5. combustion and thermochemistry · pdf fileaae 439 ch5 –3 overview important concepts...
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AAE 439
Ch5 –1
5. COMBUSTION AND THERMOCHEMISTRY
AAE 439
Ch5 –2
Overview
Definition & mathematical determination of chemical equilibrium,
Definition/determination of adiabatic flame temperature,
Prediction of composition and temperature of combusted gases as a function of initial temperature,
Prediction of amounts of fuel & oxidizer,
Thermochemical changes during expansion process in nozzle.
Performance Parameters:
CF= 2γ 2
γ −12
γ +1
⎛⎝⎜
⎞⎠⎟
γ +1γ −1
1−p
e
p0
⎛
⎝⎜⎞
⎠⎟
γ −1γ
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥+
pe− p
a
p0
⋅ ε
c* =
RT0
γγ +1
2
⎡
⎣⎢
⎤
⎦⎥
γ +1γ −1
Performance depends on: T, MW, p0, pe, pa, γ
AAE 439
Ch5 –3
Overview
Important Concepts & Elements of Analysis
Conversion of Chemical Energy to Heat
Simple Treatment of Properties of Gases
Balancing Chemical Reactions - Stoichiometry
Adiabatic Flame Temperature
Chemical Equilibrium and Gibbs Free Energy
Nozzle Expansion Effects
Thermochemical Calculations
AAE 439
Ch5 –4
5.1 THERMODYNAMICS OF GAS MIXTURES
AAE 439
Ch5 –5
Perfect Gas
Perfect Gas Law relates pressure, temperature and density for a perfect gas/mixture of gases :
Universal Gas Constant:
Gas Constant:
Calorically Perfect Gas: Internal Energy
Enthalpy
Specific Heat Relationships:
Definition of “Mole”: A mole represents the amount of gas, which contains Avogadro’s number of gas
molecules: 6.02·1023 molecules/mol.
pV = nℜT = mRT ⇔ p v = RT
ℜ = 8.314
Jmol ⋅K
R = ℜ
M
du = cvdT u
2− u
1= c
v(T
2−T
1)
dh = cp
dT h2− h
1= c
p(T
2−T
1)
c
p− c
v= R γ =
cp
cv
AAE 439
Ch5 –6
Gibbs-Dalton Law
Properties of a mixture is determined by the properties of constituents according to Gibbs–Dalton Law:
The pressure of a mixture of gases is equal to the sum of the pressure of each constituent when each occupies alone the volume of the mixture at the temperature of the mixture.
The internal energy and the entropy of a mixture are equal, respectively, to the sums of the internal energies and the entropies of its constituents when each occupies alone the volume of the mixture at the temperature of the mixture.
Temperature
Pressure
Volume
Energy
Entropy
Enthalpy
Tmix= T
1= T
2=…= T
N
p
mix= p
1+ p
2+ p
3…+ p
N= p
ii=1
N
∑
Vmix= m
mixv
mix= m
1v
1= m
2v
2=…= m
Nv
N
E
mix= m
mixe
mix= m
1e
1+ m
2e
2+…+ m
Ne
N= m
ie
ii=1
N
∑
Smix= m
mixs
mix= m
1s1+ m
2s2+…+ m
Ns
Nsmix
= Smix
nmix
Hmix= m
mixh
mix= m
1h
1+ m
2h
2+…+ m
Nh
Nh
mix= H
mixn
mix
AAE 439
Ch5 –7
Mixture of Gases
Definitions by Mass Based Molar Based
PG Law
Pressure
Fraction of Species
Enthalpy
Entropy
Equivalent Molecular Weight
piV = m
iR
iT = n
iℜT pV
i= m
iR
iT = n
iℜT
p = p
ii=1
N
∑ V = V
ii=1
N
∑
Mmixequiv
= mn= m
mi
Mii=1
N
∑= 1
yi
Mii=1
N
∑ M
mixequiv
= mn=
niM
ii=1
N
∑n
= xiM
ii=1
N
∑
x
i=
ni
nmix
= yi
Mmix
Mi
xi
i=1
N
∑ =1 y
i=
mi
mmix
= xi
Mi
Mmix
yi
i=1
N
∑ =1
h
mix= y
ih
ii∑
h
mix= x
ih
ii∑
s
mix(T,p) = y
is
i(T,p)
i∑
smix
(T,p) = xisi(T,p)
i∑
s
i(T,p
i) = s
i(T,p
ref)−R ln
pi
pref
si(T,p
i) = s
i(T,p
ref)−ℜ ln
pi
pref
AAE 439
Ch5 –8
Mixture of Gases
Definitions:
Relationship
Specific Heat
Ratio of Specific Heat
Vi
V=
pi
p= x
i= y
i
Mmix
Mi
c
p,mix= c
p,iy
ii=1
N
∑
γ
mix=
cp,mix
cv,mix
=c
p,mix
cp,mix
−Rmix
AAE 439
Ch5 –9
5.2 1st LAW OF THERMODYNAMICS
AAE 439
Ch5 –10
1st LTD - Fixed Mass
First law of thermodynamics embodies the fundamental principle of conservation of energy. Q and W are path functions and occur only at the system boundary.
E is a state variable (property), ∆E is path independent.
m, E
Q
W
System Boundary enclosing Fixed Mass
Q − W = ΔE1→2
Heat added to system in going from state 12
Work done by system on surrounding in
going from state 12
Change in total system energy in going from state 12
Q − W = dE dt
q − w = de dt
E = m u+ 1
2v2 + g z
⎛
⎝⎜
⎞
⎠⎟
AAE 439
Ch5 –11
1st LTD - Control Volume
Conservation of energy for a steady-state, steady-flow system.
Assumptions: Control Volume is fixed relative to the coordinate system.
Eliminates any work interactions associated with a moving boundary,
Eliminates consideration of changes in kinetic and potential energies of CV itself.
Properties of fluid at each point within CV, or on CS, do not vary with time.
Fluid properties are uniform over inlet and outlet flow areas.
There is only one inlet and one exit stream.
QCV WCV
Control Surface (CS) enclosing Control Volume (CV)
Q
CV− W
CV= m e
outlet− m e
inlet+ m p
ov
o− p
iv
i( )Rate of heat
transferred across the CS, from the
surrounding to the CV.
Rate of all work done by CV,
including shaft work but excluding flow work.
Net rate of work associated with pressure
forces where fluid crosses CS, flow work.
dmCV
dt= 0
dECV
dt= 0
Rate of energy flowing out
of CV.
Rate of energy flowing into
CV.
QCV
− WCV
= m ho− h
i( ) + 12
vo2 − v
i2( ) + g z
o− z
i( )⎡
⎣⎢
⎤
⎦⎥
m e + p v( )
inlet m e + p v( )
outlet
AAE 439
Ch5 –12
THERMODYNAMIC PROCESSES
Energy Equation (1st Law of TD)
Energy Change due to a process going from State 1 to State 2:
Constant–Volume (Isochoric) Process:
Constant–Pressure (Isobaric) Process:
E = U + Epotential
+ Ekinetic
= Q −Wshaft
−Wflow
ΔU = U2−U
1= Q −W
flow= Q − p V
ΔU = Q
ΔU = Q − pΔV
ΔU + pΔV = Q
ΔH = Q
AAE 439
Ch5 –13
5.3 THERMOCHEMISTRY BASICS
AAE 439
Ch5 –14
Energies in Chemical Reactions
Enthalpy of Combustion (Reactions):
Heat of Combustion:
– QCV
REACTANTS Stoichiometric fuel-oxidizer (air)
mixture at standard state conditions: Tref and pref.
PRODUCTS Complete combustion
at standard state conditions: : Tref and pref.
Hin= H
reactant Hout= H
product
Δhrxn
≡ qCV
= hprod
− hreac
ΔHrxn
= Hprod
−Hreac
ΔhC= −Δh
rxn
Graphical Interpretation
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