THEORY OF PROPULSION 3' Conservation Equations

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THEORY OF PROPULSION3. Conservation Equations

• P. M. SFORZA
• University of Florida

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Species conservation equation
ridri VdV AdA
ri V A
dri/dt
dz
-(rate of species i in) (rate of species i
out) rate of production of species I in
CV
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Species conservation equation (cont.)
global mass conservation equation
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Normalizing the species equation
introduce L and tc
L
Reaction completed
t0
tt
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Species conservation equation (concluded)
Characteristic residence time of a fluid
particle, L/V
Characteristic time for reaction to be completed
The quantity tf/tc is essentially the first
Damkohler similarity parameter for reacting flows
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Characteristic times
Slow flow and fast reaction
Fast flow and slow reaction
tf/tcgtgt1 Fluid particles move slowly
through reaction zone
tf/tcltlt1 Fluid particles pass through quickly
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Damkohler number limits
tf/tcltlt1 Here dYi 0 because the residence time
is very brief compared to the reaction time, no
appreciable reaction occurs, and dH0.

• tf/tcgtgt1
• In the limit dYi is infinite because the reaction
  time goes to zero and the reaction proceeds
  instantaneously to the equilibrium composition
  pertinent to the local temperature and pressure,
  and thus dH is determined by using the chemical
  equilibrium conditions.

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Damkohler number limits
For tf /tcgtgt1 chemical equilibrium prevails and
the heat of reaction is
For tf /tcO(1) neither limit is applicable. The
reaction in the combustor is rate-controlled and
the actual chemical kinetics must be accounted
for. This complicates the equation set and the
solution techniques required.
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Conservation of momentum
pdp
p
dFz
V
VdV
Drag force on fluid
Wall shear force on fluid
twdAw
Isentropic flow
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Conservation of momentum (concluded)
Skin friction coefficient Hydraulic
diameter, D4A/P
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Equations in standard form
Independent variables
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Equations in standard form (continued)
Dependent variables
Typical equation from influence coefficient chart
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Table of influence coefficients
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Combustion Chambers For Airbreathing Engines
A good combustor should

• Provide full combustion with minimum pressure
  loss
• Operate without significant accumulation of
  deposits
• Ignite the fuel readily
• Give reliable service over long periods of time

Good combustor design is based on the
fundamentals

• Proper mixture ratio and the 3 Ts of good
  combustion
• Temperature of the reactants
• Turbulence for good mixing
• Time for mixing and combustion to go to completion

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Constant area combustor
Equation developed from the influence coefficient
table for no change in area, power, heat
transfer, friction, or drag
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Constant area combustor (cont.)
Tt,4
Eliminating the term in square brackets and
combining equations yields
integrating
state eq.
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Constant area combustor (cont.)
The stagnation pressure change is
rVconstant V4/V3r3/r4
The stagnation temperature change is
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Constant area combustor (cont.)
Modeling the heat release as a change in Tt so
that
Tt,4
-dHcp(Tt4-Tt3)
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Can-type combustion chamber
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Constant pressure combustor
Same assumptions except that now area can change
but pressure cannot
p4p3
p3
integrating
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Constant pressure combustor (cont.)
V4V3
Here we show that Vconstant in a constant
pressure combustor
V3
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Constant pressure combustor (cont.)
Final properties are known if initial properties
are given
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Annular-type combustor
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Double-Annular type combustor
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Fuels for airbreathing engines
Paraffin hydrocarbon fuel
Conservation of atoms
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Fuels for airbreathing engines (cont.)
air
fuel
The molar fuel to air ratio
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Fuels for airbreathing engines (cont.)
Octane C8H18
Fuel to air ratio by mass
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Fuels for airbreathing engines (cont.)

• For hydrocarbon fuels of this type
• Combustion possible in the range 0.04 lt wf/wa lt
  0.25
• Maximum reaction rate occurs for wf/wa 0.073
• Complete combustion occurs for wf/wa 0.066
• Combustion is slow and erratic for wf/wa lt0.055
• Equivalence ratio

gt1 fuel-rich, CO, H2O
1 stoichiometric
lt1 fuel-lean CO2, O2
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Fuels for airbreathing engines (concl.)
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Combustor heat release
Total rate of heat addition
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Approximate combustor heat addition
25C (T-O) 150C (M2)
Tt,0(1M2/5)T0
25C (T-O) 725C (M2)
Tt,3(1M2/5)T3
Fuel to air ratio f/a0.02 Fuel temperature
Tt,fltTt,0 Tt,fltskin temperature
Cp 1.94kJ/kgK (JetA liquid at 15C)
1.51kJ/kgK (Jet A vapor at 25C) 1.00
kJ/kgK (Air at 25C) 1.14kJ/kg (Air at
750C)
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Combustor efficiency
For wf/wa w4/w3 ltlt1 the combustor efficiency
becomes
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Combustor efficiency (continued)
50 (lb-s-R/ft3 )
Combustor parameter pt,3Tt,3/V3 f (t,T,mixing)
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Combustor efficiency (concluded)
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Combustor configuration
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Apportioning the airflow
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Effect of cooling air on exit temperature
operating regime
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Curve fit for specific heat of air
Curve fits for air between 750R (417K) and 4000R
(2222K) are simply Cp(Btu/lb-R)0.0908T0.148
Cp0.242 for Tlt750R Cp(KJ/kg-K)0.415T0.148
Cp1.01 for Tlt400K
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Sample problem max heat addition in a constant
area combustor
What is maximum mf/m3?
mf
Tt,3488K M30.35
3 4
Heat addition (fuel HV44MJ/kg)
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Sample problem max heat addition in a constant
area combustor
Assume that k3k4g4/3 and note that
The maximum heat addition is that which makes
M41. Setting M41 and M30.35 yields
DTt,max1.327(488)648K and Tt4,max1136K. Then