# Temperature change at a given time

Good day to all,

Hi, i would like to ask if there is a way to calculate the temperature change at a given time. I will use this as a basis for my study which is about designing an ice container and i would like to know how long does my ice last given its own temperature and the outside temperature. The thermal conductivity of the materials will be based on the internet. The container's shape would be cylindrical with a top and bottom cap.

Sorry if this question is easy T_T I'm still quite new at this:)

## Answers and Replies

bigfooted
Gold Member
You need to study heat transfer and thermal conduction. Study for instance the first two chapters of the book of Lienhard & Lienhard:

http://web.mit.edu/lienhard/www/ahttv131.pdf

Several techniques are discussed. One of the simplest problems is to consider an ice container in the shape of a ball, then your problem becomes one-dimensional. I hope this will help you get started.

You need to study heat transfer and thermal conduction. Study for instance the first two chapters of the book of Lienhard & Lienhard:

http://web.mit.edu/lienhard/www/ahttv131.pdf

Several techniques are discussed. One of the simplest problems is to consider an ice container in the shape of a ball, then your problem becomes one-dimensional. I hope this will help you get started.
Thanks for the reply. As of now I'm still reviewing my past lessons about heat transfer. I seem to be on conduction thru a cylindrical wall.

With this topic i can calculate the heat transfer thru the differential thickness with its unit as btu/hr or kj/hr.

Btw, i assume my calculations as steady state

Am i on the right track?

Chestermiller
Mentor
Thanks for the reply. As of now I'm still reviewing my past lessons about heat transfer. I seem to be on conduction thru a cylindrical wall.

With this topic i can calculate the heat transfer thru the differential thickness with its unit as btu/hr or kj/hr.

Btw, i assume my calculations as steady state

Am i on the right track?
You also need to study natural convection heat transfer from the outside of the container to the room, heat transfer with phase change for the ice, and heat conduction through the ice and the water. It's a pretty nasty problem.

russ_watters
Mentor
It isn't difficult to test a container's insulation value --- I did it for a recent thread involving a similar issue.

You also need to study natural convection heat transfer from the outside of the container to the room, heat transfer with phase change for the ice, and heat conduction through the ice and the water. It's a pretty nasty problem.
Oh :( would there be a way just to focus on the container? Neglecting all other factors.

It isn't difficult to test a container's insulation value --- I did it for a recent thread involving a similar issue.
Could you link me to that. :) im really quite slow on this topic. :(

Chestermiller
Mentor
Oh :( would there be a way just to focus on the container? Neglecting all other factors.
Well, the contents of the container may not be all at the same temperature during the time that they are heating up. So, if you want to have a single temperature for the contents, you're out of luck. But, if you are willing to say that you are going to calculate the equilibrium temperature of the contents and container after each incremental amount of heat has entered the container, then you can come up with a number for the temperature at each time. To do this, you still need to have an idea of the convective heat transfer coefficient on the outside of the container (which is likely to be a major resistance to the heat flow). This can be estimated. To get the calculation started, you would first have to determine the initial equilibrium temperature of the contents and the container before any heat had been added. This procedure can be expected to yield a lower bound to the amount of time required to heat the system.

Well, the contents of the container may not be all at the same temperature during the time that they are heating up. So, if you want to have a single temperature for the contents, you're out of luck. But, if you are willing to say that you are going to calculate the equilibrium temperature of the contents and container after each incremental amount of heat has entered the container, then you can come up with a number for the temperature at each time. To do this, you still need to have an idea of the convective heat transfer coefficient on the outside of the container (which is likely to be a major resistance to the heat flow). This can be estimated. To get the calculation started, you would first have to determine the initial equilibrium temperature of the contents and the container before any heat had been added. This procedure can be expected to yield a lower bound to the amount of time required to heat the system.

Thanks for the reply sir chestermiller :)

now i got my initial equilibrium temperature, it is also illustrated below.
t1 = ice temperature = 0 Celsius
t3 = outside temperature or room temperature = 32 Celsius

d1 = inner diameter of the container = 8 inches
d2 = outside diameter of the container = 10 inches

k = thermal conductivity of styrofoam = .033 W/m*K <-- based on engineeringtoolbox.com My current status as of today is i finished calculating the heat loss in KJ/hr. Is there a way to change KiloJoule to Celsius?

Chestermiller
Mentor
Did you assume that the solid block of ice completely fills the chamber in the initial state? By assuming that all the material inside the container is well-mixed, you are obtaining a lower bound to the amount of time needed to raise the temperature. This will give you the minimum time needed to achieve a given temperature. This assumption transforms the problem at hand into a Newton cooling problem. To get the temperature as a function of the amount of heat added, you need to recognize that first that the contents stays at 0 C until all the ice melts and then the temperature rises. You need to work with the latent heat of melting and the heat capacity of liquid water (the latter once all the ice has melted).

• derekgabs
Did you assume that the solid block of ice completely fills the chamber in the initial state? By assuming that all the material inside the container is well-mixed, you are obtaining a lower bound to the amount of time needed to raise the temperature. This will give you the minimum time needed to achieve a given temperature. This assumption transforms the problem at hand into a Newton cooling problem. To get the temperature as a function of the amount of heat added, you need to recognize that first that the contents stays at 0 C until all the ice melts and then the temperature rises. You need to work with the latent heat of melting and the heat capacity of liquid water (the latter once all the ice has melted).
Thanks again for replying to me sir, though im sorry for the late reply.

I would like to ask this about newton's law of cooling. Does the equation factor the thermal conductivity of the material and its thickness or do i neglect it? sorry again for this simple question as i don't have any other reference to this subject matter T_T

Update: I recently read the article from wikipedia, is this the correct answer for my question sir?
https://en.wikipedia.org/wiki/Thermal_conduction#Cylindrical_shells

After this, I seem to have gotten Q though i dont know what is the exact standard unit. How will i convert this to temp per unit time?

Sorry if im a bit pushy on the questions T_T

Chestermiller
Mentor
Thanks again for replying to me sir, though im sorry for the late reply.

I would like to ask this about newton's law of cooling. Does the equation factor the thermal conductivity of the material and its thickness or do i neglect it?
It neglects it. As I said, the analysis only bounds the answer. If you want to include heat conduction in the material and its thickness, the model becomes significantly more complicated (but doable).
Update: I recently read the article from wikipedia, is this the correct answer for my question sir?
https://en.wikipedia.org/wiki/Thermal_conduction#Cylindrical_shells
This analysis would not handle the transient heat transfer analysis within the material inside the container.
After this, I seem to have gotten Q though i dont know what is the exact standard unit. How will i convert this to temp per unit time?

Sorry if im a bit pushy on the questions T_T
Show us what you did.

russ_watters
Mentor
Could you link me to that. :) im really quite slow on this topic. :(
The thread is a bit long and complicated:
How can ice cool an alcoholic drink below 0°C?

But the technique is simple: put a measured amount of hot (or cold) water in the container and use either a data logger or thermometer to measure the temperature at regular intervals for a couple of hours.

I found that hot water doesn't quite follow Newton's law of cooling because of convection and evaporation, whereas cold water does better. With cold water, you may find the external convection weak enough to neglect and just focus on the styrofoam's conduction.

It neglects it. As I said, the analysis only bounds the answer. If you want to include heat conduction in the material and its thickness, the model becomes significantly more complicated (but doable).

This analysis would not handle the transient heat transfer analysis within the material inside the container.

Show us what you did.
I think i did my calculations wrong again T_T
Below is the snap of my calculation. Another thing sir, i seem to have found a new reference, can you verify it if you have the time if this relates to my problem :)
http://www.fao.org/docrep/006/y5013e/y5013e0a.htm

The thread is a bit long and complicated:
How can ice cool an alcoholic drink below 0°C?

But the technique is simple: put a measured amount of hot (or cold) water in the container and use either a data logger or thermometer to measure the temperature at regular intervals for a couple of hours.

I found that hot water doesn't quite follow Newton's law of cooling because of convection and evaporation, whereas cold water does better. With cold water, you may find the external convection weak enough to neglect and just focus on the styrofoam's conduction.

Thanks for the reply :) Yes I will still do that kind of testing at certain time where i would put a thermometer or data logger in the container to see how long will the temperature last. But I would like to know the estimated time on how long it could last with computation :)

I would like to also ask if those commercially available ice container have a computation on how long the temperature would be inside. Since that is how i am designing it. :)

russ_watters
Mentor
Well, commercial containers don't tend to have advertised specs like that. But in my experiment with 400ml of water, I got about 0.15 W/K. If it is wrapped with 30 sq cm of 0.5 cm thick styrofoam, that would be .24 W/m - K. So that's in the ballpark of what engineering toolbox gives you.

Well, commercial containers don't tend to have advertised specs like that. But in my experiment with 400ml of water, I got about 0.15 W/K. If it is wrapped with 30 sq cm of 0.5 cm thick styrofoam, that would be .24 W/m - K. So that's in the ballpark of what engineering toolbox gives you.
Oh ok. Thanks. I'll try your experiment and observe a little later but as of now ill try sir chester's guide also about newton's cooling law

Update: i guess about now i seem to have found newtons law of cooling as a guide to answer my question as what chester said and i will try to mix it up with the link i posted above. Hopefully i can finish it within tomorrow if i would have spare time.

Once again i'd like to thank mr chester and mr. Russ for helping me. Hopefully i wouldnt encounter anymore problems.

Did you assume that the solid block of ice completely fills the chamber in the initial state? By assuming that all the material inside the container is well-mixed, you are obtaining a lower bound to the amount of time needed to raise the temperature. This will give you the minimum time needed to achieve a given temperature. This assumption transforms the problem at hand into a Newton cooling problem. To get the temperature as a function of the amount of heat added, you need to recognize that first that the contents stays at 0 C until all the ice melts and then the temperature rises. You need to work with the latent heat of melting and the heat capacity of liquid water (the latter once all the ice has melted).

Sorry to bug you again sir, in the equation of newton's law of cooling. Can i use the K for my material? if so, what unit should i be following. Assuming I have my temperatures as Celsius and my time as minutes. What would be the unit I would use for K?