Frankly, if you "don't know where to start", get someone that does. It sounds like the cost of failure is very high, so this isn't the time to experiment and learn from mistakes. Find someone that knows this stuff. You can look over their shoulder and try to do as much of the grunt work as possible so next time you're the expert. I think I underestand the basic principles pretty well and could get something working. However, for extra high reliability there is usually much lore and experience that you wouldn't guess from the physics alone. I wouldn't take this on myself without a expert to at least ask occasional questions of and to bless the ultimate design. So from a non-expert view, here are some thoughts:
First 45C isn't all that hot. It's only 113F, which is not out of line for summer shade air temperature in various parts of the world. Ordinary electronics gadgets, unless specifically for lab use or other environmental constraints, will have been designed to run at 45C ambient. Think about it. If you left your cell phone on the dashboard of your car parked in the summer sun in Phoenix, you probably wouldn't blame it for not working. Of course you probably couldn't hold it without hurting yourself either then. But, if you were standing outside under the shade of a tree mid afternoon in summer in Phoenix, you'd probably be pissed of if your cell phone didn't work. That's 113F.
Second, two other advantages you have is that this 45C is highly reliable, and essentially infinite. In other words, it's 45C all the time whether its summer or winter on the surface. Also, you can dump a few kW into a reasonable size "room" of this mine, and it will still be 45C.
So the best solution, as Russell also pointed out, is to spec things so that they run fine at 45C ambient. Space them out enough so each thing sees the 45C ambient, perferably without needing any forced air cooling. That way you don't have to worry about what happens when the coiling breaks down. Again, most off the shelf electronic gear should already be fine with that. Lifetime may go down, so derate. Get good quality stuff that should last 10s of years normally.
If you really need cooling, my first knee jerk reaction is to build a small room and put 2 or 3 air conditioners in it. Make sure any one air conditioner has the capacity to cool the room sufficiently. The multiple air conditioners are for redundancy. You seem to say it's only a few kW, which is small as airconditioners go. 1 kW is only 3400 BTU/hour. Also, it sounds like every last item doesn't need to be cooled, so some of the power you need can be dissipated outside the room and doesn't add to the air conditioner load. 113F isn't that hot and there are certainly off the shelf air conditioners intended to work under those conditions. Having 2 or 3 units always on cooling the room is for reliability. However, this is a area you really should consult an expert on.
The circuit is fine in theory.
Improvement in practice is required.
Adding a gate-source zener diode of say 12V (> Vgate_drive) is a very good idea indeed in all circuits with inductive load. This stops the gate being driven destructively high by "Miller capacitance" coupling to the drain during unexpected or extreme variations in drain voltage.
Mount the zener close to the MOSFET.
Connect Anode to source and Cathode to gate so that the zener does not usually conduct.
The 10k gate drive resistor (as shown) is large and will cause slow turn off and on and more power dissipation in the MOSFET. This is probably not a problem here.
The chosen MOSFET is very marginal in this application.
Far far far better MOSFETs available ex stock at Digikey include:
For 26c/10 Digikey IRLML6346 SOT23 pkg, 30V, 3.4A, 0.06 Ohm, Vgsth = 1.1V = gate threshold Voltage..
NDT3055 48c/10 TO251 leaded 60V, 12A, 0.1 Ohm, Vgsth = 2V
RFD14N05 71c/10 TO220 50V, 14A, 0.1 Ohm, 2V Vgsth.
ADDED
SUITABLE MOSFETS FOR 3V GATE DRIVE:
System just trashed my longer answer :-(.
So - MOSFET MUST have Vth (threshold voltage) of no more than 2V to work properly with 3V3 supply controllers.
None of the suggested FETS meet this requirement.
They may work after a fashion on the present load but are underdriven and overly lossy and the solution does not extend well to larger loads.
It seems that IRF FETS in size range concerned that have Vth (of Vgsth) <= 2 volts ALL have 4 digit numerical codes starting with 7 except IRF3708.
OK FETs include IRFxxxx where xxxx =
3708 6607 7201 6321 7326 7342 7353 7403 7406 7416 7455 7463 7468 7470
There will be others but all the ones suggested seem to have Vth = 4V or 5V and are marginal or worse in this application.
Vgsth or Vth needs to be at least one Volt less and ideally several volts less than actual gate drive voltage.
Best Answer
The junction temp rise depends on a product of switch resistance, Ron times the total thermal resistance to the internal ambient where using a heat spreader of copper substrate or a clamped case heatsink can greatly reduce the case to ambient resistance but is bulky unless forced moving air velocity is high.
Assumptions and variables for each driver must be combined to determine the total power loss and heat rise.
Choices and calculations: for temp. of jcn (Tj), case (Tc), ambient (Ta), and interface thermal resistance between each, Rjc, Rca where Rca depends on air flow and if a heatsink is used otherwise none, just convection open air Rja.
Heat loss: \$P_d= I^2* R_{on @ T_{j ~max}}~*~N_{drivers}\$
Temp. Rise = \$P_d*R_{ja}\$ unless a heatsink is used,
Absolute max energy clamp diode = \$E=1/2 ~LI^2\$.
Max Tj=85’C to 100’C depends on reliability not worst case, Ta (inside) , case design.
cost and size may be critical factors for getting lower Ron FET’s
Simulation example of transistor switch driven from an ESP at 3.3V using Ic/Ib=20
Lower Rce = Vce(sat)/Ic