increasing the specific heat capacity
 

I have seen in the Fluent manual the following information:

Global reaction mechanisms with one or two steps inevitably neglect the intermediate species. In high-temperature flames, neglecting these dissociated species may cause the temperature to be overpredicted. A more realistic temperature field can be obtained by increasing the specific heat capacity for each species. Rose and Cooper [ 362] have created a set of specific heat polynomials as a function of temperature.

362 Rose J. W. and Cooper J. R. Technical Data on Fuel. Scottish Academic Press, Edinburgh, 1977.

Unfortunately this book is out of print now. Could somebody say something about procedures and assumptions that have been used to create this set of polynomials?

Re: increasing the specific heat capacity
 

I dont have acces to the book you mentioned but tried to calculate the modified polynom for oxygen and this is what i came up with.


Consider endothermal the reaction

O2 <==> 2 O delta_H=+498,4 kj/mol

We can caculate the change of gibb free energy (G(T)) from
Delta_GR=2 * G_O - G_O2
and the equilibrium constant
Kp=exp(-Delta_GR/(R*T))

Equilibrium concentrations can be calculated from
Kp=(p_O)^2/p_O2
p_O partial pressure O in atm
p_O2 partial pressure O2 in atm
or in mole fractions
Kp*p=(n_O)^2/n_O2
p system pressure in atm

Taking one mole O2 at 1 atm pressure we get

T DGR Kp n O n O2 H(T)
K j/kmol mol/mol mol/mol j/kmol
------------------------------------------------------------------------------------
1500 3,099052E+08 1,613434E-11 4,016750E-06 9,999960E-01 4,059487E+07
1600 2,966571E+08 2,064386E-10 1,436787E-05 9,999856E-01 4,426023E+07
1700 2,833782E+08 1,961251E-09 4,428503E-05 9,999557E-01 4,795738E+07
1800 2,700722E+08 1,453539E-08 1,205555E-04 9,998794E-01 5,169142E+07
1900 2,567419E+08 8,738115E-08 2,955593E-04 9,997044E-01 5,547314E+07
2000 2,433901E+08 4,396156E-07 6,628154E-04 9,993372E-01 5,932229E+07
2100 2,300190E+08 1,898381E-06 1,376869E-03 9,986231E-01 6,327171E+07
2200 2,166307E+08 7,183696E-06 2,676652E-03 9,973233E-01 6,737193E+07
2300 2,032269E+08 2,423293E-05 4,910594E-03 9,950894E-01 7,169594E+07
2400 1,898092E+08 7,391992E-05 8,560789E-03 9,914392E-01 7,634348E+07
2500 1,763790E+08 2,063620E-04 1,426250E-02 9,857375E-01 8,144400E+07
2600 1,629375E+08 5,326311E-04 2,281402E-02 9,771860E-01 8,715727E+07
2700 1,494859E+08 1,282083E-03 3,517089E-02 9,648291E-01 9,367058E+07
2800 1,360252E+08 2,899531E-03 5,241704E-02 9,475830E-01 1,011911E+08
2900 1,225561E+08 6,200781E-03 7,570566E-02 9,242943E-01 1,099320E+08
3000 1,090795E+08 1,260921E-02 1,061630E-01 8,938370E-01 1,200918E+08

H(T)=n_O * H_O(T) + n_O2 * H_O2(T)

H(T) can be expressed as polynom with the coefficients
A0 -2,604167E+08
A1 7,824454E+05
A2 -9,267354E+02
A3 6,026670E-01
A4 -2,153371E-04
A5 3,970711E-08
A6 -2,887158E-12

cp(T) can be calculated from H(T) as cp=d(H(T)-H0)/dT. H0 is enthalpy at standard (reference)
state. This will get us the coefficients for the modified cp polynom
A0 7,824454E+05
A1 -1,853471E+03
A2 1,808001E+00
A3 -8,613484E-04
A4 1,985356E-07
A5 -1,732295E-11

The return value of cp is j/kmol/K, devide the coefficients by the molecular weight
of oxygen to get j/kg/K. Comparing the modified polynom from above (cp1) with the one
from the fluent manual (cp2) we get a good agreement between 1500K and 2700K, largest
deviation at 2400K is 8%.

T cp1 cp2 (cp1-cp2)/cp2*100%
K j/kg/K j/kg/K %
-------------------------------------------
1500 1147,9 1152,0 0,360025687
1600 1149,1 1160,9 1,017681849
1700 1159,1 1166,8 0,654172328
1800 1173,6 1170,3 -0,28382029
1900 1192,2 1173,6 -1,584541084
2000 1217,4 1180,1 -3,156017009
2100 1254,6 1196,0 -4,905009001
2200 1311,0 1229,7 -6,613965544
2300 1394,9 1293,4 -7,849970668
2400 1515,5 1403,5 -7,979328925
2500 1681,5 1581,1 -6,35188464
2600 1901,4 1853,3 -2,596428625
2700 2182,0 2253,7 3,181389523
2800 2528,2 2823,6 10,46189678
2900 2942,3 3612,9 18,56272308
3000 3423,2 4681,4 26,87600196

It should be mentioned that gas dissociation is usually pressure dependend so you will get
different polynoms for higher pressures. The Rose and Cooper values are probably calcualted
for 1 atm, applying them to higher pressures may lead to errors.

RoM

Re: increasing the specific heat capacity
 

Thanks a lot for your extended answer. I have asked Fluent for the pressure and temperature applicability range for Rose and Cooper specific heat polynomials, but I have not received good answer. I will try to find the book "Technical Data on Fuel".

In this moment I have some additional questions:

Why these polynomials could be applicable in both two cases: 1. when I am using global reaction mechanism with one steps; 2. when I am using global reaction mechanism with two steps.

And other question is following.

Now I am using methane-air-2step finite-rate/eddy-dissipation model with methane-air-2step mechanism:

2 CH4 +3 O2 -> 2 CO + 4 H2O 2 CO + O2 -> 2 CO2

Why should I use modified Cp of CH4?

And next question:

I could add third global reaction

2CO2 -> 2CO + O2

to take into account dissociation of CO2. Should I use modified Cp of CO2 in this case too? In other words is this reaction represent the main path of CO2 dissociation? If yes, maybe I should use normal Cp of CO2 in this case?

Thanks in advice.

Re: increasing the specific heat capacity
 

I would not use extended reaction mechanisms with the eddy-dissipation model. Its basically a "mixed is burnt model" so it will not catch species that are far from equilibrium (CO) or intermediate species (O,H,OH,HO2...) correct. It is best used with one global reaction mech and if the flame temperatur comes out too high, you can use the corrected cp polynoms. I would use the polynoms for O2, CO2(includes dissociation CO2->CO+O), H2O(important!) others (CH4,N2,CO) are not that important. I use this approach for CH4 combustion and it worked well so far.

Good Luck, RoM

Re: increasing the specific heat capacity
 

You are right. It seems common practice to use one-step mechanism with eddy-dissipation model, but I have seen that it is enough common practice to use 2step mechanism with finite-rate/eddy-dissipation model. I am using 2step mechanism with finite-rate/eddy-dissipation model as mechanism with minimum complexity that can provide us some minimum information about CO concentration at combustion chamber exit. I was thinking to add third global reaction controlled only by temperature but not eddy-dissipation, increasing constant A for that reaction. Maybe the same should be done for second reaction. I have seen in some work that "after the mixture is ignited and carbon monoxide is present in the flame, the rate of the reaction 2 does not depend on the eddy break-up because both O2 and CO are present at the same location. Thus, reaction 2 is taken to be controlled by chemical kinetic rate at all times by setting the coefficient A = 10e5." What do you think about? Thanks in advice.

Re: increasing the specific heat capacity
 

I did some calculation with sandia d flame and the methan/air 2 step mech from fluent. The results where not very promising. The location of the flame peak at x/d=40 is predicted correct but the temperature always came out to high (about 200K) even with the Rose/Cooper cp polynomials. Modified values for A und B could not correct this behaviour and lead to stiff chemistry. I think adding more reactions wont improve the calculation but didnt had the time to test it.

I wont vote completly against the modified A/B values. I have seen this approach in the "Coal Combustion with Eddy Break Up (EBU) Model" tutorial from fluent and it seems to work. Do you have access to the paper "Peters and Weber (1997), Mathematical Modeling of a 2.4 MW Swirling, Pulverized Coal Flame, Combustion Science and Technology, 122, 131."? I searched some online e-prints but didnt found it.

RoM

Re: increasing the specific heat capacity
 

Unfortunately, I have not access to the paper "Peters and Weber (1997), Mathematical Modeling of a 2.4 MW Swirling, Pulverized Coal Flame, Combustion Science and Technology, 122, 131."

I don't like the idea to change A/B values too. Maybe the EDC model has less case-depended constants? If yes, maybe it has some sense trying to use appropriated global 2 (or 3-5) step mechanism with EDC?

Unfortunately I have not references reported in the Fluent manual related EBM and EDC:

215 B. F. Magnussen and B. H. Hjertager. On mathematical models of turbulent combustion with special emphasis on soot formation and combustion. In 16th Symp. (Int'l.) on Combustion. The Combustion Institute, 1976.

214 B. F. Magnussen. On the Structure of Turbulence and a Generalized Eddy Dissipation Concept for Chemical Reaction in Turbulent Flow. Nineteeth AIAA Meeting, St. Louis, 1981.

118 I. R. Gran and B. F. Magnussen. A numerical study of a bluff-body stabilized diffusion flame. part 2. influence of combustion modeling and finite-rate chemistry. Combustion Science and Technology, 119:191, 1996.

Do you have this works? It would be useful to read these work to understand the range of conditions (P, T, equivalent ratio &hellip;) where these models with constant reported by Fluent could work.

The same valid for methane-air-2step mechanism. What is the range of conditions (P, T, equivalent ratio &hellip;) where this mechanism with constant reported by Fluent could work?

In addition, I have noted that second reaction of the mechanism for finite-rate/eddy-dissipation option is differ from second reaction of the mechanism for finite-rate (used with EDC) option. In later case the reaction rate depends on H2O concentration, but in former it doesn't depend on H2O concentration. Is it correct? If yes, why?