Heat transfer and time needed to reach thermal equilibrium

[Ataghan]

I was hoping someone could lend a hand with a question about thermodynamics. It's been years since I've done work on thermodynamics.

Let's say I have a plastic 275 gallon liquid container, filled with a mixture of Propylene Glycol and Water (50/50 by volume). The container is placed inside of a cold chamber in order to chill it (to -18 °C). Both the inside of the chamber and the container with liquid start at a temperature of +23 °C.

2 Questions
1. How do I find the rate of heat transfer in order to estimate the time needed to reach thermal equilibrium?
2. If I compare conditioning the room temperature container/liquid for a period of 72 hours to conditioning it for only 24 hours with a room temperature container and pre-chilled mixture, how can I determine if the plastic has reached equilibrium?

Several assumptions that I am making:
1. There is no mass exchange between the container and the environmental chamber
2. There is no mass or heat exchange between the inside of the chamber and the lab environment.
3. The container has a uniform wall thickness of 1 inch all around
4. The idealized shape of the container is a cube, with raised corners (think Lego block with 4 pegs)
5. The container has minimal contact with the chamber floor due to its design, therefore heat transfer due to conduction can be considered negligible
6. Speed of air flow from chamber cooling is 20 mph and keeps chamber at a constant -18 °C

The main problem I am having is defining the system. Should I use a lump method or determine heat transfer at the level of the container?

 

[Andrew Mason]

I was hoping someone could lend a hand with a question about thermodynamics. It's been years since I've done work on thermodynamics.

Let's say I have a plastic 275 gallon liquid container, filled with a mixture of Propylene Glycol and Water (50/50 by volume). The container is placed inside of a cold chamber in order to chill it (to -18 °C). Both the inside of the chamber and the container with liquid start at a temperature of +23 °C.

2 Questions
1. How do I find the rate of heat transfer in order to estimate the time needed to reach thermal equilibrium?

Measure it. That is the only way as there are too many variables to analyse.

2. If I compare conditioning the room temperature container/liquid for a period of 72 hours to conditioning it for only 24 hours with a room temperature container and pre-chilled mixture, how can I determine if the plastic has reached equilibrium?

Measure the temperature at frequent intervals and do a graph. You can extrapolate or interpolate the graph to see approximately when it reaches the ambient temperature, to within whatever tolerance you require.

AM

 

[Ataghan]

Andrew thank you for the reply. I actually already have collected the temperature data using twenty-two thermocouples in various locations around the container. The data trend shows that the container takes longer than 72 hours to reach thermal equilibrium, which fits my expectations. Without having sensors imbedded into the plastic wall, I can't ensure it has reached temp. The sensors are mounted on the inside and outside surfaces of the container walls. So I am really only measuring the temp at thos surfaces. I also was hoping to show that the trend would be predictable using basic thermo calcs, but got caught up in the details.

 

[CWatters]

It's not clear what you mean by equilibrium because as the temperature of the liquid falls so does the rate of heat loss. It will only reach equilibrium with the cold room at t=∞. You would need to define at what point it was considered "close enough". A small change in the definition might have a large effect on the time taken.

If we ignore that issue then...

If the contents of the tank and the air outside are stirred so that the temperatures inside and outside the container walls are uniform then the problem should get easier to model. You end up with a mass (with known heat capacity) connected to a heat sink (at a known temperature) by an insulator with known thermal properties. Should be possible to plot some sort of curve and calibrate it.

If the contents aren't stirred then as Andrew said there are too many variables. You might have to have to think about convection and conduction in the tank and how that might change the rate of heat loss.

 

[mikeph]

I'd define the system as only the liquid, and just specify the temperatures on the sides. You've got enough complicated physics going on inside the vessel.

The way I see it, you have three difficulties:

1. the spatial variation in the model, as it will be far from perfectly mixed
2. the phase change, which deposits a lot of heat at the freezing interface, and
3. natural convection and fluid mixing, as each species will have its own density/temperature curve.

I'd say start with a lumped model (ignore 1 and 3), then move on to a 3D model without phase change (ignore 2 and 3), then use your experience to try to develop and understand a 3D model with phase change (ignore 3). Convection and fluid mixing is by far the hardest part. Although you can treat it numerically, the answers from the 3D phase change model should be fairly reasonable.

 

[Ataghan]

It's not clear what you mean by equilibrium because as the temperature of the liquid falls so does the rate of heat loss. It will only reach equilibrium with the cold room at t=∞. You would need to define at what point it was considered "close enough"...

By equilibrium I mean that the temperature of the liquid and container walls was the same as the air temp in the chamber. That was of course a simplification of the actual situation. The thermal equilibrium requirement is actual not my idea, it's something imposed by testing parameters. The problem is that the person imposing is unaware of the complexity.

 

[Ataghan]

I'd define the system as only the liquid, and just specify the temperatures on the sides. You've got enough complicated physics going on inside the vessel.

The way I see it, you have three difficulties:

1. the spatial variation in the model, as it will be far from perfectly mixed
2. the phase change, which deposits a lot of heat at the freezing interface, and
3. natural convection and fluid mixing, as each species will have its own density/temperature curve.

I'd say start with a lumped model (ignore 1 and 3), then move on to a 3D model without phase change (ignore 2 and 3), then use your experience to try to develop and understand a 3D model with phase change (ignore 3). Convection and fluid mixing is by far the hardest part. Although you can treat it numerically, the answers from the 3D phase change model should be fairly reasonable.

What I have decided to do is use a CFD module in Autodesk's Inventor software. I'll combine those results with the experimental data I collected. I have seen the software in action and I should be able to model it based on your suggestions. I appreciate the feedback I've gotten, some of my suspicions have been confirmed and my memory has been jogged. Thanks folks.

 

[mikeph]

might be slightly more complicated modelling moving freezing fronts, but if the CFD software is good it's possible. And convection should be coupled automatically.