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Doglaner

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Following the recent thread about what we all do for a living, I reckon someone out there might like this little intellectual challenge and also have the skills and knowledge to answer it.

So, I am siting in a lodge in the Peak District, quite high up and hence surrounded by snow. There are some bottles chilling outside on a table, packed round with the white stuff as the fridge space has been taken up with 'more important' things like cheese, milk, trifle, cheesecake and prosecco. The beers therefore are relegated to the balcony.

So, the question is - how long should I leave a standard 500ml bottle of ale, which starts at an ambient room temp of 22c, in an outside (or fridge, if allowed) temp of, say, 3c, before it reaches a 'cellar' temperature of 10c?

Pencils sharpened? Then off you go! Extra points given for showing your workings. (That sounds wrong!)

Dog.
 
Assume a bottle is about 5cm diameter and 15cm tall, reasonable if we exclude the neck, and 5mm thick. That makes it, very approximately, 0.025msq or 250cmsq in surface area. Thermal conductivity of standard glass is approximately k=1 (cf Google). The formula for heat transfer is:

W = k * (T2-T1) * A / t

In your example

W = 1 * (22-3) * 0.025 / 0.005 = 95

So the rate of heat transfer is 95 Watts, or joules per second. It requires 4200 Joules to heat/cool one kg of water by one degree (cf Google). Let's assume beer and water are equivalent. You have 12 degrees to travel and half a kilo. So you need 4200 * 0.5 * 12 = 25,200 joules

25,200 joules at 95 per minute is 265 minutes or just under four and a half hours.

Anecdotally, I used to work in a bar where the boss reckoned on 9 hours to chill a bottle of white wine down to serving temperatures, so allowing for that probably being a little cooler, I'd say we're in the ballpark.
 
Google came up with -


So you pull out a beer that's been in the fridge for probably long enough, but it's not quite cold yet. Still got a ways to go. How long has it actually been in there versus how long you've been wanting a frosty beer?

Well, we literally just took a few beers we had lying around the office and popped them in the fridge. They were ready for consumption after two hours and not one moment before; however we did keep opening the fridge to check, letting out precious cold air. That is still a ridiculously long time, and the reason why the brews you stashed in there for half an hour aren't cold yet. Hear us? They're not cold until they're cold!
Click to expand...
 
My answer is nonsense by the way. I mixed seconds and minutes so am out by a factor of 60. Disregard.


Sent from my iPhone using Tapatalk
 
If I was forced to give this question a go with theory, I think Ajhutch is on the right track. Aside from being out by a factor of 60, (s)he assumes that the heat transfer rate is constant at 95 Watts. It is 95 Watts when the beer is at 25C, but by the time it reaches 10C that will have dropped a lot. The mathematically minded would solve this by integrating Fourier's law over time, but I'm lazy so solve it using a simulation*.

This gives a value of 14 minutes. However, this is inaccurate for two reasons. Firstly, and least importantly, the parameters are only estimates, such as the glass thickness, bottle surface area and specific heat capacity of beer; alcohol, CO2, unfermented sugars and the other solutes in the beer will all lower the amount of energy needing to be removed to get to serving temperature. However, to a first approximation they are ok. More importantly, it assumes that the temperature is even throughout the fridge and throughout the beer. This is obviously false for the fridge, especially as you don't have a fan in it. It is also false for the beer; there may be convection currents in the beer, or maybe not. Either way the 14 minute value should be treated as a lower bound on cooling time (ie “at least 14 mins”). This reminds me of the Mpemba effect, which is named after the Tanzanian school boy who made the unintuative discovery that his hot ice cream mix froze quicker than his cold ice cream mix. He was mocked by his classmates and teachers, but has become immortelised in the history of chemistry because he was right, this can happen, and the reason is because convection currents in the hotter liquid increased the rate at which it cooled, and in certain circumstances can result in a hotter liquid freezing quicker. If Mpemba teaches us anything, then it is that when it comes to rates of cooling in the real world, this isn't really a theoretical or mathematical question, but a scientific one to be sovled by experiment, so put bottles in your fridge and measure their temperature, and you'll get your answer. :thumb:


*for those who are interested:
Code: 
#Beer cooling simulation, written in Python
Tf = 3.0 # fridge temperature (C)
Tb = 22.0 # beer temperature (C)
vol = 0.5 # beer volume (L)
a = 0.025 # beer bottle surface area (m**2)
k = 1.0 # thermal conductivity of glass (W/(m.K))
cv = 4200.0 # specific heat capacity of beer (J/(kg.K))
g = 0.005 # glass thickness (m)
t = 0 # time (s)
delta_t = 0.001 # timestep (s)

while Tb > 10.0:
        w = 1.0*(Tb-Tf)*a/g # heat transfer rate, Fourier's law
        Tb -= w*delta_t/cv # new temperature after delta_t
        t += delta_t
print t/60
 
IainM said:
If I was forced to give this question a go with theory, I think Ajhutch is on the right track. Aside from being out by a factor of 60, (s)he assumes that the heat transfer rate is constant at 95 Watts. It is 95 Watts when the beer is at 25C, but by the time it reaches 10C that will have dropped a lot. The mathematically minded would solve this by integrating Fourier's law over time, but I'm lazy so solve it using a simulation*.

This gives a value of 14 minutes. However, this is inaccurate for two reasons. Firstly, and least importantly, the parameters are only estimates, such as the glass thickness, bottle surface area and specific heat capacity of beer; alcohol, CO2, unfermented sugars and the other solutes in the beer will all lower the amount of energy needing to be removed to get to serving temperature. However, to a first approximation they are ok. More importantly, it assumes that the temperature is even throughout the fridge and throughout the beer. This is obviously false for the fridge, especially as you don't have a fan in it. It is also false for the beer; there may be convection currents in the beer, or maybe not. Either way the 14 minute value should be treated as a lower bound on cooling time (ie “at least 14 mins”). This reminds me of the Mpemba effect, which is named after the Tanzanian school boy who made the unintuative discovery that his hot ice cream mix froze quicker than his cold ice cream mix. He was mocked by his classmates and teachers, but has become immortelised in the history of chemistry because he was right, this can happen, and the reason is because convection currents in the hotter liquid increased the rate at which it cooled, and in certain circumstances can result in a hotter liquid freezing quicker. If Mpemba teaches us anything, then it is that when it comes to rates of cooling in the real world, this isn't really a theoretical or mathematical question, but a scientific one to be sovled by experiment, so put bottles in your fridge and measure their temperature, and you'll get your answer. :thumb:


*for those who are interested:
Code: 
#Beer cooling simulation, written in Python
Tf = 3.0 # fridge temperature (C)
Tb = 22.0 # beer temperature (C)
vol = 0.5 # beer volume (L)
a = 0.025 # beer bottle surface area (m**2)
k = 1.0 # thermal conductivity of glass (W/(m.K))
cv = 4200.0 # specific heat capacity of beer (J/(kg.K))
g = 0.005 # glass thickness (m)
t = 0 # time (s)
delta_t = 0.001 # timestep (s)

while Tb > 10.0:
        w = 1.0*(Tb-Tf)*a/g # heat transfer rate, Fourier's law
        Tb -= w*delta_t/cv # new temperature after delta_t
        t += delta_t
print t/60
Click to expand...



Love this. Much more thorough. You're right of course, T2-T1 in my one changes all the time so the wattage isn't linear


Sent from my iPhone using Tapatalk
 
I also made a mistake, by missing the volume factor when updating the temperature, so the revised lower bound is now 7 minutes.
 
Remember the RMS Titanic when calculating or designing anything!

Some of the finest Engineering minds in Britain took many years to design and build this "unsinkable" vessel. :thumb:

They then handed it over to Operators ... :thumb:

... who managed to sink it in about a week! :whistle:
 
This is actually a complex problem that is one for a thermodynamicist, or heat transfer specialist to deal with. An accurate answer without complex mathematical modelling would be difficult to achieve.
Variables include
- thickness of snow; how large the cooling snow heat sink actually is; void created as snow melts, will the snow fill the gap or will the air then act as an insulating layer (heat transfer through gases is much worse than heat transfer through liquids by several orders of magnitude;
- thickness of glass
- shape of container
- run off of melted snow; will the snow pool or not if it does the temperature may rise and the driving force will be reduced
- will the beer be moved about occasionally
to name but a few
plus all the usual things like thermal conductivity of glass etc etc.

That said back of fag packet calcs may give a reasonable approximation.

And that comes from someone who has designed large process plant heat exchangers in another life.

But it is, after all, a bit of fun.... isn't it :whistle:
 
Dutto said:
Remember the RMS Titanic when calculating or designing anything!

Some of the finest Engineering minds in Britain took many years to design and build this "unsinkable" vessel. :thumb:

They then handed it over to Operators ... :thumb:

... who managed to sink it in about a week! :whistle:
Click to expand...

...and remember Apollo 13 returned from the far side of the moon with the computing power of a Nintendo game machine and some good maths from a female mathematician !That must have made her feel pretty good!
 

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