Optimizing Beer Temperature: A Scientific Approach

[mjc456]

A friend of mine wants help building a beer temperature calculator program. The idea is this:

He is a beer snob, and wants to drink his beer at exactly the perfect temperature. He wants to write a program that will time how long he needs to let his beer sit on his kitchen counter after pulling it out of the fridge, for it to reach this optimum temperature.

He specifies a desired temperature for his beer to reach, specifies the temperature it is stored at in his refrigerator, and any other pertinent details (bottle thickness and dimensions, ambient room temp., percentage of alcohol and water in the beer, etc.), and hits GO the second he pulls the beer out, at which time the program runs a timer that beeps at the appropriate moment.

I have tried to figure out the pure math approach to this, but I'm confused enough that I fear at this point even if I stumbled on to the right answer I wouldn't recognize it, so I'm asking for help.

I figure that Newton's Law of cooling would apply here, although it seems that for that to work, you need to take the temperature of the beer at two times during the cooling process in order to establish the value of the proportionality constant. I figure that would be the most accurate way to get the right equation, but in principle he would like to not have to measure temp. twice.

The other piece that I came up with were the two equations from my Physics book, those being:

- Energy required to heat a substance: Q = mass * specific heat * Temp. difference
- Thermal current (Rate of heat energy transfer): dQ/dt = k (proportionality constant) * A (surface area of the conducting surface) * Temp. difference between the two surfaces / thickness of the surface.

I feel like somehow all the information is there for me to solve this, but I'm not seeing how to put it together. So:

- Am I even on the right track? If so, how do I combine this stuff? I figure it should look like an exponential decay curve when I'm done, but I feel shaky with any of the solutions I've come up with.

- Should I even be using that thermal current equation, which is for heat conduction? Is this really a conduction situation, or would convection be more appropriate? (I think conduction fits, but not sure).

Anyway, I've beat my head against this for a while, going between my Physics and Calculus books, for a couple of days. So, any help anyone could provide would be greatly appreciated. Thanks.

 

[dwlink]

This will make an excellent system identification problem. You have the right equations but there are many unknown parameters.

Variables
Tb & Qb = Temp & Energy of Beer
Tg & Qg = Temp & Energy of Glass Bottle
t = time

Parameters & Constants
Cb, Mb, Kbg, Abg = Specific Heat, Mass, Beer/Glass Heat Transfer Constant, Beer/Glass Area
Cg, Mg, Kga, Aga = Specific Heat, Mass, Glass/Air Heat Transfer Constant, Glass/Air Area
Ta = Ambient air temperature (assumed to be constant)

For the Beer...
Qb = Cb*Mb*Tb
dQb/dt = Kbg*Abg*(Tg - Tb) Tg assumed to be > Tb, therefore dQb/dt > 0

For the Glass...
Qg = Cg*Mg*Tg
dQg/dt = Kga*Aga*(Ta - Tg) - Kbg*Abg*(Tg - Tb) Ta assumed to be > Tg

Now you will end up with a linear, 2nd order differential equation which has an analytic solution. Specific heat of beer will be close to that of water. Specific heat of glass will be slightly different. I believe the heat transfer constants are proportional the densities between the two mediums, but there is likely more to it than just that.

Solve the equations, make observations, update your parameters to improve the 2nd order linear models. If you have MATLAB/Simulink and know how to use it, it is very easy to construct the equations above and simulate the themal dynamics of the system.

 

[f95toli]

You could probably find an empirical formula as long as you used the same glass, placed in exactly the same place in the fridge and only drank one type of beer.
This would be impossible to solve analytically in the general case even if you had all the material parameters; you would need a good 3D FEM solver and a pretty good computer.

 

[Evil Bunny]

or you could just get a separate fridge and set the temperature where you want it... then you could improve the efficiency of your alcohol intake as well.

 

[Lsos]

I'm sure the caluclation IS possible using some complex program with large amounts of variables, not to mention real-time, accurate data of things such as the temperature of the fridge, the temperature of the room, the air velocity, etc. to not only keep track of the everchanging conditions, but to account for the extra heat that will be generated by the very computer that is trying to run the program. This would of course require some equipment such as the computer itself, multiple flow sensors and thermometers...

...thermometers, anyone of which he could just place directly on the damn beer bottle and get the most direct, efficient, and accurate reading of the variable he is interested in.

Now, while this system might not drop as many panties as a computer program that indirectly figures it all out, it IS the best method of figuring out the perfect beer drinking time, if that's what he's really interested in.

 

[mjc456]

This would be impossible to solve analytically in the general case even if you had all the material parameters; you would need a good 3D FEM solver and a pretty good computer.

Interesting. Please help me understand better why this is the case. It seems like I am going to end up with some sort of curve that looks like Temp = Ce^-kt, and I just need to then find the values of C and k, which would somehow be related, in principle to the variables like specific heat, glass thickness, temperature difference and whatnot. Where am I going wrong?

 

[dwlink]

All this stuff about needed an FEM solver is garbage. A linear approximation will do very good. Solve my equations above and for typical values for the system parameters the solution will have the form A1*e^(B1*t) + A2*e^(B2*t). I really don't know why others are making this out to be so complicated.

The need to model higher order dynamics is irrelevant to this application. Is your friends palette sensitive to a 2-3 degree difference in temperature (remember to use absolute scales - Kelvin or Rankine)?

Calculation of the surface area can be approximated by using a few cylinders with different radii. Also you'll need 2 initial conditions - Assume the beer and the glass both being with initial temperature equal to that of the refrigerator. If you are at all familiar with the Laplace transform, the differential equation will have the form A/(s^2+B*s+C) in the s domain. A, B, and C are generalized coefficients which are functions of the system parameters.

 

[nasu]

Why not use a thermometer to measure the temperature of the beer (or bottle) as it cools down?
You can connected it to your computer if you wish and write a simple program to beep (or play a song) when the temperature has reached the desired temperature.
Anyway you need a thermometer to measure the initial temperature to plug the value in your program.

 

[f95toli]

All this stuff about needed an FEM solver is garbage. A linear approximation will do very good. Solve my equations above and for typical values for the system parameters the solution will have the form A1*e^(B1*t) + A2*e^(B2*t). I really don't know why others are making this out to be so complicated.

Of course one can solve the problem approximately by first simplifying the geometry and then assume that the cooling rate is proportional to the temperature difference; this would give you a rough estimate. However, it is unlikely to give you the kind of precision (the "standard precision" for beer would be about +-1K) that would be needed here.
There is a huge difference between using an analytical model to understand the physics and actually calculating something.
Also, the FEM route is not difficult as long as you don't need to take convection cooling into account, but you do need the right software (say COMSOL or Ansys) and a good computer.

 

[mjc456]

Well, it sounds like both approaches, using the linear approximation and using the FEM software both have their advantages. Unfortunately, I think they both might be a bit out of my league, that is, at least, without further explanation. I would love to understand either process well enough to actually implement it, so if either dwlink or f95toli could elaborate a bit on what I'd need to do (or point me in the right direction so I can research it myself... I'm not afraid of doing a bit of legwork) to actually implement their solutions, that would be great.

@dwlink: I'm not really familiar with a Laplace Transform. I have a general idea of what it is, but wouldn't know how to use it to help with this problem.

@f95toli: Any idea where I could get the type of software you suggest? Also, would the software simulate the cooling every time, or would it be used to develop a general equation that could then be used in our little toy program?

Finally, barring either of those approaches, I thought of this idea. Please, anyone, let me know if it stinks. First, use the thermal conduction equation (dQ/dt = ka *delta T/delta x)to calculate how much heat energy flows over a tiny increment of time, say one-thousandth of a second. During this tiny increment, delta T would be constant. Then, I'd plug that amount of heat energy into the heat transfer equation (Q = mc delta T) to see how much temperature change that energy transfer correlates to. Then, I'd plug that delta T back into the first equation, and use it to calculate energy transfer after another thousandth of a second. Repeat this process, counting how many iterations the loop goes through until I hit the ideal temperature, and bingo, I'm done. Now, will this give me even remotely close to accurate results? Thanks.

 

[marcusl]

Pour out a glass of beer, heat it in the microwave, then give a gentle stir and measure the temperature. With a few trials, you'll figure out precisely how many seconds are needed to achieve the desired temperature.

The type of beer should make little difference, I think. If the beer always comes out of the fridge at the same temperature, you use the same type of glass (at the same temperature), and fill to the same height, then you should consistently get within a degree or two.