Page 222 - Introduction to Computational Fluid Dynamics

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6.4 APPLICATIONS
The combustion chamber walls are assumed adiabatic. The inflow bound- 11:10 201
ary specifications, however, require explanation. At the primary slot, u 1 = 0,
2 2
u 2 = 100, e = (0.005 × u 2 ) , = C µ ρ e /(µ R µ ), where viscosity ratio R µ =
−1
µ t /µ = 10, T = 773,ω fu = (1 + R stoic ) , and = 0. At secondary and dilution
2
slots, u 2 =−48 and −42.5, respectively, and e = (0.0085 × u 2 ) , R µ = 29, T =
773,ω fu = 0, and =−1/R stoic are specified. Finally, fluid viscosity is taken as
2
µ = 3.6 × 10 −4 N-s/m and specific heats of all species are assumed constant
5
at C p = 1,500 J/kg-K. The density is calculated from ρ = 8 × 10 M mix /(R u T ),
where R u is the universal gas constant, M −1 = ω fu /M fu + ω air /M air + ω pr /M pr ,
mix
and the product mass fraction is ω pr = 1 − ω fu − ω air .
In this problem, the equations are strongly coupled and an initial guess for
variables is difficult to determine a priori. To ensure convergence, therefore, the
−5
false-transient technique is used with t = 10 . Convergence is declared when
−3
residuals for all variables (except e and ) are less than 10 . Further, it is ensured
that the exit mass flow rate equals (within 0.1%) the sum of the three flow rates
specified at the slots. A total of 12,500 iterations are required.
In the middle panel of Figure 6.21, the vector plot is shown. The plot clearly
shows the strong circulation in the primary zone with a reverse flow near the axis
necessarytosustaincombustion.Allscalarvariablesarenowplottedas( − min )/
( max − min ) in the range 0–1 at a contour interval of 0.1. For turbulent viscosity,
µ t,min = 0 and µ t,max = 0.029; for temperature, T min = 773 K and T max = 2,456 K
(adiabatic temperature = 2,572 K); for fuel mass fraction, ω fu,min = 0 and ω fu,max =
0.055066, and for composite variable, min =−0.058275 and max = 0. The bot-
tom of Figure 6.21 shows that high turbulent viscosity levels occur immediately
downstream of the fuel injection slot and secondary and dilution air slots because
of high levels of mixing.
The top panel of Figure 6.22 shows that the fuel is completely consumed in the
primary zone. Sometimes, it is of interest to know the values of mixture fraction
−1
f= f stoic + (1 − f stoic ), where f stoic = (1 + R stoic ) . From the contours of
shown in the middle panel of Figure 6.22, therefore, values of f and concentra-
tions of air and products can be deciphered. The temperature contours shown on
the bottom panel of Figure 6.22 are similar to those of . This is not surprising
because although T is not a conserved property, enthalpy h = C p T + ω fu H c , like
, is conserved and ω fu
0 over a greater part of the domain. The temperatures,
as expected, are high in the primary zone and in the region behind the fuel injection
slot, but the temperature profile is not at all uniform in the exit section. Combustion
chamber designers desire a high uniformityoftemperaturein theexit sectiontosafe-
guard the operation of the turbine downstream. Such a uniformity is often achieved
by nonaxisymmetric narrowing of the exit section. However, accounting for this
feature will make the flow three dimensional and hence is not considered here.
It must be mentioned that combustion chamber flows are extensively investi-
gated through CFD for achieving better profiling of the casing, for determining