The key to temperature calculation is in obtaining the distribution of the heat generation rate q. Due to the skin effect, the heat generated by the induction is concentrated on the surface of the workpiece so the surface heat flux can be used as the input during induction heating. The heat flux model used in this paper is shown in Figure 2. The cumbersome electromagnetic calculation is unnecessary once the heat flux model is used to represent the coil.

so you can see this paper to find required mathematical expression you need

U can calculate final temperature as follows. By producing an alternating electromagnetic field due to the alternating current flowing in them, coils transfer energy from the power supply to the workpiece. The coil’s alternating electromagnetic field (EMF) creates an induced current (eddy current) in the workpiece, which generates heats due to I Squared R losses (core losses). Equate this heat generated to heat gained by steel rod which is equal to product of mass, sp heat of material and temperature rises. With this unknown parameter, final temperature can be calculated. By experiment, u can calculate actual heat lost due to radiation and convection loses by air to get accurate final temperature.

Easy (but inaccurate): If all the administred energy (P=VI) is transformed in heat in the bar, you can transform this energy to cal., then Q=mCe(T2-T1) where T2 is the final temperature, m is the mass of the bar and Ce is the specific heat of Fe.

You should be able to drive a customised equation from first principles.

Step 1 Voltage induced is proportional to strength of magnetic field and frequency. Measure the amperage (I) of the circuit. Know your resistance and work out; P=VIcos0 =I^2*R is the power input. Multiply it by t (time) to calculate energy that goes in the system. This is theoretical energy in.

Step 2 for temp rise you need to multiply specific energy of the rod material (joules/kg) x weight of the rod in kg. All this will only give theoretical delta T.

Alternatively if possible, you can setup the system as a pilot project and experiment heat transfer from coil to rod empirically. Make good clear tables and test temp rise with different energy in. Increase energy in in steps and allow the rod temp to stabilise before next step up in energy level. Interestingly your temp rise can calculate heat losses along the way.

In real world there will be a dozens of other parameters to think about eg if the place has constant draft, summer, winter etc.

Above all have fun!