Chapter 4 Verification
4.1 Preliminary Verification Study
4.1.3 Two-Species Laminar Flame Propagation
Using CANTERA to construct a reference solution, we designed a perfect gas, two- species, one-step chemistry model that approximates the propagation of a laminar flame in one dimension. For this model we matched the flame velocity and tempera- ture by changing the species’ heat release, specific heat, and molecular weight. Also, an approximate one-step reaction rate and activation energy was chosen,
˙
ω=−ATnρλe−RTEa. (4.5)
(a) Initial product mass fraction (b) Final product mass fraction
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0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68
Product Fraction Exact values
(c) Initial product mass fraction
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0.500 0.505 0.510 0.515 0.520 0.525 0.530 0.535
Product Fraction Exact Values
(d) Final product mass fraction
Figure 4.3: Verification of product/reactant mass diffusion with comparison to the heat equation.
When λ, the reaction progress variable is specified as the mass fraction of the reac- tants, then ˙ω is the mass production rate for the reactant. A is the pre-exponential factor, n the temperature factor (zero in our model), Ea the activation energy in units of (energy/mass), and R the gas constant. Note that if the common units of (energy/mol) are used for the activation energy, then the universal gas constant R must be used.
The laminar flame speed is defined as the velocity at which reactants propagate through a stationary flame front, see Williams(164) §5.1. In this case, the reactants are approaching the flame front at a constant speed, and then are accelerated to a higher speed as they react and expand. figure 4.4, shows the convention, which one notes is similar to that for a steady shock front.
Figure 4.4: Particle speeds of reactants and products in a steady laminar flame
The laminar flame velocity is determined mainly by the energy equation, where there is a balance of heat conduction and mass diffusion of the different enthalpies of the reactants and products. For a one-step model, there is the diffusion of the heat release, which is transported by the products’ and reactants’ mass diffusion mechanisms. These diffusion rates are controlled by the mass and thermal diffu- sion coefficients which are functions highly dependent on temperature and inversely dependent on pressure.
We used a simple one-step mechanism with CANTERA’s FreeFlame model and mixture transport to obtain an exemplary solution shown in figure4.5(b)for a mixture starting at standard temperature and pressure. Our parameters, which were obtained by modifying the CANTERA argon mechanism and adding a reaction are shown in figure 4.2.
Property Value Units Comments
Initial Density 1.623 kg/m3 density at 300 K, 1 atm
Reactant Specific Heat (Cp) 20.79 J/mol*K Cp/R in cti file (1st polynomical coefficient) Product Specific Heat (Cp) 20.79 J/mol*K Cp/R in cti file (1st polynomical coefficient)
Gamma γ 1.67 calculated from Cp/Cv
Heat Release q 1074.86 kJ/kg R*(Δa5o) where Δ() = ()R - ()P
Activation Energy Ea 17000 cal/mol chosen as part of one-step model
71128 J/mol common units
Pre-exponential A 9.62E+07 s-1 chosen as part of one-step model
Temperature Power n 0 no Tn dependence of reaction rate
NOTES:
1. One-Step model with 2 species: R (reactant) and P (product) 2. Both species R and P consist of one atom of Argon
8. Reaction rate parameters Ea and A chosen as part of one-step model to match flame speed 3. Constant specific heat (can be different values for R and P) - for polynomial coefficients in cti file, (a0) = Cp/R, a1-a4 = 0 (no dependence of Cp on temperature)
5. Transport properties of both R and P are that of Argon
6. Assume no temperature dependence of viscosity or thermal conductivity (constant values)
7. Values for Cp, q picked as part of one-step model, all other thermodynamic (density, viscosity, diffusion coefficients, etc.) parameters are obtained using Cantera by evaluating the gas object at 300 K, 1 atm 4. Heat release q from Reactant to Product - included in cti file in the 6th coefficient
Table 4.2: Input parameters for one-step steady flame model
0.00E+00 2.00E+00 4.00E+00 6.00E+00 8.00E+00 1.00E+01 1.20E+01 1.40E+01 1.60E+01 1.80E+01 2.00E+01
9.80E‐03 9.90E‐03 1.00E‐02 1.01E‐02 1.02E‐02 Velocity Cantera
Velocity AMROC
(a) Velocity (ms)
0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03
9.80E‐03 9.90E‐03 1.00E‐02 1.01E‐02 1.02E‐02 T Cantera
T AMROC
(b) Temperature (K)
1.40E+00 1.60E+00
Density
1.00E+00 1.20E+00
y Cantera Density AMROC
6.00E‐01 8.00E‐01
2.00E‐01 4.00E‐01
0.00E+00
9.80E‐03 9.90E‐03 1.00E‐02 1.01E‐02 1.02E‐02
(c) Density (mkg3)
1.00E+00
Product Cantera 8.00E‐01 Product AMROC
4.00E‐01 6.00E‐01
2.00E‐01
0.00E+00
9.80E‐03 9.90E‐03 1.00E‐02 1.01E‐02 1.02E‐02
(d) Product mass fraction
1.012E+05
P AMROC
1.012E+05 1.012E+05
1 012E 05 1.012E+05
1.012E+05 1.012E+05
1.011E+05
9.80E‐03 9.90E‐03 1.00E‐02 1.01E‐02 1.02E‐02
(e) Pressure (Pa)
Figure 4.5: CANTERA and AMROC comparison for the 1D laminar flame
4.1.3.1 One Dimension
Using the 1D reactive flow equations in AMROC a laminar steady flame was sim- ulated and compared the reference solution produced with CANTERA. We used a two-species model with the total energy defined by the heat release per unit mass parameter, q.
γ =γ1 =γ2, p=ρRT, R=R1 =R2, (4.6) ρ=ρ1+ρ2, ρ1 =ρY1, ρ2 =ρY2, (4.7) ρet= p
(γ−1) +1
2ρ(u2 +v2) +ρ2q, (4.8) et= p
(γ−1)ρ +1
2(u2 +v2) +qY2. (4.9)
(4.10) This is equivalent to having product and reactant enthalpies of the form
h1 =h0+cpT, (4.11)
h2 =h0+q0+cpT, (4.12)
h1−h2 =hprod−hreact= ∆hreaction =−q0. (4.13)
Also, note that in general for a two-species model, the sound speed used in the numerical simulation is the frozen sound speed, where
c=
r γp
ρ1 +ρ2, (4.14)
γ = 1 + X1
γ1−1 + X2
γ2−1, (4.15)
X1 =ρY1
W1, X2 =ρY2
W2, (4.16)
where Xi is the mole fraction and Wi is the molar mass;
Now, for the special case of a one-dimensional model assuming perfect gases, reactants and products having the same molecular weight, constant conductivity and mass diffusivities, and zero viscosity, the conservation equations for mass, momentum,
and energy are as follows,
∂(ρY1)
∂t +∂(ρuY1)
∂x = ∂
∂x
ρD∂Y1
∂x
+ ˙ω1, (4.17)
∂(ρY2)
∂t +∂(ρuY2)
∂x = ∂
∂x
ρD∂Y2
∂x
+ ˙ω2, (4.18)
∂(ρu)
∂t +∂(ρu2+p)
∂x = 0, (4.19)
∂(ρet)
∂t +∂(ρu(et+p))
∂x = ∂
∂x
ρDq0∂Y2
∂x
+ ∂
∂x
k∂T
∂x
, (4.20)
whereY1,Y2, D, andk are the reactant and product mass fractions, mass diffusivity, and thermal diffusivity. First, using constant conductivity and mass diffusivities, we started the laminar flame using a compressed and pressurized region of products. By the ideal gas law this corresponds to an increase in temperature. After a transient period, a steady propagating laminar flame resulted. Initially, a weak shock wave travels out slightly raising the temperature and pressure in front of the flame. There is an expansion fan behind the shock wave, which cools reactants close to ambient conditions.
In this first simulation using constant diffusivities corresponding the ambient val- ues, the laminar flame speed was found to be about half of that expected from CAN- TERA, which uses a temperature dependent transport model. Subsequently, another simulation was conducted using temperature dependent conductivity and tempera- ture and pressure dependent mass diffusivity that match CANTERA’s. The temper- ature dependent Sutherland law is used for the viscosity and conductivity, however, to match the CANTERA solution, the viscosity was neglected,
µ=µref T
Tref 32
Tref +sµ
T +sµ , (4.21)
k =kref T
Tref
32
Tref +sk T +sk
. (4.22)
The following equation was used for the mass diffusivity, D=Dref
D Dref
1.71 k
p
pref, (4.23)
where pref is defined as atmospheric pressure. For our temperature ranges of 300 to 2500 K, these approximate functions are very close to CANTERA’s as shown in
figure 4.6.
Figure 4.6: Comparing the Transport Properties to CANTERA’s results forT = 300 to 2500 K
Using the temperature and pressure dependent transport values, the results were much closer. The discrepancies are surmised to be the result of CANTERA using a constant pressure assumption in its solution (the momentum equation is ignored), and the slight differences in the transport properties. Since the CANTERA solution is not exact for the momentum equation, there are differences in the steady state results between the full diffusive Navier-Stokes (ignoring viscosity) equations of AMROC and the CANTERA energy and continuity based result.