Engine Analysis

The prediction of engine ThrustAvail and TSFC is done in Subroutine Engine. The PASS routine is modified to include an additional engine type (type 0), which uses data specific to the BWB engines (as listed below).
               Data for Podded Engine Configuration
                 Cruise Bypass Ratio:  ~22 to 1

               Supplied by: Bill Baumgarten
                     email: baum@lerc.nasa.gov
                     phone: 216-977-7028
                     fax:   216-977-7008

                       Gross                            Fuel
  Mach   Alt (ft)      Thrust (lb)    Ram Drag (lb)     Flow (lb/hr)
  ----   --------      -----------    -------------     ------------
  .00        .0        59334.7               .0         11556.5
  .00    1000.0        58164.7               .0         11343.1
  .20        .0        62043.4          17912.2         11411.0
  .20    1000.0        60799.7          17415.0         11208.9
  .30        .0        73533.5          29215.8         12920.8
  .30    5000.0        64696.5          24966.7         11461.6
  .40        .0        80672.7          40829.4         13061.2
  .50    5000.0        78503.4          45889.6         11831.9
  .50   10000.0        68136.0          38817.7         10492.2
  .50   15000.0        58314.5          32499.4          9086.2
  .60    5000.0        86144.1          57699.6         11654.6
  .60   10000.0        75650.1          49094.9         10605.0
  .60   20000.0        55407.5          34290.4          8094.5
  .60   25000.0        46891.5          28349.2          6951.5
  .70   20000.0        62948.0          42693.5          8516.2
  .70   25000.0        53045.0          35239.8          7304.2
  .70   30000.0        44491.0          28881.7          6284.6
  .80   25000.0        60633.0          43273.9          7749.4
  .80   30000.0        50746.4          35475.2          6646.4
  .80   35000.0        41534.9          28546.1          5608.5
  .85   25000.0        64990.1          47773.9          8000.8
  .85   30000.0        54213.7          39112.2          6836.6
  .85   35000.0        44538.0          31564.1          5792.8
  .85   40000.0        35235.8          24897.0          4614.1
Existing analyses in Subroutine Engine have a quadratic fit for Mach Number, but use linear interpolation on Altitude. We were concerned that discontinuities at Altitudes of 20000 and 30000 could give trouble for a gradient-based optimizer, so the data for the BWB engine is fitted assuming a quadratic variation for both Mach Number and Altitude. This means that we need to solve for the coefficients of the following equations:

Thrust = a1 + a2 * M + a3 * M^2 + a4 * h + a5 * h^2 + a6 * M * h

TSFC
= (b1 + b2 * M + b3 * M^2 + b4 * h + b5 * h^2 + b6 * M * h)/Thrust

where M is Mach Number and h is altitude. With 24 data points and 6 constraints, the problem is over-constrained, so a least squares solution can be found using "normal equations" (Strang "Linear Algebra and Its Applications" p.156). The procedure for the Thrust equation is outlined below.

An array [MH] is constructed with column entries:
	1   M   M2   h   h2   M*h
where M is taken from the first column of data listed above, and h is taken from the second column. [MH] will have 24 rows, one row for each line of data. Vector {a} has 6 rows, one for each coefficient in the Thrust equation, and vector {T} has 24 rows, with each entry formed by subtracting the entry in Column 4 of the data table from the entry in Column 3. The normal equations are formed by introducing [MH]T, and producing a "square" system:

[MH]T[MH]{a} = [MH]T{T}

This system is then solved for the optimal coefficients {a}.

A Matlab m-file has been written to solve this problem (listed here ).

The resulting curves for Thrust and TSFC are shown below. (Note that curves are extrapolated beyond the original data set, to low altitude/high Mach Number and high altitude/low Mach Number regions, and predictions are less reliable in these areas). The maximum error in Thrust is 4.4%, so accuracy should be adequate for preliminary design. (Initially a quartic fit was attempted, in the hope that greater accuracy would be achieved with the higher-order approximation. Matlab could not produce an accurate solution, because the system of equations was near-singular.)



The fragment of Subroutine Engine which uses the new quadratic fits for the BWB engine is listed here:
      if (Type.eq.0) then
c        BaseEngine has SLSTH = 58540 lb, SLSsfc = 0.196
         h = alt/10000.
         Thrustavail = 1.e4 * (5.854 - 6.7511 * Mach + 4.2499*Mach*Mach
     >                  - 0.6709 * h + 0.0365 * h*h  + 0.0044*h*Mach)
         TSFC = 1.e4*(1.1459 + 0.3117*Mach + 0.0166*Mach*Mach
     >            - 0.2851*h + 0.0092*h*h  + 0.0120*h*Mach)/ThrustAvail
         ThrustAvail = ThrustAvail * SLSTH/58540. * Neng
         TSFC        = TSFC * SFCratio
      end if
Subroutine Engine has also been modified to read 1 new database variables: T/Tref. This has been introduced to handle installation losses. Existing curve fits (for other engine types) were constructed using installed thrust data, which implicitly sets TInstallFactor = 1.0. Data for BWB assumes podded engines, so the new variables permit derating for losses due to buried installation.