How to calculate the operating time of steam turbine?

[O.J.Vodka]

There is 1 ton of 200 ℃ steam in the steam chamber. One Steam engine is used to absorb heat, and its condensed water returns to the steam chamber. After calculating how long the Steam engine runs, the steam temperature in the steam chamber drops to 125 ℃

[JRup]

You might want to check the trivia section on the wiki page for the ST:

https://oxygennotincluded.fandom.com/wiki/Steam_Turbine

[O.J.Vodka]

9 hours ago, JRup said:

You might want to check the trivia section on the wiki page for the ST:

https://oxygennotincluded.fandom.com/wiki/Steam_Turbine

Can you provide the formula for this question?

[degr]

My main question, why you need it at all? Use that schema:

heavy-watt wire connected to transformer, transformer connected to smart battery, smart batter connected with automation to transformer, smart battery setup - 30%-70%, AT and steam turbine connected to transformer and battery. In case if smart battery charge lower then 30, transformer enabled and charge it to 70. Rest of the time steam turbine charge smart battery (at require 1200, steam turbine prvide 850 max, so steam turbine 100% efficient).

PS AT with supercoolant produce enough heat for 1100 steam turbines power, so, you need 2 steam turbines.

[O.J.Vodka]

9 hours ago, degr said:

My main question, why you need it at all? Use that schema:

heavy-watt wire connected to transformer, transformer connected to smart battery, smart batter connected with automation to transformer, smart battery setup - 30%-70%, AT and steam turbine connected to transformer and battery. In case if smart battery charge lower then 30, transformer enabled and charge it to 70. Rest of the time steam turbine charge smart battery (at require 1200, steam turbine prvide 850 max, so steam turbine 100% efficient).

PS AT with supercoolant produce enough heat for 1100 steam turbines power, so, you need 2 steam turbines.

The formula，thats all I need. Can you provide the formula for this question? please

[blakemw]

I don't think you can calculate this kind of thing, you have to simulate it. That's because the system is not continuous so calculus can not be applied (though could be used as an estimate), you can treat it as a sum of a sequence problem, but it's neither a geometric sequence nor an arithmetic sequence so good luck with that.

Anyway as an approximation and ignoring the Aquatuner, every tick (there are 5 ticks per second) the Steam Turbine consumes 0.4 kg of Steam, it turns that steam into 95 C water, that a few dozen ticks later re-enters the steam room, if we ignored that time delay:

Basically every tick the new average steam temperature is (999.6 * temperature_last_tick + 0.4 * 95) / 1000, so at tick zero the temperature is 200 C, and after the first tick the temperature is 199.958 C.

You could run this sequence forward until the temperature is 125 C, and that's about how many ticks it would take. You could also try approximating this as continuous and applying calculus and solve for t.

A numerical simulation could do a fairly good job of estimating the time to reach 125 C. However it should be noted that there are some woolly corners of the ONI simulation that means that the results are not going to be exactly as calculated: for example the steam turbine only requires the steam under one inlet to be at least 125 C, and the exhaust water and the steam don't mix instantly, so the steam turbine actually keeps running until the average steam temperature is a bit lower than 125 C.

The Smarter Way

Giving that the system pretty much has to be simulated to get the right result, it might be best to just use the simulation we already have, the game. To do this, activate debug mode by entering "KLEIPLAY" in the main menu, this enables some useful things like ultra time acceleration and liquid sources.

Make the Steam Turbine setup. Then make a Counter setup that measures how many seconds the Steam Turbine runs for, a Counter only increments on a rising edge (when the signal turns green), and if you have a continuous green signal it doesn't increment, the easy thing is to use a Timer Sensor set to 0.5/0.5 and AND that with the primary signal - which could for example by a Wattage sensor, or a Liquid Pipe Sensor measuring whether the exhaust water is present.

Combine 4-5 Counters top-to-tail and you get a convenient decimal readout of the seconds taken.

Once it's all set up, hit ctrl-u for ultra speed acceleration and in a minute or so you'll have your answer. 

[JRup]

14 minutes ago, blakemw said:

every tick (there are 5 ticks per second) the Steam Turbine consumes 0.4 kg of Steam

There are 5 inlets per steam turbine, each inlet takes 0.4kg of steam. So for one with all unblocked inlets you'd have to consider that it continually consumes 2kg in total... The trivia section of the wiki does a nice job of illustrating such thought experiment for 10kg worth of 200ºC steam already although it still lacks some elaboration on the issue which you've kindly provided.

[blakemw]

3 hours ago, JRup said:

There are 5 inlets per steam turbine, each inlet takes 0.4kg of steam.

But it actually consumes steam in units of 80 g per inlet per tick, rather than taking it out in 400 g chunks once per second. Exhaust water though, is accumulated and emitted once per second.

[wachunga]

This feels like an extra credit question. The actual answer is pretty meaningless, but the thinking is fun. Lets give it a try! Assuming spherical dupes of course.

At time = 0 seconds, steamtemp = 200C.

At time = final, finalsteamtemp = 125C.

watertemp = 95C

 

At time = 1 second, steamtemp1 = (998 * steamtemp + 2 * watertemp) / 1000.

I don't like celsius units so let's use wachunga units instead. Wachungas are celsius - 95. steamtemp = 105W. finalsteamtemp = 30W. watertemp = 0W.

Now we have:

At t = 1, st1 = (998 * st + 2 * 0W) / 1000 = .998 st

At t = 2, st2 = .998 * st1 = .998 * .998 * st = .998^2 * st

At t = n, stn = .998^n * st.

At time = final, 30W = .998^n * 105W. Wachunga units remember. Or .998^n = 30/105. Or n = log (30/105) / log (.998).

n~= 626 seconds, or a bit over one cycle.

 

That sound right or has it been too long since I've done extra credit?

 

[JRup]

I had eyeballed 10 minutes from looking at wiki so I'll agree with @wachunga. I'll go back to making a surplus power accumulator, with blackjack and, uh, steam turbines...

Spoiler

For real, there are shall be heat accumulation! I still have to slap some more turbines on it to extract heat and do some other dumb stuff with this one...

 

[blakemw]

So now we need to test it in-game. Admittedly, it's hard to be perfect in the test setup, like how should it be primed and such? I decided to do the tests with a "fully primed" system, it is running stably on 200 C-ish steam and I just set the steam chamber to have 1000 kg of 200 C steam. This isn't exactly the same as starting with an empty system but it would like indicate the time to shut down if the heat source were abruptly cut off.

1000 kg of 200 C steam. No Aquatuner. Strong heat exchange between layers by using Aluminium Logic Gates. Good job @wachunga: 626 s is very close to 617 s.

Now with an Aquatuner, it ran almost 20% longer than without the Aquatuner which I think is more longer than would be expected from naive math. 

 

This is a variant on the no Aquatuner case, but with weak heat exchange between layers (only a single tungsten automation gate). Because of this weak heat exchange, a lot of 95 C ish water was able to accumulate in the bottom, meaning the steam layer stayed hotter. This allowed the system to run fully 35% longer than the "spherical cow in a vacuum" case where reality works as the naive maths says it should.

 

If fudging the numbers with the @wachunga approach for an Aquatuner, assuming the heat lost factor is 0.99822 to account for heat being returned, then the estimate is 703 s, compared with 735 s, however the thermal mass of the aquatuner and its general behavior in performing localizing heating of steam would tend to extend the runtime a bit.

[qda]

Looks like you really want a formula for your specific case, so here is a formula for the temperature (in Celsius) in your steam room as a function of time (in seconds) :

Temp(time)  = 105 * exp( -time / 500 ) + 95

For t = 625 seconds, Temp(625) = 105*exp(-625/500) + 95 = 125.08°C, very close to stop working, as Wachunga detailed.

This obviously does not account for thermal losses through walls, although it could be taken in account if you desire so.

Also, as blakemw said, that also doesn't take in account the fact that 95°C water does not instantly teleport back to the steam room, as this would make the calculation more complex (but not as much as needing a simulation).

If you need, you can also reverse the formula (or ask) so you can express the time needed to reach 125°C as a function of the initial temperature in your steam room.

Also, if you want to know how to get these kind of formulas for your future designs, I can detail the reasoning and calculations
 

Hope you're having fun designing stuff !

Here is also a plot of temperature vs time in your case :

[qda]

Also, I just realized your picture has an aquatuner, but your description of the problem and the subsequent question both don't mention it. My post above didn't include any aquatuner at all. However, if you wish to include the aquatuner, considering it exactly compensate the heat generated by the steam turbine (10% of deleted heat + 4kDTU/s), the formula becomes : temperature = 95.5 * exp( -time / 500 ) + 104.5, thus yielding 769 seconds to reach 125°C