Problem for heat capacity or heat ratio of rich combustion

[BX CH]

Hello all,

 

I am attempting to use Cantera for getting exhaust heat capacity (Cp) or heat ratio (gamma) from the burnt gases. 

 

One case is taken from a classic textbook, "Internal combustion engine fundamentals by Heywood," as a reference (figure shown below). However, the results from my Cantera don't fit such a trend after the fuel-air ratio is bigger than one. 

 

In the beginning, I thought the mechanism "curran_LLNL.cti" for isooctane is not suitable for showing the flow thermal property calculation. So, I extracted the portion of gas composition and imposed it into the Gri30 mechanism. But, the same result. My code for this calculation is simple:

 

######################################

gas = ct.Solution('CurranEtAl_PRF_fromLLNL.cti')

fuel = {"IC8H18":1}

 

def cp_gamma_cal(gas, fuel, lambda_c, T_c, p_c, reaction=1):

    phi = 1/lambda_c

    gas.TP = T_c, p_c*1e5

    gas.set_equivalence_ratio(phi, fuel, 'O2:1.0, N2:3.76')

    if reaction == 1:

        gas.equilibrate("TP")

    gas.TP = T_c, p_c*1e5

    return gas.cp_mass, gas.cp_mass/gas.cv_mass

######################################

 

Could anyone give me some hint? The code is also attached.

 

Sincerely thanks for your time in advance!

 

Best regard,

Beichuan

[Bryan Weber]

Hi Beichuan,

Can you please confirm whether or not Heywood used TP equilibrium or HP equilibrium? That will make a significant difference in the result.

Thanks,

Bryan

[BX CH]

Hi, Bryan,

Sincerely thanks for your kind reminding! 

I checked the original work from Heywood himself ("Correlations for the Viscosity and Prandtl Number of Hydrocarbon-Air Combustion Products, 1980"). In this paper, he used constant "TP" to investigate the in-cylinder gases property in term of the heat dissociation for temperature range 500K to 4000K.

I tried to repeat his results in this paper, but it seems my calculation in Cantera cannot reflect the heat dissociation effects (that roughly starts at 2000K). This can be seen as follows:

Still, my code is quite simple to calculate cp (both "TP" and "HP" are tried in this code, but no difference for getting heat dissociation):

####################################################

def cp_gamma_cal(gas, fuel, lambda_c, T_c, p_c, reaction=1):
    phi = 1/lambda_c
    gas.TP = T_c, p_c*1e5
    gas.set_equivalence_ratio(phi, fuel, 'O2:1.0, N2:3.76')
    if reaction == 1:

        gas.equilibrate("TP") # or "HP"

    gas.TP = T_c, p_c*1e5
    return gas.cp_mass, gas.cp_mass/gas.cv_mass

cp, gamma = cp_gamma_cal(gas, fuel, 1/phi, T_c, p_c, reaction=1)

####################################################

The full code and the mechanism are also attached.

Bryan Weber <bryan....@gmail.com> 于2021年8月9日周一 上午8:04写道：

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[Ray Speth]

Hi Beichuan,

The phrase “heat of dissociation” suggests to me that the definition of cp being used here is one where the composition follows equilibrium as the temperature changes. Cantera uses the definition where cp is defined at fixed composition.

This latter definition is the one that actually appears in governing equations for multi-species systems, such as well-stirred reactors. While the former property can be an interesting one to consider for a system at equilibrium, it is of somewhat limited use generally, since it is only defined at equilibrium.

To calculate this value with Cantera, you could do so as a finite difference with something like the following:

dT = 0.1
gas.TP = T_c, P_c
 gas.set_equivalence_ratio(phi, fuel, "O2:1.0, N2:3.76") 
gas.equilibrate("TP")
h1 = gas.enthalpy_mass
gas.TP = T_c + dT, P_c
 gas.equilibrate("TP") 
h2 = gas.enthalpy_mass
cp_eq = (h2 - h1) / dT

Regards,
Ray

​

[BX CH]

Hi, Ray and Bryan,

Super thanks for your help! 

Following your suggestion, my problem has been solved. 

Best Regards, wish you a nice summer!

Beichuan

Ray Speth <yar...@gmail.com> 于2021年8月9日周一 下午9:50写道：

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[William R. Smith]

I would argue that the more important Cp is the value when the system is constrained to be at equilibrium as T changes (it is somewhat analogous to the derivative dP*/dT along the saturation  line for phase equilibrium, which is much more important that the dP/dT value for each individual phase.)  It is the Cp value that would be observed if the system is maintained at equilibrium as T changes at constant P.  The other Cp refers to a "frozen equilibrium" (non-reacting) value of Cp.  Ray has provided a finite difference approximation for the former quantity, but its "exact" value is directly obtainable from a set of linear equations for the "sensitivity coefficients" of the equilibrium mole numbers with respect to T, (dn_i/dT)_P.  The coefficient matrix is formed from quantities available at the given equilibrium composition.  The procedure is described in Chemical Reaction Equilibrium Analysis: Theory and Algorithms, Smith and Missen, Wiley-Interscience/Malabar, 1982/1991, Section 8.4, and also described in W.R. Smith, Can. J. Chem. Eng..  47(1), 95-97 (1969).  (Interestingly, in Storey and van Zeggeren, 

Regards,

William R. Smith

[William R. Smith]

Oops, the last sentence wasn't finished.  It should be:

(Interestingly, in The Computation of Chemical Equilibria, Cambridge Un. Press, 1970" (as I recall, the book is in my university office), they say that this calculation is not possible.)

[BX CH]

Sorry for my late reply! And sincerely thanks for pointing this out.

The purpose of calculating such instantaneous values of Cp or gamma is to evaluate the change of flow energy inside cylinder during combustion process. 

I will read the books you mentioned. Again, thanks for your kindly help! 😀

Best regards,

Beichuan

William R. Smith <bils...@uoguelph.ca> 于2021年8月11日周三 下午5:35写道：

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