Chapter 18

Performance – Climb

Abstract

Fundamental relationships necessary to evaluate climb characteristics are developed in the chapter. These are used to establish the equations of motion for climbing flight. This is followed the derivation of an assortment of methods that allow the various climb characteristics of an airplane to be predicted. This is followed by introduction to climb sensitivity studies. Finally, the climb performance of selected aircraft types is presented to allow the reader an accessible reference to actual aircraft data.

Keywords

Equation of motion; climb; free-body; steady climb; horizontal airspeed; vertical airspeed; rate-of-climb; climb gradient; climb angle; best angle-of-climb; best rate-of-climb; time to altitude; service ceiling; absolute ceiling

Outline

18.1 Introduction

18.1.1 The Content of this Chapter

18.2 Fundamental Relations for the Climb Maneuver

18.2.1 General Two-dimensional Free-body Diagram for an Airplane

18.2.2 General Planar Equations of Motion for an Airplane

18.2.3 Equations of Motion for Climbing Flight

18.2.4 Horizontal and Vertical Airspeed

18.2.5 Power Available, Power Required, and Excess Power

18.2.6 Vertical Airspeed in Terms of Thrust or Power

Derivation of Equation (18-15)

Derivation of Equation (18-16)

18.2.7 Rate-of-climb

Climb Gradient

18.3 General Climb Analysis Methods

18.3.1 General Rate-of-climb

Derivation of Equation (18-18)

18.3.2 General Climb Angle

Derivation of Equations (18-19) and (18-20)

18.3.3 Max Climb Angle for a Jet

Derivation of Equation (18-21)

18.3.4 Airspeed for θmax for a Jet (Best Angle-of-climb Speed)

Derivation of Equation (18-22)

18.3.5 ROC for θmax for a Jet

Derivation of Equation (18-23)

18.3.6 Airspeed for Best ROC for a Jet

Derivation of Equation (18-24)

18.3.7 Best ROC for a Jet

Derivation of Equation (18-25)

18.3.8 Airspeed for θmax for a Propeller-powered Airplane

Derivation of Equation (18-26)

18.3.9 Airspeed for Best ROC for a Propeller-powered Airplane

Derivation of Equation (18-27)

18.3.10 Best Rate-of-climb for a Propeller-powered Airplane

Derivation of Equations (18-29) and (18-30)

18.3.11 Time to Altitude

Rapid Approximation

Linear Approximation

Derivation of Equation (18-31)

Derivation of Equation (18-33)

Derivation of Equation (18-34)

18.3.12 Absolute/Service Ceiling Altitude

18.3.13 Numerical Analysis of the Climb Maneuver – Sensitivity Studies

Altitude Sensitivity

Weight Sensitivity

Propeller Efficiency Sensitivity

18.4 Aircraft Database – Rate-of-climb of Selected Aircraft

Variables

References

18.1 Introduction

The T-O maneuver is followed by the climb maneuver. It is vital for the aircraft designer to understand how rapidly or how steeply an aircraft can climb. Great climb performance sells aircraft. The climb affects not only how quickly the airplane reaches a desired cruise altitude but also how its noise footprint is perceived. The competent designer will always try to maximize the rate-of-climb and the angle-of-climb. The study of the climb primarily involves the determination of the rate at which the airplane increases its altitude (called the rate-of-climb or ROC) and the angle its flight path makes to the ground (called the angle-of-climb or AOC). Figure 18-1 shows an organizational map displaying the T-O among other parts of the performance theory.

image

FIGURE 18-1 An organizational map placing performance theory among the disciplines of dynamics of flight, and highlighting the focus of this section; climb performance analysis.

This chapter will present the formulation of and the solution of the equation of motion for the climb, and present practical as well as numerical solution methodologies that can be used for both propeller- and jet-powered aircraft. The presentation is prepared in terms of independent analysis methods. When appropriate, each method will be accompanied by an illustrative example using the sample aircraft. The primary information we want to extract from this analysis is characteristics like maximum ROC, best (highest) AOC, the corresponding airspeeds, the ROC and AOC for a given power setting, and climb range, to name a few.

In general, the methods presented here are the “industry standard” and mirror those presented by a variety of authors, e.g. Perkins and Hage [1]; Torenbeek [2]; Nicolai [3]; Roskam [4]; Hale [5]; Anderson [6]; and many, many others.

18.1.1 The Content of this Chapter

• Section 18.2 develops fundamental relationships necessary to evaluate climb characteristics, most importantly, the equations of motion for climbing flight.

• Section 18.3 presents an assortment of methods to predict the various climb characteristics of an airplane.

• Section 18.4 presents the climb performance of selected aircraft types.

18.2 Fundamental Relations for the Climb Maneuver

In this section we will derive the equation of motion for the climb maneuver, as well as all fundamental relationships used to evaluate its most important characteristics. First, a general two-dimensional free-body diagram will be presented to allow the formulation to be developed. Only the two-dimensional version of the equation will be determined as this is sufficient for all aspects of aircraft design.

18.2.1 General Two-dimensional Free-body Diagram for an Airplane

Figure 18-2 shows a free-body diagram of an airplane moving along a trajectory, which we call the flight path. The x- and z-axes are attached to the center of gravity of the airplane such that the x-axis is the tangent to the flight path. The z-axis is perpendicular to the flight path. The airspeed is defined as the component of its velocity parallel to the tangent to the flight path. Also, a datum has been drawn on the airplane to represent the chord line of the MGC of the wing. The angle between the datum and the tangent to the flight path is called the angle-of-attack. The force (or thrust) generated by the airplane’s powerplant, T, may be at an angle ε with respect to the x-axis. The figure shows that this coordinate system can change its orientation with respect to the horizon depending on the maneuver being performed. Then, the angle between the horizon and the x-axis is called the climb angle, denoted by θ. If θ > 0, then the aircraft is said to be climbing. If θ = 0, then the aircraft is said to be flying straight and level (cruising). If θ < 0, then the aircraft is said to be descending. This chapter only considers the first scenario.

image

FIGURE 18-2 A two-dimensional free-body of the airplane in climbing flight.

The free-body diagram of Figure 18-2 is considered balanced in terms of inertia, mechanical, and aerodynamic forces. The lift is the component of the resultant aerodynamic force generated by the aircraft that is perpendicular to the flight path (along its z-axis). The drag is the component of the aerodynamic force that is parallel (along its x-axis). These are balanced by the weight, W, and the corresponding components of T. The presentation of Figure 18-2 can now be used to derive the planar equations of motion for the airplane, so called because the motion is assumed two-dimensional and assumes there is not yaw. This simplification is sufficient to accurately predict the vast majority of climb maneuvers.

18.2.2 General Planar Equations of Motion for an Airplane

Planar equations of motion (assume no or steady rotation about the y-axis) are obtained by summing the forces depicted in Figure 18-1 about the x- and z-axes as follows:

image (18-1)

image (18-2)

18.2.3 Equations of Motion for Climbing Flight

The equations of motion can be adapted to a steady climbing flight by making the following assumptions:

(1) Steady motion implies dV/dt = 0.

(2) The climb angle, θ, is a non-zero quantity.

(3) The angle-of-attack, α, is small.

(4) The thrust angle, ε, is 0°.

Equations of motion for a steady climb:

image (18-3)

image (18-4)

Equation (18-3) shows that lift-in-climb is actually less than the weight (the difference is balanced by the vertical component of the thrust):

image (18-5)

The lift coefficient at this condition is thus:

image (18-6)

The drag force, using the simplified drag model, is given by:

image (18-7)

Inserting Equation (18-6) into Equation (18-7) yields:

image (18-8)

Expanding:

image (18-9)

Note that drag, D, as calculated by Equation (18-9), should be used with Equations (18-15) and (18-17), and would ordinarily require an iterative scheme to solve for ROC and θ. However, as demonstrated by Austyn-Mair and Birdsall [7], assuming that cos θ ∼ 1 holds indeed yields an acceptable accuracy for modest climb angles. In particular, the error that results for GA aircraft is small, as their angle-of-climb is usually less than 15°. According to Figure 4.1 of Ref. [7], at the best rate-of-climb airspeed, even an angle-of-climb of 20° will deviate about 0.2° off the exact value. At an angle-of-climb of 40° the deviation is a hair short of 1.0°. The assumption is that cos θ ∼ 1 is warranted as it allows for a considerable time saving in analysis work, with low difference from the exact method.

18.2.4 Horizontal and Vertical Airspeed

The primary purpose of the methods in this section is the evaluation of the climb performance of aircraft. We want to determine characteristics like best rate-of-climb (ROC), best (largest) angle-of-climb (AOC), and the airspeeds at which these materialize. The first important step is to define the horizontal airspeed. It is important when estimating the horizontal distance covered during a long climb to altitude:

image (18-10)

Then it is possible to define the vertical airspeed, also called the rate-of-climb:

image (18-11)

Both can be derived by observation from Figure 18-3. Note that in terms of calibrated airspeed, V is the airspeed indicated on the airspeed indicator; VV is observed on the vertical speed indicator (VSI); and VH is the ground speed. Note that this VH should not be confused with the maximum level airspeed to be discussed in Chapter 19.

image

FIGURE 18-3 Airspeed components during climb.

18.2.5 Power Available, Power Required, and Excess Power

These three concepts are imperative in the climb analysis as they define the climb capability of the aircraft. Note that for aircraft propelled by jet engines, the power available is estimated by multiplying its thrust by the airspeed. For aircraft powered by propellers, the power is obtained by multiplying the engine power by the propeller efficiency. Since the engine power is usually presented in terms of BHP or SHP, this number must be converted from horsepower to ft·lbf/s by multiplying by a factor of 550, if using the UK system. If using the SI system, the horsepower number must be multiplied by a factor of 745.7 to convert to watts.

Power available:

image (18-12)

Power required:

image (18-13)

Excess power:

image (18-14)

18.2.6 Vertical Airspeed in Terms of Thrust or Power

The vertical airspeed can be estimated if thrust or power and drag characteristics of the aircraft are known.

For jets:

image (18-15)

For propellers:

image (18-16)

Note that the above expressions are some of the most important equations in the entire climb analysis methodology. Ultimately, we want to determine some specific values of VV, for instance the maximum value, or the one that results in the steepest climb possible, and so on. Knowing these are of great importance to the pilot and, as it turns out, also for the certifiability of the aircraft. Additionally, the formulation shows that in order for an airplane to increase its altitude, its thrust power (TV) or available power (PAV) must be larger than the drag power (DV) or required power (PREQ) for level flight.

Derivation of Equation (18-15)

Multiply Equation (18-4) by V/W to get:

image

QED

Derivation of Equation (18-16)

We note that power is defined as force × speed:

image

where PAV = ηp·PENG.

QED

18.2.7 Rate-of-climb

The rate-of-climb (ROC) is of great importance to the pilot, as well as a superb indicator of the airplane’s capability as it is directly dependent on its thrust and drag characteristics. If the thrust and drag can be quantified at a flight condition, the instantaneous ROC can be calculated as follows:

image (18-17)

NOTE 1: The units for ROC in Equation (18-15) are commonly ft/min or fpm, which is why we multiply it by 60 to convert the ROC in ft/s into fpm.

NOTE 2: In the SI system, ROC is usually in terms of m/s, rendering the factor 60 unnecessary. The reader must be aware of the difference in the representation of time between the UK and SI systems.

NOTE 3: Unless otherwise specified the ROC is in fpm. It is also possible that the ROC might be given in ft/s.

Climb Gradient

Climb is sometimes expressed in terms of % climb gradient. For instance, 14 CFR Part 23 refers to climb gradients in this fashion, in lieu of fpm or m/s, in order to present the regulatory requirements in a form applicable to all aircraft. The concept assumes no wind conditions and is defined as follows (see Figure 18-4):

image

image

FIGURE 18-4 Distance components during climb.

Consider an airplane whose climb gradient is 0.1 at 100 KTAS (nm/hr) in no wind conditions. Its rate-of-climb in fpm would be:

image

18.3 General Climb Analysis Methods

Armed with the equation of motion derived in the previous section, we can now begin to evaluate the climb characteristics of the new aircraft design. In this section, we will introduce a number of methods to estimate the most important climb properties of the aircraft. Note that the methods presented utilize the simplified drag model. As has been stated before, this can lead to significant inaccuracies for aircraft whose CLminD > 0, as is the case for most aircraft that feature cambered airfoils (which most aircraft do). From that standpoint the presentation is somewhat misleading. The reader should regard the methods as an introduction to concepts that are commonly used in the industry. A method that allows for a detailed climb analysis of real aircraft, with adjusted drag polars, and even ones with a drag bucket, will be presented at the end of this section. The concepts that are presented at first will then come in handy.

18.3.1 General Rate-of-climb

This general expression is used to estimate the ROC based on thrust-to-weight ratio and wing loading. It is handy for evaluating climb performance during the design stage, but is also applicable to general climb performance analyses. It assumes the simplified drag model and returns the vertical airspeed in terms of ft/s or m/s.

It is computed from:

image (18-18)

Derivation of Equation (18-18)

Insert Equation (18-9) into (18-15) and manipulate algebraically, as shown:

image

Then, simplify to get:

image

Then, complete by rearranging:

image

QED

EXAMPLE 18-1

Determine the vertical airspeed for the sample aircraft flying at 250 KCAS at S-L at maximum thrust and at a weight of 20,000 lbf.

Solution

Dynamic pressure:

image

Insert and evaluate the rate-of-climb at the condition:

image

This amounts to 6792 fpm. Using this method, the excess power and rate-of-climb can be plotted for any altitude. Figure 18-5 depicts such these using the more realistic thrust modeling:

image

FIGURE 18-5 Excess power and rate-of-climb at three altitudes.

18.3.2 General Climb Angle

The climb angle is of great importance when it comes to obstruction clearance, or when showing compliance with noise regulations (14 CFR Part 36), as well as evaluation of deck angle during climb. It assumes the simplified drag model.

Computed from:

image (18-19)

General angle-of-climb:

image (18-20)

Derivation of Equations (18-19) and (18-20)

To get Equation (18-19), divide by V on either side of Equation (18-18):

image

QED

To get Equation (18-20), divide by V on either side of Equation (18-15):

image

QED

EXAMPLE 18-2

Determine the angle of climb for the sample aircraft flying at 250 KCAS at S-L at maximum thrust and at a weight of 20,000 lbf.

Solution

Dynamic pressure:

image

image

This amounts to 15.4°.

18.3.3 Max Climb Angle for a Jet

Determining the maximum climb angle is of great importance as this can be used to evaluate the capability of the aircraft to take off from runways in mountainous regions. Typical operational procedures would require an aircraft departing a runway surrounded by high mountains to climb at or near this value, at least until threatening terrain has been cleared. But there is another and very important reason to evaluate the maximum climb angle: noise certification (14 CFR Part 36). The steeper this angle, the farther away from the sound level meter will the airplane be when it is right above it (a regulatory requirement). The maximum climb angle is computed from the following expression and assumes the simplified drag model:

image (18-21)

Derivation of Equation (18-21)

Rewrite Equation (18-15):

image

Insert Equation (18-5):

image

Assuming cos θ ≈ 1:

image

The climb angle will reach an upper limit when the L/D is maximum, LDmax. In other words:

image

Recalling Equation (19-18), LDmax can be rewritten as follows:

image

QED

EXAMPLE 18-3

Determine the maximum angle-of-climb for the sample aircraft at S-L, maximum thrust and at a weight of 20,000 lbf.

Solution

image

18.3.4 Airspeed for θmax for a Jet (Best Angle-of-climb Speed)

To continue the discussion of obstruction clearance or compliance with noise regulations from previous analyses, the pilot would establish the best angle-of-climb as soon as possible after take-off by reaching and maintaining the best angle-of-climb airspeed. In the case of a jet aircraft, this airspeed can be calculated from the following expression, which assumes the simplified drag model. Note that this result is not valid for propeller-powered aircraft. The airspeed returned is in units of ft/s if the input values are in the UK system, but m/s if the SI system is used.

image (18-22)

This airspeed is recognized by pilots and regulation authorities as VX. In short, it results in the steepest possible climb angle for a jet aircraft, or the largest gain of altitude per unit horizontal distance.

Derivation of Equation (18-22)

Rewrite Equation (18-3):

image

Lift coefficient for LDmax:

image

We insert this back into Equation (18-3), where θ and V become θmax and VX:

image

QED

EXAMPLE 18-4

Determine the airspeed for maximum angle-of-climb for the sample aircraft at S-L, maximum thrust and at a weight of 20,000 lbf.

Solution

image

This amounts to 175 KTAS (or KCAS since this is S-L).

18.3.5 ROC for θmax for a Jet

It is also of interest to calculate the ROC associated with VX. Note that this is less than the ROC associated with VY (the largest gain in altitude per unit time). For a jet climbing while maintaining its best angle-of-climb airspeed, VX, the ROC can be calculated from:

image (18-23)

Derivation of Equation (18-23)

Use θmax and VX with Equation (18-11):

image

QED

EXAMPLE 18-5

Determine the ROC at maximum angle-of-climb for the sample aircraft at S-L, maximum thrust and at a weight of 20,000 lbf.

Solution

image

18.3.6 Airspeed for Best ROC for a Jet

As the airspeed of an airplane is changed at a given power setting, so is its ROC. At a particular airspeed the ROC will reach its maximum value. At that airspeed, the airplane will increase its altitude in the least amount of time. This airspeed is particularly important for fuel-thirsty jets and allows them to reach their cruise altitude with the least amount of fuel consumed. This airspeed can be computed from the expression below that assumes the simplified drag model:

image (18-24)

Note that the above formulation can also be used to estimate the best rate-of-climb airspeed for a multiengine aircraft suffering a one engine inoperative (OEI) condition. This airspeed is denoted by VYSE. The thrust, of course, must be reduced by the contribution of the failed engine and the minimum drag CDmin must be increased to account for the asymmetric attitude of the airplane necessary to fly straight and level. Also, LDmax must be recalculated as it reduces at this condition.

Derivation of Equation (18-24)

The general rate-of-climb is given by Equation (18-18):

image

Assume cos θ ∼ 1 and differentiate with respect to V, as follows:

image

Manipulating algebraically;

image

image

As usual, maximum (or minimum) can be found where the derivative equals zero. Setting the result to zero and multiply through by V2 leads to:

image

image (i)

Then, divide through by the constant multiplied to V4:

image (ii)

Then, note that the last term resembles the expression for max L/D (see Equation (19-18)):

image

Therefore:

image (iii)

Let’s rewrite Equation (iii), noting that T/S = (T/W)(W/S):

image (iv)

For convenience, define the variables Q and x as follows:

image

Let's rewrite Equation (iv) using these definitions:

image (v)

This is a quadratic equation in terms of x (or V2) whose solution is given by:

image (vi)

Factor (T/W)Q out of the radical to get:

image (vii)

or:

image (viii)

Only the positive sign in front of the radical makes physical sense. Writing this in terms of the original definitions of Q and x leads to:

image (ix)

QED

EXAMPLE 18-6

Determine the airspeed for maximum rate-of-climb for the sample aircraft at S-L, maximum thrust and at a weight of 20,000 lbf.

Solution

Step 1: Determine LDmax :

image

Step 2: Determine T/W and W/S:

image

Step 3: Best rate-of-climb speed is therefore:

image

18.3.7 Best ROC for a Jet

It is clearly evident from Figure 18-5 that the ROC varies with airspeed. Its maximum value is called the best rate-of-climb. For a jet, the value of this ROC can be computed from the following expression. Note that the expression assumes the simplified drag model:

image (18-25)

Derivation of Equation (18-25)

The equation for the best ROC is obtained by substituting VROCmax from Equation (18-24) in Equation (18-18). In order to simplify the resulting expression, let’s define the variable Z such that:

image (i)

Then, Equation (18-24) becomes:

image (ii)

Then, substitute Equation (ii) into Equation (18-18):

image

This results in:

image

Simplifying yields:

image (iii)

Again, resorting to the term for max L/D of Equation (19-18):

image

We use this to modify the last term of Equation (iii) (and of course noting that the following arithmetic scheme holds: 6 = 12/2 = 4·3/2 = (3/2)·4):

image (iv)

Inserting this into Equation (iii) we can now rewrite:

image

Rearranging:

image

And finally:

image

QED

18.3.8 Airspeed for θmax for a Propeller-powered Airplane

The best angle-of-climb for a propeller powered airplane is found by solving for VX using the following expression:

image (18-26)

A closed-form solution of this equation is not known and its solution requires an iterative numerical scheme. Note that in order to obtain the θmax the equation is solved for VX. Then, this can be used with Equations (18-18) through (18-20) to obtain θmax.

It is important to note that the value of VX is frequently less than the stalling speed – in particular for light aircraft. Theoretically, this means the airplane cannot achieve a maximum θmax. However, it is important to keep in mind the lessons of Chapter 15, Aircraft drag analysis, regarding the simplified drag model on which Equation (18-26) is based. As is clearly demonstrated (for instance, see Section 15.2.2, Quadratic drag modeling), this drag model is invalid at low AOAs and, consequently, the airplane no longer complies with any formulation based on the model. In practice, real airplanes all have a maximum climb angle higher than their stalling speed. To obtain it analytically, a more sophisticated drag modeling must be employed, for instance using the method presented in Section 15.2.3, Approximating the drag coefficient at high lift coefficients.

Derivation of Equation (18-26)

The thrust of a propeller driven airplane is given by Equation (14-38):

image

Additionally, the climb angle is given by Equation (18-19):

image

Inserting the expression for thrust into the above equation, and writing it explicitly in terms of V, results in:

image

Then, differentiate with respect to V:

image

Set to zero to get the maximum:

image

Multiply by V3 for convenience and prepare as a polynomial:

image

Simplify further:

image

The solution can be found by an iterative scheme and may include assuming cos θ ∼ 1, but this yields the following expression, for which V is rewritten as VX:

image

QED

18.3.9 Airspeed for Best ROC for a Propeller-powered Airplane

The airspeed for best ROC for a propeller powered aircraft, assuming the simplified drag model is calculated from the following expression:

image (18-27)

The units are in terms of ft/s or m/s, depending on input values.

The expression identifies the location of the maximum excess power in terms of airspeed. Figure 18-6 plots power available and power required versus true airspeed for a typical small, propeller-powered aircraft and assumes constant engine power with airspeed. The method presented here shows that the best ROC airspeed will occur where the difference between the two is the greatest, at around 45 KTAS.

image

FIGURE 18-6 Power available and power required for a propeller-powered airplane.

As discussed in Section 18.3.6, Airspeed for best ROC for a jet, the above formulation can also be used to estimate the best rate-of-climb airspeed for a multiengine aircraft during an OEI condition. This airspeed is denoted by VYSE. This can be done by reducing the contribution of the failed engine to the total power available and the minimum drag, CDmin, must be increased and LDmax must be reduced to account for the asymmetric attitude of the airplane.

Derivation of Equation (18-27)

Equation (18-15) shows that the maximum ROC occurs when (seen graphically in Figures 18-7 and 18-4):

image (18-28)

From the power studies (see Analyses 3 through 18 in Chapter 19, Performance – cruise) we know that the maximum excess power occurs at the airspeed where required power is minimum, but this required the ratio CL1.5/CD to be at its maximum.

image

FIGURE 18-7 Best ROC (fpm) as a function of airspeed (KCAS) and propeller efficiency.

Therefore, the maximum rate-of-climb occurs at the airspeed of minimum power required, but this is given by Equation (18-27).

QED

EXAMPLE 18-7

Since Equation (18-27) is based on the simplified drag model, it is of interest to evaluate its accuracy for a real airplane. Here, determine the airspeed for best rate-of-climb for the SR22 at a weight of 3400 lbf at S-L and 10,000 ft. Compare to the POH value (VY = 101 KCAS at S-L and 96 KCAS at 10,000 ft). Use the minimum drag coefficient extracted for the airplane in Example 15-18 (CDmin = 0.02541), S = 144.9 ft2, AR = 10, and the Oswald efficiency as also calculated in the example, which amounts to 0.7566.

Solution

The density of air at S-L is 0.002378 slugs/ft3 and at 10,000 ft it is 0.001756 slugs/ft3. Also, the lift-induced drag constant k = 1/(π·AR·e) = 0.04207. We can calculate the VY by substituting the values given in the table into the Equation (18-27), first at S-L as follows:

image

And then at 10,000 ft:

image

This also amounts to 71.7 KCAS, which contrasts with 101 KCAS at S-L and 96 KCAS at 10,000 ft for the real aircraft. It can be seen that the simplified drag model yields a poor approximation to the real airplane and the reader must be aware of this shortcoming. This is in part due to the absence of the term CLminD in the simplified drag model.

18.3.10 Best Rate-of-climb for a Propeller-powered Airplane

The maximum ROC in ft/s or m/s for a propeller-powered aircraft can be calculated from:

image (18-29)

If the best rate-of-climb airspeed, VY, is known and the value is desired in terms of fpm, it can be calculated from:

image (18-30)

Note that the power, P, must be in terms of ft·lbf/s. For this reason, if power is given in BHP, it must be multiplied by the factor 550 to be converted to the proper units.

Derivation of Equations (18-29) and (18-30)

For a propeller-powered airplane the power available is given by: image

An expression for the best ROC can be obtained by inserting Equation (18-27) into Equation (18-18):

image

This is accomplished in the following manner.

Begin by expanding:

image (i)

Modify the term TV in Equation (i) by introducing power available for a piston engine and propeller efficiency:

image (ii)

Now, let’s insert Equation (18-27) in a specific manner into Equation (ii) and let’s use VY for the best ROC airspeed:

image

Further manipulation leads to:

image

Assuming cos2 θ ∼ 1 and using Equation (19-18) for LDmax we simplify further:

image

image

Finally, replacing the explicit expression for VROCmax into this equation leads to:

image

QED

EXAMPLE 18-8

Let’s evaluate the accuracy of Equation (18-30) in the same way that we did in Example 18-7. Here, determine the best rate-of-climb for the SR22 at a weight of 3400 lbf at S-L and compare to the POH value (ROCmax = 1398 fpm at S-L). Use the same constants as used in Example 18-7, but calculate and compare for three propeller efficiencies, ηp = 0.6, 0.7, and 0.8. The maximum engine power at S-L is 310 BHP. For this example, use the value of VY determined in Example 18-7, even though it has been shown to be incorrect.

Solution

The rate-of-climb will only be calculated using the first propeller efficiency, but the results for the other two will be shown. First, calculate the ROC at S-L assuming ηp = 0.6:

image

Results for the other values of the propeller efficiency over a range of hypothetical best rate-of-climb airspeeds are plotted in Figure 18-7. It indicates the propeller is a little over 70% efficient during climb, but also shows there is considerable discrepancy between the predicted and actual VY for the airplane. This is the risk of using the simplified drag model – it is convenience versus precision.

18.3.11 Time to Altitude

The time required to increase altitude, h, can be determined from the following expression. If the ROC is in terms of fpm, it will return time in minutes. If ROC is in terms of ft/s or m/s then the time will be in seconds.

image (18-31)

In this expression, h1 is the target altitude and h0 is the initial altitude (e.g. the airplane may begin a climb at 10,000 ft and level out at 30,000 ft). The minimum time to altitude is achieved if the pilot maintains the best ROC airspeed (VY) through the entire climb maneuver.

Rapid Approximation

For mathematical simplicity, it may be convenient to assume a constant value of ROC and take it out of the integral sign. The value of the ROC should be a representative value between the initial and final altitudes, denoted by the symbol ROCa and called the representative ROC. In the absence of a better value, the average of the ROC between the initial and final altitudes can be used, although the true value should be biased toward the higher altitude, as the aircraft will spend more time completing the last half of the climb than the first half. Since this approach treats the representative ROC as a constant, it can be taken out of the integral of Equation (18-31), yielding the following expression:

image (18-32)

If the ROC is known as a function of altitude and given by ROC(h) = A·h + B, then, using Equation (18-34) below, the value of ROCa can be found from the expression:

image (18-33)

If the initial and final altitudes are close, the ROCa can be approximated as the average of the ROC at the initial and final altitudes.

Linear Approximation

If a particular airspeed, such as VY, is maintained through the climb, the ROC will decrease in a fashion that is close to being linear. In this case, the ROC can be approximated with an equation of a line: ROC(h) = A·h + B. In this case, the time to altitude is given by:

image (18-34)

Derivation of Equation (18-31)

ROC is the rate of change of altitude dh/dt, that is:

image

QED

Derivation of Equation (18-33)

If ROC as a function of altitude can be approximated using the linear expression ROC(h) = A·h + B, then the representative ROC can be determined using:

image

QED

Derivation of Equation (18-34)

If ROC as a function of altitude can be approximated using the linear expression ROC(h) = A·h + B, then the time to altitude can be found from:

image

QED

EXAMPLE 18-9

Consider the ROC versus altitude graph for the Cirrus SR22 is shown in Figure 18-8. Assume the aircraft weighs 2900 lbf and is flying at 4000 ft when a climb at VY is initiated. How long will it take to get to 20,000 ft? Determine using the three following approaches: (1) ROCa is average of the ROC at the initial and final altitude. (2) Calculate ROCa using Equation (18-33). (3) Use Equation (18-34) directly.

image

FIGURE 18-8 Rate-of-climb (fpm) as a function of altitude (ft) for the Cirrus SR22 single-engine aircraft for two different weights – gross weight of 3400 lbf, and intermediary weight of 2900 lbf. (Source: SR22 POH, courtesy of Cirrus Aircraft)

Solution

(1) Calculate ROC at the initial and final altitudes using the trendline for 2900 lbf shown in Figure 18-8:

At 4000 ft:

image

At 20,000 ft:

image

Representative ROC:

image

From Equation (18-32):

image

(2) Calculate ROCa using Equation (18-33) where A = −0.0674 and B = 1731:

image

From Equation (18-32):

image

(3) Use Equation (18-34) directly:

image

18.3.12 Absolute/Service Ceiling Altitude

Two frequently referenced performance parameters are the absolute and service ceilings. The absolute ceiling is the maximum altitude at which the airplane can maintain level flight. The service ceiling is the altitude at which the aircraft is capable of some 100 fpm rate-of-climb. Generally, the designer must keep two things in mind regarding these altitudes and, thus, should treat them with caution.

EXAMPLE 18-10

Determine the absolute and service ceilings for the Cirrus SR22 at 2900 and 3400 lbf weights using the data from its POH, shown in Figure 18-9.

image

FIGURE 18-9 Determination of service and absolute ceilings.

Solution

If one has the pilot’s operating handbook (as we do for this example), it is relatively easy to determine a trendline. If working with a new design, a POH will not be ready, so the max ROC must be calculated for a number of altitudes e.g. S-L, 7000, and 14,000 ft or similar.

With respect to the SR22, using information from its POH we find that the resulting linear fit is given by:

At 2900 lbf:

image

At 3400 lbf:

image

From which we see that at the lighter weight (2900 lbf), the absolute ceiling is 25,666 ft and 23,496 ft at maximum gross weight (3400 lbf). Of course, the airplane would always weigh less than its maximum gross weight, as it would have to burn a considerable amount of fuel to get there in the first place. By the same token, it is easy to evaluate the service ceiling based on a 100 fpm climb rate:

At 2900 lbf:

image

At 3400 lbf:

image

First, each ceiling is highly dependent on the weight of the aircraft as well as atmospheric conditions and can deviate thousands of feet (or meters) from the calculated values. Second, modern-day regulations are often the arbitrator of altitudes rather than the capability of the aircraft. For instance, CFR 14 Part 23 stipulates requirements for an airplane to fly higher than 25,000 ft or 28,000 ft. These requirements have everything to do with the equipment the aircraft features and not its ability to reach those altitudes. Some business jets have theoretical service ceilings in the neighborhood of 35,000 ft or even higher, but are certified to fly only as high as 28,000 ft.

The theoretical absolute and service ceilings can be computed by the following method:

(1) Compute ROCmax at a number of altitudes.

(2) Create a trendline in the form of a line or a polynomial.

(3) Solve the trendline for ROC = 100 fpm. This is the service ceiling.

(4) Solve the trendline for ROC = 0 fpm. This is the absolute ceiling.

18.3.13 Numerical Analysis of the Climb Maneuver – Sensitivity Studies

The purpose of this section is to introduce a powerful method to calculate the ROC of a generic aircraft using numerical analysis. This will be accomplished through the preparation of an analysis spreadsheet, which here is prepared for a propeller aircraft. The spreadsheet offers far greater analysis power to the aircraft designer because it can handle all the non-linearity the preceding analysis methods cannot. It will allow the designer to estimate the ROC of an airplane whose CLminD > 0, and even aircraft whose drag polar features a drag bucket or lift-curve that becomes non-linear as a result of an early flow separation. A screen capture of the spreadsheet is shown in Figure 18-10. A description is given below:

image

FIGURE 18-10 Spreadsheet designed to estimate climb performance.

The general input values are self-explanatory in light of the preceding discussion and will not be elaborated on. The columns in the main table labeled from 1 through 13, on the other hand, require some explanation.

Column 1 contains a range of calibrated airspeeds (50 KCAS increasing by 5 KCAS to 150 KCAS). This is used to calculate the true airspeed in column 2 using Equation (16-33) as follows (using 100 KCAS as an example):

image

This is converted to ft/s in column 3 by multiplying by 1.688.

Column 4 contains dynamic pressure, calculated as follows:

image

Column 5 contains the lift coefficient, calculated as follows:

image

Column 6 contains the lift-induced drag coefficient, calculated as follows:

image

Column 7 contains the total drag coefficient, calculated as follows:

image

Column 8 contains the total drag, calculated as follows:

image

Note that, fundamentally, the method ‘doesn't care’ how the drag is determined. For instance, even though the adjusted drag model is used here, columns 6 through 8 could just as easily contain a non-quadratic drag coefficient. For instance, a lookup table containing a drag model with a drag bucket could simply replace CD (with columns 6 and 7 simply omitted). This gives the approach substantial power, because it is independent of the nature of the drag coefficient.

Column 9 contains the advance ratio, calculated using Equation (14-23) as follows:

image

The advance ratio is calculated because it is used in the expression for ηp, which was prepared using information from the propeller manufacturer. Note that the polynomial shown in the next step is prepared for this example and does not pertain to the actual aircraft.

Column 10 contains the propeller efficiency, ηp, calculated using the following hypothetical polynomial, just designed to generate reasonable token values:

image

When using the value of J calculated in column 9, this expression yields 0.7283.

Column 11 contains the propeller thrust, calculated using Equation (14-38), where the engine power at S-L (310 BHP) has been corrected to the altitude (here 5000 ft) using Equation (7-16):

image

Column 12 contains the excess power, calculated using Equation (18-14):

image

Column 13 (finally) contains the rate-of-climb, calculated using Equation (18-17):

image

By performing the same calculations for the other row, it is trivial to extract the maximum ROC using Microsoft Excel’s MAX() function, here found to equal 998 fpm at 90 KCAS (at 5000 ft). Note that more accurate calculations should take into account the weight reduction of the aircraft with altitude. For instance, the airplane will burn several gallons of fuel climbing to 10,000 ft and this will improve the ROC. Also note that the graph accompanying the spreadsheet contains a reference curve showing the climb performance at S-L. This is primarily done for convenience to help the designer realize performance degradation with altitude. It is left as an exercise for the reader to figure out how to do this (hint – it is simple). Once the spreadsheet is completed, it can be used to perform various sensitivity studies, of which three are shown below.

Altitude Sensitivity

Altitude sensitivity reveals how the design will be affected when operated at high altitudes or, worse yet, at high altitude on a hot day (high-density altitude). This is important when considering departure from high-altitude airports in mountainous regions. Such departures can pose serious challenges for pilots, particularly if the aircraft is fully loaded. For instance, the graph in Figure 18-11 shows that at 10,000 ft, the airplane is capable of climbing at about 650 fpm, less than ½ of its S-L capability.

image

FIGURE 18-11 Altitude sensitivity plot.

Weight Sensitivity

Weight sensitivity reveals how the design will be affected by deviations from the target design weight. This is important when demonstrating the importance of meeting target design weights. For instance, assume that the target gross weight of the airplane shown in Figure 18-12 is 3400 lbf. If the target is not met and the manufacturer is forced to increase it to, say, 3600 lbf, then the airplane’s best ROC is likely to drop from about 1400 fpm to 1260 fpm. This could have a significant impact on the competitiveness of the design project.

image

FIGURE 18-12 Weight sensitivity plot.

Propeller Efficiency Sensitivity

Propeller efficiency sensitivity reveals how the design will be affected if a substandard propeller is purchased, if an “aerodynamically” damaged propeller is used, or if the propeller does not meet the manufacturer’s claims of performance. For instance, if the selected propeller for the airplane shown in Figure 18-13 delivers merely 90% of the claimed propeller efficiency, then the best ROC is likely to drop from about 1400 fpm to 1170 fpm.

image

FIGURE 18-13 Propeller efficiency sensitivity plot.

It can also be helpful to extend the sensitivity evaluation to one that includes propeller and weight sensitivities.

18.4 Aircraft Database – Rate-of-climb of Selected Aircraft

Table 18-1 shows the rate-of-climb, best angle-of-climb, and best rate-of-climb airspeeds for selected classes of aircraft. This data is very helpful when evaluating the accuracy of one’s own calculations. Note that for heavier aircraft, it is practically impossible to specify a single VX or VY as these vary greatly with weight and are usually determined for the pilot on a trip-to-trip basis.

TABLE 18-1

T-O Performance of Selected Aircraft

Image

Image

Image

Unconf = unconfirmed. A reliable source has not been located for the specified value. Treat values with caution.

Variables

SymbolDescriptionUnits (UK and SI)
AOAAngle-of-attackDegrees or radians
AOCAngle-of-climbDegrees or radians
ARReference aspect ratio
bReference (typically wing) spanft or m
BHPBrake horsepowerBHP or hp
CDDrag coefficient
CDiInduced drag coefficient
CDminMinimum drag coefficient
CLLift coefficient
CL0Basic lift coefficient
CLminDLift coefficient of minimum drag
CLift curve slopePer radian or per degree
DDraglbf or N
DpPropeller diameterft or m
eOswald span efficiency factor
gAcceleration due to gravityft/s2 or m/s2
H, hAltitudeft or m
JAdvance ratio
kLift-induced drag constant
LLiftlbf or N
LDmaxMaximum lift-to-drag ratio
PEngine powerft·lbf/s or W = J/s
PAVPower availableft·lbf/s or W
PENGEngine powerft·lbf/s or W
PEXExcess powerft·lbf/s or W
PREQPower requiredft·lbf/s or W
qDynamic pressurelbf/ft2 or lbf/in2 or N/m2
RCmaxMaximum rate-of-climbft/s or fpm, m/s
ROCRate-of-climbft/min or m/s (typical)
ROCmaxBest rate-of-climbft/min or m/s (typical)
ROCXRate-of-climb at best angle-of-climb speedft/min or m/s (typical)
SReference wing areaft2 or m2
TEngine thrust (context dependent)lbf or N
tTimeseconds
TmaxMaximum engine thrustlbf or N
TREQThrust requiredlbf or N
VAirspeedft/s or m/s
VEmaxAirspeed for max endurance for a propeller airplaneft/s or knots, m/s or kmh
VHMaximum level (horizontal) airspeedft/s or m/s
VROCmaxAirspeed for best rate-of-climbft/s or m/s
VVVertical airspeedft/s or m/s
VXBest angle-of-climb airspeed (context dependent)ft/s or fpm, m/s
VXAirspeed along x-axis (see Figure 18-2) (context dependent) ft/s or m/s
VY Best rate-of-climb airspeed ft/s or fpm, m/s
VYSE Airspeed for best ROC at OEI condition ft/s or m/s
VZ Airspeed along z-axis (see Figure 18-2) ft/s or m/s
W Weight lbf or N
Z Simplification relating LDmax and T/W
α Angle-of-attack Degrees or radians
ε Thrust angle Degrees or radians
ηp Propeller efficiency
θ Aircraft climb angle (relative to horizon) Degrees or radians
θmax Maximum climb angle Degrees or radians
ρ Air density slugs/ft3 or kg/m3
ρSL Air density at sea level slugs/ft3 or kg/m3
σ Density ratio

References

1. Perkins CD, Hage RE. Airplane Performance, Stability, and Control. John Wiley & Sons 1949.

2. Torenbeek E. Synthesis of Subsonic Aircraft Design. 3rd ed. Delft University Press 1986.

3. Nicolai L. Fundamentals of Aircraft Design. 2nd ed. 1984.

4. Roskam J, Lan Chuan-Tau Edward. Airplane Aerodynamics and Performance. DARcorporation 1997.

5. Hale FJ. Aircraft Performance, Selection, and Design. John Wiley & Sons 1984; pp. 137–138.

6. Anderson Jr JD. Aircraft Performance & Design. 1st ed. McGraw-Hill 1998.

7. Austyn-Mair W, Birdsall DL. Aircraft Performance. Cambridge University Press 1992; pp. 47–49.

8. Jane’s All the World’s Aircraft. Various editors, Jane’s Yearbooks, various years.

9. Type Pilot’s Operating Handbook or Airman’s Information Manual.

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