16.50 Lecture 21
Subject: Turbofan Engines
In the lecture 19 we saw that for low Mo, η
p is not very high for turbojets. In essence
there is too much kinetic energy in the exhaust jet (per unit mass). This is the main
reason for use of the Turbofan engine. If it is designed for subsonic cruise flight it
looks like:
From Kerrebrock, Jack L. Aircraft Engines and Gas Turbines. 2nd edition. MIT
Press, 1992. © Massachusetts Institute of Technology. Used with permission.
If designed for both subsonic cruise and for supersonic flight with afterburning it
looks more like:
From Kerrebrock, Jack L. Aircraft Engines and Gas Turbines. 2nd edition. MIT
Press, 1992. © Massachusetts Institute of Technology. Used with permission.
In both of these diagrams the inlet has been greatly simplified, of course.
1
Let us now see how we can model these engines thermodynamically. In the first
(cruise engine) we have 2 jets. Up until we evaluated τt, our argument for finding
the jet velocity applies to either jet. Thus from equation (8) of Lecture 18 we get for
the core jet:
Fc
2
#
=
( t )[# o $ c $ t "1] " Mo
m˙ ao
! " 1 # o$ c
For the fan stream, there is no turbine, so let us repeat the argument.
γ−1
Tt8 = T8(1 +
Μ82) = Τοθοτf
2
and
!
!
γ−1
! "1
! "1
2
pt8 = p8(1 +
Μ8 ) = po δο πf = po (θoτf)
2
So for p8 = po
γ−1
Μ82 = θoτf
2
2
(# $ "1)
M8 =
! "1 o f
As in the turbojet, a fixed convergent nozzle is more likely to be sonic and underexpanded, namely, to have M8=1 and p8>p0, but we will here ignore the differences
this entails.
1+
We also have
T8 ! o " f
=
=1
To ! o " f
so
/
,
2
* 0+ f " 1
1
.
# u8 &
) "1
FBP
1
.
= M 0 % " 1( = M 0
"1
1
.
! 0
M0
! ma
$ u0 '
1
.
0
adding this thrust to the thrust of the core jet, we find the total thrust:
(
)
F
#
= { ! 2"1 ( #o t$c )[# o $ c $ t "1] " Mo} + % { ! 2"1 (# o $ f " 1) " Mo }
(14)
m˙ ao
Now we need τt. We will see later that most engines have two shafts, but for now we
lump their power together; at off-design conditions, this would have to be modified.
From a work balance between the compressor, fan and turbine, (both shafts),
m˙ c p (Tt4 ! Tt 5 ) = m˙ c p (Tt3 ! Tt2 ) + " m˙ cp (Tt 7 ! Tt 2 )
! t (1" # t ) = ! o (# c " 1) + $!o ( # f " 1)
2
! t = 1"
#o
[(! " 1) + $ (! f " 1)]
#t c
(15)
Substituting this gives us our result for the thrust of the turbofan. It doesn't really
help to carry out the substitution at this point. Instead let us think about how to
simplify the expressions to make them more easily understandable.
For this engine there are more parameters than for the turbojet:
θt, τc - as before
α, τf - characterizing the fan flow.
We can relate some of the parameters to others by noting that the highest propulsive
momentum
in
efficiency is realized when u8 = u6, since this maximizes the ratio of
energy
the jets. For this situation, the two
' s in the thrust equation are equal, and this
requires that:
# !t &
%
( [! " " ) 1] = ! o " f ) 1
(16)
$ ! 0" c ' o c t
(u8 = u6)
Solving for τf as a function of α using (15),
!o " c
!t
(! o " f # 1) + 1 = ! o " c #
! o " f # 1+
! o2" c
[( " c # 1) + $ ( " f # 1)]
!t
!t
= !t # ! o [(" c # 1) + $( " f # 1)]
!o "c
1+ " t # ""o !t c # " o (! c # 1) + "o $
!f =
"o + " o $
!f =
1+ " t + " o (1+ # $ ! c ) $ ""o t!c
(17)
" o (1+ # )
When this equation is satisfied, the thrust equation becomes simply
F
= (1 + ! ) " 2#1 ($ o % f # 1) # M0
m˙ ao
Now as for the turbojet there is still the choice of the compression ratio. Generally
we want to choose it for maximum power, which in this case for given turbine inlet
temperature θt and bypass ratio α means maximum thrust. To maximize F, we
choose τc to maximize τf, since F increases monotonically with τf. From the
expression for τf
[
!" f
$
= #$ o + t 2 = 0
!" c
$o " c
3
]
"t
"t
(! c ) F max =
2
"0
"o
Notice that this is precisely the same result as for the Turbojet. Substituting in the
expression for τf
!c2 =
(! f )max F =
1+ " t + " o (1+ # ) $ " t $ " t
" o (1+ # )
( " t # 1)2
(! f )max F =
+1
" o (1+ $ )
+
.
! F $
.
2 + ( *t ) 1)2
#
&
= (1+ ' )+ * o ) 10 ) M o 0
" m˙ a0 % max
/
, ( ) 1 , 1+ '
/
(17b)
(14b)
Now we have 3 parameters,
θt - which we set at the maximum feasible value
Mo - flight speed
α - the prime variable distinguishing the turbofan
As before
2
2
!propulsive = u
= F / m˙ a
6
0
+1
+2
u0
M0 (1 + " )
The variation of F/ma0 and ηp with M0 and α is shown in the figures below for
θt=6.25. Of course for the higher bypass ratios we are really only interested in the
range of M0<1, but the lower bypass and higher M0 range may be of interest for a
supersonic transport.
4
5
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16.50 Introduction to Propulsion Systems
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