Easy Peasy. Fusion reactors.

The primary challenge involved with fusion power is maintaining containment, which is a big challenge given the pressures and temperatures involved.

Not only will the neutrons deposit energy in the blanket material, but
their impact will convert atoms in the wall and blanket into
radioactive forms. Materials will be needed that can extract heat
effectively while surviving the neutron-induced structural weakening
for extended periods of time.

Also here:

To solve the issue of containment, most devices use powerful magnetic
fields to suspend the plasma in midair to prevent the scorching
temperatures from melting the reactor walls.

The TLDR is that currently a PHYSICAL containment solution is impossible, requiring magnetic solutions that suck up a lot of the power being generated.

Your Macguffin would solve this neatly, allowing a simple machined or cast sphere to be turned into a perfect containment vessel for a fusion powerplant of pretty much any size you need, and making it a lot easier to hit the breakeven point.

This still leaves the problem of how hot the reactor vessel ITSELF is going to be, but theres lots of ways to solve that including just using a magnetic field to levitate the thing. Maintaining a stable enough magnetic field to keep a solid object off the ground is a couple orders of magnitude easier than maintaining one stable enough to keep fusion plasma at 15 million degrees under control.

EDIT: Now that I'm thinking about it, it'd be a good solution for FISSION reactors as well, since a reactor vessel macguffin'd in the manner you describe wouldn't lose containment in a runaway nuclear reaction. The core could still melt down, but it'd stay in the reactor vessel. Your reactor would be destroyed, but it couldn't irradiate the entire powerplant ala Chernobyl or Fukushima.

EDIT THE SECOND: There's been some comments regarding how you MOUNT your macguffined core without causing damage to the surroundings, as well as handling the heat coming off of it. Right off the top of my head, it seems like the easiest thing to do would be to build a really tall hollow tower, Put the fusion core on the end of a tall mount that's ALSO macgufined, and put a big set of turbine generators about a hundred feet up the tower. The heat coming off the core would generate MASSIVE upward velocity of superheated air, which would drive your turbines, generating your power. You'd just want a tower tall enough that by the time the air exits, it's cooled down enough that you're not roasting ducks in-flight or creating localized thunderstorms.

How could indestructible materials be used in power generation?

Energy storage.

If you can spin a flywheel to relativistic speeds on indestructible bearings using electromagnets (in vacuum), then you can use that flywheel as a lossless energy storage device.

_{Wikipedia 2019 - CCSA License}

The energy density would be infinite (or limited by the unspecified arbitrary high amounts of energy in the question) - thus you would need a microscopic minuscule amount, a nano-flywheel mounted on gimbals - radically reducing the price per flywheel and opening it up to mass marketing, totally outclassing all battery tech available today.

Not only the obvious solution to the supply and demand issues with windpower, but for vehicles - cars/planes, phones, power-tools, toys, mobile phones and of course space exploration.

Infinite energy storage in the size of a grain of sand.

Miniature Tactical Nuke:

Of course, this section is about political power generation.

To release all that energy in one instant - perhaps an object charged with just below the threshold of it's (unspecified arbitrary potential energy) capacity, could be placed near an enemy stronghold and fed that last few joules of energy to tip it over the edge, that's the dark side, someone will find a way to weaponise it for sure, if not the leader of some isolationist sanctioned state, then a disaffected teenager.

Power of a civilisation through time travel.

Speculatively: Also it would have potential to enable time travel or at least the potential to send messages back in time as it would exhibit frame dragging. For a few hints on how this could be of tactical use see this answer to another question.

Nuclear pressure containment is a good method.

Nukes have to be held together to make fission continue for as long as possible. If you hold 20 critical masses together for a full second, you'd generate the largest nuclear explosion ever made by humans.

With indestructible materials, you could hold them together for an hour. At those high energies, there are all sorts of effects that release even more energy.

Make a box out of indestructinum. Put a nuclear bomb in it. Detonate and let it build up fusion-capable pressure. Slowly vent it out to generate power. If your material conducts heat, put it in a very effective cooling system and generate power reactor-style.

Alternatively, vent it out quickly in the direction of someone rich until they give you what you want.

Sometimes the smallest thing has the largest impact

Do you know how much wire you can extrude from a cubic meter of copper when you can trust it to be indestructible?¹

Indestructible insulating enamel + indestructible conductive wire = the perfect transformer/motor/generator.

When was the last time you opened a power supply, motor housing, generator, or anything using inductive windings, and found the transformer/motor/coil burned out. For me, it was last week (literally, it was last week). If you could make both the wire used in the windings and the enamel used to coat the wires indestructible, what you would have is the perfect transformer/motor/generator.

Yeah, but this stuff is expensive

Which is why it would make sense for large items, like turbine-style power generators where the limit to the electricity you're generating is suddenly the mechanical stress limits of the linkages and not the heat-generating characteristics of the coils. Better still, indestructible windings and enamel means you can make the coils incredibly dense — and as coil density increases, so does power output. Your efficiency might actually approach unity. Imagine a wire that is no longer a fuse if too much power is put through it. There is no longer too much power, the limitation is literally the speed electrons can be induced to move through the wire.

And if you expand to power utilization, the applications become … impressive

Miniature motors that can turn the propellers on a submarine? Dock 6. Full-size motors that push submarines at tsunami-creating speeds? Dock 2. Car alternators the size of your thumb? Aisle 14. A Dremel the size of a pencil? Aisle 1. An electric car that actually works climbing the Rockies? The display arrives next week. A residential wind turbine that actually powers an entire house? We have on the roof, you can see it as you enter the building.

The process may be expensive, but the material requirements (in terms of how much you need) drop like a rock when you can trust the wire and enamel to be indestructible. The process of making things indestructible would benefit almost any application at any price. A steam boiler the size of a Buick enjoying such high pressure that it can pull a mile-long train? On display by the front counter.

Disclaimer: at hundreds of millions of dollars per-cubic-meter there it is unlikely that any application is worth it. Unless you can jack the price through the roof, the cost recovery time at that price relegates the material to use (not necessarily power generation) in remote locations (like space) where repair costs even more. A spaceship hull would be worth that price. I frankly can't imagine any power generation/utilization solution that ever would. Not even fusion. The cost of using something less capable would be so much more economical that such a solution would only happen as a test, never a commercial solution. So, a frame challenge concerning the price.

Edit: The OP challenged my disclaimer, and he may have a point, although not for the reasons he suggests. It takes a lot of metal to make one billion-dollar plane. And that metal alone just jumped to billions of dollars. Now we have 3-4 billion-dollar planes, which only national economies can afford, and that means 25% of the planes you could have had without the indestructible hulls and infrastructure. Frankly, most nations wouldn't/couldn't justify the price (there actually are limits to what nations can pay for things. It doesn't seem that way, but there are limits nonetheless).

But...

The average car alternator only requires 0.8165 Kg of copper. With indestructible copper and enamel, it might need 20% of that (0.1633 Kg). That's 54,864 alternators at, say $200M or $3,645 above "normal" price — for an alternator that will never burn up. It would mean almost nothing to raise the price of cars by $4k. People would pay that and move the alternator from car-to-car. One alternator for the rest of their lives. Booyah.

A friend of mine once made a good point: it's easier to sell a million items for $1 each than it is one item for $1,000,000. The little things would pay off better than the big things.³

_{¹ A cubic meter of copper weighs 8,930 Kg. 40 Gauge wire weighs 0.04454 grams/meter for 200,490.6 Kilometers of wire. That's enough wire to wrap the equator 5 times.² And you might be able to use thinner wire than that. It's a lot of honking wire.}

_{² Of course, the wire is indestructible. If you wrapped the equator just once and tied the two ends to space ships, assuming a reasonable amount of thrust, could you garrote the world in half? It gets the mind wondering, doesn't it?}

_{³ An astute observer might note that making indestructible commodities eventually drives a company out of business. It's the reason antique-anything tends to last longer than the crap we buy today — because there's more money to be made with failure. This is true for power generation, too. The last thing power generating companies want — ever — is a convenient (if expensive) way to run themselves out of business. After all, eventually a class-action lawsuit will point out the fusion power plant has paid itself off and the power rates should drop to rock-bottom. U.S. President Bill Clinton won his campaign for president with the slogan, "it's the economy, stupid." In the end, the OP's indestructibility formula would revolutionize the world — if the inventor could survive to bring the formula to market.}

For a more nerdy approach, you could build indestructible turbines.

(Disclaimer: My memories of thermodynamics are fading in the mists of time, so feel free to blast me in the comments if I'm wrong).

I remember that in thermal power generation (where water is heathed into steam, whose energy is used to move a turbine), they were forced to limit the calor of the steam in output from the turbine, thus reducing the efficiency (basically, the colder the exiting steam, the better the efficiency).

The reason was that if the water steam was allowed to cool too much, it would condensate and create water droplets that would move so fast to act as bullets, damaging the turbine.

But an indestructible turbine could easily withstand this scenario, thus allowing for exploiting all the energy of the steam and generating more power.

Of course it is necessary to evaluate if the increase in efficiency is enough to compensate for the higher cost of the indestructible turbine.

Construction

A thin, unbreakable wire added to dam wall construction, would allow dams to be built cheaper, thinner, stronger and higher.

A unbreakable foil added to the overflow means it would never wear out and need replacing

Large dams for energy generation cost from twenty to thirty billion dollars so an extra cost of a couple of hundred million to make it unbreakable would easily be offset by the less concrete and steel needed not to mention to maintenance costs down the track plus the safety of an unbreakable wall.

If you want next level power generation may I present

The Space Elevator

A thin unbreakable wire running to an orbital platform with a twin on the moon, would allow the efficient harvest of He3 from the lunar surface which could power fusion reactors around the world.

Jet engine turbine blades

The limit in turbine efficiency (and why jet engines keep getting better and higher-bypass, but sloooowly) is the thermal limits of the first-stage (right behind the combustor) turbine blades. Evolution is waiting on new material-ally tech and more extreme methods of cooling(already pretty extreme and energy-robbing).

If you can make the first two stages out of indetructium, as well as a few combustor-area components that would be hard to airstream-cool, you can keep pushing up the efficiency of the engines. Now you have 20:1 or 30:1 bypass. Turboprop sub-chaser maritime patrol aircraft that can stay on station for 48 hours. Over on the ship-propulsion or terrestrial power generation side, you have ships with more range between tankers, and power plants with lower smog and cheaper power from natural gas and petroleum. It would be the deathknell of coal.

"So how could these aforementioned indestructible materials be used in conjunction with existing or near future technology to improve power generation?"

Well, if you could make copper indestructible then you could use it as I mentioned in my comment. Simply dig a very deep hole and place a copper rod in it. The heat at the bottom of the hole would conduct through the rod to boil water at ground level. The boiling water would be used in a convention steam turbine and BAM nearly infinite free and clean energy. The only reason we don't already do this is because copper would melt at the temperatures needed to get enough heat conducting through the rod to boil water on the other end. That and it would be very hard to dig a hole that deep because all the drill bits would melt but since we can make indestructible drill bits, it should be no problem... heck, we could reach the core with indestructible material.

At a minimum, if you replaced all your ball bearings with indestructible bearings, you'd be a long way towards better energy production.

Many generators in the energy industry have to be tore down periodically to have their bearings and fins/rotors replaced. Never having to do this will save that cost, including labor and the energy production to cover the "down" generator.

In fact, making the entire generator out of these indestructible materials would be a major boost. Hydroelectric dams could run at any speed, same with the huge windmills. (Ever seen a windmill with a failed break mechanism break up? YouTube that if you want to cringe.)

As much as the new forms of energy production suggested by other answers would help, simply replacing the mechanisms of the existing system would help considerably. This might be a lower investment level to get poor countries more power.

While the fusion power example is the most practical, you can connect a massive solar collector in orbit to the ground via an indestructible fibre-optic cable, and send a laser pulse down the cable to a boiler powering a steam turbine. As a bonus, since the cable is indestructible, you could use it to tether a space elevator, and get cheap rides to orbit.

Collect wind power via high-flying kites.
Make the tether cables indestructible and light, and enjoy your free renewable energy!

• https://en.wikipedia.org/wiki/Crosswind_kite_power


• https://en.wikipedia.org/wiki/Airborne_wind_turbine

The way I see it there are two main areas where your indestructibulisation process could affect power generation which will be obvious to the reader.

1. 
   

   Mega scale engineering.

   
   
   

   With super strong, super light materials previously impossible engineering projects become possible.

   
   
   

   You could tie the moon to the earth for tidal power generation, drill a hole to the earths core etc.

   
   


2. 
   

   New materials and matter states

   
   
   

   Other than the gross effects of your process, you would have to consider the micro and quantum scale effects.

   
   
   

   What if I freeze and compress hydrogen into a solid and then apply the process so it remains so at normal temperature and pressure? How does friction work between indestructible surfaces?

   
   
   

   You can imagine a range of super materials exhibiting exotic properties such as superconducting wires, frictionless bearings and perfect insulators. You might be able to construct highly efficient solar panels or perfect energy storage high capacity batteries.

   
   
   

   Do you really care how the energy is generated if your wall socket can deliver MegaWatts of power from solar farms in Africa at the flick of a switch?

   
   

There is at least some research that suggests you can build a superconductor of heat. Assuming this is true, create a tether of heat superconducting material and sink one end of it into the center of the Sun, with the other end at a power-generation space station somewhere nearish Earth (maybe, Earth-Sun L1 point). The superconductor will transport heat from the Sun to the space station, which can use the heat energy to generate power and beam it back to Earth via lasers.

Assuming your superconductor is a wire 0.1mm thick, you'd need about 15000m^3 of wire to do it. While supremely expensive, you're getting essentially free energy, forever, out of it.

Indestructible does not mean inflexible, or void of state changes. Make the equivalent of a very long, tightly wound "memory metal" out of your material, and use it to deliver energy as it unwinds via some thermal baths.

You just made a practical fusion power system.

You build a containment vessel. If your machine will permit it the best design is most of a sphere as one part and a panel that comprises the rest but overlaps so it can't be forced out. The smaller piece must be inside the bigger when it's processed.

If you can't do that you'll have to get more complex clamping the parts together, but it still can be done.

In either case there's a small hole in the side. Attached to that is a variable size orifice and a magnetohydrodynamic generator (whose inside will likewise need to be indestructible.)

Load a small atomic bomb into the device and then as much lithium deutride as you can fit while still being able to close it. Detonate. Note that you do not need the normal complexity of a fusion bomb here, that's all just to focus the energy of the fission bomb and the containment does the job perfectly well.

Now you have a container full of incredibly hot plasma. It comes out through the generator, as the pressure lowers you open the orifice more to keep the power level constant.

Your container starts out with fission products and helium (I don't know if it will get hot enough for helium burning to start but it's slow enough it won't have a substantial effect even if it does happen.) Extracting the power will greatly cool the material and then you can cool it still further by directing it into a large container of helium--the objective is to get it cool enough the fission products solidify and drop to the bottom of the container. You're left with helium and a bit of radioactive krypton and radioactive xenon as gases and the rest of the hot stuff on the floor to sweep up.

The stuff on the floor is no worse than normal nuclear reactor waste and there's a lot less of it because most of the power was from fusion and there's no plutonium left in the waste, either.

The gases are normally just vented when the wind is favorable but in this case it might make economic sense to run the gases through a fractional distillation in order to recover the helium for sale. (Note that helium has the I believe unique property that it's immune to induced radioactivity. Helium-4 that absorbs a neutron produces Helium-5 which very quickly decays back to Helium-4. This happens so fast that you can only witness it in an atom smasher, on a human timescale it simply stays Helium-4.)

You could use geothermal energy directly from magma.

Make drill bits and drill rods indestructible, make hole casings with a vacuum isolation and inner heating indestructible. Use NaK (sodium potassium alloy) as boring fluid up to 785 °C, other liquid metals as needed.

Drill at a place where magma is fluid, with casing deep into the liquid. Take care not to let the first magma cool to stone in the hole.

Now, you can run a standard geothermal power plant!

Actually, you can simply use a heat exchanger deep into the magma, and off the shelf technology as used in nuclear power plants to pump around liquid metal and use the heat. Works everywhere, no pollution.

If one tied to weights together with an indestructible string then spin the weights such that they exceed the gravitational escape velocity. You would have an anti-gravity machine. The only thing that prevents us from having transportation with almost no energy usage, moving water with no energy usage, going to space with no energy usage ... is we don't have an indestructible string. The amount of lift is determined by the speed of rotation, slowing down the rotation recovers the energy used to rotate it.

One mechanical horsepower lifts 550 pounds 1 foot in 1 second.

Such a machine may superseded the requirements for energy production but could endlessly lift water from a lower pool up to a higher pool for Hydro-power energy or numerous other forms of energy, which are derived from horsepower.

Also note: the string need not be fully indestructible. It only needs to be indestructible in that it can not be snapped by pulling on it. In theory it could still be melted or cut by bending.

Reps for Jesus(or the greater good whatever..)

Find a gym rat that loves nothing more then pumping iron. If you make him indestructible he will never age or die, this gives him and infinite time to expend energy that you can harness threw a modified smith machine.

This can also be done with a normal rat in a ball. At some point you will need to replace the treats. The advantage of the gym rat is that the end goal is a gnarly pump witch is short lasting.