# Calculated the time it takes for a liquid to freeze ?

1. ### eNathan

353
Calculated the time it takes for a liquid to freeze ??

hello fellow Mathematicians. I was just wanting to know the equation used to calculate how long it takes for something to freeze. I am assuming this equation has a constant 0c (32f) which is the freezing point. It would also be nice if the equation took into account any barriars, such as the plasic or something else in which the liquid is stored. Or the surface area and volume of the liquid. (Some liquids freez faster than others, so do we just assume that we are talking about water?)

I was just wanting to know this so I can pre-determine how long it will take my ice and to freeze :rofl: But as always, its to to gain more knoledege.

2. ### michael376071

11
Hmmm...interesting question. I guess I should break out my chemistry book :tongue: Sadly without much knowlegde in thermodynamics, I still can't connect it with time. You can easily calculate how much energy is required to freeze an amount of ice,

q= (specific heat of ice)*(mass of ice)*(delta temp)

the specific heat is just a constant that is an amount of energy used per temperature change, if you could get the rate of energy transfer into the ice, could then maybe figure out, I'm sure its probably simple though

3. ### eNathan

353
Hmn, but how exactly do you calculate how long it takes for something to freeze?

Doesn't anybody know

4. ### The Bob

You will need the equation Energy = Specific Heat Capacity * Mass * Change in Temperature and then another that will convert energy into time. I don't know what but I am looking it up.

I think it will depend on how much energy per time unit that is being given to the water.

Last edited: Apr 9, 2005
5. ### eNathan

353
Thanks, I will look forward to it :)

6. ### The Bob

I have had an idea. Lets say the fridge gives 60 Watts of power. Power is Joules per second so this means 60 joules are given to the water a second.

So I am going to predict how long a litre of water will take to freeze when it starts at 25°C.

So energy needed to freeze is:
E = cm∆T = 4.17 Jg-1 K-1 x 1000g x 25°C = 104250 J

60 J per second so $$\frac{104250}{60} = 1737.5 \ seconds$$

1737.5 seconds is about 29 minutes.

This doesn't seem quite right but I think the idea is there.

7. ### Hurkyl

16,089
Staff Emeritus
I've moved this over to the physics forum because you'll probably get better answers.

The Bob's approach won't work. Most significantly, if he was adding energy to the water, it would boil, not freeze.

Under simple circumstances, the rate that energy leaves the water will be proportional to the difference between its temperature and the ambient temperature, and it will also be proportional to the surface area of the water.

The problem is that fluids are complicated, and water moreso than most. Currents in the water will change the result, as well things dissolved in the water. The surface of the water will freeze first, insulating the interior. etc.

8. ### The Bob

Oh ********. A simple thermodymanic principle and I messed it up. Now I know I am not worthy here.

Makes sense. I think I will leave it for people that truly understand. Sorry eNathan.

9. ### eNathan

353
Hmn....I am going to make my own equation for this (and I may not post it, because mathematicians probably wont accept it). The equation will take into consideration the following...

The temperature at which you are trying to freeze the liquid
The volume of the liquid
The area of the liquid /*(obviously, if you put some water on a flat sheet it is
more exposed to the cold so it will freeze faster)*/
The substance of the liquid /*(a unit of how long it takes in hours to freeze the liquid at 0c or 32f)*/
Any median that covers the liquid. /*(If plastic covers the liquid or a thin bag or any other median)*/
How much of that median covers area of the liquid

I know it sounds crazy to include all of this within one equation, but I am sure I can derive it with enough experiments :)

10. ### Hurkyl

16,089
Staff Emeritus
Given simpilfying assumptions, the equation for the temperature of the liquid at time t should be $A + B e^{Ct}$.

11. ### Jameson

787
Could one use Newton's Law of Cooling perhaps?

12. ### eNathan

353
:surprised

And how would I do this?

13. ### eNathan

353
It seems that there is no equation to do this. C'mon nobody knows? :rofl:

14. ### eNathan

353
hmn, does anybody know of a place on the NET where I can get data on how things freeze at certian tematures, and how long it takes, so I can make an equation?

15. ### The Bob

Believe me, that was what I was looking for. I found nothing at all but you might need to search more.

16. ### Gokul43201

11,141
Staff Emeritus
Newtons Law of Cooling (the exponential profile Hurkyl suggested) will work nicely for the first part of the problem, to determine the time it takes to reach the freezing point.

Freezer Temperature = Tf
Water temperature at start = Ts

Time taken to reach the freezing point,

$$t =\frac{1}{K}~~ ln~ \frac{ T_s - T_f}{0 - T_f}~~~$$

working in deg C (ie: final temp = 0C).

K is a constant that is dependant on air flow profiles and container geometry, so it is essentially unknown until you determine it experimentally for a given geometry and set of conditions.

The second part of the solution is the crystallization time (which will be much smaller than the time till crystallization, so may be neglected), and this too is extremely sensitive to conditions such as water purity, vibration levels and container geometry, so there's no way to tell this number theoretically.

In summary : do it and find out !

17. ### edion

1
Re: Calculated the time it takes for a liquid to freeze ??

"K is a constant that is dependent on air flow profiles and container geometry, so it is essentially unknown until you determine it experimentally for a given geometry and set of conditions"

I am trying to figure out a similar problem. What experiments can you do to find a K value?

18. ### Sahman

1
Re: Calculated the time it takes for a liquid to freeze ??

From http://van.physics.illinois.edu/qa/listing.php?id=19836
Follow-Up #4: Time for water to freeze?
Q: Can someone provide an answer on how long a specif volume of water will take to freeze at a typical residential freezer temp, i.e. 8 ounces of water starting at 38 degrees farenheight takes "X" minutes to freeze, or "x" minutes per ounce at "x" degrees?
- Peter Thompson (age 29)
Minneapolis, Minnesota, USA
A:
No. This sort of problem is unanswerable because it depends on too many variables that we don't know the values of. At best one can give scaling laws, for example twice the heat transfer rate will halve the freezing time or twice the volume, at the same transfer rate, will double the time. The freezing time depends on: the air circulation rate, the surface and shape of the container, the amount of contact area, etc. The best way to answer this question is to do some experiments and vary some of the parameters. Eventually you will get some empirical idea of what's going on.

LeeH

(published on 06/20/10)

### Staff: Mentor

Re: Calculated the time it takes for a liquid to freeze ??

I disagree. This is a do-able problem, but it would take some serious math modeling to solve. We are talking about a complicated heat transport problem here. We chemical engineers and mechanical engineers are accustomed to solving complicated transient heat transfer problems like this, but in this case, I'm not sure it's worth the effort. To get an idea of how to attack transport problems, see the book Transport Phenomena by Bird, Stewart, and Lightfoot. The problem here involves phase change in a container, natural convection (fluid flow) within the container, conductive heat transfer within the water, conductive heat transfer through the container walls, and convective heat transfer to the air outside the container. All these effects can be taken into account in a math model, involving partial differential equations in space and time for the temperature and velocity variations. The formulation will involve the thermal conductivity, temperature-dependent density, heat capacity, heat of fusion, and viscosity of the water, the geometry, the thermal conductivity of the container wall, the properties and forcing of the air flow, and the geometry of the refrigerator. No Problem.