceleration term decreases with altitude gain and becomes close to zero at the ceiling (i. e., defined as when R/Caccl = 100 ft/min). (Remember that V = VeasU/° and Yeas = Ma^/ст.)
Constant EAS Climb Below Tropopause (y = 1.4, R = 287 J/kgK, g = 9.81 m/s2)
The term V (dY) can be worked out in terms of a constant EAS as follows:
V /dV VeasVeas (d(1/a) = _ V^ /da = _M2a2_ /da
g dh g^a dh 2ga2 dh 2ga dh
In SI, Equation 3.1 gives a troposphere T = (288.16 – 0.0065h), and p = 1.225 x (T/288.16)(g/00065R)-1 = 1.225 x (T/288.16)(9 81/0 0065 x 287)-1 = 1.225 x (T/288.16)4 255 derives the density ratio (up to the tropopause) by replacing T in terms of its lapse rate and h:
a = p/p0 = (288.16 – 0.0065h/288.16)4’255 = (1 – 2.2558 x 10-5 x h)4 255
This gives (da/dh) = -9.6 x 10-5 x (1 – 2.2558 x 10-5 x h)3 255 Therefore:
x [9.6 x 10-5 x (1.2558 x 10-5 x h)3 255]
M2 x 1.4 x 287 x (288.16 – 0.0065h)
2 x 9.81
x [9.6 x 10-5)/(1 – 2.2558 x 10-5 x h)] M2 1.4 287 288.16
These equations are summarized in Table 13.7.
Constant Mach Climb Below Tropopause (Y = 1.4, R = 287 J/kgK, g = 9.81 m/s2)
The term Y (dV’j can be worked out in terms of a constant Mach-number climb as follows:
Y dV MaM da aM2Y7R dVT
g dh = g dh = g dh
From Equation 3.1, the atmospheric temperature, T, can be expressed in terms of altitude, h, as follows:
T = (288 – 0.0065h)
where h is in meters. Substituting the values of y , R, and g, the following is obtained:
V /dVMVyR(dVT_ 0.00664aM2 (i3i3)
g dh = g dh = – (288 – 0.0065h) ‘
When evaluated for altitudes, the equation gives the value as shown in Table 13.7.
In a similar manner, the relationships above tropopause can be obtained. Up to 25 km above tropopause, the atmospheric temperature remains constant at 216.65 K; therefore, the speed of sound remains invariant.
With the loss of one engine at the second-segment climb, an accelerated climb penalizes the rate of climb. Therefore, a second-segment climb with one engine inoperative is achieved at an unaccelerated climb speed, at a speed a little above V2 due to the undercarriage retraction. The unaccelerated climb equation is obtained by omitting the acceleration term in Equation 13.10, yielding the following equations:
T – D = WsinY becomes sinY = (T = D)/ W
The unaccelerated rate of climb:
R/C = dh/dt = VsinY = V x (T – D)/ W (13.14)
The climb performance parameters vary with altitude. An enroute climb performance up to cruise altitude is typically computed in discreet steps of altitude (i. e., 5,000 ft; see Figure 13.10), within which all parameters are considered invariant and taken as an average value within the altitude steps. The engineering approach is to compute the integrated distance covered, the time taken, and the fuel consumed to reach the cruise altitude in small increments and then totaled. The procedure is explained herein. The infinitesimal time to climb is expressed as dt = dh/( R/Cacd). The integrated performance within the small altitude steps is written as:
At = final tinitial = (hfinal hinitial) / (R/ Caccl) ave (13.15)
AH = (hfinal – hinitial) (13.16)
Using Equation 13.8, the distance covered during a climb is expressed as:
Ax = At x Vave = At x Vcosy (13.17)
where V = the average aircraft speed within the altitude step.
Fuel consumed during a climb can be expressed as:
Afuel = average fuel flow rate x At (13.18)
The time used to climb, timeclimb = J2 At, is obtained by summing the values obtained in the small steps of altitude gain. The distance covered during a climb,
Rciimb = J2 As, is obtained by summing the values obtained in the small steps of altitude gain. The fuel consumed for a climb, Fuelclimb = J2 A fuel, is obtained by summing the values obtained in the small steps of altitude gain.
A descent uses the same equations as for a climb except that the thrust is less than the drag; that is, the rate of descent (R/Daccl) is the opposite of the rate of climb. The rate of descent is expressed as follows:
Unlike in a climb, gravity assists a descent; therefore, it can be performed without any thrust (i. e., the engine is kept at an idle rating, producing zero thrust). However, passenger comfort and structural considerations require a controlled descent with the maximum rate limited to a certain value depending on the aircraft design. A controlled descent is carried out at a partal-throttle setting. To obtain the maximum range, an aircraft should ideally make its descent at the desired minimum rate. These adjustments entail varying the speed at each altitude. To ease the pilot’s workload, a descent is made at a constant Mach number; when the Veas limit is reached, the aircraft adapts to a constant Veas descent, similar to a climb. Special situations may occur, as follows:
1. For an unaccelerated descent, Equation 13.19 becomes:
R/Dunacci = dh/dt = V[(DW T)] (13.20)
At a higher altitude, the prescribed speed schedule for a descent is at a constant Mach number; therefore, the previous tropopause Vtas is constant and the descent is maintained in an unaccelerated flight.
This indicates that at a constant V(L/D), the R/Cdescent is the same for all weights.
As in a climb, the other parameters of interest during a descent are range covered (Rdescent), fuel consumed (Fueldescent), and time taken (timedescent). There are no FAR requirements for the descent schedule. The descent rate is limited by the cabin-pressurization schedule for passenger comfort. FAR requirements are enforced during an approach and a landing. At high altitude, the inside cabin pressure is maintained as an approximate 8,000-ft altitude. Depending on the structure design, the differential pressure between the inside and the outside is maintained at approximately 8.9 lb/in.
Integrated performances for a climb to cruise altitude and a descent to sea level are computed, and the values for distance covered, time taken, and fuel consumed are estimated to obtain the aircraft payload range. Textbooks may be consulted for details of climb and descent performances.