Calculate time to defrost Prime Rib roast from freezer to refrigerator

[FredricJLowe]

Homework Statement

Trying to figure out exactly how long it will take to defrost a 8.66 lb Prime Grade prime rib roast that is in a -5 degrees Fahrenheit freezer. The refrigerator is set at 36 degrees Fahrenheit. Google just says to allow 2-3 days to defrost.

I located Newton's Calculus Law of Cooling with several examples, but am afraid I am many years removed from high school to attempt this to solve this successfully!

I'll try but I want to be sure exactly how many minutes it will take..

Would using Newton's Law of Cooling solve this and if so can someone please figure this out?

Thank you!! BTW this is for a family dinner for my 70th birthday!!!

I posed this question to my 12th grade algebra teacher I located on Facebook but he has not gotten back to me so far.

Relevant Equations

Newton's Law of Cooling

I don't know where. to even begin inserting numbers here. I know the Temperature would be -5 for the first T and 36 for T sub zero but I do not know how to solve this. I never had Calculus.

 

[PhDeezNutz]

Yo dawg you can’t figure out the cooling constant of prime rib unless you have temperature data relating to prime rib.

Like what is the change after an hour, or after 5 hours, or some other arbitrary amount of time.

If you can stick a thermometer in a frozen prime rib and note how much time has elapsed since you put it in the fridge maybe we can try to apply Newtons Cooling Law.

Not saying it will give a correct answer but we can try.

 

[pbuk]

1. There is no calculus involved here.
   

2. I know the Temperature would be -5 for the first T and 36 for T sub zero

   That's not correct, in the notation of the sheet you attached T is the final temperature you want to achieve (presumably 32°F), $T_0$ (for which the sheet has wrongly written $T_O$ with a capital O subscript) is the temperature at time $t = 0$ which is -5°F and $T_S$ (I have no idea what 'S' is supposed to represent: perhaps its 'surroundings', or maybe it's supposed to be a numeral 5 ) is 36°F.
   
3. None of this helps though because as noted in the post above all of the problems on the sheet are in two parts: first you use some data to calculate $k$ for the object, then you use $k$ to get the answer. You don't have that data.
   
4. Even worse, even though Newtons Law of Cooling also works as Newtons Law of Heating, it is not Newtons Law of Defrosting and it does not apply when there is a phase change (which is a technical term for defrosting). At a rough guess the meat will need to absorb twice the amount of heat energy to defrost as it will to change temperature by 37°F so even if you could calculate $k$ it wouldn't help.
   
5. And still more, even if there were no phase change, this simple law only applies with objects that heat up evenly all the way through (again there is a technical term for this: we say the object has a low Biot number). Clearly this is not the case for a beef rib joint otherwise when you cook it you wouldn't get that lovely rare meat in the centre while the outside is nicely caramelised.

Personally I wouldn't defrost a 9lb joint in the fridge, it will take forever. 18-24 hours at (a cool) room temperature should do it.

 

[vela]

As the others have pointed out, you're looking at a fairly complicated problem. I'd just go with what Google told you.

Happy birthday!

 

[FredricJLowe]

1. There is no calculus involved here.
2. 
   That's not correct, in the notation of the sheet you attached T is the final temperature you want to achieve (presumably 32°F), $T_0$ (for which the sheet has wrongly written $T_O$ with a capital O subscript) is the temperature at time $t = 0$ which is -5°F and $T_S$ (I have no idea what 'S' is supposed to represent: perhaps its 'surroundings', or maybe it's supposed to be a numeral 5 ) is 36°F.
3. None of this helps though because as noted in the post above all of the problems on the sheet are in two parts: first you use some data to calculate $k$ for the object, then you use $k$ to get the answer. You don't have that data.
4. Even worse, even though Newtons Law of Cooling also works as Newtons Law of Heating, it is not Newtons Law of Defrosting and it does not apply when there is a phase change (which is a technical term for defrosting). At a rough guess the meat will need to absorb twice the amount of heat energy to defrost as it will to change temperature by 37°F so even if you could calculate $k$ it wouldn't help.
5. And still more, even if there were no phase change, this simple law only applies with objects that heat up evenly all the way through (again there is a technical term for this: we say the object has a low Biot number). Clearly this is not the case for a beef rib joint otherwise when you cook it you wouldn't get that lovely rare meat in the centre while the outside is nicely caramelised.

Personally I wouldn't defrost a 9lb joint in the fridge, it will take forever. 18-24 hours at (a cool) room temperature should do it.

Thank you for your reply. I'll defrost in the fridge for 3 days as Google suggested.

 

[pbuk]

Thank you for your reply. I'll defrost in the fridge for 3 days as Google suggested.

If this is a normal domestic fridge then don't be surprised if you end up with a joint that has not properly defrosted and everything else in the fridge is partially frozen too.

You see when you put a large frozen object inside a relatively small well-insulated container the limiting factor is not how quickly the object can gain heat from the structure of the container, it is how quickly the container can gain heat from its surroundings.

At least open the door frequently to check on progress.

And have a great party!