Elements of Airplane Performance

Name: Elements of Airplane Performance
Uploaded: 2017-07-26T13:51:32+00:00
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Description: Elements of Airplane Performance

Elements of Airplane Performance
Chapter 6 Elements of Airplane Performance Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University
Simple Mission Profile for an Airplane 1 Switch on + Worming + Taxi Un-accelerated level flight (Cruising flight) 4 3 Descent Altitude Climb Landing Takeoff 5 6 1 2 Simple mission profile Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Airplane Performance Equations of Motions Static Performance (Zero acceleration Dynamic Performance (Finite acceleration) Thrust required Thrust available Maximum velocity Takeoff Power required Power available Landing Maximum velocity Rate of climb Gliding flight Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Time to climb Maximum altitude Service ceiling Absolute ceiling Range and endurance Road map for Chapter 6 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Study the airplane performance requires the derivation of the airplane equations of motion As we know the airplane is a rigid body has six degrees of freedom But in case of airplane performance we are deal with the calculation of velocities ( e.g.Vmax,Vmin..etc),distances (e.g. range, takeoff distance, landing distance, ceilings …etc), times (e.g. endurance, time to climb,…etc), angles (e.g.climb angle…etc) Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

So, the rotation of the airplane about its axes during flight in case of performance study is not necessary. Therefore, we can assume that the airplane is a point mass concentrated at its c.g. Also, the derivation of the airplane’s equations of motion requires the knowledge of the forces acting on the airplane The forces acting on an airplane are: Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

3- Thrust force T Propulsive force 4- Weight W Gravity force
1- Lift force L 2- Drag force D 3- Thrust force T Propulsive force 4- Weight W Gravity force Thrust T and weight W will be given But what about L and D? We are in the position that we can’t calculate L and D with our limited knowledge of the airplane aerodynamics Components of the resultant aerodynamic force R Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

So, the relation between L and D will be given in the form of the so called drag polar But before write down the equation of the airplane drag polar it is necessary to know the airplane drag types Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Drag Types [ Kinds of Drag ] Total Drag Skin Friction Drag Pressure Drag Form Drag (Drag Due to Flow separation) Induced Drag Wave Drag Note : Profile Drag = Skin Friction Drag + Form Drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Skin friction drag This is the drag due to shear stress at the surface. ►Pressure drag This is the drag that is generated by the resolved components of the forces due to pressure acting normal to the surface at all points and consists of [ form drag + induced drag + wave drag ]. ►Form drag This can be defined as the difference between profile drag and the skin-friction drag or the drag due to flow separation. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Profile Drag ● Profile drag is the sum of skin-friction and form drags. ● It is called profile drag because both skin-friction and form drag [ or drag due to flow separation ] are ramifications of the shape and size of the body, the “profile” of the body. ● It is the total drag on an aerodynamic shape due to viscous effects Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Skin-friction Form drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Induced drag ( or vortex drag ) This is the drag generated due to the wing tip vortices , depends on lift, does not depend on viscous effects , and can be estimated by assuming inviscid flow. Finite wing flow tendencies Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Formation of wing tip vortices Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Complete wing-vortex system Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

The origin of downwash The origin of induced drag Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Wave Drag This is the drag associated with the formation of shock waves in high-speed flight . Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Total Drag of Airplane ● An airplane is composed of many components and each will contribute to the total drag of its own. ● Possible airplane components drag include : 1. Drag of wing, wing flaps = Dw 2. Drag of fuselage = Df 3. Drag of tail surfaces = Dt 4. Drag of nacelles = Dn 5. Drag of engines = De 6. Drag of landing gear = Dlg 7. Drag of wing fuel tanks and external stores = Dwt 8. Drag of miscellaneous parts = Dms Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● Total drag of an airplane is not simply the sum of the drag of the components. ● This is because when the components are combined into a complete airplane, one component can affect the flow field, and hence, the drag of another. ● these effects are called interference effects, and the change in the sum of the component drags is called interference drag. ● Thus, (Drag)1+2 = (Drag)1 + (Drag)2 + (Drag)interference Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Buid-up Technique of Airplae Drag D ● Using the build-up technique, the airplane total drag D is expressed as: D = Dw + Df + Dt + Dn +De + Dlg + Dwt + Dms + Dinterference ► Interference Drag ● An additional pressure drag caused by the mutual interaction of the flow fields around each component of the airplane. ● Interference drag can be minimized by proper fairing and filleting which induces smooth mixing of air past the components. ● The Figure shows an airplane with large degree of wing filleting. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Wing fillets Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● No theoretical method can predict interference drag, thus, it is obtained from wind-tunnel or flight-test measurements. ● For rough drag calculations a figure of 5% to 10% can be attributed to interference drag on a total drag, i.e, Dinterference ≈ [ 5% – 10% ] of components total drag ■ The Airplane Drag Polar ● For every airplane, there is a relation between CD and CL that can be expressed as an equation or plotted on a graph. ● The equation and the graph are called the drag polar. Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For the complete airplane, the drag coefficient is written as CD = CDo + K CL2 This equation is the drag polar for an airplane. Where: CDo drag coefficient at zero lift ( or parasite drag coefficient ) K CL2 = drag coefficient due to lift ( or induced drag coefficient CDi ) K = 1/π e AR Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

e Oswald efficiency factor = 0.75 – 0.9 (sometimes known as the airplane efficiency factor) AR wing aspect ratio = b2/S , b wing span and S wing planform area Schematic of the drag polar Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Airplane Equations of Motion
Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Apply Newton’s 2nd low of motion: In the direction of the flight path Perpendicular to the flight path Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

I-Steady Level Flight Performance

Un-accelerated (steady) Level Flight Performance (Cruising Flight) Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Thrust Required for Level Un-accelerated Flight (Drag) Thrust required TR for a given airplane to fly at V∞ is given as : TR = D Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● TR as a function of V∞ can be obtained by tow methods or approaches graphical/analytical ■Graphical Approach Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

1- Choose a value of V∞ 2 - For the chosen V∞ calculate CL L = W = ½ρ∞ V2∞S CL CL = 2W/ ρ∞ V2∞S 3- Calculate CD from the drag polar CD = CDo + K CL2 4- Calculate drag, hence TR, from TR = D = ½ρ∞ V2∞S CD 5- Repeat for different values of V∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

6- Tabulate the results V∞ CL CD CL/CD W/[CL/CD ] Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

(TR)min occurs at (CL/CD)max Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Analytical Approach It is required to obtain an equation for TR as a function of V∞ TR = D Required equation Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Parasite and induced drag TR/D CDo=CDi V∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Note that TR is minimum at the point of intersection of the parasite drag Do and induced drag Di Thus Do = Di at [TR]min or CDo = CDi = KCL2 Then [CL](TR)min = √CDo/K And [CDo](TR)min = 2CDo Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Finally, (L/D)max = (CL/CD)max = √CDo/K /2CDo (CL/CD)max = 1/√4KCDo Also,[V∞](TR)min =[V∞] (CL/CD)max is obtained from: W = L = ½ρ∞[V]2(TR)minS [CL](TR)min Thus: [V] (TR)min= {2(W/S)(√K/CDo)/ρ∞}½ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

L/D as function of angle of attack α L/D as function of velocity V∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

L/D as function of V∞ : Since, But L=W Then or Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Flight Velocity for a Given TR TR = D In terms of q∞ = ½ρ∞V2∞ we obtain Multiplying by q∞ and rearranging, we have This is quadratic equation in q∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Solving for q∞ By replacing q∞ = ½ρ∞V2∞ we get Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Let Where (TR/W) is the thrust-to-weight-ratio (W/S) is the wing loading The final expression for velocity is This equation has two roots as shown in figure corresponding to point 1 an 2 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

●When the discriminant equals zero ,then only one solution for V∞ is obtained ●This corresponds to point 3 in the figure, namely at (TR)min Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Or, (TR/W)min = √4CDoK Then the velocity V3 =V(TR)min is Substituting for (TR/W)min = √4CDoK we have Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Effect of Altitude on (TR)min We know that (TR/W)min = √4CDoK This means that (TR)min is independent of altitude as show in Figure (TR)min occurs at higher V∞ V∞1 V∞2 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Thrust Available TA Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Sonic speed Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Thrust Available TA and Maximum Velocity Vmax Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For turbojet at subsonic speeds, (V∞)max can be obtained from: Just substitute (TA)max TR Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Power Required PR Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Variation of PR with V∞ Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

CD= 4CDo Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Power Available PA Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Power Available PA and Maximum Velocity Vmax The high speed intersection between PR and (PA)max gives Vmax Vmax decreases with altitude Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Minimum Velocity: Stall Velocity Airplane minimum velocity Vmin is usually dictated by its stall velocity Stall velocity Vstall is the velocity corresponds to the maximum lift coefficient (CL)max of the airplane Hence, Vmin = Vstall But, L = W = ½ρ∞ V2∞S CL V∞ = (2W/ ρ∞ S CL )½ Substitute for CL (CL)max Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Finally, Vmin= Vstall = [2W/ ρ∞ S (CL)max ]½ CL –α curve for an airplane Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

II-Steady Climb Performance

Steady Climb Assumptions: 1- dV∞/dt = 0 2- Climb along straight line, V2∞/ r = 0 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

The equations of motion in this case become: T cos ε – D – W sin ϴ = 0 L + T sin ε – W cos ϴ = 0 Assuming , ε = 0 Then, T – D – W sin ϴ = 0 L– W cos ϴ = 0 Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

[Turbojet] ,for T = constant Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

sin Turbojet aircraft Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Turbojet aircraft Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Analytical Solution for (R/C)max R/C = V∞ sin ϴ = (2W/ ρ∞ S CL )½ [ T/W- D/L] = (2W/ ρ∞ S CL )½ [T/W-CD/CL] = (2W/ ρ∞ S CL )½ [T/W-CDo +KCL2/CL] =(2W/ ρ∞ S )½ [CL-½(T/W)-(CDo+KC2L)/CL3/2] For turbojet T = const For (R/C)max d(R/C)/dCL =0 So, we get Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

So, we get: And, V(R/C)max=[2W/ ρ∞ S CL(R/C)max ]½ ( CD) (R/C)max = CDo +K C2L(R/C)max (Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max (R/C)max = V(R/C)max (sin ϴ) (R/C)max CL(R/C)max = [ -(T/W) + √ (T/W) K CD0 ] / 2K Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For Propeller Aircraft For propeller aircraft (R/C)max occurs at (PR)min Propeller aircraft Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Analytical Solution for (R/C)max V(R/C)max= V(CL3/2/CD)max ( CD) (R/C)max = CDo +K C2L(R/C)max = CDo +K [√3CDo/K ]2 = 4CDo (Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max (R/C)max = V(R/C)max (sin ϴ) (R/C)max Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

GLIDING (UNPOWERED) FLIGHT Assumptions 1- Steady gliding 2- Along straight line Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

 If PR ˃ PA the airplane will descend  In the ultimate situation when T = 0, the airplane will be in gliding Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Maximum Range For an airplane at a given altitude h, the max. horizontal distance covered over the ground is denoted max. range R Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For Rmax ϴmin Where: Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

CEILINGS Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

max Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

(R/C)-1 h Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Minimum Time to Climb tmin = Assuming linear variation of (R/C)max with altitude h, then (R/C)max = a + b h a = (R/C)max at h = 0 =1/b[ln(a+bh2)-lna] max h b =slope (R/C)max Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

III-Range and Endurance

W=Instantaneous weight Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Elements of Airplane Performance

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