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

#### eNathan

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.

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#### michael376071

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

#### eNathan

q= (specific heat of ice)*(mass of ice)*(delta temp)
Hmn, but how exactly do you calculate how long it takes for something to freeze?

Doesn't anybody know

#### 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.

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#### eNathan

Thanks, I will look forward to it :)

#### 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.

#### Hurkyl

Staff Emeritus
Gold Member
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.

#### The Bob

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

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

#### eNathan

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 :)

#### Hurkyl

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

#### Jameson

Could one use Newton's Law of Cooling perhaps?

#### eNathan

Hurkyl said:
Given simpilfying assumptions, the equation for the temperature of the liquid at time t should be $A + B e^{Ct}$.
:surprised

And how would I do this?

#### eNathan

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

#### eNathan

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?

#### The Bob

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

#### Gokul43201

Staff Emeritus
Gold Member
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 !

#### edion

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?

#### Sahman

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)

#### Chestermiller

Mentor
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)
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.

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

This problem does not have a mathematical answer due to supercooling. When water is cooled below 0°C the ice configuration is more stable than the liquid configuration. If a bit of water freezes it releases heat (just like these hand warming pads do on crystallization). This heat needs to be transported away and here all hell breaks loose. We can calculate the heat transfer through the wall of the container, and maybe we can even say something about the thermal conductivity of the water, but as soon as we have crystals in random places, we ave an inhomogeneous medium. Also the ice sheet probably starts to form at the containers surface reducing the thermal conductivity. In addition if the container has a certain minimum width most heat transport is due to convection, which produces turbulent flow and that is impossible to solve mathematically (has been for the last 200 years at least). And now we come to the worst part which is supercooling, in the absence of crystal germs water will stay liquid below 0°C so therefore water that is just cooled a bit below 0°C may never become ice or it may become so due to a cosmic ray at a random moment. It really is impossible to calculate. You can get some estimates from the thermal conductivity of water though.

#### Chestermiller

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

This problem does not have a mathematical answer due to supercooling. When water is cooled below 0°C the ice configuration is more stable than the liquid configuration. If a bit of water freezes it releases heat (just like these hand warming pads do on crystallization). This heat needs to be transported away and here all hell breaks loose. We can calculate the heat transfer through the wall of the container, and maybe we can even say something about the thermal conductivity of the water, but as soon as we have crystals in random places, we ave an inhomogeneous medium. Also the ice sheet probably starts to form at the containers surface reducing the thermal conductivity. In addition if the container has a certain minimum width most heat transport is due to convection, which produces turbulent flow and that is impossible to solve mathematically (has been for the last 200 years at least). And now we come to the worst part which is supercooling, in the absence of crystal germs water will stay liquid below 0°C so therefore water that is just cooled a bit below 0°C may never become ice or it may become so due to a cosmic ray at a random moment. It really is impossible to calculate. You can get some estimates from the thermal conductivity of water though.
You obviously don't have much experience doing math modeling of physical systems. I stand by what I said in my previous post, and guarantee that I can mathematically model this problem. Much of what you said in your paragraph above is incorrect, but I'm not going to harp on it. My experience has been that, if you think that something can't be modeled, then you will never be able to model it.

At the very least, for this problem, you can bound the answer, setting an upper limit on the amount of time it will take for all the water to freeze. For example, even though natural convection is present in the actual system, this natural convection will always enhance the rate of heat transfer. So, if you neglect natural convection, this will provide a lower bound to the rate of heat transfer, and an upper bound to the amount of time required. Neglecting natural convection within the container simplifies things considerably by reducing the behavior within the container to an unsteady state heat conduction problem (with change of phase and discontinuous heat flux at the advancing freeze front). Outside the container, the heat transfer will be determined by the heat transfer coefficient on the air side. From experience, this will have a value on the order of 1 to 10 BTU/hr-ft2-F. The calculations can be done for both these values.

The shape and size of the container will be a factor. I would start out by considering a spherical container "levitated" within the air of the freezer. This would make the heat transfer problem 1 dimensional (in the radial direction). I would assume that the water completely fills the container. One could solve this problem for several different sized containers, but it would be more efficient to reduce the equations to dimensionless form.

Another shape I would consider would be a flat slab, neglecting the heat transfer at the edges. This problem is also one dimensional.

One can see that, by making the right kinds of simplifying assumptions, the problem can be reduced to a manageable formulation. If I really wanted to solve this problem more precisely, I would resort to using computational fluid dynamics tools.

I am actually considering solving this problem to demonstrate in detail how it can be done, but I am hoping that the above allusions to the bounding approach will suffice for most readers.

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