Slightly different approach to wwfeldman.... but same questions... and agree the question isn't quite clear.
questions:
1) the 24V smps - i assume that is the output - what is the input?
2) what do you mean by a power cut? - does the input (to the smps) become open ? or short?
or so you mean the smps stops working, regardless of the stae of the input power?
How I interpret it.. You have a HMI device powered from a SMPS. The HMI device can run from a supply of 12 to 24V. At 12V the HMI device will draw a max of 500mA.
It's more than likely the HMI device will have it's own SMPS internally that allows such a wide operating voltage range. A SMPS is a constant power device provided the load remains constant which we'll assume it does.
The HMI in that case consumes 6W. Can use the capacitor equation for constant power...
C = 2 ×P ×t /(V0^2ーV1^2)
V0 -> is charge voltage V1 -> discharge voltage
C = 2 ×6W × 5s /(22^2ー12^2)
C = 0.18F
So 4 in series should do the trick nicely...
In fact... your 4x1F super caps in series will give you a run time of approx.
t = 0.5×C×(V0^2ーV1^2)/P
t = 0.5×0.25F×(22^2ー12^2)/6W
t = 7 seconds
However... it's not quite as easy as it sounds... you will need something that can charge them.. you can't just string them across your output.. it will short it out when the power returns.. the caps will be flat and want to draw as much current as they can. Even with some tricky diode ORing and current limiting resistor you can only charge 4 in series to 22V.
You could potentially do somthing like this using 5 super caps in series... They won't be fully charged.. but will still give you about 7sec as they will be charged to 24V.
D1 may not be needed... just stops backdrive into the unpowered SMPS.
D2 prevents the SMPS from dumping all it's got into flat super-caps and likely tripping it out..
You can play around with R1... it basically will charge them up from dead flat in 11sec use a max of 400mA... All diodes can be 1N4001 rectifier...
One point to note.. my calculations above don't take into account the 0.7V diode drop... but it
should still be OK.
EDIT - I did neglect to mention balancing... A 1k resistor across each cap should give reasonable balancing if the caps are all the same and of reasonable quality.
I have never used a supercap. In fact the last time I looked at them, about 30 years ago,
they were only good for holding up memory because of the high series resistance and the
resulting long charge times. (100 ohm for a 0.5 F cap, if memory serves, for a 50 second time constant)
They're pretty amazing now... you can pull 10s of amps out of even basic ones and charge them even quicker! As you move up you're talking 100s of amps... their biggest pitfall is getting them in high temp.