Challenging question on specific heat capacity?

Question

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

Assums complete mixing of water and ignore heat lossses.

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I'd appreciate detailed working please. I assure you that best answer will be chosen!

You're right! Thank you so much!

I don't get your working though :-O

Answer 1

Hi I just saw this bath deprived of requisite time:

dQ= m c dT when m and T vary
dQ/dt = m c dT/dt + m-Dot c dT

under thermal isulation from rest of universe dQ/dt=0
m c dT/dt = - m-Dot c dT (net input) - here pair of source-sink

Let T be the temp in bath (t=0, T=60). m=100 kg.
To bath:
The hot tap is giving heat at the rate of Q1-Dot with m-dot=20 kg/minute through a constant source of T1=70 C

The cold tap is giving heat (coolth - microscopically sucking out) at the rate of Q2-Dot with m-dot=20 kg/minute through a constant source of T2=10 C

the rate of change of bath temp will be given by heat balance, assuming c remains constant within temp range.

m c dT/dt = m-Dot c [70 - T] - m-Dot c [T - 10]

{in dt time net gain in heat content of bath = taken from hot source - given to cold source}

m dT/dt = 2 * m-Dot [ 80 - T]

dt = m dT / 2 * m-Dot [ 80 - T]

[t2 - t1] = m [45 - 60] / 2 * m-Dot [ 40 - 45]

[t2 - 0] = m [-15] / 2 * m-Dot [ - 5]

[t2] = 3 m / 2 * m-Dot

[t2] = 3 * 100 / 2 * (20 / minute) = 15 / 2 minute

Time taken = 7.5 minutes

Answer 2

Depends on what your goal is. I could take your same information, and change the units. Mix English and metric units. And I could use different S.I. units, and put in the requirement of significant figures. The question then really is one of more arithmetic then chemistry. But to get the answer, one has to do manipulations of units and and understanding of significant digits which has a legitimate scientific value. And still understand that it involves heat capacity. You used the units of specific heat capacity. That can be changed to asking for molar heat capacity. How about an impossible one? Instead of a gold ring, use a specific mass of a radioactive compound that generates heat through radioactive decay, and ask about the amount of temperature rise in the water over a specific period of time, given the decay rate.

Answer 3

7.5 minutes. you need to suck a total of 6,279,000 million joules of energy out of the bath of water + 2,093,000 million joules per minute of the hot tap. the cold tap sucks out 2,930,200 joules per minute. 2,930,200 - 2,093,000 = 837,200 joules being sucked out per minute. 6,279,000/837,200 gives you 7.5 minutes. all these joule numbers were gotten using q=mc(delta)t

did i get it write? i'm really curious even though im pretty sure it's right