How Long Until Bath Water Cools to 45C?

[lionely]

A bath contains 100kg of water at 60C. Hot and cold taps are then turned on to deliver 20kg per minute each at temperatures of 70C and 10C respectively. How long will it be before the temperature in the bath has dropped to 45C? Assume complete mixing of the water and ignore heat losses.

I've never done a specific heat capacity question that involves time so I'm not sure on how to even start.

 

[tiny-tim]

hi lionely!

general strategy …

start by giving everything a letter, and then translate the question into equations​

in this case, call the amount of added hot water "h", the amount of added cold water "c", the time "t", and then write out how h and c depend on t

finally apply Newton's law of cooling …

what do you get? ​

 

[lionely]

I don't know what Newton's law of cooling is I'm only in the 11th grade :(

But umm is it

h + c/t?

 

[tiny-tim]

??

what rule have you been taught for finding the final temperature of a mixture?

 

[lionely]

Ummm the basic

heat given out = heat received

mass x specific heat capacity x temp change = mass specific heat capacity x temp change.

 

[tiny-tim]

mass x specific heat capacity x temp change = mass specific heat capacity x temp change.

hmm … i don't like that …

i] it skates over the fact that one of the changes is negative

ii] it doesn't help if (as here) there's three masses​

better would be …

∑ (mass x specific heat capacity x temp change) = 0

try that!​

 

[lionely]

what is the temp change add all the temps and average and just ignore the time ?

 

[tiny-tim]

yes, ignore the time

(though you will need the time to calculate how much water is being added )

 

[lionely]

so for the temp change (60 + 70 + 10/3)-45?

 

[tiny-tim]

i've no idea what you're doing

… call the amount of added hot water "h", the amount of added cold water "c", the time "t", and then write out how h and c depend on t

 

[lionely]

∑ (mass x specific heat capacity x temp change) = 0 I was trying to follow what you said.

Is the total mass like the h + c + the 100kg water?

 

[tiny-tim]

that's later

first …

call the amount of added hot water "h", the amount of added cold water "c", the time "t", and then write out how h and c depend on t

 

[lionely]

hotter water added = h

cold = c
time = t

h = 20kg/60 seconds? Every minute another 20kg is added same for the cold.

That's how h and c depend on t.

 

[technician]

The original 100kg of water at 60C has to be cooled to 45C... Can you calculate how much energy must be removed from this 100kg of water?
This heat energy must go to warm up the mixture of (20kg at 70C + 20kg at 10C) arriving per second. This mixture ends up at 45C.
Does this help you to see what you need to sort out...ifr you had 40kg arriving per second ,what temp would it be to have the same effect as the 2 lots of 20kg?

 

[tiny-tim]

not quite

you need h on the left and a function of t on the right

(ie it needs to have t in it !)

(same for c, of course)

 

[lionely]

I'm sorry but I'm so confused is the energy needed to be removed for the 60 C water to reach 45 C = 100kg x 4200 x (60c - 45c) = 6,300,000 J?

 

[technician]

Yes !and that energy goes to warm up the mixture to 45C. Can you see what the temp of the mixture is as it enters the bath?

 

[lionely]

is it 20kg of water at 70c + 20kg of water at 10c /2 ? so 40c?

 

[technician]

YES so 40kg water at 40C per MINUTE needs to be warmed to 45C... how much energy is needed to do this... how many minutes?

 

[lionely]

Omg thank you it's 6,300,000/840,000 = 7.5 minutes I don't know how I couldn't see that before. So stupid...

 

[technician]

that is what I got 7.5 minutes

 

[lionely]

Yeah that's what it says in the back of my book.

 

[technician]

well done..it is not Newton's law of cooling ! it is straightforward heat lost = heat gained
Newton's law of cooling is something completely different and you have probably not met it yet