Electronic – MOSFET broke after attaching capacitor bank

For these kinds of things, I personally like to simulate the circuits. This not only gives you the values, it also makes it possible to play with the circuit to learn more about how it behaves. And an inductor and a capacitor in one circuit always smells like oscillation so calculations don't always turn out as simple as one might think. Also the actual form of the current pulse through the solenoid will be intresting to estimate how it would physically behave (we don't know anything about what is attached to it).

My simulator of choice is mostly LTspice IV since it is free for personal use. Lets start with a slightly modified version of your circuit:

SW1 and SW2 are voltage controlled switches which are controlled by source V2 and B1, whereas V2 is a pulse (0V to 10V 200ms on, 400ms cycle, 1ns rise/fall time). B1 is the opposite of it, so SW1 and SW2 have are always open when the other is closed.

Lets first concentrace on the pulse, for that we set R1 to a reasonably low value (10Ohm in this case) and give C1 the special value {C1}. To get a bit more realistic I usually give it some value (e.g. 4k7u) and then select a capacitor from the list that roughly matches on I would use to get a useful ESR. You can also just enter one like 60m Ohm or so. Then we add the SPICE Directive (.op toolbar button) .step param C1 100u 10000u 100u and start the simulation (simulating one second here). The resulting current through the solenoid (90mH, 6Ohm) is:

Right click on the graph, chose "Select Steps" to see what curve fits to what capacitor value.

You see that the current is of course not a square wave, so arbitrarily assuming everything above 800mA is considered part of the wanted pulse, looking at the blue one it is roughly 20ms there, so a useful value there would be 7100uF. But depending on what the solenoid should do, this may or may not be the pulse you are after.

Now we have a value to play with, so we can either replace the {C1} by that value, or if we want to later play with it again, we can comment out the .step directive by prefixing it with a ; and then lets add the line .param C1 7100u to the same directive (right click to edit).

Now lets add another directive .step param R1 1 10 1 to the circuit and change the value of R1 to {R1}. Then run the simulation again and additionally look at the current through R1 now:

Due to the way we set up the simulation (that is, we have a fixed cap loading time before we switch over) you can see that at the point we do the switch, there is for each resistor value a different current flowing, which is equivalent to a different load of the capacitor, which you can see influences the pulse on the solenoid. You can change the pulse on V2 to e.g. PULSE(0 10 100ms 1ns 1ns 400ms 600ms) to simulate a longer loading time, which then will have almost no influence on the solenoid pulse. To satisfy your "at most a couple of seconds" requirement, the 10 Ohm will sufficiently load the wanted capacitor within half a second at a peak current of ~810mA so you can probably even go a bit higher. Of course this changes when you chose a different cap.

Note that you can have at most three .step directives active at once, giving you the ability to play with a lot of value combinations, at the cost of an almost unusably crowded graph window. Sometimes using .step param list will help here to chose some better values.

Well, the datasheet of your MOSFET does try to spoonfeed you how to use it properly (as a switch):

You need 10V Vgs to get that Rds(on). You're giving it 4V Vgs and expecting the same Rds(on)? Alas it doesn't work that way. It seems to me you've used the Vgs(th) value in your design, but that only guarantees you 250uA on the drain:

That actually translates (in the datasheet test conditions with Vds=Vgs) to 4V/250uA=16Kohm Rdson at threshold. So most of the voltage drop would be on the MOSFET and almost nothing on the solenoid's coil if that's the Rdson you're actually getting. (Put your solenoid in series with a 10K or 15K resistor, straight to the source sans MOSFET and see if it still turns on. I bet it won't.) Of course this Rdson a worst case scenario since that Vgsth is the max [=worst case] value. More on how to navigate a MOSFET datasheet is found in my previous answer to a very similar question. Expecting awesome Rds(on) at Vgs(th) is a perennial newbie gotcha, it seems.

Your Amazon solenoid doesn't have anything resembling a datasheet, but you can measure the voltage drop across the solenoid's coil and/or the current through it to see exactly what's going on.

A respectable solenoid datasheet has data that lets you know exactly how much voltage you need to turn it on (and how much current it draws at that point). For example, this series:

If were using their "12V" (nominal) solenoid, at least 9.6V are needed to turn it on and it will pull at least 1.52mA at this point. (It also gives you the turn-off voltage.) Since you don't have such data available for your solenoid, you'll have to determine it experimentally (e.g. using it directly with a variable voltage source) and then decide what you need to switch it on.