# Patent

## Title
VTOL aircraft with a thrust-to-weight ratio smaller than 0.1

## Abstract

VTOL aircraft with a thrust-to-weight ratio smaller than 0.1, during vertical take-off/landing, obtains an another lift, besides a lift generated by low-temp bypass duct (15) directing the low-temp air (18) from the turbofan engine (3) to flow, through its outlet (19) in form of low-temp planar jet (20), over the upper surface of the wing and in the direction of the wingspan, by high-temp bypass duct (15) directing the high-temp air (18) from the turbofan engine (3) to flow, through its outlet (12) in form of high-temp planar jet (13), above the low-temp planar jet (20) in the direction of the wingspan but not burn up the wing and enables the ailerons (1, 2) to control the balances of the aircraft more efficiently.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
This application is claims foreign priority benefit under 35 U.S.C. §119 of Patent Application EP 14075023.3 filed on 17 Apr. 2014, the contents of which are herein incorporated by reference.
BACKGROUND OF THE INVENTION
This invention relates to aircraft, and more particularly, to aircraft with VTOL technology.
Currently, there are three kinds of traditional VTOL technologies which get vertical lift from high-speed air flowing over upper surface of the aircraft. In the first one such as patent No: US201210068020, a fan independent of a main engine increases speed of low-temp air flowing over the upper surface of wing along the direction of the chord, thereby increasing vertical lift. In this way, this fan less efficiently increases the vertical lift; In the second one such as patent No: GB792993, a high-temp bypass duct directs high-temp air by a valve from a nozzle of a jet engine to flow radially over the upper surface of the aircraft, thereby generating vertical life. In this way, the high-temp bypass duct generates the vertical lift more efficiently than the first one but the upper surface of the aircraft might be burnt up; In the third one such as patent No: GB2469612, a low-temp bypass duct directs low-temp air, with a rotatable nozzle, from a low-temp duct of a turbofan engine to flow over the upper surface of the wing along the direction of the chord, thereby generating vertical lift. But another rotatable nozzle spouts high-temp air downwardly from a high-temp duct of a turbofan engine, thereby generating another vertical lift less efficiently. In this way, the low-temp bypass duct generates the vertical lift more efficiently than above ways but thin slot outlet of the low-temp bypass duct, set in the direction of the chord, enables a part of vertical lift to lose in the long low-temp bypass duct and the problem, how the high-temp air is directed to obtain more vertical lift from the upper surface of the aircraft but not burn up it, is not solved. Because of these limitations, VTOL is impossible for aircraft, and more particularly for large aircraft with thrust-to-weight ratio smaller than 0.1.
BRIEF SUMMARY OF THE INVENTION
The present invention however, can achieve VTOL on aircraft with a thrust-to-weight ratio smaller than 0.1, the aircraft comprising: ailerons; a turbofan engine including a low-temp duct with an openable and closable propelling nozzle and a high-temp duct with an openable and closable propelling nozzle; a low-temp bypass duct including an openable and closable inlet connected to the low-temp duct, an outer wall, an inner wall and a thin slot outlet set on the upper surface of wing and in the direction of the wingspan; a high-temp bypass duct including an openable and closable inlet connected to the high-temp duct, an outer wall and a thin slot outlet set above the thin slot outlet of the low-temp bypass duct in the direction of the wingspan.
According to its first characteristic, the following are true:

    
    
        The low-temp duct spouts directly low-temp air into atmosphere to generate a thrust during forward flight when the propelling nozzle of low-temp duct is opened and the inlet of the low-temp bypass duct is closed.
        The high-temp duct spouts directly high-temp air into atmosphere to generate another thrust during forward flight when the propelling nozzle of high-temp duct is opened and the inlet of the high-temp bypass duct is closed.
        The low-temp bypass duct directs the low-temp air from the low-temp duct to flow, in form of low-temp planar jet, over the upper surface of the wing and in the direction of the wingspan, thereby generating a vertical lift and enabling the ailerons to control horizontal and vertical balances of the aircraft during vertical take-off/landing when the propelling nozzle of low-temp duct is closed and the inlet of the low-temp bypass duct is opened.
        The high-temp bypass duct directs the high-temp air from the high-temp duct to flow, in form of high-temp planar jet, above the low-temp planar jet in the direction of the wingspan, thereby generating another vertical lift and enabling the ailerons to control horizontal and vertical balances of the aircraft more efficiently during vertical take-off/landing when the propelling nozzle of high-temp duct is closed and the inlet of the high-temp bypass duct is opened.
        The thin slot outlet of the high-temp bypass duct, set above the thin slot outlet of the low-temp bypass duct, enables the high-temp planar jet not to burn up the upper surface of the aircraft during vertical take-off/landing.
    
    





BRIEF DESCRIPTION OF THE DRAWING
The attached drawings illustrate the invention:
FIG. 1 is a side view of an aircraft with this invention during vertical take-off and landing.
FIG. 2 is a top view of an aircraft with this invention during vertical take-off and landing.
FIG. 3 is a rear view of an aircraft with this invention during vertical take-off and landing.
FIG. 4 is J local view for FIG. 3.
FIG. 5 is A-A Section for FIG. 4
FIG. 6 is B-B Section for FIG. 4.
FIG. 7 is part Section for FIG. 4.
FIG. 8 is K local view for FIG. 3.
FIG. 9 is C-C Section for FIG. 8.
FIG. 10 is a side view of an aircraft with this invention during forward flight.
FIG. 11 is a top view of an aircraft with this invention during forward flight.
FIG. 12 is a rear view of an aircraft with this invention during forward flight.
FIG. 13 is M local view for FIG. 12.
FIG. 14 is D-D Section for FIG. 13.
FIG. 15 is E-E Section for FIG. 13.
FIG. 16 is part Section for FIG. 13.
FIG. 17 is N local view for FIG. 12.
FIG. 18 is F-F Section for FIG. 17.



DETAILED DESCRIPTION OF THE INVENTION
Referring to above drawings, an aircraft with this invention comprises: ailerons (1, 2); a turbofan engine (3) including a low-temp duct (6) with an openable and closable propelling nozzle (7) and a high-temp duct (4) with an openable and closable propelling nozzle (5); a low-temp bypass duct (15) including an openable and closable inlet (14) connected to the low-temp duct 6), an outer wall (16), an inner wall (17) and a thin slot outlet (19) set on the upper surface of the wing and in the direction of the wingspan; a high-temp bypass duct (9) including an openable and closable inlet (8) connected to the high-temp duct (4), an outer wall (10) and a thin slot outlet (12) set above a thin slot outlet (19) of the low-temp bypass duct (15) and in the direction of the wingspan.
Referring to FIG. (1-9), during vertical take-off/lading, the propelling nozzles (5, 7) of the low-temp duct (6) and the high-temp duct (4) of the turbofan engine (3) are closed. At same time, the inlets (8, 14) of the low-temp bypass duct (15) and the high-temp bypass duct (9) are opened. Then, the turbofan engine (3) starts to supply the low-temp air (18) to the low-temp duct (6) and the high-temp air (11) to the high-temp duct (4). Now, the low-temp bypass duct (15) directs the low-temp air (18) from the low-temp duct (6) and enables it, in form of low-temp planar jet (20), to flow over the upper surface of the wing and in the direction of wingspan. Because the velocity of the air flowing over the upper surface of the wing is higher than that over the lower surface of the wing, and according to Principle of Bernoulli stating an increase in the speed occurs simultaneously with a decrease in pressure, a vertical lift is generated on the wing due to the lower pressure on the upper surface of the wing than that on the lower surface of the wing and the low-temp planar jet (20) enables the ailerons to control horizontal and vertical balances of the aircraft. Similarly, the high-temp bypass duct (9) directs the high-temp air (11) from the high-temp duct (4) and enables it, in form of the high-temp planar jet (13), to flow above the low-temp planar jet (20) in the direction of wingspan, thereby generating another vertical lift and enabling the ailerons to control horizontal and vertical balances of the aircraft more efficiently. The thin slot outlet (12) of the high-temp bypass duct (9), set above the thin slot outlet (19) of the low-temp bypass duct (15), enables the high-temp planar jet (13) not to burn up the upper surface of the wing.
Referring to FIG. (10-18), during forward flight, the propelling nozzles (5, 7) of the low-temp duct (6) and high-temp duct (4) of the turbofan engine (3) are opened. At same time, the inlets (8, 14) of the low-temp bypass duct (15) and the high-temp bypass duct (9) are closed. Then, the low-temp air (18) and the high-temp air (11) spout directly from the propelling nozzles (5, 7) of the low-temp duct (6) and the high-temp duct (4) into atmosphere to generate thrusts.
All the formulas and calculations for this invention are listed in Annex 1.
All the formulas and calculations used to retrofit an Airbus A-380 based on this invention are listed in Annex 2.
This invention can be used to retrofit an existing aircraft to achieve VTOL or manufacture a VTOL aircraft with a thrust-to-weight ratio smaller than 0.1.
ANNEX 1














P
Atmospheric pressure
(Unit: Pa)


R
Ideal gas constant
(Unit: J. K −1. 




mol −1)


ρ0
Low-temp jet density at the outlet
(Unit:kg/m3)


ρ1
Atmospheric density
(Unit:kg/m3)


ρ
Jet density on the cross-section
(Unit: kg/m3)


ρm
Jet density of the shaft
(Unit: kg/m3)


T0
Jet temp at the low-temp outlet
(Unit: K)


T1
Atmospheric temp
(Unit: K)


T
Jet temp on the cross-section
(Unit: K)


V0
Jet speed at the outlet of low-temp
(Unit: m/s)



bypass duct



V{acute over (0)}
Jet speed at the outlet of high-temp
(Unit: m/s)



bypass duct



V
Jet speed on the cross-section
(Unit: m/s)


Vm
Jet speed of shaft
(Unit: m/s)


h0
Jet height of the outlet
(Unit: m)


hm
Jet height of the cross-section
(Unit: m)


h
Jet height on the cross-section
(Unit: m)


L
Width of the outlet
(Unit: m)


C
Gas specific volume
(Unit: m3/kg)


M
Molar mass
(Unit: kg/mol)


Q
Air inflow of turbofan engine
(Unit: kg/s)


B
Bypass ratio of turbofan engine
(Dimensionless




unit)


G
Maximum take-off weight
(Unit: T)


X
Distance between any point and outlet
(Unit: m)



of low-temp bypass duct in jet direction



X1
Distance between wing root and low-
(Unit: m)



temp outlet in jet direction



X2
Distance between low-temp outlet 
(Unit: m)



and crossover point at which jet boundary 




intersects with trailing edge of wing in 




jet direction



X3
Distance between wingtip and 
(Unit: m)



low-temp outlet in jet direction



α
Angle between jet boundary and 
(Unit: °)



trailing edge of wing



β
Angle between chord line of wing 
(Unit: °)



and long side of outlet



γ
Dihedral angle of wing
(Unit: °)


δ
Sweep forward angle of jet
(Unit: °)


ø
Included angle between axis of low-
(Unit: °)



temp jet and one of the high-temp jet



θ
Sweep back angle of jet
(Unit: °)


F
Total vertical lift
(Unit: T)


F1
Lift between wing root and low-temp outlet
(Unit: T)


F2
Lift between low-temp outlet and crossover 
(Unit: T)



point at which jet boundary intersects with




trailing edge of wing



F3
Lift between wingtip and low-temp outlet
(Unit: T)


F4
Downward pressure generated 
(Unit: T)



by slant upward deflected jet



F5
Thrust generated by jet of low-temp outlet 
(Unit: T)


F6
Thrust generated by jet of high-temp outlet
(Unit: T)


b
Constant
(Dimensionless 




unit)


d
Constant
(Dimensionless 




unit)


C
Constant
(Dimensionless 




unit)


n1
S/N of turbofan engine
(Dimensionless 




unit)


n2
S/N of turbofan engine used in vertical
(Dimensionless



Take-off/landing
unit)


n3
S/N of outlet of low-temp bypass duct
(Dimensionless 




unit)


TW R
Thrust-to-weight ratio of aircraft
(Dimensionless 




unit)









According to thermodynamic, relative values of the enthalpy of various jet cross-sections are same each other when pressures of these jet cross-sections are equal and the value of surrounding gas is starting value.
And according to the feature of planar jet, it just spread on the flat which is perpendicular to the outlet section.



  
    P
    =
    
      
        
          
            ρ
            0
          
          M
        
        ⁢
        
          RT
          0
        
      
      =
      
        
          
            
              ρ
              1
            
            M
          
          ⁢
          
            RT
            1
          
        
        =
        
          
            ρ
            M
          
          ⁢
          RT
        
      
    
  




  
    
      
        ρ
        0
      
      ⁢
      
        V
        0
      
      ⁢
      
        h
        0
      
      ⁢
      
        LC
        ⁡
        
          (
          
            
              T
              0
            
            -
            
              T
              1
            
          
          )
        
      
    
    =
    
      ∫
      
        ρ
        ⁢
        
          
        
        ⁢
        VL
        ⁢
        
          
        
        ⁢
        
          C
          ⁡
          
            (
            
              T
              -
              
                T
                1
              
            
            )
          
        
        ⁢
        
          d
          h
        
      
    
  




  
    
      
        ρ
        0
      
      ⁢
      
        V
        0
      
      ⁢
      
        
          h
          0
        
        ⁡
        
          (
          
            
              1
              
                ρ
                0
              
            
            -
            
              1
              
                ρ
                1
              
            
          
          )
        
      
    
    =
    
      ∫
      
        ρ
        ⁢
        
          
        
        ⁢
        
          V
          ⁡
          
            (
            
              
                1
                ρ
              
              -
              
                1
                
                  ρ
                  1
                
              
            
            )
          
        
        ⁢
        
          d
          h
        
      
    
  



And according to the similarity of velocity and density distribution on the various jet sections,




  
    
      
        
          
        
        ⁢
        
          
            
              
                V
                
                  V
                  m
                
              
            
            =
            
              
                
                  ρ
                  -
                  
                    ρ
                    1
                  
                
                
                  
                    ρ
                    m
                  
                  -
                  
                    ρ
                    1
                  
                
              
              =
              
                1
                -
                
                  
                    (
                    
                      h
                      
                        h
                        m
                      
                    
                    )
                  
                  1.5
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                h
                0
              
              ⁢
              
                
                  V
                  0
                
                ⁡
                
                  (
                  
                    
                      ρ
                      0
                    
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
              
            
            =
            
              ∫
              
                
                  (
                  
                    ρ
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
                ⁢
                V
                ⁢
                
                  d
                  h
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                
                  h
                  0
                
                ⁢
                
                  
                    V
                    0
                  
                  ⁡
                  
                    (
                    
                      
                        ρ
                        0
                      
                      -
                      
                        ρ
                        1
                      
                    
                    )
                  
                
              
              
                
                  (
                  
                    
                      ρ
                      m
                    
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
                ⁢
                
                  V
                  m
                
                ⁢
                
                  h
                  m
                
              
            
            =
            
              
                ∫
                
                  
                    
                      
                        (
                        
                          ρ
                          -
                          
                            ρ
                            1
                          
                        
                        )
                      
                      ⁢
                      V
                    
                    
                      
                        (
                        
                          
                            ρ
                            m
                          
                          -
                          
                            ρ
                            1
                          
                        
                        )
                      
                      ⁢
                      
                        V
                        m
                      
                      ⁢
                      
                        h
                        m
                      
                    
                  
                  ⁢
                  
                    d
                    h
                  
                
              
              =
              
                
                  
                    ∫
                    0
                    
                      h
                      m
                    
                  
                  ⁢
                  
                    
                      
                        
                          [
                          
                            1
                            -
                            
                              
                                (
                                
                                  h
                                  
                                    h
                                    m
                                  
                                
                                )
                              
                              1.5
                            
                          
                          ]
                        
                        3
                      
                      ·
                      
                        1
                        
                          h
                          m
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      d
                      h
                    
                  
                
                =
                
                  
                    
                      ∫
                      0
                      1
                    
                    ⁢
                    
                      
                        
                          [
                          
                            1
                            -
                            
                              
                                (
                                
                                  h
                                  
                                    h
                                    m
                                  
                                
                                )
                              
                              1.5
                            
                          
                          ]
                        
                        3
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        d
                        
                          (
                          
                            h
                            
                              h
                              m
                            
                          
                          )
                        
                      
                    
                  
                  =
                  0.3682
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              (
              
                
                  ρ
                  m
                
                -
                
                  ρ
                  1
                
              
              )
            
            =
            
              
                
                  h
                  0
                
                ⁢
                
                  
                    V
                    0
                  
                  ⁡
                  
                    (
                    
                      
                        ρ
                        0
                      
                      -
                      
                        ρ
                        1
                      
                    
                    )
                  
                
              
              
                
                  0.3682
                  ·
                  
                    V
                    m
                  
                
                ⁢
                
                  h
                  m
                
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




According to dynamic characteristic of jet, momentums of the various sections are same each other in case of equal pressures.




  
    
      
        
          
        
        ⁢
        
          
            
              
                ρ
                0
              
              ⁢
              
                Lh
                0
              
              ⁢
              
                V
                0
                2
              
            
            =
            
              ∫
              
                ρ
                ⁢
                
                  
                
                ⁢
                
                  LV
                  2
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  h
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                ρ
                0
              
              ⁢
              
                h
                0
              
              ⁢
              
                V
                0
                2
              
            
            =
            
              
                ∫
                
                  ρ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    V
                    2
                  
                  ⁢
                  
                    d
                    h
                  
                
              
              =
              
                
                  ∫
                  
                    
                      (
                      
                        ρ
                        -
                        
                          ρ
                          1
                        
                      
                      )
                    
                    ⁢
                    
                      V
                      2
                    
                    ⁢
                    
                      d
                      h
                    
                  
                
                +
                
                  ∫
                  
                    
                      ρ
                      1
                    
                    ⁢
                    
                      V
                      2
                    
                    ⁢
                    
                      d
                      h
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                
                  ρ
                  0
                
                ⁢
                
                  h
                  0
                
                ⁢
                
                  V
                  0
                  2
                
              
              
                
                  (
                  
                    
                      ρ
                      m
                    
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
                ⁢
                
                  V
                  m
                  2
                
                ⁢
                
                  h
                  m
                
              
            
            =
            
              
                
                  
                    ∫
                    0
                    1
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            ρ
                            -
                            
                              ρ
                              1
                            
                          
                          )
                        
                        ⁢
                        
                          V
                          2
                        
                      
                      
                        
                          (
                          
                            
                              ρ
                              m
                            
                            -
                            
                              ρ
                              1
                            
                          
                          )
                        
                        ⁢
                        
                          V
                          m
                          2
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      d
                      
                        (
                        
                          h
                          
                            h
                            m
                          
                        
                        )
                      
                    
                  
                
                +
                
                  
                    ∫
                    0
                    1
                  
                  ⁢
                  
                    
                      
                        
                          ρ
                          1
                        
                        ⁢
                        
                          V
                          2
                        
                      
                      
                        
                          (
                          
                            
                              ρ
                              m
                            
                            -
                            
                              ρ
                              1
                            
                          
                          )
                        
                        ⁢
                        
                          V
                          m
                          2
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      d
                      
                        (
                        
                          h
                          
                            h
                            m
                          
                        
                        )
                      
                    
                  
                
              
              =
              
                
                  
                    
                      ∫
                      0
                      1
                    
                    ⁢
                    
                      
                        
                          [
                          
                            1
                            -
                            
                              
                                (
                                
                                  h
                                  
                                    h
                                    m
                                  
                                
                                )
                              
                              1.5
                            
                          
                          ]
                        
                        5
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        d
                        
                          (
                          
                            h
                            
                              h
                              m
                            
                          
                          )
                        
                      
                    
                  
                  +
                  
                    
                      
                        ρ
                        1
                      
                      
                        (
                        
                          
                            ρ
                            m
                          
                          -
                          
                            ρ
                            1
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        ∫
                        0
                        1
                      
                      ⁢
                      
                        
                          
                            [
                            
                              1
                              -
                              
                                
                                  (
                                  
                                    h
                                    
                                      h
                                      m
                                    
                                  
                                  )
                                
                                1.5
                              
                            
                            ]
                          
                          4
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          d
                          
                            (
                            
                              h
                              
                                h
                                m
                              
                            
                            )
                          
                        
                      
                    
                  
                
                =
                
                  0.2786
                  +
                  
                    
                      0.3156
                      ⁢
                      
                        ρ
                        1
                      
                    
                    
                      (
                      
                        
                          ρ
                          m
                        
                        -
                        
                          ρ
                          1
                        
                      
                      )
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                0.3682
                ⁢
                
                  
                
                ⁢
                
                  V
                  0
                
                ⁢
                
                  ρ
                  0
                
              
              
                
                  V
                  m
                
                ⁡
                
                  (
                  
                    
                      ρ
                      0
                    
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
              
            
            =
            
              0.2786
              +
              
                
                  0.1162
                  ⁢
                  
                    
                  
                  ⁢
                  
                    h
                    m
                  
                  ⁢
                  
                    V
                    m
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                
                  
                    h
                    0
                  
                  ⁢
                  
                    
                      V
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          ρ
                          0
                        
                        -
                        
                          ρ
                          1
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




Substitute (1) into (2):




  
    
      
        
          
            
              
                
                  0.1162
                  ⁢
                  
                    
                  
                  ⁢
                  
                    h
                    m
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                
                  
                    h
                    0
                  
                  ⁢
                  
                    ρ
                    0
                  
                
              
              ·
              
                
                  (
                  
                    
                      V
                      m
                    
                    
                      V
                      0
                    
                  
                  )
                
                2
              
            
            +
            
              0.2786
              ⁢
              
                
                  
                    (
                    
                      
                        ρ
                        0
                      
                      -
                      
                        ρ
                        1
                      
                    
                    )
                  
                  
                    ρ
                    0
                  
                
                ·
                
                  (
                  
                    
                      V
                      m
                    
                    
                      V
                      0
                    
                  
                  )
                
              
            
            -
            0.3682
          
          =
          0
        
        ⁢
        
          

        
        ⁢
        
          
            
              V
              m
            
            
              V
              0
            
          
          =
          
            
              
                
                  0.2786
                  ⁢
                  
                    (
                    
                      
                        ρ
                        1
                      
                      -
                      
                        ρ
                        0
                      
                    
                    )
                  
                
                
                  ρ
                  0
                
              
              +
              
                
                  
                    
                      [
                      
                        
                          0.2786
                          ⁢
                          
                            (
                            
                              
                                ρ
                                0
                              
                              -
                              
                                ρ
                                1
                              
                            
                            )
                          
                        
                        
                          ρ
                          0
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      0.1711
                      ⁢
                      
                        
                      
                      ⁢
                      
                        h
                        m
                      
                      ⁢
                      
                        ρ
                        1
                      
                    
                    
                      
                        h
                        0
                      
                      ⁢
                      
                        ρ
                        0
                      
                    
                  
                
              
            
            
              
                0.2324
                ⁢
                
                  
                
                ⁢
                
                  h
                  m
                
                ⁢
                
                  ρ
                  1
                
              
              
                
                  h
                  0
                
                ⁢
                
                  ρ
                  0
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            m
          
          =
          
            
              
                1
                +
                
                  
                    1
                    +
                    
                      
                        0.1711
                        ⁢
                        
                          
                        
                        ⁢
                        
                          h
                          m
                        
                        ⁢
                        
                          ρ
                          0
                        
                        ⁢
                        
                          ρ
                          1
                        
                      
                      
                        
                          0.2786
                          2
                        
                        ⁢
                        
                          
                            
                              h
                              0
                            
                            ⁡
                            
                              (
                              
                                
                                  ρ
                                  0
                                
                                -
                                
                                  ρ
                                  1
                                
                              
                              )
                            
                          
                          2
                        
                      
                    
                  
                
              
              
                
                  0.2324
                  ⁢
                  
                    
                  
                  ⁢
                  
                    h
                    m
                  
                  ⁢
                  
                    ρ
                    1
                  
                
                
                  0.2786
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      h
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          ρ
                          0
                        
                        -
                        
                          ρ
                          1
                        
                      
                      )
                    
                  
                
              
            
            ⁢
            
              V
              0
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




Substitute (3) into (1)




  
    
      
        
          
            (
            
              
                ρ
                m
              
              -
              
                ρ
                1
              
            
            )
          
          =
          
            
              
                h
                0
              
              ⁢
              
                
                  V
                  0
                
                ⁡
                
                  (
                  
                    
                      ρ
                      0
                    
                    -
                    
                      ρ
                      1
                    
                  
                  )
                
              
            
            
              
                0.3682
                ·
                
                  V
                  m
                
              
              ⁢
              
                h
                m
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ρ
            m
          
          =
          
            
              ρ
              1
            
            -
            
              
                
                  0.2324
                  
                    0.3682
                    ×
                    0.2786
                  
                
                ⁢
                
                  ρ
                  1
                
              
              
                1
                +
                
                  
                    1
                    +
                    
                      
                        0.1711
                        ⁢
                        
                          
                        
                        ⁢
                        
                          h
                          m
                        
                        ⁢
                        
                          ρ
                          0
                        
                        ⁢
                        
                          ρ
                          1
                        
                      
                      
                        
                          0.2786
                          2
                        
                        ⁢
                        
                          
                            
                              h
                              0
                            
                            ⁡
                            
                              (
                              
                                
                                  ρ
                                  0
                                
                                -
                                
                                  ρ
                                  1
                                
                              
                              )
                            
                          
                          2
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




When X≦X1 or X≦X2




  F
  =
  
    
      P
      -
      
        
          [
          
            P
            -
            
              (
              
                P
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    ∫
                    
                      
                        ρ
                        m
                      
                      ⁢
                      
                        V
                        m
                        2
                      
                    
                  
                
              
              )
            
          
          ]
        
        ⁢
        L
        ⁢
        
          d
          x
        
      
    
    =
    
      
        1
        2
      
      ⁢
      
        ∫
        
          
            ρ
            m
          
          ⁢
          
            V
            m
            2
          
          ⁢
          L
          ⁢
          
            d
            x
          
        
      
    
  




Substitute (3), (4) into (2)




  =
  
    
      
        1
        2
      
      ⁢
      
        
          ∫
          0
          X
        
        ⁢
        
          
            
              [
              
                
                  ρ
                  1
                
                -
                
                  
                    
                      0.2324
                      
                        0.3682
                        ×
                        0.2786
                      
                    
                    ⁢
                    
                      ρ
                      1
                    
                  
                  
                    1
                    +
                    
                      
                        1
                        +
                        
                          
                            0.1711
                            ⁢
                            
                              
                            
                            ⁢
                            
                              h
                              m
                            
                            ⁢
                            
                              ρ
                              0
                            
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            
                              0.2786
                              2
                            
                            ⁢
                            
                              
                                
                                  h
                                  0
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      ρ
                                      0
                                    
                                    -
                                    
                                      ρ
                                      1
                                    
                                  
                                  )
                                
                              
                              2
                            
                          
                        
                      
                    
                  
                
              
              ]
            
            ·
            
              
                [
                
                  
                    1
                    +
                    
                      
                        1
                        +
                        
                          
                            0.1711
                            ⁢
                            
                              
                            
                            ⁢
                            
                              h
                              m
                            
                            ⁢
                            
                              ρ
                              0
                            
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            
                              0.2786
                              2
                            
                            ⁢
                            
                              
                                
                                  h
                                  0
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      ρ
                                      0
                                    
                                    -
                                    
                                      ρ
                                      1
                                    
                                  
                                  )
                                
                              
                              2
                            
                          
                        
                      
                    
                  
                  
                    
                      0.2324
                      ⁢
                      
                        
                      
                      ⁢
                      
                        h
                        m
                      
                      ⁢
                      
                        ρ
                        1
                      
                    
                    
                      0.2786
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          h
                          0
                        
                        ⁡
                        
                          (
                          
                            
                              ρ
                              0
                            
                            -
                            
                              ρ
                              1
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
              2
            
            ·
            
              V
              0
              2
            
          
          ⁢
          L
          ⁢
          
            d
            x
          
        
      
    
    ⁢
    
      

    
    ⁢
    
      
    
    =
    
      
        
          1
          2
        
        ⁢
        
          
            ∫
            0
            X
          
          ⁢
          
            
              
                {
                
                  
                    
                      
                        ρ
                        1
                      
                      [
                      
                        
                          1
                          +
                          
                            
                              1
                              +
                              
                                
                                  0.1711
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    h
                                    m
                                  
                                  ⁢
                                  
                                    ρ
                                    0
                                  
                                  ⁢
                                  
                                    ρ
                                    1
                                  
                                
                                
                                  
                                    0.2786
                                    2
                                  
                                  ⁢
                                  
                                    
                                      
                                        h
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            ρ
                                            0
                                          
                                          -
                                          
                                            ρ
                                            1
                                          
                                        
                                        )
                                      
                                    
                                    2
                                  
                                
                              
                            
                          
                        
                        
                          
                            0.2324
                            ⁢
                            
                              
                            
                            ⁢
                            
                              h
                              m
                            
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                h
                                0
                              
                              ⁡
                              
                                (
                                
                                  
                                    ρ
                                    0
                                  
                                  -
                                  
                                    ρ
                                    1
                                  
                                
                                )
                              
                            
                          
                        
                      
                      ]
                    
                    2
                  
                  ⁢
                  
                    
                  
                  -
                  
                    
                      0.2324
                      
                        0.3682
                        ×
                        0.2786
                      
                    
                    ⁢
                    
                      
                        ρ
                        1
                      
                      ·
                      
                        [
                        
                          
                            1
                            +
                            
                              
                                1
                                +
                                
                                  
                                    0.1711
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      h
                                      m
                                    
                                    ⁢
                                    
                                      ρ
                                      1
                                    
                                    ⁢
                                    
                                      ρ
                                      0
                                    
                                  
                                  
                                    
                                      0.2786
                                      2
                                    
                                    ⁢
                                    
                                      
                                        
                                          h
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              ρ
                                              1
                                            
                                            -
                                            
                                              ρ
                                              0
                                            
                                          
                                          )
                                        
                                      
                                      2
                                    
                                  
                                
                              
                            
                          
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                h
                                m
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  h
                                  0
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                }
              
              ·
              
                V
                0
                2
              
            
            ⁢
            L
            ⁢
            
              d
              x
            
          
        
      
      =
      
        
          
            1
            2
          
          ⁢
          
            
              ∫
              0
              x
            
            ⁢
            
              
                ρ
                1
              
              ⁢
              
                
                  {
                  
                    
                      
                        2
                        +
                        
                          
                            0.1711
                            ⁢
                            
                              
                            
                            ⁢
                            
                              h
                              m
                            
                            ⁢
                            
                              ρ
                              1
                            
                            ⁢
                            
                              ρ
                              0
                            
                          
                          
                            
                              0.2786
                              2
                            
                            ⁢
                            
                              
                                
                                  h
                                  0
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                              2
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            
                              1
                              +
                              
                                
                                  0.1711
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    h
                                    m
                                  
                                  ⁢
                                  
                                    ρ
                                    1
                                  
                                  ⁢
                                  
                                    ρ
                                    0
                                  
                                
                                
                                  
                                    0.2786
                                    2
                                  
                                  ⁢
                                  
                                    
                                      
                                        h
                                        0
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            ρ
                                            1
                                          
                                          -
                                          
                                            ρ
                                            0
                                          
                                        
                                        )
                                      
                                    
                                    2
                                  
                                
                              
                            
                          
                        
                      
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                h
                                m
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                
                              
                              ⁢
                              
                                
                                  h
                                  0
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                            
                          
                          ]
                        
                        2
                      
                    
                    -
                    
                      
                        (
                        
                          
                            0.2324
                            
                              0.3682
                              ×
                              0.2786
                            
                          
                          +
                          
                            
                              0.2324
                              
                                0.3683
                                ×
                                0.2786
                              
                            
                            ⁢
                            
                              
                                1
                                +
                                
                                  
                                    0.1711
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      h
                                      m
                                    
                                    ⁢
                                    
                                      ρ
                                      1
                                    
                                    ⁢
                                    
                                      ρ
                                      0
                                    
                                  
                                  
                                    
                                      0.2786
                                      2
                                    
                                    ⁢
                                    
                                      
                                        
                                          h
                                          0
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              ρ
                                              1
                                            
                                            -
                                            
                                              ρ
                                              0
                                            
                                          
                                          )
                                        
                                      
                                      2
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                      
                        (
                        
                          
                            [
                            
                              
                                0.2324
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  h
                                  m
                                
                                ⁢
                                
                                  ρ
                                  1
                                
                              
                              
                                0.2786
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                    h
                                    0
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        ρ
                                        1
                                      
                                      -
                                      
                                        ρ
                                        0
                                      
                                    
                                    )
                                  
                                
                              
                            
                            ]
                          
                          2
                        
                        )
                      
                    
                  
                  }
                
                ·
                
                  V
                  0
                  2
                
              
              ⁢
              
                d
                x
              
            
          
        
        =
        
          
            1
            2
          
          ⁢
          
            
              
                ∫
                0
              
              X
            
            ⁢
            
              
                
                  {
                  
                    
                      
                        0.1711
                        ⁢
                        
                          ρ
                          0
                        
                      
                      
                        
                          0.2324
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              h
                              m
                            
                            
                              h
                              0
                            
                          
                          )
                        
                      
                    
                    -
                    
                      1.2655
                      
                        
                          
                            [
                            
                              
                                0.2324
                                ⁢
                                
                                  ρ
                                  1
                                
                              
                              
                                0.2786
                                ⁢
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                          2
                        
                        ⁢
                        
                          
                            (
                            
                              
                                h
                                m
                              
                              
                                h
                                0
                              
                            
                            )
                          
                          2
                        
                      
                    
                    -
                    
                      
                        0.2655
                        ⁢
                        
                          
                            1
                            +
                            
                              
                                
                                  0.1711
                                  ⁢
                                  
                                    ρ
                                    1
                                  
                                  ⁢
                                  
                                    ρ
                                    0
                                  
                                
                                
                                  
                                    0.2786
                                    2
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          ρ
                                          1
                                        
                                        -
                                        
                                          ρ
                                          0
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                              
                              ·
                              
                                (
                                
                                  
                                    h
                                    m
                                  
                                  
                                    h
                                    0
                                  
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          [
                          
                            
                              
                                0.2324
                                ⁢
                                
                                  ρ
                                  1
                                
                              
                              
                                0.2786
                                ⁢
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                            
                            ·
                            
                              (
                              
                                
                                  h
                                  m
                                
                                
                                  h
                                  0
                                
                              
                              )
                            
                          
                          ]
                        
                        2
                      
                    
                  
                  }
                
                ·
                
                  V
                  0
                  2
                
              
              ⁢
              L
              ⁢
              
                d
                x
              
            
          
        
      
    
  




Because of




  
    
      h
      m
    
    
      h
      0
    
  
  =
  
    2.44
    ⁢
    
      (
      
        
          
            0.12
            ⁢
            
              
            
            ⁢
            x
          
          
            h
            0
          
        
        +
        0.41
      
      )
    
  




When X≦X2 or X≦X3



  
    
      F
      1
    
    =
    
      
        
          1
          2
        
        ⁢
        
          
            ∫
            1.0004
            
              2.44
              ⁢
              
                (
                
                  
                    
                      0.12
                      ⁢
                      
                        
                      
                      ⁢
                      
                        X
                        1
                      
                    
                    
                      h
                      0
                    
                  
                  +
                  0.41
                
                )
              
            
          
          ⁢
          
            
              
                {
                
                  
                    
                      0.1711
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      
                        0.2324
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            h
                            m
                          
                          
                            h
                            0
                          
                        
                        )
                      
                    
                  
                  -
                  
                    1.2655
                    
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        2
                      
                      ⁢
                      
                        
                          (
                          
                            
                              h
                              m
                            
                            
                              h
                              0
                            
                          
                          )
                        
                        2
                      
                    
                  
                  -
                  
                    
                      0.2655
                      ⁢
                      
                        
                          1
                          +
                          
                            
                              
                                0.1711
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ρ
                                  1
                                
                                ⁢
                                
                                  ρ
                                  0
                                
                              
                              
                                
                                  0.2786
                                  2
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        ρ
                                        1
                                      
                                      -
                                      
                                        ρ
                                        0
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                            
                            ·
                            
                              (
                              
                                
                                  h
                                  m
                                
                                
                                  h
                                  0
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        [
                        
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ·
                          
                            (
                            
                              
                                h
                                m
                              
                              
                                h
                                0
                              
                            
                            )
                          
                        
                        ]
                      
                      2
                    
                  
                
                }
              
              ·
              
                
                  
                    h
                    0
                  
                  ⁢
                  
                    V
                    0
                    2
                  
                  ⁢
                  L
                
                
                  2.44
                  ×
                  0.12
                
              
            
            ⁢
            
              
            
            ⁢
            
              d
              
                (
                
                  
                    h
                    m
                  
                  
                    h
                    0
                  
                
                )
              
            
          
        
      
      ⁢
      
        

      
      =
      
        
          1
          2
        
        ⁢
        
          
            ∫
            1.0004
            
              2.44
              ⁢
              
                (
                
                  
                    
                      0.12
                      ⁢
                      
                        
                      
                      ⁢
                      
                        X
                        1
                      
                    
                    
                      h
                      0
                    
                  
                  +
                  0.41
                
                )
              
            
          
          ⁢
          
            
              
                {
                
                  
                    
                      0.1711
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      
                        0.2324
                        2
                      
                      ⁢
                      x
                    
                  
                  -
                  
                    1.2655
                    
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        2
                      
                      ⁢
                      
                        x
                        2
                      
                    
                  
                  -
                  
                    
                      0.2655
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        2
                      
                    
                    ·
                    
                      
                        
                          1
                          +
                          
                            
                              
                                0.1711
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ρ
                                  1
                                
                                ⁢
                                
                                  ρ
                                  0
                                
                              
                              
                                
                                  0.2786
                                  2
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        ρ
                                        1
                                      
                                      -
                                      
                                        ρ
                                        0
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                            
                            ·
                            x
                          
                        
                      
                      
                        ·
                        
                          x
                          2
                        
                      
                    
                  
                
                }
              
              ·
              
                
                  
                    h
                    0
                  
                  ⁢
                  
                    V
                    0
                    2
                  
                  ⁢
                  L
                
                
                  2.44
                  ×
                  0.12
                
              
            
            ⁢
            
              
            
            ⁢
            
              d
              x
            
          
        
      
    
  




  
    
      ∫
      
        
          
            
              1
              +
              bx
            
          
          
            x
            2
          
        
        ⁢
        
          d
          x
        
      
    
    =
    
      -
      
        (
        
          
            
              
                1
                +
                bx
              
            
            x
          
          -
          
            
              b
              2
            
            ⁢
            ln
            ⁢
            
              
                
                  
                    1
                    +
                    bx
                  
                
                -
                1
              
              
                
                  
                    1
                    +
                    bx
                  
                
                +
                1
              
            
          
          +
          C
        
        )
      
    
  







  
    
      
        
          F
          1
        
        =
        
          
            
              {
              
                
                  
                    
                      0.1711
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      0.2324
                      2
                    
                  
                  ⁢
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
                +
                
                  1.2655
                  
                    
                      
                        [
                        
                          
                            0.2324
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              (
                              
                                
                                  ρ
                                  1
                                
                                -
                                
                                  ρ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      2
                    
                    ⁢
                    x
                  
                
                +
                
                  
                    0.2655
                    
                      
                        [
                        
                          
                            0.2324
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              (
                              
                                
                                  ρ
                                  1
                                
                                -
                                
                                  ρ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      2
                    
                  
                  ·
                  
                    

                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            1
                            +
                            
                              
                                
                                  0.1711
                                  ⁢
                                  
                                    ρ
                                    1
                                  
                                  ⁢
                                  
                                    ρ
                                    0
                                  
                                
                                
                                  
                                    0.2786
                                    2
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          ρ
                                          1
                                        
                                        -
                                        
                                          ρ
                                          0
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                              
                              ·
                              x
                            
                          
                        
                        x
                      
                      -
                      
                        
                          
                            0.1711
                            ⁢
                            
                              ρ
                              1
                            
                            ⁢
                            
                              ρ
                              0
                            
                          
                          
                            2
                            ×
                            
                              0.2786
                              2
                            
                            ⁢
                            
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                              2
                            
                          
                        
                        ⁢
                        ln
                        ⁢
                        
                          
                            
                              
                                1
                                +
                                
                                  
                                    
                                      0.1711
                                      ⁢
                                      
                                        ρ
                                        1
                                      
                                      ⁢
                                      
                                        ρ
                                        0
                                      
                                    
                                    
                                      
                                        0.2786
                                        2
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            
                                              ρ
                                              1
                                            
                                            -
                                            
                                              ρ
                                              0
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                  ·
                                  x
                                
                              
                            
                            -
                            1
                          
                          
                            
                              
                                1
                                +
                                
                                  
                                    
                                      0.1711
                                      ⁢
                                      
                                        ρ
                                        1
                                      
                                      ⁢
                                      
                                        ρ
                                        0
                                      
                                    
                                    
                                      
                                        0.2786
                                        2
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            
                                              ρ
                                              1
                                            
                                            -
                                            
                                              ρ
                                              0
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                  ·
                                  x
                                
                              
                            
                            +
                            1
                          
                        
                      
                    
                    ]
                  
                
              
              }
            
            
              2.44
              ×
              0.41
            
            
              2.44
              ⁢
              
                (
                
                  
                    
                      0.12
                      ⁢
                      
                        X
                        1
                      
                    
                    
                      h
                      0
                    
                  
                  +
                  0.41
                
                )
              
            
          
          ×
          
            1
            2
          
          ×
          
            
              
                h
                o
              
              ⁢
              
                V
                0
                2
              
              ⁢
              L
            
            
              2.44
              ×
              0.12
            
          
        
      
    
    
      
        (
        5
        )
      
    
  








  
    
      
        
          F
          2
        
        =
        
          
            
              {
              
                
                  
                    
                      0.1711
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      0.2324
                      2
                    
                  
                  ⁢
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
                +
                
                  

                
                ⁢
                
                  1.2655
                  
                    
                      
                        [
                        
                          
                            0.2324
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  ρ
                                  1
                                
                                -
                                
                                  ρ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      2
                    
                    ⁢
                    x
                  
                
                +
                
                  
                    0.2655
                    
                      
                        [
                        
                          
                            0.2324
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              (
                              
                                
                                  ρ
                                  1
                                
                                -
                                
                                  ρ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      2
                    
                  
                  ·
                  
                    

                  
                  ⁢
                  
                    [
                    
                      
                        
                          
                            1
                            +
                            
                              
                                
                                  0.1711
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    ρ
                                    1
                                  
                                  ⁢
                                  
                                    ρ
                                    0
                                  
                                
                                
                                  
                                    0.2786
                                    2
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          ρ
                                          1
                                        
                                        -
                                        
                                          ρ
                                          0
                                        
                                      
                                      )
                                    
                                    2
                                  
                                
                              
                              ·
                              x
                            
                          
                        
                        x
                      
                      -
                      
                        

                      
                      ⁢
                      
                        
                          
                            0.1711
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ρ
                              1
                            
                            ⁢
                            
                              ρ
                              0
                            
                          
                          
                            2
                            ×
                            
                              0.2786
                              2
                            
                            ⁢
                            
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                              2
                            
                          
                        
                        ⁢
                        ln
                        ⁢
                        
                          
                            
                              
                                1
                                +
                                
                                  
                                    
                                      0.1711
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        ρ
                                        1
                                      
                                      ⁢
                                      
                                        ρ
                                        0
                                      
                                    
                                    
                                      
                                        0.2786
                                        2
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            
                                              ρ
                                              1
                                            
                                            -
                                            
                                              ρ
                                              0
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                  ·
                                  x
                                
                              
                            
                            -
                            1
                          
                          
                            
                              1
                              +
                              
                                
                                  
                                    0.1711
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      ρ
                                      1
                                    
                                    ⁢
                                    
                                      ρ
                                      0
                                    
                                  
                                  
                                    
                                      0.2786
                                      2
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          
                                            ρ
                                            1
                                          
                                          -
                                          
                                            ρ
                                            0
                                          
                                        
                                        )
                                      
                                      2
                                    
                                  
                                
                                ·
                                x
                              
                              +
                              1
                            
                          
                        
                      
                    
                    ]
                  
                
              
              }
            
            
              2.44
              ×
              0.41
            
            
              2.44
              ⁢
              
                (
                
                  
                    
                      0.12
                      ⁢
                      
                        
                      
                      ⁢
                      
                        X
                        2
                      
                    
                    
                      h
                      0
                    
                  
                  +
                  0.41
                
                )
              
            
          
          ×
          
            1
            2
          
          ×
          
            
              
                h
                0
              
              ⁢
              
                V
                0
                2
              
              ⁢
              L
            
            
              2.44
              ×
              0.12
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




ANNEX 2
When X2≦X≦X3




  
    
      
        
          F
          3
        
        =
        
          
            1
            2
          
          ⁢
          
            
              ∫
              
                X
                2
              
              
                X
                3
              
            
            ⁢
            
              
                {
                
                  
                    
                      0.1711
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      
                        0.2324
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            h
                            m
                          
                          
                            h
                            0
                          
                        
                        )
                      
                    
                  
                  -
                  
                    1.2655
                    
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        2
                      
                      ⁢
                      
                        
                          (
                          
                            
                              h
                              m
                            
                            
                              h
                              0
                            
                          
                          )
                        
                        2
                      
                    
                  
                  -
                  
                    
                      0.2655
                      ⁢
                      
                        
                          1
                          +
                          
                            
                              
                                0.1711
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  ρ
                                  1
                                
                                ⁢
                                
                                  ρ
                                  0
                                
                              
                              
                                
                                  0.2786
                                  2
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        ρ
                                        1
                                      
                                      -
                                      
                                        ρ
                                        0
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                            
                            ·
                            
                              (
                              
                                
                                  h
                                  m
                                
                                
                                  h
                                  0
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        [
                        
                          
                            
                              0.2324
                              ⁢
                              
                                
                              
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ·
                          
                            (
                            
                              
                                h
                                m
                              
                              
                                h
                                0
                              
                            
                            )
                          
                        
                        ]
                      
                      2
                    
                  
                
                }
              
              ×
              
                
                  V
                  0
                  2
                
                ⁡
                
                  [
                  
                    L
                    -
                    
                      
                        (
                        
                          x
                          -
                          
                            X
                            2
                          
                        
                        )
                      
                      ×
                      tan
                      ⁢
                      
                        
                      
                      ⁢
                      α
                    
                  
                  ]
                
              
              ⁢
              
                d
                x
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




When force is zero in body axis direction,




  
    
      
        
          
        
        ⁢
        
          
            
              F
              4
            
            =
            
              
                F
                5
              
              +
              
                F
                6
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                (
                
                  
                    F
                    4
                  
                  +
                  
                    F
                    5
                  
                  +
                  
                    F
                    6
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              δ
            
            =
            
              
                (
                
                  
                    F
                    5
                  
                  +
                  
                    F
                    6
                  
                
                )
              
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            δ
            =
            
              
                sin
                
                  -
                  1
                
              
              ⁢
              
                1
                2
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              θ
            
          
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        F
        =
        
          
            
              
                n
                2
              
              ⁡
              
                (
                
                  
                    F
                    1
                  
                  +
                  
                    F
                    2
                  
                  +
                  
                    F
                    3
                  
                
                )
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            β
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            γ
          
          -
          
            
              n
              2
            
            ⁡
            
              [
              
                
                  
                    F
                    4
                  
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  δ
                
                +
                
                  
                    F
                    6
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        φ
                        +
                        γ
                      
                      )
                    
                  
                
                -
                
                  
                    F
                    6
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        φ
                        -
                        γ
                      
                      )
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




VTOL can be achieved once: F>G
ANNEX 2
n1=1 ̃4
n2=1 ̃2
n3=1 ̃4
M=29 kg/mol
Q(Air inflow of Trent 900 turbofan engine)=1204 kg/s
B(Bypass ratio of Trent 900 turbo fan engine)=8.5:1
G(Maximum take-off weight of A380)=560T
T0=383K
ρ1=1.293 kg/m2 
Assuming:



  
    θ
    =
    
      36.5
      ⁢
      °
    
  




  
    
      F
      4
    
    =
    
      
        
          1
          
            n
            2
          
        
        ⁢
        
          QV
          0
        
      
      =
      
        
          
            1
            2
          
          ×
          1204
          ×
          211
        
        =
        
          
            127002
            ⁢
            
              
            
            ⁢
            N
          
          =
          
            13
            ⁢
            T
          
        
      
    
  




  
    
      ρ
      0
    
    =
    
      
        
          MP
          0
        
        
          RT
          0
        
      
      =
      
        
          
            101325
            ×
            29
            ×
            0.001
          
          
            8.31
            ×
            383
          
        
        =
        
          0.9232
          ⁢
          
            
          
          ⁢
          kg
          ⁢
          
            /
          
          ⁢
          
            m
            3
          
        
      
    
  




  
    
      h
      0
    
    =
    
      0.46
      ⁢
      
        
      
      ⁢
      m
    
  




  
    L
    =
    
      6
      ⁢
      
        
      
      ⁢
      m
    
  




  
    
      V
      0
    
    =
    
      
        
          
            1
            
              n
              2
            
          
          ×
          Q
          ×
          
            8.5
            
              8.5
              +
              1
            
          
        
        
          
            ρ
            0
          
          ×
          L
          ×
          
            h
            0
          
        
      
      =
      
        
          
            
              1
              2
            
            ×
            1204
            ×
            
              8.5
              
                8.5
                +
                1
              
            
          
          
            0.9232
            ×
            6
            ×
            0.46
          
        
        =
        
          211
          ⁢
          
            
          
          ⁢
          m
          ⁢
          
            /
          
          ⁢
          s
        
      
    
  




  
    
      V
      0
      ′
    
    =
    
      
        V
        0
      
      =
      
        211
        ⁢
        
          
        
        ⁢
        m
        ⁢
        
          /
        
        ⁢
        s
      
    
  




  
    
      F
      6
    
    =
    
      
        
          1
          
            n
            2
          
        
        ⁢
        Q
        ×
        
          1
          
            8.5
            +
            1
          
        
        ×
        
          V
          0
          ′
        
      
      =
      
        
          11370.7368
          ⁢
          
            
          
          ⁢
          N
        
        =
        
          1.4
          ⁢
          
            
          
          ⁢
          T
        
      
    
  




  
    
      X
      1
    
    =
    
      11
      ⁢
      
        
      
      ⁢
      m
    
  




  
    
      X
      2
    
    =
    
      18
      ⁢
      
        
      
      ⁢
      m
    
  




  
    
      X
      3
    
    =
    
      32
      ⁢
      
        
      
      ⁢
      m
    
  




  
    α
    =
    
      14
      ⁢
      °
    
  




  
    β
    =
    
      5
      ⁢
      °
    
  




  
    γ
    =
    
      5
      ⁢
      °
    
  




  
    θ
    =
    
      36.5
      ⁢
      °
    
  




  
    φ
    =
    
      2
      ×
      16.3
      ⁢
      °
    
  







  
    
      
        
          
            
              
                F
                1
              
              =
              
              ⁢
              
                209832.582
                [
                
                  
                    2.9246
                    ⁢
                    
                      
                    
                    ⁢
                    ln
                    ⁢
                    
                      
                    
                    ⁢
                    x
                  
                  +
                  
                    0.1488
                    x
                  
                  +
                  
                    0.0312
                    ·
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ⁢
                            x
                          
                        
                      
                      x
                    
                    -
                    
                      9.621
                      ⁢
                      
                        
                      
                      ⁢
                      ln
                      ⁢
                      
                        
                          
                            
                              1
                              +
                              
                                19.2419
                                ⁢
                                x
                              
                            
                          
                          -
                          1
                        
                        
                          
                            
                              1
                              +
                              
                                19.2419
                                ⁢
                                x
                              
                            
                          
                          +
                          1
                        
                      
                    
                  
                  ]
                
                
                  2.44
                  ×
                  0.41
                
                
                  2.44
                  ⁢
                  
                    (
                    
                      
                        
                          0.12
                          ⁢
                          
                            
                          
                          ⁢
                          
                            X
                            1
                          
                        
                        
                          h
                          0
                        
                      
                      +
                      0.41
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                209832.582
                ⁢
                
                  {
                  
                    
                      2.9246
                      ⁢
                      
                        
                      
                      ⁢
                      ln
                      ⁢
                      
                        
                          2.44
                          ⁢
                          
                            (
                            
                              
                                
                                  0.12
                                  ×
                                  11
                                
                                0.46
                              
                              +
                              0.41
                            
                            )
                          
                        
                        
                          2.44
                          ×
                          0.41
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  0.1488
                  ⁡
                  
                    [
                    
                      
                        1
                        
                          2.44
                          ⁢
                          
                            (
                            
                              
                                
                                  0.12
                                  ×
                                  11
                                
                                0.46
                              
                              +
                              0.41
                            
                            )
                          
                        
                      
                      -
                      
                        1
                        
                          2.44
                          ×
                          0.41
                        
                      
                    
                    ]
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                0.0312
                ·
                
                  [
                  
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ×
                            2.44
                            ⁢
                            
                              (
                              
                                
                                  
                                    0.12
                                    ×
                                    11
                                  
                                  0.46
                                
                                +
                                0.41
                              
                              )
                            
                          
                        
                      
                      
                        2.44
                        ⁢
                        
                          (
                          
                            
                              
                                0.12
                                ×
                                11
                              
                              0.46
                            
                            +
                            0.41
                          
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      1
                      +
                      
                        19.2419
                        ×
                        2.44
                        ×
                        0.41
                      
                    
                  
                  
                    2.44
                    ×
                    0.41
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  9.621
                  ⁢
                  
                    
                  
                  ⁢
                  ln
                  ⁢
                  
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ×
                            2.44
                            ⁢
                            
                              (
                              
                                
                                  
                                    0.12
                                    ×
                                    11
                                  
                                  0.46
                                
                                +
                                0.41
                              
                              )
                            
                          
                        
                      
                      -
                      1
                    
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ×
                            2.44
                            ⁢
                            
                              (
                              
                                
                                  
                                    0.12
                                    ×
                                    11
                                  
                                  0.46
                                
                                +
                                0.41
                              
                              )
                            
                          
                        
                      
                      +
                      1
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                ⁢
                
                  9.621
                  ⁢
                  
                    
                  
                  ⁢
                  ln
                  ⁢
                  
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ×
                            2.44
                            ×
                            0.41
                          
                        
                      
                      -
                      1
                    
                    
                      
                        
                          1
                          +
                          
                            19.2419
                            ×
                            2.44
                            ×
                            0.41
                          
                        
                      
                      +
                      1
                    
                  
                
                ]
              
              }
            
          
        
        
          
            
              =
              
              ⁢
              
                209832.582
                ⁢
                
                  
                
                [
                
                  6.0811
                  -
                  0.1301
                  -
                  0.0918
                  -
                  0.0873
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                1211132.68
                ⁢
                
                  
                
                ⁢
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                124
                ⁢
                
                  
                
                ⁢
                T
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  








  
    
      
        
          
            
              F
              2
            
            =
            
              209832.582
              ⁢
              
                
              
              ⁢
              
                {
                
                  
                    2.9246
                    ⁢
                    
                      
                    
                    ⁢
                    ln
                    ⁢
                    
                      
                        2.44
                        ⁢
                        
                          (
                          
                            
                              
                                0.12
                                ×
                                18
                              
                              0.46
                            
                            +
                            0.41
                          
                          )
                        
                      
                      
                        2.44
                        ×
                        0.41
                      
                    
                  
                  +
                  
                    0.1488
                    [
                    
                      
                        1
                        
                          2.44
                          ⁢
                          
                            (
                            
                              
                                
                                  0.12
                                  ×
                                  18
                                
                                0.46
                              
                              +
                              0.41
                            
                            )
                          
                        
                      
                      -
                      
                        1
                        
                          2.44
                          ×
                          0.41
                        
                      
                    
                    ]
                  
                  +
                  
                    0.0312
                    ·
                    
                      [
                      
                        
                          
                            
                              1
                              +
                              
                                19.2419
                                ×
                                2.44
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        0.12
                                        ×
                                        18
                                      
                                      0.46
                                    
                                    +
                                    0.41
                                  
                                  )
                                
                              
                            
                          
                          
                            2.44
                            ⁢
                            
                              (
                              
                                
                                  
                                    0.12
                                    ×
                                    18
                                  
                                  0.46
                                
                                +
                                0.41
                              
                              )
                            
                          
                        
                        -
                        
                          
                            
                              1
                              +
                              
                                19.2419
                                ×
                                2.44
                                ×
                                0.41
                              
                            
                          
                          
                            2.44
                            ×
                            0.41
                          
                        
                      
                      ]
                    
                  
                  -
                  
                    9.621
                    ⁢
                    
                      
                    
                    ⁢
                    ln
                    ⁢
                    
                      
                        
                          
                            1
                            +
                            
                              19.2419
                              ×
                              2.44
                              ⁢
                              
                                (
                                
                                  
                                    
                                      0.12
                                      ×
                                      18
                                    
                                    0.46
                                  
                                  +
                                  0.41
                                
                                )
                              
                            
                          
                        
                        -
                        1
                      
                      
                        
                          
                            1
                            +
                            
                              19.2419
                              ×
                              2.44
                              ⁢
                              
                                (
                                
                                  
                                    
                                      0.12
                                      ×
                                      18
                                    
                                    0.46
                                  
                                  +
                                  0.41
                                
                                )
                              
                            
                          
                        
                        +
                        1
                      
                    
                  
                  +
                  
                    9.621
                    ⁢
                    
                      
                    
                    ⁢
                    ln
                    ⁢
                    
                      
                        
                          1
                          ⁢
                          
                            
                              
                                +
                                19.2419
                              
                              ×
                              2.44
                              ×
                              0.41
                            
                          
                        
                        -
                        1
                      
                      
                        
                          
                            1
                            +
                            
                              19.2419
                              ×
                              2.44
                              ×
                              0.41
                            
                          
                        
                        +
                        1
                      
                    
                  
                
                ]
              
            
          
          }
        
        =
        
          
            209832.582
            ⁡
            
              [
              
                7.3757
                -
                0.1368
                -
                0.1015
                -
                0.0969
              
              ]
            
          
          =
          
            
              1477326.2936
              ⁢
              
                
              
              ⁢
              N
            
            =
            
              151
              ⁢
              T
            
          
        
      
    
    
      
        (
        6
        )
      
    
  








  
    F
    3
  
  =
  
    
      1
      2
    
    ⁢
    
      
        ∫
        
          2.44
          ⁢
          
            (
            
              
                
                  0.12
                  ⁢
                  
                    X
                    2
                  
                
                
                  h
                  0
                
              
              +
              0.41
            
            )
          
        
        
          2.44
          ⁢
          
            (
            
              
                
                  0.12
                  ⁢
                  
                    X
                    3
                  
                
                
                  h
                  0
                
              
              +
              0.41
            
            )
          
        
      
      ⁢
      
        
          {
          
            
              
                0.1711
                ⁢
                
                  ρ
                  0
                
              
              
                
                  0.2324
                  2
                
                ⁢
                
                  (
                  
                    
                      h
                      m
                    
                    
                      h
                      0
                    
                  
                  )
                
              
            
            -
            
              1.2655
              
                
                  [
                  
                    
                      0.2324
                      ⁢
                      
                        ρ
                        1
                      
                    
                    
                      0.2786
                      ⁢
                      
                        (
                        
                          
                            ρ
                            1
                          
                          -
                          
                            ρ
                            0
                          
                        
                        )
                      
                    
                  
                  ]
                
                ⁢
                
                  
                    (
                    
                      
                        h
                        m
                      
                      
                        h
                        0
                      
                    
                    )
                  
                  2
                
              
            
            ⁢
            
              
            
            -
            
              
                0.2655
                ⁢
                
                  
                    1
                    +
                    
                      
                        
                          0.1711
                          ⁢
                          
                            ρ
                            1
                          
                          ⁢
                          
                            ρ
                            0
                          
                        
                        
                          
                            0.2786
                            2
                          
                          ⁢
                          
                            (
                            
                              
                                ρ
                                1
                              
                              -
                              
                                ρ
                                0
                              
                            
                            )
                          
                        
                      
                      ·
                      
                        (
                        
                          
                            h
                            m
                          
                          
                            h
                            0
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  [
                  
                    
                      
                        0.2324
                        ⁢
                        
                          ρ
                          1
                        
                      
                      
                        0.2786
                        ⁢
                        
                          (
                          
                            
                              ρ
                              1
                            
                            -
                            
                              ρ
                              0
                            
                          
                          )
                        
                      
                    
                    ·
                    
                      (
                      
                        
                          h
                          m
                        
                        
                          h
                          0
                        
                      
                      )
                    
                  
                  ]
                
                2
              
            
          
          }
        
        ×
        
          [
          
            L
            -
            
               
              
                
                  
                     
                    
                      
                        (
                        
                          
                            
                              
                                h
                                0
                              
                              
                                2.44
                                ×
                                0.12
                              
                            
                            ×
                            
                              (
                              
                                
                                  h
                                  m
                                
                                
                                  h
                                  0
                                
                              
                              )
                            
                          
                          -
                          
                            
                              0.41
                              ⁢
                              
                                h
                                0
                              
                            
                            0.12
                          
                          -
                          
                            X
                            2
                          
                        
                        )
                      
                      ×
                      tan
                      ⁢
                      
                        
                      
                      ⁢
                      α
                    
                    ]
                  
                  ⁢
                  
                    
                      
                        h
                        0
                      
                      ⁢
                      
                        V
                        0
                        2
                      
                    
                    
                      2.44
                      ×
                      0.12
                    
                  
                  ⁢
                  
                    d
                    
                      (
                      
                        
                          h
                          m
                        
                        
                          h
                          0
                        
                      
                      )
                    
                  
                
                =
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∫
                      
                        2.44
                        ⁢
                        
                          (
                          
                            
                              
                                0.12
                                ⁢
                                
                                  X
                                  2
                                
                              
                              
                                h
                                0
                              
                            
                            +
                            0.41
                          
                          )
                        
                      
                      
                        2.44
                        ⁢
                        
                          (
                          
                            
                              
                                0.12
                                ⁢
                                
                                  X
                                  3
                                
                              
                              
                                h
                                0
                              
                            
                            +
                            0.41
                          
                          )
                        
                      
                    
                    ⁢
                    
                      
                        {
                        
                          
                            
                              0.1711
                              ⁢
                              
                                ρ
                                0
                              
                            
                            
                              
                                0.2324
                                2
                              
                              ⁢
                              x
                            
                          
                          -
                          
                            1.2655
                            
                              
                                
                                  [
                                  
                                    
                                      0.2324
                                      ⁢
                                      
                                        ρ
                                        1
                                      
                                    
                                    
                                      0.2786
                                      ⁢
                                      
                                        (
                                        
                                          
                                            ρ
                                            1
                                          
                                          -
                                          
                                            ρ
                                            0
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                                2
                              
                              ⁢
                              
                                x
                                2
                              
                            
                          
                          -
                          
                            
                              0.2655
                              ⁢
                              
                                
                                  1
                                  +
                                  
                                    
                                      
                                        0.1711
                                        ⁢
                                        
                                          ρ
                                          1
                                        
                                        ⁢
                                        
                                          ρ
                                          0
                                        
                                      
                                      
                                        
                                          0.2786
                                          2
                                        
                                        ⁢
                                        
                                          
                                            (
                                            
                                              
                                                ρ
                                                1
                                              
                                              -
                                              
                                                ρ
                                                0
                                              
                                            
                                            )
                                          
                                          2
                                        
                                      
                                    
                                    ·
                                    
                                      (
                                      
                                        
                                          h
                                          m
                                        
                                        
                                          h
                                          0
                                        
                                      
                                      )
                                    
                                  
                                
                              
                            
                            
                              
                                [
                                
                                  
                                    
                                      0.2324
                                      ⁢
                                      
                                        ρ
                                        1
                                      
                                    
                                    
                                      0.2786
                                      ⁢
                                      
                                        (
                                        
                                          
                                            ρ
                                            1
                                          
                                          -
                                          
                                            ρ
                                            0
                                          
                                        
                                        )
                                      
                                    
                                  
                                  ·
                                  x
                                
                                ]
                              
                              2
                            
                          
                        
                        }
                      
                      ·
                      
                        [
                        
                          
                        
                        ⁢
                        
                          
                            L
                            -
                            
                              
                                 
                                
                                  
                                    (
                                    
                                      
                                        
                                          
                                            h
                                            0
                                          
                                          
                                            2.44
                                            ×
                                            0.12
                                          
                                        
                                        ⁢
                                        x
                                      
                                      -
                                      
                                        
                                          0.41
                                          ⁢
                                          
                                            h
                                            0
                                          
                                        
                                        0.12
                                      
                                      -
                                      
                                        X
                                        2
                                      
                                    
                                    )
                                  
                                  ×
                                  tan
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  α
                                
                                ]
                              
                              ⁢
                              
                                
                                  
                                    h
                                    0
                                  
                                  ⁢
                                  
                                    V
                                    0
                                    2
                                  
                                
                                
                                  2.44
                                  ×
                                  0.12
                                
                              
                              ⁢
                              
                                d
                                x
                              
                            
                          
                          =
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              
                                ∫
                                
                                  2.44
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          0.12
                                          ⁢
                                          
                                            X
                                            2
                                          
                                        
                                        
                                          h
                                          0
                                        
                                      
                                      +
                                      0.41
                                    
                                    )
                                  
                                
                                
                                  2.44
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          0.12
                                          ⁢
                                          
                                            X
                                            3
                                          
                                        
                                        
                                          h
                                          0
                                        
                                      
                                      +
                                      0.41
                                    
                                    )
                                  
                                
                              
                              ⁢
                              
                                〈
                                
                                  
                                    
                                      0.1711
                                      ⁢
                                      
                                        ρ
                                        0
                                      
                                    
                                    
                                      
                                        0.2324
                                        2
                                      
                                      ⁢
                                      x
                                    
                                  
                                  -
                                  
                                    1.2655
                                    
                                      
                                        
                                          [
                                          
                                            
                                              0.2324
                                              ⁢
                                              
                                                ρ
                                                1
                                              
                                            
                                            
                                              0.2786
                                              ⁢
                                              
                                                (
                                                
                                                  
                                                    ρ
                                                    1
                                                  
                                                  -
                                                  
                                                    ρ
                                                    0
                                                  
                                                
                                                )
                                              
                                            
                                          
                                          ]
                                        
                                        2
                                      
                                      ⁢
                                      
                                        x
                                        2
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
  







  
    
      
        
          
            
              
                -
                
                  
                    0.2655
                    ⁢
                    
                      
                        1
                        +
                        
                          
                            
                              0.1711
                              ⁢
                              
                                ρ
                                1
                              
                              ⁢
                              
                                ρ
                                0
                              
                            
                            
                              
                                0.2786
                                2
                              
                              ⁢
                              
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                                2
                              
                            
                          
                          ·
                          x
                        
                      
                    
                  
                  
                    
                      
                        [
                        
                          
                            0.2324
                            ⁢
                            
                              ρ
                              1
                            
                          
                          
                            0.2786
                            ⁢
                            
                              (
                              
                                
                                  ρ
                                  1
                                
                                -
                                
                                  ρ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      2
                    
                    ⁢
                    
                      x
                      2
                    
                  
                
              
              }
            
            ·
            
              [
              
                1
                +
                
                  
                    
                      (
                      
                        
                          
                            0.41
                            ⁢
                            
                              h
                              0
                            
                          
                          
                            0.12
                            ×
                            L
                          
                        
                        +
                        
                          
                            X
                            2
                          
                          L
                        
                      
                      )
                    
                    ·
                    tan
                  
                  ⁢
                  
                    
                  
                  ⁢
                  α
                
              
              ]
            
          
          -
          
            
              
                
              
              ⁢
              
                {
                
                  
                    
                      0.1711
                      ⁢
                      
                        ρ
                        0
                      
                    
                    
                      0.2324
                      2
                    
                  
                  -
                  
                    1.2655
                    
                      
                        
                          [
                          
                            
                              0.2324
                              ⁢
                              
                                ρ
                                1
                              
                            
                            
                              0.2786
                              ⁢
                              
                                (
                                
                                  
                                    ρ
                                    1
                                  
                                  -
                                  
                                    ρ
                                    0
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                        2
                      
                      ⁢
                      x
                    
                  
                  -
                  
                    
                      0.2655
                      ⁢
                      
                        
                          1
                          +
                          
                            
                              
                                0.1711
                                ⁢
                                
                                  ρ
                                  1
                                
                                ⁢
                                
                                  ρ
                                  0
                                
                              
                              
                                
                                  0.2786
                                  2
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      
                                        ρ
                                        1
                                      
                                      -
                                      
                                        ρ
                                        0
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                            
                            ·
                            x
                          
                        
                      
                    
                    
                      
                        
                          [
                          
                            
                              
                                0.2324
                                ⁢
                                
                                  ρ
                                  1
                                
                              
                              
                                0.2786
                                ⁢
                                
                                  (
                                  
                                    
                                      ρ
                                      1
                                    
                                    -
                                    
                                      ρ
                                      0
                                    
                                  
                                  )
                                
                              
                            
                            ·
                          
                          ]
                        
                        2
                      
                      ⁢
                      x
                    
                  
                
                }
              
            
            ⁢
            
              
                
                  h
                  o
                
                ⁢
                tan
                ⁢
                
                  
                
                ⁢
                α
              
              
                2.44
                ×
                0.12
                ×
                L
              
            
          
        
        〉
      
      ⁢
      
        
          
            h
            0
          
          ⁢
          
            V
            0
            2
          
          ⁢
          L
        
        
          2.44
          ×
          0.12
        
      
      ⁢
      
        d
        x
      
      ⁢
      
        ∫
        
          
            
              
                1
                +
                
                  d
                  x
                
              
            
            x
          
          ⁢
          
            d
            x
          
        
      
    
    =
    
      
        2
        ⁢
        
          
            1
            +
            
              d
              x
            
          
        
      
      -
      
        ln
        ⁢
        
          
            
              
                1
                +
                
                  d
                  x
                
              
            
            -
            1
          
          
            
              
                1
                +
                
                  d
                  x
                
              
            
            +
            1
          
        
      
      +
      C
    
  




  
    
      F
      3
    
    =
    
      209832.582
      ⁢
      
        {
        
          
            1.8133
            ×
            
              [
              
                
                  2.9246
                  ⁢
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
                +
                
                  0.1488
                  x
                
                +
                
                  0.0312
                  ·
                  
                    (
                    
                      
                        
                          
                            1
                            +
                            
                              19.2419
                              ⁢
                              x
                            
                          
                        
                        x
                      
                      -
                      
                        9.621
                        ⁢
                        ln
                        ⁢
                        
                          
                            
                              
                                1
                                +
                                
                                  19.2419
                                  ⁢
                                  x
                                
                              
                            
                            -
                            1
                          
                          
                            
                              
                                1
                                +
                                
                                  19.2419
                                  ⁢
                                  x
                                
                              
                            
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              ]
            
          
          -
          
            
              0.0653
              [
              
                
                  2.9246
                  ⁢
                  x
                
                -
                
                  0.1488
                  ⁢
                  ln
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
                -
                
                  0.0312
                  ⁢
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          
                            1
                            +
                            
                              19.2419
                              ⁢
                              x
                            
                          
                        
                      
                      -
                      
                        ln
                        ⁢
                        
                          (
                          
                            
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ⁢
                                    x
                                  
                                
                              
                              -
                              1
                            
                            
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ⁢
                                    x
                                  
                                
                              
                              +
                              1
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
              
              }
            
            12.4578
            21.3691
          
        
      
    
  







  
    
      
        =
        
          
            209832.582
            ⁢
            
              {
              
                
                  1.8133
                  ×
                  
                    [
                    
                      
                        2.9246
                        ⁢
                        
                          
                        
                        ⁢
                        ln
                        ⁢
                        
                          21.3691
                          12.4578
                        
                      
                      +
                      
                        0.1488
                        ⁢
                        
                          (
                          
                            
                              1
                              21.3691
                            
                            -
                            
                              1
                              12.4578
                            
                          
                          )
                        
                      
                      +
                      
                        0.0312
                        ⁢
                        
                          (
                          
                            
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ×
                                    21.3691
                                  
                                
                              
                              21.3691
                            
                            -
                            
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ×
                                    12.4578
                                  
                                
                              
                              12.4578
                            
                            -
                            
                              9.621
                              ⁢
                              ln
                              ⁢
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        21.3691
                                      
                                    
                                  
                                  -
                                  1
                                
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        21.3691
                                      
                                    
                                  
                                  +
                                  1
                                
                              
                            
                            +
                            
                              9.621
                              ⁢
                              ln
                              ⁢
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        12.4578
                                      
                                    
                                  
                                  -
                                  1
                                
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        12.4578
                                      
                                    
                                  
                                  +
                                  1
                                
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
                -
                
                  0.0653
                  ⁡
                  
                    [
                    
                      
                        2.9246
                        ⁢
                        
                          (
                          
                            21.3691
                            -
                            12.4578
                          
                          )
                        
                      
                      -
                      
                        0.1488
                        ⁢
                        ln
                        ⁢
                        
                          21.3691
                          12.4578
                        
                      
                      -
                      
                        0.0312
                        ⁢
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ×
                                    21.3691
                                  
                                
                              
                            
                            -
                            
                              2
                              ⁢
                              
                                
                                  1
                                  +
                                  
                                    19.2419
                                    ×
                                    12.4578
                                  
                                
                              
                            
                            -
                            
                              ln
                              ⁢
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        21.3691
                                      
                                    
                                  
                                  -
                                  1
                                
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        21.3691
                                      
                                    
                                  
                                  +
                                  1
                                
                              
                            
                            +
                            
                              ln
                              ⁢
                              
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        12.4578
                                      
                                    
                                  
                                  -
                                  1
                                
                                
                                  
                                    
                                      1
                                      +
                                      
                                        19.2419
                                        ×
                                        12.4578
                                      
                                    
                                  
                                  +
                                  1
                                
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
              
              }
            
          
          =
          
            
              209832.582
              ⁢
              
                {
                
                  
                    1.8133
                    ×
                    
                      [
                      
                        1.5781
                        -
                        0.005
                        +
                        
                          0.0312
                          ⁢
                          
                            (
                            
                              
                                -
                                0.2953
                              
                              -
                              0.2934
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                  
                    0.0653
                    ⁡
                    
                      [
                      
                        26.062
                        -
                        0.0803
                        -
                        
                          0.0312
                          ⁢
                          
                            (
                            
                              9.5748
                              -
                              0.0305
                            
                            )
                          
                        
                      
                      ]
                    
                  
                
                }
              
            
            =
            
              
                239636.5129
                ⁢
                N
              
              =
              
                24
                ⁢
                T
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                δ
                =
                
                  
                    sin
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        1
                        2
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
                
                  sin
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      1
                      2
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    36.5
                    ⁢
                    °
                  
                  )
                
              
            
          
        
        
          
            
              =
              
                17.3
                ⁢
                °
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




According to this sweep forward angle, low-temp planar jet enables the aircraft to keep balances during vertical take-off/landing.




  
    
      
        
          F
          =
          
            
              
                
                  
                    n
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        F
                        1
                      
                      +
                      
                        F
                        2
                      
                      +
                      
                        F
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                β
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                γ
              
              -
              
                
                  n
                  2
                
                ⁡
                
                  [
                  
                    
                      
                        F
                        4
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      δ
                    
                    +
                    
                      
                        F
                        6
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            φ
                            +
                            γ
                          
                          )
                        
                      
                    
                    -
                    
                      
                        F
                        6
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            φ
                            -
                            γ
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
            =
            
              
                
                  2
                  ⁢
                  
                    (
                    
                      124
                      +
                      151
                      +
                      24
                    
                    )
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  5
                  ⁢
                  °
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  5
                  ⁢
                  °
                
                -
                
                  2
                  ⁡
                  
                    [
                    
                      
                        13
                        ⁢
                        
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        17.3
                        ⁢
                        °
                      
                      +
                      
                        1.4
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ×
                                16.3
                                ⁢
                                °
                              
                              +
                              
                                5
                                ⁢
                                °
                              
                            
                            )
                          
                        
                      
                      -
                      
                        1.4
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                2
                                ×
                                16.3
                                ⁢
                                °
                              
                              -
                              
                                5
                                ⁢
                                °
                              
                            
                            )
                          
                        
                      
                    
                    ]
                  
                
              
              =
              
                568
                ⁢
                
                  
                
                ⁢
                T
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            F
            -
            G
          
          =
          
            
              568
              -
              560
            
            =
            
              
                8
                ⁢
                T
              
              >
              0
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        TWR
        =
        
          
            
              
                n
                3
              
              ⁡
              
                (
                
                  
                    F
                    5
                  
                  +
                  
                    F
                    6
                  
                
                )
              
            
            G
          
          =
          
            
              
                
                  n
                  3
                
                ⁢
                
                  F
                  4
                
              
              G
            
            =
            
              
                
                  4
                  ×
                  13
                
                560
              
              =
              
                0.09
                <
                0.1
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




It is clearly demonstrated above that VTOL is achievable on Airbus A380 once remodeled as shown, and more particularly, in case of using just two turbofan engines and thrust-to-weight ratio smaller than 0.1.

## Claims

1. An aircraft capable of vertical take-off/landing, comprising: ailerons (1, 2); a turbofan engine (3) including a low-temp duct (6) with an openable and closable propelling nozzle (7) and a high-temp duct (4) with an openable and closable propelling nozzle (5); a low-temp bypass duct (15) including an openable and closable inlet (14) connected to the low-temp duct (6), an outer wall (16), an inner wall (17) and a thin slot outlet (19) set on the upper surface of the wing and in the direction of the wingspan; and a high-temp bypass duct (9) including an openable and closable inlet (8) connected to the high-temp duct (4), an outer wall (10) and a thin slot outlet (12) set above the thin slot outlet (19) of the low-temp bypass duct (15) in the direction of the wingspan.

2. The aircraft according to claim 1 wherein the low-temp duct (6) spouts directly low-temp air (18) into the atmosphere to generate a thrust during forward flight when the propelling nozzle (7) of low-temp duct (6) is opened and the inlet (14) of the low-temp bypass duct (15) is closed.

3. The aircraft according to claim 1 wherein the high-temp duct (4) spouts directly high-temp air (11) into the atmosphere to generate another thrust during forward flight when the propelling nozzle (5) of high-temp duct (4) is opened and the inlet (8) of the high-temp bypass duct (9) is closed.

4. The aircraft according to claim 1 wherein the low-temp bypass duct (15) directs the low-temp air (18) from the low-temp duct (6) to flow, in form of low-temp planar jet (20), over the upper surface of the wing and in the direction of the wingspan, thereby generating a vertical lift and enabling the ailerons (1, 2) to control horizontal and vertical balances of the aircraft during vertical take-off/landing when the propelling nozzle (7) of low-temp duct (6) is closed and the inlet (14) of the low-temp bypass duct (15) is opened.

5. The aircraft according to claim 1 wherein the high-temp bypass duct (9) directs the high-temp air (18) from the high-temp duct (4) to flow, in form of high-temp planar jet (13), above the low-temp planar jet (20) in the direction of the wingspan, thereby generating another vertical lift and enabling the ailerons (1, 2) to control horizontal and vertical balances of the aircraft more efficiently during vertical take-off/landing when the propelling nozzle (5) of high-temp duct (4) is closed and the inlet (8) of the high-temp bypass duct (9) is opened.

6. The aircraft according to one of claims 1-5 wherein the thin slot outlet (12) of the high-temp bypass duct (9), set above the thin slot outlet (19) of the low-temp bypass duct (15), enables the high-temp planar jet (13) not to burn the upper surface of the aircraft during vertical take-off/landing.