**3. Zhukovsky curves**

Let us now consider steady horizontal flight. The scheme on **Figure 2** will be transformed to the following form (**Figure 9**):

In steady horizontal flight, we have the following equation of the force balance:

$$\begin{cases} L = \mathcal{G}, \\ T = D. \end{cases} \tag{5}$$

Based on the above result, we can state that the required thrust *Tr* for the steady horizontal flight should be equal to the sum of zero lift drag *D*<sup>0</sup> and lift-induced

<sup>2</sup> *Sref* <sup>¼</sup> <sup>1</sup>

As we can see, zero lift drag *D*<sup>0</sup> is proportional to the square of the air speed, while lift-induced drag *Di* is inversely proportional to the square of air speed. Let us now define the conditions of minimal drag or, which is the same as,

*Sref* � <sup>2</sup> <sup>1</sup>

*Sref* <sup>¼</sup> <sup>1</sup> *πλe*

It is obvious that the left-hand side of the above is zero lift drag *D*<sup>0</sup> and the right-

*D*<sup>0</sup> ¼ *Di*,

*cD*<sup>0</sup> ¼ *cDi:*

*cD* ¼ *cD*<sup>0</sup> þ *cDi* ¼ 2*cD*0*:*

As we remember from the drag polar, the same condition is true for maximal lift-to-drag ratio, so the conditions for minimal required thrust and maximal lift-todrag ratio are the same. To complete the calculations of all parameters for minimal required thrust, let us derive the expressions for lift coefficient and air speed:

*cDi* <sup>¼</sup> *cD*<sup>0</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup>

*cL* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi *πλecD*<sup>0</sup> p ,

> 4 s

*v*ð Þ *<sup>L</sup>=<sup>D</sup>* min ¼

*L πλe* ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 *πλecD*<sup>0</sup>

4*G*<sup>2</sup> *ρ*<sup>2</sup>*S*<sup>2</sup> *ref*

*:*

*πλe*

2*G*<sup>2</sup> *ρv*<sup>3</sup>*Sref*

2*G*<sup>2</sup> *ρv*<sup>2</sup>*Sref* ¼ 0*:*

*:* (6)

*πλe*

2*G*<sup>2</sup> *ρv*<sup>2</sup>*Sref :*

*Tr* ¼ *D*<sup>0</sup> þ *Di*,

*ρv*<sup>2</sup> 2 *Sref* ,

*ρv*<sup>2</sup>

*ρv* 2

*D*<sup>0</sup> ¼ *cD*<sup>0</sup>

8 >>>>><

>>>>>:

minimal required thrust at steady horizontal flight:

*∂Tr*

The above expression can be rewritten as:

hand side is lift-induced drag *Di*:

Thus, the drag coefficient is equal:

or

so,

**161**

and from Eq. (6)

*<sup>∂</sup><sup>v</sup>* <sup>¼</sup> <sup>2</sup>*cD*<sup>0</sup>

*cD*<sup>0</sup> *ρv*<sup>2</sup> 2

*Di* ¼ *cDi*

drag *Di*, which are defined as:

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

*Flight Vehicle Performance*

which is same as:

$$\begin{cases} \mathcal{c}\_L \frac{\rho v^2}{2} \mathcal{S}\_{\text{ref}} = \mathcal{G}, \\\\ T = \mathcal{c}\_D \frac{\rho v^2}{2} \mathcal{S}\_{\text{ref}} = \left( \mathcal{c}\_{D0} + \frac{\mathcal{c}\_L^2}{\pi \lambda \mathcal{e}} \right) \frac{\rho v^2}{2} \mathcal{S}\_{\text{ref}}. \end{cases}$$

From the first equation of the above system, we can find that

$$c\_L = \frac{2G}{\rho v^2 S\_{\eta f}},$$

and by substituting the value *cL* in the second equation we get:

$$T = c\_{D0} \frac{\rho v^2}{2} \mathcal{S}\_{r\circ f} + \frac{1}{\pi \lambda e} \frac{2G^2}{\rho v^2 \mathcal{S}\_{r\circ f}}.$$

**Figure 9.** *Force balance at steady horizontal flight.*

*Flight Vehicle Performance DOI: http://dx.doi.org/10.5772/intechopen.92105*

Examined material is one of core bases of aircraft performance, and the results obtained through the above analysis are used in studies of different flight paths and patterns and will be referred in next subsection dedicated to the Zhukovsky curves.

Let us now consider steady horizontal flight. The scheme on **Figure 2** will be

In steady horizontal flight, we have the following equation of the force balance:

*L* ¼ *G*, *T* ¼ *D:*

<sup>2</sup> *Sref* <sup>¼</sup> *cD*<sup>0</sup> <sup>þ</sup>

*cL* <sup>¼</sup> <sup>2</sup>*<sup>G</sup> ρv*<sup>2</sup>*Sref* ,

*Sref* þ

1 *πλe*

2*G*<sup>2</sup> *ρv*<sup>2</sup>*Sref :*

*c*2 *L πλe* � � *ρv*<sup>2</sup>

<sup>2</sup> *Sref :*

(5)

(

**3. Zhukovsky curves**

*Aerodynamics*

which is same as:

**Figure 9.**

**160**

*Force balance at steady horizontal flight.*

transformed to the following form (**Figure 9**):

*cL ρv*<sup>2</sup>

8 >>><

>>>:

*T* ¼ *cD*

<sup>2</sup> *Sref* <sup>¼</sup> *<sup>G</sup>*,

*ρv*<sup>2</sup>

From the first equation of the above system, we can find that

and by substituting the value *cL* in the second equation we get:

*ρv*<sup>2</sup> 2

*T* ¼ *cD*<sup>0</sup>

Based on the above result, we can state that the required thrust *Tr* for the steady horizontal flight should be equal to the sum of zero lift drag *D*<sup>0</sup> and lift-induced drag *Di*, which are defined as:

$$\begin{cases} T\_r = D\_0 + Di, \\ D\_0 = c\_{D0} \frac{\rho v^2}{2} S\_{r\text{rf}}, \\ D i = c\_{Di} \frac{\rho v^2}{2} S\_{r\text{rf}} = \frac{1}{\pi \lambda e} \frac{2G^2}{\rho v^2 S\_{r\text{rf}}}. \end{cases}$$

As we can see, zero lift drag *D*<sup>0</sup> is proportional to the square of the air speed, while lift-induced drag *Di* is inversely proportional to the square of air speed.

Let us now define the conditions of minimal drag or, which is the same as, minimal required thrust at steady horizontal flight:

$$\frac{\partial T\_r}{\partial v} = 2c\_{D0} \frac{\rho v}{2} \mathbf{S}\_{\rm ref} - 2 \frac{\mathbf{1}}{\pi \lambda e} \frac{2G^2}{\rho v^3 \mathbf{S}\_{\rm ref}} = \mathbf{0}.$$

The above expression can be rewritten as:

$$
\omega\_{D0} \frac{\rho v^2}{2} \mathcal{S}\_{\text{ref}} = \frac{1}{\pi \lambda e} \frac{2G^2}{\rho v^2 \mathcal{S}\_{\text{ref}}}.\tag{6}
$$

It is obvious that the left-hand side of the above is zero lift drag *D*<sup>0</sup> and the righthand side is lift-induced drag *Di*:

*D*<sup>0</sup> ¼ *Di*,

or

$$c\_{D0} = c\_{Di\cdot}.$$

Thus, the drag coefficient is equal:

$$
\mathcal{c}\_{D} = \mathcal{c}\_{D0} + \mathcal{c}\_{Di} = \mathcal{2}\mathcal{c}\_{D0}.
$$

As we remember from the drag polar, the same condition is true for maximal lift-to-drag ratio, so the conditions for minimal required thrust and maximal lift-todrag ratio are the same. To complete the calculations of all parameters for minimal required thrust, let us derive the expressions for lift coefficient and air speed:

$$
\mathcal{c}\_{Di} = \mathcal{c}\_{D0} = \frac{\sigma\_L^2}{\pi \lambda \mathcal{e}},
$$

so,

$$c\_L = \sqrt{\pi \lambda \overline{\operatorname{acc}\_{D0}}},$$

and from Eq. (6)

$$\nu\_{(L/D)\text{ min}} = \sqrt[4]{\frac{1}{\pi \lambda \text{ec}\_{D0}} \frac{4G^2}{\rho^2 \mathcal{S}\_{\text{ref}}^2}}.$$

We can also find the characteristics of available thrust *Ta* provided by manufacturers of engines or the estimations of available thrust from the sources of literature. In [7, 8], the forms of dependency of available thrust on the air speed for several types of engines are presented. Particularly, in [3] we can find available thrust versus air speed at several altitudes for the jet aircraft L-39 (**Figure 10**).

Based on the available information, Nikolay Zhukovsky developed a graphical method for the analysis of the range of the horizontal flight speeds at different altitudes. His method is based on the plotting curves of zero lift drag and liftinduced drag versus air speed at different altitudes, graphically calculating their sum, and plotting the dependencies of available thrust on air speed at corresponding altitudes for graphoanalytic estimation of ranges of available air speeds for horizontal flight at different altitudes. The set of curves obtained through the abovedescribed procedure in memory of him are called Zhukovsky curves. The graph with curves of required and available thrusts or powers at certain altitude is also called performance diagram.

Let us implement the method proposed Zhukovsky for any altitude, for example, do a plot for the altitude of zero meters above sea level (**Figure 11**) and define the speed characteristics of the horizontal flight.

From Zhukovsky curves at sea level altitude, we can find the maximum speed *v*max corresponding to the right point of intersection of curves for required thrust *Tr*ð Þ*v* and available thrust *Ta*ð Þ*v* . At speed *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min for which zero lift drag becomes equal to lift-induced drag, *<sup>D</sup>*<sup>0</sup> *<sup>v</sup>*ð Þ *<sup>L</sup>=<sup>D</sup>* min <sup>¼</sup> *Di v*ð Þ *<sup>L</sup>=<sup>D</sup>* min , we get minimal required thrust *Tr* min <sup>¼</sup> <sup>2</sup>*D*<sup>0</sup> *<sup>v</sup>*ð Þ *<sup>L</sup>=<sup>D</sup>* min <sup>¼</sup> <sup>2</sup>*Di v*ð Þ *<sup>L</sup>=<sup>D</sup>* min . This speed is a very important

> quantity from the point of horizontal flight, as the flight at lower speeds is not stable to the accidental changes of speed and requires attention of pilots. On the other hand, the flight at higher speeds than *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min is stable to accidental changes of speed, as if speed decreases drag force also decreases and as the thrust was at initial value it accelerates the aircraft till there is a balance between thrust and drag. We get a similar picture for accidental increase of speed at stable region II, for which the drag force decelerates the aircraft till drag-thrust balance. For unstable region I, the dangerous case is the accidental decrease of speed, which increases drag leading to decelerating the aircraft till stall. To fly at unstable region of speeds, the pilot needs always to work with the throttle to increase thrust when it is required. Minimum speed *v*min at sea level flight is not defined from the above curves and refers to the

stall speed, which can be found from the condition of required lift:

or

**163**

**Figure 11.**

*Zhukovsky curves at sea level altitude.*

*Flight Vehicle Performance*

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

available thrust *Ta*ð Þ*v* .

*cL* max

*v*min ¼

*ρv*<sup>2</sup>

<sup>2</sup> *Sref* <sup>¼</sup> *<sup>G</sup>*,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*G cL* max *<sup>ρ</sup>Sref* <sup>s</sup>

For flight at higher altitudes, we can get conditions when the required thrust at the above minimum speed is much higher than the available thrust at that altitude (**Figure 12**). In such cases, minimum speed defined as minimum thrust speed *vT* min corresponds to the left point of intersection of curves for required thrust *Tr*ð Þ*v* and

The idea of plotting Zhukovsky curves at sea level flight allows us to have the

same graph at any altitude *H* for the required trust *Tr*ð Þ *vIAS* versus indicated

*:*

**Figure 10.** *Available thrust for jet aircraft L-39.*

We can also find the characteristics of available thrust *Ta* provided by manufacturers of engines or the estimations of available thrust from the sources of literature. In [7, 8], the forms of dependency of available thrust on the air speed for several types of engines are presented. Particularly, in [3] we can find available thrust versus air speed at several altitudes for the jet aircraft L-39 (**Figure 10**).

Based on the available information, Nikolay Zhukovsky developed a graphical method for the analysis of the range of the horizontal flight speeds at different altitudes. His method is based on the plotting curves of zero lift drag and liftinduced drag versus air speed at different altitudes, graphically calculating their sum, and plotting the dependencies of available thrust on air speed at corresponding altitudes for graphoanalytic estimation of ranges of available air speeds for horizontal flight at different altitudes. The set of curves obtained through the abovedescribed procedure in memory of him are called Zhukovsky curves. The graph with curves of required and available thrusts or powers at certain altitude is also

Let us implement the method proposed Zhukovsky for any altitude, for example, do a plot for the altitude of zero meters above sea level (**Figure 11**) and define

From Zhukovsky curves at sea level altitude, we can find the maximum speed *v*max corresponding to the right point of intersection of curves for required thrust *Tr*ð Þ*v* and available thrust *Ta*ð Þ*v* . At speed *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min for which zero lift drag becomes

<sup>¼</sup> *Di v*ð Þ *<sup>L</sup>=<sup>D</sup>* min

<sup>¼</sup> <sup>2</sup>*Di v*ð Þ *<sup>L</sup>=<sup>D</sup>* min

, we get minimal required

. This speed is a very important

called performance diagram.

*Aerodynamics*

thrust *Tr* min ¼ 2*D*<sup>0</sup> *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min

**Figure 10.**

**162**

*Available thrust for jet aircraft L-39.*

the speed characteristics of the horizontal flight.

equal to lift-induced drag, *D*<sup>0</sup> *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min

**Figure 11.** *Zhukovsky curves at sea level altitude.*

quantity from the point of horizontal flight, as the flight at lower speeds is not stable to the accidental changes of speed and requires attention of pilots. On the other hand, the flight at higher speeds than *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min is stable to accidental changes of speed, as if speed decreases drag force also decreases and as the thrust was at initial value it accelerates the aircraft till there is a balance between thrust and drag. We get a similar picture for accidental increase of speed at stable region II, for which the drag force decelerates the aircraft till drag-thrust balance. For unstable region I, the dangerous case is the accidental decrease of speed, which increases drag leading to decelerating the aircraft till stall. To fly at unstable region of speeds, the pilot needs always to work with the throttle to increase thrust when it is required. Minimum speed *v*min at sea level flight is not defined from the above curves and refers to the stall speed, which can be found from the condition of required lift:

$$c\_{L\max} \frac{\rho v^2}{2} \mathcal{S}\_{r\circ f} = G\_\*$$

or

$$v\_{\min} = \sqrt{\frac{2G}{c\_{L\max} \rho \mathcal{S}\_{ref}}}$$

For flight at higher altitudes, we can get conditions when the required thrust at the above minimum speed is much higher than the available thrust at that altitude (**Figure 12**). In such cases, minimum speed defined as minimum thrust speed *vT* min corresponds to the left point of intersection of curves for required thrust *Tr*ð Þ*v* and available thrust *Ta*ð Þ*v* .

The idea of plotting Zhukovsky curves at sea level flight allows us to have the same graph at any altitude *H* for the required trust *Tr*ð Þ *vIAS* versus indicated

**Figure 12.** *On minimum thrust speed.*

airspeed (IAS), which is usually measured on aircraft and related to the true airspeed (TAS) *vTAS* via the expression:

$$
\upsilon\_{\text{IAS}} = \upsilon\_{\text{TAS}} \sqrt{\frac{\rho\_H}{\rho\_0}},
$$

where *ρ*<sup>0</sup> is the air density at sea level and *ρ<sup>H</sup>* is the air density at altitude *H* above sea level (ASL).

Based on the Zhukovsky curves based on IAS, we can define the theoretical or static ceiling of horizontal flight (**Figure 12**), which is an altitude where the horizontal flight is possible only with IAS equal to *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min . The service ceiling has a more practical meaning, as it is the altitude where rate of climb (ROC) becomes less than 0.5 m/s [6, 7].

There are also defined such concepts as speed of maximum endurance *ve*, cruise speed *vcr*, and speed of maximum range *vr*. The maximum endurance speed and maximum range speed depend on fuel consumption characteristics of engine. The speed corresponding to the minimum hourly fuel consumption of engine is called speed of maximum endurance or economical speed. On the other hand, the speed corresponding to the minimum per-kilometer consumption of engine is the speed of maximum range and it is very close to the cruise speed (slightly more than cruise speed for real aircraft). The cruise speed is the speed at which the ratio of drag to speed is minimal, and can be found using Zhukovsky curves by drawing a tangent to the required thrust graph from the origin (**Figure 13**).

*L* ¼ *G* cosj j *θ* , *D* ¼ *G* sin j j *θ :*

tan j j *<sup>θ</sup>* <sup>¼</sup> *<sup>D</sup>*

The flattest glide corresponds to the minimum magnitude of flight path angle j j *θ min*, which is case when the tan j j *θ* is minimum, thus, we can write down the

*<sup>L</sup>* <sup>¼</sup> <sup>1</sup> *k :*

(

*Force balance for descending and ascending flights. (a) Gliding flight; (b) ascending flight.*

where *θ* is the flight path angle. Based on the above we can get:

following:

**165**

**Figure 13.**

**Figure 14.**

*On cruise and other speeds.*

*Flight Vehicle Performance*

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

Let us now consider descending flight or glide (**Figure 14a**) and ascending (**Figure 14b**) flight of aircraft.

In descending flight, the throttle is usually set to minimum, so the thrust can be neglected and consider the gliding flight (**Figure 14a**), for which we can write down the following equations of force balance:

*Flight Vehicle Performance DOI: http://dx.doi.org/10.5772/intechopen.92105*

**Figure 13.** *On cruise and other speeds.*

airspeed (IAS), which is usually measured on aircraft and related to the true

*vIAS* ¼ *vTAS*

where *ρ*<sup>0</sup> is the air density at sea level and *ρ<sup>H</sup>* is the air density at altitude *H*

Based on the Zhukovsky curves based on IAS, we can define the theoretical or static ceiling of horizontal flight (**Figure 12**), which is an altitude where the horizontal flight is possible only with IAS equal to *v*ð Þ *<sup>L</sup>=<sup>D</sup>* min . The service ceiling has a more practical meaning, as it is the altitude where rate of climb (ROC) becomes less

There are also defined such concepts as speed of maximum endurance *ve*, cruise speed *vcr*, and speed of maximum range *vr*. The maximum endurance speed and maximum range speed depend on fuel consumption characteristics of engine. The speed corresponding to the minimum hourly fuel consumption of engine is called speed of maximum endurance or economical speed. On the other hand, the speed corresponding to the minimum per-kilometer consumption of engine is the speed of maximum range and it is very close to the cruise speed (slightly more than cruise speed for real aircraft). The cruise speed is the speed at which the ratio of drag to speed is minimal, and can be found using Zhukovsky curves by drawing a tangent

Let us now consider descending flight or glide (**Figure 14a**) and ascending

neglected and consider the gliding flight (**Figure 14a**), for which we can write

In descending flight, the throttle is usually set to minimum, so the thrust can be

to the required thrust graph from the origin (**Figure 13**).

down the following equations of force balance:

ffiffiffiffiffi *ρH ρ*0 r ,

airspeed (TAS) *vTAS* via the expression:

above sea level (ASL).

**Figure 12.**

*Aerodynamics*

*On minimum thrust speed.*

than 0.5 m/s [6, 7].

(**Figure 14b**) flight of aircraft.

**164**

**Figure 14.** *Force balance for descending and ascending flights. (a) Gliding flight; (b) ascending flight.*

$$\begin{cases} L = G \cos |\theta|, \\ D = G \sin |\theta|. \end{cases}$$

where *θ* is the flight path angle. Based on the above we can get:

$$\tan|\theta| = \frac{D}{L} = \frac{1}{k} \,\mathrm{s}$$

The flattest glide corresponds to the minimum magnitude of flight path angle j j *θ min*, which is case when the tan j j *θ* is minimum, thus, we can write down the following:

$$\left| \theta \right|\_{\min} = \text{atan} \frac{1}{k\_{\max}}$$

*:*

The above means that the flattest glide, resulting the longest gliding distance, is also related to maximum lift-to-drag ratio *k*max.

For the ascending flight the force balance is presented as follows:

$$\begin{cases} L = G \cos \theta, \\ T - D = G \sin \theta. \end{cases} \tag{7}$$

which corresponds to

*DOI: http://dx.doi.org/10.5772/intechopen.92105*

*Flight Vehicle Performance*

combat use.

**Figure 16.** *Performance limits.*

**167**

flight for passengers.

for aircraft to climb to the required altitude.

**4. Flight envelope and operational limits of aircraft**

*<sup>θ</sup>*max <sup>¼</sup> asin ð Þ *Ta* � *Tr* max

In ideal jet aircraft case, the angle of steepest climb is defined at the speed of maximum lift-to-drag ratio. The most economical ascending flight is defined by the operational regime of aircraft engine at which the minimum fuel will be consumed

For any aircraft, performance and operational limits are defined. Performance limits (**Figure 16**) are mostly defined by the aerodynamic configuration of aircraft. On the other hand, operational limits are based on the type of aircraft, structural and engine limits, wind resistance parameters, and maximum Mach number. All these limits are presented in diagrams of altitude *h* versus airspeed *v* and the final diagram, which represents the intersection of all limits, is called flight envelope (**Figure 17**). The flight envelope can be represented in graphs of altitude versus true airspeed, altitude versus indicated airspeed, or altitude versus Mach number.

All the speeds presented on **Figure 16** were described in the previous subsection. Operational limits related to the type of aircraft are described from the point of view of aircraft application: if an aircraft is a passenger jet, it should apply to the requirements of comfort for passengers and not exceed load factors of comfortable flight; on the other hand, if an aircraft is a jet fighter its operational limits from the point of view of load factors should be derived from compromise between the prevention of health issues of pilot that may occur and maneuverability for the

Structural limits are mostly related to aircraft strength, while the engine limits can be the result of its design and performance at higher altitudes. Wind resistance limits can be derived from the requirements of operational use or comfortability of

*G :*

As we know, the rate of climb is the projection of airspeed to the vertical plane:

$$ROC = v \sin \theta.$$

From second equation of Eq. (7) we can get:

$$
\sin \theta = \frac{T - D}{G},
$$

or which is same as:

$$ROC = v \sin \theta = \frac{Tv - Dv}{G} = \frac{P\_a - P\_r}{G},$$

where *Pa* ¼ *Tv* is the available power, *Pr* ¼ *Dv* is the required power. From the above we can find the maximum of ROC, that is, *ROC*max

corresponding to maximum of excess power Δ*P*max ¼ ð Þ *Pa* � *Pr* max (**Figure 15**).

The angle *θ*max of steepest ascending flight can be found by dividing the ROC value by speed [9]:

$$
\theta\_{\text{max}} = \text{asin}\left(\frac{ROC}{v}\right)\_{\text{max}},
$$

**Figure 15.** *Power diagram for jet aircraft.*

*Flight Vehicle Performance DOI: http://dx.doi.org/10.5772/intechopen.92105*

which corresponds to

j j *θ min* ¼ atan

For the ascending flight the force balance is presented as follows:

(

is also related to maximum lift-to-drag ratio *k*max.

From second equation of Eq. (7) we can get:

or which is same as:

*Aerodynamics*

value by speed [9]:

**Figure 15.**

**166**

*Power diagram for jet aircraft.*

The above means that the flattest glide, resulting the longest gliding distance,

*L* ¼ *G* cos *θ*,

*T* � *D* ¼ *G* sin *θ:*

As we know, the rate of climb is the projection of airspeed to the vertical plane:

*ROC* ¼ *v* sin *θ:*

sin *<sup>θ</sup>* <sup>¼</sup> *<sup>T</sup>* � *<sup>D</sup>*

*ROC* <sup>¼</sup> *<sup>v</sup>* sin *<sup>θ</sup>* <sup>¼</sup> *Tv* � *Dv*

where *Pa* ¼ *Tv* is the available power, *Pr* ¼ *Dv* is the required power. From the above we can find the maximum of ROC, that is, *ROC*max corresponding to maximum of excess power Δ*P*max ¼ ð Þ *Pa* � *Pr* max (**Figure 15**). The angle *θ*max of steepest ascending flight can be found by dividing the ROC

*<sup>θ</sup>*max <sup>¼</sup> asin *ROC*

*v* � �

*<sup>G</sup>* ,

*<sup>G</sup>* <sup>¼</sup> *Pa* � *Pr*

max , *<sup>G</sup>* ,

1 *k*max *:*

(7)

$$\theta\_{\text{max}} = \text{asin}\left(\frac{(T\_a - T\_r)\_{\text{max}}}{G}\right).$$

In ideal jet aircraft case, the angle of steepest climb is defined at the speed of maximum lift-to-drag ratio. The most economical ascending flight is defined by the operational regime of aircraft engine at which the minimum fuel will be consumed for aircraft to climb to the required altitude.
