Why does Nitrogen gas appear in the equation for making producer gas?
As the comments allude, the coefficient is 4 to represent the approximate 4:1 ratio of oxygen to nitrogen in air (more precisely 21:78 or 7:26).  This added nitrogen must be heated along with the oxygen and carbon.  From a stoichiometric perspective, nitrogen is just a spectator and could be excluded, but from a thermal/total system perspective as nitrogen increases the specific heat of the system requiring more heat to be inputted for heating the mixture which subsequently requires more heat dumping to cool the system, reducing the efficiency of the system compared to a system of pure reactants.