Actual Working Fluid

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Actual Working Fluid

• Fuel-air cycle analysis

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Effect of Variation in specific heat and gamma

• Because of variation in specific heat and gamma,
  the cycle analysis will be different.
• We cannot use the standard formulas for
  determining the air standard efficiencies.
• We must determine the temperatures and pressures
  taking into account the variation in cp and ? and
  determine the net work and heat supplied or heat
  supplied and heat rejected to determine the
  efficiencies.

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Correlations for air

• A number of correlations are available for
  determining the specific heat of air (at constant
  pressure or at constant volume) as function of
  temperature. Some correlations for gamma are
  also available.
• For example
• ? 1.4 7.18 x 10-5T
• A few of the correlations are given below

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Correlations for air (1)

• Krieger and Borman (Internal energy)
• u 0.6919943T 0.3917296x10-4T2
  0.5292534x10-7T3 0.2286286x10-10T4
  0.277589x10-14T5 kJ/kg
• cv du/dT and
• cp R cv
• T is in Kelvin

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Correlations for air (2)

• cp 0.9211 0.0002306 T kJ/kg-K
• T is in Kelvin
• Other properties can be obtained.
• A third order equation was proposed by Partha
  Pratim Saha, 89085,ex student of This course
• cp 26.430213692 8.443567110-3T
• 2.156769249610-6T2
• 1.946195410 -10T3 kJ /kmole K
• T is in Kelvin
• Molecular weight of air is 29

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Correlations for air (3)

• According to Lucas, the cp of any gas is given as
  follows
• cp aij (T/1000)i-1
• where i 1 to 7 and j represents the particular
  species, isooctane, oxygen or nitrogen. The units
  are kJ kmole-1 K-1
• Values of aij are available

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Other correlations
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Correlations for Fuel as Isooctane

• Obert1 has reproduced enthalpy data for
  isooctane (2,2,4 trimethyl pentane) in calories
  per gram above zero Kelvin. These values are
  tabulated as function of temperature. Empirical
  equations were obtained by Vivek Saxena, B.Tech.
  student in 1981, from these data as follows
• h A (2.099210-4T 1.838) (41200/114.232)
  (For the range of 300-700 K)
• h A(5.776310-4T1.6826) (41200/114.232)
  (For the range of 701-1000 K)
• where T is in Kelvin. A1000/114.232. Units of h
  are kcal/kg
• 1 Internal Combustion Engines and Air
  Pollution, Intext, 1973, p 724

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Correlations for Fuel as Isooctane

• Another formula developed by K.S.Reddy, former
  research scholar in the department, in 1982, is
• h C1T C2T2 .. C8T8 - 172.46
• where the values of C1, C2 etc are given below.
• C1 0.139149126610-1
• C2 0.304526706310-3
• C3-0.600428753610-7
• C40.369024438910-9
• C5 -0.942875933710-12
• C6 0.934904827610-15
• C7 - 0.423628830310-18
• C8 0.735264743410-22

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Correlations for Fuel as Isooctane

• These correlations have been given by Ferguson
• Cp /R a0 b0 TC0 T2
• h/RT a0 (b0 /2)T(C0 /3)T2 d0/T
• s0/R a0 lnT b0 T(c0 /2)T2 e0
• Values for the coefficients are given in the book
  or in the text handout

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Another correlation
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Composition of unburned mixture

• The total mass of charge trapped in the cylinder
  at the end of the intake stroke, m, is the sum of
  the fresh charge, ma, and residual charge, me.
  The residual gas fraction, f, is given by
• This value is large at part load in an SI engine
  and relatively small at full load. Figures for CI
  engines are lower, and more or less constant
  because the intake is unthrottled.

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Burned Gas Fraction

• If the inducted mixture is fuel and air (as in an
  SI engine) or air alone (as in a CI engine),
  then, the burned gas fraction, xb, in the
  unburned mixture during compression equals the
  residual gas fraction.

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Recirculated Exhaust Gas

• If EGR is used for NOx control, the percentage of
  exhaust gas recycled is the percent of the total
  intake mixture which is recycled exhaust, thus
• Here megr is the mass of exhaust gas recycled.
  The burned gas fraction in the fresh mixture is

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Burned Gas Fraction

• Since up to 30 of the exhaust gas can be
  recycled, the total burned gas fraction in the
  fresh mixture during compression can be as high
  as 40 of the total charge.
• The calculation of the composition of this burned
  gas fraction becomes important in order to obtain
  the properties of the charge during compression.

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Composition of burned gas fraction in unburned
mixture

• Assuming a typical hydrocarbon fuel, CxHy to burn
  in air, we can write the combustion equation per
  mole of fuel
• CxHy m(O2 3.76N2) ? n1CO2 n2H2O
  n3CO n4H2 n5O2 n6N2
• m is the number of moles of oxygen per mole of
  fuel

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Stoichiometric case

• Under stoichiometric conditions, CO, H2 and O2
  are zero. Also, m ms, the stoichiometric moles
  of oxygen per mole of fuel.
• The equivalence ratio, F FA/FAs AFs/AF ms/m
• The combustion equation will be
• CxHy ms(O2 3.76N2) ? n1CO2 n2H2O
  n6N2

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Lucas Charts for Isooctane-air mixtures

• Lucas prepared a set of charts for isooctane-air
  mixtures. Five such charts are available, for
  equivalence ratios 0.8 to 1.2 in steps of 0.1.
  These charts use thermodynamic data obtained from
  the JANAF tables. The description of the Before
  Combustion chart is given below.

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Before Combustion Charts

• The composition of the charge before combustion
  is assumed to be gaseous isooctane and air. The
  chemical equation, therefore, is
• n1C8H18 n2O2 n3N2 ? Products .
  (1)
• The stoichiometric equation for this reaction is
• C8H18 12.5(O2 3.76N2) ? 8CO2 9H2O
  3.76N2 (2)

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Before Combustion Charts

• The stoichiometric fuel-air ratio of isooctane is
  0.0665141. If the equivalence ratio (also called
  relative fuel-air ratio or fuel-air equivalence
  ratio) is denoted by ?, the actual fuel-air ratio
  is 0.066514?, which is the mass of fuel in kg.
  The mass of the charge is given by 10.066514?,
  that is for 1 kg of air.

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Before Combustion Charts

• The mass equation of the reaction is
• 0.066514? kg C8H18 1 kg Air ? Products
  (3)
• Now, 1 kg air contains 0.232 kg oxygen of
  relative atomic mass 32 and 0.768 kg atmospheric
  nitrogen of relative atomic mass 28.161. The
  relative molecular mass of isooctane is 114.232.
  Hence the molar equation is given by

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Before Combustion Charts

• 0.066514?/114.232 C8H18 0.232/32 O2
  0.768/28.161 N2 ? Products (4)
• 0.000582271? C8H18 0.00725 O2 0.0273 N2
  ? Products (5)
• ?/1717.42 C8H18 12.5/1717.42 O2
  (79/21)(12.5/1717.42) N2 ? Products (6)

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Before Combustion Charts

• Thus, in Eq, 1
• , n1 ?/1717.42
• n2 12.5/1717.42
• n3 (79/21)(12.5/1717.42)
• Hence, the total number of moles of gaseous
  reactants is

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Specific heat formula

• According to Lucas, the cp of any gas is given as
  follows
• cp aij (T/1000)i-1
• where i 1 to 7 and j represents the particular
  species, isooctane, oxygen or nitrogen. The units
  are kJ kmole-1 K-1
• Values of aij are given in next slide

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The enthalpy of the residual gas is given by

• where nTP is the total number of moles of
  products,
• f is the residual gas fraction,
• mp is the number of the product species in the
  residual gas,
• xj is the concentration of each of these
  species, and
• HFj is the corresponding enthalpy.

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The sensible internal energy, Es, of the charge
in kJ/kg of air, is given by

• G is the universal gas constant
• 8.3143 kJ/kgmolK

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The sensible enthalpy, Hs, of the charge, in
kJ/kg of air, therefore, is given by
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The entropy, S, of the charge in kJ/K per kg of
air, is given by

• p is the pressure.
• A typical chart for F1 is now given

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Relation between Unburned and Burned Mixture
Charts

• Given the thermodynamic properties of the
  unburned mixture before combustion, we would like
  to know the state of the burned mixture following
  adiabatic (i) constant-volume and (ii) constant
  pressure combustion.
• The datum for the internal energy and enthalpy
  for the unburned mixture is zero internal energy
  and enthalpy at 298.15 K. For the burned mixture,
  zero enthalpy for the gaseous species oxygen and
  nitrogen at 298.15 K is assumed.

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Enthalpy of formation

• If ?hof,u is the enthalpy of formation of the
  unburned mixture at 298.15 K, per kilogram of air
  in the original mixture and ?hof,i is the
  enthalpy of formation of species i at 298.15 K,
  per kilomole, then
• ?hof,u Sni?hof,i
• where ni is the number of kilomoles of species i
  per kilogram of air.

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• The unburned mixture enthalpy, hu, with the same
  datum as the unburned mixture enthalpy, is
  therefore given by the sensible enthalpy, hs,u
  and ?hof,u, thus
• hu hs,u ?hof,u
• and similarly, the internal energy uu is given by
• uu us,u ?uof,u
• and ?uof,u can be obtained either from the
  equation
• ?uof,u Sni?uof,i
• or from ?uof,u thus
• ?uof,u ?hof,u (nP nR)RoT
• For constant volume adiabatic combustion, uu ub
  and for constant pressure adiabatic combustion,
  hu hb.

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Chemical Internal Energy

• The incoming charge is deemed to consist of air
  plus residual combustion products left over from
  the previous cycle. If the volumetric residual
  gas is f (assuming the densities of both to be
  the same), the composition of the new charge may
  be written as

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• where nTP is the total number of moles of the
  product gases resulting from the combustion of
  fuel in 1 kg of air, mp is the number of product
  species Ap,j and xj is the concentration of each
  of these species.
• This means that the nominal 1 kg of air is
  contaminated with a small proportion of residual
  gas.

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• At the datum temperature of 298.15 K, therefore,
  the enthalpy of the charge per kg of air is
• The enthalpy of formation of isooctane has been
  taken from the table and the concentrations of
  the main product species together with the sum of
  their enthalpy values are also given in the
  table.

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• At any other temperature, the enthalpy of the
  charge is given by
• where mR is the number of species in the charge,
  that is, h hc hs where hs is the sensible
  enthalpy as shown earlier.

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• The internal energy of the charge per kg of air
  at the datum temperature of 298.15 K is
• where the term in the brackets is the total
  number of moles of the charge per nominal 1 kg of
  air and G is the universal gas constant.
• At any other temperature T, the internal energy
  is given by e ec es so that

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The constants for the evaluation for the chemical
internal energy, ec, are given in the table
below. The sensible internal energy, es, may be
read from the vertical axis of the before
combustion charts.
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The After Combustion Charts

• The after combustion charts were obtained, taking
  dissociation into account, assuming that thirteen
  species to exist in the products. These are H2O,
  H2, H, O2, O, OH, CO2, CO, N2, N, NO, NO2, and
  N2O. The last two oxides are taken because
  exhaust pollution research has shown them to
  exist, albeit in small quantities. Their absence
  makes little difference to the charts.

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After combustion charts

• The after combustion charts are plots of internal
  energy against entropy for lines of constant
  pressure, constant temperature and constant
  volume.
• In the low temperature range, where the constant
  temperature lines are practically horizontal, a
  subsidiary scale of sensible enthalpy, hs, may be
  constructed on the vertical axis to some
  advantage. This will help in estimating the
  temperature of the new charge after it has mixed
  with the residual.

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