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.
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
18.1.1 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)
18.3 General Climb Analysis Methods
Derivation of Equation (18-18)
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 θ_{max} for a Jet (Best Angle-of-climb Speed)
Derivation of Equation (18-22)
18.3.5 ROC for θ_{max} for a Jet
Derivation of Equation (18-23)
18.3.6 Airspeed for Best ROC for a Jet
Derivation of Equation (18-24)
Derivation of Equation (18-25)
18.3.8 Airspeed for θ_{max} 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)
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
Propeller Efficiency Sensitivity
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.
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.
• 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.
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.
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.
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.
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:
(18-1)
(18-2)
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.
Equations of motion for a steady climb:
(18-3)
(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):
(18-5)
The lift coefficient at this condition is thus:
(18-6)
The drag force, using the simplified drag model, is given by:
(18-7)
Inserting Equation (18-6) into Equation (18-7) yields:
(18-8)
Expanding:
(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.
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:
(18-10)
Then it is possible to define the vertical airspeed, also called the rate-of-climb:
(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; V_{V} is observed on the vertical speed indicator (VSI); and V_{H} is the ground speed. Note that this V_{H} should not be confused with the maximum level airspeed to be discussed in Chapter 19.
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·lb_{f}/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.
The vertical airspeed can be estimated if thrust or power and drag characteristics of the aircraft are known.
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 V_{V}, 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 (P_{AV}) must be larger than the drag power (DV) or required power (P_{REQ}) for level flight.
We note that power is defined as force × speed:
where P_{AV} = η_{p}·P_{ENG}.
QED
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:
(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 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):
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:
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 C_{LminD} > 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.
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:
(18-18)
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.
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:
(18-21)
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.
(18-22)
This airspeed is recognized by pilots and regulation authorities as V_{X}. In short, it results in the steepest possible climb angle for a jet aircraft, or the largest gain of altitude per unit horizontal distance.
It is also of interest to calculate the ROC associated with V_{X}. Note that this is less than the ROC associated with V_{Y} (the largest gain in altitude per unit time). For a jet climbing while maintaining its best angle-of-climb airspeed, V_{X}, the ROC can be calculated from:
(18-23)
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:
(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 V_{YSE}. The thrust, of course, must be reduced by the contribution of the failed engine and the minimum drag C_{Dmin} must be increased to account for the asymmetric attitude of the airplane necessary to fly straight and level. Also, LD_{max} must be recalculated as it reduces at this condition.
The general rate-of-climb is given by Equation (18-18):
Assume cos θ ∼ 1 and differentiate with respect to V, as follows:
Manipulating algebraically;
As usual, maximum (or minimum) can be found where the derivative equals zero. Setting the result to zero and multiply through by V^{2} leads to:
(i)
Then, divide through by the constant multiplied to V^{4}:
(ii)
Then, note that the last term resembles the expression for max L/D (see Equation (19-18)):
Let’s rewrite Equation (iii), noting that T/S = (T/W)(W/S):
(iv)
For convenience, define the variables Q and x as follows:
Let's rewrite Equation (iv) using these definitions:
(v)
This is a quadratic equation in terms of x (or V^{2}) whose solution is given by:
(vi)
Factor (T/W)Q out of the radical to get:
(vii)
or:
(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:
(ix)
QED
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:
(18-25)
The equation for the best ROC is obtained by substituting V_{ROCmax} from Equation (18-24) in Equation (18-18). In order to simplify the resulting expression, let’s define the variable Z such that:
(i)
Then, Equation (18-24) becomes:
(ii)
Then, substitute Equation (ii) into Equation (18-18):
This results in:
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):
(iv)
Inserting this into Equation (iii) we can now rewrite:
QED
The best angle-of-climb for a propeller powered airplane is found by solving for V_{X} using the following expression:
(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 θ_{max} the equation is solved for V_{X}. Then, this can be used with Equations (18-18) through (18-20) to obtain θ_{max}.
It is important to note that the value of V_{X} is frequently less than the stalling speed – in particular for light aircraft. Theoretically, this means the airplane cannot achieve a maximum θ_{max}. 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.
The thrust of a propeller driven airplane is given by Equation (14-38):
Additionally, the climb angle is given by Equation (18-19):
Inserting the expression for thrust into the above equation, and writing it explicitly in terms of V, results in:
Then, differentiate with respect to V:
Set to zero to get the maximum:
Multiply by V^{3} for convenience and prepare as a polynomial:
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 V_{X}:
QED
The airspeed for best ROC for a propeller powered aircraft, assuming the simplified drag model is calculated from the following expression:
(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.
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 V_{YSE}. This can be done by reducing the contribution of the failed engine to the total power available and the minimum drag, C_{Dmin}, must be increased and LD_{max} must be reduced to account for the asymmetric attitude of the airplane.
Equation (18-15) shows that the maximum ROC occurs when (seen graphically in Figures 18-7 and 18-4):
(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 C_{L}^{1.5}/C_{D} to be at its maximum.
Therefore, the maximum rate-of-climb occurs at the airspeed of minimum power required, but this is given by Equation (18-27).
QED
The maximum ROC in ft/s or m/s for a propeller-powered aircraft can be calculated from:
(18-29)
If the best rate-of-climb airspeed, V_{Y}, is known and the value is desired in terms of fpm, it can be calculated from:
(18-30)
Note that the power, P, must be in terms of ft·lb_{f}/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.
For a propeller-powered airplane the power available is given by:
An expression for the best ROC can be obtained by inserting Equation (18-27) into Equation (18-18):
This is accomplished in the following manner.
Modify the term TV in Equation (i) by introducing power available for a piston engine and propeller efficiency:
(ii)
Now, let’s insert Equation (18-27) in a specific manner into Equation (ii) and let’s use V_{Y} for the best ROC airspeed:
Further manipulation leads to:
Assuming cos^{2} θ ∼ 1 and using Equation (19-18) for LD_{max} we simplify further:
Finally, replacing the explicit expression for V_{ROCmax} into this equation leads to:
QED
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.
(18-31)
In this expression, h_{1} is the target altitude and h_{0} 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 (V_{Y}) through the entire climb maneuver.
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 ROC_{a} 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:
(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 ROC_{a} can be found from the expression:
(18-33)
If the initial and final altitudes are close, the ROC_{a} can be approximated as the average of the ROC at the initial and final altitudes.
If a particular airspeed, such as V_{Y}, 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:
(18-34)
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:
QED
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:
QED
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.
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:
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 C_{LminD} > 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:
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):
This is converted to ft/s in column 3 by multiplying by 1.688.
Column 4 contains dynamic pressure, calculated as follows:
Column 5 contains the lift coefficient, calculated as follows:
Column 6 contains the lift-induced drag coefficient, calculated as follows:
Column 7 contains the total drag coefficient, calculated as follows:
Column 8 contains the total drag, calculated as follows:
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:
The advance ratio is calculated because it is used in the expression for η_{p}, 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, η_{p}, calculated using the following hypothetical polynomial, just designed to generate reasonable token values:
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):
Column 12 contains the excess power, calculated using Equation (18-14):
Column 13 (finally) contains the rate-of-climb, calculated using Equation (18-17):
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 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.
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 lb_{f}. If the target is not met and the manufacturer is forced to increase it to, say, 3600 lb_{f}, 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.
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.
It can also be helpful to extend the sensitivity evaluation to one that includes propeller and weight sensitivities.
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 V_{X} or V_{Y} as these vary greatly with weight and are usually determined for the pilot on a trip-to-trip basis.
Symbol | Description | Units (UK and SI) |
AOA | Angle-of-attack | Degrees or radians |
AOC | Angle-of-climb | Degrees or radians |
AR | Reference aspect ratio | |
b | Reference (typically wing) span | ft or m |
BHP | Brake horsepower | BHP or hp |
C_{D} | Drag coefficient | |
C_{Di} | Induced drag coefficient | |
C_{Dmin} | Minimum drag coefficient | |
C_{L} | Lift coefficient | |
C_{L0} | Basic lift coefficient | |
C_{LminD} | Lift coefficient of minimum drag | |
C_{Lα} | Lift curve slope | Per radian or per degree |
D | Drag | lb_{f} or N |
D_{p} | Propeller diameter | ft or m |
e | Oswald span efficiency factor | |
g | Acceleration due to gravity | ft/s^{2} or m/s^{2} |
H, h | Altitude | ft or m |
J | Advance ratio | |
k | Lift-induced drag constant | |
L | Lift | lb_{f} or N |
LD_{max} | Maximum lift-to-drag ratio | |
P | Engine power | ft·lb_{f}/s or W = J/s |
P_{AV} | Power available | ft·lb_{f}/s or W |
P_{ENG} | Engine power | ft·lb_{f}/s or W |
P_{EX} | Excess power | ft·lb_{f}/s or W |
P_{REQ} | Power required | ft·lb_{f}/s or W |
q | Dynamic pressure | lb_{f}/ft^{2} or lb_{f}/in^{2} or N/m^{2} |
RC_{max} | Maximum rate-of-climb | ft/s or fpm, m/s |
ROC | Rate-of-climb | ft/min or m/s (typical) |
ROC_{max} | Best rate-of-climb | ft/min or m/s (typical) |
ROC_{X} | Rate-of-climb at best angle-of-climb speed | ft/min or m/s (typical) |
S | Reference wing area | ft^{2} or m^{2} |
T | Engine thrust (context dependent) | lb_{f} or N |
t | Time | seconds |
T_{max} | Maximum engine thrust | lb_{f} or N |
T_{REQ} | Thrust required | lb_{f} or N |
V | Airspeed | ft/s or m/s |
V_{Emax} | Airspeed for max endurance for a propeller airplane | ft/s or knots, m/s or kmh |
V_{H} | Maximum level (horizontal) airspeed | ft/s or m/s |
V_{ROCmax} | Airspeed for best rate-of-climb | ft/s or m/s |
V_{V} | Vertical airspeed | ft/s or m/s |
V_{X} | Best angle-of-climb airspeed (context dependent) | ft/s or fpm, m/s |
V_{X} | Airspeed along x-axis (see Figure 18-2) (context dependent) | ft/s or m/s |
V_{Y} | Best rate-of-climb airspeed | ft/s or fpm, m/s |
V_{YSE} | Airspeed for best ROC at OEI condition | ft/s or m/s |
V_{Z} | Airspeed along z-axis (see Figure 18-2) | ft/s or m/s |
W | Weight | lb_{f} or N |
Z | Simplification relating LD_{max} and T/W | |
α | Angle-of-attack | Degrees or radians |
ε | Thrust angle | Degrees or radians |
η_{p} | Propeller efficiency | |
θ | Aircraft climb angle (relative to horizon) | Degrees or radians |
θ_{max} | Maximum climb angle | Degrees or radians |
ρ | Air density | slugs/ft^{3} or kg/m^{3} |
ρ_{SL} | Air density at sea level | slugs/ft^{3} or kg/m^{3} |
σ | Density ratio |
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