# How is the reaction progress variable $c$ coupled with Temperature equation while solving the SIMPLE algorithm in CFD?

While solving the Navier-Stokes equation using SIMPLE algorithm, we solve the transport equation for additional scalar at the last step according to the SIMPLE's procedure.

We need to add a new scalar, reaction progress variable $c \ \epsilon \ [0,1]$, to model the extent of burning in case of diffusion flames, but I am not able to understand how is this variable coupled with the temperature equation?How do we recalculate the temperature once we have solved for $c$ ?

If we proceed according to cycle below, can someone please expand upon the transport equation part (STEP 4)--

Do we solve for Energy (Enthalpy) equation first and then find out $h$ and then, $c$ or is it something else? Because as far as I understand, I know that $c$ is used to calculate Temperature of the cell.

Please correct me if I am wrong.

EDIT: $c$ is the reaction progress variable denoting the extent or fraction of reaction, $h$ is enthalpy, $u$ is speed, and the $2nd$ and $3rd$ term on RHS below denote source terms independent of $c$.

$$\frac{\partial (\rho c)}{\partial t} + \frac{\partial(\rho u c)}{\partial x} = \frac{\partial(\rho D_{eff} \frac{\partial(c)}{\partial x})}{\partial x} + \omega_{deflagration} + \omega_{detonation}$$

• It would be useful if you explain the notation: what quantities do the symbols and subscripts represent? Jun 7 '16 at 19:00
• More importantly, can you include the (differential?) equation for your variable "c" in your question? Also, what is "h"?
– Paul
Jun 7 '16 at 20:12
• I am extremely sorry for the late edit. I was unable to reply as I had some personal matters to look after, so I couldn't come online. I hope I have provided the rest of the required information now. Please ask if any other clarification needed. Jun 13 '16 at 2:22