THEORY OF PROPULSION 2' Basic Flow Equations

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THEORY OF PROPULSION2. Basic Flow Equations

• P. M. SFORZA
• University of Florida

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Analysis of propulsion system components

• Equations for flow through a duct in the steady
  quasi-1-D approximation
• Applications extend to propulsion subsystems,
  such as combustors, nozzles, inlets, and
  turbomachinery cascades.
• Equations are simple, yet they are reasonably
  accurate
• The equation set is particularly useful for
  preliminary design purposes.

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Quasi-one-dimensional flow theory accounts for

• Changes in duct cross-sectional area
• Chemical reactions
• Variations in gas molecular weight and specific
  heat
• Exchange of heat with the surroundings
• Drag caused by internal bodies or solid particles
  in the flow
• Losses due to friction on the combustor walls

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Equation of state
ZW0/W is called the compressibility
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Acoustic speed
k is the isentropic exponent
In logarithmic differential form this becomes
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Mach number
In logarithmic differential form this becomes
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Mass conservation equation
r
z
For axisymmetric flow this becomes
Vn
A(z)
Integrating over the area A(z)prw(z) 2 leads
to
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Mass conservation equation (continued)
wall
Wall boundary condition
V
drw
Vr
Vz
dz
The mass conservation equation reduces to
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Mass conservation equation (concluded)
Cross-sectional area A
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Energy conservation equation
Rate of heat transfer in across the boundary
Total energy change within boundary
Power taken out across the boundary
Heat input rate - Power output Total
energy change
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Conservation of energy (continued)
Enthalpy change
Differential species enthalpy
Specific heat of mixture
Composition change
Temperature change
Change in mixture enthalpy
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Conservation of energy (continued)
Within the control volume
Across the control surface
Change in composition due to chemical reaction
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Conservation of energy (continued)
General chemical reaction among N species
dH (molar basis)
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Energy conservation equation (concluded)
Heat of reaction of a chemical reaction is given
by
(Finite changes)
(Infinitesimal changes)
To find the heat of reaction we must find the
chemical composition of the flow.