flyback Transformer core gap

What will I compute for the air gap between my core, if I am planning to build a flyback transformer for 3 watts power supply?

You can calculate Ipk peak current(A) = 5.5*Pout/Vin
L = primary inductance in H ( L = AL value in H * Turns *turns
Ac is cross sectional area of ferrite core (cm2)
Bmax is Flux density in tesla = 0.25 to 0.4T

Gap in mil = (0.4*π*L*Ipk*10^8) / (Ac*Bmax^2)

A simple suggestion: please do not use formulae blindly. Please understand each step and convince yourself that the application in that particular step is legitimate.

Gaps are added to increase the magnetic reluctance of the circuit. Higher reluctance will need higher current before the core saturates. But that in turn will also need larger number of turns in the coils. You can follow the steps in simple manner. If you provide more details, we can try better.

The transformer will function at a frequency of 34KHz at 45Vdc input, the total power must be 3W, will have two additional forward windings (with output voltages of 10 and 15 volts) at the primary section and another two at the secondary( with outputs of 5 and 15 Volts).

How does the gap relates with the primary inductance value?

Basically the inductor current must ramp up to a peak in the time available, at the lowest expected input voltage, to store enough energy to get the job done.

So a rough as guts estimate of inductance might go something like this:
Switching frequency 34 Khz, max on time (50% duty) 14.7uS.

Now assuming our input voltage is 45v dc minimum, input power might be about 4 watts (allowing for efficiency). Average input current 89mA.
Peak current might be about four times, maybe about 356mA.

Inductance required to achieve that = 45v x 14.7uS divided bu .356A micro henries.
Call it 1.86mH

In practice you would load it up to 3 watts output and monitor the current ramp and the on time.

If your inductance is too high, it will not reach 3 watts, but run out of on time first.

If the inductance is too low, it will reach 3 watts much sooner, with a shorter on time, but the current peak will be higher than 356mA.
That is not altogether bad, but it will hurt efficiency slightly.

So basically you test it and tweak the air gap to get it running how you want it to run.

The practical way beats trying to work it out mathematically to huge precision, because there will be some unknowns, like final efficiency. Use the maths to get reasonably close, then test it.
The final gap size required may be a slight surprise, but real bench testing never lies.

Please follow the suggestions in post #5. However, if I were you, I would start with the available core and not the switching frequency. I generally would like to build around the transformer because you will need a small core and several windings and fine wire (your current demands are modest). However, you need to fit the transformer around your design.

Select the bobbin and compute the number of turns for the primary and the secondary. At this time you need to decide about the core gap. Some cores come with gaps already fixed (you cannot do much about them because the gap is fixed and grinding the gap to your own specification is messy) or you can use a thin sheet of mylar or teflon as a gapping material. Recalculate the primary inductance using manufacturer's data about the core.

Rewinding the transformer again and again is messy and you tweak the frequency instead. It is going to be a iterative process but rewarding.

There is a lot of logic, and huge amount practical experience demonstrated in the above post #6.

Work out all your windings first "assuming" a mid frequency.
But you might find that the number of turns required no not fit neatly into convenient single full layers. You may have to fudge and add 20% of turns, or remove 15% of turns or whatever, to reach some kind of logical rational layout that fits best on the available bobbin.

Then you can tweak your frequency up or down to suit the turns you end up deciding on.