Effective Date: 15 June 98

Propulsion System

The ideal gas flow parameter is calculated from the following equation,

1. W#T = PT #r - 1 K (D-l)

   Equation .1-l can be found in any Gas Dynamics textbook, (The K term is for units conversion).

Nozzle Areas

For the generalized case, let the nozzle area be represented by the flow through an annulus. The cold nozzle area is
1. AC =-##(rl - r22) ( -2)

   If the metal is heated, the hot area is

2. AH =-##(rl + #rl# (r2 + #r#)2 ( -3)

   The incremental change in radius is the product of the radius, coefficient of linear expansion, and the temperature change.

3. #rl = rl##(TH - TC) = rlc# #T ( -4)

4. #r2 = r2c#. #T ( -5)

   From equation ( -3) and ( -4)

5. (rl + #r,)2 = rl + 2rl #rl + (#r,)2 ( -6)

6. = rl + 2rl rl#=# #T + (r,####T)2 ( -7)

   The coefficient of expansion,#,#, is of the order 2 x 10-5 per oC; therefore, (c#)2 is negligible and equation ( -7) reduces to:

7. (rl + #r,)2 = rl + 2rl### #T ( -8)

   Similarly, for the inner radius, r2'

8. (r2 + #r)2 = r2 + 2r2#=##T ( -9)

   Substituting ( -8) and ( -9) into equation ( -3),

9. AH = #####T) --# (r2 + 2r2 #=##T) ( -lO)

   Using the equation ( -2) and ( -lO), the hot area is

10. AH = AC(l + 2#,##T) ( -ll)

Gross Thrust Parameter

The ideal gross thrust can be computed as the total momentum plus the force acting at the exit,

F = WV + (PS - g ( -12)

Bernoulli's equation for compressible flow is,

## P + V2 = Constant##-1 ## 2 ( -13)

Writing equation ( -13) between ambient and a stagnation point,

=0 ## ( -14)

PAM V2 PT + 2 = ##-1 +/2 #T

For a perfect gas,

)AM PAM_ '# ( -15)

Multiplying Equation ( -14) by # AM,

V2 AM PAM + 2 .AM = #.-1 PT ( -16)

Substituting Equation ( -15) into ( -16),

#.-1 PAM + 2 .AM = ##-1 PT\_P# ( -17)

Using the gas law relationship #> = P/gRT in the above equation,

V2PAM PAM \ ##.##-1 PAM +2gRTAM = ##-1 PT# PT / ( -18)

Solving Equation ( -18) for V2,

PT /PAM)V2 = 2## (gRT) ___, ( -19)

Substituting

TAM = ##&&

into equation ( -19), and solving for V,

V = #$% (gR) TT 1 - ( -20)

Substituting Equation ( -20) into Equation ( -12), assuming complete expansion at the exit (PS = PAM), the gross thrust parameters is,

### = F PAM 1 - 2.443 ( -21)

Ram Drag

The Ram Drag is numerically equal to the inlet momentum, or the product of the inlet airflow and the free stream velocity.

FRD = g ( -22)

Multiplying top and bottom of equation ( -22) by ###gRT ,

FRD = W2 g # ##gRT ( -23)

Using definition of Mach No.,

# #-gRT FRD = W2 M g ( -24)

Substituting in consistant units in ( -24) as follows,

# = 1.4

g = 32.174 ft./ sec.2

R = 53.35(1.8) ft.-lbm/lbf oK

T = Deg K

#AM = T/288.16

FRD = 372.7 W2 M##AM ( -25)