How Torque varies with Air to Fuel Ratio

For my Stellaris Launchpad engine simulator, I wished to add a calculation of the Air to Fuel Ratio (AFR) or lambda/equivalence ratio for the current engine conditions. I did this crudely by using the throttle position, engine speed (this->s) and current fuel pulse width (this->F) as factors in calculating the AFR:

this->AFR = (0.32666 * this->s * this->throttle)/(this->F);

The equivalence ratio, or lambda (inverse of each other) can then be calculated from the AFR and the stoichiometric ratio which for gasoline is 14.7 units of air to 1 unit of fuel:

this->equiv = this->AFR/14.7;

Then I was interested in knowing how the Air to Fuel Ratio affects the maximum Torque output of an engine. In chapter 4 of Introduction to Internal Combustion Engines by Richard Stone the affect of equivalence ratio and ignition timing on BMEP are discussed. I used the figures given here to calculate a rough equation for relating lambda and a torque scale factor (0 to 1). An explanation and some good diagrams are also shown on this website

Pinched from

Pinched from

And so my scale factor is calculated by

float scale = -1.1536*pow(this->equiv,2) + 2.6954*this->equiv - 0.5695; // Scale torque for equivalence ratio

I actually had to restrict the effect of this scaling factor though because at engine start up, the ratios are all wrong which would cause the engine to produce 0 torque and never get going.

if(scale < 0.7){
scale = 0.7;

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