Installation of preheater and efficiency

We install a preheater to preheat air from the heat captured by incoming products of combustion as shown in the figure:

The efficiency of the preheater is defined as

Preheating of air brings sensible heat into the furnace in addition to the heat of combustion. Two possibilities are considered to use the additional amount of sensible heat of air. In one, the fuel input rate is not changed which means H_in is unchanged. Revised Heat balance of the furnace:

It must be noted that the furnace has to operate at temperature  which means that T_POC and T_l remains also unchanged. As a consequence, extra amount of charge is to be added to utilize the sensible heat in air. Now the heat output by the charge will be  such that

Thermal efficiency of the furnace increases in proportion to amount of sensible heat in air.     

In another, we keep the amount of load constant so that heat carried by the load is H_u. Sensible heat in air will increase the temperature of the furnace, if the furnace is operated at the same fuel input (H_in) as was without preheating air. Since the temperature of the furnace should remain at , thus fuel input  has to be reduced such that

 is fuel saving and can be expressed in percent of H_in

Fuel saving (%) =