I have a transformer with 220 V primary and three secondaries;
1. 24 V | 1.2 A
2. 24 V | 600 mA
3. 12 V | 200 mA
Each secondary have 4400uF capacitor and 100mH inductor after the bridge rectifier.
How can I estimate the inrush current and pick the proper thermistor value for limiting the inrush current?
There are two inrush currents, capacitor charging, and transformer saturation.
Unfortunately, they both depend on the resistance of the transformer, and the saturation inrush depends on the detail of the transformer magnetising curve, so it's somewhat easier to measure them than to gather enough good data to calculate them.
It's possible to measure the transformer saturation current independently of the capacitor charge current, as it exists even when the transformer is unloaded.
A transformer is designed for the flux to swing between nearly saturated in one direction, and nearly saturated in the other. At switch on, a good transformer will have zero flux in the core. If the transformer is switched on at a point in the mains voltage for when the flux would normally be zero, so maximum voltage, then it will start up with no inrush. If it's switched on a point when transformer flux would be maximum, at the voltage zero crossings, then it will attempt to swing to nearly 2x the saturation flux, which is not possible. The inductance collapses, and a huge current in drawn, causing an extra voltage drop in the transformer resistance which serves to correct the flux a little. The higher the transformer resistance, the more the flux will be corrected each cycle, and the quicker it will reach its eventual balanced flux operating condition. Paradoxically therefore, the 'better' the transformer, the longer it will spend drawing this saturation inrush current. Note that 'zero crossing' switches, normally ideal for resistive loads, are exactly the wrong thing for transformer loads.
If you don't have test gear capable of timing the switch-ons, then just switch on and off repeatedly, and record the maximum inrush current.
Once you've connected the rectifier/capacitor load, switch on and off repeatedly, and record the minimum inrush current. The capacitor inrush will be the same every switching cycle, so the total will be minimum when there is zero saturation inrush current.
In practice, inrush tends to be a manageable problem, at least for smallish power supplies. Silicon rectifier diodes tend to have surge capabilities an order of magnitude or more greater than their steady ratings (1N400x 30A versus 1A, 1N540x 200A versus 3A for half a mains cycle) so shrug off the initial capacitor charging spike. Time delay mains fuses will usually ride out the transformer saturation spike. Your PSU ratings are definitely in the 'smallish' range.
You have two reasons for the inrush:
1) the iron core has been left to saturated state since the last usage.
2) the initial charging of the capacitors.
Reason 1 causes an inrush pulse which must be measured. Your linked story that you said to be too advanced is the way to find a way around it. Do not have any load when you test the inrush caused by saturation
Reason 2 can be estimated with a circuit simulator if you have a short circuit test data of the transformer. The losses and stray inductance limit the inrush to empty capacitors.
How these 2 inrush mechanisms interfere when there are rectifiers + capacitors connected is difficult to estimate. I guess you can sum the energies of the pulses and place the initial charging pulse to happen after the demagnetization has happened at least 50%.
Unfortunately I cannot calculate anything exact of the transition between the demagnetization and initial charging, to be sure it should be measured with parts that can take the punishment.
ADD due a comment: Hot NTC resistor and repowering the system is a risk of inrush. If inrushes must be absolutely prevented you must have a soft-start circuit which doesn't get fooled by short off-periods.
