Can someone explain why the fuel mass reaction rate expressed using the arrhenius law is highly non-linear. I have read that this is because it cannot be easily represented as a funciton of the mean mass fractions and density. I assume this is for RANS? The Arrhenius law looks fairly linear to me. What am I missing? What is the case in LES?

Thank you.

it has Temperature T in exponential term. how can you make mean of a term which is exponential

Why not? I am afraid I don't understand why I cannot calculate the mean of an exponential term. It doesn't seem to me to be non-linear. What is the mathemiatical reason for it being on-linear?

Thanks Z.

expand the exponential terms using taylor series and then perform the mean. or check this in Poinsot book

Thank you. Can you tell where to find this in Poinsot's book...which page?

Its o.k., I found it on page 146-147 of Poinsot's book.

Thanks again.

In a turbulent flow, the RANS-predicted mean temperature may be 1500 K. But the real temperature encountered by the chemical term fluctuates and may be 1600 K or 1400 K (even if in mean T ~ 1500 K). Compute exp (-Ta/1400 K) and exp (-Ta/1600 K) with Ta very high (as in combustion), and take the mean. It is really different from exp (-Ta / 1500 K): the mean of the exp. is really different from the exp. of the mean...

Surely we are only interested in the mean of the exp though? So why would we care what the exp of the mean was?

Hi,

I think there are 2 fundamental issues here.

1. The Arrhenius theory for reaction rates. This is just a simple thery to explain what is happening on a mollecular level however fits many reactions over limited temperature ranges. As pointed out above this is non-linear in temperature. Any reaction mechanism applied strictly speaking to a homogenous system.....so no turbulence.

2. How do we inlude the above effects in a turbulent flow where we do not resolve all scales and times of the fluctuations of velocity, temp, mass fraction, etc. If we do a RANS simulation we only have mean values available (and a few statistics on our turbulence...k, eps, maybe u'T', etc). THAT is where the issue of the mean of a non-linear operator IS NOT the same as the operator applied to the mean of the independant variables. Many of the modern texts cover this well ie Warnatz, Peters, etc

Hope this helps,

Bak_Flow

You are interested in the mean of the exp., but your code will provide the exp. of the mean because this is what is computed (mean of T, U, Y...)

Is it possible to have a transport equation for the mean of the exp?

Sounds like turbulent modelling, does not it?

More exactly, concerning the (non-linear) reaction term, this is not a transport equation but a closure, sometimes based on transported quantities (CMC, Pdf,...)