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I'm a microbiologist and my physics is not up to par for figuring this out. I've requested our engineering department to perform a temperature mapping of the vessel, but they can't do it for me. I have a cylinder shaped stainless steel tank. I don't know it's thickness, but estimate it to be 1/2 cm to 1 cm if that is important. It will be filled with $1600$ L of fermentation fluid (I'd guess water or beer would be similar fluid to use). I estimate the tank to have a holding volume of $2000$ L ($2\times 10^6$ cm$^3$). The "water" will be at ~$22°$C and will sit in an incubator (air heating) set to $35°$C. I need an estimate on how long it will take these fluids to reach at least $34°$C in the air bath incubator.

I tried to figure this out on my own and ran across the formula $A/t=KA(t_h-t_c)/d$. But solving it is a different ball game. I don't know how to figure out what each letter means to solve this puzzle.

Are there other variables or equations I need to figure this out?

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    $\begingroup$ How does the incubator heat it? Does it blow warm air over the water? Does it blow warm air around the tank? I doubt your generic formula will apply too well here. Generally; heat transfer like this has dependence on geometry. In this case; where it is using convective heat transfer; the directions and speeds of the air will also play a crucial factor. Basically you need to figure out how accurate you need to be; that will determine how you have to model it. $\endgroup$ – JMac Jan 10 '18 at 14:59
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    $\begingroup$ If the hardware already exists, do the experiment! $\endgroup$ – Bill N Jan 10 '18 at 15:01
  • $\begingroup$ I agree with doing the experiment, however, it's another department that has all the equipment (several states away) and I'm at their mercy. They haven't agreed to run it for me. $\endgroup$ – Crystal Day Jan 11 '18 at 16:40
  • $\begingroup$ JMac, It is warm air blowing over/around a cylindrical tank. It's a heated room set to 35°C with constant airflow. There is not airflow inside the tank. I pondered if it would be too complicated to figure out. $\endgroup$ – Crystal Day Jan 11 '18 at 16:43
  • $\begingroup$ This question might be more suitable for engineering SE. $\endgroup$ – user1583209 Jan 16 '18 at 11:17
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Best case all the energy from the incubator goes into the water. Divide the energy needed to raise the temperature of the water your target amount by the incubator's power requirement/power draw (in Watts) for a best-case estimate.

Water's heat capacity is $4.18\ J/g°C$ and since its density is $1000\ g/L$ then the heat capacity is $4180\ J/L°C$ or 4180 Joules per [Liter * temperature difference in Celcius]. You have 1600 L of water that you want to raise 12 degrees Celcius so:

$$4180\ J/LK\ \times 1600\ L\ \times 12\ °C\ =\ 80\ MJ$$

As a worked example, I found this manual for the Thermo Scientific Precision Model 815 Refrigerated Incubator - Microprocessor Controlled. It has a capacity of 566 L:

$$4180\ J/LK\ \times 566\ L\ \times 12\ K\ =\ 28\ MJ$$

and max energy usage of 600 W:

$$\frac{28\ MJ}{600\ W} =\ 13\ hrs$$

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